pattern recognition problems in geology and paleontology
TRANSCRIPT
Lecture Notes in Earth Sciences
Edited by Gerald M. Friedman and Adolf Seilacher
2
UIf Bayer
Pattern Recognition Problems in Geology and Paleontology
Springer-Verlag Berlin Heidelberg New York Tokyo
Author Dr. UIf Bayer Instltut f(3r Geologie und Pal~ontologie der Unlversit~t T0bingen S~gwartstr. 10, D-7400 TfJbmgen, FRG
ISBN 3-540-13983-4 Spnnger-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13983-4 Sprmger-Verlag New York Heidelberg Berlin Tokyo
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To
Dorothee
Julia and Vincent
P r e f a c e
The research on m a t h e m a t i c a l methods and compute r
appl icat ions in geology since 1977 was supported by the "Sonder-
forschungsbere ich 53, PalOkologie" T{ibingen, d i rec ted by A.
Sei lacher . During the years, several "Tei lprojekte" were in-
volved: "Konstrukt ionsmorphologie , Fossi ldiagenese, Fossi lver-
gesel lschaf tungen, Foss i l -Lagers ta t t en" . During the last period
of the "Sonderforscbungsbereich" a special pro jec t "Quan t i t a t ive
Methoden der PalOkologie" was establ ished: Chapte rs 1 to 3
serve as a final repor t of the sc ien t i f ic ac t iv i t ies . Fur the r
in format ion is avai lable in the repor t s of the "Sonderforschungs-
bere ich 53".
The ideas on the seismic record in chap te r 4 arose during
ac t iv i t i e s on Leg 71 of the DSDP-program in 1980, and I am
indebted for valuable discussions to W. Gtl t t inger , G. Dangel-
mayr, D. Armbrus te r , H. Eikenmeier of the "Ins t i tu t far Infor-
mat ionsverarbe i tung" , Tt~bingen.
During the years, a considerable number of people was
engaged somewhere in the research ac t iv i t ies . Here I want
to express my special thanks to E. A l the imer and W, Deutschle ,
which were ac t ive in programming problems during several
years.
Ttlbingen Ulf Bayer
C O N T E N T
1. INTRODUCTION
1.1 Ma thema t i ca l Geology and Algor i thmiza t ion
1.2 Syntax and Semant ics
1.3 Stabi l i ty
2. NOISY SYSTEMS AND FOLDED MAPS
2.1 Recons t ruc t ion of Sed iment -Accumula t ion
2.1.1 Accumula t ion Ra tes and Deformat ions of the Time-Scale
2.1.2 Es t ima t ion of Original Sediment Thickness
2.1.3 Underconsol idat ion of Sediments - - a History Ef fec t
2.2 In t raspec i f ic Variabi l i ty of Paleontological Species
2.2.1 Al lomet r i c Relat ionships
2.2.2 The ~Ontogenetic Morphospace ~
2.2.3 Discont inui t ies in the Observed Morphospace
2.3 Analysis of Direc t ional Data
2.3.1 The Smoothing Error in Two Dimensions
2.3.2 Stabi l i ty of Local Ex t rema
2.3.3 Approximat ion and Averaging of Data
2.3.4 A Topological Excursus
2.3.5 Densi t ies , Folds and the Gauss Map
2.4 Recons t ruc t ion of Surfaces from Sca t t e r ed Da ta
2.4.1 The Regular Grid
2.4.2 Global and Local Ext rapola t ions
2.4.3 Linear In terpola t ion by Minimal Convex Polygons
2.4.4 Stabi l i ty Problems with Minimal Convex Polygons
2.4.5 Cont inuat ion of a Local Approximat ion A) A local cont inuous approximat ion B) Cont inua t ion of a local sur face approximat ion
3, NEARLY CHAOTIC BEHAVIOR ON FINITE POINT SETS
3.1 I t e r a t ed Maps
3,1,1 The Logist ic Di f fe rence Equat ion
3,1.2 The Numerical Approximat ion of a Pa r t i a l Di f fe ren t ia l Equat ion
3.1.3 Inf ini te Series of Caust ics
3.2 Chi2-Tes t ing of Direc t ional Da ta
8
10
i1
12
16
19
21
23
26
29
30
35
40
45
46
51
51
54
56
57
62 62 65
70
72
73
76
79
82
Vt
3.3 Problems with Sampling S t ra teg ies in Sedimentology 86
3.3.1 Markov Chains in Sedimentology 86 A) Disc re te Signals 88 B) Equal In terval Sampling 89
3.3.2 Art i f ic ia l P a t t e r n Format ion in S t ra t ig raph ic Pseudo-Time Series 92 A) Sampling of periodic funct ions 92 B) The analysis of 'bed th ickness ' by equal d is tance samples 95
3.4 Cent ro id Clus te r S t ra t eg ies - - Chaos on Fin i te Point Sets 98
3.4.1 Binary Trees 99
3.4.2 Image Concepts t00
3.4.3 Stabi l i ty Problems with Cent ro id Cluster ing 103 Cent ro id c lus te r s t r a t eg ies 104 Ins tabi l i t ies be tween c lus te rs t06 I n s t a b i l i t i e s w i t h i n c l u s t e r s 106
3.5 Tree P a t t e r n s be tween Chaos and Order 112
3.5.1 Topological Proper t i es of Open Network P a t t e r n s 115
3.5.2 P a t t e r n Genera to r s for Open Networks 120 A) Algebra ic models -- pro to types of branching pa t t e rn s 120 B) A met r i c model -- the Honda t r ee 123
3.5.3 Morphology of Branches in Honda Trees 127 A) Length of b ranches 127 B) Branching angles -- s imi lar i ty and se l f - s imi la r i ty 129 C) Branches and b i furca t ions - - a quasi-cont inuous approximat ion 131
3.5.4 Evolution of Shape 134 A) Trees, Peano and Jordan curves 135 B) The out l ine of Honda t rees 138 C) Chance and de te rmin ism 141
4. STRUCTURAL STABLE PATTERNS AND ELEMENTARY CATASTROPHES 144
4.1 Image Recogni t ion of Three-Dimensional Objec ts 146
4,1,1 The Two-Dimensional Image of Three-Dimensional Objec ts 147
4.1.2 The Skeleton of P lane Figures 151
4.1.3 Theore t ica l Morphology of Worm-Like Objec ts 153
4.1.4 Continuous Trans format ions of Form 155
4.2 Surface Inversions in the Seismic Record -- the Cusp and Swallowtail Ca t a s t rophes 157
4.2.1 Compute r Simulat ions of Rays, Wave Fronts and Trave l t ime Records 160
Linear rays 160 Successive wave f ronts 161 Trave l t ime record 163
4.2.2 Local Surface Approximat ion 165
4.2.3 Linear Rays, Caust ics and the Cusp Ca ta s t rophe 167
4.2.4 Wave Fronts and the Swallowtail Ca tas t rophe 172
4.2.5 Wave Front Evolution and the Trave l t ime Record 174
4.2.6 The Trave l t ime Record as a Plane Map 176
VII
4.2.7 Singular i t ies on the Ref l ec to r Line
4.2.8 Genera l i zed Ref lec t ion P a t t e r n s in Two and Three Dimensions A) The deformed c i rc le and the dual cusp B) Three-d imens ional p a t t e r n s -- the double cusp
4.2.9 Dis t r ibu ted Rece ive r s
4.3 "Para l le l Systems" in Geology
4.3.1 Some Examples of Para l le l Systems
4.3.2 Similar and Paral le l Folds
4.3.3 Bending a t Fold Hinges - - the Hyperbolic Umbil ic
4.3.4 Nota t ion of S t ra in
4.3.5 Genera l ized Plane Strain in Layered Media
4.4 SUMMARY
REFERENCES
179
183 183 189
191
198
199
201
206
210
211
214
217
INDEX 226
1. I N T R O D U C T I O N
Theoret ical modelling and the use of mathemat ical methods are presently gaining
in importance since progress in both geology and mathematics offers new possibilities
to combine both fields. Most geological problems are inherently geometr ical and morpholog-
i c a l , and, therefore, amenable to a classification of forms from a "Gestalt point of
view". Geometr ical objects have to possess an inherent stability in order to preserve
their essential quality under slight deformations. Otherwise, we could hardly conceive
of them or describe them, and today's observation would not reproduce yesterday's result
(DANGELMAYR & GOTTINGER, 1982). This principle has become known as ' s t ructural
s tabil i ty ' (THOM, 1975), i.e. the persistence of a phenomenon under all allowed perturba-
tions. Stability is also, of course, an assumption of classical Newtonian physics, which
is essentially the theory of various kinds of smooth behavior (POSTON &STEWART, 1978).
However, things sometimes "jump". A new species with a different morphology appears
suddenly in the paleontological record (EI.DREDGE & GOULD, 1972), a fault develops,
a landslide moves, a computer program becomes unstable with a certain data configura-
tion, e tc . It is, surprisingly, the topological approach which permits the study of a broad
range of such phenomena in a coherent manner (POSTON &STEWART, 1978; LU, 1976;
STEWART, 1982). The universal singularit ies and bifurcation processes derived from the
concept of structural stabiIity determine the spontaneous formation of quali tat ively similar
spat io-temporal s tructures in systems of various geneses exhibiting cri t ical behavior
( DANGELMAYR & GI~TTINGER, 1982; THOM, 1975; POSTON & STEWART, 1978; GI21T -
TINGER & EIKEMEIER, t979; STEWART, 1981). In addition, this return to a ' g eome t r i z a -
tion of p h e n o m e n a " - - af ter decades of a lgo r i t hmiza t i on - - comes much closer to the
geologist 's intuitive geometr ic reasoning. It is the aim of this study to elucidate, by
examples, how the quali tat ive geometr ical approach allows one to classify forms and
to control the behavior of complex computer algorithms.
1.1 MATHEMATICAL GEOLOGY AND ALGORITHMIZATION
The geometr ical approach dominated the "mathematizat ion" of geology until recently
the computer "changed the world". As VISTELIUS (1976) summarized in his discussion
of mathemat ica l geology:
"the restoration (or 18th century) of axiomatic ideas is due to maturation
of @eoloEical sciences .... The more mature the geological ideas in the problem
are, the more the mathematical tool is determined by the Ecological meanin E
of the problem. Less mature geological problems make it necessary to introduce
more routine mathematical means with restricted foundation to form geology".
Here, two types of "ma thema t i ca l applicat ion" occur: The m a t h e m a t i c a l a t t e m p t
to "model" a speci f ic o b j e c t - - the classical method of theore t i ca l physics which can
lead to the formulat ion of physical l a w s - - and "rout ine m a t h e m a t i c a l means" which
can most ly be t r ans l a t ed as " s ta t i s t i ca l methods". And, case s tudies by compute r are
widely viewed as the " m a t h e m a t i c a l methods". Usually, descr ip t ive s t a t i s t i ca l resul t s
are t r e a t ed like "physical laws", a s i tuat ion s trongly cr i t ic ized by THOM (1979). He
gave the following reasons why the tool of m a t h e m a t i c s looses i ts s t reng th as one goes
down the scale of sciences:
"... the first is that those sciences which do not have as efficient tools
at hand as physical laws would like to be like physics and try to appear in
the eyes of other people as precise as physics. Every science wants to become
mathematized because it believes that way it would be put on the same footin E
as fundamental physics .... The second, internal reason now works in the re-
verse sense: Inasmuch as a given science does not allow for precise mathemati-
zation it opens practically indefinite workin E possibilities to scientists
in that field, because they can make models of all kinds, with approximations,
statistical hypotheses, and so on, and there is practically no limit to the
possibility of buildin E models in situations which actually do not allow for
specific, exact quantitative models .... And the third reason, of course,
is the computer industry's lobby: Every laboratory wants to have its own com-
puter workin E even in situations where a priori there is no reason to believe
that you can extract any kind of useful information out of the things you
have put into the computer. "
It was not Thom's aim to b lame those sciences which are not as precise as phys-
ics. R a t h e r this was d i rec ted against the degradat ion of the m a t h e m a t i c a l t o o l - - may
be the reason is tha t topologists like Thom "want q u a l i t i e s - - though these somet imes
acquire a fearsomely a lgebraic , even numerical , expression" (POSTON & STEWART, 1978).
Of special in t e res t is Thorn's second argument , the indef ini te working possibili t ies. It
is always a very s tr iking exper ience in applying "computer methods" tha t some of these
methods allow for various and con t rad ic to ry in te rp re ta t ions of the same d a t a - - and,
fu r the rmore , tha t some methods can even be inf luenced by the ordering of the input
data . Such observat ions were the s ta r t ing point to analyze the qual i ta t ive behavior of
propagated a lgor i thms in geology.
In geology and paleontology s t a t i s t i ca l and approximat ion methods are general ly
used as s t r a t eg ies of p a t t e r n recogni t ion. A densi ty dis t r ibut ion is e s t ima ted from a
sample, a sur face is r econs t ruc t ed from s c a t t e r e d da ta points, per iodici ty pa t t e rns of
profi les are analyzed by means of s t a t i s t i ca l t ime series analysis, e tc . Al te rna t ive ly ,
data are sorted, grouped and classified by using factor analysis, cluster or discriminant
analysis, and so forth. These are the fields where "routine mathemat ica l methods" domi-
nate, and it is the field where the computer allows one to analyze everything without
regard to any a priori scient i f ic meaning and without the formulation of a scient i f ic
hypothesis. During several years of work with the computer, and implementing computer
programs at the 'Sonderforschungsbereich 53, PalOkologie -- University Tt~bingen', it
was a challenge to accept that ra ther sophist icated pat tern-recogni t ion programs may
become ra ther unstable if some initial conditions, e.g. the input data, do not sat isfy
the proper conditions, and that it is, in general, not known what the "proper conditions"
are. On the other hand, such computer work allowed me to collect and to analyze exam-
ples of instable procedures and problems of in terpreta t ion. A collection of such examples
is presented here together with the 'qual i tat ive ' analysis of instabilities.
1.2 SYNTAX AND SEMANTICS
Af te r decades of algorithmization in science the computer provides a valuable
and indispensable tool. Much work has been invested in computer science to find rules
for the verif icat ion of program correctness . The idea is to solve the programming problem
"by decomposing the overall problem into precisely specified
subproblems and then verifying that if each subproblem is solved
correctly, and if the solutions are fitted together in a speci-
fied way, then the original problem will be solved correctly"
(ALAGACIC & ARBIB, 1978}.
Thus, it seems not very difficuIt to construct "correct" programs -- as far as the
syntax is concerned (WIRTH, 1972). The other problem, however, is a semant ic one:
The meaning of a computer output is not defined -- no ma t t e r how cor rec t the syntax
may be -- until the meaning of the input is defined and until the input is consistent
with the operations within the algorithm. In the same sense, the formulation of a program
is usually not only a syntac t ic problem, as in most cases semantics is initially involved to
some extent . The problem, however, is not res t r ic ted to computer applications in a narrow
sense: It occurs whenever "formulas" are applied to data. In addition to the "correctness"
of algorithms, therefore , the problem of the cor rec t application of algori thms arises and,
fur thermore, the question of how to "control" the computat ions. These are quali tat ive
problems because semantics i tself is qualitative.
The problem and the necessi ty of a lgori thm-control in the field of geological appli-
cat ions will be elucidated by a collection of examples. The material is ordered in three
chapters . These a t t empt to re la te the observed instabili t ies with current areas of research
in topological, i.eo geometr ical , areas. Somet imes the examples are only weakly connected
with the theoret ica l introduction to each chapter , as a theoret ical classif icat ion is not
yet available for f inite point sets from which most of the examples arose. However, it
will be e lucidated that it is commonly a question of the viewpoint - - the question what
we assume as variables and what as parameters - - if we classify a problem as a d iscre te
or a d i f ferent iable system. Such systems are commonly accounted whenever ~stability v
problems arise:
Branching solutions, i.e. bifurcations, can be de tec ted in many classical procedures:
like Chi2-test ing of directional data, surface reconst ruct ion from sca t t e red data points
and equal distance sampling in sedimentology. The widely used centroid clustering methods
turn out to provide an excel lent example of chaot ic behavior on finite point sets, Their
s ta t is t ica l value is strongly questioned because they lack structural stability. Smoothing
of directional data on a sphere and the classical Chi2- tes t for or ientat ion data fur ther
provide examples of a degenera ted bifurcation problem. A br ief discussion of i te ra ted
maps gives a connect ion to present areas of research.
The application of the concepts of s t ructural stabili ty and of ca tas t rophe theory
to ref lec t ion seismics provides a classif icat ion of s t ructural ly stable singularities in two
dimensions. The analysis of image inversions in t e rms of the local curvature of the re-
f lector and its depth produces a catalogue of images which allows a detailed, semiquanti-
ta t ive on-si te survey of the t rave l t ime record. For the geologist it can provide a f rame-
work for his quali tat ive s t ructural in terpreta t ion.
The concept of s t ructural s tabil i ty also provides new insights in paleontological
and evolutionary problems, e.g. the analysis of the "morphospace" of paleontological
species or the "bifurcation" of species. However~ such models are quali tat ive in the
narrow sense of this term and provide ra ther a framework for further analyses which
may te rmina te in models which can be t e s t ed exper imental ly or s tat is t ical ly .
Mathemat ica l details are ignored as far as possible: The object is to convey the
Vspirit~ of s t ructural s tabil i ty and re la ted fields, and its application to geological pa t te rn
recognition problems. As far as mathemat ics is required, it is kept to a minimal l e v e l -
examples of various fields are thought to be of more interest than the mathemat ica l
theory which has been summarized in various textbooks.
1.3 STABILITY
~Pattern recognition problems v as used here, cover a wide field of ~deformations t
and VinstabilitiesL Various types of pa t te rn recognition problems -- which are usually
solved by computer methods - - are analyzed in te rms of ttopological stabil i ty v. The term
ttopological stabil i ty ~ or ~structural s tabil i ty t means that the pa t te rn does n o t drast ical ly
change under a small disturbance (ANDRONOV et aL, 1966; THOM, 1974; NICOLtS &
PRIGOGINE; 1977). However, pa t t e rn recognit ion problems may result even if the disturb-
t 0
i m
m
0
0
a ~ - - m i g r a t i o n ~ [..) ~ " - - ~
Fig, 1.1: The phylogenet ic history of horses: (a) the classical gradual phylogeny a f t e r SIMPSON {1951); (b) the same phylogeny redrawn along a modern t ime scale. In t e rms of evolut ionary veloci t ies two quite d i f fe ren t "modes of evolu- tion" are represen ted by the two figures. However, in t e rms of p h y l o g e n y - the relat ionship be tween s p e c i e s - - the t r ans format ion is s t ruc tura l ly s table as all pa thways remain the same.
a n c e , the t rans format ion , is s t ruc tura l ly s table . Such an example is given in Fig. 1.1. The
phylogeny -- the evolut ionary history -- of horses is one of the most ce leb ra ted examples
of gradual Darwinian evolution, and to some ex ten t of direct ional se lect ion {Fig. 1.1 a).
However, if the t ime scale used by SIMPSON (1951 and others) is replaced by the absolute
t ime scale under cur ren t use, the phylogenet ic pa t t e rn changes dramat ica l ly with respec t
to mode and veloci ty of evolution (Fig, 1.1 b). All s ignif icant "evolut ionary events" are
now concen t r a t ed within very narrow t ime intervals . What does not chang% is the princi-
pal s t ruc tu re of the phytogenet ic lineages, i.e. the ances to r -descendan t relat ionships
are s t ruc tura l ly s table . The ttime~ axis in Fig. 1.1 b is deformed like a rubber s t r ip
which is d i f fe ren t ia l ly s t r e t ched without folding - - a purely topological deformat ion .
Although this deformat ion is purely topological and s t ruc tura l ly s table , it changes
the ' s eman t i c ' i n t e rp re ta t ion of the evolut ionary mode. While Fig. 1.1 a indicates a
slow gradual evolut ion under a long- term changing envi ronment , Fig. 1.1 b indica tes
periods of ' s tas is ' with l i t t le morphological evolution which are in te r rup ted by short t e rm
in tervals of rapid evolut ionary change and associa ted speciat ion events . A simple, however
not t r ivial , t r ans fo rmat ion of the scale, thus, may t ransform a gradual is t ic p ic ture into
a punc tua ted one (cf. STANLEY, 1979).
S t ruc tura l s tabi l i ty in a more precise sense can be r e l a t ed to the topological simi-
lar i ty of the " t ra jec to r ies" of a process in this phase-space (NICOLIS & PRIGOGINE, 1977;
HAKEN, 1977). The " internal dynamic" of a process is usually descr ibed by d i f fe ren t ia l
equat ions which usually depend on some pa rame te r s . In many physical i n t e rp re t a t ions
these p a r a m e t e r s can be ident i f ied with some s t a t e of the env i ronment of the system,
i.e. they depend on various kinds of d is turbance act ing continuously on the system (HA-
KEN, 1977). As the sys tem and/or its env i ronment evolves, some of these p a r a m e t e r s
can change smoothly or suddenly, and during such a change the principal behavior of
the system can change.
homogeneous ~ ~" ed
A B strain stra/n
Fig. 1.2: Stress-strain diagrams of deep-sea sediments: a) homogeneous sedi- ments; b) s t r a t i f i ed sediments . Within each sequence compac t ion and overload increase(modi f ied from BAYER, 1983).
In physical sys tems not uncommonly a threshold occurs which, when passed, causes
a sudden change of the behavior of the system. The most d ramat i c change in a dynamic
sys tem is tha t its t r a jec to r ies in the phase-space change the i r topological configurat ion.
By a small pe r tu rba t ion of a pa rame te r , the system then devia tes widely from the initial
s i tuat ion. Fig. t .2 i l lus t ra tes this s i tuat ion roughly by the " t ra jec to r ies" of a s t r ess - s t ra in
diagram. The exper iments were per formed with a ro ta t ing vane (with the vanes inser ted
paral lel to the bedding planes of sediments; cf. BOYCE, 1977; BAYER, 1983). In the
s t r e s s - s t r a in diagram -- the 'phase plane ' of the process -- two qual i ta t ive ly very d i f fe r -
ent p a t t e r n s were observed (Fig. 1.2; BAYER, 1983) depending on the " s t r a t i f i ca t ion"
of the sediments . In homogeneous sediments the s t ress - s t ra in curves are smooth while
in s t r a t i f i e d s e d i m e n t s a sudden b reak occu r s . Wi th in t he r a n g e of m e a s u r e m e n t s t h e s e
two t y p e s a r e i n d e p e n d e n t o f t h e c o m p a c t i o n (preloading) o f t h e s e d i m e n t s , i .e. w i t h in
e v e r y se t t h e t r a j e c t o r i e s evo l ve s m o o t h l y and s t r u c t u r a l l y s t a b l e wi th i n c r e a s i n g c o m -
pac t ion . However , t h e o t h e r o b s e r v e d p a r a m e t e r , t h e l a m i n a t i o n or s t r a t i f i c a t i o n o f t h e
s e d i m e n t s , c a u s e s an e s s e n t i a l c h a n g e in t h e m o d e o f fa i lu re : A ve ry d i s t i n c t po in t o f
f a i l u re a p p e a r s in w e l l - s t r a t i f i e d s e d i m e n t s wi th a " sudden j ump" in t h e s t r e s s va lues .
T h e r e is no c o m m o n s e n s e w i t h r e g a r d to t he t e r m ' s t a b i I i t y ' . As H O C H -
STADT (1964} no tes :
"Often i t . is not necessary to determine the explicit solution
to a problem, but it is important to be able to say something
about the solution ... In many physical problems one is motivated
by the feeling that a small change in the conditions of the
problem should result in a comparably small change in the solu-
tion .... The word stability is a very tricky word. No one defi-
nition seems to be adequate for all purposes."
. • . " . % 0 . ° . ~
, A " ~ ' ~ " ~ ' " - . 4 " t .~.:--...
2. N O I S Y S Y S T E M S A N D F O L D E D M A P S
Many geological and paleontological problems are r e l a t ed to the recons t ruc t ion
of some anc ien t s t a t e from the present remains. The present s ta te , however, is usually
noisy as various fac tors may have inf luenced the sys tem during its history. This s i tuat ion
comes very close to the r econs t ruc t ion of de formed signals in informat ion theory. In
the theory of informat ion the d is turbance of signals by random (white) noise pIays an
impor tan t role (YOUNG, 1975). The s tab i l i ty problems of such noisy systems can be
NOISE
X
Fig. 2.1: Dis tu rbance of signals by random, whi te noise. The init ial signals are well s epa ra ted points or suff ic ient ly small c i rc les on the {x,y)-plane at t ime t=0. With increasing t ime, whi te noise is added, and the area increases where the signals are found with some probabil i ty. Where these areas in te rsec t , two signals are in Competi t ion for the r econs t ruc t ion process.
visual ized in a three-d imens ional model like Fig. 2.1. An init ial signal is cha rac t e r i zed
as a point (or as a suff ic ient ly small circle) on a space plane (x,y). On its way to the
r ece ive r whi te noise is added. As a result , the signal is dr iven out of its original position.
When the random noise sums up during t ime, the signal will be found with a specif ic
probabi l i ty within a ce r t a in area which surrounds the original position of the signal.
A serious recogni t ion problem occurs when the probabi l is t ic neighborhoods of d i f fe ren t
signals s t a r t t o overlap. Indeed, a signal found within an in te r sec t ion area cannot be
r econs t ruc t ed with ce r ta in ty .
The signals, discussed so far, are isolated points which are originally located in
disjunct areas. Now, if we cover parts of the (x,y)-plane densely with signals so that
their initial areas of definition are connected along boundary lines, then another way
to formulate the recognition problem is more appropriate. The evolution of the signal--
space along the t ime axis can be described as a map
source ---> receiver .
The overlapping of the probability areas, in which a signal will be found {Fig. 2.1}, can
then be described as local folding of the original definition space. Fig. 2.2 i l lustrates
the local folding of the (x,y)-plane. Within the folded areas it is not possible to solve
" 7
: : . ' . .
a Fig. 2.2: A folded sheet as a model of a locally folded map (a). A continuous curve on a sheet may develop se l f - in tersect ions in the project ive plane of a folded sheet (b).
uniquely the inverse problem, the reconstruct ion of the 'original signal'. Fig. 2.2b illus -
t ra tes the distortion which can be caused by a local fold. A regular curve on the original
plane develops a se l f in tersect ion on the projection of the local fold. However, deforma-
tions of this type will be discussed in more detail in the last chapter .
In an informat ion- theore t ic approach the disturbance of signals is due to random
forces, and the reconstruct ion process is mainly a s tochast ic problem. The possible drift
of signals or par t ic les is governed by a probability d i s t r i b u t i o n - - the classical example
is the Brownian movement of part icles in a fluid. In geology and paleontology problems
of this type arise mainly if global propert ies of distributions are recons t ruc ted from
s tochas t ic samples by local es t imat ion methods. In this first chapter , pa t t e rn recognition
problems will be elucidated by examples which are related to the superposition of densi ty
functions and to double- (multiple-) valued local solutions of reconstruct ion processes.
In the first example s ta t is t ica l problems are discussed in terms of the reconstruct ion
of original sediment volumes and sediment accumulation rates. The system bears three
10
types of disturbances: Uncer ta in ty about the datum points; s tochast ic components , and
a sys temat ic t rend to under-consolidation of sediments with low overburden.
Then the analysis of ' in t raspeci f ic variability of paleontological species ~ is discussed
in te rms of a probabilistic ontogenet ic morphospace. It will turn out that the covarianee
s t ructure between (measurable) features within the ontogenet ic morphospace can be help-
ful for the taxonomist . But, we will see further tha t one cannot expect linear relat ion-
ships between the f e a t u r e s - - a nonlinear theory seems appropriate ra ther than just
an analysis by the so-ca l led higher s ta t is t ica l methods.
In a quali tat ive discussion of the analysis of directional data the problem will be
to find an optimal weighting function for the reconst ruct ion of a smooth density distribu-
tion from sca t t e red data. It will become clear that the cri t ical areas of the reconst ruc-
tion are in tersect ions of the areas~ on which the weighting function is defined,
In the next example the reconstruct ion of surfaces from sparse point pa t te rns by
computer methods is discussed. It will be shown that the problems which arise in this
con tex t are mainly of a geometr ical nature. Therefore , the algori thmizat ion of the recon-
s t ruct ion process is not trivial. The local es t imat ion methods in use provide no unique
solution, i.e. they are very sensit ive to small changes of the initial conditions. The
example, therefore , leads over to the next chap te r where this type of instabil i t ies is
discussed in more detail .
2.1 RECONSTRUCTION OF SEDIMENT-ACCUMULATION
The problem to reconst ruct accumulat ion- and sedimentat ion ra tes arises in sedimen-
tology in order to gather information about sea-level changes, c l imat ic changes, and
the evolution of basins. Fur thermore, accumulation and sedimentat ion ra tes clearly indi-
ca te hiatuses in the sedimentary sequence on the base of which local and global events
of the past are recognized (e.g. VAIL et aI., t977). However, several processes and as-
sumptions are involved in the reconstruct ion of which the most important ones are
** the dating of the sediment sequence
** the es t imat ion of the original sediment thickness without compact ion.
Both reconst ruct ions are biased and usually involve specif ic assumptions about the datum
points, the initial porosit ies and the consolidation s t a te of the sediments.
11
2.1.1 Accumulation Rates and Deformations of the Time-Scale
The computat ion of accumulation ra tes requires es t imates of the absolute t ime
scale, i.e. a suff ic ient number of datum points along the sediment column. As soon as
the datum points are given, the computation of the accumulation ra tes is ra ther simple,
age s tages accumulation rate cm/kyr stages accumul, ra te cm/kyr
0 I 2 3 4 t
14o [ K I M M E R I D - / GIAN
_1 O X F O R D I A N
15o
CALLOVIAN
160 BATIqONIAN
BAJ OCIAN
I70
AALENIAN
TOARCIAN
-- 180 PLIENS- BACHIAN
S1NEMURIAN
- - 1 9 0 1 HETTANGIAN
- . J r -
t
I _ _ J
m
m
l _ _ _ I
I - - - -
t- a
K I M M E R I D G I A N
OXFORD,
! L . . . . . . . . . , ,
I . . . . . . . . . . . . . . '
CALLOV0
BATHON.
BAJOCIAN
AALENIAN
TOARCIAb
PLIENS- BACHIAN
SINEMUR.
HETTANG.
[]
HI
t l . , .
~ Cyclic accumulation rates of sediments in the South German Jurassic. on the t ime scale of VAN HINTE (1976); b) based on an a l tered
Jurassic t ime scale {see text for explanation). On both scales a cycl iei ty is obvious, however, in (b) the cycles are much more regular (about 4 Ma), and a superimposed megatrend appears.
b
12
i.e. it is the quot ient
sediment thickness
t ime interval of depos i t i on .
Fig. 2.3 gives such accumula t ion ra tes for the South German Jurass ic and i l lus t ra tes
how they subdivide the Jurass ic sequence into gener ic deposit ional cycles. However,
the pa t t e rn is not invar ian t against deformat ions of the t ime scale. Essent ial ly the same
s i tuat ion arises as was discussed for the revolution of horses v if the t ime scale is a l te red
(Fig. 2,a A,B).
The accumula t ion r a t e s in Fig. 2.3A have been ca lcu la ted using the t ime scale
of VAN HINTE (1976). However, for the t ime- in te rpo la t ion within s tages a more r ecen t
b ios t ra t ig raphic subdivision (COPE e t al., 1980) was used (McGHEE & BAYER, in press}.
A deformat ion of this t ime scale (Fig. 2.aB) a l te rs the cycl ic i ty drast ical ly, and the
pa t t e rn becomes much more regular . In addition, a well pronounced megacycle becomes
visible (Fig. 2.3 B).
Van Hin te ' s Jurass ic t ime scale is based on e s t ima tes of the upper and lower bound-
ary of the Jurass ic and of one addit ional ' ca l ib ra t ion point v at the middle of the Jurass ic
(base of the Bathonian). Between these points he divided the t ime scale l inearly by the
number of ammoni t e Z o n e s - - with the resul t of an average durat ion t ime of IMy for
each ammon i t e Zone. Now, s ince he published his t ime scale, the b ios t ra t ig raphic scheme
has been a l tered, and the re fo re the durat ion t ime var ies from Zone to Zone. However,
by shif t ing the addit ional -- middle Jurass ic -- ca l ibra t ion point into the Callovian the
original assumption of 1My/ammoni te Zone can be res tored for the Lower and Middle
Jurassic . This has been done in Fig. 2.3B, and this simple t r ans fo rmat ion genera tes the
except ional cycl ic pa t t e rn which agrees with o therwise es tabl i shed cycles of s imilar
phase length (cf. EINSELE or McGHEE & BAYER in BAYER & SEILACHER, eds., in
press}.
As discussed ear l ie r for the evolut ion of horses, the cycl ic i ty per se is a s t ruc tu r -
al s table p a t t e r n which is preserved under topological t r ans fo rmat ions of the t ime scale
(even if o the r proposed scales are used, e.g. HARLAND et al., 1978), while the regular i ty
of the cycles, the i r phase length and m a g n i t u d e - - i.e. all p roper t ies from which we
can ga ther in format ion about the veloci ty of the p r o c e s s - - change as the scale is
changed.
2.1.2 Es t ima t ion of Original Sediment Thickness
One impor tan t process ~ which a l t e r s the physical proper t ies of sediments~ is compac-
13
Sediment Water Content (w) Bulk Coml~sition and Porosity (p) Density
{%) (%) (g/cm 3)
0
c~ 4OO
5 0 0 ~ ~ 1
Fig. 2.4: Depth-logs for sediment
20 40 60 80 100 20 40 60 80 1.4 1.8 2.2 I I ¥ ] I I r "
Computed Grain
Density (g/cm 3}
2.4 2.7 3.0
fi
Sonic Velocity
(m/s)
1.4 2.0 2.6
composi t ion and physical proper t ies for DSDP-si te 511 (adapted from BAYER, 1983). Data are mean values for cores (D: diatoms, N: nannofossils, c: clay content , O: o ther components) .
tion under the overburden of la te r deposits. Especially in clays the physico-chemical evolu-
t ion is dominated by compact ion. Fig. 2.4 i l lus t ra tes how the physical proper t ies in a
sediment column change with depth {i.e. overburden). In the example given, the sediment
column below 200 m depth is dominated by clay, and within this column the physical
pa rame te r s porosity and wa te r - con t en t decrease continuousIy as the sed iment becomes
increasingly compacted . In the same course the densi ty of the sediment increases and
tends slowly towards the mean grain density of the sediment .
tf one assumes tha t the void volume of the sediments is in equilibrium with the
overburden, then a f irst approximation for the equilibrium curve of compact ion can be
given by the equat ion
dn dp + cn = 0 (2.1)
{where n: porosity = re la t ive void space; p: pressure or overburden). The cons tan t 'c '
can be in te rp re ted as a coef f ic ien t of volume change (TERZAGHI, 1943), which is specif-
ic for par t icu lar mater ia ls . In tegra t ion gives a simple declining exponent ia l funct ion
n = no exp (-cp). (2.2)
14
n 1.o
0.8
0.5
0.,~
0 2
0 .0 m
0 100 200 300 ZOO 500 500 200 800
Fig. 2.5: Porosi ty data from Fig. 2.4 with least square f i t ted t rend lines (see tex t for explanat ion) .
This model has been used in Fig. 2.5 to e s t i m a t e t h e decl ine in porosity with respec t
to depth (i.e. the change in bulk densi ty has been neglected, cf. Fig. 2.4). Empir ical ly
the da ta are well approximated. Besides the mean t rend line some more t r a j ec to r i e s
are given in Fig. 2.5~ which have been cons t ruc ted under the assumption tha t the coeff i -
c ient of volume change is cons tan t while the init ial porosity of the sediments may have
been var iable and, thus, cause the s ca t t e r ing of the da ta points.
If we assume tha t the sed iment is everywhere in equil ibrium with the overload,
it is no problem to e s t i m a t e the thickness of the sediment column which would resul t
wi thout compac t ion (e.g. MAGARA, 1968; HAMILTON, 1976). Fig, 2.6 i l lus t ra tes how
the two principal componen ts of a sediment change under pure compact ion: The volume
vn
vs
Fig, 2.6: The two components of a s e d i m e n t - - voids and s o l i d s - - during
compact ion. Vn: volume of voids, Vs: volume of solids, p: pressure = overload.
15
of solids remains constant while the volume of voids decreases. The porosity is defined
as the re la t ive volume of the voids so that
V n = nV and Vs= ( 1 - n ) V . (2.3)
where n: porosity, Vn: volume of voids, and Vs: Volume of solids. Because the volume
of the solids is not changed by compaction, one has
and, therefore ,
Vs(t=O ) = Vs(t )
(l-no)Vo = (l-n)V
from which we immediately have the compact ion number
V 1-no C = Vo - 1 , n (2.4)
and the decompact ion number
D = Vo l-n V - 1-no -
The decompaction number allows to compute
layer if we know its original porosity no, i.e.
1 C ' (2.5)
the original thickness of any sediment
V o = Dr. (2.6}
The thickness of the ent i re sediment column then can be computed by summing up all
sediment layers or, if regression curves are used as in Fig. 2.5 by evaluation of the inte-
gral
! - n V f z l - n ( z ) dz (2.7) i=l l-no i or zo 1-no
Both techniques have been used in Fig. 2.7 whereby the original porosities of the samples
have been es t imated from the intersect ion of their associated t ra jec tory (from Fig. 2.5)
with the zero-depth line. In a s ta t is t ical sense the t ra jec tor ies of Fig. 2.5 are error
bounds to the mean regression line (probabilities can be a t tached to them by standard
s ta t is t ica l techniques}, and so the curves in Fig. 2.7 can be interpreted. Thus, the recon-
s truct ion of the original sediment amount is simply a s ta t is t ica l process. However, it
closely resembles the si tuation of Fig. 2.1. The data are biased by the sampling tech-
nique as well as by the laboratory technique. Now, if we add a small error to a data
point, it will not a f fec t the results much if the overburden (or depth) is small. However,
as the overburden increases, the t ra jector ies in Fig. 2.5 come closer and closer. An error
of the same magnitude, therefore , biases the results increasingly.
16
15
km t 4
1.2
1.0
0.8
0.6
0.4
0 2
0.0
thickness
n = 0 9
/
depth
o 1oo 20o 3oo 4oo 500 6o0 zoo m
0 8 5
0.8
075
Fig. 2.7: Decompac ted sediment thickness based on the data of Fig. 2.5 (m: mean trend, n=0.9 etc . : in tegrals of the t rend lines in Fig. 2.5.
While the recons t ruc t ion of the init ial s t a t e of the sediments depends on how far
the "compact ion machine was run", we can, on the o ther hand, atways find the output
if the "machine would work unti l infinity". This s table limit, of course, is simply the
volume of solids, and the "dry sed imenta t ion ra tes" (of. SWIFT, 1977) are, of course,
only biased by the t ime scale and the labora tory technique.
2.1.3 Underconsol idat ion of Sediments -- a History Ef fec t
Es t imat ions of the original sed iment volume are usuatly based on the assumption
tha t the consol idat ion s t a t e of the sediment is in equilibrium with the overburden. In
this case, as was pointed out, the e r ror of the es t imat ion should increase with increasing
overburden. However, if burial depth is small, then the t ime-dependen t flow of the pore-
wa te r cannot be neglected; it bears on our unders tanding of the widespread underconsol-
idat ion of r ecen t sediments , which is observed even under slow sed imenta t ion ra t e s (MAR-
SAL & PHILIPP, 1970; EINSELE, 1977).
The consol idat ion of sed iments is descr ibed by Terzaghi ' s model (e.g. TERZAGHI,
1943; CHILINGARIAN & WOLF, eds., 1975, 1976; DESAI & CHRISTIAN, 1977). In a
one-dimensional sed iment column and under the assumption of a cons tan t coef f ic ien t
of consol idat ion Terzaghi ' s model takes the form
17
3p = a2p
- - m -
a t 3x 2 ' (2.8)
where p: the excess pore water pressure due to overload, m: the consolidation coeff i -
cient, and t: t ime. If this model is discret is ized in space, i.e. if the sediment column
is divided into small discrete e lements , then the partial different ial equation is t rans-
formed into a set of ordinary different ia l equations:
dP x d--~ = m(Px-Ax- 2Px + Px+fXx)" (2.9)
Now, if one reduces the system to a single e lement -- a situation which occurs in labora-
tory exper iments - - then we can rewri te equation {2.9) as
dp --+ cp = i(t), (2.10) dt
where the right side describes the " i n p u t " - - i.e. the f l u x e s - - at the boundaries of the
e lement as a function of time, and with free boundary conditions (I(t)=O) a suddenly
imposed pressure declines exponentially with time.
The idea of Terzaghi 's model is that a sudden imposed load increases initially the
pore-water pressure (excess hydrostat ic pressure) and that this pressure decreases a f t e r -
ward due to a loss of pore-water from the e lement whereby the excess hydrostat ic pres-
sure is t ransformed into a pressure at grain contacts . Associated with the loss of pore--
water is an increase in the number of grain contacts . Therefore, the sediment approaches
a new equilibrium s ta te a f te r compression which, of course, is usually not reversible.
The reduction of volume is res t r ic ted to the volume of voids, and the change in pore
volume is simply proportional to the decline in the excess hydrostat ic pressure:
~n ap __dz = m--dz . (2.11) at ~t
Thus, we can solve equation (2.10) in terms of the pore volume, which in case of free
boundary conditions takes the form:
-ct V(t) = Ve+ (Vo- V e) e (2.12)
for a load which is suddenly applied. The load is here represented by the equilibrium
volume V e (cf. equation 2.2), and the excess hydrostat ic pressure is proportional to the
reducible void volume (Vo-Ve). Now, if at t ime t=t 1 an additional load is applied, then
equation (2.12} takes the form
V(t) = Ve2+ (V(t I) - Ve2) e-C(t-tl) , (2.13)
18
which can be r ewr i t t en if V(t l ) is inser ted from equat ion (2.t2):
V(t) = Ve2+ (Vel- Ve2)eCt2 e-at+ (Vo-Vel)e -ct " (2.14)
As this equat ion shows, the re is some remaining reducible porevolume from the f irst
loading even t , which has to be taken into considera t ion. If fur ther load is added in dis-
c r e t e s teps, we ar r ive finally at
I"I
V ( t ) = V i + ( [ ( V i _ l - V i ) e C t i ) e - c t
±=1
(2.15)
which i l lus t ra tes how the ear l ie r loading s t a t e s con t r ibu te to l a t e r s ta tes . The equil ibrium
can only be approached if the t ime in tervals be tween Ioading are suff ic ient ly long, o ther -
wise the sed iment layer will be underconsol idated. This history e f f e c t of loading is illus-
t r a t e d in Fig. 2.8 for various t ime in tervals be tween loading events . The excess hydros ta t -
ic pressure {p in Fig. 3) develops c lear ly a maximum which degene ra t e s to a s imple
declining exponent ia l funct ion for a single loading event and to a sequence of such single
events , as the t ime in tervals be tween loading become large.
10ad. ~ t i i, i i 1 i j i i i i l i i i i i
V
\
i
k
m , . .
t t 1 t
Fig. 2.8: Responce of a single sed iment layer under s tepwise loading when loads are apptied a t d i f f e ren t t ime intervals : t ime in terva l dec rease from lef t to r ight; r ight: a single load. V: void volume; P: momen ta ry reducible void volume which will vanish even if no addi t ional load is applied; C:equi l ib- rium void volume for every loading event . At the top the loading in te rva ls are marked, the to ta l applied load is cons tan t for all ' exper imen t s ' .
With respec t to the previous discussion we have, the re fore , to expect tha t e s t ima te s
of original sediment volume are biased by the t ime-de lays in the consol idat ion process,
the p a r a m e t e r s t i in equat ions (2.13) and (2.14) have, of course, the s t ruc tu r e of a t ime
delay. Fur the rmore , if the pressures at the boundaries of the sed iment layer are not
zero, i.e. if the sediment layer is a segment within a sediment column, then the t ime-
19
delay e f f ec t increases further . In case, the permeabil i ty of the sediment is low, the
excess hydrosta t ic pressure will s tay for ra ther long t ime near the values of the over-
burden, and the t ime lack between loading and equilibrium consolidation causes a cont in-
uation of pore-water flow when sedimentat ion has stopped. On the other hand, if we
consider a two- or three-dimensional system of strongly underconsolidated sediments,
any spatial disturbance like unequal loading can initialize an instable flow of pore-water ,
which may lead to fluidization or Iiquidization of the upper sediment layers.
2.2 INTRASPECIFIC VARIABILITY OF PALEONTOLOGICAL SPECIES
~n 1966, WESTERMANN observed that in several ammonite stocks -- a group of
cephalopods (Fig. 2 . 9 ) - - a specif ic intercorrelat ion of morphological fea tures occurs:
~Of particular interest is the inter-correlation between costa-
tion, whorl section, and coiling which has been observed in dif-
ferent, unrelated ammonoid stocks and cannot he satisfactorily
explained" (WESTERMANN, 1966).
Fig. 2.9: Ectocochl ia te cephalopods, lef t recent Nautflus and two ammonites with well marked ontogenet ic changes in morphology.
Because BUCKMAN (1892) observed, probably for the first t ime, this part icular type
of covariat ion (intercorrelation) between the ornament and the whorl section in ammonites,
Westermann named this relationship 'B u c k m a n' s 1 a w o f c o v a r i -
a n c e ' . In some cases, the 'covariance ' extends to other features:
"in general the complexity of the suture-line increases in pro-
portion to the decrease of ornament"
This caused Westermann to establish 'Buckman's second law of covariance'. Proceeding
in this way, any correlation between features, which cannot be satisfactorily explained,
would lead to a new 'law', and 'experiments' with other ammonite stocks would soon
disprove the specific correlation sufficiently 'to be a law'.
20
1 D 1-8
1.0
• e e • • •
• • • •
, e e , ~ •
• e e e •
• • • •
• , • e
e
• j
,I o •
e • •
o
• • •
DSP. b ' ~ - - - ~ ' ~ - ~ - ~ - ~ 2 6 o I 50 100 m m
Fig. 2.10: Covar ia t ion of o rnamen t and c ross -sec t ion of Sonninia (Euhoploceras) -
adicra (Waagen), modified from WESTERMANN (1966). The s c a t t e r g r a m shows tha t the morphotypes cover a cont inuous area in the p a r a m e t e r space; D: Raup 's morphological p a r a m e t e r " ra t io of whorl height to whorl width", DSP: end d i ame te r of the spinous s tage.
On the o ther hand, Wes te rmann was able to show, by means of the eovar ia t ion s t ruc-
ture, t ha t 80 descr ibed species of the subgenus S o n n i n i a ( E u h o p l o c e r a s ) belong to a single
species and tha t the observed var iabi l i ty must be viewed as an in t raspec i f ic property .
His b iomet r ica l study {cf. Fig. '2.10) shows tha t the specimens of this lumped species
fill a cont inuous area in the p a r a m e t e r space (DSP, I/D) and tha t the cos ta t ion types
or ' f o rma ' are regular ly a r ranged within this p a r a m e t e r space (of. Fig. 2.10 for explanat ion
of parameters ) .
The covar ia t ion pa t t e rn described by WBuckman's law w is not unique within the ammo-
ni tes , but it is also not universal . Addit ional s tudies (e.g. BAYER, t977) show tha t in
some cases Wage e f f ec t s w may play some role and tha t there are some special condi t ions
which make 'Buckman ' s law' easi ly visible. One of these condi t ions is t ha t the morphology
changes s t rongly during ontogeny (cf. Fig. 2.9 for cases of r a t h e r s t rong on togene t i c
changes). The avai lable in format ion makes i t likely tha t the observed cor re la t ion is due
to oblique sec t ions through the on togene t i c morphospaee because t ime is not accessible .
The problem tha t age is not avai lable in paleontology is well known; GOULD (1977} discus-
ses in deta i l the problems, which arise, if equal sizes but d i f fe ren t ages of specimens (and
species) are compared by the a l lomet r ic relat ionship.
21
Evidence for an age control of 'Buckman's law' comes from additional features of
the s h e l l s - - the spacing of growth lines and s e p t a - - which both are likely formed in
ra ther regular t ime intervals. Especially the spacing of septa (which is more easily ana-
lyzed) shows a close relationship to cross-sect ion and sculpture in cer tain ammonites
(BAYER, 1972, 1977). Fig. 2.11 i l lustrates such a relationship between spacing of septa
and shell morphology.
S
70°
5o
3o
to
0:2 oL5
:g
r l i i
"t 5 lo 20 m m
Fig. 2.11: Relationship be tween spacing of septa and morphology in ammoni tes (modified from BAYER, 1972). s: angular distance of septa; r: radius of the shell.
2.2.1 Allometric Relationships
If one accep ts the hypothesis that 'Buckman's law' describes a phenomenon of intra-
specif ic variation, we should be able to deduce it from more basic biological principles.
Everyday exper ience on living organisms shows that most morphological features change
with age and that the relationship be tween two morphological features (which can be
quantified) leads usualiy to an al lometric relationship, i.e. a relationship of the form:
y = ax b or log(y) = log(a) + bx. {2.16)
Actually, the al lometr ic relationship can be t raced fur ther down to the, say,
' f i r s t p r i n c i p 1 e s o f g r o w t h' (HUXLEY, 1932), The term ' f i rs t
22
principle ' is here used in the sense tha t i t is very likely to observe such an a l lomet r i c
re la t ionship. As HUXLEY not iced, two measu remen t s (organs etc.) are in an a l lometr ic
re la t ionship when they both grow exponential ly, i.e. let Yl' Y2 be the two measurements ,
which grow exponent ia l ly
Yl = a l e e l t ; y2 = a 2 e C 2 t , (2.17}
then by e l imina t ing t ime we find the a l lomet r i e re la t ionship
= (Y2) c l / c 2 Yl a l ~ '2 " (2.18)
Now, s t r ic t ly a l lomet r i c growth resul ts also in more sophis t ica ted growth models like
the "Oomper tz model". In this model one assumes tha t the p a r a m e t e r ' c ' is not cons tan t
but decreases with age. Growth then can be descr ibed by a pair of d i f fe rent ia l equat ions
dy dc d--'t + c ( t ) y = 0 and an equat ion like d--t = -c . (2.t9)
The growth p a r a m e t e r ' c ' can be any funct ion of t ime, which goes to zero for large t ime
values {ideally as t ime approaches infinity). Especially, any s tab le output of a l inear con-
t rol system (e.g. homogeneous l inear d i f fe ren t ia l equations) provides a possible input for
the growth pa rame te r . A pe r fec t a l lomet r ic re la t ionship resul ts whenever the two organs
under cons idera t ion are cont ro l led by the same mechanism, i.e. if the i r growth equat ions
take the form:
dy dx dt c(t)*ay = O; d-'t- c(t)*by = O; (2.20)
by e l imina t ing t ime one finds the pe r fec t a l lomet r ic relat ionship
dy a y o r y = XoX a / b . dx - bx
In both cases considered so far the a l lomet r ic re la t ionship descr ibes the re la t ionship b e -
tween two growing organs in the phase-plane, i.e. the t r a jec to r ies of growth without con-
s idera t ion of the veloci ty of growth. Indeed, we may still fu r ther genera l ize the re la t ion-
ship to pairs of l inear d i f fe ren t ia l equat ions like
dy dx f(t)~-~ = ax + by; f(t)~ T = cx + dy, (2.21)
and the re la t ionship be tween the two measu remen t s takes the form
dy ax + by (2.22) d-x = cx + dy
23
- (a + d)
ters
Fig.2.12: Relat ionship be tween type of equilibrium a n d coef f ic ien t s of a pair of f i rs t order d i f fe ren t ia l equat ions {equation 2.21}. The type of equilibrium depends on the eigenvalues of the coef f ic ien t mat r ix of equat ion 2.21. The e igenvatues are given by the root
),l,)t2 = { (a+d) -+/((a+d) 2 - 4(ad-cb)) } /2
(e.g. HOCHSTADT, 1964; JACOI3S, 1974; HADELER, 1974).
which provides a l lometr ic relat ionships for a wide range of p a r a m e t e r values (cf. Fig.
