1. Introduction

Latent heat flux (LE) is the key variable that provides a link between energy and water budgets at the land surface. Since much of our understanding of the complex feedback mechanisms between the earth surface and the atmosphere is focused on quantifying these budgets, there is considerable interest in developing methods that routinely predict this variable. Local- and regional-scale estimates of LE would offer insight into hydroecological processes, aid in improving irrigation efficiency, and would provide a valuable tool for water resource management. Accurate estimation at large scales is required to improve our understanding of the global climate and its spatial and temporal variability (Miller et al. 1995). However, the prediction and validation of LE across all scales remains problematic.

The conventional methods to estimate LE are based on point measurements of energy balance components or turbulent surface fluxes and are representative only for very local scales. Recently, a new class of techniques based on satellite remotely sensed (RS) information has been developed to compute LE at scales from 1 km to a continent. Despite their theoretical attractiveness, especially for regional and global hydrological applications, “satellite derived” LE_sat usually does not compare well with “in situ measured” LE_is. Both proxies of LE, however, contain information about the true variability of this quantity. The difficulty in inferring this information from data is due to different sources of uncertainty involved (e.g., measurement errors, support scale, heterogeneity of land surface). In this context it is therefore natural to treat LE as a joint probability density function (pdf) over a spatially distributed set of bivariate random variables comprised of both proxies. Although this “true” high-dimensional joint pdf is a purely theoretical object, we are able to observe the bivariate samples from its marginal pdfs at particular locations in space. The purpose of this paper is threefold: to investigate a new nonparametric methodology for fitting these marginal LE pdfs to the experimental data from six sites, to pose the (preliminary) hypothesis of regionalization of the estimated pdfs through land use alone and additional parameters like degree of vegetation cover using again the six datasets, and to show the theoretical option of using these pdfs in spatiotemporal interpolation and data assimilation. Technically, the last objective is accomplished by first estimating the marginal pdf of bivariate LE and then using it to derive a conditional pdf of LE_is given LE_sat. The motivation for the latter asymmetry is that since in situ measurements of LE are derived from observations of physical processes at the land surface that determine natural variability of LE, they provide best estimates of LE at local scale. On the other hand, satellites observe states that are affecting this flux (as, e.g., surface temperature) at coarser pixel scales. Moreover, when RS information is used to derive LE there are extra sources of uncertainty as compared to in situ measurements due to inadequacies in retrieval algorithms, nonlinear measurement error propagation through RS models for LE, influence of cloud cover, and errors in land-use classification (Hipps and Kustas 2000). In this paper we therefore choose to condition local LE_is on LE_sat. The conditionals are modeled by a nonparametric class of continuous pdfs referred to as mixtures of Gaussians (MGs; see McLachlan and Peel 2000). The attractive property of MGs is that they do not require any arbitrary assumptions on the form of an underlying pdf (like, e.g., Gaussian assumption). This implies that as compared to the classical parametric approaches MGs can adapt to the local geometry of data ensembles (e.g., points distributed in multiple modes or points distributed on a low-dimensional surface in a high-dimensional space) and are able to approximate any continuous density to an arbitrary precision. Moreover, it is easy to simulate ensembles of points from parameterized MGs, which makes them useful in ensemble Kalman filtering.

The marginal MGs can further be regionalized and used for spatiotemporal interpolation of the LE proxies. In this article we investigate a practical method for regionalization of MGs that utilizes land use and vegetation cover as discriminatory variables. The interpolation is performed by deriving the conditional pdfs from a particular regionalized marginal pdf and then calculating their (conditional) expectation. Such nonlinear interpolation is particularly suitable for the LE flux, which does not aggregate/disaggregate linearly (Braud 1998). Another benefit from having the regionalized conditionals parameterized by MGs is that they can be assimilated into land surface models by the recently developed nonlinear ensemble Kalman filter (see Anderson and Anderson 1999; Torfs et al. 2002).

The LE_is data in this paper are obtained from energy balance Bowen ratio (EBBR) systems from southern Great Plains (SGP) region in the United States. The LE_sat proxies are estimated with Surface Energy Balance System (SEBS) developed by Su (2002).

2. Mixtures of Gaussians

a. Definition

To estimate the conditional uncertainty of LE_is given LE_sat a pdf needs first to be fitted to a bivariate sample {LE_sat,k; LE_is,k}^K_k=1. In this work the focus is on the use of MGs, which are defined as a linear combination of Gaussian densities (see Fig. 1), called components:

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where x is a D-dimensional vector of variables, N_c is the number of components, and g_(mn,Cn) stands for the Gaussian density with mean m_n and covariance C_n.

