calibration of an evapotranspiration model to simulate soil water dynamics in a semiarid rangeland

HYDROLOGICAL PROCESSESHydrol. Process. 22, 4655–4669 (2008)Published online 16 July 2008 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.7087

Calibration of an evapotranspiration model to simulate soilwater dynamics in a semiarid rangeland

M. P. Maneta,1,2* S. Schnabel,2 W. W. Wallender,1 S. Panday3 and V. Jetten4

1 Dept Land, Air and Water Resources, University of California, Davis, One Shields Avenue, 95616, Davis, CA, USA2 Grupo de Investigacion Geoambiental, Departamento de Geografıa y Ordenacion del Territorio, Universidad de Extremadura, avda de la

Universidad s/n, 10071, Caceres, Spain3 Geomatrix Consultants Inc, 620 Herndon Pkwy, Ste 200, 20170, Herndon, VA, USA

4 Department of Earth Systems Analysis, ITC, P.O. Box 6, 7500 AA Enschede, The Netherlands

Abstract:

Soil is the largest reservoir of water in a catchment and evapotranspiration is the largest yearly output flux of water fromthe soil in semiarid environments at the catchment scale. Hydrologic stress is one of the typical characteristics of semiaridenvironments. The shortage of water affects all the biophysical processes and increases the nonlinearity in the rainfall–runoffrelationship. A proper description of semiarid watersheds should include an accurate description of the wetting–drying cycleof the soil, which is critical to properly describe important processes such as the connection between the surface and the soilsaturated zone, the production of runoff or the growth of plants. Evapotranspiration is typically modelled in comprehensivedistributed hydrologic models (e.g. MIKE-SHE or MODHMS) using functions that relate evapotranspiration to soil moistureand plant biomass through a set of empirical coefficients, which typically adopt some default values regardless of the conditionsof the area where the model is applied. In this work, using a global and a local search algorithm, a three-dimensional soil waterflow model coupled to an evapotranspiration model is calibrated in a soil profile covered with grass in a semiarid environment.Four parameters for the water flow model and three parameters for the evapotranspiration model are calibrated. An existingtrend between daily average soil moisture and daily maximum temperature is used as extra information during the calibrationprocess. The results showed that the known relationship between maximum temperature and average soil moisture includesinformation on the seasonality of the atmospheric demand. Although total evapotranspiration is similar in all calibrations, thetrade-off between evaporation and transpiration is sensitive to this extra information mainly through one model parameter.Further research is necessary to identify the value of the parameter that produces the correct trade-off. Copyright 2008 JohnWiley & Sons, Ltd.

KEY WORDS evapotranspiration; model calibration; semiarid rangeland; root water uptake

Received 25 October 2007; Accepted 1 May 2008

INTRODUCTION

In semiarid environments, evapotranspiration is thelargest output flux in the water-mass balance in a catch-ment at seasonal and yearly time scales. Several studieshave calculated that evapotranspiration may account formore than 95% of the outputs in the annual water bud-get at the catchment scale (Pinol et al., 1991; Schnabel,1997; Wilcox et al., 2003).

In dry environments, atmospheric water demandexceeds precipitation most of the year, causing a char-acteristic hydrologic stress that affects most of the pro-cesses in the ecosystem. From a hydrologic point of view,evapotranspiration affects the amount of precipitationavailable for runoff and soil/aquifer recharge and howatmosphere–soil–plant systems interact. Those processesin turn affect how soil moisture and vegetation patternsare spatially distributed (Breshears and Barnes, 1999;Gomez-Plaza et al., 2001; Wilcox, 2002), the growth ofmicrobial organisms in the soil able to mineralize organic

* Correspondence to: M. P. Maneta, Dept Land, Air and Water Resources,University of California, Davis, One Shields Avenue, 95616, Davis, CA,USA. E-mail: [email protected]

matter and fix nutrients (Rodrıguez-Iturbe, 2000; Porpo-rato et al., 2003) or the leaching of nutrients and par-ticles. Whereas at the event scale evapotranspiration hasoften been regarded as negligible, continuous models thatsimulate hydrologic processes beyond the event have tosimulate the recovery of soil potentials in the interstormperiods and therefore should include it as a componentof the mass balance.

Owing to the difficulty of observing the evapotranspi-ration process and the amount of data required to inves-tigate it (which are usually unavailable when conductingwatershed scale investigations), evapotranspiration is per-haps one of the least well known hydrologic processesfor researchers that deal with watershed scale hydrol-ogy. Many watershed hydrologic models deal with errorsin the potential evapotranspiration estimates by numer-ical readjustment (calibration) of the model parameters(Andreassian et al., 2004). This numerical readjustmentof parameters will fit the mass balance according to thecomponents that are most commonly measured (e.g. out-put discharge, soil moisture), leaving evapotranspirationas the closure term of the balance. Because the errors inthe estimates of potential evapotranspiration are absorbed

Copyright 2008 John Wiley & Sons, Ltd.

4656 M. P. MANETA ET AL.

by the model structure in the calibration process, spe-cific research on the evapotranspiration component ofthe model has not been a top priority for watershed-scale hydrologic modellers. The problem arises whenmodel errors due to the effect of hydrologic processesnot included (or not properly described) in the model for-mulation, non-measurable characteristics of the basin orimperfect knowledge of other boundary conditions to thesystem are all treated by adjusting the model parameters.In that case, the number of calibration parameters mayincrease and show a correlation structure with respectto the experimental data used for calibration that willmake the identification of their optimal parameter valuesa complicated process (Vrugt et al., 2004).

The reference crop evapotranspiration method recom-mended by FAO (Allen et al., 1998), is one common wayof calculating optimal evapotranspiration from vegetatedsurfaces. In semiarid lands where plants are commonlyunder hydrologic stress, the evapotranspiration is typ-ically suboptimal so a routine is needed to transformpotential evapotranspiration to actual evapotranspiration.

Different methods are used in the literature to calcu-late water limited evapotranspiration rates. The simplestformulations involve a set of efficiency factors derivedfrom functions that depend on biomass density, portionof soil covered by vegetation and other scaling factorsthat are functions of the root density and water poten-tial for different soil depths (Belmans et al., 1983; Kempet al., 1997). In more complex formulations calculationof reduced transpiration and evaporation can be sepa-rated. Reduced transpiration can include detailed physicsaccounting for the effects of changes in canopy and stom-atal conductance due to soil moisture and vapour pressuredeficit (Campbell, 1977) while transpiration can explicitlyinvolve vapour pressures between phases, vapour diffu-sivity and resistance to diffusion as functions of soil prop-erties and atmospheric conditions (Kemp et al., 1997).

To be operative in watershed scale models, the rou-tines to calculate reduced evapotranspiration must beparsimonious while still flexible and physically mean-ingful. The Kristensen and Jensen model (Kristensenand Jensen, 1975) is the method of choice to calculateactual evapotranspiration in some of the most compre-hensive and widely used physics based models such asMIKE-SHE (Danish Hydraulic Institute, 2004; Refsgaardand Storm, 1995) or MODHMS (Panday and Huyakorn,2004). These models are used to conduct hydrologicresearch in watersheds covering a wide range of environ-ments (Christiaens and Feyen, 2002; McMichael et al.,2006; Werner et al., 2006) but no attempt for specificcalibration of the evapotranspiration routine parametershas been made and the default parameter values are oftenused. This is because, given the large number of param-eters involved in a watershed scale model, only a fewproperties usually related to the surface or subsurfaceflow and storage components are chosen for calibration.

