Advances in Water Resources 27 (2004) 521ndash531

wwwelseviercomlocateadvwatres

Layer averaged Richardrsquos equation with lateral flow

Praveen Kumar

Environmental Hydrology and Hydraulic Engineering Department of Civil and Environmental Engineering

University of Illinois Urbana IL 61801 USA

Received 29 October 2003 received in revised form 19 November 2003 accepted 5 February 2004

Available online 8 April 2004

Abstract

In this paper a formulation for layer averaged sub-surface moisture transport that models the lateral flow is developed using the

Richardrsquos equation This formulation is consistent with the approach currently adopted in contemporary models used in landndash

atmosphere interaction studies Explicit expressions are derived for layer averaged lateral transport contribution from diffusion

gravity dispersion and convergence due to landndashsurface curvature It is argued that lateral contribution can be a significant

component of the total soil-moisture flux Ignoring these contributions can result in significant model error leading to inaccurate

prediction or unrealistic calibration of parameters that compensate for these errors

2004 Elsevier Ltd All rights reserved

Keywords Land-surface parameterization Richardrsquos equation Lateral transport

1 Introduction

Models of terrestrial soil-moisture transport are anintegral part of SVAT (soil-vegetation-atmosphere-

transfer) schemes [11123242737485160] for the

simulation of weather and climate Although some re-

cent models based on basincatchment formulation uti-

lizing the Topmodel framework [9] account for the

lateral sub-surface moisture transport [153852] most

characterize them as an one-dimensional process in the

vertical with little or no accounting of the lateral flowThis is due to the challenge of developing simplified

models that incorporate both vertical and lateral trans-

port while still being computationally manageable

Consequently heavy reliance has been developed with

one-dimensional models which have seen various level

of success in their ability to partition the incident radi-

ation into sensible latent and ground heat fluxes

However with increasing emphasis on the use of thesemodels for the assessment of the impact of climate on

terrestrial hydrologic and ecologic systems the required

accuracy for the partitioning of the water budget into its

various components (streamflow and lateral and verti-

cal sub-surface transport) is significantly higher Fur-

ther satellite observation derived soil-moisture is needed

E-mail address kumar1uiucedu (P Kumar)

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

doi101016jadvwatres200402007

for assimilation in land surface models to improve pre-

dictability This dictates the need to better incorporate

the lateral flow for improved model prediction of nearsurface moisture (and temperature) Ability of the

models to predict the moisture gradients well determines

the success in the assimilation of near-surface soil-

moisture It seems unlikely that this will be a successful

endeavor without the incorporation of lateral sub-sur-

face transport in the models In addition there is an

increasing trend in hydroclimatological and hydrome-

teorological studies to model the land-surface at smallerscales going from several 10 s of kms to a few kms This

is facilitated by the increase in the resolution of regional

climate and mesoscale atmospheric models supported by

availability of increasing computational power This

necessitates the need to incorporate processes that

manifest at the smaller scales lateral sub-surface mois-

ture flow being one of them

In this paper I develop a formulation using theRichardrsquos equation for layer averaged sub-surface

moisture transport that incorporates the lateral flow

This formulation is consistent with the approach cur-

rently adopted in contemporary models In addition I

provide an analysis using this formulation to illustrate

that the lateral transport is not negligible No specific

numerical implementation is proposed with the hope

that the formulation can be adapted in a variety of waysin existing models to improve their performance

522 P Kumar Advances in Water Resources 27 (2004) 521ndash531

2 Background

The present analysis takes advantage of the work of

Albertson and Montaldo [244] in using Reynolds

averaging rules to obtain soil-moisture content in adifferential volume We assume the validity of the

Richards equation for flow in the unsaturated zone and

use the BrooksndashCorey [12] approximation for soilndashwater

retention We write the Richardrsquos equation for the

evolution of the moisture content h in the form

ohot

frac14 o

oXiKiethwTHORN

ohoXi

L eth1THORN

where the summation over Xi 2 fX Y Zg is implied Z is

the vertical coordinate and XndashY represents the hori-

zontal plane Any sink such as evapotranpiration loss ischaracterized through the loss term L Using

h frac14 w thorn Z where w is the suction head the above may

be written as

ohot

frac14 o

oXiDiethhTHORN

ohoXi

thorn SiKiethhTHORN

L eth2THORN

where DiethhTHORN frac14 Kiowoh is the diffusivity and the local

slope Si 2 fSX SY SZg where SX frac14 oZoX SY frac14 oZ

oY SZ frac14oZoZ frac14 1 Let us write the soil-moisture h at any point as

hethx y z tTHORN frac14 hethx y z tTHORN thorn ~hethx y z tTHORN where the h repre-

sents a mean in the ensemble sense and the ~h representsperturbations from the mean with Efrac12~h frac14 0 The above

equation may then be written as

ohot

thorn o~hot

frac14 o

oXiDiethh

thorn ~hTHORN ohoXi

thorn o~hoXi

thornSiKiethhthorn ~hTHORN

L

eth3THORNUsing Taylor series expansion of D and K and neglecting

second and higher order terms we get

Diethh thorn ~hTHORN frac14 DiethhTHORN thornoDi

oh

h

~h eth4THORN

Kiethh thorn ~hTHORN frac14 KiethhTHORN thornoKi

oh

h

~h eth5THORN

Substituting these in Eq (3) and expanding we get

ohot

thorn o~hot

frac14 o

oXiDiethhTHORN

ohoXi

thornDiethhTHORN

o~hoXi

thornD0iethhTHORN~h

ohoXi

thornD0iethhTHORN~h

o~hoXi

thorn o

oXiSiKiethhTHORNh

thorn SiK 0iethhTHORN~h

iL

eth6THORN

where D0iethhTHORN oDi

oh jh and K 0iethhTHORN oKi

oh jh Taking expected

values and noting that

~ho~hoXi

frac14 1

2

o~h2

oXi

the above reduces to

ohot

frac14 o

oXiDiethhTHORN

ohoXi

thorn 1

2D0

iethhTHORNor2

h

oXithorn SiKiethhTHORN

L eth7THORN

where r2h Efrac12~h2

Noting that

or2h

oXifrac14 or2

h

ohohoXi

eth8THORN

we can write Eq (7) as

ohot

frac14 o

oXiDiethhTHORN

ohoXi

thorn 1

2D0

iethhTHORNor2

h

ohohoXi

thorn SiKiethhTHORNL

eth9THORN

We notice that on the RHS the first term is the diffusion

term the second represents dispersion due to the vari-ability and the third is the gravitational term Alterna-

tively we may also write the above as

ohot

frac14 o

oXiDiethhTHORN

ohoXi

thorn SiKiethhTHORN

thorn o

oXi

1

2D0

iethhTHORNor2

h

ohohoXi

L eth10THORN

In this form the first term on the RHS represents themoisture flux due to the mean profile while the second

