on the physical geometry concept at the basis of space...

On the physical geometry concept at the basis of space/timegeostatistical hydrology

G. Christakos a,*, D.T. Hristopulos a, P. Bogaert b

a Environmental Modeling Program, Department of Environmental Science and Engineering, Center for the Advanced Study of the Environment,

University of North Carolina at Chapel Hill, 111 Rosenau Hall, CB#7400, Chapel Hill, NC 27599-7400, USAb Facult�e des Sciences Agronomiques, Unit�e de Biom�etrie et Analyse des Donn�ees, Universit�e Catholique de Louvain, Louvain-la-Neuve, Belgium

Received 4 November 1999; received in revised form 17 March 2000; accepted 17 March 2000

Abstract

The objective of this paper is to show that the structure of the spatiotemporal continuum has important implications in practical

stochastic hydrology (e.g., geostatistical analysis of hydrologic sites) and is not merely an abstract mathematical concept. We

propose that the concept of physical geometry as a spatiotemporal continuum with properties that are empirically de®ned is im-

portant in hydrologic analyses, and that the elements of the spatiotemporal geometry (e.g., coordinate system and space/time metric)

should be selected based on the physical properties of the hydrologic processes. We investigate the concept of space/time distance

(metric) in various physical spaces, and its implications for hydrologic modeling. More speci®cally, we demonstrate that physical

geometry plays a crucial role in the determination of appropriate spatiotemporal covariance models, and it can a�ect the results of

geostatistical operations involved in spatiotemporal hydrologic mapping. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Spatiotemporal; Random ®eld; Hydrology; Mapping; Geostatistics

1. Introduction

Spatiotemporal random ®eld (S/TRF) modeling ofhydrologic phenomena has led to considerable advancesover the last few decades, e.g., [3,30,34,35]. The fol-lowing question can now be posed: Which are the fun-damental concepts responsible for the success of S/TRFmodeling? From our perspective, there are three fun-damental concepts [8]: (a) the spatiotemporal continuumconcept (i.e., a set of points associated with a continuousspatial arrangement of events combined with theirtemporal order), (b) the ®eld concept (which associatesmathematical entities ± scalar, vector, or tensor ± withspace/time points), and (c) the complementarity concept(according to which uncertainty manifests itself as anensemble of possible ®eld realizations that are inagreement with what is known about the hydrologicphenomenon of interest). In this work, we will discusscertain features of the spatiotemporal continuum con-cept (a), including suitable coordinate systems and

metric structures. We will show that these features canhave important consequences in geostatistical analysisand mapping of hydrologic processes.

2. Spatiotemporal continuum and its physical geometry

The majority of applied scientists today view space/time as a continuous spatial arrangement combined witha temporal order of events. In other words, space rep-resents the order of coexistence and time represents theorder of successive existence. In the natural sciences,space/time is viewed as the union of space and time,de®ned in terms of their Cartesian product. Spatiotem-poral continuity implies an integration of space withtime and is a fundamental property of the mathematicalformalism of natural phenomena [6]. The continuumidea implies that continuously varying spatiotemporalcoordinates are used to represent the evolution of asystem's properties. The operational importance of thespatiotemporal continuum concept is its book-keepinge�ciency that permits ordering hydrologic measure-ments and establishing relations among them by meansof physical theories and mathematical expressions. This

www.elsevier.com/locate/advwatres

Advances in Water Resources 23 (2000) 799±810

* Corresponding author. Tel.: +1-919-966-1767; fax: +1-919-966-

7911.

E-mail address: [email protected] (G. Christakos).

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 2 0 - 8

description of space/time su�ces for data analysis andmapping of macroscopic processes in hydrologic geo-statistical applications.

The systematic study of a spatiotemporal continuumrequires the introduction of two important entities: (a) asuitable coordinate system with a measure of space/timedistance (metric), and (b) models and techniques thatestablish linkages between spatiotemporally distributedhydrologic data. These entities require the developmentof a physical geometry model, i.e., a spatiotemporalcontinuum that has a structure with empirically de®nedproperties. In geostatistical studies of hydrologicphenomena, one may consider di�erent coordinate sys-tems that allow representations of spatiotemporal ge-ometry based on the underlying symmetry of thehydrologic processes involved, the topography, etc. Inaddition to coordinate systems, an important issue is themeasurement of distances (metrics) in space, or moregeneral, in space/time. The de®nition of an appropriatemetric depends on both the local properties of space andtime (e.g., the curvature of space/time) as well as on thephysical constraints imposed by the speci®c hydrologicprocess (e.g., many-scale obstacles on fractal structures).Mathematical models that establish linkages betweenspatiotemporally distributed data include covariancefunctions of various forms (ordinary and generalizedcovariances, structure functions, etc.). These covariancefunctions need to satisfy certain permissibility criteria[6,8]. The permissibility conditions depend crucially onthe space/time metric, as we further discuss in Section 5.The de®nition of a space/time metric is important informulating parametric models for these covariancefunctions, which are then used in hydrologic estimationand simulation studies. A metric may be de®ned ex-plicitly or implicitly. Explicit expressions for the space/time metric are generally obtained on the basis ofphysical considerations, invariance principles, etc. Ifsuch expressions are not available, it is still possible toobtain the covariance functions for speci®c hydrologicvariables from numerical simulations or experimentalobservations (variables that occur in fractal spaces arean example of the latter).

3. Spatiotemporal coordinate systems

It is important to identify points on a continuum bymeans of an unambiguous address. However, it is oftentaken for granted because it seems so obvious. The in-troduction of a coordinate system is essential in deter-mining the `addresses' of di�erent points in aspatiotemporal continuum. Generally speaking, a co-ordinate system is a systematic way of referring toplaces, times, things and events. The choice of the co-ordinate system depends on the pertinent informationabout the system (natural laws, topographical features,

etc.), as well as on the mathematical convenience re-sulting from a particular choice of the coordinate system(e.g., a spherical spatial coordinate system may simplifycalculations in the case of an isotropic problem). Below,we discuss two types of coordinate systems, Euclideanand non-Euclidean, which are of interest in geostatisticalapplications.

