Initial Soil Water Content as Input to Field-ScaleInfiltration and Surface Runoff Models

Renato Morbidelli & Corrado Corradini &Carla Saltalippi & Luca Brocca

Received: 9 March 2011 /Accepted: 13 January 2012 /Published online: 1 March 2012# Springer Science+Business Media B.V. 2012

Abstract Evidence is given of the role of initial soil moisture content, θi, in determining thesurface runoff hydrograph at field scale, that is a crucial element when distributed models forthe estimate of basin response to rainfall have to be formulated. This analysis relies uponsimulations performed by a model that, because of the necessity of representing theinfiltration of surface water running downslope into pervious saturated or unsaturated areas,uses a coupled solution of a semi-analytical/conceptual approach for local infiltration and anonlinear kinematic wave equation for overland flow. The model was applied to actualspatial distributions of θi, earlier observed over different fields, as well as to a uniform valueof θi assumed equal to the average value or to the value observed in a site characterized bytemporal stability. Our results indicate that the surface runoff hydrograph at a slope outlet ischaracterized by a low sensitivity to the horizontal heterogeneity of θi, at least in the cases ofpractical hydrological interest. In fact, in these cases the correct hydrograph can be simulatedwith considerable accuracy replacing the actual distribution of θi by the correspondingaverage value. Moreover, the surface hydrograph is sufficiently well reproduced eventhough a single value of θi, observed at a site anyhow selected in the field of interest, isused. In particular, this extreme simplification leads to errors in magnitude on peak runoffand total volume of surface water with values typically within 10% and 15%, respectively.

Keywords Hydrology . Surface runoff modeling . Soil water content

1 Introduction

The estimate of rainfall infiltration at field scale is a fundamental topic in hydrologicalapplications based on the use of rainfall-runoff transformations. In fact, the latter is frequently

Water Resour Manage (2012) 26:1793–1807DOI 10.1007/s11269-012-9986-3

R. Morbidelli (*) : C. Corradini : C. SaltalippiDepartment of Civil and Environmental Engineering, University of Perugia,via G. Duranti 93, 06125 Perugia, Italye-mail: renato@unipg.it

L. BroccaResearch Institute for Geo-Hydrological Protection, National Research Council,Via Madonna Alta 126, 06128 Perugia, Italy

represented through abstractions of the basin geometry into networks of elements with planesand channels (Hager 1984; Woolhiser et al. 1990; Singh 1996; Melone et al. 1998). Further, aconceptual approximation of the hydrological response from this basin structure requires toidentify: (1) an objective correspondence of each element with a specific watershed region, (2)an appropriate formulation of effective rainfall over the planes, and (3) the transformation ofeffective rainfall to direct runoff, through the mechanisms of overland flow and channel flow.

In this context, infiltration is generally the main loss that limits the development ofeffective rainfall at field scale. Therefore, the estimate of the areal-average infiltration rate, I,is required. That becomes complicated by the spatial variability of soil hydraulic properties(Zhao et al. 2010), rainfall rate and initial soil water content. The spatial variability ofsaturated hydraulic conductivity, Ks, is generally assumed as the most important in compar-ison with those of the other soil hydraulic properties that influence the infiltration process(Russo and Bresler 1981; Dagan and Bresler 1983). At the same scale the spatial variabilityof rainfall rate, r, is considered less significant, also in the case of convective rainfalls(Goodrich et al. 1995; Morbidelli et al. 2006). Many studies have shown that the spatialvariability of Ks, assumed as a stochastic variable with a lognormal probability densityfunction (Nielsen et al. 1973; Warrick and Nielsen 1980; Sharma et al. 1987; Loague andGander 1990), can significantly affect the response of a slope to a uniform rainfall (Binley etal. 1989a,b; Saghafian et al. 1995; Corradini et al. 1998; Merz et al. 2002). Under theseconditions, several semi-analytical (Sivapalan and Wood 1986; Govindaraju et al. 2001) andsemi-empirical models (Smith and Goodrich 2000; Corradini et al. 2002) were proposed forthe estimate of the areal-average infiltration rate. Some formulations concerning the effectsof the joint spatial variability of Ks and r were also developed. Wood et al. (1986) used thetwo-term Philip equation to find approximate expressions for areal mean and variance ofinfiltration rates, however, their averaging operation occurred in space over a single reali-zation. Their relationship for areal average infiltration was validated by comparison withMonte-Carlo simulations, but the magnitude and the trend of the resulting errors were notspecified. Furthermore, Castelli (1996) developed a theoretical approach but under verysimplified conditions, particularly for the formulation of local infiltration rate. Govindarajuet al. (2006) and Morbidelli et al. (2006) proposed a more complete modelling to obtain theareal average infiltration, with a semi-analytical formulation and an additional componentthat describes empirically the run-on process consisting in infiltration of surface waterrunning from saturated areas downslope over a pervious soil. The possibility to replacethe spatially varying infiltration rate by the average infiltration rate was investigated byTayfur and Singh (2004) but specifically in the context of models for sediment transport.However, all these models don’t consider the possible effects of spatial variability of initialsoil water content, θi, that has to be appropriately clarified.

