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DIGITAL SIMULATION OF SEISMIC RAYS Supplement to Final Report
1 June 1966 Through 30 May 1970
Prepared for Geophysics Division Air Force Office of Scientific Research Arlington, Virginia 22209
By P. L. JACKSON
August 1970
SEP 20'm
GEOPHYSICS LABORATORY
INSTITUTE OF SCIENCE AND TECHNOLOGY
Sponsored by Advanced Research Projects Agency Nuclear Monitoring Research Office Project VELA UNIFORM ARPA Order No. 292, Amendments 32 and 37 Contract AF 49(638)-1759
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8071-33-Fi
DIGITAL SIMULATION OF SEISMIC RAYS Supplement to Final Report
1 June 1966 Through 30 May 1970
Prepared for Geophysics Division
Air Force Office of Scientific Research Arlington, Virginia 22209
By P. L. JACKSON
Sponsored By Advanced Research Projects Agency Nuclear Monitoring Research Office
Project VELA UNIFORM iRPA Order No. 292, Amendments 32 and 37
August 1970
Gcophytici Loberotory
THI INSTITUtj Of SCItNCj AND ttCHNOt.TQT
T H I UNIVIISItT Of wiCHiC*N
Ann Arbef, Michigan
'
FOREWORD
The research described in this report was conducted by the Geophysics Labora- tory of Willow Run Laboratories, a unit of The University of Michigan's Institute of Science and Technology. The work was performed as part of Project VKLA UNIFORM, sponsored by the Advanced Research Projects Agency and monitored by the Air Force Office of Scientific Research under Contract No. AF 49(638)-1759. The research period extended from 1 June 1966 through 30 May 1970; the Project Scientist is Mr. William .1. Best.
The principal investigators for this project were P. Jackson, R. Turpening, and D. Willis. This report was submitted for publication in July 1970. The Wtllow Run Laboratories' report number is 8071-33-F|.
ili
PREFACE
The investigation reported in this dissertation was conducted at the
Geophysics Laboratory of the Institute of Science and Technology, The
University of Michigan, over a three-year period. During this period the
preliminary results of this work have been described in presentations,
reports, and a journal article.
On April 5, 1968, a slide presentation of computer-drawn plots was
made to the Geophysical Advisory Committee of the U. S. Air Force Office
of Scientific Research, Alexandria, Virginia.
On April 13, 1968, a paper concerning this work was presented to
the Annual Meeting of the Seismological Society of America, in Tucson,
Arizona (Jackson, 1968).
A major report, including listings and flow diagrams of computer
programs was submitted In September, 1968 to the Air Force Office of
Scientific Research (Willis and Jackson, 1968).
On October 5, 1969, a paper was presented to the Annual Meeting of
the Eastern Section of the Seismological Society of America in Blacks-
burg, Virginia (Jackson, 1969).
A Journal article describing the application to a spherical earth
appeared in the Bulletin of the Seismological Society of America in
June, 1970 (Jackson, 1970).
During this three-year period numerous progress reports on this
investigation have been submitted to the Air Force Office of Scientific
Research and the American Petroleum Institute.
Ttii' latlior would like to thank David E. Willis, James T. Wilson,
Charles C. Ruf.", Henry N. Pollack, Paul W. Pomeroy, Timothy C. Swanson,
and Rom.>r M. Turpening for helpful suggestions and encouragement. He
also would like to thank Mrs. Clara M. Randazzo for Invaluable help in
preparing the manuscript.
Vl
ABSTRACT
Simulation of seismic rays for a spherical earth and a flat earth has been
achieved in highly complex models. Travel times and approximate amplitudes of
seismic waves can be found for both two- and three-dimensional models of portions
of the earth. In seismology and other disciplines ray construction customarily has
been applied to simplified geometries. It has been necessary to assume that the
seismic wave velocity distribution of the earth was relatively uniform and symmetric.
Recently, however, the earth has been found to be more complex and non-
uniform than formerly assumed. A need has thus arisen in seismology to test highly
heterogeneous models of seismic velocity distribution. At the same time the devel-
opment of the modern digital computer has provided a means of performing the
necehsary ray constructions and numerical calculations.
The problem at complicated seismic velocity distributions was therefore investi-
gated in terms of the most appropriate use of the digital computer. For this investi-
gation a velocity field was set up, and the propagation mmputatlnns made for short
segments of rays within this field. Total travel timt's are found by adding the travel
times of connected ray segments. Essentially, the nature of propagation was dupli-
cated on the computer, In that, at the location of fach Kigiiifnt along the path of propa-
gation, the initial condition and effect of the surrounding* dclrrmini- the »ucrording
direction of the following segment.
Both visual aiid numerical results have shown that this simulation method can
be usefully applied to Investigation of seismic velocity distributions of portions of
the earth of any size or complexity.
vll
CONTENTS
Prefaco li
List of Fi^utvs v
Introduction 1
Two-Dimensional Simulation 9 Flat Earth 9 Spherical Earth 12
Three-Dimensional Simulation 14
Auxiliary Computations 18 Travel Time 18 Approximation of Amplitude 19 Reflection 20 Multiple Reflections and Refractions 20 Interface Location 22
Computer Programs 22
Resul tH 24 Accuracy 24 Plane Waves 26 Spherical Earth 27 Three-Dimensional 27 Potential Application to Seafloor Spreading Problem 29
Conclusions • 33
Appendix: Listings of Computer Programs ••• 57
ix
FIGURES
1. Reference Frame for Incrementing Rays. 37
2. Computer Output Using Direct Ray Simulation for 38 Five P-Rays,
3. Propagation of P Through Hypothetical Crust ind Upper 39 Mantle Structure.
4. Propagation of PcP Through Hypothetical Crust and Upper 40 Mantle Structure Shown in Figure 3.
5. Plots of P. PcP, PKP, PKIKP, PKKP, and PKIKKIKP Rays 41 Generated with a Spherical Earth Program.
6. Multiple Reflection. PcP and PKP Rays up to PKKKKKP. 42
7. Uniformity of Plotting and Travel Times Symmetrically 43 Plotted from 700 km Depth.
8. Three-Dimensional Ray Tracing in a Region Defined by 44 30 x 30 x 30 Velocity Samples.
9. Line Source at Depth for Three-Dimensional Representation 45 as Shown in Figure 8,
10. Three-Dimensional Ray Tracing with Velocity Distribution 46 Defined by Continuous Mathematical Function.
11. Three-Dimensional Ray Tracing with Velocity Functions 47 Specified for Different Regions.
12. Ray Paths From Earthquake Above Underthrusting Lithosphere 48 at Continental Plate Boundary. Depth of Focus: 50 km.
13. Ray Paths From Earthquake Above Underthrusting Lithosphere 49 at Continental Plate Boundary. Depth of Focus: 100 km.
14. Ray Paths Frcm Earthquake Within Underthrusting Lithosphere 50 at Continental Plate Boundary. Depth of Focus: 60 km.
15. Ray Paths From Earthquake Within Underthrustlng Lithosphere 51 at Continental Plate Boundary. Depth of Focus: 110 km.
16. Earthquake Above Underthrustlng Lithosphere at Continental 52 Plate Boundary. Depth of Focus: 50 km.
Xi
17. Earthquake Above Underthrusting Lithosphere at Continental 53 Plate Boundary. Depth of Focus: 100 km.
18. Earthquake Within Underthrusting Lithosphere at Continental 54 Plate Boundary. Depth of Focus: 60 km.
19. Earthquake Within Underthrusting Lithosphere at Continental 55 Plate Boundary. Depth of Focus: 110 km.
xii
t
Introduction
Possibly no artificial construction has been more useful to science
than the ancient concept of the ray. Not only in seismology, but in all
scientific fields concerned with the transfer of energy, reasoning with
and construction of rays have resulted in fundamental advances.
The firm basis of rays in science is most dramatically shown in
optics. From ancient times to the present, light rays have been used
for intuitive visualization and have resulted in advances in optics and
related fields. Archimedes, Aristotle, Roger Bacon, Kepler, Newton,
Young, Fresnel, Rutherford, and Conpton all used the ray concept. Newton's
Optiks (1730; 1952) is probably the most lucid example of rays as both
a rationale and a means of intuitive understanding and imaginative dis-
covery; almost every figure in Newton's book is a ray tracing.
In seismology, with which this thesis is concerned, the use of rays
aided Mohovoricic (1910) in discovering the discontinuity between the
mantle and the crust; Gutenberg (1914) inferred the existence and esti-
mated the size of the earth's core with the aid of rays; and, in theor-
etical analysis, Zoeppritz (1919) employed the ray concept to derlvt the
relationships of reflected and refracted dilatational and distortion«!
waves at an interface.
Current selsmologlcal contributions to the knowledge of the earth
continue to rely on the concept of rays. Current issues of scientific
journals in seismology usually contain several articles which are il-
lustrated by and rely on seismic rays.
In scismics r^st ray tracing has been based on Herglotz-Wiechert
formulation as extended by Bullen (1963), Helbig (1965) used this geo-
metrical construction for a graphical method for spherical shells,
Sph ically symmetric ray tracing has also been described by Julian and
Anderson (1968) and shown by Lewis and Meyer (1968). Engdahl, et al,
(1968) employed the formulation for the 1968 P-Phase Tables (Taggart,
et al., 1968).
Iyer and Punton (1963) broke away from rigid geometrical limita-
tions in constructing successive wavefronts by applying Huygens' princi-
ple involving complicated logic. Yacoub, et al, (1968) also departed
from rigid geometry by considering regions of constant velocity in which
the interfaces can be oriented In any direction. He also computed the
amplitudes by Zoeppritz's equations for rays crossing an Interface.
The ray simulation described In this dissertation Is based upon a
new concept leading to the treatment of heterogeneous structures and the
travel-time solutions for any discretely specified and/or analytically
represented velocity distributions.
An advance in the art of tracing rays Is thus a significant contri-
bution to the science of seismology; and, by extension, to other sciences.
Currently two aspects of recent scientific and technological development
provide, first, a new requirement and, second, a practical means of
satisfying this requirement.
The requirement to Investigate non-slmpllfled, heterogeneous velo-
city distributions, has arisen from the discovery that the earth Is less
symmetric and more heterogeneous than previously supposed. More accurate
and extensive seismic measurements are currently being made. The need
for greater understanding has made simplifying assumptions of the velo-
city distribution of the earth less useful. Tracing rays through
heterogeneous velocity structures and determining their travel times
would be useful in gaining detailed understanding of the earth.
The practical means to achieve the tracing of rays and determina-
tion of travel times through complex velocity distributions is the em-
ployment of the modern digital computer. Complicated velocity distribu-
tions imply unpredictable velocity gradients. In large regions a ray
must be constructed with many different computations because of changes
within relatively small regions. With a digital computer one can carry
out successive computations, which enables one to treat the complicated
velocity distributions indicated by modern seismology.
With these two aspects in mind—the requirement and the apparent
means of fulfilling the requirement—a preliminary trial was made to
develop a new method of ray tracing. The point of departure was to
somehow propagate rays through some type of mathematical or representa-
tional cross-section of a velocity distribution. The approach was to use
a sampled structure into which a short segment of a ray could be intro-
duced at a given location and with which the directions and positions of
successively added segments could be found as functions of the velocity
distribution represented by the structure.
A rectangular two-dimensional grid of equally-spaced points was
considered. The grid points represented positions on a vertical cross-
section of the earth, each point corresponding to a given horizontal and
viTtical (list.im.- from a reference point. A "sampled" velocity and slope
would tnrr.spond to each point of the Rrid. A means was then required
of usin^ the velocities and slopes, and the relationships between neigh-
boring samples of their values, to "propaßate" short successive segments
of the rays through the sampled velocity distribution. This means was
found. An introductory description follows.
Hie aforementioned grid was considered as a matrix in which the
rows represented discrete horizontal distances along the cross-section,
while the columns represented discrete vertical distances, as shown in
Figure 1. Such a matrix can be represented as a vector with two Indices.
Call the vector V , in which i represents the row, and j Che column of
the matrix. For each set of indices (i,j) a position on Che cross-secCion
can bo identified. For example, let Che inCerval bcCween Che samples be
10 km. Then the indices (i ■ 1, j ■ 1) could represent Che reference
point (0,0) at the surface of the earth and on Che cross-secdon; Che in-
dices (i - 10, j - 1) represent a distance 90 km from Che reference polnc
on the surface; the indices (I - 5, j ■ 3) a horizontal distance 40 Ion
from the reference point at a depth of 20 km. The indices must be inte-
gers greater than zero to represent Che macrix in Che computer. For Chls
reason the smallest pair of indices (1,1) are taken Co represenC an ori-
gin (0,0). This bias is easily handled, as Che source can be anywhere
within the bounds of the matrix, provision for measuring from the source
location must be made.
