WATER RESOURCES RESEARCH, VOL. 34, NO. 5, PAGES 1251–1263, MAY 1998
Stochastic analysis of the relationship between topography
and the spatial distribution of soil moisture
Pat J.-F. Yeh and Elfatih A. B. Eltahir
Ralph M. Parsons Laboratory, Department of Civil and Environmental Engineering
Massachusetts Institute of Technology, Cambridge
Abstract. This paper deals with the issue of the spatial horizontal variability of soil
moisture in the root zone of a shallow soil at the large scale. The problem of water flow in
the unsaturated zone is formulated so that topography appears explicitly as a forcing for
the movement and redistribution of soil moisture. This formulation emphasizes the role of
the lateral redistribution of water that is induced by topography. A stochastic theory is
developed to relate the statistical distribution of soil moisture to that of elevation. This
approach will ultimately facilitate the use of the readily available data sets describing
topography for the purpose of defining the large-scale distribution of soil moisture. The
steady state horizontal distribution of soil moisture under homogeneous bare soil
conditions is regulated by three distinct factors: topography, climate, and soil properties.
First, topography, forces a distribution of soil moisture that tends to mimic the elevation
field at large scales. The other two factors are the vertical divergence of water in response
to the climate forcing (evaporation) and the capillary resistance to water movement. The
climate forcing tends to smooth the spatial distribution of soil moisture. However, the
capillary forces exerted by the soil matrix tend to resist displacement of water and hence
exert adverse effects against the topography and climate forcings. The variance of the soil
moisture distribution increases with the variance of the elevation field and decreases with
the correlation scale of the elevation field and the magnitude of the climate forcing. The
impact of capillary forces on the vertical fluxes of water is more significant than their
impact on the topographically induced horizontal fluxes, owing to the larger hydraulic
gradient in the vertical direction resulting from the disparity in scale between the vertical
and horizontal directions.
1.
Introduction and Motivation
The accurate specification of soil moisture distribution is a
critical step in two important and related research areas: (1)
representation of land and surface processes in climate models
[Entekhabi and Eagleson, 1989] and (2) the development of
large-scale models of vegetation dynamics [Solomon and Shugart, 1993]. Under certain conditions, soil moisture variability
may control land surface processes such as evaporation, runoff,
and vegetation growth. The relations between soil moisture
and land surface processes are, in general, nonlinear, which
suggest that the scaling up of these processes from small scales
to large scales has to be carried out with careful consideration
to the spatial variability in soil moisture distribution over large
areas.
Topography, soil type, vegetation, and climate are the key
physical factors that control soil moisture distribution over
large scales. Although significant information is readily available about topography (for example, from digital elevation
maps (DEMs) at a resolution of about 30 m), soil type (from
soil survey maps), vegetation (from global land cover data
sets), and rainfall (from surface stations and radar networks),
only very little information is usually available about the largescale distribution of soil moisture. Some of our ignorance is
due to the difficulty in obtaining synoptic measurements of soil
Copyright 1998 by the American Geophysical Union.
Paper number 98WR00093.
0043-1397/98/98WR-00093$09.00
moisture over large areas using standard equipment such as
neutron probes. As a potential solution, microwave remote
sensing techniques have been developed for measuring soil
moisture from space [Jackson and Schmugge, 1989]. Although
these techniques are promising because they offer the potential for coverage of large areas, they would provide wetness
information about only the upper few centimeters of the soil
profile.
The focus of this paper is soil moisture averaged over the
total depth of the upper soil layer. The emphasis is on developing a general theory that would ultimately facilitate the use
of the significant amounts of data on topography, together with
information about rainfall and soil type, for the purpose of
defining the large-scale distribution of soil moisture. Although
physical intuition suggests that for a given soil type and climate, areas of topographic convergence (divergence) are favorable locations for relatively wet (dry) soil moisture conditions, there is no rigorous theory relating this degree of soil
wetness to the corresponding topographic conditions. The primary objective of this paper is to develop a theoretical framework for relating the spatial distribution of soil moisture to
topography. We are interested in the spatial distribution of soil
moisture (measured in terms of the level of soil saturation
relative to the storage capacity of the soil layer), at the following scales: (1) horizontal spatial scale of ⬃10 km and (2)
vertical spatial scale (soil depth) of the upper 1 m. The 10-km
horizontal scale is considered as a first step before addressing
the problem of soil moisture distribution at larger scales (⬃100
km) compatible with the typical resolution of a climate model.
1251
1252
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
Figure 1. Methodology.
However, the techniques developed in this paper are applicable to the latter problem.
The classical treatment of water flow in the unsaturated zone
[e.g., Philip, 1957] considers the one-dimensional vertical infiltration into the horizontal surface of an isotropic and homogeneous soil. By definition this classical theory deals with the
vertical flow of water. In a more recent study, Giorgini et al.
[1984] considered the more general case of two-dimensional
flow into sloping surface of anisotropic homogeneous soil.
Their theory describes the situations in which lateral flow
would be significant. Several recent studies suggested that lateral flow of water is a significant process in the unsaturated
zone. McCord and Stephens [1987] observed significant lateral
unsaturated flow in a hillslope located in New Mexico. They
concluded that [McCord and Stephens, p. 225] “there is strong
lateral component to unsaturated flow on a hillslope, even in
the absence of apparent sublayers of much lower permeability.” On the basis of theoretical considerations and observations, Zaslavasky and Sinai [1981] indicated that the slope of
the soil surface is a significant cause of lateral flow in unsaturated zone. Overall, the results of these studies suggested that
the primary causes for lateral flow are the anisotropy of the
medium and the gradient of the surface (topography). Hewlett
and Hibbert [1963] investigated the flow of water in a sloping
soil mass using an experimental approach and found a significant lateral flow component. Many of the observations about
lateral flow come from runoff generation studies. Genereux and
Hemond [1990] estimated that about 70% of the water flowing
into the reach of a small stream in central Massachusetts is
flowing from the unsaturated zone. In this paper we emphasize
the role of lateral flow in the redistribution of soil moisture
under topographic forcing.
A simple and popular approach for relating the spatial distribution of soil moisture to topography was suggested by
Beven and Kirkby [1979] and is known as the TOPMODEL.
The basic idea in this model assumes that the groundwater
table intersects with the surface at those locations where the
capacity of the saturated soil profile to transport water is
smaller than the flux of water. The relation to topography
comes from the assumption that the hydraulic gradient for this
saturated flow is equivalent to the elevation gradient at the
surface. Hence this model is based on considerations of flow in
the saturated zone. It neglects lateral flow of water in the
unsaturated zone. Using the method of characteristics, Hurley
and Pantelis [1985] extended the theory of kinematic subsur-
face stormflow proposed by Beven [1981] to include the downslope lateral flow in a porous layer overlying on an impervious
bedrock. Their numerical model can be used to calculate the
recharge and subsequent drainage following a rainfall event.
Using an analytical approach, Stagnitti et al. [1986] proposed a
simple hillslope hydrological model for the prediction of the
drainage from a sloping shallow soil. The experimental data
collected by Hewlett and Hibbert [1963] were used to test their
model.