2.12).
Huxley's a l lomet r i c relat ionship, there fore , appears as a r a the r likely f irst order
approximat ion of the relat ionship be tween growing organs or measu remen t s taken on a
growing organism. However, t he re are numerous except ions especial ly in ontogeny. Such
an example is given in Fig. 2.13 -- the non-l inear on togenet ic t rend in a Paleozoic ammo-
ni te which, however, can be approximated by a l lomet r ic relat ionships in d i f fe ren t intervals .
2.2.2 The tOntogenetic Morphospace f
If one picks individuals of a ce r t a in age class from a species~ then the morphological
24
lo
r
I i i t I i t I
2 4 6 8 w h o r l N °
Fig. 2.13: Nonlinear ontogenet ic relationships in a Paleozoic ammonite which can be s tepwise approximated by simple a l lometr ic relationships (modified from KANT & KULLMANN, 1980).
fea tures show usually a typical intraspecif ic variability, and in most cases the d i f fe rent
fea tures are cor re la ted within every age class, e.g. size and weight are corre la ted and
can be described by a two-dimensional Gaussian distribution for every age class. In the
most simple case one needs two sources of variation to describe the ontogenet ic mopho-
space of a species:
a) for every age class a description of the variabili ty of all features under consider-
ation and their covariances. As a first approximation one can assume that the
t ime sect ions through the ontogenet ic morphospace are multi-dimensional Gaussian
distributions;
b} a description how the mean of these distributions moves with increasing age
through the morphospace. This gives a charac te r i s t i c {mean) ontogenet ic t r ace for
the ent i re species -- for measurements , the mean {multidimensional) a l lometr ic
relationship.
Fig. 2.14 i l lustrates this description of the morphospace whereby the 'mean ontogenet -
ic t race ' is approximated by a s traight line {e.g. an ideal al lometric relationship in loga-
r i thmic coordinates), and the age sect ions are idealized as ellipsoids {ideal Gaussian distr i-
bution). It is obvious that this description cannot be used only for continuous ontogenet ic
development {as in the ammoni te example), it also holds for growth in f ini te s teps like
in crus tacea . Thus, this kinematic model provides a relat ively general description of the
25
IU N'," . - -
, • ~ - ~ o • • • • t , : m @@
• ; '~:" 4 , • . • •
Oj
O
l m O
$
Fig. 2.14: A linear modeI for the 'ontogenet ic morphospace' of a species. The variables u,v,s are parameters or measurements which charac te r ize the morphol- ogy. The ellipsoids are t ime sections, i.e. they are the probability distributions for a cer ta in age cIass. They are dislocated within the (u,v,s)-space with t ime t e i ther continuously or in d iscre te steps. The hull of these ellipsoids {in the linear model a cone) is the probabilistie boundary of the ontogenet ic morpho- space. Sections through this morphospace by another variable than time, e.g. size (s), are ellipses which contain various age classes which may appear to be strongly corre la ted.
ontogenet ic development as well as a definition of a probabilistic morphospace for the
whole ontogeny.
If this ontogenet ic morphospace is now sect ioned by another variable than by age,
e.g. by constant size which is an accessible con t ro l -parameter in p a l e o n t o l o g y - - then
the section contains parts of the ontogenet ic trend. Thus, even if the fea tures under
consideration are uneorrelated within an age-sect ion, it is possible to find a strong corre la-
tion within the s ize--sect ions (Fig. 2.14). l~low strong this correlat ion will be, depends
on the specif ic ontogenet ic t race, on the correlat ion of fea tures in the age-sect ions and
on the angle between' the principal axis of the age distribution and the ontogenet ic t race .
26
'Buckmanfs law' was observed in those ammoni tes which show specially strong mor-
phological changes through ontogeny, and the observed variability for a constant size
consists of morphotypes which are found as ontogenet ic growth s ta tes in all specimens.
Therefore , it is likely that this Vlaw' results simply from the oblique sect ions through
the age dependent morphospaces; whereby a high correla t ion be tween fea tures on the
level of the age sect ions may inforce the strong correla t ion within the size sections.
2.2.3 Discontinuit ies in the Observed Morphospace
So far, the mean ontogenet ic t race has been assumed to be a s traight line or can
be t ransformed into a s traight line (i.e. if it is ideally allometric). However, even al lometry
is only an idealized first order approximation. Especially in ontogeny more complex rela-
tionships commonly occur,, which only allow an a l lometr ic approximation through cer ta in
intervals (Fig. 2.13, cf. KANT & KULLMANN; 1980). Non-linear relationships are usually
found if morphology is described by some index numbers -- as it is the case in t theoret ical
morphology ~ (e.g. RAUP, 1966). Thus, in the general case one has to expect that the
mean ontogenet ic t race is a th ree- or more-dimensional curve. This causes complicat ions
if t ime is not available as the controll ing variable; e.g. the size sect ions will show defor-
mations as a function of age. A mathemat ica l description of the morpho-space without
t ime can, therefore , lead to ra ther complicated nonlinear equations.
In addition, one has to expect complicat ions in any project ion of the n-dimensional
ontogenet ic morphospace (all possible relationships) onto a subspace, say the two-dimension-
al subspace of a point plot. Fig. 2.15 gives a sketch of such a curved ontogenet ic morpho-
space. In the convex area of its hull a singularity appears due to the projection into
......... . ......... :..:.:.:z::,-:::::.::.::
Fig. 2.15: The hull of a curved ontogenet ic morphospace, a single ontogenet ic t race and the probabilistic neighborhood of this t race. Other trajectories~ which s ta r t close to the sketched trace, will be within this probabilistic neighborhood. In the concave area of the hull a swallowtail singularity appears, which will be discussed in chapter 4.
27
the plane. Such structural ly stable singularities will be discussed in detail in chapter 4,
however, some aspects of the deformations in subspaces can be already discussed here
by the analysis of the ontogenet ic t races of single specimens.
If one picks a cer ta in set of ontogenet ic t races for single specimens from the proba-
bilistic ontogenet ic morphospace, then, by experience, one can expect that they evolve
in a regular manner and that they do not depart too much from their original relat ive
position within the age section:
Experience shows that a juvenile ' pyknic ' human will, in general,
not turn into a FleptosomeF one during its ontogeny.
Now, we can describe the evolution of the ontogenet ic morphospace as an i te ra ted
(or continuous) map which describes the change of the age dependent probability distribu-
tion and the dislocation of its mean. And, one can assume that the map, which genera tes
the probabilistic ontogenet ic morphospace of a species from some initial distribution,
also describes the ontogenet ic t races of single specimens up to some error term. If one
neglects the error term, which causes the representa t ion of ontogenet ic t races by tubes
rather than by lines (Fig. 2.15), then a significant regular disturbance within a family
of ontogenet ic t races can result only from the projection of the multi-dimensional space
onto a subspace. What then reasonably can be expected, without further analysis, are
local folds of the map (Fig. 2.2).
A simple model of such a fold in two dimensions is the tangent space of a parabola
(Fig. 2.16c) whereby the tangents are local linear approximations of the ontogenet ic t ra-
jectories . The concave side of the fold line, the parabola, is empty, no tangents pass
through this area. In contrary, on the convex side of the fold line two tangents pass
through every point of the plane. Naturally, such a fold model can be valid only as a
local model. In this sense Fig. 2.16 provides a paleontological example of a local fold
in the ontogenet ic morphospace.
The ontogenet ic t races of several individuals of the ammonite genus Hyperlioceras
are drawn in a two-dimensional parameter space (non-al lometric) which includes size (=Dm).
The specimens belong to d i f ferent species of this genus (BAYER, 1970)~ but this should
not be a serious problem because the idea is only to show that local folds can be expected
in paleontological 'growth ' d a t a - - under the aspects of the previous discussion these
species may well be lumped into a single species. During late ontogeny, measured by
size, an inversion of the morphological trend occurs (Fig. 2.16). Specimens with ra ther
high relat ive whorl height (N) turn into forms with moderate values of this parameter
and vice versa. This pa t tern is very regular with respect to the precision of the measure~
ments, and the inversion occurs within a relat ive small size interval. Thus, the local behav-
28
50"
~0"
r..-..: ...... :; /
/ u r n i 16 2b mm
Fig. 2.16: Ontogenet ic t ra jec tor ies of ammoni tes of the genus Hyperlioceras (a: H. desori, b: H. subsectum, c: H. d~d t e s ) , modified from BAYER (1969). N: relat ive height of whorl, Dm: d iameter of the shell. During the late ontoge- ny, measured by size, an image inversion occurs, which can be in terpre ted as a local fold. In the model t he fold causes local in tersect ions of the t ra jec to- ries and an empty area. If age (t) is used as an additional variable, one can expect that the t ra jec tor ies are well separated, i.e. that the in tersect ions are due to the projection onto the two-dimensional pa ramete r space.
ior of the morphological t ra jec tor ies can well be compared with a local fold. If age
could be added as an independent variable, the t ra jec tor ies would be l if ted into the third
dimension. However, if age is re la ted to the ear l ier development in the pa ramete r space
(Dm,N), then the t ra jec tor ies will be arranged in a more or less regular manner within
the three-dimensional space {Dm,N,t). The local singularity, where the t ra jec tor ies inter-
sec t , may then appear like a piece of a ruled hyperbolic surface (Fig. 2.16). The rulings
model locally the ontogenet ic t races, and their projection onto the (Dm,N)-ptane is the
discussed tangent space of a parabola.
This is not the place to say that this is the way to study and to describe the pat-
tern of Fig. 2.16. But it is a way to i l lustrate and perhaps to overcome the diff icul t ies
which arise from singular s t ructures like the regular in tersect ion of the t ra jec tor ies . In
chap te r 4 it will be shown that singularity theory or, more specific, e lementary ca tas t rophe
theory provides a very elegant method to analyze such pat terns . Anyway, it became clear
that the ontogenet ic development of morphology cannot always be considered to be linear,
nei ther on the probabilistic level of the ontogenet ic morphospace nor on the level of
individual ontogenet ic t races. The ce lebra ted analysis of morphology by the s o - c a l l e d
higher s ta t i s t ica l methods {like factor analysis) has, therefore , to be used with caution.
Pa t t e rns like in Fig. 2.16 cannot be l inearized within the observed paramete r space, and,
therefor% they cannot be analyzed with linear models. On the other hand, the ear l ier
29
discussion of the on togene t ic morphospace shows that , even within the most simple l inear
model, the on togene t ic t rend cannot be ruled out for a l inear f ac to r analysis as is some-
t imes assumed (BLACKITH & REYMENT, 1971). If the covar iance s t ruc tu re is a l t e red
by age e f f ec t s within the size sect ions, then we cannot r econs t ruc t the original dis t r ibut ion
from this sect ions without addit ional informat ion -- in paleontology qual i ta t ive informat ion
will then be pre fe rab le against any quan t i t a t ive measurement .On the o ther hand, the discussed
models provide tools for the taxonomist . They give qua l i ta t ive a rguments for the var iabi l i -
ty of species and, there fore , for the defini t ion of a species. In addition, they allow to
fo rmula te speci f ic quan t i t a t ive models.
2.3 ANALYSIS OF DIRECTIONAL DATA
The analysis of three-dimensional direct ional da ta by means of the ' s te reographic
projec t ion ' (Fig. 2.17) is a s tandard procedure in tec ton ics and sedimentology. The aim
of the procedure is usually to e s t ima te a densi ty funct ion of unknown form from data
points on the sphere (el. MARSAL, 1970). The recons t ruc t ion of the densi ty dis t r ibut ion
requires a smoothing process, in general a moving average. The classical hand method
works with a count ing a rea (circle) of 1% of the sur face of the half sphere (or of i ts
C
Fig. 2.17: a) Represen ta t ion of a t angen t plane in the unit sphere: by the ' c i rc le of in te r sec t ion ' wi th the sphere, i ts unit normal and a point on the sphere ( in tersec t ion of the normal with the sphere), b) a pair of idealized shear planes and a system of real shear plains in the s te reographic projec- tion: r epresen ta t ion by the 'c i rc les of in te r sec t ion ' and the normals, c) Two s te reographic project ions of the same set of joints; above: Schmidt ' s grid (equal area); below: Wulf's grid (equal angles).
30
projection into the plane). When the first computer programs for the analysis of direction-
al data appeared (e.g. SPENCER & CLABAUGH, 1967; ADLER et al., 1968; BONYUM
& STEPHENS, 1971; ADLER, 1970), they did not only simplify the analysis of directional
data, but they added new 'degrees of f reedom' : to choose the size of the counting circle,
to use various weighting functions or projections of the sphere (Fig. 2.17), and the com-
puter allows to handle a ra ther large number of data. A question, which arose early
{KRAUSE, 1970), was, therefore , whether there exists an optimaI size of the counting
area with respec t to the number and to the distribution of data points on the sphere.
Alternat ively, new ' inf luence functions ' like an exponential decay function have been
introduced {BONYUM & STEVENS, 1971).
The problems associated with the smoothing process can be divided into more quan-
t i t a t ive and more qual i ta t ive ones. The variation of the influence area (either by chang-
ing the d iamete r of the 'counting c i rc le ' or by d i f fe rent 'weighting functions') a l ters
the total number of expec ted values at a grid point. The classical way to s tandardize
this number to a percen tage of all observed data points causes deformat ions of the
distribution in the way that the maxima are s t r e t c h e d - - the sum over all grid points
is g rea te r than 100%. The counted data need to be normalized into 'densi t ies per unit
area ' , or the area of influence has to be replaced by a weighting function for which
the integral over the area of influence equals one (BAYER, 1982), A more quali tat ive
aspect is that the smoothing process a f fec t s the variance of the distribution (GEBELEIN,
1951). This de fec t is mainly a function of the size of the area of influence. These prob-
lems are briefly discussed in the first section. However, while they are important in
a s ta t is t ica l sense, they are less significant for the geological in terpre ta t ion of orienta-
tion data. In geology only the position of ex t rema may play a role for the s t ructural
in terpreta t ion, and in this case the described deformat ions of the global distribution
do not a f fec t the local in terpreta t ion. Therefore , most of the following discussion will
focus on the question whether the local ex t rema are stably es t imated by the methods
current ly in use. In the final sect ion we will return to a more general problem and
analyze under which conditions we can suspect a density distribution at all.
2.3.1 The Smoothing Error in Two Dimensions
The es t imat ion of a density function from sca t t e r ed data on the sphere or its
project ion onto the plane involves a moving average. For one-dimensional histograms
the result ing errors and the deformat ion of the moments have been discussed in detail
by GEBELEIN (1951). Fig. 2.18 i l lustrates how a one-dimensional histogram is deformed
if a moving average is used. Two-dimensional data and data on the sphere behave in
the same way (Fig. 2.19), and what we will do here is to e s t ima te the error of the
smoothing process, i.e. the expec ted d i f fe rence between the true and the computed
density distribution. Technically this requires Taylor expansions and integrations, however,
31
1 f = ~Xf i J
/ '7= 3
I
f = Xf i
' 1
d
~ 2 . 1 8 : Smoothing a histogram by a moving average: a to c: normalized averages; d: not normalized histogram of a three point moving average.
the mathemat ic remains ra ther simple.
The way to es t imate the error is to compare the observed densities with a theoret ica l den-
sity function f(x,y) which is analytic (i.e. continuous and differentiable) with the values which
result from averaging over a small interval. The error is the d i f ference between the
true value of the density function and the average. In the plane we choose an interval
{Ax,~y) in the way that its c e n t e r - - t h e ar i thmet ic m e a n - - has coordinates (0,0). We
can do this for any interval by simply shifting the coordinate system. To find the mean
density within the interval we have to sum over all points within the interval and to
divide by the area of the interval, i.e. -Ay <=y __<A~2 2
f(O,O) = _I__ Y'fX; f(x,y)dxdy; AxAy -Ax ~X =<A~2 2 (2.23)
32
Fig. 2. t9: Smoothing a ' h i s togram' (a) on the sphere, b-d: increasing ' a rea of inf luence ' ( t r iangular weight ing funct ion -- see tex t for discussion).
We assume fur ther tha t the densi ty funct ion f(x,y) can locally be developed in a Taylor
series:
f(O+h,O+k) = f(O,O) + hf x + kfy + (h2fxx+2hkfxy+k2fyy)/2 + "'" (2.24)
If f(x,y) in equat ion (2.23)is replaced by the approximat ion (2.24}, we can eva lua te the
in tegral and find
2 ~.~4 2 (2.2a) ~ ( 0 , 0 ) = f ( o , o ) + -~{ f xx + fyy + "'"
whereby it is worth noting tha t all in tegrals over odd p o w e r s in equat ion (2.24) vanish,
the la rges t e r ror term, the re fore , depends on the local curva tu re of the densi ty funct ion
33
f(x,y) and is approximately
8f = f(O,O)-f(O,O) 2 2
= - { 2 ~ f x x + -~¼ f y y ) " (2.26)
The error te rms have a negative sign, i.e. maxima are lowered, minima are f i l l e d - -
depending on the local curvature and on the area of influence. Thus, if we have enough
information about the curvature of the density function, a good s t ra tegy would be to
use a small e lement for averaging where the curvature is strong, and a large one where
the function is flat, in order to combine a good approximation with fast computation.
In the empirical problem, however, this will not be possible.
If the normalization by the area of the interval {~x.~y) is ignored, then it makes
not even sense to speak about an error {cf. Fig. 2.17d), the averaging process then gener-
ates a totally new distribution as defined by equation (2.25). A safe s t ra tegy, which
easily can be used by the computer, is to transform the initial data first into percent -
ages (densities} and to normalize them again by the counting area a f te r averaging.
In the case of a circular counting area the error is of similar form. In this case
one has to evaluate the integral
1 f(O,O) .... x2 f ; f(x,y)dydx;
~ XG
or
-/(Ax2-x2)= < G < /(Ax2-x 2)
(2.27)
f(O,O) .... 1 2 ~r }~Irf(rcos@,rsin~)rdrd~)
0 -r
using again a Taylor approximation, the integral can be evaluated, and the error is of
order d 2
~f = f(O,O)-r(O,O) = ~ (fxx+fyy)
where d: the diameter of the 'counting area'. (2.28)
Even for averaging on the sphere the error is of the same magnitude. Averaging on
the sphere is easily evaluated with a computer if the data points are t rea ted as vectors .
The angular dis tance (a spherical cap) can be defined by the scalar product of vector
pairs. The computat ion of the smoothing error proceeds as before, however, it is slightly
more tricky to evaluate the integrals. We assume that the density function can be wri t ten
as f(x,y) = g(x,y,z}, because we are dealing with unit vectors, we have z = (x2+y2) 1/2.
Thus, we can again use the Taylor expansion (2.24}, and the mean density within the
area F (area e lement dr) can be wri t ten
34
Fig. 2.20: The ' a r ea of inf luence ' on the unit sphere a f t e r i ts c e n t e r has been ro t a t ed to coincide with the z-axis (see tex t for discussion),
t Z
F F F I f(O,O) = ~ {f(O,O)fdf + fxfhdf + fyfkdf
+ + + 2- (fxx h2df + 2fxy yy
Transforming the coordina te system to spherical coord ina tes yields the integral
(2.29)
i 2Y a
f(0,0) - 2~(l-cosa) { f(0,0~]o] sin~d~d~ +
+ fx ~ ~ cos~sin26d~d~ + fy~ ~ sin~sin2~d~d~
I + ~ [fxx ~ ~ cos2~sin3~d~d~
(2.30)
+ 2fxyl; cos~sin~sin3~d~d~
+ fyy] ~ sin2~sin36d6d@ ) + ...}
The coord ina te system has been chosen so tha t the cent ro id of the spherical ' count ing
cap ' coincides with the z-axis of the sphere. The angle ' a ' defines a cone with the
z-axis i ts cen t r a l line (Fig. 2.20} 7 whereby all in tegrals conta ining cos a and sin a vanish.
The in te r sec t ion of this cone with the sphere bounds the averaging area. The e s t ima ted
densi ty value is of magni tude
I f (O,O)2~(l-cosa) ~(0,0) =~(1 -c0sa )
1 c o s 3 a ) {2.31) + f x x l r ( - c o s a + ~-
2 + fyy~(~ - rosa + } cos3a)
or f(O,O) = f(O,O) + (fxx+fyy)(# o - u I cosa(l + rosa)).
35
The averaging error is approximately
1 1 ~T(0 ,O) = (Tj - g c o s a (1 + c o s a ) ) ( f x x + f y y ) ,
(2,32)
2.3.2 Stability of Local Extrema
The major problem of a geological analysis of directional data is not so much
the "stat is t ical stability", but it is the stabili ty of local extrema. However, in this context
it turns out that the classical approach causes problems, i.e. the smoothing process by
a s tep function over an influence area or by a rectangular weighting function. Let the
area of the rectangular weighting function increase until it reaches the surface area
of the half sphere, then the whole distribution becomes equalized, independent of the
original data pat tern. In other words, the ext rema are smeared out with increasing area
of influence. The instability of the local maxima becomes more obvious if one studies
sparse data s t ructures . If the area of influence, the counting circle, is varied on such
a data set (Fig. 2.21), then new maxima are generated whenever two areas of influence
I . . . . . . . . !
Fig. 2.21: Two stereographic projections for the same sparse data pa t tern but with d i f ferent sizes of the 'counting c i rc le ' or ' rectangular weighting function' (5% and 20% of the area of the half sphere). The location of max- ima (dark areas) depends strongly on the size of the counting circle. The lower figures i l lustrate how the maxima arise over the intersect ion area of the rectangular weighting functions (Wcounting circles ') .
36
begin to overlap. Furthermore, the transit ion to such a new maximum is not a smooth
proeess~ but it is a 'sudden jump'. The smoothing process is, therefore , highly instable
with respect to the position of the local maxima and to small size changes of the in-
f luence area.
However, it is not only the size of the influence area which may cause the sudden
occurrence and disappearance of local maxima. The smoothing process is executed on
a finite net e i ther on the sphere or on a two-dimensional project ion of the sphere. There-
fore, any change of the grid s t ruc ture will locally change the overlapping pa t te rn of
the influence areas, and the identical instabil i t ies will arise. Because there exists no
'equally spaced grid ~ on the sphere, even a rigid rotat ion of the data may al ter the
distribution of maxima. Thus, the position of local maxima is not invariant against (even
small) changes of the grid s t ruc ture or of the size of the counting circle. Indeed, the
classical method has only two stable s ta tes with respect to local extrema:
a) the si tuation where the counting area is identical with the surface area
of the half sphere, in which case all ex t rema vanish, and there fore all
information is lost,
b) the case where the counting area goes to zero and the distribution resem-
bles a plot of the original data points. This case, however, does not summa-
rize the s t ructural information.
Now, one could go back to the old hand method and hope that all problems can
be solved by a ' s tandardized ' counting area and grid s t ruc ture because then the system
is forced to be without variations or irregulari t ies. However, this would only cover the
problem because any addition or subtract ion of a data point (or of a group of data
points} can change the local pa t te rn of ex t rema in the same way as discussed above.
The reason is that the method does not preserve a data point as the smallest possible
maximum, but that the maxima appear in the intersect ion areas of the counting circles
(Fig. 2.21). The equivalence between the grid and the data can be i l lustrated by two
s t ra tegies~which can be used for computer -a lgor i thms (Fig. 2.22):
1) The classical method t r ea t s a data point as a point and defines a grid on
the sphere (or its projection). To every grid point a counting circle is assigned,
and one has to find the number of data which fall within this circular area,
i.e. aIl points are summed up at a grid point which are within a dis tance limit.
2) Another s t ra tegy is to assign the counting circle to every data point. Then
one has to find all grid points which fall inside the distance limit (which surrounds
the data point} and to add the ( w e i g h t e d ) value of the data point to the
grid points which sat isfy the dis tance property.
37
Fig. 2.22: Two viewpoints of the smoothing process, a ) A data point belongs to a grid point with a ce r t a in probabili ty. Several grid points are in 'compe- t i t ion ' , b} One or several da ta points have a - c e r t a i n probabil i ty to be co r r ec t - ly l o c a t e d - - the spherical cap, which is associated with a data point, is an area of confidence. In the project ion the grid points have to be found which belong to the area of confidence.
Both s t ra teg ies are possible implementa t ions . In the first case, one usually has
to s tore all data in the cen t ra l memory while the grid points can be t r e a t ed one a f t e r
the other; in the second case, one stores the grid and can call one data point a f t e r
the o ther from some ex te rna l memory, tt is this second method which i l lus t ra tes tha t
the da ta points are not preserved as the smal les t possible maximum. Actually, they
are replaced by a rec tangular (density) function over a f ini te area. Variat ion of the
area of the count ing circle, thus, resembles the d is turbance of a signal {Fig. 2.1). The
count ing c i rc le defines the area in which the data point will be found with a ce r ta in
probabili ty, and this probabil i ty is everywhere equal (within the area of influence). The
rec tangu la r weighting function over the count ing circle, therefore , can be in t e rp re t ed
as a rec tangula r probabil i ty distr ibution. A consequence is tha t one can not e i the r assume
tha t ano ther (sparse) sample from the identical universe shows the same ex t rema.
It turnes out tha t the smoothing process by a r e c t a n g u l a r weighting function is
very sensi t ive and locally highly unstable with respect to the position of the ex t rema .
The s tr iking point is tha t o ther weighting functions give much b e t t e r resul ts with regard
to the considered s tabi l i ty problem. Such a family of non-s tandardized weighting funct ions,
which includes asymptot ica l ly the point plot and the rec tangula r weighting function, is
given by the family of polynomials (Fig. 2.23)
w(r) = (i - r)n; 0_- < r= < I
w(r) = 0 r>l (2.33)
r = R/Ro, n = 1,2,3 ..... 1/2, 1/3 ....
38
(l-x) n
Fig. 2.23: The family of functions y=(l-x) n in the interval 0 <x _-< 1 provides a possible family of weighting functions (not s tandard- ized). They include asymptot ical ly the point plot and the rectangular weighting function.
X
where Ro is the d iameter of the counting circle, and R is the distance of a grid point
from the data point. For every grid point one has to form the sum E w(r) over the
data points which are inside the dis tance R o. In the case of a rectangular weighting
function one has w=l within the counting circle and w=0 outside. For the point plot
one has ro=0, w=l at the data point, To normalize the es t imated distribution one has
to sum over all counts and then to divide the grid point values by this total sum,
Figs. 2.24 and 2.25 i l lustrate that the major maxima are well preserved, even for
large counting areas if one of these functions is used, especially if the exponent 'n '
is chosen small e n o u g h - - while for large 'n' the weighting function resembles the rec-
tangular weighting function. A similar result was found by BONYUM & STEVENS (1970).
They used an exponential decay function as a weighting function which was defined
over the ent i re grid area, i.e. every data point contr ibuted to every grid point. Their
computer studies showed that the resulting frequency distributions are as useful as those
genera ted by the classical ~hand ~ method. The point is that the stabil i ty of the locaI
maxima increases if the weighting function has a weII pronounced maximum at the cen te r
of the counting circle in local coordinates or at the data points in global coordinates.
As this maximum of the weighting function vanishes - - like in the case of a rectangular
weighting f u n c t i o n - - the es t imated ex t rema become instable in the sense that they
do not fur ther resemble the original data points. Fig. 2.26 i l lustrated how a triangular
weighting function preserves the local data s t ruc ture in contras t to the rectangular weight-
ing function of Fig. 2.21.
In summary, it seems worthwhile to discuss the two technical s t ra tegies given
above in more general terms. If the counting circle is associated with the grid points~
then we are working in local coordinates. The question is whether a data point belongs
with some probability to this grid point. In general, one assumes that this probability
39
Nr 1.8 stEP 1.6 NORM m x l
Fig. 2.24:_ Estimated frequency distributions on the sphere by use of various weighting functions: point plot (data); std: rectangular weighting function (classical counting circle);
sqrt: w=(l-r /ro)l /2; sqr: w=(1-r/ro)2; cub: w=r~r2(3-2r). The size of the counting circle is 1.8% of the surface area of the half sphere for all smoothed distributions.
0 ~ N r l ~
\
1 /
. . . . . 9 0
Fig. 2.25: Estimated frequency distribution on the sphere for counting circles of different size and for various weighting functions, Data and symbols like in Fig. 2.24. Counting areas 10% and 90% of the surface area of the half sphere,
40
- ,
Fig. 2.26: Tr iangular weight ing funct ions over a c i rcular area of inf luence preserve the da ta point as a local maximum. As the overlapping of the circles of inf luence increases , e i the r by increasing densi ty of data points or by a larger area of influence, the new maxima develop smoothly be tween the grid points.
dec reases cont inuously wi th the d i s tance from the grid point. The o the r possibility is
to assign the count ing c i rc le to the data point. In this case, we are working in global
coordinates , i.e. we have to find the grid points which fall inside the count ing circle.
The quest ion is now what is the probabi l i ty for the data point to fall on a specif ic
grid point -- and again the probabi l i ty should decrease with increasing d i s tance be tween
the da ta point and the grid point. In any case, the smoothing procedure focuses on the
"sampling error" , i.e. the problem tha t we have only a small sample from a large universe
of usually unknown s t ruc ture .
In a s t a t i s t i ca l sense, the weight ing funct ion should assure tha t a random sample:
which is t aken from the smoothed distr ibut ion, wilt not depar t too much from the original
da ta set , a t least if the e s t ima ted dis t r ibut ion is co r r ec t ed for the variance(GEBELEIN,
1951). MARSAL {1970) even t r ied to in t roduce a t e s t - s t a t i s t i c s on this argument . For
the r ec tangu la r weight ing funct ion this condit ion does obviously not hold, as Fig. 2.21
i l lus t ra tes . .
2.3.3 Approximat ion and Averaging of Da ta
Another useful aspect for the previous discussion can be developed from the
methods for local sur face f i t t ing. The local Shepard method (e.g. SCHUMAKER, 1976)
uses a weight ing funct ion for the approximat ion of surfaces:
w(r;R) =
i/r ; O< r~ R/3
27 r 2 R (~ - !) ; 3- <=r <R (2.34)
0 ;R< r
4]
and the local es t imat ion at any point (x,y) is given by
[ ~ Fi(w(ri;R))~ ; ri ~ 0
(w(r;R)) U f(x,y)
t F. ; r . = 0 ( 1 1
(2.35)
where the F i are the observed z-values at the data points (or frequencies); r i is the
dis tance between a grid point and a data point; la : is a parameter (a 'metr ic ' ) which
can be freely chosen; R: is the radius of the area of influence (the 'counting circle ') .
Formula (2.35) is defined at all points (x,y) in the plane. It interpolates the ob-
served values correc t ly at the data points (the values F. at r.=0) while the values at 1 1
non-data points are weighted averages of the data points which lie within a distance
R of the grid point. The weights are defined by equation {2.34}. The local Shepard
method provides an approximation method which is based on a 'counting circle ' as
discussed above. SCHUMAKER (1976) shows that the local Shepard method is an optimal
approximation s t ra tegy for the local surface reconstruct ion. The relatively complicated
definition of the weighting function ensures that the observed F.-values, i.e. the data 1
points, are preserved.
The local surface fit t ing method can be used to es t imate a density function if
the samples become ra ther large. For a large sample, with space coordinates measured
on a discrete scale (with fixed precision), one can expect that the resulting histogram
is a good es t imat ion of the s ta t is t ical universe. The local surface approximation (with
normalization over the area of influence R) gives a first order approximation of the
density function. As the sample size decreases, the histogram becomes noisy, and the
local surface approximation produces local ex t rema which are due to the sampling error.
A smoothing process is then required, and this procedure should be capable to el iminate
the sampling noise. On the other hand, the smoothing process should not deviate too
much from the local approximation. If one takes Shepard's method as a model for the
local approximation, then the transition to a smoothing procedure requires to drop 1/r
term for the interval 0 < r < R/3 in equation 2.34 because this term would cause infinite
values (if ri=0). The remaining parts are
r (g - 1) ; o _<-r_-<R
w(r) = (2.36)
0 ; R< r
This is the earl ier discussed polynomial weighting function (2.33), which together with
the first term in equation (2.35) provides a smoothing procedure. It is not hard to see
42
tha t the rec tangu la r weight ing funct ion is the most degenera ted case of the family
of weight ing funct ions defined by equat ion (2.36); it is approached as lJ ~ .
Another question is how to choose the p a r a m e t e r lJ , or which funct ion of the
family w{r; !J) is opt imal . In order to see what we can achieve by the p a r a m e t e r ]j
of equat ion (2.36) or by the p a r a m e t e r 'n ' of equat ion (2.33), one may discuss the most
general form of these weight ing functions:
w(x) = (I - x) ; x =<i (2.37)
x = r/R
Equation (2.3,7) takes the values w(r=0)=l at the data point and w(r=R)=0 at the boundary
of the a rea of inf luence, as required. The first de r iva t ive takes the values
w'(X) = -~/(I - x) ~-I with I/>0
with t l>0 (2.38)
w'(O) = -~
w'(1) = -I if ~=0 else w'(1) = O.
The p a r a m e t e r !J allows to adjust the slope of the weighting funct ion at the grid point.
However, to use this addit ional degree of freedom requires addit ional informat ion about
the proper v a l u e for the slope a t x=0. From a general viewpoint, the re fore , the s t ra igh t
line approximat ion (n=l) is nearly opt imal (DeBOOR, 1978). It does not require any
addit ional assumptions, and it solves the problem to connec t the da ta point with the
boundary of the a rea of inf luence by a cont inuous function.
However, a na tu ra l boundary condit ion for the weight ing funct ion could be tha t
the f i rs t de r iva t ives vanish at the data point and at the boundary of the count ing circle .
If one considers the weighting funct ion as a probabi l i ty dis t r ibut ion which assigns a
data point to a grid point , then the Gaussian dis t r ibut ion would, of course, be a model
with an inf in i te a rea of inf luence. In t e rms of polynomial weight ing funct ions this condi-
t ion cannot be sa t is f ied by equat ion {2.37), at least we need a cubic polynomial like
w(x) = (x 3 + ax 2 + bx + c). (2.39)
The boundary condi t ions w(0 )> 0, w(R) = 0, and w'(0)=w'(R)=0 are only sat isf ied by
the polynomial
3 3 2 i w(x) = x - ~'x + ~ (2.40)
or
w(x) = s(l + x2(2x - 3)).
43
--R-- R Fig. 2.27: Interpola t ion by spline functions: a) the l inear Euler spline; b) the cubic spline with vanishing der iva t ives at the grid point and the boundary of the area of inf luence (R).
It turns out tha t there is a l imited number of opt imal weighting functions depending
on the boundary condit ions (Fig. 2.27). Equation (2.39) allows a l t e rna t ive ly to adjust
the slope of the weighting function in an arb i t ra ry way at the grid point and at the
boundary of the area of influence. It is the cubic Hermi te in terpola t ion and can be
used for cubic spline in terpola t ion (De BOOR, 1978). The ear l ie r discussed t r iangular
weight ing funct ion is simply the l inear Euler spline.
The t inear and the cubic spline provide a simple in te rp re ta t ion of the radius of
inf luence (R). Assume the s i tuat ion of two data points with d is tance R: Without loss
of genera l i ty we can locate them on the real line and s i tua te one of them a t the origin,
the o ther at the point x=l {i.e. x=r/R). The weighting funct ions for the two points
are
w1(x ) = (I - x)
(2.41) w 2 ( x ) = (1 - (1 - x ) ) = x
for the l inear spline and
W l ( X ) = (1 + x 2 ( 2 x - 3 ) ;
w2(x) = (i + (l-x)2(2(l-x) - 3) = x2(3 - 2x) (2.42)
for the cubic spline. In both cases we find tha t Wl+W2=l within the interval (0,R).
The approximated values within this in terval are
f ( x ) = f ( 0 ) w 1 + f ( 1 ) w 2 (2.4a)
= f(O)(l - w2) + f(1) w 2
= f(O) + (f(1) - f(O))w2 ~
tha t is a s t ra ight line in the case of the l inear spline and a cubic funct ion in the second
case. However, if f{0}=f(R} (e.g. single measurement s at both points), then the connect ion
be tween the two data points is simply the hor izontal line f(0), cf. Fig. 2.27. The radius
44
of inf luence R, there fore , def ines a threshold of resolution: Data points with dis tance
less or equal R are subsumed in a single maximum while data points with dis tances
larger than R are preserved as dis t inct maxima. As far as it is possible to r e l a t e R
to the number of da ta (cf. KRAUSE, 1970), the spline funct ions provide opt imal approxi-
mat ion and smoothing capabi l i t ies .
It is ins t ruc t ive to inver t the above argument . If one defines the threshold proper ty
as the opt imal solution~ then one can discuss any polynomial weight ing function in t e rms
of the opt imal solution, i.e. we have to find coef f i c ien t s so t ha t w(x)+w{1-x) = cons tan t .
In the case of an a rb i t r a ry polynomial
w I = a 0 + alx + a2 x2 + ... + an xn
w 2 = a 0 + al(l-x ) + ... + an(l-x)n
(2.44)
the condi t ion can only be sa t is f ied i f a(-x) n = -ax n, tha t is, if the leading power te rm
is an odd number . The polynomials of equat ions (2.33) and (2.36), the re fore , are not
opt imal . Now, we may analyze the special case of the cubic weight ing function:
w I
w 2
3 bx 2 = ax + + cx + d
= a ( 1 - x ) 3 + b ( 1 - x ) 2 + c ( 1 - x ) + d
3 =-ax + (3a+b)x 2 - (3a+2b+c) +,(a+b+c+d).
(2.45)
To sa t i s fy the condit ion a l l t e rms which conta in powers of x have to vanish in the
sum Wl+W 2. This gives the following re la t ions be tween the coeff ic ients :
b + (3a+b) = 0 --~ 3a = -2b
c - (3a+2b+c) = 0 --~ 3a = -2b, (2.46)
and we have twice the same re la t ionship be tween ' a ' and 'b ' which, of course, appeared
a l ready in equat ion (2.39}. The Pa rame te r ' c ' is a rb i t r a ry -- if a=b=0 and c~0, the cubic
funct ion degenera tes to the l inear Euler spline.
The general solution for a cubic spline under the condi t ion w(x,R)+w(R-x,R)=constant
is lust the l inear combina t ion of the special cubic spline of equat ion (2.39) and the
l inear Euler spline. We can rewr i te equat ion {2.45) with the p a r a m e t e r s of equat ion
(2.46) as d d +cx} w ( x ) = { a ( x 3 3 x 2 ) + ~} + { ~ , (2.47)
45
i.e. as a l inear combinat ion of the spline functions. To see what happens at the special
points x=0 and x=R we take the der iva t ive of equat ion (2.47) at these points
w' = 3ax 2 - 3ax + c
w ' ( O ) = +- c (2.48}
w ' ( 1 ) = +- c
and find tha t the weighting function has identical slopes at the points under considera-
tion. Some weighting functions~ which sa t is fy the discussed condit ions, are given in Fig.
2.28~ and it turns out tha t these functions are no proper weight ing functions as they
Fig. 2.28: Symmet r ic cubic weighting funct ions which sa t is fy the condit ion w(x)+w(1-x)=constant.
have ex t r ema inside the 'count ing c i rc le ' and may even assume negat ive values. The
only remaining weighting funct ion is the cubic spline of equat ion (2.39), however, this
function tends to produce local ' p la t fo rms ' at the data points (cf. Fig. 2.27). Thus,
the l inear Euler spline remains the opt imal solution for a smoothing procedure.
2.3.4 A Topological Excursus
The previous discussion focussed on the problem to find a proper weighting funct ion
which takes values g rea t e r than zero inside a ce r t a in area and the value zero outside
this area. A similar problem occurs in topology and is solved there by Urysohn's
l e m m a - - for deta i ls see J~NICH (1980). Formal ly we can formula te our problem in
the following way (of. Fig. 2,29):
What we are looking for is a function f: x--* [0,1] on U ( V (~W which has
value 1 on U and value 0 on W. On V = W\U we are looking for a cont inuous
connection° The simple idea is (J~NICH, 1980) to cons t ruc t a s tep funct ion which
46
wbz,. ....... -J/" ,1
Fi~. 2.29: (a) The problem is to find a continuous connect ion be tween the areas U and W. (b) A possible solution is to const ruct a s tep function in V = W U. (c) However, if the boundaries of the subsets A,, A2, ... are Iocally in contac t , a further re f inement of the s tep function is not possible.
decreases from U to W (Fig. 2.29 B), i.e. one has to find a chain of sets
A = A I ( . . . ( A n ( W\U.
The s tep function takes the values
1 on U, 1 - i/n on A. and 0 on W. 1
By stepwise ref inement one can const ruct the continuous connection between
V and W. The only problem is that the boundaries of U i and Ui_ 1 do not meet
(Fig. 2.29 C). In this case, it would not be possible to const ruct a continuous
connection, i.e. we cannot insert an additional s tep between the boundaries
(JNNICH, t980).
If one applies this to the discussion of weighting function G it becomes immediately
clear that the rectangular weighting function is the ex t reme solution which does not
allow any fur ther re f inement between the areas V and W because U = V\W = 0. On
the o ther hand, the linear Euler spl int is a possible connection, even when V shrinks
to a point.
2.3.5 Densit ies, Folds and the Gauss Map
So far, the reconst ruct ion of density distr ibutions was discussed without regard
to the question whether there exis ts a density distribution for or ientat ion data. In the
case of current or iented obstacles in sedimentology and of lineations and cleavages in
47
tectonics , we can use standard arguments. One can expect that in these cases the orien-
tat ion is control led by a potential (BAYER, 1978). The systems stay at the minima of
the potentials , i.e. in the position of minimal drag in a current; c leavage will occur
in the direction of maximal shear s t ress (minimal normal stress) e tc . To arrive at a
probability distribution or a density distribution, one assumes that the objects are driven
out of the minimum position by random forces. One way to find the density distribution
is provided by the s ta t ionary solution of the Fokker-Planck equation (e.g. HAKEN, 1977}.
The situation is d i f ferent when orientat ion data are taken from surfaces, e.g. from
deformed bedding planes in tectonics . The orientat ion data are then the normals of the
surface, and therefore the surface needs to be regular, that is, there exists a tangent
plane at every point of the surface. If the surface is given as a map R2 ~ R 3, e.g.
as
X ( u , v ) = { x ( u , v ) , y ( u , v ) , z ( u , v ) } , (2.49)
then the unit normal vector at each point p of the surface is given by
N ( p ) = (X u A X v ) / ( I X u A X v l ) ;
(2.50) A is the vector product.
Now, in different ia l geometry the map N: S - - R 3 (S: the surface) is studied which
takes its values on the unit sphere
S 2 = { ( x , y , z ) R 3 [ x 2 + y 2 + z 2 = 1}. (2.51)
The map N:S ~ S 2 is called the Gauss map of the surface S (DoCARMO, 1976), and
this map is equivalent to the standard representa t ion of or ientat ion data on the unit
sphere. The Gauss map has various propert ies , which can be interest ing for geological
in terpre ta t ion of directional data. Detai led discussions are given by DoCARMO (1976)
and DANGELMAYR & ARMBRUSTER {1983).
The first point to be discussed is how a surface e lement ~S' maps onto the unit
sphere. If only local propert ies of a surface are considered, that is a small neighborhood
of a surface point, then typically three situations o c c u r - - the local surface s t ruc ture
is e i ther elliptic, parabolic or hyperbolic as i l lustrated in Fig. 2.30. With regard to the
positive normals of the surface one finds that the Gauss map preserves orientat ion at
an elliptic point and reverses it at a hyperbolic point (Fig. 2.30) - - at a parabolic point
a degenera ted situation arises, the normal vectors are all aligned in a plane which inter-
sec ts the sphere. In geological problems one has also to consider the inverse surfaces
{synclines). In this case, the closed pathways around the surface points in Fig. 2.30 are
inverted in the Gauss map.
48
e l l i p t i c
@@ p a r a b o l i c
r. , ~ , , ' ; : . ~ ,¢." ;..:...¢." ...
Fig. 2.30: The local s t r uc tu r e of a surface is e i the r of ell iptic, parabolic or hyperbolic type. The o r ien ta t ion of a closed pathway (surrounding the c r i t i - cal po in t ) i s preserved at an el l ipt ic point and reversed at hyperbolic and para- bolic points (a: posi t ive normals, anticl ines) . For synclines (b) the en t i re pat- t e rn is reversed. Al te rna t ive ly , (a,b) can be in t e rp re t ed as project ions in the upper and lower h e m i s p h e r e - - both methods are used in geology. Grids: in- cl ined L a m b e r t ' s equal area project ion; adapted from HOSCHEK, 1969).
Af t e r these geomet r i ca l considera t ions we analyze how a surface e l e m e n t ' S ' maps
onto the unit sphere, and which value the area of the surface e l ement takes on the
unit sphere. If we choose a very small sur face e l ement dB, the expec ted densi ty of
normals on the sphere is proport ional to dB/dS (where dS is the corresponding surface
e l emen t on the sphere). The area of a small sur face e l emen t is
B = / . f l X u ^ x v I du d r , (2.52) R
and the image of B on the unit sphere has a rea
S = f f i N u A NvJ du d v . R (2.53)
(R is the area in the (u,v)-plane which corresponds to the surface e l ement B). *S* can
49
be expressed by
s = ; ; K lXu A Xvl du dv 12.541 R
where K is the Gaussian curvature of the surface. The relation between the surface
e lements B and their image on the unit sphere S is finally
l i m B/S = ( IX u A X v l ) / ( K J X u A X v l ) = 1 / K . {2.55) B + 0
For a detai led proof see DoCARMO (1966). The local density on the sphere is propor-
tional to l /K, or for a larger surface area we expect the density at a point p to be
I Xu A Xvl du dv (2.56)
that is the inverse 'weighted average ' of the Gaussian curvatures. There is one point
one has to take care of. The Gaussian curvature K can assume positive and negative
v a l u e s - - thus, it is necessary that the Gaussian curvature does not change sign on the
surface e lement under consideration. Otherwise we cross a point where K=0, and at
that point the density is not defined (equation 2.56).
However, if K=0 everywhere at the surface, one has to distinguish two cases. If
the surface is planar~ then there exists no density distribution; all normals are mapped
onto the identical point on the sphere. In the case of a parabolic surface e lement , e.g.
a cylindrical or conical one, we can define densities if the Gaussian curvature K is
replaced by the curvature k of the generating curve of the surface; the surface e lements
are replaced by arc length. The concept of a density distribution, therefore , is ra ther
complex for or ientat ion data from surfaces. One has to distinguish various cases, and
one can handle only finite surface e lements that sat isfy cer tain conditions.
Anyway, it is as usually, the most interest ing situations are those which cause
trouble. In this case, interest ing situations arise when the surface contains points at
which the Gaussian curvature changes sign. Such situations arise in the most simple
cases, e.g. if the surface is a sinusoidal cylinder - - in this case, the Gaussian curvature
K changes sign at the inflection points of the generating sine function. Somewhat more
complicated surfaces are given in Fig. 2.31. The lines of parabolic points, which divide
the surfaces, appear also on the Gauss map (stereographic projection) where they bound
the area of normal vectors . The density of normals on the sphere is especially high
at (or near) the boundary of parabolic points which define a fold line of the Gauss map
(cf. Fig. 2.2). The concept of folded maps applies also to the s tereographic projection
in the vicinity of parabolic points on a surface. The high ( theoret ical ly infinite) density
50
Fig. 2.31: Complex "sinusoidal" s u r f a c e s - - the sets of parabol ic points on the sur faces map onto ' caus t i c lines' {with high densit ies) bn the unit sphere. These ~caustics ~ are typical ~eatastrophe s ingular i t ies ~.
of normals at these s ingular i t ies has i ts analogy in caust ics , and an in te res t ing appl ica t ion
to phonon focusing {cf. DANGELMAYER & ARMBRUSTER, 1983, for a thorough theo re t i -
cal discussion). These pa t t e rn s likely apply to the focal mechanism da ta in seismology
with minor modif icat ions . Fur ther , such s ingular i t ies occur at leas t in the theore t i ca l
s imulat ion of p las t ic deformat ions {Fig. 2.32) as they were studied by LISTER e t al.
{1977). The singulari t ies , which bound the dis t r ibut ion of normals are closely re la ted
to the s ingular i t ies s tudied in ca t a s t rophe theory (DANGELMAYR & ARMBRUSTER,
t983), and some of them will be analyzed in more deta i l in the 4 th chapte r . In geology
a common problem is to find the ' fo ld-axis ' from a set of t angen t planes. The l inear
approach is to find a "regression c i rc le" under the assumption tha t the t ec ton ica l folds
are cyl indric {or conic). The approach by "caust ic lines" probably allows to ga the r addi-
t ional in format ion as soon as the surfaces are more compl ica ted .
Fig. 2.32: c-axis pole f igures for model qua r t z i t e under plane s t ra in {modified from LISTER e t al. 1978).
51
2.4 RECONSTRUCTION OF SURFACES FROM SCATTERED DATA
Contouring by compute r became a widely used method in geology (e.g. DAVIS &
McCULLAGH, 1975; KRUMBEIN & GRAYBILL, 1965; HARBAUGH et al., 1977; FREE-
MAN & PILRAU, 1980}. The use of computers , however, causes special problems which
do e i the r not occur within the classical hand method or are simply not recognized when
contours are in te rpola ted from a hand made t r iangula t ion net . Fig. 2.3a i l lus t ra tes with
a ' ca tas t rophic ' example how di f fe rent compute r resul ts may be dependent on the es t ima-
t ion method. Fig. 2.34 gives a much ear l ie r hand made es t imat ion for the identical da ta
set (BAYER, 1975).
The t r iangula t ion method (cf. CAVENDISH, 1974) is, in general , not used for
compute r procedures because it needs too much ' in tu i t ion ' to c r e a t e a t r iangula t ion
net from s c a t t e r e d data points. On the o ther hand, it is re la t ive ly simple to in te rpo la te
and draw contours from a regular grid {e.g. SCHUMAKER, 1976). Therefore , contouring
procedures for computers use mostly a two pass process:
in a f irst step, the surface data are e s t ima ted for the points of a regular
grid,
** in the second step, the contour lines are in te rpola ted from the regular grid.
The final computa t ion of the contour lines is a r a the r s table process. Useful methods
are well known from f ini te e l ement s (ZIENKIWFI'Z, 1975) and from spline interpolat ion.