At each location defined by two indices there corresponds a velocity
V . and a slope S... These values at discrete points can be referenced i j ij
fron locations closest to then. A ray, however, is a succession of
short segments of possibly differing directions. These sepients may be
Joined together nt non-discrete locations, say the coordinates (h,d),
where h is horizontal distance and d is depth in a cross-section of the
earth. At each matrix location (h,d) a computation must be made to de-
temine the direction and hence end position of a succeeding segment.
The ray segnent midpoints could Chen be referred to the nearest discrete •
(itJ) location co determine Che velocity V and slope S. For example,
consider the scale, or stapling Incerval as shown In Figure 1 was 10 km,
ard Che ray segnent midpoint was located wich respect to Che matrix at
h • 4.75, d • 3.25 corresponding Co H - 37*5 ka and D - 22.5 ka on Che
referenced cross-secdon of Che earch. The velocity V and slop« S would
be found ac Che locadon IndlcaCed by Che Indices (1 • 5, J ■ 3), and
could be easily referenced; If desired, one could Incerpolace Che values
beewcen Chose found aC (1 - 5, j ■ 3) and (i - 4, J • 4).
AC Che nidpoint of every ray segnent a value for velocity and slope
Is found. For geonecrlcal ray tracing, Snell's law requires the knowl-
edge of an Incident direcdon wich respect to an Interface and Che velo-
clcy of Che nedlun on elcher side of Che interface. The representational
structure described above supplies this Infomation. The direction of the
IncldenC ray segnenc and Che slope of Che Incerface a.c known, therefore
Che IncldenC angle for Snell's law can be found. The velocity V. of Che
nediun on Che IncldenC side of Che Incerface Is found as described In Che
lasc paragraph. The veloclcy V. on Che energlng side can be found by ex-
Cending Che ray segnenc in Che saie direction as the -ncldent ray for
the Httinmeni , .i^ain using the technique described in the last paragraph.
Fur .1 fixed rectangular Cartetiian coordinate eystea, a horizontal
distance can he represented at» L sin 6 and a vertical distance as L cos 0,
where I. U a length of a ray segment and 6 Is the angle fro« the up-
ward vertical. 0 is positive in the counterclockwise direction. Now
consider that the head of an incident ray sagaent of angle 0. located at
(h. td. )t and, hy invoking Snell's law the anerging angle is found to he
** , a» shown In Figure 1. The location of the tail end of the eaerging n
ray segment is found to be h ■ h. + 0.3 L sin 6, d ■ d. ♦ 0.5 L cos 8.
As the direction and location of the eacrging ray sagaent as wall aa
the velocities and slopes of the isaediate surrounding regions arc known
whenever they may he within the defined aatrix, the eaerging ray sapient
can he redefined as the new incident ray aapwnt, and succaading seg-
ments generated and Joined end-to-end until a coopletc ray is forasd.
Travel times of the radiation of seisalc waves froa the disturbance
to the HeiNioometer are of fundauntal importance in seiaaology. The in-
formation to determine travel times is available with this ray tracing
method. Each segment of the ray is located in a region In which the
velocity is known. The travel time for each segaent can be found by di-
viding the length of rhe seprntt by the average velocity along its
length. The travel cime for the ray is then the sum of Che travel tiaes
of the sepnents. The distance of travel la found by suaalng Che lengths
of the segments. This distance is useful for computing spherical spread-
ing and logarithmic attenuation.
6
The approach described above was teated with a conputer prograa.
The initial reaulta were proaising, and Che ideas seemed Co be valid.
A aeiamic ray could he simulated through a matrix representing a sampled
velocity distribution in a vertical cross-section of the earth. The in-
veatigation to produce a useful, accuratec and versatile seismic ray
simulation was then commenced. The ensuing Investigation waa mainly de-
voted to attacking the following problems:
1. Accommodating ray angles through the entire rangt of 360* ( and
of using any defined slope In aasociation with any given ray angle,
2. Reflection from an interface,
3. Both critical and non-critical multiple reflections,
4. Precisely locating an Interface between matrix-designated dis-
crete locationa and extending the ray to thia interface,
5. Developing and testing a method to obtain high accuracy, of
determining travel times and approximate amplitude in a multiply-reflec-
ting model,
6. Adding the capability of apecifying regions of velocity charac-
teristics defined by an analytical mathematical function,
7. Treating both curved and flat earth,
8. Applying the concept f a throe-dimensional model, and
9. Making the method general so that one program can be used for
many different regions of the earth.
A versatile tool has been developed for the scientific investiga-
tion of velocity distribution in the earth. Any conceivable velocity
distribution can be almulated and cohered to actual travel times in
investiiMtiuna where Reoaetrlcal ray approxlaatlons are valid. In addi-
tion, the tifthod 1M expandable, and can be uaed aa a baae for «ode con-
version, the .iddition of Zoepprics'a equation! for anplltude, extenalon
to the jtmoNphere where winds provide a Moving aediu«, and, possibly,
for Reoaetrlc diffraction.
The simulation of seisaic rays can be performed with sicher two» or
three-dimensional models. Flat ssrth modsls naturally accowodate them-
selves to a rectangular Cartesian coordinate system. Curved earth
models, which at first glance would appear to be better treated in polar
coordinates, are also trested in a rectangular Carteaim system becauss
the velocities and slopss for a spherical earth are easily referer.ccd.
Also the compatibility of coordinate systems between flat and spherical
earth enables one to inssrt ssmpled values for anomaloua regions within
a spherical earth model.
Aa first conceived, it wss thought thst the two-dimensional techni-
que of ray tracing could be incorporated into the three-dimensional
technique, and two-dimensional tracing performed when holding the values
along one of the axes of the three dimensions constant. However, the
use of three-dimensional models required s sufficisntly different tech-
nique that the two types sre better explained ssparatsly. The descrip-
tion of the use of the three-dimensional modal is not self-contained.
To avoid repetition of identical material, pertinent matter presented in
the two-dimensional section will be referred to when describing its use
In three dimensions.
8
Two-Dlacnstonal Sliulatlon
FUt Earth
Both discrete values assigned to a matrix and continuous mathemati-
cal fisictions may be employed to reference velocity. To aid in visuali-
zation, the following exposition is based on the discrete case.
Construct a two-dimensional scalar field representing the seismic
velocity characteristics and slopes along a plane in a medium, such that
V - f(xft) fl)
S - g(x(t) - h(V) (2)
where V is the velocity (actually the speed with which seismic P-waves
propagate, but to be referred to henceforth as velocity, in keeping with
common practice In seismology), f, g, and h are functions, and x,s represent
distances parallel to the axes of a rectangular Cartesian coordinate sys-
tem. The functions f and g may be defined differently for specified regions
of the field, and can Include both continuous mathematical functions and
references to tables of arbitrary and discretely changing values. The
function g may be derived directly from f as the normal to the gradient,
or specified separately. One might term S a vector, as it corresponds
to a direction and Is based on the gradient. However, It is defined as
an angle and Is used as a scalar quantity in the computation.
Consider a matrix of discrete values, the rows of which represent
equally spaced horizontal positions within the scalar field. For each
element of the matrix a velocity and slope Is assigned which corresponds
to the location represented In the scalar field. Values for velocities
and slopes can be found In the scalar field at any point not corresponding
9
to thf lii.. i.tf loc.itlons represented by liie n..itrix elements. These
values art* foiiiid bv either USIMR those of the matrix elements to which
the location is closest or Interpolating betwi-en the nearest horizontal
and vertir.il elements of the matrix. Thus for any position in the sca-
lar field, the value for the velocity and slope can be found by refereme
to the nearest element of the matrix.
As arbitrary values can be stored in the matrix, any type of velo-
city distribution can be referenced to the scalar field. Through such
a reference system a completely heterogeneous velocity distribution can
he used for ray tracing; the limits of heterogeneity are limited only by
the detail with which the matrix elements represent the scalar field.
That Is, by the distance represented by two adjacent elements of the ma-
trix. The matrix with the velocities and slopes represented as its ele-
ments is necessary to arbitrarily represent velocity distributions, but
not to perform ray tracing through the scalar field.
Select a position x. , z. for the head end of an incident ray seg-
ment of length L and an angle 0. (O^eOöO*). The midpoint of this seg-
ment is the location which is used to determine the incident velocity
V. in the manner described above. Reference 6. to the negative z-axls,
counterclockwise positive. The z-axis, which represents depth in the
earth cross-section, is positive downward. Extend the ray segment a
distance I. in the direction 9,. k
The Initial emerging location P. is found by extending the ray seg-
ment at the incident angle 6. , shown in Figure 1, using the two following
equat inns:
10
x = x, + L sin 0. (3a) p k k
z » z. + L cos 9. (3b) p k k
The midpoint of this extended ray segment is the location which is used
to determine the initial emerging velocity V . m
Sufficient information (position and angle of the ray, and the
velocities and slopes surrounding the region of the ray) is available
to construct the ray continuation by Snell's law. The equation used for
this purpose Is
em - ARCSIN[(Vm/Vk) sin (6^)] + Sk (A)
where 0 is the angle of emergence, V is the emerging velocity, V. is
the incident velocity, and Sk(-90<,<S$90o) is the slope of the Interface
referenced counterclockwise positive from the x-axis. In case of con-
tinuous velocity functions S. is the normal to the direction of the
velocity gradient. The extension of this segment is termed "probing"
in the sense that one must repetitively probe for the proper emerging
angle 6 . For the first repetition 6 replaces 6 in Equation (3)
and P , P are recomputed. A new 6 is determined using the newly-found
V in Equation (A). Equations (3) and (4) are repeated in succession, m
each time using the previously determined value of 6 for 6. in Equation
(3). The repetition is terminated when successive values of 6 are suf- m
ficlently close in value (Id - 6 ,|<c, where c is a predefined small m m—i
value).
This repetition is necessary because the location of the extension
computed with the incident angle is in general different from the loca-
tion computed with the emerging angle. If the ray is penetrating into
11
a region of increasing velocity the emerging angle between the ray segment
and the normal to the interface will be too large. If penetrating into
a region of decreasing velocity this angle will be too small. These
improperly evaluated angles are found because the tentative ratio V /V. m K
may be a different value from the ratio as found when extending the ray
with the correct emerging angle. One can asymptotically converge to the
correct emerging angle bv repeating these computations. These repeti-
tions are performed to improve accuracy for computations using any given
ray segment length L. Linearity of the velocity function is assumed with-
in the distance L. Also, to improve accuracy one might reduce the seg-
ment length L, as the accuracy is an inverse function of L, and L is not
required to be equal length for all ray segments. However, it still
must hold that the difference between two successively-determined emer-
ging angles 0 would have to be a small value.
When the sufficiently accurate 6 is found by repeatedly applying m
Equations (3) and (4), the emerging ray segment and angle is determined.
This emerging segment and angle is then treated as the incident segment
with a new location and angle. Segments are repeatedly computed, the
tail of the "emerging" segment joined to the head of the "incident" seg-
ment repeatedly until the ray, which is the sunmation of all the seg-
ments, penetrates to the surface or the boundary of the defined cross-
section of the earth.
Spherical Earth
Consider a velocity distribution with depth for a symmetric earth.
12
One can compute the velocity and slope for any position (x,y) within a
circle representing the cross-section of the entire earth. The velocity
is found as follows:
Vk - f(R) = f[(xk2 + yk
2)?'2] (5)
where f is a function of the radius R; V. is found for any location de-
fined by (x, ,y.) from either a table and interpolating, or by computing
some mathematical function of the radius R ■ (x2 + y2) 2.
For each location defined by (x, ,y, ) a slope S. can be found,
Sk - ARCTAN (yk/xk) ± TT/2 (6)
where y and x are the coordinates of a given location, y-0, x*0 are the
coordinates of the center of the earth, and the sign of 71/2 depenas upon
the quadrant In which y and x are found.
The angular distance at which a ray emerges Is found by
A - ARCTAN (yk/xk) (7)
where the proper quadrant is determined from the relationships of the
signs of x and y, and where x and y are located within a small predefined
distance from the surface of the earth.