Here D = 2, x = [LE_is, LE_sat]^T and the w_n’s are the component weights that “∀_n w_n ≥ 0” and Σw_n = 1. Densities of the MG type inherit a lot of interesting properties from their Gaussian components: for example, the conditional densities p(x₁|x₂),

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are again MGs and can be calculated analytically (for technical details see Sharma 2000; Torfs and Wójcik 2001), Monte Carlo sampling is easy and fast, and, if needed, the conditional expectation (regression curve),

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can be derived from (1) and again is given by an analytic expression (Torfs and Wójcik 2001).

b. Fitting procedure

Fitting (1) to data requires optimizing weights w_n and the parameters of the components θ_n = {m_n, C_n}. The most commonly used optimization criterion is the maximum likelihood (ML) criterion implemented as expectation maximization (EM) algorithm (McLachlan and Krishnan 1997). Standard EM for mixtures, however, exhibits some weaknesses. It requires knowledge of N_c and good initialization is essential for reaching a good local optimum. To overcome these difficulties we use the approach of Figueiredo and Jain (2002) based on the minimum message length (MML) criterion. The rationale behind MML is that if one can built a short code describing one’s data that means that one has a good data generation model (Bishop 1995). Mathematically, the MML criterion for MG pdfs consists of minimizing with respect to θ, where θ ≡ {θ₁. . .θ_Nc,w₁. . .w_Nc}, the following cost function:

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where N = dim(θ_n), and K is the number of sample points. An attractive property of the algorithm of Figueiredo and Jain (2002) is that it is coupled with the model selection procedure that automatically determines the number of components N_c. Thus, MG can be initialized with a large value of N_c, alleviating the need for careful initialization. Because of this, a component-wise version of EM (Celeux et al. 2001) is adopted in Figueiredo and Jain (2002) to minimize (4).

3. Model structure of SEBS

The estimates of LE_sat were computed with SEBS (Su 2002). This model calculates atmospheric turbulent heat fluxes using satellite earth observation data. The SEBS consists of three components—land surface parameters, sensible heat flux estimation, and the energy balance—that are described briefly below [see Su (2002) for additional details].

Required land surface parameters include albedo, emissivity, surface temperature, fractional vegetation coverage, leaf area index, and the height of the vegetation from which displacement height and roughness height are derived. All this information is usually derived from remote sensing radiance data [e.g., the Moderate Resolution Imaging Spectroradiometer (MODIS)] in conjunction with other surface-related data [as, e.g., those from the Land Data Assimilation System (LDAS) database; http://ldas.gsfc.nasa.gov/LDAS8th/MAPPED.VEG/LDASmapveg.shtml]. The sensible heat flux H is based on the aerodynamic profile method and, because of the use of surface temperature, the determination of the two roughness lengths for heat and momentum transfer, as described in (Su et al. 2001). Required observations [or from four-dimensional data assimilation (4DDA) analysis fields] include air pressure, air temperature, humidity, and wind speed at a reference height (the measurement height for local-scale applications). For the local-scale SEBS obtains the friction velocity, the sensible heat flux, and the Monin–Obukhov stability length by solving iteratively a system of nonlinear equations. For field measurements performed at a height of a few meters above ground, where the surface fluxes are related to surface variables and variables in the atmospheric surface layer, all calculations involve the Monin–Obukhov similarity (MOS) functions given by Brutsaert (1999). The fluxes are based on the energy balance relations and utilize measured net radiation (or its equivalent derived through satellite-based observations of incoming radiation and surface temperature) and an estimate of the ground heat flux (see Su 2002). Latent heat flux can be estimated now from the energy balance equation:

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where R_n stands for net radiation and G for soil heat flux, respectively.

4. MGs in the generalized ensemble Kalman filter

Apart from considering the conditionals in (2) as pure uncertainty descriptors, they can be assimilated into land surface models [as e.g., variable infiltration capacity (VIC) model in Liang et al. 1996] by the generalized ensemble Kalman filter (GEnKF). This algorithm [originally inspired by Anderson and Anderson (1999) and cast by Torfs et al. (2002) into a broader probability theoretical framework] goes beyond the classical linear ensemble Kalman filter (EnKF; Evensen 1994) in the sense that it does not require any a priori assumptions on the form of pdfs for state and observational noise, nor does it presumes linearity of state and/or output equations to get optimal state estimates. Accordingly, when new observations become available the state updates are not restricted to assimilating Gaussian pdfs of these observations as in the EnKF, but allow any nonparametric pdfs (e.g., MGs) to be incorporated. This new idea extends the work of Torfs et al. (2002) and makes GEnKF competitive with more traditional data assimilation schemes. Since the overall objective of this research direction is the implementation of GEnKF we find it useful here to give a brief mathematical description of the algorithm with emphasis on state updates given new observations.

