Impact of Covariance Localization on Ensemble Estimation of Surface Downwelling Longwave and Shortwave Radiation Fluxes

B. A. Forman Department of Civil and Environmental Engineering, University of Maryland, College Park, College Park, Maryland

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S. A. Margulis Department of Civil and Environmental Engineering, University of California, Los Angeles, Los Angeles, California

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Abstract

Accurate estimates of terrestrial hydrologic states and fluxes are, in large part, dependent on accurate estimates of the spatiotemporal variability and uncertainty of land surface forcings, including downwelling longwave (LW) and shortwave (SW) fluxes. However, such characterization of land surface forcings does not always receive proper attention. This study attempts to better estimate LW and SW fluxes, including their uncertainties, by merging different sources of information while considering horizontal error correlations via implementation of a 2D conditioning procedure within a Bayesian framework. A total of 25 experiments were performed utilizing four different, readily available downwelling radiation products. The localized region of space used to constrain horizontal error correlations was defined using an influence length, , specified a priori. Quantitative comparisons are made against an independent, ground-based observational network. In general, results suggest moderate improvement in cloudy-sky LW fluxes and modest improvement in clear-sky SW fluxes during certain times of the year when using the 2D framework relative to a more traditional 1D framework, but only up to a certain influence length scale. Beyond this length scale the flux estimates were typically degraded because of the introduction of spurious correlations. The influence length scale that yielded the greatest improvement in LW radiative flux estimation during cloudy-sky conditions, in general, increased with increasing cloud cover. These findings have implications for improving downwelling radiative flux estimation and further enhancing existing Land Data Assimilation System (LDAS) frameworks.

Corresponding author address: B. A. Forman, University of Maryland, Department of Civil and Environmental Engineering, College Park, MD 20742-3021. E-mail: baforman@umd.edu

Abstract

Accurate estimates of terrestrial hydrologic states and fluxes are, in large part, dependent on accurate estimates of the spatiotemporal variability and uncertainty of land surface forcings, including downwelling longwave (LW) and shortwave (SW) fluxes. However, such characterization of land surface forcings does not always receive proper attention. This study attempts to better estimate LW and SW fluxes, including their uncertainties, by merging different sources of information while considering horizontal error correlations via implementation of a 2D conditioning procedure within a Bayesian framework. A total of 25 experiments were performed utilizing four different, readily available downwelling radiation products. The localized region of space used to constrain horizontal error correlations was defined using an influence length, , specified a priori. Quantitative comparisons are made against an independent, ground-based observational network. In general, results suggest moderate improvement in cloudy-sky LW fluxes and modest improvement in clear-sky SW fluxes during certain times of the year when using the 2D framework relative to a more traditional 1D framework, but only up to a certain influence length scale. Beyond this length scale the flux estimates were typically degraded because of the introduction of spurious correlations. The influence length scale that yielded the greatest improvement in LW radiative flux estimation during cloudy-sky conditions, in general, increased with increasing cloud cover. These findings have implications for improving downwelling radiative flux estimation and further enhancing existing Land Data Assimilation System (LDAS) frameworks.

Corresponding author address: B. A. Forman, University of Maryland, Department of Civil and Environmental Engineering, College Park, MD 20742-3021. E-mail: baforman@umd.edu

1. Introduction

An ensemble-based land data assimilation system (LDAS) is an attractive framework to systematically merge modeled and measured estimates of land surface states and fluxes in order to advance scientific understanding (e.g., assessment of hydrologic balance over regional and global scales) while benefiting societal needs (e.g., flood forecasting). However, one significant challenge in constructing an LDAS is the generation of ensemble forcings [e.g., downwelling longwave (LW) and shortwave (SW) radiation at the earth’s surface] based on deterministic, remotely sensed estimates. The quality of the forcing ensemble, in large part, dictates the efficacy of the prior, land surface state ensemble (and its corresponding error structure), which ultimately drives the quality of the LDAS estimates (Pan and Wood 2009b). Therefore, it is important that an ensemble-based LDAS utilize forcing fields that contain an accurate estimate of the forcing uncertainty and do so with the least amount of error in order to yield the highest quality estimates.

A multitude of radiation products (Pinker et al. 2003; Cosgrove et al. 2003; Gupta et al. 1992) currently exist that are derived from space-based instrumentation and have been created at spatial and temporal scales relevant to land surface modeling. These deterministic products all have value. For example, remotely sensed measurement products are typically better at observing clouds than large-scale models are at accurately predicting their occurrence and/or properties, and hence have a greater potential for capturing cloud-related processes operating on downwelling radiative fluxes. However, each radiation flux product is derived using different methods and employs different remotely sensed measurements, and as a result, each product has its own spatiotemporal characteristics and uncertainties. Making the most of these different products requires merging them in a way that extracts the most information while appropriately accounting for differences in uncertainty structure. Merging these deterministic products into a single ensemble such that it contains the appropriate uncertainty structure (ideally with reduced error) would not only be beneficial in LDAS applications, but would also add value to the existing products. One such technique that achieves this is a Bayesian merging framework applied to the land surface forcings that has been shown to increase utility associated with merging readily available, satellite-based radiation measurements into a bulk, physically based radiation model (Forman and Margulis 2010a,b).