corresponds to the flux due to the fluctuations

Land-surface parametrizations in climate models

typically use a layer averaged (in the vertical) equation

Consider a layer of thickness DZj between vertical

coordinates Zj and Zjthorn1 Following Wetzel and Boone

[55] and Boone and Wetzel [10] we multiply the above

equation by dZ and integrate over the layer thickness toget

Z Zjthorn1

Zj

ohot

dZ frac14Z Zjthorn1

Zj

o

oXiDiethhTHORN

ohoXi

dZ

thornZ Zjthorn1

Zj

SioKiethhTHORNoXi

dZ thornZ Zjthorn1

Zj

KiethhTHORNo2ZoX 2

idZ

thorn 1

2

Z Zjthorn1

Zj

o

oXiD0

iethhTHORNor2

h

ohohoXi

dZ

Z Zjthorn1

Zj

LdZ eth11THORN

Notice that we have used oXiethSiKiethhTHORNTHORN frac14 Si

oKiethhTHORNoXi

thornKiethhTHORN oSi

oXifrac14 Si

oKiethhTHORNoXi

thorn KiethhTHORN o2ZoX 2

ito arrive at the second and

third terms in the equation Separating the vertical (Z)

and implied summation over the lateral terms ethXi 2fX Y gTHORN and integrating we obtain

Table 1

Hydraulic decay constant obtained from published results (see also

[46])

c (m1) Sitesregions Refs

)235ndash915 Various sites [8]

151ndash517 FIFE Kansas [28]

326 Sleepers River Vermont [52]

70 Walnut Gulch Arizona [2634]

100 Creek Alaska [53]

80 Red Arkansas River [61]

18 North America [15]

20 Global [23]

20ndash60 Ovre Abiskojokk and

Ovre Landsjarv Sweden

[46]

Soil type

151 Alluvial land [28]

374 BenfieldndashFlorence complex

517 ClimendashSogn complex

222 DwightndashIrwin complex

211 Irwin silty clay loam

211 Irwin silty clay loam(eroded)

218 Ivan and Kennebec silt loams

218 Reading silt loam

491 Stony steep land

235 Tully silty clay loam

P Kumar Advances in Water Resources 27 (2004) 521ndash531 523

ohot

DZj frac14Z Zjthorn1

Zj

o

oZDZethhTHORN

ohoZ

dZ frac12Vertical diffusion

thornZ Zjthorn1

Zj

oKZethhTHORNoZ

dZ frac12Vertical gravitational

thorn 1

2

Z Zjthorn1

Zj

o

oZD0

ZethhTHORNor2

h

ohohoZ

dZ

frac12Vertical dispersion

thornZ Zjthorn1

Zj

o

oXiDiethhTHORN

ohoXi

dZ frac12Lateral diffusion

thornZ Zjthorn1

Zj

SioKiethhTHORNoXi

dZ frac12Lateral gravitational

thorn 1

2

Z Zjthorn1

Zj

o

oXiD0

iethhTHORNor2

h

ohohoXi

dZ

frac12Lateral dispersion

thornZ Zjthorn1

Zj

KiethhTHORNo2ZoX 2

idZ frac12Lateral convergence

Z Zjthorn1

Zj

LdZ

eth12THORN

where the average soil-moisture h within a layer is de-

fined as h frac14 1DZj

R Zjthorn1

Zjhdz Notice that as expected there

is no vertical convergence term since o2ZoZ2 frac14 0 Using one

dimensional approximations (as in climate models) is

tantamount to retaining the first two term in the RHS

(along with the sink term) and neglecting the othersThis results in an approximation of the form

DZjohot

frac14 DZethhTHORNohoZ

Zjthorn1

DZethhTHORNohoZ

Zj

thorn KZethhTHORNjZjthorn1 KZethhTHORNjZj L eth13THORN

where L is the total water loss from the layer The

above is usually written in the flux form [55]

DZjohot

frac14 Qjthorn1 Qj L eth14THORN

where Qj frac14 DZethhTHORN ohoZ jZj thorn KZethhTHORNjZj is the net flux through

the interface of the layer at Zj

No assessment is available in the literature regarding

the relative significance of the neglected terms (thatcorrespond to vertical dispersion as well as lateral dif-

fusive dispersive convergence and gravitational trans-

port) relative to the vertical transport and how that

changes with soil properties and topographic attri-

butes such as slope and curvature We explore this issue

next

3 Layer averaged model

To simplify other terms in Eq (12) we begin by

assuming that the saturated hydraulic conductivity KSi

in any direction Xi decreases with depth Z and is equal toK0i at some reference depth Z that is

KSi frac14 K0iecethZZTHORN eth15THORN

This formulation is consistent with the Topmodel for-

mulation [9] although other forms of the functional

dependence of hydraulic conductivity with depth havebeen studied [3] Table 1 provides a summary of some

published values of c which generally lie between 1 and

10 We further assume that the lateral saturated

hydraulic conductivity is larger than the vertical by a

factor f to account for anisotropy in the two directions

that is

K0X frac14 K0Y frac14 fK0Z eth16THORN

Table 2 provides a list of published values of the

anisotropic ratio f which are generally of the order of

10ndash100 Using the BrooksndashCorey [12] parameterization

of the form

wethhTHORN frac14 ws

hhs

13b

KiethhTHORN frac14 KSi

hhs

132bthorn3

eth17THORN

where b is an empirical constant and hs the saturated

soil-moisture content we can easily obtain the followingrelationships

Table 2

Anisotropic ratio obtained from published results

f Testmeasurement methods Aquifersoil types Aquifer thickness (b)

Core types

Test sitesreigions Refs

042ndash269 Laboratory test Mineral Cylinder core A frac14 26

cm2

Oyster Virginia [13]

26ndash69 Laboratory test Peat Cylinder core A frac14 23

cm2

Quebec Canada [50]

05ndash24 Laboratory test Limestone Cylinder core A frac14 57

cm2

Boliver Australia [58]

11ndash2363 Laboratory test Peat Cube core A frac14 56 cm2 Humberhead

Peatlands England

[56]

72ndash155 Pumping test Sand and gravel b frac14 824 m Saint Pardon de

Conques Gironde

France

[45]

33 Pumping test Borden Ontario

Canada

[47]

11ndash108 Tracer and permeameter Fluvial sand aquifer b frac14 02ndash3 m Twin Lake Ontario

Canada

[36]

2ndash350 Pumping test Alluvial aquifer Susquehanna River

New York USA

[59]

2ndash5 Tracer test Sand and gravel b frac14 30 m Cape Cod

Massachusetts USA

[39]

12 Flowmeter and permeameter Sand and gravel b frac14 30 m Cape Cod

Massachusetts USA

[32]

2 Pumping test Sand and gravel b frac14 30 m Cape Cod Massachu-

setts USA

[43]

9ndash63 Pumping test Alluvial aquifer Vekol Valley Arizona

USA

[41]

1ndash100 Pumping test Sand aquifer b frac14 15 m [49]

29ndash452 Pumping test Glacio fluvial sand b frac14 5 m Vejen Denmark [35]