The focus of this paper is the root zone of a shallow unsaturated soil where the groundwater table is deep below the
surface. The interactions between the root zone and the location of the shallow groundwater table add more complexity to
the problem and will not be addressed in this paper. These
interactions will be considered as a subject for future research.
This paper is organized in six sections. A general methodology
for studying the relationship between topography and the distribution of soil moisture is described in section 2. The problem of unsaturated flow of water in the top soil layer is formulated and presented in section 3. A stochastic analysis of the
relationship between topography and soil moisture distribution
is presented in section 4. The last two sections of the paper
include the discussions and conclusions.
2.
Methodology
This section describes a framework that is proposed for
relating the distribution of soil moisture to topography, soil
type, and climate. This methodology is summarized in Figure 1.
The input is statistical information about topography, soil type,
and climate. This information serves as forcing for the equation describing flow of water in the unsaturated zone. The
output is statistical characterization of soil moisture distribution at the large scale. This methodology follows a statisticalanalytical approach. Analytical techniques are applied to address the soil moisture distribution problem. The same
problem can be addressed by applying numerical techniques
using distributed models of catchment processes. These two
different approaches are complementary; we chose the analytical approach in order to develop a theory that is general
enough to provide some insight regarding the role of topography in determining the spatial distribution of soil moisture.
Statistical representations are used to describe the natural
heterogeneity of environmental variables. The same approach
has been applied successfully to stochastic theories of ground-
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
water hydrology [Gelhar, 1993]. In this paper, similar techniques will be applied to derive the statistical distribution of
soil moisture in space. The product of the proposed analysis
will be to relate the statistical distribution of soil moisture to
the statistical distribution of elevation. This product is adequate for the purposes defined in the introduction of this paper.
The horizontal scale considered in this paper is of the order
of kilometers, and the vertical scale considered is of the order
of meters. This comparison of horizontal and vertical scales
suggests that the aspect ratio between the horizontal scale and
the vertical scale is of the order of 103. Hence for simplicity we
will address the problem in two dimensions, considering the
two horizontal coordinates. The treatment of the problem as a
two-dimensional horizontal problem brings a significant advantage. In a two-dimensional description the horizontal derivatives of elevation (the contribution of gravity to hydraulic gradient) will no longer be zero, and hence the role of topography
becomes explicit. This point will become clear following the
formulation of the problem, which will be developed in the
next section. The vertical fluxes of water (evaporation and
percolation) can be parameterized in terms of soil moisture
state as a forcing term in the two-dimensional equation. In the
following section the unsaturated flow problem will be formulated following this general methodology.
3.
Formulation of the Problem
It is widely accepted that the flow of water in the unsaturated
zone may be described by
q i ⫽ ⫺K共h兲
⭸h
⭸ xi
(1)
where q i is specific flow in the ith direction, K is unsaturated
conductivity, and x i is distance. The variable h denotes hydraulic head defined by h ⫽ ⫺ ⫹ z, where is capillary pressure
(a negative sign indicates hydraulic suction) and z is elevation
(positive upward). Equation (1) is a nonlinear equation, which
complicates treatment of the unsaturated flow problem.
Several empirical relations have been suggested in literature
to relating K with and to relate both variables with , defined
as available water content per unit total volume. Following
Gardner [1958] and Mantoglou and Gelhar [1987], we assume
that
K ⫽ K se ⫺␣
(2)
⫽ s ⫺ C
where K s is the saturated conductivity, s is the saturated soil
moisture content, and C is the specific moisture capacity,
which is the slope of the soil-water retention curve (i.e., the
rate of change of soil moisture content with respect to capillary
pressure head). Here ␣ represents the relative rate of decrease
of hydraulic conductivity with increasing capillary pressure
head and is associated with the width of soil pore size distribution, and ␣⫺1 is the thickness of capillary fringe, a measure
of the relative importance of capillary force to gravity force for
soil moisture movement in a specific soil. Fine-textured soils in
which capillary force tends to dominate have greater thickness
of capillary fringe than coarse-textured soils, in which gravity
effects manifest themselves most readily [Philip, 1969]. According to published literature (for a review, see Pullan
[1990]), the range of ␣⫺1 covers 0.01–10 m. However, 0.2–5 m
seems to be the typical range of values for ␣⫺1 [Philip, 1969].
1253
The substitution of (2) into (1) results in a significant simplification
qi ⫽ ⫺
⭸z
1 ⭸K
⫺K
␣ ⭸ xi
⭸ xi
(3)
Equation (3) is a linear equation on K which represents a
significant advantage when compared with (1). Topography is
represented by the elevation gradient, which appears explicitly
in (3). (Note that since the problem is considered in two dimensions, the partial derivatives of z are no longer zero.)
Conservation of water mass requires
D
⭸
⭸q i
⫺s⫹R
⫽ ⫺D
⭸t
⭸ xi
(4)
where D is the depth of root zone (assumed to be constant)
and s is the sink of water mass due to vertical fluxes. The
variable R is effective rainfall, defined as the fraction of rainfall
that infiltrates into the soil. Although the value of this fraction
depends on the soil moisture conditions at the surface, for
simplicity we assume that effective rainfall is a constant fraction of surface rainfall.
In most natural settings the top layer of the soil (1–2 m) has
significantly larger hydraulic conductivity than the underlying
layers [Beven, 1982]. Here we assume that the underlying layer
is completely impermeable. Hence the sink of moisture due to
vertical divergence, s, represents the evaporation flux upward
to the atmosphere. Vertical fluxes of transpiration and percolation are not considered in this study. This evaporation flux
can be parameterized as s ⫽  K, assuming that the evaporation flux is proportional to the unsaturated hydraulic conductivity of the soil. Note that physically,  is equivalent to the
vertical hydraulic gradient near the ground surface. From the
Gardner [1958] relationship in (2), the evaporation flux s for
ⱕ s , can be written as
s ⫽  K ⫽  K se 关␣共⫺s兲兴/C
(5)
which increases exponentially to approach its maximum of  K s
as increases from the residual moisture content to the saturation. Note that the exponential dependence in (5) resembles
the widely recognized pattern of the observed relationships
between evaporation and soil moisture level of a bare soil [see
Lowry, 1959, Figure 1; Rodriguez-Iturbe et al., 1991, Figure 2].
This fact provides an observational basis for the proposed
parameterization as a rough description of evaporation flux.
For a specific soil type the magnitude of the gradient  depends on climate and land cover. However, this parameterization may not be appropriate for a deep soil layer where variability in the vertical soil moisture distribution is significant.
For deep soils the rate of evaporation decays with depth, and
the vertical flow induced by gravity becomes significant. This
vertical flow exerts additional influences on both the vertical
climate flux and horizontal soil moisture redistribution, which
needs the consideration of a three-dimensional description.
Therefore a shallow, unsaturated soil overlying an impervious
soil layer or bedrock is the condition considered in this paper.
The more complicated treatment of three-dimensional flow in
deep soils is left as a topic for future research.