Given the importance of evapotranspiration in semiaridenvironments, a specific calibration of the evapotranspi-ration model parameters at a scale that allows a more

specific treatment of the process is advised. In addition,further knowledge of the behaviour of the evapotranspira-tion model and on the range of values that the parametersmay take for the specific conditions of the site where themodel is applied should improve the confidence on thedescription of the system, especially in cases where evap-otranspiration is expected to be a major output componentin the mass balance.

The purpose of this study is to test the performance andto calibrate a root water uptake model composed of theKristensen and Jensen evapotranspiration model coupledto a solution of Richard’s equation that governs flowin variably saturated soils. Seven of the most relevantparameters are calibrated: three empirical parametersof the Kristensen and Jensen model, two parametersdefining the van Genuchten soil moisture retention curve(van Genuchten, 1980) and two parameters defining afunction that characterizes the distribution of saturatedhydraulic conductivity with depth. The calibration uses acombination of global and local search algorithms.

The parameters are calibrated and tested with 2 yearsof soil moisture measurements taken with a set of time-domain reflectometer (TDR) probes spanning the entiredepth of a soil profile 1 m deep. In addition, the effects ofadding some extra information to the calibration processin the form of prior information on soil water content atfield capacity and maximum grass coverage as well asthe information provided by a known linear relationshipbetween maximum daily temperatures and soil moisturewill be explored. The goal is to see how a differentbalance of the different sources of information on thecalibration problem affects the parameter estimates andthe model output. The study is done in a semiaridrangeland, which will give the opportunity to test theKristensen and Jensen model and the originally suggestedparameter values in a different environment from whichthe original model was developed and calibrated.

MODEL DESCRIPTION

To simulate evapotranspiration a form of the Kristensenand Jensen model (Kristensen and Jensen, 1975) wasused. In the model implementation, evapotranspiration ismodelled as the sum of evaporation and plant transpi-ration. The actual transpiration Tp [LT�1] at node i isrelated to the potential evapotranspiration of the crop interms of its optimal water requirement Etp [LT�1] by alinear function f1 of leaf area index (which is a measureof the biomass density and may be replaced by any othermeasure such as dry weight of biomass as used in thisresearch), a nonlinear function of soil moisture f2 androot extraction capacity calculated using a root distribu-tion function, RDF (Kristensen and Jensen, 1975; Pandayand Huyakorn, 2004):

Tpi D [f1�LAI�f2��i�][RDFi][Etp] �1a�

In which

f1�LAI� D maxf0,min[1, �C2 C C1LAI�]g �1b�

Copyright 2008 John Wiley & Sons, Ltd. Hydrol. Process. 22, 4655–4669 (2008)DOI: 10.1002/hyp

CALIBRATION OF AN EVAPOTRANSPIRATION MODEL 4657

f2��i� D

0 for �i � �wp

1 �[�fc � �i�fc � �wp

]C3

/Etp

for �wp < �i � �fc

1 for �fc < �i�1c�

RDFi D∫ z2

z1rF�z�dz

/ ∫ L1

0rF�z�dz �1d�

where the subscript i indicates a location in the soilprofile (node), LAI is the leaf area index [L2L�2], C1 [�],C2 [�] and C3 [LT�1] are empirical parameters of themodel, �wp and �fc, are soil moisture contents [L3L�3]for wilting point and field capacity, respectively, rF�z� isthe root extraction function that varies with depth z [L].The RDF function indicates the proportion of the totaltranspiration extracted between depth 0 and the maximumroot length L1 [L] removed between depths z1 and z2.The RDF function states that the amount of energy usedto extract water in the soil profile will never exceed theavailable energy or that the effective transpiration is thesum of the transpiration extraction between the surfaceand maximum root length L1.

Evaporation Esi is calculated as

Esi D ˛Łi �Etp��1 � f1�LAI��EDFi �2a�

In which

˛Łi D

� � �e2�e1 � �e2

for �e2 � � � �e11 for �i > �e10 for �i < �e2

�2b�

and �e1 and �e2 are, respectively, the water content thresh-olds at maximum evaporation and at no evaporation.EDF is the function that distributes the evaporation upto the maximum energy penetration depth below whichno evaporation occurs.

Water flow within the soil is simulated using a 3Dform of Richard’s equation adapted to variably-saturatedflow (HydroGeoLogic Inc, 1996; Panday and Huyakorn,2004):

∂

∂x

(Kxxkrw

∂hG∂x

)C ∂

∂y

(Kyykrw

∂hG∂y

)

C ∂

∂z

(Kzzkrw

∂hG∂z

)�W

D �∂Sw∂t

C SwSs∂hG∂t

�3�

where x, y and z [L] are the Cartesian coordinates; Kxx,Kyy and Kzz [LT�1] are the components of the hydraulicconductivity tensor along the x, y and z axes; krw [�] isrelative permeability as a function of the degree of watersaturation (see below); hG is the hydraulic head; W [T�1]is a source-sink term represented as a volumetric flux perunit volume of the subsurface; ϕ is the drainable porosity(specific yield); Sw is the degree of saturation of water as

a function of the pressure head (see below) and Ss [L�1]is specific storage.

Relative permeability, which depends on soil moisture,is defined by (van Genuchten, 1980):

krw D S0Ð5e

[1 � (

1 � S1/�e

)�]2�4�

and the soil retention function (Panday and Huyakorn,2004; van Genuchten, 1980)

Se D Sw � Swr1 � Swr

D{ [

1 C �˛ �ˇ]��

� 01 > 0

�5�

where Se is the relative degree of saturation, Swr is thedegree of saturation of soil with residual water content,˛ and ˇ are fitting parameters of the function, is thepressure head ( D hG � z) and � is related to ˇ by� D 1 � 1/ˇ.

DESCRIPTION OF THE EXPERIMENTAL PLOTAND THE EXPERIMENTAL DATA

The study plot is located in the valley bottom of the typ-ical savanna-like rangeland termed dehesas (Plieningeret al., 2004) that is common in the entire Mediterraneanbasin. Soil moisture data was obtained from 16 TDRprobes installed in an open area (no trees or shrub, onlypasture cover) at the valley bottom. The probes have threestainless steel rods 25 cm long and 0Ð3 cm in diameter.

To install the probes a 3 m long trench was duguntil bedrock was reached (1 m deep). The probes wereinstalled parallel to the soil surface in four columnsand four rows spanning the entire soil depth at 20, 40,70 and 90 cm from the surface. With this distributionfour measurements were obtained for each depth, whichpermitted an estimation of the horizontal soil moisturevariability for the different layers as well as an estimationof the moisture changes with depth. Once the probeswere installed the trench was filled and several monthswere allowed to permit the soil to settle and the grassto recover. Measurements were taken manually usinga Tektronic 1502C TDR cable tester as indicated byDirksen (1999). The available database included 768measurements of the soil moisture profile at 48 timesduring the years 2003–2004 and 2004–2005.