384ndash918 Pumping test Phreatic aquifer b frac14 49 m Cosenza Italy [25]

11 Pumping test Alluvial aquifer b frac14 235 m Wood River Platte

River valley Nebraska

USA

[21]

155 Pumping test Alluvial aquifer b frac14 348 m Scottsbluff North

Platte River valley

Nebraska USA

[18]

597 Pumping test Alluvial aquifer b frac14 305 m Grand Island Platte

River valley Nebraska

USA

[16]

25ndash562 Pumping test Alluvial aquifer b frac14 135 m Shelton Platte River

valley Nebraska USA

[4]

91 and 20 Pumping test Alluvial aquifer b frac14 13 m MSEA Platte River

valley Nebraska USA

[42]

108 Pumping test Alluvial aquifer b frac14 146 m Palisade Frenchman

Creek Nebraska USA

[20]

69 Pumping test Alluvial aquifer b frac14 104 m Bloomington Republi-

can River Valley

Nebraska USA

[20]

33 Standpipe method Alluvial aquifer b frac14 32 cm Bloomington Republi-

can River Valley

Nebraska USA

[17]

228 Pumping test Alluvial aquifer b frac14 204 m McCook Republican

River Valley Nebraska

USA

[20]

49 Standpipe method Alluvial aquifer b frac14 255 cm McCook Republican

River Valley Nebraska

USA

[17]

2000 Land model calibration North America [15]

122 Land model calibration Ovre Abiskojokk and

Ovre Landsjarv

Sweden

[46]

Order of 100 or gt Large scale field studies [30]

1ndash100 Stream depletion analysis b frac14 305 m [19]

524 P Kumar Advances in Water Resources 27 (2004) 521ndash531

00 01 02 03 04 05

Mean

00

05

10

15

Coe

ffici

ento

fVar

iatio

n

01 03 05 07

Mean

00

02

04

06

08

10

Coe

ffici

ent o

f Var

iatio

n

Famiglietti et al(1999)

Hills amp Reynolds(1969)

Bells et al(1980)

Bells et al(1980)

Hawley et al(1983)

Chapentier amp Groffman(1992)

Loague(1992)

Western et al(1998)

Wilson et al(2003)

Fig 1 (Top) Plot of coefficient of variation (Cv) versus mean soil-

moisture ethhTHORN for the SGPrsquo97 experiment as published in Famiglietti et

al [29] The regression curve corresponds to Cv frac14 a=hb with a frac14 00362

and b frac14 127 and the coefficient of determination is 076 (Bottom) Plot

of coefficient of variation (Cv) versus mean soil-moisture from ob-

served data reported in the literature (see Table 3)

P Kumar Advances in Water Resources 27 (2004) 521ndash531 525

KiethhTHORN frac14 K0iecethZZTHORN h

hs

132bthorn3

eth18THORN

K 0i ethhTHORN frac14

eth2bthorn 3THORNK0i

hs

ecethZZTHORN hhs

132bthorn2

eth19THORN

DiethhTHORN frac14 bwsK0i

hs

ecethZZTHORN hhs

13bthorn2

eth20THORN

D0iethhTHORN frac14 bethbthorn 2THORNwsK0i

h2s

ecethZZTHORN hhs

13bthorn1

eth21THORN

We now study each of the remaining terms in Eq (12)

31 Vertical dispersion

Let us assume that V ethhTHORN or2

h

oh (see Eq (8)) can be

parametrized as a function of h To find the function

V ethhTHORN we first study the coefficient of variation Cv frac14 rhh as

a function of the mean moisture content h Intuitivelywe expect that Cv is a decreasing function of h to reflect

that moisture shows more relative variability when the

soil is drier Fig 1 shows the plot of Cv versus h for data

obtained from Southern Great Plains Experiment 1997

(SGP97) as published by Famiglietti et al [29] All the

data from Table 2 of Famiglietti et al [29] which reflects

a variety of soil properties and moisture conditions is

used without distinction of any kind (see also Fig 5 inFamiglietti et al [29]) As shown in Fig 1(Top) the

functional form

Cv frac14a

hb eth22THORN

fits the data quite well and the least squares estimates are

a frac14 00362 and b frac14 127 Table 3 provides a summary of

a and b obtained from published soil-moisture data

which are plotted in Fig 1(Bottom) Hence using the

definition of Cv

r2h frac14 a2h2eth1bTHORN eth23THORN

and consequently

V ethhTHORN or2

h

ohfrac14 2a2eth1 bTHORNh12b eth24THORN

Using this we may write the vertical dispersion term

(third term) in Eq (12) as

1

2

Z Zjthorn1

Zj

o

oZD0

ZethhTHORNor2

h

ohohoZ

dZ

frac14 1

2D0

ZethhTHORNV ethhTHORNohoZ

Zjthorn1

D0

ZethhTHORNV ethhTHORNohoZ

Zj

eth25THORN

Substituting for D0ZethhTHORN and V ethhTHORN from Eqs (21) and (24)

we obtain

1

2

Z Zjthorn1

Zj

o

oZD0

ZethhTHORNor2

h

ohohoZ

dZ

frac14 K0Z bethbthorn 2THORNa2eth1 bTHORNws

h3thornbs

ecethZZTHORNhb2bthorn2 ohoz

Zjthorn1

ecethZZTHORNhb2bthorn2 ohoz

Zj

eth26THORN

32 Lateral diffusion

Recognizing that

ohoXi

frac14 ohoZ

oZoXi

frac14 SiohoZ

eth27THORN

and using Eq (20) we can write the lateral diffusive term

in Eq (12) as

Table 3

Regression results for Cv versus mean soil-moisture for published data

a b No of

data

R2 Sampling

depths

Sitesregions Area No of

samples

Sampling dates Refs

0088 056 15 011 5ndash8 cm Chew Stroke

Bristol UK

24 m2ndash6 km2 60field June 21 1966 and

July 8 1966

Hills and Rey-

nolds [33] Fig 6

0119 028 60 035 0 cm April 1974ndashOctober

13 1976 (6 dates)

[7] Table 2a

0059 065 60 039 1ndash2 cm

0037 077 60 025 2ndash5 cm

0042 057 60 009 5ndash9 cm Phoenix

Arizona USA

16 ha

(1 field)

36

0058 028 60 002 9ndash15 cm

0042 066 300 051 All of above Jefferson

County

Kansas USA

16 ha

(29 fields)

19field

0113 029 59 035 0ndash1 cm [7] Table 2b

0076 049 59 048 0ndash2 cm Finney County

Kansas USA

16 ha

(24 fields)

9ndash35field

0032 083 59 056 0ndash5 cm

0030 078 59 039 0ndash9 cm

0041 053 59 015 0ndash15 cm

0041 065 295 062 All of above

0090 062 32 021 0ndash25 cm Chickasha

Oklahoma

USA

51ndash179 ha

(8 basins)

16ndash92basin May 1 10 12 and

30 1978 (4 dates)

