Weighted objective function selector algorithm for parameter estimation of SVAT models with remote sensing data

Authors

  • Joseph A. P. Pollacco,

    1. Department of Biological and Agricultural Engineering, Texas A&M University, College Station, Texas, USA
    Search for more papers by this author
  • Binayak P. Mohanty,

    Corresponding author
    1. Department of Biological and Agricultural Engineering, Texas A&M University, College Station, Texas, USA
    • Corresponding author: B. P. Mohanty, Department of Biological and Agricultural Engineering, Texas A&M University, College Station, TX 77843-2117, USA. (bmohanty@tamu.edu)

    Search for more papers by this author
  • Andreas Efstratiadis

    1. Department of Water Resources and Environmental Engineering, School of Civil Engineering National Technical University of Athens, Zographou, Greece
    Search for more papers by this author

Abstract

[1] The objective function of the inverse problem in Soil-Vegetation-Atmosphere-Transfer (SVAT) models can be expressed as the aggregation of two criteria, accounting for the uncertainties of surface soil moisture (θ) and evapotranspiration (ET), retrieved from remote sensing (RS). In this context, we formulate a Weighted-Objective-Function (WOF) with respect to model effective soil hydraulic parameters, comprising of two components for θ and ET, respectively, and a dimensionless coefficient w. Given that the sensitivity of θ is increased by omitting the periods when soil moisture decoupling occurs, we also introduce within the WOF a threshold, θd, which outlines the decoupling of the surface and root-zone moisture. The optimal values of w and θd are determined by using a novel framework, weighted objective function selector algorithm (WOFSA). This performs numerical experiments, assuming known reference conditions. In particular, it solves the inverse problem for different sets of θ and ET, considering the uncertainties of retrieving them from RS, and then runs the hydrological model to obtain the simulated water fluxes and their residuals, ΔWF, against the reference responses. It estimates the two unknown variables, w and θd, by maximizing the linear correlation between the WOF and maximum ΔWF. The framework is tested using a modified Soil-Water-Atmosphere-Plant (SWAP) model, under 22 contrasting hydroclimatic scenarios. It is shown that for each texture class, w can be expressed as function of the average θ and ET-fraction, while that for all scenarios θd can be modeled as function of the average θ, average ET, and standard deviation of ET. Based on the outcomes of this study, we also provide recommendations on the most suitable time period for soil moisture measurements for capturing its dynamics and thresholds. Finally, we propose the implementation of WOFSA within multiobjective calibration, as a generalized tool for recognizing robust solutions from the Pareto front.

1. Introduction

[2] In the hydrological community, there is a growing interest to make suitable usage of data retrieved from remote sensing (RS), to be employed within physically based models. Two of the most typical variables, which are of key importance in hydrological modeling, are surface soil moisture, θ [Sun et al., 2007; Wang et al., 2008; Zhan et al., 2008; Naeimi et al., 2009; Entekhabi et al., 2010] and actual evapotranspiration, ET [e.g., Wang et al., 2008; Wu et al., 2008; Hong et al., 2009; Ramos et al., 2009; Teixeira et al., 2009a]. In particular, RS data of this type have been used to invert the soil hydraulic parameters of Soil-Water-Atmosphere-Plant (SVAT) models [e.g., Mohanty and Zhu, 2007; Ines and Mohanty, 2008a, 2009; Gutmann and Small, 2010]. Recently, Pollacco and Mohanty [2008a] performed numerical experiments under 18 contrasting hydroclimatic scenarios to estimate the uncertainties of computing the water fluxes (WF) through a modified SVAT model, by inverting its soil hydraulic parameters from θ and ET. They found that the predictive capacity of the model against its simulated fluxes strongly depends on the hydroclimatic conditions; specifically, the uncertainty increases under dry climates, coarse textures, and deep rooted vegetation.

[3] In this paper, we provide a novel methodological framework, termed Weighted Objective Function Selector Algorithm (WOFSA), to improve predictions by SVAT models, by ensuring the most appropriate combination of these two types of information (θ and ET), for a wide range of hydroclimatic conditions and soil texture patterns. In the simulations, we use a modified SWAP 3.2 model, for which we are interested in inverting the effective soil hydraulic parameters, while the vegetation parameters are assumed known. The modified SWAP 3.2, introduced by Pollacco and Mohanty [2008a] and next termed SWAPinv, is briefly described in section 2.1.

[4] In the proposed framework, the inverse problem is expressed in multiobjective terms, by formulating a Weighted Objective Function (WOF) of two criteria, OFθ and OFet, which account for the deviation of the simulated to the “reference” surface soil moisture θ and evapotranspiration ET, i.e.,

display math(1)

where w is a dimensionless weighting coefficient. Multiobjective approaches have been widely documented in all aspects of hydrological modeling, starting from the late 1990s [e.g., Mroczkowski et al., 1997; Gupta et al., 1998; Yapo et al., 1998; Bastidas et al., 1999; Gupta et al., 1999]. The rationale is that as more information is embedded within calibration, it is expected that the identifiability of parameters is improved, thus also ensuring an improved predictive capacity. These advantages have been demonstrated in several applications involving SVAT and land surface models [e.g., Bastidas et al., 1999; Franks et al., 1999; Gupta et al., 1999; Demarty et al., 2004, 2005; Coudert et al., 2006; Mo et al., 2006]. In this respect, conditioning the hydraulic parameters of SVAT models against both θ and ET data is generally accepted, although not all researchers found advantageous of calibrating SVAT models simultaneously with θ and ET data [Ines and Droogers, 2002; Jhorar et al., 2002, 2004; Ines and Mohanty; 2008b].

[5] In order to increase the information embedded in calibration, the WOF is further parameterized by introducing a threshold soil moisture θd, which indicates the period when soil moisture θ can be calibrated, in order to avoid decoupling between surface and subsurface θ. The concept of θd is one of the novelties of our framework, as explained in section 2.4.2. It is well known that by tuning the weighting coefficient w and next solving the inverse (calibration) problem for a given value of θd, we can obtain different sets of optimized hydraulic parameters. The later are called nondominated or Pareto-optimal and lie in the boundary of the feasible objective space (Figure 1). By assigning a specific value to w and θd, we assert that the solution obtained by minimizing WOF ensures an acceptable compromise between OFθ and OFet. In this respect, the “optimal” combination of θ and ET data is mathematically represented as the determination of the weighting coefficient w and the decoupled soil moisture θd. The Weighted Objective Function Selector Algorithm (WOFSA) is a novel numerical procedure, which allows for identifying the optimal values of both the control variables of the multiobjective function (i.e., w and θd) and the model hydraulic parameters. The suitability of w and θd is evaluated on the basis of the information provided by the simulated water fluxes (model outputs), in terms of uncertainty, in an attempt to constrain the feasible parameter space. In contrast to the classic calibration paradigm, which merely aims to achieve the smallest departure between the observed and simulated model responses, the WOFSA also takes into account the uncertainties due to errors in input data. For convenience, in the investigations, we use synthetic data provided by numerical experiments with known parameter sets, in order to eliminate the impacts of other sources of uncertainty, e.g., structural (model) errors. In this context, WOFSA assumes that the uncertainties of the water fluxes are only caused by prescribed uncertainties of the observed θ and ET.

Figure 1.

Graphical example illustrating the objective space, the Pareto front and characteristic solutions of a hypothetical problem of simultaneous minimization of two criteria (in the specific case OFθ and OFet). Vector e = [e1, e2] indicates limits of acceptability (in the specific case uncertainty bounds), for distinguishing feasible solutions. Shown are the extreme solutions of the Pareto front (corresponding to w = 0 and w = 1), the solution that has the minimum distance from the origin and the solution provided by WOFSA, corresponding to w = 0.75.

[6] Specifically, we consider that the top 5 cm soil moisture retrieved from remote sensing has an average accuracy of root mean square error (RMSE) of 0.04 m3 m−3, in terms of volumetric soil moisture [e.g., Kerr et al., 2001; Simmonds et al., 2004; Davenport et al., 2005; Choi et al., 2008; Das et al., 2008; Sahoo et al., 2008; Verstraeten et al., 2008; Vischel et al., 2008]. This has been validated with field campaigns, typically under low vegetated area for which the biomass is up to 4–8 kg m−2 (for example, under mature corn and soybean), by using passive microwave remote sensing [e.g., Jackson and Schmugge, 1991; Bindlish et al., 2006; Li et al., 2006; Njoku and Chan, 2006]. On the other hand, the procedures for retrieving the actual evapotranspiration from remote sensing exhibit an average relative error of 20%, as also validated from field campaigns. This value is suggested by Kalma et al. [2008], from a compilation of 30 publications [e.g., Zhang et al., 2006; Gao and Long, 2008; Opoku-Duah et al., 2008; Bashir et al., 2009; Ramos et al., 2009; Teixeira et al., 2009b]. We note that the uncertainties of retrieving θ are different when compared to the uncertainties of ET, and therefore have different implications on the uncertainties of the modeled/inverted water fluxes. Moreover, the behavior of the uncertainties of θ and ET retrieved from RS with increasing θ and ET is still poorly understood [e.g., Fernández-Gálvez, 2008]. For this reason, we also assume that the uncertainties of θ and ET linearly increase with increasing θ and ET, thus suggesting that the WOF and the corresponding residuals are correlated. Under this premise, the optimal w and θd are those which achieve the maximum linear correlation between the WOF and the residuals of the simulated water fluxes. This is a key point of the methodology, which is analytically presented in section 3.