Typical values of  can be estimated from typical values of
evaporation. The range of evaporation is assumed to be 50 –
100 cm yr⫺1; typical values for saturated hydraulic conductivity
K s are 8 ⫻ 10⫺3 cm s⫺1 for sand and 3 ⫻ 10⫺5 cm s⫺1 for clay
1254
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
Figure 2. (a) Elevation distribution and (b) correlation function for a 3.2-km transect (5-m resolution) from
the Mahantango Creek drainage basin, Pennsylvania. The correlation function is fitted by two different
theoretical correlation functions: exponential function in (22) and hole-type function in (23). The dotted line
in the figure marks the correlation scale corresponding to the correlation of e ⫺1 .
[Bras, 1990, p. 352]. Therefore the range of , which is an
indicator of the strength of evaporation, is approximated as
10⫺4–10⫺1 in this paper.
Substituting for q and s in (4) results in a general formulation of the problem
D
⭸
⭸
⫽D
⭸t
⭸ xi
冉
冊
1 ⭸K
⭸z
⫺ K ⫹ R
⫹K
␣ ⭸ xi
⭸ xi
(6)
The basic variable in this formulation is the unsaturated hydraulic conductivity K, which serves as a surrogate for soil
moisture content. Even if soil properties are constant in space,
K would vary in space depending on soil moisture variability.
All the variables and parameters considered in this study are
effective variables and parameters representing a block of soil
with a vertical dimension of about 1 m. The root zone in a
natural environment is composed of heterogeneous soil, roots,
and possibly macropores. In this study it will be assumed for
simplicity that classical theories describing flow of water in the
unsaturated zone (such as (1) and (2)) are valid for characterizing the behavior of the root zone. Similar assumptions are
made in most field applications that deal with groundwater
problems.
In the following, the steady state case will be considered. It
is reasonable to assume steady state when considering long
timescales. Equation (4) can be averaged in time to obtain
D
⭸E关 兴
⭸E关q i兴
⫽ ⫺D
⫺ E关s兴 ⫹ E关R兴
⭸t
⭸ xi
(7)
where E[ ] denotes the expected value or time average. Since
large-scale soil moisture distribution is of interest, and since
evaporation is the only process responsible for the vertical
divergence of soil moisture (i.e. monotonic drying), the hysteresis effect is disregarded in this paper. At steady state, the
left-hand side of (7) is negligible. Substituting for q and s into
(7) results in
D
⭸
⭸ xi
冉
冊
1 ⭸E关K兴
⭸z
⫹ E关K兴
⫺  E关K兴 ⫹ E关R兴 ⫽ 0
␣ ⭸ xi
⭸ xi
(8)
Equation (8) is an ordinary differential equation on E[K].
Time-averaged effective rainfall, E[R], is almost uniform over
the scale of interest (i.e., order of kilometers). Hence the main
forcing in (8) is spatial variability in topography, which is represented by the gradients of elevation. This equation relates
unsaturated hydraulic conductivity K, which is a surrogate for
soil moisture content, to topography, described by elevation z.
These two variables will be considered as random processes in
the stochastic analysis of the relationship between soil moisture and topography presented in the next section.
4. Stochastic Analysis of the Relationship
Between Topography and Soil Moisture
Observations of elevation fields indicate significant variability for a range of spatial scales. S. Lancaster (unpublished
report, 1993) analyzed several transects of elevation extracting
from DEMs for different regions of the continental North
America. The resolution of DEMs is 30 m, and the length of
these transects is about 30 km. After appropriate detrending,
the correlation scales of these elevation distributions are found
to lie within the range from several hundred meters to several
kilometers (i.e., 102–104 m). Note that the estimated correlation scale is closely dependent on the resolution of data: finerresolution data have the impact of shortening the correlation
scale estimated from a coarser one. We also have analyzed the
statistical distribution of elevation which provides an adequate
representation of observed topography. Figure 2 shows one of
the transects of elevation distribution and the corresponding
autocorrelation function from the Mahantango Creek drainage basin, Pennsylvania. The resolution of the pixels is 5 m.
The standard deviation and correlation scale of this elevation
transect are roughly 20 m and 500 m, respectively. Then the
correlation function is fitted by two frequently used theoretical
models of correlation [Gelhar, 1993, p. 44]: exponential and
hole-type function (which are given later in (22) and (23)).
Following Bakr et al. [1978], the correlation scale of hole-type
correlation function is taken as 2.5 times of that of the exponential correlation function for a consistent fit of a given data.
The assumption that the elevation field, which appears in
(8), is a random field transforms the equation into a stochastic
differential equation. In this paper we investigate the solutions
of this stochastic differential equation when forced by statistical descriptions of the elevation fields. The mathematical technique that will be used in solving this equation is the perturbation method. Application of this method results in equations
relating small perturbations in unsaturated hydraulic conductivity to perturbations in elevation field. The solutions of such
equations will be explored using spectral representations of
stationary random fields. The final result will be to relate the
statistical distribution of unsaturated conductivity to the statis-
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
tical distribution of elevation. Following this step, the statistical
distribution of will be estimated from the statistical distribution of K through (2).
4.1.
⭸
⭸ xi
冉
䡠
冊
冋
⭸ z
1 ⭸K
⭸K⬘ ⭸ z⬘
⫹ R ⫹ DE
⫹K
⫺ K
␣ ⭸ xi
⭸ xi
⭸ xi ⭸ xi
册
⫽ R/ 
K
That is, the large-scale, steady state average unsaturated
hydraulic conductivity is proportional to effective rainfall and
inversely proportional to vertical hydraulic gradient (i.e., evaporation). Subtracting the mean equation (9) from (8), the perturbation equation is given by
冉
冊
冕
冕
K⬘ ⫽
In order to relate soil moisture content (hereinafter referred
as SM for brevity) and topography, we analyze the relationship
between SM and K. Combining Gardner’s relationships from
(2), we get
⫽ s ⫹
(12)
⬁
e i共kxx⫹kyy兲 dZ K共k x, k y兲
⬘ ⫽
S K共k x, k y兲
␣ 2K 2共k 2x ⫹ k 2y 兲 2
S 共k , k 兲
共k ⫹ k ⫹ ␣ /D兲 2 ⫹ ␣ 2关k x共⭸ z /⭸ x兲 ⫹ k y共⭸ z /⭸ y兲兴 2 z x y
2
x
2
y
(13)
Therefore the variance of K, 2K , is related to that of z, 2z , by
the following relation:
2K ⫽
冕冕
⫺⬁3⬁
where
2 20 2z
S K共k x, k y兲 dk x dk y ⫽ ␣ 2K
C
␣ K
K⬘
(14)
兩 兩⬍⬍1
(16)
(17)
Using the spectral representation of ⬘
冕
⬁
e i共kxx⫹kyy兲 dZ 共k x, k y兲
(18)
⫺⬁
(17) leads to the following relations between the spectrum and
variance of SM, S (k x , k y ) and 2 , and those of the unsaturated hydraulic conductivity:
S 共k x, k y兲 ⫽
C2
S K共k x, k y兲
␣ 2K 2
C2
2 ⫽ 2 2 2K
␣K
(19)
The goal is to relate the variance in SM to the variance of
elevation. This is achieved by combining (14) and (19) to obtain
2 ⫽ C 2 20 2z
where k x , k y are wave numbers in x and y directions, and dZ
is the spectral amplitude of a random process. By substituting
the spectral relationships (12) into (11), we find that the spectral density function of K, S K (k x , k y ), is related to that of z,
S z (k x , k y ), through
(15)
is used with its first-order term only to deduce that
⫺⬁
⫽
2 3
⫹ ⫺ ⫹···
2
3
ln 共1 ⫹ 兲 ⫽ ⫺
(11)
e i共kxx⫹kyy兲 dZ z共k x, k y兲
C
K
ln
␣
Ks
⫹ K⬘, and K⬘ ⬍⬍ K
, the
Assuming that ⫽ ⫹ ⬘, K ⫽ K
Maclaurin series
⬁
⫺⬁
2
S z共k x, k y兲
dk x dk y
2z
⬘ ⫽
In the derivation of (11) it has been assumed that the second-order term, (⭸K⬘/⭸ x i )/(⭸ z⬘/⭸ x i ), which includes the
product of perturbation terms, is equal to its expected value in
brackets in the mean equation (9).