Soil characteristics

During the installation of the probes, bulk and coresamples were collected to characterize the soil in the pro-file. Porosity, texture and field capacity were determinedin the vicinity of the TDR probes (approximately 20, 40,70 and 90 cm from the soil surface, Table I). The soiltexture at all depths is silt-loam with a low organic mat-ter content that decreases with depth. The upper layerhas a higher content of fine material (silt C clay) andwith depth coarse particles (>2 mm) and sand contentincrease. Porosity was determined from the core samples,and shown to decrease with depth to 70 cm (Table II).Field capacity increased linearly with soil porosity in the



Table I. Texture, organic matter content and porosity from theditch where soil moisture probes are installed

Depth(cm)

Particles>2 mm (%)

Sand(%)

Silt(%)

Clay(%)

Orgmatter (%)

Texture

20 16Ð0 16Ð5 50Ð2 15Ð2 2Ð1 Silt-loam40 30Ð0 14Ð4 40Ð8 13Ð5 1Ð3 Silt-loam70 23Ð0 20Ð0 38Ð7 17Ð5 0Ð8 Silt-loam90 24Ð0 19Ð2 39Ð9 16Ð1 0Ð8 Silt-loam

Table II. Soil physical characteristics from the ditch where soilmoisture probes are installed

Depth (cm) Porosity (%) Field capacityŁ (%)

20 44Ð22 26Ð6340 38Ð46 19Ð5260 32Ð34 12Ð4090 37Ð67 19Ð52

Ł Obtained from porosity values using a linear model relating porosityand field capacity build on 86 samples (R2 D 0Ð82)

Figure 1. Soil moisture in the profile for the hydrologic years 2003/2004and 2004/2005. Each line represents the arithmetic mean of the four

probes located at each depth

range 0Ð35–0Ð55 measured for 86 samples (�fc D 1Ð19porosity— 0Ð25, R2 D 0Ð82) in the study area (Maneta,2006).

Soil moisture measurements

Figure 1 shows the average volumetric soil moistureat each depth for the 2 years of monitoring. Averagemoisture close to the surface was highest in the wettestmonths and lowest in the driest months as the influenceof the atmospheric demand is strongest near the surface.At deeper locations demand is less and soil moistureshowed a smaller range of variation from the wet to thedry season.

The amplitude of the standard deviation of soil mois-ture increases with depth and is highest at 70 and 90 cmfrom the soil surface (Figure 2). The spatial variabilityof the probes installed close to the surface was smallerprobably because they are wetted by precipitation and

Figure 2. Standard deviation of soil moisture for the four probes locatedat each depth

dried by evaporation and transpiration more homoge-neously than soils at lower depths where the effect of soilheterogeneities is stronger in the redistribution of waterwithin the soil and plant water extraction is more hetero-geneous. Variability of the deepest probe (90 cm) dropsdrastically in the dry season (summer) when the soil isalso at its driest state and the plants have scavenged forsoil moisture.

MODEL CONCEPTUALIZATION, BOUNDARYCONDITIONS AND DOMAIN

CHARACTERIZATION

It is a characteristic of the semiarid lands that the soilsare below field capacity most of the year. By definition,water below this threshold is held in the soil by tensionforces and is dominantly removed by evapotranspiration.This also indicates that soil water redistribution is mainlyvia vertical movements (i.e. downwards redistribution ofinfiltrated water in excess of field capacity (gravitationalwater) until equilibrium is reached and thereafter evap-otranspiration extracts water and returns it back to theatmosphere). The lack of horizontal water transfers belowthe subsurface in semiarid areas produces the hydrologicdisconnection of a point and its upslope area (Gomez-Plaza et al., 2001; Western et al., 1999).

This concept was used to calibrate the Kristensen andJensen model by considering a one-dimensional soil col-umn in a flat area, as soil moisture is below field capac-ity during a large part of the year and the horizontalmovement of water is considered to be small. Althoughthe model used implements a governing equation thatdescribes a 3D transient solution of the flow in soilsEquation (3), which allows for further research in morecomplex cases with lateral flows, the problem at handcould be sufficiently addressed using a 1D solution of thegoverning equation. The lateral and bottom boundaries ofthe domain are no flow boundaries. Therefore, exchangeof water occurs only through the soil surface drivenby incident rainfall and crop potential evapotranspirationfor the specific condition of the soil cover (Figure 3).Infiltration from run-off is neglected in the flat areas.



Figure 3. Upper face boundary conditions for the soil column. Verticallines are incident rainfall and the intervening line represents crop potential

evapotranspiration in terms of optimal crop water requirements

Crop-specific potential evapotranspiration (Etp) was cal-culated multiplying reference evapotranspiration for agrass field (Eto) obtained from the Penman–Monteithequation by a crop coefficient KC (see the following sub-section) that adjusts the reference evapotranspiration tovegetation conditions and growth stage as recommendedby FAO (Allen et al., 1998). Rainfall is added as rechargeto the soil water storage in the upper soil layer. When themaximum water storage is reached (soil is saturated), anyextra rainfall is assumed to form runoff. No ponding inthe surface is considered.

In the present case, each soil column has an area of1 m2 and is 1 m deep (soil depth) discretized in 20 layersof 0Ð05 m; therefore the entire domain has an area of9 m2 (9 elements of 1 m2) and a depth of 1 m with 20layers resulting in 180 nodes. To calibrate the model theyear with lowest soil moisture content was chosen solateral water transfers were considered to be minimum.This control volume was located in an open space (notrees) with a light herbaceous cover. This does not meanthat there are no deep roots from nearby trees, whichis in fact confirmed by the drying cycle at the bottom ofthe soil profile where the effect of evaporation is low andtherefore water below field capacity can only be retrievedby transpiration.

Domain characterization

Soil hydraulic properties needed in the model are fieldcapacity �fc, residual moisture �r and wilting point �wp.Porosity was obtained for each node by interpolation andextrapolation of the porosity data in Table II. The wiltingpoint soil moisture used was that at 4Ð2 pF (log10 of soilwater tension head in cm) of the current retention curve.A value of 0Ð05 for all depths was used for �r as it isapproximately the lowest soil moisture content measuredin the study area during a dry summer.

Because there is variation in soil characteristics withdepth (Tables I and II), vertical hydraulic conductivityKzz was assumed to decay from the value at the surfacefollowing an exponential curve (Beven and Kirkby,1979):

Kzzi D Kzz0e�fz �6�

Figure 4. Crop coefficients (straight solid line), reference optimal waterrequirements (intervening line) and stress periods (numbered 1 to 7

between vertical lines) for the pasture growth cycle

where Kzz0 is the hydraulic conductivity in the first nodei D 1 and f is an empirical decay parameter for hydraulicconductivity.

Similarly, an exponential decay with depth functionwas used to calculate the root distribution, choosing adecay parameter so that approximately 50% of the rootsare located in the first 25 cm of the soil as observed dur-ing the soil profile description and observed by Morenoet al. in the same area (2005). The maximum depthof direct evaporation (EDF) was set to 25 cm, whichis larger than the 15 cm recommended by Allen et al.(1998) to account for the deeper diffusion of water vapourexpected in sparsely cover soils in semiarid climates.Experimental research on average pasture production inthe study area determined the existence of seven periodsin the yearly pasture growth cycle (Figure 4). Starting atthe beginning of the hydrologic year, from September toNovember, pasture growth is at a minimum after the sum-mer. November to February is the initial stage of pasturegreening. Production increases gradually with autumnrains. The pasture development stage spans from Febru-ary to March, when pasture production rapidly increasesto its peak. The mid-season where pasture production ishigh is from March to mid-April. From April to Maypasture production decays rapidly to enter the late seasonstage from May to June. From June to August pastureproduction is again at a minimum. The cycle describedis used to identify the different growth stages and cropcoefficients.