Because of the relatively simple covariance localization technique used in Forman and Margulis (2010a,b), however, horizontal error correlations may not be adequately considered. That is, the merging procedure only conditions estimates one at a time localized to the spatial extent of a single, coarse-scale measurement pixel. This type of procedure is referred to herein as one-dimensional (1D) conditioning. Borrowing language from Janjić et al. (2011), this type of localization falls under the realm of domain localization and is commonly employed because of its relative ease of implementation and computational efficiency (Forman and Margulis 2010b). An alternative approach is sometimes referred to as direct localization, which employs a Schur product (i.e., elementwise multiplication) of the ensemble covariance matrix with a localization matrix defined by a chosen correlation function (Janjić et al. 2011). Both methods have their respective merits, but only the domain localization method is explored here as to maintain a tractable project scope.

Even though the findings of Forman and Margulis (2010b) showed considerable improvement via the 1D system when verified against independent observations, there is potential for further improvement via consideration of horizontal error correlations beyond the 1D spatial domain. Here we consider the impact of horizontal error correlations by employing a two-dimensional (2D) conditioning scheme as a follow on to the 1D results presented in Forman and Margulis (2010b). A 2D procedure not only utilizes the spatial domain of a single pixel as in the 1D procedure, but also utilizes neighboring pixels spread out over 2D space. Multidimensional data assimilation schemes (e.g., 3D procedures in Reichle and Koster 2003 and De Lannoy et al. 2010) have been shown to outperform their 1D counterparts when using synthetically generated measurements. These 3D procedures were applied to soil moisture estimation, and not only utilize measurements spread out across horizontal space as in the 2D approach, but also incorporate vertical information from the soil column. When applying these methods to radiative flux estimation at the earth’s surface, however, only the 1D and 2D procedures are applicable as there is no vertical dimension in the bulk, vertically integrated radiation estimate. The study presented here expands on the findings of Forman and Margulis (2010b) by demonstrating a “real-world” application of a multidimensional Bayesian merging framework using real measurements and independent, ground-based observations (herein referred to as validation data) to verify the results.

2. Science questions

The overarching hypothesis of this work is that conditioning a prior radiative flux model using radiative flux measurements will improve estimates within a regionalized space defined by a characteristic length scale. Beyond this length scale, the flux errors are effectively uncorrelated with one another. This representative length scale, or influence length scale, should vary according to regional climatology as well as be strongly dependent on local atmospheric (i.e., clear versus cloudy sky) conditions. It has been shown that radiative flux estimates (and their uncertainty) are strongly dependent on cloud conditions (Forman and Margulis 2010a) and that proper accounting of radiative flux uncertainty within a Bayesian framework is often ignored in more simple radiative flux perturbation schemes (Forman and Margulis 2010b). The work presented here not only carefully accounts for different sources of uncertainty, including cloud conditions, but also considers the impact of horizontal error correlations. Additional science questions considered in this study include the following:

  • Can a reduction in posterior (conditioned) error be achieved by simultaneously merging more than one coarse-scale, satellite-derived radiative flux measurement pixel?

  • Are there differences when merging LW versus SW fluxes?

  • Is the amount of error reduction solely a function of influence length scale? Or is it more dependent on the spatial resolution of the merged measurements?

3. Methodology

a. Prior ensemble model

The prior ensemble model used in this study is a bulk, physically based model of downwelling LW and SW radiation reaching the earth’s surface (Forman and Margulis 2010a). Required inputs to the model are satellite-based measurements of cloud, atmospheric, and land surface states derived from both geostationary and polar orbiting platforms. By virtue of its data-driven nature, the model is diagnostic in form, which has implications on the Bayesian merging framework described below (Forman and Margulis 2010b). The model could be made prognostic through the inclusion of a cloud resolving model, for example, but is left in diagnostic form for reasons of computational efficiency and the intent of being applied in a reanalysis construct.