Equation (11) will be solved using spectral techniques by
defining the spectral representations of the two dimensional
stationary random fields K⬘ and z⬘ as follows [Gelhar, 1993]:
z⬘ ⫽
2
y
(9)
(10)
⭸ z⬘
⭸ z
1 ⭸K⬘
⫺  K⬘ ⬇ 0
⫹K
⫹ K⬘
␣ ⭸ xi
⭸ xi
⭸ xi
共k 2x ⫹ k 2y 兲 2
共k ⫹ k ⫹ ␣ /D兲 ⫹ ␣ 2关k x共⭸ z /⭸ x兲 ⫹ k y共⭸ z /⭸ y兲兴 2
2
x
⫽0
i ⫽ 1, 2
can be
Assuming constant large-scale averages K and z , K
derived if the second-order product term in the bracket of (9)
is neglected:
⭸
D
⭸ xi
冕冕
⫺⬁3⬁
Preliminary Analysis
We first write (8) in two dimensions and drop the implied
expected value notation for simplicity. Each of K and z is
decomposed into two terms, a spatial mean and a perturbation
, K⬘, z , z⬘). Then by taking the average of (8), we
term (K
obtain the following partial differential equation:
D
20 ⫽
1255
(20)
where 20 is defined in (14).
4.2.
One-Dimensional Analysis
The one-dimensional unsaturated flow in a varying elevation
field is considered. First, by using the one-dimensional version
of the spectral relationships, (13) and (19), and assume that the
trend of elevation does not exist, we derive
S 共k兲 ⫽
C 2k 4
S 共k兲
共k ⫹ ␣ /D兲 2 z
2
(21)
In order to evaluate the variance and covariance function of
SM, the exponential covariance and hole-type covariance functions plotted in Figure 2b are selected to describe the elevation
field. They will be compared to investigate the impact of the
functional form of input statistics on the analytical results. The
covariance-spectrum pair for the exponential covariance function is
1256
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
Figure 3. One- and two-dimensional normalized SM variances 2 / 2z C 2 in (25), (27), and (34), resulting
from the input of one-dimensional exponential and hole-type covariance functions of elevation and from the
two-dimensional Bessel-type covariance function of elevation, respectively. (The correlation scale of holetype covariance function is taken as 2.5 times of that of the exponential covariance function, according to Bakr
et al. [1978]).
R z共 兲 ⫽ 2z exp
冉 冊
⫺
in (23) can be substituted into (21), and the covariance function R ( ) is evaluated as (see Appendix A)
兩兩
(22)
2 2z
S z共k兲 ⫽
共1 ⫹ 2k 2兲
R 共 兲 ⫽ 2z C 2
and the covariance-spectrum pair for the hole-type covariance
function is
R z共 兲 ⫽ 2z 共1 ⫺ 兩 兩/ 兲 exp 共⫺兩 兩/ 兲
2
z
(23)
3 2
2 k
S z共k兲 ⫽
共1 ⫹ k 2 2兲 2
⫹
冋
1/ 2
g
exp 共⫺g 1/ 2 兲共1 ⫹ g 1/ 2 兲
2共1 ⫺ g 兲
g 1/ 2共 g ⫺ 2兲
1
exp 共⫺g 1/ 2 兲 ⫹
exp 共⫺ 兲
共 g ⫺ 1兲 2
共 g ⫺ 1兲 2
g ⫽ ␣ 2/D
册
(24)
⫽ 兩 兩/
where g is a dimensionless parameter and is the lag normalized by the correlation scale .
The variance of SM is obtained when ⫽ 0
2 ⫽
共 g 3/ 2 ⫺ 3g 1/ 2 ⫹ 2兲 2 2
zC
2共 g ⫺ 1兲 2
⫺g 3/ 2
exp 共⫺g 1/ 2 兲共1 ⫹ g 1/ 2 兲
共 g ⫺ 1兲 2
2g 3/ 2共 g ⫺ 3兲
1
exp 共⫺g 1/ 2 兲 ⫺
共 g ⫺ 1兲 3
共 g ⫺ 1兲 2
䡠 exp 共⫺ 兲共1 ⫹ 兲 ⫹
2共3g ⫺ 1兲
exp 共⫺ 兲
共 g ⫺ 1兲 3
g ⫽ ␣ 2/D
where and are the correlation scale of the exponential and
the hole covariance functions, respectively. By substituting (22)
in (21), the covariance function of SM distribution is derived in
terms of the inverse Fourier transform of S(k) (see Appendix A):
R 共 兲 ⫽ 2z C 2
⫹
冋
(25)
Using the same approach, the hole covariance-spectrum pair
册
(26)
⫽ 兩 兩/
The variance of is obtained if ⫽ 0
2 ⫽
共 g 5/ 2 ⫺ 5g 3/ 2 ⫹ 5g ⫺ 1兲 2 2
zC
共 g ⫺ 1兲 3
(27)
The SM variances (normalized by 2z C 2 ) in (25) and (27)
corresponding to the input of exponential and hole-type covariance functions of elevation, are graphed in Figure 3. Figure
4 shows the sensitivity of SM variance to the parameters , ,
␣⫺1, and D. Moreover, the correlation function of SM (i.e.,
covariance function R in (24) and (26) divided by the variance
in (25) and (27), respectively) corresponding to the exponential and hole-type function inputs are graphed in Figure 5 for
several values of . Figure 6 shows the sensitivity of the SM
correlation function to , ␣⫺1, and D (Figures 6a, 6b, and 6c,
respectively), as well as the correlation scales of SM distribution for several typical values of , ␣⫺1, and D. These results
will be discussed in section 5.