LAI was used in the original formulation of themodel to represent the amount of transpiring biomass. Nomeasurements of the LAI of the herbaceous cover existfor the study plot; however there is information on thedry biomass of pasture that can be used to establish thisdependency. The dry biomass was obtained by cuttingand drying pasture from an area protected from grazingactivity with cages. Three cuts were available for thegreening, development and mid-season (peak) stages withvalues of 176Ð8, 227Ð3 and 397Ð1 g m�2, respectively.An average minimum value of 100 g m�2 was assumedfrom September to November and from June to August.



Table III. Time periods, dry biomass values and crop coefficientsused in the simulations. Bold values of biomass indicate experi-

mental values

Period Dry biomass (g m�2) Kc

1 Sept to 31 Oct 100 0Ð201 Nov to 31 Jan 176Ð8 0Ð301 Feb to 29 Feb 227Ð3 0Ð511 March to 31 March 397Ð1 0Ð731 April to 14 April 273Ð6 0Ð7315 April to 31 May 150 0Ð751 June to 31 August 100 0Ð20

This is because although pasture is almost nonexistent inthe study area during the summer period, there is stillwater uptake by the roots of nearby trees (Holm oaks,Quercus rotundifolia) that depletes the soil water contentin the soil beyond the direct evaporation depth (Morenoet al., 2005). For the periods where no information exists,the dry biomass has been estimated proportionally tothe known pasture production curve through the yearand the three experimental values (Table III). The cropcoefficients shown in Table III for the different growthstages were calculated using the FAO tabulated cropcoefficients for pasture and adjusted numerically for theaverage wind speed and relative humidity conditionsfor each growth period in the area (Allen et al., 1998).Because no pasture growth model is included in theformulation, the pasture growth and senescence cyclesare prescribed in the model using the crop coefficientsto calculate reference optimal crop water requirements(Etp) at different growth stages used in the Kristensenand Jensen model.

MODEL CALIBRATION

Data from the hydrologic year 2004–2005 was chosento calibrate the model because it was dryer than theyear 2003–2004 and hence it better supports the assump-tions of vertical movement of water when soil moistureis below field capacity. Data from year 2003–2004 wasused for validation. A total of seven parameters: threesoil parameters ˛, ˇ and Kzz for the topsoil node, thehydraulic conductivity exponential decay parameter fand three evapotranspiration parameters C1, C2 and C3

were included in the optimization process. In high dimen-sional inverse problems, the simultaneous estimation of alarge number of parameters may result in numerical insta-bility due to a non-smooth solution space or insensitivityof the parameters derived from parameter correlation. Inthat case, a robust global search calibration technique andthe inclusion of additional information to regularize theproblem (i.e. lay a better posed inversion problem) areadvised. In this research we explore the entire parame-ter space to locate the global minima using the shuffledcomplex evolution (SCE) algorithm (Duan et al., 1992).The bounds of the parameter space allowed in the globalsearch are shown in Table IV. The final set of param-eters identified by the SCE algorithm is used as the

Table IV. Search space boundaries for the SCE (global) andPEST (local) algorithm

Parameter Units Global search Local search

Lower-upper bound Lower-upper bound

C1Ł g�1 m2 1Ð0E-06–0Ð01 0Ð001–0Ð01

C2 — 1Ð0E-06–0Ð01 0Ð001–0Ð5C3 cm h�1 1Ð0E-04–0Ð1 1Ð0e-7–1˛ cm�1 0Ð001–0Ð01 0Ð0005–1ˇ — 1Ð2–3 1–10Kzz0 cm h�1 0Ð01–10 0Ð001–15f cm�1 0Ð001–0Ð2 1Ð0E-06–0Ð2Ł Units for parameter C1 are when LAI is replaced by aerial dry biomassin Equation 1(b)

starting point for a gradient based, local search algo-rithm to further refine the calibration. A larger parameterspace is allowed during the local search (Table IV). TheLevenberg–Marquardt algorithm, as implemented in thePEST optimization suite (Doherty, 2002), was chosen toperform the local search.

To regularize the inversion problem, further informa-tion is included during the local search part of the cal-ibration process. This extra information includes priorknowledge on the grass coverage, prior knowledge on thevalue of field capacity and the requirement (constraint)to fit a known empirical trend given by a linear modelassociating water content and daily maximum tempera-tures.

In practice a multi-criteria calibration approach withfour objective functions was followed. One objectiveF1�b� was the minimization of the sum of the squareweighted errors between the vector of n simulated (qsim)and the vector of n observed (qobs) soil water content

F1�b� D �qobs � qsim�TQ1�qobs � qsim� �8�

in which Q1 is a nð n diagonal matrix with measurementweights inversely proportional to the measurement error.

The second objective F2�b� was to bring the residualsof a simple linear model associating soil water contentand maximum daily temperatures as close to zero aspossible

F2�b� D �0 � r�TQ2�0 � r� �9�

where 0 is the zero matrix, r is the n-dimensional vec-tor of residuals between observed and model-generatedmaximum daily temperatures (r = Tobs − Tsim) and Q2

is a nð n diagonal matrix of weights that are inverselyproportional to the standard deviation of r. The linearregression model (see below) is used with the simulatedsoil water content to generate the vector of associatedmaximum daily temperatures.

The third objective function F3�˛, ˇ� was to generatea retention curve that would produce a value for fieldcapacity (soil moisture corresponding to 2Ð5 pF in thecalibrated retention curve) as close as possible to the



value of field capacity predicted from porosity qfc(h)

F3�˛, ˇ� D �qfc�h�� qfc�˛, ˇ��TQ3�qfc�h�� qfc�˛, ˇ��

�10�

where qfc(h) is field capacity obtained from the linearregression relating porosity and field capacity, qfc�˛, ˇ�is the field capacity obtained from the retention curve atpF 2Ð5 and Q3 is a nð n diagonal matrix of weightsinversely proportional to the square root of the unex-plained variance of the regression model.

The fourth objective function F4�C1, C2� includessome prior knowledge on the grass coverage in thecalibration process. Soil is bare or only partially coveredby grass during most of the year. Full grass coverage isonly achieved when the amount of biomass is at its peak.In that case, the sun does not hit directly the bare groundand evaporation is reduced to a minimum. This effectis controlled by Equation (1b), in that case the equationshould be equal to unity when the amount of biomass isat a maximum (397Ð1 g m�2, Table III)

F4�C1, C2� D f[f1�397Ð1�� 1]Q4g2 �11�

where Q4 is a dimensionless scalar weight determinedexperimentally.

All objective functions are weighted and summedtogether to form a final pooled objective function (F�b�)to be minimized:

F�b� D w1F1 C w2F2 C w3F3 C w4F4 �12�

In this equation w1, w2, w3 and w4 are relative weightsmodulating the importance of each objective in the finalpooled function.