Cloud state information is obtained from a high-resolution cloud product—the Visible Infrared Solar-Infrared Split-Window Technique (VISST)—produced by Minnis et al. (1995, 2008) at the National Aeronautics and Space Administration (NASA) Langley Research Center. Atmospheric and land surface state estimates are obtained from a variety of Moderate Resolution Imaging Spectroradiometer (MODIS) and Atmospheric Infrared Sounder (AIRS) products. For brevity only the essential details of the prior model are provided here. The ensemble-based framework begins with a simple vector representation of model output, y(x, t), that is explicitly dependent on both space, x, and time, t, as
e1
where is LW flux, is SW flux, is the radiation model operator, and u(x, t) is the vector of model inputs. Basic details pertaining to are provided in Forman and Margulis (2009). Similarly, considerable detail regarding the formulation of u(x, t) is found in Forman and Margulis (2010a) and not repeated here. Suffice it to say that u(x, t) is composed of satellite-derived measurements as well as empirical parameterizations of atmospheric and land surface states that dictate the first-order uncertainty of LW and SW fluxes.
We assume u(x, t) is a random vector that explains the uncertainty in y(x, t) and that u(x, t) follows some underlying (unknown) probability density function (pdf) upu(u). In principle, given the underlying pdf, we can sample it to generate ensemble fluxes such that
e2
where j represents a single replicate from an ensemble of size N and ujpu(u). Since the underlying pdf is generally unknown, we postulate that it can be represented by
e3
where is the nominal estimate, γj(x, L) is a perturbation replicate sampled from the distribution of γ(x, L), L is the characteristic length vector that defines spatial correlations in u(x, t), and γ(x, L) ~ LN[1, Cγ(x)] is a mean unity, lognormal distribution with covariance Cγ(x) where L is implicit in its formulation. Here L approximates a typical characteristic length for each variable in . The values used for L are reproduced in Table 1 and are based on the analysis of satellite-based measurements of atmospheric, land surface, and cloud states that comprise u(x, t) for the simulation period used in this study. The use of correlated perturbations is designed to enhance the physical consistency of individual ensemble replicates (Forman and Margulis 2010a) whereas excluding cross correlations from the perturbation framework can result in physically unrealistic results. That is, including cross correlations ultimately yields results whereas in some replicates SW flux is attenuated while LW flux is simultaneously amplified in the presence of clouds. Excluding cross correlations from the perturbation framework often results in SW and LW being simultaneously amplified (or attenuated), which is physically inconsistent in the presence of clouds. Further, a series of sensitivity analyses were conducted (results not shown) that suggest the inclusion of cross-correlated perturbations does not degrade the performance of the Bayesian merging procedure and can, at times, lead to an improved posterior estimate as compared to results when cross correlations in the perturbations are neglected. Details on the development of L, including a discussion of the application of computed variograms, are provided in Forman and Margulis (2010a). It is important to note that L dictates a significant amount of the spatial variability in the prior ensemble, but that first-order control over LW and SW spatial variability is dominated by the unperturbed structure of the atmospheric and cloud inputs contained in u(x, t). The multiplicative γ(x, L) formulation used in this study assumes u(x, t) is unbiased; however, this assumption could be relaxed.
Table 1.

Model states and approximated characteristic length vector, L.

Table 1.

b. Merged products

The satellite-based measurement products of LW and SW selected for conditioning y(x, t) were chosen because each is readily available to the public and each is representative of an advanced radiative flux k estimate using spaceborne instrumentation. The products used in this study are the same as those used in Forman and Margulis (2010b), which includes a discussion of their performance relative to an independent, ground-based observational network. The spatial resolution of each product is listed in Table 2 when the number of merged measurements is equal to 1.

Table 2.

Maximum number of measurements, , as a function of (n/a = not applicable; − = not tested).

Table 2.

1) Longwave products

Two different LW products of instantaneous, downwelling LW radiation at the earth’s surface are used in the experiments. The products are 1) the NASA Global Energy and Water Cycle Experiment (GEWEX) Surface Radiation Budget (SRB) Project (herein referred to as SRB-LW) and 2) the North American Land Data Assimilation (NLDAS) Project (herein referred to as NLDAS-LW). SRB-LW is derived globally at a 3-h temporal resolution originally computed on a quasi-equal-area grid. The original product was subsequently regridded to a 1° equal-area grid (Mlynczak et al. 2006). Additional details regarding the SRB-LW algorithm are provided in Gupta (1989) and Gupta et al. (1992). NLDAS-LW is derived for the continental United States at an hourly temporal resolution and a 0.125° spatial resolution. Additional details regarding the NLDAS-LW algorithm are provided in Cosgrove et al. (2003).

2) Shortwave products

Analogous to the LW products, two different instantaneous, downwelling SW products are used in the experiments. The products are 1) the GEWEX SRB Project (herein referred to as SRB-SW) and 2) the GEWEX Shortwave Radiation Budget (Pinker et al. 2003) (herein referred to as Pinker-SW). SRB-SW contains the same spatial and temporal resolutions as the SRB-LW product described above, the details of which are found in Pinker and Laszlo (1992). Pinker-SW is produced independently from SRB-SW. Both SW products effectively integrate radiative processes over the entire atmospheric column and account for cloud interactions using a radiative transfer model. Pinker-SW, however, is produced at a finer spatial (0.5° versus 1°) and temporal (hourly versus 3 h) scale.

