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

Advances in Water Resources 23 (2000) 799±810
www.elsevier.com/locate/advwatres

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
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 important 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 following question can now be posed: Which are the fundamental concepts responsible for the success of S/TRF
modeling? From our perspective, there are three fundamental 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 concept (a), including suitable coordinate systems and
*

Corresponding author. Tel.: +1-919-966-1767; fax: +1-919-9667911.
E-mail address: george_christakos@unc.edu (G. Christakos).

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 represents the order of coexistence and time represents 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. Spatiotemporal 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 measurements and establishing relations among them by means
of physical theories and mathematical expressions. This

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

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

description of space/time suces for data analysis and
mapping of macroscopic processes in hydrologic geostatistical 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 systems that allow representations of spatiotemporal geometry 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 explicitly or implicitly. 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 introduction of a coordinate system is essential in determining the `addresses' of di€erent points in a
spatiotemporal continuum. Generally speaking, a coordinate system is a systematic way of referring to
places, times, things and events. The choice of the coordinate system depends on the pertinent information

about the system (natural laws, topographical features,

etc.), as well as on the mathematical convenience resulting 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 s ˆ …s1 ; . . . ; sn † in Rn (i.e., s 2
S  Rn ), and the time coordinate t along the time axis
T  R1 , so that
p ˆ …s; t†:

…1†

For example, the `address' of a point in an aquifer over
time is characterized by n ‡ 1 numbers (n ˆ 2 or 3) that

depend on the coordinate system. For many applications it is sucient to investigate the temporal evolution
after an initial time, set equal to zero, so that T  ‰0; 1†.
Depending on the choice of the spatial coordinates
s ˆ …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 si ˆ
…s1 ; . . . ; sn †i and ti are the orthogonal projections of a
point Pi on the spatial axes and temporal axis, respectively, so that the following mapping is de®ned:
Pi ! …si ; ti † ˆ …s; t†i ˆ pi :

…2†

In an alternative notation, a point is denoted by Pij ,
where its spatial coordinates are si 2 S and its time coordinate is tj 2 T , i.e., a point Pij in Cartesian space/time
is de®ned as
…si ; tj † ˆ pij ! Pij :

…3†

In a non-Cartesian environment, the Euclidean curvilinear spatial coordinates are de®ned by means of a
spatial transformation of the form
s1 ; . . . ; sn †; ti ;
T : si ˆ Ti …

…4†

where the …
s1 ; . . . ; sn † denote the rectangular spatial coordinates (note that the time coordinate tj is not a€ected
by the transformation). In the polar coordinate system:
n ˆ 2 and s ˆ …s1 ; s2 † ˆ …r; h† with r > 0. In cylindrical
coordinates: n ˆ 3 and s ˆ …s1 ; s2 ; s3 † ˆ …r; h; s3 †. In
spherical coordinates: n ˆ 3 and s ˆ …s1 ; s2 ; s3 † ˆ
…q; u; h†. Physical data must be associated with a space/
time coordinate system that is appropriate for the observed process. Geographic coordinates are used in
some water resources management systems which involve the latitude / and the longitude h of a point P on
the surface of the earth (both expressed in radians). The
latitude is de®ned as the angle between P and the

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

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 Greenwich, UK) in the plane of the equator.
In practice, one may need to establish a transformation of the original coordinate system into one that
provides the most realistic representation of the hydrologic 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 system, the rectangular grid of the Euclidean space is replaced by an arbitrary dense grid of ordered curves
(Fig. 1) generated as follows: Fixing the value of one
coordinate, s1 or s2 , 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 the s2 -curves are uniformly spaced. This type of
grid permits locating points, but not a direct measurement of the distance between them. If a global

Fig. 1. A Gaussian coordinate system.

801

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-dimensional 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 includes 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 twodimensional space or manifold (note that in Cartesian
coordinates a relation of the form f …s1 ; s2 ; s3 † ˆ 0 is
required to describe such a manifold). Riemann extended Gauss' two-dimensional (n ˆ 2) surface to n-dimensional manifolds (n > 2) in Riemannian coordinate
systems. Thus, the Riemannian coordinate system consists of a network of si -curves (i ˆ 1; . . . ; 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 systems 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 automatically provide a measure of the distance between points.
If explicit relations or measures are required, the concept of spatiotemporal metric structure should be
introduced.