In order to investigate the degree of correlation between
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
1257
topography and SM distribution, the cross-covariance function
between and elevation z is derived as follows:
Exponential function
R z共 兲 ⫽ 2z C
冋
g 1/ 2
1
exp 共⫺g 1/ 2 兲 ⫺
exp 共⫺ 兲
g ⫺ 1
g ⫺ 1
册
(28)
Hole-type function
R z共 兲 ⫽ 2z C
冋
2g 3/ 2
1
exp 共⫺g 1/ 2 兲 ⫹
共 g ⫺ 1兲 2
g ⫺ 1
䡠 exp 共⫺g 1/ 2 兲共1 ⫹ g 1/ 2 兲 ⫺
2共2g ⫺ 1兲
exp 共⫺ 兲
共 g ⫺ 1兲 2
册
(29)
The derivations of (28) and (29) are similar to those presented in Appendix A and hence are not included in this paper.
The one-point (i.e., lag zero) cross covariance between and z
can be derived if ⫽ ⫽ 0 in (28) and (2):
Exponential function
z ⫽
2z C
1 ⫹ g 1/ 2
(30)
Hole-type function
z ⫽ 2z C
2g 3/ 2 ⫺ 3g ⫹ 1
共 g ⫺ 1兲 2
(31)
The one-point (lag zero) cross correlations derived from dividing (30) and (31) by the corresponding z , are displayed
in Figure 7, and the cross-correlation functions (i.e., cross-
Figure 5. (a) One-dimensional and (b) two-dimensional correlation functions of SM distribution, R / 2 , for several values
of elevation correlation scale ( ⫽ 10⫺3, ␣⫺1 ⫽ 0.5 m, D ⫽
1 m). The dotted line in the figure marks the correlation scale
corresponding to the correlation of e ⫺1 .
covariance functions in (28) and (29) divided by the corresponding z , respectively) resulting from the input of exponential and hole-type functions are shown in Figure 8. These
figures are discussed in section 5.
4.3.
Two-Dimensional Analysis
The spectral approach can be extended to the twodimensional unsaturated flow case by using multidimensional
spectral representation of elevation and soil moisture. A
Bessel-type spectrum-covariance pair for two-dimensional isotropic spatial processes was suggested by P. Whittle [see
Gelhar, 1993, p. 47]
R z共 兲 ⫽ 2z
冉冊
K
1
(32)
2z 2
S z共k兲 ⫽
共1 ⫹ 2k 2兲 2
where k 2 ⫽ k 2x ⫹ k 2y . K 1 is the modified Bessel function of
first order. The covariance function of SM can be obtained by
introducing (32) into (21) (see Appendix B):
R 共 兲 ⫽ C 2 2z
冋
䡠 K 0共 g 1/ 2 兲 ⫹
1
4g
K 共 兲
⬘K 共 兲 ⫺
共 g ⫺ 1兲 2 1
共 g ⫺ 1兲 3 0
g 3/ 2
4g
K 共 g 1/ 2 兲 ⫹
共 g ⫺ 1兲 2 1
共 g ⫺ 1兲 3
册
(33)
where the dimensionless parameters g and are defined in
(24). The corresponding SM variance is (see Appendix B)
2 ⫽
Figure 4. Sensitivity of (a) elevation correlation scale , (b)
vertical hydraulic gradient , (c) capillary fringe thickness ␣⫺1,
and (d) soil depth D on the variance of SM distribution for
typical ranges of these parameters: ⫽ 103 m, ␣⫺1 ⫽ 0.5 m,
 ⫽ 10⫺3, D ⫽ 1 m, z ⫽ 30 m. The variances of SM
distribution for three different values of the standard deviation
of elevation are shown in Figure 4a.
冋
册
g ⫹ 1
2g
ln g 2z C 2
2⫺
共 g ⫺ 1兲
共 g ⫺ 1兲 3
(34)
The two-dimensional SM variance in (34) is displayed in
Figure 3 as a function of dimensionless parameter g along
with the SM one-dimensional variances. The correlation function of two-dimensional SM distribution in (33) is shown in
Figure 5b for several values of .
The cross-covariance function between and z can be derived as follows:
R z共 兲 ⫽ C 2z
⫹
冋
⫺2g
1
K 共 g 1/ 2 兲 ⫺
⬘K 共 兲
共 g ⫺ 1兲 2 0
共 g ⫺ 1兲 1
2g
K 共 兲
共 g ⫺ 1兲 2 0
册
(35)
1258
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
Figure 6. One-dimensional correlation functions of SM for several values of (a) vertical hydraulic gradient
, (b) capillary fringe thickness ␣⫺1, and (c) soil depth D, and the corresponding correlation scales ( ⫽ 103
m).
The corresponding one-point cross covariance (i.e., at lag zero) is
z ⫽ 2z C
冋
2g
1
ln g ⫺
共 g ⫺ 1兲 2
g ⫺ 1
册
(36)
The one-point cross correlation ((36) divided by z ) is
plotted in Figure 7 for comparison with the corresponding
one-dimensional solutions. The cross-correlation function (i.e.,
(35) divided by z ) is shown in Figure 8 along with the
corresponding one-dimensional solutions.
5.
Discussion
Three distinct mechanisms can be identified as the dominant
factors in determining the steady state horizontal distribution
of soil moisture assuming a bare homogeneous soil condition.
First, topography forces a large-scale distribution of SM that
mimics the elevation field. Specifically, the amplitude and correlation scale of SM distribution are proportional to those of
elevation distribution if only topographic forcing is considered.
Figure 7. One- and two-dimensional one-point (i.e., lag zero) cross-correlations between SM and elevation z as functions of dimensionless parameter g .
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
Figure 8. (a) One-dimensional and (b) two-dimensional
cross-correlation functions between SM and elevation z for
several values of elevation correlation scale ( ⫽ 10⫺3,
␣⫺1 ⫽ 0.5 m, D ⫽ 1 m).
Generally, areas of topographic convergence (valleys) are favorable locations for the accumulation of SM, while areas of
topographic divergence (ridges) usually experience dry SM
conditions. Second, the climate forcing (evaporation)
smoothes the SM distribution by differentially extracting water
from the relatively wet regions. This results in a reduced variance of SM distribution. Finally, the soil properties affect the
SM distribution through capillary forces, which can be quantified by the thickness of the capillary fringe, ␣⫺1. For a thick
capillary fringe (␣⫺1 large, for example, clayey soils), water
molecules are strongly bound to the soil and resist gravity
drainage, mainly as a result of the smaller pores and hence
larger surface area. Relatively, sand and gravel contain a high
percentage of larger pores such that water is only weakly
bound to the soil by capillary forces. Therefore SM in sandy
soils with smaller capillary resistance can be removed by the
topographic forcing or climatic forcing relatively easier than is
the case in clayey soils. Since the vertical divergence fluxes (i.e.,
evaporation) have the effect of smoothing the SM distribution,
this capillary resistance has an adverse effect on the vertical
fluxes, and hence would tend to enhance the variance of the
SM distribution. Contrarily, in the horizontal direction, the
variance of SM distribution induced by topography should be
reduced because of the capillary resistance. However, as will be
illustrated later, the impact of capillary resistance on the vertical fluxes of water is more significant than its impact on the
topography-induced horizontal fluxes, which leads to a net
increase of SM variance as the capillary resistance increases.
