Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue8.2000:

On the physical geometry concept at the basis of space/time

geostatistical 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, USA

b

Facultedes Sciences Agronomiques, Unitede Biometrie et Analyse des Donnees, UniversiteCatholique 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 of hydrologic phenomena has led to considerable advances over 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/TRF modeling? From our perspective, there are three fun-damental concepts [8]: (a) the spatiotemporal continuum concept (i.e., a set of points associated with a continuous spatial arrangement of events combined with their temporal order), (b) the ®eld concept (which associates mathematical entities ± scalar, vector, or tensor ± with space/time points), and (c) the complementarity concept (according to which uncertainty manifests itself as an ensemble of possible ®eld realizations that are in agreement with what is known about the hydrologic phenomenon of interest). In this work, we will discuss certain features of the spatiotemporal continuum con-cept (a), including suitable coordinate systems and

metric structures. We will show that these features can have important consequences in geostatistical analysis and 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 with a temporal order of events. In other words, space rep-resents the order of coexistence and time reprep-resents the order 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 with time and is a fundamental property of the mathematical formalism of natural phenomena [6]. The continuum idea implies that continuously varying spatiotemporal coordinates are used to represent the evolution of a system's properties. The operational importance of the spatiotemporal continuum concept is its book-keeping eciency that permits ordering hydrologic measure-ments and establishing relations among them by means of physical theories and mathematical expressions. This

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*

Corresponding author. Tel.: 1767; fax: +1-919-966-7911.

E-mail address:george_christakos@unc.edu (G. Christakos).

description of space/time suces for data analysis and mapping of macroscopic processes in hydrologic geo-statistical applications.

The systematic study of a spatiotemporal continuum requires the introduction of two important entities: (a) a suitable coordinate system with a measure of space/time distance (metric), and (b) models and techniques that establish linkages between spatiotemporally distributed hydrologic data. These entities require the development of a physical geometry model, i.e., a spatiotemporal continuum that has a structure with empirically de®ned properties. In geostatistical studies of hydrologic phenomena, one may consider di€erent coordinate sys-tems that allow representations of spatiotemporal ge-ometry based on the underlying symmetry of the hydrologic processes involved, the topography, etc. In addition to coordinate systems, an important issue is the measurement of distances (metrics) in space, or more general, in space/time. The de®nition of an appropriate metric depends on both the local properties of space and time (e.g., the curvature of space/time) as well as on the physical constraints imposed by the speci®c hydrologic process (e.g., many-scale obstacles on fractal structures). Mathematical models that establish linkages between spatiotemporally distributed data include covariance functions of various forms (ordinary and generalized covariances, structure functions, etc.). These covariance functions need to satisfy certain permissibility criteria [6,8]. The permissibility conditions depend crucially on the space/time metric, as we further discuss in Section 5. The de®nition of a space/time metric is important in formulating parametric models for these covariance functions, which are then used in hydrologic estimation and simulation studies. A metric may be de®ned ex-plicitly or imex-plicitly. Explicit expressions for the space/ time metric are generally obtained on the basis of physical considerations, invariance principles, etc. If such expressions are not available, it is still possible to obtain the covariance functions for speci®c hydrologic variables from numerical simulations or experimental observations (variables that occur in fractal spaces are an example of the latter).

3. Spatiotemporal coordinate systems

It is important to identify points on a continuum by means of an unambiguous address. However, it is often taken 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 a

spatiotemporal continuum. Generally speaking, a co-ordinate system is a systematic way of referring to places, times, things and events. The choice of the co-ordinate system depends on the pertinent information about 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 simplify calculations in the case of an isotropic problem). Below, we discuss two types of coordinate systems, Euclidean and non-Euclidean, which are of interest in geostatistical applications.

3.1. Euclidean coordinate systems

In the classical Euclidean space, a point p in the spatiotemporal continuum is identi®ed by means of the spatial coordinates sˆ …s1;. . .;sn† in Rn (i.e., s2

S Rn), and the time coordinate t along the time axis

T R1_{, so that}

p_ˆs;t_†: …1_†

For example, the `address' of a point in an aquifer over time is characterized byn‡1 numbers (nˆ2 or 3) that depend on the coordinate system. For many applica-tions it is sucient to investigate the temporal evolution after an initial time, set equal to zero, so thatT ‰0;1†. Depending on the choice of the spatial coordinates

sˆ …s1;. . .;sn†, Eq. (1) suggests more than one way to de®ne a point in a spatiotemporal domain as described in the following. In the commonly used Euclidean rectangular (Cartesian) coordinate system, the siˆ