3.1. Euclidean coordinate systems

In the classical Euclidean space, a point p in thespatiotemporal continuum is identi®ed by means of thespatial coordinates s � �s1; . . . ; sn� in Rn (i.e., s 2S � Rn), and the time coordinate t along the time axisT � R1, so that

p � �s; t�: �1�For example, the `address' of a point in an aquifer overtime is characterized by n� 1 numbers (n � 2 or 3) thatdepend on the coordinate system. For many applica-tions it is su�cient to investigate the temporal evolutionafter an initial time, set equal to zero, so that T � �0;1�.Depending on the choice of the spatial coordinatess � �s1; . . . ; sn�, Eq. (1) suggests more than one way tode®ne a point in a spatiotemporal domain as describedin the following. In the commonly used Euclideanrectangular (Cartesian) coordinate system, the si ��s1; . . . ; sn�i and ti are the orthogonal projections of apoint Pi on the spatial axes and temporal axis, respec-tively, so that the following mapping is de®ned:

Pi ! �si; ti� � �s; t�i � pi: �2�In an alternative notation, a point is denoted by Pij,where its spatial coordinates are si 2 S and its time co-ordinate is tj 2 T , i.e., a point Pij in Cartesian space/timeis de®ned as

�si; tj� � pij ! Pij: �3�In a non-Cartesian environment, the Euclidean curvi-linear spatial coordinates are de®ned by means of aspatial transformation of the form

T : si � Ti��s1; . . . ; �sn�; ti; �4�where the ��s1; . . . ; �sn� denote the rectangular spatial co-ordinates (note that the time coordinate tj is not a�ectedby the transformation). In the polar coordinate system:n � 2 and s � �s1; s2� � �r; h� with r > 0. In cylindricalcoordinates: n � 3 and s � �s1; s2; s3� � �r; h; s3�. Inspherical coordinates: n � 3 and s � �s1; s2; s3� ��q;u; h�. Physical data must be associated with a space/time coordinate system that is appropriate for the ob-served process. Geographic coordinates are used insome water resources management systems which in-volve the latitude / and the longitude h of a point P onthe surface of the earth (both expressed in radians). Thelatitude is de®ned as the angle between P and the

800 G. Christakos et al. / Advances in Water Resources 23 (2000) 799±810

equator along the meridian (meridians are lines de®nedby the intersection of the Earth's surface and any givenplane passing through the North and South poles). Thelongitude is de®ned as the angle between the meridianthrough P and the central meridian (through Green-wich, UK) in the plane of the equator.

In practice, one may need to establish a transfor-mation of the original coordinate system into one thatprovides the most realistic representation of the hyd-rologic phenomenon under consideration. While theuse of a speci®c coordinate system is determined fromthe physical processes involved, the mathematicalconvenience a�orded by the speci®c system will alsoplay a role. For example, in the case of a hydrologicprocess that has cylindrical symmetry (e.g., ¯ow in awell), the cylindrical coordinate system captures theunderlying symmetry and is thus more convenient formathematical analysis than a rectangular coordinatesystem. In this case the latter is ine�cient, but it is notruled out.

3.2. Non-Euclidean coordinate systems

A rectangular Euclidean coordinate system is notappropriate for physical processes that occur in curvedspaces. Non-Euclidean coordinate systems are notconstrained to rectangular coordinates. For a curvedtwo-dimensional surface, a Gaussian coordinate systemmay be appropriate. In the Gaussian coordinate sys-tem, the rectangular grid of the Euclidean space is re-placed by an arbitrary dense grid of ordered curves(Fig. 1) generated as follows: Fixing the value of onecoordinate, s1 or s2, produces a curve on the surface interms of the free coordinate. In this way, two familiesof one-parameter, non-intersecting curves are generatedon the surface. Only one curve of each family passesthrough each point. The s1-curves intersect the s2-curves, but not necessarily at right angles. Neither thes1- nor the s2-curves are uniformly spaced. This type ofgrid permits locating points, but not a direct mea-surement of the distance between them. If a global

coordinate system does not su�ce to entirely cover agiven surface, local coordinate systems should be usedinstead.

For natural processes that take place on the Earth'ssurface, a Cartesian coordinate system with origin at thecenter is not convenient. In addition, a rectangular gridis not appropriate for processes a�ected by the earth'scurvature (see, global hydrological modeling, climaticprocesses, etc.; e.g., [19,21,22]). Instead, the two-di-mensional continuum of the earth's surface is describedby a non-Euclidean geometry of the Gaussian type.Gaussian geometry o�ers an internal visualization of theearth's surface (to visualize a surface internally isequivalent to living on such a surface; to visualize asurface externally is to view it from a higher dimensionalspace that includes it). In this case, things are simpli®edconsiderably by using curvilinear coordinate systems. Inthis case, straight lines are replaced by arcs, for these arethe shortest distances between points (geodesics). Atriangle consists of three intersecting arcs, and the sumof its angles is greater than 180°. Every surface has a setof properties, called intrinsic (or internal), that remaininvariant under transformations preserving the arclength (e.g., [24]). The above example points out animportant consideration in the choice of a coordinatesystem for a natural process: it is more e�cient to useinternal, as opposed to external, properties of thephysical space.

Riemann generalized Gauss' analysis by introducingthe concept of a continuous manifold as a continuum ofelements, such that a single element is de®ned by ncontinuous variable magnitudes. This de®nition in-cludes the analytical conception of space in which eachpoint is de®ned by n coordinates. Since two Gaussiancoordinates, (s1; s2), are required to locate a point on asurface in three-dimensional space, the surface is a two-dimensional space or manifold (note that in Cartesiancoordinates a relation of the form f �s1; s2; s3� � 0 isrequired to describe such a manifold). Riemann ex-tended Gauss' two-dimensional (n � 2) surface to n-di-mensional manifolds (n > 2) in Riemannian coordinatesystems. Thus, the Riemannian coordinate system con-sists of a network of si-curves (i � 1; . . . ; n). A detailedmathematical presentation of the Riemannian theory ofspace may be found in [5]. Other types of non-Euclideancoordinate systems are discussed in [7], including sys-tems of coordinates with particular physical propertiessuch as the geodesic, the Glebsch, and the toroidalsystems. For geostatistical applications, it is importantto realize that the Riemannian coordinates specify theposition by consistently assigning to each point on amanifold a unique n-tuple, but they do not automati-cally provide a measure of the distance between points.If explicit relations or measures are required, the con-cept of spatiotemporal metric structure should beintroduced.Fig. 1. A Gaussian coordinate system.