Figure 5 shows a scheme for the entire calibrationgroup. The purpose of this multi-objective procedure is toinclude enough information so that all the parameters aresensitive throughout the calibration and the search spaceis smoothed to improve the performance of the searchalgorithm.

Figure 5. Model ensemble optimization scheme. Pest runs the root wateruptake model. The simulated soil water content (�sim) is used to calculatethe sum of the squared weighted errors between observed and simulatedsoil moisture (F1) and the residual between predicted and observedmaximum temperature using the linear model (F2). F3 and F4 use priorinformation on parameters that can be compared to the current calculatedparameters. The residuals of all four functions are weighted and added

together into Fpool

Figure 6. Linear regression model relating measured daily maximumtemperature and measured average soil water content

Model ensemble for second objective function. Theexisting linear relationship between daily maximum tem-perature and soil water content may be used to obtainan estimate of soil water content (Figure 6, R2 D 0Ð71).An important characteristic is that this relationship has aconsistent trend that is followed at low and high watercontents. This trend was used, together with the measuredsoil water content, to calibrate the evapotranspirationmodel. The rationale behind the second objective func-tion was to produce a robust calibration using informationthat is not specific to the conditions of a hydrologic yearbut is a long-term trend. The hypothesis was that soilmoisture information reflects the specific conditions ofthe measurement period, introducing a potential bias inthe parameters. This bias may be reduced by introducingsome information that reflects a more general behaviorof the system, even though it may result in a reducedperformance of the model in the calibration period. Fur-thermore, the use of a multiple objective approach withthe inclusion of this fuzzy information may provide a bet-ter posed problem introducing more information that canbe crucial to help the calibration algorithm in the properidentification of parameters.

To use this information in the calibration process,the linear model was inverted so that daily maximumtemperature is written as a function of soil water content.The inverse of the model presented in Figure 6 is Tsim D�87Ð893� C 48Ð04. Each time the water uptake modelis run during the calibration process, the modelled soilwater content time series is averaged and used as aninput in the linear model to produce a time series ofcalculated maximum daily temperatures. The modelledtemperature time series was compared with the timeseries of measurements and the residuals were calculated.The objective of the calibration algorithm was bringingthose residuals as close as possible to zero as explainedabove.

Weights. Choosing the weights embedded in the Qmatrix (Equations (8) to (11)) to be inversely propor-tional to the standard deviation of the measurements isimportant to produce a group of dimensionless residualsthat can be squared and summed together in a meaningfulway when working with measurements having different



dimensions (Hill, 1998). In addition, such a set of chosenweights equalizes the different magnitudes of the residu-als in terms of their errors, so less accurate measurementswill not have undue influence in the pooled objectivefunction (Hill, 1998; Doherty, 2002). Furthermore, sucha set of weights makes sure that the minimization ofthe least squares will produce the maximum likelihoodestimation of the parameters (Aster et al., 2005). TheTDR probes used in this research have an estimated stan-dard error of measurement of 0Ð026 volumetric soil watercontent (Martınez Fernandez and Ceballos, 2001). Invert-ing this value a weight of 38Ð46 (1/0Ð026 D 38Ð46) isobtained and was used for each soil moisture observationQ1. The value in Q2 used to weigh the residuals of thelinear model was similarly the inverse of the standarderror of the estimation of the maximum daily temper-ature: 1/4Ð627 °C D 0Ð216). In the case of Q3 and Q4,their values were set to 88Ð56 and 10, respectively. Inboth cases, the values were chosen so that their contri-bution of all the objectives to the pooled function wassimilar when all the w weights Equation (12) were setto 1.

The effect of the external forcing represented bytemperature in the calibration process is studied bygradually increasing the relative importance of F2 in thefinal pooled objective function. For this, a set of trade-off weights w1 and w2 were used for seven differentcalibrations starting with a value of 1 for w1 and 0 for w2

for the first calibration and then increasing the importanceof F2 using the pair of weights 0Ð95–0Ð05, 0Ð92–0Ð08,0Ð9–0Ð1, 0Ð87–0Ð13, 0Ð85–0Ð15 and 0Ð8–0Ð2 for w1 andw2, respectively, in the subsequent six calibrations.

Although w3 and w4 are fixed to a value of 1 for allcalibrations, a trade-off between w1 and w2 implies thatthe relative influence of w3 and w4 in the pooled objective

function also change, allowing the exploration of a widepart of the solution space but maintaining the focus onF1 and F2. Recall that an objective of this researchis to investigate how an extra calibration criterion (F2)holding information on some external forcing affects thecalibration process compared with including only theinformation about soil moisture content represented byF1. In that scheme, w3 and w4 are weights for priorinformation about the values of the parameters while w1

and w2 balance the influence of information on the statevariables and external forcing to the system respectively.

RESULTS AND DISCUSSION

Calibration results

The final calibrated parameters after running the SCEalgorithm and then PEST for the different w1 and w2

weights are given in Table V. Each calibrated set ofparameters produces a root mean square error (RMSE)in the calibration period as shown in Table VI.

The RMSE at each individual depth as well as theoverall RMSE for the different combinations of w1 –w2

is similar although there is a downward trend for the20 cm layer and an upward trend for the 90 cm layeras w2 increases. Furthermore, the RMSE is higher forthe shallower probes, where temporal soil moisturevariability is higher and decreases for deeper probes,where soil moisture is more stable over time. Exceptfor parameters ˇ and C3, the final parameter values forall the calibration runs are generally in the vicinity ofthe parameters calibrated with the SCE algorithm. In thecase of ˇ and C3, the calibrated values were limited bythe predefined search space used in the SCE algorithm.The wider parameter space prescribed for PEST allowed

Table V. Calibrated parameter values

SCE PEST calibration for different combinations of w1 –w2

1Ð0–0Ð0 0Ð95–0Ð05 0Ð92–0Ð08 0Ð9–0Ð1 0Ð87–0Ð13 0Ð85–0Ð15 0Ð8–0Ð2

C1 0Ð0056 0Ð0014 0Ð0029 0Ð0031 0Ð0028 0Ð0034 0Ð0036 0Ð0048C2 0Ð0041 0Ð5000 0Ð0089 0Ð0013 0Ð2556 0Ð0012 0Ð0349 0Ð0010C3 1Ð04E-04 1Ð22E-06 2Ð94E-07 1Ð00E-07 1Ð00E-07 1Ð00E-07 1Ð00E-07 5Ð22E-07˛ 0Ð0035 0Ð0032 0Ð0034 0Ð0036 0Ð0033 0Ð0037 0Ð0032 0Ð0034ˇ 2Ð998 6Ð442 3Ð754 3Ð446 4Ð101 3Ð214 4Ð858 3Ð986Kzz0 9Ð620 13Ð2458 9Ð6333 9Ð6076 9Ð6171 9Ð6092 9Ð5980 9Ð5840f 0Ð093 0Ð0984 0Ð0948 0Ð0965 0Ð0983 0Ð1039 0Ð1027 0Ð1008

Table VI. Root mean square error for soil moisture prediction (calibration period)

Depth (cm) SCE PEST calibration for different combinations of w1 –w2

100–0 95–5 92–8 90–10 87–13 85–15 80–20

20 0Ð052 0.053 0.050 0.050 0.050 0.048 0.049 0.04950 0Ð052 0.048 0.055 0.056 0.051 0.057 0.054 0.05370 0Ð050 0.042 0.052 0.052 0.048 0.050 0.047 0.04890 0Ð041 0.036 0.040 0.041 0.041 0.052 0.046 0.044Overall 0Ð048 0.044 0.048 0.049 0.046 0.050 0.048 0.047



adjustment of these parameters to a larger value duringthe local search runs.