…s1;. . .;sn†i and ti are the orthogonal projections of a point Pi on the spatial axes and temporal axis, respec-tively, so that the following mapping is de®ned:

Pi ! …si;ti† ˆ …s;t†iˆpi: …2†

In an alternative notation, a point is denoted by Pij, where its spatial coordinates are si2S and its time co-ordinate istj2T, i.e., a pointPijin Cartesian space/time is de®ned as

…si;tj† ˆpij ! Pij: …3†

In a non-Cartesian environment, the Euclidean curvi-linear spatial coordinates are de®ned by means of a spatial transformation of the form

T :siˆTi…s1;. . .;sn†;ti; …4†

where the s1;. . .;sn†denote the rectangular spatial co-ordinates (note that the time coordinatetjis not a€ected by the transformation). In the polar coordinate system:

n_ˆ2 and s_ˆs1;s2† ˆ …r;h† with r>0. In cylindrical

coordinates: n_ˆ3 and s_ˆs1;s2;s3† ˆ …r;h;s3†. In

spherical coordinates: nˆ3 and sˆ …s1;s2;s3† ˆ …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 in some water resources management systems which in-volve the latitude/and the longitudehof a pointP on the surface of the earth (both expressed in radians). The

equator along the meridian (meridians are lines de®ned by the intersection of the Earth's surface and any given plane passing through the North and South poles). The longitude is de®ned as the angle between the meridian through 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 that provides the most realistic representation of the hyd-rologic phenomenon under consideration. While the use of a speci®c coordinate system is determined from the physical processes involved, the mathematical convenience a€orded by the speci®c system will also play a role. For example, in the case of a hydrologic process that has cylindrical symmetry (e.g., ¯ow in a well), the cylindrical coordinate system captures the underlying symmetry and is thus more convenient for mathematical analysis than a rectangular coordinate system. In this case the latter is inecient, but it is not ruled out.

3.2. Non-Euclidean coordinate systems

A rectangular Euclidean coordinate system is not appropriate for physical processes that occur in curved spaces. Non-Euclidean coordinate systems are not constrained to rectangular coordinates. For a curved two-dimensional surface, a Gaussian coordinate system may 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 one coordinate, s1 ors2, produces a curve on the surface in

terms of the free coordinate. In this way, two families of one-parameter, non-intersecting curves are generated on the surface. Only one curve of each family passes through each point. The s1-curves intersect the s2

-curves, but not necessarily at right angles. Neither the

s1- nor thes2-curves are uniformly spaced. This type of

grid 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 a given surface, local coordinate systems should be used instead.

For natural processes that take place on the Earth's surface, a Cartesian coordinate system with origin at the center is not convenient. In addition, a rectangular grid is not appropriate for processes a€ected by the earth's curvature (see, global hydrological modeling, climatic processes, etc.; e.g., [19,21,22]). Instead, the two-di-mensional continuum of the earth's surface is described by a non-Euclidean geometry of the Gaussian type. Gaussian geometry o€ers an internal visualization of the earth's surface (to visualize a surface internally is equivalent to living on such a surface; to visualize a surface externally is to view it from a higher dimensional space that includes it). In this case, things are simpli®ed considerably by using curvilinear coordinate systems. In this case, straight lines are replaced by arcs, for these are the shortest distances between points (geodesics). A triangle consists of three intersecting arcs, and the sum of its angles is greater than 180°_{. Every surface has a set}

of properties, called intrinsic (or internal), that remain invariant under transformations preserving the arc length (e.g., [24]). The above example points out an important consideration in the choice of a coordinate system for a natural process: it is more ecient to use internal, as opposed to external, properties of the physical space.

Riemann generalized Gauss' analysis by introducing the concept of a continuous manifold as a continuum of elements, such that a single element is de®ned by n

continuous variable magnitudes. This de®nition in-cludes the analytical conception of space in which each point is de®ned by n coordinates. Since two Gaussian coordinates, (s1;s2), are required to locate a point on a

surface in three-dimensional space, the surface is a two-dimensional space or manifold (note that in Cartesian coordinates a relation of the form f…s1;s2;s3† ˆ0 is

required to describe such a manifold). Riemann ex-tended Gauss' two-dimensional (nˆ2) surface ton -di-mensional manifolds (n>2) in Riemannian coordinate systems. Thus, the Riemannian coordinate system con-sists of a network of si-curves (i_ˆ1;. . .;n). A detailed mathematical presentation of the Riemannian theory of space may be found in [5]. Other types of non-Euclidean coordinate systems are discussed in [7], including sys-tems of coordinates with particular physical properties such as the geodesic, the Glebsch, and the toroidal systems. For geostatistical applications, it is important to realize that the Riemannian coordinates specify the position by consistently assigning to each point on a manifold 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 be introduced.