G. Christakos et al. / Advances in Water Resources 23 (2000) 799±810 801

4. The spatiotemporal metric structure

Central among the quantitative features of a physicalgeometry is its metric structure, that is, a set of math-ematical expressions that de®ne spatiotemporal dis-tances. These expressions cannot always be de®nedunambiguously. The expression for the metric in anycontinuum depends on two entirely di�erent factors: (a)a ÔrelativeÕ factor ± the particular coordinate system; and(b) an ÔabsoluteÕ factor ± the nature of the continuumitself. The nature of the continuum is imposed byphysical constraints, such as the geometry of the spacein which a given process occurs (i.e., whether it is aplane, a sphere, or an ellipsoid). Other constraints areimposed by the physical laws governing the naturalprocesses. If a natural process takes place inside athree-dimensional medium with complicated internalstructure, the appropriate metric for correlations is sig-ni®cantly in¯uenced by the structure of the medium. Wefurther investigate this issue in relation with fractalspaces in Section 6. Below, we discuss the separate andthe composite metric structures which are often used inspatiotemporal geometry.

4.1. Separate metric structures

These metrics may be more convenient for geosta-tistical applications, because they reat the concept ofdistance in space and time separately. The separatemetric structure includes an in®nitesimally small spatialdistance jdsjP 0 and an independent time lag dt, so that

dp : �jdsj; dt�: �5�In Eq. (5), the structures of space and time are intro-duced independently. For a ®xed point in space `dis-tance' means `time elapsed', while for a ®xed time itdenotes the spatial `distance between two locations'. Thedistance jdsj can have di�erent meanings depending onthe particular topographic space used. In Euclideanspace the jdsj is de®ned as the length of the line segmentbetween the spatial locations s1 and s2 � s1 � ds, i.e., theEuclidean distance in a rectangular coordinate system isde®ned as

jdsj ��Xn

i�1

ds2i

s: �6�

Non-Euclidean distance measures may be more appro-priate for particular applications. For example, thedistance between points P1 and P2 with spatial coordi-nates s1 and s2 � s1 � ds, respectively, can be de®ned by

jdsj �Xn

i�1

jdsij: �7�

The distance measure of Eq. (7) may represent, e.g., thelength of the shortest path traveled by a ¯uid particle

moving from point P1 to point P2, if the particle isconstrained by the physics of the situation to movealong the sides of the grid. This distance measure is thusmore appropriate for processes that actually occur on adiscrete grid or network of some sort (this is notnecessarily true for continuous processes simulated onnumerical grids, since in this case the grid is only aconvenient modeling device and does not change thespace/time metric). We consider the impact of the metric(7) on the permissibility of covariance functions inSection 5, and we investigate the di�erence between themetrics of Eqs. (6) and (7) from a mapping perspectivein Section 7. Yet another distance metric jdsj is de®nedby

jdsj � max�jdsij; i � 1; . . . ; n�: �8�The distance jdsj between two geographical locations onthe surface of the earth (considered as a sphere withradius r) is de®ned by

jdsj � r��d/2 � �cos2/�dh2

q; �9�

where d/ and dh are the latitude di�erence and longi-tude di�erence, respectively (both expressed in radians).Note that the spatiotemporal metric and the coordinatesystem in which the metric is evaluated are independent.An exception is the rectangular coordinate system, thede®nition of which involves the Euclidean metric. Thefollowing example illustrates how the metrics consideredabove can lead to very di�erent geometric properties ofspace. In the geostatistical analysis of spatial isotropy inR2 one needs to de®ne the set H of points at a distancer � jdsj from a reference point O. In Fig. 2 it is shownthat in the case of the metric (6) the set H is a circle ofradius r, while in the case of the metric (7) H is a squarewith sides

��2p

r. Note that the Hs may represent isoco-

Fig. 2. The set H of points at a distance r � jdsj from O: (a) when r is

the Euclidean distance of Eq. (6) with n � 2; and (b) when r is the

absolute distance of Eq. (7) with n � 2. The set H de®nes an isoco-

variance contour. Such isocovariance contours may be associated with

the spatial distribution of a hydraulic head ®eld in an aquifer, etc.


variance contours associated with the spatial distribu-tion of a hydraulic head ®eld in an aquifer, etc.

A general form for distance metrics ± Euclidean ornon-Euclidean ± can be summarized in terms of theRiemannian distance de®ned as

jdsj ��Xn

i;j�1

gij dsi dsj

vuut ; �10�

where gij are coe�cients that, in general, depend on thespatial location. The tensor g � �gij� is called the metrictensor. Although from the di�erential geometry view-point the metric tensor gives in®nitesimal length ele-ments, the mathematical form of Eq. (10) may be usedto de®ne ®nite distances as well (see, e.g., Eq. (22)). Ametric tensor satis®es certain physical and mathematicalconditions [5]. A metric of the form (10) is Euclidean if acoordinate transformation exists such that Eq. (10) isexpressed in Cartesian form. The Euclidean metric in arectangular coordinate system is a special case ofEq. (10) for gii � 1 and gij � 0 (i 6� j). In a polar coor-dinate system, the metric is obtained from Eq. (10) forn � 2, g11 � 1, g22 � s2

1 and gij � 0 (i 6� j). Eq. (10) forn � 3, g11 � g33 � 1, g22 � s2

1 and gij � 0 (i 6� j) providesthe metric in a cylindrical coordinate system. In aspherical coordinate system, the metric is obtained fromEq. (10) for n � 3, g11 � 1, g22 � s2

1, g33 � �s1 sin�s2��2 andgij � 0 (i 6� j). The metric structures of Gaussian andRiemannian coordinate systems are also represented bymeans of Eq. (10). For n � 2, Eq. (10) gives the localdistance on a curved surface (e.g., a hill); the metric co-e�cients gij are functions of the spatial coordinates si

(i � 1; 2), and g12 � g21. Thus, the curvature of a Gauss-ian (or Riemannian) surface is re¯ected in the metric.