The products described here vary considerably in terms of their methods of generation, the satellite inputs used, and their model formulations. Therefore, each product has its own space–time error structure and resolution. The conditioning procedure described below effectively integrates these differences, and as a result, provides a convenient method for merging multiple products produced at multiple resolutions (in both space and time) where each product has its own, unique spatiotemporal characteristics.

c. Conditioning procedure

Given an individual, unconditioned (prior) replicate from Eq. (2), we condition that estimate using a Bayesian framework posed as
e4
where the superscript “+” represents the updated (conditioned) flux, the “−” superscript represents the prior (unconditioned) flux, is the measurement vector within the regionalized space is the random measurement error vector, is the predicted measurement model operator, and is the gain matrix defined as
e5
where Cyz is the sample covariance between the prior fluxes and the predicted measurements, Czz is the sample covariance of the predicted measurements, and Cυυ is the measurement error covariance. The update equation shown in Eq. (4) is a function of the gain matrix as well as the difference between the actual measurement and the predicted measurement. The predicted measurement model, , involves projection of a prior model flux into measurement space. Since the prior model and the measurements in this study are both estimates of instantaneous, downwelling radiative flux, this transformation is relatively simple. Namely, the predicted measurement model is a mapping operator that spatially aggregates the prior model estimate of downwelling radiation into the spatial scale of the corresponding measurement product. That is, computes the spatial average of many finescale prior pixels corresponding to a single, coarse-scale measurement pixel, and does so at each individual measurement pixel of interest.

The formulation for the 2D procedure is similar to that of 1D except that the regionalized space explicit in the covariance formulation extends beyond the spatial resolution of the measurement, r, to a distance dictated by an influence length, . When , the 2D formulation is equivalent to that of the 1D formulation. Conceptually, examples of 1D and 2D procedures are shown in Fig. 1. The leftmost subplot is the 1D application whereas the middle and rightmost subplots are 2D applications. Increasing causes more measurements to be used in the Bayesian update scheme at a given location designated by the solid, black square. As increases in size, the length of the measurement vector, , in (4) increases accordingly in size as do the covariance matrices shown in (5). That is, the measurement model described above is now also a function of . No spatial averaging of the measurements is conducted as increases; rather, the measurement vector merely increases in length.

Fig. 1.
Fig. 1.

Examples of (left) 1D and (middle),(right) 2D measurement setups for , , and merging measurements spaced on a 1° × 1° grid. The size of the influence length (black ellipse) dictates which measurements are included (black × symbols) and which are excluded (gray × symbols) for the local region of conditioning (black square).

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

d. Measurement error model

The second term listed inside the brackets of Eq. (4) requires specification of measurement error, which is characterized by the measurement error covariance, Cυυ, shown in Eq. (5). Measurement error models for the instantaneous LW and SW products are similar to those used in Forman and Margulis (2010b) except that errors were derived on a finer, monthly time scale rather than the 14-month averaging period used in Forman and Margulis (2010b). Use of a monthly error model allows for measurement error to reflect the effects of seasonality whereas the original error model does not.

Each measurement error model was derived by direct comparison against 1-min averages of independent, validation data from the Atmospheric Radiation Measurement (ARM) Solar Infrared Radiation Stations (SIRS) program. The basic premise of the error model formulation is that uncertainty (and hence error) is a function of cloud cover fraction (CCF). CCF is computed from the finescale VISST cloud product used in the prior model formulation. Cloud presence in VISST is aggregated in space to match the spatial characteristics of the given measurement product. The measurement error, which is computed as the root-mean-square error, is then estimated via comparison to all available, nonzero SIRS measurements during the simulation period used in this study. The use of root-mean-square error as a surrogate for the measurement error standard deviation is reasonable assuming the SIRS validation data are relatively unbiased. With the linear approximation of the measurement error covariance as a function of cloud cover fraction, measurement error for a given replicate, vj, is modeled as a random Gaussian process with mean zero noise of covariance Cυυ such that measurement error increases with increasing cloud cover.

Horizontal measurement error correlations were not considered in this study. The measurement models (e.g., radiative transfer models) and the corresponding inputs to these measurement models were not available for use. Hence, the same methodology outlined in Forman and Margulis (2010a) that approximates horizontal error correlations could not be applied. Rather, we assume the horizontal measurement errors are uncorrelated in a similar manner as conducted in Reichle and Koster (2003) and De Lannoy et al. (2010). Accounting for horizontal measurement error correlations effectively reduces the number of degrees of freedom in the measurements and, as a result, has the effect of reducing the available information content during conditioning. A similar impact could be achieved by increasing the measurement error (thereby reducing the information content), but that approach was excluded in favor of utilizing the data-driven measurement error model described above.

e. Conditioning application

The region selected for use in this study is the southern Great Plains (SGP) region of the United States shown in Fig. 2. This region was selected because it allows for verification against an extensive ground-based observational network. In addition, the relatively homogeneous terrain in the SGP allows one to treat the satellite pixel field of view (on the order of kilometers) as a homogeneous region, which helps minimize much of the scale difference between the satellite-scale estimates and the point-scale validation data. A simulation period from 1 January 2003 to 1 January 2005 was selected for analysis.

Fig. 2.
Fig. 2.

Map showing the SGP topographical domain. Markers represent ARM-SGP SIRS (•) and Oklahoma Mesonet (▿) observation stations.