In a simulation study, using the Coweeta catchment topography, North Carolina, Graysonet al. (1995) considered two patterns of soil moisture with the same properties of mean,variance and correlation length. They showed that, assuming a pattern spatially random andthe other organized by a wetness index, very different responses to given rainfall inputs wereproduced. Similar results were also obtained by Merz and Plate (1997), Merz and Bárdossy(1998) and Bronstert and Bárdossy (1999). However, rather different results were derived inother investigations (Goodrich et al. 1994; Aubert et al. 2003; Longobardi et al. 2003;Corradini et al. 2008; Brocca et al. 2009b). In particular, for two sub-basins of the WalnutGulch experimental watershed, Goodrich et al. (1994) found that the knowledge of theremotely sensed average initial soil moisture was sufficient as input to rainfall-runoff modelsin semi-arid regions, provided the rainfall was known with a great spatial resolution.Grayson and Western (1998) suggested that a network with a limited number of moisture

1794 R. Morbidelli et al.

sensors could provide soil moisture time series potentially usable as antecedent conditiondata. Therefore, these indicative wetness conditions derived by satellite, and/or a few localmeasurements at surface (see also Tombul 2007), might be sufficient to establish antecedentconditions for a rainfall-runoff event in alternative to detailed spatial measurements (Korenet al. 2008). Recently, Brocca et al. (2009a, 2010) from spot measurements carried out indifferent experimental plots located in the Upper Tiber River (Central Italy) investigated thespatial pattern of the volumetric soil moisture at the surface. The results stated that, at fieldscale, θi can be usually assumed as a random variable characterized by a probability densityfunction of normal type, set equal to zero for negative values, and coefficient of variationequal to≈0.1. Moreover, they found that a limited number of sampling points representativeof all the values observed in each experimental field was sufficient to determine averageconditions at basin scale (see also Vachaud et al. 1985), but they did not quantify theinfluence of this approximation on overland flow generation.

Considering the results aforementioned someway contrasting, the main objective of thispaper is to improve the knowledge of the role of spatial variability of initial soil moisture onthe overland flow generated at the field scale. The proposed study takes into account theinfiltration of rainfall and, in addition, of overland flow running downslope into pervioussaturated or unsaturated areas (run-on process) that none of the above-mentioned inves-tigations considered earlier. Theoretical results have been obtained starting from the Broccaet al. (2009a, 2010) experimental data and combining a local infiltration model earlierproposed for complex rainfall patterns (Corradini et al. 1997) with a nonlinear kinematicwave approximation for overland flow. The second objective is to address the problemconcerning the number of soil moisture measurements to be sampled for an appropriateestimate of the field scale surface runoff hydrograph.

2 Statement of the Problem

As a geometrical simplification of a natural hillslope we represent it by a single plane ofslope S0. The soil is assumed to be vertically homogeneous, but the initial soil water contentis considered to vary through the soil surface according to experimental observations byBrocca et al. (2009a, 2010). Given a time varying rainfall rate, r, the generation of surfacerunoff will be governed by the combined effects of the processes of rainfall infiltration andrun-on. Figure 1 provides a simple scheme of the interacting processes along a strip of the

rainfall

saturated cell

unsaturated cell

runon

infiltration

overland flow

slope outlet

t1

t2

t3

rainfall

saturated cell

unsaturated cell

runon

infiltration

overland flow

slope outlet

t1

t2

t3

Fig. 1 Schematic representationof the mechanisms contributingto the generation of the surfacerunoff hydrograph at the fieldscale. A single strip at threedifferent times t1< t2<t3 is shown

Initial Soil Water Content as Input to Field-Scale Infiltration 1795

slope at three different times t1< t2< t3. As it can be seen, in this example at the time t1 thereare only unsaturated cells subjected to rainfall infiltration, while at the time t2 there aresaturated cells with rainfall infiltration and unsaturated cells with infiltration due to bothrainfall and run-on; lastly, at the time t3 all the strips contribute to the overland flow at theslope outlet.