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

4. The spatiotemporal metric structure
Central among the quantitative features of a physical
geometry is its metric structure, that is, a set of mathematical expressions that de®ne spatiotemporal distances. 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 ÔrelativeÕ factor ± the particular coordinate system; and
(b) an ÔabsoluteÕ 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 signi®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 geostatistical applications, because they reat the concept of
distance in space and time separately. The separate
metric structure includes an in®nitesimally small spatial
distance jdsj P 0 and an independent time lag dt, so that
dp : …jdsj; dt†:

…5†

In Eq. (5), the structures of space and time are introduced independently. For a ®xed point in space `distance' 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 jdsj is de®ned as the length of the line segment
between the spatial locations s1 and s2 ˆ s1 ‡ ds, i.e., the
Euclidean distance in a rectangular coordinate system is
de®ned as
s
n
X
ds2i :
…6†
jdsj ˆ

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 jdsj is de®ned
by
jdsj ˆ max…jdsi j; i ˆ 1; . . . ; n†:

…8†

The distance jdsj between two geographical locations on
the surface of the earth (considered as a sphere with
radius r) is de®ned by
q
…9†
jdsj ˆ r d/2 ‡ … cos2 /† dh2 ;

where d/ and dh are the latitude di€erence and longitude 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
R2 one needs to de®ne the set H of points at a distance
r ˆ jdsj from a reference point O. In Fig. 2 it is shown
that in the case of the metric (6) the set H is a circle of
radius r, while
p in the case of the metric (7) H is a square
with sides 2r. Note that the Hs may represent isoco-

iˆ1

Non-Euclidean distance measures may be more appropriate for particular applications. For example, the
distance between points P1 and P2 with spatial coordinates s1 and s2 ˆ s1 ‡ ds, respectively, can be de®ned by
jdsj ˆ

n
X
iˆ1

jdsi j:

…7†

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

Fig. 2. The set H of points at a distance r ˆ jdsj from O: (a) when r is
the Euclidean distance of Eq. (6) with n ˆ 2; and (b) when r is the
absolute distance of Eq. (7) with n ˆ 2. The set H de®nes an isocovariance contour. Such isocovariance contours may be associated with
the spatial distribution of a hydraulic head ®eld in an aquifer, etc.

803

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

variance contours associated with the spatial distribution 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
v
uX
u n
…10†
jdsj ˆ t gij dsi dsj ;
i;jˆ1

where gij are coecients that, in general, depend on the
spatial location. The tensor g ˆ …gij † is called the metric
tensor. Although from the di€erential geometry viewpoint the metric tensor gives in®nitesimal length elements, 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) for gii ˆ 1 and gij ˆ 0 (i 6ˆ j). In a polar coordinate system, the metric is obtained from Eq. (10) for
n ˆ 2, g11 ˆ 1, g22 ˆ s21 and gij ˆ 0 (i 6ˆ j). Eq. (10) for
n ˆ 3, g11 ˆ g33 ˆ 1, g22 ˆ s21 and gij ˆ 0 (i 6ˆ j) provides
the metric in a cylindrical coordinate system. In a
spherical coordinate system, the metric is obtained from
2
Eq. (10) for n ˆ 3, g11 ˆ 1, g22 ˆ s21 , g33 ˆ ‰s1 sin …s2 †Š and
gij ˆ 0 (i 6ˆ j). The metric structures of Gaussian and
Riemannian coordinate systems are also represented by
means of Eq. (10). For n ˆ 2, Eq. (10) gives the local
distance on a curved surface (e.g., a hill); the metric coecients gij are functions of the spatial coordinates si
(i ˆ 1; 2), and g12 ˆ g21 . Thus, the curvature of a Gaussian (or Riemannian) surface is re¯ected in the metric.
4.2. Composite metric structure

i;jˆ1

A composite metric structure requires a higher level
of physical understanding of space/time, which may involve theoretical and empirical facts about the investigated 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 generate. This is expressed by the following de®nition: In
the composite metrics the structure of space/time is interconnected by means of an analytical expression, i.e.,
dp : jdpj ˆ g…ds1 ; . . . ; dsn ; dt†;