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

Independent, ground-based validation data available for model comparison/verification include the SIRS observational network as well as the Oklahoma Mesoscale Network (OKMESONET; Brock et al. 1995) that was used to develop the uncertainties in u(x, t) associated with reference-level air temperature and humidity (Forman and Margulis 2010a). Each measurement product listed in Table 2 was merged independently into both the LW and SW estimates. Multiple products could have been simultaneously merged into the flux estimates as done in Forman and Margulis (2010b), but this was avoided in an effort to focus on the effects of covariance localization for a single product without the confounding, synergistic effects often witnessed when merging more than one product at a time. Finally, no cumulative distribution function (CDF) matching was conducted between the different products because each product was relatively unbiased (i.e., typically less than 1%) when compared against the SIRS validation data over the 24-month simulation period. An exception to this rule is the NLDAS-LW product, particularly during clear-sky conditions, which can contain a negative bias up to 4%. However, no CDF matching was performed on any of the products in order to maintain consistency between different experiments as well as to investigate the potential for information transfer using the original, unaltered measurement products in a manner consistent with that conducted in Forman and Margulis (2010b).

4. Results

a. Posterior error statistics

Comparison of 1D and 2D conditioning results begins with computation of root-mean-square difference (RMSD) and correlation coefficient (ρ) of the ensemble mean relative to the SIRS ground-based validation data. Calculations of mean difference (MD) were also performed, but are excluded from this discussion as the results show both the prior and posterior estimates are relatively unbiased [majority of results had −2 ≤ MD ≤ 2 (W m−2)] and hence add relatively little to the conclusions beyond what RMSD and ρ already provide. In addition, an investigation of replicatewise statistics (i.e., computed statistics of individual replicates relative to the SIRS validation data) was also conducted, but is omitted because the results simply reinforce the findings from the ensemble mean analysis.

In general, the 2D procedure offered some improvements to the estimates during both clear- and cloudy-sky conditions during certain times of the year. Tables 3 and 4 list RMSD values relative to the SIRS validation data for LW and SW posterior estimates, respectively. Each table lists the minimum RMSD achieved (across all values of ) where the number in parentheses denotes the corresponding in degrees for a given product. Over the course of the entire 2-yr simulation period (far-right column), 2D conditioning offers little improvement beyond that of the 1D procedure. During the months of September–November (SON), however, modest improvement is achieved in all-sky LW fluxes as well as clear-sky SW fluxes. Improvements in LW flux during SON are in large part due to the presence of large-scale cloud systems that are common in the SGP during the fall season, which often results in increased rainfall relative to the summer and winter seasons (Garbrecht et al. 2004). The increased cloud cover enables more information transfer from the measurements to the conditioned estimates (see further discussion in section 4d). Limited improvements are also seen in SW fluxes during December–February (DJF), but to a lesser degree than those witnessed in SON. Most of the improvements via application of the 2D procedure were achieved when merging the finest-resolution LW or SW product (i.e., NLDAS-LW and Pinker-SW) because these products more adequately captured finescale features that operate on downwelling radiation processes, and hence allowed for a more pronounced covariance structure that was beneficial toward information exchange.

Table 3.

Minimum RMSD (across all values of ) for LW fluxes during clear-sky and cloudy-sky conditions where the number in parentheses represents the corresponding in degrees. (Full = 2-yr period.)

Table 3.
Table 4.

As in Table 3 except for SW fluxes.

Table 4.

It is worth noting that the greatest SW improvement (albeit small in magnitude) during cloudy-sky conditions in March–May (MAM) and June–August (JJA) occurred through conditioning on the NLDAS-LW product. This was not witnessed in Forman and Margulis (2010b) in part because a different (longer) simulation period was used here as well as the use of a monthly averaged measurement error model. The study in Forman and Margulis (2010b) covered 14 months with only one complete MAM and JJA period. The 24-month study period used here utilized multiple MAM and JJA periods that allowed for a more robust investigation, which included interannual variability. The longer study period coupled with the monthly averaged measurement error model contributes to the improvement (relative to the prior) in cloudy-sky SW flux during MAM and JJA as shown in Table 4. A reasonable explanation for this behavior is the preponderance of small-scale convective cloud systems that are better captured in NLDAS-LW (~12.5-km resolution) compared to Pinker-SW (~50-km resolution), and hence better enable information transfer from the measurement product into the conditioned estimate. Granted, the improvement occurred while using the 1D procedure instead of the 2D procedure, but it is worthwhile highlighting the ability of the Bayesian merging framework to yield an improved estimate via information transfer from a LW measurement product into a SW estimate in the presence of cloud cover.

Returning to the topic of 2D conditioning, a graphical example of 2D performance is shown in Fig. 3 for computed statistics of RMSD (top row) and ρ (bottom row) for the months of SON. The gray lines represent the ensemble mean from the prior (unconditioned) ensemble. The black lines represent the ensemble means from the experiments independently merging each of the measurement products as a function of . The leftmost point for each experiment is representative of the 1D procedure. The remaining points represent the different experiments at the different values of . As is clearly shown, LW fluxes during both clear- and cloudy-sky conditions as well as clear-sky SW fluxes are improved beyond the 1D conditioning result by expanding the region of covariance localization.