Local infiltration is described by assuming one-dimensional flow into independentvertically homogeneous columns, following the scheme adopted by Corradini et al.(1998). The model proposed earlier by Corradini et al. (1994) and then reformulated byCorradini et al. (1997) is selected to determine the infiltration rate in each column, consid-ering that it was found to be very accurate by using the Richards equation as a benchmark.The flow problem is simplified by assuming an initial soil water content, θi, invariant withdepth, z, and approximating the dynamic wetting profile, θ(z), by a distorted rectanglerepresented in functional form through a shape factor β (≤1) which depends on the surfacewater content, θ0. Combining the continuity equation and a depth-integrated form of theDarcy law, the following relation is obtained:

dθ0dt

¼ θ0 � θið Þ b θ0ð ÞI 0 θ0 � θið Þ db θ0ð Þ

dθ0þ b θ0ð Þ

h i q0 � K0 � θ0 � θið Þ G θi; θ0ð Þ b θ0ð Þ p K0

I 0

� �ð1Þ

where t is the time; q0 is the downward water flux at the surface; K0 is the hydraulicconductivity at the surface; I′ is the cumulative dynamic infiltration depth; p is a quantitydepending on the profile shape and linked with β; G(θi, θ0) is the net capillary drivedepending on both the suction head, ψ, and hydraulic conductivity, K, as:

G θi; θ0ð Þ ¼ 1

K0

Zy θ0ð Þ

y θið Þ

K dy ð2Þ

The model requires the knowledge of the functional forms of hydraulic soil properties,which are expressed according to the parameterization adopted by Smith et al. (1993) as:

K θð Þ ¼ Ksθ� θrθs � θr

� �3þ2 l=

ð3Þ

y θð Þ ¼ ybθ� θrθs � θr

� ��c l=" #1 c=

þ d ð4Þ

where θs is the volumetric water content at natural saturation; θr is the residual volumetricwater content; 1 is the pore size distribution index; yb is the air entry head, which is given ina tabular form for soil texture classes (Rawls et al. 1983); and c and d are empiricalcoefficients. The downward water flux at the soil surface, q0 , may be formally expressed as:

q0 ¼ r þ v0 unsaturated surface ð5Þ

q0 ¼ f saturated surface ð6Þwhere v0 represents the run-on in terms of discharge per unit surface, estimated as specifiedin the following. The quantity f denotes the infiltration capacity derived from Eq. 1 with

1796 R. Morbidelli et al.

dθ0/dt00. All the parameters in Eqs. 1–4 are considered to be constant through the slope,except the initial soil water content.

Surface runoff is routed over the plane by the kinematic wave equation with flowresistance expressed by the Manning law (Woolhiser and Goodrich 1988; Singh 1996;Venkata et al. 2008). We have:

@h

@tþ @ahm

@x¼ r � q0 ð7Þ

where h is the depth of flow at time t and position x; a ¼ 1=nð ÞS1=20 , with n Manningroughness coefficient; m05/3 and !hm at the slope outlet represents the discharge per unitwidth. The upper boundary condition and the initial condition are:

h 0; tð Þ ¼ 0 and h x; 0ð Þ ¼ 0 ð8ÞBy Eq. 5 we observe that over an unsaturated elementary area r-q0 of Eq. 7 is equal to -v0,

while on a saturated surface r-q0 represents the effective rainfall rate. The quantity v0 isobtained through the solution of Eq. 7 in the adjacent upstream area. With the support ofprevious numerical simulations (Schmid 1989), we neglect the influence of overland flowdepth on the infiltration rate.