[7] Our methodology is validated by employing numerical experiments with SWAPinv. Following the recent research study by Pollacco and Mohanty [2008a], we used as reference states/fluxes the surface and root-zone soil moisture, groundwater recharge, actual evapotranspiration, actual evaporation, and actual transpiration. In order to investigate the variability of the optimized w and θd, we formulated 22 contrasting hydroclimatic scenarios, which are composed as combination of five climates across the USA, three soil textures, and two rooting depths. The need for investigating different rooting depths is justified by Ines and Mohanty [2008b], who found that the predictions of the hydraulic parameters of SVAT models are much more sensitive to rooting depths than other vegetation parameters. In the numerical experiments, we assumed that the soil hydraulic parameters are unknown and that the vegetation parameters are not subject to calibration, since these can be readily retrieved from MODIS (MODerate resolution Imaging Spectroradiometer) [e.g., Huete et al., 2002; Simic et al., 2004; Nagler et al., 2005; Vegas Galdos et al., 2012]. In all simulations, we assumed that the soils are homogeneous, based on the work by Jhorar et al. [2004], who found that, in most cases, a reliable water balance can be obtained by replacing the heterogeneous soil profile by an equivalent single one. Finally, we selected a deep water table, since Pollacco and Mohanty [2008a] showed that inverting the soil hydraulic parameters with ET in the presence of shallow water table causes extra uncertainties.

[8] The goals of this study include:

[9] 1. Development of the weighted objective function selector algorithm (WOFSA) for determining the best-compromise weights of a WOF.

[10] 2. Application of WOFSA within SWAPinv, in order to investigate the variability of the optimal coefficient w and threshold θd under contrasting hydroclimatic conditions, on the basis of synthetic data obtained through numerical experiments, i.e., by inverting the soil hydraulic parameters.

[11] 3. Determination of the most suitable calibration period (in terms of soil moisture thresholds) to take full advantage of the information provided simultaneously by θ and ET retrieved from remote sensing.

[12] 4. Development of empirical relationships correlating w and θd against typical statistical metrics of θ and ET.

[13] 5. Comparison with the minimum Euclidian distance approach, which is usually employed in multiobjective calibration problems.

[14] 6. Discussion of future research perspectives, for implementing WOFSA within a multiobjective calibration framework, and on the basis of actual (i.e., field) data.

2. Modeling Framework and Set-Up of Numerical Experiments

2.1. Soil-Water-Atmosphere-Plant Hydrological Model

[15] We introduce a modified version of the so-called Soil-Water-Atmosphere-Plant (SWAP 3.2), which is a physically based Soil-Vegetation-Atmosphere-Transfer (SVAT) water flow model for representing the unsaturated zone soil water fluxes of vegetated land [e.g., Van Dam et al., 2008; Kroes et al., 2000; Van Dam et al., 1980]. SWAP has been extensively used to calibrate the hydraulic parameters by matching θ and/or ET retrieved from remote sensing [e.g., Ines and Mohanty, 2008a-2008c, 2009; Shin et al., 2012]. The governing equation solves the mixed form of the Richards' equation, combined with a sink term for root water extraction, to simulate the variably saturated soil moisture movement in the soil profile

display math(2)

where θ is the volumetric water content (L3 L−3) or the fraction of water-filled pore space, h is the capillary pressure head (m), t is the time (T), z is the vertical coordinate (L) defined as positive upward, K(θ) is the unsaturated hydraulic conductivity (L T−1), and S(h) is the soil water extraction rate by plant roots (L3 L−3).

2.1.1. Soil Water Retention and Unsaturated Hydraulic Conductivity

[16] The model accuracy depends on two functions, the soil-moisture characteristic curve h(θ) and the unsaturated hydraulic conductivity K(θ). The analytical function of h(θ) is provided by the van Genuchten model [1980]

display math(3)

where θe is the normalized volumetric water content (L3 L−3), θr and θs are the residual and saturated water contents (L3 L−3), respectively, with 0 ≤ θr < θ < θs, h is the capillary pressure head (m), hae(1/α) is associated to the air-entry matrix potential (m−1), n (>1) is a shape parameter related to the pore-size distribution (dimensionless), and m is another shape parameter. The two parameters m and n are interrelated via the expression m = 1 − 1/n, following the assumption by Mualem [1976].

[17] The unsaturated hydraulic conductivity function K(θ) is given by Mualem [1976] and van Genuchten [1980]

display math(4)

where L is a dimensionless shape factor and Ks is the saturated hydraulic conductivity (m d−1). The shape factor L is not a sensitive parameter and it is normally kept fixed to 0.5. Similarly, θr does not affect the goodness-of-fit of the characteristic curve and it is typically eliminated [e.g., Russo, 1988; Luckner et al., 1989; Tietje and Tapkenhinrichs, 1993; Boufadel et al., 1998; Schaap and Leij, 1998; Ines and Droogers, 2002]. Hence, in this study, θs, hae, n, and Ks are the sole hydraulic parameters to be inverted. The expected range of the above parameters is provided in Table 1; this range was computed by taking the 90% confidence interval of the combined datasets of GRIZZLY [Haverkamp et al., 2005] and UNSODA [Leij et al., 1996]. In particular, the minimum range of θs is determined for each hydroclimate by calculating the maximum range of the reference θ.

Table 1. Expected Parameter Space of the van Genuchten Hydraulic Parameters Computed by Taking the 90% Confidence Interval of the Combined Data Sets of GRIZZLY [Haverkamp et al., 2005] and UNSODA [Leij et al., 1996]a
 θs (m3 m−3)hae (cm)n (–)Ks (cm d−1)
  1. a

    The minimum range of θs is computed from the maximum value of reference θ.

MinimumMAX(θref)7.61.090.48
Maximum0.543752.3465

2.1.2. Modified Sink Term of SWAP 3.2 (SWAPinv)

[18] Building parsimonious SWAP models by reducing the number of input vegetation parameters, without decreasing their predictive capacity and their physical concept, are a challenging task. In this context, we modified the evaporation, transpiration, and rainfall interception modules of SWAP, next termed SWAPinv, in order to use a reduced number of input parameters, namely the Leaf Area Index (LAI), the extinction coefficient of solar radiation (Kg), the rooting depth, and the saturated (θs) and residual (θr) water contents. In this respect, we use the Beer-Lambert law that partitions potential evaporation, potential transpiration, and potential evaporation of a wet canopy by using LAI and Kg [e.g., Ritchie, 1972; Goudriaan, 1977; Belmans et al., 1983]. In addition, LAI, Kg, and the potential evaporation of a wet canopy are also used to compute the interception, based on the works of Noilhan and Lacarrere [1995] and Varado et al. [2006]. Thus, the sensitivity of LAI and Kg is increased since they control multiple processes. The modified SWAP evaporation module does not require extra parameters, since it is directly estimated from the soil moisture, the potential soil evaporation, and the hydraulic parameters [e.g., Eagleson, 1978; Milly, 1986; Simmons and Meyer, 2000; Romano and Giudici, 2007, 2009]. Consequently, the sharing of the hydraulic parameters, which computes soil moisture and evaporation, increases the sensitivity of the hydraulic parameters when they are inverted simultaneously from soil moisture and evapotranspiration. The general shape of the roots in SWAP is entered manually in tabular form. Nevertheless, for large-scale modeling, a detailed description of roots is not required, thus we introduced an empirical power-law root density function [Gale and Grigal, 1987], that was further modified by Pollacco et al. [2008b]. The root density function requires two parameters, the maximum rooting depths and the percentage of roots in the top 30 cm. A detailed mathematical description of SWAPinv is provided in Appendix Appendix.

2.2. Generation of Reference Data for Numerical Experiments

[19] The numerical experiments were carried out for 22 hydroclimatic scenarios, derived by combining three soil types, two rooting depths, and five climates (Table 2). In order to provide realistic simulations, deep roots (DR) were not assigned to subtropical climates and shallow roots (SR) were not allocated to arid climate [Schenk and Jackson, 2002]. Moreover, in semiarid climates, only loamy sand was modeled. More precisely:

Table 2. Contrasting 22 Scenarios Composed of Three Soil Types, Two Rooting Depths, and Five Climatesa
  Temp. AridSemi-Mediter.Temp. ContinentalTemperateSubtropical
  1. a

    To maintain the simulations more realistic deep roots were not assigned to subtropical climates and shallow roots were not allocated to arid climate.