For the evapotranspiration model, parameters C1 andC2 are, respectively, the slope and intercept of the lin-ear function that controls transpiration as a function ofbiomass Equation (1b). While parameter C1 has consis-tent values across the different calibrations, the valuesfor parameter C2 drifts over two orders of magnitudewith no apparent trend, suggesting low sensitivity ofthis parameter to the data used for calibration. Withthe calibrated values for those parameters, full tran-spiration (Equation 1b reaches 1) occurs in calibrationw1 –w20Ð8–0Ð2 with the lowest amount of biomass (210 gm2) and needs a larger amount of biomass as the influenceof F2 decreases (maximum of 350 g m2 for calibra-tion 1Ð0–0Ð0). Parameter C3 controls the shape of thefunction relating transpiration to soil moisture content(Equation (1c)). It consistently achieves very low valuesthat produce small C3/Etp ratio values. It indicates thatduring periods of high evapotranspiration demand (sum-mer period or during daytime) the model quickly inhibitstranspiration in favour of evaporation as soon as the watercontent falls below field capacity.

Regarding soil parameters, except for the calibrationwith no influence of F2 (1Ð0–0Ð0), the different calibratedparameters ˛ and ˇ produce similar soil retention curvesshown in Figure 7a. Figure 7b presents the relative soilpermeability (krw) curves as a function of the degree ofsaturation. Also in this case, calibration 1Ð0–0Ð0 presentsa slightly more permeable soil (least steep curve on top)although there is little influence of F2 on the steepness ofthe permeability curve. Figure 7c shows the distributionof saturated vertical hydraulic conductivity with depth(parameters Kzz and f). Similarly to the previous case,hydraulic conductivity at the topsoil and its decay is dif-ferent for calibration 1Ð0–0Ð0 and similar for the rest ofthe calibrations although it converges rapidly as depthincreases. This is an indication of correlation betweenparameters Kzz0 and f Equation (6) whose effects can-cel each other out by producing a slightly steeper decayin hydraulic conductivity with depth (parameter f) forhigher values of hydraulic conductivity at depth zero(Kzz0). Looking at the parameter for the different calibra-tion schemes, a clear influence of the second objectivefunction is only apparent in parameters C1 and ˇ. In thefirst case, the calibrated value increases along with theinfluence of F2 (Figure 8) while for ˇ it acquires a lowercalibrated value as soon as F2 enters the calibration pro-cess. The rest of the parameters fluctuate close to a meanvalue unrelated to F2.

Validation period

Simulated soil moisture at the monitored depths forthe independent validation year 2003–2004 is shown inFigures 9a–d and the RMSE for each individual depthas well as the overall RMSE is shown in Table VII.Calibrations generally produce very similar curves forthe validation period. All solutions tend to underestimate

Figure 7. Set of calibrated retention curves with the different calibrationschemes (a); Set of calibrated relationship between relative permeabilityand soil saturation (b); Calibrated distribution of soil hydraulic conduc-tivity with depth according to the different calibration schemes. The line

with circles indicate calibration 1.0–0.0

Figure 8. Value of parameter C1 affected by weighting of F2

the sudden rise in soil water content triggered by the highrainfall event at the end of October 2003, especially at the20 cm depth location (Figure 9a). During the wet periodof the year (December through May) the soil is close tosaturation. Simulated recession of the curve at the end ofthe period matches measured values well for all depths



Figure 9. Prediction (lines) and measured (dots) volumetric soil water content (SWC) for the validation year 2003–2004 at (a) 20 cm, (b) 40 cm,(c) 70 cm, (d) 90 cm depth. ‘Whiskers indicate š10% of the SWC value’

Table VII. Root mean square error for soil moisture prediction (validation period)

Depth (cm) SCE PEST calibration for different combinations of w1 –w2

100–0 95–5 92–8 90–10 87–13 85–15 80–20

20 0Ð055 0.054 0.050 0.049 0.051 0.047 0.048 0.04950 0Ð026 0.026 0.028 0.027 0.025 0.026 0.025 0.02570 0Ð032 0.025 0.033 0.033 0.030 0.032 0.029 0.03090 0Ð046 0.045 0.048 0.050 0.050 0.060 0.056 0.053Overall 0Ð041 0.039 0.040 0.041 0.040 0.043 0.041 0.041

Figure 10. Components of the mass balance during the validation year. Etp is crop potential transpiration in terms of optimal crop water requirements.Eta is actual evapotranspiration

except at 90 cm, where soil moisture is overestimated. Interms of RMSE, the simulations of soil moisture contentsat 20 and 90 cm are simulated with less accuracy thanthe soil moisture contents in the middle of the profile(Table VII).

The components of the water balance for the vali-dation period for the different calibrations are shownin Figure 10. The differences between the calibrationsare small, producing very similar cumulative values. Atthe end of the period, total recharge for all calibrationsaveraged 366 mm; average actual evapotranspiration was382 mm. Soil water storage quickly increases from Octo-ber to December, when recharge increases in the autumn

and winter rains and actual evapotranspiration are low.Soil storage stays relatively stable during the wet periodfrom December through March. During those months,cumulative potential and actual evapotranspiration (Eta)increase slowly and there is little water stress on vegeta-tion. Actual evapotranspiration runs at about 95% of theoptimal crop water requirement in June when hydrologicstress starts to increase rapidly. Also, by the end of Jan-uary, the soil is almost saturated and part of the incidentprecipitation does not recharge the soil. At the end of theyear, actual recharge to the soil profile is 75% of the totalprecipitation. From March on, potential and actual evap-otranspiration increase rapidly and water storage depletes



until it reaches similar October storage levels around July.From July on, cumulative actual evapotranspiration slowsdown and diverges faster from optimal crop evapotran-spiration, indicating that there is an increasing hydrologicstress on the pasture. At the end of the year, actual evapo-ration is about 83% of the calculated annual optimal cropwater requirement.

DISCUSSION OF THE RESULTS

All calibration sets produced a very similar mass balancein the soil column, even when some of the parameterswere substantially different. This indicates that the effectsof some parameters are correlated and cancel out oneanother’s effect in the prediction of soil water content.Soil moisture content measurements, through F1 pro-vides the calibration process with information about themass balance. This objective conditioned the parametersso that the model produced a soil moisture predictionthat matched as closely as possible the soil moistureobservations. Unfortunately, a correct mass balance canbe achieved through different combinations of the sub-components of the mass balance, which is reflected inparameter correlation in a calibration context.

