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  • View in gallery

    Daytime-averaged (top) latent and (bottom) sensible heat flux scatterplots from the (left) triangle and (right) variational methods for FIFE 1987.

  • View in gallery

    Same as in Fig. 1, but for FIFE 1988 data.

  • View in gallery

    Mean diurnal cycle of (top) latent and (bottom) sensible heat fluxes for (left) 1987 and (right) 1988 (observations are shown with a solid line with open circles, triangle method estimates are shown with a dashed line with open triangles, and variational assimilation estimates are shown with a dashed line and “+” symbols).

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A Comparison of the Triangle Retrieval and Variational Data Assimilation Methods for Surface Turbulent Flux Estimation

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  • 1 Department of Civil and Environmental Engineering, University of California, Los Angeles, Los Angeles, California
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Abstract

Future operational frameworks for estimating surface turbulent fluxes over the necessary spatial and temporal scales will undoubtedly require the use of remote sensing products. Techniques used to estimate surface fluxes from radiometric surface temperature generally fall into two categories: retrieval-based and data assimilation approaches. Up to this point, there has been little comparison between retrieval- and assimilation-based techniques. In this note, the triangle retrieval method is compared to a variational data assimilation approach for estimating surface turbulent fluxes from radiometric surface temperature observations. Results from a set of synthetic experiments and an application using real data from the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) site indicate that the assimilation approach performs slightly better than the triangle method because of the robustness of the estimation to measurement errors and parsimony of the system model, which leads to fewer sources of structural model errors. Future comparison work using retrieval and data assimilation algorithms will provide more insight into the optimal approach for diagnosis of land surface fluxes using remote sensing observations.

Corresponding author address: Steven Margulis, Department of Civil and Environmental Engineering, UCLA, 5732D Boelter Hall, Los Angeles, CA 90095. Email: margulis@seas.ucla.edu

Abstract

Future operational frameworks for estimating surface turbulent fluxes over the necessary spatial and temporal scales will undoubtedly require the use of remote sensing products. Techniques used to estimate surface fluxes from radiometric surface temperature generally fall into two categories: retrieval-based and data assimilation approaches. Up to this point, there has been little comparison between retrieval- and assimilation-based techniques. In this note, the triangle retrieval method is compared to a variational data assimilation approach for estimating surface turbulent fluxes from radiometric surface temperature observations. Results from a set of synthetic experiments and an application using real data from the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE) site indicate that the assimilation approach performs slightly better than the triangle method because of the robustness of the estimation to measurement errors and parsimony of the system model, which leads to fewer sources of structural model errors. Future comparison work using retrieval and data assimilation algorithms will provide more insight into the optimal approach for diagnosis of land surface fluxes using remote sensing observations.

Corresponding author address: Steven Margulis, Department of Civil and Environmental Engineering, UCLA, 5732D Boelter Hall, Los Angeles, CA 90095. Email: margulis@seas.ucla.edu

1. Introduction

Diagnosis of the surface moisture and energy fluxes is important in land–atmosphere interactions and has application in many disciplines. Evaporation is one of the key fluxes in all water budget studies, while the surface turbulent sensible and latent heat fluxes are the primary forcings of the atmosphere by the land. Despite the importance of these fluxes, in situ measurement is generally limited to sparse ground-based monitoring sites that are often associated with short-term field experiments or extremely localized regions. To create operational frameworks for estimating these fluxes over longer time periods and larger spatial extents will undoubtedly require the use of remote sensing products. While there are many potentially useful remote sensing observations that can be used to estimate surface fluxes, radiometric surface temperature is often used (e.g., McNider et al. 1994; Norman et al. 2000; Diak et al. 2004; etc.) because it is a variable that is implicit in the surface energy balance. Further, it is a variable that is measured operationally from several orbiting platforms [e.g., the Geostationary Operational Environmental Satellite (GOES), the Moderate Resolution Imaging Spectrometer (MODIS), the Advanced Very High Resolution Radiometer (AVHRR), the Special Sensor Microwave Imager (SSM/I), etc.] that provide excellent spatial and temporal resolution and coverage (multiple measurements per day at resolutions up to 1 km2 over most of the globe).