The depth of penetration, or, correspondingly, the minimum radius
of the ray upon reflection is found by computing the minimum radius along
a given ray, and retaining that radius to correspond with the other pre-
viously described data of the emerging ray.
Any region which is defined by minimum and maximum values
or functions of x, y, or R can be postulated so that a different function
can be invoked. For example, one might wish to investigate the travel
times of rays which penetrate an anomalous region in an otherwise symmetric
13
region of the earth. Within the limits defined, any sampled value can
be inserted for the positions defined by x and y, as in the flat earth
case.
Three-Dimensional Simulation
Construct a three-dimensional scalar field representing the seismic
velocity characteristics of a medium, such that
V - f(x,y,z) (8)
Sj^ - g(x,y,z) (9)
S2 - h(x,y,z) (10)
where, as in the two-dimensional fields V Is the velocity, S. represents
the slope with respect to the z-axls, S« the slope along planes parallel
to the x,y plane, and f, g, and h are functions. Positions within the
field are referenced in a three-dimensional orthogonal Cartesian coordi-
nate system. Directions within the field are referenced In a spherical
coordinate system, in which the angle from the positive z-axls is
4)(0<(|»<-rT) and the direction from the a-axls in a'plane parallel to the
x,y plane is the angle 9(0<9<2Tr), the same as that described for two di-
mensions. A segment L oriented in a direction defined by (j) and 6 pro-
jects a distance P on the z-axis z
and on the x-axis as
and on the v-axis as
P - L cos (() (11) z
P - L sin (j) sin 6, (12) x
P - L sin (J) cos 9. (13) y
14
To visualize the role of the two slopes ^j (-TT/2<S1<7T/2) and S.,(0. S2<2TT)
more clearly, consider an interface or a plane normal to the gradient
which lies obliquely to all of the axes. Then the slope S Is the di-
hedral angle between the z-axis and the plane of the interface
(when interface is mentioned it also refers to the normal plane to the
gradient). The slope S. is found by the angle of the line which defines
the intersection between the x,y plane and the plane of the Interface,
and is referenced as in the two-dimensional case.
The coordinate system is a right-handed system in which the z-axis
is downward. Thus when a region of the earth is represented positive z
values correspond to depth from the surface of the earth, or from an
arbitrary horizontal boundary. Although an orthogonal Cartesian coordi-
nate system is employed for position, a new position can hs determined
by employing a length L, and the angles $ and 6. The two coordinate
systems are complementary.
We see from the foregoing definitions that, as in the case of the
two-dimensional system, we have sufficient information to compute Snell'«;
law In fhe three-dimensional system: L, ^, 0, x, y, z, V, S., and S..
However, in the three-dimensional system the algebraic and trigonometric
processes become much more Involved.
Consider a velocity distribution in which the gradient is everywhere
parallel to the z-axis as defined above. Such a distribution would cor-
respond to a region of the earth in which all interfaces were parallel
to the surface of the earth. Therefore, no velocity change would occur
along a horizontal direction. With such a velocity distribution, the
15
i<ri,(in.tl .iiv.l>' ". of » ray would never change» althoufth It* «ign night.
This Is Sft-jiwe V • V, and, therefor« P • 6. , a« aeen by reference
to Equation (4). The «traightnefta of the ray ia illustrated in the
orthoRraphii- projection in the x.y plane of Figure 8. The total change
In direct ion will be in the angle 4, which can be computed aa
♦B • ARCSIH|(Vi|/Vlt) aln^-S^J ♦ Slk (14)
where the Ruhacripta ■ and k refer to emergence and Incidence aa in the
two-dimensional caae.
Uc now consider a scalar field in which the interface* «ay be any
orientation in three-dimensional apace; Che condition described in Che
last paragraph would not hold for this caae. If Che coordinaces were
oriented in such a manner that Che t-axi* waa perpendicular Co Che inter-
face, then the condition holds where 6 does not need Co be recompuced
and a simple expression for ♦, Equation (14), can be employed. AC any
Riven location within Che field Che anglea 9. and ♦. are known, aa well
as the »lopes S . and S. . If a coordinate system can be found in which
the r-.ixls Is perpendicular to the interface defined by S.. and S,. , Chen
the calculation of Snell's law for 4 in Equation (14) needs only Co
he used.
Such a coordinate system can be found by rotating the axes as a
function of S.. and S... After rocation Che positions x,y,s are referenced
by x'.y^z' and the angles $ and 6 by 4* and 6* to Che new oriencations
of the coordinate system axes. Snell's law can then be invoked with para-
meters referenced to the new coordinate system. The new position is
found as fellows.
16
Let S* b« Che »igle by which the x- and y-asee are rotated around
the s-aKis, and S! the angle by which the x'-axia ia rotated about the
y'-axia (or, viewed differently, the angle with which the s-axia ia
tilted to becoae normal to the interface.) For a continuoua ■atheaatl-
cal function angle S* ia 2
SJ • (ii/2) - arctan(AX/AY) (15)
where AX and AY are directional derlvativea along the x-asia and v-axia
respectively. S* is found aiailarly aa
8J - (»/2) - arccos(AZ/AX> ♦ AY2 ♦ AZ1)1* (16)
where AZ is the directions! derivstive along the s-axia. For ssapled
functions S! and S^ are
S'j- (ii/2) - SjU.j.k) (17)
and
S'j- S2(l.J.k) (18)
respectively.
The coordinste sxss can then be rotated, and correaponding location«
in the new coordinate syataa found by rotscing first sround Che s-axia
and next sround the y'-sxls:
X- • X cos (S*2) ♦ Y sin (Sj) (19)
Y' - Y cos (SJ) - X ain (Sj) (20)
X» - XM cos (S|) - Z sin (SJ) (21)
Z* - Z cos (SJ) ♦ X- sin (SJ) (22)
The angles 6* and +* are found in a aiailar Manner by coaputing a
position with trigonoBstric functions, rotating the coordinate axea, and
caking Inverse trlgonoastrlc functions of the sngles of a directed vector
froa the origin to the posicion In Che new coordinste systea.
17
Wh*n f.K-h Hfitpient of the ray is computed, the new position can be
round bv rotating th« coordinates back into Che originally specified
i-oordinatf system. From this position in the given coordinate system
the incident velocity V Is known, and by repeated computation in the
manner deticribed in the section on two-dimensional tracing Che proper
emerginK .wu;!«— •> and ; can be asymptotically approached Co within a fli wk
predetermined small value.
In returning the rotated coordinate system, the values V* and V* k m
required tc compute Snell's law to determine the emsrging angle A*, are
VJ - Vk (23)
'i " Vk * (Vm " Vk) co• ^ (24)
where, again, the prims refers to values In the new coordinate system,
.md .*' refers to the repetitively probed value to correctly determine V . m m In the rotated coordinate system we are concerned with only one
direction of reflection, so this can be handled in the sams manner aa in
F.quation (30) of the next section. The only angle affected is A', aa 6*
will he unaffected. However, in the originally given coordinate system,
the direction of the ray may reverse itself in its projection on the
x,y pl.m«-.
Auxiliary Computatlona
Trawl Times
AH each segment of the ray is found and added on to the preceding
Hegmentfl, the travel time can be determined. The time of travel for
each *ean<>nt is simply the length of the individual segment L divided
18
by the average velocity V at which It la propagating( ao that the total
travel time TT of a ray la n
TT - [ L /V (25) aM
where n la the total nunber of aegaents fro« the lay's origin to ita
eaergence on the surface.
ApproKlaatlon of ^»plltude
The total diatance D a ray haa travelled la n
D - J L (26) a-1 a
Knowing chia diatance, one can coapute the aaplitude dinenalon due to
apherical apreading and the exponential attenuation to energy dlaaipa-
tlon aa
^ - Aa(l/D) axp(-cD) (27)
what« A la the aaplitude at the receiver, A la the aaplitude at the
aourca, and c la the attenuation coefficient.
In caae of non-critically reflected raya and the enaulng refracted
raya, the partition of energy la rigoroualy coaputed by Zoepprits's
aquatlona. The approxlaat.ve approach uaed here la that of Praanal'a
reflection coefficient at notaal Incidence. At a reflective interface
where non-critleal reflection occura the approxlaate reflected aaplitude
Arl.
Ar • \(Vk * Vl>/(Vk 4 Vl> (28)
and the aubaequantly refracted «plltude la
Vl • \ - Ar (29)
19
iU-f loction
Reflrction is required whenever an interface of auffir lent velocity
difference Is encountered by the ray, or whenever the critical angle it
exceeded. At such a boundary the reflected angle 8 la
9t - w - ei ♦ 28J 1 2nir (30)
where n is -1, 0, or 1, ao that the condition 0*d<2ir la fulfilled.
Multiple Reflections and Mfractlona
When the ray is both reflected and refracted at an interface, it
splits into two separate raya. To accosaodate thla fact, and to trace
out both ray branches, the inforaation of the angle, poaition, total
travel time and distance to the interface, and approxiaate aaplltudes
cosiputed by Fresnel'-. reflection coefficient at nons«! incidence auat
he stored for one ray branch while the inforaation for the other is
employed for further ray tracing. Ihia is accoaplished by continuing
with the reflected ray until it eaerges or strikes another interface.
In emerRing at the surface all the required data, auch aa distance,
total travel tiae, approxiaate aaplitude, and the depth of penetration
of the ray is available. The depth of penetration is found by coaparing
the depth at all segaents of the ray, and retaining the poaition aost
distant from the surface. One then returns to the previous interface
from which non-critical reflection occurred, and, uaing Che inforaation
listed in the laat paragraph which haa been retained, continues with the
refracted ray. As an enseable of inforaation can be retained froa pre-
20
vlouM reflections, on« Is sble to accoModst* «s many aultiple reflec-
tions as desired.
The Multiple reflections nay be accomodated in the following Ban-
ner. Consider arrays of any arbitrary number of element« for each para-
•eter needed to construct a ray, say
H(l),V(l)te(i).TT(i),Tl(l), and A(i), (31)
where H Is horizontal position, V is vertical position, 6 Is angle of
the first refracted segment, TT is trsvel time to the interface under
consideration, TL is total path length to the interface, and A is aapli-
tude; the index i in each value in (31) represents the reflection nuaber.
On« starts at the inception of a ray with ial. At the first non-critical
reflection, the refracted values are stored under the index 1"1, and the
reflected under the index i>2; aiailarly, if another interface with
non-critical reflection is encountered, the ref acted ray is given the
index l"2, while the ref» *y is given the ndex i"3, and so on.
For each non-critical reflected ray the index is increaented by 1, while
the corresponding refracted ray is referred to by the previous index.
It is seen that any arbitrary nuaber of reflected rays can be acconao-
dated and traced by increasnting indlcea in this aanner. After any
given reflected ray haa reached ita destination the index i is reduced
by 1, and the previously refracted ray la then continued by using the
initial conditions of the paraasters Hated in (31) with the next lower
index nuaber. It can be seen that all refracted rays will be traced out
until the index reverts ro 1*1, which is the index nuaber of the inci-
dent ray. Also, because previoualy refracted rays nay subsequently
21
encounter .1 new set of multiple reflections, all multiple reflections
of the originally reflected or refracted rays will be accommodated.
Interface l.ocatlons
Because interface locations are so critical In affecting the loca-
tion of the emergence of a ray, a means was developed of precisely lo-
cating the interface and extending or retracting the ray to terminate
precisely on the interface. This is accomplished in the tollowlng man-
ner. The location of a point Is found in the square defined by four
matrix elements and located on the Interface. The horizontal and verti-
cal distance from the ray segment head end to this point is determined.
Knowing the slope of the interface, the angle of the ray, and the
horizontal and vertical distances to the defined point, a series of
computations In which the law of sines is invoked is then made. Many
different angular conditions are involved as a result of combinations of
ray angle, slope, and positive or negative horizontal distances. For
details, one can consult the subroutine GINMAD in the Appendix, where
the computations are given in Fortran.
Computer Programa
Three main programs have been developed. The first is for two-
dimensional flat earth, the second for two-dimensional spherical earth,
and the third for three dimensions. All have been written in the most
general computer language, Fortran. The version is Fortran IV-G, each
program of which has been compiled and run on the IBM 360/67 computer
22
of The University of Michigan. The programs Include plotting instructions
which have been used for the Illustrations.