Shallow rootsLoamy Sand 
Silt Loam 
Silty Clay 
Deep rootsLoamy Sand 
Silty Loam  
Silty Clay  

[20] 1. The hydraulic parameters for the three contrasting benchmark soils (loamy sand, silty loam, and silty clay) are given in Table 3. These soil textures were selected from Carsel and Parrish [1988] and Ines and Mohanty [2008b], and they ensure a large variability of annual evapotranspiration and groundwater recharge.

Table 3. Reference Values of the Mualem [1976] and van Genuchten [1980] Hydraulic Parameters
TextureAcronymθs (m3 m−3)θr (m3 m−3)L (–)hae (cm)n (–)Ks (cm d−1)Sources
Loamy sandLS0.410.0570.582.28350.2Carsel and Parrish [1988]
Silty loamSiL0.430.0610.5831.3930.5Ines and Mohanty [2008b]
Silty claySiC0.360.070.52001.090.48Carsel and Parrish [1988]

[21] 2. For the two contrasting benchmark-rooting depths (i.e., shallow and deep), the rooting depths and the percentage of roots for the top 30 cm are given in Table 4. These contrasting rooting depths were selected to depict shrubs, and they are provided by Schenk and Jackson [2002] and Jackson et al. [1996]. Forested land use was not selected, because remote sensing platforms using passive microwave still cannot retrieve soil moisture under dense canopy, the biomass of which is higher than 8 kg m−2 (e.g., vegetation denser than mature corn) [e.g., Jackson and Schmugge, 1991; Bindlish et al., 2006; Li et al., 2006; Njoku and Chan, 2006].

Table 4. Contrasting Scenarios of the Percentage of Roots in the Top 30 cm, ΔRDF30 [Jackson et al., 1996], and Maximum Rooting Depths, zroot [Schenk and Jackson, 2002]
DescriptionAcronymzroot (cm)ΔRDF30 (%)Vegetation Type
Shallow rootsSR4080Meadows
Deep rootsDR13050Semidesert

[22] 3. The values of the typical vegetation parameters that remain constant for all simulations are provided in Table 5 and explained in Appendix Appendix. It is assumed that all these parameters can be retrieved from MODIS remote sensing [e.g., Huete et al., 2002; Simic et al., 2004; Nagler et al., 2005; Vegas Galdos et al., 2012].

Table 5. Values of Vegetation Parameters that Remains Constant, Where h1, h2, h3high, h3low, and h4 are the Capillary Pressure Head that Regulate the Water Uptake Model, LAI is the Leaf Area Index, β is the Crop Factor, and Kg is the Extinction Coefficient of Solar Radiation (–)a
h1 (cm)h2 (cm)h3high (cm)h3low (cm)h4 (cm)LAI (m3 m−3)Kg (–)β (–)
  1. a

    Refer to Appendix Appendix for further information.

−1−22−1000−2200−16,00020.50.9
[Singh et al., 2006][Brutsaert, 2005][Varado et al., 2006][Pollacco, 2012]
WheatScrublandUniversalGrassland

[23] 4. To formulate the hydroclimatic scenarios, we used daily precipitation time series and meteorological data for computing the potential evapotranspiration through the Penman and Monteith formula [1965], which were compiled from AmeriFlux http://public.ornl.gov/ameriflux/ (Table 6). The contrasting climates correspond to typical mainland Southern United States conditions, for which snowfall is scarce. The forcing data was selected by combining a dry, a normal, and a wet water year (1 October to 30 September).

Table 6. Sources of Reference Hydroclimate Data Compiled From AmeriFlux (http://public.ornl.gov/ameriflux/)
ClimateAcronymSiteStateLat.Long.IGBP Classif.
Temperate semiaridTsaKendall GrasslandAZ32−110Grasslands
MediterraneanMTonzi RanchCA38−121Woody Savannas
Temp. continentalTcWalnut riverOK37−97Cropland
TemperateTMead RainfedNE41−96Croplands
SubtropicalSKennedy Space Center Scrub OakFL29−81Closed shrublands

[24] A summary of the 22 reference water fluxes computed with SWAPinv is presented in Figure 2. The scenarios provide satisfactory high variability of the model fluxes. Specifically, the annual groundwater recharge ranges from 30 to 800 mm, the annual transpiration ranges from 120 to 370 mm, and the annual evaporation ranges from 7 to 144 mm.

Figure 2.

The 22 hydroclimatic scenarios depicted by average yearly groundwater recharge Q, transpiration T, evaporation E, interception Pint computed from SWAPinv. For visualization, the gross precipitation Pg= Q + T + E + Pint with the long-term storage computed to 0. The acronyms are provided in Table 3 for the soil texture, in Table 4 for the roots, and in Table 6 for the climate.

2.3. Boundary Conditions and Discretization

[25] Within the simulations, the soil column was discretized for deep roots of a total depth of 1.80 m and for shallow roots of a total depth of 0.90 m. Finer discretization (0.25 cm) near the land atmospheric boundary and coarser discretization (5 cm) at deeper depths were employed. For all scenarios, the soil columns were initialized uniformly at h = −0.1 m and SWAPinv run for 90 days (spin up time) ahead of the experiment, to tune the state of the initial soil moisture profile. For the bottom boundary condition of the soil columns, the free drainage was selected. The upper boundary condition was determined by the daily net precipitation, which was computed with the interception model, and the potential evapotranspiration, estimated by the Penman-Monteith equation. The potential evapotranspiration was partitioned into potential soil evaporation, potential evaporation of wet canopy, and potential transpiration by using the Beer-Lambert law [e.g., Ritchie, 1972; Goudriaan, 1977; Belmans et al., 1983]. Finally, a maximum of 2 cm of ponding water is permitted with any overflow lost as runoff.

2.4. Formulation of the Inverse Problem

2.4.1. The Weighted Objective Function

[26] Within the inverse problem, we use a WOF comprising two fitting criteria, OFθ and OFet, and two control variables, w and θd. In order to account for the differences in magnitude between the individual criteria, it is preferable that all the components of the WOF are either dimensionless or normalized. The WOF is derived by dividing the mean absolute error by the typical observation error (uncertainty) of the corresponding reference state or flux, i.e.,

display math(5)

where θ [L3 L−3] is the top 5 cm surface soil moisture where decoupling does not occur and Nθ and Net are the lengths of daily soil moisture and evapotranspiration time series, respectively. When OFθ or OFet is greater than 1 indicates that the errors of simulations are greater than the uncertainties of retrieving the observation from remote sensing. We highlight that for both functions, all model outputs which provide values greater than 1 are considered as nonacceptable. Hence, a trial set is rejected if OFθ > 1 or OFet > 1.

[27] To provide a proper configuration of the multiobjective calibration problem, it is essential to ensure that the two fitting criteria, OFθ and OFet, are approximately uncorrelated. Indeed, Pollacco and Mohanty [2008a] showed that for contrasting hydroclimatic conditions the related processes θ and ET are rather independent. This is because the surface θ is influenced by the evaporation and decouples between the surface and root-zone soil moisture, while ET is a signature of the whole root-zone θ, since ET results in the uptakes of water stored at depth. In addition, the storage of θ in the root-zone profile is dependent on the past weather events, whereas the near-surface θ reflects the present weather condition.

2.4.2. Introducing Decoupling Within WOF

[28] One of the peculiarities when calibrating hydrological models against surface soil moisture is that soil moisture is prone to decoupling. This originates from the significantly faster drying of the surface compared to the root zone, due to evaporation and shallow root water uptake, causing a sharp vertical soil water gradient near the surface. When this occurs, the surface θ is no more representative of the soil moisture dynamics in the rooting zone [Capehart and Carlson, 1997; Walker et al., 2002; De Lannoy et al., 2007; Pollacco and Mohanty, 2008a]. For instance, large-scale decoupling was evidenced in New Zealand by Wilson et al. [2003] between 0–6 cm and 0–30 cm in situ. Decoupling is more prominent when surface θ is in the drying phase and it is below the threshold θd (L3 L−3), which is computed by

display math(6)

[29] On the basis of equation (6), we modified OFθ such that to increase its sensitivity by omitting the period when surface and root-zone decoupling occurs. If θd = 0, decoupling is not taken into account within WOF.

