Parameter estimation of coupled water and energy balance models based on stationary constraints of surface states



[1] We use a conditional averaging approach to estimate the parameters of a land surface water and energy balance model and then use the estimated parameters to partition net radiation into latent, sensible, and ground heat fluxes and precipitation into evapotranspiration and drainage plus runoff. Through conditional averaging of the modeled fluxes with respect to soil moisture and temperature, we write an objective function that approximates the temperature- and moisture-dependent errors of the modeled fluxes in terms of atmospheric forcing (e.g., precipitation and radiation), surface states (moisture (S) and temperature (Ts)), and model parameters. The novelty of the approach is that the error term is estimated without comparison to measured fluxes. Instead, it is inferred from the deviation of the conditionally averaged tendency terms (expectation equation image and expectation equation image) from zero since each of these terms equals zero in stationary systems but diverges from zero in the presence of misspecified parameters. Minimization of the approximated error yields parameters for model applications. This strategy was previously studied for simple water balance models using soil moisture conditional averaging. Here we extend the idea to include energy balance fluxes and surface temperature conditioning. The method is tested at two AmeriFlux sites, Vaira Ranch (California) and Kendall Grassland (Arizona). The estimated fluxes (using only observed forcing and state variables) are in reasonable agreement with field measurements. Because this method is based on conditional averages, it can be applied to situations with subsampled or missing data; that is, continuous integration in time is not required.

1. Introduction

[2] Accurate modeling and estimation of water and energy fluxes at the land surface is significant for climate and weather modeling, for predicting the impact of land use changes, and for agricultural and water resource planning [French et al., 2005]. The relative magnitudes of the turbulent latent and sensible fluxes, which are largely determined by the land surface moisture and temperature states, impact the development of the boundary layer and the dynamics of the lower troposphere [Fisher et al., 2008; Caparrini et al., 2004a].

[3] Recently, seasonal field experiments and permanent observation networks have been used to study these fluxes and their variability over diurnal, seasonal, and interannual time scales. Examples include experiments at First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) [Kanemasu et al., 1992], the Boreal Ecosystem-Atmosphere Study (BOREAS) [Sellers et al., 1997], the Northern Hemisphere Climate Processes Land Surface Experiment (NOPEX) [Halldin et al., 1998, 1999], and the FLUXNET [Baldocchi et al., 2001] network. These direct observations, either using eddy covariance systems, Bowen ratio methods, sap flow measurements, or other methods, are necessary for understanding the system and creating and validating models [e.g., Chen et al., 1997]. However, they are costly and not necessarily representative of landscape-scale fluxes because of the complex heterogeneity of the land surface [Stannard et al., 1994]. Research into scaling observations and models for land surface-atmospheric fluxes, including developing methods that can use satellite observations to improve spatial coverage, has been ongoing for years [e.g., Sellers, 1991; Wood and Lakshmi, 1993; Molod et al., 2004, Fisher et al., 2008].

[4] Accurate spatially distributed estimates of surface fluxes require physically based models driven by accurate field or remote sensing observations of surface variables [French et al., 2005]. Some models for estimating fluxes use remotely sensed data to take characteristics of surface heterogeneity into account. Examples include the Two-Source Energy Balance model (TSEB) [Norman et al., 1995], disaggregated atmosphere land exchange inverse model (DisALEXI) [Norman et al., 2003], the Surface Energy Balance Algorithm for Land (SEBAL) [Bastiaanssen et al., 1998], and the Surface Energy Balance System (SEBS) [Su, 2002].

[5] Regardless of their specific model structure, all of these models and methods require parameters to be specified, and there is a rich history of literature on methods of parameter estimation [e.g., Sellers et al., 1989; Franks and Beven, 1997]. Recently, a multicriteria parameter optimization method has been adopted by Gupta et al. [1999] to estimate the optimal parameter values in the Biosphere-Atmosphere Transfer Scheme (BATS) and thus improve model performance over the use of standard look-up tables based on biomes. Xia et al. [2002] used the same multicriteria calibration method on six modes of the Chameleon Surface Model (CHASM) and again found improvement of model performance. However, most parameter optimization methods require measurements of the fluxes (e.g., latent heat), limiting application to areas where such data exist (e.g., at AmeriFlux sites and in intensive field campaigns such as FIFE). The methods derived here are distinct in that they do not require measured fluxes for calibration, but rather infer parameters from the degree to which modeled tendency terms (the rate of change of water and energy storage) are stationary.

[6] The approach we propose is also distinct from data assimilation techniques [e.g., Reichle et al., 2001; Crow, 2003], which acknowledge errors in measured states, such as soil moisture and temperature, and optimally adjust the state variables using model forecasts. Most related to our work is that of Pan and Wood [2006], who developed a constrained ensemble Kalman filter (CEnKF) that allows the merging of land surface state and flux observations assimilated into a land surface model while maintaining closure of the water budget (which can be considered a stationarity constraint). The quality of inferred fluxes from data assimilation depends on both the underlying model and the parameters specified in that model. The methods presented here for parameter estimation could be followed by data assimilation, but we have not attempted to do that here.

[7] Other strategies combine parameter estimation and surface temperature assimilation to estimate surface heat fluxes and energy partitioning [e.g., Castelli et al., 1999; Boni et al., 2001; Caparrini et al., 2004b]. Recently, Sini et al. [2008] extended these efforts by introducing a simplified soil wetness model into a data assimilation scheme and thus enhanced the estimation of surface energy balance partitioning. The governing parameters estimated in these studies are evaporative fraction and bulk transfer coefficient, making the cost function relative easy and feasible to compute. However, evaporative fraction, which represents the surface control on latent heat flux, can only be assumed to be approximately constant for near-peak radiation hours on days without precipitation [Crago and Brutsaert, 1996], which imposes restrictions on the applicability of the method.

[8] Salvucci [2001] developed a method to extract information about the soil moisture dependence of surface hydrologic fluxes from soil moisture and precipitation data. The basis of the method is that under stationary conditions, the expected value of the change in soil moisture storage over an interval of time, conditioned on the storage during that interval, is zero. In conjunction with the water balance equation, the disappearance of the storage change term implies that precipitation measurements over an interval time, conditionally averaged according to the soil moisture storage, can be used as a surrogate measure of moisture-dependent total outflow (evapotranspiration plus runoff and drainage). Under these constraints, model parameters can be estimated by matching the soil moisture conditional expectation of modeled fluxes to the soil moisture conditional expectation of precipitation. The novelty of the approach is that parameters controlling evaporation and drainage are estimated without measurements of evaporation and drainage. Instead, the parameters are estimated on the basis of the information provided by the covariance of precipitation and soil moisture, as expressed in the conditionally averaged precipitation equation image. A further strength of this method is that the required data (precipitation and soil moisture) do not need to be measured continuously in time. However, while the method robustly estimates the combined sensitivity of evapotranspiration and drainage to soil moisture, it has difficulty distinguishing evaporation from drainage and runoff, particularly for relative dry surface soils when the root zone can draw moisture up from deeper stores to satisfy evaporative demand [Salvucci, 2001].