Subroutines have been developed to precisely reflect at an interface,
to extend the ray precisely to the surface, to compute Snell's law for
any incident angle across an interface of any slope, to increment initial
horizontal and vertical positions and angles, and to compute reference
travel times for each ray making up an onset ting plane wave.
The indexing for the spherically symmetric case has been made com-
patible with that for velocity distributions with a rectangular reference
system, in such a manner that the data for horizontal layering structures
can be read in by a Fortran Namellst. In this way anomalous regions
which are very heterogeneous can be included.
The programs have been made general, so that a different number of
samples and size of sampling interval, number of multiple reflections,
incremented or changed output, number of initializations, etc., can be
entered without recompiling the programs. Thirty to forty values are
read from Fortran Namelists; the values can be changed individually.
Fach ray increment involves calling trigonometric functions approxi-
mately 7 to 25 times (usually about 15 times). For an indication of the
time required, see Figure 7. The computation for Figure 7, including
loading, 20 travel-time printouts, and plotting instructions, required
10 seconds of IBM 360/67 computer time. It is felt that this is reason-
able and economical time for such ray tracing.
The three main programs and their subroutines are listed in the
Appendix.
23
Results
Accuracy
The method of ray simulation was developed primarily with synthetic
data. First very simple data were used for testing and developmental
purposes. As the work progressed, it was felt that a test was necessary
to determine its accuracy.
For this reason ehe method for tracing rays in spherically symmetric
earth was attempted. For at least 60 years the earth has been investigated
as a spherically symmetric structure in which the velocity gradient every-
where points directly at the center of the earth. Much of seismology
has been concerned with this model. Because of the continued investiga-
tion of this model over most of the world for such a long period of time,
many travel times from individual disturbances at many distances have
been recorded.
From these travel times a velocity-depth distribution of the earth
has been inferred. This distribution is one of the fundamental problems
of seismology, and most seismologists have been concerned with it.
Recently Herrin, et al. (1968) published the 1968 Seismological
Tables for P Phases in a special issue of the Bulletin of the Seismologlcal
Society of America (BSSA). These tables represented the latest refine-
ments of the contributions of the science of seismology. In this
special issue a velocity vs. depth distribution of the mantle was given.
As both the velocity distribution and travel times were given in this
special issue of BSSA, a model was available to test this method of
ray tracing.
24
As one might expect, the ray tracing method was initially shown to
be inaccurate in the first tests. Two improvements were made in the
technique as it then existed. The first was to take the incident velo-
city at the midponnt of the inciden1: ray segment and the emergent velo-
city at the midpoint of the emergent ray segment. The second change
was to repeat the computation of Snell's law, as explained in the two-
dimensional modelling section, until the proper emerging velocity was
found. It was found with a testing program that the proper value of
emerging velocity was approached in a damped, oscillating fashion. It
was then a simple expedient to compare successive values until the dif-
ference between them was less than an arbitrary value.
This latter change, for which the need was not formerly apprehended,
was the key to achieving as much accuracy as desired, without reducing
the length of the ray segments to require an uneconomlcally large amount
of computation. As Snell's law is the basis of both this method and
that used for the 1968 P-Phase Tables (Engdahl, et al. 1968) the two
methods should result in identical relationships between velocity distri-
bution and travel times. Differences should be due only to the size of
the sampling intervals, length of the incremented segments, and method
of interpolation.
The equivalence between the two methods has been found. As the
sampling Interval and segment length were reduced, the travel times
shown in the 1968 P-Phase Tables have been approached more and more
closely when using the same velocity distribution.
25
The avuracy is Illustrated in Figure 2, where the velocity distri-
bution used for the 1968 P-Phase Tables was approximated with samples
taken at 15 km intervals with 9 km segments. Figure 2 is a reproduction
of a computer output using this ray tracing method and also shows cor-
responding distances and travel times from the 1968 P-Phase Tables, The
average difference between the two methods in travel times for the five
P-rays shown in Figure 1 is .252 seconds, or an average difference in
travel times between methods of 1/100 of 1%. As the standard deviation
of the P-Phase Tables is 1 second, .252 seconds is considered satisfac-
tory. Every indication is that the travel time vs. velocity distribution
values could be made closer by further reduction of the sampling interval
and segment length.
Plane Waves
Figures 3 and 4 were drawn to investigate the simulation of a plane
wave. The velocity structure is based upon a hypothetical velocity dis-
tribution within a cross section from the Pacific Ocean across the Coast
Ranges, Central Valley of California, the Sierras, and part of Nevada.
Figure 3 represents a PcP wave from A • 30° and Figure 4 a P wave from
the same distance. The simple subroutine Planit was used as a reference
for travel time across the wavefront. These figures were obtained in
May 1969 as an aid in Charles G. Bufe's investigation of PcP waves (1969).
Bufe was able to estimate PcP-P travel-time differences, and anticipate
arrival anomalies.
26
Spherical Earth
Figures 5 through 8 illustrate two-dimensional seismic ray tracings
of an entire cross-section through the earth. In Figure 5 only a single
reflection was allowed, enabling one to obtain PcP, PKKP, PKIKKIKP, and
P rays. Had no reflection been allowed below the critical angle of in-
cidence, only the PKP, PKIKP, and P would have been drawn in Figure 5.
With four reflections allowed one can generate PKP type waves up to
PKKKKP, as shown in Figure 6.
In Figure 7 a source at 700 km depth was simulated. Rays were
traced from this source at 18° intervals, and each ray traced inde-
pendently of the others. Note that the second K leg of the initial
18° ray is overtraced by the second K leg of its symmetric counter-
part the 360° - 18°, or 342°, ray. This overtrace indicates high
positional accuracy, in that A and travel times differ by less than
.01% between symmetric rays.
Three Dimensional
Illustrations of computer plots In three dimensions are shown in
Figures 8 through 11. To clearly show the paths of the rays in three
dimensions a computer program was devised to show a perspective and
three orthogonal projections. Figures 8 through 11 show a single-point
perspective in the upper left portion of the figures, and three or-
thogonal projections along the principle planes in the remaining portions
of the figures. Consider that the (x,y) plane represents the surface of
the earth, and the positive z-axis corresponds to depth In the earth.
27
In Figure 8 and Figure 9 a velocity distribution increasing with
depth .mJ (laviiu; zero slope everywhere with respect to the surface is
shown. In Figure 8 cones of rays from a point source at the «surface
are shown; ravs of a sufficiently large initial angle undergo critical
reflection and return to the surface (xty plane). In Figure 9 a line
source at depth is shown. Points are chosen along the line, and fans
of rays are propagated toward the surface. It should be pointed out
here that any type of initializing rays may be chosen. For example, in
Figure 9 the points of origin along the line could have been made much
closer together, and the angular extent of the fan decreased, with any
choice of angles between individual rays chosen. As described in the
section on the computer programs, the values for the increments of these
points and angles can be entered in the Nanellst.
Figures 10 and 11 illustrate an oblique gradient with respect to
the z-axis. The mathematical distribution of the velocities in Figure
10 is
V - 6.0 + X + 2Y + 3Z (32)
These illustrative figures make use of velocity values which exceed nor-
mal earth velocities. These velocities were used for the purposes of
demonstrating the capability of tracing such a distribution. In Figure
11 a sampled layer extending from a level corresponding to 40 km depth to
190 km depth is placed within the field described for Figure 9. The
gradient is parallel to the z-axis within this slab, and from 140 km to
190 km depth the velocities are constant. As shown in Figure 10, one can
define regions in three-dimensional space which can be either analytical
mathematical functions or sampled data. The number, sizes, or shapes of
the regions are not restricted.
28
Potential Application to the Seafloor Spreading Problem
In the theory of seafluor spreading, the earthquake zones are
associated with upward veiling zones, such as the Mld-Atlantlc Ridge,
where surface material Is being produced, and the boundary between
large lithospherlc pistes as In the Clrcum-Paclf1c zone.
The problem of these boundaries, one of which Is Impinging on the
other, is treated by laacks, Oliver, and Sykes (1968). They postulate
that one plate underthrusts the other, particularly in island arc re-
gions. At this boundary the lithosphere (the upper 100 kB of the earth's
surface) of the oceanic plate is bent downward into the underlying
asthenosphere of the continental plate. The precise geometry in which
the two plates Join is not known, and differs between regions. Isacks,
Oliver and Sykes (1968) postulated several configurations as a function
of local conditions and rate of underthrusting. Mitronovas, Isacks, and
Secber (1969) have investigated this problem in the Tonga Islands arc.
Murdock (1969) has discussed velocity distribution under the Aleutian
Region. Carder, et al. (1967) have documented travel-time anomalies
from the L0NGSH0T explosion.
Figures 12 through 15 were drawn with an arbitrarily-chosen geometry
baaed on the models postulated. In the geometry chosen the 6.75 km/sec
layer is taken as the topmost layer which bends downward and the surflcial
6.00 km/sec layer is unaffected by bending. These figures show different
earthquake locations and the directions of seismic waves propagating from
them. TVo of the earthquakes are directly above and two directly beneath
the upper boundary of the underthruating lithosphere. The two earthquakes
29
.»bow t h« ! .'unJarv .ire located In reKion» where volcanic activity and
mlnui s«is^i. .utivltv Is Indicated (Mitronova«, l*-.iiks, and Seeber,
I'»'«'»*. Tin iwv> i II t liquakes beneath the boundary are located where most
seismi, at n-ss and the lilgheHt probability of earthquakes are thought
to occur. T1K- rays (or thehe plots were generated at W radial incre-
ments from a point source.
AltiiiHi.-.!-. the frequency of earthquakes with sources above is less
than the frequency of those within the underthrusting llthosphcre, both
source regions have been shown for contrast. This contrast arise« fro«
trapped waves within the low velocity layer when the source is within
the layer. Ray» from within the layer critically reflect at the interface
at angles greater than 55* from the normal, to the Interface. Rays from
sources outside the low velocity layer can be critically reflected only
when the inclination of the interface through which the rays have entered
Is oriented differently from the interface through which they would have
otherwise emerged. Parallel interfaces bounding a low velocity layer
cannot trap rays whicn enter from outside the layer. In the model used
for Figures 12 through 13 critical reflection from externally generated
rays can only occur where the Interface of the low-velocity layer is
curved, as shown by the one critically reflected ray In Figure 12.
The effects of the low-velocity layer are clearly neen in Figures
12 through IS. Shadow zones are found at the surface when the source is
above the layer. These occur because the ray« penetrating the layer are
bent toward the normal of the interface. These rays are displaced from
the paths they wouldhnve taken in the absence of th« low-velocity layer.
30
and they emerge at a greater distance from the source than would other-
wise be the case. Fron sources within the low-velocity zone, the influ-
ence of critical reflections is shown in Figures 14 and 15. Three
separate groups of rays can be seen propagating up the low-velocity layer:
Chose initially reflecting from the upper interface, those which do not
Couch either Che upper or lower interface, and those Initially reflecting
fro« chc lower interface.
Travel times from each of chc four sources are shown in Figures 16
through 19. In Figures 16 and 17, the focal depchs are SO km and 100 km
respecdvely, with sources above Chc layer. The shadow tones caused by
Che ray dlsplacemencs are evidenc. Travel-dme anomalies above 2 seconds
occur near Che limit of Che shadow tones nearesC Che source.
Travel cimes for sources wichit Che downward bending layer are shown
in Figures 18 «id 19, where Che focal depchs are 60 and 110 km respecdve-
ly. Cricical reflecdons occur with a large angular excenc from Che
source. The Chrec separace arrival sequences are produced by Che three
groups of rays described above. The three separate legs of the travel-
time curves occur at distances approximacely Che same as Chose where Che
shadow tones are produced for Che sources above Che layer as shown in
Figures 17 and 18.
These resulcs indicate chac large effeccs upon travel times are
found within SO km of the poinc where Che underchruscing lichosphere
starts Co bend downward from Che unpercurbed oceanic place. A percinenc
invescigacion of seismic propagation in the Tonga Islands arc
(Hictonovaa, at al. 1969) was conducted with five seismometer locations
31
sp.ii t-j .it .:; '.nu is ,>i US km t.. 200 Un from this jiolnt toward the under-
thrust nu*. 1 i' no .ptiftf. rh< lu'art'st s«'i sraoneter location toward the oceanic
plat«- w.i> ipp'-cxim.itcl v JJ(» km distant 011 R.ir.it ..a,-.! in the Giok Islands.