3. Outline of the Weighted Objective Function Selector Algorithm (WOFSA)

3.1. Identification of the Best-Compromise Parameter Set in Multiobjective Calibration: Approaches and Drawbacks

[30] Equation (1) is a specific case of aggregated objective functions that represent an overall measure of the model performance, in which the characteristics of the best-compromise solution, which also reflect the relative importance of the individual criteria, are specified a priori. The later are expressed in terms of multipliers (e.g., weighting method), target values combined with distance metrics (e.g., goal programming and ε-constraint methods) [e.g., Laumanns et al., 2002; Reed et al., 2003] or priorities (e.g., lexicographic ordering). Besides, the detection of the best-compromise parameter set remains an open issue in hydrological calibration, which has not been thoroughly addressed in the literature [e.g., Dumedah et al., 2010].

[31] Most approaches employ hybrid strategies, based on combined objective and subjective criteria, to support the manual identification of the “most prominent” parameter values [e.g., Efstratiadis and Koutsoyiannis, 2010]. In particular, a well-accepted technique for detecting the best-compromise parameters, which is usually employed in subsurface flow modes, is by minimizing the Euclidean distance of the Pareto set to the origin [e.g., Refsgaard and Storm, 1996; Madsen, 2003; Twarakavi et al., 2008]. Although this methodology, which is a subcase of goal programming, appears to be statistically reasonable, its hydrological meaning is not well understood. On the other hand, few are the procedures for recognizing effective nondominated solutions a posteriori, through systematic filtering of the Pareto set. Some of the proposed approaches are preference ordering and compensation between model objectives [e.g., Khu and Madsen, 2005] as well as cluster analysis [e.g., Taboada and Coit, 2006; Crispim and de Sousa, 2009; Dumedah et al., 2010]. For instance, Dumedah et al. [2010, 2012a, 2012b] used cluster analysis to evaluate the distribution of solutions on the trade-off surface, to find relationships in both objective space and parameter space. The linkage between the two spaces describes the level of robustness for the parameter sets (according to Deb and Gupta [2005], robust solutions are less sensitive to variable perturbations in their vicinity). They also showed that the use of criteria that are based on a compromise between representative pathways in the parameter space and a dominant variability in the objective space provides solutions that remain nondominated across different validation subsets.

[32] The above, rather subjective, approaches for detecting the best-compromise parameter set in multiobjective calibration problems, ignore uncertainties that are due to errors in input data, which prevent providing robust solutions. In this respect, we are proposing a systematic procedure, called weighted objective function selector algorithm (WOFSA), which identifies the most appropriate weighted objective function (WOF), by performing inverse modeling, where the uncertainties in retrieving θ and ET data are directly accounted for. Next are described the key assumptions of the methodology as well as the detailed computational procedure.

3.2. Key Assumptions of WOFSA

[33] The key idea of WOSFA is based on the postulation that the optimal weighting between the individual objectives is the one ensuring the maximum linear correlation between the residuals ΔWF of the computed model responses of interest (water fluxes) and the WOF. The rationality is that if the inverse modeling is well posed, then an increase in the OF should cause the error of each specific simulated flux to also increase and vice versa [Pollacco et al., 2008b]. If the later is insensitive against to variations of θ and ET, the problem is ill-posed as the modeled flux cannot be calibrated solely from the observed θ and ET; thus, additional observations should be included into the WOF.

[34] This assumption is further illustrated in Figure 3, where we plot three hypothetical relationships between a normalized WOF* and a dimensionless residual metric (e.g., relative bias) ΔQ*, which is a measure of uncertainty of the corresponding water flux (in the specific case, the groundwater recharge). It is assumed that the optimal relationship is the 1:1 line (intermediate curve of Figure 3, e.g., 2), which indicates that a specific change of the WOF* value results to an equal change of the model uncertainty ΔQ*. Therefore, this expression is the most suitable to be used in calibration. Any other relationship, derived by different combinations of weights, is suboptimal. For instance, the right curve of Figure 3 (e.g., 1) demonstrates a weighted function that initially has limited sensitivity against the model uncertainty (a significant change of the WOF* results to a much less significant change of ΔQ*), followed by sharply varying model uncertainty for small changes of the WOF*. On the other hand, the left curve of Figure 3 (e.g., 3) represents an opposite performance, which is also far from desirable. This feature forms the basis for our linearity assumption between WOF* and ΔQ* in WOFSA.

Figure 3.

Examples of relationship between a normalized weighted objective function (WOF*) with the normalized uncertainty of the water flux (ΔWF*) error represented by ΔQ*. The ideal is a linear correlation between WOF* and ΔQ*.

3.3. Description of Computational Procedures

[35] The algorithm is applied to a SWAPinv model, using the objective functions of section 2.4. The model runs on daily basis. The water fluxes of interest are groundwater recharge Q (mm d−1), evaporation E (mm d−1), transpiration T (mm d−1), evapotranspiration ET (mm d−1), while the modeled state variables are the root-zone soil moisture θrz (m3 m−3) and the surface soil moisture θ (m3 m−3). The method is performed in three successive steps, as also shown in the flowchart of Figure 4

Figure 4.

Flowchart of the weighted objective function selector algorithm (WOFSA) where HYDRAUref and VEGETATIONref are a known set of reference parameter values; WFref are the modeled reference water fluxes outputs that are computed from the SVAT hydrological model requiring a priori known sets of hydraulic parameters (HYDRAUref) and gross precipitation (Pg) and potential evapotranspiration (ETp) as forcing data; HYDRAUsim are the trial set of parameter values that are obtained from the OPTIMIZATION ALGORITHM; WFsim are the simulated fluxes and ΔWF is their residuals, derived by a posteriori estimated hydraulic parameters (HYDRAUsim) by minimizing the weight w (between the fitting criteria based on soil moisture θ and evapotranspiration ET) and the decoupling threshold θd of WOF. All the trials are stored in the STORAGE that are filtered such that the uncertainties in θsim and ETsim are not greater than the uncertainties of retrieving θ and ET from remote sensing (Δθrs and ΔETrs). WOFSA is performed in two separate parts: Part A generates uncertainties in the fluxes ΔWF as if they were available from independent measurements, while Part B optimizes w and θd by minimizing the OFlin such that to ensure the maximum linearity between the normalized WOF* and the normalized maximum uncertainty ΔWFmax*. The different loops are colored coded with blue for loop 1, red for loop 2, and green for loop 3.

3.3.1. Step 1: Generation of Reference Runs

[36] WOFSA performs numerical experiments to determine the optimal control variables w and θd of the WOF, which requires that the soil moisture θ, evapotranspiration ET, and water fluxes (as well as state variables) WF, are known a priori. The later will be next called “reference” data, symbolized θref, ETref, and WFref, respectively. In particular, WFref are computed by inputting known sets of hydraulic parameters (HYDRAUref), vegetation parameters (VEGETATIONref), and daily forcing (precipitation, potential evapotranspiration) data into SWAPinv (Figure 4, Loop 1). We remark that the vegetation parameters are treated as known properties of the model (cf. section 2.2), while the soil hydraulic parameters are to be inverted through optimization.

3.3.2. Step 2: Monte Carlo Simulation and Calculation of Uncertainties

[37] In order to assess the uncertainties in retrieving θref and ETref from remote sensing, we use different sets of θsim and ETsim, provided through Monte Carlo simulation (Figure 4, Loop 2). Each trial set is formulated on the basis of different values of soil hydraulic parameters (HYDRAUsim), which are generated by SWAPinv to provide the corresponding simulated time series WFsim, θsim, and ETsim. The “unknown” constrained HYDRAUsim are estimated by minimizing the WOF. To initialize the search procedure, the typical values w = 0.5 and θd = 0 are assigned to WOF, which are updated after the completion of Step 3. The simulations are carried out by employing the Shuffled Complex Evolution University of Arizona (SCE-UA) algorithm, developed by Duan et al. [1992, 1994]. The customized global optimization can be seen as a restrained Monte Carlo simulation that seeks for different combinations of “compromise” parameter sets (HYDRAUsim), in the vicinity of the global minimum [van Griensven and Meixner, 2006; Pollacco et al., 2008b, 2009].

[38] For each trial set (i.e., hydraulic parameters and resulting fluxes), the model uncertainties, in terms of residuals ΔWF, are computed by

display math(7)

where Nwf is the time length of simulations (days).

[39] During the Monte Carlo procedure, all the different trials of HYDRAUsim and the corresponding WFsim and ΔWF are stored in the STORAGE archive (Figure 4). At the end of Step 2, the trial sets are sorted in increasing order of WOF values. Figure 5a depicts the relationship between WOF and the residuals of the groundwater recharge ΔQ, for one of the experiments that are examined next (i.e., loamy sand, temperate climate, and short rooting depth).

Figure 5.