[9] To address this weakness, we take the energy balance into account and apply a similar conditional averaging approach to net radiation with conditioning on surface temperature. Because the water and energy balance are linked through evapotranspiration, more robust parameter estimation and flux partitioning can be expected.

[10] We describe the conditional averaging methodology and a simplified coupled water and energy balance model in section 2. Then we present and discuss results of applying the proposed parameter estimation technique at a synthetic case in Charlotte, North Carolina, in section 3 and two AmeriFlux sites (Vaira Ranch in California and Kendall Grassland in Arizona) in section 4. The results are contrasted with a traditional parameter estimation on the basis of matching the observed and modeled fluxes. This allows discrepancies between the estimated and measured fluxes to be attributed both to weaknesses of the underlying model and to uncertainties introduced by the proposed parameter estimation strategy.

2. Parameter Estimation Methodology and Land Surface Model

2.1. Method Development

[11] The parameter estimation strategy is based on the method developed by Salvucci [2001]. The detailed derivation is provided for the water balance equation, and then an analogous result is presented for the energy balance equation.

[12] For a unit area of land surface, the water balance equation can be written as

equation image

[13] In equation (1), equation image represents the rate of change of moisture stored (L T−1) in a layer of soil, starting at the surface and extending to some depth z. The variables on the right-hand side represent instantaneous fluxes (L T−1): P is precipitation, ET is evapotranspiration, and Q is the combined water losses resulting from surface runoff and drainage out of (or capillary rise into) the surface layer extending to depth z.

[14] We treat the runoff and drainage losses as a term that is dependent solely on a single soil moisture storage (S) and denote it as equation image, where equation image represents a vector of model parameters (e.g., one component of equation image will be the saturated hydraulic conductivity). This simplified model does not account for multiple soil layers, or for topography. Similarly, evapotranspiration will be modeled as dependent on the same single-layer moisture variable, S, and also on a bulk surface temperature (Ts), meteorological forcing variables (e.g., wind speed (u) and vapor pressure (eair)), and further parameters that control aerodynamic and surface (combined soil and canopy) conductance. We denote this as equation image, where equation image represents the augmented vector of the model parameters. At this stage of the derivation, we are only focusing on identifying state variables and parameters and not on particular models.

[15] We can rewrite the rate of storage change, with these general formulations of the dependence of fluxes on states and parameters, as

equation image

[16] In equation (2), equation image represents model structural error, that is, the part of ET and Q that cannot be explained by the models equation image and equation image because of structural deficiencies in the models themselves. For the purpose of simplifying the derivation, we assume that the forcing and states are measured without error. Including measurement error in the forcing (e.g., P, u, eair) and states (S, Ts) and deriving the impact of those errors on the parameter estimates are beyond the scope of this work.

[17] For an estimate of the parameter vector, denoted equation image, the model-estimated storage change rate can be written as

equation image

[18] We can rewrite equation image in terms of model error and an added parameter specification error by subtracting equation (2) from equation (3), as

equation image

[19] In equation (4), equation image represents error in the estimated rate of storage change due to misspecification of the parameter vector (i.e., using equation image instead of the true model parameters equation image):

equation image

[20] Our purpose here is to minimize the parameter specification errors without direct measurements of ET, Q, or equation image. There are other ways to estimate the parameters. For example, if we can observe ET and Q directly, the parameters could be simply estimated by least squares fitting of the model to observation. Or, if we had continuous measurements of S, then we could take its time derivative and minimize the squared difference between the observed equation image and equation image. Here, however, we are trying to infer these parameters indirectly from information contained in S (and Ts) and in the future from remotely sensed indices that are related to these states but that may not be measured continuously. The method developed here allows for both subsampling and for S to be an index of (as opposed to a direct measure of) moisture storage.

[21] To achieve this goal, we first partition equation image into a term related only to soil moisture and a remaining term related to other sources. Taking the conditional expectation of the error with respect to soil moisture, the moisture-dependent error term can be written as equation image. This conditioning extracts those components of the error in equation image that arise from parameter misspecification in flux terms related to soil moisture. For example, an incorrect soil hydraulic conductivity will create moisture-dependent errors in drainage, which ultimately lead to moisture-dependent errors in equation image. With the error term conditionally averaged, we can conditionally average equation image and equation image and rearrange equation (4), yielding

equation image

[22] Now we exploit the stationarity condition on the moisture storage term. Salvucci [2001] demonstrated (through mathematical derivation, Monte Carlo studies, and observations) that equation image. Given that equation image vanishes, and under the assumption that equation image, that is, assuming our models are flexible enough with respect to moisture that model structural error will vanish, equation (6) becomes

equation image

[23] Equation (7) implies that we can minimize the magnitude of equation image with respect to the parameter vector equation image by minimizing equation image with respect to equation image. The key strength is that we can express the objective function solely in terms of observed forcing (P, u, eair), surface states (S, Ts), and estimated parameters equation image, thus allowing estimation of equation image through minimizing the objective function.

[24] To extend the method for energy balance, we write an energy conservation equation that expresses the relationship between forcing and states in similar form to the water balance equation. We assume there is a bulk medium that stores, absorbs, and releases energy. That is, the fluxes and the temperature state are, for an effective medium, representing the combined soil and vegetation components. Under this representation, conservation of energy is written as

equation image

[25] In equation (8), equation image represents the rate of change of heat storage per unit area (J m−2 s−1) in the bulk medium, and equation image (J m−2 K−1) is the effective heat capacity of the medium, which we approximate as a constant but acknowledge that it should vary with soil and vegetation water content and biomass. The terms on the right side of the equation represent instantaneous energy fluxes at the land surface. Rn is the net radiation. Here it is taken as a measured forcing, but equation (8) could readily be rewritten with model-estimated upwelling shortwave and longwave radiation. H is the sensible heat flux, presumed to depend only on surface and air temperature and wind speed. LE is the latent heat flux, presumed to depend on surface moisture and temperature, air temperature, vapor pressure, and wind speed. Finally, G is the ground heat flux leaving the bulk medium, presumed to be dependent only on the surface temperature and a deeper, time-averaged surface temperature. The specific model formulations used for these fluxes are discussed in section 2.2.

[26] Following the derivation of the moisture-dependent error term, conditional averaging and stationarity of surface temperature yield

equation image

[27] Note that equation image is now augmented to include the parameters specific to the energy balance flux components.

[28] Combining the error terms from both the water (equation (7)) and energy balance (equation (9)), we set up an objective function in terms of the two stationarity terms equation image as follows:

equation image

In equation (10), the two tendency terms that approximate the moisture- and temperature-dependent errors have been expressed in common units of W m−2, multiplying the moisture term by the product equation image where equation image is latent heat of vaporization (taken as 2.47 × 106 J kg−1) and equation image is density of water (taken as 1000 kg m−3).