I'nfortunat il v, tin- location of the si-isnuioeters were dictated by the
positions ot tht- .iv.iil.ihU' islands, and wer»- not within Che region in
which r iv ttiiiu»; HIKIWH shadow zones and multiplo arrivals.
To invest tgalc tht travol times '.. v.<iid the region shown in Figures
1J through I'}, tin- sphericjl earth p: .»•.i ■n was attached as a subroutine
to Che fl.it earth program. Many rays exceeded Che boundaries shown in
Figures '.: through 1'i wlchout reaching the surface. These rays wer«
used as sources for the spherical earth program. This was accomplished
by saving the location, angle, travel time, and distance cravelled as an
input to tlu- spherical earth subroucint*. Travel times out co approxi-
Riatetv HO0 were found. These closely paralleled the 1968 P-Phase Tables
for corresponding depths of focus. The anomalies found were a maximtai
of .,» seconds, in.I were .it the maximum in the region near 25*. No
conclusions wer«» drawn ft^m these results.
Finur«"» 1J throunh 1" illustrate the diagnostic capabillCy of Che
method prt-sonted in this study. The travel times are particularly ra-
vcalink; at distance* within 50 km of the point at which Che iichosphere
starts to herd downward. They depend upon the location of Che source,
varv with direction, and produce distinctive features in Che ray dia-
grams and travel-time curves. Civen a set of seismograas at suicablc
distances from the earthquake in a continental place boundary area, an
invest IKat ion .'f the velocity distribution and source depth can be aided
32
with this method. A more detailed investigation would also Include azi-
muthal effects, which could he found by projecting the given model on
cross-sections at various angles with the cross-section shown.
It Is not the purpose of this study to investigate a portion of the
earth in detail, but rather to present a method to aid in such investi-
gations. For example the travel tines and anomalies found with curved
earth out to 60* war« not pursued as Co chair meaning, but rather a method
to obtain chem using a spherical «arch subroutine was demonstrated. Also,
Che model chosen for Che «mderthrusting lichosphere did noc Include Che
6.00 km/sec layer. Many variations of the model would be made during a
decalled Invesdgaclon—variadons such as chlcknesses of Che layers,
effaces of depch, and, cherefore, heat, on Che velocity of Che layers,
angle of downward banding, extent of Che downward thruating layer,
effects upon the overlying layers, departures from horizontal layering,
and many ocher variadons. It is apparent that such changes can be
accommodaced by chis ray simulation method. The results of these changes
are given hoch visually and numerically.
Conclualona
This study has resulted In an economical and versatile method to
aid In seismic analysis. Geometric ray tracing can be performed with
models representing any conceivable veloclcy distribution wlchln Che
earch. Refreceive aa well aa reflecclve compucstions can be performed.
Including multiple reflections. Travel clmes and approximace ampllcudea
can be decemlned for theae veloclcy distributions.
33
>Hu' »M i <• primary goals of seismology Is to model the velocity
ilist rlhut i.nis within the earth. Hypotheses of velocity distributions
•iff . i<i. t iiMious ! v being produced by seismologists. An important test
tor the valiJity of these models has been developed.
Cumputi-r drawn plots .md numerical output of travel times have
been us«, d to illustrate the capabilities of this simulation method.
The travel times can he made arbitrarily close to those published by
the Seismological Society of America. This method of geometrical ray
tracing performs as accurately and with more versatility Chan previous-
ly published ray tracing techniques.
It is hoped and expected that this technique will be used by the
»eismological profession and chat it will be extended and inproved, as
any uneful, new contribution should.
34
References
Bufe, C. C. (l^ft'J), "An estimate of the configuration of the surface of the earth's core from the consideration of surface focus PcP travel times. PhD. Dissertation, The University of Michigan.
Carder, D. S., D. Tocher, C. G. Bufe, S. W. Steward, J. Eisler, and E. Berg (1967), "Seismic wave arrivals from LONGSHOT, 0° to 21* " Bulletin of the SeismoloRical Society of America. Vol. 57, No. 4, pp. 573-590.
Bullen, K. E. (1963), An introduction to the theory of seismology. Third Edition, Cambridge University Press.
Engdahl, E. R., J. N. Taggart, J. L. Lobdell, E. P. Arnold, and G. E. Clawson (1968), "Computational methods." Bulletin of the Selsmolo- gical Society of America. Vol. 58, pp. 1339-1344.
Gutenberg, B. (1914), "Über Erdbebenwellen, VIIA. Beobachtungen an Registierungen von Fernbeben in Gottingen und Folgerungen über die Konstitution des Erdkorper," Got tinger Nachrichten, pp. 1-52.
Heibig, K. (1965), "A graphical method for the construction of rays and travel times In spherically layered media," Part I: Isotropie Case, Bulletin of the Selsmologlcal Society of America. Vol. 55, 641-651
Herrin, E.( E. P. Arnold, B. A. Bolt, G. E. Clawson, E. R. Engdahl, H. W. Freedman, D. W. Gordon, A. C. Hales, J. L. Lobdell, 0. Nuttli, C. F Ronney, J. N. Taggart, and W. C. Tucker (1968), "1968 Selsmologlcal tables for P-phases," Bulletin of the Selsmologlcal Society of America. Vol. 58, pp. 1193-1241.
Isacks, B.( J. Oliver, and L. R. Sykes (1968), "Seismology and the new global tectonics," Journal of Geophysical Research. Vol. 73, No. 18, pp. 5855-5899.
Iyer, H. M. and V. W. Punton (1963), "A Computer program for plotting wavefronts and rays from a point source in dispersive mediums," Journal of Geophysical Research. Vol. 68, No. 11.
Jackson, P. L. (1968), "Two- and Three-dimensional ray tricing through inhomogeneous media," Paper presented to the Annual Meeting of the Selsmologlcal Society uf America, Tucson, Arizona, April.
Jackson, P. L. (1969), "Seismic ray simulation," Paper presented to the Eastern Section, Selsmologlcal Society of America, Blacksburg, Va., October.
Jackson, P. L. (1970), "Seismic ray simulation in a spherical earth," Bulletin of the Selsmologlcal Society of America. Vol. 60, No. 3, pp. 1021-1025.
Jacob, K. H. (1970), "P-residuals and global tectonic structures investi- gated by three-dimensional seismic ray tracing with emphasis on L0NGSHOT data," Paper presented to the Annual Meeting of the Ameri- can Geophysical Union, Washington, D. C.» April.
35
.Iftfn-vs, il. uul K. K. Bullen (1^48), Seismological tables. British A sis i 11ion for the Advancement of Science, London.
Julian, B. R. and D. L. Anderson (1968), "Travel times, apparent velo- ritiis and amplitudes of body waves," Bulleti.i of the Seismologlcal SiHi.'t y of America. Vol. S8, pp. 339-366.
Lewis, B. T. K. .md R. P. Meyer (1968), "A Seismic Investigation of the upper mantle to the west of Lake Superior," Bulletin of the Seismological Society of America. Vol. 58, pp. 565-569.
Mitronovas, W., B. Isacks, and L. Seeber (1969), "Earthquake locations and seismic wave propagation In the upper 250 km of the Tonga Island arc," Bulletin of the Seismologlcal Society of America. Vol. SO, pp. 1115-1135.
Mohoroviciif, A. (1910), "Jahrbuch des meteorologischen," Observatoriums in Zagreb (Agram) fur das Jahr 1909. Vol. 9, pt. 4, pp. 1-63.
Murdock, J. M. (1969), "Crust-mantle system in the Central Aleutian res Ion—A hypothesis," Bulletin of the Seismologlcal Society of America. 59, 1543-1558,
Newton, I. (1730; 1952). Qptlks. or a treatise of the reflections. refractions, inflections, and colours of light. Reprint, based on the Fourth Edition, London, Dover Publication, Inc., New York
Willis, 0. E. and P. L. Jackson (1968), Collection and Analysis of Seismic wave propagation data. University of Michigan, Willow Run Laboratories, Geophysics Laboratory, Annual Report Number 8071-15-P.
Yacouh, N. K,, J. H. Scott, and F, A. McKeown (1968), Computer technique for tracing seismic rays In two-dimensional geological models. Geological Survey, Open-File Report, U. S. Department of the Interior.
Zoeppritz, K. (1919), "Über Erdbebenwellen Vllb." Goettinger Nachrichten. pp. fih-R4,
36
(Matrix Elements Indicating Positions On Earth Cross-Section
FIGURE 1. REFERENCE FRAME FOR INCREMENTING RAYS. Seismic velocity and slope are associated with each matrix element. Referencing for the slope is shown at element (2, 1). An Interface between two velocity layers can be positioned between matrix elements as shown by the dotted line through B. The scale, or sampling inter- val, is the distance between matrix elements as represented in kilometers. An incident ray segment of length L and initial angle \ extends from A to B. Coordinates of B are found by extension from A: Bx = Ax ♦ L sin \, Bz * Az + L cos ^ The segment BPj is the first "probing" segment to obtain initial values of seismic velocity at midpoint 1. The angle »m is computed with Snell's law, using emerging velocity at midpoint 1. in- cident velocity at midpoint k, and Incident angle \. "Probing" segments are recom- puted using successive values of ')m until the absolute value of ('<m - ''m.i) is less than a predetermined small value. The velocity and slope associated with the segment BPm and the last computed value of "m are then used as the Incident ray parameters, and a subsequent emerging angle computed for the ray segment proceeding from point Pm.
37
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40
FIGURE 5. PLOTS OF P, PcP, PKP, PKIKP, PKKP, AND PHKKIKP RAYS GENERATED WITH A SPHERICAL EARTH PROGRAM. Jeffreys-Bullen core and 1968 P-phase mantle velocity distributions. Radial velocity distribution sampled at 70 km Intervals. Rays seg-
ment lengths 70 km.
41
FIGURE 6. MULTIPLE REFLECTIONS. PcP and PKP rays up to PKKKKKP. Four multiple reflect ions preset Into computer program. Any number of multiple reflec-
tions can be preset into program.
42
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KU.l HK H. THREE DIMENSIONAL RAY TRACING IN A REGION DEFINED BY 30 MO 30 VELOCITY SAMPLES Constant velocity in planes parallel to the x. y plan«', inrreasini: velocity in z-direction for IS sample points from the x. y plane. Four i uns nf rays originating from point in the x, y plane. Rays emerging at tin \. v plane shown by slash marks with orientation ot angle of emergence. I ppt i left one-point perspective; remaining views: labelled orthographic pro-
jections.
44
M
JIM JW JW J I It
HGURE 9. LINE SOURCE AT DEPTH FOR THREE-DIMENSIONAL REPRESENTA- TION AS SHOWN IN FIGURE 8. Plotting defects In upper right (x, z) projection: all
lines should extend to x-axls.
45
%1;' \
.J KUilUF 10. TURKF-DIMENSIONAL RAY TRACING WITH VELOCITY DISTRIBU- IION DFUNKD BY CONTINUOUS MATHEMATICAL FUNCTION. Velocity 4.0
0 l )\ ■ 0.3y • 0.6/.. whero v(»lumr is defined as in Flpure 8.
46
--n
—A
FIGURE 11. THREE-DIMENSIONAL RAY TRACING WITH VELOCITY FUNCTIONS SPECIFIED FOR DIFFERENT REGIONS. Sampled values for intermediate layer be- tween A and D. Velocity = 4.0 * 0.15x ♦ 0.3y ♦ 0.6z for layers above A and below P
Volume is defined as in Figure 8.
47
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51
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20 Critically
Reflected Rays
Shadow Zone Due to Critical
Reflections
ll I I.
{Toward Underthrustlng Llthosphere
Away From UndcrthrusUnn Llthosphere
Portion of Travel Time Curve Unaffected by Llthosphere Bending
I I I I I I I I I I I I I I I I 20 40 60 100 120 140 160 180 200
DISTANCE FROM DISTURBANCE (km)
Z 2.5
2 2.0 u I 1.8 BE U E S 1.0
0.5
0
H 0.5
p
Critical Reflection Region
Shadow Zone
.L_J I I I L 20 40 60 80 100 120 140 160 180
DISTANCE FROM DISTURBANCE (km)
FIGURE 16. EARTHQUAKE ABOVE UNDERTHRUSTING LITHOSPHERE AT CON- TINENTAL PLATE BOUNDARY. Depth of focus: 50 km. Above: Travel times to- ward and away from underthrustlng llthosphere. Below: Differences between travel times toward and away from underthrustlng llthosphere. See Figure 12 for ray paths
and velocity distribution.