Fig. 3.
Fig. 3.

Computed statistics of (top) RMSD and (bottom) ρ for longwave and shortwave fluxes during clear- and cloudy-sky conditions for SON.

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

Improvements to LW flux occur for two reasons: 1) the LW measurement products during all-sky conditions are superior to the prior estimates (Forman and Margulis 2009) and hence there is potential for improvement via information transfer from the measurements into the posterior ensemble, and 2) characteristic lengths in states that dictate much of the LW flux (e.g., air temperature) are typically longer than the spatial resolution of the merged measurements (Forman and Margulis 2010a). Therefore, information transfer is further enabled by increasing the influence length.

Behavior of the 2D procedure for LW flux can be better understood via inspection of Table 1. Specifically, LW flux is dominated by the near-surface and cloud base temperature. Table 1 shows the typical characteristic length for these temperatures are 2.5° and 1.5°, respectively. It is important to note here that these lengths increase or decrease based on local atmospheric and cloud cover conditions for a given day, and can differ from the typical values shown in Table 1, which are derived from computed variograms. That being said, improvements to LW flux generally occur using values of , which is roughly equal to or less than the characteristic lengths of the first-order controls on LW flux. Conversely, 2D conditioning, in general, fails to improve conditioned estimates of LW flux when is greater than the characteristic length of the temperature states that dominate LW flux processes. It should also be noted that the increase in clear-sky LW RMSD coincident with little or no change in ρ when merged with the NLDAS-LW product is due to the introduction of a small but nonnegligible bias, which is briefly mentioned in section 3e. Additionally, the increase in LW flux correlation via conditioning on NLDAS-LW is partly due to its hourly temporal resolution, which provides more opportunities for beneficial information exchange over the diurnal cycle. The 3-hourly SRB-LW product, on the other hand, provides fewer opportunities to improve the flux estimate on the subdiurnal time scale and, as shown in the Fig. 3 examples, can degrade the temporal correlation of the conditioned estimate when compared against the SIRS validation data.

A similar argument can be made for the behavior of the SW flux estimation procedure. Table 4 shows that clear-sky SW flux estimates were improved only when . The most sensitive controls on the SW clear-sky formulation are 1) column-integrated water vapor, 2) aerosol scattering, and 3) Rayleigh scattering (Forman and Margulis 2010a). All three of these controls have typical characteristic lengths of 2.0°, which are lengths greater than the improvements shown in Table 4. During cloudy-sky conditions, the best performance via 2D conditioning occurred when . These findings coincide with the characteristic length of the dominant controls on cloudy-sky SW flux (i.e., cloud hydrometeor size and cloud water path; Forman and Margulis 2010a) such that the best-performing 2D conditioning experiment occurred using the finest-scale measurement product for . The one exception to this was the utilization of the SRB-LW during DJF where a value of yielded the best results for that particular product. However, it should be noted that the results for the other values used in that experiment differed only by 0.1 (W m−2) (results not shown), hence there is no substantial difference between the SW flux estimates conditioned on the SRB-LW product over the range . In addition, the improvement to the posterior estimate of SW flux is small relative to LW flux because the prior SW model performs quite well (Forman and Margulis 2010a); hence, there is relatively little room for improvement. Accurate estimation of SW flux, particularly in the presence of clouds, is largely dependent on the accurate estimation of local cloud conditions (Forman and Margulis 2009) and is less dependent on neighboring cloud conditions.

b. Posterior uncertainty statistics

Performance metrics highlighted in the previous section discuss ensemble error relative to the SIRS validation data. An additional consideration of model performance is that of ensemble uncertainty. Uncertainty is investigated here using both the RMSD values discussed in the previous section as well as applying a metric for ensemble spread, which here is defined as the LW ensemble standard deviation, σLW, and the SW ensemble standard deviation, σSW. The conditioning procedure reduces the ensemble spread; however, the question remains whether the ensemble spread contains an amount of variability that is comparable to the natural variability of the validation data.

Figure 4 shows the domain-average, 2-yr temporal average of the posterior ensemble standard deviations via conditioning on each of the four radiation products. As is clearly shown, ensemble standard deviation decreases with increasing . Rationale for this behavior is discussed below in section 4d. The point made here is that the uncertainty contained within the ensemble, in most cases, is a reasonable approximation of the uncertainty of the ground-based SIRS validation data, which is on the order of 10 (W m−2) (Li et al. 2005). However, the conditioning procedure utilizing specific products at relatively large values of can result in reduced ensemble spread that does not contain a reasonable amount of variability. For example, the top subplot in Fig. 4 shows that 2D conditioning of LW estimates using the NLDAS-LW product results in a dramatic reduction in σLW for . Similarly, the bottom subplot shows that 2D conditioning of SW estimates using the Pinker-SW product results in a significant reduction in σSW for . Both cases illustrate how the conditioning procedure at relatively large values of can result in the undesirable characteristic where ensemble uncertainty is less than the natural variability of the ground-based validation data. Fortunately, both instances of a large reduction in posterior ensemble spread occur at values larger than where the greatest improvements via 2D conditioning are typically achieved (Fig. 3).