Techniques used to estimate surface fluxes from radiometric surface temperature generally fall into two categories: retrieval-based (e.g., Price 1982, 1990; Taconet et al. 1986; Carlson et al. 1981, 1995a; Carlson and Buffum 1989; Diak and Stewart 1989; Diak 1990; Diak et al. 1995; Norman et al. 1995; Kustas and Norman 1996; Kustas et al. 1999, 2003) and data assimilation approaches (e.g., Castelli et al. 1999; Boni et al. 2000, 2001; Caparrini et al. 2003, 2004; Margulis and Entekhabi 2003). Retrieval is a general term for techniques that invert an empirical or physically based model that relates the measured quantity and the variable of interest. The type of model and required inputs varies greatly and can range from a simple regression approach to one that incorporates a dynamic physical model. These procedures generally produce an instantaneous estimate from an instantaneous observation input. Hence, estimates are at the same temporal and spatial resolution of the satellite instrument and subject to measurement errors in the radiometric temperature. Data assimilation approaches differ from retrieval methods in that they combine a physically based model with a sequence of remote sensing observations and attempt to merge the two to produce an optimal estimate (based on the relative uncertainty of all inputs). Depending on the application, this approach can be more computationally demanding than retrievals, but has distinct advantages including the ability to provide estimates between observations (interpolation) and increased robustness with respect to measurement errors.

Up to this point, there has been little comparison between retrieval- and assimilation-based techniques to assess the side-by-side performance of these two different approaches. In this note we compare the popular triangle retrieval method (Carlson et al. 1995b; Gillies and Carlson 1995; Gillies et al. 1997) to a variational data assimilation approach (Caparrini et al. 2003, 2004) for estimating surface turbulent fluxes from radiometric surface temperature observations. Diak et al. (2004) recently reviewed various retrieval methods that estimate land surface fluxes from remote sensing data (mostly via radiometric temperature and/or vegetation indices). Methods range from simple empirical models (Price 1990) to atmospheric boundary layer (ABL)–soil–vegetation–atmosphere transfer (SVAT) based methods (Carlson et al. 1994; Moran et al. 1994; Gillies et al. 1997) to methods that couple soil and vegetation components to an ABL scheme (Norman et al. 1995). Newer techniques use more complex time-rate difference schemes (e.g., Anderson et al. 1997; Mecikalski et al. 1999; Norman et al. 2000). The triangle method was chosen for comparison in this analysis because it can be used with any model. This allows for its application with a model similar in form to that used in the variational approach and therefore allows for a more meaningful one-to-one comparison. Future studies will include other retrieval and/or assimilation approaches. In section 2 the two approaches are briefly introduced. In section 3 the field application is described, and the results are presented and discussed in section 4. Finally, conclusions are discussed in section 5.

2. Methodology

a. Variational data assimilation

The variational approach applied in this study is based on the recent work of Caparrini et al. (2003, 2004). For a system model, this approach uses the surface energy balance component of the force-restore (FR) model (Noilhan and Planton, 1989) that can be recast in terms of the evaporative fraction EF (ratio of latent heat flux to available energy) as
i1525-7541-6-6-1063-e1
The evaporative fraction has been shown to be approximately conserved during the daytime hours (Crago 1996; Crago and Brutsaert 1996). Hence, in this approach, EF is treated as a constant parameter to be estimated for each day. The model requires the following forcing variables: net radiation, micrometeorological air temperature and humidity, wind speed, and surface pressure. The estimation is performed using a variational approach, which consists of deriving the corresponding adjoint model for Eq. (7) (see below) and the parameter update equation for EF. For brevity the adjoint model is not shown here and the reader is referred to Caparrini et al. (2003, 2004) for more details. This technique is an iterative one which involves integrating the forward model for each day, integrating the adjoint model backward in time (which is forced by model-measurement misfits), and using the adjoint state variables to update the EF parameter until convergence is achieved (Caparrini et al. 2004). The key inputs to the algorithm (in addition to the model forcing) are the prior mean and error variance of EF and the standard measurement error for Ts. The approach implicitly weighs these two uncertainties to determine a posterior estimate for EF that fits the data in a statistically optimal way. A more general discussion of the variational approach is given in Margulis and Entekhabi (2001). Once EF has been estimated for a given day, it can be used with the predicted values of H, to determine the latent heat flux throughout the daytime window. It should be noted that the most general application of the method (Caparrini et al. 2004) can be used to simultaneously estimate the EF and a scalar heat transfer coefficient parameter. The model used below for the triangle method requires specification of this parameter, so it is specified for both methods in this application.