Figure 11 shows the evaporation and transpirationcomponents for four different calibrations. The trade-off between evaporation and transpiration is mainlycontrolled by parameters C1, which is sensitive to F2, andby parameters C2 and C3. When F2 is introduced, someinformation beyond the mass balance (provided by F1)is included in the calibration process. F2 can be regardedas a rough source of information on atmospheric demandand seasonality. F2 forces the parameters to produce anoutput that matches average soil moisture to maximumtemperature. Since the soil water storage extractable byevaporation and transpiration are different (evaporation

reaches 25 cm while transpiration reaches the entireprofile depth) and the search algorithm is trying to matchthe averaged soil moisture content, it affects the partitionof evaporation and transpiration. Increasing the influenceof F2 in the calibration process increases exponentiallythe transpiration to evapotranspiration ratio except forcalibrations 1Ð0–0Ð0 and calibration 0Ð9–0Ð1, which areoutliers to the trend presenting a larger contribution bytranspiration.

In calibration 1Ð0–0Ð0 (Figure 11a), transpiration dom-inates over evaporation through the year accountingfor 57% of annual evapotranspiration. Those values arewithin the range of variation in the reported transpi-ration ratios for semiarid lands (Ng and Miller, 1980;Paruelo and Sala, 1995; Ferreti et al., 2003; Laurenrothand Bradford, 2006; Scott et al., 2006). The partition ofevapotranspiration appears to be very sensitive to manyfactors identified in various studies although differentauthors rank them in different order of relevance. Lau-renroth and Bradford (2006) report biomass and annualcycles of vegetation as the most sensitive factors over soilcharacteristics and precipitation while others indicate thatprecipitation pulses or root distribution is more deter-minant for transpiration (Kemp et al., 1997; Kurc andSmall, 2004). Although different factors may play differ-ent roles in the transpiration ratios, there is an agreementthat the transpiration share of actual evapotranspirationis larger during the grass growing season. Ferreti et al.(2003) report for a grassland in Colorado a transpira-tion to actual evapotranspiration ratio of 0Ð6 during thegrowing and up to less than 0Ð1 during the dormant sea-son depending on the wetness of the year. In situationswhere pasture does not fully cover the soil, evaporationfrom bare soil patches can overrun transpiration in draw-ing water from the topsoil, where most of the pasture

Figure 11. Components of actual evapotranspiration. Thick line is transpiration; thin line is evaporation. (a) calibration 1Ð0–0Ð0; (b) calibration0Ð9–0Ð1; (c) calibration 0Ð08–0Ð92; (d) calibration 0Ð2–0Ð8



roots concentrate. If evaporation rates are high, vegeta-tion may enter the senescence period to reduce transpira-tion while evaporation may still continue extracting waterfrom the soil. This behaviour is reproduced in calibration0Ð9–0Ð1 (Figure 11b). In this case, the transpiration toevapotranspiration ratio is 0Ð53 during winter and springalthough at the year scale, the system is dominated byevaporation, accounting for 56% of total evapotranspi-ration. The difference between the components becomeslarger in late spring, when grass coverage is still abundantand atmospheric demand is high. Once summer startsevaporation competes harder with vegetation for water,evaporation increases while transpiration decreases. Thedifferent rooting depths may cause a progressive declinein grass density and thus in overall transpiration needsbefore the soil water storage is depleted and while evap-oration rates are still high. Furthermore, in competitiveenvironments with poor soils, grass may enter this loweractivity period to reduce water use and because in poorsoils they do not have the vigour to extract water athigh tension. The relationship between grass productiv-ity and competition for water in semiarid rangelandsand grasslands is complex (Mueller-Dombois and Sims,1966; Moyes et al., 2005) and sometimes senescence mayarrive as early as April/May, such as reported by Moyeset al. (2005) for Ripgut brome (Bromus diandrus) inCalifornia, leaving soil moisture still available for evap-oration compared with forbs, which senesced later andfor a longer period. Calibrations other than 1Ð0–0Ð0 and0Ð9–0Ð1 produce scenarios where evaporation dominates(exemplified in Figure 11c and d) as the influence ofF2 decreases. The evaporation ratios range from 61%(calibration 0Ð2–0Ð8) to 76% (calibration 0Ð08–0Ð92).Those are also possible but unlike scenarios in the studyarea. Certainly, in a semiarid grazed rangeland if grasscover is sparse enough and the sun hits the bare soilfor long periods transpiration may not be the dominat-ing component at the annual scale (Ng and Miller, 1980;Paruelo and Sala, 1995) although for the environmentwhere this study was carried out complete dominationby evaporation throughout the year is unlikely since sucha case is reported typically in extreme conditions whereno deep recharge exists (Evans et al., 1981). Unfortu-nately, with the available information it is not possible toknow the optimal weighting scheme between the objec-tive functions that would produce the correct componentsof evapotranspiration and further research is necessaryto reach sound conclusions. Experimental data for vali-dation is especially important when the components ofevapotranspiration are sensitive to many factors and thewide variability of responses as reported in the literatureis possible.

A further refinement of calibration was possible usinga local search algorithm after ensuring the region of theglobal minimum was identified using a global searchon the parameter space (Vrugt et al., 2001a; 2001b).By running the different local search optimizations, thechange in the position on the optimal vector of parametersindicates how the parameter space is affected by the

balance of objectives in the pooled function. The optimalparameter vector shifts mainly driven by small changesin C1, ˇ and Kzz0 while the large mobility of C2 indicatesthat the parameter space in that direction is flat over mostof its range, with no clear gradient that may help thesearch algorithm to find the optimal value. Parameter C1

is plant dependent. The value suggested by Kristensenand Jensen (1975) and the Danish Hydraulic Institute(2004) for parameter C1 is 0Ð3. This value for C1 isrecommended when LAI is used as a measure of biomassin Equation (1b). Because dry grass weight used in thisstudy to measure biomass is about a factor of 100 largerin magnitude than the typical values for LAI in pasturesand grasslands, the converted value for C1 should beabout a factor of 100 smaller. That value is in the rangeof calibrated values for the parameter. Parameter C2 isthe intercept of Equation (1b) and is interpreted as a basicevaporation taking place regardless of vegetation densityand soil dryness resulting from the process of diffusionfrom the moist soil atmosphere to the generally drieratmosphere (Kristensen and Jensen, 1975). For cropsand grass on clayey loamy soils the suggested value isaround 0Ð2. This parameter achieves a similar value tothat recommended in calibration 0Ð9–0Ð1, generally beinga small value for other calibrations. The inconsistent driftof its value across calibrations is an indication of lowsensitivity with respect to the experimental data used.

Parameter C3 Equation (1c) shapes the function thatmodels the actual to potential evaporation ratio withrespect to the soil moisture content. Kristensen andJensen (1975) suggest a value of 0Ð042 cm h�1 whereasthe Danish Hydraulic Institute (2004) recommend ahigher value of 0Ð083 cm h�1. The calibrated value ismuch lower in all cases and in four of the calibrationsit hits the lower allowed limit for the parameter Smallvalues of C3 increase the effect of soil dryness on theactual to potential evapotranspiration ratio. The suggestedvalues in the literature are based on experiments donein colder regions and on longer (daily) time stepsthat smooth out hourly evapotranspiration maxima andminima. Using potential evapotranspiration at hourly timesteps in Equation (1c) requires a lower value for C3 toproduce appropriate C3/Etp ratios for daily minimum Etpvalues. When applied to small time steps Equation (1c)produces inhibited transpiration during the periods whenpotential evapotranspiration is high (small C3/Etp ratio)and enhances transpiration during the night or duringperiods of low potential evapotranspiration ratios. Thisformulation may be inadequate for southern climateswhere high evapotranspiration rates are expected for longperiods, over-inhibiting transpiration and thus prone to asplit of evapotranspiration dominated by evaporation assoon as the water content falls below field capacity. It issuggested that the C3 parameter be set to lower valuesfor soils with high retention capabilities and sparse roots(DHI, 2004).