4.2. Composite metric structure

A composite metric structure requires a higher levelof physical understanding of space/time, which may in-volve theoretical and empirical facts about the investi-gated hydrologic process. The metric is determined bythe geometry of space/time and also by the physicalprocesses and the space/time structures that they gen-erate. This is expressed by the following de®nition: Inthe composite metrics the structure of space/time is in-terconnected by means of an analytical expression, i.e.,

dp : jdpj � g�ds1; . . . ; dsn; dt�; �11�where g is a function determined by the availablephysical knowledge (topography, physical laws, etc.;[7]). Consider, e.g., a point P in the space/time contin-uum R2 � T with coordinates p � �s1; s2; t�. A hydrologicprocess that varies within this continuum is denoted byX �p� � X �s1; s2; t�. If the separate metric structure isused, the distance jOP

!j is de®ned in terms of two inde-

pendent space and time distances forming the pair

�jsj; t�, where jsj has one of the spatial forms discussedabove. If, however, the composite metric structure isused, the function g should be determined by means ofthe dynamic structure of the hydrologic processX �s1; s2; t�. Concerning the representation of physicalknowledge, the Euclidean and non-Euclidean ge-ometries display important di�erences. Euclidean ge-ometry determines the metric, which constrains thephysics. In this case, a single coordinate system implyinga speci®c metric structure covers the entire spatiotem-poral continuum. Non-Euclidean geometries clearlydistinguish between the spatiotemporal metric and thecoordinate system, thus allowing for choices that aremore appropriate for certain physical problems.

In several problems the separate metric structure (5)is adequate. In other cases, however, the more involvedcomposite structure (11) is necessary. In the latter case,considering the several existing spatiotemporal ge-ometries that are mathematically distinct but a prioriand generically equivalent, the spatiotemporal metricstructure (i.e., function g) that best describes physicalreality must be determined. Mathematics describes thepossible geometric spaces, and empirical knowledgedetermines which best represents the physical space.Axiomatic geometry is not su�cient for physical appli-cations in space/time, and it is required to establish arelationship between the geometric concepts and theempirical investigation of space/time as a whole. Theterm `empirical' includes all available physical knowl-edge bases (observational data, covariance functions,physical laws, etc.). A special case of Eq. (11) is thespace/time generalization of the distance (10) that leadsto the spatiotemporal Riemannian metric

jdpj ��Xn

i;j�1

gij dsi dsj � 2dtXn

i�1

g0i dsi � g00 dt2

vuut ; �12�

where the metric coe�cients gij (i; j � 1; . . . ; n) arefunctions of the spatial location and time.

We can learn about the nature of the spatiotemporalcontinuum by studying the characteristics of the physi-cal system it describes. Hydrologic processes are subjectto constraints imposed in the form of physical laws.Assume that the distribution of a hydrologic ®eld X �p� isexpressed by the law

X �p� � L�m;BC; IC; p�; �13�where m � �m1; . . . ; mk� are known coe�cients, BC and ICare given boundary and initial conditions, p are space/time coordinates, and L�� is a known mathematicalfunctional. The law (13) can play an important role inthe determination of a physically consistent spatiotem-poral metric form. Often Eq. (13) leads to an explicitexpression for the metric

jdpj � jp0 ÿ pj � g�v0; v; m;BC; IC�; �14�


where the g�� has a functional form that depends on theL��-operator. Eq. (14) restricts the number of possiblemetric models. It determines the metric jdpj of the space/time geometry from the hydrologic ®eld values at p andp0, the coe�cients m, the BC, and the IC. Assuming thatEq. (14) is valid, one cannot specify both the spatio-temporal metric and the hydrologic ®eld values inde-pendently, since they are connected via Eq. (14). In someother cases, the metric form is obtained indirectly fromthe ®eld equations. This happens if the solution of thephysical law is such that

X �p� � �X �g�s1; . . . ; sn; t��: �15�Solution (15) puts restrictions on the geometrical fea-tures of space/time and suggests a metric of the formjpj � g�s1; . . . ; sn; t�, where jpj de®nes the space/timedistance from the origin. It is possible that the physicallaw could lead to a solution (14) that o�ers informationabout the coe�cients gij of the metric (11). Thesepossibilities are demonstrated with the help of the fol-lowing examples. Consider the hydraulic head ®eldh�s1; s2�, the spatial distribution of which is governed bythe Laplace equation

r2h�s1; s2� � 0: �16�In the case of radial ¯ow, Eq. (16) admits a solution ofthe form h�s1; s2� � �h�g�s1; s2�� h�

��s2

1 � s22

p�. Hence,

the spatial metric suggested by Eq. (16) is the Euclideanjsj � ��

s21 � s2

2

p. In the case of two-phase ¯ow in a porous

domain the governing equations for phases a (�waterand oil) are [9]

dfa

dla� /�ea;Ka�fa � 0; �17�

where ea is the direction vector of the a-¯owpath tra-jectory, fa the magnitude of the pressure gradient in thedirection ea, Ka denotes the intrinsic permeabilities of thephases, and / is a function of ea and Ka. The solution ofEq. (17) is of the form fa � fa�jsj�, where the corre-sponding metric jsj � la is the distance along the a-¯owpath. Next, let us assume that the geophysical ®eldX �s1; s2; t� is governed by the ¯ux-conservative equation

oX=ot � m � rX � 0; �18�where m � �m1; m2� is an empirical velocity to be deter-mined from the data. By means of a coordinate trans-formation from the rectangular Euclidean system (si) tothe system of coordinates de®ned by �si � si ÿ mit, thesolution of Eq. (18) has the form

X �s1; s2; t� � �X �s1 ÿ m1t; s2 ÿ m2t�; �19�i.e., it depends on the space/time vector p � sÿ mt.Therefore, in the rectangular coordinate system a geo-physical ®eld governed by Eq. (18) may have a metric ofthe Riemannian form (12), where n � 2, g00 � �m2

1 � m22�,

g11 � g22 � 1, g10 � g01 � ÿ2m1, g20 � g02 � ÿ2m2, and

g12 � g21 � g01 � g10 � g02 � g20 � 0. This expressiondemonstrates how the physical law determines the geo-metric metric through the empirical vector m � �m1; m2�T.The adoption of the spatiotemporal metric above couldbe usefully exploited in Eulerian/Lagrangian schemes ofhydrodynamics. Below, we discuss how the covariancefunction can be instructive in determining the appro-priate geometry in a spatiotemporal continuum.