Even more complex methods like global or semilocal surface f i t t ing can be used to de-
rive contour lines from regular grids {SCHUMAKER, 1976). Anyway, on the level of
the local grid cell the es t imat ion of contour lines is not uniquely de te rmined , an aspect
which wilt be discussed in some detail .
The major problems, however, derive during the es t imat ion of the grid values,
and this will be the major theme of the following sections. The cen t ra l example is an
es t imat ion method by minimal convex polygons, which is capable to i l lus t ra te the prob-
lems arising during the es t imat ion of the grid values.
2.4.1 The Regular Grid
The main problem with s ca t t e r ed data is to eva lua te the surface values for the
regular grid. In geology, in par t icular , this process can be compl ica ted because the data
commonly show a natural ordering along outcrop lines such as r iver beds, t ec ton ic zones
e tc . Therefore , very sophis t ica ted methods have been developed for the gridding process.
For ins tance , weight ing funct ions are used tha t include di rect ional searching of data
points; also r a the r compl ica ted s t a t i s t i ca l methods have been developed for the prior
52
E
\ C !
Fig. 2~Y3: 'Catastrophic' contour maps from a data set with strongly fluctuating surface values, a,b: regular grids of different roughness; minimal polygon method (see text). c,d: the same estimations on an irregular spaced rectangular grid, every data point is located on a grid point, e: grid point estimation by Shepard's weighting function (see section 2.3.3). f: point distribution and bounding polygon.
Ca "H" und +~'
l,,ig"H'_~j b e:t,- schu~
r~val
2®°
Fig. 2.34: A ~hand' estimation of contour ' ~ ' ~ / ~ ' ~ lines for the data set of Fig. 2.33. I~ / ' < ~
53
i ........ l
Fig. 2.35: Three possible solutions of isolines for the ident ical da ta set° a} Approximat ion by Shepard 's method; b} minimal polygons; c) minimal poly- gons when the grid is genera ted from the data points. Grid points are marked. Modified from ALTHEIMER et al., 1982.
and poster ior analysis of the grid values (JOURNEL & HUIJBREYTS, 1978; HARBAUGH
et al., 1977; HUIJBREYTS, 1975).
In principle, the gridding technique is a map from one f ini te point set onto another
f ini te point set , whereby no one to one correspondence exists. There may be more grid
points than data points, or there could be less grid points than data points. This re la t ion-
ship may also change locally within the global gridding s t ruc ture , as i l lus t ra ted in Fig.
2.35 in t e rms of the grid points. Therefore, one may expect s tabi l i ty problems to occur,
i.e. t ha t the map becomes locally folded. Fig. 2.33 shows a ' ca t a s t roph ic v resul t of com-
puter contouring, due to the special, very inhomogeneous data configurat ion, the grid
s t ruc tu re and the gridding methods. This conf igurat ion was chosen to i l lus t ra te the prob-
lems tha t arise from computer contouring. As far as s tabi l i ty problems are involved,
they will be discussed below.
A minor problem in gridding is the special s t ruc tu re of the regular grid (Fig. 2.35).
In general , the grid (or net} is chosen in such a way tha t the d is tances be tween the
net points are constant ; at least they are cons tan t for every coordinate direct ion. There-
fore, the resul t ing contours will depend on the init ial decision about the grid s t ruc tu re
(Fig. 2.35}. If the grid is too rough, data points will be lost, or they are not cor rec t ly
recorded. On the o ther hand, if the grid is fine enough to provide locally the required
54
Fig. 2.36: Two isoline representa t ions of foraminiferal diversity (entropy) in Todos Santos Bay, California. a) Shepardts method; b) minimal polygons. Adapted from ALTHEIMER et al., 1982; data from WALTON, 1955.
resolution, the net may be too fine in other areas where the data points are sparsely
sca t t e red (cf. Fig. 2.35 c). For a computer program the la t te r case may be more impor-
tant than lost data points because a very fine grid can cause an extensive need of
memory and unreasonably long computat ion times.
Fur thermore, the original data points may not be recorded correct ly, even on a
very fine grid if they have in termedia te position between the grid points. This can be-
come important if the es t imat ion procedure has a smoothing e f f ec t like it is the case
with Shepardts local method (Fig. 2.36 a, cf. sect ion 2.3.3). This problem can be solved,
to some ex ten t , if it is possible to choose the grid points in such a way that every
data point becomes a grid point and that the grid is still a rectangular one {Fig. 2.35
c; ALTHEIMER et al., 1982). A really satisfying solution of this problem would require
that the grid is formed over the data points. The classical way to do this is the triangu-
lation method, but a good triangulation net is hard to establish within the computer
{SCHUMAKER, 1976; CAVENDISH, 1974).
2.4.2 Global and Local Extrapolat ions
Another problem is to bound the es t imated surface values to a reasonable area
in order to avoid extrapolat ions into areas where no data points are available. Most
55
gridding techniques allow such extrapolations, and most computer programs in use do
not tes t this simple problem. As can easily be derived from the classical tr iangulation
method (ALTHEIMER et al., 1982), the most extensive boundary giving reasonable est i -
mates is the convex polygon that surrounds all data points, though it does not ensure
a reasonable solution, as the Shepard es t imat ion in Fig. 2.33 il lustrates. Somet imes
a be t t e r res t r ic t ion of the solution space can be forced by additional const ra ints like
a limit dis tance between data points and grid points. But such restr ic t ions a f fec t all
es t imat ions, not only the boundary -- and a typical result are 'holes ' within the working
area.
There are several possible s t ra tegies to establish a convex boundary for the data
points (cf. ALTHEIMER et al., 1982). However, we shall see in the next section that
a very simple method results from 'minimal polygons', which can be used to el iminate
the interior points of the bounding polygon.
Indeed, the problem to find a reasonable global boundary for the working area
has its local equivalent. One has to ensure that a grid point, for which a surface value
is evaluated, is located within a closed polygon of data points and that these points
are used to e s t ima te the surface value. Otherwise, local extrapolat ions may occur, which
cause maxima or minima, which are not represented in the data or 'p la t forms ' result
if the local es t imat ion uses weighting functions.
The sectorial search method is an approach to solve this problem (HARBAUGH
et al., 1977)o The problem, of course, does not occur with the classical triangulation
method because, in this special case, the whole area in question is densely covered
by triangles or by convex polygons with the data points at the corners. Again, most
of the gridding techniques in use do not necessari ly tes t these conditions, especially
the weighting average methods do not. For the usually complex geological data the
(1
•
/ / '/
/ b
• / / /
/ ' / / ~
\
Fig. 2.37: a) the oc tant search method. A data point must be found in every octant for the es t imat ion of a grid point value. This avoids local extrapola- tions, b) Surface reconstruct ion by an octand search method genera tes sinusoi- dal contours even from a simple cylindrical surface.
56
problem has been well recognized (HARBAUGH et al., 1977), and s t ra teg ies like quadrant
and oc t an t search pa t t e rns have been developed to overcome the problem (Fig. 2.37).
These methods require t ha t for every grid point a se t of da ta points must be found
in a ce r t a in number of radial sec tors (Fig. 2.37). Although the procedure secures tha t
the grid point is surrounded by a - - n o t necessar i ly c o n v e x - - polygon of data points,
it may produce o ther defec t s . A local gap will be produced if one sec tor is empty,
even if t he re exis ts a convex polygon which incIudes the grid point and which would
secure a useful local es t imat ion . From this viewpoint, the method is not f lexible enough
because it requires locally a specif ic data conf igurat ion. On the o ther hand, the process
tends to i r regular contours because the sur face values e s t ima ted from the moving sec to r
system do not change smoothly, but they change r a the r suddenly when a data point
en te r s or leaves the system (Fig. 2.37), at least , if the required number of data per
sec to r is small . Conversely, a higher number of required da ta causes an increasing
number of gaps within the solution area.
2.4°3 Linear In terpola t ion by Minimal Convex Polygons
The discussed problems caused us in 1980 to develop in the 'Sonderforschungsbere ich
53, PalOkologie' (ALTHEIMER et al., 1982) a gridding technique tha t sa t i s f ies all the
so far discussed condi t ions in a very simple w a y - - the grid values are e s t ima ted from
the minimal bounding polygon of a grid point, by a t r iangle. The method works r a t h e r
well up to the point tha t somet imes s t range local ex t r ema appear in areas where no
da ta points are available, i.e. the ex t r ema cannot be explained by the da ta s t ruc ture .
A l a t e r s tab i l i ty analysis provided some remarkab le resul t s which also holds for o the r
gridding techniques, especial ly for the discussed sec to r search methods and for the
classical t r iangula t ion method. It will turn out tha t the problems are mainly geomet r i ca l
ones and tha t our gridding technique is not a real ly good way to e s t ima te surface points,
but t ha t it i l luminates very c lear ly the problems of contour ing from s c a t t e r e d data .
As was pointed out above, the locally s tab le e s t ima t ion of grid point values requires
tha t the grid point is loca ted within a polygon spanned by a subset of the original da ta
points. For a plane mapping problem the smal les t possible polygon is a t r iangle with
da ta points at i ts corners . As is well known from polygon theory and l inear opt imiza t ion
(COLLATZ & WETTERLING, 1971), this minimal polygon has the proper ty t ha t the sur face
values can be simply e s t ima ted by a l inear in terpola t ion for any point within the t r iangle
and along its boundaries (cf. ZIENKIEWlTZ, 1975; SCHUMAKER, 1976). Geomet r ica l ly
the th ree points es tabl ish a plane sur face e lement . Fur the rmore , the principle of ' n ea r e s t
neighborhood' can be easi ly sat isf ied. 'Neares t neighborhood' means tha t a grid point
value should be e s t i m a t e d from closest da ta points (as far as this does not cause local
57
extrapolation). If there exists a polygon at all that encloses the grid point in question,
then there exists a convex polygon, especially a triangle, which both includes the grid
point and has the point of nearest neighborhood at one corner. The proof of this s t a t e -
ment can he outlined in the following way:
If the point of neares t neighborhood is connected with every point on the convex
boundary of a l l data points, then the whole area inside the global convex boundary
is densely covered by non-overlapping triangles, and the grid point must be e i ther
an interior point or a boundary point of one of these triangles.
This secures that we find a bounding triangle. The minimal triangle without an interior
point is found with the following strategy:
Start at the point of nearest neighborhood and find the second and third nearest
points which form, together with the point of nearest neighborhood, a bounding
tr iangle for the grid point.
We can call this triangle the minimal convex polygon -- minimal because these are points
of minimal dis tance with respect to the convexity condition. One should expect that
such a triangle is a locally optimal form for the approximation of the grid point value.
The approximation turns into a local interpolation. In addition, the local propert ies of
the tr iangles project onto the global problem to find the bounding polygon of all data
points. The local triangulations are bounded to the interior of the convex polygon bound-
ary. This gridding technique has, therefore , the additional advantage that it can be
implemented in a very compressed way on a computer (ALTHEIMER et al., 1982) and
that it provides automatical ly control over the boundaries of the working area. On the
other hand, one can use the method to el iminate all points inside the global convex
boundary of the data set. The points to be el iminated are simply those for which a
minimal polygon exists which has data points on its c o r n e r s - - or the global convex
polygon consists of the data points which are not interior points of a triangle with data
points at its corners.
2.4.4 Stabil i ty Problems with Minimal Convex Polygons
Having done all these analyses of local and global properties, it was surprising
for us that the implemented process became unstable with cer ta in data configurations
(cf. Fig. 2.33). The main anomalies are local ex t rema which are not justified by the
data. They occur mostly within relat ively large areas without data points, and they are
especially strong if the surface values of the nearby data points are very irregular.
/ / /
Fig. 2.38,: The subdivision of a r ec tang le by the local t r iangula t ion method {minimal polygons). The blank areas belong to the same t r iangula t ion (as the shadowed do).
58
In addition, the anomal ies are very sens i t ive to small changes of the grid s t ruc ture .
Some of these pa t t e rn s resemble very closely the problems which may ar ise from sec to-
rial search methods, To explain these anomal ies it is necessary to study the local s t ruc-
ture of the polygons, and to see how t r iangles may come into compe t i t ion during the
gridding process,
Four points form the minimal polygon, which can be fur ther divided into t r iangles.
For any grid point located within this polygon two possible t r iangula t ions exist {Fig.
2.a8). Which t r iangle will be chosen, depends on the d is tance funct ions be tween the grid
points and the polygon c o r n e r s - - i.e. on the nea res t neighborhood rule. The minimizing
condit ion for the d i s tance funct ion divides the polygon into four regions (Fig. 2.38).
The opposite r ec tang les belong, thereby, to the same t r iangula t ion, and, there fore , have
s tab le and near ly smooth solutions in the i r inter ior . But the two possible t r iangula t ions
give d i f fe ren t solutions which are not connec ted in a smooth way (Fig. 2,39). If the
grid point moves over the t r iangula t ion boundaries, then the solution jumps from one
in te rpola t ion sur face onto the a l t e rna t ive solution. The instabi l i t ies , the re fore , occur
Fig. 2,39: A ruled sur face over a rec tangu la r grid e l emen t and its approxima- tion by the two possible t r iangula t ions which yield d i f fe ren t solutions.
59
. . . . . . . . . . [ . ! " ~:~. ~,
-" .-" 5 , : : . . . . . . . : "
Fig. 2.40: The bifurcation surface which corresponds to the two possible triangulations of Fig. 2,a9 (cf. Fig. 2.38). The instability is independent of the density of the grid, it results only from the local triangulation method. On a suff icient ly fine grid the contours re f lec t the bifurcation of the surface (b).
in two ways:
1) Closely related grid points may deviate strongly in the es t imated surface values
because they belong to di f ferent local triangulations.
2) Even small changes in the s t ructure of the regular grid can locally cause ra ther
dramatic changes in the es t imated surface v a l u e s - - e.g. for two independent con-
touring processes over the identical data set.
These e f f ec t s are especially strong within relat ively large areas without data points,
i.e. whenever the approximation involves data points which are far apart.
Fig. 2.40 i l lustrates in more detail how the local solution over a rectangle bifur-
ca tes into two disconnected surfaces with discontinuous contours. From which interpola-
tion surface the es t imated value will be taken, depends, as discussed, only on the position
of the grid point(s). The situation becomes even worse if one considers higher polygons,
i.e, a larger number of data points in competi t ion. The possible number of local triangula-
tions increases rapidly with the number of polygon corners {F{g. 2.41). The approximation
process will become more and more instable as the number of data points in compet i t ion
i n c r e a s e s - - a situation favored by large empty areas within the data space. A curious
situation is that in such cases a ref inement of the grid increases the instability in the
way that the resulting surface approximates the local discontinuities of the triangulation
ra ther than a continuous surface (cf. Fig. 2.40 h). The situation is nearly the same with
the octant search method where the triangles are replaced by open angular sectors.
60
Z
~
Fig. 2.41: The various possibilities for the tr iangulation and linear surface approximation of a f ive-point polygon (1 to 5: the numbers z=l ... indicate the height of the corner points; r: the cen te red i n t e r p o l a t i o n - - the central point is the a r i thmet ic mean of the corner points.
One could argue that the discussed instabil i t ies are only problems of the local
approximation method. But if we t ransfer the results to the classical tr iangulation meth-
od, which covers the data space densely with triangles, then it i s not hard to see that
we have, in principle, the same problems. Let several people draw a map from the iden-
tical data set with a f ree choice of triangulation, then the results can diverge to a large
ex ten t . The d i f ferent solutions for a f ive-point polygon in Fig. 2.41 can be taken as
an example. What we find is that the instabil i t ies appear now exclusively between di f fer -
ent maps.
We can go a s tep further . The same problems encounter us again when we draw
contour lines from the e s t ima ted regular grid. A simple way to do this is again a triangu-
lation of the g r i d - the contour lines are uniquely de termined on a triangle. However,
there are several possibilities for this triangulation, some of them are i l lustrated in
Fig. 2.42. Every possible triangulation gives another solution, and we can only hope that
these soIutions deviate not too widely because the surface is simple enough. Besides this
triangulation method other s t ra teg ies are in use which draw the contour lines direct ly
from the rectangles , e.g. such a s t ra tegy is used in the program package SURFACE
II (SAMPSON, 1975). Within this s t ra tegy problems arise when two opposite corners of
a cell are higher and the two others are lower than the value for the contour line en te r -
61
/ / / / / / / / / / / / / / / / o
\ \ \ \ \ \ \ \ o o \ \ \ \ \ \ \ \
\ / \ / \ / \ / \ / ~ / \ /
\ / \ / ~ o / \ / \
N/'•/ /\/\
Fig. 2.42: Four a l ternat ive triangulations of a regular grid,
ing the cell. Fig, 2.43 il lustrates this situation, and it becomes immediately clear that
the cases (b) and (c) are just the two a l ternat ive triangulations of the grid cell, the
decision problem is the same as discussed earlier. Case (a) is slightly different , it re-
presents a cen te red grid e lement where the centra l value could have been es t imated
as the a r i thmet ic mean of the corners (cf. Fig. 2.41). This approximation will be dis-
+ - - 4 - - 4 -
/ /
a b + - + -
,> "> c
4-
Fig. 2.43: Possible paths of contour lines through a grid c e l l - - (+) corners higher and (-) corners lower than average of corner values. Modified from SAMPSON (1975).
cussed in the next section. In the SURFACE II program a decision is made be tween
the solutions (b) and (c) in Fig. 2.43: (b) is chosen if the average of corner values is
higher than the entering contour line while (c) is chosen if the average is lower than
the value of the contour line. This choice is arbitrary, however, it ensures that contour
lines do not in tersec t within the grid e lement {Fig. 2.43 a) -- this switch causes a jump
from the lower to the upper surfaces in Fig. 2.40 when the average height of the corner
points is passed,
62
2.4.5 Continuation of a Local Approximation
It turned out that the method of minimal polygons or of a lbcal or global triangu-
lation is instable with respec t to small changes of the initial conditions. In the case
of the 'hand method ' , the initial condition is the choice of the triangulation, in the
' computer method ' , it is the choice of the grid. The same problem extends to o ther
local gridding techniques, to the seetorial search methods and even to the approximations
by weighting functions. They all are very sensit ive to small changes of the initial condi-
tions and to changes of the pa ramete r set t ing. A major problem arises if there are
large areas without data points. In this case, the interpolat ion process can be somewhat
stabil ized if one does not use the minimal convex polygons but t r ies to find the locally
maximal convex polygon. However, compet i t ion be tween polygons can be only avoided
if the en t i re interior of a locally bounded polygon is t r ea ted as a local continuum,
and if all grid points inside the polygon are es t imated from its corner points by some
smooth process. The compet i t ion during the formation of local polygons can be avoided
if the local solution projects continuously into the neighborhood, i.e. no overlapping
polygons are allowed until they have the same solution inside the in tersect ing areas
and on the common boundaries. The problem has a formal analogy in the analytic cont in-
uation of a function in the complex plane. This analogy suggests that one could s tar t
from a local solution, a local contour line, and then const ruct its continuation through
the data space by use of some convergence cr i ter ia . The convergence circle of the
analytic problem could thereby be replaced by convex polygons over the finite data
set.
The previous remarks lead to a geometr ica l problem, which is hard to solve in
the case of randomly sca t t e r ed data. Nevertheless , it seems useful to discuss finally
how the linear interpolat ion over tr iangles can be general ized for any convex polygon
and how a local solution over a regular grid e lement can be extended throughout the
global data space,
A) A Local Continuous Approximation
In the case of a rectangle , the simplest approach toward a stable continuous surface
approximation is to cons t ruc t a bilinear function over the corner points (SCHUMAKER,
1976)
f(x,y) = a I + a2x + a3Y + a4xY ' (2.57)
The corner values of the grid e lement have to be used to de te rmine the coef f ic ien ts .
Now, any rec tangle can be s tandardized to a square of unit area by the map
68
/" C_-- - - -
/ . ._-- - - - -"< / x
Fig. 2.44: Surface interpolat ion over a rectangle by use of a bilinear function (above); for details see text. The bilinear interpolat ion can be approximated by a linear interpolat ion if an additional centra l point is used, which can be computed as the a r i thmet ic mean of the corner points.
x ---- (Xi+ I - x)/(Xi+ 1 - Xi)
Y ---- (Yi+l - Y)/(Yi+I - Yi )'
where X. and Y. are the coordinates points, i i
(2.58)
of the corner
64
Besides s tandard iza t ion , the map (2.58) t rans forms the global grid coordina tes into local
oneS. Using the new local coordinates , the l inear in terpola t ion along the boundaries
of the r ec tang le can be expressed as
f(x) = w2F(x=O ) + WlF(X=l), (2.59)
where the F-values are the surface height at the corner points and the w i are weight ing
functions: Wl=X, w2=l-x , x in local coordina tes {for the y d i rec t ion x has to be replaced
by y).
The approach by a bi l inear funct ion implies to cons t ruc t a two-dimensional weight-
ing funct ion from the product w(x,y)=w(x)w(y) (e.g. PFALTZ, 1975; DeBOOR, 1978).
tf the weight ing funct ions for the boundaries are inser ted, one finds
and
w1(x=O,y=O) = (1-x)(1-y)
w2(x=O,y=l ) = (l-x)y
W3(x=l,y=l ) = xy
w4(x=l,y=O) = x(l-y)
w(x,y)=xy,
= w(l-x,l-y)
= w(l-x, y )
= w( x ,l-y)
= w( x ,i-y)
(2.60)
a very s imple pa t t e r n of pe rmuta t ions of the coordinates , which easily can be pro--
g r amme& It is easy to prove t ha t ~ wi=l, and t ha t z(x,y)= ~ Fiwi(x,y) is just the
ear l ie r no t iced bi l inear funct ion which provides a cont inuous sur face approximat ion over
the grid e l ement . The weight ing funct ions have the proper ty tha t
Fiwi(0.5,0.5 ) = ~Fi/4 , (2.61)
i.e. t he re exis ts one point on the sur face which is simply the a r i t h m e t i c mean of the
corner points. This observat ion allows a f irst order approximat ion of the bi l inear surface
over a r ec t ang le by a s imple t r iangula t ion. If one adds the cent ro id of the corner points
to the data points, then t h e r e exis ts locally a unique t r iangula t ion of the grid e l emen t
which is given by the connec t ions of the cen t r a l point with the corner points. The
sur face e s t i m a t e d from this t r iangula t ion is a l inear approximat ion of the sur face , which
was def ined by the bi l inear equat ion (2.57). Figs. 2.41 and 2.44 provide examples for
this approximat ion. It is easy to see t ha t a unique t r iangula t ion and, there fore , a unique
local sur face approximat ion can be cons t ruc ted for any convex polygon with n corners .
The addi t ional cen t r a l point is given by
(Xc,YcZc) = (l/n) [ (Xi,Yi,Zi). (2,62)
i t may be useful to in t roduce a meaning for this in terpola t ion scheme. The bi t inear
model is a harmonic function, and this allows a physical in te rp re ta t ion . If a shee t of
65
rubber is s t r e t ched over the rec tangula r boundary, the resul t ing surface equals the sur-
face described by the bi l inear equation. For the general ized convex polygon with n
corners one can cons t ruc t such a surface in the following way (BETZ, t948): The convex
polygon is mapped onto the unit c i rc le by means of the Schwarz-Chr is tophel formula.
The boundary values are then evaluated in t e rms of a Fourier series. The required
harmonic funct ion on the unit c i rc le is finally expressed by the equat ion
f(r,~)=ao/2+ Z rn(anC°S(n~) + bnsin(nq0)), and a first approximation on the original polygon
is given again by the cen te red t r iangulat ion.
Fig. 2.45: A ten t s t ruc tu re provides an example of a continuous surface with discont inui t ies at the poles.
B) Cont inuat ion of a Local Surface Approximation
The cen te red grid e lement , as defined above, leads in a r a the r na tura l way to
a cont inuous solution over the global regular grid s t ruc ture . If we add the computed
cent ra l grid points to the grid, we have simply a r e f inemen t of the grid, and we can
repea t this process infinitely. In the second i tera t ion, addit ional grid points and values
are computed at the boundaries be tween the original grid e l ement s and provide a con-
tinuous approximation be tween grid e lements . In general terms, a regular approximat ion
within grid e l emen t s occurs at every odd in terpola t ion s tep while at even s teps over lap-
ping grid e l ement s are eontinuous!y connected. What we find, is a surface which every-
where sa t is f ies the Laplace equat ion
= O, (2.63) Uxx + Uyy
a surface, which is everywhere smooth, only at the grid points local discont inui t ies
A
\/\/
I •
66
Fig. 2.46: I t e ra t ive r e f inemen t of the grid s t ruc tu r e by recurs ive averaging. Only a single pathway is i l lus t ra ted , which asymptot ica l ly approaches a corner of the cen t r a l grid e lement . Any o the r point, the original grid points and the averaged ones, cause s imilar cascades: The original cen t ro id grid e l emen t is subdivided into smal ler and smal ler r ec tang les which, in the l imit, cover the area densely, however wi thout being cont inuous in a d i f fe ren t i ab le sense. Left : a regular or thogonal grid, r ight: a cen te red regular grid which actual ly consis ts of two overlapping grids as indicated.
appear (Fig. 2.45). The re la t ion be tween the recurs ive averaging process and the Laplace
equa t ion can easily be shown if the Laptace equat ion is approximated on a f ini te grid.
In t e rms of a f ini te grid, equat ion (2.6a) reads
(Ui_l,j_2Ui,j+Ui+l,j) + (Ui,j_l-2Ui,j+Ui,j+l) = 0 (2.64)
providing a f ini te approximat ion, which can be r ewr i t t en as
Ui,j= (I/4)((Ui_l,j+Ui+l,j) + (Ui,j_l+Ui,j+ 1 ),
and this is simply the average discussed above. Thus, our cont inuat ion
analogue to the analy t ic cont inua t ion in the complex plane.
(2.65)
process is a f ini te
The cont inua t ion process by recurs ive averaging is, in addition, opt imal in the
sense tha t it is s tab le under small d i s turbances of the grid p a t t e rn and is opt imal in
t e rms of computa t ion costs, To see, why the l a t t e r remark holds, let change the view-
point again to a single grid e lement . We want to find a local approximat ion which
is continuously connec ted with the neighborhood. To find such a local approximat ion
67
we need in to ta l i ty 16 grid points like in Fig. 2.46a, and this e lementary grid allows
re f inement to any level within the central grid e lement . An al ternat ive would be to
use initially a cen te red grid (Fig. 2.46b) which, of course, provides an initial triangula-
tion. However, if we request a continuous connection with neighboring elements , we
need 21 grid points. The increased number of necessary grid points can be related to
the fact that the cen te red grid is not unique, i.e. that there exist two a l ternat ive grid
s t ructures as indicated in Fig. 2.46b. A continuous solution requires that these a l ternat ive
grids are superimposed, and this causes the higher number of required grid points.
However, the continuation problem can be solved in a quite d i f ferent way: We
can request that the local surface e lement has continuous derivat ives along its bounda-
ries and, thus, can continuously be connected with the neighboring elements . Such an
approximation requires at least cubic splines, and first we consider the case that the
first derivat ive vanishes along the boundaries of the grid e lement . A useful approxima-
tion is given by the weighting function
and w(y) = y2(3-2y).
w ( x , y ) = w ( x ) w ( y
w ( x ) = x 2 ( 3 - 2 x ) ;
(2.66)
The height of a surface point can be expressed as weighted average of the height of
corner points
z(x,y) = ZlW(X,y ) + Z2w(1-x,y ) + Z3w(l-x,l-y ) + Z4(x,l-y). (2.67)
If we use equations (2.66), we can rewri te equation (2.67) as
z(x,y) = ((ZI+Z4)--(Z2+Z3)Xx2(3-2x)Xy2(3-2y)) + (Z3-Z4Xx2(3-2x)) + (z2-z4)(y2(3-2y)), (2.68)
an equation which looks ra ther complicated. However, if we introduce the abbreviations
2 2( u = x ( 3 - 2 x ) ; v = y 3 - 2 y ) , ( 2 . 6 9 )
equation (2.68) turns into a simple bilinear equation
z ( x , y ) = auv + bu + cv (2.70)
with obvious parameter identif ications for 'a t, 'b t, and 'c ' . Thus, we are still dealing
with equation (2.57), with the only d i f ference that the coordinates (x,y) are replaced
by functions of these coordinates. Equations (2.70) and (2.69) provide a system of equa-
tions consisting of two parts: The interpolat ion equation, which is simply a bilinear
equation, and a map {x,y) ~ (u,v), which defines a deformation of the original c o o r d i -
68
es, thus , t h a t t h e y s a t i s f y c e r t a i n c o n d i t i o n s a t t h e b o u n d a r i e s o f t h e gr id e l e m e n t .
Our a p p r o x i m a t i o n p r o b l e m t u rn s in to t h e p rob l em to f ind a p rope r m a p (x,y) -~ (u,v)
wh ich s a t i s f i e s t he r e q u i r e d cond i t ions : T he m a p
3 2 u = alx 3 + blX + ClX + d 1
v a2Y + b2 y2 + c2Y + d 2 (2.71)
for i n s t a n c e a l lows to ad jus t t h e f i r s t d e r i v a t i v e a long t he b o u n d a r i e s o f t he gr id
e l e m e n t ; h o w e v e r , it is no t t he m o s t g e n e r a l c a s e (for a d i s cus s ion of sp l ines , s e e
DeBOOR, 1978). A n y w a y , e v e n an a p p r o x i m a t i o n by e q u a t i o n 2.71 is r a t h e r s e n s i t i v e
to s m a l l c h a n g e s in t h e gr id s t r u c t u r e . T h e d i s cus s ion o f t h e cub i c sp l ine in s e c t i o n
2 .a .3 c an e a s i l y be e x t e n d e d to t h e t w o - d i m e n s i o n a l c a s e - - a s m a l l d i s t u r b a n c e o f
t he e s t i m a t e d s lope a t t he gr id b o u n d a r i e s c a n t o t a l l y c h a n g e t h e e s t i m a t e d s u r f a c e
p a t t e r n , wh ich m a y s w i t c h f rom a r idge to a va l l ey or v i ce ve r sa . As eas i ly c an be
seen , t he s t a b i l i t y o f our a p p r o x i m a t i o n p r o b l e m d e p e n d s on ly on t he m a p {2.69), and
th i s r e l a t e s it to s i n g u l a r i t y t h e o r y . We c a n t r a n s f o r m the m a p {2.69) to a m o r e c o n v e n -
i en t f o r m s by a s i m p l e
dislocation of the origin x --- x + 1/2; y -- y + 1/2
and u -- u - 1/2, v --- v - 1/2, which yields
u= 2x 3 - (3/2)x; v = 2y 3 - (3/2)y,
a rotation x -- x + y; y -- x - y yields
3 3 x 2 y ~ y 2 3 u = x + + x + y 3 3 v = x - 3 x 2 y + 3 x y 2 - y
and a final rotation in the (u,v)-space u --~ u+v, v--- u-v
transforms our original map into
u = 2 x 3 + 6 x y 2 ; v = 2y 3 + 6 x 2 y , ( 2 . 7 2 )
a m a p w h i c h r e p r e s e n t s a spec i a l fo rm of t he double cu sp c a t a s t r o p h e . In c a t a s t r o p h e
t h e o r y such a m a p is e m b e d d e d in a p o t e n t i a l , in th i s c a s e t h e p o t e n t i a l would be
V = x4/2 + y4/2 + 3x2y 2 -ux -vy, (2.73)
and t he m a p r e s u l t s f rom t h e cond i t i on t h a t t h e pa r t i a l d e r i v a t i v e s van ish , i .e. f r om
t h e e q u a t i o n s
V = 0 = 2x 3 + 6xy 2 - u x
V = 0 = 2 y 3 + 6 x 2 y - v . Y
69
If we now re turn to the more general case {the der iva t ives are de te rmined from the
da ta points), we need again addit ional pa ramete r s . Ca tas t rophe theory implies tha t the
potent ia l (2.73} has general unfolding
V = x 4 + y4 + a x 2 y 2 + b x 2 y + c y 2 x + dx 2 + e x y + f y 2 _ ux - v y , (2.74)
an expression which provides us with 6 f ree pa rame te r s to adjust the boundary con-
ditions. This expression, however, is in local coordinates~ in global coordinates we would
have to unfold the map (2.72} and to consider all possible pa rame te r s inclusively the
cons tan t ones.
The Double Cusp is ex t remely unstable, the s table regions are ex t remely narrow,
and even small d is turbances cause switching solutions, in this special case switches
from ridges and hills to valleys and depressions. This gives us a d i rec t re lat ionship
to ca t a s t rophe theory; however, the problems encounte red through these sect ions are
connec ted with ca tas t rophe theory in a much wider sense: The approximation problem
in surface recons t ruc t ion is usually associa ted by some opt imizing problem, i.e. to es t i -
ma te the grid point from the neares t data points. A common problem with such opt imiz-
ing s t ra teg ies is tha t during a smooth change of the dis tance function ' the opt imum
solution changes with a jump, t ransfer r ing from one compet ing maximum to the o the r '
(ARNOLD, 1984). That, of course, is what we observed throughout the discussion of
surface reconst ruct ion . The connect ion of the observed instabi l i t ies with ca t a s t rophe
theory may not necessar i ly be obvious because we usually think about d i sc re te and non--
d i f fe ren t iab le systems in approximation processes. However, d iscre te are only the grid
points, which turned out to be pa rame te r s of a smooth interpolat ion surface. By a smooth
change of these pa rame te r s the approximat ion may reac t with sudden jumps in the
geomet r ica l solution. A change of the grid point values, however, is equivalent to a
change of the boundary conditions, and these clear ly a f fec t the opt imizing function.
in t e rms of var iable boundary condit ions we can, therefore , apply ca t a s t rophe theory
to the surface approximation problem. A discussion of more general opt imizing prob-
lems is given in ARNOLD (1984).
one parameter
F(y) = I Y l
Table 2.1 Singulari t ies in opt imiza t ion problems
Normal forms of a maxima funct ion F {ARNOLD, 1984):
two parameters
IyT or
F(y) = max(Yl' Y2' YI+Y2 ) o r
2 mxaX(-X4 + y l x + Y2 x)
3. N E A R L Y C H A O T I C B E H A V I O R
O N F I N I T E P O I N T S E T S
Chaos implies to ta l ly and apparen t ly i r remediable lack of organizat ion. In physics,
a classical example for chaos is turbulence. In a turbulent system, the pathway of a
par t ic le cannot be predic ted at all, and two par t ic les , which are init ial ly close toge ther ,
may depar t in a shor t t ime interval . The t rans i t ion from a de te rmin i s t i c {laminar) behav-
ior to chaos ( turbulence) can be usually described by a b i furca t ion t ree (Fig. 3.1). "Af te r
the f i rs t b i fu rca t ion the flow becomes periodic, a f t e r the second b i furca t ion the flow
is quasi periodic with two periods, and so on" (RICHTMYER, 1981). Af t e r a suff ic ient ly
high number of b i furca t ions the chaot ic aspect of the flow is so highly developed tha t
s t a t i s t i ca l methods are the proper way to study i ts behavior . It is c lea r t ha t the behav-
ior of such sys tems during the course of t ime depends very sensi t ively on the init ial
condi t ions {HAKEN, 1981), and t ha t the b i furca t ions are not a dynamical fea ture , but
appear in the s t a t e space of the system, i.e. they are a topological proper ty of the
system.
During the last decade, another way to study chaos has a t t r a c t e d much a t ten t ion :
the behavior of d i f fe rence equat ions in ca lcu la tors (MAY, 1974; ROSSLER, 1979; THOMP-
SON, 1982). In this case, the dynamical system is rep laced by an i t e ra t ed map describing
the ou tcomes in f ini te t ime intervals . MAY's (1974) favor i te example was the s tandard ized
form of the logist ic d i f f e rence equat ion. A s h o r t review of the behavior of this equat ion
will be given in the f irst sect ion to in t roduce the concepts of b i furca t ions and of chaos
more precisely. The explici t numer ica l approximat ion of a par t ia l d i f fe ren t ia l equat ion
then e luc ida tes once more the concep t of bi furcat ion, and the concept of i t e r a t ed maps
is used to study inf ini te sequences of caust ics in r e f r ac t ion seismics.
Fig. 3.1: A b i furca t ion cascade or a genera l ized ca t a s t rophe (THOM, 1975), as it resul ts e.g. from the logist ic d i f fe rence equation.
71
Fig. 3.2: Two versions of Galton's m a c h i n e - - a small and a large o n e - - which produce the binomial distribution.
As was noticed above, s ta t is t ica l methods are the usual way to study chaot ic
systems. The 'Galton machine ' (Fig. 3.2) i l lustrates the relationship between the chaot ic
t ra jec tor ies of particles, which cannot be predicted, and the well predictable outcome
if enough part icles are considered. In this case, our impression of chaot ic motion within
the machine will not at least depend on its size (Fig. 3.2). In addition, the form, the
internal geometry of the machine, a f fec t s the type of the s ta t is t ical outcome. The bifur-
cation t ree of Fig. 3.1 can be taken as another machine of this type. It will produce
a uniform distribution. An interest ing case occurs if the internal configuration of such
machines depends on some parameters , or if the initial conditions can a f fec t the outcome
of the machine.
The first example is a brief review of the logistic d i f fe rence equation. A more
interest ing example, from the geological viewpoint, is the instability of the explicit
approximation of a partial different ia l equation. The bifurcation, which is caused by
a smooth change of a parameter , can be nicely visualized by the uncoupling of the grid
into two independent substructures. The concept of i te ra ted maps is finally applied to
series of caust ics in ref rac t ion seismology.
72
The concept of bifurcations and chaos is then applied to several computer methods.
The problem is that chaos in such cases is not obvious. In most exampies, a small change
of parameters will strongly influence the outcome, but with a computer procedure this
sensi t ivi ty will normally not be de t ec t ed because the data are only processed with a
cer ta in pa ramete r set t ing. The first of these examples is the usual Chi2-test ing of direc-
tional data. The tes t is commonly performed against a uniform distribution, and it is
unstable with respect to an arbi t rary choice of the sectorial pa t te rn on which the com-
putation of the tes t s ta t i s t ic is evaluated. The striking point is that the stabil i ty of
the tes t decreases with increasing sample size.
In the third sect ion, problems with sampling s t ra tegies in sedimentology are dis-
cussed. One goal of the s ta t is t ica l analysis of profiles is to de tec t periodicity pat terns .
Two methods are in use, the analysis versus transit ional probabilit ies and the classical
t ime-ser ies analysis. In both cases it is a typical s t ra tegy to take samples at equal dis-
tances. In this case, the transit ion matrix becomes dominated by singular loops, and
the so -ca l l ed ' t ransi t ional probabil i t ies ' are not fur ther free of dimensions. In the case
of a t ime series anaIysis, the identical approach can cause art i f icial pa t te rn formation.
The example is closely related to the generici ty problem of maps, an aspect which is
briefly mentioned. The main result will be that geometr ical and geological reasoning
cannot be replaced by a formal, pseudo-object ive sampling s t ra tegy.
Then, we shall deal with various aspects of classical centroid cluster s t ra tegies .
Again a si tuation is encountered where an increase of the sample size does destabil ize
a ' s ta t i s t ica l ' pa t t e rn recognition process, and it will turn out that these methods provide
excel lent examples of chaot ic behavior on finite point s e t s - - they show the discussed
proper t ies of chaos, especially the ex t remely high sensi t ivi ty to small changes in the
initial data.
Finally, the bifurcation of t ree- l ike bodies is analyzed. The basic model is ent irely
determinis t ic ; never theless the bifurcation pa t te rns gene ra ted are rather chaotic. From
this chaot ic pat tern , however, a well de termined shape a r i s e s - - an analogy found in
the shape of t rees , which is typical on the species level. The analysis is based on a
modificat ion of HONDA's (1971) computer model and takes up the geometr ical analysis,
which roots in D'Arcy Thompson's and even Leonardo da Vinci's work.
3.1 ITERATED MAPS
Classically, stabil i ty is the most important concept for the numerical solution of
different ia l equations. The typical way to solve dif ferent ia l and partial different ia l equa-
tions numerically is to t ransform them into an ' i t e r a t ed map' by use of Taylor's theorem,
73
It is well known tha t there are somet imes several choices for the t rans format ion , and
tha t the various possible approximations behave d i f ferent ly with respect to the quali ty
of the approximation, to the convergence and to o ther s tabi l i ty problems. Here some
aspects of i t e r a t ed maps are brief ly discussed under topologicaI aspects because this
approach may give some insight not only in those problems, which occur with d i f fe rence
equations, but aIso in the concepts of b i furca t ions and chaos.
3.1.1 The Logist ic Di f fe rence Equation
The logistic growth function plays some role in biology and in paleontology. The
d i f fe rence formulat ion of this equat ion was MAYas (1975) favored example for b i furca t ions
and chaot ic behavior. In the meant ime, it became an impor tan t example for b i furca t ion
cascades and chaos in various fields (e.g. HAKEN, ed. 1982). The d i f fe ren t ia l equat ion
of the logistic equat ion is given by
y' = ay(b-y), (3.1)
which has a wei1 known explici t solution. A simple s t ra ight forward d i f fe rence approxi-
mat ion is given by
Yi+l = Yi + dtaYi(b-Yi)" (3.2)
For a special p a r a m e t e r se t t ing of 'a ' and 'b ' , the solution of this d i f fe rence equat ion
depends only on the p a r a m e t e r At, which represen t s a f ini te t ime interval . As Fig. a.a
shows, the upper boundary is only approached for small values of a t . As this p a r a m e t e r
increases, one finds tha t the solution f luc tua tes around the sa tura t ion level. For larger
values of the discre te t ime intervals , the long t ime output of the d i f fe rence system
resembles much more the Lotka-Vol tera model (LOTKA, 1956) of a preda tor -prey system
than the original logistic growth model.
By some e l emen ta ry coordinate t r ans format ions (e.g. ROSSLER, 1979) the logistic
d i f fe rence equat ion can be s tandardized to the form
Yi+l = rYi(l-Yi)' (3.3
which allows to analyze tlle behavior of this model in a general way. The relat ionship
be tween the Yi+l and the Yi values can be p lo t ted as the graph of a funct ion for which
the Yi values are the values of the independent variable. For the f irst i tera t ion, the
graph of this funct ion is a parabola (Fig. 3.4, it I). The sa tura t ion value is exact ly
reached if Yi+l = Yi' and this defines a s t ra ight line in the (yi,Yi+l) coordinates . In the
74
Fig. 3.3: Numeric solutions of the logist ic d i f fe ren t ia l equa- tion for various d i sc re te t ime intervals .
N
At
. . . . . . . . . . t
graph of funct ion (3.3) the sa tura t ion point is given by the in te rsec t ion of this line with
a specif ic parabola, which is de te rmined by the p a r a m e t e r r. Fig. 3.4 (I) shows how
one can use these proper t ies to analyze which values of r allow for a s table solution.
The equivalent a lgebraic expression would be
Yi = rYi(l-Yi)' (3.4)
which can be solved for Yi" Higher i te ra t ions are capable to produce periodic solutions.
i t I Yi÷1 i t I I
Yi+t
Y,. Fig. 3.4: Fi rs t (I) and second (II) i t e ra t ion system of the logistic equat ion in s tand- ardized form. The curves correspond to d i f fe ren t values of the p a r a m e t e r ' r ' ; the i r in te r sec t ions with the s t ra igh t line are the equilibrium values.
75
The first one occurs for Yi+2 = Yi' i.e. every second i terat ion takes the same value.
Again one can find a graphic representat ion as well as an algebraic one. If one rewri tes
equation {3.4) in te rms of Yi+2 and of Yi' one finds a quartic polynomial
Yi+2 = rYi+l(l-Yi+l)
= (rYi(l-Yi))(l-rYi(l-Yi)).
(3.5)
The stable points are found in the same way as before {Fig. 3.4, it II) by set t ing Yi+2=Yi ,
and we find up to four equilibrium points, but not all of them are stable. As the para-
meter r varies, one finds up to three intersect ions between the polynomial and the equi-
librium line (except the trivial solution Yi = Yi+2 = 0). In the same way we find an in-
creasing number of periodic solutions for every relationship Yi+k = Yi' or, as k increases,
we get an infinite number of periodic solutions or a bifurcation cascade like in Fig. 3.1.
This type of chaot ic behavior was analyzed by MAY (1974), who showed that the logistic
equation has an infinite number of possible periodic t ra jec tor ies and is of chaot ic
behavior.
However, the logistic equation is only the special ce lebra ted example. OSTER &
GUGGENHEIMER (1976) showed that any convex function can replace the parabola in
equation (a.al and drew connections to the Hopf bifurcation. Even a linear spline
approximation causes such bifurcations and periodic solutions {Fig. 3.5a). Probably models
based on exponential functions are more biological than the finite logistic model because
they have no sharp upper limit. In the case of the finite logistic equation, there is a
limit for the 'height ' of the parabola: It cannot exceed its 'width ' , otherwise the process
escapes into negative values without bounds, i.e. the i terat ion simply breaks down. There-
l V I I
Fig. 3.5: Any convex function can be used to define an i te ra ted map, which posses- ses periodic solutions. Dashed lines indicate pathways which te rminate in a cyclic motion.
76
fore, we are not free in choosing the parameter r which is bound to values 0 <r <4
(y(0.5)=r/4 ---> rmax=4), and a second limit is given when the parabola is so shallow that
it does not in tersect with the line Yi+l=Yi . Some possible a l ternat ive models are (OSTER
& GUGGENHEIMER, t976):
Yi+ 1 = Yi exp(r( 1-ay i)
and
Yi+ 1 = Yi/{ I +exp(-b(1-aYi)).
For r << 1 the last equation can be approximated by equation (3.3} (see OSTER & GUG-
GENHEIMER, 1976). In general, the logistic equation provides a 'prototype ' of chaotic
behavior of i tera ted maps, which easily can be analyzed, and, therefore, it is the most
ce lebrated example.
3.1.2 The Numerical Approximation of a Part ial
Different ia l Equation
In geology the partial differential equation u t = Uxx plays some role as a ' t ransport
equation' . It describes e.g. the flow in porous media and the compaction of sediments
(TERZAGHI, 1943; DESAI & CHRISTIAN, 1977}. A straight forward approximation by
differences leads to the explici t scheme
(Ux,t+l-Ux,t)/At = (Ux_l, t - 2Ux+l, t + Ux+l,t)/Ax2. (3.6)
This equation can be rewri t ten as an i tera ted map
Ux,t+ 1 = (l-2(At/Sx2)Ux,t + (Ux_l, t + Ux+l,t)At/Ax 2. (3.7)
It is well known from numerical mathemat ics (MARSAL, 1976) that the stability of this
approximation requires that
1 - 2(At/Ax 2) ~ O. (3.8)
Fig. 3.6: Stabili ty region of equation (3.8).
Ax
" i l At=Ax2/2
4 f "
if1 table cr i t ical
1
The first r ight hand te rm of equat ion (3.7) needs to be positive, and this de te rmines
the s tabi l i ty condition. If the left hand side of equat ion (3.8) is set to zero, the equat ion
describes a parabola (Fig. 3.6) in the control space (~t,kx), and the in teres t ing point
is what happens in this case with equat ion (3.7). If the control pa r ame te r (3.8) is zero,
the local solution of the map (3.7) does not fur ther depend on the Ux, t values, and
the grid separa tes into two disconnected subs t ruc tures (Fig. 3.7}. The solution then does
77
Fig. 3.7: Two represen ta t ions of the grid b i furca t ion for the explici t d i f fe rence scheme of the t ranspor t equat ion u t = Uxx. The bi furcat ion occurs if the control
p a r a m e t e r is zero (I - 2(kt/Ax 2} = 0).
not fu r ther descr ibe the original ' t r anspor t equat ion ' , but two independent solutions of
this type arise. In consequence, the solution depends strongly on the init ial da ta config-
uration, a fac t which is somet imes not recognized (MARSAL, 1976}. To see how the
solution depends on the init ial conditions, the degenera ted version of equat ion (3.7) can
be wr i t t en as
Ux,t+l = (Ux_l, t + Ux+l,t)/2 (3.9)
Now, we take the example of a subs tance spreading from a source of cons tan t
in tensi ty into an empty medium, and we compute this by use of equat ion (3.9):
t 0 0 0 0 0 0 0
t i l I 0.5 0 0 0 0 0 t I 0.5 0.25 0 0 0 0
t I 0.625 0.25 0.125 0 0 0
78
The numerical approximation looks ra ther well, and, as can be shown (MARSAL, 1976),
it really approximates the different ial equation. Next, we take the same boundary condi-
tions but d i f ferent initial conditions:
t = O
t = I
t = 2
t = 3
t = 4
1 0 1 0 1 0
1 1 0 1 0 1
1 0 .5 1 0 1 0
1 1 0.25 I 0 I
1 0.625 1 0.125 1 0
O W g
b O ~
This time, the solution is nei ther numerically nor physically reasonable, we get fluctua-
tions which cannot approximate the t ransport equation. What happens, becomes clear
if one sketches how the successive values are connected:
..'% ><....'>< 0 I 1 . . . j I ~ . . . . j ~ . . . 0 . . . j I
" " " " 7 ""0 1 U , ; ~ 1 st] ~ •
J" "~ s • f ..'% 5< 5 < ' > 5 I x _ I . . 0 . 2 5 _I . 0 . I -><. ->< I ~ I O. 1 0
Clearly, the bifurcation of the grid into two independent subsets causes two independent
solutions -- one is constant because the initial conditions are constant on this subset
(the diagonal series of ones}, the other one resembles the solution of the first example,
i.e. a spreading process from the left boundary into an empty medium. It is not hard
to see that the stable solution of the first example is not really stable, as somet imes
is assumed in texts on numerical methods (MARSAL, 1976), but that it also consists
of two independent solutions, which under the special conditions become identical.
In case the cri t ical pa ramete r (3.8) takes values less than zero, the result f luctuates
and assumes negative values on one of the bifurcation grids. In this case, the grids are
again connected, but the negat ive control pa ramete r causes a l ternat ing signs of the
U values so tha t one, in pr inc ip le , has still two di f ferent solutions of the discussed x,t
type. In addition, we observe a close relationship to the discussion of regular and cen te red
grids in sect ions 2.4.4-5. Indeed, as the cri t ical value of the parameter (3.8) is ap-
proached, the stable regular grid {Fig. 3.6-1) evolves into two grids which resemble the
cen te red grids of the previous discussion. These grids are disconnected and provide two
79
independent solutions. The stability problem, therefore, has a strong topological
component. Another analogy provides the discussion of linear systems in section 2.2.1,
where we observed a similar parabolic stability boundary. Of course, the partial d i f feren-
tial equation ut+Uxx=0 can be approximated by a set of differential equations
Y'I = allYl + a12Y2 + "'" + alnYn
Y'2 = a21Yl + a22Y2 + "'" + a2nYn etc.,
(3.1o)
which provide a discontinuous spatial but continuous temporal approximation.
To general ize this result, we can briefly analyze the difference approximation of
first derivatives. There are three approximations in use (e.g. DESAI & CHRISTIAN, 1977),
the
forward difference
backward difference
central difference
(Ui+l, j-ui, j) /Ax + O(Ax)
(ui, ]-Ui_l, j)/Ax + O(bx)
(Ui+l, j-Ui_l, j)/(gkx) + O((Ax)2).