This is due to the larger hydraulic gradient in the vertical
direction resulting from the disparity in scale between the
vertical and horizontal directions.
In the following we discuss the results presented in the
Figures 3–9. In Figure 3 the normalized variance of SM is
plotted as a function of dimensionless parameter g , which is
1259
defined by g ⫽ ␣ 2 /D. (Unless stated otherwise, typical
values such as correlation scale of elevation ⫽ 103 m, vertical
hydraulic gradient  ⫽ 10⫺3, capillary fringe thickness ␣⫺1 ⫽
0.5 m, soil depth D ⫽ 1 m, specific moisture capacity C ⫽ 0.1
m⫺1 and variance of elevation 2z ⫽ (30 m)2 are taken as the
nominal values in producing these figures). Figure 3 shows that
the variance of SM distribution decreases as g increases.
Large g results from large and , as well as small ␣⫺1 and
D. The value of g ranges over several orders of magnitude
mostly due to the wide range of (⬵ 10 2 –10 4 m) and
(⬵ 10⫺4–10⫺1). From Figure 3 it can be seen that using the
exponential and the hole-type covariance function of elevation
virtually leads to a similar magnitude of SM variance and a
similar rate of decrease with g . Therefore the statistical properties of SM distribution are not sensitive to the functional
form of elevation covariance. The two-dimensional SM variance resulting from the Bessel-type covariance is approximately one order of magnitude smaller than the onedimensional SM variance at small g , and 3– 4 orders of
magnitude smaller at large g . That is, two-dimensional SM
variance decreases with increasing g at a rate faster than the
one-dimensional variance. This stems from the increased freedom of SM movement in two dimensions than in one dimension. In one dimension a low-conductivity zone (i.e., dry soil)
can essentially prohibit SM movement along its gradient direction. However, in two dimensions, soil water movement can
take place laterally, passing around the low-conductivity region
and consequently causing a smoother distribution of SM. Similar conclusions have been reached in the published literature
on stochastic groundwater hydrology. [e.g., Bakr et al., 1978;
Gelhar, 1993].
The sensitivity of SM variance to , ␣⫺1, , and D is shown
in Figure 4. Importantly, the correlation scale of topography
has a direct impact on SM distribution. As shown in Figure 4a,
a smooth elevation field with large correlation scale leads to a
smooth SM distribution, while a surface with significant topographic undulation leads to a highly variable SM distribution.
The variance of SM distribution is proportional to the variance
of elevation distribution (see the expressions in (25) and (27)).
This trend remains unaltered although not presented in Figures 4b, 4c, and 4d. In addition, a larger variance of SM
distribution arises from a smaller vertical divergence of SM
(small ), a fine-textured soil (large capillary resistance), and a
deep soil layer. (The elevation variance of (30 m)2 is used in
producing Figure 4). A large vertical gradient  creates a
substantial vertical moisture sink that eventually reduces the
variance of horizontal SM distribution. Large capillary fringe
thickness ␣⫺1 constrains the moisture movement in both horizontal and vertical direction. Because of the large aspect ration between the horizontal and vertical scales (i.e., ⬃103)
considered, a large disparity in the hydraulic gradient exists in
these two directions. The effect of ␣⫺1 on limiting the efficiency of vertical climatic fluxes and hence smoothing the SM
distribution is thus more significant than the effects of the
capillary resistance on the horizontal fluxes induced by topographic forcing. This explains the slowly rising trend of SM
variance with the capillary fringe thickness in Figure 4b. Further, soil depth, which can be viewed as a characteristic scale of
the soil reservoir, regulates the magnitude of vertical climatic
(evaporative) fluxes in terms of hydraulic gradient in the vertical direction. A relatively deep soil layer alleviates the rate of
vertical fluxes and as a result provides a smaller reduction of
the variance of SM horizontal distribution.
1260
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
Figure 5 shows the one- and two-dimensional correlation
functions of SM distribution for several values of elevation
correlation scale with typical value  ⫽ 10⫺3. The dotted line
marks the correlation scale corresponding to the correlation of
e ⫺1 . In general, smaller leads to a SM horizontal distribution
with a shorter correlation scale, and the hole phenomenon
occurs at small lags (i.e., 10 –30 m) in the one-dimensional
correlation function. The one-dimensional SM correlation
functions resulting from the exponential and hole-type covariance functions of elevation are similar. This again suggests that
the statistical properties of SM distribution are not sensitive to
the functional form of the input covariance. The overlap of the
SM correlation for ⫽ 103 m and ⫽ 104 m indicates that the
correlation structure of topography has little impact on the SM
distribution for larger than 103 m (for the typical value  ⫽
10⫺3). For two-dimensional analysis the overlap of the SM
correlation function for large is similar to that in one dimension. However, for the range of (103–104 m), the twodimensional SM distribution tends to correlate over longer
distances (i.e., 100 –200 m) in comparison with the onedimensional distribution (i.e., 20 –30 m). Further, the hole phenomenon occurring in one dimension at small lags does not
occur in two dimensions for typical values of . In two dimensions, soil water has greater freedom in choosing a flow path
and usually avoids the relatively low conductivity zones, which
leads to a smoother SM horizontal distribution.
The influences of vertical hydraulic gradient , capillary
resistance ␣⫺1, and soil depth D on the one-dimensional SM
correlation function and its correlation scale are illustrated in
Figure 6. The vertical climatic fluxes have the effect of shortening the correlation scale of SM horizontal distribution. For
the homogeneous soil system investigated in this paper, if the
elevation field is flat, the correlation scale of SM horizontal
distribution would be infinitely large and the variance of SM
distribution would be zero. Because of the absence of horizontal unsaturated flow, this situation remains unchanged regardless of the magnitude of . For a rolling topography, SM tends
to accumulate at the valleys, which results in a nonuniform
horizontal distribution of SM. Under this situation, water is
removed by vertical climatic fluxes from the high-SM regions,
which leads to a smaller amplitude and a smaller correlation
scale of horizontal distribution. This is demonstrated in Figure
6a such that the increase of  results in the decrease of SM
correlation scale. As was stated in the preceding section,  is
the vertical hydraulic gradient driving water away from a
high-SM region to the atmosphere by evaporation. This process balances the SM differences between wet and dry regions
in soils and thus has the effect of reducing the variance and
correlation scale of SM distribution. On the other hand, the
capillary force of soil matrix tends to resist both the horizontal
topographic forcing and vertical climatic forcing; that is, it has
a negative effect on the tendency of climate to reduce the
correlation scale. This impact of capillary resistance is shown in
the rising trend of SM correlation scale in Figure 6b. Further,
the soil depth D regulates the magnitude of vertical fluxes.
Large D (resulting in weaker vertical hydraulic gradient) tends
to dampen the effect of , whereas small D tends to enhance
that effect (Figure 6c).