[29] In the applications in sections 3 and 4, we further expand each of the tendency terms equation image and each of the conditioning variables (S and Ts) into a seasonal term and a perturbation. This is done by regressing the terms onto Fourier components of the Julian day (using three harmonics). Because the seasonal component and perturbation are both stationary, each conditionally averaged tendency term equation image will be zero, and the objective function becomes

equation image

[30] Using equations (7) and (9), the parameters of the land surface energy and water balance model can be estimated by minimizing the weighted objective function (equation (11)). The novel characteristic of this approach is that the parameters governing the individual water and energy fluxes can be estimated without measuring each flux. Instead, only “forcing” terms (net radiation, precipitation, wind speed, humidity, and air temperature) and state variables (moisture and temperature) are required. By coupling the water and energy balance in one model, we expect more robust estimation of the parameters than by using only the water balance.

2.2. Model Formation

[31] The stationarity-based objective function we propose in this paper is meant to be generally applicable. In this paper, we apply it to a simplified model which itself is derived, in part, from components of the Noah land surface model (LSM) [Chen et al., 1997], the MOSAIC model [Koster and Suarez, 1996], and the Simple Biosphere Model [Sellers et al., 1996].

2.2.1. Water Balance

[32] In the water balance equation as described in section 2.1, precipitation (P) is measured directly. The net drainage and runoff term (Q) is represented as follows:

equation image

[33] In equation (12), k, w, c, and n are parameters which account for the properties of water flow in the soil. This formation allows both downward (when soil is wet) and upward (when soil is dry) flow to and from the root zone.

[34] The evapotranspiration term (ET) in equation (1) couples the water and energy balance equations. Here we relate it to the latent heat flux, and in section 2.2.2 we specify a model for it.

equation image

2.2.2. Energy Balance

[35] In the current development and testing of this approach, we use measured net radiation (Rn) directly instead of measured incoming radiation and modeled outgoing radiation. This is feasible since the net radiation was measured and recorded in the data sets we use in this paper. More parameters (e.g., albedo, cloud cover, and atmospheric emissivity) would be required to apply the model with incoming solar radiation as the forcing term.

[36] The following simple model is adopted for the ground heat flux:

equation image

[37] In this force restore type ground heat flux model, Ts is bulk surface temperature. To be consistent with the temperature used in turbulent fluxes, we use the skin (i.e., radiometric) temperature in our model, which is estimated from outgoing longwave radiation, i.e., equation image, where Lout is the observed outgoing longwave radiation (reflected longwave is neglected here), equation image is the Stefan-Boltzmann constant (5.67 × 10−8 W m−2 K−4), and ɛ is the emissivity parameter to be estimated. Td is deep soil temperature. Td is estimated in this paper as the time average of Ts; Kd is the apparent conductivity of the soil, calibrated as done by Garratt [1992] for seasonal time scale fluctuations as equation image, where equation image is angular velocity of the Earth, Cs is volumetric heat capacity, and equation image is the soil thermal diffusivity. Reasonable upper and lower limits on Kd can be estimated according to typical parameter values of Cs and equation image for different soil textures.

[38] The sensible (H) and latent (LE) heat fluxes are modeled using bulk difference equations [e.g., Garratt, 1993]:

equation image
equation image

[39] In equations (15) and (16), equation image is the density of air (approximated here as 1.24 kg m−3), cp is the heat capacity of the air (1004 J kg−1 K−1), Pa is the surface atmosphere pressure, ua is the wind speed measured at some height z (typically, 2 m), Ta is the air temperature at height z, es and ea are the vapor pressure at the same levels as Ts and Ta, and qa and qs are the corresponding specific humidity values. The surface vapor pressure (es) is assumed to be saturated at the surface temperature (Ts).

[40] Ch and Ce are the bulk transfer coefficients for heat and water vapor, respectively. They are intended to represent complex exchanges (involving molecular diffusion and turbulent mixing) between the three-dimensional land surface (vegetation, soil, open water) and atmosphere at the scale of the measured (or modeled) state variables. Given the complex processes involved, and the variety of scales (tens of meters to hundreds of kilometers) at which such equations are applied (depending on model resolution and measurement support scale), they are probably best defined implicitly by equations (15) and (16) instead of assigning too much physical realism to them. Chen et al. [1997] found that differences in land surface parameterization schemes and the resulting bulk transfer coefficients did not result in significant differences in model-simulated surface fluxes, after testing three widely used parameterization schemes (Mellor-Yamada level 2, Paulson, and modified Louis) in both uncoupled model runs and when coupled to the mesoscale ETA model of National Centers for Environmental Prediction (NCEP). However, they did find that the models are very sensitive to the relations between bulk transfer coefficients and roughness lengths. Their testing showed that the Zilitinkevich [1995] formulation, relating the ratio of roughness length for heat to the roughness length for momentum to the Reynolds number, improved model performance. Below we adopt this formulation.

[41] Here we use the modified Louis model suggested by Chen et al. [1997]: Ch is parameterized following the modified Louis approach given by Mahrt [1987] for the stable boundary layer case and Holtslag and Beljaars [1989] and Chen et al. [1997] for the unstable case, i.e.,

equation image

[42] In equation (17) the function equation image is given by

equation image

for stable conditions and by

equation image

for unstable conditions.

[43] For the calculation of the Richardson number (Ri), we use the work of Dingman [2002]:

equation image

[44] In equations (18)(20), a is a constant (set equal to 1 in the model); kv is the Von Karman constant, set to 0.4; and z is the height at which the wind speed and air temperature are measured. Following common practice [Dingman, 2002], the zero-plane displacement zd is approximated as 0.7 zveg, and the roughness height for momentum z0m is approximated as 0.1 zveg. In our model, we estimated vegetation height zveg as a free constant parameter. We also evaluated the model with vegetation height dependent on leaf area index (LAI), but the improvements were limited. The roughness height for heat z0h is more complicated and will be discussed below.

[45] When estimating surface fluxes using bulk difference equations (15) and (16) in numerical models or with data, issues arise concerning the relationship between the aerodynamic surface temperature (appearing in equation (15)) and the skin temperature. The differences can be accounted for by explicitly modeling each temperature or by using a single temperature but with different roughness lengths for heat (z0h) and momentum transfer (z0m) [Chen et al., 1997; Sun and Mahrt, 1995; Garratt, 1992]. Relating these lengths (z0h and z0m) is a challenging problem that has been studied widely in the past decades [Owen and Thomson, 1963; Thom, 1972; Verma, 1989; Garratt, 1992; Stewart et al., 1994; Braud et al., 1993; Beljaars and Viterbo, 1994; Kubota and Sugita, 1994; Zilitinkevich, 1995; Blumel, 1999]. Suggestions from the literature range from simply setting z0h as a fixed fraction of z0m, to defining z0h as a function of radiation, soil type, canopy height, atmospheric stability, and vegetation cover fraction. For operational numerical weather prediction models, Chen et al. [1997] compared a fixed ratio method with a Reynolds number–dependent formulation suggested by Zilitinkevich [1995] and found that the Zilitinkevich approach improved the surface heat flux simulations in their one-dimensional column model and reduced forecast precipitation bias in the NCEP mesoscale Eta model. Following Chen et al. [1997], we specify the relation between z0h and z0m as

equation image
equation image

[46] In equation (22), equation image is the kinematic molecular viscosity (1.44 × 10−5 m2 s−1), Re is the roughness Reynolds number, u* is the surface friction velocity, calculated through equation image, and C is a free parameter (see Table 1 for upper and lower limits).

