WATER
RESOURCES
RESEARCH,
VOL.
23, NO.
11, PAGES
2135-2143,
NOVEMBER
1987
Effects of Spatial Variability on Annual Average Water Balance
P. C. D. M!LLY
Department of Civil Engineering, Water ResourcesProgram, Princeton University, Princeton, New Jersey
P.S.
EAGLESON
Department of Civil Engineering,MassachusettsInstitute of Technology, Cambridge
Spatial variability of soil and vegetation causesspatial variability of the water balance. For an area in
which the water balanceis not affectedby lateral water flow, the frequencydistributionsof storm surface
runoff, evapotranspiration, and drainage to groundwater are derivable from distributions of soil hydraulic parametersby means of a point water balance model and local application of the vegetal
equilibrium hypothesis.Means and variancesof the componentsof the budget can be found by Monte
Carlo simulationor by approximatelocalexpansions.
For a fixedsetof meansoilparameters,
soilspatial
variability may induce significantchangesin the areal mean water balance,particularly if storm surface
runoff occurs.Variability of the pore size distribution index and permeability has a much larger effect
than that of effectiveporosity on the means and variancesof water balance variables.The importance of
theporesizedistribution
indeximpliesthatthemicroscopic
similarity
assumption
mayunderestimate
the effectsof soil spatial variability. In general, the presenceof soil variability reducesthe sensitivityof
water balance to mean properties. For small levels of soil variability, there exists a unique equivalent
homogeneoussoil type that reproducesthe budget componentsand the mean soil moisture saturation of
an inhomogeneousarea.
INTRODUCTION
The local hydrologic responseof the land surface to atmosphericforcing (precipitation, solar and atmosphericradiation,
etc.) is determined by the properties of the surfaceand by the
forcing. Both of thesefactors may vary significantlyat spatial
scalessmaller than the scale of practical hydrologic analyses.
Since analysesof small-scalephysical processesin catchments
suggestthat the local responseis a nonlinear function of these
variables, we infer that the problem of spatial integration to
the catchment scale is not trivial. It is not at all clear, for
example, under what conditions a one-dimensional model of
the soil boundary layer, grounded in soil water hydrodynamics, can represent the average behavior of a spatially variable
natural catchment. This problem of local variability has been
one of the major impedimentsto the successfulapplication of
Darcy scalephysical theory to explain hydrologic responseat
catchment
scales.
Variability of atmospheric forcing is most apparent in the
heterogeneityof rainfall fields. Individual storms have significant internal spatial and temporal structure and are generally
in motion relative to the ground. The result is that rainfall
intensity histories, and even aggregate quantities such as total
storm depth, are sometimes variable at scales smaller than
those of the catchment of interest. With increasingly large
averaging periods (i.e., many storms), the spatial correlation
naturally increases.However, the short-term hydrologic responseis not necessarilydetermined uniquely by such longtime averages, whose uniformity is therefore not especially
helpful in the analysisof singleevents.On the other hand, the
time average water balance can be parameterizedin terms of
rainfall statisticsthat are quite uniform spatially.
Variability of the land surface (itself, the soil texture and
Copyright 1987 by the AmericanGeophysicalUnion.
structure,the vegetationtype and density,and geomorphological parameters),is the secondmajor source of variability of
hydrologic fluxes. Soil propertiesaffect the ability of the soil
surfaceboundary layer to transmit water into or from the soil.
The vegetation alternately enhancesor hinders the transmission process.Geomorphologicalfactorsof importanceinclude
thoseinducinglateral subsurfaceflows, which complicate(and
often govern) the direct runoff generationprocess.The spatial
variability of the surface,unlike that of the forcing,is a factor
that cannot be eliminated by averaging in time; surface
properties of untilled soils are so persistentin time as to be
effectively static.
In this paper we examine the effect of spatial variability of
soil and vegetation on water balance. In particular, we are
concerned with the extent to which their areal variations affect
the temporal-spatialaveragesof the major hydrologicfluxes.
We are also interestedin the question of the existenceof an
equivalent homogeneous soil capable of reproducing the
averagebehaviorof an inhomogeneous
area.
Theoretical analysesof water balance on spatially variable
soilshave been describedby Peck et al. [1977] and by Sharma
and Luxmoore [1979]. They employed dynamic simulation
modelsin analysesof the water balanceof a particular catchment on a monthly time scale,accountingfor storagechanges.
Soil spatialvariabilitywasrepresented
usingthe conceptof
microscopic scale similarity. Sharma and Luxmoore demonstrated the importance of soil variability and noted the difficulty of making generalizationsin light of the dependenceof
resultson the soil-plant-weathercombination. They observed
that their neglect of variations of vegetation properties associated with soil variability was unrealistic.