Under numerical aspects the central difference should be the best one to approximate
a first partial derivat ive because its discretization error is only of order (Ax) 2. But nearly
all approximations using the central difference are instable (MARSAL, 1976). The previous
discussion has shown that this is not a numerical problem but a topological one. The
central di f ference causes a grid bifurcation as in the previous example, i.e. one computes
two independent solutions, and, therefore, the approximation can only be used for very
special initial conditions. Actually, the problems, which arise here, are very close to
those discussed in the last chapter, especially the surface approximations from scat tered
data.
3.1.3 Infinite Series of Caustics
In refract ion seismology one is sometimes interested in the so-called 'higher arrivals '
and in the caustic formed by the rays. The caustic is the envelope of rays, which, in
its totali ty, can be writ ten as
F(x,y,p) = O. (3.11)
The equation of the envelope of this family of rays is obtained by eliminating the ray
80
p a r a m e t e r p f rom e q u a t i o n (3.11) and f rom i ts pa r t i a l d e r i v a t i v e
(3.12) 3 F(x,y,p) = 0
3p
(e.g. BEN-MENAHEM & SINGH, 1981). For a general overview one can choose special
conditions, i.e. the situation where the source is located at the reflector. For a medium
with linear velocity increase, the rays are circles (e.g. OFFICER, 1974). Because the
aim here is only to demonstrate the use of iterated maps, a still more simple model
will be used , pa r abo l i c r ays . If t h e c u r v a t u r e o f t h e r a y s u n d e r c o n s i d e r a t i o n is l a rge ,
t h e p a r a b o l i c r a y s a p p r o x i m a t e t h e c i r c u l a r ones to s o m e e x t e n t . If all pa rabo l i c r ay s
h a v e a c o m m o n s o u r c e point , e q u a t i o n (3.11) b e c o m e s
y - ( x 2 - b x ) = 0. (3.13)
A s p e c i f i c r ay wi th r ay p a r a m e t e r b p a s s e s t h r o u g h t h e po in t s x = 0 and x = b if y = 0.
A t x = b t h e r ay is r e f l e c t e d , and b e c a u s e t he s o u r c e is l o c a t e d a t t h e r e f l e c t o r , it
r e a c h e s t he r e f l e c t o r a s e c o n d t i m e a t x = 2b. This g ives t h e g e n e r a l i t e r a t e d m a p
for t h e r e f l e c t i o n po i n t s
or
x i = xi_ I + b
x. = xo + ib. 1
(3.14)
T h e r ay i t s e l f is d e f i n e d in t he i n t e rva l (0,b) and m a p s i t e r a t i v e l y in to t he i n t e r v a l s
(b,2b), (2b,3b), . . . . Or if a c e r t a i n i n t e r v a l is g iven , t h e e q u a t i o n for t h e r ays {3.13)
t a k e s t h e fo rm (by u se o f e q u a t i o n (3.14)
y - ( ( x i - i b ) 2 - b ( x i - i b ) ) = 0 . ( 3 . 1 5 )
Fig. 3.8: Caus t i c ( s ) o f a s i m p l e p a r a b o l i c r ay s y s t e m w i t h r e f l e c t i o n a t t h e s u r f a c e . T h e s o u r c e is l o c a t e d a t t he r e f l e c t o r .
81
By use of equations (3.11) and (3.12) one finds the equation of the caust ics by eliminating
the ray paramete r b:
y - xi2(1-(l+2i)2/(4i(i+l))) = 0
or (3.16)
y + xi/(4i(l+i)) = O.
Thus, the caust ics are an infinite series of parabola with increasing curvature, which
all pass through the point {0,0). Fig. 3.8 gives the first i terat ion as an example. More
complicated ray systems like circular rays or arbitrary positions of the source can be
t rea ted in the same way.
Of some interest is the case of a source located below the ref lec tor . This situation
causes a bifurcation, which can be qualitatively described in the following way. The
source is now located at depth 'a ' and the ref lec tor , as before, at depth zero. Then
the ray equation (3.16) becomes
( y + a ) - ( x 2 - b x ) = 0, ( 3 . 1 7 )
i.e. all rays are passing through the source point (-a,0). Now, the rays are re f l ec ted
at y = 0, and their horizontal position is then
2 - x + bx +a = 0 . ( 3 . 1 8 )
To find the ref lect ion points, one has to solve the quadratic equation (3.18), and, in
general, this will give two ref lect ion points because the ray propagates into the positive
and into the negative x-direct ion from the source point (o,-a). Because of symmetry
Fig. 3.9: Caustic of parabolic rays. The source is located below the ref lec tor .
82
reasons, we have to consider the absolute values of the ref lec t ion point; both solutions
propagate into the positive and into the negative direction -- we have to fold the solution
space. Only in this case, the whole semiinfini te solution space is covered with rays.
The consequence is that the ray parameter 'b' is not further uniquely defined; it describes
two rays, and both ray systems are capable to produce i tera t ively caustics. The two
caust ics are connected at a cusp point, i.e. one finds cuspoid caust ics (POSTON &
STEWART, 1978; Fig. 3.9). The bifurcation of the solution arises also with other ray
systems, such as circular rays. It is not a property of a special ray system, but it only
depends on the position of the source; it is a s t ructural ly stable topological proper ty
of ray systems.
3.2 CHI2-TESTING OF DIRECTIONAL DATA
Within the analysis of or ientat ion data one has to prove whether the observed
distribution has pronounced ext rema. The usual way is to tes t the contrary, whether
the data differ significantly from a uniform distribution. The propagated method in text -
books {e.g. MARSAL, 1979) is the Chi2-test . BALLENTYNE & CORNISH {1979) observed
tha t the Chi2-value obtained from such a tes t depends on the arbi t rary select ion of
the of f se t point for the sector system, in which the direct ional data are grouped. By
an extensive numerical analysis they showed that the Chi2-values f luctuate as the sector
pa t te rn is ro ta t ed over the data. In their analysis, the Chi2-values passed thereby several
t imes the s ignif icance leveh Therefore they concluded:
"The hitherto widespread use of the test in this way cannot, therefore, be
considered a valid mode of analysis, and the results of previous studies
employin E this methodolgy must be treated with extreme caution. "
While the numerical analysis of BALLENTYNE & CORNISH (1979) shows that this
is a problemat ic tes t , it does not e lucidate why the uniform distribution as a zero-hypoth-
esis behaves in such an unpredicted way. Ballentyne & Cornish not iced that the expec ted
frequency E is a constant for all sec tors under the special condition of a uniform distri-
bution:
E = N / k; N: number of data (3.19)
k: number of sectors.
The equation for the Chi2-value can, therefore , be wr i t ten as
1 k k X2= ~ i~=l ((Oi-E)2 = i=l ~ (Oi-Ei)2/Ei
if E. = constant; I
0.: number of observed data in the sector i. i
(3.20)
83
But this simplified version contains still a constant value inside the sum. Further evalua-
tion yields
X2= 1 }]0i 2 _ 2 }]0 i + kE. ( 3 . 2 1 )
The indices of the sums are the same as in equation (3.20), they will not be repea ted
in the following equations. By use of equation (3.19) one has the relations
O = N and kE = N , 1
and this gives finally
N 2 - N. (3 .22 ) X = = ~- [ 0 i
Thus the obtained Chi2-value depends only on the sum of squares of the observed values
in the sector system. Because N and k are constants for a cer tain data set and a given
sec tor pat tern , one can define a modified tes t s t a t i s t i c , which simplifies the fur ther
analysis:
k O. 2 (3.23) ~x%1= X • •
The important point is that the tes t s ta t i s t ic does not really depend on the expecta t ion
values Ei; the only remaining variables are the Oi's , and the Oi's can be al tered by
a dislocation of the sector system or by another spacing of the sectors . The same situa-
tion arises in normal histograms and two-dimensional data if they are t es ted against
the uniform distribution. The derived folmulae also hold in these cases. To study the
behavior of the tes t s ta t i s t ic it will be sufficient to al ter the offse t point of the sector
system.
Rotat ion of the sector pa t tern causes jumps of data points from one sector into
the neighboring one when the sector boundary passes a data point. Such a jump modifies
sum ~Oi 2 locally: the
2 + 2 (0i-I)2 + (0i+I)2 = 0 i Oi+ 1 + 2(Oi+l-Oi) + 2. (3.24)
For an arbitrary number of jumps between two classes one finds
(OI-J)2 + (02+J)2 = 012 + 022 + 2(02-01 ) + 2J 2 (3.25)
J: number of jumps.
Therefore, the tes t s ta t i s t ics is changed by a value
84
2(J(02_01) + j2). (3.26}
The general equation for a dislocation of the sector pa t te rn is found by summing up
all local changes, and the modified tes t s ta t i s t ic (3.21) can be wri t ten as
k Xa 2 + 1 = I 0 i + 2 10i(Ji-Ji+l) + 2 ~Ji 2 - 2 [JiJi+l , (3.27)
where the sec tor k + 1 is identical with the sec tor i. The equation shows that any
jump of a data point from one sector into another, under rotat ion of the sec tor pat tern ,
will a l ter the tes t value, as was numerically found by Ballentyne & Cornish. But equation
(3.27) allows a more detai led analysis. To have no change of ~ Oi 2 under a rotat ion
of the sec tors requires
o = X o i ( a i - J i + 1) + XJi 2 - X J i a i + l or alternatively
2 0 = XJi(Oi+l - Oi) + ~ Ji - XmiJi+l"
( 3 . 2 8 )
These conditions can only be sat isf ied if J l = J2 . . . . = Jk' or if Ji = 0 for atl i. In
all o ther cases, the original sum is al tered, and one gets d i f fe rent tes t values. Now,
the condition to have an equal number of jumps for all sectors is mainly a geometr ica l
problem.
The const ra in ts given by equation (3.28) require that the data are equally spaced
on the circle and that they have unique frequency. The unique frequency within every
sec tor is what one suspects to be proved by the tes t method, but now spacing appears
as a new parameter , which a f f ec t s the tes t value. In addition, on a suff icient ly fine
scale the unique distribution does not play any role fur thermore . The data are measured
on a scale of real numbers, and, therefore , any local c lus ter is due to the roughness
of the measurement ; as the scale becomes finer and finer, the cluster will be divided
into single points. Therefore , on a suff ic ient ly fine scale the data are single points on
the circle {Fig. 3.10). The only remaining variable, then, is the spacing of the data
points. One can in terpre t any theore t ica l distribution in the way that the density over
Fig. 3.10: Equally spaced and uniform distr ibuted points on a circle.
85
a cer ta in interval gives a measurement of the (infinitesimal} spacing of points on the
real line. In the case of the uniform distribution, the density describes an (infinitesimal}
uniform spacing. But what we expect , are not equally spaced data. Our opinion is that
the data result from a random process, which selects the data with equal probability
from a cer ta in interval of the real numbers. Therefore, one cannot expect to find equally
spaced data in a finite size s a m p l e - - one has, of course, the combinational problem
to arrange N data on M points on the real line where M is given by the roughness of
the measurement .
Now, one may ask how strong the a l terat ions of the test value for d i f ferent distri-
bution pa t te rns are. The most simple system are two sectors , and 100 data give a likely
sample size. In a f irst case, we may have nearly equal spacing, then every sec tor con-
tains approximately 50 data points. The rotat ion of the sector system causes only jumps
of a few data points at once, say, in the order 1 to 10. We find that ~ 0 i takes values
like (equation (3.24)):
502 + 502 5000
492 + 512 5002
452 + 552 = 5050
402 + 602 = 5200.
The change of the test value is not very impressive. Changes of this magnitude are
found if random numbers a r e , u s e d to provide a numerical test . Now, assume in the
second case that the distribution has a single well pronounced maximum so that it
is possible to locate all data in one of the two sectors. On the other hand, there exists
a rotat ion of the sector system, which divides the data into two sets with nearly equal
frequency; therefore , under the rotat ion one will find sums in the range
02 + 1002 = 10 000
5() 2 + 502 = 5 000,
and the ex t rema diverge quite clearly. For this data configuration the behavior of the
system becomes ra ther chaotic with respect to an arbitrary location of the sec tor system.
In addition, the possible outcomes diverge rapidly with increasing sample size. Thus,
we have the striking situation that , in contradict ion to the general s ta t is t ical opinion,
an increase of sample size does not stabil ize the result, but that the contrary is true:
The uncertainty about the result of the test increases as the
deviation of the sample from the uniform distribution approaches
certainty.
86
These observat ions allow a final discussion of the causes for the instabili ty of
the tes t . Take a t w o - s e c t o r sample from the normal distribution over the circle (e.g.
MARDIA, 1972)~ then there are two ex t r eme cases: In the first, one sec tor contains
nearly all data, in the second, both sec tors contain an equal number of data. This is
again the si tuation considered above. But now, with the normal distribution, the expecta-
tion values are ne i ther independent of the data s t r u c t u r e - - they depend on the mean
and variance of the d a t a - - nor are they independent of the choice of the of f se t point
of the sec tor system. As the sec tor ro ta tes , the observed frequency within a sec tor
is a l tered in the same way as the expecta t ion values for this sec tor and vice versa.
To use a term from synerget ics (HAKEN, 1977), both values are ' s laved' by the position
of the sec tor system. In the case of a uniform distribution, the expecta t ion values break
out of this 'slaving' , and this allows the system to f luctuate f ree as described above.
The ' revolt of the slaved paramete rs ' becomes especially strong when the observed fre-
quencies are strongly dependent on the position of the sector system, i.e. when the
distribution has a well pronounced maximum.
3.3 PROBLEMS WITH SAMPLING STRATEGIES IN SEDIMENTOLOGY
The analysis of pseudo-t ime series (strat igraphic thickness against some variable)
by means of classical methods like polynomial curve fit t ing, moving averages, cross
correla t ion e tc . is well known (FOX, 1975; SCHWARZACHER, 1974). Besides these
methods, random models have been used, and here especially the concept of transitional
probabili t ies and of Markov chains (KRUMBE1N, 1975}. In sedimentology and s t rat igraphy
the general problem with these techniques is that the ' t ime series ' or the ' sequence
of signals' is usually too short because the profiles are of l imited length. Other problems
result from special, propagated sampling techniques like equal dis tance sampling. These
problems are of special in teres t in the present context because they cause problems
of convergence, and they are capable to genera te ar t i f icial pat terns: They are, in some
way, the inverse problem of i t e ra ted maps. It will turn out that the problems are again
geometr ica l ones, and that it is not advisable to replace geological (morphological} reason-
ing by a sampling formalism.
3.3.1 Markov Chains in Sedimentology
There arise special problems if transit ional probabil i t ies are used to study periodici ty
pa t te rns of profiles. These problems are mainly of a classif icatory nature; they resemble
closely the problem to define the geometry of a bed or facies unit. Fur thermore , they
are re la ted to the process of sedimentat ion and from this to the type of ' s i g n a l s ' - -
87
a b c
T ° 7 m e
Fig. 3.11: Graphic representa t ion of d i f ferent scales for definition of probabili- t ies on profiles, a) The profile in classical representat ion, b) the probability to find a cer ta in lithology measured in terms of bed thickness, c) the probability of a lithotogy to be deposited during a t ime interval.
whether the sedimentat ion process consists of dist inct events or ref lec ts ,a continuously
changing environment .
Transitional probabilities are just one possible definition of a large set of probability
measurements to be defined on. profiles. Some other types of probability measurements
on profiles may be briefly reviewed as a base for the later discussion of transitional
probabilities. From a sedimentologicat viewpoint, we have the total probability of a
cer tain facies type, the likelihood to take a sample from the profile and to find a sand-
stone, carbonates , a claystone e tc . In order to define the probability measurement , the
profile has to be classified into dist inct facies units or beds, and the probabilit ies result
from bed-thicknesses. Still the same probability s t ructure but with other probability
values results if the depth scale of the profile is t ransformed into a t ime scale. So,
any smooth deformation of the scale will give new probability values, but it will not
disturb the special probability algebra (Fig. 3.11). Another type of probabilities with
somewhat d i f ferent algebraic rules (RENYI, 1977; FISZ, 1976) results if two objects
are analyzed simultaneously. In this case, one works with conditional probabilities, e.g,
the probability to find a cer ta in fossil in a specif ied lithology. If one can define some
ordering relation like before and af ter , then a series of d iscre te signals can be analyzed
in terms of the conditional probabilities: to find a cer ta in signal before or a f te r another
one. If the positional relationship is reduced to ' just before ' or 'just a f te r ' , then the
conditional probabilit ies become the classical transitional probabilit ies of a Markov chain.
The s t ructure of the conditional probability space depends not only on the scale used
for the measurements but also on the definition of the relationship between the two
88
sets of objects.
In sedimentological prac t ice the main problem is to define dist inct signals, i.e.
dist inct sedimentological units. In a very narrow sense, this is only possible if the sedi-
mentat ion process is not continuous. But long series of sedimentat ion by events usually
are r es t r i c t ed to turbidites, in which the sedimentological s t ruc ture is very homogene-
o u s - - commonly without transit ions be tween a larger number of lithologies. In all o ther
cases, the transit ions be tween beds or facies units are more or less continuous. But,
as sharp boundaries be tween the 'signals ' disappear, the definition of a probability space
becomes more and more subjective, and this cont radic ts the aim of an objective s ta t i s t i -
cal analysis. For this reason, a modified sampling technique is frequently used to establish
the empirical t ransit ion probabili t ies of a ' sedimentological ' Markov chain (MIALL, 1973).
Instead of d iscre te signals, which have to be defined subjectively, one takes small samples
in some regular distance. The reason is that the samples can be more easily classified.
It seems worthwhile to analyze how the d i f fe rent sampling methods may influence the
probability s t ructure .
A) Discre te Signals
First some additional features of the classical approach of Markov chains m a y
be discussed with respect to the sedimentological questions. A sedimentary unit charac te r -
ized by its lithological, sedimentological , paleontological etc . content needs to be defined
as a signal, which is an event e i ther a p r i o r i - - separa ted from the events ' just below '
and 'just above' by dist inct b o u n d a r i e s - - or which can become a 'd is t inct event ' by
a useful definition of its boundaries. Because the event is defined by its s t ructure , the
transit ion matrix is independent of bed thickness and profile depth. The transit ion matrix
is of the form
(3.29)
nij: number of transit ions from signal Sj to Si;
Nj : total number of occurrences of the signal Sj.
Repet i t ions of identical units such as a sequence sandstone - - - sandstone are
usual events in classical Markov chains. In sedimentology they are only possible if there
exists some boundary which can be recognized, a fea ture which usually is re la ted to
banking. Now, the boundary between two beds, no ma t t e r how small, means some change
in sedimentat ion, e.g. a short interval of lowered sedimentat ion ra te that caused the
physical boundary. Thus, if we do not identify the boundary between beds as a separa te
89
Fig. 3.12: A sequence of three sedimentary units a, b, c and different possible transit ion graphs. Above: transitions if the sedimentary units are defined as 'dis- t inct events t, middle: one possible transition graph for equal interval sampling; the number of singular loops depends on the spacing of the samples. Below: occur- rence of repetitions, i.e. singular loops, due to lumping of lithologies; tat and 'b' are not distinguished.
event, this may be due to a high threshold that has disturbed the record. This view
can be formalized in terms of a map from the transition matrix of degree n onto a
transition matrix of degree n - k, e.g.
l 0 P12 Pl3N~ 0 / $3 - - ~ $2 Q P l l P 1 2 ) P21 0 P 2 3 ] . . . . . . . . . . . . . . . . . . . . . . . - ~
P21 P22 " P31 P32
(3.30)
A more instructive representation of this map can be given by a transition graph
{Fig. 3.12) which shows how singular loops, like a sandstone --- sandstone sequence,
arise due to ignorance or to lumping of intermediate signals (e.g. sandy shales). As will
be discussed below, the same structure with singular loops arises if the samples are
taken in equal intervals. Within sedimentological problems, the occurrence of singular
loops usually indicates that a transition state (such as non-deposition) is missing.
B) Equal Interval Sampling
If we now go on to the second sampling method, the sampling in discrete intervals,
90
we have to prove whe the r it real ly t e rmina t e s into a Markov chain. MIALL (1973) wri tes
tha t this
"method can give rise to a much more accurate measure of the
relative frequencies of the lithotypes present, but at the
expense of accuracy in measuring step-by-step depositional
changes".
This remark shows t ha t one has to take care tha t the probabi l i t ies are well defined,
i.e. t ha t one does not produce a mix ture out of condi t ional and to ta l probabi l i t ies (see
above). Miall fu r the r emphas izes a sampling in terva l slightly less than the average bed
thickness. This addit ional r ecommenda t ion will be studied in detai l in the next sect ion.
In order to analyze the in te rva l - sample s t r a t egy one may assume to have a ser ies of
well def ined even t s wi th t rans i t iona l probabi l i t ies
0 P j i . . . n j / N j
P i j 0 = n i j / N i
o . , . . . . . . . . 0
( 3 . 3 I )
The samples are t aken in equal in tervals from the identicaI universe. As long as
the sampling d i s tance is larger than the average bed thickness, one will miss qui te a
lot of t rans i t ions . As the sampling dis tance becomes less than the smal les t bed, the
counts of t rans i t ions s tabi l ize , but the sampling d is tance can be fu r ther reduced, in
the e x t r e m e case down to an inf in i tes imal small size. From the point, where the t rans i -
t ion counts (not yet probabil i t ies) become s table , one will find new t rans i t ions only along
the diagonal of the count ing matr ix . The counts along the diagonal will increase with
decreas ing sampling d is tance unti l the values equal the to ta l thickness of every l i thotype.
The sum of the diagonal e lements , the re fore , gives the length of the profile:
With respect to the content of the diagonal elements, therefore,
every interval-sampling process generates nothing but a measurement
of the total thickness of the different lithotypes within a profile --
measured as meter-intervals, cm-intervals or~ generally, in units
of the sampling distance.
The e l emen t s of the diagonal line, divided by the i r sum, provide the to ta l probabi l i ty
to find a l i thotype within the profile. If one computes t rans i t ion probabi l i t ies from this
count ing matr ix , then the probabi l i ty ma t r ix reads
91
(:: . . . . mi i / (mi i +Ni
. , o
n j i / ( m j j + N j ) (3 .32)
m..: th ickness of the l i t ho type 1; 11
n..: counts of t r ans i t ions f rom l i tho type j to i; jl N.: to ta l number of t r ans i t ions f rom l i tho type j to ano the r s t a t e .
l
This p robab i l i ty ma t r i x has a suspicious s t ruc tu r e . F i rs t , the p robab i l i t i e s a re not
d imens ion less numbers because the n . . ' s a re coun t s while the m . . ' s have the d imens ion 1J II
of a length measurement, and this is independent of the roughness of the intervals--
but, per definition, probabilities are dimensionless numbers. On the other hand, if we
take a small sampling distance on a profile, where the bed thickness is not small relative
to profile length, the diagonal elements approach one, and the transition graph (Fig. 3.12)
is dominated by singular loops.
The suspicious s t r u c t u r e of this p robab i l i ty ma t r i x b e c o m e s c l ea r if one s e p a r a t e s
t h e coun t ing ma t r ix into i ts i ndependen t pa r t s
O O
(3.33)
. . . . . . . . ° ' ° ° i] nil nj. i = j . . 0 . . . . + mii 0
. . . . . . I. 0 0 . . .
The t. . a re the real t r ans i t ions b e t w e e n l i tho types , the m.. m e a s u r e t he th i ckness of U 1l
the individual l i tho types .
F rom the m a t r i c e s on the r ight s ide one finds two independen t p robab i l i ty s y s t e m s - -
the t r ans i t iona l p robabi l i t i es f rom the t. . and the probabi l i ty or r e l a t i ve f r equency of li
a l i t ho type mi i / ~mii. It s eems , t h e r e f o r e , r a t h e r dangerous to t r a n s f e r the geologica l
or s e d i m e n t o l o g i c a l dec is ion ' w h a t can be def ined as a d i s c r e t e l i thological s ignal ' to
a p s e u d o - o b j e c t i v e sampl ing s t r a t e g y . What is the mean ing of the ' p robab i l i ty m a t r i x '
n.. when the th ickness of t he beds var ies s t rongly? In the case tha t t he beds have near ly 1j t he s a m e th ickness , t he addi t ional rule t ha t the sampl ing d i s t a n c e should be t aken near
the ave r a ge bed th ickness secures tha t the e s t i m a t e d ma t r i x a p p r o x i m a t e s t he t r ans i t ion
ma t r i x to some e x t e n t . In t he ca se of highly va r iab le bed th ickness , the ma t r i x n.. will q
no t make much sense . At leas t , if one can c o m p u t e a mean bed th ickness , one has some
idea wha t a bed looks like in the prof i le under s tudy, and why should one then go this
doubtfu l way?
92
3.3.2 Art i f ic ia l Pa t t e rn Formation in Strat igraphie
Pseudo-Time Series
It is known for a long t ime that several as t ronomic cycles in the order of 20 000
to 400 000 years may cause c l imat ic changes, and there has been a large number of
a t t empt s to re la te geological phenomena to these astronomic cycles, e.go there are several
good arguments in the case of marl - l imestone rhythms, coming from the proposed cl imat ic
changes (EINSELE & SEILACHER, eds. 1982). However, it is very hard to give a s ta t is t ica l
proof of the correlat ion between bedding phenomena and astronomic cycles. To do this
would require to have true t ime series of carbonate production and of clay influx under
control led conditions. But already the relation between t ime and sediment accumulation
is not known down to suff ic ient ly small intervals in any profile. Therefore, this problem
is usually ignored and a fairly constant ra te of sedimentat ion assumed {SCHWARZACHER
& FISCHER, 1982). An additional handicap is that most computer procedures for t ime- -
series analysis require equal interval samples. SCHWARZACHER & FISCHER (t982)
recommend equal interval samples at a distance which is not smaller than the thinnest
beds encountered with some frequency. Surely, the bed is the {possible} unit for an analysis
of cycles. It is the smallest visible f luctuation within the profile. If one has enough in-
formation about the cycles, one will get a good picture of the periodic process if one
chooses the sampling distance exact ly as half or as one wavelength. If, in addition, the
sampling points are located at the ex t rema of the periods, one gets a very simple linear-
ized approximation. For the s t ra t igraphic problem this would require to locate the sampling
points at the ex t rema of a l imes tone- sha le sequence at the boundaries and the centers
of beds, in order to describe the f luctuations of the carbonate content properly. But,
if the data have to be analyzed with a computer algorithm, equal distant sampling points
are necessary while bed thickness usually f luctuates . So, the computer obtrudes a cer ta in
s t ra tegy, whe ther we like it or not. The question is what we have to do in order to
get a physically or sedimentologically reasonable result and not only a meaningless product
of the computer . To elucidate some problems, two d i f ferent aspects will be discussed,
the sampling of periodic functions and the analysis of bed thicknesses. In the first case,
it will turn out that the propagated sampling distances can cause art if icial pa t tern forma-
tion, in the second exampl% we will see that a not properly defined s ta t is t ica l hypothesis
causes in terpre ta t ional problems.
A) Sampling of Periodic Functions
SCHWARZACHER & FISCHER (1982) recommend equal interval samples at a distance
which is not smaller than the thinnest beds encountered with some frequency, If we assume
the bed to represent approximately one wavelength of the smallest periods, we can study
the influence of the sampling distance for an idealized periodic process. A simple cosine
93
J
..V. TV_.V w,--->v v .-.
Fig. 3.13: Art i f ic ia l pa t t e rn format ion due to intervaI sampling (intervaI width near ~r) on a cosine signal. The cosine signal and the sampling pa t te rns , which resul t from intervals of exact ly the width ~r (but with d i f fe ren t s t a r t ing point), are drawn enlarged.
funct ion provides a model for, e.g., the f luc tua t ion of the ca rbona te con ten t (Fig. 3.13).
Sampling at d is tances of exact ly ~r will be successful if the s ta r t ing point of the samples
does not coincide with the inf lect ion point of the cosine function. In this special case,
no per iodici ty will be d e t e c t e d - - anyway, if the sampling dis tance is chosen as 2 ~ ,
then the periodici ty will vanish for every s ta r t ing point. For all o ther s ta r t ing points,
one finds ampl i tude f luctuat ions which cor rec t ly represen t the wavelength, but the t rue
ampli tude is only found if the sampl ing-po in t s coincide with the maxima of the cosine
function.
Now, it is unlikely tha t the sampling intervals have exact ly the length w ( o r 2Tr)
94
even in a computer simulation. Therefore , we disturb the sampling distance slightly, i.e.
we take dis tances Jr +~ {or 2 Jr+ ~); What happens, demonst ra tes Fig. 3.13. The original
cosine function is modulated, and new periodic pa t te rns of higher order occur which only
depend on the disturbance pa ramete r E . In the ex t r eme case, a new simple periodic
curve appears with several t imes the wavelength of the original signal. The error term
causes the sampling points to Vmove' along the periodic function, and depending on
the error pa ramete r this shift ing process adds an amplitude modulation to the cosine
function. Already for small values of the error term we get an infinite number of possible
higher periods and of pure chaot ic b e h a v i o r - - this depends on the quotient njr / {~+E );
if there exists a ~n' such that the quotient is a rational number, then we have a period
of n ~ ; otherwise, we have chaot ic behavior. Now, in a numerical sense, the sampling
distance near Jr is much too large to approximate the periodic function; an analysis
by equally spaced samples would require sampling distances of only a fract ion of J r
As the sampling distance becomes small in relation to phase length, we can safely use
equal dis tance sampling as well as the mathemat ica l 'machiner ies v of t ime-ser ies analysis
and control theory. In geology, however, we have to consider the sampling scale, and
this re la tes the problem to i te ra ted maps, where similar problems with distances occur --
like in the example of the finite logistic equation. The relation to i t e ra ted maps can
be i l lustrated in more detail.
The cosine function is a well defined map of a rotat ing radius vector of unit length
onto a Cartesian coordinate system and vice versa. The rotat ing radius vec tor can be
identif ied with a mass point, which moves with constant velocity on the circle. Equal
dis tance sampling, then, is equiwatent to a uniform motion of the mass point inside the
circle with regular e last ic collisions with the circular wall. The points of collision are
the sample points. This is a classical example of chaot ic motion (GRENANDER, 1978}:
If the first angle of collision is denoted by e {measured from the radius vector of the
circle), then the arc length between successive impacts is Jr -2 e (from planimetry). The
Fig. 3.14: Elastic motions in a circle.
95
resul t ing p a t t e r n depends on the ra t io 2~r /(~T-2@ }. If 2 @ and 7r are incommensurable ,
the system tends to long t e rm chaot ic behavior {Fig. 3.14); however, the system depends
also on the magni tude of @ . If @ is large, we have a high frequency of collisions, and
the pathway of the mass point approximates the circle. If @ is small, a qui te d i f fe ren t
pa t t e rn appears, like in Fig. 3.11b, where the pathway consis ts of quasi tr iangles, which
slowly ro ta te . The lines of motion envelop again a circle, and in t e rms of the approxima-
tion problem the re la t ion of the radii of the outer and inner c i rc le provides a measure-
ment for the quali ty of approximation (the outer c i rc le is the function which should be
recorded).
B} The Analysis of 'Bed Thickness ' by Equal Dis tance Samples
In the previous example we had two variables, profi le length and a second independ-
ent variable, for ins tance ca rbona te conten t . If rhythms are studied, usually the only
var iable is bed t h i c k n e s s - - the sum of this var iable being profi le length. Therefore ,
it is classically assumed tha t the beds have been deposi ted in equal t ime intervals , i.e.
the number of beds is t aken as a re la t ive measu remen t of t ime. Then bed thickness
is a measu remen t of the varying sed imenta t ion ra te , and these two var iables form an
independent f rame. An a l t e rna t ive sedimentological model is (SCHWARZACHER &
FISCHER, 1982) tha t the sed imenta t ion ra t e was fair ly cons tant , and tha t the bedding
phenomenon is due to f luctuat ions in the l i thological composit ion, as discussed in the
last example. To analyze this model in t e rms of bed thickness by compute r algori thms,
Schwarzacher & Fischer used a special sampling technique. They posit ioned the i r equally
spaced sampling points along the profi le and then measured the thickness of the bed
which was hidden by a sampling point (Fig. 3.15a,b). Therefore , some beds are lost while
o thers are recorded several t imes. The method has the e f f e c t tha t i t appears more simply
to decide which of several possible bedding planes has to be taken as the boundary of
the bed {there can be secondary fea tures due to solution). In addition, the method allows
to hope t ha t e r rors will be smoothed out by the ob jec t ive sampling procedure. In the
next step, they re la ted bed thickness to profi le length. The resul t ing graph (Fig. 3.15c)
can well be analyzed by methods like au tocor re la t ion and spect ra . Now, one can ask
again what happens if the sampling dis tance becomes very small. For an ' inf in i tes imal ly '
small sampling dis tance, the graph resembles a step function (Fig. 3.15d), and 'bed thick-
ness ' appears twice in this graph, i.e. the beds are represen ted as squares along the
profile. It is hard to ident ify this r ep resen ta t ion with a useful in te rp re ta t ion . The only
s igni f icant pa t t e rns are the s teps at the boundaries of the bed. Using only these in te r rup-
t ions {Fig. 3.1~e} one gets a ser ies of phase modulated signals along the profile. These
signals - - the bedding planes -- could be recorded di rect ly along the profi le by sed imen-
tological reasoning, and then, of course, with higher precision than by an 'equal in terval
method ' , where the dis tances are chosen by a rule like ' t ake the th innest beds which
96
Fig. 3.15: The analysis of 'bed thickness ' along a profi le {a). The points, at which bed thickness is measured, are taken in equal in tervals (b) and p lo t ted against prof i le length (c). The t rans i t ion to inf in i tes imal ly small sampling in tervals gives a sequence of ' squares ' (d) - - the remaining informat ion are the in te r rupts be tween beds (e). The assumption of fair ly cons t an t sed imen ta t ion ra tes allows to t ransform the phase modula ted signals (e) into ampl i tude modulated ones (f).
are encoun te red by some f requency ' . Anyway, the decision is necessary, what is and
what is not a bedding plane.
The only point, which is con t ra ry to the geological method ( the def ini t ion of bedding
planes), is tha t the signals are not equally spaced, and tha t they, there fore , cannot be
analyzed with s tandard compute r programs for t ime series. On the o ther hand, t ha t
can be easi ly done if the equal d i s tance sampling method is u s e d - - but can this be
a reason to use the less sui ted method, are we real ly ' s laved ' by the compute r? By
a s imple manipulat ion, the phase modulated system of bedding planes (Fig. 3.15e) can
be brought into a form which allows us to analyze the data by an 'equal s tep ' au to-
cor re la t ion program. The approach, so far, was tha t the sed imen ta t ion r a t e was considered
fair ly cons tan t , i.e. the d i s tance be tween the bedding planes provides a measu remen t
of the durat ion t ime of a sedimentological ' s ignal ' - - of a bed. Therefore , one can give
a plot ' number of the ini t ial signal ' or ' number of the in te r rup t ' against the 'dura t ion
97
t ime of the signal' (Fig. 3.15f). This t ransforms the phase modulated signal of bedding
planes into an amplitude modulated one with equal dis tances between the data points.
But now, it turns out that the two di f ferent sedimentological models become identical.
The assumption that the sedimentat ion ra te is fairly constant cannot be distinguished
from the model that the beds are deposited as events in equal t ime intervals because
the duration t ime, as defined, is proportional to bed thickness. This, of course, holds
only for the ' s ta t i s t ica l approach'; by geological reasoning it is usually possible to dis-
tinguish these two cases (EINSELE & SEILACHER, eds. 1982). Thus, one comes out with
the result that the connection of the depositional models with a s ta t is t ica l formalism
can cause a worse defined s ta t is t ical problem, in the sense that the s ta t i s t ics cannot
prove which of the sedimentological models is the cor rec t one while a successful s ta t i s t i -
cal tes t always seems to prove the a priori sedimentological model. The problem whether
an initial model is consis tent with the s tat is t ical analysis is not a trivial one; in topology
the analogous problem is genericity.
"To many scientists the 'genericity' problem has always been
interesting. What we are looking for is the following: given
a mapping f:U --~ R m, where U is an open set in R n, how can we
perturb f slightly to obtain a nicer and simpler mapping?" (LU, 1976).
The idea to analyze the sedimentological problem in terms of a mapping leads
to the following diagram
~ s 2 It'
The set of observations (location and thickness of beds: P) maps under the sedimentological
hypothesis s 1 (event sedimentation) onto the discrete space D 1 (bed number and bed thick-
ness). Under the hypothesis s 2 (constant sedimentat ion rate) the same set maps onto an
a l ternat ive d iscre te space. As turned out during the earl ier discussion, there exists a
map I which t ransforms D 2 into Dt, but only if D 2 is cons t ructed from infinitesimally
small sampling intervals, i.e. if D 2 contains all bedding planes. Otherwise, if the sampling
distance becomes suff icient ly large, D 1 cannot be const ructed in all details from D 2.
On the other hand, from D 1 we can construct all possible outcomes of the model and
sampling s t ra tegy s2(n) (n for the number of sampling points). The reason is that P can
be recons t ruc ted in all details from D 1 but not from D 2. Thus, diagrammatical ly
98
s 1 p "., D I
Sl -1
D2(o~)
and it turns out t ha t only D I is a gener ic r ep re sen ta t ion of the data , i.e. m a t h e m a t i c a l
analysis jus t i f ies the way via geological reasoning. In re t rospec t ion , the problems concern ing
Markov chains a re the same -- geologically and mathemat ica l ly .
3.4 CENTROID CLUSTER STRATEGIES -- CHAOS ON
FINITE POINT SETS
A wide field of geological and paleontological research focuses on c lass i f ica t ion
problems. A set of o b j e c t s - - samples, specimens e t c . - - has to be classif ied in such
a way tha t the e l ement s of a c lus te r show a maximum of ' s imi la r i ty ' while d i f fe ren t
c lus ters have a maximal dissimilari ty. As the numbers of objec ts and var iables become
large, it is convenien t to use compute r procedures to solve the c lus te r p a t t e rn recogni t ion
problems. The class ical approach into this d i rec t ion b e c a m e known as 'numer ica l t axono-
my' {SOKAL & SNEATH, 1964). Nowadays, the re exis ts a large number of a lgor i thms
which t ry to solve the p a t t e r n recogni t ion problem in a s t ra igh t forward way by use
of various s imi lar i ty and d i s tance measu remen t s {e.g. HARTIGAN, 1975; STEINHAUSEN
& LANGER, 1977; VOGEL, 1975). Clus ter ing s t r a t eg ies produce, in general , no unique
solution; an a l t e rna t ive dis tance measu remen t or even a d i f fe ren t input sequence of
the data can change the local and global s t ruc tu re of the c lus ters (VOGEL, 1975). In
addit ion, our opinion about the image of the s t a t i s t i ca l universe can fair ly diverge from
the s imi lar i ty s t ruc tu re which is gene ra t ed by a c lus te r ing s t ra tegy . There a re t h r ee
points to be discussed: the r ep re sen ta t ion of c lus te r s in binary t rees , image concepts ,
and s tabi l i ty problems with d u s t e r s t ra teg ies . The binary t rees imply t ha t they give
a c lass i f icat ion, but the comparison with c lass i f ica t ion t rees shows tha t this is not the
case. The image concept of most c lus te r s t r a t eg ies is far from our geomet r i ca l intui t ion.
99
It will turn out that this discrepancy explains much of the instable b e h a v i o r - - because
the clusters are not geometr ical objects, they cannot be used for a true classification,
and any additional e lement destroys the local s t ructure . Besides the instability between
clusters, there appears an instability within clusters, which is more striking because
it does not depend on some metadefini t ion of a cluster and leads to true chaotic behavior
on finite point sets.
3.4. I Binary Trees
The c luster s t ructure found by some algorithm is usually represented as a binary
tree, which i l lustrates the similari t ies or dis tances between the hierarchies of e lements
and clusters. The binary t rees in cluster analysis are just a picture of one possible simi-
larity s t ructure (VOGEL, 1975); ~hey are not at all a classif icat ion by some ordering
sequence. Thus, they do not allow to insert an additional object without changing the
s t ructure , at least locaiIy, nor do they allow to search an e lement by some decision
rule, as it is possible with binary t rees which represent a relation or ordering between
data. Binary t rees are commonly used as classif icat ion s t ructures that allow to search
and to insert e lements very quickly in sophist icated computer programs (WIRTH, 1975;
DENERT & FRANK, 1977), for a mathemat ica l discussion see e.g. SCHREIDER (1975).
The d i f ference between binary t rees and cluster t rees can be i l lustrated in the following
way:
classif icat ion trees:
(I)
B F t N
1\ /" /" C E G I K
(II)
(L 0/\i j , o'\ , / \ , / \ , I ) ) I I I ~ I
000 001 010 OII IO0 I0I IIO III
100
clus ter t ree:
!
{ A,B,C,D,E,F,G,H,I}
I
I {A,B,C,D}
{E 'F 'G} I
{A,B} { C,D} {E,F} r~ r-%
A B C D E F
{ A,B,C,D,E,F,G,H,I, J, K,L,M,N,O}
! { J,K,L,M,N,O}
{ E,F,G,H,I } t l 'i
[ { J,K,L} { M,N,O } i
,
{H,I} { {
G H I J K L M N O
The classif icat ion t rees represent an ordering relat ionship like 'before ' in t ree (I}:
A < B < C < ... < O. Therefore, it provides an optimal s t ra tegy to search e lements .
In the same way, objects can be classif ied by a sequence of binary decisions ' to have
or to have not a cer ta in property ' . Tree (II) gives an example for such a classif icat ion
of the binary numbers. This t ree allows to find e lements by the decision rules at the
nodes, and it allows to insert new e lements at its proper position, e.g. it is no problem
to insert a number with four digits. Tree (I) provides a minimal classif icat ion while
t ree (II) is an optimal dynamic classif icat ion because it allows to insert new e lements
without any a l te ra t ion of the present t ree s t ructure .
In contrary, the c lus ter t ree does not describe how to arrive at any one of the
e lements if one s ta r t s at the highest hierarchical level. As far as some distance meas-
urement exists be tween A, B, ..., 0, the cluster t ree simply s ta tes that A is more similar
to B than to C. But because the binary t ree cannot describe a higher dimensional variable
space, it does also not ensure that the dis tance be tween A and t:3 may not be identical
with the d is tance be tween A and C. This proper ty will la ter turn out to be the main
source for instable clustering within clusters. The cluster t ree only gives a possible struc-
ture of similari t ies and a possible grouping into s u b s e t s - - and not at aI1 a rule how
to distinguish the subsets.
3.4.2 Image Concepts
"Cluster analysis is one of the Pa t t e rn Recognition techniques and should be appreci-
a ted as such" DIDAY & SIMON (1980) write, and
"the general idea of all these methods is to build an identifica-
tion function (clustering) from two types of information.
101
0 O
O
Q O
O
O
O O
Fig. 3.16: A point pa t tern in the plane (left} and the intuitive geometr ical inter- pre ta t ion as a bounded object (right).
on one hand, the experimental result,
on the other, the a priori representation of a class or
cluster."
Although the concepts of clustering analysis can be well described in a formal mathemat i -
cal notat ion (DIDAY & SIMON, 1980; HARTIGAN, 1975), these concepts have some 'weak w
points. As Diday & Simon summarize:
"In fact these (the cluster) representations rely heavily on
the concept of potential or inertial functions. The 'classifier
specialist' selects this representation from the knowledge of
the properties of his experimental universe."
Indeed, to work properly with such techniques does not only require to understand the
mathemat ica l formulae, but also to understand the quali tat ive behavior of a cluster
s t ra tegy and the concept of 'c lus ters ' underlying a cer tain clustering process.
Given a point set in the plane (Fig. 3.16), our intention will be, in general, that
these points are the image of a two-dimensional object with a well defined boundary.
In a more general sense, one expects that the points are a sample from a n-dimensional
density distribution which is continuously defined in R n, and, therefore , one suspects
102
Fig. 3.17: Al te rna t ive pa t te rn concepts for the point set of Fig. 3 . t 6 - - a line {left) and the binary fusion s t ruc ture of a weighted centroid method {right},
that an increasing sample size gives a s ta t is t ica l universe which can be bounded by
smooth probabilistic surfaces. A consequence is that one identif ies automatical ly ' c lus ter
recognit ion ' with classif ication. Indeed, if one has found a cluster and has bounded it
by a closed surface, then any additional data point is accepted as a member of the
point set if it is located inside the bounding hypersurface. A point outside of this 'ob jec t '
will be associa ted with it if its inclusion does not disturb our concept of the image
too much. Thus, the geometr ica l objects, of our opinion provide classif icat ion rules, and
these are possibie because the geometr ical objects are stable.
Besides this intuitive concept , there are other possibilities to define 'c lusters ' .
The points of Fig. 3.16 can be connected by a line; the result ing image approximates
a two-dimensional curve (Fig. 3.17). The resulting 'objec t ' can be accepted, but it is
not the kind of things one expects in a two-dimensional space. The binary fusion pa t te rn
of a centroid cluster method (Fig. 3.17) diverges still fur ther from our opinion. That
these pa t te rn concepts appear somewhat strange, may have its reason in the feeling
that they are not stable, and, in fact , they are not stable in a s tructural sense. Any
additional point al ters these pat terns , at least locally, in an unpredictable way. Fur ther-
more, these objects genera te no classif icat ion for additional points, there is no relation
like ' inside' or 'outside ' , like 'belonging to ' e tc . Classif icatory objects for points in R n
103
require tha t they are bounded by a surface. Therefore , an object in R 3 must be t h r ee - -
dimensional i tself . Points, lines and open surfaces in R 3 are degenera ted objec ts which
can exist ne i the r physically nor s ta t i s t ica l ly , they are objects which cannot ' con ta in '
something. The defini t ion of a c lass i f ica tory object in a d iscre te n-dimensional var iable
space requires , the re fore , at leas t n+l data points or samples. These n+t points then
descr ibe just one object , a t r iangle in R 2, a t e t r ahedron in R 3, a hyper te t rahedron in
R n, A c lus te r and c lass i f ica t ion concept requires tha t the number of e l emen t s to be
classif ied exceeds c lear ly the number of var iables defining the position of the e l ement s
in a hyperspace. There should not be more than (n DIV m) o b j e c t s - - n: number of
e lements , m: number of variables, DIV: division of integers .
In the appl icat ions of c lus te r analysis it is commonly the case t ha t the number
of var iables equals, or even exceeds the number of e l ement s or samples. The only s t ruc-
ture one can find under such condit ions is an ordering by some s imilar i ty rule, but one
cannot expect tha t this provides a classif icat ion, as it is usually done. In a s ta t i s t i ca l
sense, ' c lass i f ica tory objects ' are closely re la ted to the convex hull of a f ini te point
se t {EFRON, 1965). In a topological sense, they are just h igher-dimensional analogues
of a smooth closed surface, a manifold. Because the point sets are finite, the cont inuous
d i f fe ren t i ab le s t ruc tu res are replaced by simplices and the i r unions, polygons and polyhe-
dra.
3.4.3 Stabi l i ty Problems with Cent ro id Cluster ing
Within the various types of c lus ter s t ra tegies , h ierarchica l methods are the ones
most commonly used, and within this subgroup the cent ro id methods are most popular.
The t e rm ' cen t ro id ' is here used in a very general sense. It simply expresses tha t two
fused e l emen t s are fu r ther represen ted by a single e lement , the eent ro id of the original
points. It is known from a respec tab le number of c lus te r s t ra teg ies tha t the cons t ruc ted
d is tance t r ee depends, to some exten t , on the init ial ordering of the input data {VOGEL,
1975). As Vogel found by extens ive tes t ing with one special da ta set, the cen t ro id
methods become par t icu lar ly sens i t ive to the ordering of the da ta if an en t ropy d is tance
measu remen t is used. Clus ter s t r a t eg ies with o ther d is tance measu remen t s showed this
sens i t iv i ty only on the lower h ierarchica l levels. In palecology one has very nice homoge-
neous data s t r u c t u r e s - - the f requencies of species within a sample. In this special case,
the usual me t r i c d is tance measurement s make not much sense while the entropy measure-
men t provides a ' na tu ra l ' d i s tance measurement for these f requency data of species
(BAYER, 1982). This d is tance measu remen t has, in addition, the proper ty tha t it approxi-
ma tes the Chi2- tes t for homogenei ty of the samples (DIDAY & SIMON, I980); homogenei -
ty of the samples being of special in te res t in palecological studies. Therefore , a program
104
~ t
-%.
Fig. 3.18: Class i f ica t ion and dis t r ibut ion of the foramini fe ra fauna of Todos Santos Bay. Left : Class i f ica t ion by hand (WALTON, 1955); right: Ordering by a c lus te r program {KAESSLER, 1966).
was implemented tha t al lowed to analyze faunal da ta by the ' en t ropy method ' (BAYER,
1982). The da ta of a *hand made ' ecological analysis (WALTON, 1955), which had la te r
been reeva lua ted by c lus te r analysis {KAESSLER, 1966), were r e - r eeva lua ted again to
see how the s t r a t egy works. Fig. 3.18 gives the spat ia l faunal dis t r ibut ion pa t t e rn of
foramini fera , which had been found by the previous workers. Already the first few runs
wi th the ' en t ropy analysis ' showed a s t rong dependence of the resul t on the ordering
of the input data . The nice s t a t i s t i ca l proper t ies of the en t ropy d is tance measu remen t ,
the re fore , were useless. But the use of o ther d is tance measu remen t s with r ep resen ta t ion
of the c lus te rs by the i r cent roids did not e i the r s tabi l ize the pa t te rn , in con t ras t to
VOGEL's (1975) claim. In fact , dependenc e from the init ial ordering of the data was
near ly the same. Fig. 3.19 gives some examples of various t ree s t ruc tures , which resul t
from sl ightly d i f fe ren t ini t ia l condi t ions and p a r a m e t e r se t t ings . The observat ion t ha t
it is not the choice of a special d i s tance measurement , t ha t causes the ins tabi l i t ies ,
was the s t a r t ing point for a more de ta i led analysis of the s tab i l i ty proper t ies of the
cen t ro id c lus te r s t ra teg ies . This s tab i l i ty analysis will now be outl ined.
The widespread use of cen t ro id c lus te r s t r a t eg ies is, in par t , due to the i r proper ty
to produce nice binary h ie ra rch ica l s t ruc tu res from every da ta set . The points found
to belong to a c lus ter are replaced by the i r centroid , which can be e i the r weighted
or unweighted. 'Weighted ' means tha t the cent ro id is computed from all da ta points
within a c lus te r while 'unweighted ' means tha t a computed cent ro id is fur ther t r e a t e d
like a da ta point, or tha t it has equal weight as a da ta point. In both eases, the progress
of c lus te r ing causes a concen t r a t i on of the original da ta points into fewer and fewer
r ep resen ta t ives , the centroids . Thus, every single fusion of two points evacua tes locally
the point space. As this concen t r a t i on of the points proceeds in a ce r t a in area, it forces
105
c / / °
t !
a
F
Fig, 3.19: Various classif ications with centroid cluster s t ra tegies of WALTONIs (1955) foraminifera data. Lower left and upper left: Two of a large number of possible cluster t rees which result only of an a l tered input sequence of the data. Upper right: a l tered metr ic , Lower right: d i f fe rent clustering algorithm,
106
the process to branch into o ther areas which are still more densely covered by data
points, and the same process is repeated. This coupled process of local evacuation and
subsequent branching into another area of the data space is the reason why every data
set is mapped onto a nice binary tree. The t ree s t ruc ture is due to the clustering proc-
ess, and it is not a property of the data set. The same process causes, in addition,
that the dis tances be tween the clusters increase monotonically, a ce lebra ted proper ty
of these s t ra teg ies (STEINHAUSER & LANGER, 1977) - - but an a l ternat ive viewpoint
is that these s t ra teg ies impose a binary t ree pa t te rn onto every data set .