…jsj; t†, where jsj has one of the spatial forms discussed
above. If, however, the composite metric structure is
used, the function g 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 geometries display important di€erences. Euclidean geometry determines the metric, which constrains the
physics. In this case, a single coordinate system implying
a speci®c metric structure covers the entire spatiotemporal 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 geometries 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 applications 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 knowledge 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
v
uX
n
X
u n
…12†
jdpj ˆ t gij dsi dsj ‡ 2dt g0i dsi ‡ g00 dt2 ;

…11†

where g is a function determined by the available
physical knowledge (topography, physical laws, etc.;
[7]). Consider, e.g., a point P in the space/time continuum R2  T with coordinates p ˆ …s1 ; s2 ; t†. A hydrologic
process that varies within this continuum is denoted by
X …p† ˆ X …s1 ; s2 ; t†. If! the separate metric structure is
used, the distance jOP j is de®ned in terms of two independent space and time distances forming the pair

iˆ1

where the metric coecients gij (i; j ˆ 1; . . . ; n) are
functions of the spatial location and time.
We can learn about the nature of the spatiotemporal
continuum by studying the characteristics of the physical system it describes. Hydrologic processes are subject
to constraints imposed in the form of physical laws.
Assume that the distribution of a hydrologic ®eld X …p† is
expressed by the law
X …p† ˆ L‰m; BC; IC; pŠ;

…13†

where m ˆ …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 spatiotemporal metric form. Often Eq. (13) leads to an explicit
expression for the metric
jdpj ˆ jp0 ÿ pj ˆ g‰v0 ; v; m; BC; ICŠ;

…14†

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

where the g‰
Š has a functional form that depends on the
L‰
Š-operator. Eq. (14) restricts the number of possible
metric models. It determines the metric jdpj of the space/
time geometry from the hydrologic ®eld values at p and
p0 , the coecients m, the BC, and the IC. Assuming that
Eq. (14) is valid, one cannot specify both the spatiotemporal metric and the hydrologic ®eld values independently, 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 features of space/time and suggests a metric of the form
jpj ˆ g…s1 ; . . . ; 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 following examples. Consider the hydraulic head ®eld
h…s1 ; s2 †, the spatial distribution of which is governed by
the Laplace equation
r2 h…s1 ; s2 † ˆ 0:

…16†

In the case of radial ¯ow, Eq. (16) admits
a solution
of

p
 1 ; s2 †Š ˆ h…
 s2 ‡ s2 †. Hence,
the form h…s1 ; s2 † ˆ h‰g…s
1
2
the spatial
metric
suggested by Eq. (16) is the Euclidean

p
jsj ˆ s21 ‡ s22 . In the case of two-phase ¯ow in a porous
domain the governing equations for phases a ( ˆ water
and oil) are [9]
dfa
‡ /…ea ; K a †fa ˆ 0;
dla

…17†

where ea is the direction vector of the a-¯owpath trajectory, fa the magnitude of the pressure gradient in the
direction ea , Ka denotes the intrinsic permeabilities of the
phases, and / is a function of ea and Ka . The solution of
Eq. (17) is of the form fa ˆ fa …jsj†, where the corresponding metric jsj ˆ la 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
oX =ot ‡ m
rX ˆ 0;

…18†

where m ˆ …m1 ; m2 † is an empirical velocity to be determined from the data. By means of a coordinate transformation from the rectangular Euclidean system (si ) to
the system of coordinates de®ned by si ˆ si ÿ mi t, the
solution of Eq. (18) has the form
X …s1 ; s2 ; t† ˆ X…s1 ÿ m1 t; s2 ÿ m2 t†;

…19†

i.e., it depends on the space/time vector p ˆ s ÿ mt.
Therefore, in the rectangular coordinate system a geophysical ®eld governed by Eq. (18) may have a metric of
the Riemannian form (12), where n ˆ 2, g00 ˆ …m21 ‡ m22 †,
g11 ˆ g22 ˆ 1, g10 ˆ g01 ˆ ÿ2m1 , g20 ˆ g02 ˆ ÿ2m2 , and

g12 ˆ g21 ˆ g01 ˆ g10 ˆ g02 ˆ g20 ˆ 0. This expression
demonstrates how the physical law determines the geoT
metric metric through the empirical vector m ˆ ‰m1 ; m2 Š .
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 appropriate 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 k is sought such that
cx …h1 ; . . . ; hn ; s† ˆ cx …k†:

…20†

The metric may be viewed as a transformation
k ˆ T …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
v
uX
n
X
u n
…21†
k…h; s† ˆ t gij hi hj ‡ 2s g0i hi ‡ g00 s2 :
i;jˆ1

iˆ1

While the ®nite space/time distance (21) has the same
form as the in®nitesimal Riemannian distance (12), the
gij s 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 coordinates, that is gij ˆ gij …hi ; hj †, g0i ˆ g0i …s; hi †, i; j ˆ
1; . . . ; n, and g00 ˆ g00 …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 following set of equations:
Pn
ocx =ohi
jˆ1 gij hj ‡ g0i s
ˆ Pn
ocx =ohj
iˆ1 gij hi ‡ g0j s
and

Pn
ocx =ohi
jˆ1 gij hj ‡ g0i s
:
ˆ Pn
ocx =os
iˆ1 g0i hi ‡ g00 s

…22†

For illustration, consider a covariance function in
R1  T that satis®es the following physical model:
ocx =oh
h
ˆ 2 ;
ocx =os m s

…23†

where h ˆ Ds1 ˆ s01 ÿ s1 > 0, s ˆ Dt ˆ t0 ÿ t > 0, m ˆ a=b,
a and b are empirical covariance coecients. Note that
determining the covariance from physical equations,
whenever possible, avoids common problems of empirical covariance estimation, and eliminates the circular
problem of standard geostatistics (i.e., estimating the
covariance from the same dataset that is also used to
obtain the kriging estimates). We seek a metric form

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

k such that cx …h; s† ˆ cx …k†. In view of Eqs. (22) and (23),
the metric coecients are such that
g11 h ‡ g01 s
h
ˆ 2 :
g01 h ‡ g00 s m 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 n ˆ 1,
g00 ˆ m2 , g01 ˆ 0 and g11 ˆ 1; i.e.,
p p
…25†
k…h; s† ˆ h2 ‡ m2 s2 ˆ h2 ‡ a2 s2 =b2 :

Eq. (25) shows how the covariance coecients determine the spatiotemporal metric. A function which is a
permissible covariance model and has a metric of the
form (25) is cx …h; s† ˆ c0 exp …ÿh2 ÿ m2 s2 †. 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 signi®cant consequences in geostatistical analysis. One
such consequence is related to the permissibility of a
covariance model cx …h; s† in Rn  T : 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. Indeed, a covariance that is permissible for one spatiotemporal 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
Z
Z
…26†
c~x …k; x† ˆ dh dseÿi…k
hÿxs† cx …h; s†

805

means of a numerical calculation of the Fourier transform 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 component cx …h†. The
covariance cx …h† is related to the spectral density c~x …k† as
follows:
Z
dh exp …ÿik
h†cx …h†:
…29†
c~x …k† ˆ
R2

Hence, cx …h† is a permissible covariance if the c~x …k† is
non-negative. In the case of the spatial component
2
cx …h† ˆ c0 exp …ÿjhj † 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 integration function `dblquad' with the adaptive±recursive
Newton Cotes algorithm that allows relative and absolute error control (we set both to 1
E ÿ 5). Thus, the

be a real-valued, integrable and non-negative function
of the spatial frequency k and the temporal frequency x.
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 covariance 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, in R2  T the Gaussian
function
cx …h; s† ˆ c0 exp…ÿjhj2 ÿ m2 s2 †;

…27†

jhj ˆ jh1 j ‡ jh2 j

…28†

where the spatial distance is de®ned as

is not a permissible covariance model (a mathematical
proof can be found in [7]). This result is veri®ed by

Fig. 3. The Fourier transform c~x …k† of Eq. (29) using the metric of
Eq. (28). Note the islands of negative values at the four corners of
the frequency domain.

