Functional relationship to describe temporal statistics of soil moisture averaged over different...

Advances in Water Resources 28 (2005) 553–566

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Functional relationship to describe temporal statistics ofsoil moisture averaged over different depths

Michael J. Puma a,*, Michael A. Celia a, Ignacio Rodriguez-Iturbe a, Andrew J. Guswa b

a Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USAb Picker Engineering Program, Smith College, Northampton, MA 01063, USA

Received 4 March 2004; received in revised form 28 July 2004; accepted 5 August 2004

Available online 21 January 2005

Abstract

Detailed simulation studies, highly resolved in space and time, show that a physical relationship exists among instantaneous soil-

moisture values integrated over different soil depths. This dynamic relationship evolves in time as a function of the hydrologic inputs

and soil and vegetation characteristics. When depth-averaged soil moisture is sampled at a low temporal frequency, the structure of

the relationship breaks down and becomes undetectable. Statistical measures can overcome the limitation of sampling frequency,

and predictions of mean and variance for soil moisture can be defined over any soil averaging depth d. For a water-limited ecosys-

tem, a detailed simulation model is used to compute the mean and variance of soil moisture for different averaging depths over a

number of growing seasons. We present a framework that predicts the mean of soil moisture as a function of averaging depth given

soil moisture over a shallow d and the average daily rainfall reaching the soil.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Soil moisture; Remote sensing; Richards equation; Root-water uptake; Plant model; Modeling

1. Introduction

Soil moisture is the critical variable that dynamically

links plants to the overall water balance, thereby influ-

encing feedbacks to the atmosphere [33]. Soil moisture

is controlled by complex interactions involving soil,

plants, and climate. Plants connect the soil to the atmo-

sphere through their active roots, which provide path-

ways for water transport from the root zone to the

atmosphere [9]. Therefore, knowledge of soil moisturewithin the root zone, the region where active roots reside

in the soil, is essential for estimation of fundamental

hydrological and atmospheric processes. Accurate esti-

mation of these processes is important for large-scale cli-

mate models as well as for ecohydrological models.

0309-1708/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2004.08.015

* Corresponding author. Tel.: +1 609 258 7819; fax: +1 609 258

2799.

E-mail address: [email protected] (M.J. Puma).

Soil moisture can be estimated by in situ measure-

ments, by remote sensing, or by hydrological modeling.For large-scale applications, in situ methods cannot be

used because an in situ measurement network does not

exist over large land surface areas and the technique is

expensive [16]. Microwave remote sensors have been

successful because they are sensitive to soil moisture

through the effects of moisture on the dielectric constant

and, consequently, the soil�s emissivity [30,23]. However,

remote sensing has uncertainty that depends on the sen-sor type (active or passive), vegetation cover, landscape

roughness, and soil type. Yet the primary shortcoming

of remote sensing is that soil moisture is inferred only

for the top few centimeters of the soil column e.g.,

[8,22]. Consequently, remote sensing must be used in

conjunction with some other information to estimate

soil-moisture values over the entire root zone.

Hydrological models often have uncertainpredictions that result from model assumptions and

mailto:[email protected]

554 M.J. Puma et al. / Advances in Water Resources 28 (2005) 553–566

parameterization [16]. Kostov and Jackson [19] sug-

gested that combining remotely sensed data with

hydrological models is the most promising approach

for soil-moisture estimation. Accordingly, recent re-

search has focused on this combination e.g., [28,6,16,

27,14] with a general strategy of guiding hydrologicalmodels with periodic remote sensing of soil moisture.

The instantaneous soil-moisture data inferred from

remote sensing observations are assimilated into hydro-

logical models with the hope that model biases will be

corrected. Fundamental to this strategy is the relation-

ship between the measured instantaneous soil moisture

and instantaneous soil moisture at deeper locations.

This paper first investigates the relationship betweeninstantaneous values of soil moisture over different aver-

aging depths, d, using a detailed simulation model with a

focus on temporal resolution of measurements. This

instantaneous soil-moisture analysis illustrates the diffi-

culties in relating instantaneous soil-moisture values

averaged over the top 5 cm to the instantaneous soil-

moisture values averaged over the top 30 cm. We then

consider an alternative approach to relating instanta-neous soil-moisture values over different averaging

depths using statistical measures of soil moisture. In par-

ticular, we investigate statistical soil-moisture measures

as functions of depth, d, focusing on the relationships

among the soil-moisture means over different d as well

as the variances over different d. Based on this approach,

a methodology is presented that enables prediction of

the mean of soil moisture as a function of averagingdepth, d, as a function of soil and climate parameters.

2. Simulation description

A one-dimensional model, based on the Richards

equation, in combination with a model of water uptake

by plants and stochastically generated rainfall, is used tosimulate soil-moisture dynamics in a water-limited eco-

system, which we take to be a savannah. Assumptions

have been made regarding the model�s resolution and

complexity with some processes simplified. For example,

plant growth and nutrient uptake are not modeled,

because we assume that these processes are not impor-

tant to estimate soil moisture for a mature plant in a

savanna. We take the output from our highly resolved(in space and time) model to represent actual field

conditions.

Data for the model correspond to measurements ta-

ken at Nylsvley, South Africa [34]. Since this model only

considers the vertical spatial dimension of the soil, an

inherent assumption is that only vertical soil-moisture

dynamics are important, and, therefore, lateral soil-

moisture dynamics do not have to be resolved. The rootzone of the vegetation (typically 30–100 cm) is resolved

into 1 cm layers. We use highly resolved time discretiza-

tion, with a maximum time step of approximately 3 min,

and finer resolution around storm events. The model

represents water uptake by the plant using the so-called

�Type I� model [15,13], in which water uptake is con-

trolled by differences in fluid potential between the soil

and the plant [1,38,24,12,13,15,17]. The model may beinterpreted as representative of either a single plant or

a homogeneous stand of vegetation.