Let us denote the state of the system at time n as s_n, the state at time n + 1 as s_n+1, and the observation at time n + 1 as o_n+1. These three variables can be scalars or vectors. We use ϕ_n to denote the joint pdf and f_n, f_n+1 and h_n+1 their respective marginals. Given a new observation o_n+1 the solution to GEnKF problem is given by

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Assuming conditional independence of s_n and o_n+1 given s_n+1 and making use of the Bayesian approach the following relations hold (Torfs et al. 2002):

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Equation (7) is referred to as the prediction step and (8) as the Kalman gain or analysis step, respectively. When the observation is not known deterministically, but only its probability density ϕ is given,¹ this last formula is to be replaced by

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With preliminary knowledge of f_n,n+1(s_n, s_n+1) and ϕ_n+1(s_n+1, o_n+1), the integrals above are calculated recursively until the time of the latest observation.

Equations (6)–(9) describe an abstract setting for Kalman filtering, regardless of how the densities involved are given. In this paper we propose to approximate them by MGs. MGs are particularly well suited for this: because simulating from them is extremely fast, all integrals above can be evaluated by Monte Carlo sampling. Figure 2 illustrates the steps involved in computing the product in (9). First, a number of starting points is simulated from known observational ϕ_n+1 pdf (Fig. 2a; the marks stand for the simulated points). From this ensemble, we calculate a statistically relevant set of posterior state pdfs f^g_n+1 (on the lines of Fig. 2b). For this we use our knowledge of ϕ_n+1 (Fig. 2c) at the simulated starting points. Then, the posteriors are again sampled, which is visualized in Fig. 2d. The joint pdf ϕ*_n+1 is then fitted in Fig. 2f to the sample in Fig. 2e. Finally, ϕ*_n+1 is marginalized (Fig. 2g), resulting in the analysis pdf F^g_n+1.

When dimensionality of state/observation space is high, as is the case of spatially distributed hydrologic models, the performance of the algorithm above would be suffering the curse of dimensionality (Gershenfeld 1992). That term was coined by Bellman (1961) to describe the problem that occurs when searching in or estimating pdfs on high-dimensional spaces. This problem may become intuitively clearer by looking at the example of fitting multidimensional histograms. Given a fixed number of M grid lines per dimension D, the number of independent cells grows as M^D. Furthermore, if the density function is to be estimated based on a set of high-dimensional samples, the number of samples required for accurate histogram estimation also grows as M^D. The same is true for MG models—here the components are continuous equivalents of cells used in histograming, and their weights can be viewed as histogram values at those cells. A pragmatic way to tackle this problem is to regionalize the pdfs. In other words, instead of using high-dimensional joint pdfs representative of the entire spatial domain of a particular model, we propose to use a few lower-dimensional marginal pdfs that are representative only of subdomains. For marginal LE pdfs, identification of these subdomains might be based on two discriminatory variables described in section 7c.

5. Control run and the surrogates

When estimating MG pdfs for bivariate LE ensembles, there are three potential difficulties that should be addressed:

• Undersampling: For optimization of 2D MGs, six parameters have to estimated for each component [see Eq. (1)]. The algorithm in Figueiredo and Jain (2002) is only guaranteed to be robust with regard to the initialization procedure if the sample size is “large enough.” As mentioned in section 4 the number of points required for reasonable density estimation in D-dimensional space scales roughly as M^D. So gaps in data might result in the higher uncertainty in MG parameters. There are a few common reasons for missing records in satellite-derived data. For example, the performance of the RS retrieval algorithms for surface temperature T_s is influenced by cloud cover. Thus, SEBS predictions are available only for cloud-free or partly cloud-free days. Moreover, satellites provide only a snapshot view of spatial variability in T_s and as a consequence indirectly the spatial variability in LE_sat. Finally, accidental technical failures in in situ or RS instruments are responsible for gaps in data.