Table 1. Definition, Lower, and Upper Boundary for Parameters Used in the Model
ParameterPhysical MeaningLower BoundaryUpper Boundary
KdApparent soil conductivity for heat flux (W m2 K−1)0.21
kSoil hydraulic conductivity (cm d−1)0300
cPercolation exponent in net drainage model (dimensionless)315
wCapillary rise (cm d−1)010
nCapillary rise exponent in net drainage model (dimensionless)01
equation imageFirst parameter in vegetation evaporation efficiency (dimensionless)0Minimum of soil moisture index
equation imageSecond parameter in vegetation evaporation efficiency (dimensionless)Minimum of soil moisture index1
AMaximum leaf conductance (mm s−1)0.225
zvegVegetation height (cm)1100
CParameter in roughness length formulation (dimensionless)01
equation imageFirst parameter in soil evaporation efficiency (dimensionless)0Minimum of soil moisture index
equation imageSecond parameter in soil evaporation efficiency (dimensionless)Minimum of soil moisture index1
ɛEmissivity (dimensionless)0.951
BMaximum soil conductance (mm s-1)0.01100
gParameter in VPD stress term controlling transpiration (1 KPa−1)00.3
ToptThe optimal air temperature for transpiration (°C)1040

[47] Similar issues arise concerning the differences between the aerodynamic surface temperature and the bulk surface temperature used in the ground heat flux equation (14) and the heat storage term (left-hand side of equation (8)). We assume that the parameter estimation of Kd in equation (14) will compensate for the first problem. Because the storage term becomes zero under the conditional averaging applied here, the latter issue is assumed to be irrelevant.

[48] The above bulk transfer coefficient (Ch) is applied for H. The bulk transfer coefficient for water vapor (Ce) is further modified on the basis of the formulation of Ch (i.e., equation (17)). For LE, we need to consider canopy and soil resistance to water vapor and evapotranspiration dependence on soil moisture, vapor pressure deficit (VPD), and temperature conditions. To keep the model as simple as possible, we define another surface conductance Csurf, which is composed of canopy conductance and soil conductance for the bulk surface area,

equation image

[49] In equation (23), Ccanopy represents canopy conductance, and we apply a Jarvis-type stomatal conductance [Jarvis, 1976], which is then scaled by LAI and associated with three environmental stress terms:

equation image

[50] Here A is a free parameter accounting for maximum leaf conductance. The first environmental stress function characterizes the influence of soil moisture, defined as

equation image

which is a transpiration efficiency meant to represent stomatal conductance limitations under dry soil conditions. Two parameters equation image and equation image are used to represent F1 as a piecewise linear function. Restrictions on the vales of equation image and equation image are listed in Table 1. The soil moisture stress function is evaluated relative to the dimensionless soil moisture index equation image, where equation image represents the measured soil moisture data and equation image is the maximum of soil moisture data in the studied time series. This two-parameter model for soil moisture control on stomatal conductance is flexible, allowing for control of the slope (i.e., sensitivity of conductance to moisture and for control of the range of moisture over which F1 limits conductance).

[51] The second stress term, representing the influence of VPD, is defined as the same formation in the Noah land surface model [Ek et al., 2003; Alfieri et al., 2008],

equation image

[52] In equation (26), g is a species-dependent empirical parameter. In the Noah LSM, g is taken as a constant. Here it is a free parameter (see Table 1 for upper and lower limits).

[53] The third environmental stress term is related to temperature, and following Jarvis [1976], Matsumoto et al. [2005], and Samanta et al. [2008], it is defined as

equation image

[54] In equation (27), Topt, Tl, and Th represent the optimal, lowest, and highest temperature for transpiration, respectively. Tl and Th are fixed as 0°C and 50°C, while Topt is a free parameter (see Table 1 for bounds).

[55] In equation (23), Csoil represents soil conductance, defined as

equation image

where B is a free parameter accounting for maximum soil conductance. Csoil is scaled by effective soil area index (SAI), which is fixed to be equal to 1. F4 is an evaporation efficiency meant to represent bare soil evaporation limitations under dry soil conditions. Similar to the soil moisture stress function in canopy conductance, F4 is dependent on the dimensionless soil moisture index (S), but with two distinct parameters equation image and equation image to be estimated, i.e.,

equation image

[56] Conceptually, soil evaporation is treated like an additional layer of leaf area, but with its own moisture dependence and maximal conductance. This is clearly an approximation, but alternative, more realistic formulations (e.g., as by Shuttleworth and Wallace [1985] or Choudhury and Monteith [1988]) would require specification of canopy and soil temperatures, whereas our model is written in terms of a single bulk temperature. Using the circuit analogy often employed in soil-vegetation-atmosphere parameterizations, the soil and vegetation flow paths are being treated as if in a parallel circuit. Koster and Suarez [1996] applied the same single temperature, dual conductance approximation in the MOSAIC land surface model.

[57] For the combined soil and vegetation area, LE can now be written as

equation image

[58] Applying equations (12)(30), all terms in the coupled land surface water and energy balance model are expressed in terms of measured forcing (e.g., precipitation, net radiation, wind speed, air temperature, and humidity) and the bulk state variables S and Ts. Thus, after calculating the conditional average of these terms with respect to measured surface states (soil moisture and skin temperature), the parameters (see Table 1) of the coupled model can be estimated by minimizing the objective function, as stated in section 2.1 Following parameter estimation, the fluxes at land surface can be obtained by reapplying equations (12)(30) with the observed forcing, states, and estimated parameters.

[59] In the testing reported in sections 3 and 4, we use soil moisture measurements from the top few centimeters of soil, instead of root zone average moisture, so that the resulting estimation errors will be more similar to those expected when using remotely sensed soil moisture (which is the ultimate goal of this work). Through numerical simulation and analysis of AmeriFlux data, we have found that the soil moisture of the top few centimeters is correlated well enough with the deeper soil moisture that the shallow estimate works well in the estimation procedure. We have found that when using soil moisture averaged to deeper depths, the estimated parameters in the drainage and evaporation models are different but predict similar fluxes.