b. Triangle retrieval method

The triangle method was introduced by Carlson et al. (1994) and has been used to estimate surface fluxes from radiometric surface temperature (Ts) and fractional vegetation cover (Vc), both of which can be estimated from remotely sensed observations. It is named after the general pattern seen when plotting observed values in VcTs space, where a triangle or trapezoid is usually formed by a dry edge corresponding to higher temperatures and a wet edge corresponding to lower temperatures. In this method, an empirical function is then fit describing the relationship between evaporation versus vegetation content and temperature. Gillies et al. (1997) suggest a second-order polynomial can be used to adequately describe the instantaneous ratio of latent heat (LE) to net radiation (Rn) as a function of fractional vegetation cover and surface temperature:
i1525-7541-6-6-1063-e2
A similar expression for the sensible heat flux is given by
i1525-7541-6-6-1063-e3
Hence, given estimates of vegetation fraction, surface temperature, and net radiation (all of which can be estimated from remote sensing) the instantaneous latent and sensible heat fluxes can be estimated. The implicit assumption in the triangle method is that the surface latent heat flux is primarily dependent on soil moisture and vegetation fraction. Since simultaneous measurements of temperature, vegetation fraction, and surface fluxes over the range of expected values are rarely available, the above relationships are constructed by running a model over the full range of possible soil moisture and vegetation fraction values. These vegetation fraction and soil moisture inputs (along with Rn) and the model outputs (Ts, LE, and H) are then used to construct the second-order polynomials.
As described above, the triangle method retrieval equations are inherently model dependent. The original application of the triangle method used the parsimonious model described in Taconet et al. (1986). In this study we chose to use the FR model (Noilhan and Planton 1989), which is not only relatively parsimonious, but is very similar in form to the one used in the data assimilation approach outlined above. The FR model is a simple two-layer SVAT scheme. The state equations for the model include a prognostic equation for the surface temperature:
i1525-7541-6-6-1063-e4
where CT is the surface heat capacity, τ is the diurnal period, and Td is the deep temperature, which can be represented essentially as a low-pass-filtered version of the surface temperature:
i1525-7541-6-6-1063-e5
The model also has two prognostic equations for surface (W1) and root zone (W2) soil moisture:
i1525-7541-6-6-1063-e6
i1525-7541-6-6-1063-e7
where θs is the soil porosity, C1 and C2 are parameters dependent on soil hydraulic properties, d1 and d2 are the two soil layer thicknesses, P is precipitation, and Eg and Etr are bare soil evaporation and transpiration, respectively. For further details about the model the reader is referred to Noilhan and Planton (1989).

To apply the triangle method, several approaches can be taken. In its most parsimonious form, a coupled land surface–atmospheric boundary layer model can be used (e.g., Taconet et al. 1986) such that the primary inputs are representative atmospheric conditions (via radiosonde data) and specification of soil and vegetation type. The model can then be run over a range of soil moisture initial conditions and vegetation-cover values to generate the polynomials. Gillies et al. (1997) suggest that a single set of polynomials (as a function of a normalized temperature) for representative model inputs (i.e., mean hourly forcing) can be used for application to other days. Of course, when available, using a particular day’s forcing inputs are preferable as the use of representative inputs can introduce errors. In fact, initial tests in our study also found that in cases where latent heat flux is a function of factors other than soil moisture and vegetation fraction (e.g., solar radiation, air temperature, and vapor pressure deficit in the stomatal resistance parameterization), variability in forcing can introduce large errors if polynomials generated from representative inputs are used. Our investigations also revealed that using a normalized temperature did not prevent these types of errors. However, the impact of these errors may depend on the particular application, since other studies have shown lesser sensitivity to model inputs (e.g., Gillies et al. 1997). Since the variational method described above requires forcing inputs on an hourly time scale, the triangle method was used in this context.

Given this framework, the FR model is run using forcing for a given day over the full range of initial soil moisture conditions and fractional vegetation cover values to generate a set of outputs (LE, H). For the particular satellite overpass times for that day, the necessary inputs (Vc, Rn) and outputs (Ts, LE, H) are used to determine the polynomial coefficients shown in Eqs. (2) and (3). Once the coefficients are determined for the particular measurement times, the observed surface temperature and vegetation fraction are used in the retrieval equations to determine the surface fluxes. The results are instantaneous estimates of LE and H at each observation time. In the application discussed below we will focus on the daytime window when these fluxes are large and the surface temperature is sufficiently sensitive to surface fluxes.