4. The spatiotemporal metric structure

Central among the quantitative features of a physical geometry is its metric structure, that is, a set of math-ematical expressions that de®ne spatiotemporal dis-tances. These expressions cannot always be de®ned unambiguously. The expression for the metric in any continuum depends on two entirely di€erent factors: (a) a_Ôrelative_Õfactor ± the particular coordinate system; and (b) an _Ôabsolute_Õ factor ± the nature of the continuum itself. The nature of the continuum is imposed by physical constraints, such as the geometry of the space in which a given process occurs (i.e., whether it is a plane, a sphere, or an ellipsoid). Other constraints are imposed by the physical laws governing the natural processes. If a natural process takes place inside a three-dimensional medium with complicated internal structure, the appropriate metric for correlations is sig-ni®cantly in¯uenced by the structure of the medium. We further investigate this issue in relation with fractal spaces in Section 6. Below, we discuss the separate and the composite metric structures which are often used in spatiotemporal geometry.

4.1. Separate metric structures

These metrics may be more convenient for geosta-tistical applications, because they reat the concept of distance in space and time separately. The separate metric structure includes an in®nitesimally small spatial distance_jds_jP0 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 it denotes the spatial `distance between two locations'. The distance_jdsj can have di€erent meanings depending on the particular topographic space used. In Euclidean space the_jds_jis de®ned as the length of the line segment between the spatial locationss1ands2ˆs1‡ds, i.e., the

Euclidean distance in a rectangular coordinate system is de®ned as

jdsj ˆ

 Xn

iˆ1

ds2

i s

: …6_†

Non-Euclidean distance measures may be more appro-priate for particular applications. For example, the distance between points P1 and P2 with spatial

coordi-natess1ands2ˆs1‡ds, respectively, can be de®ned by

jdsj ˆX

n

iˆ1

jdsij: …7†

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

moving from point P1 to point P2, if the particle is

constrained by the physics of the situation to move along the sides of the grid. This distance measure is thus more appropriate for processes that actually occur on a discrete grid or network of some sort (this is not necessarily true for continuous processes simulated on numerical grids, since in this case the grid is only a convenient modeling device and does not change the space/time metric). We consider the impact of the metric (7) on the permissibility of covariance functions in Section 5, and we investigate the di€erence between the metrics of Eqs. (6) and (7) from a mapping perspective in Section 7. Yet another distance metric_jds_jis de®ned by

jdsj ˆmax…jdsij; iˆ1;. . .;n†: …8†

The distance_jdsjbetween two geographical locations on the surface of the earth (considered as a sphere with radiusr) is de®ned by

jds_{j ˆ}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 coordinate system in which the metric is evaluated are independent. An exception is the rectangular coordinate system, the de®nition of which involves the Euclidean metric. The following example illustrates how the metrics considered above can lead to very di€erent geometric properties of space. In the geostatistical analysis of spatial isotropy in

R2one needs to de®ne the setHof points at a distance

rˆ jds_j from a reference pointO. In Fig. 2 it is shown that in the case of the metric (6) the set His a circle of radiusr, while in the case of the metric (7)His a square with sides p2r. Note that theHs may represent

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

A general form for distance metrics ± Euclidean or non-Euclidean ± can be summarized in terms of the Riemannian distance de®ned as

wheregij are coecients that, in general, depend on the spatial location. The tensorgˆ …gij†is called the metric tensor. 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 used to de®ne ®nite distances as well (see, e.g., Eq. (22)). A metric tensor satis®es certain physical and mathematical conditions [5]. A metric of the form (10) is Euclidean if a coordinate transformation exists such that Eq. (10) is expressed in Cartesian form. The Euclidean metric in a rectangular coordinate system is a special case of Eq. (10) forgii ˆ1 andgijˆ0 (i6ˆj). In a polar coor-dinate system, the metric is obtained from Eq. (10) for

n_ˆ2,g11ˆ1,g22ˆs21 and gij ˆ0 (i6ˆj). Eq. (10) for

n_ˆ3,g11ˆg33ˆ1,g22ˆs21andgijˆ0 (i6ˆj) provides the metric in a cylindrical coordinate system. In a spherical coordinate system, the metric is obtained from Eq. (10) fornˆ3,g11ˆ1,g22ˆs21,g33ˆ ‰s1sin…s2†Š

2

and

gij ˆ0 (i6ˆj). The metric structures of Gaussian and Riemannian coordinate systems are also represented by means of Eq. (10). For nˆ2, Eq. (10) gives the local distance on a curved surface (e.g., a hill); the metric co-ecients gij are functions of the spatial coordinates si (iˆ1;2), andg12ˆg21. Thus, the curvature of a

Gauss-ian (or RiemannGauss-ian) surface is re¯ected in the metric.