The one- and two-dimensional one-point (i.e., lag zero)
cross correlations z are plotted in Figure 7 as a function of
dimensionless parameter g . As seen in the figure, the onepoint cross correlation between elevation and SM distribution
decreases as g increases. This trend is similar to that of SM
variance demonstrated in Figure 3. Large and , as well as
small ␣⫺1 and D, reduce the cross correlation of topography
and SM. The one-point cross correlation approaches zero as
increases to a very large value, which indicates the independence of topography and SM distribution for the situation of a
uniform elevation field. For small g (small and ), Figure 7
implies a relatively high correlation between and z. Note that
a perfect correlation relation between and z corresponds to
 ⫽ 0 (i.e., unit cross correlation can be derived from substituting  ⫽ 0 in (25) and (30)), which describes the situation
when topography is the only dominant forcing. Furthermore,
for small g the one-point cross correlation in two dimensions
is higher than that in one dimension. This regime is dominated
by the topographically induced horizontal fluxes. Under such
conditions, water has more freedom to move in two dimensions
and hence tends to closely mimic topography, resulting in a
higher cross correlation. As g increases ( or  increases), the
two-dimensional one-point cross correlation between and z
approaches the one-dimensional cross correlation as the role
of topographically induced horizontal fluxes becomes less and
less important in comparison to the climate forced vertical
fluxes.
Figure 8 shows the one- and two-dimensional crosscorrelation function between and z for several values of
with  ⫽ 10⫺3. Generally, the trends of one- and twodimensional cross correlations between and z are similar
except that in two dimensions, SM distribution correlates with
the topography for a larger distance (i.e., hundreds of meters)
than it does in one dimension (i.e., tens of meters). For a small
the cross correlation between and z is relatively larger for
small lags and then decays faster than the corresponding cross
correlation for a larger . Overall, the degree of cross correlation between and z decreases with the . As was mentioned
earlier, water in a hilly topography tends to accumulate at the
topographic convergence areas (bottom of the valleys) which
usually lie next to areas of topographic divergence (ridges).
This implies the negative cross correlation between and z for
the separation distance at the order of . Since is representative of the average scale of topographic undulation, a small
in Figure 8 describing a rolling topography suggests that the
SM distribution has higher positive correlation with the elevation at lags smaller than , while the same distribution has
higher negative correlation with the elevation at lags larger
than . The distance of the transition from positive to negative
cross correlation increases with at the expense of the decreasing magnitude of the cross correlation. Therefore the
cross correlation between the topographic forcing and SM
distribution is inversely proportional to the correlation scale of
elevation. For ⬎ 103 m the topography might not have
apparent effects on the horizontal redistribution of SM. Although this argument seems intuitively reasonable, it should be
noted that depends on the resolution of elevation data. Finer
resolutions of data (usually resulting in smaller ) are capable
of capturing the variation of microtopography, which is shown
to have significant impacts on SM distribution (see Figure 7).
Finally, the combined impact of varying  and on the cross
correlation between and z is shown in the contour plot of
Figure 9. This figure indicates the range of significant cross
correlation between and z lies in ⬍ 103 m and  ⬍ 10⫺3.
The cross correlation between and z decreases as  increases.
In summary, the magnitude and scale of the cross correlation
between topography and SM horizontal distribution are dependent on the correlation scale of elevation and the magni-
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
1261
Figure 9. Contours of the one-dimensional one-point cross correlation for varying values of correlation scale
of elevation and vertical hydraulic gradient  (␣⫺1 ⫽ 0.5 m, D ⫽ 1 m).
tude of vertical evaporative flux. Microtopography with a correlation scale smaller than 103 m has a significant impact on
the horizontal SM distribution.
6.
Conclusions
An equation explicitly relating the variability of soil moisture
distribution to the variability of topography under steady state
homogeneous soil conditions is proposed and solved using the
perturbation method. Both the one- and two-dimensional unsaturated flow equations are analyzed by assuming the exponential form and hole-type form of elevation covariance function in one dimension, and the Bessel-type covariance function
of elevation in two dimensions. The covariance and variance of
a soil moisture distribution, as well as the cross correlation
between elevation and soil moisture distribution, are derived in
closed form expressions for one and two dimensions.
The steady state soil moisture distribution under homogeneous bare soil conditions is regulated by three distinct factors:
topography, climate, and soil properties. First, topography
forces a distribution of soil moisture that tends to mimic the
elevation field at large scales. The other two factors are the
vertical divergence of water in response to the climate forcing
and the capillary resistance to water movement. The climate
forcing, which is represented by a simple parameterization of
vertical divergence of moisture flux, tends to smooth the spatial
distribution of soil moisture. However, the capillary force exerted by the soil matrix tends to resist the displacement of
water and hence exert adverse effects against the topography
and climate forcings. The impact of capillary forces on the
vertical fluxes of water seems to be more significant than their
impact on the topographically induced horizontal fluxes. In
addition to these three factors, soil depth regulates the impact
of the soil moisture vertical flux such that a relatively large soil
depth results in a relatively larger soil moisture variance.
The variance of SM distribution decreases with elevation
correlation scale and vertical hydraulic gradient , and in-
creases with variance of elevation 2z , capillary fringe thickness
␣⫺1, and soil depth D. Moreover, the two-dimensional analysis
indicates that the SM variance is smaller than that in one
dimension. Additionally, the hole phenomenon occurring in
one-dimensional SM correlation function vanishes in two dimensions for typical range of elevation correlation scales.
These two findings result from the greater freedom of soil
moisture movement in two dimensions than in one dimension.
The cross correlation between soil moisture horizontal distribution and topography decreases as the elevation correlation scale and the vertical evaporative fluxes increase. This
trend corresponds to a shift of the dominant forcing on the soil
moisture distribution from the topographically induced horizontal fluxes to climate-forced vertical fluxes. In contrast to the
variance of soil moisture distribution, the cross correlation
between soil moisture and elevation in two dimensions is
higher than that in one dimension, especially for the condition
of small elevation correlation scale and small vertical evaporative fluxes.
This paper considered the steady state distribution of soil
moisture assuming the homogeneous soil condition. Future
research will focus on (1) the corresponding transient problem,
(2) incorporating the variability of rainfall and (3) the impacts
of soil heterogeneity and vegetation on the topographically
induced soil moisture distribution. To test some of the theoretical concept proposed in this paper, a field experiment has
been designed to measure soil moisture along a hillslope using
neutron probe technology. The experimental site is at Harvard
Forest, located in central Massachusetts. Results from this
experiment will be reported in a forthcoming paper.
Appendix A: Derivations of Equation (24) and (26)
A1.