[31] Table III

0099 032 32 007 0ndash15 cm

0085 055 64 014 All

0045 077 18 055 0ndash5 cm Manhattan

Kansas USA

4356 m2

(3 plots)

49plot June 28 1987ndash

August 4 1989

(6 dates)

[14] Table 3

0106 105 35 051 015 m Chickasha

Oklahoma

USA

01 km2

(34 sites)

8 depthssite January 1971ndashJune

1974 (84 dates)

[40] Table 3

0040 151 35 075 030 m

0024 176 35 057 045 m

0053 103 35 026 060 m

0094 046 35 005 075 m

0077 051 35 006 090 m

0058 060 35 006 105 m

0079 027 35 001 120 m

0011 241 140 057 015ndash060 m

0070 036 13 035 0ndash30 cm Tarrawarra

Australia

105 ha 500ndash2000 September 27 1995ndash

November 29 1996

(13 dates)

[54] Table 1

0003 393 18 076 0ndash30 cm Mahurangi

New Zealand

5ndash60 ha

(3 sites)

275ndash480site November 1998

May 1999

November 1999

[56] Table 2

526 P Kumar Advances in Water Resources 27 (2004) 521ndash531

Z Zjthorn1

Zj

o

oXiDiethhTHORN

ohoXi

dZ

frac14Z Zjthorn1

Zj

o

oXiSi DiethhTHORN

ohoZ

13 dZ

frac14 K0i bws

hbthorn3s

Z Zjthorn1

Zj

o

oXiSi hbthorn2 oh

oZecethZZTHORN

13 dZ

frac14 K0i Sibws

hbthorn3s ethbthorn 3THORN

o

oXi

thorn vi

Si

13Z Zjthorn1

Zj

ohbthorn3

oZecethZZTHORN dZ

eth28THORNwhere vi frac14 oSi

oXifrac14 o2Z

oX 2i

is the local curvature Recognizing

that the curvature vi is not a function of Z and inte-

grating by parts we obtain

Z Zjthorn1

Zj

o

oXiDiethhTHORN

ohoXi

dZ

frac14 K0i Sibws

hbthorn3s ethbthorn 3THORN

o

oXi

thorn vi

Si

13

ecethZZTHORNhbthorn3jZjthorn1

ecethZZTHORNhbthorn3jZj

thorn cZ Zjthorn1

Zj

hbthorn3ecethZZTHORN dZ

eth29THORN

Closed form solution for the integral on the RHS can be

obtained using the Mathematica package [57] as

P Kumar Advances in Water Resources 27 (2004) 521ndash531 527

I1 cZ Zjthorn1

Zj

hbthorn3ecethZZTHORN dZ

frac14 ch ecZ

Z4thornbhCeth4 thorn b cZTHORNethcZTHORNeth4thornbTHORNjZjthorn1

Zj

i eth30THORN

where C is the incomplete gamma function

33 Lateral gravitational

Similarly the lateral gravitational term in Eq (12)