2.1. Infiltration and redistribution

We model vertical infiltration and redistribution,

including evapotranspiration, using the one-dimensional

Richards� equation with suitable sink terms to accountfor evapotranspiration. The governing equation has

the following form,

oð/SÞot

� o

ozKoWoz

� �þ oK

oz¼ �e0 � u0 ð1Þ

where S is the relative soil-moisture content or satura-

tion (L3 water/L3 voids), / is the porosity (L3 voids/L3

soil), K is the unsaturated hydraulic conductivity (L/

T), W is the fluid pressure head (L), e 0 is the rate of evap-

oration (L3 evaporated water/L3 soil/T), u 0 is the rate of

plant uptake (L3 plant-extracted water/L3 soil/T), z isthe vertical dimension designated to be positive down-

ward (L), and t is time (T) e.g., [3,4,37]. In unsaturated

soil, the fluid pressure head, W, is negative and is often

referred to as the soil matrix potential. Suction is defined

for unsaturated soil as the absolute value of W.

2.2. Rainfall model and boundary conditions

Rainfall input is treated as an external random forc-

ing that is independent of soil moisture. The rainfall

input is modeled with the storm occurrence, depth,

and duration represented as random variables. Storm

occurrence is modeled as a Poisson process with rate kwith a duration and intensity associated with each

occurrence. The duration is obtained from a beta distri-

bution. The total storm rainfall depth is taken from anexponential distribution with mean rainfall depth a.

The aboveground portion of plants intercepts a sig-

nificant amount of rainfall, especially in arid and semi-

arid ecosystems where rainfall duration is short and

evaporation demand is high e.g., [21,7]. Following the

simplified approach of Laio et al. [21], interception is

modeled by setting a fixed threshold rainfall depth, D.If a simulated storm produces a total storm rainfalldepth less than D, then no rain reaches the soil due to

interception. For storm depths greater than D, the depthof rainfall reaching the soil surface is simply the total

storm rainfall depth minus D. The rate of arrival of

storms with rainfall reaching the soil surface, k 0, be-

comes [32,21]

M.J. Puma et al. / Advances in Water Resources 28 (2005) 553–566 555

k0 ¼ ke�D=a ð2ÞSince both the evaporation and transpiration functions

do not depend on whether it is raining, k is important

in this model only as it relates to k 0.

Two types of boundary conditions, fixed pressure

head and flux, are imposed at the top of the soil column

in this model. The flux is set to zero during periods whenno precipitation reaches the soil. When a storm with rain

reaching the soil surface occurs, the top boundary condi-

tion is changed to a non-zero flux condition. This flux

boundary condition is equal to the rainfall rate at the soil

surface, as long as the surface remains unsaturated. If the

top boundary becomes fully saturated, then the bound-

ary condition switches to a pressure head condition and

remains so as long as full-saturation or ponding condi-tions exist at the surface [12]. The bottom boundary con-

dition for the soil is a fixed pressure head condition. The

location of the bottom boundary of the soil is set to a

depth where the bottom boundary condition has minimal

impact on soil-moisture dynamics in the root zone [12].

2.3. Discretization and solution

The governing equation, Eq. (1), is highly non-linear,

because it must be combined with its constitutive rela-

tionships for the soil�s hydraulic properties, evaporation,and transpiration. Consequently, Eq. (1) must be solved

using numerical methods. The soil column is discretized

into layers, or grid cells, with Dzi denoting the thickness

of layer i. Following the procedure of Celia et al. [3], a

backward Euler approximation in time with a modifiedPicard iteration scheme is applied to the governing equa-

tion, using uniform grid cell sizes. In the simulations,

evaporation and transpiration are calculated explicitly

in time, and unsaturated conductivities are determined

using upstream weighting.

2.4. Hydraulic properties

An empirically derived constitutive equation for the

soil–water retention curves approximates the relation-

ship between W and S. The Brooks and Corey formula-

tion [31] is used here,

WðSÞ ¼ We

S � Sh

1� Sh

� ��b

ð3Þ

where We is the air-entry pressure head (L), Sh is the

hygroscopic saturation, and b is a parameter that con-

trols the curve�s shape. We use a hydraulic conductivity

function that is also based on the formulation by Brooks

and Corey [31],

KðSÞ ¼ Ksat

S � Sh

1� Sh

� �2bþ3

ð4Þ

where Ksat is the saturated hydraulic conductivity (L/T).

2.5. Evaporation and transpiration

The sink terms in Eq. (1) are represented empirically

and account for water loss from the soil column due to

evaporation and transpiration. These two processes are

governed by the atmospheric demand, the amount ofwater in the soil, and the characteristics of the vegeta-

tion. We model a completely vegetated surface. There-

fore, transpiration is dominant relative to evaporation,

so we assume that evaporation accounts for 10% of total

evapotranspiration. Consequently, we model evapora-

tion in a simplistic manner relative to evaporation mod-

els that take into account heat and moisture transport,

but this simplification is justified given the dominanceof transpiration.

2.5.1. Evaporation

In our model, evaporation is constrained to occur

over depth Ze with the greatest evaporative losses from

soil closest to the surface. The local evaporation rate is a

function of depth, saturation, and time, and is repre-

sented by the following functional form [12,21],

ei ¼ Dzi � e0i ¼ Dzi � ewi �EðSi; tÞPme

i¼1

Dzi � ewi

ð5Þ

where ei is the extraction rate in cell i, ewi is a depth-

dependent weighting function for evaporation evaluated

at the centroid of cell i, Si is saturation in cell i, E(Si, t) isan empirical evaporation function, and me is the number

of soil layers over which the evaporation is non-zero.

The evaporative weight, ew(z), constrains the evapora-

tion function so that shallow depths have the greatest

evaporation potential. It is approximated with a beta

distribution [12] as

ewðzÞ ¼ CðAþ BÞCðAÞCðBÞ z

A�1ð1� zÞB�1 ð6Þ

where C(Æ) is the gamma function and A [-] and B [-] are

positive parameters. E(Si, t) is given by [21]

EðSi; tÞ ¼0 Si 6 Sh

Si�ShSw�Sh

� EmaxðtÞ Sh < Si < Sw

EmaxðtÞ Si P Sw

8><>:

9>=>; ð7Þ

where Emax (L3 evaporated water/L2 soil/T) is the max-

imum instantaneous evaporation rate for an average

growing season day, Sw is the wilting saturation, and

Sh is the hygroscopic saturation. Emax is varied toapproximate atmospheric demand with a sinusoidal

function over 12 h representing daytime. The maximum

value of Emax(t) is reached at noon, and it is zero for the

12 night hours. The wilting saturation, Sw, is the satura-

tion at which the plant can no longer extract soil water,

because the suction required to do so is so high that the

plant tissue is damaged e.g., [21,10]. While Sw as a


threshold value for evaporation is arbitrary, dynamics

of evaporation are not crucial for the simulations in this

paper because the soil is covered by vegetation, and

transpiration is the dominant process.