Exponential Function
From the S z (k) in (22) and the spectral relationship of soil
moisture (SM) and elevation in (21), the spectrum of SM can
be derived as
1262
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
S 共k兲 ⫽
⫽
2 2z C 2
2 2z C 2
冉
冋
k4
␣
k2 ⫹
D
冊
Appendix B: Derivation of Equation (34)
2
By substituting the Whittle spectrum (equation (32)) into
(21), we have (i.e., k 2 ⫽ k 2x ⫹ k 2y )
共1 ⫹ 2k 2兲
a
b
c
⫹
⫹
共k ⫹ A兲 2 k 2 ⫹ A k 2 ⫹ B
2
册
(A1)
S 共k兲 ⫽
2z C 2 2
冉
冋
where
1
B⫽ 2
␣
A⫽
D
⫺A 2
a⫽
A⫺B
⫽
k2 ⫹
冕
A⫽
⫹
b
a
e ⫺ 冑A兩兩共1 ⫹
2 A 3/ 2
e ⫺ 冑A兩兩 ⫹
冑A
c
冑B
冑A兩 兩兲
e ⫺ 冑B兩兩
册
(A2)
⫽
a
b
d
c
⫹
⫹
⫹
共k 2 ⫹ A兲 2 k 2 ⫹ A 共k 2 ⫹ B兲 2 k 2 ⫹ B
where
d⫽
⫺2 AB
共 A ⫺ B兲 3
a⫽
⫺A 3
共 A ⫺ B兲 2
⫺B 3
c⫽
共 A ⫺ B兲 2
b⫽
2 2z C 2
2
e ikS 共k兲 dk
2 冑A
c K 1共 冑B 兲
2 冑B
⫹ bK 0共 冑A 兲
⫹ dK 0共 冑B 兲
册
g 3/ 2
4g
K 共 g 1/ 2 兲 ⫹
K 共 g 1/ 2 兲
共 g ⫺ 1兲 2 1
共 g ⫺ 1兲 3 0
⫹
1
4g
K 共 兲 ⫺
K 共 兲
共 g ⫺ 1兲 2 1
共 g ⫺ 1兲 3 0
冕
⬁
kS 共k兲 dk
0
a ⫺ 冑A兩兩
e
共1 ⫹
A 3/ 2
冑A兩 兩兲 ⫹
冑B兩 兩兲 ⫹
d
冑B
2b
冑A
⫽
e ⫺ 冑A兩兩
e ⫺ 冑B兩兩
册
(A6)
冋
⫺⬁
c ⫺ 冑B兩兩
e
共1 ⫹
B 3/ 2
a K 1共 冑A 兲
册
(A7)
Since the SM variance cannot be evaluated directly from (A7),
it should be derived by letting ⫽ 0 in the second line of (A6):
B 2共3A ⫺ B兲
d⫽
共 A ⫺ B兲 3
2 ⫽ 2
冋
冋
By defining g ⫽ ␣ 2 /D and ⫽ 兩 兩/ , (A6) can be written
into the following dimensionless form as in (33):
A 2共 A ⫺ 3B兲
共 A ⫺ B兲 3
⬁
2z C 2
⫽
e i共kxx⫹kyy兲S 共k x, k y兲 dk x dk y
共 2 ⫽ 2x ⫹ 2y 兲
R 共 兲
⫽
C 2 2z
1
B⫽ 2
␣
A⫽
D
⫹
B2
共 A ⫺ B兲 2
kJ 0共k 兲S 共k兲 dk
⫹
册
(A3)
冕
c⫽
⫺⬁
By the same procedure except taking S z (k) from (23) into
(21), (A1) and (A2) becomes
R 共k兲 ⫽
2 AB
共 A ⫺ B兲 3
⬁
Hole-Type Function
冋
1
2
⫺⬁3⬁
A2.
2 C
S 共k兲 ⫽
B⫽
b⫽
冕冕
冕
R 共 x, y兲 ⫽
⫽
2
册
The SM covariance function can be derived as
By defining the non-dimensional parameter g ⫽ ␣ 2 /D
and substituting A, B, a, b, c from (A1) into (A2), (24) is thus
derived.
2
z
␣
D
A2
共 A ⫺ B兲 2
e ikS 共k兲 dk
冋
共1 ⫹ 2k 2兲 2
a⫽
⬁
2z C 2
2
where
⫺⬁
⫽
冊
b
d
a
c
⫹
⫹
⫹
共k 2 ⫹ A兲 2 k 2 ⫹ A 共k 2 ⫹ B兲 2 k 2 ⫹ B
The decomposition by partial fraction in (A1) is for the
convenience of integration. The SM covariance function can be
obtained by taking inverse Fourier transform of (A1):
R 共k兲 ⫽
␣
D
(A5)
B2
c⫽
共 A ⫺ B兲 2
A共 A ⫺ 2B兲
b⫽
共 A ⫺ B兲 2
2z C 2 2
k4
2 2z C 2
2
冕 冋
⬁
k
b
a
⫹
共k 2 ⫹ A兲 2 共k 2 ⫹ A兲
0
⫹
(A4)
Thus (26) is derived and expressed in terms of the dimensionless parameter g ⫽ ␣ 2 /D.
⫽
c
d
⫹
共k 2 ⫹ B兲 2 共k 2 ⫹ B兲
2z C 2
2
冉
册
dk
a
c
⫹ b ln A ⫹ ⫹ d ln B
A
B
冊
YEH AND ELTAHIR: TOPOGRAPHY AND SOIL MOISTURE DISTRIBUTION
⫽
⫽
2z C 2
2
2z C 2
2
⫽ 2z C 2
冉
冉
冋
a
c
k2 ⫹ A
⫹ ⫹ b ln 2
A B
k ⫹B
a
c
B
⫹ ⫹ b ln
A B
A
冏冊
⬁
0
冊
g ⫹ 1
2g
⫺
ln g
共 g ⫺ 1兲 2 共 g ⫺ 1兲 3
册
(A8)
Notice that b ⫽ ⫺d from (A5). Thus (34) is obtained.
Acknowledgments. The authors are grateful to the hydrology program of the National Aeronautics and Space Administration (NASA)
for their support of the work under grant NAGW-4707. We also thank
Guleid Artan (University of California, San Diego), Gabriel Katul
(Duke University), and an anonymous reviewer. The manuscript benefited greatly from their helpful comments.
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E. A. B. Eltahir and P. J.-F. Yeh, R. M. Parsons Laboratory, Department of Civil and Environmental Engineering, Massachusetts
Institute of Technology, Cambridge, MA 02139. (e-mail: eltahir@
mit.edu; patyeh@mit.edu)
(Received April 18, 1997; revised January 5, 1998;
accepted January 7, 1998.)
WATER RESOURCES RESEARCH, VOL. 34, NO. 8, PAGE 2075, AUGUST, 1998
Correction to “Stochastic analysis of the relationship
between topography and the spatial distribution of soil moisture”
by Pat J.-F. Yeh and Elfatih A. B. Eltahir
In the paper “Stochastic analysis of the relationship between topography and the spatial distribution
of soil moisture” by Pat J.-F. Yeh and Elfatih A. B. Eltahir (Water Resources Research, 34(5), 1251–1263,
1998), the negative sign is missing in the cross-correlation equations (28), (29), (30), (31), (35), and (36).
Accordingly, the sign in the ordinate of Figures 7 and 8, as well as in the contours of Figure 9, should
be reversed. The cross correlation between elevation distribution and soil moisture distribution at lag
zero should be negative.
(Received May 20, 1998.)
Copyright 1998 by the American Geophysical Union.
Paper number 98WR01876.
0043-1397/98/98WR-01876$09.00
2075