can be obtained as

Z Zjthorn1

Zj

SioKiethhTHORNoXi

dZ frac14 Si

Z Zjthorn1

Zj

oKiethhTHORNoZ

oZoXi

dZ

frac14 S2i

Z Zjthorn1

Zj

oKiethhTHORN

frac14 S2i KiethhTHORNjZjthorn1

h KiethhTHORNjZj

i eth31THORN

Using the BrooksndashCorey representation (18) the above

simplifies to

Z Zjthorn1

Zj

SioKiethhTHORNoXi

dZ frac14 S2i K0i

h2bthorn3s

ecethZZTHORNh2bthorn3jZjthorn1

h

ecethZZTHORNh2bthorn3jZji eth32THORN

34 Lateral dispersion

Similarly the lateral dispersion term may be obtained

as

1

2

Z Zjthorn1

Zj

o

oXiD0

iethhTHORNor2

h

ohohoXi

dZ

frac14 1

2

Z Zjthorn1

Zj

o

oXiD0

iethhTHORNV ethhTHORNohoZ

oZoXi

dZ

frac14 1

2

Z Zjthorn1

Zj

o

oXiSi D0

iethhTHORNV ethhTHORNohoZ

13 dZ eth33THORN

Using Eqs (21) and (24) the above reduces to

frac14 K0i Sibethbthorn 2THORNwsa2eth1 bTHORN

hbthorn3s ethb 2b thorn 3THORN

o

oXi

thorn vi

Si

13

Z Zjthorn1

Zj

ohb2bthorn3

oZecethZZTHORN dZ eth34THORN

where we have used hb2bthorn2 ohoZ frac14 1

ethb2bthorn3THORNohb2bthorn3

oZ Inte-

grating we obtain

1

2

Z Zjthorn1

Zj

o

oXiD0

iethhTHORNor2

h

ohohoXi

dZ

frac14 K0i Sibethbthorn 2THORNwsa2eth1 bTHORN

hbthorn3s ethb 2b thorn 3THORN

o

oXi

thorn vi

Si

13

hb2bthorn3ecethZZTHORNjZjthorn1

hb2bthorn3ecethZZTHORNjZj

thorn cZ Zjthorn1

Zj

hb2bthorn3ecethZZTHORN dZ

eth35THORN

Closed form solution for the integral on the RHS can be

obtained using the Mathematica package as

I2 cZ Zjthorn1

Zj

hb2bthorn3ecethZZTHORN dZ

frac14 ch ecZ

Zb2bthorn4hCethb 2b thorn 4 cZTHORN

ethcZTHORNethb2bthorn4THORNjZjthorn1

Zj

i eth36THORN

35 Lateral convergence

This term can be computed as

Z Zjthorn1

Zj

KiethhTHORNo2ZoX 2

idZ frac14 vi

Z Zjthorn1

Zj

KiethhTHORNdZ

frac14 viK0i

h2bthorn3s

Z Zjthorn1

Zj

h2bthorn3ecethZZTHORN dZ

frac14 viK0i

h2bthorn3s

I3 eth37THORN

where

I3 Z Zjthorn1

Zj

h2bthorn3ecethZZTHORN dZ

frac14h ecZ

Z2bthorn4hCeth4 thorn 2b cZTHORNethcZTHORN2b4jZjthorn1

Zj

i eth38THORN

36 Layer averaged soil-moisture transport

Collecting all the relevant terms in Eq (12) from Eqs

(13) (26) (29) (32) (35) and (37) and algebraically

simplifying we can write the layer averaged form of Eq

(12) as

528 P Kumar Advances in Water Resources 27 (2004) 521ndash531

DZjohot

frac14 ecethZjthorn1ZTHORN (

A1 hbthorn2 ohoZ

Zjthorn1

ecDZjhbthorn2 oh

oZ

Zj

frac12Vertical diffusion

thorn A2 h2bthorn3jZjthorn1

h ecDZjh2bthorn3jZj

ifrac12Vertical gravitational

thorn A3 hb2bthorn2 ohoZ

Zjthorn1

ecDZjhb2bthorn2 oh

oZ

Zj

frac12Vertical dispersion

thorn A4

o

oXi

thorn vi

Si

13hbthorn3jZjthorn1

h ecDZjhbthorn3jZj thorn ecethZjthorn1ZTHORNI1

ifrac12Lateral diffusion

thorn A5 h2bthorn3jZjthorn1

h ecDZjh2bthorn3jZj

ifrac12Lateral gravitational

thorn A6

o

oXi

thorn vi

Si

13hb2bthorn3jZjthorn1

h ecDZjhb2bthorn3jZj thorn ecethZjthorn1ZTHORNI2

ifrac12Lateral dispersion

thorn A7ecethZjthorn1ZTHORNI3

)frac12Lateral convergence

L

eth39THORNwhere summation over Xi 2 fX Y g is implied and the

coefficients are given as

A1 frac14K0Z bws

hbthorn3s

ethDimension L2=T THORN

A2 frac14K0Z

h2bthorn3s

ethDimension L=T THORN

A3 frac14K0Z bethbthorn 2THORNwsa

2eth1 bTHORNhbthorn3

s

ethDimension L2=T THORN

A4 frac14K0Z fSibws

ethbthorn 3THORNhbthorn3s

ethDimension L2=T THORN

A5 frac14K0ZfS

2i

h2bthorn3s

ethDimension L=T THORN

A6 frac14K0Z fSibethbthorn 2THORNwsa

2eth1 bTHORNhbthorn3

s ethb 2b thorn 3THORNethDimension L2=T THORN

A7 frac14viK0Zf

h2bthorn3s

ethDimension L=T THORN

Notice that A1 A3 A4 and A6 have the dimensions of a

dispersion coefficient whereas A2 A5 and A7 have the

dimensions of velocity One may also consider Ai as

weights for each term in the evolutionary equation for

the mean moisture profile h

37 Relative contribution of lateral flow

To compare the lateral terms with the vertical we

examine ratios of the form

Aij frac14Ai

Aj eth40THORN

In particular the following dimensionally consistent

ratios shed light on the relative contributions

Vertical dispersion=Vertical diffusion

frac14 A31 frac14A3

A1

frac14 ethbthorn 2THORNa2eth1 bTHORN eth41THORN

Lateral diffusion=Vertical diffusion

frac14 A41 frac14A4

A1

frac14 fSibthorn 3

eth42THORN

Lateral gravitational=Vertical gravitational

frac14 A52 frac14A5

A2

frac14 fS2i eth43THORN

Lateral dispersion=Vertical diffusion

frac14 A61 frac14A6

A1

frac14 fSiethbthorn 2THORNa2eth1 bTHORNb 2b thorn 3

eth44THORN

Lateral dispersion=Vertical dispersion

frac14 A63 frac14A6

A3

frac14 fSib 2b thorn 3

eth45THORN

Lateral convergence=Vertical gravitational

frac14 A72 frac14A7

A2

frac14 vif eth46THORN

The BrooksndashCorey parameter b typically ranges in value

from about 4 for sand to about 11 for clay [22] How-

ever a2 is of the order of 104 (see Table 3) Conse-quently A31 is small reflecting that the vertical dispersion

term is significantly smaller than the vertical diffusion

The contribution of lateral diffusion relative to that of

vertical depends on the product of the anisotropic con-

stant f and slope Si Given that both these parameters

take on a range of values we study it by plotting the

product fSi as a function of f and Si Fig 2 shows the

plot of log10ethfSiTHORN for 0016 Si 6 1 and 16 f6 2000covering a wide range of realistic values (see Table 2)

Use of f as high as 2000 has been reported in land-

surface modeling studies [15] Recognizing that contour

values of )1 )03 and 1 for log10ethfSiTHORN corresponds to a

ratio of 10 50 and 100 respectively for A41 (ex-

cept for a factor of 1=ethbthorn 3THORN) we see that the lateral

diffusion can be quite significant as compared to the

528 P Kumar Advances in Water Resources 27 (2004) 521ndash531

DZjohot

frac14 ecethZjthorn1ZTHORN (

A1 hbthorn2 ohoZ

Zjthorn1

ecDZjhbthorn2 oh

oZ

Zj

frac12Vertical diffusion

thorn A2 h2bthorn3jZjthorn1

h ecDZjh2bthorn3jZj

ifrac12Vertical gravitational

thorn A3 hb2bthorn2 ohoZ

Zjthorn1

ecDZjhb2bthorn2 oh

oZ

Zj

frac12Vertical dispersion

thorn A4

o

oXi

thorn vi

Si

13hbthorn3jZjthorn1

h ecDZjhbthorn3jZj thorn ecethZjthorn1ZTHORNI1

ifrac12Lateral diffusion

thorn A5 h2bthorn3jZjthorn1

h ecDZjh2bthorn3jZj

ifrac12Lateral gravitational

thorn A6

o

oXi

thorn vi

Si

13hb2bthorn3jZjthorn1

h ecDZjhb2bthorn3jZj thorn ecethZjthorn1ZTHORNI2

ifrac12Lateral dispersion

thorn A7ecethZjthorn1ZTHORNI3

)frac12Lateral convergence

L

eth39THORNwhere summation over Xi 2 fX Y g is implied and the

coefficients are given as

A1 frac14K0Z bws

hbthorn3s

ethDimension L2=T THORN

A2 frac14K0Z

h2bthorn3s

ethDimension L=T THORN

A3 frac14K0Z bethbthorn 2THORNwsa

2eth1 bTHORNhbthorn3

s

ethDimension L2=T THORN

A4 frac14K0Z fSibws

ethbthorn 3THORNhbthorn3s

ethDimension L2=T THORN

A5 frac14K0ZfS

2i

h2bthorn3s

ethDimension L=T THORN

A6 frac14K0Z fSibethbthorn 2THORNwsa

2eth1 bTHORNhbthorn3

s ethb 2b thorn 3THORNethDimension L2=T THORN

A7 frac14viK0Zf

h2bthorn3s

ethDimension L=T THORN

Notice that A1 A3 A4 and A6 have the dimensions of a

dispersion coefficient whereas A2 A5 and A7 have the

dimensions of velocity One may also consider Ai as

weights for each term in the evolutionary equation for

the mean moisture profile h

37 Relative contribution of lateral flow

To compare the lateral terms with the vertical we

examine ratios of the form

Aij frac14Ai

Aj eth40THORN

In particular the following dimensionally consistent

ratios shed light on the relative contributions

Vertical dispersion=Vertical diffusion

frac14 A31 frac14A3

A1

frac14 ethbthorn 2THORNa2eth1 bTHORN eth41THORN

Lateral diffusion=Vertical diffusion

frac14 A41 frac14A4

A1

frac14 fSibthorn 3

eth42THORN

Lateral gravitational=Vertical gravitational

frac14 A52 frac14A5

A2

frac14 fS2i eth43THORN

Lateral dispersion=Vertical diffusion

frac14 A61 frac14A6

A1

frac14 fSiethbthorn 2THORNa2eth1 bTHORNb 2b thorn 3

eth44THORN

Lateral dispersion=Vertical dispersion

frac14 A63 frac14A6

A3

frac14 fSib 2b thorn 3

eth45THORN

Lateral convergence=Vertical gravitational

frac14 A72 frac14A7

A2

frac14 vif eth46THORN

The BrooksndashCorey parameter b typically ranges in value

from about 4 for sand to about 11 for clay [22] How-

ever a2 is of the order of 104 (see Table 3) Conse-quently A31 is small reflecting that the vertical dispersion

term is significantly smaller than the vertical diffusion

The contribution of lateral diffusion relative to that of

vertical depends on the product of the anisotropic con-

stant f and slope Si Given that both these parameters

take on a range of values we study it by plotting the

product fSi as a function of f and Si Fig 2 shows the

plot of log10ethfSiTHORN for 0016 Si 6 1 and 16 f6 2000covering a wide range of realistic values (see Table 2)