52
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55
r
APPENDIX
Listings of Computer Programs
56
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r MA fM si.i|TrH in tNr.L'inF r.dMTiMiinii«; MATMFMATir.Ai r K'Mlil T MHMUFP OF Mlli.TTPIFBPFI.Fr.TTnM«: Al.l fUlPh r K'PB NIIMMPH OF |MITIA( ANr,|. F IMf.PFMFMT5; f MHM MIIMPF (IF t^ITIAl HfiPI/dMTAI IMT.PFMFMT^ f MHTM TdTAI MHMUFP IIF IMITIAI BAY«;
FHMf. TIMNi
9 '
11 T Niriy/ MIIMHFH nF IMITIäI V'FBTir AL r'yrwFMFMT«;
r. Mill IMF MIIMMFfi (IF rALCriMP l.tMFS MFCFSSABV Til nilTLlNC STMIir.TIIPF r PBir.w IMJTIAI H(i«l7nMTAi pnsttldM IIF PAY
r OBICV IMITIAI VFBTTfAl Pn^ITfdM MF BAY r PMM IMITIAI AMP|. ITIIPF \/AlllF r oFF\(F| PFFFPFNTF WFinCITV Td MAKF N'(n/IF
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A A A A A A T" t A A A
17 '•*■>
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SIIMPnilTIMF TO ^iiupnilTrM^Tfr SlintdlMT IMF Td dl^TAMTF FRdM
IMCRFMFMT IMITIAI AMCI.F dF IMC I HP MCE IMCRFMI« MT ( NTTTAI. HhH \ /WH k'C PHSltldM IMfBFMFMT IMITIAI. VFBTK.SL PUSlTldM <niiR"rF 'Td PMISBCPMCF dH «AV
HfSTAMr.F FROI* FOr.F (IF F(FLn Td FMFPnFMCF HF BAY r FFF CIMF dF FMFBGIMT. AM(;t F ( ARd\/F 1.00 IF CRITICAL AMCLF) r r.jMMAd siippdintMF Td uBiMr, BAY FMH PBFCISFLY Tfi IMTgRPACP r r.r. UFRTKAI iMtRPMC^t PPOR penRIMr. AMCI.F" " r r,r,r, WFRTKAI IMCRFMFMT
r MM MOBWdMTAI IMCRFMFMT FdB FMFRr-IMr, ANr,LF "S9
57
r r r r r
TTiTn «wTTrTTTrri^nirATF fpjTir.fti. .«^FlFf.'t irtf4 I I I (NTFRI M \/ M i IP HF PIP i i M r IJW i T ' " ni i Mn i r A T P » P TnoN FBrfM"H?FIR TTimV" (M HIlHl/ilMTAI POnHPO IMPIFV P(KITI(1M iMAMi SUIITCM TI\ iMnifATF (I«;F IIF ^FcrnMÖ" MAMF'I I ST IM i AwwAv pnn v/Hnri T v vAiii F <; AFTFB «FFLFCTION
IM li Af
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vuMTrw fii IMMjrATF IMTFMFAr.F IMIMMFU MF TJM»;«; THwnnr.M INNFO LOOP
^0
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pwnHpn MIIPI7(IMTAI pn<;iT|nN FOB roMPHTiMr. vFinf.ITv rnmiAMT: I,S7'>B
AUMAv POP \/AI um MF VFBTIfAl POSITIOM
A 74 A no
TTWmrWSTTfH OHAUFIS Pn^lTIAKi pnmi pwoHpn vcuTirAi pnsiTinM FOB roMPiiTINr. VFLUCITV P| AMI! «iimunilTIMP FHB IMfinFM't TBAVfl tTMFTfW PLAN? "WAVF roi v nmir FMMrTiriM TU enMPMTF VFinr.tTV u ir.u nu ir.iMAi on iC;M
CFr TUAV'PI TIMF IM SPniMnS AWQVF tMTFT.FB MINUTES
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(;( np IMTFM^POI ATF" { <ÄVFnT"\/ALIIF HF SLOPF " I 9T SPFFH PHUKPfi \(AI MF OF v/FinciTV A 9* IMIM T«A\'FI TiMP IN' MiKiiifFS Ä TBF^rM miMWMHTiMP Tfi rriMPMTF VFLflCITIFS FOB IINOEBTHBIISTIMP LITHflSP A
95
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ntMFNSIflN X(1f>f>0l« v(innn) 102 10^
n» TA P i T, i r « I T , P ( T T_, IMIIM, i«;tiBF/^.14169?7,0tl.S70B,()»0/ MAMCI i <;T /MAUI / MniN!,nF|.M,HFr.IM,NMIILT,nBir,VftlBir,HtrM lPrmSH,JPORSHt
irwmAr t\/Pi niF, emi TP .npr . lNAML,NnBtNnv.NnMtnEl VtnELMtTFSTtMATHt ICY ?r i p , WTTVFI , l^nwiF, t Px&rT,PMM,AniF,X,Y,NULlNF,Sr,A,IPtANE.IPLnT/NAM2
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'joi TP n,n)
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58
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IT (T7.i P.m r.n'Tn X] MP| T = MP| T*1 iMMMr tMllM+l ip i iMiiM.r.f;, jcvri p) r,Al.i Awn Pi =AMr,i p?
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IP (MPI T.MP.I) RM TO 7 V(MI>| 7|=1>.H t M V(MP| Tl=P!(i;i Aip| T = MP| T*1
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««apurr r IMI; iM^TPIirT IIIN**«■ ( IPI jit.pn.o» t P (IPI MT.PO.OI r.n rn a
IF ( IVFI .FO,1,AMn.MP(.T.r,P./.l y(MPl. Tl = P.i( 17 1 V(MP| T1=PI{I71
MP|.T=NP| T-1 w 71 7? 7^
« rnMTlMnp A r »«« A*« A
IF fP.M IZJ.ßP.tiH 60 TO H "A" q t<;ilBP = l _ A
TF ( IFKAri.Pn.l ) f.AI I RlNM»n (ÄNf;LFl",Sl.nP,PlTl7.i,PJ( 17 1 ,1M,IS,AT( I A
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ip ( i PI "i .pn.iM r,n TH ]■? r'UST MPI r = ^'•,l r-i A ?0^ rm PI IMP I K( ->! ,y( 7) ,MP| T,\ ,0,0,1» MP| T-n
1 7 r OM T I MI i. r •** •«* IT rriMTtMiip
i."3 I TF (7.^Bl I'l ( T / ) ,P.I{ I 7 ) .AMr,|.Fl,nN( 17 1 . ATI 17 )
A ?04 A ?0S Ä ?0^ A ?07 A ?0H A 20«»
r PI (IZ »»PI IIZ1*1t-nHIRH r r.Al i I-'MPI H(PI .P I.HM, AMT, AT, AMr.i F?.17tXtYtL> 14 IP ( I 7 .FP.1 1 r.n TO ^i
i \/ u ■ n AMTI I7-1 \-{ l-AMT( I/ ) 1*AMT( I>^-'n I 7^1 7-1
A 210 A 211 A 212 A 818 A 214 A 21^ TTTT
A 21B A 219
1 s
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f.n rn s I ■ P I ( I M*.S ,l«P.t( 17 ) + , S
A 220 A 221
PI PP lIMr P I P*HMM P ipwim-p i i + r.r.r, I M - P I P u n M ♦ . s I C - P,| PUHH* . ^ IP ( ti-'p ! T^.pn.n i r,n rn i ^ WRTTF M.^l P M 17 1 .P! ( 17 ).V/FL.cLOP,ANr,LFl
1^ IP MMTPAT .PH. i i r.n Til 1H IF IMATH.pn.n r.n tn TT Ic I 1 St L T, 1 1 rn rn q ^ IP ( l^.r.T, iPnpc,M,nR. i^,r.T.,jp'nRSW.nR, IM.LT.l t Gh TU 13 I I IrPIP-f.S
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17 r.ALI TUPNTK ( PIP,PI,I,VF| ,<;inPF,QIFRAr.) IP n IM T.PM. n üB|_ ■ INT MTI
A 222 A 223 A 22* A 22* A 22^ A 227 A >2P A 22«, A 230 A 231 A 232 A 233 A 23* A 23* A 23fr A 237 A 23«
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A ?S? A ?c.^
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A P^A A ?BS
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f mnmn CHHBOHTIME ir PBFriSFiv Wfjfi BAY AT SIIBFACF
r TT 1 ABBAY FfB VAM'FS OF ANA^FS TO HF BfTllBNF» Til AFTf IgFLfrTt f n BAiifAl vFinriTY <t <.TB|HiiT|llN OF A SPMFPIC»L FA^TM r KFPTM HftV|HIIW IFPTM BFACMFn HY BAY (4^7| HM-BADIISI f n«« > tv ■ i IM Y*
49
«- »>
%4
*7
§«»
■ »
66
T"
nj<;T«
wnp jMf MDB7 KM trim 11»'!»-«
nunprri BUR iPPnacRiiii HTWFRr mMK dt- f-McorMMr, Awri wjARnvF JL.ort IF r.BITICAL^ ANf.l.F) wnOI^IlMTAi TMrfiMMlPKitATII»! OF BAY SF^MFNT" MOPI/flMTAI pfKfTIOM IM K || flHFTt-BS ^wiTfw Tn tMOlfATt: r«ITir.»l. «FFL Ff.T IHM ^l-'ITrH F'lP MA«IMr. HFACMFn IMTFUFAfF
f-d
iMFar,F ^uiirw FOU MAviwr. FMFor.Fn AT ^imFAfF WITH KAV
iMtiM NiiMnFe nc T|MF^ TMBfmr.w |n»iFB_tri(>p \\¥r TPAVFITIMF (M IMTFr.FB MIMMFH nFHlKinTFS ' IWFI nr swirrH jr\_i»i»irATF BFFi.Fr.TtnM AT i^TFBFAr.F IVW «TuitrM FTP BFAfH|Mr, BFFCFrf int« IWWU rntlMT FfU» MUMHFB fiF ^F"HFNT1AI
(CBITKAL I»« MlM-fP ITIC4H r.BITirAI BFrtFf TKIN^
»>7 » •
71) 71 7?
17 XBJT lüppir KWfl
I
\M\r-t FOB MMMqFB OF BAY HFF|.Fr.T |I)N «wirrM FOB HAWIMR HFAr.HFn CRITICAL Awr.i F TCTrr:FF"vrALiie «F ftP»TM 7TF pftrtkPo «AtiTT VFBTIfAl BO^ITIMN |M INTFCFR MIMftFft OF SAMPLE PHINTV
TBTTTTR MPI flT
-ftH" 0! PM PIT
VFBftrAI PO^ITJOM IM kiFARF^T (NTFr.Fa NUMMFM l»F ^AI»P| FS «wTTrw Tn BFAO HAT* nn FI»<;T r»Lt WMFM HSFO |g simnnitT|MF
71 7t 7-. 7». 77 TU
TUTU wwKw iw mm^ni NiiMRFB OF PinTTINf. INrRFüfNTS
AB8AV FOB UAI IIF«. OF MOBt^ONTAI MOB ( IMIII. wi^rri?fli iv UNPir r.nmTAWTt «.i4|«a
POSITION
BO
«1 B? Ml
TTTT—HIWPW mi UP nr —WWWI WlfTIWI PW IfPWWCI Pit» Mn«|7nMTA| pn<(|TinM |» %AMPI F POINTS FariM HtFFBfBNr.F PIPBOB P»»WFIT"V*M««" n» MiwurnNTAL wsinnu— PIBFF MomynwTAi pnsiTin* FOB nMgi»|gif|qg OF vti:nr,|Tv_ P.I ABBAV ^PB «Ai lift f»F WOlfK'.Al PftTTTIllN P.M WFBTirAI POSITIIW IM MMgif POIMTS FBflM f.PMFB OF FABTw
»- BC
0(l
pmin WAIHP if ulTTfS ^n^iTinN nvi plffl rti ntra vFHTirAi. pn^innN IM SAHFLP POINT« FBHM CFNTF« OF FABTM PBftÄ*irv«rnrTfr"vi6*TtCAi NlllTTnl VFBTirAl POSITIfW FOB t>F TFBK|N*T ION OF WFLOCITV
P.M1 P.I.I7 PIPBOB P.IBFF BAD I nisfANff F«n""f**^B" of FABT^'TN'^üMPLF^TMTS
hlSTAMfF FBON fFMTFB OF FABTM
«1 a? Ql •14 oV o».
wmiw wtifü nwi ffiw nr um w WPCT nwwfi BAOllS M|N|MI|M BAIMlH BFACMFO BV «AY |N (CfLOMFTFRS ■mm WWPWIIW In mi Ipcvniinici nrwn iNtisrAff SFf TRAVFI T|MP |N «FrONni AMOVF |NTFr.FB MfNllUS tiNnFT <ii«7VnTTB» "Ti» C/WBiiTr rsfwnnrrw.ir rWi ANVT.TVI:«- AV...L»r
AND SIOPP
-ST t)A CM
inn
in? Q OPF «I HPFB <PFFn 1««|N
WFI VFIOI
■TW! SAWFO WAIHF OF <|.nPF IP*
tos in«.