Our treatment follows the basic approach first laid down by
Peck et al. [1977-1,but offersseveralnew insights,primarily as
a result of our different approachto the point water balance.
By restrictingour attention to the annual averagewater bal-
Paper number 7W4939.
0043-1397/87/007W-4939505.00
ance, we are able to employ the results of Eagleson[1978a, b,
2135
2136
MILLYANDEAGLESON:
EFFECTS
OFSPATIAL
VARIABILITY
ONANNUAL
AVERAGE
WATER
BALANCE
c, d, e,f, g], whichconcisely
express
themajorcomponents
of
the water balance as functions of soil and atmospheric param-
eters and which account for the dependenceof equilibrium
Climate
Parameters
d•, average
rateofpotential
evaporation;
mto meantimebetween
storms;
vegetationdensitieson soil properties.The computational
mtr meanstormduration;
simplicityof the water balancemodel allowsus to obtain
m• meandurationof rainy season;
detailedresultsregardingthe sensitivityof averagehydrologic
fluxesto the varianceof soil parametersfor a wide range of
soil typesand climaticconditions.It alsopermitsus to examine the effectof departuresfrom microscopic
similarityin the
mpA meanannualrainfall;
•c parameterof gammadistribution
of stormdepth'
•a meanannualair temperature.
analysis.Just as important,the parametricfrugalityof the
Other Parameters
model allows us to infer from those results some relatively
generalconclusions
regardingthe effectsof soilspatialvariability. In what followsthe term "waterbalance"will referto
the annual average water balance, which will be further
averagedhere in space.
h0 surfaceretentioncapacity;
Z
effectivedepth to water table;
kv potentialtranspiration
efficiency;
M
vegetalcanopy density.
Given theseparameters,the water balancemodel yieldsexpectedvaluesof the followingvariables'
FRAMEWORK
So averagesoil moisturesaturation;
We suppose
that an inhomogeneous
areaA may be represented,to first order, as a battery of parallel,independent, ETA annualevapotranspiration;
one-dimensionalsoil columns,each describedby a determinis-
tic modelyieldingoutputsas functionsof local inputsand
parameters.
In orderto derivethearealaverage
waterbalance,
weintegratethepointwaterbalanceovertheregionA, using
thejoint densityfunctionof thesoilproperties
asa weighting
function.Thusif y is any outputdependent
on the soilproperties to with joint frequencydistributionf(to) in A, then the
areal average(y), definedby
RsA annualsurface
runoff;
RgA annualrecharge
to groundwater.
Furthermore,invocationof the vegetalequilibriumhypothesis
[Eagleson,
1978f-]allowsus to solvethe waterbalancewithout specifying
M;in that case,the waterbalancemodelalso
provides
thevalueof M asan output.According
to theequilibrium hypothesis,natural systemsof vegetationevolve
towarda canopydensitythat maximizesthe averagesoil
moisture saturation.
dx
(1)
is given by
With respectto climate,onlycertainlong-termstatistics
are
prescribed;
the modelimplicitlyperformsa time averageof
the dynamics[Eagleson,1978a,b, c, d, e, f, g]. In contrast,
constantsoil parametersare specified,with the implication
(Y)
=•,og(o•)f(o•)
do•
(2)neitywithinthemodeledarea.Directapplicationofthemodel
that the model is basedon a physicalconceptionof homoge-
in whichx is the spatialcoordinate,g( ) is the outputfunction relatingto to y, and fl is the set of all to foundin A.
(Notation followsthe main text of thispaper.)
In principle,the arealintegrationshouldincludenot only
the soil parameters,
but alsothe inputs,suchas rainfalland
potentialevaporation
rate.Whilethesemaybe highlyvariable
in spaceat anytime,it is onlyclimaticstatistics
that enterthe
water balancemodelwe useto representg( ). Thesestatistics
exhibitconsiderably
greatercorrelationin spacethan the instantaneo•svalues,but, of course,may also have significant
spatialvariability.We are therefore
ignoringsuchtemporally
constant,spatiallyvariableeffectsas orographicinfluences
on
to a heterogeneous
areaimplicitlyrequiresan assumption
that
"effective"
soil parameters
may be defined.This assumption
is
examined in a later section of this paper.
One of the critical assumptions underlying the onedimensional water balance model used here is that surface
runoff is generatedonly by the infiltration-excess
mechanism.
Thereis no provisionfor runoffproducedat riparianseepage
faces. Our conclusions are therefore limited to catchments for
whichthis assumption
is valid or to thoseportionsof catchmentsin which the water table doesnot approachthe surface.