Geostatistical analysis usually includes a covariancemodel ®tted to the data or derived from a physicalmodel. The covariance can be helpful in determining thespace/time geometry. In particular, the form of themetric k is sought such that

cx�h1; . . . ; hn; s� � �cx�k�: �20�The metric may be viewed as a transformationk � T �h1; . . . ; hn; s� of the original coordinate system,where T has a Riemannian structure and the forms ofthe coe�cients gij are sought on the basis of physicaland mathematical facts. In particular, let k be of theform

k�h; s� ��Xn

i;j�1

gijhihj � 2sXn

i�1

g0ihi � g00s2

vuut : �21�

While the ®nite space/time distance (21) has the sameform as the in®nitesimal Riemannian distance (12), thegijs do not necessarily coincide with the metric coef-®cients of (12). In Eq. (21) the gij denote functions ofthe spatial and lag distances rather than the local co-ordinates, that is gij � gij�hi; hj�, g0i � g0i�s; hi�, i; j �1; . . . ; n, and g00 � g00�s�. Clearly, the determination ofthe gij may require additional assumptions based ontheoretical and experimental facts. If the gij are spaceand time independent, Eqs. (20) and (21) give the fol-lowing set of equations:

ocx=ohi

ocx=ohj�Pn

j�1 gijhj � g0isPni�1 gijhi � g0js

and

ocx=ohi

ocx=os�Pn

j�1 gijhj � g0isPni�1 g0ihi � g00s

:

�22�

For illustration, consider a covariance function inR1 � T that satis®es the following physical model:

ocx=ohocx=os

� hm2s

; �23�

where h � Ds1 � s01 ÿ s1 > 0, s � Dt � t0 ÿ t > 0, m � a=b,

a and b are empirical covariance coe�cients. Note thatdetermining the covariance from physical equations,whenever possible, avoids common problems of empir-ical covariance estimation, and eliminates the circularproblem of standard geostatistics (i.e., estimating thecovariance from the same dataset that is also used toobtain the kriging estimates). We seek a metric form


k such that cx�h; s� � �cx�k�. In view of Eqs. (22) and (23),the metric coe�cients are such that

g11h� g01sg01h� g00s

� hm2s

: �24�

Therefore, a geometric space/time metric that satis®esthe last relationship and, thus, is consistent with thephysical equation (23) is of the form (21) with n � 1,g00 � m2, g01 � 0 and g11 � 1; i.e.,

k�h; s� ��h2 � m2s2

p�

��h2 � a2s2=b2

p: �25�

Eq. (25) shows how the covariance coe�cients deter-mine the spatiotemporal metric. A function which is apermissible covariance model and has a metric of theform (25) is cx�h; s� � c0 exp�ÿh2 ÿ m2s2�. In light of theabove analysis, the choice of a space/time geometrymust be compatible with the `natural' geometry ± asrevealed by the physical equations.

5. Spatiotemporal geometry and permissibility criteria

The choice of the spatiotemporal geometry has sig-ni®cant consequences in geostatistical analysis. Onesuch consequence is related to the permissibility of acovariance model cx�h; s� in Rn � T : The permissibilitycriteria ± that determine if a function can be used as acovariance, semivariogram, generalized covariance, etc.model ± depend on the assumed metric structure. In-deed, a covariance that is permissible for one spatio-temporal geometry may not be permissible for anothergeometry. According to Bochner's theorem (e.g., [8]) anecessary and su�cient condition for a spatiotemporalfunction cx�h; s� to be permissible is that its spectraldensity

~cx�k;x� �Z

dh

Zdseÿi�k�hÿxs�cx�h; s� �26�

be a real-valued, integrable and non-negative functionof the spatial frequency k and the temporal frequency x.An important issue is whether the type of the coordinatesystem or the distance metric considered modify thepermissibility of a function.

As we saw above, in relation with Eq. (20), the co-variance model is a function of the spatiotemporalmetric, which may have a variety of forms (Euclidean ornon-Euclidean). In the following, we will show that thespatiotemporal metric a�ects the permissibility of thecovariance model. For example, in R2 � T the Gaussianfunction

cx�h; s� � c0 exp�ÿjhj2 ÿ m2s2�; �27�where the spatial distance is de®ned as

jhj � jh1j � jh2j �28�is not a permissible covariance model (a mathematicalproof can be found in [7]). This result is veri®ed by

means of a numerical calculation of the Fourier trans-form that gives the spectral density. Since Eq. (27) isa separable space/time covariance, i.e., cx�h; s� �cx�h�c0x�s�, we focus on the spatial component cx�h�. Thecovariance cx�h� is related to the spectral density ~cx�k� asfollows:

~cx�k� �Z

R2

dhexp�ÿik � h�cx�h�: �29�

Hence, cx�h� is a permissible covariance if the ~cx�k� isnon-negative. In the case of the spatial componentcx�h� � c0 exp�ÿjhj2� of the covariance (27), the spectraldensity (29) is negative in parts of the frequency domain(see Fig. 3). We have calculated the Fourier transformusing a Gauss±Legendre quadrature method [31] with 80abscissas in each direction. This involves a total of25,600 function evaluations using double-precisionarithmetic. The Fourier transform exhibits negativevalleys near the corners of the spatial frequency domain.We used di�erent numbers of abscissas to verify that thenegative areas are true features of the Fourier transformand not artifacts of the numerical integration due tooscillations of the integrand. We have also veri®ed thatthe FT is accurate using the MATLAB double integra-tion function `dblquad' with the adaptive±recursiveNewton Cotes algorithm that allows relative and abso-lute error control (we set both to 1 � Eÿ 5). Thus, the

Fig. 3. The Fourier transform ~cx�k� of Eq. (29) using the metric of

Eq. (28). Note the islands of negative values at the four corners of

the frequency domain.


Gaussian function (27) is not a permissible covariancefor the distance metric (28), even though it is permissiblefor the Euclidean metric. Next, we consider the expo-nential function in R2 � T

cx�h; s� � c0 exp�ÿjhj ÿ ms�; �30�where the spatial distance is de®ned as in Eq. (28). Thespectral density ~cx�k;x� of the function (30) is non-negative for all k 2 R2, x 2 R1. Hence, the exponentialcovariance is permissible for the metric of Eq. (28).

In conclusion, the permissibility of a covariancemodel cx�h; s� with respect to the Euclidean metric doesnot guarantee its permissibility for a non-Euclideanmetric. Hence, the permissibility of each model cx�h; s�must be tested with respect to the corresponding non-Euclidean metric.