2.5.2. Transpiration

The effect of vegetation is incorporated into the

model through the plant water uptake rate, u 0, which

accounts for plant transpiration T(S). Plant water up-

take is represented by a simplified mathematical expres-

sion based on experimental and field observations. We

use a semi-mechanistic (Type I) model where water up-

take is assumed proportional to the difference between

the soil and plant water potentials with no change instorage within the plant. This formulation assumes that

an Ohm�s law analogy for water uptake may be applied

to the transport of water from the soil to the atmosphere

through the plant [1,24,15,38,13]. Water uptake is lim-

ited by the resistance of water flow through the soil to

the root surface and from the root surface to the root

xylem:

ui ¼ Dzi � u0i ¼ Dzi �Wi �Wp

rs;iðtÞ þ rr;ið8Þ

where Wi is the local fluid pressure head (L), Wp is the

fluid potential in the plant (L), rs,i is the local soil resis-

tance (T/L), and rr,i is the local root resistance (T/L).

Fluid potential is assumed to be constant throughout

the plant. The soil resistance can be expressed as [12]

rs;i ¼Cs

rwi �RW0 � KðSiðtÞÞð9Þ

where Cs is a dimensionless constant that accounts for

root diameter, geometry and arrangement, rwi is the rel-

ative root density as a function of depth, and RW0 is the

average root-length density in the root zone (L roots/L3

soil). Similarly, the root resistance is [12]

rr;i ¼Cr

rwi �RW0

ð10Þ

where Cr is a constant parameter of the plant (T/L). As

for ewi, the relative root density, rwi, is approximated by

as beta distribution as

rwðzÞ ¼ CðAþ BÞCðAÞCðBÞ z

A�1ð1� zÞB�1 ð11Þ

The extraction rate ui is constrained such that it cannot

be negative, and, consequently, hydraulic lift is not rep-

resented explicitly in this model e.g., [25]. Transpiration

is the sum of the water uptake from all layers i in the

root zone and is expressed as

T actðtÞ ¼Xzi6Zr

ui ¼Xzi6Zr

Wi �Wp

rs;iðtÞ þ rr;ið12Þ

where Tact is the actual transpiration and Zr is the max-

imum depth of active roots. Since Eq. (8) contains three

unknowns, Wp, Cr, and Cs, additional constraints are

necessary to solve the system of equations.

2.5.3. Solution of transpiration expression

Maximum total transpiration, Tmax, is dependent

upon atmospheric conditions, while actual transpira-tion, Tact, is controlled by soil moisture in the root zone,

soil properties, and water uptake characteristics of the

plant with an upper limit of Tmax. We approximate Tmax

in the same manner that Emax is approximated, with a

diurnal sinusoidal variation. The plant is modeled as a

single point described by Wp, which must be constrained

based on plant physiology. First, the lower limit for Wp

is the plant fluid potential at wilting, Ww, which preventsthe plant from extracting water from a layer if Wi is

more negative than Ww. The other constraint restricts

the total transpiration, so that it cannot exceed Tmax.

Consequently, Wp is either Ww or the fluid potential in

the plant for which the water uptake from the root zone

equals Tmax, whichever is larger (i.e., less negative) [12].

The parameter, Cr, a coefficient in the root resistance

term, can be estimated experimentally, and some valuescan be found in the literature. Yet these estimations are

species specific, are known for a small number of plants

under conditions that do not necessarily exist in the

field, and are sometimes different for different root resis-

tance formulations. In addition, no reliable data exist

for the parameter Cs. Therefore, we seek two constraints

that will enable solution for these unknown parameters

Cr and Cs.The first constraint considers the plant�s physiologi-

cally controlled ability to extract water from the soil.

We define a plant compensation factor, c, as a specific

characteristic of the plant�s ability to uptake water. It is

derived using the Ohm�s law formulation for transpira-

tion by considering the minimum fraction of the root

zone, 1/c, that needs to be saturated in order for the plant

to meet Tmax. If a hypothetical situation is consideredwhere soil layers are either at W = 0 (full saturation) or

below Ww, then water uptake occurs only from the satu-

rated layers. In order to find the minimum fraction of the

root zone that must be saturated, the fluid potential in the

plant should be at the minimum value, Ww, which pro-

duces the greatest difference in potential between the soil

and the plant (and hence the greatest force the plant can

produce for water uptake). Consequently, 1/c becomesthe minimum fraction of roots that must be saturated

for the plant to withdraw enough water to meet the tran-

spiration demand if extraction from elsewhere in the soil

column is zero [12]. This scenario is expressed as

Tmax ¼Xzi6Zr

ui ¼Zr

c� ð�WwÞ

Cs

KsatRW0þ Cr

RW0

ð13Þ

When the soil is at or near full saturation, the soil resis-

tance is assumed to be small relative to the root resis-

0Weight

Dep

th

Z r

Z e

ew(z)

rw(z)

124 8

Fig. 1. Plot of evaporation and root-length-density weights as

functions of depth for Burkea africana in Nylsvley soil.


tance. Therefore, the plant compensation factor, c, iscontrolled by the root resistance term, rr = Cr/RW0.

The plant compensation factor, c, is physically

meaningful in that it is a measure of the plant�s ability

to extract water at high rates from wet sections of the

soil to compensate for dry sections [12]. A plant witha value of c greater than one can increase water uptake

from wet portions of the soil column when soil mois-

ture is non-uniformly distributed in the root zone. If

c equals one, then the plant cannot compensate, and

it would need the entire root zone to be saturated to

meet atmospheric transpiration demand. Since the

model only includes a static root system, root response

to available soil moisture through growth or changes inroot resistance, rr, is not considered. Therefore, c can

also be considered an empirical way to account for root

growth or for rr changes in response to available soil

moisture.