3. Field application

We applied the two methods described above to a grassland prairie in Kansas [site of the First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE)]. The FIFE dataset consists of high-quality half-hourly data for the model forcing (Betts and Ball 1998), significant ground-based observations of model parameters (Collelo et al. 1998), as well as radiometric surface temperature observations for input to the two methods and surface sensible and latent heat fluxes for validation (Betts and Ball 1998) over two full summer seasons. All model auxiliary parameters for the FIFE site were obtained from Collelo et al. (1998). Specifically the fractional vegetation cover is needed for Eqs. (2) and (3). A representative value of Vc = 0.8 was used for the FIFE site (Collelo et al. 1998; Betts and Ball 1998). Caparrini et al. (2004) found that for the simultaneous estimation of EF and the heat transfer coefficient, multiday assimilation windows were necessary. In this application (with a specified heat transfer coefficient), each day can be treated as a single assimilation window in the variational method. A representative value for the neutral heat transfer coefficient was taken from Caparrini et al. (2004) and used along with the stability correction factor of Louis et al. (1982) in both methods. In terms of other model parameters, the variational approach only requires the specification of the heat transfer coefficient and average surface heat capacity (CT). The other key inputs are the prior mean value for EF (set to 0.5) and its error covariance (standard error set to 0.2). Crow and Kustas (2005) recently found that the variational results can be sensitive to prior values specified for EF, when the heat transfer coefficient is simultaneously estimated. In this application with the specification of the heat transfer coefficient, the variational results were not found to be sensitive to prior values of EF. In previous work using FIFE data, Gillies et al. (1997) applied the triangle method for a handful of days, while Caparrini et al. (2004) applied the most general application of the variational approach (simultaneous EF and heat transfer coefficient estimation) for the summers of 1997 and 1998.

In this work, we apply the two methods in parallel for the entire summers of 1987 and 1988 using identical inputs. The two methods were applied sequentially to each day during the daytime window of 0900–1600 local time (LT), which is the window over which EF is generally constant (Caparrini et al. 2004) and surface fluxes are high. To represent typical temporal sampling from satellites we specified the observation times to be at hourly intervals during this window. We first performed synthetic experiments using the FIFE forcing to assess the impact of measurement error on the flux estimates. We then used the real radiometric surface temperature observations to estimate the latent and sensible heat fluxes, which were compared to the observed values. Results from these case studies are shown below.

4. Results and discussion

a. Synthetic experiments

First, a set of synthetic experiments was performed using the 1997 FIFE forcing and parameter set. In these experiments, each approach was tested using surface temperature observations that were synthetically generated from the respective model used in each method. The main goal was to test the robustness of each approach for different surface temperature measurement error scenarios. We tested the methods for measurement error standard deviations of 0.0, 0.5, 1.0, and 2.0 K. (Theoretically there cannot be zero measurement error in the variational assimilation approach, so instead we use a very small 0.01-K measurement error.) In retrieval methods like the triangle method, measurement errors directly propagate to the estimates because data is a direct input to the algorithm [Eqs. (1) and (2)]. In the assimilation approach, the data is not a direct input, but instead the estimates are obtained by trying to fit the data over a daily window. While correlated temperature measurement errors can be accounted for in the variational assimilation approach, here we assume uncorrelated measurement error. Hence, variational estimates should generally be more robust to measurement errors than those obtained from direct retrieval methods (e.g., triangle method). However, retrieval methods based on time-differencing methods will generally be less sensitive to these types of errors. Summary results for instantaneous (hourly) and daytime-averaged latent and sensible heat flux estimates from both methods are shown in Table 1. In the case of perfect measurements (zero measurement error), the triangle method results outperform the variational estimates as both flux estimates contain smaller bias and root-mean-square errors (rmses). The two primary assumptions of the triangle method are 1) that model output can be fit to a second-order polynomial and 2) that this surface will realistically capture the relationship between surface temperature and the ratio of turbulent fluxes to net radiation. By using the model to generate synthetic observations, the first assumption can be tested since the second is automatically satisfied. The synthetic experiment results indicate that the first assumption is reasonable for the FR model because if the second-order polynomial could not reasonably fit the model output, the results would be poor despite the fact that the surface temperature observations were generated by the FR model. In fact, the average rmse associated with the polynomial fit to the model output for both flux equations is on the order of 1%–2%, which translates to a turbulent flux error of less than 10 Wm−2 at the FIFE site. Therefore, sources of error in this method are more likely to be caused by measurement errors (which get propagated through the retrieval equations) and/or when the true relationship between temperature, fractional vegetation cover, and surface fluxes does not fall on the fitted surface (a result of model errors). These issues will be discussed in the next section.