2.2.3. Shortcomings of the Simple Water and Energy Balance Model

[60] Keeping the number of free parameters relatively small (16) and writing all equations in terms of two bulk state variables (S and Ts) clearly comes at the cost of physical realism. Physically more realistic models (e.g., the community Noah land surface model [Ek et al., 2003]) could be described to account for multiple vegetation and soil layers, each with a prognostic temperature and moisture content, and each with its own contribution to latent and sensible heat fluxes. In doing so, however, the number of free parameters grows beyond that which can realistically be estimated even under the most ideal conditions of long-term field experiments (e.g., see discussion by Rosero et al. [2009]), and the conditioning methods proposed here for parameter estimation become increasingly complex.

2.3. Parameter Estimation

[61] Here we test our model and estimation method with daily averaged forcing and surface data. The original AmeriFlux data are half hourly, with some missing values, which can cause anomalous daily values when we aggregate the half-hourly data to daily ones. In the analysis in section 4, we have filled in missing data using a regression model containing seasonal and diurnal cycles. We also treat measurements that are more than four standard deviations from the regression estimate as outliers and replace them with the data from similar conditions. For example, we search the data within one hour surrounding the missing point in the same day and same year. If there are values at those times, we replace the missing point with the existing ones; if those values are also missing, we continue to search the data within one hour in the nearest three days and same year; if there are still no suitable values, continue to search data within one hour in the nearest three Julian days in any year. When one forcing variable is replaced, every other forcing is replaced at the same time to keep the covariance among forcing and state variable realistic. Very few points (<1%) were identified as outliers.

[62] As discussed, the parameters of the physical model (equations (12)(30)) are estimated by minimizing the stationarity-based approximation of the model error given in equation (11). Here we take equation (10) as an example to demonstrate the calculation procedure. The first step required to minimize (10) is to choose an estimator of the conditional average terms, i.e., an estimator of equation image. Here we do this by dividing the range of the conditioning variable (the observed soil moisture, S, in this example) into 32 bins chosen such that each bin has the same number of observations. We then calculate the arithmetic mean of all the values of equation image (given by equation (3) for some set of parameters), for which S falls in a given bin. In this way the entire time series of equation image, for example 1000 daily values found by evaluating equation (3) for the specific models described by equation (12)(30), is reduced to 32 conditionally averaged values for use in equation (10). The same set of steps is taken for the energy balance equation to estimate equation image. While using equation (11) in the application, we simply follow the above procedure but expand the tendency terms equation image and the conditioning variables (S and Ts) into a seasonal term and a perturbation, respectively.

[63] With equation (11) thus evaluated, minimizing it with respect to the parameter vector can be accomplished through traditional grid search methods, but this is extremely time consuming for such high-dimensional optimization problems, and the results are sensitive to the grid spacing. Gradient-based methods (e.g., fmincon of MATLAB) are fast but often return local minima. We found that gradient methods with random initial guesses yielded repeatable results as long as they were applied hundreds of times, but again, they cannot guarantee a global minimum. Considering efficiency and safety (i.e., avoiding local minima), we ultimately chose the generalized pattern search (GPS) algorithm, found in the Genetic Algorithm and Direct Search Toolbox in MATLAB, as our main optimization strategy. Unlike most traditional optimization algorithms, which use the gradient or higher derivatives information to search for an optimal point, the GPS algorithm guarantees a global convergence without requiring directional derivative or enforcing a notion of sufficient decrease. The pattern search method handles optimization problems with nonlinear, linear, and bound constraints and does not require functions to be differentiable or continuous [Torczon, 1997; Lewis and Torczon, 1999, 2000; Audet and Dennis, 2003]. In our testing, results from a single application of the GPS algorithm were essentially identical to those from applying fmincon with hundreds of random initial points.

3. Model Testing Against Benchmark Synthetic Data

[64] A simulated data set with meteorological forcing from Charlotte, North Carolina, is used as a first test of the theoretical underpinning of the estimation approach since, in such a synthetic case, perfect knowledge of the states and fluxes are available. The synthetic time series of surface moisture and temperature states were simulated using a simple three-soil-layer soil vegetation atmosphere transfer (SVAT) model with turbulent fluxes modeled as in equations (12)(30) but with equation image in equation (17) forced to equal 1 to represent neutral conditions. The model numerically solves the diffusion equations for soil moisture and temperature as forced by incoming solar and longwave radiation; wind speed, air temperature, and vapor pressure at 2 m; and precipitation. It is not the most complex SVAT model, but it has the key ingredients necessary to serve as a test bed for our parameter estimation methodology. The forcing data used to simulate the test case are from the Solar and Meteorological Surface Observation Network (SAMSON), for the 30 year period 1961 to 1990, at a station in Charlotte, North Carolina. The forcing data are hourly, and the model is run at subhourly time steps, thus capturing diurnal processes. In the tests below, the output of the simulations and the forcing data are aggregated to daily values. In this synthetic test case, the LAI was held constant and equal to 3, and the soil hydraulic parameters were typical of a silty-loam soil. Bare soil evaporation was not modeled in the synthetic case.

[65] The estimated and benchmark energy and water fluxes are plotted in Figure 1 as conditional averages against moisture and skin temperature. The fluxes are segregated into 12 bins with respect to surface states (soil moisture and skin temperature), with the mean and standard error of each flux (and tendency term) plotted. For the water balance case, a soil moisture index (the top 10 cm water content scaled to its maximum value) is used on the x axis instead of soil moisture content. The measured and estimated fluxes match very well for each term of the balance equations. As found by Salvucci [2001] for water balance, the conditional average of the tendency terms (energy balance (EB) residual and water balance (WB) residual) fluctuate within a small range around zero. Note that the residual of the conditionally averaged moisture tendency is largest in the wettest bins, where it approaches a few millimeters per day. Residuals of this magnitude are not unusual after conditional averaging at the wet end of the soil moisture range because the variance of the conditional mean decreases approximately as the variance of the tendency term divided by N, where N is the number of data points in that bin of moisture values. The variance at the wet end is dominated by the variance of precipitation, which is large, and the system does not spend a long time at the wet end (because of rapid drainage); thus, the net effect is to divide a large variance (of precipitation) by a small N. In this case (Figure 1), for example, only about 0.5% of all the data points fall in the last three bins of water balance. Across the entire range of soil moisture, the water balance residual is smaller than the error bars of the conditional mean precipitation.

Figure 1.

Comparison of estimated and simulated fluxes for a synthetic test case using meteorological forcing from Charlotte, North Carolina. The solid lines are obtained from the estimated model (equations (12)(30)), and dashed lines with error bars represent the simulated (i.e., benchmark) fluxes through a three-soil-layer soil vegetation atmosphere transfer model. (a) Energy balance and (b) water balance.