JOINT DENSITY OF SOIL PARAMETERS
In order to proceedwithin the proposedframework,we
rainfallstatistics,
topographic
influences
onnetradiation,
and requirea statistical
description
of the distributions
andcorre-
so forth.
lationsof the three independentsoil parametersne, k(1), and
m, whichwe denotecollectivelyby o•.The natureof thejoint
POINT WATER BALANCE
variabilityof theseparameters
hasnot beenwell-defined
and
is an areaof ongoingresearch.
The marginaldistributions
of
somehydraulicparameters
havebeenstudied,eitherdirectly
The analysesare
transformationof local hydrologicinputs and surfaceproper- or indirectly,by a numberof investigators.
from
tiesto hydrologic
fluxes.The inputparameters
to the model mainlyof two basictypes.In the first,data are collected
We employthe statistical-dynamic
waterbalancemodelof
Eagleson
1-1978a,
b, c, d, e,f, g] to explain,to firstorder,the
are defined as follows.
Soil Parameters
ne effectiveporosityof soil;
k(1) soil permeability;
m pore sizedistributionindexof soil.
numeroussourcesof wide geographicorigin and groupedby
soil texturalclassification.
The resultingstatisticscharacterize
variabilitywithina textureclassoverthe setof all soils.Such
analyses
havebeenreportedby ClappandHornberger
[19783,
by Brakensiek
et al. [1981],and by Rawlset al. [1982].The
secondtypeof analysisusesa largesetof data from a single
MILLY AND EAGLESON:EFFECTSOF SPATIAL VARIABILITY ON ANNUAL AVERAGEWATER BALANCE
location, field, or watershed to develop the densities, and
therefore seemsmore relevant to the current analysis. Warrick
and Nielsen [1980] provide a review of the earlier work in this
area, and numerousworks have been publishedsubsequently,
particularly in the soil scienceliterature.
It is rather well-establishedthat the hydraulic conductivity,
and hence the intrinsic permeability, is often lognormally distributed in the field. Effective porosity is sufficiently wellrepresented by the normal distribution. The parameter m,
which is the exponent in the Brooks-Corey [Brooks and
Corey, 1964] equation for the soil moisture characteristic,was
found to be lognormal by Brakensieket al. [1981] within a
textural class,but its local variability within a field or a catchment has not received much attention. Our investigations of
available soil moisture retention data support the use of the
lognormal distribution at the local scale.
On the basis of the foregoing, we adopt the multivariate
normal distribution to describesoil variability in this work.
We treat he,In [k(1)], and In (m) as the three variates,denoted
by
(T o
(0.6)
BYERS AND STEPHENS (1983)
x TRANSECT
(I.2)
....
ANDERSON
ANDCASSEL
(1986)
A HORIZON
(I. I)
COELHO(1974)
300 mm DEPTH
(0.8)
AHUJA et ol. (1984)
O- 150mm
(0.9)
IO)il
[1ne
]1
ro2
=
n [k(1)
(3)
[ln(m)_]
305mm
DEPTH
LOAMY
SAND
"]•_•
(2'0)(2•)7)
•
SANDY
LOAM
[•, ,,,o
LOAM
(I.8)
(2.1)
.•__•____•___•
CLAY
LOAM
[ m
(2.._5_)••_____•SlLTY
CLAY
[ ua
(2.1) /
•
SILTY
CLAY
LOAM)
-2,5
-20
ALL
TEXTURES
-15
-IO
-,5
tn[k(I),mm
2]
Then the joint density function of m is
f(to) = (2zr)3/2[Q[
•/2 exp [-«(to -- III)TQ- •(0}-- IIl)]
DEPTH
NIELSEN et ol. (1973)
(2.5) •
m=
2137
(4)
Fig. 1. Fitted lognormal frequencydistributions of k(1). Parenthetical numbersat left are valuesof ao.
in which
(5)
Q = EE(to- p)(to- p)r]
a•
a•a2P•2
(6)
o'10'3p 1
g3•lP31 g392P32 •32
The vector g is the mean value of •, the matrix Q is the
covariance
of •, a• is the standarddeviationof w•, and Pu is
the correlationof w• and w•. For convenience,
we scalethe
standard deviations by a parameter ao,
ffi =•ffO
i= 1, 2, 3
(8)
We setf2 equal to unity, so ao is equivalentto the standard
deviation of the logarithm of permeability. We fix the values
off• andf3 as constantsfor the entire investigation.Basedon
our interpretation of some of the limited available data, we
takef• to be 0.05 andf3 to be 0.4. Theseshouldbe considered
as representativevalues with no universal significance.The
parameterao is usedas an indicatorof the levelof variability
in the soil parameters.