Instabil i t ies be tween clusters are re la ted to the problem which images or which
point configurat ions can be accep ted as a cluster, i.e. one needs a metalanguage definit ion
of a cluster . Although the proper t ies of clusters and part i t ions, their homogeneity, are
usually defined in te rms of the metr ic , the idea of a 'good' c luster is mainly a geomet-
rical one. The question 'what is a good cluster ' is commonly reduced to the problem
how distant two clusters must be to be separable and whether concave clusters are
allowed or not. In clustering analysis the terms convex and concave are replaced by
homogenei ty and chain-homogenei ty (DIDAY & SIMON, 1980). At the moment , we may
use the ear l ier discussed geometr ical image concept to define 'good' clusters. Then,
concavi ty means simply that every line connect ing two points of a cluster is bound to
the interior of an hyperpolyhedron, that forms the boundary of the cluster. Consequently
the centroid is not necessari ly bound to the interior of a concave cluster (Fig. 3.2O).
And because every fusion of two points increases locally the distances, the process can
reach a s t a t e where the image breaks into disconnected pieces, and where these pieces
are connec ted with data points which form separa te clusters under the geometr ical image
concept (Fig. 3.20). The striking point is that in such cases the centroid method does
not follow the 'rule of neares t neighborhood'; not the clusters, which appear closest
because their boundaries are closest , are fused, but the more distant clusters are connec t -
ed (Fig. 3.20). This behavior elucidates again that such a pa t te rn representa t ion cannot
work as a classif icat ion, at least not as a 'natural c lassif icat ion ' ; in some respects ,
this pa t te rn concept is even cont rad ic tory to our understanding of similarity.
So far, a metadescr ip t ion of clusters was necessary. One can also simply accept
that a c lus ter formed by a centroid process has nothing in common with our geometr ica l
imagination. If it can be uniquely cons t ruc ted by an algorithm, it may be a totally ab-
s t r ac t s t r u c t u r e - - but why should it not be as real as geometry? That clustering does
not produce such a well defined abs t rac t pat tern, wiil be shown by a discussion of insta-
bil i t ies within clusters . To avoid any assumption about the s t ruc ture of clusters, one
has to res t r i c t the a t ten t ion to local s t ruc tures as they are defined by the clustering
process i tself . Then one can analyze how a local disturbance propagates into the es tab-
lished hierarchy. Fig. 3.21 gives two cluster pa t te rns over a given point set (compare
107
E3
121
Q . . . . . . . . i.
Fig. 3.20: Branching between clusters. The smaller c luster will be fused with the ' rectangular points ' because the formation of centroids changed the local topology of the point space. Consequently the larger cluster will be fused with the ~triangu- lar points ' in a later step.
Figs. 3.17- 3.18). This time, an unweighted centroid method was used to find the similari-
ty s t ructure . This method causes a local decision problem because there are two choices
for the same point to be fused, two neighbors have equal distance. Dependent on this
choice, two di f ferent local pa t te rns are generated {Fig. 3.21). These local a l ternat ives
project onto the global t ree s t ructure changing the distances or similari t ies all the way
through the cluster t ree (Fig. 3.21). The di f ference is strong enough to change the overall
t ree pat tern . Thus, one of the two t rees appears more compact , and, therefore , it will
be more easily accepted as a larger cluster if the t rees are embedded in a more complex
cluster t ree. On the lower cluster level the substructure appears also more compact
in the right t ree of Fig. 3.21 while in the lef t one two clusters can still be distinguished
on the lower level. The decision, which one of the two clusters will be produced, depends
only on the ordering of the data. The computer algorithm has ei ther to take the first
pair of data points with smallest distance or the last one for f u s i o n - - a consequence
of the binary t ree representat ion. Once made the decision, the local topology of the
point pa t te rn has changed, and, therefore , the local decision projects onto all la ter fusions
contact ing this area. Thus, a true bifurcation of the solution occurs, and it depends
108
/ %. %.
Fig. 3.21: A single local decision problem (arrows) between two equally distant points a l ters the local t r ee s t ruc ture and the dis tances within the ent i re hierarchy.
only on the initial conditions (ordering of the data) which branch of the solution will
be taken.
The next question is whether there exist s i tuations that cause cascades of bifurca-
tions. In this case, the clustering process would become chaotic, as far as this is possible
on a finite point set. Such a sequence of bifurcations can be easily const ructed by just
109
a
C
Fig. 3.22: Centroid clustering as a decision game. The figure gives two possible cluster solutions for the identical, equally spaced point set. a) Every fusion causes the bifurcation into two identical subsystems; b) the associated fusion tree; c) the cluster trees.
choosing the most degenerated case of input data -- equally spaced data points. It does
not ma t t e r whether these points are arranged on a straight line, on a circle or on a
regular (hyper-)grid. Equally spaced data points on a straight line provide the most simple
case; Fig. 3.22 i l lustrates what happens in this special case from various viewpoints.
First , the clustering process has a dynamical aspect - - at the same time, only two points
can be fused, and the fusion of these two points al ters the local dis tance s t ructure
between the points. In the case of Fig. 3.22, the initial dis tance of izhe equally spaced
points, ax, is a l tered to (3ax)/2 between the centroid and its neighbors. Therefore, the
data space is divided into two subsets of points which have still the original s t ructure .
But the two substructures are now independent; they can be t r ea ted as parallel processes
of identical behavior. This bifurcation process proceeds until there are no data points
lef t with the original spacing. Within the limit of a finite point space, any fusion on
one of the subsets genera tes new identical subsets, every bifurcation of the dynamical
system genera tes two dynamical systems of the identical type. In addition, any fusion
is a random choice be tween several possibilities, and the decision, which pair of data
points is taken, depends only on the initial data configuration. One has a per fec t ly chaot ic
110
system on a finite point set . Because the point set is finite, the process te rmina tes
finally. There are, for every subsystem, two main possible outcomes which depend on
some initial choices of the fusion process {Fig. 3.22). One possibility is that during t ime
all points are fused into pairs. Then the resulting centroids are again equally spaced
(Fig. 3.22 left}, only with a l tered distances. Thus, the process is cyclic. The a l ternat ive
is that already the initial fusions determine, to some extent , the further progress because
there remain some points of smallest dis tance on every level which dominate the process
{Fig. 3.22 right}. The centroid clustering techniques, therefore , provide excel lent examples
of chaot ic behavior on finite point sets. In the case of equally spaced data, every slight
change of the input data will cause another cluster t ree, i.e. the system is ex t remely
sensit ive to the initial conditions. Because the number of data points is finite, the number
of possible outcomes is f inite as well, it is the number of disjunct permutat ions of the
data. Therefore, the chaot ic aspect of the clustering process will increase with the num-
ber of (equally spaced} data, i.e. one has the same si tuation as with the Chi2- tes t against
a uniform distribution or the t ra jec tor ies of part icles in Galton's machine.
Now, a prac t i t ioner could argue that the chaot ic behavior of the centroid c lus ter
s t ra teg ies requires a very special and degenera ted data s t ructure . On the other hand,
it is well known that most c luster s t ra teg ies depend to some ex ten t on the input sequence
of the data {VOGEL, 1975}, and it seems likely that this phenomenon is related to the
discussed instability. The remaining question, therefore , is how this instability can arise
in a s ta t i s t ica l sample. There are two s t ruc tures which can enforce the chaot ic behavior
with any kind of data. The first one is that the data are measured with a cer ta in preci-
sion. Thus, with increasing number of data one will get an increasing number of equally
dis tant data. Because multiple values at the same point do not change the local topology
if they are fused under the centroid condition, the data pa t tern tends, at least locally,
towards equally spaced data. And once more we have the si tuation that an increase
of data does not stabil ize the process, as one should expect from s ta t is t ica l reasoning,
but that it destabi l izes the clustering process. In addition, once more one finds that
a uniform or an equally spaced distribution leads to instability. This s t ruc ture probably
is the main reason why the palecological data cause particularly instable clustering.
The frequencies are natural numbers which enforce equal spacing locally. The second
s t ructure , which can enforce chaot ic behavior, is a numerical one. Every computat ion
with a f inite number of digits causes numerical rounding or truncation errors. An excel- 10000
lent example due to WIRTH (1972) is the computat ion of the sum i_~l/i , which takes
d i f fe rent values if it is computed forward and b a c k w a r d - - the numerical error a f fec t s ,
e.g., 3 of 12 valid digits of a 48-bit machine. The same problem occurs for the compu-
tat ion of dis tances between data points (and centroids). If the number of variables is
high enough, one has to expect that the dis tance be tween two points is not symmetr ic ,
e.g. in the case of the Euclidian distance one has
111
}~ (Xli-X2i)2 ~ ~ (X2i-Xli)2 .
Therefore , this e r ror will also depend on the input sequence of the data , and this t ime
the error increases with the number of var iables and with the complexi ty of the d is tance
measurement . This can explain why the ' en t ropy dis tance measu remen t ' is especial ly
instable. Logar i thms are necessary to compute this d is tance measurement , and this en-
forces numerica l errors .
Thus, the centroid c lus ter methods turn out to be highly instable due to the i r pat-
t e rn recogni t ion concept . They behave in a chaot ic manner with respec t to sl ight changes
of the ini t ial conditions, namely the ordering of the input data . The chaot ic behavior
is t r iggered by the tendency of measured data to form, at least locally, regular spacings,
and by the numer ic error of the dis tance measurement , mainly by i ts a s y m m e t r y - -
whereby the computa t ion order depends again on the input sequence of the data . in
to ta l i ty , it turned out during the last two chapte rs tha t most operat ions on f ini te point
sets with the compute r have to be handled with special care. The local methods, which
domina te this field, are usually ex t r eme ly ins table due to small changes of the boundary
condit ions and of the init ial conditions. What we, in general , would need, are r a the r
more rigid topological methods than the highly sophis t ica ted numerical procedures.
(3C
Fig. 3.23: Class i f ica t ion by probabi l is t ic neighborhoods. Points with overlapping
neighborhoods belong to a c lus te r with a well defined boundary (right).
112
Such a - - p r o b a b l y m o r e - - r i g i d method was proposed by GRENANDER (1981}. The
idea is similar to the single linkage methods; however, it has a more geometr ic back-
ground, and this returns to the discussion in sect ion 3.4.2. A probabilistic neighborhood
is associated with every point: e.g., circles in the plane, spheres, hyperspheres or a l terna-
t ively polyhedrons (Fig. 3.23t. Every point has a cer ta in probability to be found elsewhere
in this neighborhood, and a possible assumption about the probabili ty is ' tha t the curvature
at the boundary is proportional to the density of the probability measure, ~ GRENANDER
(1981), or the probabili ty decreases , as the d iameter of the probabilistic neighborhood
increases. Two points are grouped in a similari ty c lus ter if their probabilistic neighbor-
hoods overlap. If the radius e of the neighborhood is fixed to a cer ta in value, then a
unique part i t ion of the data set arises, and we get part i t ions of d i f ferent roughness
for d i f fe rent values of e. The resulting s t ructure is not a t ree because two or more
e lements may fuse at once for a cer ta in value of e. However, the resulting 'c lus ters '
have now well defined boundaries as i l lustrated in Fig. 3.23. The point set of a ' c lus ter '
is bounded by a ' tubular neighborhood ~, and 'c lus ters ~ are d i f ferent on a cer ta in e - l eve l
if their boundaries do not overlap. Thus, given such a classification, we can later add
samples and question whether they belong to a ' c lus ter ~ for a cer ta in e -va lue . Clearly,
the classif icat ion is not s table in the s e n s e that an additional object may cause two
or more ' c lus ters ' to fuse without changing the e-level. However, that is a quite more d i f ferent
instabili ty than those discussed above. The classif icat ion cer ta inly stabil izes as the number
of data points increases and approaches the universe. This approach returns to the exam-
ples of the second chapter and is closely re la ted to the continuation problem discussed
there .
3.5 TREE PATTERNS BETWEEN CHAOS AND ORDER
The previous presentat ion focussed at pract ical problems which arise from unstable
algorithms. Here, the viewpoint is much more theoret ical , and the questions are mainly
conceptual . The sect ion serves as a kind of summary of the previous chapters and con-
nec ts them with the following one. The morphology of t ree- l ike s t ructures wilt be the
topic of this section, especially the connection between branching pa t te rn and overall
shape, which may arise under cer ta in conditions: If one closely studies the crown of
a t ree, the branching pa t te rn appears ra ther irregular while an ent i re view of the crown,
from some distance, is charac te r i s t i c on the spec i e s - l eve l (Fig. 3.24). On this basis
HONDA (1971) a t t empted to describe the multifarious form of t rees by a few parameters
of the branching pat tern . This a t t empt , however, was not new: Already D'Arcy Thompson
discussed the cymose inf lorescence of the botanists and noticed that these botanical
pa t te rns are 'analogous in a curious and instruct ive way to the equiangular spiral'
(DVARCY THOMPSON, t952); and, as G.L. Steucek informed me, branching pat terns
of t rees were already studied by Leonardo da Vinci.
113
Fig. 3.24~" Tree images.
Rather similar branching pat terns occur in geomorphology, the network of drainage
systems (Fig. 3.25). Trees and networks are equivalent on a cer ta in level, e.g. a t ree
genera tes a network if two branches~ which come into contact , fuse ra ther than over-
l a p - - a situation which necessarily occurs if the branching process is bounded to a
Fig. 3.25: The simplified drainage nets of the Amazon (left} and the Ganges del ta
(right).
114
surface or is res t r i c ted to a nearly two-dimensional object . The network in leaves
provides an example for the la t te r case. As a concession to the l i terature, the t e r m
' t r ee s ' will be equivocally used as the term 'open networks ' .
The botanical a t t emp t s toward a description of branching pa t te rns in t rees are
mainly de terminis t ic and have, therefore , be cr i t ic ized. Thus NIKLAS (1982) wri tes in
a study on plant branching simulations:
"Unless the extent of apparent order that can arise from random
processes can be determined, there is no valid basis for assert-
ing that a particular structure implies deterministic causes."
The meaning of ' random' , however, is ra ther vague. If we a t t r ibu te a probability to
a growing system to branch or to branch not during a growth step, we may al ternat ively
formulate a de terminis t ic system which branches at the beginning of every growth step,
but may be disturbed by some external cause which inhibits branching, and such external
events may be truly s tochast ic . This aspect was clearly expressed by OSTER & GUCKEN-
HEIMER (1976):
"If a series of . .. cerises are collected . .. , and they appear
chaotic, exhibiting no perceivable regularities, then we conclude
one of three things:
(a) the system is truly stochastic--dominated by random influ-
ences
(b) experimental error is of such a magnitude that all regulari-
ties are obscured~
(c) a very simple deterministic mechanism is operating, but
is obscured " (by the phenomenon described here in section
3.1.1).
This is a viewpoint which now seems to be common in physics (e.g. HAKEN, 1981;
THOMPSON, 1982). Examples of points (b) and (c) have been given in the previous
sections, and experimental ly it seems more appropriate to exclude points (b) and (c)
than to prove point (a); the failure of the Chi2- tes t in connection with a uniform distri-
bution i l lustrates this point: There is still enough to be de tec ted on the de terminis t ic
level in which s tochas t ic e lements may en te r in te rms of disturbances.
Branching pa t te rns of t ree- l ike bodies are of some in teres t because they are not
r e s t r i c t ed to trees, they occur commonly in biology and paleontology, on the level of
organisms and on the level of organs. Besides this, they are even interest ing objects
in geology, geomorphology, and physics. The discussion of met r ic t rees is, to some extent ,
based on a study by Bayer & McGhee (An Analytic Approach to Branching Systems,
115
prepr int 1984), a study which was ini t ia l ized by a paper given by G.L. Steucek at Tfi-
bingen in 1984o
3.5.1 Topological Proper t ies of Open Network P a t t e r n s
In geomorphological studies open networks and t ree pa t t e rns natura l ly or ig inate
at the sources of s t reams, a viewpoint which is opposite to the bo tan i s t ' s one where
the branches usually or ig inate at the trunk. However, in t e rms of morphomet ry botanis t s
t ake a viewpoint ident ical to the geomorphological one. The s t r eams or b ranches are
ordered by the descr ip t ive "s t ream number", which is due to S t rah le r {e.g. BARKER
et al., 1973; GRENANDER, 1976):
¢,./ , =",A ,-. / k
Fig. a.26: Trees with branches ordered by S t rah le r numbers. All t rees have equal numbers of end branches.
Defini t ion 1: The S t r a h I e r n u m b e r ' s ' , or the order of a
single branch, is given by the following rules:
(1) end branches have order s=l
(2) two branches of order s produce a branch of order s+l
(3) two branches of d i f fe ren t order (Smax,Smi n) produce a branch of order Sma x,
The highest S t rah le r number S involved in an open network is called the
o r d e r o f t h e n e t w o r k .
The St rahler number {Fig. 3.26) of branching sys tems has special proper t ies (e.g. BARKER
et al., 1973; GRENANDER, 1976). In par t they are based on empir ical observat ions,
in par t they are deducible from the definit ion. Thus, if the numbers of b ranches of
each order are counted, then the logar i thm of the number of branches in each order
p lo t ted against the St rahler number gives a l inear plot: i.e. log n s = a+bs; where Us:
the number of branches of order s {BARKER et al., 1973); or, an observat ion from
116
g e o m o r p h o l o g y (HORTON, 1945), t h e b i f u r c a t i o n r a t i o s a r e a p p r o x i m a t e l y c o n s t a n t . Le t
aga in n s be t h e n u m b e r o f b r a n c h e s o f o r d e r s, t h e n t h e b i f u r c a t i o n
r a t i o
n s _ l / n s = R s (3.34)
is a p p r o x i m a t e l y c o n s t a n t , i .e t h e n u m b e r s ns_ 1 and n s a r e in a g e o m e t r i c a l r e l a t i o n -
ship. T h e s e ' e m p i r i c a l l aws ' , h o w e v e r , a r e no t u n e x p e c t e d , t h e y r e s u l t f r om t h e de f i n i t i on
o f t h e S t r a h l e r n u m b e r . By de f i n i t i on e v e r y n e t w o r k o f o r d e r s m u s t c o n t a i n a t l e a s t
two b r a n c h e s o f o rde r s - I , and t h e s a m e holds for any s u b s t r u c t u r e in t he t r e e w i th
s > 1. T h e r e f o r e ,
n s -2ns_ I > 0 or ns_i/n s > I/2. (3.35)
In the case ns_ l = I/2, the bifurcation tree is a perfect symmetric tree. The 'empirical
laws' are expected whenever the deviation from the binary tree is random, i.e. whenever
branches are randomly suppressed and added with a zero-mean.
A n o t h e r m e a s u r e m e n t , wh i ch is s o m e t i m e s use fu l , is t h e n u m b e r o f end b r a n c h e s
o f a t r e e or a b r a n c h ( G R E N A N D E R , 1976):
1 1 \1 1 /
1 1 1 1 1 1 1 2 1 2 1
, 7 2 1
3 1 8 8 1
Fig. 3.27: The t r e e s o f Fig. 3.26 wi th b r a n c h e s e n u m e r a t e d by ' n u m b e r s o f c o m - p l e x i t y ' .
D e f i n i t i o n 2: The n u m b e r o f c o m p 1 e x i t y
o f o rde r s is t h e n u m b e r o f i t s end b r a n c h e s .
N of a b r a n c h c
Fig. 3.27 i l l u s t r a t e s t h e c h a r a c t e r i z a t i o n of trees by t he i r n u m b e r s o f c o m p l e x i t y . The
n u m b e r o f end b r a n c h e s and t h e o r d e r o f t h e n e t w o r k a r e not i n d e p e n d e n t . T h e r e l a t i o n -
sh ip b e t w e e n t h e n u m b e r o f c o m p l e x i t y a n d t h e S t r a h l e r n u m b e r is s imp ly
N > 2 S - l , (3.36) c
I I7
and the equal i ty holds only if the t r ee is a pe r fec t symmet r ic t ree . To prove this rela-
t ionship it is helpful to introduce
Defini t ion 3: The s k e 1 e t o n of an open network is i ts largest symmet r i c
subst ructure .
The skeleton is cons t ruc ted by e l iminat ing the branches with S t rah le r number Smi n
at nodes where SL,i_ 1 ~ SR,i_ 1,
The remaining t ree conta ins only nodes where the S t rah le r number increases and, the re -
fore, is pe r fec t ly symmetr ic . Along a pathway from an end branch to the trunk the
Nc 4 s ' 3 2 1
SKELETONS S=1,...,5
, ,r - V -
Fig. 3.28: Trees of complexi ty Nc = 3,4, and 5 (s:Strahler numbers) with dis t inct pe rmuta t ions of branch pa t te rns . Note tha t these pe rmuta t ions are not all topolo- gically d is t inct as indicated by brackets . Lower left : e l emen ta ry skeletons of S t rahler numbers 1 to 5.
118
St rah le r number simply counts the number of b i furcat ions . If we inver t this pathway,
the number of branches of equal S t rah le r number increases monotonously like 20 , 21 , . . . . . . . 2 S-1 2 2 , , 2 S-s, Arr iving a t an end branch the number of end branches is Nc= .
Rela t ion (a.a6) now is easi ly proved: If the t r ee is symmetr ic , then the equal i ty holds
as was shown, o therwise branches have been e l imina ted during the cons t ruc t ion of the
skeleton, and, the re fore , end branches have been e l iminated , i.e. t he inequal i ty in equat ion
(3.36) holds. This discussion provides us fur ther with a topological i n t e rp re t a t ion of the
S t rah le r number: It uniquely def ines the e 1 e m e n t a r y s k e 1 e t o n of
a t r ee or i ts subs t ruc tures (Fig. 3.28), where ' e l e m e n t a r y ' means tha t the length of
b ranches is not considered (empty nodes of the skeleton).
Another quest ion is how many d i f fe ren t t rees with ident ical number of end branches
do exist , or general ly: Given a ce r t a in number of end branches , how many d is t inc t ne t -
works can be genera ted? GRENANDER (1976) gives the following solution, which is due
to SHREVE (1966):
The number of open networks N(n) for any branching sys tem with n end branchings
is
N(n) = (2n-2)! (n! (n-l)!) -1, (3.37)
and the number N(nl ,n2, . . . ,nS_l , l} of topological dis t inct networks of order S with
nS_l , t b ranches of order 1 .. S is n 1
S-1 N(nl,n 2 .... nS_l,l ) = 11 2(ns-2ns+l) ((ns-2)! ((ns'2ns+l)! (2ns+l-2)!)-l. (3.38)
s=l
These two equat ions provide us with a probabi l i ty measu remen t of finding an open
network with S t rah le r numbers nl ,n2, ... ,ns_l,1; when the number of end branches
is given, the d i sc re te probabi l i t ies are
P(nl ,n2, . . . ,ns_l , l ) = N(nl ,n 2 ..... ns_l , l ) / N(n). (3.39)
Fig. 3.28 i l lus t ra tes the networks which correspond to N(3)=2, N(4)=3, N(5)=14; i.e. the
various combina t ions of branches . However, these t rees are not all topologically dis t inct ,
some of them are simple symmet r i c inf lect ions and cannot be distinguished: Let the
t r ees be the two-dimensional projec t ion of a th ree-d imens iona l image, then the inflec-
t ions mean simply tha t an observer walked around the object by 180 ° and then classif ied
it d i f fe ren t ly because the symmet r i e s have changed. The number of topological d is t inct
t rees is only
t19
N T = N{n) div 2 + N(n) mod 2. {3.40)
The d i f fe rence of the two concepts is not tr ivial , e.g. the probabi l i t ies will be d i f fe rent
under the two hypotheses: The probabil i t ies for a t r ee with four end branches {Fig. 3.28)
are p(3,2,1) = N(3,2,1)/N(4) = 1/3 and p(2,1) = 2/3. Based on topological similari ty, how-
ever , we find the probabi l i t ies PT{3,2,1) = PT(2,1) = 1/2, and the geomet r ica l hypothesis
inf luences the conclusions we may draw from the probabil i t ies .
Finally it seems worthwhile to emphasize tha t the probabi l is t ic aspect , ment ioned
briefly, is based on the comparison of an observed pa t t e rn with a to ta l ly de te rmin i s t i c
pattern~ the symmet r i c t r e e - - a possible question would be: How strong do we have
to disturb a binary t ree to ar r ive at the observed t r ee pa t te rn . The concept of skeletons
can be used to emphas ize this point. If the symmet r ic t rees wi thout repe t i t ions of empty
nodes -- the e l emen ta ry s k e l e t o n s - - are defined as pr imi t ives of our t r e e s t ruc tures ,
then any t r ee can be defined as 'p roduc t ' of e l emen ta ry skeletons: A node is the product
{Si,S]}, where the f irst t e rm means left , the second r ight branching {for symmet ry reasons
this no ta t ion is arbi t rary) and Si defines an e l emen ta ry skeleton of order (Strahler
number) s=i. The t rees of Fig. 3.26 can be described as l ists (Fig. 3.29)
",/ N/ x / V v v V-
"9' <L V,,/_v V , , / ..... /
Fig. 3.29: The t rees of Fig. 3.26 decomposed into e l emen ta ry skeletons.
(a) S3: (($2: ($2,0), O, ($2: (0, $3: ((S1,$2), O, O, $2), O)
or (($2,0), 0 , (0, (S1,$2), 2*0, S2), O)
(b) S3: ((S2, S2, S2: (0, S3: ((Sl,S2), O, O, S2)), O)
or (2 '$2, (0, (SI,$2), 2 ' 0 , $2), O)
(c) $4: (0, O, O, O, S2: (0,S2), $2, $2, $2)
or (4*0, (0,$2), 3"$2)
where the b racke t s conta in descr ipt ions of the end branches clockwise; the t e rm
I20
~O ~ indica tes t e rmina t ion of the branch, and t.~ means repet i t ions . The descr ipt ion is
recurs ive and, of course, could be reduced to binary decisions or s-expressions which
form the basic s t ruc tu re of the programming language LISP. E lementa ry LISP and the
lambda-ca lcu lus provide ano ther conceptua l f ramework to handle such s t ruc tu res (e.g.
DENERT & FRANK, 1977).
3.5,2 P a t t e r n Gene ra to r s for Open Networks
The theme of the previous sect ion was the descr ipt ion of t r ee pa t te rns . A common
problem, however, is the s imulat ion of t r e e pa t t e rn s and the quest ion how many dif fer-
ent pa t t e rn s can be gene ra t ed from some simple rules .Essent ial ly , the re are two ap-
proaches: t r ans fo rma t ions of qua l i ta t ive proper t ies and me t r i c models. Two examples
will be discussed here. In both cases, the process envolves some pr imi t ive e l emen t s
and de te rmin i s t i c product ion rules. However, one can adapt probabi l i t ies to the bi furca-
t ion events and then s imula te randomly dis turbed pa t te rns ; such sys tems were eog.
used by Raup to ana lyze whe the r phylogenet ic t rees are random s t ruc tu res (RAUP
et al., 1973).
A) Algebra ic Models -- P ro to types of Branching P a t t e r n s
One class of models, which genera tes regular s t ruc tu res inclusively t r ee pa t te rns ,
is based on the t r ans fo rma t ion of ~alphabetsV: A s t r ing of symbols like (a,b~c,...) charac-
t e r i zes qua l i t a t ive proper t ies of an object , and a se t of t r ans fo rmat ion rules like
(a ~ a d , b ~ c, ...) def ines the evolut ion of these propert ies , for ins tance how they
t r ans fo rm during t ime. Such sys tems have been extens ively applied to deve lopmenta l
sys tems by LINDENMAYER (1975). The theory of such t r ans fo rma t ion groups is much
older and closely r e l a t ed to the theory of permuta t ions ; for a de ta i led discussion see
ASHBY (1956, 1974). A typical sys tem, which produces t ree pa t te rns , is (LINDEN-
MAYER, 1975):
the a lphabe t or se t of pr imit ives: (a,b,c,d,e)
the product ion rules: (a --~db, b ~ c, c - ~ d , d ~ e , e --- a),
s t a r t ing with the e l emen t ~a ~ gene ra t e s the t r ee
121
a
A i I e c
I I a d
I I I e c a
ll \ b a d d
I I t I d e e c
I n s p e c t i n g t h e t r e e one f inds t h a t it c o n s i s t s o f t he r e g u l a r r e p e t i t i o n s of s t r i n g s (d,e,a)
and (b,c). The r e d e f i n i t i o n A = (d,e,a) , B = (b,c) and t h e mod i f i ed p roduc t i on ru l e s
(A - ~ AB, B - - - A ) s i m p l i f y t he t r e e p a t t e r n
A
A A B
/ \ \ A B A /! f \ \
A A B
I 1 \ t",,,,\ A A B A [3 A
which , h o w e v e r , c o n t a i n s t h e s a m e i n f o r m a t i o n as t h e or ig ina l t r e e b e c a u s e we can
r e s t o r e it by t he i n v e r s e s u b s t i t u t i o n . The two t r e e s a r e e q u i v a l e n t , and we can p r o d u c e
m o r e e q u i v a l e n t t r e e s by s u b s t i t u t i o n s o f t h e t y p e A = ( a l , a 2 . . . . . an) , B={bl ,b 2 . . . . . bn} ,
w h e r e t he s e t s o f p r i m i t i v e s in A and B a r e d i s junc t , i .e. do not c o n t a i n i den t i c a l
e l e m e n t s . Two t r e e s c a n now be de f i ned as e q u i v a l e n t if it is poss ib le to r e d u c e t h e m
to t h e s a m e p r i m i t i v e fo rm; t h e r e d u c t i o n s , wh ich a r e a l lowed, a r e t h o s e wh ich do
not a l t e r t h e s i ngu l a r p a t t e r n o f t h e p roduc t i on ru les , i .e. wh ich p r e s e r v e
t e r m i n a t i o n po i n t s l ike A ~ A,
b i f u r c a t i o n po i n t s A - - B C ,
and loops A -*- B ~ A.
122
The problems, which arise if o ther reductions are used, were briefly mentioned during
the discussion of Markov chains in sedimentology.
The reduct ion process may be i l lustrated by a somewhat more compl ica ted
system:
the alphabet: (a,b,c,d,e, f,g,h,i,k)
the production rules: (a - - b c , b - - kd, c -,- Ik, d --~ gb, e --- cf, f - - ih,
g - - hi, h - - de, i - - k, k --- k),
which are ra ther compl ica ted and genera te an evenly compl ica ted pa t te rn
a
bc
kdek
kgbcfk
khikdekihk
kdek kgbcfk kdek
kgbefk khikdekihk kgbcfk
khikdekihk kdek kgbcfk kdek khikdekihk
The system was used by LINDENMAYER {1975) to describe the s t ruc ture of compound
leaves. The strings of symbols were in te rpre ted as ceils along the margin of the leaf
which posses cer ta in morphological propert ies . We try to reduce the complicated produc-
tion rules and observe first that the e lement 'a v occurs only once; indeed, it serves
only as an ent ry point. Regular repet i t ions of substrings in the t ree occur from the
third production downward, these are A=(kdek), B={kgbcfk), C=(khikdekihk), and the
production rules can be simplified {A -~ B, B -~- C, C -- ABA)
B
J C
/IN A B A
/ I \ B C B
/ / t \ \ C A B A C
A B A B C B A 13 A
The reduced system produces a nice t ree with triple points at C, a s t ruc ture which
123
was not obvious from the original production rules, but, clearly, the reduced system
is equivalent to the original o n e - - if we subst i tute the original values for A,B, and
C, we have the original productions.
Inspecting the e lements A,B,C it turns out that they all are of the form (k ..o
k), i.e. the ' k ' - e iements act as brackets of the original strings. Only in the e lement
C, k's occur within the string. If we want that this fine s t ruc ture is not lost in the
reduced version, we can introduce the radicals H=(k,h,i) and HT=(i,h,k) replacing the
e lement C, which now reads C=HAH T, and the productions are
A
J B
I HAH T
/ 1 \ A B A
/ \ B B / J \ \
A B A HAH 'F
which again comprise the s t ructure of the original rules.
This sect ion a t t empts to i l lustrate that the reduction of complicated systems
to equivalent simple ones is an essential step in pa t tern recognition problems. However,
such reductions can only involve those t ransformat ions and redefinit ions which do not
a f fec t the singular s t ructure of the original system. Using the concept of equivalence
and proper reduction processes may also save much investigation t ime because a ra ther
complicated system may turn into a well known simple one, and the classif icat ion of
such s t ruc tures reduces to a (not necessari ly finite) catalogue of prototypes.
B) A Metric Model - - the Honda Tree
HONDA's (1971) approach was to formulate a pa t t e rn -genera to r which allows to
study the form of t rees by means of the computer, based on a minimal set of parameters
{Fig. 3.30). Here only his 'geometr ica l assumptions ' are of importance, they are repeated
in a slightly modified form:
I The branches are always straight, and their girth is not considered;
II the mother branch produces two doughter branches with each branching event;
124
Fig. 3.30: Two-dimensional images of Honda t r ees in the (x,y}-plane. The r ight f igure has in par t been comple ted to approximate tr ipling at branching points. Still more rea l i s t ic b i fu rca t ion pa t t e rns for plants can be cons t ruc ted if the branches fork off under a ' d ivergence angle ' which follows a Fibonacci ser ies (cf. HONDA, 197t).
III the length of each successive doughter branch is shor tened in a ce r t a in ra t io
wi th respec t to the mothe r branch;
IV the doughter branches form cons tan t angles with the i r mother branch through-
out the t ree; however, lef t and r ight branching angles may be d i f ferent ;
V the mother branch forks off into doughter branches in the plane which con-
ta ins the mothe r branch and whose s teepes t gradient line coincides with the
d i rec t ion of the mother branch.
From these assumptions Honda derived a formula which allows to compute the
successive coord ina tes of branching points and, thus, can be p rogrammed for graphical
compu te r output . Point (V), which is not a sepa ra te point on Honda's list, is responsible
for a r a t h e r compl ica ted s t ruc tu r e of Honda 's formula, which has to be applied separa te ly
for l e f t and r ight b ranches (d i f ferent p a r a m e t e r s R and {3 ):
x = x B + R(u cos0 - (Lv sin@)/(u2+v2) I/2 (3.41)
Y = YB + R(v cos8 + (Lu sine)/(u2+v2) I/2
z = z B + R w cose
where R: ratio between subsequent branches, @ : branching angle
and u = XB-XA, v = yB-YA, w = ZB-Z A, L = (u2+v2+w2) I/2
125
Fig. 3.31: l londa 's
bifurcat ion.
nota t ion of
XA, x B etc . are coordinates of the branching points (P) as indicated in Fig. 3.31. Honda 's
formula is easily p rogrammed and evaluated wi th a computer ; however, equat ion (3.41)
is not of the proper form for an analyt ic analysis; it takes a much more convenien t
form if one does not work in global coordinates but considers the dislocations of branch-
ing points which are
[ Z - Z b J ~ 0 c o s 0 LZB-ZAJ
where b = L(u2+v2) I/2 = (i + w2/(u2+v2)i/2.
(3.42)
The matrix, which appears on the r ight side, describes a ro ta t ion and a simple compres-
sion (or di latat ion) . We can separa te the two components by dividing all ma t r ix e l emen t s
by a fac to r
R I = (cos~ + b2sin28 )I/2= (I + w2(u2+v2)I/2sin28)I/2),
and the dislocat ions can be wr i t t en
I II = RIR k sin@O co~@ 0 itlJ 0 c
where tan = b(sin 8)/(cos e)
and c = (cos0)/(cos 2 @+ b2sin20 )1/2.
(3.43)
The fac to r R 1 defines an addit ional shortening ra t io which depends like the mat r ix
e l ement 'c ' on the ro ta t ion angle and the length of the mother branch. However, only
the length of the mothe r branch is var iable (cf. assumption IV). The re levan t coef f ic ien t ,
Fig. 3.32:
bifurcation.
Vector notation
126
of a
r L
P r L
which depends on this parameter , is w2/(u2+v2), Cfo equations (3.42) and {3.43). In case
this quotient is constant , equation (3,42) is a simple linear system. We shall not discuss
Honda's original model in more detail; however, condition (II) can only be sat isf ied if
w2/(u2+v2)=constant. Only in this case, the number R defines the shortening ratio of
branches, and Honda~s model can be replaced by equat ion (3.431.
Honda~s model provides a framework which captures essential aspects of branching
pa t te rns in t ree- l ike bodies although already Honda needed some additional assumption for
th ree -d imens iona l t rees. However, it can be simplified leading to a description by two
sets of vectors: r, the dislocations of branching points, and P , the radius vector of
the branching points in global coordiantes {Fig. 3.32):
The h o d o g r a p h of a H o n d a t r e e is defined by the pair
of i te ra t ive equations
rR, i+1 = aRARri (3.44)
rL, i+ 1 = aLALri ,
and the space coordinates of the Honda tree are given by
PR, i+I = P i + rR,i+l (3.45)
PR,i+I = P i + rL,i+l;
r, p will denote vectors throughout this section; R,L are the indices for left and
right branching; 'A t is an orthogonal matr ix (cf. equation (a.4a)); 'a t is a scalar
and denotes the ratio be tween subsequent branches.
While in a Honda t ree s. str. 0 < a R , aL < 1, we call a t ree with arbitrary values
of aR, a L a g e n e r a t i z e d H o n d a t r e e , and, where not necessary,
the a t t r ibu te ~generalized ~ is dropped.
A Honda tree, thus, is a binary tree, and if we superimpose some probability that
127
branches may be lost, i t genera tes an open network, as discussed in the last section,
with all i ts propert ies . The Honda t ree, developed for botanica l models, possesses a
proper ty which is a classical observat ion in geomorphology: the cons tan t ra t io of branch-
es. The s t r eam length sa t i s f ies the approximate re la t ion (GRENANDER, 1976)
ls_l/1 s = RL, s = 2,3,...,S (3.46)
1: length of branches, s: S t rahler number, RL:length rat io,
an empir ica l observat ion, which coincides with Honda's assumption (tI). We shall derive
some proper t ies of Honda t rees , which are only based on this assumption, and, therefore ,
can be applied to s t r eam pat terns .
3.5.3 Morphology of Branches in Honda Trees
"DVArcy Thompson's Problem"
Honda t rees , as defined in the previous sect ion, possess special propert ies: con-
s t an t branching rat ios and branching angles. These proper t ies allow to compute the length
and morphology of ce r t a in b r a n c h e s - - under special c i r cums tances even for inf ini te
numbers of branching events . Fur thermore , the d i sc re te model will turn out to be iso-
morphic with a d i f fe ren t i ab le system.
A) Length of Branches
By defini t ion the branches of a general ized Honda t r ee are in a geomet r ica l rela-
t ionship, i.e. the successive length of branches is given by the i t e ra t ed map
I r i ]= a [ri_l [ (3.47)
where 'a t may be e i the r the branching ra t io for lef t or for r ight turns, or b e t t e r a m a x
or amin, a no ta t ion used now because ief t and r ight are of l i t t l e meaning (cf. sect ion
3.5.1). If one considers a continuous sequence of turns in one direct ion, the length of
the b ranch is given by
]r i[= Jr0] ~ a 1, (3.48)
a geomet r i ca l ser ies with the well known sum
L =1 r 0 1 (1-an+l)( t -a) -1. (3.49)
In case of a convergen t Honda t r ee , the branching rat ios have to sa t is fy
0 < a . < a <l, and we can compute the length of such a branch even for n - 7 ~ : mln max
128
lim] ro} (1-an+l)(1-a) -1 = ]to] (l-a) -1, n ~ co
(3.50)
From this observa t ions we es tabl ish the
Proposi t ion: In a Honda t r ee the sums
Lma x = I ro l (1-amaxn+l}(1-amax )-1 and
Lmi n = ]ro](1-aminn+l)(1-amin )-1
provide upper and lower bounds for the length of any pathway through the t ree
with fixed n (n = 1,2 . . . . . k) and ini t ial length r 0 . In case amax=amin , all possible
pathways are of equal length. In case 0 <ami n <amax < 1, this holds even for
n - ~ o o .
Consider any o ther pathway, it necessar i ly includes at least one turn opposite to the
cont inuous path, i.e. an e l emen t a lmaxalmin . However, for any combinat ion of i and
j we have
a i+j <= a i a j < a i+j (3.51) max max min min"
The sum over the length of e l ement s along an a rb i t ra ry pa thway with init ial length
r 0 conta ins only products of this type, and the length is bounded for fixed n because
every e l emen t is bounded. The equal i ty in the equat ions holds if amax=amin , and independ-
en t of the pathway, i ts length is cons tan t for fixed n.
These resul ts can immedia te ly be applied. Consider a d is t r ibut ive system which
has to be op t imized in the way tha t the lengths of pathways with common origin are
equal. This op t imiza t ion problem is solved by any t r ee with amax=amin , independent
of i ts morphology, e.g. branching angles.
If one considers genera l ized Honda t rees , one cannot expect tha t the length of
b ranches converges for n - - ~ , however, t he re may be special pathways with f ini te length.
To get an idea what branches converge we consider pathways consist ing of regular repe t i -
t ions of 'm ~ lef t and ~n' r ight turns, for which we wri te
( m ' R , n 'L) , ( m ' R , n 'L ) . . . . ; m,n ~ (0,I,2 ..... ).
The sum of such a ser ies is
(L-] r d )= 2 m
a + a + . . . + a
+ am{b + b 2 + ... + b n)
+ , . . +
+ akmb(k ' l )n (b + b 2 + ... +b n)
; the f irst sequence of r ight turns
; the f irst sequence of lef t turns
; the k th and final repet i t ion .
129
This sum can be r e w r i t t e n as
L - I roI = (~ ai + am ~ bi)(l + ambn + (ambn)2
or
L lrol = (g ai + an~ bi)(ambn)k / (1 - ambn),
and b ranch leng th c o n v e r g e s if
0 < arab n < 1.
+ ... + (ambn)k (3.52)
(3.53)
Given a c e r t a i n area x and ami n we have to find numbers m,n so tha t equa t ion (3.53)
holds; if they exis t , t he b ranch is of f in i t e length . In s o m e sense , t h e s e spec ia l pa t hways
provide us again wi th upper l imits: If ( m ' R , n ' L ) c o n v e r g e s and a R > aL, then all p a t h -
ways ( m ' R , (n+i)*L) c o n v e r g e even if ' i ' is some random var iab le but s a t i s f i e s i _-< 0.
B) Branching Angles - - S imi la r i ty and Se l f -S imi la r i ty
The second var iab le in Honda ' s model is the b ranch ing angle . However , this var ia -
ble is no t a p roper m e a s u r e m e n t because t h e t r e e p a t t e r n depends s t rong ly on the b ranch
ra t ios . If one wan t s to c o m p a r e two t r ees , a m e a s u r e m e n t is neces sa ry which t akes
t he b ranch ing angles and the b ranch ra t ios in to cons ide ra t ion . This leads to t he fol lowing
Def in i t ion : The s i m i 1 a r i t y i n d i c e s o f a Honda t r e e a r e t h e
two number s S m a x l m i n = (log amax lmin ) /0 ; 0 in radians . Two t r e e s a r e abso-
lu te ly s imi la r if bo th ind ices a r e iden t ica l , t h e y a re pa r t i a l ly s imi la r i f they a g r e e
in one s imi la r i ty index.
To m o t i v a t e this def in i t ion we cons ider t he hodograph of the Honda t r ee , i .e . the v e c t o r s
r i = aAr i_ l , and we cons ide r again a con t inuous pa thway which cons i s t s e n t i r e l y o f l e f t
or r ight turns . Then the v e c t o r s r i evolve like r0, aAr0, (aA)2r0, ... (aA)nr0 . Because =anA n A is an o r thogona l ma t r i x (cf. equa t ion 3.43), t he powers r n can be w r i t t e n as
The s imple r ede f in i t i on ~ = n (? t r a n s f o r m s this equa t ion into
130
r n =
I cos t9 =sin ~ 0 e c '~ sin ~ cos tp 0
0 0 c n r 0 (3.55)
c = (log a)/0
that is, the vectors r are located on a logarithmic spiral in the (x,y)-plane or on a n
trochospiral in three dimensions, whereby the term 'spiral ' includes circles and straight
lines. To see this more clearly consider ro=(XoYoZo) in its most general form, and equation
(3.55) can be rewr i t t en as
r = E!o yo o I Eco: 1 o : % 0 e c $ sin
0 Z o C n
(3.56)
The term in brackets is clearly a logari thmic spiral if we allow ~ to vary continuously.
Equations (3.54-56) provide a continuous approximation for the location of dislocation
vectors in a monotonous sequence of left or right turns, and the similari ty indices are
based on the similari ty of these spirals taking into account that a change of the magni-
rude of the branching ratio can be compensated by the branching angle so that the
vectors are still located on the identical spiral. Another aspect is that a Honda t ree
is everywhere selfsimilar in te rms of its hodograph (F~ig. 3;33). Equation (3.56) shows
that we can build the hodograph ent irely of sequences of continuous left and right turns,
i.e. of spirals in the (x,y)-plane. Along such spirals the branching points are defined
by ~=0n , and they give rise to a spiral in opposite direction. In case the Honda t ree
is absolutely convergent , the two initial spirals bound the hodograph, i.e. all o ther disloca-
tion vec tors are located inside these leading spirals. The Honda model thus describes
an ideal self similar system with possible infinite repet i t ions.
F_~ig. 3.33: The hodograph of a Honda t ree and its continuous approximation.
i
/ / /
I j
%
J • •
, s l a • • / # J ~ l
i s ~ " I
131
C) Branches and Bi furca t ions - - a Quas i -Cont inuous Approx ima t ion
So far , we g a t h e r e d i n fo rma t ion about the branching p a t t e r n wi thout r ega rd to
the form of b ranches ; however , it will turn out t ha t mos t work was a l r eady done. We
have seen tha t t he hodograph of a Honda t r e e cons i s t s o f s imi la r spirals which o r ig ina te
a t a ' l ead ing ' spiral , and in the ca se of a c o n v e r g e n t Honda t r e e they all approach
the s a m e coi l ing c en t e r :
Def in i t ion: A 1 e a d i n g b r a n c h is a con t inuous s equence of e i t he r
l e f t or r ight turns , and it is t he cont inuous app rox ima t ion o f a leading branch .
The t e r m 's p i r a 1' inc ludes t rochosp i ra l s and c i r c l e s and s t r a igh t lines.
The b r a nc he s and the b i fu rca t ion p a t t e r n a re found if it is poss ib le to ' i n t e g r a t e '
t he hodograph, i .e. one has to sum the d i s loca t ion v e c t o r s along e v e r y leading spiral:
P i = (i-~0aiAi)r0 ' (3.57)
tha t is again a g e o m e t r i c se r ies , but this t ime it involves a mat r ix . However , if one
uses t he inverse of a m a t r i x - - deno t ed by M -1 - - this se r ies can be summed like
an ord inary power se r ies (e.g. ZURMOHL, 1964), and the sum takes the form
Pi = (I - aA)- l ( I + an+lAn+l) ro
w h e r e I: the i den t i t y mat r ix ,
(3.58)
or in a m o r e e x t e n s i v e form
P i = ( I -aA)- l ro . ( i_aA)- l (an+lAn+l) ro .
This equa t ion desc r ibes the success ion of b i fu rca t ion poin ts along our leading spiral
if we use 'n ' as var iab le . It t u rned a l ready out t ha t a t e r m (aA)nro desc r ibes a s e q u e n c e
of v e c t o r s (cf. equa t ion 3.54) with the v e r t i c e s l oca t ed on a spiral , and r e tu rn ing to
t h e s e a r g u m e n t s again i t b e c o m e s c l ea r t ha t t he va r iab le t e r m in equa t ion (3.58) de-
s c r ibes again tile spiral of the hodograph. The t e rm (I-aA) -1 is cons t an t and can be
d e c o m p o s e d into a ro t a t i on and an e longat ion , as i l l u s t r a t ed in s ec t ion 2.5.2 B. The
leading spi ra ls of t he hodograph, t h e r e f o r e , map on to s imi la r spirals , t he b ranches .
The r e p r e s e n t a t i o n of a b ranch by a s equence of spira ls is i l l u s t r a t ed in Fig. 3.34.
The form of b ranches , or more prec ise ly , the loca t ion of b i fu rca t ion points , thus is
well de f ined in a Honda t r e e and r e s t r i c t e d to a s ingle fami ly of func t ions . The spi ra ls
a re t he s a m e as in t h e hodograph up to s imi l a r i ty t r a n s f o r m a t i o n s .
The b i fu r ca t i on p a t t e r n thus can be a p p r o x i m a t e d by spira ls which o r ig ina te a t
132
Fig. 3.34: A Honda t ree as a sequence of spirals.
a leading spiral and give rise to further spirals and so on. The Honda t ree provides
a model for
"D'Arcy Thompson's" theorem: A branch system with continuous branching angles
and geomet r ic relationships between mother and doughter segments of branches
is isomorphic with a system of leading spirals: The first generation of branch-
es is a t tached to the leading spiral in regular, usually geometr ical ly increasing
or decreasing distances measured by arc length and give rise to a second genera-
tion, and so on.
This pa t tern has been i l lustrated above, here we give some more propert ies which will
be needed later, Thereby we res t r ic t ourself to the two-dimensional image of Honda
t rees in the (x,y)-plane (of. equation 3.44):
Proposi t ions concerning the form of branches in Honda trees:
(i) There are at most two d i f ferent spirals in a bifurcat ion t ree of Honda type.
All others are similar to these two spirals. Similar spirals have identical coiling
133
directions.
(2) All branches of a ce r t a in generat ion are d i rec ted to the same side of the leading
s p i r a l
(3) The length of a spiral arc be tween two branching points is proport ional to the
length of the d iscre te branch segment be tween these points,
(4) The lengths of any two spirals with equal numbers of branching points are in
an a l lomet r ic relat ionship. If the two spirals belong to the same family of leading
spirals, the i r lengths are simply p ropor t iona l
Proposition (I) follows from the fact that a Honda tree contains only bifurcations with constant branching angles and branch ratios. The two spirals are defined by equations (3.54-56). Proposition
(2) follows from the definition of a leading spiral which is a continuous pathway of only left or right
turns, and from the constancy of branching angles. To prove proposition (3) we choose the coiling center of a leading spiral as origin of the coordinate system and assume the bifurcation points are at regular
angular distances @=constant. The succession of these points is given by p i=exp(-i¢ ). The length of a spiral segment is
L a e-l (c~+l) l /~(tPi l - IPI_IL? (3.5g) c -I (c2+i)I/2(I _eC)e -el.
The length of the discrete branch segment connecting the same bifurcation points is
Lb = lP i - Pi-~ I= ri (3.60) (e-2Ci+e-2C(i-1)_2e2Ci+C((cosCi)(cos~(i_1 )) + (s:'w~i)(sin(b(i-1)
(I +e2C-2eCcos~)e -ci.