The other constraint specifies that if the soil column is

uniformly saturated at the saturation where stomata clo-

sure commences, S* in the model of Laio et al. [21], then

the total plant uptake is equal to the transpirationdemand when the plant fluid potential is at its lowest

value, Ww:

Tmax ¼ Zr �ðW� �WwÞCs

KðS�ÞRW0þ Cr

RW0

ð14Þ

where W* is the fluid pressure head that corresponds to

S*. While the condition in Eq. (14) facilitates compari-

son with the bucket model of Laio et al. [21] and Guswa

et al. [12], it is a constraint not based on the physical sys-

tem and made only for convenience. Specifying c, com-

bining Eqs. (13) and (14) with the conditions imposed on

Wp, and using the constitutive relationships in Eqs. (3)–(11) enable solution of Eq. (1).

2.6. Description of soil, plant, and climate in simulations

As an initial investigation, the analysis of this paper

will be applied to the savanna in Nylsvley, South Africa,

which is a water-limited ecosystem that has been exten-

sively studied [34] and has been the focus of previousmodeling efforts e.g., [20,12]. The Nylsvley site has a cli-

mate typical of a savanna, characterized by a hot and

rainy growing season and a warm and dry dormant sea-

son [34]. The simulations in this paper will focus on the

soil-moisture dynamics during the hot and rainy grow-

ing season.

This analysis will consider only one vegetation type,

Burkea africana, which is a dominant broad-leafed woo-dy species in this savanna [34]. The roots of B. africana

generally extend to a depth of 100 cm and are distrib-

uted with the root density having a minimum (equal to

zero) at the soil surface and at all depths greater than

100 cm, and a maximum at a depth of 50 cm [18,34].

To match this root-distribution data, in Eq. (11) we

set A and B both equal to 2 with RW0 equal to

0.02 cm/cm3 and Zr equal to 100 cm [20]. The depth of

the bottom boundary is set to 200 cm for the simula-

tions. For the evaporative weights in Eq. (6), A is 0.9

and B is 5 [12]. This distribution assures that water loss

from the soil due to evaporation is highest from the soillayers closest to the surface. Fig. 1 shows the two

weighting functions, rw(z) and ew(z).

The water usage of B. africana is reported in Scholes

and Walker [34], which enables estimation of Tmax, Ww,

W*. Laio et al. [20] adjusted the maximum instantaneous

transpiration rates found in Scholes and Walker [34] to

evapotranspiration values that are representative for

average conditions for a 24-h period over a growing sea-son, hETmaxi1 day, where hÆi1 day denotes averaging over a

day. In Laio et al. [20], hETmaxi1 day for B. africana is

0.475 cm/day. Because our model has a temporal resolu-

tion finer than a day, evapotranspiration is partitioned

into two separate functions. The maximum depth from

which evaporation can occur, Ze, is estimated to be

20 cm and the average daily evaporation for a mean

growing season day under well-watered conditions,hEmaxi1 day, is approximated as 0.046 cm/day for a com-

pletely vegetated surface. This hEmaxi1 day value is based

on the assumption that if the soil is at full saturation for

an entire day, then evaporation accounts for 10% of

evapotranspiration. The maximum transpiration,

hTmaxi1 day, is then the difference between hETmaxi1 dayand hEmaxi1 day. However, because both maxima (evapo-

ration and transpiration) are dependent upon atmo-spheric conditions, the instantaneous maximum values,

Emax and Tmax, should change over a day. Fig. 2 shows

the diurnal maximum instantaneous evapotranspiration

function over a day for B. africana. The sinusoidal func-

tion is subject to the constraint that if the instantaneous

evaporation and transpiration are meeting atmospheric

demand for an entire day, then 0.475 cm is the maxi-

mum amount of water loss over a day from the soildue to evaporation and transpiration.

The plant fluid potential at wilting, Ww, for B. afri-

cana is �31,600 cm, and W* is �730 cm [34]; from these

0

0.4

0.8

1.2

1.6

0 0.2 0.4 0.6 0.8 1Time [days]

ET

max

[cm

/day

]

ET max = constant

ET max(t )

Fig. 2. Plot of maximum instantaneous evapotranspiration, ETmax,

over a day for Burkea africana. ETmax is assumed diurnal and is

approximated with a sinusoidal function for 12 h each day.

Table 1

Soil parameters values for Nylsvley, South Africa [34,20]

Parameter Symbol Value

Saturated conductivity Ksat 109.8 cm/day

Porosity / 0.42

Retention curve parameter b 2.25

Air-entry pressure head We �3.0 cm

Hygroscopic saturation Sh 0.02

0

10

20

30

0 0.1 0.2

S

Dep

th [

cm]

0.35S30S


it follows that Sw is approximately 0.06, and S* is 0.105

[20]. The fixed interception threshold, D, is estimated to

be 0.2 cm for B. africana [20]. The Nylsvley soil is rela-

tively homogeneous and generally has a high sand con-

tent (85–90%). Following Laio et al. [20], weighted

averages of the properties of the A and B soil horizons

reported in Scholes and Walker [34] are calculated and

applied uniformly. Table 1 presents these soil parametervalues [34,20,5].

The rainfall in the Nylsvley savanna occurs primarily

as convective storms with high intensity and short dura-

tion [34]. Therefore, a marked Poisson process is used to

simulate the Nylsvley rainfall [20]. For the instantaneous

soil-moisture analysis in Section 3, average climate

parameters typical of Nylsvley are used. Following Laio

et al. [20], storm arrival rate, k, is 0.167 day�1, and thetotal depth of rainfall produced by a storm is generated

from an exponential distribution with a mean of 1.5 cm.

The duration is from a beta distribution with shape

parameter A equal to 2 and B equal to 4.67, assuming

a mean storm duration of 0.0625 days, a minimum of

0.0125 days, and a maximum of 0.25 days.

40

Fig. 3. Sketch of saturation as a function of depth to show

instantaneous saturation values for two averaging depths, 5 cm and

30 cm.

3. Instantaneous soil-moisture analysis

Let Sd denote a spatial average of instantaneous soil

moisture with the average taken between the soil surface

and a depth d. A special value of this average corre-

sponds to observations of soil moisture, denoted by

Sobs, which are measurements taken over depth dobs.

An example of Sobs derives from remote sensing mea-

surements, which provide estimates of soil moisture for

the upper few centimeters of soil at an instant in time.The corresponding averaging depth, dobs, depends on

sensor type, vegetation cover, landscape roughness,

and soil type. For our purposes, we take dobs equal to

5 cm as a characteristic depth for remote-sensed data,

although the depth is often less than this value. Fig. 3

shows a soil-moisture profile demonstrating vertical

averaging of instantaneous soil moisture for S30 and S5.