4.2. Composite metric structure

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

dp:_jdpj ˆg…ds1;. . .;dsn;dt†; …11†

where g is a function determined by the available

physical knowledge (topography, physical laws, etc.; [7]). Consider, e.g., a pointP in the space/time contin-uumR2_{T with coordinatespˆ …s1;s2;t†. A hydrologic}

process that varies within this continuum is denoted by

X…p† ˆX…s1;s2;t†. If the separate metric structure is used, the distance_jOP!j is de®ned in terms of two inde-pendent space and time distances forming the pair

…jsj;t†, wherejsjhas one of the spatial forms discussed above. If, however, the composite metric structure is used, the functiong should be determined by means of the dynamic structure of the hydrologic process

X…s1;s2;t†. Concerning the representation of physical knowledge, the Euclidean and non-Euclidean ometries display important di€erences. Euclidean ge-ometry determines the metric, which constrains the physics. In this case, a single coordinate system implying a speci®c metric structure covers the entire spatiotem-poral continuum. Non-Euclidean geometries clearly distinguish between the spatiotemporal metric and the coordinate system, thus allowing for choices that are more appropriate for certain physical problems.

In several problems the separate metric structure (5) is adequate. In other cases, however, the more involved composite structure (11) is necessary. In the latter case, considering the several existing spatiotemporal ge-ometries that are mathematically distinct but a priori and generically equivalent, the spatiotemporal metric structure (i.e., function g) that best describes physical reality must be determined. Mathematics describes the possible geometric spaces, and empirical knowledge determines 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 a relationship between the geometric concepts and the empirical investigation of space/time as a whole. The term `empirical' includes all available physical knowl-edge bases (observational data, covariance functions, physical laws, etc.). A special case of Eq. (11) is the space/time generalization of the distance (10) that leads to the spatiotemporal Riemannian metric

jdpj ˆ functions of the spatial location and time.

We can learn about the nature of the spatiotemporal continuum by studying the characteristics of the physi-cal system it describes. Hydrologic processes are subject to constraints imposed in the form of physical laws. Assume that the distribution of a hydrologic ®eldX…p†is expressed by the law

X…p† ˆL‰m;BC; IC;pŠ; …13†

wheremˆ …m1;. . .;mk†are known coecients, BC and IC are given boundary and initial conditions, p are space/ time coordinates, and L‰Š is a known mathematical functional. The law (13) can play an important role in the determination of a physically consistent spatiotem-poral metric form. Often Eq. (13) leads to an explicit expression for the metric

where theg‰Šhas a functional form that depends on the

L‰Š-operator. Eq. (14) restricts the number of possible metric models. It determines the metricjdpjof the space/ time geometry from the hydrologic ®eld values atp and

p0_{, the coecientsm, the BC, and the IC. Assuming that}

Eq. (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 some other cases, the metric form is obtained indirectly from the ®eld equations. This happens if the solution of the physical law is such that

X…p_{† ˆ}X‰g…s1;. . .;sn;t†Š: …15†

Solution (15) puts restrictions on the geometrical fea-tures of space/time and suggests a metric of the form

jp_{j ˆ}gs1;. . .;sn;t†, where jpj de®nes the space/time distance from the origin. It is possible that the physical law could lead to a solution (14) that o€ers information about the coecients gij of the metric (11). These possibilities are demonstrated with the help of the fol-lowing examples. Consider the hydraulic head ®eld

h…s1;s2†, the spatial distribution of which is governed by

the Laplace equation

r2

h…s1;s2† ˆ0: …16†

In the case of radial ¯ow, Eq. (16) admits a solution of the form h…s1;s2† ˆh‰g…s1;s2†Š ˆh… the spatial metric suggested by Eq. (16) is the Euclidean

jsj ˆ 

s2 1‡s22 p

. In the case of two-phase ¯ow in a porous domain the governing equations for phasesa (_ˆwater and oil) are [9]

dfa

dla‡

/ea;Ka†faˆ0; …17†

where ea is the direction vector of the a-¯owpath tra-jectory,fathe magnitude of the pressure gradient in the

directionea,Kadenotes the intrinsic permeabilities of the

phases, and/is a function ofeaandKa. The solution of

Eq. (17) is of the form faˆfa…jsj†, where the

corre-sponding metric jsj ˆla is the distance along the a

-¯owpath. Next, let us assume that the geophysical ®eld

X…s1;s2;t†is governed by the ¯ux-conservative equation

o_X=o_{t‡m rX ˆ0; …18†}

where mˆ …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) to the system of coordinates de®ned by siˆsiÿmit, the solution of Eq. (18) has the form