Fig. 4.
Fig. 4.

Domain-averaged, temporal-averaged ensemble standard deviation of (top) LW and (bottom) SW radiative fluxes as a function of .

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

c. Optimal influence length

To better illustrate the behavior of the 1D and 2D procedures during clear- and cloudy-sky conditions, Fig. 5 shows a 1-month time series during November 2004 for the merging of NLDAS-LW. One can clearly see that the RMSD based on 1D conditioning is, in general, significantly larger than its 2D counterpart when the cloud cover is greater than 50%. The RMSD increases with increasing cloud cover in both the 1D and 2D applications because of the added variability that clouds introduce. However, because of the existence of horizontal error correlations associated with the cloud system, the 2D application at is able to extract more information from the measurements during conditioning in order to improve the estimates and subsequently reduce the RMSD relative to the SIRS validation data. The 1D procedure does not take advantage of the presence of horizontal error correlations, and hence the RMSD remains relatively large after conditioning.

Fig. 5.
Fig. 5.

LW RMSD examples of 1D (0.125°) and 2D (0.50°) conditioning of NLDAS-LW during November 2004. Cloud covered percent is shown as the thick line. Dates are labeled as month/day/year.

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

Further attempts were made at determining an optimal influence length in the sense that the “optimal” yielded the lowest RMSD when compared against SIRS. Figure 6 shows a 2-week time series of cloudy-sky LW estimates during the end of August 2004 after conditioning by the NLDAS-LW product. The left-hand axis displays the optimal as a thin, solid line. The right-hand axes show the corresponding cloud cover percentage as a thin, dashed line as well as the approximate characteristic cloud length scale of the cloud base temperature as a thick, dashed line. The characteristic length scale is defined in an analogous manner as that done in Forman and Margulis (2010a). Note that there is reasonable agreement between the two dashed lines when increasing cloud cover amount (or characteristic cloud length scale), in general, corresponds to a larger optimal . The correlation coefficient between the optimal time series and the cloud cover is 0.66. Similarly, the correlation between the optimal and the characteristic length scale is 0.68. Both estimates suggest each can serve as a reasonable proxy for estimating the optimal prior to running the 2D conditioning procedure in order to yield the best results. The use of the characteristic length scale over that of cloud cover percentage should, in theory, provide more information on cloud structure and their properties that modulate downwelling radiative fluxes. For example, characteristic length scales for 2D conditioning of SW could be based on cloud particle size whereas LW could be based on cloud base temperature, which is often much longer in scale than cloud particle size (Forman and Margulis 2010a). A rigorous method for defining characteristic length scales of clouds, however, must first be defined and is well beyond the scope of this manuscript, but the results presented here show promise for future work.

Fig. 6.
Fig. 6.

Influence length achieving a minimum RMSD (left-hand axis) as a function of time in late August 2004. The corresponding cloud cover percentage (thin dashed line) and cloud characteristic length scale (thick dashed line) are shown along the right-hand axis. Dates are labeled as month/day/year.

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

d. Integrated gains

Additional information on the performance of 2D conditioning may be found by investigating the gain matrix shown in (5). The gain matrix is a relative measure of the covariance structure, which can serve as an indicator of conditioning behavior (Forman and Margulis 2010b), and can help elucidate information as to the limits of potential information transfer from the measurement products into the posterior ensemble. When flux measurements of similar type are merged with the prior estimate (i.e., LW estimates conditioned on LW products or SW estimates conditioned on SW products), the upper limit of the computed gain is typically ~1. This suggests the sum of the sample covariance of the predicted measurements and the measurement error covariance, Czz + Cυυ, is comparable to the sample covariance between the prior fluxes and the predicted measurements, Cyz. In addition, the covariance structure found in Cyz suggests strong, positive correlations between the prior fluxes and predicted measurements, which is reasonable given that conditioning is on the same flux type as the estimate. If the computed gain is ~0, it suggests little or no correlation between the two exists. Computed gains significantly less than 0 are indicative of an anticorrelation structure within Cyz, and is most likely to occur in the case where the measured flux and predicted flux are of different type (e.g., attenuated SW flux coincident with amplified LW flux in the presence of clouds).