Use of f as high as 2000 has been reported in land-

surface modeling studies [15] Recognizing that contour

values of )1 )03 and 1 for log10ethfSiTHORN corresponds to a

ratio of 10 50 and 100 respectively for A41 (ex-

cept for a factor of 1=ethbthorn 3THORN) we see that the lateral

diffusion can be quite significant as compared to the

100

101

102

103

10ndash2

10ndash1

Anisotropic Constant

Slo

pe

log10

(Anisotropic Constant Slope)

296562634623035

1972516415

13104

097937

064833

031729

ndash0013751

ndash034479

ndash067583

ndash10069

ndash13379ndash1669

Fig 2 Plot of log10ethfSiTHORN for different values of the anisotropic constant f and slope Si

P Kumar Advances in Water Resources 27 (2004) 521ndash531 529

vertical for a realistic range of combinations of slope

and anisotropic constant The term A52 varies as fS2i

Fig 3 shows the plot of log10ethfS2i THORN again for the same

range 0016 Si 6 1 and 16 f6 2000 Following an

argument similar to that for A41 we see that the lateralgravitational term can also be quite significant as com-

pared to the vertical for a wide and realistic range of Siand f Since A61 is dominated by a2 following the earlier

100

101

10ndash2

10ndash1

Anisotrop

Slo

pe

log10(anisotropic

010192

080

ndash12654

ndash17212

ndash21769

ndash26327

ndash 30885ndash

ndash

35442

Fig 3 Plot of log10ethfS2i THORN for different values o

argument for A31 we may say that this term is small

However unlike A31 different gradients are associated

with the terms A1 and A6 namely ooZ and eth o

oXithorn vi

SiTHORN

respectively and therefore the conclusion relates only to

the coefficients and not to the term for lateral dispersion(this issue also relates to A41) Similarly variation of A63

as a function of fSi showsthat the lateral dispersion can

be comparable to the vertical dispersion However the

102

103

ic Constant

constant slope2)

2836523808

1925

14692

10135

055769

035385

962 ndash

f the anisotropic constant f and slope Si

530 P Kumar Advances in Water Resources 27 (2004) 521ndash531

caveat regarding different gradients mentioned above for

A61 holds again The ratio A71 shows that the relative

contribution of flow convergence is determined by the

curvature and the rate of decay of the hydraulic con-

ductivity with depth

4 Summary

In this largely theoretical paper I use a small pertur-

bation approach along with Reynolds averaging using

the Richardrsquos equation to develop a formulation of layer

averaged soil-moisture transport This formulation ac-counts for lateral flow as well as dispersion due to var-

iability The analysis shows that the lateral flow can be

quite significant for certain ranges of slope and soil-

properties The dispersion terms are small but I con-

jecture that they can account for significant flux when

integrated over large areas particularly in regions with

heterogeneous soil properties We also see that the

curvature of the land-surface contributes to the lateralmoisture flux Ignoring these contributions can result in

significant model error leading to inaccurate prediction

or unrealistic calibration of parameters that compensate

for these errors It is quite likely that the lateral flux may

not have significant contribution for all physiographic

regions depending upon the model scale however the

formulation presented here can be used when they are

importantNumerical schemes can be developed for incorporat-

ing the formulation in existing land-surface schemes with

appropriate boundary conditions so that they are con-

sistent with other aspects of the model It is envisioned

that Eq (39) can be numerically implemented using a

time lagging scheme where the lateral moisture fluxes and

gradients from the previous time step may be used for

predictions at the current time step thereby eliminatingthe need for significant computational complexity arising

from the introduction of the new terms

Acknowledgements

This research has been supported by NASA Grant

NAG5-8555 and NSF Grant EAR02-08009 The authorwould like to thank Francina Dominguez and Hyun Il

Choi for carefully checking the derivation of the equa-

tions and generating the summary reported in the Ta-

bles

References

[1] Abramopoulos F Rosenzweig C Choudhary B Improved

ground hydrology calculations for global climate models (GCMs)