It is helpful to considerthe range of values that ao may
assume.Typical fitted frequencydistributionsof the logarithm
of permeabilityare plotted in Figure 1 for representativetextural classes and for selected field sites; Table 1 summarizes
creaseswith the horizontal dimension, but there are obviously
other important factors determining the soil variability. The
data provided by Brakensieket al. [1981] can be processedto
yield an estimateof ao equal to 2.5 for the entire United States
of America. The hypothetical extreme caseof an area having a
bimodal soil distribution with permeabilities differing by a
factor of 1000 yields a value of about 3.5. For areas of interest
for the water balanceproblem, it seemsreasonableto focuson
valuesof aoin the rangefrom about 1-2.5.
WATER BALANCE SENSITIVITY TO THE SOIL VARIANCE
Monte Carlo Analysis of the Means
The evaluationof (2) is complicatedby the fact that g( ) is
not known as an explicit function; a numerical integration
scheme is required. Since to is three-dimensional and the
evaluation of g( ) is not simple, a straightforward discretization method is computationally unattractive. We instead employ a Monte Carlo techniquefor evaluation of (2).
We use a random sampling schemeto generate numerous
equally probablevaluesof to. With a set of N toi valuesso
generated,we approximate(2) accordingto
1
N
<y>
=• ,•a(to,)
(9)
the experimental findings of some investigators.For sites For all resultsreportedin this paper, we usedN equal to 500,
which was found through numeroussensitivityanalysesto be
whose characteristic horizontal dimension is 1000 m or less,
sufficient for accurate estimation of the means. We also exam-
the value of ao typically ranges from 0.5 to 1.0, but may
becomeeven larger. For instance,Nielsenet al. [1973] found ined the variancesof the outputs,estimatedaccordingto
the smaller values on a relatively homogeneousfield, while
1 N
Andersonand Cassel[1986] found valuesas high as 2.6, which
they speculatedmay have been causedby the presenceof tree
We have chosenthe two climatic data setsusedby Eagleson
root channelsin the lower soil horizons. In general, ao in-
O'y
2---• i•1[•](O)/)
--<y>]2
(10)
2138
MILLY
AND EAGLESON: EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE
TABLE 1. Representative
Valuesof So,the StandardDeviation ofln [k(1)]
Characteristic
Horizontal
Length of Site,
m
Soil
ao
Byers and Stephens[1983]
Cassel[1983]
Bresler et al. [1984]
Ahuja et al. [1984]
Andersonand Cassel[1986]
Coelho [1974]
Nielsen et al. [1973]
15
60
90
300
500
1000
1500
5 x 106
5 x 106
fluival sand
Norfolk loamy sand
Hamra Red Mediterranean
three silt loams
Portsmouth sandy loam
Pima clay loam
Panoche soil series
0.38-0.58
0.53-1.1
0.35-0.70
0.75-1.2
1.2-2.6
0.74-1.2
0.55-0.87
1.8-2.7
2.5
Brakensieket al. [1981]
Brakensieket al. [1981]
[1978f];
Estimated
Sourceof Data
the data for Clinton, Massachusetts,and for Santa
Paula, California, are summarized in Table 2. For the soil
parameters, the mean values !s are assignedthe values given
by Eagleson['1978e] as typical for each of four textural classes
of soil; these are given in Table 3. We considerthe casewhere
there is no cross-correlationamong the soil parameters.
We consider only the case when the surface retention capacity is negligible and the water table is so deep that there is
no capillaryrise to the surface.We take kv to be unity, and we
employ the vegetalequilibrium hypothesisto evaluateM.
Figure 2 displays the solutions for mean soil water saturation so of the four soilsat both locations.The parameterso
is plotted as a function of variability of soil type. One striking
feature of theseplots is the relatively small dependenceof so
on the soil variance.
The
second
remarkable
feature
is the
small sensitivityto climate in comparisonwith the larger sensitivity to soil type. This is apparently explained by the fact
that soil parameters such as permeability vary over a wider
range than the climatic parameters such as precipitation and
potential evaporation.
Figure 3 shows the major water balance components for
each soil-climate combination as functions of soil variability.
All plotted quantities are normalized by averageprecipitation.
As is seenin Figure 2, the effectof soil type dominatesthat of
climate. In contrast to Figure 2, significantsensitivityto ao is
visible. In general, it may be said that variability tends to
equalize the magnitudesof the three components.For example, surfacerunoff appearsin the caseof sandy loam and silt
loam as ao2 growsfrom 0 to 6. In contrast,it decreases
from a
very large fraction in the case of clay. Whereas only the clay
and the clay loam soils yield storm surface runoff in the homogeneouscase,all inhomogeneoussoilsproduceit.