The discharge at the slope outlet is given by:

QðtÞ ¼ a hm x ¼ L; tð Þ B ð9Þwith L and B length and width of the slope, respectively. In principle Q(t) is dependent on thespatial distribution of θi through Eq. 1. We note that, in principle, for natural surfaces withirregular microtopography, flow dynamics should be represented in two dimensions by thediffusion wave approximations (Govindaraju et al. 1992; Tayfur et al. 1993; Tayfur andSingh 2004), while the use of the bi-dimensional kinematic wave approximation (Tayfur2001) would be inappropriate. However, for the case of a plain with smooth surface it iswidely recognized that the one-dimensional kinematic wave approximation provides anappropriate representation of the surface runoff hydrograph (see for example Giakoumakisand Tsakiris 2001). Our mathematical formulation is supported by the fact that our primaryobjective is not to maximize the accuracy in simulating the surface runoff hydrograph but toinvestigate its sensitivity to different spatial representations of soil moisture content. On theother hand, it is expected that the choice of a slope with erratic microtopography determinesan irregular bi-dimensional overland flow and therefore mitigates the role of the spatialvariability of θi.

3 Numerical Approach, Study Soils and Selected Rainfall Events

Numerical solutions of the kinematic wave equation coupled with the infiltration equationwere carried out over two different planes: the first-one, 60 m long and 50 m wide, thesecond-one 120 m long and 90 m wide, both with 4% slope along a specific direction.Equation 7 was solved by the Lax-Wendroff finite difference scheme using n00.15 sm−1/3.A square grid of 10 m×10 m was used for representing the soil moisture spatial variability.For a given soil moisture distribution we computed the overland flow response to a specificrainfall pattern at the slope outlet. Because of the one-dimensional flow paths along thex-direction, for a given set of θ values the contribution to discharge from each strip wastreated independently and the contributions of all the strips summed to form Q(t). The

Initial Soil Water Content as Input to Field-Scale Infiltration 1797

computations were performed by adopting a 1.0 m long and 10 m wide grid to meet stabilityrequirements of the numerical procedure. This size of the grid was found to be the minimumfor which Q(t) changed appreciably as a result of the distortion associated with therepresentation of the run-on process. In fact, for the numerical solution we assumed thatthe discharge at the upper boundary of a grid element wetted immediately the entire element,while it physically takes some time before the wetting front advances over the same element.

We selected a soil representative of a clay loam that is typically found in a region ofCentral Italy where measurements of spatial distribution of θi were earlier carried out byBrocca et al. (2009a, 2010). The main hydraulic features of this soil that are required for theestimate of infiltration rate are given in Table 1.

For this study we selected as θi the values of soil water content observed by Brocca et al.(2009a, 2010) by a portable Time Domain Reflectometry. Many sequences of measurementsin different fields located inside the Vallaccia basin (area 60 km2) were carried out. In a givensampling day, soil moisture measurements were usually collected over regular grids in eachfield, characterized by uniform structure and use of soil, of extension between 3000 m2 and10800 m2. In this study we have selected data sets representative of a very large range ofinitial conditions. The main characteristics of the selected fields (Castel Rigone, Molino,Preggio and Colorso, hereafter denoted as CRI, MOL, MON and COL, respectively) andsoil moisture measurements in different periods are summarized in Table 2.

The surface runoff hydrographs were derived for a variety of synthetic experimentsinvolving real rainfall patterns and design hyetographs referred to Central Italy.

4 Analysis of Results

In principle, distributed or semi-distributed rainfall-runoff models should consider the spatialvariability of soil hydraulic properties, rainfall rate and initial soil moisture at the field scale.The problem has been widely examined with results that substantially agree for the first twoquantities. On the other hand, the role of the last quantity does not appear sufficiently clearand a possible representation based on the assumption of θi as a random variable wouldmake the models considerably complex. Therefore, in the light of this problem we haveexamined the effects of the spatial variability of initial soil moisture on the overland flowhydrographs produced in each experimental field under different time-varying rainfall rates.Many observed soil moisture spatial distributions, together with their simplified representations,have been considered as potential initial distributions of θi. In particular, a simple representationrelies on the observed soil moisture in a site that, through a temporal stability analysis, was earlierfound to be representative of a specific field (Brocca et al. 2010).

Table 1 Hydraulic quantities ofthe study soil (for symbols see text) Property Study Soil

θs 0.55

θr 0.08

yb (mm) −400Ks (mmh−1) 0.75

1 0.2

c 5

d (mm) 100

1798 R. Morbidelli et al.

Tab

le2

Maincharacteristicsof

theselected

fields

andobserved

soilmoisturedistributio

ns(Broccaet

al.20

09a,20

10)

Field

SoilTexture

LandUse

Field

Dim

ens.