6. Fractal geometry

Many physical processes that take place in non-uni-form spaces with many-scale structural features (e.g.,within porous media) are better represented by fractalrather than Euclidean geometry. In fractal spaces it isnot always possible to formulate explicit metric expres-sions, such as Eq. (12), since the physical laws may notbe available in the form of di�erential equations. Geo-metric patterns in fractal space/time are self-similar (orstatistically self-similar in the case of random fractals)over a range of scales (e.g., [11,26]). Self-similarity im-plies that fractional (fractal) exponents characterize thescale dependence of geometric properties.

A common example is the percolation fractal (e.g.,[11,37]) generated by the random occupation of sites orbonds on a discrete lattice. In the site percolation model,each site is occupied with probability p and empty withprobability 1ÿ p. Similarly, in the bond percolationmodel, conducting and non-conducting bonds are ran-domly assigned with probabilities p and 1ÿ p. Themedium is permeable if p > pc, where pc is a criticalthreshold that depends on the connectivity and dimen-sionality of the underlying space (for a table of pc valueson di�erent lattices see [20]). The percolation model hasapplications in many environmental and health pro-cesses that occur at various scales. These applicationsinclude single and multiphase ¯ow in porous media[1,2,12,15], the geometry [27,32] and the permeability ofhard and fractured rocks [23,25,28,39,40]). Percolationmodels are also used to model the spread of forest ®resand epidemics [18,33], tumor networks [14], and anti-gen±antibody reactions in biological systems [38].

Length and distance measures on a percolation clus-ter, denoted by l�r�, scale as power laws with the Eu-clidean (linear) size of the cluster. Power-law functionsare called fractal if the scaling exponents are non-in-

teger. The fractal functions are homogeneous (e.g., [4]),i.e., they satisfy

l�br� � bd0 l�r�; �31�where r is the appropriate Euclidean distance, d0 thefractal exponent for the speci®c property, and b a scalingfactor. In practice, scaling relations like Eq. (31) onlyhold within a range of scales bounded by lower andupper cuto�s. For a small change db, the scaling factorbecomes b0 � b� db and Eq. (31) leads to

l�b0r� � �b� db�d0 l�r�: �32�Expanding both sides around b � 1, we obtain

dl�r�dr� d0

l�r�r: �33�

By integrating Eq. (33), we obtain

l�r�l�rco� �

rrco

� �d0

: �34�

where rco is the lower cuto� for the fractal behavior. Forexample, the length of the minimum path on a perco-lation fractal scales as lmin�r� / rdmin , where r denotes theEuclidean distance between the points. The fractal di-mension dmin of the percolation fractal on a hypercubiclattice satis®es 16 dmin6 2, where dmin � 1:1; 1:3 ford � 2; 3 [17,36]. Thus, if the minimum path length be-tween two points at Euclidean distance r is on average2 miles, the length of the minimum path between twopoints separated by 2r is, on average, more than 4 miles.In Fig. 4, we show the minimum path length betweentwo points separated by a Euclidean distance r in Eu-clidean space (curve 1) and in a fractal space withd0 � 1:15 (curve 2). The path length in the Euclideanspace is a linear function of the distance between the twopoints, for all types of paths (e.g., circular arcs, or linearsegments). The fractal path length increases nonlinearly,because the fractal space is non-uniform and obstaclesto motion occur at all scales.

Space/time covariance functions in fractal spaceshave dynamic scaling forms (e.g., [10]) that can be quitedi�erent than Euclidean covariance functions. The self-similarity of fractal processes implies that covariancefunctions decay as power laws. This means that the tailof the covariance function carries more weight than thetail of short-ranged models (e.g., exponential, Gaussian,spherical, etc.) An example of a fractal process thatgenerates power-law correlations is invasion percolation(e.g., [13]) in which a defending ¯uid (e.g., oil) is dis-placed from a porous medium saturated by an invading¯uid (e.g., water).

Below, we investigate an example of a compositespace/time covariance model for fractal spaces. Withinthe fractal range, we consider a covariance function ofthe form


cx�h; s� / ra�s=rb�z; �35�where r � jhj, s0 � s� sm and r0 � r� rm de®ning thespace/time fractal ranges. General permissibility con-ditions for unbounded fractals [20], i.e., s0 � r0 � 0 andrm � sm � 1 impose certain constraints on the expo-nents a, b and z. The permissibility conditions can berelaxed by using ®nite cuto�s. In addition, cuto�s ensurethat cx�h; s� tends to a ®nite variance at zero lag anddrops o� faster than a power-law for lags that exceed thecuto�s. As we discussed above, Bochner's theorem re-quires that the spectral density ~cx�k;x� be a monotoni-cally decreasing function of bounded variation. Thespectral density of the function (35) is given by

~cx�k;x� �Z

dheÿik�hraÿbz

Zdseixssz: �36�

It follows that the permissibility conditions areÿ1 < z < 0 and ÿ�n� 1�=2 < aÿ bz < 0. If b > 0, thelast inequality implies that a < 0.

As is shown in Appendix A, a covariance functionthat has the fractal behavior of Eq. (35) and a ®nitevariance r2 is given by

cx�h; s; uc;wc� � r2f̂z�s=rb; uc�f̂a�r; wc�; �37�where r2 is the variance. The covariance function isplotted in Fig. 5 for r � 1, z � a � ÿ1=2, b � 1:1, andcuto�s uc � 25; wc � 25. The axes used in the pictureare r and s=rb. The function f̂z�s=rb; uc� has an unusualdependence on the space and time lags through s=rb.

For large s, the ratio s=rb is close to zero if r is su�-ciently large, and the value of the function f̂z�s=rb; uc� isclose to one. With regard to f̂z�s=rb; uc� two pairs ofspace/time points are equidistant if s1=rb

1 � s2=rb2 .

Hence, the equation for equidistant space/time contoursis s=rb � c. This dependence is physically quite di�erentthan the one implied by, e.g., a Gaussian space/timecovariance function. In the latter, equidistant lags satisfythe equation r2=n2

r � s2=n2s � c. The di�erence is shown

in Fig. 6, in which we plot the equidistant space/timecontours for f̂z�s=rb; uc� (solid lines) and forexp�ÿr2=n2

r ÿ s2=n2s� (dots) as a function of the space

and time lags. The contour labels represent the values ofc0s=rb (solid lines), and r2=n2

r � s2=n2s (dots), obtained

using c0 � 62:95, nr � 10 and ns � 5.