A consequenceof the behavior noted above is that there is
less sensitivity of the water balance to the average soil type
when soil variability is considered. We illustrate this more
directly by considering our clay loam soil and varying the
TABLE
Parameter
2.
Climatic
mean value of In [k(1)], with ne and In (m) remaining fixed.
Figure 4 shows,for the Clinton climate, the sensitivityof the
water balance to In [k(1)] for a homogeneoussoil and for a
soil with ao•- equal to 6. Not only is the sensitivityof the
balanceto the mean generallyreducedby inhomogeneity,but
additionally the curves for the inhomogeneous case are remarkably linear over the range of possible means. Several
similar calculations revealed that the effect of averaging is
similar for In (m)and for the other soil-climatecombinations.
Monte Carlo Analysis of the Variances
We
calculated
the variances
mtb, S
mr,, S
mpA, mm
mT, S
K
1.74 x 10-5
2.59 x 105
2.76 x 104
Santa
balance
Figure 5 showsvariancesof so as functionsof ao•- for the
two cases.The relations are roughly linear. In all casesexam-
ined, the standard deviation of so was between0.03 and 0.1
when6o2 wasunity.
Figure 6 showsvariancesof the normalized componentsof
thewaterbalanceasfunctions
of ao2.In thecaseof thesandy
loam,variances
are quitesmallfor ao2 lessthan unity.This
seems to be associated
with
the virtual
absence of surface
runoff in this region; in all of our results, flux variances are
disproportionatelysmall when the surfacerunoff componentis
very small. This suggeststhat much of the variability in fluxes
induced by soil variance is controlled by storm surfacerunoff.
In the caseof the clay loam soil, variancesare much larger
and approach maximum values asymptotically.The order of
magnitude of the maxima can be estimated by consideringthe
hypothetical case where variability of a particular flux is so
great that the flux is uniformly distributed between zero and
one. In that situation, the variance would be about 0.08. This
is consistentwith Figure 6 and other calculationsnot reported
Paula
3.12 x 10-5
8.99 x 105
1.21 x 105
1113
3.
Mean
Soil Parameters
Clay
Clay
Loam
Silt
Loam
Sandy
Loam
ne
k(1),mm2
0.25
1.0 x 10-8
0.35
2.8 x 10-8
0.35
1.2 x 10-7
0.45
2.5 x 10-7
m
0.222
0.286
0.667
2
544
3.15 x 10 ?
1.83 x 10 ?
0.5
0.25
281.5
water
soils.
TABLE
•,•,,mms-1
of the various
outputs according to (10). In our discussionof the results,we
shall focus on the cases of sandy loam and clay loam soils
with the Clinton climate; the first is qualitatively representative of all caseswith sandy loam or silt loam soils, while
the secondis representativeof all caseswith clay loam or clay
here.
Parameters
Clinton
10 USDA textureclasses
all texturescombined
286.9
MILLY ANDEAGLESON:
EFFECTS
OF SPATIALVARIABILITY
ON ANNUALAVERAGE
WATERBALANCE
2139
1.0
CLAY
08
CLAY LOAM
•
•o
2=6
0.6-
<So>
SILT LOAM
.
O,4-
SANDYLOAM
0.2
0.4
--
CLINTON
SANTA PAULA
øo''
E
•
0.2
/
Fig. 2. Areal meansof so as functionsof %2 for varioussoilsand
!E
[Rga
]/mpa
climates.
10-9
Approximate Analysis of the Means
10-7
k(I) ,mm2
loam soil with Clinton
climate.