As measurement error increases to a more typical value of 1.0 K, the average rmse for instantaneous (hourly) turbulent flux estimates for the triangle and variational methods are approximately 60 and 20 Wm−2, respectively. This would seem to confirm the hypothesis that the assimilation approach as applied here is more robust under erroneous observations. This has significant implications for flux retrieval algorithms using real remote sensing observations where measurement errors of order 1.0°C are common. For the same level of measurement error, the daytime-averaged rmse for the triangle method is reduced to approximately the same value for the variational results (∼20 Wm−2). For a large value of measurement error (2.0 K), the error propagation to the instantaneous flux estimates is magnified further. However, a significant fraction of the error seems to be random due to the sharp reduction in daily averaged fluxes that are again comparable to the variational results.

b. Application with FIFE surface temperature observations

The two methods were applied over the summers of 1987 and 1988 at FIFE and the estimated surface fluxes were compared to the ground truth observations. Using real data allows for the assessment of the relative weight of model error in the two schemes (which was by construction not considered in the synthetic case studies). For the variational approach, a measurement error of 1.0 K was assumed. Summary statistics for estimated hourly and daytime-averaged fluxes for both methods over both years are shown in Table 2.

As expected, both sets of results are generally worse when compared to the synthetic experiment. For both methods, a significant component of the estimation error is in the form of systematic biases. For the instantaneous (hourly) estimates from the triangle method, the average biases in latent and sensible heat fluxes are 56 and −28 Wm−2, respectively. These biases are likely due to structural model errors and are the most problematic since they will not be reduced under temporal averaging. The triangle method average rmse values for hourly latent and sensible heat are 82 and 68 Wm−2, respectively. The primary source of error for the triangle method appears to be related to the second assumption discussed above. While the second-order polynomial fits the FR model output reasonably well (∼2% rmse), the empirical model-generated surface does not appear to fit the observed values very well, with rmses between the retrieval equation and observed ratios on the order of 20% for a given temperature input. These model-related errors are undoubtedly due to the inability of the FR model to properly represent the true relationship between surface temperature, vegetation cover, and observed fluxes. In this case, overreliance on the model translates to large errors in the instantaneous flux estimates. Daytime-averaged results show reduced average random error values of 70 Wm−2 (latent heat) and 54 Wm−2 (sensible heat). Results for the 15 August 1997 “golden” day (clear sky) examined by Gillies et al. (1997) were similar in magnitude to those obtained here (∼40 Wm−2), suggesting that the different models used do not lead to significantly different results. The larger values obtained by examining the entire summers over 1987 and 1988 are likely due to the inclusion of days where the model is unable to accurately capture the true relationship between surface temperature, vegetation fraction, and surface fluxes and/or where this relationship is not as strong. The magnitude of the errors from our application is also similar to those seen in Jiang and Islam (2003) who used several methods to estimate surface latent heat flux in the southern Great Plains and found biases and rmses for individual days on the order of approximately 20–50 and 50–80 Wm−2, respectively.

Results from the variational approach show significant, but somewhat smaller errors for the hourly flux values. The average bias in latent and sensible heat fluxes is 44 and −10 Wm−2, with corresponding rmse values of 73 and 57 Wm−2. For the daytime-averaged results, the rmse values are reduced by approximately 10 Wm−2 (see Table 2). The primary sources of error for the variational approach are also likely due to structural model errors related to the constant EF assumption and/or in the potential misspecification of (and lack of seasonality in) the heat transfer coefficient. In Caparrini et al. (2004), where the heat transfer coefficient was simultaneously estimated with evaporative fraction, the daily averaged rmse values for latent and sensible heat fluxes were 49 and 45 Wm−2, respectively. While simultaneous heat transfer coefficient estimation improved the results for the FIFE site, Crow and Kustas (2005) illustrate how results from the most general application of the variational approach can lead to a degradation of the results in some cases.