The influence of F2 and F3 in the calibration processesis noticed in the calibrated values of ˇ. Its large valuein calibration 1Ð0–0Ð0 produces a steep curve typical of



coarse textured soils. Its value decreases when functionF2 is included in the calibration process. Because theweights for F1 and F2 compete in a trade-off, theinclusion of F2 implies a reduction of the weight of F1

so the relative influence of F3 with respect to F1 alsoincreases in the pooled objective function. The combinedinfluence of the different objectives in the pooled functionproduced retention curves more akin to those of siltloam textures. Still, the calibrated values are slightlylarger than the average values suggested in look-up tablesfor silt-loam soils (Schaap et al., 2001). A reason forthis is that while the databases are based on laboratoryanalysis using core samples that only consider flowthrough the soil matrix, the values calibrated using thisinverse modelling approach calibrate for field conditionsincluding the effects of macroporosity and coarse soilmaterial (stones). This scale effect is more noticeable forfiner textures, as is shown by Pachepsky et al. (2001),who report smaller measured values of water retention atthe field scale than at the laboratory scale.

A calibrated larger value for parameter ˇ indicates theneed for a steeper retention curve and more permeablesoils (Equation (4)) in order to match the fast dryinglimb of the measured soil moisture time series. Amore retentive curve, such as those produced if usingthe suggested parameter for silty-loam soils, cannotreproduce the observed drying cycle. The reason maylie in the effect that macroporosity has in the waterand energy fluxes, allowing faster water circulation andflushing and more efficient air and energy penetration thatmay encourage evaporation. Thus, the calibrated retentioncurve parameter may be regarded as effective parameterscompensating for those processes not explicitly capturedin the model.

A similar consideration is valid for the calibrated valueof Kzz, which is larger that the 0Ð3 cm h�1 saturatedhydraulic conductivity measured on soil cores in a nearbyarea (Ceballos and Martınez Fernandez, 2002). The largecalibrated values for hydraulic conductivity may be seenas compensation for rapid macropore fluxes. However,this large value of hydraulic conductivity is only forthe upper layers of the soil as it decreases rapidly withdepth given the calibrated decay parameter f of the Kzz0distribution function (Figure 7c). A value of about 0Ð3 cmh�1 is already achieved at 35 cm depth.

Without the combined influence of the different objec-tives, the calibration model is free to assign unrealisticvalues of parameters. The search algorithm may do so tocompensate for processes not explicitly included in themodel structure. This may be one of the reasons why cal-ibration 1Ð0–0Ð0 seems to perform slightly better in termsof RMSE than the rest, although this improvement wouldbe at the expenses of simulating less realistic retentioncurves and perhaps overestimated transpiration ratios.

Although some differences exist, in general, all thecalibration schemes perform similarly in terms of soilmoisture prediction during the validation year, so thecalibrations are functional if the partition of evapotranspi-ration is not needed. With the calibrated parameters, the

model can reproduce the general pattern of observed soilmoisture (Figure 9a–d). The most important inaccuraciesare at shallow depths (Figure 9a) and during recessionat deeper levels (Figure 9d). At 20 cm, soil moisture ismore variable as it is more exposed to atmospheric con-ditions and to possible uncontrolled water inputs fromrun-off. In the case of the deeper probe values, most ofthe mismatch occurs during the drying cycle, which mayindicate that the calculated effective root density at thatpoint may be too low so that not enough transpirationextraction is at the lower levels. An appropriate choiceof the root distribution within the soil profile is criticalfor improving the accuracy of predictions (Kemp et al.,1997; Kurc and Small, 2004). The accuracy of a dis-tributed two-dimensional root water uptake model with 9calibration parameters, six of which are related to the rootdistribution function, can predict the soil water storagewith an accuracy similar to the standard error of the watercontent measurements (Vrugt et al., 2001a). In the casepresented, the performance of the Kristensen and Jensenmodel calibrated for a semiarid rangeland is similar tothe 10% accuracy reported by these authors (Kristensenand Jensen, 1975). The results of the model are betterassessed if we consider the standard errors of the TDRprobes (0Ð0226 m m�1). In the best case the prediction ofsoil moisture during the validation period has an RMSEsimilar to the standard errors of the probes (predictionat 40 and 70 cm from the topsoil) and on average theprediction RMSE is smaller than twice the TDR probes’accuracy.

CONCLUSIONS

Three parameters of the Kristensen and Jensen model(1975) and four parameters of the flow model werecalibrated using soil moisture measurements taken ina 1 m deep soil profile. The calibration was doneusing a global search algorithm to estimate the globalminima and a local search algorithm to refine the search.Prior knowledge of the values of field capacity andmaximum grass coverage as well as the trend in a knownrelationship between daily maximum temperatures andaverage soil moisture for the column were used duringthe local search stage of the calibration process to testif the effects of that extra information could improvethe parameter search process. The relative novelty ofthis approach is that it uses a trend as a ‘softer’ sourceof information that may help to formulate a betterposed calibration process. The results of the differentcalibrations were validated by predicting soil moisturefor an independent hydrologic year.

The results showed that complementing a global searchcalibration method with a gradient based local searchalgorithm can refine the calibration in the restricted vicin-ity of the search space indicated by the result of theglobal search algorithm. Also, the known relationshipbetween maximum temperature and average soil moistureincludes some extra information on seasonality of atmo-spheric demand. This extra information brings the model



parameters to values that produce a trade-off betweenevaporation and transpiration describing different massbalance scenarios. All calibrated scenarios were satis-factory in terms of simulation of the volumetric soilwater content and produce operative models when onlyintegrated evapotranspiration demand is needed. Withoutfurther experimental information it was not possible todecide which scenario produced the correct partition oftranspiration and evaporation. Furthermore, including thedifferent objectives in the calibration process producedparameters for the retention curve closer to the valuesexpected for the type of soil.

The ratio of evaporation to transpiration, partiallycontrolled by C1, was sensitive to F2. The influence ofparameters C2 and C3 on the results were more limitedand their values across the different calibrations wereless consistent. In spite of the difference in parametervalues between the calibrations and from the valuessuggested in the literature, the performance of the modelwas satisfactory suggesting an appropriate range of valuesfor C1 if aerial dry biomass is used instead of LAIin the Kristensen and Jensen model. Also, alternativevalues for parameter C3 are offered when the model isused in semiarid rangelands using hourly time steps thatinclude daily maximum and minimum evapotranspirationrates. In those conditions the values for C3 have to besignificantly lower than those suggested in the literatureto produce adequate C3/Etp ratios in the transpirationefficiency function of the Kristensen and Jensen model.

ACKNOWLEDGEMENTS

Research funded by the Spanish Ministry of Science andTechnology through the projects PROHISEM (REN2001-2268-C02-02) and CANOA (CGL-2004-04919-C02-02).

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calibration of an evapotranspiration model to simulate soil water dynamics in a semiarid rangeland

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