Recent research has focused on use of the instanta-neous measurement Sobs in conjunction with a model

or other information to improve prediction of Sd , where

d is deeper than dobs e.g., [28,6,16,27]. A recent field

study by Wilson et al. [39] investigated the relationship

between S6 and S30 using in situ measurement tech-

niques, but those authors were unable to find a clear

relationship between the two variables.

The relationship among instantaneous soil-moisturevalues integrated through different soil depths is dy-

namic, and it evolves in time as a function of hydrologic

inputs, atmospheric conditions, soil properties, and

plant characteristics. Because of its dynamic nature,

the time interval between soil-moisture measurements

is an important quantity. To demonstrate the impor-

tance of this time interval, we simulate a 200-day grow-

ing season for B. africana in the Nylsvley savanna,assuming an average Nylsvley climate realization with

k equal to 0.167 storms/day and a equal to 1.5 cm. Addi-

tionally, the plant compensation factor, c, is set equal to2, which means the plant has a moderate ability to ex-

tract water at high rates from wet sections of the root

zone to compensate for dry sections.

Fig. 4a shows a scatter plot of instantaneous soil

moisture for S30 versus S5. We plot S30 versus S5 (both

0

0.2

0.4

0.6

0 0.2 0.4 0.6

30S

5S

(a)

0

0.2

0.4

0.6

0 0.2 0.4 0.6

30S

5S

(b)

0

0.2

0.4

0.6

0 0.2 0.4 0.6

30S

5S

(c)

Fig. 4. Scatter plots of instantaneous saturation for 30 cm versus 5 cm averaging depth for Burkea africana (c = 2) from 200-day growing season

simulation with (a) dt equal to 0.1 day, (b) dt equal to 1 day, and (c) dt equal to U{0.1,5}.


from model output at the same instant in time) to gen-

erate each data point with a temporal interval between

output data, dt, of 0.1 day. This interval between output

data, dt, is interpreted to represent the time between soil-

moisture measurements, Sobs. As shown in Fig. 4a, bothS30 and S5 immediately increase when a storm with rain-

fall reaching the soil occurs. The amount of this increase

depends on the storm�s intensity, duration, and the ini-

tial soil conditions. The S5 value declines much faster

than S30 following a large storm event, which is a conse-

quence of the infiltrating front advancing downward in

the soil column coupled with drying in the upper layers.

Drying curves for individual storms are evident in Fig.4a with points forming a path from right to left in the

figure. This relationship is a function of the soil, plant,

and climate characteristics of the system.

In Fig. 4b, dt is increased to 1 day. We observe that

the relationship between S30 and S5 is still detectable.

However, because the instantaneous soil-moisture data

are available less frequently, detail is lost from the dry-

ing curves. Consequently, it becomes difficult to deter-mine to which drying curve some of the data points

belong, especially immediately after large storm events.

If dt is randomly chosen from a uniform distribution be-

tween 0.1 and 5 days, then the structure of the relation-

ship breaks down and becomes undetectable, as seen in

Fig. 4c. Clearly, a relationship is evident only when dt isconsistently small enough to capture the infiltration

dynamics of the system.

Wilson et al. [39] could not find a structured relation-

ship between instantaneous soil-moisture values with d

equal to 6 cm and 30 cm. One probable reason for the

unstructured relationship is that the dt between mea-

surements was not small enough. The dt necessary to de-tect the relationship between different instantaneous

soil-moisture averaging depths will be a function of

the soil, plant, and climate characteristics. A second rea-

son for the unstructured relationship is that the scatter

plots of the S6 versus S30 field measurements were for

different horizontal spatial locations in each of the

watersheds studied. Undoubtedly, soil profile, vegeta-

tion, and infiltrating water is variable over a watershed.Consequently, their finding of no clear relationship be-

tween pairs of S6 and S30 measurements is consistent

with the variability among the different site locations.

Notwithstanding the challenge of measurement fre-

quency, it is worth exploring the relationship between

S30 and S5 in Fig. 4a. In Fig. 5, the ratio of S5 to S30

is plotted over a 200-day growing season for a dt equalto 0.1 day. The ratio increases during and immediatelyfollowing a storm event, where the increase depends

on the storm�s intensity, duration, and the initial soil

conditions. From Fig. 5, a storm with a high intensity

and long duration will lead to S5 : S30 ratios greater than

3:1. For less intense and shorter storms with dry initial

conditions, the ratio of S5 to S30 is between 1:1 and

2:1. If initial soil-moisture conditions and the storm�sintensity and duration are known, then, in theory, one

0

1

2

3

4

0 50 100 150 200Time [days]

S /S 5

30

Fig. 5. Trace of the ratio of instantaneous saturation averaged over

5 cm to instantaneous saturation averaged over 30 cm for a 200-day

growing season with dt equal to 1 day for Burkea africana in Nylsvley

soil.


should be able to predict the relationship between S30

and S5. In addition, if we have frequent remote sensing

measurements, then data assimilation techniques, such

as the Kalman filter technique, would be able to estimate

accurately soil-moisture profiles from the remote sensingdata as demonstrated by Galantowicz et al. [11].

In summary, two major challenges prevent the use of

Sobs measurements to estimate Sd , where d is deeper than

dobs. First, dt must be sufficiently small relative to the

infiltration dynamics, as demonstrated in Fig. 4a–c. A

primary constraint for remote sensing or any other data

collection scheme is the measurement frequency. Cur-

rent remote sensing measurement intervals are greaterthan daily [30], while field measurements are often

obtained at irregular intervals when large horizontal

spatial areas are studied [39]. Second, regarding horizon-

tal spatial scale, instantaneous soil-moisture values can

only be related if hydrologic inputs, soil properties,

and plant characteristic are similar for locations being

considered. In light of these challenges in relating instan-

taneous soil-moisture measurements, we propose a sta-tistical approach to address the problem of the data�stemporal resolution.

0

20

40

60

80

0.15 0.2 0.25 0.3λ [1/day]

Extreme Savanna Parameters

Simulation Parameters

Tot

al R

ainf

all f

or 2

00-D

ayG

row

ing

Seas

on [

cm/2

00da

ys]

Fig. 6. Climate sample space for model simulations of Burkea africana

in Nylsvley soil used in the statistical soil-moisture analysis.