The only variable is ti~, all other terms are constant. Therefore, we carl express the length of discrete branch segments in terms ef the length of the spiral arcs
L b = a Ls. (3.61)
The length of a branch for a given ~i Y is the sum over subsequent segments. The constant factors, how- ever, need not to be summed, the sum involves only terms axp(-ic) which are identical in beth expres-
sions: Equation (3~I~ therefore, holds for any fixed number of branch segments Ti~. In both equations
(3.59+60) the ratio between subsequent segments is Lb,i+I/Lb,i = Ls,i+I/Ls,z" = exp(-c), that is
the geometrical relationship in D'Arcy Thcmpsons theorem. To prove proposition (4) we start with the ailometric relationship between two logarithmic spirals:
c 1 ¢ e 2 ¢ P 1 = ale ; P 2 = a2e '
c 2 can be expressed i n terms of o 1 by an equat ion c2=bc 1 or p2 = a2exp(bc 1 ), and, the re fo re ,
P 2 = ( a 2 P l / a l ) b ' (3,62)
i.e. the radii are in an allometric relationship. The length of a logarithmic spiral is proportional
to the length of its radii (equation 3.59), and the allomatric relationship holds alse for the length of spiral arcs for any fixed angular interval i C. If the two spirals belong to the same family, the coefficient b=1.
As long as we consider branches of f inite length, the resul ts hold for any values
of branching ratios. There are some special cases, e.g. if one branching ra t io equals
134
Fig. 3.35: The plane image of a Honda t r ee if amax=l ,
0 <ami n < 1 are regular poly-
hedrons.
lm
....
/
one. In this case, the images of b ranches in the (x,y)-plane are circles as i l lus t ra ted
in Fig. 3.3~ for amax=l , ami n <1. If such a special p a r a m e t e r se t t ing is chosen, the
net pa t t e rn depends strongly on the b i furca t ion angles, e,g. if amin=0, then the system
resembles the ' e l a s t i c collision in a c i rc le ' as discussed in sect ion 3.3.2 (Fig. 3.14). If
both branching ra t ios equal one and the branching angles are 2~/3, w / 2 . . . . , the Honda
t r ee degene ra t e s into regular t r iangular , r ec tangu la r and hexagonal grids which cover
the en t i r e (x,y)-plane. Usually, however, we will find sequences of logar i thmic spirals
which, in the case of convergent Honda trees, are bounded in length. This causes the
chao t ic .appearance in the two (and three-) dimensional images of the t rees , the end
branches c lus te r and overlap prohibi t ing the recogni t ion of regular s t ructures .
3.5.4 Evolut ion of Shape
'Shape ~ or ~outline ~ means tha t an object is bounded by some kind of surface .
Simple b i furca t ion diagrams, e.g. the skeletons S1-$4 in Fig. 3.28, do not possess a
shape in this sense. However, as the number of b ranches increases, an out l ine evolves,
at least subject ively. From previous work we know tha t a convergen t Honda t r ee needs
to have a maximum out l ine or l imit ing shape for a suff ic ient ly high number of b i furca-
tions. This l imit ing shape exis ts and cannot exceed a ce r t a in boundary because all
b ranches are of f ini te length. On the o ther hand, the number of end branches increases
with the number of b i furca t ion even ts like n=2 s (s=Strahler number because the t r ee
is symmetr ic ) and c lear ly goes to inf ini ty if the number of b i furca t ion even ts is not
135
bounded. BARKER et al. (1973) es t imated the mean number of buds arising from the
highest order branch in a birch t ree to be >7000.
As the number of end branches increases, we expect increasing density of end
points and a t rend towards a cont inuous outline. Fig. 3.36 elucidates this point where
only the end points of the t rees of Fig. 3.30 are marked. Clearly, the end points clus-
ter in cer ta in areas and give the illusion of continuous lines which together give the
Io i
\
Fig. 3.36: Distribution of branch endings in the t rees of Fig. 3.30.
illusion of a continuous outline. One question to be discussed is whether these points
may define a continuous curve or even fill some space, as the number of bifurcation
points goes to infinity. The o ther question is if there exists some defined envelope
or hull of a convergent Honda t ree which can be considered as its limiting shape.
A) Trees, Peano and Jordan Curves
To i l lustrate the concept of shape in more detail we consider a quite d i f ferent
system, continuous curves which are nowhere different iable . A special example, which
is useful in this context , is the Koch curve {cf, MANGOLDT-KNOPP, 1968). The closed
version of a Koch curve can be const ructed in the following way (Fig. 3.37): A regular
tr iangle (equal angles) is inf lected and superposed on i tself so that the sides are divided
into three segments of equal length, the resulting pa t te rn is a regular star . The corners
of this s tar are again regular triangles, similar to the original ones, and with each
t36
vS J ig d, 13tsaTrelC~nstTt~Otr:: ~hatteKrOCh cu
of these triangles the process is repeated, The Koch curve is the outer boundary of
the resulting pat tern as i l lustrated in Fig, 3.37. If the process is repeated infinitely,
the triangles shrink to points and give rise to a continuous, non-differentiable curve.
If we now connect the centers of successive triangles by straight lines, these form
a t ree pat tern with a triple point at each branching event. The branching angles and
the shortening ratios are constant, i.e. if we dele te one of the branch directions, the
resulting binary t ree is a Honda tree.
.Fig, 3.38: Construction of a binary t ree as sequence of similar triangles.
137
a /
b
I 1 I
#
C
~ -If '4 '
J(÷ Ji
Fig. 3.39: Triangulation of a rectangle by i tera t ive averaging (a; cf. Fig. 2.46) reconsidered as a space filling Peano curve (b). Connection of subsequent centroids generates t ree pat terns, c: bifurcation and tripling with ' l imit shape'; d: space filling t ree pat tern (see text).
The cen te r of every triangle along the boundary is the terminat ion point of a
end branch. As the i terat ion process goes to infinity, the triangle shrinks to its centroid
which, by definition, is the terminat ion point of an end branch, and we expect a con-
tinuous curve as limit shape for the tree. In detail, this assumption is not quite correct .
If we only consider branches as indicated in Fig. 3.37, then there are no branches
terminat ing in the concave intervals of the Koch curve while the original construct ion
produces triangles in these areas. In the t ree pat tern an empty interval remains between
terminal points, and new ones are genera ted with every i terat ion. The points belonging
to the t ree are only a subset of the Koch curve. If the points, however, are replaced
by objects of f inite area, the boundary becomes again densely covered. Numerically
this has been studied by HONDA & FISHER (1978, 1979) in terms of the most equitable
distribution of leaf clusters. Fig. 3.3a shows a similar construct ion of a binary bifurca-
tion pat tern with terminal triangular ' leaves ' which tend toward a continuous outline,
but again there are areas which are not covered, and in addition the leaves overlap.
The relationship to the discussion of cluster t rees is obvious; however, let us
return for a moment to a problem of the first chapter , the reconst ruct ion of a continu-
138
ous surface by i te ra t ive
e i ther way i l lustrated in
centroids as unique limit
MANGOLD-KNOPP, 1968;
we have a space filling
averaging (section 2.4.5). If we draw the successive nets in
Fig. a.ag, we get a nested sequence of squares with their
point. The sequence of centroids can be paramet r ic ized (e.g.
GUGGENHEIMER, 1977}, and, as the process goes to infinity,
curve or a Peano curve. Again we can connect a sequence
of centroids in the way indicated in Fig. 3.39 generat ing another special t ree pa t te rn
with its terminat ion points a subset of the Peano curve. As the process goes to infinity,
the terminat ion points of a binary t ree cluster; however, they are always well separated.
The t ree becomes more dense if we allow triple points (Fig. 3.39), and finally we can
turn the branching process into a space filling process by the condition that branches
of any generat ion sprout into areas of least density (with maximal dis tance from neigh-
boring branches).
B) The Outline of Honda Trees
The brief excursus to such s trange s t ruc tures as Jordan and Peano curves provides
us with two contras t ing aspects: It encourages the previous view that there are t r ee
s t ruc tures which genera te an outline, and it discourages the a t t emp t to search a descrip-
tion of this outline by inspecting the distribution of branch terminat ions. A conclusion
could be that we now need s ta t i s t ica l methods to study the distribution of branch t e rmi -
nations more deeply; the si tuation is quite similar to the problem to find a closed bound-
ary for a f inite point set as discussed in te rms of cluster s t ra tegies . However, in the
special case of Honda t rees we have continuous di f ferent iable functions which approximate
the branches, and, what we ar looking for, is another continuous approximation of the
most extended outline. Therefore , we replace the d iscre te model by a continuous approxi-
mation hoping to replace the multivarious outputs of a computer program by a theorem
about the shapes, which possibly can be cons t ruc ted from the model under consideration,
similar to the model 's original a t t emp t to describe the essent ial s t ructures of complex
natural pa t te rns by a few parameters .
Let us consider a leading branch and replace it by its spiral approximation. The
spirals of the second generat ion are then most economically described in local coordinates
of the leading spiral, i.e. in te rms of its local moving frame or Frenet trihedron. If
we consider s traight branches of second order, we can e.g. express them in terms of
the tangent vec tor of the leading spiral and ro ta t e this vec tor into the proper position.
The local descript ion of the d iscre te system is analogous to equation (3.57), only the
vec tor ro is replaced by the ro ta t ed tangent vec tor bBt:
Pi = (i~=O aiAi)(bBt)" (3.63)
139
If we move this spiral of second order along the heading spiral, the branching angle
is everywhere preserved. Consider a convergent Honda t ree with infinite number of
branching points, then by proposition {4) of the previous section the length of the second
order spiral is always in an al lometr ic relationship with the length of the leading spiral,
and the same holds for the radii of the two spirals. We take the coiling center of the
leading spiral as the origin of the global coordinate system, and the global coordinates
of a point fixed on the second generation branching pat tern takes the form:
PB =Ps + a lps Ib (bBt)
where the index s denotes the leading spiral.
(3.64}
Before discussing this equation in more detail, we observe that b=l if we res t r ic t ourself
to leading spirals of the same system because these are all similar. Furthermore, if
we consider any fixed point on a spiral of higher generation, this point is described
by a vector which originates at the leading spiral. This holds for any point, even for
the most distant one. Although it is cumbersome to evaluate the most distant point,
it will turn out that we can give a quali tat ive answer about the outline.
Remaining in a system of leading spirals equation (3.64) is simplified and can
be wri t ten in a more extensive form as
I: :1] I or (3.65)
l p B = I + bB sin~ -cos~j Lsin~,
and once more it turns out that this equation describes the original spiral which is ro-
ta ted around its coiling center . Every family of leading spirals, therefore , is bounded
by a similar s p i r a l - - because a t ree consists of two systems of leading spirals, the
shape results from superposition of the two systems. On the other hand, within every
system of leading spirals there exists an infinity of identical subsystems which differ
only in size, and we can consider the maximal outline as result of the superposition
of smaller subsystems as i l lustrated in Fig. 3.40. Concerning the density of terminat ion
points we note that the distance of branching points decreases regularly along the leading
spiral, the density of branching points is proportional to the curvature of the leading
140
Fig. 3.40: A Honda t ree is se l f -s imi lar on all levels. Superposit ion of images on a ce r t a in level genera tes the ident ical but enlarged image.
spiral, and the same holds for any secondary leading spiral, etc. . Thus, the densi ty in-
c reases towards the coiling cen te rs , and the densi ty of coiling cen te r s is proport ional
to the cu rva tu re of the leading spiral. Densi ty dis t r ibut ions of this type were a l ready
encoun te red with the ' c lus t e r s t r a teg ies ' .
Finally we consider b ranches of f ini te length. The proposit ions of the previous
sec t ion sti l l hold, the only d i f f e rence is tha t the leading branches do not coil to infinity.
Equat ion 3,64, there fore , takes the form
PB =Ps + a(JPsJ -IP 0 ])b(bBt), (3.66)
and the b ranches t e r m i n a t e for Jpsl=JOOJ. Far from this s ingular point and for b=l
the shape is qui te close to the spiral of the leading branch. The system, there fore , is
real ly se l f - s imi la r on each level.
141
C) Chance and Dete rmin i sm
The previous discussions showed tha t the Honda model genera tes ideal se l f -s imilar
and de te rmin i s t i c s t ructures : The morphology of b ranches is (infinitely) repea ted and
impresses i ts pa t t e r n onto the shape of the en t i r e s t ruc ture , and the shape is again
repea ted on the level of every genera t ion of b ranches differ ing only in size, With these
proper t ies it is unlikely tha t the Honda model re f lec t s r e a l i t y - - it is a s trongly ideal-
ized model, and in a la te r study HONDA & FISHER (1979) int roduced addit ional sources
of var ia t ion to approximate real t rees more closely. However, se l f - s imi lar i ty and f rac ta l
sys tems are cu r ren t fields of in te res t (MANDELBROT, 1977), and the Honda t r ee provides
a simple l inear model al though the basic model is r a t h e r unreal is t ic . However, t he re
is some possibili ty tha t models "nearby" approximate real i ty, i.e. t ha t dis turbed Honda
t rees provide b e t t e r approximations. There are several ways to disturb the original model,
which involve s tochas t ic and de te rmin i s t i c aspects .
A simple approach is ' chance ' , i.e. some ex te rna l and perhaps in ternal noise is
added which a l t e r s branching ratios, branching angles, and may even inhibit branching.
However, ' chance ' involves also events as ' to evolve at a ce r ta in place ' , to 'evolve
within a ce r t a in t ime interval ' , or to 'evolve under ce r t a in and for the individual s t ruc tu re
not pred ic tab le env i ronmenta l condit ions ' , even if they are not random but exhibi t only
a complex spa t io - tempora l dis tr ibut ion. The random aspect "if the re would not have
been t ha t par t icu lar thunders torm at t ha t par t icu lar t ime, then those branches would
not have been broken off and those buds would not have been inhibited" is only a very
special viewpoint.
Other aspects are ex te rna l and internal cons t ra in ts , the system has to fulfill cer -
tain boundary condit ions and cannot grow freely. A possible resul t could be tha t under
ce r t a in cons t ra in t s only one or few of the topologically dis t inct t rees discussed in sect ion
3.5.1 are s table. Or, consider the drainage systems of Fig. 3.25; al though they are ir-
regular, they are not random; the geology de te rmines much of the drainage system:
So the inf luence of the Andes obviously de te rmines s t r eam direct ions at the wes te rn
margin of the Amazon drainage system, and sea- level f luctuat ions through glacia t ion
periods de te rmined much of the dra inage pa t t e rn of the Ganges del ta . If we consider
such systems as random or par t ia l ly formed by chance, the reason is simply tha t we
cannot r econs t ruc t the his tor ical evolution of boundary condit ions to a suff ic ient pre-
cision.
A third group of factors, which may be of in teres t , are in ternal const ra ints , or
in ternal regulat ion systems, The Honda model is a special case of the family of l inear
maps
Xi+l = a lx i + blYi Yi+l = a2xi + b2Yi (3,67)
142
which is equivalent to the d i sc re t iza t ion of a pair of coupled l inear d i f fe ren t ia l equat ions
dx/dt = a l l x + a22Y; dy/dt = a21 x ÷ a22Y (3.68)
which gene ra t e spiral pa t t e rns in the phase space for p a r a m e t e r se t t ing equivalent to
Honda 's model (cf. sect ion 2.2.1). Another possible extension is tha t the local evolu-
tion depends not only on the mother branch (a "Markovian" s i tuat ion) but on o the r branch-
es nearby, e.g. t ha t the inhibi t ion of a branch inf luences growth in some neighborhood.
If the resul t ing regula t ion is l inear, then the map (3.67) comes close to a t r anspor t
equat ion as discussed in sec t ion 3.1.2. Any local d is turbance then would a l t e r the system
in some neighborhood. Adding diffusion t e rms to equat ion (3.68) and assuming tha t the
coef f i c ien t s aij may depend on (x,y) or I (x,y)], leads to the (kto)-models which play
some role in spiral chemica l waves (e.g. DUFFY et al., 1980; VIDAL & PACAULT, 1982).
Still more compl ica ted sys tems ar ise if non- l inear feedback mechanisms or au toca ta ly t i c
sys tems are in t roduced (cf. MEINHARD, 1984; NICOLIS & PRIGOGINE, 1977). Such a
sys tem is e.g. the H~non map (cf. THOMPSON, 1982)
Xi+l = Yi + l -ax i2 ' Yi+l = bxi (3.69)
which exhibi ts a s t range a t t r a c t o r and chaot ic behavior . THOMPSON (1982) concluded
from the random response of such a de te rmin i s t i c modeh
"Strange attractors may thus have a profound effect on our
modellin E of seemingly random behaviour, since it is now seen
that a stochastic modellin E may no longer be essential in all
cases. For simple deterministic mechanical systems, they mean
that computer results of their non-linear dynamics must be in-
spected with care (as must any result of a conventional Krylov--
Bogoliubov averaging technique), since one feature of a strange
attractor is that a sudden leap in response may occur after
a lon E period of apparent quiescence. "
Another point is also e luc ida ted by this sect ion: Somet imes it is worthwhile to
study compu te r models analy t ica l ly to de t ec t the i r in ternal s t ruc ture . We usually think
the o the r way, we fo rmula te a problem analyt ica l ly and then use the compute r to solve
it. Compute r modelling and s imulat ion thus b e c a m e decoupled from classical ana ly t ic
m a t h e m a t i c s to some ex ten t , providing us usually wi th an enormous or inf ini te number
of possible solutions. In this respect , na tu re is r a the r closely approximated, the da ta
a re not summarized, but we can es tab l i sh an inf in i te ca ta logue of forms wi thout the
d isadvantage to go to the field.
143
A major point throughout this section was self-s imilar i ty which leads to the aspect
that it is somet imes worthwhile to decompose complex s t ructures into smaller units
and to ex t rac t the 'pr imit ives ' which allow to describe and to analyze the complex global
pat tern. Fig. 3.41 i l lustrates this in the case of ammonite sutures which are close to
t ree pat terns , and which are self-similar to some exten t . By turning a global problem
into a local one the analysis of a s t ruc ture commonly is s i m p l i f i e d - - however, the
inverse procedure is not always possible as was elucidated by various examples.
The analytic approach, however, usually condenses the possible pat terns and allows
to discover unexpected s t ructures which may evolve under cer ta in conditions. Thus, the
previous discussion of shape was incomplete, as new pa t te rns may arise if the second
order branches are d i rec ted to the concave side of the leading spiral. Straight branches
e.g. then overlap and are bounded by their evolute, and this pa t te rn does not require
the convergence of the Honda tree. Such p a t t e r n s - - evolutes and caust ics - - will be
the topic of the next chapter .
Fig. 3.41: Ammoni te suture lines are comparable to t ree pat terns . Commonly they are also self-s imilar (including reflections) on various levels. The enlarged 'pr imit ives ' can repeatedly be found within the complex suture lines.
4. S T R U C T U R A L S T A B L E P A T T E R N S A N D
E L E M E N T A R Y C A T A S T R O P H E S
In the preceding discussion it became clear that most of the observed instabili t ies
are due to the fact that the pa t te rn recognition process lacks an inherent s tabi l i ty prop-
erty. As LU (1976) states:
"In any branch of science, it is always a challenge to try to classify the
objects under study. Unfortunately, it is often extremely difficult to carry
out this classification. It becomes much easier if one tries to classify only
the stable objects .... stable objects have boundaries where discontinuities
appear. We all know that mathematics used in almost all sciences so far is
based on the differential calculus, which presupposes continuity. There is
a @rest demand, therefore, for a mathematical theory to explain and predict
(if possible) the occurrence of discontinuous phenomena."
A very instruct ive example of s t ructural s tabil i ty and singularit ies we owe to CAL-
LAHAN (1974): Take a piece of fabric at one corner and put it onto a f lat surface.
What forms can occur locally on the sheet {Fig. 4.1)? There are three possibilities:
a) the sheet lies flatly and smoothly, these points are called the regular ones;
b) a fold line appears on the sheet;
c) a pleat forms at the end of a fold line.
' ~ L o c a l forms on a folded sheet (after CALLAHAN, 1974}; (a) a regular sheet lies flat; (b) a singular point on a fold line; (c) a singular
point where the fold turns into a pleat; (d) a singular point which is not s tructural ly stable.
The points (b) and (c) are cal led the singular points because of their special nature;
in part icular , they are s tructural ly stable singular points. To see this, take a point such
as (d) in Fig. 4.1. This is an additional type of singular points, but it is not a s tructural ly
145
s tab le one because any dis turbance, slight as it may be, turns this point into one of
type (a), (b} or (c). On the o ther hand, points of types (a), (b) or (c) cannot be made
to disappear by a small per turbat ion . One can dis locate a fold or pleat by a small per tur-
bat ion, but one cannot a f fec t i ts presence. The discussed s ingular i t ies yield in addit ion
a s t ruc tu ra l informat ion. Put a s t i f f shee t of paper beside the fabr ic in the same way:
It will only consis t of regular and perhaps fold points. It is dist inguished from the piece
of fabr ic by the singular points which may appear under this exper iment . It is usually
the set of s tab le s ingulari t ies which allows us to classify objects .
It was THOM (1975) who pushed forward these concepts in ma thema t i c s and thei r
applicat ions. He has shown tha t the concept of s t ruc tura l s tabi l i ty , i.e. the insensi t ivi ty
to small per turbat ions , is re la ted in one contex t to s table singulari t ies . These s table
s ingular i t ies have then been classif ied by Thorn in his "seven e l emen ta ry ca tas t rophes" .
Here, examples are given which demons t r a t e tha t these concepts are useful to under-
s tand pa t t e rns and to de t e rmine the geomet r i c s ingular i t ies tha t an unknown surface
gener ical ly impresses on a sensing wavefield (DANGELMAYR & GOTTINGER, 1982).
The examples are derived from Huygens' cons t ruc t ion of wave fronts and envelopes. A
deta i led discussion of the m a t hem a t i ca l machinery and of mainly physical appl icat ions
can be found in LU (t976}, POSTON & STEWART (1978), GOTTINGER & EIKEMEIER
(eds. 1979), STEWART (1981, 1982).
Fig. 4.1 repea t s Fig. 2.1 with the addit ion of the typical singular points which
uniquely de t e rmine the local s t ruc tu re of the s u r f a c e - - i.e. the se t of singular points
on surfaces provides a skeleton of essent ia l s t ruc tu ra l informat ion. In the previous chap-
te r s such sets of singular points were viewed as instabi l i t ies , in this chap te r the inverse
problem will be studied, the c lass i f ica t ion of "form" by the intr insic set of singular
points.
In the f irst sect ion singulari t ies are discussed, which occur as discont inui t ies in
the two-dimensional images of three-dimensional objects . The typical s ingulari t ies provide
useful in format ion in pic ture processing. The concept of skeletons then re la tes the analy-
sis of two-dimensional images to the previous discussion of op t imiza t ion problems on
one hand~ and to D'Arcy Thompson's classical Vtransformations of form ~ on the o ther .
In the second sect ion the theory of e l emen ta ry ca tas t rophes is applied to the l inear
ray model in re f l ec t ion seismology. Most of the discussion remains r e s t r i c t ed to the two--
dimensional t rack line problem, and the concepts are der ived from simple assumptions
with basic ma themat i c s . Instead of dealing with waves direct ly, the l inear ray pa t t e rns
and the i r caust ics are studied. The wave fronts then are analyzed in t e rms of a cont inu-
ous plane map. Finally, the t r a v e l t i m e record is analyzed as a map which t r ans fo rms
the th ree-d imens iona l spa t io - tempora l system into the t r a v e l t i m e record. It will turn
146
out tha t the t r ave l t ime record is locally equivalent to {oblique) sect ions through a swal-
lowtail c a t a s t rophe which is loca ted on an oblique line in the d e p t h / t i m e coordinates ,
In the third sect ion some aspects of folds and faul ts are discussed in t e rms of
'para l le l sur faces ' . This discussion takes up the ' p r e - compu te r ' analysis of large scale
deformat ions in t ec ton ics and reviews this ' ea r ly ' k ine t ic approach in t e rms of r ecen t
developments in s ingular i ty theory, The la te ra l cont inua t ion and the depth l imit of folds
will be discussed in d i f fe ren t ways and cont inues in some respec t the problems of chap-
t e r 2.
4.1 IMAGE RECOGNITION OF THREE-DIMENSIONAL OBJECTS
A common problem in pa t t e rn recogni t ion is the r econs t ruc t ion and c lass i f ica t ion
of th ree-d imens iona l ob jec ts from two-dimensional p ic tures . Typically, this problem arises
in t ransmiss ion microscopy. Fig.4.2 shows the larva of a medusa in t r an s mi t t i n g light.
Locally t r iangular pa t t e rns of increased densi ty appear, which can be r e l a t ed to a bound-
Fig. 4.2: The larva of a medusa in t r ansmi t t i ng light. A swallowtail s ingulari ty occurs in the two-dimensional image. The ident ical p a t t e rn is found in the two-dimensional image of a t r anspa ren t canal surface {modified a f t e r WUN- DERLICH, 1966).
147
ary e f fec t . An ident ical pa t t e rn is well known from two-dimensional perspec t ive views
of locally convex surfaces , especial ly from canal surfaces in cons t ruc t iona l geomet ry
(WUNDERLtCH, 1966).
Another field, where similar pa t t e rns play some role, are tec tonica l ly f l a t t ened
and deformed images. The proper recons t ruc t ion of such images yields useful in format ion
for the paleontologis t as well as for the de te rmina t ion of the local t ec ton ica l s t ress
field. The normal approach in such cases are af f ine t ransformat ions . These describe
the deformat ions suff ic ient iy if the dimension of the object , i.e. of i ts image, is not
a l tered: if one has maps R 2 ~ R 2 or R3 --~ R 3. In the case of a map R3 ~ R 2,
new pa t t e rns occur, d iscont inui t ies at the boundary lines of the image which are closely
r e l a t ed to the local surface s t ruc ture .
Fig. 4.3: Tectonical ly deformed and f l a t t ened bivalves show surface discontinui- t ies s imilar to those in Fig. 4.2: In this case, the objects are not t ransparent , and, therefore , only par ts of the surface discont inui t ies are visible {modified a f t e r ROLLIER, 1918}.
4.1.1 The Two-Dimensional Image of Three-Dimensional Objec ts
Locally hyperbolic surfaces are capable to project onto swallowtai l - l ike images
in two dimensions (Fig. 4.4). These images are closely re la ted to Thom's swallowtail
ca t a s t rophe (see e.g. POSTON & STEWART, 1978). This re la t ionship secures t ha t the
pa t t e rn is s t ruc tura l ly s table , and, therefore , it can be used to de t ec t locally concave
surface e l emen t s or holes in the two-dimensional images of three-d imensional objects .
148
• . ° °
°,°• ...:.. " . ' . . . . . . . . . . . . . . : . .
::.'..::...-.:..'.......:...:.. : , .. : ; . : : ' . . . . . . ' : : . . : , . .
• .- • o • •
. • . . . . . : . . ' . . : . , : : . • . . . . . • . . , , • . . : . . . ' . ~ .
. . . . .
.:.:';:!".':"
Fig• 4.4: Two projec t ions of a hyperbolic sur face e lement• An oblique project ion causes the occur rence of a surface d iscont inui ty in the two-dimensional image space. The th ree-d imens iona l surface e l emen t can be ident i f ied wi th the ca t a s - t rophe manifold of the swallowtai l ca tas t rophe• The discont inui ty is the two-- dimensional image of the b i furca t ion se t of the swallowtail (modified a f t e r POSTON & STEWART, 1978).
How such sur face discont inui t ies develop in the two-dimensional image space, can
be analyzed most simply by use of canal sur faces (WUNDERLICH, 1966)• A canal sur face
is the envelope of a (one-parameter ) family of spheres of cons tan t radius (GUGGEN-
HEIMER, 1977). The canal surface, there fore , can be genera ted by moving the c e n t e r
of a sphere along a ( three-dimensional) curve; the envelope sur face of the spheres then
defines the canal surface• The most s imple case of such a canal surface is the torus
where the leading curve is a circle• Under an oblique project ion i n t o the plane the
c i rcular leading curve t rans forms into an ellipse• This de format ion of the leading
curve can be descr ibed by an af f ine t r ans fo rma t ion which takes the c i rc le into an ellipse•
But the boundary lines of the torus in the two-dimensionaI pro jec t ive space behave dif-
ferent ly . The genera t ing spheres project onto R 2 as c i rc les independent of a rigid ro ta t ion
of the torus• The resul t is tha t the apparen t boundaries of the t r anspa ren t torus develop
into a curve with a se l f - in te r sec t ion and with two cuspoid edges, the earI ier not iced
' t r i ang le ' (Fig• 4•2)• Fig• 4•5 gives th ree d i f fe ren t views of the torus and of the local
sur face discont inui t ies in the p ro jec t ive plane.
The ident ical pa t t e rn appears if one keeps the leading curve cons tan t and a l t e r s
the d i a m e t e r of the canal sur face (Fig. 4.6)• In this case, classical cons t ruc t iona l geome-
t ry (WUNDERLICH, 1966) tel ls us tha t the locally discontinuous image appears when
the c i rcu la r project ions of the spheres e n t e r the evolu te of the leading curve, along
which the cen te r s of the spheres are located• A family of such surfaces with var iab le
149
I>cI
Fig. 4.5: Three perspect ive views of a (transparent) torus and of its apparent surface discontinuities. The la t ter develop at the interior hyperbolic surface points, as the projection becomes oblique.
canal d iameter maps, therefore , onto a family of involutes or of parallel curves, which
unfold inside the evolute of the leading curve into swallowtaiI-like curves {Fig. 4o6}.
The la t te r observations allow to relate the evolution of the image boundaries to
Huygens ~ principle (Fig, 4.7). If the image of the canal surface is studied in two dimen-
sions~ then the envelope of the spheres reduces to the envelope of circles with a constant
radius. A change of the d iameter of the canal surface corresponds to a change of the
radii of these circles. In two dimensions, this is identical with a continuous transport
of the boundary lines along the normals of the leading curve. The image changes
locally when the area is reached where the normals in tersec t {Fig. 4.7). This construct ion-
al principle allows to study all possible images of canal surfaces that can occur in two
, : : % : ..:., " ~ ~,:~!..'....:~"
Fig. 4.6: Canal surfaces with identical generating curve but of d i f ferent dia- meter . The boundaries of the surface map onto a family of parallel carves in the two-dimensional image space.
150
Fi~. 4.7: Construct ion of the two-dimensional images of canal surfaces by use of Huygens v principle. The rays indicate the dislocation lines for the sur- face boundary in the project ive plane.
dimensions (Fig. 4.8). The only discontinuities, which appear, are the swallowtail-l ike singu-
lar curves. The same holds for the perspect ive views of the torus; the only d i f fe rence
is that , in this case, not the d iameter of the canal surface is a l tered but the local
curvature o f the leading curve. The result is the same: As the curvature decreases ,
the surface ~moves into the evolute v of the leading curve, and the boundary lines
deform in the identical way discussed above.
The cri t ical points of a canal surface are the hyperbolic points. They behave under
the project ion onto a two-dimensional image in the discussed way. The occurrence of
the ~swallowtait' is typical for locally convex s tructures , as will be shown in sect ion
4.2. The relation to hyperbolic surface e lements allows for a simple construct ion of
\ \
Fig. 4.8: A family of boundary lines for canal surfaces which have a common parabolic generat ing curve.
151
Fig. 4.9: Cons t ruc t ion of the swallowtail s ingulari ty as a ruled surface over a folded polygon.
the swallowtail . A hyperbolic surface, which locally takes the form z = xy, can be gener-
a ted as a ruled surface. If these rulings are pro jec ted into two dimensions (Fig. 4.9),
then the envelope of the s t ra igh t lines gives again the various possible images of the
surface in two dimensions including the swallowtail discontinuity.
The concept of parallel curves, which are genera ted by a t ranspor t along the i r
normals, can be re la ted to the concept of potent ia l s and to e l emen ta ry ca tas t rophes .
tn the l a t t e r case, the original three-d imensional obiec t resembles the ca t a s t rophe mani-
fold. If the leading curve is continuous, it can be locally described as an implici t
funct ion f(x,y) = 0. In addition, one can suspect t ha t the family of paral lel curves, which
is gene ra t ed by the t ranspor t along the normals , can be given in implici t form as
c = fix,y). But this l a t t e r formulat ion can be in t e rp re t ed as a potent ia l . The di rect ional
der ivat ives , the gradient of this local potent ia l , def ine the dislocat ion field. Ca ta s t rophe
theory then gives a c lass i f ica t ion of the s tab le s ingular i t ies of such ' loca l ' potent ia ls .
One of the ca tas t rophes , which appear under the map R3 ~ R 2, is the swallowtail .
In the present con tex t it is a s table discontinuity, which allows to ident i fy local hyper-
bolic sur face e l emen t s in the two-dimensional image. A more deta i led discussion of this
s ingular i ty will follow in the next sect ions.
4.1.2 The Skeleton of P lane Figures
In P ic tu re Recogni t ion the problem arises to represen t objec ts economical ly in
the computer , and to classify objects as equivalent even if they are disturbed to some
ex ten t . Especial ly the l a t t e r problem leads to the concept of skeletons or 'media l axes '
(BLUM, 1973; BLUM & NAGEL, 1977). BOOKSTEIN (1978) gives the following defini t ion
152
of the skeleton of a plane figure:
The skeleton of a plane figure is a ce r t a in graph inside the figure, toge the r with
a funct ion on the graph. The skeleton is the locus of all points which do not
have a unique neares t boundary point upon the shape; the function is the d is tance
to any of the set of equally d is tant neares t points.
Thus, the skele ton is lust the set of s ingular points discussed in sect ion 2.4.5, which
arose from the problem to find the neares t boundary point. The most na tura l way of
finding the skeleton of a given object is by shrinking it until it reduces to i ts skeleton
(ROSENFELD & WESZKA, 1980). This shrinking c a n - - t h e o r e t i c a l l y - - be done by a
wave front , which s t a r t s ins tan taneous ly at every point along the boundary and moves
with cons tan t v e l o c i t y - - i.e. by Huygens' principle, as discussed in the previous sect ion
(for t echn ica l deta i ls of computa t ion see e.g. ROSENFELD & WESZKA, 1980). Still more
i l lus t ra t ive is the "grassf ire" model (BOOKSTEIN, 1978):
"We imagine a shape boundary to be 'drawn' on the dry grass of a prairie,
and fired (simultaneously). The fire will burn evenly in all directions from
its starting locus until it encounters points at which it arrives simultane-
ously from two directions, whereupon it quenches itself~ as grass cannot
burn twice. Such loci comprise the skeleton, and the function we seek is
the time it takes the fire to arrive there and Eo out. "
The c r i t i ca l set of singular points we encounte red in the opt imiza t ion problem
{section 2.4.5) has now a to ta l ly d i f fe ren t quali ty: Toge ther wi th a vec tor valued funct ion
(which defines the d i rec t ion to the neares t boundary points) and a d is tance measu remen t
{which def ines the locat ion of the original boundary line) the se t of s ingular points or
the Vmedial axis ' gener ical ly defines our object , Fig. 4.10 shows various examples of
plane f igures and the i r 'media l axes ' .
Fig. 4.10: Skele tons (medial axes) of various two-dimensional objects ,
153
4.1.3 Theore t ica l Morphology of Worm-Like Objects
The concept of skele tons becomes especia l ly simple in the case of cylindrical ob-
jec ts when par ts of the boundary line and the skeleton are parallel curves. Fig, 4.11
i l lus t ra tes the case of a cylinder with spherical caps t e rmina t ing its ends. We may think
about such a cylindrical object in t e rms of a wiggly worm or a chromosome if we allow
qg7 Fig. 4.11: Morphology of a wiggly worm, which preserves length, width, c i r cum- ference, and area when it bends (a,b). The cu rva tu re of coiling, however, is l imited by the width of the worm (c).
X- and Y-like pa t te rns . Now, if our wiggly object s t r e t c he s or changes its form in some
way, "our intui t ion expects tha t the width of its form will s tay pract ica l ly the same
and we expec t in addition its length to be invariant" (BOOKSTEIN, 1978). What we then
need are precise defini t ions of width and length and a map which preserves these proper-
t ies if the 'worm' bends.
Let the cylindrical object be s t ra ight , then we always can find a coordinate sys tem
so tha t the boundaries are described by the map
x = s if I S l < a y = X (4.1a)
and
x = -+(a + cos(s)) if a< Is I <a+k y = ~sin(s) (4. Ib)
In this case, the medial axis is the line y=0. If the medial line is bended to some arbi-
t ra ry form, we can describe this deformat ion by a map
154
x ---> ( F ( x ) , G ( x ) )
or (4.2)
u = f(s)
v = g(s) ,
whereby we have to secure t ha t the length of the medial line does not change (even
locally). This requires tha t the dis turbed image of the medial line is pa rame t r i c i zed
by arc length, what will be assumed for the following discussion. The next condi t ion
to be sa t i s f ied is cons tan t width. In the s t ra igh t object (equation 4.1) width is defined
perpendicular to the medial axis. If we t r ans fe r this def ini t ion to the wiggly worm,
then width is measured along the normals to the medial axis, i.e. the de format ion (4.2)
def ines the de format ion of the en t i re object which is given by the map
(x,y) ---> (F(x),G(x)) -+ X~(F(x),G(x))
or (4.3)
u = f ( s ) - X g ( s )
v = g ( s ) + x ~ ( s ) ,
where N is the normal to the medial line. The spherical caps (4.1b) are still loca ted
at the endpoints of the medial line and are not deformed {cf. Fig. 4.11).
Finally, we are i n t e r e s t ed how the c i r cumfe rence of the object and its area are
a l tered. From equat ion (4.3) we find the length inc remen t ds of the paral lel pieces of
the boundary to be
• 2 ds B = (I - Xk(s)) - "- -~/(f(s) 2 + g(s) ) ds ,
where k(s) is the curvature of the medial line and • 2 /(~(s) 2 + g(s) ) = I because the medial line is parametri-
cized by arc length ,
and the length of the boundary is
LB= f(l-Xk)ds + f(l +%k)ds + 2Lspherica I caps
(4.4)
(4.5) = 2 fds + 2Lspherical caps ; k: curvature of medial axis.
That is, the length of the boundary is simply twice the length of the medial axis, inde-
pendent of the deformation of the medial line! By definition, the length of the medial
line does not change under arbitrary deformations, and the length of the circumference
is an invariant property of the map (4.3). The area of the object is given by
155
A = ff(l-Xk)dsdl + ff(l +Xk)dsdl
= If dsdX + 2A spherical caps,
+ 2A spherical caps
and again the deformat ion does not a f fec t the area which, there fore , is another invari-
ant for any fixed value of the p a r a m e t e r )~. The map (4.2) describes ideal de forma-
t i o n s - - bending and c o i l i n g - - of worm-like objects , which preserve the length of the
medial axis, the length (surface) of the boundary, and the area (volume) of the objec t - -
proper t ies which -- by intui t ion -- can be expected for biological objects .
However, if we re turn to the discussion of the preceding section, it is c lear tha t
there are l imits for the deformat ion of such worm-like objects . As the curva ture or
the width of the object increases (Fig. 4.6), the surface would develop se l f - in tersec t ions ,
which cannot be real ized. These singulari t ies, which occur at the convex areas of the
wiggly worm (Fig. 4.11), are ' s tandard ca tas t rophes ' , which will be discussed in more
detai l in the next sect ions. For our worm-like object , however, we find a s trong corre la-
tion be tween width and the curva tu re of coiling.
4.1.4 Continuous Trans format ions of Form
D'Arcy Thompson introduced the concept of shape t r ans fo rmat ions to descr ibe the
morphological evolut ion in phylogeny and ontogeny (THOMPSON, 1952). BOOKSTEIN
(1978) summarizes:
"The formal theme of D'Arcy Thompson's method is this: to represent a change
of one shape into another by the single mathematical object which is the
map of one shape onto the other, and then to visualize this mathematical
object. "
This program has been followed with much emphasis, and a common goal was to
find the ' ideal family of t r ans format ions ' , which solves the biological problem, as the
ideal hydrodynamic problem is solved by the conformal mappings. Many published
a t t empts , of course, r e la te D 'Arcy Thompson's problems to classical physical m e t h o d s - -
conformal mappings (Fig. 4.12; RICHARDS & KAVANAGH, 1947), biorthogonal grids
(BOOKSTEIN, t978), and to the Navier -Stoke equat ion as a general solution (GRENAN-
DER, 1976).
The methods, based on D'Arcy Thompson's program, compare d i f fe rent objects
and morphological s ta tes . They describe a map from one s t a t e to the o ther but not
a continuous deformat ion as it is usually obvious in ontogeny and appears likely for
156
A
, - - - V - - ' '. ', ~ - - : - - - I
F- - " ; " ; - "-'/ t J , t L 4 - ~ _ _ , j s
. - - . J , - - - ~
.... 4--- ,>'-;
" i B
Fig. 4.12: Transformation of form in a bivalve (A) and a brachiopod (B}. Both mappings are produced by the conformaI map w=a(z + l /z) + b log z. (Adapted from BAYER, 1978).
phylogeny if one does not believe in macro-mutat ions . Further, the use of 'Car tes ian
grids' implies a continuous, a topological deformation. However, in the real systems
we find somet imes ra ther sudden changes although the deformation is continuous.
Equation (4.3) provides a 'p ro to type ' of such continuous deformations, which are
capable of sudden morphological changes as soon a s the surface enters the 'caust ic '
of normals. Of course, it is not hard to argue that real biological t ransformat ions are
much more complicated than the map (4.3). However, if we do not a t t empt to describe
the ent i re morphology at once but res t r ic t the study to interest ing local pat terns , then
Fig. 4.13: The early ontogeny of Sepia (after NAEF, 1928 and BLIND, 1976) and the c icatr ix of Nautilus (after BLIND, 1976) compared with a swallowtail surface (after THOM, 1970).
157
the map (4°3) provides a f irst order approximation for the evolution of a locally cylindri-
cal surface e l e m e n t - - what may happen, is the evolution of folds on the surface (cf.
Figs. 4.1--4.9). Fig. 4.13 i l lus t ra tes folding processes in the ear ly ontogeny of cephalopods,
which can be re la ted to the evolution of a wave f r o n t - - the s u r f a c e - - which folds
and genera tes singular lines.
The map (4.3) describes a simple case of a wave front and is equivalent to THOM's
(1970,1975} approach to morphogenesis, as soon as not the morphology i tsel f but the
singular se ts are emphasized. In the simple example of equat ions (4.3) the in teres t ing
s ingular i t ies are the cusp and swallowtail ca tas t rophes . How these e l emen ta ry ca tas t rophes
are re la ted to the map (4.3), will be discussed in the next sect ion.
4.2 SURFACE INVERSIONS IN THE SEISMIC RECORD --
THE CUSP AND SWALLOWTAIL CATASTROPHES
In a wide field of applicat ions geologists are concerned with the problem to in te r -
pre t the records of re f lec t ion seismology in a qua l i ta t ive way. The morphological {geometri-
cal} i n t e rp re t a t i on is here usually much more impor tan t than the recons t ruc t ion of t rue
depth r e l a t i o n s h i p s - - a major problem for the seismologist . To solve the in t e rp re t a t ion
problem quan t i t a t ive ly requires to solve the full dynamics of the wave equation, which
cap tures the process of r emote sensing. In the most general sense, waves are spreading
processes which sa t is fy a to ta l hyperbolic d i f fe ren t ia l equat ion (COURANT & HILBERT,
1968). The t rouble is tha t one does not only need the init ial condit ions but also the
f
Fig. 4.14: Sketches of t r ave l t ime records, which have been in te rp re ted as sal t domes with se l f - in te rsec t ing re f lec tors (above: a f t e r DRIVER & PARDO, 1974; below: a f t e r BIJU-DUVAL et al., 1974).
158
ent i re boundary conditions to solve the general equation. Instead of solving the total
hyperbolic equation one can pick a visible s t ruc ture from the wave field, say a cres t
or trough line or, in a general notation, a wave front (WRIGHT, 1979)o To solve the
spreading process of the wave front one can use Huygens ~ principle (COURANT & HIL-
BERT, 1968; OFFICER, 1974). If a wave front is known at a cer ta in t ime, one studies
the evolution of the wavelets that spread from every point of the generat ing wave front.
The successive wave fronts are the envelopes of the wavelets . In the three-dimensional
case the wavelets are spheres, and the successive wave fronts are surfaces F(x,y,z) =
constant . If the propagation of the wavelets is fairly constant along the generat ing wave
front, the successive wave fronts can be approximated by a transport of the original
surface along its no,reals with constant velocity. In two dimensions the spherical wavelets
reduce to circular ones, and the problem is moderate ly simplified. In a mathemat ica l
sense this construct ion of the propagating wave fronts can be described as a continuous
map, and, at least locally, one can suspect that this map can be summarized in a poten-
tial equation V = w(x,y,z).
The in teres t ing s t ruc tures of these potent ia ls are their s table singularities. The
geometr ica l singularit ies can welt be studied by topological methods. DANGELMAYER
& GUTTINGER (1982) discussed the physical aspects of remote s e n s i n g - - Fresnel-zone
topographies, d i f f ract ion pa t te rns e tc . - - careful ly in te rms of ca tas t rophe theory. They
showed that the topological approach yields reasonable results for the inverse sca t te r ing
problem as well as for the on-s i te survey:
"Since, in practice, the analytic analysis is of little interest to the seis-
mologist -- because he needs an overall picture, a 'Gestalt' point of view
to classify his forms-- tackling the inverse problem at its topological
roots comes much closer to the interpreter's intuitive geometric9 i.e. quali-
tative, approach" (DANGELMAYER & GUTTINGEB, 1982).
This "quali tat ive approach" is, indeed, still more important for the geologist than for
the seismologist . Interpret ing seismograms from areas with salt domes it may be a
reasonable question whether in tersect ing re f l ec to r s are real is t ic or singular pa t te rns
within the seismogram. Shadow zones, double ref lec t ions and 'hyperbolic re f lec t ions '
are well known pa t t e rns of the record. Figs. 4.14 and 4.15 give some examples how
such 'anomalies ' show up in the t rave l t ime record. Much more impressive s t ruc tures
of these types have been described from 3.5 to 12 kHz records (echograms) {JOHNSON
& DAMUTH, 1979; DAMUTH, 1980;FLOOD, 1980; EMBLEY, t980). The in terpre ta t ion
of echograms and of pa t te rns like 'hyperbolic ref lec t ions ' became of interest during
the last decade, when sedimentologists s ta r ted to in terpret the deep sea morphology
for their purposes.
In the case of high frequency echograms, the wave front evolution can be well
159
Fi, g. 4.15: Trave l t ime records with in te rsec t ing ref lect ions , ' extensions of sed imenta ry layers into the basement s ' and high energy zones within an isotrop- ic sed iment cover,
approximated by l inear rays (FLOOD, 1980). tn the case tha t the model is reduced to
the 1seismic t rack ' , the problem is fur ther simplified. The re la t ion be tween a (cylindrical)
sur face and the t r ave l t ime record along the t rack line can be viewed as a map R 2 . . . .
R 2 or ( x , y ) ~ (u,2t), where (x,y) are the spat ia l coordinates of a sect ion through the
sur face and (u,2t) are the space - t r ave l t ime coord ina tes of the record. In this notat ion,
the in t e rp re t a t ion of the seismogram becomes identical with the problem to de te rmine
the map tsL This approach was recent ly s t ressed by FLOOD (1980), who analyzed periodic
wavefields. He found tha t 'hyperbol ic re f lec t ions ' depend on the wavelength of sinusoidal
surfaces and wate r depth. But the approach by any global sur face approximation is much
160
too general : There is no real chance to es tabl ish a suff ic ient c a t a l o g u e of global morphol-
ogies. One problem, the re fore , is whe the r one can classify sur face points in such a way
t ha t the i r image on the seismic record can be uniquely ident if ied. This, of course, is
a topological problem. It is the purpose of this sec t ion to der ive a c lass i f ica t ion of
t r a v e l t i m e images for the two-dimensional (seismic track} problem. This c lass i f ica t ion
will be one which re l a t e s the t r a v e l t i m e pa t t e rn s to local topological proper t ies of the
re f lec tor . There are two ways to discuss the re la t ion be tween sur face proper t ies and
the t r a v e l t i m e image, versus the wave fronts or versus the plane map approach. Both
methods will be discussed to demons t r a t e d i f fe ren t aspec ts of ca t a s t rophe theory.
4.2.1 Compute r Simulat ions of Rays, Wave Fron ts and
Trave l t ime Records
Ray theory becomes dras t ical ly s implif ied if one assumes tha t the rays are s t ra igh t
lines, and t ha t the source and the rece iver are located in a single point. This s i tua t ion
is near ly rea l ized for deep sea echograms (FLOOD, 1980), but it can also be used as
a f irst approximat ion for o ther re f lec t ion seismograms. Under these special condit ions,
the s u r f a c e of the r e f l ec to r can be viewed as the envelope of the wavelets , it forms
a 'wave f ront ' , and the normals of the r e f l ec to r are the rays along which the r e f l ec t ed
wave f ront propagates (e.g. GRANT & WEST, 1965). If the sec t ion through the r e f l ec to r
along the t rack line is given analyt ical ly , one can immedia te ly wr i te down the (linear)
ray equat ions.
tf the re f l ec t ing sur face is given in expl ic i t form as y = f(x), then the l inear rays
a re given in pa rame t r i c i zed form as
u = Xo - Tf'(xo)
w = f(xo) + T
(4.6)
where (Xo, f(Xo)) def ines a point on the re f l ec to r , (u,w) are the spat ia l coordina tes of
the ray which passes through the point (Xo, f(Xo)), and T is a p a r a m e t e r which genera tes
the ray. It turns out t ha t the pa t te rns , which can be found, are not a proper ty of the
r ays , but t h a t they need to be formed by the morphology of the r e f l ec to r (WRIGHT,
1979). The r e f l ec to r impresses its pa t t e rn gener ica l ly onto the sensing wave system
(DANGELMAYER & GOTTINGER, t982), in this case, onto the family of rays.
Equat ion (4.6) allows to draw the l inear ray p a t t e rn for any d i f fe ren t iab le function.
This was done for a sinusoidal funct ion in Fig. 4.16, and it becomes c lear tha t the seis-
mic record will be dis turbed in the areas of ray overlap. In these areas one finds double
and t r iple ref lec t ions . The two p ic tures of Fig. 4.16 show the farf ie ld p a t t e rn and, en-
161
Fig. 4.16: Computer simulations of the linear ray system of a sinusoidal re f lec- tor. Left: farfield pat tern, right: nearfield pat tern.
larged, the nearfield pat tern (close to the surface). Fig. 4.17 i l lustrates that the ra ther
complicated farfield pat tern results from superpositions of various nearfield pat terns .
If the rays, which pass through the surface e lement of a wave front, are known,
then the evolution of the wave front along the ray family can be simulated, a f ter the
ray equations have been normalized by arc length, tn the case of linear rays one finds
the successive wave fronts from the continuous map
u = x - Tf'(x)//(l+f'(x) 2 ) (4.7)
w = f ( x ) + " r / / ( l + f ' ( x ) 2 ) .