806

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

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 exponential function in R2  T
cx …h; s† ˆ c0 exp…ÿjhj ÿ ms†;

…30†

where the spatial distance is de®ned as in Eq. (28). The
spectral density c~x …k; x† of the function (30) is nonnegative for all k 2 R2 , x 2 R1 . Hence, the exponential
covariance is permissible for the metric of Eq. (28).
In conclusion, the permissibility of a covariance
model cx …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 nonEuclidean metric.

6. Fractal geometry
Many physical processes that take place in non-uniform 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 expressions, such as Eq. (12), since the physical laws may not
be available in the form of di€erential equations. Geometric 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 implies 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 probability p and empty with
probability 1 ÿ p. Similarly, in the bond percolation
model, conducting and non-conducting bonds are randomly assigned with probabilities p and 1 ÿ p. The
medium is permeable if p > pc , where pc is a critical
threshold that depends on the connectivity and dimensionality of the underlying space (for a table of pc values
on di€erent lattices see [20]). The percolation model has
applications in many environmental and health processes 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 antigen±antibody reactions in biological systems [38].
Length and distance measures on a percolation cluster, denoted by l…r†, scale as power laws with the Euclidean (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† ˆ bd0 l…r†;

…31†

where r is the appropriate Euclidean distance, d0 the
fractal exponent for the speci®c property, and b a 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 change db, the scaling factor
becomes b0 ˆ b ‡ db and Eq. (31) leads to
d

l…b0 r† ˆ …b ‡ db† 0 l…r†:

…32†

Expanding both sides around b ˆ 1, we obtain
dl…r†
l…r†
ˆ d0
:
dr
r
By integrating Eq. (33), we obtain
 d0
l…r†
r
ˆ
:
l…rco †
rco

…33†

…34†

where rco is the lower cuto€ for the fractal behavior. For
example, the length of the minimum path on a percolation fractal scales as lmin …r† / rdmin , where r denotes the
Euclidean distance between the points. The fractal dimension dmin of the percolation fractal on a hypercubic
lattice satis®es 1 6 dmin 6 2, where dmin  1:1; 1:3 for
d ˆ 2; 3 [17,36]. Thus, if the minimum path length between two points at Euclidean distance r is on average
2 miles, the length of the minimum path between two
points separated by 2r is, 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 Euclidean space (curve 1) and in a fractal space with
d0 ˆ 1:15 (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 selfsimilarity 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 displaced from a porous medium saturated by an invading
¯uid (e.g., water).
Below, we investigate an example of a composite
space/time covariance model for fractal spaces. Within
the fractal range, we consider a covariance function of
the form

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

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 ˆ 1:15 (curve 2).

cx …h; s† / ra …s=rb †z ;

…35†

where r ˆ jhj, s0  s  sm and r0  r  rm de®ning the
space/time fractal ranges. General permissibility conditions for unbounded fractals [20], i.e., s0 ˆ r0 ˆ 0 and
rm ˆ sm ˆ 1 impose certain constraints on the exponents 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 requires that the spectral density c~x …k; x† be a monotonically decreasing function of bounded variation. The
spectral density of the function (35) is given by
Z
Z
ds eixs sz :
…36†
c~x …k; x† ˆ dh eÿik
h raÿbz
It follows that the permissibility conditions are
ÿ1 < z < 0 and ÿ…n ‡ 1†=2 < a ÿ bz < 0. If b > 0, the
last inequality implies that a < 0.
As is shown in Appendix A, a covariance function
that has the fractal behavior of Eq. (35) and a ®nite
variance r2 is given by
cx …h; s; uc ; wc † ˆ r f^z …s=r ; uc †f^a …r; wc †;
2

b

…37†

where r2 is the variance. The covariance function is
plotted in Fig. 5 for r ˆ 1, z ˆ a ˆ ÿ1=2, b ˆ 1:1, and
cuto€s uc ˆ 25; wc ˆ 25. The axes used in the picture
are r and s=rb . The function f^z …s=rb ; uc † has an unusual
dependence on the space and time lags through s=rb .

807

Fig. 5. Plot of the fractal correlation function of Eq. (37) for
z ˆ a ˆ ÿ1=2 and b ˆ 1:1. The correlation function is de®ned as the
covariance normalized by the variance.