4. Statistical soil-moisture analysis

This alternative analysis investigates the relationship

between the temporal mean of soil moisture and averag-ing depth, d, as well as between the variance of soil mois-

ture and d. The overall goal is to predict these

relationships from the statistics of soil-moisture mea-

surements over averaging depth dobs and from known

parameters that characterize the soil, plant, and climate

system. Although issues such as scaling of soil proper-

ties, vegetation, and precipitation in horizontal space

need to be considered, we propose that if these statisticalrelationships can be predicted, then this information

could assist in the validation of soil-moisture estima-

tions produced by large-scale models. That is, if the

statistics of the large-scale model�s soil-moisture estima-

tions do not match the statistics of the estimations ob-

tained through our methodology for a given location

in horizontal space, then the validity of the large-scale

model�s estimations are questionable.As in Section 3, we will use our model to simulate

soil-moisture dynamics during the growing season at

Nylsvley. Herein we consider two basic statistical mea-

sures: the mean of soil moisture, lSd, and the variance

of soil moisture, r2

Sdover 1000 consecutive growing sea-

son days. We calculate the mean and variance of soil

moisture from instantaneous soil-moisture output

during 1000-day simulations for various climate realiza-tions. Since the dormant season is dry, the soil-moisture

conditions at the beginning of a growing season are sig-

nificantly changed after the first rainfall event. That is,

the initial soil-moisture conditions for each growing

season do not last long and do not change the soil-

moisture statistics of the growing season. Consequently,

lSdand r2

Sdare obtained by simulating a continuous

1000-day growing season with our model.Because instantaneous values of soil moisture are

dependent on hydrologic inputs, atmospheric condi-

tions, soil properties, and plant characteristics, the sta-

tistical characteristics of soil moisture throughout the

soil profile must depend on these same variables. For

this initial investigation, we vary rainfall frequency

and intensity, while soil and plant properties remain

constant. The range of values for k and total growingseason rainfall (assuming a 200-day growing season)

commonly found in a water-limited ecosystem are

shown in Fig. 6. We sample the parameter space system-

atically to define 25 different simulations. These 25 pairs

of parameters are shown in Fig. 6. Specification of k and

total growing season rainfall allows calculation of a. Theprobability distribution of storm duration is not chan-

ged, and, therefore, the analysis focuses only on the role


of k and a in a statistical analysis of soil moisture versus

averaging depth.

From model output of instantaneous soil moisture

with a dt of 0.1 day over 1000 growing season days,

we calculate lSdand r2

Sdfor averaging depths ranging

from dobs to the total root-zone depth, Zr, taken in5 cm increments. In all of the simulations, dobs is consid-

ered equal to 5 cm, while Zr is 100 cm. Fig. 7a shows a

plot of lSdversus d for four of the simulations that are

representative of the relationships found among the 25

simulations, where the markers are the simulation data.

Fig. 7b shows the same simulations for r2

Sdversus d.

Piecewise linear interpolation in Fig. 7a and b produces

smooth and well-behaved curves that reveal the exis-tence of relationships among lSd

for a range of d and

r2

Sdfor a range of d. If these relationships can be related

to soil, plant, and climate parameters, then soil-moisture

mean and variance over any d might be estimated from

easily measured environmental parameters.

A functional form describing the relationships in Fig.

7a and b is sought. In particular, we wish to relate sta-

tistics for the depth of observation, dobs, to statisticsfor other depths of interest. We begin by focusing on

the behavior of the means. We assume that we know

0

0

0.05

0.1

0.15

0.2

0.25

0.3

5d [cm]

αλ' = 0.27 cm/dayαλ' = 0.21 cm/dayαλ' = 0.16 cm/dayαλ' = 0.10 cm/day

0.005

0.01

0.015

0.02

0.025αλ' = 0.27 cm/dayαλ' = 0.21 cm/dayαλ' = 0.16 cm/dayαλ' = 0.10 cm/day

20 35 50 65 80 95

5d [cm]

20 35 50 65 80 95

(a)

(b)

µ sd

σ sd2

Fig. 7. Plots of (a) the mean of saturation over 1000 growing season

days as a function of averaging depth and (b) the variance of

saturation over 1000 growing season days as a function of averaging

depth for Burkea africana in Nylsvley soil for four climate scenarios

from model simulations.

dobs, and that we can use a large number of remote sens-

ing observations (for example, every 2 days for several

years) to estimate lSobs. The curves in Fig. 7a can be fit

with the following expression [35]:

lSd¼

lSobsðdj

obs þ vÞ1=j

ðdj þ vÞ1=jð15Þ

where lSobsis the mean of soil moisture over observation

averaging depth, dobs, and v and j are parameters that

affect the curvature and asymptote of the function.

For this analysis, we assume that lSobsis known, and,

consequently, model output is used to compute lSobs.

Eq. (15) provides excellent fits to the data in Fig. 7a

through implementation of the Levenberg–Marquardt

algorithm [29], which solves for the parameters v and

j. The values of the parameters v and j from the four

climate realizations in Fig. 7a are presented in Table 2.

Fig. 8 shows the data and fits for the mean saturation

versus averaging depth from these climates, where thesolid lines represent the fits using Eq. (15). The parame-

ters v and j were also calculated for the other climate

realizations. The goal is to relate the unknown parame-

ters v and j to measurable properties of the ecosystem

under consideration.

Since soil and plant properties remain constant in

the simulations presented, the parameters v and j,which determine the shape of the curve relating the

Table 2

Parameter values of Eq. (15) for the four climate realizations in Fig. 7a

Average daily rainfall reaching

the soil, ak0 [cm/day]

Parameter v Parameter j

0.27 2890 2.06

0.21 652 1.67

0.16 181 1.46

0.10 32.7 0.91

The mean of saturation versus averaging depth curves, which are

produced using these parameters in Eq. (15), are shown in Fig. 8.