X…s1;s2;t† ˆ_{X…s1ÿm1t;s2ÿm2t†; …19†}

i.e., it depends on the space/time vector pˆsÿmt. Therefore, in the rectangular coordinate system a geo-physical ®eld governed by Eq. (18) may have a metric of the Riemannian form (12), wherenˆ2,g00ˆ …m21‡m22†,

g11ˆg22ˆ1, g10ˆg01ˆ ÿ2m1, g20ˆg02ˆ ÿ2m2, and

g12ˆg21ˆg01ˆg10ˆg02ˆ g20ˆ0. This expression

demonstrates how the physical law determines the geo-metric geo-metric through the empirical vector mˆ ‰m1;m2Š

T

. The adoption of the spatiotemporal metric above could be usefully exploited in Eulerian/Lagrangian schemes of hydrodynamics. Below, we discuss how the covariance function can be instructive in determining the appro-priate geometry in a spatiotemporal continuum.

Geostatistical analysis usually includes a covariance model ®tted to the data or derived from a physical model. The covariance can be helpful in determining the space/time geometry. In particular, the form of the metric kis sought such that

cx…h1;. . .;hn;s† ˆcx…k†: …20†

The metric may be viewed as a transformation k_ˆT…h1;. . .;hn;s† of the original coordinate system, where T has a Riemannian structure and the forms of the coecients gij are sought on the basis of physical and mathematical facts. In particular, let k be of the form

While the ®nite space/time distance (21) has the same form as the in®nitesimal Riemannian distance (12), the

gijs do not necessarily coincide with the metric coef-®cients of (12). In Eq. (21) the gij denote functions of the spatial and lag distances rather than the local co-ordinates, that is gij ˆgij…hi;hj†, g0iˆg0i…s;hi†, i;jˆ 1;. . .;n, and g00ˆg00…s†. Clearly, the determination of

the gij may require additional assumptions based on theoretical and experimental facts. If the gij are space and time independent, Eqs. (20) and (21) give the fol-lowing set of equations:

For illustration, consider a covariance function in

R1_{T that satis®es the following physical model:}

o_cx=o_h o_cx=o_sˆ

h

m2_s; …23†

whereh_ˆDs1ˆs01ÿs1>0,sˆDtˆt0ÿt>0,mˆa=b,

ksuch thatcx…h;s† ˆcx…k†. In view of Eqs. (22) and (23), the metric coecients are such that

g11h‡g01s

g01h‡g00sˆ

h

m2_s: …24†

Therefore, a geometric space/time metric that satis®es the last relationship and, thus, is consistent with the physical equation (23) is of the form (21) with nˆ1,

g00ˆm2,g01ˆ0 andg11ˆ1; i.e.,

kh;s_{† ˆ}ph2_‡m2_s2_ˆp_h2_‡a2_s2_=b2_{: …25†}

Eq. (25) shows how the covariance coecients deter-mine the spatiotemporal metric. A function which is a permissible covariance model and has a metric of the form (25) iscx…h;s† ˆc0exp…ÿh2ÿm2s2†. In light of the

above analysis, the choice of a space/time geometry must be compatible with the `natural' geometry ± as revealed by the physical equations.

5. Spatiotemporal geometry and permissibility criteria

The choice of the spatiotemporal geometry has sig-ni®cant consequences in geostatistical analysis. One such consequence is related to the permissibility of a covariance model cx…h;s† in RnT: The permissibility criteria ± that determine if a function can be used as a covariance, 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 another geometry. According to Bochner's theorem (e.g., [8]) a necessary and sucient condition for a spatiotemporal function cx…h;s† to be permissible is that its spectral density

~

cx…k;x† ˆ Z

dh

Z

dseÿi…khÿxs†_c

x…h;s† …26†

be a real-valued, integrable and non-negative function of the spatial frequencykand the temporal frequencyx. An important issue is whether the type of the coordinate system or the distance metric considered modify the permissibility of a function.