Different steps of the WOFSA given as an example for loamy sand, temperate climate, and short rooting depth. (a) An ensemble of generated parameter sets HYDRAUsim with the relationship between WOF and the residuals between reference and simulated WFsim given as an example for groundwater recharge ΔQ. (b) From each generated WOFi, described in Figure 5a, the maximum corresponding error ΔQmax is selected. (c) Selection of feasible parameter sets ΔQmax to reproduce the uncertainties in retrieving θref and ETref from remote sensing. (d) Correlation between normalized WOF* and normalized WFmax* for top soil moisture SM (θ), root-zone soil moisture SMrz (θrz) evapotranspiration ET, evaporation E, transpiration T, and groundwater recharge Q.

3.3.3. Step 3: Estimation of w and θd

[40] As explained in section 3.2, in order to determine the best-compromise values of w and θd, it is essential to ensure the greatest linearity between the so-called normalized WOF and the normalized maximum uncertainties ΔWFmax*. This linearity is obtained by minimizing an “auxiliary” objective function OFlin through the SCE-UA method, using the ensemble sets that are generated in Step 2. The computational procedure is the following.

[41] From each generated WOFi, the maximum corresponding error of WFsim (ΔWFmax) is selected and plotted. As shown in Figure 5b (where ΔWFmax is ΔQmax), the key asumption is that the relationship between WOF and ΔWFmax monotonically increases is reasonable. The computation of ΔWFmax is mathematically expressed as

display math(8)

where index i corresponds to the ith simulation, classified by an increasing order of WOF. We remind that here we only consider the uncertainties of the reference data that are used in calibration, and we do not take into account other error sources, such as structural errors of the model. In order to implicitly account for the later, we use the upper envelope uncertainties of the water fluxes ΔWFmax and not, for instance, their average values.

[42] For each flux, in order to evaluate the linearity between WOF and ΔWFmax, the two variables are normalized, thus taking values in the range [0, 1]. This is performed by selecting the corresponding “envelopes” of simulated ΔWFmax such that the following condition is fulfilled

display math(9)

where Δθrs and ΔETrs are typical values of the uncertainties in retrieving θ and ET, respectively, from remote sensing. In the case study, we generated 7000 sets of θsim and ETsim which comply with equation (9). Preliminary investigations indicated that generating more sets improve the optimal values of w and θd only marginally. On the basis of literature data already mentioned in section 1, for the soil moisture we assigned a volumetric root-mean-square error Δθrs = 0.04 m3 m−3 while for the evapotranspiration we set a relative error ΔETrs = 20% (apparently, in a particular study, different values can be employed, taking advantage of uncertainty estimations based on local data). The simulated ΔWFmax values that comply with equation (9) are depicted in Figure 5c, through the nonshaded area. WOF and ΔWFmax are normalized and symbolized with (*), using the maximum feasible simulated value that complies with equation (9), which is annotated with the circle in Figure 5c.

[43] As explained in section 3.2, the optimal WOF is determined such that to ensure the maximum linearity between WOF* and ΔWFmax*. The linearity is quantified by means of the auxiliary objective function OFlin (Figure 3), which is computed separately for each water flux, as follows:

display math(10a)

where the index i corresponds to the ith simulation, classified by an increasing order of ΔWF*max and math formula is only used for graphical reasons, i.e., in order to normalize OFlin thus being equal to half the diagonal of a unit square.

[44] The value of OFlin* depicts the maximum deviation from the 1:1 line composed of ΔWFmax* and WOF*, as described in Figure 3. The value OFlin = 0 denotes a perfect linearity, while OFlin = 1 corresponds to the greatest deviation from the desirable line 1:1. The final value of OFlin is computed by averaging OFlin, which is calculated for each individual water flux and state variables of interest (root-zone soil moisture, groundwater recharge, evapotranspiration, evaporation, and transpiration) by using the following expression:

display math(10b)

where index j corresponds to jth water flux of interest and NOFlin is the number of water fluxes of interest. An example of the relationship between the optimal WOF* and ΔWFmax* for all water fluxes is provided in Figure 5d.

[45] The SCE-UA optimization algorithm is next used to minimize the auxiliary function (equation (10b)) against w and θd. After getting the optimal values of w and θd, the initial objective function (WOF) is updated and Steps 2 and 3 are repeated. The iterative procedure continues until the values of w and θd are stabilized, thus WOFi ≈ WOFi−1. Typically, four runs are enough to achieve convergence.

4. Results

4.1. General Outcomes

[46] An overview of the WOFSA capabilities is provided by investigating the three representative scenarios, which are presented in Table 7 and plotted in Figure 6. The figure shows the relationship between WOF* and ΔWFmax*, which is computed for ET, T, E, θ, θrz, and Q. The following general outcomes are drawn:

Table 7. Detailing the Different Scenarios Used in Figure 6
TextureSpecificationOFlin (%)Figure 6
Loamy sandDecoupling equation21A1
No decoupling17A2
Sandy clayShallow roots13B1
Deep roots10B2
Silty clayCalibrated with OFet13.9C1
Calibrated with WOF14.5C2
Figure 6.

Relationships between normalized optimized WOF* and normalized ΔWFmax* for the scenarios described in Table 7.

[47] 1. The strength of linearity between WOF* and ΔWFmax* can vary greatly with hydroclimate conditions (Figure 6).

[48] 2. The usage of the decoupling algorithm (equation (6)) increases the linearity between WOF* and ΔWFmax* (e.g., for loamy sand; Figure 6a).

[49] 3. Deep roots compared to shallow roots tend to increase the discrepancy in the predictions of transpiration (e.g., for sandy clay; Figure 6b).

[50] 4. The usage of WOF instead of a single OFet did not improve the linearity between WOF* and ΔWFmax* (e.g., for silty clay under a Mediterranean climate for which we will show that it is a special case; Figure 6c).

[51] Next, we further investigate how the optimized values of w and θd vary under different hydroclimatic conditions.

4.2. Correlating Soil Moisture Decoupling With Hydroclimatic Variables

[52] The weighting coefficient w and the decoupling threshold θd (m3 m−3) were optimized by minimizing OFlin (equation (10b)). As already mentioned in section 2.4.2, to account for the observed θ within WOF we only used periods when soil moisture decoupling does not occur. We remind that decoupling only occurs when the soil is drying and the soil moisture falls below θd. For loamy sands, an example of decoupling is given in Figure 7a, where the reference time series of soil moisture θ are plotted at different depths. Figure 7a suggests that during the drying period and when θ < θd = 0.07 m3 m−3 (where 0.07 m3 m−3 is the optimal value obtained through the WOFSA, for the specific combination of soil texture and climate), the surface moisture is decoupled from root-zone moisture.

Figure 7.

Reference time series θ plotted at different depths: (a) for coarse soils, showing that the top layer is decoupled from the deeper layer when θ is drying and θ < θd and (b) for fine texture soils under dry climate, showing that the top layer gets gradually decoupled from the deeper layer.

[53] For each hydroclimatic scenario, we employed preliminary simulations to express θd as function of average surface soil moisture math formula, average evapotranspiration math formula and its standard deviation σET (Figure 8). The scatter plots indicate a negative correlation between math formula and math formula. The ratio math formula is a climatic indicator which increases as the climate gets wetter, since there is a positive correlation (r2 = 0.70) between math formula and the evapotranspiration fraction math formula math formula (results not provided here). On the other hand, the ratio math formula can be viewed as a normalized expression of θd, where math formula is representative of the soil texture, which is lower for coarse texture and higher for fine texture. To understand the correlation we rewrite θd model as

display math(11)
Figure 8.

For all hydroclimatic conditions a relationship is obtained between average ET divided by the standard deviation of ET (σET) with (θd/θ)0.3. The scenarios are wetter as ET/σET increases.

[54] From the above equation, it results that when the soil moisture storage math formula in the root zone increases, math formula also increases, which is reasonable. An increase in math formula also generates a decrease in soil moisture decoupling, which is represented by a decrease in θd. However, a high value of σET indicates more pronounced periods of drying and wetting, which in turn produces an increase in soil moisture decoupling θd, due to differences in the soil moisture storage between the surface and the root zone. Figure 8 shows that for dry hydroclimates math formula, while for wetter hydroclimates math formula. Thus, math formula is negatively correlated with math formula. The conclusion that soil moisture decoupling is more pronounced in drier climates is in line with the results of Capehart and Carlson [1997].

4.3. Correlating Weighting Coefficient With Hydroclimatic Variables

[55] A major objective of this study is to relate the weighting coefficient w with easily obtainable predictors. The optimal value of w is a complex trade-off between the information gathered by OFθ and OFet. When more weight is assigned to OFθ, then the errors in Δθrs influences more the computation of the water fluxes (WF) compared to ΔETrs. On the other hand, when more weight is assigned to OFet then the errors in ΔETrs influences more the computation of the WF compared to Δθrs. In fact, the inverse modeling will favor the weighting to OFθ, since θ is a better predictor of the hydraulic parameters than ET [Pollacco et al., 2008b].