• Instrumental and model errors: Although LE_is observations are the closest approximation to natural variation of LE at local scales, the techniques used to measure LE_is, as, for example, EBBR systems, are themselves not without error. Indeed, in the heterogeneous landscape of the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) campaign, LE_is predictions were often as high as 20% in error (Nie et al. 1992). On the other hand, LE_sat estimates are prone to errors in RS inputs to SEBS and limitations of SEBS itself to reproduce the complicated physical situation in the surface layer of air.

• Scaling: The footprint over which an LE_sat is determined is rarely at the same scale as LE_is, making direct comparison difficult.

The above-mentioned problems are usually responsible for strong scattering effects that blur the dependency structure underlying the bivariate LE ensembles. Recalling that the second objective of this work is to investigate the robustness of a practical methodology for regionalization of marginal LE pdfs—or, in other words, to investigate whether for classes of visually similar regions particular pdfs can be representative—such a situation needs to be resolved. As a pragmatic approach we propose the following course of action. We first pose the hypothesis that p(LE_is, LE_sat)’s have a structure that depends on land use and vegetation cover, equivalent to visually similar regions. Next, the hypothesis is verified by creating “idealized” bivariate LE samples for a variety of environmental conditions (in terms of water supply, available energy, saturation deficit, turbulent transport, and vegetation characteristics). These hourly, daylight-based samples are referred to as the control run. The LE_is in the control run are taken “as is,” and LE_sat proxies are obtained from SEBS forced with R_n estimated from in situ–measured radiation fluxes and T_s derived from the longwave radiation:

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where LW_out denotes the outgoing longwave radiation, LW_in refers to the incoming longwave radiation, ε is the emissivity of the surface, and ς stands for the Stefan—Boltzmann constant. Note that in SEBS R_n and H, which specify the total available energy and sensible heat flux, respectively, depend on T_s. Next, the LE_sat in the control run is perturbed by Monte Carlo propagation of “satellite” error in T_s through R_n and H in SEBS (see section 7a for technical details of this procedure). This way, keeping again LE_is unchanged, we obtain another bivariate sample referred to as the surrogate data. It is expected that introduction of error sources blurs the structure in the control run but does not let it disappear. Thus, MG pdfs fitted separately to control run and surrogate data should be similar. To quantify the strength of this similarity we use the L² correlation in Scott and Szewczyk (2001):

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where p₁ and p₂ are L² integrable pdfs for the control run and surrogate data, respectively. This measure is 0 if two pdfs show no similarity and 1 if two pdfs are just the same. For MGs (11) can be calculated analytically.

To summarize the above procedure: undersampling in LE_sat data is tackled by creating an hourly surrogate data; by creating a control run of hourly data, error propagation is controlled and scaling is accounted for automatically by obtaining the conditional pdfs p(LE_is|LE_sat) in (2) from regionalized marginal pdfs p(LE_is, LE_sat) and using these conditionals to compute the regression curves in (3).

6. Data

The measurements of LE_is used in this study come from six EBBR U.S. Department of Energy’s Atmospheric Radiation Measurement Program Cloud and Radiation Testbed (ARM/CART) stations (E15, E4, E9, E20, E7, E25) distributed across the SGP region of the United States (see Fig. 3). ARM is aimed at obtaining field measurements and developing models to better understand the processes that control solar and thermal infrared radiative transfer in the atmosphere and at the earth’s surface. The SGP CART site was the first field site established by ARM and consists of in situ and remote sensing instrument clusters across north-central Oklahoma and south-central Kansas.

The LE_is proxies are based on 30-min averaged observations. The LE_sat estimates in the control run were obtained with SEBS forced with the input set displayed in Table 1.

Both types of LE data were obtained at 1-hourly resolution in the period of 1 July 2001–30 September 2001. We further restricted the data to 8-hourly sections of a day (0900–1700 local time) so we only considered unstable and neutral conditions in the atmospheric surface layer. Figure 4 shows the control run ensembles of the bivariate LE data.

These ensembles can be thought of being discrete samples from the unknown “true” marginal pdfs. Looking at the ensembles in Fig. 4 it is clear that simple parametric families of pdfs as Gaussians are not flexible enough to capture the geometry of the problem. For this reason in section 7b we fit MGs to describe the LE data.