As we saw above, in relation with Eq. (20), the co-variance model is a function of the spatiotemporal metric, which may have a variety of forms (Euclidean or non-Euclidean). In the following, we will show that the spatiotemporal metric a€ects the permissibility of the covariance model. For example, inR2T the Gaussian function

cx…h;s† ˆc0exp…ÿjhj 2

ÿm2s2†; …27†

where the spatial distance is de®ned as

jh_{j ˆ j}h1j ‡ jh2j …28†

is not a permissible covariance model (a mathematical proof 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) is a separable space/time covariance, i.e., cx…h;s† ˆ

cx…h†c0x…s†, we focus on the spatial componentcx…h†. The covariancecx…h†is related to the spectral densityc~x…k†as follows:

~

cx…k† ˆ Z

R2

dhexp_ÿikh†cx…h†: …29†

Hence, cx…h† is a permissible covariance if the c~x…k† is non-negative. In the case of the spatial component

cx…h† ˆc0exp…ÿjhj 2

†of the covariance (27), the spectral density (29) is negative in parts of the frequency domain (see Fig. 3). We have calculated the Fourier transform using a Gauss±Legendre quadrature method [31] with 80 abscissas in each direction. This involves a total of 25,600 function evaluations using double-precision arithmetic. The Fourier transform exhibits negative valleys near the corners of the spatial frequency domain. We used di€erent numbers of abscissas to verify that the negative areas are true features of the Fourier transform and not artifacts of the numerical integration due to oscillations of the integrand. We have also veri®ed that the FT is accurate using the MATLAB double integra-tion funcintegra-tion `dblquad' with the adaptive±recursive Newton Cotes algorithm that allows relative and abso-lute error control (we set both to 1E_ÿ5). Thus, the

Gaussian function (27) is not a permissible covariance for the distance metric (28), even though it is permissible for the Euclidean metric. Next, we consider the expo-nential function inR2_T

cx…h;s† ˆc0exp…ÿjhj ÿms†; …30†

where the spatial distance is de®ned as in Eq. (28). The spectral density ~cx…k;x† of the function (30) is non-negative for allk₂R2,x₂R1. Hence, the exponential covariance is permissible for the metric of Eq. (28).

In conclusion, the permissibility of a covariance modelcx…h;s†with respect to the Euclidean metric does not guarantee its permissibility for a non-Euclidean metric. 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 fractal rather than Euclidean geometry. In fractal spaces it is not always possible to formulate explicit metric expres-sions, such as Eq. (12), since the physical laws may not be available in the form of di€erential equations. Geo-metric patterns in fractal space/time are self-similar (or statistically 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 the scale dependence of geometric properties.

A common example is the percolation fractal (e.g., [11,37]) generated by the random occupation of sites or bonds on a discrete lattice. In the site percolation model, each site is occupied with probabilitypand empty with probability 1ÿp. Similarly, in the bond percolation model, conducting and non-conducting bonds are ran-domly assigned with probabilities p and 1_ÿp. The medium is permeable if p>pc, where pc is a critical

threshold that depends on the connectivity and dimen-sionality of the underlying space (for a table ofpcvalues

on di€erent lattices see [20]). The percolation model has applications in many environmental and health pro-cesses that occur at various scales. These applications include single and multiphase ¯ow in porous media [1,2,12,15], the geometry [27,32] and the permeability of hard and fractured rocks [23,25,28,39,40]). Percolation models are also used to model the spread of forest ®res and 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 functions are called fractal if the scaling exponents are

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

l…br† ˆbd0_{l…r†; …31†}

where r is the appropriate Euclidean distance, d0 the

fractal exponent for the speci®c property, andba scaling factor. In practice, scaling relations like Eq. (31) only hold within a range of scales bounded by lower and upper cuto€s. For a small changed_{b, the scaling factor}

becomes b0ˆb‡d_{band Eq. (31) leads to}

l…b0r† ˆ …b‡d_b†d0

l…r†: …32_†

Expanding both sides aroundb_ˆ1, we obtain

dl…r†

dr ˆd0 l…r†

r : …33†

By integrating Eq. (33), we obtain

l…r†

l…rco†ˆ

r rco d0

: …34†

wherercois the lower cuto€ for the fractal behavior. For

example, the length of the minimum path on a perco-lation fractal scales aslmin…r† /rdmin, whererdenotes the

Euclidean distance between the points. The fractal di-mension dmin of the percolation fractal on a hypercubic

lattice satis®es 16dmin62, where dmin1:1; 1:3 for

d ˆ2; 3 [17,36]. Thus, if the minimum path length be-tween two points at Euclidean distance ris on average 2 miles, the length of the minimum path between two points separated by 2ris, on average, more than 4 miles. In Fig. 4, we show the minimum path length between two points separated by a Euclidean distance r in Eu-clidean space (curve 1) and in a fractal space with

d0ˆ1:15 (curve 2). The path length in the Euclidean

space is a linear function of the distance between the two points, for all types of paths (e.g., circular arcs, or linear segments). The fractal path length increases nonlinearly, because the fractal space is non-uniform and obstacles to motion occur at all scales.