7. Metric structure and spatiotemporal mapping

Space/time estimation and simulation depend on themetric structure assumed, since the covariance (orsemivariogram, etc.) are used as inputs in most mappingtechniques (e.g., kriging estimation of precipitationdistribution, turning bands simulation of hydraulicconductivity). Hence, the same dataset can lead to dif-ferent space/time maps if estimation is performed usingdi�erent metric structures.

Fig. 5. Plot of the fractal correlation function of Eq. (37) for

z � a � ÿ1=2 and b � 1:1. The correlation function is de®ned as the

covariance normalized by the variance.

Fig. 4. Minimum path length between two points separated by a

Euclidean distance r in Euclidean space (curve 1) and in a space with

fractal length dimension d0 � 1:15 (curve 2).


For example, consider a two-dimensional ®eldX �s1; s2� with a constant mean and an exponentialcovariance

cx�h� � exp�ÿ1:5jhj�; �38�

where h � �h1; h2�. The X �s1; s2� may denote, e.g., theconcentration of a groundwater pollutant. Since thesubsurface is a medium with complicated internalstructure, it is likely that a non-Euclidean metric is amore appropriate measure of distance. The metricshould in principle be derived based on a physical modelof the subsurface medium and the dynamics of trans-port. For the sake of illustration, assume that the ap-propriate metric for this ®eld is the non-Euclidean formjh1j � jh2j. Spatial estimates using this metric were gen-erated on the basis of a hard dataset vhard using a geo-statistical kriging technique [29]. This led to the contourmap of Fig. 7(a). Practitioners of geostatistics or spatialstatistics often favor a theory-free approach which fo-cuses solely on the dataset available and ignores physicalmodels. The standard commercial software for geo-statistics restricts the user to the Euclidean metric��

h21 � h2

2

pfor covariance estimation and kriging. If this

metric were used, the same dataset vhard as above resultsin the contour map in Fig. 7(b). As expected, the twomaps show considerable di�erences. The Euclidean-based map (Fig. 7(b)) assumes a convenient but inade-

quate choice of metric, while the correct one (Fig. 7(a))accounts for the underlying physical geometry.

8. Discussion and conclusion

In this paper, we investigated the important role ofspace/time coordinate systems and distance metrics inthe geostatistical analysis of hydrologic systems. In

Fig. 7. Maps obtained using: (a) the appropriate non-Euclidean

metric; (b) the inappropriate Euclidean metric.

Fig. 6. Equidistant contours for fractal space/time dependence (solid

contours) and for Gaussian dependence (dotted contours).


particular, we presented several examples for metricsand covariance functions in Euclidean and non-Euclid-ean spaces, including fractal spaces. We showed thatthese metrics can lead to very di�erent geometric prop-erties of space. We also showed that covariances whichare permissible for one type of metric are not de factopermissible for a di�erent metric. We investigated acomposite fractal covariance model with a new de®-nition of space/time metric. A characteristic of this co-variance function is that the correlations decayasymptotically much slower than short-range modelswith the usual Euclidean metric. Finally, we showed thatunder di�erent assumptions about the type of the met-ric, the same dataset can lead to very di�erent maps ofthe hydrologic processes under consideration. In suchcases, the physical models governing the hydrologicprocesses could play a pivotal role in determining theappropriate space/time metric. These considerationsalso imply that users of commercial geostatistical soft-ware should be aware of the limitations of these pack-ages. One of these limitations, discussed in this work, isthat the Euclidean metric is chosen by default regardlessof the physical situation.

Acknowledgements

We would like to thank the four anonymous ref-erees for their helpful comments. We are also gratefulto Dr. Cass T. Miller for reading an earlier version ofthe paper and making valuable observations. Thiswork has been supported by grants from the ArmyResearch O�ce (Grant nos. DAAG55-98-1-0289 andDAAH04-96-1-0100), the Department of Energy (Grantno. DE-FC09-93SR18262), and the National Instituteof Environmental Health Sciences (Grant no. P42ES05948-02).

Appendix A

Our aim is to construct a fractal covariance that has a®nite variance and behaves like Eq. (35) within a fractalrange. A power-law function of a general argument x(where x stands for the spatial lag, the time lag, or somecombination thereof) with a negative exponent m < 0can be expressed as

xm � 1

C�ÿm�Z 1

0

dy eÿyxyÿ�m�1�; �A1�

where C is the gamma function. Because m < 0, thefunction xm is singular at r � 0. The singularity is tamedby imposing an upper cuto� yc on the integral, thusleading to the function

fm�x; yc� � 1

C�ÿm�Z yc

0

dy eÿyxyÿ�m�1�: �A2�

This function has power-law behavior for x� xc � 1=yc,and at the same it is ®nite at the origin. The power-lawbehavior can be shown as follows: For xyc � 1, Eq. (A2)can be replaced with Eq. (A1) without considerable er-ror, due to the fact that the exponential term in the in-tegral is essentially zero. Then, it is straightforward toshow by a change of variables x ! kx in Eq. (A1) thatfm�kx; yc� � kmfm�x; yc�, which characterizes a power lawwith exponent m. We express fm�x; yc� as fm�x; yc� �xmc�ÿm; xyc�=C�ÿm�, where c denotes the incompletegamma function [16]. The integrand of Eq. (A2) has anintegrable singularity at y � 0. By means of the trans-formation y � 1=y 0, we avoid the singularity and obtain

fm�x; yc� � 1

C�ÿm�Z 1

1=yc

dy 0y 0mÿ1eÿx=y0 : �A3�

Eq. (A3) is more convenient than (A2) for numericalcalculations. In view of (A3), the value at the origin isfm�0; yc� � �ÿmC�ÿm��ÿ1yÿm

c . In Fig. A1, we plot the nor-malized function

f̂m�x; yc� � fm�x; yc�=fm�0; yc� �A4�for yc � 1 and the exponential function exp�ÿx=n� withn � 6:5. Note that the power-law decays asymptoticallymuch slower than the exponential. In Fig. A2, we plotf̂m�x; yc� for di�erent values of 1=yc. Increasing yc (that is,

Fig. A1. Plot of the fractal correlation function f̂m�x; yc� with m � ÿ1=2

and yc � 1 (solid) and the exponential correlation function exp�ÿx=n�with n � 6:5 (dots) vs. the lag x.


decreasing 1=yc) leads to a steeper slope of f̂m�x; yc� at theorigin. Based on the above, the covariance functiongiven by Eq. (37) has the fractal behavior of Eq. (35) anda ®nite variance r2.