(u,w} are the new spatial coordinates of a point {x,f(x)) on the original wave front, and
T is the distance between these two points { T = vt, v: velocity, t: time). The equations
(4.7)) define a continuous map from the plane into the plane
{(u,~) ~(x,y)]S(~):(~,v)}, (4.8)
where (Xo,Y o) are the points on the (cylindrical) ref lector . Fig. 4.17 i l lustrates how the
wave fronts evolve from sinusoidal surfaces, and how they are folded within the areas
in which the rays in tersect . As was noted earlier, the ref lec tor impresses its s t ructure
onto the family of rays and onto the wave fronts. This is i l lustrated in Fig. 4.18 where
the re f l ec to r was simulated by a cycloid. When the general ized cycloid develops a cusp,
a local singular point, then the ray and the wave front pa t te rn change dramatical ly.
Now, it is not hard to see that this is not a s t ructural ly stable situation. First one
162
. . . . . - _ ~ = - = - = - = - = - :
• ~ ~ ¢ ~ , ~ , ~ ~
Fig. 4.17: Co mpu te r s imula t ions of the evolut ion of rays and wave f ronts from sinusoidaI re f lec tors .
Fig. 4.18: Linear ray sys t em and wave f ronts of a cycloid. The degenera ted case occurs when the cycloid develops a cusp.
163
can argue that the cusp is morphologically not stable. Any small disturbance turns it
into a pa t te rn like Fig. 4.17. In addition, in this degenera ted case, the ray approach
does not fit Huygens' principle. Fig. 4.19 i l lustrates how the envelope of the wavelets
evolves near the singular surface point. It turns out that the successive wave fronts
' ignore ' the singularity of the ref lector . They are again continuous functions, which can
be approximated by a surface model like the general ized cycloid of Fig. 4.18. This obser-
vation allows to ignore such singularities for most of the following discussion.
Fig. 4.19: Huygens' wave front construction by wavelets near a singular surface point.
The last point to be discussed is how the surface maps onto the t ravel t ime record.
To find the map (Xo,Y o) -{u,vt /2) one has to section the ray pa t te rn in a cer tain
distance of the ref lector , i.e. along the survey track line. The modified horizontal coor-
dinate 'u' (Fig. 4.20) is a function of the inclination of the rays. If the track line is
taken as the zero level, y = 0, then the horizontal dislocation is
U X 0
Fig. 4.20: The parameter system for the linear ray model: (x,y,): surface coordinates of the ref lector , u: horizontal coordinate of source and receiver at y = 0, r: distance between source (receiver) and surface.
164
AX = (x - u) = -f'(x)f(x). (4.9)
The horizontal dislocations can be easily derived from equations (4.6}} and the condition
y = w = 0 if the pa ramete r T is el iminated. Now, the t ravel t ime is proportional to the
distance be tween the shot point and the re f lec tor point {Fig. 4.20) or proportional to
r = ¢ ' ( ( u - x ) 2 + f ( x ) 2 ) • (4.10)
Thus, if all variables are expressed in te rms of x and f(x), one finds the map (FLOOD,
1980) {x,f(x)) ~ {u,w) as
u = x - f'(x)f(x)
r = f(x)/(l-f'(x) 2) = vt/2.
{4.11)
The equations allow to simulate t rave l t ime record numerically for any di f ferent iable
surface t race . Fig. 4.21 gives some examples of such simulations. The figures include
the ray systems and the simulated t rave l t ime records. The surface models are all convex,
and this gives 'hyperbolic ref lec t ions ' on the t ravel t ime record. The mapping equations,
which have been discussed so far, allow to simulate the linear ray pa t te rn , the evolution
of wave fronts and the t rave l t ime record for any d i f ferent iable surface e lement . Indeed,
Fig. 4.21: Computer simulations of rays and of the t rave l t ime record for various s u r f a c e models {fourth order polynomials}. The saddle points of the t rave l t ime record have been set to zero depth.
165
.- ..... ~?.:.'.' . . . . . ".::,a . . . . . . :~c."..:';:iI'~':'.':..~.-.x.
L.'- . . . . . . t " ' "~'bx" .- . ' :'-,::,>
S, 7 • | - , , :
Fig. 4.22: Simulations of the t ravel t ime record like in Fig. 4,21 but as point plots for ' layered media' .
these approximations can be extended to three-dimensional surface s t ructures . But they
are still much too general. There is an infinite number of possible functions~ which can
be used to approximate a re f lec tor surface, and these functions may depend on a large
number of parameters so that we are not able to catalogue the t ravel t ime pat te rns
within the pa ramete r space. The major aim of the next sect ion is, therefore , to analyze
the local propert ies of ref lectors , and to show how these local propert ies impress their
s t ruc ture generically on a sensing wavefield and, therefore , on the t ravel t ime record.
The computer simulation has the advantage that more complicated systems can
be simulated. Thus, one can t ransform the geometr ica l simulation into a point plot
(Fig. 4.22) which resembles the received energies to some extent , i.e. the observed travel-
t ime record. The comparison of such a plot for a mult i layer-system (Fig. 4.22) with
t rave l t ime records {e.g. Fig. 4.15) shows that not only 'hyperbolic re f lec tors ' may arise
from local concave surface elements . High energy zones, which are somet imes found
in otherwise nearly isotropic areas, may well be re la ted to the surface morphology ra ther
than to a property of the sediment cover.
4,2.2 Local Surface Approximation
tn the preceding discussion it became clear that the ref lect ing surface impresses
its s t ruc ture onto the rays, onto the wave fronts and, therefore , finally onto the t ravel-
t ime record. It is this generic situation which makes remote sensing a s tructural ly stable
p r o c e s s - - and s t ructural s tabil i ty alone secures that one has a chance to recons t ruc t
the surface propert ies. On the other hand, it turned out that it is unreasonable to work
with global surface s t ructures . Therefore, at first one needs a classif icat ion of the cri t i-
166
cal po in t s on t h e r e f l e c t i n g s u r f a c e , This c l a s s i f i c a t i o n was done by D A N G E L M A Y E R
& G U T T I N G E R {1980, 1982) in de t a i l for t h r e e - d i m e n s i o n a l p rob l ems . Here t h e d i s cus s ion
will be r e s t r i c t e d to p l ane c u r v e s , i .e . to a f i r s t a p p r o x i m a t i o n o f cy l ind r i ca l r e f l e c t o r s
a long t h e t r a c k l ine. In th i s c a se , t he usua l s i t u a t i o n will be t h a t t he func t i on , wh ich
d e s c r i b e s t h e s u r f a c e , is o f bounded v a r i a t i o n (GUGGENHEIMER, 1977). For m o s t rea l
s u r f a c e s we c a n e v e n suppose t h a t a s u r f a c e l ine is o f sma l l v a r i a t i o n , and, t h e r e f o r e ,
t h e usua l s i t u a t i o n will be t h a t t h e c u r v e can be loca l ly a p p r o x i m a t e d by a Tay lo r s e r i e s
f(x) = a 0 + alx + a2x2 + ... + higher terms. (4.12)
Fig. 4.23: R a y s y s t e m s (x,z) and ' t r a v e i t i m e r eco rd ' (x,t) o f l inea r s u r f a c e e l e m e n t s . The s t r a i g h t l ines m a p on to s t r a i g h t l ines, bu t t he h o r i z o n t a l d i s loea - t ion c a n c a u s e on lapp ing f e a t u r e s and shadow zones .
Now, t h e r e a r e two i n t e r e s t i n g cases : If t he c o e f f i c i e n t s a 2 = 0 and a 1 ~ 0, t he r e f l e c t o r
equa l s loca l ly (near x = 0) a s t r a i g h t l ine, This s t r a i g h t r e f l e c t o r e l e m e n t m a p s on to
a s t r a i g h t l ine on t h e t r a v e l t i m e r e c o r d (Fig. 4.23), as c an be p roved by u se of e q u a t i o n
(4.11). The only s p e c t a c u l a r p a t t e r n s a r e s u m m a r i z e d in Fig. 4.23. An inc l ined s t r a i g h t
l ine is d i s l o c a t e d a long t h e h o r i z o n t a l c o o r d i n a t e . Th i s h o r i z o n t a l d i s l oca t i on can c a u s e
shadow zones and on lapp ing p a t t e r n s in t h e t r a v e l t i m e r eco rd , E x a m p l e s for th i s d i s t u r b -
a n c e o f t he r e c o r d a r e t he i n t e r s e c t i o n s b e t w e e n s e d i m e n t c o v e r and b a s e m e n t in
Fig. 4.15. On t he o t h e r hand , t h e g e o m e t r i c a l p r o p e r t y ' t o be a s t r a i g h t l ine ' is p r e s e r v e d
u n d e r t h e map , i .e . t h e r e is no s p e c t a c u l a r d e f o r m a t i o n on t he t r a v e l t i m e r eco rd .
T h i n g s b e c o m e m o r e i n t e r e s t i n g if t h e p a r a m e t e r a 2 is n o n - z e r o . In th i s c a s e , t h e
Tay lo r e x p a n s i o n ca n be s i m p l i f i e d in t he fo l lowing way: One l o c a t e s a new c o o r d i n a t e
s y s t e m at x = a ° by use o f t h e m a p x - - ~ x - a o. T h e n one r o t a t e s t h e new c o o r d i n a t e
s y s t e m in such a w a y t h a t t h e t r a n s f o r m e d x - a x i s c o i n c i d e s wi th t h e t a n g e n t a t x = 0,
and t h a t t h e y - a x i s c o i n c i d e s w i t h t h e n o n - o r i e n t e d n o r m a l a t t h i s po in t . T h e t r a n s f o r m a -
t ion r e d u c e s t h e T a y l o r s e r i e s to t he fo rm
1 f(x) = ~ kx 2 + ... + higher terms.
The p a r a m e t e r k is t h e local c u r v a t u r e o f t h e c u r v e a t t h e po in t x = 0
(4.13)
167
k = f " ( 0 ) / ( 1 - f ' ( 0 ) 2 ) 3 / 2 • (4.14)
Dependent on the sign of k the point is e i ther a maximum or a minimum in the local
coordinates. The discussed t ransformation re la tes the local s t ructure of the re f lec tor
to its curvature at the cr i t ical point, a well known procedure from different ial geometry
(GUGGENHEIMER, 1977; DO CARMO, 1976). If the Taylor series s ta r t s with higher 2
te rms than x , one has a degenerated situation, and one will find pa t te rns like in the
case of the cycloid of Fig. 4.18. Such situations will be avoided during most of the
following discussion.
In the case 0 <k< co the only spectacular points are local minima, i.e. a locally
concave ref lec tor . Under these constraints there exists an area where the rays in tersect ,
and one will receive ref lect ions from several surface points (Fig. 4.16). Therefore, the
stable spatial pa t te rns of rays will be analyzed in the next section.
4.2.3 Linear Rays, Caustics and the Cusp Catas t rophe
The interest ing s t ructures in ray geometry are caustics (e.g. BEN-MENAHEM &
SINGH, 1981) -- the envelopes of the rays or the boundary line of the area where rays
overlap (Fig. 4.16). In geometrical optics a caust ic appears as a line of high intensity
(e.g. NYE, 1979), in ref lect ion seismics it separates those areas, which are covered by
a single family of rays, from those areas with two or more intersect ing ray systems.
Within the linear ray model the caust ic is found as the locus of the radii of curvature
of the re f l ec to r line. If the ref lect ing curve is given explicitly, one has the classical
formulae (e.g. GUGGENHEIMER, 1977)
u = x - f'(x)(l+f'(x)2)/f"(x)
w = f ( x ) + ( l + f ' ( x ) 2 ) / f " ( x ) (4.15)
or by use of the curvature k = 1/R
u = x - R f ' ( x ) / ( l + f ' ( x ) 2) (4.16)
w = f ( x ) + R / ( l + f ' ( x ) 2
The second set of equations re la tes the caustics to the continuous map (4.7) for the wave
front evolution. From the relation T (t) = R(x) one finds the t ime t, at which a cer tain
point of the wave front arrives at the caustic.
Now, if the re f lec tor line is locally approximated by a parabola y = bx 2, the equa-
tions for the caust ic take the form
168
or i m p l i c i t l y
u = 4b2x 3
I w = 3bx 2 +-~
(4.17)
= 1 3 27u 2 16b(w - -2--~) .
This is a s e m i c u b i c pa r abo l a h a n g i n g ove r t he g e n e r a t i n g c o n v e x r e f l e c t o r l ine. I ts on ly
p a r a m e t e r is t h e local c u r v a t u r e o f t h e r e f l e c t o r (b = k/2). T h e r e f o r e , t h e c a u s t i c is
a s t a b l e s p a t i a l p a t t e r n wh i ch is un ique ly d e t e r m i n e d by t he local p r o p e r t y o f t h e r e f l e c -
tor .
The s e m i c u b i c pa rabo la , wh i ch a p p e a r s h e r e as t h e c a u s t i c , is wel l known as t he
c r i t i c a l s e t o f T H O M ' s (1975) cu sp c a t a s t r o p h e . Indeed, for t he r ay p a t t e r n s d i s c u s s e d
ea r l i e r t h i s c u r v e bounds t h e a r e a w h e r e t h e i n t e r s e c t i o n o f r ay s c a u s e s a b n o r m a l r e f l e c -
t ion p a t t e r n s . To s e e how c a t a s t r o p h e t h e o r y is invo lved we c h a n g e t he v i e w p o i n t s l igh t ly .
In t he s e i s m i c r e c o r d one p i cks up t h e r e f l e c t i o n s a long a l ine wh ich is l o c a t e d in a
c e r t a i n h e i g h t above t h e r e f l e c t o r . T h e h o r i z o n t a l d i s l o c a t i o n o f a r e f l e c t o r po in t on
t h e r e c o r d c a n be found f r o m t h e e q u a t i o n s o f t h e n o r m a l s (4.6) by e l i m i n a t i o n o f t h e
p a r a m e t e r • , and one f inds t h e n e w h o r i z o n t a l c o o r d i n a t e on a t r a c k l ine o f h e i g h t
w to be
u = x - f'(x)(w - f(x)). (4.18)
By s e t t i n g w = c o n s t a n t ( the t r a c k line) one ha s a r e l a t i on b e t w e e n t he s t a r t p o i n t o f
t he r ays and t h e poin t w h e r e t h e y hi t t he t r a c k l ine. Now, one can i n se r t t h e e q u a t i o n
for a local p a r a b o l i c r e f l e c t o r a p p r o x i m a t i o n , y = bx 2, and f rom (4.18) one f inds t he
po in t w h e r e t h e r ay i n t e r s e c t s t h e t r a c k line:
u = x - 2 b x ( w - b x 2 )
or (4.19)
u = (1 - 2 b w ) x - 2 6 2 x 3 .
B e c a u s e t h e c u r v a t u r e shou ld no t v a n i s h a t t h e s p e c t a c u l a r po in t , one c a n s t a n d a r d i z e
th i s e q u a t i o n :
u _ (i - 2bW)x - x 3
2b 2b 2
or (4.20) 3
U = x - sx
wi th obvious p a r a m e t e r i d e n t i f i c a t i o n s .
169
The new p a r a m e t e r ' s ' is a c o m p o s i t e s t r u c t u r e of t he local c u r v a t u r e o f t he r e f l e c -
tor (b) and o f t he h e i g h t of t he t r a c k line (w) above t h e s p e c t a c u l a r point ; thus , a va r i a -
t ion of Vs' c o r r e s p o n d s e i t h e r to a c h a n g e o f t he h e i g h t o f t he t r a c k line o r / a n d to
a c h a n g e o f t h e local c u r v a t u r e . The r e s u l t i n g cub ic e q u a t i o n (4.20) c an be s imp ly
ana lyzed . It a l lows for t h r e e poss ib le s i t u a t i o n s . If s > 0, t hen t h e cub ic is m o n o t o n o u s l y
i n c r e a s i n g (or dec r e a s i ng ) , and t he m a p x ~ u is one to one , i.e. t h e r e e x i s t s only one
f a m i l y of n o n - i n t e r s e c t i n g rays . If s <0 , t h e n t h e cub ic has a m a x i m u m and a m i n i m u m ,
and t h e r e e x i s t s an a r e a w h e r e t he m a p x ~ u is no t un ique ly d e t e r m i n e d , i .e. t h e r ay s
i n t e r s e c t . The c a s e s = 0 d e f i n e s t he t r a n s i t i o n s t a t e b e t w e e n t h e s e two poss ib i l i t i e s ,
it d e f i n e s t h e loci w h e r e t h e c a u s t i c i n t e r s e c t s t he t r a c k line.
Now, t he c r i t i c a l a r e a in t he (x , s}-space is g iven by t he e x t r e m a of t he cub ic
e q u a t i o n . T h e s e e x t r e m a de f i ne t he loci on a t r a c k line, w = c o n s t a n t , w h e r e t h e c a u s t i c
i n t e r s e c t s t h e t r a c k line. T he e x t r e m a a re g iven by
U = 0 = s + 3 x 2 X
or (4.21)
s = -3x 2 •
F r o m e q u a t i o n (4.20) and (4.21) one f inds t h e c r i t i c a l l ine in t he (u , s ) - space by e l i m i n a t i o n
o f t he v a r i a b l e x:
U2/4 = -s3/27 (4.22)
i.e. up to a proper parameter setting the same equation as before. If one relates a
change of the parameter 's' to a change of track height (b = constant), then the semi-
cubic parabola describes just the earlier discussed caustic. On the other hand, we have
a l r e a d y s e e n t h a t t h e m e a n i n g now is m u c h m o r e g e n e r a l b e c a u s e t h e p a r a m e t e r ' s '
i nc ludes a lso c h a n g e s o f t he local c u r v a t u r e of t he r e f l e c t o r l ine. Thus , e q u a t i o n (4.20)
c a p t u r e s t he poss ib le spa t i a l p a t t e r n s in the i r m o s t g e n e r a l s e n s e by a m i n i m a l s e t o f
p a r a m e t e r s . One can use e q u a t i o n (4.20) to d raw a p i c t u r e o f t he c r i t i c a l s u r f a c e in
t h e t h r e e - d i m e n s i o n a l s p a c e {u,x,s). This fo lded s u r f a c e is shown in Fig. 4.24. Eve ry
s e c t i o n s = c o n s t a n t t h r o u g h th i s fo lded s u r f a c e d e s c r i b e s t he d i s loca t ion of t h e h o r i z o n t a l
r e f l e c t o r c o o r d i n a t e a long a poss ib le t r a c k l ine (for a f ixed local c u r v a t u r e ) .
To a r r ive a t t he f inal c a t a s t r o p h e r e p r e s e n t a t i o n , t he v i ewpo in t ha s to be c h a n g e d
once more . The cub ic e q u a t i o n (4.20) c an be d i s c u s s e d in t e r m s of i t s d i s c r i m i n a n t or
in t e r m s of t h e n u m b e r o f i t s roots . This g ives the add i t iona l i n f o r m a t i o n how m a n y
r ays m a y i n t e r s e c t a t a poin t in t he spa t i a l c o o r d i n a t e s . The d i s c r i m i n a n t t a k e s t he
fo rm
D = u2/4 - s3/27. {4.23)
170
K
Fig, 4.24: The ca t a s t rophe set of the cusp ca tas t rophe ,
For D = 0 one has the points which sepa ra te the p a r a m e t e r se t t ings leading to a single
root (D > 0) from those tha t cause t r iple roots (D < 0). Now, the quest ion how many
roots a cubic equat ion has is ident ical with the quest ion how many ex t r ema a quar t ic
equat ion may have. The cubic can, the re fore , be embedded into the ca t a s t rophe potent ia l
V = x4/4 + ux2/2 + sx, (4.24)
which has been published as the cusp ca t a s t rophe (Rieman-Hugoniot ca t a s t rophe in THOM,
1975). This c a t a s t ro phe po ten t ia l cap tu res the discussed two-dimensional ray pa t t e rn s
in the i r most general topological behavior ,
The previous discussion of the cusp ca t a s t rophe allows to classify the t r a v e l t i m e
record in t e rms of depth and local cu rva tu re of the r e f l ec to r line. To do this explici t ly
one can wr i te the local parabol ic approximat ion as y = - a + bx 2, where the p a r a m e t e r
' a ~ indica tes depth, The t rack line is then located at depth zero. From the equat ions
(4.11) one finds the local t r a v e l t i m e image:
u = (l-2ab)x + 2b2x 3
r = (-a+bx2)(l-4b2x 2) 1/2
(4.25)
As turned out from the analysis of the cusp ca tas t rophe , the c r i t i ca l set is given by
s = (l-2ab)/(2b 2) = O. (4.26)
The p a r a m e t e r iden t i f ica t ion a = - w re la tes this r ep resen ta t ion to equat ion (4.20). Now
the cr i t ica l se t can be r ewr i t t en in t e rms of the pa rame te r s (a,b) as
171
/k
O - - K ~
i
O _ _
b
A w - W - ~?
e-% 7- " T Y
m •
Fig. 4.25: The morphology of local elements on the ref lec tor line (upper graph) and their image on the t ravel t ime record. The surface elements and their images are located in the parameter system depth (a) and local curvature of the ref lec tor line (b). Inside the hyperbolic boundary the ref lec tor image is inverted, i.e. it is the domain of hyperbolic reflections.
172
1 - 2ab = 0 . {4.27)
Equation {4.27) descr ibes a hyperbolic boundary line in the (a,b)-space, and equation (4.25)
allows to compute the image of various parabolas dependent on the choice of the parame-
ters 'a ' and tbL Fig. 4.25 i l lustrates the relationship between the local morphology of
the re f l ec to r and its image on the t rave l t ime record within the pa ramete r space (a,b).
From the discussion of the cusp ca tas t rophe we know that the pa ramete r 's ' a f f ec t s
the in tersec t ion of the caust ic with the track line. In analogy to the pa ramete r 's ' one
can vary equation {4.27):
c - 2ab = 0 . (4.28)
This defines a family of hyperbolae in the (a,b)-space of identical t ravet t ime image
(different depth location), which result from di f ferent conditions (Fig. 4.25).
Thus, the previous discussion provides us with some pract ical results, at least for
the in terpre ta t ion of echograms. The analysis of the t rave l t ime record, in te rms of local
propert ies of the re f lec tor line and of caustics, allows to classify the t rave l t ime images
by a minimal set of parameters , and it becomes clear that these p a r a m e t e r s - - depth
and local c u r v a t u r e - - are not independent with respect to the e f f e c t s they produce
on the t rave l t ime record. In addition, it becomes ctear that the ex t rema of a ref lec t ing
surface are s tabte points. In this case, one can approximate the re f l ec to r line locally
by a parabola without any rotat ion of the local coordinate system, and the point (0,f(x))
is recorded at its cor rec t ly horizontal position as well as with the cor rec t t rave t t ime.
This allows to e s t ima te the wavelength of sand waves and similar s t ructures along the
track line from the original t rave l t ime record. Fur thermore, the amplitude can he esti-
mated as well in te rms of t ravel t ime, and the relation to the cusp ca tas t rophe allows
to draw charts , from which the local curvature can be es t imated .
4.2.4 Wave Fronts and the Swallowtail Ca tas t rophe
The next point of in teres t is how the wave fronts evolve near a cuspoid caust ic .
Within the linear ray model, a wave front is given as a set of points {on the family
of rays} which have equal dis tance from the r e f l ec to r
{(u,w) E (x,y) 1 ((x-u) 2 + (Y-W)2) I/2 = r = const.}. (4.29)
This two-dimensional relationship can be extended to the three-dimensional case
(DANGELMAYR & GOTTINGER, 1982). Here it is more appropriate to re turn to the
continuous map for the wave fronts, as it was derived in equation (4.7). From these
173
equations one finds a wave front by se t t ing • = cons tant . The only cr i t ica l set for the
rays is the cuspoid caust ic . Therefore , as WRIGHT (1979) points out, we should "expect
the c r i t i ca l value graph of the cusp ca tas t rophe , which is the b i furca t ion se t of the
swallowtail". Indeed, if one draws the wave fronts for several values of the p a r a m e t e r
T to s imula te thei r evolut ion from a locally parabolic ref lec tor , then they take the
form of sect ions through the swallowtail ca t a s t rophe (Fig. 4.26) as far as they are located
inside the caust ic . To see in detai l how the swallowtail is r e la ted to wave fronts one
/~il'l//llll/l/ltlllllllltlllltllfllll'llllllllllllll~ IIIIIIIIIIItltlIItlfltt'tIII'tlIIItlIIHIIlItt~ / /ltfft/////llltlltflliltttt
174
s = (l-2bT)/(b3T).
On the o ther hand, the ca tas t rophe potent ial of the swallowtail is defined as
V = x 5 / 5 + a x 3 / 3 + b x 2 / 2 + c x . {4.32)
Its cr i t ical value graph in the pa ramete r space (a,b,c) is defined by its der ivat ives
(THOM, t975}
V xxx
V = x 4 + ax 2 + bx + c = 0 x
V = 4x 3 + 2ax + b = 0 xx
= 12x 2 + 2a = 0.
(4.33)
If one uses the first two derivat ives to solve for the parameters b and c in te rms of
a and x, one finds the map
b = -4x 3 - 2ax
c = 3x 4 + ax.
(4.34)
By a proper choice of the signs (take x - - -x) this map becomes equivalent to the
local Taylor expansion {4.31) if one takes the foIlowing pa ramete r ident if icat ions
IJ - c, W ~- b, a - s.
At least locally (by an approximation up to order 4), one finds that the wave fronts
are equivalent to sections a = s = constant through the catastrophe set of the swallow-
tail. Again, one finds that a standard catastrophe on THOM's (1975) list gives a good
approximation to the ray model. In this case, the swallowtail catastrophe describes rather
pretty the evolution of wave fronts.
4.2.5 Wave Front Evolution and the Travel t ime Record
An examinat ion of the original Taylor approximations for the wave fronts (equations
(4.30)) shows that the sect ions s = const , through the swallowtail are located on a line
w = T. The project ions onto the {u,w)-plane { {u,w)~(x,y) ) of these sect ions through
the modified swallowtail (4.30) give the typical evolution pa t te rn for the wave fronts
{Fig. 4.26}. The sect ions through the swallowtail are si t t ing one behind the other in
the caust ic . Now, the pa ramete r T can be wr i t ten as ~r = vt (v: velocity, t: time}, and
we find that the way, in which the swallowtail is si t t ing above the cuspoid caustic,
depends on the sonic velocity of the medium or on the velocity of the wave front dislo-
cation. On the other hand, the velocity cannot a f fec t the spatial pa t te rn -- the caust ic --
as turned out during the discussion of the cusp catas t rophe. The caust ic is a s tructural ly
175
s t a b l e spa t i a l p a t t e r n , wh ich only depends on t h e local s u r f a c e s t r u c t u r e , i .e. t h e local
c u r v a t u r e .
Now, t h e ana lys i s of w a v e f ron t s adds a th i rd d imens ion , t i m e , to t h e t w o - d i m e n -
s iona l s p a t i a l c o o r d i n a t e s . T he m a p f rom t he r e f l e c t o r to t he t r a v e l t i m e r eco rd , t h e r e -
)f x
Fig. 4,27: Two v iews of t he m o d i f i e d swa l lowta i I c a t a s t r o p h e . The swa l lowta i l s i t s on a l ine y = vt , T he s e c t i o n s y = c o n s t a n t t h r o u g h th i s c a t a s t r o p h e s e t a r e t he r e c o r d e d r e f l e c t i o n s n e a r a c o n c a v e s u r f a c e e l e m e n t .
176
fore, turns out to be a map R 3 ~ R 2 or ( x , y , t ) ~ (u,t). From the caus t ic we know
tha t it is a s table pa t t e rn in the (x,y)-plane. In addition, we know tha t the images of
the sect ions through the swallowtail need to have s tab le posit ions inside the caus t ics
(Fig. 4.27}. A change of the sonic velocity, the re fore , cannot a l t e r these spat ia l pa t te rns ,
i t can only a f f ec t the recorded t r ave l t ime , i.e. the spreading veloci ty of the wave front.
For the t r a v e l t i m e record this means tha t the t ime-axis is s t r e t ched or compressed.
In the space - t ime coordina tes the sonic veloci ty can only a f f ec t the t i m e - a x i s . The
only allowed t r ans fo rma t ion of the ca t a s t rophe se t {Fig. 4.27) by a change of the ve loc i ty
is, the re fore , pure shear in the (y, t)-plane with equat ion t = ay. This t r ans fo rma t ion
does not a l t e r the spat ia l coordina tes of the sect ions through the modified swallowtail
(4.30)), i.e. the i r projec t ions onto the spat ia l (x,y)-plane.
How does the local surface pa t t e r n map onto the t r ave l t ime record? To study
this, one has to sect ion the modified swallowtail (4.30) by a plane w = y = cons tan t
(in the equat ions (4.30) 'w' means depth). This gives the image of the local surface
in the space - t ime coordinates , in the (u,vt /2)-plane. Fig. 4.27 gives two views of this
modified swallowtail , which have been sect ioned by a plane w = cons tan t . Again the
ca t a s t rophe approach summar izes the pa t te rns , which can ar ise from a local concave
r e f l ec to r a rea in a very condensed way. By comparison of the observed record with
plane sec t ions through the th ree-d imens iona l ca t a s t rophe set , one can get reasonable
qua l i t a t ive in format ion about the local surface s t ruc ture . Especially the 'hyperbol ic r e f l ec -
t ions ' turn out to represen t local surface inversions, which are re la ted to the wave
fronts , which have en t e red the local caus t ic .
4.2.6 The Traveltime Record as a Plane Map
A second approach to analyze the re la t ion be tween the local r e f l ec to r geomet ry
and the t r a v e l t i m e record is versus the plane map (x,y} -~ (u ,v t /2 ) , which has been defined
by equat ions (4.11}. This method is very close to FLOOD's (1980) study of 'hyperbol ic
r e f l ec t ions ' in deep sea echograms. Again the ca t a s t rophe approach versus local proper t ies
of the r e f l ec to r will provide general results .
Firs t , one has to specify the mapping equat ions (4.11). To int roduce depth explici t ly,
the r e f l e c t o r line is locally approximated by a parabola f(x) = a + bx 2 like in equat ions
{4.25). The Taylor expansion of ' r ' (equation (4.25)) up to order 4 gives the local map
u = (l+2ab)x + 2b2x 3
r = a + b(l+2ab)x 2 + 2b3(l-ab) x4. (4.35)
177
Although this map is very similar to the evolution equation of the wave fronts
(4,30), it is not possible to t ransform it into the standard form of the swallowtail (4,34)
by means of simple transformations, which preserve the topological s t ructure , Indeed,
as follows from the previous discussion, we should expect arbi trary sect ions through
the swallowtail ra ther than its standard form,
Now, instead of r we can use r 2 = v2t2/4 as the dis tance measurement between 2 the source and the ref lect ion point. The square r ms a monotonous function of r because
r > 0 (Fig. 4.20). This t ransformat ion is not unusual to a seismologist (e.g. KERTZ, 1969},
and it allows to formulate the dis tance r as
2 (f(x)2 )2 r = + (n-x
2 x 2 2 = a + (l-2ab) + u - 2ux + b2x 4.
(4.36)
This equation can be rewri t ten as a ' ca tas t rophe potent ia l ' if b ¢ 0,
V -- (r 2 - a2)/b 2
4 (l-2ab)x2 U 2 = x + - 2Ux + b 2
or
V = x 4 + 2vx 2 - 2Ux + U 2
with obvious parameter identifications.
(4.37)
The first derivat ive of this ' ca tas t rophe potential ' defines U:
V x = 0 = 4x 3 + 4vx - 2U,
i.e. the original cusp catastrophe (eq. 4.20).
(4.38)
This ca tas t rophe potent ial does not appear in Thorn's list of e lementary ca tas t ro-
phes, but he discusses it as a selfreproducing singularity or as the stopping potent ial
of the cusp ca tas t rophe (THOM, 1975), In terms of ca tas t rophe theory this potent ial
is the universal unfolding of the cusp catas t rophe, and we can embed it into a local
potent ial
V 1 = x5/5 + v x 3 / 3 + u x 2 / 2 + u2x (4.39)
by a proper choice of the parameters . This is a swallowtail with a degenera ted paramete r
space. The parameters ' c ' and 'b' from equations (4.33) are now rela ted by b = c 2. The
cr i t ical set appears in the (V,U,v)-space (Fig. 4.28), and the t rave l t ime record is re la ted
to sect ions v = constant through the cri t ical set . Thereby one has to keep in mind that 2
V means r , not r. The sect ions v = constant have locally a swallowtail-l ike appearance,
but, in addition, they have two maxima where the curves bend down again (in the repre-
sentat ion of Fig. 4.28).
i V
v
U
178
Fig. 4.28: The stopping potent ial of the cusp ca tas t rophe (a} in the pa ramete r space (V,U,v). The positive V-axis is drawn downward for the convenience in comparing it with the standard swallowtail (b) and the hyperbolic ref lec t ion of the t rave l t ime record.
The appearance of the two additional maxima above the point of se l f in tersec t ion
in Fig. 4.28 needs an explanation because we cannot expect this pa t tern from the simple
parabolic approximation of the ref lec tor . Similar pa t te rns can be found in the simulated
record of Fig. 4.21, but it will turn out that these pa t te rns are of a very d i f ferent
type because they are really re la ted to the surface s t ructure . What happens with the
stopping potential , i l lustrates Fig. 4.29. There, the parabolic re f lec tor line extends over
the track line (S). Now, one can const ruct the image of this abs t rac t surface on the
t rave l t ime record in a very simple way. One has just to project the length of the rays,
which connect the rece iver with the ref lec t ion points s t ra ight downward from the point
where they in tersec t the track line. This gives the curve (r), i.e. the image on the t ravel -
t ime record. This construct ion can also be done for those parts of the re f l ec to r line
which extend above the track line. Because t rave l t ime is measured without a directional
component , i.e. it can only assume positive values, the curve (r) bends down again, as
one moves away from the in tersect ion point of (S} and (r). Therefore, one has to choose
s s A
Fig. 4.29: The abs t rac t si tuation that the t rack line (S) in te rsec ts the re f lec tor line. In this case, the t rave l t ime record (r) reaches the track line at the in tersect ion point and bends then down again because r can assume only positive, values.
179
carefully the correc t interval if the stopping potential is used as a model for the t ravel-
t ime record. The cor rec t interval is, in any case, located between the two maxima of
the sect ions v = constant of Fig. 4.28.
If one analyzes the cr i t ical surface of Fig. 4.28 with the noted res t r ic t ions in
mind, then it turns out that the typical 'hyperbolic ref lec t ions ' with a swallowtail-like
appearance are res t r ic ted to a limited range of the parameter v. If v is positive, one
has a convex ref lector , which in a topological sense is recorded correct ly . As v assumes
suff icient ly large negative values, the 'hyperbolic ref lect ions ' turn smoothly into a more
parabolic appearance, which, like the 'hyperbolic ref lect ions ' , is an inversion of the local
re f lec tor topology -- a concave surface e lement turns into a convex image. Those 'para-
bolic ref lec t ions ' are also well known from echograms (FLOOD, 1980), but, more common-
ly, they are found within basement ref lect ions (Figs. 4.14, 4.15).
As was shown in the last section, the approach versus wave fronts provides another
f rame to summarize the images on the t ravel t ime record. The advantage of the plane
map approach is that the 'stopping potential ' represents the images in a still more con-
densed way.
4.2.7 Singularities on the Ref lector Line
So far, a very simple re f lec tor model was used. In the case of faults, folds and
flexures the situation may become more complicated although a local parabolic approxi-
mation with rotat ion of the coordinate axes may be still possible. The most simple case,
where one can find such a cri t ical situation, are flexures and folds. A first impression
ii f );
Fig. 4.30: First order approximation of ray systems and wave fronts near flexures.
180
of what may happen near a fault gives the simple linear model of sect ion 4.2.2
(Fig. 4.23). What is actually new in this linear approximation, is the appearance of a 3
shadow zone. A flexure can be simulated by a cubic equation x = y + ay which also
includes simple folds. Fig. 4.30 gives a rough approximation of rays and wave fronts
which arise from the cubic re f l ec to r line model with a > 0, a = 0 and a < 0. For a < 0,
the caust ic pa t t e rns can be approximated by a parabolic approximation at the ex t rema
of the cubic equation, but only parts of the wave fronts sca t t e r back to the t rack line,
i.e. only one branch of the caust ic in te rsec t s with the t rack line. Fig. 4.31 trys to
capture the behavior of the caust ic over a family of cubic re f lec tor lines. For compari-
Fig. 4.31: The caust ics of a family of cubic re f lec tor lines. Left: The family of caust ics of only one ex t remum {the lower one). Right: Separation of the caust ics into their relevant parts, i.e. the branches which reach the track line.
son, the family of caust ics for only one ext remum is also shown. These graphical methods
only give a very rough idea of what happens near such s t ructures , but it is not the
scope here to analyze these problems in detail . In this context it becomes at least neces-
sary to study di f f ract ion pat terns . For this approach see DANGELMAYR & GUTTtNGER
(1982).
Similar problems arise if the re f lec tor has singular points like
Fig. 4.18 in sect ion 4.2.1. The cycloid can be described by the map
x = t - sin(t)
y = I - cos(t).
By taking a Taylor expansion near the cusp point, one finds
the cycloid of
(4.40)
181
x = t 3
Y -- t 2 (4.41)
w h e r e t h e c o n s t a n t s h a v e been abso rbed in x and y for c o n v e n i e n c e . If t he p a r a m e t e r 3 2
t is e l i m i n a t e d , one f inds t h a t t h e cusp po in t e q u a l s t h e s e m i c u b i c pa r abo l a y = x .
The ma in point , h o w e v e r , is no t t h a t we h a v e a cusp , bu t t h a t t h e s i n g u l a r i t y is an
i so l a t ed po in t of t he r e f l e c t o r l ine a t wh ich dx / d t = 0 and d y / d t = 0. The c a u s t i c n e a r
t h e c r i t i c a l po in t is g iven as t he loci of t he radi i of c u r v a t u r e on t he n o r m a l s o f t he
s e m i e u b i c pa rabo la :
x -- 4t 3 + o4--t (4.42) C .3
Yc --~ ~ t4 - t 2 .
Thus, not even too close to the isolated singular point (t = 0) the caustic behaves like
the map
u = t 2
(4.43) V = t ,
i .e. i t is a fold c a t a s t r o p h e (Fig. 4.32; LU, 1976). The t e r m 'no t too c l o s e ' m e a n s t h a t
t 3 is m u c h s m a l l e r t han t and t h a t t 4 is m u c h s m a l l e r t han t 2. A t a fold c a u s t i c t he
r ays a r e only l o c a t e d on one s ide o f the c a u s t i c s and cause , t h e r e f o r e , local ly a shadow
zone. A d e t a i l e d ana lys i s o f such s ingu la r po in t s on t h e r e f l e c t o r l ine would r e q u i r e
a t opo log i ca l c l a s s i f i c a t i o n , and i t would be n e c e s s a r y to s t u d y t h e w a v e f i e l d r a t h e r
t h a n t h e r ay s y s t e m .
m m
Fig . 4.32: The fold c a t a s t r o p h e (caus t ic ) n e a r a s ingu la r poin t on t he r e f l e c t o r c a u s e s a s h a d o w zone .
182
Table 4-1: Summary of the Ray Model
The t r ave l , l ine record in i ts most c r i t i ca l case corresponds to sect ions y = cons tan t (y:depth) through a swallowtai l c a t a s t r ophe which is located on a l ine y = vt in the th ree -d imens iona l space {x,y,t). The various types of specia l ized deformat ions depend on t he locaI cu rva tu re of the r e f l ec to r line, on the d i s tance be tween the t r ack line and the c r i t i ca l point on the r e f l ec to r line, and on the sonic ve loc i ty of the medium. In the p a r a m e t e r space depth of the c r i t i ca l point (a) and local cu rva tu re (b), the c r i t i - cal boundary line for an image inversion, i.e. for the occur rence of ~hyperbolic re f lec - t ions ' , is given by the hyperbola 1 - 2ab = 0. This hyperbol ic equat ion simply compares the local cu rva tu re of the r e f l ec to r with a c i r cu la r wave f ront a t depth ' a ' . In detai l , one finds t h a t these p a r a m e t e r s a f f e c t the t r a v e l t i m e record in the following way:
I) Spatial pa t te rns , the cusp ca t a s t rophe
1) The local cu rva tu re of the r e f l ec to r line:
Only convex a reas of the r e f l ec to r line are spec tacu la r (cause t rouble within the record) because a cuspoid caus t ic evolves. Two special s i tuat ions occur:
a) The local approximat ion of the r e f l ec to r line requires a ro ta t ion of the local coordi- na te sys tem wi th respec t to the global one. The sect ions through the ca t a s t rophe se t becomes oblique. This p a t t e r n can be de t ec t ed on the t r a v e l t i m e record because the 'hyperbol ic r e f l ec t ions ' are asymmetr ic .
b) Di f fe ren t local cu rva tu res (b = k/2) or the r e f l ec to r cause a dis locat ion and s t r e t ch ing (compression) of the caus t i c in the spat ia l coordinates . This de format ion can only be dist inguished from (2) if the t rue dep th posit ion of the spec tacu la r point on the r e f l ec to r l ine is known.
2) The hei_~.h_t_of ,_he t r ack l ine above the r e f l ec to r line:
Because t he re f l ec t ion p a t t e r n depends on the re la t ion be tween the cu rva tu re of the inc ident wave f ront and the cu rva tu re a t the spec tacu la r point on the r e f l ec to r line, this case canno t be dis t inguished from a change of the local cu rva tu re of the r e f l ec to r wi thout addi t ional in format ion (e.g. a m eas u r emen t of t rue depth). This p a r a m e t e r chooses a special line through the ca t a s t r ophe se t of the cusp which is s tably located in the space coordinates . Because the cusp ca t a s t rophe is the b i fu rca t ion se t for the swallowtai l and the discussed s topping potent ia l , th is p a r a m e t e r also appears in the o the r ca ta s t rophes .
3) Ex t r ema of curva ture :
In case the r e f l ec to r has a local minimum of curva ture , it can be approximated by a parabola , and the discussion of sect ions 4.2.1-7 holds: Typical pa t t e rn s inside the caus t ic are 'hyperbol ic re f lec t ions ' . However~ if the r e f l ec to r has a local maximum of curva ture , the caus t ic p a t t e r n is inversed, as discussed in sec t ion 4.2.8. Anyway, the previous discussion remains valid if the propagat ion of wave f ronts is inversed. A f t e r ~ the wave f ronts have passed through the caus t ic , a parabol ic re f lec t ion pa t t e rn r e s u l t s which allows to dist inguish this case from the ' s t anda rd s i tua t ion ' .
I!) S p a c e - t i m e pa t t e rns : the swallowtai l c a t a s t rophe
The sonic ve loc i ty of the medium does only a f f e c t the t r ave l t ime . This p a r a m e t e r can, the re fo re , cause only those t r ans fo rma t ions which le t t he space p a t t e r n i n v a r i a n t - - pure shear in the ( y , t ) - - plane. The ca t a s t r ophe set , which descr ibes the evolut ion of the wave fronts , is a modif ied swallowtai l which is loca ted on a l ine y = vt . The t r ave l , l ine images a re plane sec t ions through this ca t a s t rophe set . Al te rna t ive ly , the t r a v e l t i m e image can be descr ibed by the unfolding of the cusp ca tas t rophe , i.e. by i ts s topping potent ia l . The l a t t e r approach gives a descr ipt ion in the coordina tes
(x,y,v2t2).
183
Table 4-2: Summary of s t ra teg ies in the analysis of t r ave l t ime records
"wave f ront approach" "plane mapping method"
Cons t ruc t ion of the ray syste m (normals of the local r e f l ec to r e lement )
The caus t ic or the envelope of the rays cen te r s of curvature)
Evolution of the wave f ronts along the rays, the swallowtail ca t a s t rophe
t r a v e l t i m e sec t ions through the ca t a s t rophe set of the wave f ron t s - - t he modif ied swallowtail
The ca tas t rophe map along the t rack line, the cusp ca t a s t rophe
Unfolding of the cusp ca tas t rophe , the ' s topping po ten t i a l '
i¢ The local image of the t r a v e l t i m e record
4.2.8 Genera l ized Re f l ec to r P a t t e r n s in Two and Three Dimensions
In case the r e f l ec to r can be described by an explici t funct ion y=f(x), the previous
discussion provides a f ini te c lass i f ica t ion of r e f l ec to r pa t t e rns as long as a l inear ray
modeI is suf f ic ient and ca t a s t rophe theory provides a f rame for this c lass i f icat ion, as
summar ized in tables 4-1 and 4-2. However, the appl icat ion of ca t a s t rophe theory requires
local coordinate changes, which somet imes may be assumed inadequate for the problem.
tn the previous discussion it turned out tha t the t r ave l t ime record depends on a p a r a m e t e r
s= 1-2ab which appears in alI equat ions -- for the caust ic , the wave fronts and the t r ave l -
t ime record. The p a r a m e t e r 'a ' is equivalent to the depth of the re f lec tor , and '2b=k'
is i ts local curva tu re (cf. equat ion 4.13). The p a r a m e t e r 's ' , the re fore , provides a simple
in te rp re ta t ion , i t measures the re la t ion be tween an incident wave f ront wi th radius ' a '
(depth} and the radius of cu rva tu re of the re f lec tor . Image inversion occurs for a > l/(2b),
i.e. if the radius of the incident 'wave f ront ' is larger than the radius of curvature , mul-
t iple re f lec t ions arise locally because the curva tu re of the r e f l ec to r increases, as one
depar ts from the c r i t i ca l minimum. Fig. 4.33 i I lus t ra tes this viewpoint .
A) The Deformed Circle and the Dual Cusps
A na tura l quest ion is what happens if the r e f l ec to r has a d i f fe ren t s t ruc ture , i.e.
184
~ J
t I •
~ t
Fig. 4.33: The contac t be tween the incident wave front and the circle of curva- ture de te rmines the possible number of received ref lect ions: In the case of a parabolic ref lec tor , multiple ref lect ions result only if the curvature of the incident wave front is larger than the local curvature of the ref lec tor , i.e. if the shotpoint is located inside the ' caus t ic ' of normals. The usual si tuation is a fold point on the caust ic (b); a cusp point appears only at a local ex t remum of curvature .
if the curvature decreases , as one depar ts from the minimum. This causes a d i f fe rent
type of con tac t be tween the circle of curvature and the ref lector : The ref lec tor is total ly
bound to the convex side of the circle of curvature, a si tuation which cannot arise in
the case of a locally 'parabolic re f lec to r ' . An appropriate way to study both si tuations
simultaneously is to consider a per fec t circular arc, and to t ransform it by a simple
aff ine t ransformat ion
{ ,)= r[10 co ,1
which takes the circle into an ellipse. Fig. 4.34 i l lustrates how the re f lec tor e lement ,
its contac t with the circle of curvature and the caust ic are a l tered by a smooth change
of the pa ramete r 'e ' :
In the case 0 < e < 1 the ellipse has a local minimum of curvature. The circle of
curvature is bounded to the concave side of the ref lec tor , which, therefore , can
be approximated by a parabola, and the previous discussion can be applied.
For e=0, the r e f l ec to r is a pe r fec t circular arc. All rays pass through a single
p o i n t - - a singularity with indefinite codimensions. This si tuation is s tructural ly
unstable, as any small disturbance t ransforms the singular point into a caustic.
If e >1, the circle of curvature is located on the convex side of the ref lector ,
and a new pa t te rn arises. However, the caust ic is again cuspoid, similar to the
caust ic of a cycloid (Fig. 4.18); but the cusp points into the opposite direct ion
than in the case of a parabolic ref lec tor .
I85
Fig. 4.34: R a y s and w a v e f r o n t s f r om an e l l i p t i c r e f l e c t o r , a: 0 < e <t , b: e=0, c: e > 1. See t e x t for d i scuss ion .
The t ype o f c a u s t i c t hus d e p e n d s on t he t ype o f c o n t a c t b e t w e e n t he c i r c l e of c u r v a t u r e
and t h e r e f l e c t o r . T he e l l ipse s t i l l p rov ides a r a t h e r spec ia l e x a m p l e . A m o r e g e n e r a l
v i e w p o i n t and c l a s s i f i c a t i o n can be de r i ved if t he a r g u m e n t s o f s e c t i o n 4.2.2 a re appl ied
to m o r e g e n e r a l c u r v e s .
Loca l ly , t he c i r c l e o f c u r v a t u r e p rov ides a r a t h e r good a p p r o x i m a t i o n o f a t w o - -
d i m e n s i o n a l c u r v e . C h o o s i n g i ts c e n t e r as t he or ig in o f a po la r c o o r d i n a t e s y s t e m we
c a n d e s c r i b e t h e r e f l e c t o r by an e q u a t i o n
r = R + f ( e ) , (4.45}
w h e r e R is t h e local r ad ius of c u r v a t u r e and f(0} d e s c r i b e s t he d e v i a t i o n o f t he c u r v e
f rom the p e r f e c t c i r c u l a r a rc (of. Fig. 4.35). The q u e s t i o n is w h a t we can in fe r abou t
t h e f u n c t i o n f(0). The r ad i u s o f c u r v a t u r e in polar c o o r d i n a t e s is g iven by
186
d Fig. 4.35: The three possible con tac t s be tween a two-dimensional r e f l ec to r and its c i rc le of curvature.
R = ( r 2 + r ' 2 ) 3 / 2 2 r + 2 r ' 2 - r r " (4.46)
At O =0 the re f l ec to r has curvature R, and this is the case if f{0) sat isf ies the three
conditions f(0)=0, f'(0)=0, and f"(0)=0 as can easily be verif ied from the standard equation
(4.46). If we use a power series to approximate fie), then this series cannot involve
powers less than three, i.e. we need at least a function frO)= fla+...+higher terms. Such
functions, of course, are really flat at the origin, their curvature vanishes at 8 =0.
However, f(fl)= ( 3 is an odd function, and if we insert it into equation (4.45), it
becomes clear that the circle of curvature in te rsec ts the re f lec tor in some neighborhood
of e =0; the local re f lec tor model is a 'spiral arc ' with monotonously increasing (decrea-
sing) curvature in a suff ic ient ly small neighborhood of e =0 (Fig. 4.35a). A sign change
of the leading term (f(e)=-+e a) simply re f l ec t s the intersect ion pa t te rn at the ray fl =0;
the pat tern , however, does not change.
The situation becomes d i f fe ren t if the power series s ta r t s with a fourth order
term. Then the re f lec tor deviates symmetr ical ly from the circle of curvature , and a
sign change of the leading fourth order term changes the type of contact : For +8 4 the
c i rc le of curvature is ent i re ly on the concave side of the re f l ec to r while for -0 4 it
is on the convex side (Fig. 4.35).
The two a l te rna t ive power series with leading te rms of order three or four are
really dist inct and exclude one another, as now will be shown. A local re f lec tor approxi-
mation involving both te rms could always be brought to the form
f(8) = O 3 + ae 4 + ... + higher terms. (4.47)
187
[ However, by a redefinit ion of the zero angle ( 8 - 8 - ~-a) ,equat ion (4.47) can be t rans-
formed into
3 2 04/(4a) - ~0 + 2a20 + (a4-a3). (4.48)
tn equation (4.48) the radius of curvature is given by (R+c), and f~) has again to sat isfy
f(0)=f'(0)=f"(0)=0, i.e.