For large s, the ratio s=rb is close to zero if r is suciently large, and the value of the function f^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 =r1b ˆ s2 =r2b .
Hence, the equation for equidistant space/time contours
is s=rb ˆ c. 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 =n2r ‡ s2 =n2s ˆ c. 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 =n2r ÿ s2 =n2s † (dots) as a function of the space
and time lags. The contour labels represent the values of
c0 s=rb (solid lines), and r2 =n2r ‡ s2 =n2s (dots), obtained
using c0 ˆ 62:95, nr ˆ 10 and ns ˆ 5.
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 different space/time maps if estimation is performed using
di€erent metric structures.

808

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

Fig. 6. Equidistant contours for fractal space/time dependence (solid
contours) and for Gaussian dependence (dotted contours).

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 h ˆ …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 transport. For the sake of illustration, assume that the appropriate metric for this ®eld is the non-Euclidean form
jh1 j ‡ jh2 j. Spatial estimates using this metric were generated on the basis of a hard dataset vhard using a geostatistical 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 focuses solely on the dataset available and ignores physical
models. The standard commercial software for geostatistics
p restricts the user to the Euclidean metric
h21 ‡ h22 for covariance estimation and kriging. If this
metric were used, the same dataset vhard as above results
in the contour map in Fig. 7(b). As expected, the two
maps show considerable di€erences. The Euclideanbased map (Fig. 7(b)) assumes a convenient but inade-

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

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

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

particular, we presented several examples for metrics
and covariance functions in Euclidean and non-Euclidean spaces, including fractal spaces. We showed that
these metrics can lead to very di€erent geometric properties 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 covariance 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 metric, 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 software should be aware of the limitations of these packages. 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 referees 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).

809

This function has power-law behavior for x  xc  1=yc ,
and at the same it is ®nite at the origin. The power-law
behavior can be shown as follows: For xyc  1, Eq. (A2)
can be replaced with Eq. (A1) without considerable error, due to the fact that the exponential term in the integral is essentially zero. Then, it is straightforward to
show by a change of variables x ! kx in Eq. (A1) that
fm …kx; yc † ˆ km fm …x; yc †, which characterizes a power law
with exponent m. We express fm …x; yc † as fm …x; yc † ˆ
xm c…ÿm; xyc †=C…ÿm†, where c denotes the incomplete
gamma function [16]. The integrand of Eq. (A2) has an
integrable singularity at y ˆ 0. By means of the transformation y ˆ 1=y 0 , we avoid the singularity and obtain
Z 1
1
0
dy 0 y 0mÿ1 eÿx=y :
…A3†
fm …x; yc † ˆ
C…ÿm† 1=yc
Eq. (A3) is more convenient than (A2) for numerical
calculations. In view of (A3), the value at the origin is
ÿ1
fm …0; yc † ˆ ‰ÿmC…ÿm†Š ycÿm . In Fig. A1, we plot the normalized function
f^m …x; yc † ˆ fm …x; yc †=fm …0; yc †

…A4†

for yc ˆ 1 and the exponential function exp …ÿx=n† with
n ˆ 6:5. Note that the power-law decays asymptotically
much slower than the exponential. In Fig. A2, we plot
f^m …x; yc † for di€erent values of 1=yc . Increasing yc (that is,

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
(where x stands for the spatial lag, the time lag, or some
combination thereof) with a negative exponent m < 0
can be expressed as
Z 1
1
dy eÿyx y ÿ…m‡1† ;
…A1†
xm ˆ
C…ÿm† 0
where C is the gamma function. Because m < 0, the
function xm is singular at r ˆ 0. The singularity is tamed
by imposing an upper cuto€ yc on the integral, thus
leading to the function
Z yc
1
dy eÿyx y ÿ…m‡1† :
…A2†
fm …x; yc † ˆ
C…ÿm† 0

Fig. A1. Plot of the fractal correlation function f^m …x; yc † with m ˆ ÿ1=2
and yc ˆ 1 (solid) and the exponential correlation function exp…ÿx=n†
with n ˆ 6:5 (dots) vs. the lag x.

810

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

Fig. A2. Plot of the function f^m …x; yc † with m ˆ ÿ1=2, vs. the lag x for
three di€erent values of the cuto€ 1=yc .

decreasing 1=yc ) leads to a steeper slope of f^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 variance r2 .
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