[66] Plotting the fluxes conditionally averaged according to moisture and temperature is useful for testing the basic idea behind the stationarity-based estimation method. That is, the conditionally averaged (by temperature) surface energy fluxes (H, LE, G) sum to the conditionally averaged radiative forcing (Rn) and the conditionally averaged (by soil moisture) surface water fluxes (ET, Q) sum to the conditionally averaged precipitation forcing (P).

[67] This method of plotting also reveals model behavior and, as shown in the field tests in section 4, moisture- and temperature-dependent model errors. Note, for example, that the drainage is negative (i.e., upward capillary flow) for dry soil conditions and strongly positive (i.e., percolation) for wet conditions. In the synthetic data, this behavior arises naturally from the simulation of water movement between layers. In the simplified model that is fit to the synthetic data, it is represented by the relative magnitude of the parameters in equation (12). Also, note the interesting behavior of the sensible heat flux with respect to surface temperature. It is relatively large both when the soils are hot (and evapotranspiration is low because of dry moisture conditions) and when the soils are cold (when evapotranspiration is inefficient because of a large equilibrium Bowen ratio [Raupach, 2001] and the ground heat flux is toward the surface).

4. Model Testing at Two AmeriFlux Sites

[68] With confidence that the underlying stationarity-based parameter estimation works well under ideal conditions, we tested the methodology using field measured data from the AmeriFlux network. The network provides continuous observations of carbon dioxide, water vapor, and energy exchanges at the ecosystem level spanning diurnal, seasonal, and interannual time scales. It includes sites throughout the American continents. AmeriFlux is part of FLUXNET, which coordinates regional and global analysis of observations from micrometeorological tower sites [see Baldocchi et al., 2001].

4.1. Study Sites

[69] We chose two flux tower sites with different micrometeorological conditions for our study. One is Vaira Ranch (38.4067°N, 120.9507°W, elevation of 129 m) in California. The soil type is rocky silt-loam. The site is a grazed C3 grassland opening in a region of oak and grass savanna. The climate is Mediterranean. Forcing and surface state data from 2004 to 2007 are used in the analysis. The annual mean precipitation and air temperature are 565.8 mm and 16.0°C during the study period. The second site is the Kendall Grassland (31.7365°N, 109.9419°W, elevation of 1531 m) in Arizona. The tower is located in a small intensively studied area within Walnut Gulch Experimental Watershed. The soil is coarse-loamy, mixed, and contains some limestone fragments. The climate is temperate and semiarid and the landscape is a warm season C4 grassland with a few shrubs interspersed. Forcing and surface state data from 2006 to 2007 are available and used for testing, with the annual mean precipitation and air temperature at 293.6 mm and 17.3°C during the period.

[70] At Vaira Ranch, the field measured LAI are available and linearly interpolated to daily for use in this model. At Kendal Grassland site, Moderate Resolution Imaging Spectroradiometer (MODIS)-derived LAI estimates (MOD15A2, are used to represent the seasonality of the vegetation cover, with the original 8 day composite products disaggregated to daily values to match the daily forcing and surface state data. In both cases, the first layer volumetric moisture content (equation image, representing the top 3 cm at Vaira Ranch and 5 cm at Kendall) is transformed and used to represent the dimensionless soil moisture index S (i.e., equation image).

4.2. Results and Discussion of the Field Tests

[71] The GPS optimization scheme introduced in section 2.3 was applied to minimize the stationarity-based estimate of the error terms (equations (7), (9), and (11)) for the coupled water and energy balance model (equations (12)(30)) at the two sites. The parameters that minimized equation (11) were then used to calculate the water and energy fluxes and compared with the observed fluxes.

[72] Note that generally there is an observed energy balance closure problem at AmeriFlux sites [Wilson et al., 2002]. Here the observed half-hourly turbulent fluxes are corrected using the Bowen ratio method [Twine et al., 2000], except for times of negative Bowen ratio, when that procedure can yield huge, offsetting latent and sensible heat fluxes (specifically if the Bowen ratio approaches minus one). For those conditions, we add half of the energy residual (net radiation minus the sum of ground, sensible, and latent heat flux) to the latent heat flux and half to the sensible heat flux. These adjusted fluxes are used to evaluate our model performance and methodology. At the Vaira Ranch site the impact of the adjustments was minimal. At the Kendall Grassland site the energy imbalance is larger and more consistently biased (net radiation exceeds the sum of ground, sensible, and latent heat by approximately 25%), and thus, the adjustments are larger. The results are discussed in sections 4.2.1 and 4.2.2.

4.2.1. Vaira Ranch in California

[73] The model-estimated and observed fluxes, conditionally averaged with respect to surface states both for the energy and water balance, are illustrated in Figure 2. The averaging and parameter estimation was done using 32 bins, but the results are plotted in 8 bins for clarity in the figures. We can see that the model captured the sensitivity of the fluxes to moisture and temperature, and the overall magnitude of the fluxes, relatively well. Note again that the parameters used in the model-estimated fluxes (the solid lines) were estimated without any knowledge of the drainage flux or the latent, sensible, and ground heat fluxes. They were estimated only from the measured forcing (P and Rn) and surface states (S and Ts). The estimated parameters are listed in Table 2 and discussed later in this section.

Figure 2.

Comparison of modeled and measured conditionally averaged fluxes at Vaira Ranch, Ione, California. The solid lines are the modeled fluxes using parameters found by minimizing the stationarity-based objective function (equation (11)), and the dashed lines (with error bars) represent the field measured fluxes. The parameters used in the model-estimated fluxes (the solid lines) were estimated only from the measured forcing (P and Rn) and surface states (S and Ts) without any knowledge of the drainage flux or the latent, sensible, and ground heat fluxes. (a) Energy balance and (b) water balance.

Table 2. Parameter Values Chosen in Best Fit Comparisons and the Proposed Stationarity-Based Objective Functions for Vaira Ranch.a
Best fit 10.89----7.220.6800.310.07125.0501013.263.630.18
Best fit 20.89----7.542.110.040.310.07127.030.0510.3015.793.180.56

[74] The physical behavior underlying the statistical relationships between fluxes and surface states in Figure 2 can be understood by considering both the seasonal controls on surface temperature and moisture and also the controls due to random weather patterns. Because of the Mediterranean climate in California, the site experiences clear skies, high temperatures, and low humidity during summer, while in the winter it is relatively cold with precipitation occurring mostly from October to May. The grass cover is abundant and actively transpiring during the winter months when temperatures are relatively cool and rainfall is frequent. Transpiration is greatly reduced because of arid conditions during the hot, dry summer months. Since the wet winter season receives the least solar radiation with the most rainfall, the water supply exceeds atmospheric demand for water. The latent heat flux LE (asterisks in Figure 2a) is then relatively low for the colder surface temperatures when the atmospheric demand (i.e., potential evaporation) is low. The grasslands receive the most solar radiation with the least rainfall during the dry season, causing the potential water demand to greatly exceed water supply [Ryu et al., 2008] and also causing high surface temperatures.