"tOT" to«
pnm VFin? wTi'M vFin* wiinfv VFBINf
TOJSTS-Rr-SATOfifmnrÄTTilN TMAVF) TtMF tw MlNllTFS VFitlf ÄT*n|<"TANfF FBriM CFMFR OF FARTM |N SAMPLF POINTS VflOrtTV vriiVtTv HW* mm 6">rcTiniii VFlOTITV -TNTrirnATinw nr neBTM-Tr"Ut«»ir »rtitnir vifliVlTv
TÖ5- lin
ABBAV TO MOm PRFWIÖII« VPtor.jTV VALliM AFftH «FFL^CT t HNS" VfTtfAl INfBFMFNTATinill OF BAY 5Fr.MFNT VFBTBM OFPTM |N KlLOMfTFtS
nr It? m it*
cimannTINF urwi n |P| ,i».ltON( ANT,ftf,~ANf.LF?»t 7 t««Y.t I n I MF N K t ON PH|ft |j Rjnwt, f,|.M10|, ONIIQI. ANT 11 n» , „AlQOl, JMJ 00 0
H. wfi nrPYUm
67
HfMt-NMf'* » ( """M , vjionni A l?n
r.AM .1 i..'i T >, irui i ,i»i.-,n,,|v.iw/-<.i4,i«)<j,i.s7nM,(j,ntn,ntn/ ' & )?i »»- (1 .'T.I 1 ..\ TU fc 4 1??
MMMMIM mATAIfi/ ^^r.M(.».,f^AMP,| IMUFB.l IMMOO.^IMPKM.WFLniF.nFCllir,, A l?3 »MMMiil B.Kvn ...Fur ^,^MM»-.T<iA«P,PNM,t IMIMCf IWBITFtPHhCftSCA^.IHCl.C? A 1?* »prj.r lur^.MiuMM.nnuMftw.v^MMiti.wFUMAii.l^r.KM.inAT.lPLnT.IPP.HINnÄT, A 1?H ■»MAi(f>AT,»-«ri>AT, iMimt. i AAP.Tri-r ,MiMnFr,tMAxnfr.,r.M|N.r.MA«fMiMnKM,HAii A I^«.
*""«• A 1?/ BPAO Jfc.l'AIAI'") A I?»* H-AI1 »»..-»M (n( 1 t,|.1 ,1 IMWFB1 A 1?«>
nn 1 fr»lni<*T,MA«nAT A Ml IM 1 l»(i» f lul-ArriAT A 191
1 rriMfiMii». A 1*1 7 rnNftMli A 1 »<.
r ••»••••««•■•«•PI HTTItur. |M«TR|ir.T|IIM*««««**««*««****««* A n*> «* i |PI nT.m.iM r.n TO 4 A 117 r*ii PI Tnn c. ,^«"(i,n, .^rA,*^ ,«.,» A n* IF ( ir |pr,».n,i»i r.n Til ^ A 11Q
rAii ..riuri 1 T^AM^.I..rM|Ki,rMA»tr.iRr.i,riRr.ifo«i 1 A 1*0 r^M priHn iT<<\Mp.i..r*f*.r.MAvtf|Af.*.ciBC3*ntli i 1*1 r»M .ri-ri -'^"•'.i. .r"|k..rMA«,f iBC?,f.|Rr,?tOtll 4 I*?
t r. » ii-..,. A ,fc1
4 fnwIIWIt- A 1*4 VtliT«M«|P.)M;)-l.lo«AMPKM A 14* MnB;i,M.|P|(in-l,l««;A»"P«»» A 14». *i»l T.o Ä 147 MlM M- ••' A 14« vw(.| ■».l| tn A |4q IF IKVH .r.T.I |MVF»-H KVF| ■?.*^AMPKM.|-KVFl A 1^0 •run fap || IM»,* A |f| vt| • I^AfP-PM171 A \%7
. ■ II7»VM IW A l^ J • i. •-|. ....... 11 1 / 1 A 1S«> ..gtt-.t «.1 ».-^ t» ( ». .1 i ? 1 A ISf«
•rut f,n A 19» II»-TI.W.H A I*«» IMSBr.»..»« A 140 tww.n A 14| l«wB«»» A 14? ntr.F<i»r«, A 141
* rm-TiMi». A 144 f ••••••••«•••••»( OTT l^r. |»'^T8llf ttnh*«««*«*a*««*B*««»« A 149
1»- I ffl <'T,FO,n| r.n rn T A 144 MPI T.^-PI T.I A 147 IF IMM t,«"1.!! AH TP 4 A 14* «i»P| TlcP.lf III A IM» YI^PI TI»P|1171 A 170 . .. T.^PI?♦! A 171
4 , . rf , A 17? T riWTtMNi A 179 r ••••••«••.. ••.•••••••••••••(«••••••••■••••• A 174
IMlM.fklllM*! A I" IF llMMM.r.i-.irvri Fl BFTMRIK A [74 ^»«r.t PfaMNM *■•> k 17> praFF*P|i f/i».v*MnB|«ir A 17B . . .i .. •• i ' i ..'•.■ tu «*r A |7«
9
»I(f7 >■•»!( 17 1 ♦MOB INC A IBO
!*= «Pt( r/».i T.HnBMiM.n«.p| (i; ».r,T,HnRMAx.nR.p.i( 17 KLT.WFPMIN.OR.PJ
A 1«) a 1R?
r. u i7).r,T.«F»MAxt r,n rn >s A 1H3
A IBiV tF ( IPI OT.FO.O» r.n TO H X(WPI T)-P,I(T7I
A 1«S A IHl"!
ft VIMPI TI-PT(T7I rOMTINIIF
A 1HV A IRR
r. IF t IWRITF.r.F.l) WRITF «7,?«) P H ! 7 » , P.t( 1 7 1 . »< FLdl , ANGLFl
A IRQ A 140
?F «PCI 17».1 T.OIBF» »FTMRW PII.PIBFF-1,
A 141 A IP?
P.M»T^AMP-P.IBFF RAni>^nBT(PII»PII«P.I.|«P.|.n
A I'n A 1 "A
IF (BAnnK.r.T.Pini» nanTii^>B>ni WFI ■T^AMP-B»ni
A ms A 196
KVFI «WF| «VFI 1»VF| ».S A IQH
P.I.I?-T^aMP.P.M|7l A 1QQ A ?()0
♦ f «Mt>.LT..ftAAH PM>.,Artftl IF IPIf.LT.O.» PTI?"-PII?
A ?ni A ?n?
ABR*P.M7/P||7 IF («Rc.r.F.n.) on rn 9
A ;>cn A 704
knwfsnwtAiRuviT^ GO Tn in
A ?0^ A ?0h
0 f nUTINIIF «LOPF-ATANIABRI-PIT?
A ?07 A ?0R
nr IF (ivFinr.Fo.ni r,n rn n
A ?0U A ?10
IF (IVW.FO.n WFIOl.VFiniH A 311 A ?1?
n rww.A IF iwFini,LT,,ftSi r,n rn i?
A ?1* A ?1A
vriiU«vFirt^ Alt tn-ATf |7U^Fr,MFN*S*MPKM/VFLn«
A ?IS A ?16
1? IF IRADI.r.T.TCAMP.t. ) OP TO ?^
. A ?I7 A ! R
TV BAn^.BAOl irB|T>n A ??0 IF nwFinf.Ffl.rt» vpiniaVFin? ivFinr-n
A"7?T 1 ???
nn 1* n-l.A piPRnn.pnin*HnRtNr/?,
A ffj A ??<.
P.tpRnft.P.U]7t>\>TRIHf/>, PlllaPIPBOft-l. A ??A
" p.ui»Tiu«w-p.iBBft*r— - - B«n>»«;oHTiPin«Pin*p.i.n*p.Mi)
A ?T7 A ??R
IF (ftAn?.RT.T*AMP.1.| RH TH ?1 IF i«n«(B«m-R«n?i.i.F..no^i r,n rn |f
A ??<* A ?10
OAM.BAh? ^PFFnaT^AMP-RAO?
ATTl " A ?1?
««"FFn.^BFfh vFin?-M«KSPFFnt*iSPFFo-KSPFFO»"»<niKSPFFn*n-n(KSPtfnn
A ?M A ?^4
r.tll MNHFT »ANRl FUANftl F?,WFLni,VFLn?,^LC»PF,ICR!T,FFF) A ?1S A ?^8
.irBiT-irBiT^.ir.BtT IF i.ir.BiT.RF.^i r,n TO ?S
A ?^7 A ?■**
IF iirBiT.Fo.M r.n in 1*, A 2^9
69
>. „ ... .,■■..; ■.■.,;■ ■■ .: ■ . ■ ■ —.
IS
Mnp i Mr i^K^i-Mft^ i M ( ft Mr, i (-7 >
VPB TMT ä TTlRffrTJwTTfTTÄTTTTTT^T ^n^| TlMi it- r OM ri MI if
IF ( («WH.r.T.i ) |VWRsTVWR-1
v P 1 1 n = \/ n n ?
A ?4()
TTTST A ?»? A lut A 244 A PA«? A ?4(S
1 T
I (■ 1 P
IP
■>} ( 1 IMTPH.PO, 1) r,n Td
t |wmTp,r.p.A) WBITF (7.^0) vptm .vFLn?,st.nPF, i? (|«WB,NP',n) Rfi TO 17
( ^PFHri.r,T.\/i 1 . AMII. flns(n(KVFi.ii-n(KVPLl*l) ).r,T.veLi)lF) GO in 19 (VH .'.T .^Pt-m.AMn.ftRsi iV(KVFL 1 )-n(KVFLl-in .GT.VtLDlF) 0,0 TO 19'
A ?47 A 24H
rnMT IMlF
A ?49 A 250_ A ?Sl A 25?
I«
1«
IP ( If.HIT.FO.fM <;i ilPPu ■ ^1 OPF r.ii Tu ->? rnM TiMuh c.n rn '1 niM TI^lll^-
r,n Tn 1 n A 253 •A 25« A ?'6h A 25^ A 2,57 A 25«
vFl in-vVi ni TftM wi'i'IMT ( ftMf.l F| .91 (1PF.PH 17 ) ,P.|( 17 ) «TSAMP.FOntrtVeLtnNI 1Z).AT( 1
171 .«^n.KM.vH (i'.PH ,PI T ?, 4R r,, ^PF F n ,TTM 1 NC . R An AH, PK tC 1. EORF Sf KVFLI) • «««««««»DiitiitimPI nTTIMT, IM<;THHrTI(1N*<"l'**»'l'**«*«*«*«'t'W* 1 (- MPMU.pn.ru r,n Tn 5n "" IP I fPI I1T .Pn.ri) x |Mp| T)«P,I( 17 )
70
r
vi MPi 1) =PI ( 171 "' ' "•" ■ rftM P| IMP (<( 1 ) ,Y(1 ) .MPLT.1»D.Otl )
r nw T JMMP
• « :s« * i:t u A « « >» 4 a >» 4 A « « i * * 4 # *.» a « * « « * «'i * «in*!»* A « « 4141 ^c iji i|i >|i«1» ^
IP ( I'-iMTP.r.t-.M WRTTf (7.^01 PI( 17),ANr,Lei.SLnPF ^i nppu = >;i npp ip 1 ir.wiT,Fo, 1) Rn 1 INTFB«I »/PI tl^aVfl m VH m w =UM ni nn TU 1 <
Tn ??