Space/time covariance functions in fractal spaces have dynamic scaling forms (e.g., [10]) that can be quite di€erent than Euclidean covariance functions. The self-similarity of fractal processes implies that covariance functions decay as power laws. This means that the tail of the covariance function carries more weight than the tail of short-ranged models (e.g., exponential, Gaussian, spherical, etc.) An example of a fractal process that generates 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).

cx…h;s† /ra…s=rb†z; …35†

whererˆ jhj,s0ssmandr0rrmde®ning the space/time fractal ranges. General permissibility con-ditions for unbounded fractals [20], i.e.,s0ˆr0ˆ0 and

rmˆsmˆ 1 impose certain constraints on the expo-nents a, b and z. The permissibility conditions can be relaxed by using ®nite cuto€s. In addition, cuto€s ensure that cx…h;s† tends to a ®nite variance at zero lag and drops o€ faster than a power-law for lags that exceed the cuto€s. As we discussed above, Bochner's theorem re-quires that the spectral densityc~x…k;x†be a monotoni-cally decreasing function of bounded variation. The spectral density of the function (35) is given by

~

cx…k;x† ˆ Z

dheÿikh_raÿbz Z

dseixssz_{: …36†}

It follows that the permissibility conditions are

ÿ1<z<0 and ÿ…n‡1†=2<aÿbz<0. If b>0, the last inequality implies thata<0.

As is shown in Appendix A, a covariance function that has the fractal behavior of Eq. (35) and a ®nite variancer2_{is given by}

cx…h;s;uc;wc† ˆr2f^z…s=r

b_;

uc†f^a…r;wc†; …37†

where r2 _{is the variance. The covariance function is}

plotted in Fig. 5 for rˆ1,zˆaˆ ÿ1=2,bˆ1:1, and cuto€s ucˆ25; wcˆ25. The axes used in the picture

arerands=rb_{. The functionf}^

z…s=rb;uc†has an unusual

dependence 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 functionf^z…s=rb;uc†is

close to one. With regard to f^z…s=rb;uc† two pairs of

space/time points are equidistant if s1=rb1ˆs2=r b 2.

Hence, the equation for equidistant space/time contours iss=rb_{ˆc. This dependence is physically quite di€erent}

than the one implied by, e.g., a Gaussian space/time covariance function. In the latter, equidistant lags satisfy the equation r2_=n2

r‡s

2_=n2

sˆc. The di€erence is shown

in Fig. 6, in which we plot the equidistant space/time contours for f^z…s=rb;uc† (solid lines) and for

exp_ÿr2=n2_rÿs2_=n2

s† (dots) as a function of the space

and time lags. The contour labels represent the values of

c0s=rb (solid lines), and r2=n2r‡s

2_=n2

s (dots), obtained

usingc0ˆ62:95,nrˆ10 andnsˆ5.

7. Metric structure and spatiotemporal mapping

Space/time estimation and simulation depend on the metric structure assumed, since the covariance (or semivariogram, etc.) are used as inputs in most mapping techniques (e.g., kriging estimation of precipitation distribution, turning bands simulation of hydraulic conductivity). Hence, the same dataset can lead to dif-ferent space/time maps if estimation is performed using di€erent metric structures.

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

z_ˆa_{ˆ ÿ}1=2 andb_ˆ1:1. The correlation function is de®ned as the covariance normalized by the variance.