1 3 a 0 - 302 + 2a 2 = 0
3 e 2 - 6(? -. 0 . (4.49) a
These two equations, however, are usually not zero, and the function f(O) is dominated
by the first and second order te rms with non-vanishing first and second order derivat ives
and, therefore , does not sat isfy the requested approximation.
Therefore, our problem is, locally, strongly equivalent to a power series which
s ta r t s e i ther with a third or a fourth order term, and ca tas t rophe theory implies a fold
or cusp catas t rophe. The cr i t ical point in our problem is the point r=R, the cen te r of
the circle of curvature which, of course, is a point on the evolute of the rays, i.e.
a point on the caustic. Sufficiently close to 0 =0, the radii of our polar coordinate system
coincide with the rays. The t ransformat ion p =r-R maps the re f lec tor {the wave front}
to the cr i t ical point. Near this point, we take the re f lec tor as f{0)=04 or more conven-
iently, we use the unfolding
0 = -+04/4 + u02/2 + v@. (4.50)
We cannot choose u and v
to the set of equations
freely because f(O) has to satisfy f'(O)=f"(O)=O. This leads
v = ¥0 3 - uO
and (4.51)
u = $3@ 2
If we solve for u and v in terms of fl and insert this in equation (4.50), this equation
simplifies to a fourth order term as required. However, if we use u and v as local or tho-
gonal coordinates, then we can el iminate fl and find one of the dual cusps
u 3 v 2 (g) = ~(~) (4.52)
188
i.e. the caus t i c we expect . In a spatial in te rpre ta t ion , u is the (negative) first, v the
second der iva t ive of the funct ion p . In te rpre ted as vec tors , they provide a local o r tho-
gonal f r ame and cap tu re qua l i ta t ive ly the dislocat ion of rays close to the cr i t ica l point
r=R. Similar a r g u m e n t s can be applied to the case f ( 9 ) = 0 3 , the cr i t ica l points are
fold points.
If we re s t r i c t our a t t en t ion to local s t ruc tu res , there is not more than fold and
cusp points on a caus t ic . Their occur rence is a funct ion of the c on t a c t be tween the
re f l ec to r and its local c i rc le of cu rva tu re , as i l lus t ra ted in Fig. 4.35. In t e rms of r e f l ec -
t ion pa t t e rns , however, ' local t is r a the r re la t ive . In this con tex t , a fold point is a point
where two rays in te rsec t ; however, this is only the case on the caus t i c i tself . In the
inter ior of a caus t i c (cf. Fig. 4.34), which is not re la ted to a s ingular point on the re-
f lec tor {e.g. Fig. 4.32), we find tha t th ree rays i n t e r sec t at every point. Thus, fold points
are not such impor tan t f rom a less tocal viewpoint . Impor tant , however , is the d i f f e rence
be tween the dual cusps because they provide an essent ia l source for the se i smic in ter -
pre ta t ion .
In one sense the dual cusps are not d i f fe ren t , they are s imply dual re f l ec t ions
at the x-axis which resul t from a sign change of the leading power t e rms , and thus
are all r e la ted pa t t e rns . This is obvious because any wave front can be considered as
a r e f l ec to r and vice v e r s a - - here a wave front is a map of the r e f l ec to r along the
rays preserv ing angles. The wave front pa t t e rn s of the two dual cusps, therefore , are
ident ical , only the di rect ion of propagat ion is inversed. This is a nice resul t because
it shows tha t the previous discussion holds also for the dual cusp if the di rect ion of
wave propagat ion is inversed; and the ear l ier discussion provides really a ca t a logue of
the essen t ia l re f lec t ion pa t t e rns as far as a l inear approach is suf f ic ien t .
On the o ther band, the re r emains a d i f f e rence be tween the dual cusps. In the case
of a locally parabolic ref lector , the wave f ronts are sec t ions through the swallowtail
wi th its cusps and se l f in t e r sec t ions , and the t r ave l t ime records in the cr i t ica l case are
F i ~ 4.36: Wave f ronts of the dual cusps.
189
'hyperbol ic re f lec t ions ' , again with cusps and se l f in tersec t ions . In the case of the dual
cusp, the re f l ec to r is an e l l ip t ic arc which is bounded to the in ter ior of the cuspoid
caust ic; as soon as the image passes through the cusp point, the wave fronts have a
'parabol ic ' appearance , and thus has the t r ave l t ime record; cusp points and se l f in te r sec -
t ions then are missing. A typical pa t t e rn , which commonly arises, is a ser ies of parabolae
which a l t e rna t ive ly correspond to synclines and ant icl ines, and which in te r sec t on the
t r a v e l t i m e record, but wi thout cusp points. In the case the t rack line sect ions the
caust ic , swallowtail pa t t e rns may arise, but they are inver ted with respect to the pa t t e rns
arising from a 'parabol ic r e f l ec to r ' (Fig. 4.36).
In summary, the various re f lec t ion pa t te rns , which may arise, are well c lass i f iable
in t e rms of the con tac t be tween the re f lec tor , i ts c i rc le of cu rva tu re and the incident
wave front (dis tance from the source). Usually there should be enough informat ion
avai lable for a qual i ta t ive ly cor rec t in terpre ta ion . The l inear ray model, of course, is
only a f irst approximation, but the principal re lat ionships remain s table even if the sonic
veloci ty of the medium is not a cons tan t .
B) Three-Dimensional Pa t t e rn s -- The Double Cusp
At least, a few remarks shall be made in what respec t the simplif ied model of
l inear rays and especial ly of a two-dimensional r e f l ec to r line gives insight into a larger
class of images which may resul t from compl ica ted r e f l ec to r topologies. The two-dimen-
sional approach extends wi thout di f f icul t ies to cylindrical surface e l ement s or, more
generally, to parabolic surface points. Fig. 4.37 gives two e x a m p l e s - - a cyl indrical
and a conical surface -- tha t show how the caus t ic and a single wave f ront are located
over the surface . In such cases, the t r a v e l t i m e record will depend on the re la t ion be-
tween the axis of the syncline and the t rack line -- one may find 'hyperbol ic re f lec t ions ' ,
onlapping pa t te rns , doubted or t r ipled re f lec t ions (Fig. 4.38). Thus, an i r regular topo-
graphy, which impresses i ts s t ruc tu re onto the wavefield, can cause nice mult iple re f lec-
t ion pa t t e rns which look like pe r fec t ly s t ra t i f i ed sediments; and, therefore , one may
ask how much onlapping fea tures in Fig. 4.15 are real, and which ones are due to the
rough topography of the basement . The complexi ty of these e f f ec t s increases if one
considers far f ie ld e f f ec t s or more compl ica ted sur face e l ement s like hyperbolic and
el l ip t ic surface points. Represent ing the surface near (Xo,Yo,Zo) by z=f(x,y) the evolut ion
equat ion for the wave fronts becomes
{(U,V,w) E (x,y,z) I ((x-u)2 + (Y-V)2 + (w-f(x'y))2) I/2 = r = const. }. (4.53)
In the case of a parabol ic or hyperbolic surface point, the family of rays is given by
the (vector) equat ion
190
Fig. 4.37: The caust ic and a single wave front over a cylindrical (above) and conical (below) surface.
r = (x, y, x 2 ± ay 2) + k(2x, +2ay, -i), (4.54)
and a point on the track line may be given as (Xo,Yo,Zo). To see, which surface points
map onto the track line, one has to solve the equation
(Xo' YO' Zo) = (x, y, x 2 + ay 2) + ),(2x, +2ay, -i). (4.55)
Let the track line be located at Zo, then by elimination of the parameter ), , one finds
the relationship 2 = x 2 -+ ay - z 0
x 0 = (1-2Zo)X + 2x 3 + 2ay2x
YO = (1 -~- 2z 0 + 2ay 3 +- 2x2y, (4.56)
191
Fig. 4.38: Sketch of the t ravel t ime record of a cylindric syncline with track line sect ions parallel and oblique to the syncline axis.
a map which is a special degenera ted case of the double cusp ca tas t rophe including
the standard umbilic catas t rophes . The caust ic pat terns , which result from the double
cusp, are ra ther complicated. A full discussion of three-dimensional phenomena is above
the scope of this discussion; however, a detailed study in terms of standard ca tas t rophes
was given by DANGELMAYER & GUTTINGER (1983).
4.2.9 Distr ibuted Receivers
Seismic shooting rarely resembles the idealized situation that source and receiver
are at the same place. However, as turned out during the previous discussion, the results
found from ra ther idealized assumptions hold for a much wider class of 'dis turbed'
problems. It will be shown here that the principal results still hold if source and re-
ceiver are at d i f ferent places, or if a chain of receivers is used. In the la t ter case,
not a single ref lect ion but the re f lec ted and deformed wavelet is recorded. What we
shall do here is, therefore , to study how the re f lec ted wavelet deforms.
The ref lect ion of a wavelet is governed by Snell 's law of equal angles, i.e. the
angle an incident ray forms with the normal of the ref lect ing surface is the same as
192
the angle the re f l ec ted ray forms with the same normal. A convenient way, therefore ,
is to view the incident and the r e f l ec t ed rays in te rms of the ref lec tor . Let the ref lec-
tot be given in te rms of its local curvature, i.e. with the cen te r of the global coordi-
nate system at the cen te r of its local circle of curvature (cf. equ. 4.45): Locally the
re f l ec to r can be wr i t ten
and the normal rays are
Yn = r (sin 0 + X -r Lsin + r ( cos
(4.57)
(4.58)
Now, in a plane problem we can express the incident rays in local coordinates by means
of the tangent (t) and normal (n) vectors at the surface:
r . = r + X ( - a n + g t ) , (4.59) 1
and the re f l ec ted rays are simply the ref lec t ions of incident rays at the normals
r = r + X(-ccn - B t ) . (4.60) r
The coef f ic ien t s ' a ' and 'b ' can be de termined to sat isfy special conditions of the
source, e.g. in the case of a point source, equation (4.59) leads to a pair of linear
equations from which the coef f ic ien t s can be determined. A very simple system arises
if the re f l ec to r is locally a pe r fec t circular arc. The equations for the incident and
re f l ec ted rays then simplify to the pair of equations
cos [-sin
[~] -- [ c°s Ab [ -sin o001. (4.61)
The condition that the incident rays originate from a point source requires that these
rays pass through the source point for some value of I . Without loss of generali ty,
we can choose the value k=l , and the values for the parameters 'a ' and 'b ' can be
de termined from the linear equations
(r-a)cose - bsinO = x 0
(r-a)sin@ + bcose = YO
to be
193
a = r - (YosinO + xncosO) b YoCOSO - x~sinO (4.62)
,xi] and the r e f l ec t ed rays are
Because of the symmet ry of the c i rcular arc a s imple ro ta t ion allows to locate the
source formal ly at (Xo,0) so t ha t the previous equat ions simplify fur ther . If one inser ts
' a ' and 'b ' from equat ion {4.62) into equat ions {4.61), one finds a s implif ied equat ion
for the incident rays
= r ( 1 - X + )` (4.63} sin
(Xrl [ fc°81 II )`x r cos2e + (4.64)
= r ( 1 - ) , ) 0 s i n 2 0 Yr [sin @
As previously, the caus t ic of the r e f l ec ted ray system is of special in teres t . If we
consider equat ion (4.61) as a map, the caus t ic is equivalent to i ts singular set , which
can be de t e rmined from the condit ion tha t the Jacobian of the map vanishes, i.e. tha t
I xe xk I J = = xey ~ - xky ~ = O.
YO YX
From this condit ion and equat ions (4.61) and (4.64) we de te rmine the c r i t i ca l se t in
t e rms of )`:
= a I - xocose =
2(a2+b2)_a l+2x2_3xocos 8 if yo=O. (4.65)
If we inser t these values for X into equat ion (4.64), we find an equat ion for the caus t ic
= r 2 x° ....... - x o c o s e xo - XoCOSe2 (4.66)
Yr l + 2 x 2 - 3 x c o s e { s i n e J l + 2 x 2 - a x o c o s O ~ . s i n 2 e j J
which looks r a the r compl ica ted . However, a simple observat ion is impor tant . Let us
compute the values ( l - X ) and )`Xoat e=0:
2 2 (I-~) -- 2 Xo - xo Xo - x~
i+2xo2 ; kxo = - I +2x 2
194
F_ig. 4.39: The cardioid caust ics of a circular r e f l ec to r and their relat ion to point sources.
Locally, near 0=0, we have the simple relationship 2(1-%)= kxo, and this means that
near this special point the caust ic behaves like a cardioid independent of the complexi-
ty of our original equation. The cardioid, however, has a cusp point at O =0, and this
is a s tandard cusp point, as can easily be shown by developing the equations
x = r(2cos8 - cos2e);
in Taylor series near the critical point e =0:
x n, 1 + 82; y ~ nl----~0e3
y = r(2sin9 - sin2e)
- - - > ( x _ l ) 3 = ( . ~ . y ) 2
What we now can do with the source point, is to dis locate it along the x-coordinate
{Fig. 4.39). Clearly, a cr i t ical si tuation arises if the source is located at {0,0), the
cen te r of the c i rc le of curvature. In this case, all rays pass through the origin, the
caust ic degenera tes to a singular point, and we would not rece ive any ref lec t ions at
points besides this degenera ted singularity.
If 0 <lXoi> R, we find that the caust ic has formally two cusp points if we consid-
er not simply a circular arc but a full circle. These cusp points are given by 0=0
and 8 =~r, tn addition, we observe that these cusp points are simple inversions of the
corresponding source locations x o ~ - x o . A somewhat striking point is that we always
have the same type of a cusp (what we called the dual cusp of the ref lec t ion prob-
lem) independent of the radius of the incident wavelet . The caust ic pat tern , therefore ,
does not depend on the con tac t between the re f l ec to r and the (circular) incident wave-
let.
Another special si tuation occurs if tXoi =R. In this case, we have only one cusp
point, the other one degenera tes into a fold point with its tangent coincident with
195
the t angent of the c i rcular r e f l e c t o r - - the caus t ic becomes a pe r fec t cardioid, in the
case the source point is located outside the circle, there remains only one cusp point,
but in addition we find two cr i t ical fold points where the deformed cardioid has tangen-
tial co n t ac t with the c i rcular re f lec tor . In some sense, the s i tua t ion Xo=l def ines a
b i furca t ion point. However, if Xo~m, we find again a s y m m e t r i c solution, the caus t i c
is now a nephroid (cf. POSTON & STEWART, 1978} whereby the s y m m e t r y re fe r s to
the two sources Xo=+~.
The caus t ic phenomena associa ted with a point source and a per fec t c i rcular ref lec-
tor, therefore , can be summar i zed as cont inuous de format ions of a cardioid. The s table
pa t t e rn is the cusp point of the cardioid, which locally remains the identical cusp caus-
t ic independent of the location of the source. Now, the ci rcular r e f l ec to r is unstable ,
and the quest ion ar ises what happens if it is deformed. Before going in detai ls , we
Fig. 4.40: The vir tual sources of a planar and circular ref lec tor . The wave f ronts provide vir tual re f lec tors .
f irst observe tha t there exis ts a vir tual surface , for which the re f l ec ted rays of the
wavele t are normals . In the case of a plane ref lec tor , this vir tual sur face is again
a point source, a s tandard example in se ismology (Fig. 4.40). tn a more genera l sense ,
every wave front is a potent ia l ly virtuaI r e f l ec to r su r f ace because the wave f ronts in ter-
sec t the rays orthogonatty. In the case of linear rays, the wave fronts are found from
the normal ized equat ion (4.61), i.e. from
196
[Xr} IcosO [afc°sO r s,n011 Yr = r [sinej (a2+b2) I/2 [sinOj -b [ cosOJ. (4.67)
The t r av e l t im e is equiva lent to the sum of the length of the incident and re f l ec t ed
rays. If the rece ive rs are on the s ame x-level as the source , , t he t r a ve l t ime is given
by
2t = (a2+b2)I/2(l x__~o _- r cos 8 - acos ~ bsinO )' (4.68)
and the identical t r ave l t ime record would be rece ived in a sys t em where source and
rece iver coincide, e i ther from a vir tual source or a vir tual r e f l ec to r which, of course,
is s imply a wave front (Fig. 4.40).Now, we can use the discussion of the last sect ion.
A cr i t ica l s i tua t ion ar ises if the vir tual re f lec t ing su r face becomes a circle. Any small
de fo rmat ion then de fo rms i t , and the s ingular caus t i c point evolves in e i ther one of
the dual cusps. We consider this degene ra t ed s i tua t ion and dis turb the re f l ec ted rays
(the normals) by a not necessar i ly cons t an t ro ta t ion
[XrJ = riO@sO} - It°s01 + Yr [ainOJ [sinSJ [ coseJ
(4.69)
where
f COSe (0) -sins (8)] A = [sins(e) cose(O)J. (4.70)
The cr i t ica l se t can again be found from the Jacobian to be
-X = (p2+p'2)e°se
{l+c~, } p2 {2+e,) p, 2_pp,, (4.70)
The deformat ion of the original ray sys tem, therefore , cons is t s of a ro ta t ion as defined
by the m a t r i x 'A ' and a dislocat ion along the rays which is proportional to cos ~ . In
the case cos e vanishes , the re f lec tor becomes identical with its caust ic , and the image
is inversed, as cos e a s sumes nega t ive values. However, this would require ra the r s t rong
de format ions . We conclude, the re fore , t ha t the c a us t i c pa t t e rn formed by the normals
r emains s t ab le as long as the de fo rma t ions are of reasonable order.
We note finally tha t we can r e fo rmu la t e cos c~ A as
1 [l+cos2e -sin2e 1
(cose)A = ~_ [sin2e l+cos2~J .
The caustic formed by the rotated norrnals can now be written
(4.71)
197
Fig. 4.41: Normal and re f l ec ted ray sys tem and caus t i cs at a parabolic and hyper- bolic re f lec tor . A point source does not change the caus t ic pa t te rn .
Fi G . 4.42: Ro ta t ed normals of a circular ref lec tor . The s ingular point is t r ans - formed into a caus t ic . The ro ta ted rays can be cons t ruc ted as average of the normals and rays ro ta ted twice the original angle (but which still have the length of the normals).
198
r r = r - f ( r , r ' ) ~- [ s i n 2 u c o s 2 ~ j n (4.72)
The resul t ing p a t t e r n is the average of two vec tor fields which di f fer by a ro ta t ion .
Fig. 4.42 e luc idates this point and i l lus t ra tes once more the ins tabi l i ty of a c i rcular
re f lec tor . Even a cons tan t ro ta t ion of the normals deforms the singular point of the
c i rc le into a c i rcular fold line. Fig. 4.42 e luc idates in addit ion tha t only a ro ta t ion
with angles larger than ~r/2 can real ly change the caus t ic pa t t e rn , as can also be infer red
from equat ion {4.73). Thus, we can finally conclude tha t the caus t ic pa t t e rn of the
normals of a sur face remains s tab le even if r e f l ec t ed wavele t s are rece ived because
the ro ta t ion of the normal at a c i rcu la r r e f l ec to r is equivalent to a deformat ion of
this re f lec tor . This point is especial ly e lucidated by equat ion (4.73), which s t a t e s tha t
a monotonous de format ion of t h e re f lec t ion angle within reasonable bounds cannot real ly
change the original caus t ic and the re la ted pa t te rns . Therefore , the caus t i c formed
by the normal rays must be accep t ed as a s t ruc tura l ly s tab le pa t t e rn , even under reason-
able d is turbances .
4.3 "PARALLEL SYSTEMS" IN GEOLOGY
Structural geology describes and analyzes the "geometry" of deformed rocks. The
procedure is mainly geometrical, and the relations to the physical processes are estab-
lished by "classification procedures" (GZOVSKY et al., 1973). The base for these relation-
ships is developed from various physical, exper imen ta l and numer ica l methods for which
a wide var ie ty of m a t h e m a t i c a l methods has been used (BAYLY, 1974; MATTHEWS et
al., 1971; JOHNSON & POLLARD, 1973; BEHZADI & DUBEY, 1980; COBBOLD et al.,
1971; DIETRICH, 1970; FLETCHER, 1979; SMITH, 1975 to give a few examples). Most
of the geological s t ruc tu res are the resul t of complex s t ra in fields. These s t ra in fields,
in general , are not the resul t of s imilar complex global fields of forces, but the complex
and inhomogeneous s t ra in field resul ts from the var iab le e las t ic , plast ic , and viscous
behavior of rocks, i.e. from the i r pr imary inhomogenei t ies . The deformat ions of rocks
can be very large, and then they are outs ide of the scope of classical d i f fe ren t ia l calcu-
lus. This is especial ly t rue if t rans i t ions from e las t ic to plast ic and viscous behavior
occur, if the boundary condi t ions are not k n o w n - - in general they are n o t - - and if
dis locat ions of mate r ia l by solution and recrys ta l l i za t ion play an impor tan t role during
the de format ion process (STEPHANSON, 1974; TRURNIT & AMSTUTZ, 1979). A classical
approach to s tudy deformat ions of rocks, the re fore , is the geomet r i ca l analysis. A common
way is to apply the methods of f ini te s t ra in analysis {RAMSAY, 1967; JAEGER, 1969;
HOBBS, 1971) to regions for which a near ly homogeneous s t ra in can be assumed. Basically,
this type of analysis is the study of some special mappings, and some of them will be
br ief ly discussed here.
199
4.3.1 Some Examples of Para l le l Systems
Much of the previous discussion focussed on systems of quasi-paral lel layers, which
posses a formal geomet r ica l s imilar i ty with deformed s t ruc tu res in geology. Considering
a three-d imensional space such a paral lel system can be wr i t t en
F(u,v;t) = x(u,v) + tN(u,v)
where N(u,v) = (Nx,Ny,Nz) , the unit normal vec tor at x(u,v).
(4.73)
If F(u,v;0) is everywhere d i f ferent iable , then the Jakobian de t e rminan t of such a system
is nowhere zero (DoCARMO, 1976):
det J(F)= = I ( F u) (F v) (F t) l = ]XuA x v ] ~ 0 (4.74)
where F u e tc . are column vectors of the Jacobian mat r ix (see sect ion 4.3.4 for details).
Equation (4.74) shows tha t there exists a tubular neighborhood to the surface x(u,v)
which is uniquely defined. Given a solution for a surface x(u,v) under ce r t a in conditions,
we can extend this solution into a small but f inite neighborhood of x(u,v). In a conceptual
sense this secures tha t the solutions can be applied to a real physical system where
a surface is always of f ini te thickness. Assume equat ion (4.73) is applicable as a l inear
first order approximation, then we immedia te ly get an e s t ima te of the maximal local
ex ten t of the tubular neighborhood, i.e. the area into which we may extend the solution.
This area is hounded by the ' focal sur faces '
Xl(U,V) = x(u,v) + p "IN(U,V)
x2(u,v) = x(u,v) + 92N(u,v)
where 0 1' P2 are the principal
1976; GUGGENHEIMER, 1977).
(4.75)
curva tu res of the sur face (cf. DoCARMO,
The assumption of parallel layers has a long t radi t ion in the recons t ruc t ion of
folds in t ec ton ics (e.g. HILLS, 1963 for an overview). The recons t ruc t ion of folds from
surface measurement s is i l lus t ra ted in Fig. 4.43 a f t e r an example of GILL (1953). A
point, obvious from Fig. 4.43, is tha t the fold cannot be extended continuously into
depth~ as several segments vanish along the ~caustic of the normal rays ' as discussed
in the previous sect ion; and it has been assumed (e.g. BUSK, 1956) tha t t h e s e ' l i n e s (or
sufaces) evolve into faults. Of course, equat ion (4.74) is only a f irst order approximat ionl
however, r a t h e r s imilar a rguments hold for the 'normal var ia t ion ' of a sur face x{u,v)
which can be wr i t t en (DoCARMO, 1976)
200
~ Recons t ruc t ion of paral lel folds from sur face data. Modified a f t e r GILL
F(u,v;t) = x(u,v) + th(u,v)N(u,v)
where h(u,v) is some sca lar var iable .
(4.76)
Such a formula t ion provides us with the possibil i ty to adapt some condi t ions which have
to be sa t i s f ied by h(u,v), and equat ion (4.76) can be considered as a var ia t ional problem,
or we may consider equat ion (4.76) as the dis turbed l inear problem descr ibed by equat ion
(4.73)•
Para l le l sys tems are encoun te red in various areas. With respec t to geology~ an impor-
t an t one is the concep t of sl ip-l ines in the theory of pe r fec t plas t ic i ty , which is closely
r e l a t ed to evolu tes and involutes as s t a t ed by Hencky~s and Prandt l t s theorems (LING,
1973):
HENCKY's theorem: The angle formed by the t angen ts of two fixed shear l ines
of one family at the i r points of in te r sec t ion wi th a shear line of the second family
does not depend on the choice of the in te r sec t ing shear line of the second family.
PRANDTL~s theorem: Along a fixed shear line of one family, the cen te r s of curva-
ture of the shear l ines of the o the r family form an involufe of the fixed shear
line.
Given one non- l inear shear line, the t l inear system t of normals and involutes provides
a f i rs t approximat ion for the sl ip-l ine field. The most s imple cases are or thogonal ly
in t e r sec t ing s t r a igh t lines and t cen te red fans ' of c i rcu la r arcs~ which provide reasonable
f irst order approximat ions of p las t ic de fo rmat ion (e.g. LING, 1973). Fig• 4.44 i l lus t ra tes
P rand tPs solut ion for sl ip-l ines below a s t r ip load. A more general soIution consis ts of
20I
Fig. 4.44: Prandt l ' s solution for slip-lines below a s tr ip load above a homogeneous hal fspace.
cen te red arcs of logar i thmic spirals: Consider x(u,v) a genera l ized logar i thmic spiral,
as discussed in sect ion 3.5.3, and h(u,v) to be proport ional to the curva tu re of the leading
spiral x(u,v), then equat ion (4.76} describes a family of possible solutions, from which
we have to choose the locally valid one which then can be ex tended to neighboring areas
by connect ing local solutions along the s t ra ight charac te r i s t i c s .
ODE (1960) applied the slip-line theory to the format ion of faul ts in sand and
clay under the condi t ions of plane s t rain. By a s imilar a t t emp t , also more geometr ica l ly ,
FREUND (1974) s tudied the t e rmina t ion of t r anscu r ren t faul ts by splaying; from his
analysis the curva' ture of t r anscu r ren t faul ts can be re la ted to the format ion of an
evolute of a fan of faults. Evolutes, as lines {or surfaces) of discontinuity, occur fu r ther
under unidirect ional glide in solid crys ta ls (e.g. KLEMANN, 1983).
4.3.2 Similar and Paral le l Folds
Concerning geologically 'shallow' deformat ions (without phase transitions} HOEPPE-
NER (1978) found from exper iments tha t most folds can be t raced back to the following
types:
1) similar folds
2) paral lel folds
a) concen t r i c folds
b) box folds.
Para l le l folds occur usually near the f ree surface or near shear planes while else-
where the more energy consuming s imilar folds develop. The d i f fe rences be tween the
202
=
A Fig. 4.45: A) Ideal parallel folds (kinks or box folds) and B) ideal similar folds (chevron folds},
two types of folds are schemat ical ly i l lustrated in Fig. 4.45. Parallel folds are of finite
depth range, i.e. they resemble the parallel wave fields discussed in the last section.
Similar folds in contrary continue (ideally) infinitely. The strain in folded layered systems
has extensively be studied by HOBBS (1971), here we consider only volume preserving
systems. Similar folds with constant divergence are described by maps of the form
X = ax + f(y,z) {4.77)
Y = by + h(z)
Z = cz
where a,b,c: constants; f,g,h: arbi t rary functions.
The Jacobian de te rminant
3 fi Net J= axTI =abc
is constant and by choosing a,b,c in ratios such that abc=l, the deformation described
by equation (4.77) is volume conserving, locally and globally. If we consider cylindrical
folds, equation (4.77) reduces to a two-dimensional system and describes a two-dimension-
al dislocation field as i l lustrated in Fig. 4.46. A special proper ty of this case is that
the principal strains are identical along every s t ra ight ' shear line' of the dislocation
field. As these parallel dislocation lines never in tersect , the fold extends ideatly into
infinite depth, and laterally the local fold pa t te rn can easily be continued if we connect
local solutions along a straight dislocation line (cf. Fig. 4.46c). Of course, along such
lines the solution is discontinuous with respect to the curvature, a discontinuity which
occurs in sinusoidal systems at the inflection points. JOHNSON & ELLEN (1973) pointed
out that such lines of discontinuit ies may be of some value in the analysis of folds
203
a b
I
1
Fig. 4.46: Deformat ion of a homogeneous hal f -space (a) into s imilar folds (b,c). A local solution (c) can be extended la teral ly by cont inuat ion along a slip-line.
and compared them with ' cha rac t e r i s t i c s ' as they occur in the slip-line theory of plast i -
city.
The possibili ty to cont inue local solutions la tera l ly is common for both fold types.
In the case of paral lei cylindrical folds, a local solution can be cont inued along any
*normal ray v as i l lus t ra ted by Fig. '4.47. However, for a paral lel system the re exists
no solution with cons tan t Jacobian, and thus i t cannot descr ibe a deformat ion which
preserves volume locally. On the o ther hand, we have already seen in sect ion 4.1.3 tha t
it is possible to connect deformed pieces in such a way tha t the en t i r e volume of the
systems is not a l te red (Fig. 4.47}. Concerning global volume changes, s imilar and parallel
folds provide comparable solutions. Paral le l folds are best considered as lamina ted systems
which allow the laminae to glide one above the o ther as i l lus t ra ted in Fig. 4.47. Within
Fig. 4.47: Ideal (concentr ic) folds lateralIy cont inued along ' rays ' or ' l ines of disconti- nuity ' . Heavy lines indica te intervals of equal length for the layers, Black grid e lements : deformed 'volume e lements ~ of originally rec tangula r grid e lements .
204
Fig, 4.48: Buckling of a card deck under la tera l s t ress conf i rmed by ver t ica l plates. Right: deta i ls of kink format ion.
205
laminae the ideal model allows only for membrane stresses, a s i tuat ion somet imes appli-
cable to deformat ions in liquid crys ta ls {KLEMAN, 1983). The deformat ions be tween
subsequent layers then are proport ional to the change of surface e l ement s {rather than
volume elements) :
let F(u;t) = x(u) + zN(u)~ (4,78)
then ds/dt = (1-zk)
where k: the local curva ture of the, leading curve,
and the deformat ion is simply proport ional to the curva tu re of the surface e lement ,
We find tha t the sur face e l ement s vanish along the evolute of the normals or a t
the focal surface, which w% therefore , can expect to be par t of the shear surface sepa ra t -
ing successive paral lel folds.
The two fold types are both idealized systems, and exper imenta l ly t ransi t ions occur
be tween the two types. JOHNSON & HONEA {1975) concluded from the i r mul t i layer
exper iments tha t the common assumption tha t one can e s t ima te depth of folding by
the ' ray method ' is of l imited value. Fig. 4.48 i l lus t ra tes some phases of mul t i layer
folding under two-axial stress. The t rans i t ion from paral lel to s imilar folds is again a
cont inua t ion problem. Assume tha t the solution is known a t the f ree surface of a half
space and tha t this solution is give n by a box fold: The range of the parallel fold solution
is of l imited depth, and it is bounded by a cuspoid focal line (Fig. 4.49). We are in t e re s t -
ed to ex tend the d is turbance into depth and require tha t the fold lines are cont inuous
along the focal line. To cont inue the d is turbance we project the focal line into depth,
i.e. assume the focal line is given by a function g(x,y) = 0, then we consider the family
Fig . 4.49: Cont inu- at ion of paral lel folds into s imilar folds by propaga t - ing the cusp discont inui ty into depth.
206
of functions
g(x,y) = c. (4.79)
The fold lines of the parallel system in tersec t the focal line perpendicular as was dis-
cussed in te rms of wave fronts. The extended fold lines, therefore , have also to in tersec t
the original focal line by right angles. A possible continuation, therefore , are the or tho-
gonal t ra jec tor ies of the family g(x,y) = c which are found by solving the di f ferent ia l
equation
+ gxy' = O. --gy
2 In the case of a cuspoid focal line, x
t ra jec tor ies are the family of functions
y = (8/9)x 4/3 +c
(4,80)
(y-c) 3 = 0 or y - x 2/3 = c, the orthogonal
(4.8t)
which clearly provide a set of similar folds (Fig. 4.49). Usually the solution will be bound-
ed to a str ip of f inite length, however, the str ip can be continued laterally as discussed
previously, and if we consider a layer of f inite thickness, this continuation can be adjust-
ed to preserve volume globally. Clearly, the discussed models are only first order approxi-
mations which, however, allow graphical analysis of even complicated large scale deforma-
tions and which capture some essential quali tat ive proper t ies of exper iments .
4.3.3 Bending at Fold Hinges
The previous discussion focussed on systems composed of layers of vanishing thick-
ness, or of negligible thickness with respect to the ent i re system. Concerning a compact
layer of finite thickness one has to consider the deformations near the fold hinge, as
schematicaIly i l lustrated in Fig. 4.50. The previously discussed linear approach of paraI-
Fig. 4.50: Idealized parallel folds (kinks) composed of layers of d i f ferent f inite thickness.
207
lel layer reveals Bernoulli 's theorem, i.e. undeformed cross-sect ions in the deformed
s ta te . A more realist ic model provides St. Vernant~s solution for bending of a bar by
couples. The deformed s ta te is described by the map
c X = x(1 + ~ z) (4.82)
Y = y(1 - ~ z)
c Z = z + ~ ( o ( y 2 - z 2) - x 2)
where c: s t rength of couples; E: Young~s modulus, o : Poisson~s ratio and
c /E = R-I; R: radius of curvature (see e.g. BUDO, 1974; LOVE, 1944).
In engineering the usual procedure is to study the deformation of an object under
specif ic s tress configuration. In geology we usually know li t t le about the original s t ress
field. Therefore, it is worthwhile to work with models, and the question is not mainly
how the object deforms within a cer tain s t ress field but how far the model is applicable.
One question, which can be pushed forward by mathemat ica l analysis, is how the various
parameters in teract and whether the solution is bounded to some region, i.e. concerning
the bending model we are in teres ted if the thickness of the bar is unlimited.
The limits of the solution are given by the condition that the Jacobian of St. Ver-
nant ' s map vanishes; however, in this case we can simplify the analysis by reducing
the map to a standard ca tas t rophe on Thomas list. If we slide the bar along the line
y=0, i.e. by a vert ical plane along the long axis (z), equation (4.82) simplifies to
-Z = ~ E ( o z 2 + x 2) + z (4.83)
c X = E x z + x,
and by means of the t ransformat ion oz 2 ~ z 2 this equation simplifies to the standard
form
* 2 2 2E Z = z + x +c'--o¢ z (4.84)
* 2E X = 2xy + - - x . c
The only assumptions involved are that the couples ~c' do not vanish {we consider only
deformed states) and that o ¢0. The singular set of this map is i l lustrated in Fig. 4.51
by use of the standard form of the hyperbolic umbilic. Fig. 4.52 i l lustrates a single
sect ion (E,c, =constant), and it becomes clear how the solution space is limited by a
cusp and a fold line, i.e. even for small deformations of this type the bar cannot exceed
a cer ta in thickness. Fig. 4.52a ° i l lustrates the shape of the undeformed area, i.e. the
boundaries defined by J=0 {J: Jacobian determinant) . By set t ing J=c one finds lines of
208
. .. ;i !. • . , ' £ •
Fig. 4.51: The s tandard form of Thomas hyperbolic umbilic. Isolines inside c i rc les ind ica te the local tpotential~.
a ~" b i ......... "
Fig. 4.52: a) The non-local sec t ion through a bent bar along i ts long axis (see text) . The c r i t i ca l se t ( se l f - in te rsec t ions of parabolas) corresponds wi th the sect ion through the hyperbol ic umbilic. Parabolas indicate lines which are paral lel in the undeformed s ta te , a °) assoc ia ted undeformed s ta te : Only the blank a rea can be deformed to the image indica ted in (a). b) The same bar with lines of cons tan t values of the Jacobian de te rminan t , b °) the associa ted undeformed image.
209
Fig. 4.53: The 'hyperbolic umbilic' as a sheet of paper folded in its plane.
210
"equal volume change" in the undeformed s t a t e (Fig. 4.52b °) and by means of the
mapping (4.83 or 4.84) the i r image in the deformed s t a t e (Fig. 4.52b). Fur ther proper t ies
will be analyzed in a more general sense in the next sect ions. The reduct ion of the
original map to a two-dimensional problem, clearly, gives only an idea how the en t i r e
sys tem reacts~ however, the solution is co r r ec t for the plane se lected, and it allows
to r e Ia t e the de format ion to a r a t h e r simple exper imen t {Fig. 4.53): A shee t of paper
'bended ' in i ts plane i l lus t ra tes in a r a t h e r s imple way how the l imit ing fold line evolves.
To connec t this sect ion with the fu r ther analysis of paral lel sys tems we observe
t ha t equat ion (4.83) can a l t e rna t ive ly be wr i t t en (using vec tor notation):
= y + ~z -(~y - 2--E z
e 2 2 (oY -x E/c
(4.85)
where the f irst t e rm is just the descr ipt ion of the "neut ra l surface" and the second
t e rm is the non-normal ized normal of the surface e lements . Thus, the f irst two t e rms
on the r ight side descr ibe the 'normal va r ia t ion ' of the surface, i.e. h(u,v) = I Xu A Xv{
in equat ion (4.76). The final t e rm on the r ight side can be taken as a non- l inear dis turb-
ance of the quasi-paral le l sys tem. This non- l inear t e rm depends only on z such tha t
the bending equat ion is properly approximated by the quasi-paral lel system if z is suff i-
c ient ly small.
4.3.4 Nota t ion of S t ra in
Fll X 2 =
3
deformed
Whenever e las t ic or p las t ic de format ions are considered, the problem is usually
fo rmula ted in t e r m s of s t resses and s t ra ins . The procedure is to solve a given problem in
t e r m s of dis locat ions {e.g. LOVE, 1944). The deformed s t a t e then is given in the form
discussed with s imilar folds, i.e. by a map
• = f ( x l , x 2 x2 + ' ~i 'x3)
undeformed dislocations
The elements of strain are related to the Jacobian matrix of the dislocations (e.g. LOVE,
1944).
211
J(E) =
"agl agl. ag~- axl ax 2 ax 3
ag= aga a~a ax 1 ax 2 ~x 3
a~a ~ 3 3,~3 ax 1 ax 2 a x 3 a
where (Ex) etc. are the column vec tors of the Jacobian matrix.
= [<, <, <,] (4.87)
Now, the re exists a simple relat ionship be tween the Jacobian mat r ix of the map (4.86) I
which will be denoted by J(X)~ and the Jacobian mat r ix of the dislocat ions
(4.88) J(X) = J(E} + I
where I: the ident i ty matrix.
If we now consider the more general map
[iil I: X 2 = V(Xl,X2,X3) [ ,
(Xl'X2'X3)]
and the Jacobian matrix of the dislocations is
J(E) = J(X) - I.
The mat r ix of s t ra in e lements in the l inear theory of e las t ic i ty then is given by (e.g.
LOVE, 1944; MEANS, 1976)
(4.90)
i~['¢ SXi~ + (_~)] _ I}. ( eij,] = ~ Lt-3-£-x. J ~ . , , d
l l
In the nonl inear theory of f inite s train, however, the mat r ix of s t ra in e lements can
be wr i t t en
= l { ~ 3 X i , faXj )~ - I } (4.91) ( ) L , a x i
g
where < , > denotes the scalar product of the column vectors of the Jacobian matr ix .
We shall need these nota t ions because they simplify the following work.
4.3.5 Genera l ized Plane Strain in Layered Media
Geological problems are commonly solved under the assumption of plane strain.
In this case, the three-d imensional problem simplifies to a two-dimensional one, which
212
causes less computa t iona l problems even in the computer . Here we consider the case
of genera l ized plane s t ra in and define i t by the condit ion tha t e =e =e =0. If we zz xz yz
apply this to the equat ions of s imilar folds (equation 4.77), we find tha t the s t a t e of
plane s t ra in is achieved by a map
X = ax + f{y)
Y = by + g(x)
Z = z
(4.92)
where not only the s t ra in componen ts involving the z-d i rec t ion vanish but also the asso-
c ia ted dislocations, Special cases of this type have been discussed by HOBBS (1971)
in detail , A more in te res t ing s i tuat ion arises if one considers paral lel folds. We take
the discussion of the bending of bars as mot iva t ion and consider the quasi-paral lel system
X = ~ +z (4.93)
(x,y
o r x = x(u,v) + z]x u A x v I N
and app!y the l inear theory of s t rain. The s t ra in e l emen t s are found from equat ion (4.90)
exx=-Z fxx; eyy=-Z fyy; exy= Zfxy; (4.94)
ezz=exz=eyz=0;
i.e. the quasi-paral le l system descr ibes a s t a t e of genera l ized plane s t rain. Next we
consider the pe r f ec t paral lel system
X = x(u,v) + zN(u,v) (4.95)
and apply the non- l inear theory of f ini te s t rain. The Jacobian ma t r ix of this map can
be wr i t t en
J(X) = ((x u + ZNu), (% + ZNv), (N)). (4.96)
The f ini te s t ra ins are defined by the sca lar products of the column vectors of J (equ.
4.91) which simplify i f one applies the or thogonal i ty re la t ions <Xu, Nu > =0;< Xu,N> =0
etc . . The f ini te s t ra ins are
2~< (Xu+ ZNu),(Xu + ZNu) > 1 z2< Nu,N Cxx = -1) = ~e-(<x ,x >+ 2z<N x > + u >) (4.97) 2 u u u u
@<(Xv+ZNv)'(Xv+ZNv )> -I)= <Xv,Xv>+ 2z<N x > + z2< -I) E YY v v Nv'Nv>
213
1 ~ z2<Nu,Nv>) xy = ~{2<(Xu+ZNu)'(Xv+Nv)> -1) = <Xu,Xv> + ZkNu, Xv> + <NvXu> ) +
z z = ~xz = ~ z = O .
The two equations, thus, are isomorphic with respect to the applied theory of strain,
and because the linear theory is an approximation of the non-linear one, equation (4.93)
provides an approximation of equation (4o95). Indeed, if we consider only the "neutral
surface" X = x(u,v) which is everywhere different iable , then we can approximate it locally
by an explicit function z=f(x,y) in te rms of its moving frame. There is another impor-
tant aspect: If we consider the case x(u,v)=f{u,v), then the focal surfaces, which bound
the solution space, are identical for equation (4.93) and equation (4.95) because we can
define them as the evolute (surface) of the normal rays, and the only d i f ference be tween
these equations is the magnitude of dislocation along the normals. Thus, even the solu-
tions deviate to some ex ten t in the linear and non-linear model, the cr i t ical set of focal
domains remains identical.
Both equations studied here describe families of surfaces, and this fact has its
expression in the equations for the strains, which can be expressed in te rms of the
fundamental forms of the "neutral surface". Following DoCARMO (1976) we note that
E = < Xu~X u> ; G = < Xv, Xv> ; F = < Xu~X V>
-e = <Nu,Xu > ; -g =<NvlXv> ; -2f = <Nu,Xv>+< Nv,Xu> ,
and the e lements of strain can be wri t ten
2 xx = E - 2ze + z2<Nu,Nu>
Cyy G 2zg + z < Nv,Nv>
Cxy = F - 2zf + z 2 <Nu,Nv>
(4.98)
providing the base for a potential further analysis.
The study of parallel system thus provides us with ra ther general models for gener-
alized plane strain and with s t ra tegies to find more general solutions in te rms of finite
strain than usually: If we are able to find a solution in te rms of the linear theory, the
close relat ions discussed here allow to t ransfer it to the non-linear theory as far as
quasi-parallel systems are concerned. The equations governing bending deviate from the
quasi-parallel linear model only by an additional non-linear term. In the finite strain
model a ra ther similar s t ruc ture is achieved if the variable 'z ~ is replaced by a function
h(z), e.g. h(z} = h(1-z).
214
The quasi-paral lel sys tems provide a family of funct ions of po ten t ia l value for
the analysis of large scale deformat ions , c lass i f iable by the s t ruc tu re of the i r Jacobian
mat r ix (or the re la ted s t r a i n s ) - - an aspect which links this study with ca tas t rophe
theory. The problems, which have been discussed here, are those which can be solved
by 'hand methods ' , In layered media the physical proper t ies usually a l t e rna t e (or change
gradually). Replacing the scalar funct ion h(x,y) in equat ion (3.76) by a mat r ix H(x,y;z)
allows to study more compl ica ted systems in t e rms of i t e ra t ed maps.
4.4 SUMMARY
The discussed examples are manifold, cover ing various geological and "applied
m a t h e m a t i c a l " methods. This calls for a sys t ema t i za t ion of the various forms of ins table
behavior and p a t t e r n format ion. There are two aspects: First , we have th ree major
m a t h e m a t i c a l o b j e c t s - - surfaces , t ra jec tor ies , and d is tance f u n c t i o n s - - which, under
ce r t a in maps, develop discont inui t ies or become instable in some sense. Secondly, we
have the ins tabi l i t ies themse lves which cause branching solutions.
A first group of objects , which occured repea ted ly throughout the examples, are
surfaces which can locally be described as F(x 1 . . . . , xi)=0. The in te res t ing deformat ions
occur under the t r ans fo rma t ion F=F(Xl, ... , x i) + sN(Xl, . . . . xi). The surface t rans forms
like a wave front , a t leas t locally, which is cons t ruc t ed by Huygens' principle. This
implies t ha t the family of sur faces can locally be descr ibed as F(Xl, ... , xi; s)=0, and
the concep t of s t ruc tura l s tab i l i ty can be used to study the geomet r i ca i s ingulari t ies .
The evolut ion of wave f ronts is, of course, the typical example al though the discont inu-
i t ies, which ar ise in the two-dimensional project ion of three-d imens ional objects , provide
still more geomet r i ca l examples. In addition, we can loca te here the probabi l is t ic hull
of the on togene t i c 'morphospace ' of chap te r 2, and if we allow for more general disloca-
tions, then the whole example of the 0n togene t ie 'morphospace ' belongs to this class
of problems. Clearly, the "pre -compute r" analysis of paral lel folds and the slip-line theory
are closely r e l a t ed in geomet r i ca l t e rms al though the physical p a r a m e t e r s are quite
d i f fe rent .
The second group of ob jec ts were t r a j e c t o r i e s - - on togene t i c t r aces in chap te r 2,
rays and sl ip-l ines in chap te r 4. Along these t r a j ec to r i e s a dynamics can be es tabl ished --
the spreading of a wave front , the on togene t i c deveIopment e tc . - - by an equat ion like
ds/dt=f(s) (s:arc length). But the t r a j ec to r i e s or rays themse lves are s t a t i c ob jec ts like
the previously discussed famil ies of surfaces , and they are descr ibed by the identical
equat ions up to a normal iza t ion fac to r of the evolut ion pa rame te r . If one has especial ly
a gradient system, then e l emen ta ry ca t a s t rophe theory provides a c lass i f ica t ion of the
in te res t ing geomet r i ca l s ingulari t ies . However, the gradient r es t r i c t ion is not so essent ia l
215
because e lementary ca tas t rophe theory is a local theory, i.e. it is suff ic ient if we can
t ransform such a system locally, near a cri t ical point, into a gradient system. The
interest ing singularities are usually d i f ferent from the ' involutes of rays'; however, they
are re la ted as discussed in terms of seismic reflect ion.
A further group of examples can be col lected under the term "distance functions",
and this is the most comprehensive group. Again, we can easily return to e lementary
ca tas t rophe theory. The wave fronts on a given set of rays have to sat isfy the dis tance
function r=I(x,y)-(Xo,Yo) I. Similar formulations are possible for the slip-line theory and
e las t ic i ty theory using the concept of strain and s t ress potentials. It is commonly the
approach via a dis tance or energy function, which allows to study problems more deeply
in topological terms. In principle, the problems encountered in surface reconstruct ion
and the convex hulls of point sets (chapter 2, Honda trees) belong to this group.
If we now take the example of cluster analysis, the situation is d i f ferent . What
we are doing in this case is essentially that we map a n-dimensional space onto a one--
dimensional one, the space of distances. This allows to represent clustering results from
a n-dimensional space as a binary tree. The binary t ree is nothing than the graphical
representa t ion of the pair (X,d) where 'X' is a one-dimensional point set and 'd' is a
relation be tween the points which defines an order. In the case of c luster t rees, the
problem is that the dis tance 'd' does not imply an ordering sequence and that the map
R n - - R can be multi-valued. This returns the problem to singularity theory as encoun-
tered with the two-dimensional images of three-dimensional objects.
What does hold together all examples, is that we can describe them all as maps
and that the observed instabili t ies are related to singularities of these maps whereby
the identical geometr ical singularity may have di f ferent physical meaning concerning
d i f ferent objects and may be di f ferent with respec t to any chosen subspace, i.e. in spatial
and spat io- temporal coordinates. Maps in their most general sense are not very specific.
Insofar, the previous discussion provides examples of more specific maps, which are
of some in teres t in geological applications.
Concerning the bifurcation of solutions the preceding discussion remains within
kinemat ic models. A bifurcation point is defined as the value of an evolution paramete r
at which the local topology changes. Thus, the cusp ca tas t rophe is the bifurcation set
of the swallowtail, it separa tes the area with continuous wave fronts from that one
with se l f in tersec t ing wave fronts which, of course, are of d i f ferent topological type,
and in a similar sense the transit ion from parallel to similar folds can be interpreted,
etc . . The ca tas t rophe approach "implies that a non-bifurcation point is one at which
the topology does not change, i.e. a point at which the system is s tructural ly stable.
... bifurcat ion is seen as a loss of (structural) stabil i ty of the system, ra ther than the
fo ld
loss of s t a b i l i t y o f a p a r t i c u l a r so lu t ion" (STEWART, 1982). To avoid n o m e n c l a t u r e
c o n f l i c t s , t h e s e t opo log ica l b i f u r c a t i o n s a re c a l l ed c a t a s t r o p h e s . B e c a u s e of t he i r wide
m e a n i n g and t h e i r t opo log ica l o b j e c t i v e s t h e y a p p e a r wel l s u i t e d for new a p p r o a c h e s
o f g e o m e t r i c a l r e a s o n i n g in geo logy .
E L E M E N T A R Y C A T A S T R O P H E S
P O T E N T I A L S A N D C R I T I C A L SETS
c u s p
v ( x ) = ½~3 + ux ~I
vlxl -- ~x4, ~x~ + V X
f / \
s w a l l o w t a i l
v~xl~ ~s+ ~3+ ~x~+ ~
216
h y p e r b o l i c u m b l l i c
V(x,y) = x 3 + y3 + w x y - ux -
e l l i p t i c u m b i l i c vY u
V(x,y) = x 3 - 3xy 2 + w{x 2 +y2) _ x -
T h e s e a r e on ly t h o s e e l e m e n t a r y c a t a s t r o p h e s wh ich a r e e a s i l y g r aphed , For a d e t a i l e d
l is t s e e one o f t h e t e x t b o o k s .
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