or
f22+63ro32
f32
(Y)=g(P)
+ao
2 •,63-•
2f•2+63ro•_2
set
+ •,•,0•,02
f,f•,•2+ •0•,0-----•
y=g(•o)
=g(•)
+ • • (•o
- •)'•-d•g(•) (••)
+ &o2&o-------•
Taking the expectedvalue, and neCectingterms for which n is
3 or greater, we find
(Y)=g(•)
+•
i=t
•,•% Qu
SANDY LOAM
(•2)
632•]
f2f3P23)]
SILT LOAM
(o)
SANTA
CLINTON
-
PAULA
CLAYLOAM I,
_
0.8
( c)
SANTA
PAULA
(d)
CLAY
ItE[ETa
]/mp
A
0.6
I
IIE[Rsa
]/mpa
CLINTON
SANTA
O.4
0.2
o
PAULA
, E[R•a]/mpA
I
2
4
(13)
in which all partial derivativesare taken at m equal to p. This
equation tells us, to a first approximation, that the effect of
soil spatial variability of any input parameter on any output
variable is directly proportional both to the variance of the
parameter and to the second derivative of the output with
CLINTON
I.O
10-6
Fig. 4. Water balance as function of mean In [k(1)] for the clay
It is difficult to infer from application of (9) the manner in
which individual soil parametersaffect the average water balance. Furthermore, the effects of correlation are not easily
discerned.In this sectionwe produce an alternative expression
for (y) that shows the interaction of parameters explicitly,
albeit only approximately. Assuming that soil variability is
small, we develop second-orderapproximations for the mean
values of the output variables of the water balance. We
expand the output y in a Taylor seriesaround the mean parameter
10-8
6
o- o
Fig. 3. Arealaverage
expected
waterbalancecomponents
asfunctions
of ao2for varioussoilsandclimates.
2140
MILLY AND EAGLESON'EFFECTSOF SPATIALVARIABILITYON ANNUAL AVERAGEWATER BALANCE
0.04
TABLE 4.
o
o
SANDY LOAM
CLAY LOAM
Sensitivityof Mean Evapotranspiration to Soil
Variability
1 02___g.
0.03
Soil
Type 2&0
x2fx2
1 02g
f22
2 0(_D22
1 02g
--' f32
2 0c032
Clinton
Sandy loam
o-s%0.02
Silt loam
O.OI
-0.002
-0.006
0
-0.004
- 0.001
- 0.001
-0.006
-0.001
-0.217
0.077
Clay loam
Clay
-0.002
0
Sandy loam
-0.003
Silt loam
-0.003
-0.002
-0.028
Clay loam
Clay
-0.001
0
-0.012
- 0.001
-0.021
0.112
Santa Paula
o
o
o
I
o
I
I
0
I
4
6
0.003
0
Soil
Type &ox
0092
fzf2 Owz
009
3flf3 0092
0093
Fig. 5. Varianceof so asfunctionof ao2 for two soilswith Clinton
climate.
Clinton
respect to the parameter. Effects may also arise due to the
interaction of two parameters when they are correlated with
each other
Sandy loam
Silt loam
Clay loam
Clay
0.005
0.001
- 0.004
0.001
Sandy loam
Silt loam
Clay loam
Clay
-0.003
0.002
0.002
0.002
ForeachOfthecombinations
ofsoilandclimate
already
discussed,and for each of the major output variables,we have
computed
theterms(32g/3ooi3ooj)f•f•.
Table4 summarizes
the
results for evapotranspiration. These numbers are representative of results for all other outputs, which are therefore
not presented here. Each term in Table 4 represents the
change in the mean output resulting from a particular term in
0.004
0
- 0.033
0.017
-0.010
0
- 0.050
0.024
Santa Paula
-0.003
-0.001
0.019
0.019
-0.002
-0.008
0.005
0.042
Note {fox,ro2,c03}= {ne,In [k(1)],In (m)}.
(13) when a0 is unity. In assessing
the importanceof a given
term, it is helpful to recall that each of the outputs considered
is dimensionlessand is confined to the interval [0, 1].
The relative importance of the different terms in (13) varies
from case to case, but some generalizations can be made. Let
us consider first the terms associateddirectly with the variances of the three parameters. For the casesin which no sur-
0.05
0.02
SANDY
LOAM
• E[ETA
]/mpA
_ o E[RsAi/mpA
[] E
o
o
o
0.01
o
c
O.lO
CLAY LOAM
i
i
o
o
I
0.08
0.06
0.04
face runoff is generated in the homogeneoussituation, the
effectof spatial variability of any of theseparametersis minimal. This is consistent with our earlier observation
that storm
surfacerunoff is an important source of variability. For the
soils of finer texture, variability of m is most effective in
changingthe outputs,and variability of k(1) also has a significanteffect.The effectof variabilityof tie is negligible.
The importance of the terms that contain the crossderivativeswill dependon the magnitudeof the correlation between
parameters,but generally appears to be minimal. The table
entries representthe maximal effect, which occurs when two
parametersare perfectlycorrelated.In any case,the tie- k(1)
term is consistentlysmall.The tie- m term is generallysomewhat larger, yielding absolute changesof the outputs potentially (for IP•31- 1) as high as 3% for the situationsin which
surfacerunoff occurs. The ,largest cross-derivativeterm is the
m - k(1) term.