[75] This behavior is clearly reflected in Figure 2, in both the water balance and energy balance plots. For example, LE increases along with the surface temperature in the cold season since there is no water stress. In the warm season, however, soil moisture becomes limiting, coincident with increasing temperatures, causing the grasses to close their stomata to prevent water loss and ultimately leading to a decrease of LE with further increases of surface temperature. In the summer season, the grasses are mostly dead (the green LAI is close to zero), and LE is close to zero as the surface temperatures reach about 30°C. The sensible heat flux H (pluses in Figure 2a) increases with surface temperature, mostly because of a low temperature gradient in cold seasons and high temperature gradient in warm seasons. As seen in the work of Salvucci [2001], the dramatic increase of conditionally averaged precipitation with soil moisture, as soil moisture increases, is consistent with percolation to deeper soil. Also evident is the upward capillary rise necessary to maintain evapotranspiration under moderately dry moisture conditions.

[76] There does appear to be some structural (i.e., nonrandom) error in the estimated model. For example, LE is underestimated somewhat at low temperature and overestimated at high temperature, and the mismatch also happens for H at some bins. Note that usually there are fewer observations in the extreme bins (e.g., about 5% of observations fall in the last two hottest bins in this case), so the mismatch at those bins may not contribute much to the overall root mean square deviation of the predicted fluxes.

[77] To explore whether these structural errors arise from the parameter estimation methodology or from the model formulations, we estimate a new set of parameters by minimizing the sum of squared errors between the measured and modeled conditional averaged energy fluxes. That is, we use the measured flux data directly to determine a best fit case, relative to temperature-dependent errors, by minimizing the following:

equation image

[78] The resulting conditionally averaged model fluxes are compared with the measured fluxes in Figure 3. Again, the estimated parameters are listed in Table 2.

Figure 3.

Comparison of modeled and measured conditionally averaged fluxes at Vaira Ranch, Ione, California. The solid lines are the modeled fluxes using parameters found by minimizing the temperature-dependent error as expressed in equation (31), and the dashed lines (with error bars) represent the field measured fluxes.

[79] Comparing Figures 2a and 3 demonstrates that some misestimation of parameters is due to the stationarity method: When conditionally averaged, the overestimated LE for high temperatures and underestimated H fit the net radiation slightly better than the best fit case and thus are chosen in optimization. Focusing on Figure 3 reveals structural errors. In the best fit case, the conditional averages of modeled and measured LE are in almost perfect agreement, but H still shows some biases at the colder bins. There are many possible sources of structural error in the model (e.g., in the simplified parameterizations in equations (14)(30)) and in our treatment of parameters. For example, periodic removal of brush for cattle grazing and the death of grasses may influence the surface roughness in ways that not be reflected in our treatment of vegetation height as a constant. Similar analysis cannot be made for the moisture-dependent error because we do not have direct measurements of drainage.

[80] For comparison with standard parameter estimation methods, we also estimate model parameters by minimizing a mean square error between the three measured and modeled energy fluxes as

equation image

[81] The angle brackets in equation (32) denote averaging over all observations. The parameter estimates from minimizing equation (32) are similar, but not identical, to those found from equation (31). Equation (32) is more standard, but equation (31) allows, in our opinion, for better identification and minimization of structural errors.

[82] The estimated parameters are reported in Table 2 for the Vaira Ranch case from minimizing the three different objective functions (equations (11), (31), and (32)). The estimated parameters are within typical ranges (see Table 1), and the chosen values are normal and typical for the site. For example, the chosen values for maximum stomatal conductance (A) are within the range of directly measured values [e.g., Schulze et al., 1994]. The best fit comparisons (equations (31) and (32)) could only be performed for the energy fluxes because drainage was not measured, and so Table 2 does not include values for parameters controlling soil water flow for those cases (k, c, w, and n). Parameters representing leaf and effective soil conductance as well as their dependence on environmental factors are time averaged and reported below as bulk canopy and soil conductance (Ccanopy and Csoil) additionally.

[83] Root-mean-square deviations (RMSD) for energy fluxes are reported in Table 3 for the Vaira Ranch case with parameters estimated from the three different objective functions (equations (11), (31), and (32)). Compared with the best fit case (equation (32)) and with the conditional averaging objective function (equation (31)), the RMSD for the stationarity objective function (equation (11)) is only slightly different. The small differences indicate that the proposed methodology works well and that the mismatch of the measured and modeled fluxes is mostly due to inherent problems with the underlying model itself (rather than the methodology). For perspective, an assessment of some 30 published validations of remote sensing based estimated flux against ground based measurements of evapotranspiration shows an average RMSD value of about 50 W m−2 [Kalma et al., 2008]. Also reported in Table 3 are normalized root-mean-square deviations (NRMSD), which can be calculated as the RMSD divided by the range of the observations (e.g., the maximum minus the minimum flux for H, LE, and G).

Table 3. RMSD and NRMSD Comparison for Fluxes Calculated Using Best Fit Parameters and Those Estimated Using the Proposed Stationarity-Based Objective Function for Vaira Rancha
StationarityBest Fit 1Best Fit 2StationarityBest Fit 1Best Fit 2StationarityBest Fit 1Best Fit 2
RMSD (W m−2)23.6022.2921.9315.7014.1713.064.714.374.37
NRMSD (%)111111121010101010

[84] The turbulent fluxes (LE and H) are the most important terms in any land surface model. The time series comparison and scatterplots of these fluxes calculated using the parameters estimated from the stationarity-based objective function are shown in Figure 4. It is observed that the model tracks well for both LE and H. The fluxes tend to be simulated better when temperature is relatively high, which is also seen in the conditional average plots (Figure 2a). Negative LE values are not captured well, possibly because of averaging half-hour data to daily values. H is overestimated from the second half of year 2004 until beginning of year 2005 (around days 180–450) and underestimated from the end of year 2005 until beginning of year 2006 (around days 760–810). Most of these days fall into the first three bins in Figure 2a, corresponding to the mismatch of the conditional averages. Note that even for the best fit case (Figure 3), the sensible heat flux is biased at cold temperatures, implying that the problem is not parameter estimation, but either a structural model deficiency or a data quality issue. Overall, however, LE and H are simulated quite well (r = 0.87 for LE and r = 0.88 for H) considering that the fluxes are estimated only with knowledge of the forcing and state variables.

Figure 4.

Time series comparison and scatterplots for H and LE in the stationary case at Vaira Ranch, Ione, California. Red line shows modeled fluxes, and blue line shows observed flux.