References

[1] Acuna JA, Yortsos YC. Application of fractal geometry to the

study of networks of fractures and their pressure gradients. Water

Resour Res 1995;31(3):527±40.

[2] Berkowitz B, Balberg I. Percolation theory and its application to

groundwater hydrology. Water Resour Res 1993;29(4):775±94.

[3] Bell TL. A space-time stochastic model of rainfall for satellite

remote-sensing studies. J Geophys Res 1987;92(D8):9631±43.

[4] Bunde A, Havlin S, Fractals and disordered systems. New York:

Springer, 1996.

[5] Chavel I. Riemannian geometry: a modern introduction. New

York: Cambridge University Press, 1995.

[6] Christakos G. Random ®eld models in earth sciences. San Diego,

CA: Academic Press, 1992.

[7] Christakos G. Modern spatiotemporal geostatistics. New York:

Oxford University Press, 2000.

[8] Christakos G, Hristopulos DT. Spatiotemporal environmental

health modelling: a tractatus stochasticus. Boston, MA: Kluwer

Academic Publishers, 1998.

[9] Christakos G, Hristopulos DT, Li X. Multiphase ¯ow in

heterogeneous porous media from a stochastic di�erential ge-

ometry viewpoint. Water Resour Res 1998;34(1):93±102.

[10] Family F. Dynamic scaling and phase transitions in interface

growth. Physica A 1990;168:561±80.

[11] Feder J. Fractals. New York: Plenum Press, 1988.

[12] Ferer M, Sams WN, Geisbrecht RA, Smith DH. Crossover from

fractal to compact ¯ow from simulations of two-phase ¯ow with

®nite viscosity ratio in two-dimensional porous media. Phys Rev E

1993;47(4):2713±22.

[13] Furuberg L, Feder J, Aharony A, Jossang T. Dynamics of

invasion percolation. Phys Rev Lett 1988;61(18):2117±20.

[14] Gazit Y, Berk DA, Leunig M, Baxter LT, Jain RK. Scale

invariant behavior and vascular network formation in normal and

tumor tissue. Phys Rev Lett 1995;75(12):2428±31.

[15] Guyon E, Mitescu CD, Hulin JP, Roux S. Fractals and

percolation in porous media and ¯ows. Physica D 1989;38:172±8.

[16] Gradshteyn IS, Ryzhik IM. Table of integrals, series and

products. New York: Academic Press, 1994.

[17] Hermann HJ, Stanley E. J Phys A 1988;21:L829±833.

[18] Hermann HJ. Growth: An introduction. In: Stanley H, Ostrowsky

N, editors. Growth and form. Boston, MA: Kluwer Academic

Publishers, 1986. p. 3±20.

[19] Houghton JT. The physics of atmospheres. Cambridge: Cam-

bridge University Press, 1991.

[20] Isichenko MB. Percolation, statistical topography, and transport

in porous media. Rev Mod Phys 1992;64(4):961±1043.

[21] Jones JAA. Global hydrology. Essex, UK: Longman, 1997.

[22] Kalma JD, Sivapalan M, Scale issues in hydrological modelling.

New York: Wiley, 1995.

[23] Katz AJ, Thompson AH. Quantitative prediction of rock per-

meability in porous rock. Phys Rev B 1986;34(11):8179±81.

[24] Lipschutz M. Di�erential geometry. New York: McGraw-Hill,

1969.

[25] Makse HA, Davies GW, Havlin S, Ivanov PCh, King PR, Stanley

HE. Long-range correlations in permeability ¯uctuations in

porous rock. Phys Rev E 1996;54(4):3129±34.

[26] Mandelbrot BB. The fractal geometry of nature. New York:

Freeman, 1982.

[27] Margolin G, Berkowitz B, Scher H. Structure, ¯ow and general-

ized conductivity scaling in fracture networks. Water Resour Res

1998;34(9):2103±21.

[28] Mourzenko VV, Thovert JF, Adler PM. Percolation and conduc-

tivity of self-a�ne fractures. Phys Rev E 1999;59(4):4265±84.

[29] Olea RA. Geostatistics for engineers and earth scientists. Boston,

MA: Kluwer Academic Publishers, 1999.

[30] Polyak I, North GR, Valdes JB. Multivariate space±time analysis

of PRE-STORM precipitation. J Appl Meteor 1994;33:1079±87.

[31] Press WH, Flannery BP, Teukolsky SA, Vettering WT. Numerical

recipes. Cambridge: Cambridge University Press, 1986.

[32] Radlinski AP, Radlinska EZ, Agamalian M, Wignall CD, Linder

P, Randl OG. Fractal geometry of rocks. Phys Rev Lett

1999;82(15):3078±81.

[33] Rhodes CJ, Anderson RM. Power laws governing epidemics in

isolated populations. Nature 1996;381:600±2.

[34] Rodriguez-Iturbe I, Mejia JM. The design of rainfall networks in

time and space. Water Resour Res 1974;10(4):713±28.

[35] Rouhani S, Hall TJ. Space-time kriging of groundwater data. In:

Armstrong M, editor. Geostatistics. Boston, MA: Kluwer Aca-

demic Publishers, 1989;2:639±51.

[36] Stanley E. An introduction to self-similariry and fractal behavior.

In: Stanley H, Ostrowsky N, editors. Growth and form. Boston,

MA: Kluwer Academic Publishers, 1986. p. 21±53.

[37] Stau�er D, Aharony A. Introduction to percolation theory.

London, UK: Taylor & Francis, 1992.

[38] Stau�er D. Celluar automata. In: Bunde H, Havlin S, editors.

Fractals and disordered sytems. New York: Springer, 1996,

p. 351±59.

[39] Thompson AH, Katz AJ, Krohn CE. The microgeometry and

transport properties of sedimentary rock. Adv Phys 1987;36:

625±94.

[40] Wong P, Koplik J, Tomanic JP. Conductivity and permeability of

rocks. Phys Rev B 1984;30(11):6606±14.

Fig. A2. Plot of the function f̂m�x; yc� with m � ÿ1=2, vs. the lag x for

three di�erent values of the cuto� 1=yc.


on the physical geometry concept at the basis of space...

Documents