[56] For the three soil texture classes subdivided climatically, Figure 9a depicts the relationship of w against the average evapotranspiration fraction math formula and Figure 9b shows the correlation of w against the average measured surface soil moisture math formula. For every texture class, the empirical linear equations of Figures 9a and 9b are described in Table 8. No correlation of the rooting depth with w was found since the former influences indirectly w through math formula. The hydroclimates that are enclosed in ovals are the ones for which little difference arises in math formula, given that OFet is used instead of WOF. These hydroclimates are depicted by arrows, representing threshold values of math formula and ETf.

Figure 9.

For the three soil textures class subdivided climatically: (a) relationship of w with average evaporative fraction ETf, and (b) correlation of optimal w with the measured average θ. The empirical linear equations of each texture classes are described in Table 8. The enclosed hydroclimates are those for which a single OFet can be used instead of a WOF. These hydroclimates are depicted by arrows which represent threshold values of θ and ETf. Figure 9c schematizes the Feddes et al. [1978] plant water stress response function (ETf) as a function of soil water pressure. The position of the parameter h3 depends on the intensity of the potential transpiration (Tp < 1 mm d−1 or Tp ≥ 5 mm d−1). The interpolation of h3 is between the interval h3low, h3high for which their values are provided in Table 5.

Table 8. Empirical Relationship for the Three Texture Classes Which Relates w With Average math formula (ET/ETp) and w With Average math formula
Texturew =
Figure 10aFigure 10b
Loamy sand math formula math formula
Sandy clay math formula math formula
Silty clay math formula math formula

4.3.1. Loamy Sand

[57] For coarser texture soils (loamy sand and sandy clay), w is negatively correlated to both math formula (Figure 9a) and math formula (Figure 9b). It is to be noted that math formula is small for coarse soils because, as shown in Figure 7a, there are long periods of droughts for which θ ≈ 0. Therefore, for dry hydroclimates, represented by low values of math formula, more weight is assigned to OFθ, and for wetter hydroclimates, represented by larger math formula, more weight is assigned to OFet.

[58] These results can be explained in terms of the sensitivity of OFet against ΔWF, which depends on math formula. The later is computed from the water uptake function of Feddes et al. [1978], as shown in Figure 9c. The sensitivity of OFet is considerably reduced when the vegetation is under arid conditions, with math formula as low as 10%. This is due to the closure of the stomata, thus more weight is assigned to OFθ. For wetter hydroclimatic scenarios characterized by an increase of math formula, the sensitivity of OFet increases but at the same time OFθ weakens due to the enhanced surface and root-zone decoupling caused by evaporation. As math formula further increases (hydroclimates enclosed in ovals in Figures 9a and 9b), math formula remains invariant if either OFet or WOF is used independently. Therefore, under these hydroclimatic conditions, it is preferable to use OFet instead of WOF. OFlin remains invariant when math formula and θ > 0.035 L3 L−3, approximately (refer to arrows in Figures 9a and 9b), where the relationships between w versus math formula and w against θ change slope. These outcomes explain why Ines and Droogers [2002], Ines and Mohanty [2008b], and Jhorar et al. [2002, 2004] did not find advantages of using a WOF instead of a single OF to optimize the hydraulic parameters.

4.3.2. Sandy Clay

[59] The behavior of sandy clay soils is very similar to the loamy sands described above. Nevertheless, sandy clays are less coarse than loamy sand and thus the average drainage and the evaporation rate is moderated. Therefore, for the nonarid hydroclimatic scenarios math formula, w is clustered around 0.60.

4.3.3. Silty Clay

[60] For finer texture soils (silty clay), w is positively correlated with both math formula (Figure 9a) and math formula (Figure 9b). Therefore, for dry hydroclimates more weight is assigned to OFet and for wetter hydroclimates more weight is given to OFθ. The correlation between w with math formula and w with math formula of fine texture soils is positive, while for coarse texture soils it is negative (Table 8). This difference arises because the vegetation under moist soils do not experience much stress, thus ETf remains close to unity (Figure 9c). Under this premise, h is free to vary between h2 and h3 (equation (A8)), which reduces the sensitivity of OFet. Thus, for wet hydroclimates, more weight is assigned to OFθ.

[61] On the other hand, for drier hydroclimates, more weight is assigned to OFet due to another type of decoupling, which occurs for fine texture soils termed as fine texture decoupling. An example is provided in Figure 7b, where the reference time series θ are plotted at different depths. Figure 7b suggests that for drier climates the top soil dries up progressively and decouples with the root zone, for which there is a substantial amount of water stored at depth. Under these conditions, ET is more representative of the root-zone soil moisture than the surface soil moisture, thus more weight should be assigned to OFet.

5. Discussion

5.1. Selection of the Most Suitable Calibration Period

[62] It is widely accepted that the information which is embedded in calibration data plays much more important role than the length of observations themselves. However, most of the existing hydrological calibration approaches do not provide any guidance about which sets of measurements are most informative for specific model parameters [e.g., Vrugt et al., 2002]. In particular, for SVAT models, an additional quest is to determine the “optimal” period to calibrate the hydraulic parameters from reference surface θ and ET retrieved from remote sensing. The use of a multiobjective function, by means of the WOF, can adequately represent the errors that may be incurred due to the inverted parameter sets and may also help to recognize the structural errors much easier than when using a single fitting criterion. Therefore, to reduce ΔWF we need to select the period where the optimal w is theoretically around 0.6 (more weight is assigned to OFθ, since θ is a better predictor of the hydraulic parameters), thus taking full advantage of the information provided simultaneously from OFet and OFθ.

[63] Feddes et al. [1993], Ines and Mohanty [2008c], Jhorar et al. [2002], and Van Dam [1997] suggested that the identifiability of the parameters increases with the ranges of the data from very dry to very wet. Nevertheless, these results are partly supported by our study, which showed that better predictions are obtained when optimization is performed during periods where soil moisture decoupling does not occur. In this respect, given that soil moisture decoupling is accentuated under dry conditions (equation (6)), inverse estimations should be avoided during dry periods. Our investigations also indicated that under dry conditions ETf is reduced and therefore OFet, driven by the Feddes et al. [1978] model, becomes less significant. In section 4.3, it was also shown that for wet periods, during which ETf remains close to unity (Figure 9c), the sensitivity of OFet is reduced. Thus, the common belief that one requires a period such that θ goes from saturated to residual water content is not supported by this study.

[64] In practical terms, it is recommended that the hydraulic parameters should be preferably optimized after heavy rainfall events, when the soil moisture profile is homogeneous. Nevertheless, the measurements should only start after the plant is starting to experience stress and stopped when the roots are experiencing excessive stress. This finding suggests that the inverse modeling should be performed during the period where evaporation is not at its maximum, to avoid soil moisture decoupling.

5.2. Comparison of the WOFSA With the Minimum Distance From the Origin

[65] In section 3.1, we mentioned that a well-accepted technique for detecting the optimal value of w, which is a complex trade-off between the information gathered by OFθ and OFet, is by Minimizing the Euclidean Distance of the Pareto front to the origin (MEDP). Apparently, this requires determining the shape of the Pareto front. In nonlinear spaces, this is only achievable by running a suitable multiobjective evolutionary optimization algorithm, which can provide representative nondominated solutions that are uniformly distributed across the objective space [Efstratiadis and Koutsoyiannis, 2010]. For a given shape of the front, the computation of its minimal distance from the origin is trivial. In particular, as illustrated in Figure 1, when OFθ and OFet are normalized this method results to w = 0.5, independently of the values of Δθrs and ΔETrs, and also independently of the hydroclimatic conditions.

[66] The major drawback of the MEDP approach is the erroneous assumption that the magnitude of Δθrs is similar to the one of ΔETrs and that the impact of Δθrs and ΔETrs on the WF is similar. Indeed, our extended investigations within this paper concluded that w is far from constant; on the contrary, it is highly dependent on both the soil texture and climate (Figure 9). Moreover, MEDP fails to take into consideration that when more weight is assigned to OFθ, then the errors in Δθrs influences more the computation of the water fluxes and state variables WF, compared to ΔETrs. On the other hand, when more weight is assigned to OFet, the errors in ΔETr have more influence to the simulated WF, if compared to Δθrs. Hence, the only advantage of MEDP against WOFSA is the simplicity of the computational procedure, but only under the premise that the shape of the Pareto front is well approximated.