"PT 1 IMTPP=^ —
1 V/M nr.«i IP ( I / .pn.MiiMin T 1 nn TU s 1 7 = 1 ;*i it- M 7 .r.T .MiiMni r 1 «pTiiRTT-
nw( 171 nfiNi i7-M A T( PI ( P.M ir.B AMT
I F
I / 1 = A
17 ) ^P
t; »«p 1 T»n (17 1 = ( pllP
TM 7-M 1(17-1) M I ;-M
AMI ( (7-1 )«( iniKVFl.l )-n(KVFI.2n/(n<KVFLl )+L1(KVFL2) ) ) |l«;,Ffi,1 ) ftMTI 17 )=.S#AMT( 17-1 )
«Fl r I J IP
r riM IMP
nr v ( 1
( 17-1 ( W All 1
P = 1 TI Ml II- H(;p = f
7-1 ) rVFI '17 1 = AMM P? ns.r.t ,P AHART R AfMH^^RAIlA»
M|l|
\/C|
AMC,
IP
I «F
Tl P = M ni k = \/ I P7 = P ( n-- p 1
1 nr r 1
III TIP*! PI ni IT-AMf.l p 1+"?.'*^TiPFR TF,r,F,4) WRITF (7,30) ANr,LFUSLnPFR,AMRLF2
A 2,>9 A 2^0 ~A 2AI " A 262 A 2iS3 A 2(S4 A 265 A 2^ A HI A 26« A 269 A 270 A 271 A 272 A 273 A 274 A 275 A 27C« A ?77. A 278 A 279 A 2*0 A 111 A 2R2 A fRI A 2ft4 A 2R5 A 2B«S A 2B7 A ?Rn A JAO A 290 A ?<»1 A 292 A 293 A 294 A m A 29(S A 297 A 29« A m
70
'
.
:,:•.:-,,■,.:■:•. ..
i IF (flNr,|.F?.r.T.7.«PIT) 4Mr,LF? = ANr,LF?-?,«PIT TF (AMftl FJ.l.T.n.» ÄNr,LF9 = 4Nr,LP?+?.*Pn HnRiNr,= (<;F';MFN.t.Fnr,F»;)*siN( ANRI.F?) -vTRTMrsKPr-MTN+FnftF^Ur.nsiANfiLF?) aT(ln=AT( I7) + SAMPKM«FnrFS/VFLn4
A 300 ä tm A 302
A 304 nM( in=nM(i7 )+Fnr,FS«SAMPKM vFLn3=vFi.n4
305 3nc
Fnr,FS=n. I\/W=l
>07 30B
IVWR=H r,n in 5
3oq 310
?3 rOMTINIIF TF (IWRITE.r,F.3) WRITF (7,?P) AT( IZ ) ,nN{ I Z » , VFLOl, SAMPM, ANGLE 1,PIT
'311 312
1 I7),Ptl(IZ) CALL RDRD (PT(TZ),P.MlZI,ANRLFl,ISAMP,FOr,E,nN{ IZ)fAT(IZI ,SAMPKM,D(
3li 314
IKVFLUni'JTA» " IF {PKm.LT.l. ) niST4«0ISTA + lR0.
^15 31(S
nKM=nFr,KM«ni<TA APPVFI =nKM/AT(17»
517" 31«
AMPi s| UNM*äKIT( m/nwnz n*FXP(nPCLnr,*nN( m i TMIN=AT(IZ)/ftO.
A ^1«J 320
ISFC=TMIN «;Fr = ftO.*(TMIN-ISFC)
'321 322
TF (IWRlTfi.RF.?) WtttTP <?.**) ATlintONIlII RAnilSsSAMPKM<OiRAniUS
423 324
nFPTH=«;»MPKM*(TS4MP-l J-RAnilü IF (OKM.LF.MTNnKM.nR.nKM.RF.MAXOKM) GO TO 24
A 3?5 326
WRITE (7,?B» niSTÄ,nKHfTMTN,Si:r,,AT(IZJ,RADUSfnPPTHtAWPL. IF (IA4P.F0.1» WRJTF (7,31) APPV6L
WJLTIP 1 A 327 32«
?4 rnMTiNiiF TVFi.nr.si "i 32<?""
330 TMFRRFsl <;LnPFR«si.npF
A 331 332
r. ?6
«**ft«**««***»**«p(.nTTTNr: INSTRUCTinN*«*****««»***i"i>* IF (IPLOT.FO.O) RO TO 26
333 334
X(NPI.TTrP.MIZ) Y(MP|.T)«PI(IZ)
33!) 33(S
MPI.TsO 'A 337
33« ?6 r.
rnMTiNiiF «««««««««««nil**«*****«»***««******«****»«»*«***«**»*«
33<5 340
IF (fMpRftp.Pö.u en rn ?? IF (IPP.FO.l.ANn.lMFRGE.FO.l) Rfl TO 22
3*1 342
Mill TIP-MIILTIP-1 IF (IZ.EO.l) RETURN
A 3Z.3 344
I7-IZ-1 .ir.RiT»n
3^5 34(S
ANr,i.E?»r.ij(i7» VFLn3«VELnr.Y(lZ) ;
347 34«
HnRINC»SERMFN«SIN(ANRLE2) A 344
350 wPl>lNr»SFRMPN*CnSUNr,LE9» ivFi nr,«i
351 352
r. r,n Tn s 354
354 ?7 FORMAT (inF4.?)
FORMAT (•',F(S.7,FlO,?,F7.?,Ffc,?,Ffl.2,2F9.2,F8.2,I6) "A i*5
35* ?p FORMAT MOINITIALAMfiLF,DEPTH,f.niSTANC6'/'ntLTAtfiKMMlNS6CTOTSECftAO
11HSnEPTHAMPL» , iMIILTIPLFS' ) 35« 30 FORMAT |»TfIPi,f,ffl) 3M
71
11 FRfiMAT { ' APPAWPNT\/PLnr,ITY=',PS.?) A ^^0 pNfi A ^h\«
SimnnilTINE Rnsn (PIR,P,IR,ANir,LFl,ISAMPtFnr,F,nN,AT,l)llFRAC,A,niSTA) R 1 RAMP = i<;aMP-), _JL-_2
^AMt-r T SAMP b S nn j»i.3o R 4 Ms,^ R S PTWr.!»pnfie*STN(AN(»LPl) P t Piwr.,i = Fn'-.F*r.n';( AMCI.FI) ' n 7 POPsSAMP-PJR __5 ?_ PMaPfR-l. — " H" Q RAni=SQRT(PRR*PRR*PII»PII ) R 10 IF (RAni.LT.RAMPi r,n rn i H ll PTNri=-PiNri R \? PTKff;) = -PTwr.l R1 I» M«-l . R 1» PIRVpfR + PlNr, 1 " ~ R IT P,IR = p,|R + PINr..l R 1A ATrAT+ntFRAfcH^PrtftFM R—rr nNanN*niFRAr«Fnr,F*H H 18 IF ( R AD rrr^/RAMP^PTiRFVAKtri.RAnf.LT.ftAMP+EftfiEl ftn Tu 5 PT'TT rnMTIMUF R ?n ! F ( A R < (FfR - 1. ) . L T7 I'M ) PIRsl.01 R ?!' ÄRr,s(SAMP-P.IRl/(PIR-l . ) R 22 TMFTÄ = ÄTAKi(ÄRftl R—7T nr<;TA=(1.57nR-AR<;(THFTAn*S7.?P'>R R 2*
"fT" (TMPTä.LTTöT» OHTA=n.570ft*ARS<THETA»»*,?7.245<!. U W RFTIIRN R 2ft
r. FND R 27*
SIIRRnilTINF RPHIMT ( ANGLF],SLP,PIR,PJR.TSAMP,FORE.VEL.ON,AT.SAMPKM, IVFI ni,piT,piT?,«RrMSPFFn,LiMiNC,RAni,PRECi,pnr,ES,Ki)
c c
1 2
NF = 0. P!?=PIR-1.
t e 4
PRHaTSAMP-p.lB RÄni=SORT(PRH*PRR*PlI«PI 1)
c c
5
r «♦•«•»•♦»«C.HFrK TO SFF IF RAY IS INWARO OR niJTWARn***««****«*»* KVFI=K1
t c
7.
IF (SPFFD.IT.VFL) KVFL=K1-1 C 9 RAnTFS=TSAMP-KVFI.-PRFr,I C 10 IF (SPFFn.i.F.vFi.) r,n TrTT C TF IF (RADTFS.I T.RAm i r,n Tn ? c. 12 r.n TO i C TT" -.flMTINUF c 14 IF (RÄni.l.T.HADTFSl GO TO 2 C 15 flfl Tfl ^ C 1ft r.nfotiNiiF r TT H=I. C IB _____ J J_
H=-l. C 20 r.nNTlNllF C 21 ««**«*«*<;PT HDRIZDNTAL f, VERTICAL INCREMFNTS OF RAY««*«***«»* C 22 PlSCl«FnRF*<«IN(ÄN6LFl) C 23 PIMr.i=Fnr.E«rn';(ANRLFl I C 2* nn s I=I,IIMINC C W PRRgTSAMP-P.IB f. 2ft PII=PIR-1. C 27 R A nIs SOR T(PRR«PRR»PII»PI1 | C 2B ««■**«*«««*«TF<;T FhR CLnSFNFSS TO RADIUS OF RFFLECT10N C 29
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I UNCLASSIFIED Sm-uritv CtttNHilu'atiun
DOCUMENT CONTROL DAI A -RAD i.Sfiurilv i litstifit atiun uf lillr hinli uf al>\ttmt iim/ intlrkiiu; unnuliilinn must hv t-nli-fi-J uhin tht uirttill trptttt i» • IM\\*I%< *h
I ORIGINATING AC1IVITY l( nr/mru/i' aulluit I
Willow Run Laboratories of Hit« Institute of Science and Technoloior, The University of Michigan, Ann Arbor. Michigan
2U HLI'OHI SICUKlT. Cl »'.Sil II »TlON
UNCLASSIFIKI) 2A. CROUP
3 REPORT TITLE
DIGITAL SIMULATION OF SEISMIC RAYS
4 DESCRIPTIVE NOTES llypr nf rrpotl anj inr/uoVr Jultst
Scientific Final. Part 2 S AUTHORISI llitsl numr. miilJIr inilial, luil name t
P. L. Jackson
6 REPORT DATE
August 1970 7a. TOTAL NO OF PAGES
Xii ♦ 85 7* NO or Rcrs
23
BO.CONTRACT OR GRANT NO.
AF i9(638)-n59 6. PROJECT NO
8652 c.
62701D
9a.ORIGINATOR S REPORT NUMtUHlSI
8071-33-F
• t OTHER REPORT NOISI (Any ..»*re numlttt Ihul mmthramtrnJ Ihn trpotl I
AFOSR-70-2391TR 10 DISTRIBUTION STATEMENT
1. This document has been approved for public release and sale; Its distribution Is unlimited.
1. SUPPLEMENTARY NOTES
TECH, OTHER
12 SPONSORING MILITARY ACTIVITY
AF Office of Scientific Research (SRPG) 1400 Wilson Boulevard Arlington. VA 22209
13 ABSTRACT
Simulation of seismic rays for a spherical earth and a flat earth has been achieved in hiuhly complex models. Travel times and approximate amplitudes of seismic waves can be found for both two- and three- dimensional models of portions of the earth. In seismology and other disciplines ray construction customarilv has been applied to simplified geometries. It has been necessary to assume that the seismic wave velocity distribution of the earth was relatively uniform and symmetric.
Recently, however, the earth has been found to be more complex and non-uniform than formerly assumed. A need has thus arisen in seismology to test highly heterogeneous models of seismic velocity distribution. At the same time the development of the modern digital computer has provided a means of performing the necessary ray constructions and numerical calculations.
The problem of complicated seismic velocity distributions was therefore investigated in terms of the most appropriate use of the digital computer. For this investigation a velocity field was set up, and the pmpauation computations made for short segments of rays within this field. Total travel times are found by adding the travel times ui connecieu ray begmeuib. Eba^niiany, lite nature of propagation was duplicated an the com- puter. In that, at the location of each segment along the path of propagation, the initial condition and effect of the surroundings determine the succeeding direction of the following segment.
Doth visual and numerical results have shown that this simulation method can be usefully applied to in- vestigation of seismic velocity distributions of portions of the earth of any size or complexity.
14 KEY WORDS
Seismology Digital Simulation
Seismic Rays Seismic Travel Times
Seismic Velocity DistributIons
DD A1473 UNCLASSIFIKI) Security Classification