The approximation implied by (12) correspondsto straight
lines that are tangent to the curvesin Figure 3 at zero variance; this correspondencewas verified numerically in the
course of our work as a check on the consistencyof the calculations. The range of validity of the approximation (12) is not,
however,readily apparentfrom Figure 3 for all cases,sincethe
plots do not resolve some of the changesin slope for small
variance.
0.02
A
A
OOA
AA
o(•
I
I
2
i
i
4
I .
6
o- o
Fig. 6. Variancesof normalizedwater balancefluxesas functionsof
ao2 fortwosoilswithClintonclimate.
In summary, this approximate analysis suggeststhat soil
spatial variability has a significanteffect on the areal average
water balance only for casesin which surfacerunoff is generated. In those cases,the output is most strongly affected by
variability of m, next by that of k(1), and only slightly by
variability of tie' Cross-correlationof parametersis unimportant.
MILLY
AND EAGLESON' EFFECTS OF SPATIAL VARIABILITY
2141
by gk(o•),then the requirementfor equivalenceis
ApproximateAnalysisof the Variances
As in the case of the mean, we may use (11) to derive an
approximate result for the variance that offers insightsabsent
from the Monte Carlo approach. For sufficientlysmall values
of ao2 wefind
i=••=3&øi
&o•
Qu
ON ANNUAL AVERAGE WATER BALANCE
<yk>= gt,(o•e)
k = 1, 2, .'-, K
(17)
If o•esatisfies(17), then we may say that o• is an equivalent
parameter set with respectto the K outputs.
Several questions naturally arise concerning the existence
and uniquenessof o•, as well as its functional dependenceon
f(o•) and other parameters. These questionsare most directly
(14)
addressedby viewing(17) as a systemof equationsdefining%.
Direct, exact solutionof (17) for any usefulset of gndoesnot
or
appear feasible,given the complex nature of the equations. In
this section we consider only an approximate case, the case
wherea0 is small.
We approximate the left side of (17) using (12), and represent the right side of (17) using (11), in both casesreplacing g
by gn,
1• • c32g•
Q0 = g•,(P)
•=•
in which the partial derivativesare evaluated at o• equal to It.
We see, to a first approximation, that the variance of any
j=•
k = 1, 2,...,
waterbalanceoutputvariableis directlyproportionalto a0e.
K
(18)
=
We employed (15) to estimate the relative contributions of
eachof the threesoilparameters
to varianceof waterbalance For ao sufficiently
small,o• will be closeenoughto It that we
components.
Thefindings
parallel
theobservations
madein maydropall buttheleading
termin theinfinite
sum.This
discussion
of Table 4. In brief,varianceof In (m) contributes yieldsa newsystemof equationsfor o•e
the most to output variance, and variance of In [k(1)] contrib-
02g•,
Qij
__
(O,)e
__
It).•• g•
•1• • Ocøic%•
utes the bulk of the remainder.
The range of validity of (15) is variable; departures from
i=1
j=l
proportionality
in therelation
between
a0eandayeareindica-
k=
(19)
1,2,...,K
tive of failure of (15). Referring to Figure 6, we see that the
approximationis not bad for the balancecomponentsin the where derivatives are evaluated using the mean soil. This is a
caseof sandyloam when a02 is lessthan unity, mainly be- set of K linear equations for the three unknown components
cause
thereis no surface
runoffandhenceno variability
of of o)e. In the casewhereK is 3 andthe matrixOg/3o•
is
fluxes.
However,
in thecase
ofclayloam,therangeofvalidity nonsingular,
thereexists
a uniqueequivalent
soilparameter
is muchsmaller.For the varianceof so, the situationis similar.
vector
For sandyloam,(15) is accurateup to aoe near 2, as can be
me-- It -[Qij
(20)
seenin Figure 5. Lessapparent in Figure 5 is the inadequacy
of (15) for the clay loam' the slopeis quite small at the origin,
but increasesseveraltimes over near the origin.
If •/•
is singular or if K is 1 or 2• then there is an infinite set
of equivalentparametersets.If K is •reater than 3 and t•ere is
ON THE EXISTENCE OF AN EQUIVALENT HOMOGENEOUS
no redundancy in (l•), then there can be no equivalent paramSOIL TYPE
eter
set.