soil water movement and evapotranspiration J Climate

19881921ndash41

[2] Albertson JD Montaldo N Temporal dynamics of soil moisture

variability 1 Theoretical basis Water Resour Res

200339(10)1274 doi1010292002WR001616

[3] Ambroise B Beven K Freer J Toward a generalization of the

TOPMODEL concepts topographic indices of hydrological

similarity Water Resour Res 199632(7)2135ndash45

[4] Ayers JF Chen XH Gosselin D Behaviour of nitratenitrogen

movement around a pumping high-capacity well a felid example

Ground Water 199836(2)325ndash37

[5] Beckwith CW Baird AJ Heathwaite AL Anisotropy and depth-

related heterogeneity of hydraulic conductivity in a bog peat I

Laboratory measurements Hydrol Processes 20031789ndash101

[6] Beckwith CW Baird AJ Heathwaite AL Anisotropy and depth-

related heterogeneity of hydraulic conductivity in a bog peat II

Modelling the effects on groundwater flow Hydrol Processes

200317103ndash13

[7] Bell KR Blanchard BJ Schmugge TJ Witczak MW Analysis of

surface moisture variations within large field sites Water Resour

Res 198016(4)796ndash810

[8] Beven KJ On subsurface stromflow an analysis of response

times Hydrol Sci J 198227505ndash21

[9] Beven KJ Kirkby MJ A physically based variable contributing

area model of basin hydrology Hydrol Sci Bull 197924(1)43ndash69

[10] Boone A Wetzel PJ Issues related to low resolution modeling of

soil moisture experience with the PLACE model Global Planet

Changes 199613161ndash81

[11] Bonan GB A land surface model (LSM version 10) for

ecological hydrological and atmospheric studies technical

description and userrsquos guide NCAR Technical Note NCAR

TN_417+STR National Center for Atmospheric Research

Boulder Colorado 1996 Available from lthttpwwwcgducar

educmslsmindexhtmlgt

[12] Brooks RH Corey AT hydraulic properties in porous media

Fort Collins CO Colorado State University 1964 27 pp

[13] Burger RL Berlitz K Measurement of anisotropic hydraulic

conductivity in unconsolidated sands a case study from a

shoreface deposit Oyster Virginia Water Resour Res

199733(6)1515ndash22

[14] Charpentier MA Groffman PM Soil moisture variability within

remote sensing pixels J Geophys Res 19929718987ndash95

[15] Chen J Kumar P Topographic influence on the seasonal and

inter-annual variation of water and energy balance of basins in

North America J Climate 2001141989ndash2014

[16] Chen XH Assessment of hydraulic properties in an unconfined

alluvial aquifer near Grand Island Nebraska J Am Water Resour

Assoc 199824(3)603ndash16

[17] Chen XH Measurement of streambed hydraulic conductivity and

its anisotropy Environ Geol 200039(12)1317ndash24

[18] Chen XH Ayers J Utilization of the Hantush solution for the

simultaneous determination of aquifer parameters Ground Water

199735(5)751ndash6

[19] Chen XH Yin YF Evaluation of stream depletion for vertical

anisotropic aquifer J Environ Syst 199927(1)55ndash70

[20] Chen XH Goeke J Summerside S Hydraulic properties and

uncertainty analysis for an unconfined alluvial aquifer Ground

Water 199937(6)845ndash54

[21] Chen XH Goeke J Ayers J Summerside S Observation well

network design for pumping tests in unconfined aquifers J Am

Water Resour Assoc 200339(1)17ndash32

[22] Clapp RB Hornberger GM Empirical equations for some soil

hydraulic properties Water Resour Res 197814601ndash4

[23] Dai Y Zeng X Dickinson RE Baker I Bonan GB Bosilovich

MG et al The common land model (CLM) Bull Am Meteorol

Soc 200384(8)1013ndash23

530 P Kumar Advances in Water Resources 27 (2004) 521ndash531

caveat regarding different gradients mentioned above for

A61 holds again The ratio A71 shows that the relative

contribution of flow convergence is determined by the

curvature and the rate of decay of the hydraulic con-

ductivity with depth

4 Summary

In this largely theoretical paper I use a small pertur-

bation approach along with Reynolds averaging using

the Richardrsquos equation to develop a formulation of layer

averaged soil-moisture transport This formulation ac-counts for lateral flow as well as dispersion due to var-

iability The analysis shows that the lateral flow can be

quite significant for certain ranges of slope and soil-

properties The dispersion terms are small but I con-

jecture that they can account for significant flux when

integrated over large areas particularly in regions with

heterogeneous soil properties We also see that the

curvature of the land-surface contributes to the lateralmoisture flux Ignoring these contributions can result in

significant model error leading to inaccurate prediction

or unrealistic calibration of parameters that compensate

for these errors It is quite likely that the lateral flux may

not have significant contribution for all physiographic

regions depending upon the model scale however the

formulation presented here can be used when they are

importantNumerical schemes can be developed for incorporat-

ing the formulation in existing land-surface schemes with

appropriate boundary conditions so that they are con-

sistent with other aspects of the model It is envisioned

that Eq (39) can be numerically implemented using a

time lagging scheme where the lateral moisture fluxes and

gradients from the previous time step may be used for

predictions at the current time step thereby eliminatingthe need for significant computational complexity arising

from the introduction of the new terms

Acknowledgements

This research has been supported by NASA Grant

NAG5-8555 and NSF Grant EAR02-08009 The authorwould like to thank Francina Dominguez and Hyun Il

Choi for carefully checking the derivation of the equa-

tions and generating the summary reported in the Ta-

bles

References

[1] Abramopoulos F Rosenzweig C Choudhary B Improved

ground hydrology calculations for global climate models (GCMs)