Table 4. Parameter Values Chosen in Best Fit Comparisons and the Proposed Stationarity-Based Objective Functions for Kendall Grasslanda
Best fit 10.48----18.623.1501011000.650.950.310.630.500.48
Best fit 20.48----17.511.910100.931000.710.950.2812.050.560.31

4.2.2. Kendall Grassland, Arizona

[85] The climate at Kendall is temperate and semiarid with cool winters and warm summers. The site receives about 30 cm yr−1 of precipitation in a bimodal pattern, with 60% arriving in the July–September summer monsoon season and with the remainder arriving as gentler, longer duration frontal winter systems [Scott et al., 2000]. Similar to the presentation of the Vaira Ranch results, the observed and model-estimated fluxes are conditionally averaged with respect to surface states and compared with one another in 8 bins (Figure 5). The results show that energy fluxes are partitioned very well, and the trend and magnitude of each flux are captured nicely, although the conditional mean H is slightly overestimated in the coldest bins and the dependence of G on Ts is slightly overestimated (i.e., the slope is too steep). Note only 2% of data fall in the first (coldest) bin and 12% of data fall in the first two bins.

Figure 5.

Comparison of modeled and measured conditionally averaged fluxes at Kendall Grassland, Arizona. The solid lines are the modeled fluxes using parameters found by minimizing the stationarity-based objective function (equation (11)), and the dashed lines (with error bars) represent the field measured fluxes. The parameters used in the model-estimated fluxes (the solid lines) were estimated only from the measured forcing (P and Rn) and surface states (S and Ts) without any knowledge of the drainage flux or the latent, sensible, and ground heat fluxes. (a) Energy balance and (b) water balance.

[86] The dependence of the fluxes on temperature and moisture can be understood through the interactions of meteorological variables, vegetation, and soil conditions. For example, H basically has a positive relationship with skin temperature. It grows slowly with temperature at low temperatures and increases dramatically because of larger temperature gradients in the warm season. LE is very small for low temperatures, partly because the low temperatures occur in the winter when precipitation is low and soil moisture is limited and partly because of low available energy (net radiation) in winter. At intermediate temperatures during the warm season, LE peaks. The temperatures coincide with abundant soil moisture coming from monsoon precipitation and peak LAI. For skin temperatures above 25°–30°, soil moisture tends to be low (not shown), limiting evapotranspiration and requiring H and G to balance most of the net radiation.

[87] The result of best fit case based on conditional averaging (equation (31)) is shown for comparison in Figure 6. Comparing Figures 5a and 6, the estimated energy fluxes improve somewhat for the best fit case. For example, the conditionally averaged LE are better predicted in the hottest and wettest bins as well as G in most bins, but overall the differences between the stationarity and best fit estimates are small. These insignificant changes and good estimations by each method indicate both that the physical model is reasonable and that the stationarity-based estimation methodology works well.

Figure 6.

Comparison of modeled and measured conditionally averaged fluxes at Kendall Grassland, Arizona. The solid lines are the modeled fluxes using parameters found by minimizing the temperature-dependent error as expressed in equation (31), and the dashed lines (with error bars) represent the field measured fluxes.

[88] The estimated parameters for the stationarity objective function and best fit cases (equations (11), (31), and (32)) are reported in Table 4. Most parameters fall in a reasonable range, except for zveg (100 cm), which is too high for this grassland. The role of zveg, however, is in setting the aerodynamic roughness, and the larger value could be accounting for roughness due to rolling hills and interspersed shrubs.

Table 5. RMSD and NRMSD Comparison for Fluxes Calculated Using Best Fit Parameters and Those Estimated Using the Proposed Stationarity-Based Objective Function for Kendall Grasslanda
StationarityBest Fit 1Best Fit 2StationarityBest Fit 1Best Fit 2StationarityBest Fit 1Best Fit 2
RMSD (W m−2)19.4419.3518.6615.8915.6815.167.085.795.79
NRMSD (%)131212999161313

[89] The RMSD and NRMSD values for energy fluxes are calculated and showed in Table 5. The resulting values found from the three objective functions (equations (11), (31), and (32)) are very similar, indicating that our proposed stationarity method works very well. Comparing with the Vaira Ranch site, we find similar RMSD for LE and much smaller RMSD for H. Note that while fitting with equation (32) is best from the point of view of prediction error, structural errors are better identified with equation (31).

[90] The time series comparison and scatterplots of LE and H calculated using the parameters estimated from the stationarity-based objective function are shown in Figure 7. From the time series, we see LE and H both track the observations quite well on most days. H is overestimated at the end of year 2006 and in some parts of year 2007, corresponding to the overestimated conditional average value of H in the first two bins in Figure 5a. For LE, the model underestimates the data for days 450–500 and overestimates near days 650–700. But the conditional average of the modeled and measured fluxes (Figure 5a) does not show a corresponding mismatch. This can occur if the data that are underestimated and the data that are overestimated fall in the same temperature-dependent bin, such that the conditionally averaged error cancels out. Again, it is significant that when using the stationarity-based objective function for parameter estimation, which only depends on the forcing data and the state variables, the model predicts the measured fluxes with a correlation coefficient of 0.88 for both LE and H.

Figure 7.

Time series comparison and scatterplots for H and LE in the stationary case at Kendall Grassland, Arizona. Red line shows modeled fluxes, and blue line shows observed flux.

5. Summary and Future Work

[91] In this paper, we have demonstrated a conditional averaging method [Salvucci, 2001] to estimate parameters for a simple coupled water and energy balance land surface model. After performing conditional averaging to the fluxes in the water balance equation (using soil moisture as a conditioning variable) and in the energy balance equations (using surface temperature as a conditioning variable), an objective function for temperature- and moisture-dependent errors can be estimated. This objective function is only dependent on atmospheric forcing, surface states, and model parameters. It does not contain the observed fluxes, and thus, it can be used for indirect estimation of water and energy balance terms. In summary, it offers an alternative to direct calibration of land surface models and could be useful in locations where surface fluxes are not measured. In this work, the atmospheric forcing and surface state data were field based, but the method is amenable to remote sensing. Current research is exploring the use of MODIS land surface temperature and Advanced Microwave Scanning Radiometer–EOS (AMSR-E) soil moisture in place of field data.

[92] Promising results were found applying the proposed method to a synthetic case and to two AmeriFlux sites. One important feature of this method is that it does not require unbiased and calibrated measurements of the surface states. Because the method is based on statistical conditional sampling, as long as the conditioning indices for surface temperature and moisture are monotonic and continuous functions of the true surface temperature and moisture, the method will yield results. In the context of remote sensing, this relieves the burden of translating radiometric moisture estimates to precise water contents for application to soil physics based land surface models. Another feature of the method is that the data can be sparse, as long as the missing data is not missing in a systematic way that would cause bias. Furthermore, since this method is derived only from stationarity and conservation statements (of energy and water), it is scale free and thus can be applied at the scale appropriate to the land surface model whose parameters are being estimated.


[93] This study was funded by the NASA Terrestrial Hydrology Program under grant NNG06GE48G and NAG5-11695. The AmeriFlux data used in this paper can be downloaded at Dennis Baldocchi is the primary investigator at the Vaira Ranch site, and Russell Scott is the primary investigator of the Kendall Grassland. These data, and the investigators' willingness to answer questions, are greatly appreciated.