(mxm)

Spacing

(m)

Num

berMeasur.

Points

Date

MeanValue

(%)

Stand

.Dev.(%

)MoistureDistribution

Peculiarity

CRI

Siltyclay

andsand

Grass

60×50

1030

Feb

22,20

0746

.31.1

HMV

May

25,20

0712

.21.7

LMV

MOL

Gravel

Grass

60×50

1030

Mar

30,20

0729

.51.3

HMV

May

25,20

0710

.11.5

LMV

MON

Sandy

loam

Grass

60×50

1030

Feb

16,20

0749

.13.3

HMV

May

25,20

0720

.13.8

LMV

Apr

26,20

0721

.25.8

HCV

COL

Sandy

loam

Grass

120×90

1010

8May

5,20

0524

.86.7

LMV

Dec

2,20

0539

.74.8

HMV

HMV,Sam

plingdaywith

thehigh

ervalueof

themeanaveragesoilmoistureforthespecific

field(see

Broccaet

al.20

10);

LMV,Sam

plingdaywith

thelower

valueof

themeanaveragesoilmoistureforthespecific

field(see

Broccaet

al.2

010);

HCV,Set

ofvalues

with

thehigher

coefficientof

variationforallsamplingdays

andfields

(see

Broccaet

al.20

10).

Initial Soil Water Content as Input to Field-Scale Infiltration 1799

Representative results produced by the same observed rainfall pattern of convective type,selected among the natural events occurred in the Vallaccia River Basin, are shown in Figs. 2and 3. The surface runoff hydrographs of Fig. 2 were obtained for the CRI field. As it can beseen, the hydrograph is practically independent of the spatial distribution of θi. In fact, thecurves obtained for the real distribution or for constant values of θi, equal to the average inthe field and to the measurement in the site characterized by temporal stability, are verysimilar. We note that the results of Fig. 2 were obtained starting from a very humid soil.Under particularly dry soil initial conditions, Fig. 3 illustrates the results obtained for theMOL field where the observed moisture content was generally lower than those in the otherfields of Table 2. The shapes of the hydrographs are rather similar to those shown in Fig.2.

A similar analysis was performed at the MON field where the average soil moisture wastypically the maximum observed at the field scale. Two sample events, with surface hydro-graphs generated by a low intensity natural rainfall of frontal type, are represented in Figs. 4and 5. The hydrographs of Fig. 4, simulated starting from a high average soil moisture,indicate that the approximation of the natural spatial distribution of θi by the average valueleads to a shape of the hydrograph rather similar to the actual one, while the peak dischargeand the total volume of surface water become underestimated of 13% and 23%, respectively.On the other hand, the adoption of the θi observed in a temporally stable site produces acurve with minor errors. In addition, Fig. 5 shows that much greater relative errors areassociated with experiments involving a very low value of initial soil moisture. Specifically,the hydrograph derived through the value of θi measured in a temporally stable site ischaracterized by values of peak discharge and total volume of surface water overestimated ofabout 123% and 130%, respectively, while for θi assumed equal to the areal-average valuethe same quantities are underestimated of 54% and 75%, respectively. In any case, the lasterrors are linked with an experiment that generates a very limited volume of surface water.

Some simulations were also carried out for design hydrographs. As it could be expected,because in these conditions the design hyetographs generally involved rainfall rates largerthan those considered in the experiments of Figs. 2 and 3, the representation of the spatialheterogeneity of θi was found to be needless, even for return periods typical of hydraulicstructures of minor importance (equal to 2–5 years).

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

Dis

char

ge (m

3 /s)

θi,v

θi,m

θi,t0.6

16

0

5

10

15

20

Time (h)

Rai

nfa

ll R

ate

(mm

)

0.5

0 0.5 1 1.5 2Time (h)e

θi,v

θi,m

θi,t

16

Time (h)

1.0

Fig. 2 Comparison of simulated surface runoff hydrographs at the study slope outlet (60 m long and 50 mwide) for different initial soil water distributions: values spatially variable, θi,v; average value of θi,v at eachsite (0.463), θi,m; and value observed at a site characterized by temporal stability (0.457) considered constantthrough the field, θi,t. Soil moisture observed on February 22, 2007 at CRI experimental field by Brocca et al.(2010). Hydrographs obtained under a natural rainfall pattern

1800 R. Morbidelli et al.

An overall analysis of our results obtained over fields of 3000 and 10800 m2 indicatesthat the spatial variability of the initial soil moisture does not affect appreciably thegeneration of surface runoff at the field scale. In fact, infiltration under intense storms ofshort duration can be accurately described by adopting the soil moisture obtained as averagevalue through the field or that measured at a site characterized by temporal stability.Furthermore, in the events with rainfalls of moderate intensity and considerable duration asimilar simplification can be used, because the actual spatial distribution of θi is crucial onlywhen the volume of surface water is very limited and the rainfall-runoff transformationbecomes not useful in the hydrological practice.