soil water movement and evapotranspiration J Climate

19881921ndash41

[2] Albertson JD Montaldo N Temporal dynamics of soil moisture

variability 1 Theoretical basis Water Resour Res

200339(10)1274 doi1010292002WR001616

[3] Ambroise B Beven K Freer J Toward a generalization of the

TOPMODEL concepts topographic indices of hydrological

similarity Water Resour Res 199632(7)2135ndash45

[4] Ayers JF Chen XH Gosselin D Behaviour of nitratenitrogen

movement around a pumping high-capacity well a felid example

Ground Water 199836(2)325ndash37

[5] Beckwith CW Baird AJ Heathwaite AL Anisotropy and depth-

related heterogeneity of hydraulic conductivity in a bog peat I

Laboratory measurements Hydrol Processes 20031789ndash101

[6] Beckwith CW Baird AJ Heathwaite AL Anisotropy and depth-

related heterogeneity of hydraulic conductivity in a bog peat II

Modelling the effects on groundwater flow Hydrol Processes

200317103ndash13

[7] Bell KR Blanchard BJ Schmugge TJ Witczak MW Analysis of

surface moisture variations within large field sites Water Resour

Res 198016(4)796ndash810

[8] Beven KJ On subsurface stromflow an analysis of response

times Hydrol Sci J 198227505ndash21

[9] Beven KJ Kirkby MJ A physically based variable contributing

area model of basin hydrology Hydrol Sci Bull 197924(1)43ndash69

[10] Boone A Wetzel PJ Issues related to low resolution modeling of

soil moisture experience with the PLACE model Global Planet

Changes 199613161ndash81

[11] Bonan GB A land surface model (LSM version 10) for

ecological hydrological and atmospheric studies technical

description and userrsquos guide NCAR Technical Note NCAR

TN_417+STR National Center for Atmospheric Research

Boulder Colorado 1996 Available from lthttpwwwcgducar

educmslsmindexhtmlgt

[12] Brooks RH Corey AT hydraulic properties in porous media

Fort Collins CO Colorado State University 1964 27 pp

[13] Burger RL Berlitz K Measurement of anisotropic hydraulic

conductivity in unconsolidated sands a case study from a

shoreface deposit Oyster Virginia Water Resour Res

199733(6)1515ndash22

[14] Charpentier MA Groffman PM Soil moisture variability within

remote sensing pixels J Geophys Res 19929718987ndash95

[15] Chen J Kumar P Topographic influence on the seasonal and

inter-annual variation of water and energy balance of basins in

North America J Climate 2001141989ndash2014

[16] Chen XH Assessment of hydraulic properties in an unconfined

alluvial aquifer near Grand Island Nebraska J Am Water Resour

Assoc 199824(3)603ndash16

[17] Chen XH Measurement of streambed hydraulic conductivity and

its anisotropy Environ Geol 200039(12)1317ndash24

[18] Chen XH Ayers J Utilization of the Hantush solution for the

simultaneous determination of aquifer parameters Ground Water

199735(5)751ndash6

[19] Chen XH Yin YF Evaluation of stream depletion for vertical

anisotropic aquifer J Environ Syst 199927(1)55ndash70

[20] Chen XH Goeke J Summerside S Hydraulic properties and

uncertainty analysis for an unconfined alluvial aquifer Ground

Water 199937(6)845ndash54

[21] Chen XH Goeke J Ayers J Summerside S Observation well

network design for pumping tests in unconfined aquifers J Am

Water Resour Assoc 200339(1)17ndash32

[22] Clapp RB Hornberger GM Empirical equations for some soil

hydraulic properties Water Resour Res 197814601ndash4

[23] Dai Y Zeng X Dickinson RE Baker I Bonan GB Bosilovich

MG et al The common land model (CLM) Bull Am Meteorol

Soc 200384(8)1013ndash23

P Kumar Advances in Water Resources 27 (2004) 521ndash531 531

[24] Dickinson RE Henderson-Sellers A Kennedy PJ Wilson MF

Biospherendashatmosphere transfer scheme (BATS) for the NCAR

community climate model Technical Note TN-275+STR Na-

tional Center for Atmospheric Research Boulder Colorado

1986 69 pp

[25] Fallico C Mazzuca R Troisi S Determination of confined

phreatic aquifer anisotropy Ground Water 200240(5)475ndash80

[26] Famiglietti JS Aggregation and scaling of spatially-variable

hydrological processes local catchment-scale and macroscale

models of water and energy balance Dissertation Department of

Civil Eng and Oper Res Princeton University Princeton New

Jersey 1992

[27] Famiglietti JS Wood EF Evapotranspiration and runoff from

large land areas land surface hydrology for atmospheric general

circulation models Surv Geophys 199112179ndash204

[28] Famiglietti JS Wood EF Sivapalan M Thongs DJ A catchment

scale water balance model for FIFE J Geophys Res

19929718997ndash9007

[29] Famiglietti JS Devereaux JA Laymon C Tsegaye T Houser PR

Jackson TJ et al Ground-based investigation of spatial-temporal

soil moisture variability within remote sensing footprints during

SGP97 Water Resour Res 199935(6)1839ndash51

[30] Freeze RA Cherry JA Groundwater Prentice-Hall 1979

[31] Hawley ME Jackson TJ McCuen RH Surface soil moisture

variation on small agricultural watersheds J Hydrol 198362179ndash

200

[32] Hess KM Wolf SH Celia MA Large-scale natural gradient

tracer test in sand and gravel Cape Cod Massachusetts 3

Hydraulic conductivity variability and calculated macrodispersiv-

ities Water Resour Res 199228(8)2011ndash27

[33] Hills TC Reynolds SG Illustrations of soil moisture variability in

selected areas and plots of different sizes J Hydrol 1969827ndash

47

[34] Houser PR Shuttleworth WJ Famiglietti JS Gupta HV Syed

KH Goodrich DC Integration of soil moisture remote sensing

and hydrologic modeling using data assimilation Water Resour

Res 199834(12)3405ndash20

[35] Hvilshoj S Jensen KH Barlebo HC Madsen B Analysis of

pumping tests of partially penetrating wells in an unconfined

aquifer using inverse numerical optimization Hydrol J

19997(4)365ndash79

[36] Killey RWD Moltyaner GL Twin Lake tracer tests setting

methodology and hydraulic conductivity distribution Water

Resour Res 198824(10)1585ndash612

[37] Koster RD Suarez MJ Modeling the land-surface boundary in

climate models as a composite of independent vegetation stands J

Geophys Res 1992972697ndash715

[38] Koster RD Suarez MJ Ducharne A Stieglitz M Kumar P A

catchment based approach for modeling land-surface processes in

a GCM part 1ndashndashmodel structure J Geophys Res

2000105(D20)24809ndash22

[39] LeBlanc DR Garabedian SP Quadri RD Morin RH Teasdale

WE Paillet FL Hydrogeologic controls on solute transport in a

plume of sewage-contaminated ground water In Ragone SP

editor US Geological Survey Program on Toxic Waste and

Ground Water Contamination Proceedings of the Second Tech-

nical Meeting Cape Cod Massachusetts US Geol Surv Open File

Rep 86-481 1988 p B7ndashB12

[40] Loague K Soil water content at R-5 Part 1 Spatial and temporal

variability J Hydrol 1992139233ndash51

[41] Marie JR Hollett KJ Determination of hydraulic characteristics

and yield of aquifers underlying Vekol Valley Arizona using

several classical and current methods US Geol Surv Water-

Supply Paper 2453 Menlo Park California 1996

[42] McGuire VL Kilpatrick JM Hydrogeology in the vicinity of the

Nebraska Management Systems Evaluation Area (MSEA) site

central Nebraska US Geol Surv Water-Resour Invest Rep 1998

p 97-4266

[43] Moench AF LeBlanc DR Garabedian SP Preliminary type-

curve analysis of an aquifer test in an unconfined sand and gravel

aquifer Cape Cod Massachusetts US Geol Surv Water-Resour

Invest Rep 1995 p 94-4015

[44] Montaldo N Albertson JD Temporal dynamics of soil moisture

variability at the landscape scale 2 Implications for land surface

models Water Resour Res 200339(10)1275 doi101029

2002WR001618

[45] Neuman SP Analysis of pumping test data from anisotropic

unconfined aquifer considering delayed gravity response Water

Resour Res 197511(2)329ndash42

[46] Niu G-Y Yang Z-L The versatile integrator of surface atmo-

spheric processes (VISA) Part 2 Evaluation of three topography-

based runoff schemes Global Planet Changes 200338191ndash208

[47] Nwankwor GI Cherry JA Giilham RW A comparative study of

specific yield determinations for a shallow sand aquifer Ground

Water 198422(6)764ndash72

[48] Pitman AJ Yang Z-L Gogley JG Henderson-Sellers A Descrip-

tion of bare essentials of surface transfer for the bureau of

meteorological research centre AGCM BMRC Australia

BMRC Research Report No 32 1991

[49] Schafer DC Determining vertical anisotropy ratio using a

graphical iterative procedure based on the Hantush equation

Ground Water 199836(2)293ndash304

[50] Schlotzhauer SM Price JS Soil water flow dynamics in a

managed cutover peat field Quebec field and laboratory inves-

tigations Water Resour Res 199935(12)3675ndash83

[51] Sellers PJ Mintz Y Sud YC Dalcher A A simple biosphere

model (SiB) for use within the general circulation models J Atmos

Sci 198643505ndash31

[52] Stigelitz M Rind D Famiglietti JS Rosenzweig C An efficient

approach to modeling the topographic control of surface hydrol-

ogy for regional global climate modeling J Climate 199710118ndash

37

[53] Stieglitz M Hobbie J Giblin A Kling G Hydrologic modeling of

an arctic tundra watershed toward Pan-Arctic predictions J

Geophys Res 1999104(D22)27507ndash18

[54] Western AW Grayson RB The Tarrawarra data set soil

moisture patterns soil characteristics and hydrological flux

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[55] Wetzel PJ Boone A A parameterization for land-atmosphere-

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[56] Wilson DJ Western AW Grayson RB Berg AA Lear MS

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[57] Wolfram S The mathematica book 5th ed Wolfram Media Inc

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[58] Wright M Dillon P Pavelic P Peter P Nefiodovas A Measure-

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