7. Results

b. MG density fitting

Bivariate MGs were fitted to both the control run and surrogate data. We initialized mean m_n vectors in (1) to 30 randomly chosen data points. The initial covariances were made proportional to the identity matrix 𝗖_n = σ²_init𝗜 with the diagonal entries σ²_init equal to 1/10 of the mean of the variances along each dimension of the data:

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where m = (1/K)Σ^K_i=1x⁽ⁱ⁾ is the global data mean (see Figueiredo and Jain 2002). This step was meant to assure the initial density on each data point be reasonably higher than 0. Figure 6 displays the fitted MGs. Comparing these pdfs to discrete underlying ensembles in Figs. 4 and 5 shows that MGs are a smoothed continuous representation of the underlying points. It is also clear from the figure that MGs can capture the particular local features of the ensemble while standard parametric densities (as, e.g., Gaussians) are unable to do this.

To quantify the similarity between pdfs for the control run and those for the surrogate data, the L² correlation in (11) was calculated for each pair of MGs. These results are given as numbers in Fig. 6. Since the value of the correlation is high (0.84–0.97), the error in T_s did not have much influence on the control run pdfs of the bivariate LE data. It refines the control run pdfs, one may say. Note that the error in T_s has, however, a pronounced impact on LE_sat estimates from SEBS, which can be seen by comparing horizontal spread of the ensembles in Figs. 4 and 5.

Calculation of the conditional MGs from the marginal MGs fitted to the surrogate data provides a basis for potential applications in spatiotemporal interpolation and data assimilation. The former can be achieved by making probabilistic predictions of LE_is by either resampling from the conditional density p(LE_is|LE_sat) or calculating the conditional expectation E[LE_is|LE_sat]. The latter requires the knowledge of p(LE_is|LE_sat) and implementation of the algorithm described in section 4. For two grassland sites in the SGP region, marginal and conditional pdfs together with corresponding regression curves and standard deviation envelopes are presented in Fig. 7.

It is clear that the form of the conditional MG pdfs alters with an increase of LE_sat values and reveals a variety of shapes: from Gaussian to highly non-Gaussian (e.g., multimodal or skewed). This nonstationary behavior influences the geometry of conditional expectation and standard deviation envelopes, which, for E15, are clearly nonlinear. Interestingly, the conditional pdfs for the E4 site appear to flatten (or technically, to have higher entropy) with increasing value of LE_sat. The opposite is true for the E15 site. This is in accordance with the scatter patterns for both sites in Fig. 4. Both sites are situated in the grassland area and show a similar range of LE_is values. However, it can be seen from Figs. 4 and 5 that hydrometeorological conditions, as produced by SEBS, for E15 suggest that it is dryer than E4. Therefore, our first conclusion with respect to the objectives of this study is that land-cover type alone cannot be used to regionalize LE pdfs. In the next section we identify an additional control parameter in SEBS that makes the regionalization feasible.

8. Discussion and outlook

In this paper we have proposed a new procedure for describing bivariate LE ensembles as MG pdfs. This procedure is able to produce nonparametric pdfs that are fully data driven and are more flexible to describe local geometry of LE ensembles than classical parametric pdfs like, for example, Gaussians. Moreover, the procedure offers a vehicle for uncertainty analysis due to undersampling, error sources, and scaling problems, which are notorious when comparing LE_is to LE_sat data. We have shown that the conditional pdfs p(LE_is|LE_sat) can theoretically be useful in novel data assimilation schemes and spatiotemporal interpolation of bivariate LE data. The essential prerequisite for these applications is the ability to regionalize MG pdfs. The preliminary results in this work have demonstrated that it is feasible to regionalize the pdfs using land cover and vegetation fraction as discriminatory variables.

There are a number of issues that need to be investigated in order to implement the above methodology in hydrologic practice. Additional research is needed to quantify errors in LE_is data for various measuring techniques and see how they influence pdfs of bivariate LE ensembles. Progress in this direction, for EBBR/ARM-CART sites, is discussed in Stricker and Wójcik (2005, unpublished manuscript). An attractive option would also be to use scintillometeric measurements. The scintillation technique is one of the few techniques that can provide LE_is fluxes at scales of several kilometers (up to 10 km), making them more comparable to LE_sat fluxes (Meijninger et al. 2002). Additional work is further required to fine-tune the regionalization of LE pdfs by investigating the use of other environmental variables and influence of various uncertainty sources on pattern of bivariate LE ensembles. As demonstrated in section 7c an environmental variable to conceive here is the soil moisture. Finally, the parallel implementation of GEnKF for high-dimensional hydrologic systems is a challenging research problem. If successful, the assimilation of p(LE_is|LE_sat) into land surface models promises to enhance significantly the quality of water balance estimates over a range of spatial and temporal scales.