In spite of our aforementioned results indicate that distributed rainfall-runoff models canavoid considering the heterogeneity of θi at field scale, an alternative representation of soil

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

0 0.5 1 1.5 2Time (h)

Dis

char

ge (m

3 /s)

θi,v

θi,m

θi,t0.6

16

Time (h)

0.5 1.0

Time (h)

θi,v

θi,m

θi,t

16

Rai

nfa

ll R

ate

(mm

)

0

5

10

15

20

Fig. 3 Comparison of simulated surface runoff hydrographs at the study slope outlet (60 m long and 50 mwide) for different initial soil water distributions: values spatially variable, θi,v; average value of θi,v at eachsite (0.101), θi,m; and value observed at a site characterized by temporal stability (0.121) considered constantthrough the field, θi,t. Soil moisture observed on May 25, 2007 at MOL experimental field by Brocca et al.(2010). Hydrographs obtained under a natural rainfall pattern

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

9.00E-03

0 1 2 3 4 5 6 7 8

Dis

char

ge(m

3 /s)

θi,v

θi,m

θi,t

0.6 0.6 0.6 0.6

1.6

3.8

0.4

2.42.4 2.2

1

0.20.2

6

0123

4567

Time (h)

Rai

nfa

llR

ate

(mm

)

.

.

.

.

.

.

.

7.00E-03

.

Time(h)

ir

θi,v

θi,m

θi,t

0.6 0.6 0.6 0.6

1.6

3.8

0.4

2.42.4 2.2

1

0.20.2

6

)

1.0 2.0 3.0 4.0 5.0 6.0

Fig. 4 Comparison of simulated surface runoff hydrographs at the study slope outlet (60 m long and 50 mwide) for different initial soil water distributions: values spatially variable, θi,v; average value of θi,v at eachsite (0.491), θi,m; and value observed at a site characterized by temporal stability (0.537) considered constantthrough the field, θi,t. Soil moisture observed on February 16, 2007 at MON experimental field by Brocca etal. (2010). Hydrographs obtained under a natural rainfall pattern

Initial Soil Water Content as Input to Field-Scale Infiltration 1801

moisture through the areal-average value or the value observed at a site earlier characterizedas temporally stable is required. Therefore while the models can be simplified, extendedexperimental observations have to be in any case performed. In this context, we haveexamined the possibility of reducing the experimental burden by selecting the essentialmeasurements to be carried out.

Figure 6 shows, for very dry initial conditions, the sensitivity of the surface runoffhydrograph generated under a rainfall event of convective type to the number of sensorsused for estimating the areal-average value of θi. The curves obtained using 30 sensors, and10 or 3 sensors anyhow selected among them are substantially superimposed. On this basis,

0.00E+00

1.00E-05

2.00E-05

3.00E-05

4.00E-05

5.00E-05

6.00E-05

7.00E-05

8.00E-05

9.00E-05

1.00E-04

Time (h)

Dis

char

ge (m

3 /s)

θi,v

θi,m

θi,t

0.6 0.6 0.6 0.6

1.6

3.8

0.4

2.42.4 2.2

1

0.20.2

6

01234567

Time (h)R

ain

fall

Rat

e (m

m)

1.0 2.0 3.0 4.0 5.0 6.0

0 1 2 3 4 5 6 7 8

θi,v

θi,m

θi,t

0.6 0.6 0.6 0.6

1.6

3.8

0.4

2.42.4 2.2

1

0.20.2

6

Time (h)

Fig. 5 Comparison of simulated surface runoff hydrographs at the study slope outlet (60 m long and 50 mwide) for different initial soil water distributions: values spatially variable, θi,v; average value of θi,v at eachsite (0.201), θi,m; and value observed at a site characterized by temporal stability (0.199) considered constantthrough the field, θi,t. Soil moisture observed on May 25, 2007 at MON experimental field by Brocca et al.(2010). Hydrographs obtained under a natural rainfall pattern