5.3. Implementing WOFSA Within a Pareto-Optimization Framework

[67] Forthcoming research needs to address how we can integrate WOFSA within global multiobjective calibration procedures (e.g., MOSCEM, MOPSO, MOHBMO) [Barros et al., 2010], by using real observations. Moreover, it can provide guidance for the selection of the most robust solution, among the mathematically equivalent Pareto optimal alternatives. Indeed, the best-compromise solution of the multiobjective calibration problem is theoretically found in the cross section of the optimally weighted objective function (WOF) and the Pareto-front. Yet, the task of implementing the above idea is nontrivial, since the true water fluxes and state variables (WFref) are unknown. In the following, we propose preliminary guidelines on how to use WOFSA in a multiobjective calibration setting by assuming that the inverse problem is well posed, thus exhibiting relatively steep trade-offs and that an increase in WOF would produce an increase in WFsim.

5.3.1. Step A: Run Multiobjective Optimization

[68] Perform multiobjective optimization by simultaneous minimizing OFet and OFθ, for which w does not need to be provided. On the other hand, θd which depends on the climate data can be estimated from Figure 8. During the optimization, all the feasible HYDRAUsim and WFsim which complies with equation (9) are kept in storage which will give the subset of acceptable Pareto-optimal solutions (Figure 1).

5.3.2. Step B: Selection of Temporary Reference Water Fluxes

[69] A first guess of the reference parameters (WFref, HYDRAUref) is obtained from the cross section of the weighted objective function (WOF) and the subset of Pareto optimal solutions. To obtain a first guess of WOF, w is approximated from Table 8 and θd is provided from Figure 8. Next, ΔWF is computed for the subset of acceptable solutions.

5.3.3. Step C: Dividing the Subset of Acceptable Solutions

[70] WOFSA is performed independently on different parts of the subset of acceptable solutions, i.e., the Pareto front (Figure 1). The area is divided on the basis on w. For instance, if the subset of acceptable solutions are divided into four subareas, then the ranges of w are [0; 0.25], [0.25; 0.5], [0.5; 0.75], and [0.75; 1.0]. For each subareas, the WOFSA runs from Step 3, (section 3.3.3), thus obtaining the corresponding math formula.

5.3.4. Step D: Refining the Results

[71] The WFref is updated with the new value of w based on the group which exhibits the lowest math formula. Thus, the best-compromise solution is in the cross section of the optimal WOF and the Pareto front (Figure 1). Steps b and c are repeated until convergence occurs between the new optimal w and the previously computed value.

[72] We should remark that although in this study we used two fitting criteria, the WOFSA can be performed with more criteria. In the current version, we suggest using a maximum of four fitting criteria, thus allowing the calibration of up to three weights within the minimization of math formula (equation (10b)). The introduction of more criteria would result in a significantly extended Pareto front, tending to cover a large part of the entire objective space. Evidently, this is far from desirable, for both theoretical (i.e., increased uncertainty) and practical reasons (i.e., poor understanding of the generated trade-offs). Nevertheless, very limited are the cases where more than four independent criteria have been applied in real-world applications [Efstratiadis and Koutsoyiannis, 2010]. Forthcoming research will investigate whether is it practical to increase the number of fitting criteria, taking into account that the WOFSA enables to constrain the feasible Pareto front, as depicted in Figure 1, thus significantly facilitating the multiobjective searching procedure.

5.4. The Need for Validation Experiments With Field Data

[73] The proposed WOFSA methodology, which was thoroughly tested on the basis of synthetic data for a wide range of soil texture and climatic conditions, provided consistent and reasonable results. By using synthetic data, we also explicitly ignored uncertainties that are related to field observation errors, thus only focusing to uncertainties due to retrieval of surface soil moisture and evapotranspiration from remote sensing. Evidently, in real-world conditions, inherent modeling and measurement errors and uncertainties cannot be neglected.

[74] Yet, for a full validation of the methodology, and in order to quantify the gain in accuracy would require the collection of field data. This is by far nontrivial, due to the extent of in situ and remote sensing data requirements as well as potential scaling problems. In fact, performing measurements of effective large-scale water fluxes is considered infeasible because typically θ and ET are retrieved at a scale of several square kilometers. Without considering the scale issues, a way forward can be by using precise weighing lysimeters for which all the water fluxes are continuously monitored (storage, drainage, and evapotranspiration). The surface θ determined (for example) by neutron probe or time-domain reflectometer needs to be monitored. To mimic the uncertainties in retrieving θ and ET from remote sensing, noise can be introduced into the measurements of surface θ and lysimeter ET.

[75] The different lysimeters experiments should contain contrasting textures and climate as described in Figure 2. Preferably, the lysimeters should be filled with representative soils and vegetation. Too dry climates may be avoided since it causes strong surface and root-zone θ decoupling for which these periods can be recognized through the newly introduced threshold θd, which is computed from equation (6).

[76] During the validation phase, it is also important to recognize that nondaily information for observed θ and ET is retrieved from thermal-band land surface temperature retrievals, which to date are limited to cloud-free atmospheric conditions [e.g., Anderson et al., 2011]. This implies that the collected data from remote sensing is skewed toward drier conditions.

6. Conclusions

[77] The inversion of the hydraulic parameters of a one-dimensional physically based SVAT model by taking advantage simultaneously of surface soil moisture (θ), and evapotranspiration (ET), requires to take into consideration the uncertainties of retrieving θ and ET from remote sensing and the decoupling of the surface and root-zone θ. To increase the sensitivity of θ, the optimization should not be performed during dry periods, i.e., when decoupling of the surface and root-zone soil moisture occurs. These periods can be recognized through the newly introduced threshold θd, which is computed from equation (6).

[78] The proposed multiobjective approach, by means of a weighted objective function (WOF), provides a suitable compromise between fitting criteria against θ and ET, also taking into consideration the contrasting uncertainties in retrieving θ and ET from remote sensing. As shown in the simulations, the uncertainties of θ have different implications in the computation of the water fluxes of interest compared to the uncertainties of ET. WOF comprises of two control variables, namely a weighting coefficient (w) and the decoupling threshold θd.

[79] In order to determine the best-compromise values of w and θd, we developed a novel inverse modeling framework, called weighted objective function selector algorithm (WOFSA). WOFSA aims to minimize the uncertainties of the computed water fluxes and state variables, following a systematic and as much as objective procedure, in terms of a theoretical framework for formulating an optimal WOF, on the basis of synthetic data. WOFSA performs forward simulations in order to ensure the greatest linearity between the optimized WOF and the maximum uncertainties of the generated water fluxes ΔWF. The ΔWF are derived by mimicking the typically recommended uncertainties of retrieving θ and ET from remote sensing.

[80] To determine how the optimal w and θd of WOF vary under different hydroclimatic conditions, 22 contrasting hydroclimatic scenarios were formulated, by combining five climates, three soil textures, and two different rooting depths. Based on the results provided by WOFSA, we established relationships between the optimized values of w and θd. In particular, for all scenarios, we provided empirical relationships to compute θd from the average values of θ and ET, and the standard deviation of ET. Moreover, for each texture class, we correlated w with average evaporation fraction and with average surface soil moisture, for which we also provided empirical linear equations. All results are interpreted in terms of hydrological evidence, which is a strong justification of the proposed WOFSA methodology. For instance, we found that θd increases for drier hydroclimates and that the rooting depths indirectly influence w through the average evapotranspiration fraction. We remark that typical multiobjective calibration approaches, such as the well-known minimization of the Euclidean Distance of the Pareto set, erroneously assume that the magnitude of Δθrs is similar to the one of ΔETrs and that the impacts of Δθrs and ΔETrs on the simulated model responses are not affected by soil and climate conditions.

[81] In practical terms, it is recommended to employ soil moisture measurements preferably after heavy rainfall, when the soil moisture column is homogenized to avoid soil moisture decoupling. Nevertheless, the measurements should be performed only after the plant is starting to experience stress since it was found that the fitting criteria of ET reduces the sensitivity when the Feddes plant water stress response function equals the potential evapotranspiration. The measurements should also not be taken when the plant is experiencing excessive stress, since it reduces the sensitivity of the fitting criteria of ET and causes soil moisture decoupling. It is also advised to perform the study during the season where evaporation is not at its maximum to avoid soil moisture decoupling.

[82] The proposed framework, which was thoroughly tested on the basis of synthetic data for a wide range of soil texture and climatic conditions, provided consistent and reasonable results. Yet, for a full validation of the methodology, and in order to quantify the gain in accuracy without considering the scale issues, a number of calibration experiments with real data are necessary. Evidently, this task is not trivial, mainly because it is very demanding in terms of in situ data measurements, e.g., through high-precise weighing lysimeters.

[83] Our next research step is the implementation of WOFSA within a multiobjective optimization context, taking into account the preliminary ideas of section 5.3. This will enable to reduce the range of the Pareto set in a hydrological perspective, on the basis of real (observed) data across a specific study area. The results of these investigations will be reported in due course.