0

0.05

0.1

0.15

0.2

0.25

0.3

5d [cm]


20 35 50 65 80 95

µ sd

Fig. 8. Plot of the fits of Eq. (15) (solid lines) to the simulation data

(markers) of Fig. 7a, which shows the mean of saturation versus

averaging depth.


soil-moisture mean to averaging depth, are controlled by

rainfall frequency and intensity. We would like to use

these climate parameters to predict the parameters vand j. We expect that v and j will depend on the aver-

age daily rainfall reaching the soil for a growing season,

which is given by a Æ k 0. Alternatively, these parameterscould depend on the index of dryness, DI, from Milly

[26], which is defined as DI = ETmax/(ak 0). Since ETmax

does not change because the vegetation is not varied in

this study, we focus on relating the parameters v and jto a Æ k 0. Fig. 9a presents the relationship for v versus

a Æ k 0, while Fig. 9b shows j versus a Æ k 0. The relation-

ship in Fig. 9a can be fit by the equation

v ¼ 1:85� 105 � ða � k0Þ3:90; a � k0 < 0:17 cm=day

1:40� 107 � ða � k0Þ6:59; a � k0 P 0:17 cm=day

(

ð16Þwhile for Fig. 9b the equation is

j ¼ 1:20 lnða � k0Þ þ 3:57 ð17ÞEqs. (16) and (17) were obtained empirically through

regression. Knowledge of a Æ k 0, used in conjunction with

Eqs. (16) and (17), allows for estimation of the parame-

1

10

100

1000

10000

0 0.1 0.2 0.3αλ'

αλ' = 0.17

(a)

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 0.1 0.2 0.3

αλ'

κχ

(b)

Fig. 9. Plots of (a) the parameter v versus ak 0 and (b) the parameter jversus ak 0 for the 25 climate scenarios for Burkea africana in Nylsvley

soil. The parameters v and j are obtained from fitting Eq. (15) to the

mean of saturation versus averaging depth curves.

ters v and j. By incorporating the estimated v and j into

Eq. (15), we can predict the lSdversus d relationship for

a specific vegetation and soil. That is, if we have accu-

rate knowledge of the average daily rainfall reaching

the soil, a Æ k 0, for B. africana in Nylsvley soil and obtain

lSobsfrom remote sensing observations, then we can pre-

dict the mean soil-moisture value for any averaging

depth d.

We test this prediction procedure by considering the

four climate realizations plotted in Fig. 7 and attempt-

ing to predict the lSdversus d curves. We know a Æ k 0

and lSobsfor each of these realizations, while Eqs. (16)

and (17) estimate the parameters v and j. Then Eq.

(15) predicts the lSdversus d curves shown in Fig. 10.

We find close approximations for the lSdversus d rela-

tionship for three of the four climate realizations. The

exception is the climate with a Æ k 0 equal to 0.21 cm/

day, where the predicted curve provides a somewhat

poorer fit to the data. The reason is that the parameters

v and j for that climate realization are not well esti-

mated by Eqs. (16) and (17).

Eqs. (16) and (17) are not based on any physical char-acteristics of the system but were chosen to balance

complexity and prediction accuracy. Future research

could focus on increasing the number of simulations

and improving the fits for Eqs. (16) and (17) to improve

prediction ability. Another concern is accurate knowl-

edge of a Æ k 0 for a given ecosystem. If the actual a Æ k 0

does not match the predicted a Æ k 0, problems could arise

when seeking a relationship between parameters of Eq.(15) and the parameters of the climate realizations.

Alternatively, we attempted to predict the parameters

v and j from either the mean rainfall depth, a, or the

mean storm arrival rate, k. However, the relationship

between a Æ k 0 and the parameters was much stronger.

It is also possible to predict the parameters v and j

650

0.05

0.1

0.15

0.2

0.25

0.3

5d [cm]


20 35 50 80 95

µ sd

Fig. 10. The curves for mean of saturation versus averaging depth are

predicted from ak 0 using the fits in Fig. 9a and b to estimate the

parameters of Eq. (15). The Eq. (15) predictions (solid lines) are

compared to the simulation data (markers).


from, as an example, both a Æ k 0 and k, but we focused

on using one variable for simplicity in this initial inves-

tigation. Despite the challenges, the results in Fig. 10

demonstrate that this methodology has the potential to

estimate effectively the lSdversus d relationship using

readily available data on soil, plant, and climatecharacteristics in a model of appropriate resolution

and detail.

The curves for variance of instantaneous soil mois-

ture versus averaging depth in Fig. 7b require a fitting

equation other than Eq. (15), because this equation fails

to provide close fits to the data. Therefore, further work

must be done to obtain a function that accurately fits the

variance versus averaging depth curves. An alternativeapproach to looking for a particular functional form

for the curves in Fig. 7a and b is to look at lSdversus

a Æ k 0 for different d and r2

Sdversus a Æ k 0 for different d.

Fig. 11a shows that lSdincreases linearly as a Æ k 0 in-

creases for two averaging depths, 30 cm and 100 cm.

In Fig. 11b, r2

Sdincreases as a Æ k 0 increases, but the

spread of the simulation data increases as a Æ k 0 in-

0

0.05

0.1

0.15

0.2

0.25

0.3

0

αλ' [cm/day]

d = 30 cmd = 100 cm

0

0.004

0.008

0.012

0.016

0.02

d = 30 cmd = 100 cm

0.1 0.2 0.3

0

αλ' [cm/day]0.1 0.2 0.3

µ sd

σ sd2

(a)

(b)

Fig. 11. Plots of (a) the mean of saturation for different averaging

depths as a function of average daily rainfall reaching the soil, ak 0, and

(b) the variance of saturation for different averaging depths as a

function of ak 0 for Burkea africana in Nylsvley soil.

creases. Fig. 11 highlights the opportunity for explora-

tion of the statistical properties of soil moisture in the

root zone through a variety of approaches.

5. Discussion

The statistical analysis presents a methodology that

allows prediction of the mean of soil moisture, lSd,

for any averaging depth in the root zone. We consider

a specific vegetation and soil, while simulating a range

of climate realizations. We would like to extend the

current analysis to predict the relationships between sta-

tistical measures of soil moisture and averaging depthfor a range of vegetation and soil subjected to various

climates. Extension of this analysis highlights the impor-

tance of understanding the effect of the different pro-

cesses considered in soil, plant, and climate models on

lSd. In fact, it is important to recognize that the model

used for a given soil, plant, and climate system depends

on the characteristics of the system and the model out-

put of interest [36]. Therefore, while this model is appro-priate for simulating a completely vegetated area of B.

africana in Nylsvley soil, it might not be appropriate

for another system.