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

9.00E-03

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (h)

Dis

char

ge

(m3 /

s)

θi,v

θi,3

θi,100.6

16

0

5

10

15

20

Time (h)

Rai

nfa

llR

ate

(mm

)

0.5 1.0

θi,v

θi,3

θi,10

16

0.5 1.0

Fig. 6 Comparison of simulated surface runoff hydrographs at the study slope outlet (60 m long and 50 mwide) for different initial soil water distributions: values spatially variable, θi,v; values assumed invariant withposition and obtained by averaging 3 (θi,3) and 10 (θi,10) random measurements, respectively. Soil moistureobserved on May 25, 2007 at CRI experimental field by Brocca et al. (2010). Hydrographs obtained under anatural rainfall pattern

1802 R. Morbidelli et al.

for the same rainfall event the extreme hypothesis with 1 sensor has been also taken intoaccount. Figure 7 compares the surface hydrograph estimated through the complete set ofobservations available in a representative field and those obtained by using only themaximum and the minimum values of θi. The errors in magnitude on peak discharge andvolume of surface water are rather limited and in any case with values within 10% and 15%,respectively. A similar analysis has been carried out for surface runoff hydrographs producedby a natural rainfall event of frontal type involving much lower intensities (see Fig. 8). Theuse of a single extreme value of observed initial soil moisture leads to relative errors inmagnitude up to 22% for the peak value and 66% for the total volume of surface water,however the quantity of surface water involved is very reduced. Our results suggest that thewater content observed in a site randomly selected in the field study can be used as initialuniform value of θi in rainfall-runoff modelling.

This analysis showing the small role of spatial heterogeneity of θi gets stronger if spatialvariability of saturated hydraulic conductivity, Ks, is also considered. In fact, in natural soilsthe spatial variabilities of θi and Ks were found considerably different, with a coefficient ofvariation of Ks (typically between 0.3 and 1.0) much larger than that of θi coming out fromour earlier measurements. Furthermore, the spatial variability of Ks was found to have asignificant role on areal infiltration for a wide range of conditions (Corradini et al. 2002).From these elements, it can be deduced that the spatial variability of Ks leads to mask theeffects of the heterogeneity in initial soil moisture on the surface water hydrograph.

Further, we note that the results we have obtained at the COL field are similar to those wehave shown in this section for the other fields of smaller dimensions and in addition thesediffer from the earlier ones presented in some other investigations. However, such differ-ences can be ascribed to well-defined elements as the representation of the run-on processand the main characteristics of the study regions. In any case, at least for heavy rainfall rates,the results are not contrasting because our simulations refer to fields with uniform character-istics of both structural type and soil use, while Grayson et al. (1995), Merz and Bárdossy(1998) and Bronstert and Bárdossy (1999) used simulations at a larger scale where thesecharacteristics were spatially variable and associated with large coefficients of variation of

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

9.00E-03

0 0.5 1 1.5 2

Time (h)

Dis

char

ge(m

3 /s)

θi,v

θi,1M

θi,1m0.6

16

0

5

10

15

20

Time (h)

Rai

nfa

llR

ate

(mm

)

θi,v

θi,1M

θi,1m

16

Time (h)

0.5 1.0

Fig. 7 Comparison of simulated surface runoff hydrographs at the study slope outlet (60 m long and 50 mwide) for different initial soil water distributions: values spatially variable, θi,v; values assumed invariant withposition and equal to the maximum (θi,1M) and minimum (θi,1m) observed values, respectively. Soil moistureobserved on May 25, 2007 at CRI experimental field by Brocca et al. (2010). Hydrographs obtained under anatural rainfall pattern

Initial Soil Water Content as Input to Field-Scale Infiltration 1803

θi. The last statement is also supported by an investigation on the role of run-on. We havefound that, for any experimental situation examined here, the surface runoff hydrographsestimated with and without run-on are practically coincident. On the other hand, for largecoefficients of variation of θi the same comparisons reveal that the run-on is still unimportantwhen heavy rainfall rates are involved, while, as it can be seen in Fig. 9, it reducesconsiderably the effects of spatial variability in soil water content for low intensity rainfalls.The representative event shown in Fig. 9 is referred to a coefficient of variation of θi equal to0.4 (see, for example, Minet et al. 2010).