Appendix A

[84] The appendix describes the sink term and the interception module of SWAPinv which is substantially different than the ones implemented into SWAP.

A1. Potential Evapotranspiration

[85] The potential evapotranspiration ETp (mm d−1) is estimated by the Penman and Monteith [1997]equation that was further modified by Allen et al. [1998] and is computed by

display math(A1)

where Δv is the slope of the vapor pressure curve (M L−1 T−2θ−1), λw is the latent heat of vaporization of water (L2 T−2), Rn is the net radiation flux density (M T−3) above the canopy, G is the soil heat flux density (M T−3), p1 accounts for unit conversion (86,400 s d−1), ρair is the air density (M T−3), Cair is the heat capacity of moist air (L T−1 θ−1), esat is the saturation vapor pressure (M L−1 T−2), ea is the actual vapor pressure (M L−1 T−2), rair is the aerodynamic resistance (L−1 T), γair is the psychrometric constant (M L−1 T−2 θ−1); and rcrop = 70 s m−1 is the crop resistance [Allen, 1986].

[86] ETp is partitioned into potential evaporation of the wet canopy EPW (mm d−1), potential soil evaporation Ep (mm d−1) and potential transpiration Tp (mm d−1). The partitioning is performed using the leaf area index LAI (m3 m−3) and the fraction of the canopy, 1 − Fw that is not wet. It is to be noted that Fw is computed differently in SWAPinv (equation (A.15)). SWAP assumes that the net radiation inside the canopy decreases exponentially and that the soil heat is negligible. The partitioning is performed by using a Beer-Lambert law [e.g., Ritchie, 1972; Goudriaan, 1977; Belmans et al., 1983]

display math(A2)
display math(A3)
display math(A4)

where Fs (dimensionless) is the interception of solar radiation that will also be used in the interception model, Kg (−) is the extinction coefficient for solar radiation that is set to 0.5 [Varado et al., 2006; Wang et al., 2009]. ETp decreases with increasing Kg and increasing LAI. Epo (mm d−1) is the potential evaporation of bare soil, computed for albedo equal to 0.1. For further information on the computation of ETp, EPW, and Epo the readers are referred to the SWAP manual (http://www.swap.alterra.nl/).

A2. Sink Term

[87] To take into account tree physiology and the reduction of transpiration by soil water stress, the actual transpiration T is distributed by the sink term S(hi) over the whole root zone and is calculated for each cell by Feddes et al. [1978]. The sink term is computed by

display math(A5)

where β is the transpiration fraction or crop factor (−), the value of which is provided in Table 5, Tp (mm d−1) (equation (A2)) is the potential transpiration estimated for short grass, ΔRdfi is the vertical fraction of the root density function per cell i (%) (equation (A6)), and G(hi) is the reduction of root water uptake at pressure head h per cell i (−) (equation (A8)). All these variables except for Tp are dimensionless.

A2.1. The Root-Density Distribution

[88] In SWAP, the vertical fraction of the root-density function per cell iRdfi), which defines the general shape of the roots, is entered manually in tabular form. In SWAPinv, the root distribution is modeled with an empirical function of Gale and Grigal [1987] that was modified further by Pollacco et al. [2008b]. The model requires the rooting depth and the percentage of root density in the top 30 cm (ΔRdf30). It is to be noted that in this literature the percentage of root density is often stated for the top 30 cm, but the user can specify any other depth. The values of the parameters for the two contrasting scenarios used in this study, composed of shallow and deep rooted plants, are provided in Table 4. For each cell i, the fraction of roots ΔRdfi between the top depth zup and the bottom depth zdown is computed as

display math(A6)

where zup and zdown are, respectively, the top and bottom depth of each cell which is positive downward (cm). Ec is the “extension coefficient” parameter, zroot is the rooting depth (cm), and imax is the last cell of the root zone. Ec varies between 0.700 and 0.9999, such that when Ec is close to 0.7 all the roots are distributed in the top cell, and when Ec is close to 1, the roots are distributed evenly within the root zone.

[89] The value of Ec is computed from the percentage of roots. For example, in the top 30 cm, ΔRdf30 is estimated by solving the following equation:

display math(A7)

where Ec is the “extension coefficient” parameter and zroot is the rooting depth (cm).

A2.2. Root Water Uptake

[90] When the capillary pressure head hi per node i is reduced, the vegetation closes their stoma and decreases transpiration, by using the Feddes et al. [1978] stress function computed as follows:

display math(A8)

[91] Water uptake below |h1| (oxygen deficiency) and above |h4| (wilting point) is set to zero. Between |h2| and |h3|, g(hi) = Tp maximal. The value of h3 varies with Tp. For different values of Tp, h3 is linearly interpolated between h3low and h3high. The values of h1, h2, h3high, h3low, and h4 are provided in Table 5.

A3. Evaporation From Bare Soil

[92] The evaporation module of SWAP was simplified. Under wet soil conditions, the actual soil evaporation E [mm d−1] equals the potential soil evaporation Ep. During interstorm period SWAP computes E by using the empirical evaporation method of Black et al. [1969] that requires two fitting parameters. Nevertheless Eagleson [1978], Milly [1986], Simmons and Meyer [2000], and Romano and Giudici [2007, 2009] showed that good results can be achieved by relating evaporation with θ. We therefore used the Romano and Giudici [2007, 2009] evaporation model that does not require any extra parameters

display math(A9)

where the maximum θ is taken from the highest soil moisture between the surface and the depth to 15 cm; θr and θs are the residual and saturated water contents (L3 L−3), respectively, defined earlier by equation (3).

A4. Rainfall Interception Model

[93] SWAP computes rainfall interception following Braden [1985] and Von Hoyningen-Huene [1981]. These interception models require extra parameters and do not use potential evaporation of a wet canopy Epw (mm d−1). We introduced in SWAPinv a physically based interception model, following the work of Noilhan and Lacarrere [1995] and Varado et al. [2006] described in Pollacco and Mohanty [2008a]. In this model, Epw is used as a predictor, while the leaf area index LAI (−) and the extinction coefficient of solar radiation Kg (−) are assumed as parameters. The values of the LAI and Kg are provided in Table 5. The gross precipitation Pg (mm d−1) defined as the amount of water which reaches the canopy is computed following Rutter et al. [1971]

display math(A10)

where Pfree (mm d−1) is the free throughfall that is the fraction of precipitation that reaches the ground surface through gaps in the canopy and Pint (mm d−1) is the intercepted precipitation.

[94] The foliage of the canopy is considered as a water reservoir filled up to a depth of Wr (mm), with a maximum storage capacity Wmax (mm). When the canopy is fully saturated (Wr = Wmax), then any excess of Pint overflows Pover (mm) to the ground such that according to Valante et al. [1997]

display math(A11)

[95] The amount of water that reaches the ground is the net precipitation Pnet (mm d−1)

display math(A12)

[96] A fraction of the water from the reservoir Wr will be evaporated at the rate of the actual evaporation of a wetted canopy EAw (mm d−1) during and after a rainfall event. Wr is calculated following Deardorff [1978]

display math(A13)

[97] The maximum quantity of water that can be evaporated during a time step is computed as

display math(A14)

where Epw is the potential transpiration of a wet canopy.

[98] According to Rutter et al. [1971], evaporation from wet canopies is assumed to be proportional to the fraction of the canopy that is wet Fw (0–1) that is computed following Deardorff [1978]

display math(A15)

[99] Wmax is related to LAI based on the empirical relationship of Varado et al. [2006] and Von Hoyningen-Huene [1981]. Varado et al. [2006] assumes that the interception of water of a canopy is similar to the interception of solar radiation Fs (0–1) (equation (A.4)). Combining Varado et al. [2006] and Von Hoyningen-Huene [1981], Wmax is computed as

display math(A16)

[100] Wmax increases with increasing LAI and Kg. The partitioning of Pg and Pfree is computed as

display math(A17)
display math(A18)
display math(A19)

Acknowledgments

[101] We acknowledge the partial support of National Science Foundation (CMG/DMS grant 062113 and 0934837) and NASA THPs (NNX08AF55G and NNX09AK73G) grants. We are thankful for Kroes Joop (ALTERRA, Netherlands), Jos van Dam (ALTERRA, Netherlands), Isabelle Braud (CEMAGREF, France), Rafael Angulo (ENTPE, France), and Edwin Norbeck (Dept. of Physics and Astronomy, University of Iowa) for their suggestions during the course of this study, as well as Martha Anderson (USDA-ARS Hydrology and Remote Sensing Lab, Beltsville, MD, USA), for her assistance with questions on the uncertainties in retrieving soil moisture from remote sensing. We would also like to thank the three reviewers as well as the Associate Editor, Alberto Montanari, for their constructive comments and critique, which helped us providing a much improved paper.

Ancillary