We have shown that the areal average water balance deThe value of K equal to 3 seemsto be especiallyrelevant
pendson spatial variability of soil parameters.In applications here. Of the three normalized water balance fluxes(evapotranof the water balance model, however, it is usually desirable to spiration, surface runoff, groundwater recharge),only two of
work with a singlehomogeneous
soiltype.In sucha situation, them yield independentconditionson •e since their sum is
it is often argued that the single one-dimensionalcolumn may
capture the essentialdynamics and that the soil type used is
an "effective"soil type, functionally dependentupon the distribution of actual soil types within the modeled area. In this
sectionwe proposea precisedefinition of an equivalenthomogeneoussoil and explore the conditionsfor its existencein the
case of low soil variance.
precipitation, which is independentof •. A third independent
outputis the meansoilmoisturesaturation.Equation(20)can
therefore be used, for a0 small, to constructthe equivalent
homogeneous soil with respect to the three water balance
fluxes and the mean soil moisture
saturation.
We note that the
equivalentsoil dependsnot only on the densityfunctionfor
the soil parameters,but also on the climate parameters.
The generalidea of an equivalenthomogeneoussoil is that
SUMMARY AND CONCLUSIONS
it should yield the same responseas the spatially variable
ensembleof soils.If we take the measureof responseto be the
Our study exploressomeeffectsof soil spatial variability on
areally averagedoutput y, then we require that the equivalent annual average water balance. We use • idealized dynamic
soil type havesoil parameterso.}
e satisfying
model, and our findings should be interpreted in light of the
simplifying assumptionsmade. The most important of these
(16)
are the following.
1. The spatially variable land area behavesas a battery of
Typically, we are concernedwith a set of K outputs yn (k - 1,
2, ..-, K). If the correspondingoutput functionsare denoted parallel homogeneoussoil columns without dynamic interac-
2142
MILLY
AND
EAGLESON'
EFFECTS
OFSPATIAL
VARIABILITY
ONANNUAL
AVERAGE
WATER
BALANCE
tion. Excessinfiltration capacityat one location cannot compensatefor a deficiency
elsewhere.
2. The spatialvariabilityof climateis negligible
withinthe
modeled area.
3. Theannual
average
waterbalance
at a pointin the area
is described
by the modelof Eagleson
[1978a,b, c, d, e,f, g],
the majorassumptions
of whicharesummarized
by Eagleson
[1978a].
4. Thevegetal
equilibrium
hypothesis
ofEagleson
[1978f]
applies.
5. The effectiveporosity,the logarithmof permeability,
and the logarithmof the poresizedistributionirldexof the
soil follow a joint normal distribution.
6. The watertableis deep,the surfaceretentioncapacityis
negligible,
andthepotential
transpiration
efficiency
isunity.
Keepingin mindtheselimitations
of thestudy,wehavethe
following conclusions.
M
vegetalcanopy density.
m pore sizedistributionindexof soil.
mpA meanannualprecipitation.
mtb meantimebetweenstorms.
mr, meanstormduration.
m, meandurationof wet season.
N numberof samplesin Monte Carlo computations.
ne effectiveporosityof soil.
Q
Covarianceof to.
RgA annualrecharge
to groundwater.
RsA annualsurface
runoff.
so timeaveragesoilmoisturesaturation.
•'a meanannualtemperature.
x
location
vector in A.
y a setof waterbalancevariables.
y any water balancevariable.
yk kth componentof y.
1. The arealmeanof the time averagesoil moistureso is
Z
depth to water table.
insensitive
io the varianceof the soil hydraulicparameters,
despiteits strongdependence
on themeansoilparameters.
2. The averagedivisionof precipitationamongsurface
runoff,groundwater
recharge,
andevhpotranspiration
maybe
significantly
affectedby the varianceof soil properties.
The
significance
of theeffectdepends
uponthemeansoilparameters. Soilsof high averagepermeabilityexhibita slightin-
•
parameterof gammadistribution
of stormdepth.
is mean of to.
Pij cross
correlation
of coiand
ao standarddeviationof In [k(1)].
a• standarddeviationof
aso standard
deviation
of so.
ay standard
deviation
of y.
creasein surfacerunoff due to soil variance,while thoseof low
fl
averagepermeability
exhibitdecreased
surface
runoffandincreased
drainageto groundwater.
As a generalrule,increased
varianceof soil typetendsto equalizethe magnitudes
of the
to soilparameter
vector{tie,In [k(1)],In (m)}.
to• ith samplevalueof to usedin Monte Carlo runs.
threemajor water balancecomponents.
3. A corollaryof 2 is that the sensitivity
of thewaterbalanceto the meansoil propertiesis markedlyreducedby the
presence
of spatialvariability.
4. Whenthe soilhydraulicproperties
are parameterized
in
set of all to values found in A.
co• ith componentof to.
( )
El- ]
areal averageon A.
expectedvalue at a point in A.
Acknowledgments.
Thiswork wassupported
by the NationalScience Foundationunder grantsATM-7812327, ATM-8114723,and
CEE-8307282.
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