Alternatively, we have verified that the application of the diffusion hydrodynamic modelproposed by Hromadka and Yen (1986) (see also Corradini et al. 1989) does not modify therole of the spatial variability of initial soil moisture content.

0.00E+00

5.00E-05

1.00E-04

1.50E-04

2.00E-04

2.50E-04

3.00E-04

3.50E-04

8543210

Time (h)

Dis

char

ge(m

3 /s)

θi,v

θi,1M

θi,1m

0.6 0.6 0.6 0.6

1.6

3.8

0.4

2.42.4 2.2

1

0.20.2

6

0123

4567

Time (h)R

ain

fall

Rat

e(m

m) θi,v

θi,1M

θi,1m

0.6 0.6 0.6 0.6

1.6

3.8

0.4

2.42.4 2.2

1

0.20.2

6

1.0 2.0 3.0 4.0 5.0 6.0

6 7

Fig. 8 Comparison of simulated surface runoff hydrographs at the study slope outlet (60 m long and 50 mwide) for different initial soil water distributions: values spatially variable, θi,v; values assumed invariant withposition and equal to the maximum (θi,1M) and minimum (θi,1m) observed values, respectively. Soil moistureobserved on March 30, 2007 at MOL experimental field by Brocca et al. (2010). Hydrographs obtained undera natural rainfall pattern

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

0 1 2 3 4 5 6 7 8

Time (h)

Dis

char

ge (m

3 /s)

0.6 0.6 0.6 0.6

1.6

3.8

0.4

2.42.4 2.2

1

0.20.2

6

01234567

1 3 5 7 9 11 13

Time (0.5 h)

Rai

nfa

ll R

ate

(mm

) without run-on

with run-on

Fig. 9 Comparison of surface runoff hydrographs computed at the study outlet (60 m long and 50 m wide)with and without run-on. Initial soil moisture distribution of normal type with average value of 0.274 andcoefficient of variation equal to 0.4. Hydrographs obtained under a natural rainfall pattern

1804 R. Morbidelli et al.

5 Conclusions

In the representation of surface runoff hydrographs at the field scale, both as an independentmodel and a fundamental component of distributed modeling for estimating the basinresponse, the infiltration process requires to consider the random spatial variability both ofsaturated hydraulic conductivity, Ks, and initial soil moisture content, θi. The role of the firstquantity is widely recognized and many specific empirical and conceptual/semi-analyticalformulations have been proposed in the scientific literature in the last 10 years. On the otherhand, contrasting results concerning the weight of the spatial heterogeneity of θi have beenpresented and thus a modelling describing the interaction between the spatial variability ofKs and θi has not been sufficiently investigated. In this context, an interesting approach thatneglects the run-on process has proposed by Craig et al. (2010). In any case, the first issue tobe addressed in advance is to extend the analyses earlier carried out on the effects of thevariability of θi for a given uniform Ks. This work accomplishes this objective.

Our results indicate that for the estimate of the surface runoff hydrograph at a slope outletthe actual spatial distribution of θi can be generally approximated with good accuracy by thecorresponding areal-average value or the θi value observed at a site characterized bytemporal stability. In addition, our simulations show that, at least in the situations of majorinterest, the hydrograph is sufficiently well reproduced through a single measurement of θiperformed at a site anyhow selected in the field of interest. These results point out that thesurface runoff hydrograph is characterized by a low sensitivity to spatial variations of θi atleast when rainfalls of high intensity are involved. A similar behaviour has been found evenfor very humid soils under natural frontal rainfalls of low intensity. On the other hand, thissensitivity is much more pronounced for particularly dry soils under natural frontal rainfallof low intensity even though these situations determine a small amount of surface water andare therefore of minor interest in the hydrological practice.

An overall analysis of our results suggests that, at the field scale, the spatial heterogeneityof θi can be disregarded and therefore the fundamental quantity to be described as a randomvariable is the saturated hydraulic conductivity. On the other hand, when a larger scale thatincludes several fields with different structures and/or soil uses has to be considered, thevariability of both Ks and θi should be represented by adopting as basic element a componentof deterministic type linked with the main trend of these two quantities.

Acknowledgment This research was mainly financed by the Italian Ministry of Education, University andResearch and by the Cassa di Risparmio di Perugia Foundation.

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