For example, consider a two-dimensional ®eld

X…s1;s2† with a constant mean and an exponential

covariance

cx…h† ˆ exp…ÿ1:5jhj†; …38†

where hˆ …h1;h2†. The X…s1;s2† may denote, e.g., the

concentration of a groundwater pollutant. Since the subsurface is a medium with complicated internal structure, it is likely that a non-Euclidean metric is a more appropriate measure of distance. The metric should in principle be derived based on a physical model of 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 form

jh1j ‡ jh2j. Spatial estimates using this metric were

gen-erated on the basis of a hard datasetv_hard using a geo-statistical kriging technique [29]. This led to the contour map of Fig. 7(a). Practitioners of geostatistics or spatial statistics often favor a theory-free approach which fo-cuses solely on the dataset available and ignores physical models. The standard commercial software for geo-statistics restricts the user to the Euclidean metric



h2 1‡h22 p

for covariance estimation and kriging. If this metric were used, the same datasetv_hardas above results in the contour map in Fig. 7(b). As expected, the two maps 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 of space/time coordinate systems and distance metrics in the geostatistical analysis of hydrologic systems. In

Fig. 7. Maps obtained using: (a) the appropriate non-Euclidean metric; (b) the inappropriate Euclidean metric.

particular, we presented several examples for metrics and covariance functions in Euclidean and non-Euclid-ean spaces, including fractal spaces. We showed that these metrics can lead to very di€erent geometric prop-erties of space. We also showed that covariances which are permissible for one type of metric are not de facto permissible for a di€erent metric. We investigated a composite fractal covariance model with a new de®-nition of space/time metric. A characteristic of this

co-variance function is that the correlations decay

asymptotically much slower than short-range models with the usual Euclidean metric. Finally, we showed that under di€erent assumptions about the type of the met-ric, the same dataset can lead to very di€erent maps of the hydrologic processes under consideration. In such cases, the physical models governing the hydrologic processes could play a pivotal role in determining the appropriate space/time metric. These considerations also imply that users of commercial geostatistical soft-ware should be asoft-ware of the limitations of these pack-ages. One of these limitations, discussed in this work, is that the Euclidean metric is chosen by default regardless of the physical situation.

Acknowledgements

We would like to thank the four anonymous ref-erees for their helpful comments. We are also grateful to Dr. Cass T. Miller for reading an earlier version of the paper and making valuable observations. This work has been supported by grants from the Army Research Oce (Grant nos. DAAG55-98-1-0289 and DAAH04-96-1-0100), the Department of Energy (Grant no. DE-FC09-93SR18262), and the National Institute of Environmental Health Sciences (Grant no. P42 ES05948-02).

Appendix A

Our aim is to construct a fractal covariance that has a ®nite variance and behaves like Eq. (35) within a fractal range. A power-law function of a general argument x

(wherexstands for the spatial lag, the time lag, or some combination thereof) with a negative exponent m<0 can be expressed as

xm ˆ_C 1

…ÿm_† Z 1

0

dyeÿyx_yÿ…m‡1†_{; …A1†}

where C is the gamma function. Because m<0, the functionxm_{is singular atrˆ0. The singularity is tamed}

by imposing an upper cuto€ yc on the integral, thus

leading to the function

fm…x;yc† ˆ

1 C_ÿm_†

Z yc

0

dyeÿyx_yÿ…m‡1†_{: …A2†}

This function has power-law behavior forxxc1=yc,

and at the same it is ®nite at the origin. The power-law behavior can be shown as follows: Forxyc1, 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 to show by a change of variables x! kxin Eq. (A1) that

fm…kx;yc† ˆkmfm…x;yc†, which characterizes a power law

with exponent m. We express fm…x;yc† as fm…x;yc† ˆ

xm_c…ÿm;xy

c†=C…ÿm†, where c denotes the incomplete

gamma function [16]. The integrand of Eq. (A2) has an integrable singularity at y_ˆ0. By means of the trans-formationy _ˆ1=y0_{, we avoid the singularity and obtain}

fm…x;yc† ˆ

1 C…ÿm†

Z 1

1=yc

dy0y0mÿ1eÿx=y0

: …A3_†

Eq. (A3) is more convenient than (A2) for numerical calculations. In view of (A3), the value at the origin is

fm…0;yc† ˆ ‰ÿmC…ÿm†Šÿ 1

yÿm

c . In Fig. A1, we plot the

nor-malized function

^

fm…x;yc† ˆfm…x;yc†=fm…0;yc† …A4†

forycˆ1 and the exponential function exp…ÿx=n†with nˆ6:5. Note that the power-law decays asymptotically much slower than the exponential. In Fig. A2, we plot

^

fm…x;yc†for di€erent values of 1=yc. Increasingyc(that is,

Fig. A1. Plot of the fractal correlation function_f^_m…x;yc†withm_{ˆ ÿ}1=2 andycˆ1 (solid) and the exponential correlation function exp…ÿx=n†

decreasing 1=yc) leads to a steeper slope off^m…x;yc†at the

origin. Based on the above, the covariance function given by Eq. (37) has the fractal behavior of Eq. (35) and a ®nite variancer2_.

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.