As the simulations of B. africana in Nylsvley soil

demonstrate, the average rainfall reaching the soil,

a Æ k 0, has a significant impact on lSdfor all d in the root

zone. The average amount of water in the root zone in-

creases for higher values of a Æ k 0. Consequently, all lSdmust increase. The shape of each curve depends on the

average infiltration and redistribution characteristics of

the system, which are controlled by the rainfall, soil,

and vegetation. We would like to identify parameters

that are dominant in controlling the shape of these

curves. Measurable parameters that have a significant

impact on lSdinclude Ksat, ETmax, a, k 0, average rainfall

intensity, and root distribution. The dimensionlessgroups of parameters discussed by Guswa et al. [12]

could be useful measures in determining what controls

lSd. For example, the spatial infiltration index is defined

as [12]

I I;x ¼Zi

Zr

¼ a=½/ðSfc � ShÞ�Zr

ð18Þ

where Zi is a measure of the depth of infiltration of an

average storm and Sfc is the saturation at field capacity(0.29 for Nysvley soil) [12]. This index is a measure of

the infiltration depth relative to the depth of the root

zone. II,x values for the simulations range from 0.03 to

0.15, which means that, on average, water is infiltrating

to very shallow depths relative to Zr. This finding is con-

sistent with the results of Fig. 7a, which show that lSddecrease rapidly as d increases.


When considering different soil, plant, and climate

systems, the proposed statistical approach may vary

depending upon the properties of the system. For exam-

ple, if an ecosystem has high interannual climatic

variability, then using a Æ k 0 to predict lSdmight not be

accurate. It is necessary to identify the parameters thatprovide meaningful information on lSd

for each system.

Then, in terms of extending the proposed analysis to

other systems, simulations can be run for a variety of

soil and vegetation types to obtain statistical soil-mois-

ture data for different averaging depths. A function

can be fitted to the data from the simulations, and we

can then look for relationships between the parameters

of the fitting equation and measurable soil, plant, andclimate parameters (or dimensionless groups of these

parameters).

Next, we consider the uses of these statistical soil-

moisture measures for different averaging depths. To ad-

dress this issue, we must again consider the suggestion

that the most promising approach to estimate soil mois-

ture for large-scale applications is through combination

of remotely sensed data and hydrological models [19].The instantaneous soil-moisture analysis (Fig. 4a–c)

highlights the challenges that prevent the use of shallow

instantaneous soil-moisture values, inferred from re-

mote sensing observations, in the estimation of instanta-

neous soil moisture for deeper averaging depths. The

methodology that we propose, which uses the statistics

of these shallow instantaneous soil-moisture values,

can be used as a diagnostic tool to evaluate the estima-tions of soil moisture made by large-scale land surface

models. Because computation time typically constrains

the complexity and resolution of large-scale models, a

large-scale model might not be sufficiently resolved in

the vertical spatial dimension or might not include all

processes necessary to estimate accurately soil-moisture

dynamics. In our methodology, the appropriate com-

plexity and resolution of the model depends on the soil,plant, and climate characteristics of the system under

consideration. The statistics of soil-moisture estimations

for each horizontal grid cell from a coarse and simplified

large-scale model could be validated through compari-

son with the soil-moisture statistics estimated using

our methodology, where the estimations will be from

statistics of remotely sensed shallow-layer soil moisture

and a model of proper complexity and resolution forthe system under consideration. Through our statistical

methodology, we avoid the difficulties associated with

using instantaneous soil-moisture values and demon-

strate a technique that could assess the validity of the

soil-moisture estimations from large-scale models. We

do not guide model estimations through assimilation

of soil-moisture values from remote sensing. However,

the proposed validation scheme could elucidate sourcesof inaccuracy in large-scale models. In addition to large-

scale model validation, statistics of soil moisture ob-

tained through the proposed analysis are important for

applications in ecohydrology. For example, if we want

to estimate plant stress in a water-limited ecosystem

due to climate change, soil-moisture statistics would be

useful. On the other hand, soil-moisture statistics and,hence, this methodology would not be very useful for

applications such as weather forecasting or real-time

flood predictions, because they do not provide event-

based information on soil moisture.

Finally, it is important to discuss two issues regarding

soil-moisture values obtained from remote sensing.

First, we consider the case where dobs is extremely shal-

low, less than 5 cm, corresponding to cases where asharp vertical gradient in the top few centimeters of soil

occurs [2]. The methodology presented can be adjusted

easily by using a dobs less that 5 cm with a smaller inter-

val between values of d in the simulations. The function

in Eq. (15) might change, but the overall approach will

remain the same. In this manner, we overcome the prob-

lematic situation of very shallow dobs. Second, the hori-

zontal scale of soil-moisture values inferred from remotesensing measurements is typically much larger than the

horizontal scale of the model used in this initial investi-

gation. Therefore, we must make certain that the hori-

zontal spatial scale of the soil-moisture values from

remote sensing measurements is consistent with the

horizontal scale of the model in our methodology.

Future work must examine the validity of our method-

ology when the model, used to simulate different soil,plant, and climate systems, has a large horizontal spatial

scale.

6. Conclusions

It is not generally feasible to obtain remote sensing

measurements at the highly resolved temporal scalesnecessary to see a structured relationship among Sobs

and Sd for different values of d. Because of this limi-

tation, we have presented and tested a statistical

methodology. The results of this initial investigation

demonstrate that a simulation model, combined with

known climate parameters and soil moisture from re-

mote sensing measurements, allows accurate prediction

of the mean of root-zone soil moisture for any averagingdepth. Initial attempts to predict the r2

Sdversus d rela-

tionship have been less successful than those reported

herein for the mean, yet there is evidence for optimism

that further work will enable accurate prediction of this

relationship as well. While much research needs to be

done to validate these soil-moisture predictions, the

methodology presented provides a promising pathway

to obtain statistical information about soil moistureover the entire root zone.


Acknowledgments

The authors gratefully acknowledge the support of

the William Clay Ford, Jr �79 and Lisa Vanderzee Ford

�82 Princeton Graduate Fellowship, and I.R-I. gratefully

acknowledges the support of the National ScienceFoundation through grants DEB-0083566 (Biocomplex-

ity) and EAR-0120914 (National Center for Earth Sur-

face Dynamics).

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