Figures 1 and 2 show scatterplots of the estimated versus observed daytime-averaged fluxes for 1987 and 1988. For both years, the two methods perform similarly with respect to the sensible heat flux estimates. This is not surprising because this flux is computed identically in both models with the primary input parameter being the surface heat transfer coefficient. As a result, first-order errors in this flux are most likely due to misspecification of this parameter. The additional errors in the triangle method are likely due to the second-order polynomial fitting and the direct insertion of measured surface temperature (which includes measurement error). The primary difference between the two methods lies in the latent heat flux estimates. In both years, the triangle method has higher biases and rmse in the daytime-averaged latent heat flux compared to the variational approach. This is most likely due to the inherent assumptions made in each method. For the variational approach, the partitioning of available energy is estimated simply by adjusting the evaporative fraction, that is, given the computed sensible heat flux, the EF is adjusted until the measurement-model misfit is minimized. This approach has fewer potential sources of parameter and/or structural model error. In contrast, for the triangle method the latent heat flux equation is a complicated function of model parameters such as canopy resistance. Any misspecification of the parameters and/or atmospheric conditions (i.e., air temperature, vapor pressure deficit) that determine this quantity will introduce systematic errors into the model output and ultimately the retrieval equations. In both years, the triangle method predicts too much evaporation, which may be a result of a negative bias in canopy resistance. While parameters can be calibrated to obtain better results, the ultimate application of any retrieval approach over large scales may likely have to rely on typical default land surface parameters and therefore could suffer from these types of errors.

The mean diurnal cycle of the two estimation methods, along with flux observations, is shown in Fig. 3 for both 1987 and 1988. The two techniques perform similarly in estimating the sensible heat flux during midday with the triangle method having a tendency to have large prediction errors in the morning and late afternoon. This may be partly explained by the fact that the polynomial fit is generally less robust during the hours when radiative forcing is low and therefore the dynamic range of surface temperatures over which the fitting occurs is small. For the latent heat flux, the variational method results are generally closer to observations than those from the triangle method throughout the course of the day. Both methods appear to estimate evaporation better during 1988. We hypothesize that this is due to the fact that 1987 was a rather wet year where the evaporation rate was near the potential rate for much of the summer (Caparrini et al. 2004), while in 1988 the evaporation was primarily surface controlled and therefore had a stronger signal in the surface temperature sequences.

5. Concluding remarks

In this note, retrieval-based and assimilation-based methods for estimating surface turbulent fluxes from surface temperature observations are compared over two summer seasons at the FIFE site. Results from a synthetic experiment indicate that the triangle method’s assumption of a second-order polynomial fit to the (modeled) ratios of latent and sensible heat fluxes to net radiation are reasonable. However, while the triangle method compares favorably in cases with very low surface temperature measurement error, estimation errors are generally larger than that of the variational method in cases with measurement errors on the order of 1.0 K or greater. This would seem to confirm, at least at the FIFE site, the hypothesis that the assimilation approach (when estimating EF only) is more robust under erroneous observations. However, the propagation of this random error in temperature observations to surface flux estimation generally average out at the daily time scale where estimates from the two methods are comparable. Other retrieval algorithms based on time differences in temperature sequences (e.g., Diak 1990; Diak and Whipple 1993) may not be as susceptible to these types of errors. For the tests using real radiometric surface temperature observations, both methods contain significant flux estimation errors; with especially noticeable systematic biases. These biases are likely due to structural model errors in both methods and are a general problem found in most estimation methods. The variational method appears to perform slightly better largely due to the simpler parameterization of the relationship between surface fluxes (the EF parameter) and surface temperature observations. Future work should be done to include more retrieval algorithms and applications to different conditions to determine the optimal approach to diagnose land surface fluxes using remote sensing observations.

Acknowledgments

This study was supported by the National Science Foundation (NSF) under Grant EAR0333133 and the National Aeronautics and Space Administration (NASA) under grant NNG04GP71G. The authors would also like to thank three anonymous reviewers for their helpful comments on the manuscript.

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Fig. 1.
Fig. 1.

Daytime-averaged (top) latent and (bottom) sensible heat flux scatterplots from the (left) triangle and (right) variational methods for FIFE 1987.

Citation: Journal of Hydrometeorology 6, 6; 10.1175/JHM451.1

Fig. 2.
Fig. 2.

Same as in Fig. 1, but for FIFE 1988 data.

Citation: Journal of Hydrometeorology 6, 6; 10.1175/JHM451.1

Fig. 3.
Fig. 3.

Mean diurnal cycle of (top) latent and (bottom) sensible heat fluxes for (left) 1987 and (right) 1988 (observations are shown with a solid line with open circles, triangle method estimates are shown with a dashed line with open triangles, and variational assimilation estimates are shown with a dashed line and “+” symbols).

Citation: Journal of Hydrometeorology 6, 6; 10.1175/JHM451.1

Table 1.

Error statistics (in W m−2) for hourly and daytime-averaged flux estimates in synthetic experiments for both the triangle and variational assimilation method.

Table 1.
Table 2.

Same as in Table 1, but with observed surface temperature observations for both the triangle and variational assimilation method.

Table 2.
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