Since the sizes of the covariance matrices change as a function of both the spatial resolution of the measurements and the influence length scale, it is difficult to compare the results from one experiment to another using the standard formulation for shown in Eq. (5). This is conceptually understandable via inspection of Table 2 where the number of measurements, , differs dramatically depending on the measurement product used and the influence length applied. Therefore, in order to compare different experiments, we introduce the concept of an integrated gain. In essence, the integrated gain, , is the summation of all the gains within the spatial domain captured by , averaged over the localized space defined by a single measurement pixel, Ω = xr. In other words, is the summation of the gains within an ellipse depicted in Fig. 1 averaged across the region denoted by the solid, black square. During the analysis, this process is repeated at each measurement location within the study domain. In compact notation this is defined for a given measurement location at a given measurement time, t, as
e6
The term inside the brackets accounts for differences due to varying . The summation sign within the brackets accounts for differences during matrix multiplication associated with increasing sizes of Cyz, Czz, and Cυυ as increases, which would otherwise be obscured if only single measurement pixel gains were computed via Eq. (5). The term outside the brackets accounts for differences in measurement pixel scale (Table 2) where Nr represents the number of prior model pixels relative to a single measurement pixel. Collectively, these two terms allow for a level comparison between different experiments regardless of the scale of the measurements or the scale of the influence length applied.

Using the integrated gains computed via Eq. (6), the daily averaged, domain-averaged time series of integrated gains (as a function of ) are shown in Figs. 7 and 8 for LW and SW merging, respectively. Each subplot shows the gains for a particular flux type associated with the merger of a particular measurement product. For example, Fig. 7a shows the LW-integrated gain matrix, , as a function of and time for the merger with the SRB-LW product (i.e., ). The left-hand axis shows the influence lengths used in the experiment whereas the right-hand axis shows the corresponding number of measurements. Collectively, Figs. 7 and 8 display the integrated gains for all of the experiments outlined in Table 2.

Fig. 7.
Fig. 7.

Daily-averaged, domain-averaged integrated gain for (a),(b) LW flux, , and (c),(d) SW flux, , as a function of influence length, , and time for (a),(c) SRB-LW and (b),(d) NLDAS-LW. Dates are labeled as month/day/year.

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

Fig. 8.
Fig. 8.

As in Fig. 7 except for (a),(c) SRB-SW and (b),(d) Pinker-SW.

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

Inspection of these figures, in general, shows consistent behavior between all experiments such that 1) gains increase when merging fluxes of the same type, 2) gains increase in the summer relative to the winter, and 3) the absolute magnitude of the gains increase with increasing influence length. Item 1) mirrors the findings of Forman and Margulis (2010b) where the covariance between the prior fluxes and predicted measurements, Cyz, is greatest between fluxes of the same type. The covariance structure is small in magnitude (less pronounced) when SW/LW estimates are conditioned on LW/SW measurement products, but that information content is still available for use during conditioning. Additionally, when SW estimates are conditioned on LW measurement products (or LW estimates conditioned on SW measurement products) the gain is generally less than zero, which suggests the presence of negative correlations. Negative gains increase in magnitude when cloudy conditions persist, which is due to the anticorrelation behavior of clouds attenuating SW flux while simultaneously amplifying LW flux. Item 2) occurs as a consequence of seasonality. As the range of values within the prior model and predicted measurement ensembles increases, in general, so does the sample covariance that is used to represent the error covariance. The covariance Cyz typically increases during the summer because of the increased variability in both the prior estimates and the predicted measurements associated with increasing flux magnitude. The covariance Czz increases, too, but to a smaller degree because the measurement model is aggregating the finescale estimates across space, which effectively smooths out much of the finescale variability, in turn causing a smaller increase to Czz relative to Cyz. As a result, Cyz tends to increase more than (Czz + Cυυ)−1, which has the effect of introducing seasonality into the computed gains. Item 3) occurs because Cyz and Czz become increasingly similar when viewed across increasingly larger regions of space while at the same time Czz can increase relative to Cυυ. Recall that Cyz is the sample covariance between the prior fluxes and the predicted measurements and that Czz is the sample covariance of the predicted measurements. Further, recall that the predicted measurements are a spatial aggregation of the prior fluxes into the spatial scale of the corresponding measurement product. When viewed from the scale of the predicted measurement, Cyz and Czz can be significantly different from one pixel to the next. However, when one increases the vantage point out across larger regions of space (i.e., increases ), the prior fluxes and predicted measurements begin to look increasingly similar. As a result, the covariance structure between the two becomes more similar [i.e., Cyz and (Czz + Cυυ)−1 become more equivalent], which ultimately leads to an effective gain close to 1.

An example of this last item is most clearly seen in Fig. 7b where the effective gain asymptotically approaches ~1 with increasing . This behavior is attributable to two factors. First, a maximum value of is reasonable since the measurements and model are in the same space. A gain close to 1 results in a large reduction in posterior ensemble spread because the individual replicates are all effectively updated to the value of the measurement product. This situation is not witnessed when the LW model is conditioned on the SW measurements (or SW model conditioned on LW measurements) because Cyz and Czz are comparing different types of fluxes, and hence the magnitude of the two covariance matrices can be vastly different. Secondly, the asymptotic behavior of is indicative of a lack of additional spatial correlation structure beyond a given influence length scale. This behavior is not only seen in Fig. 7b between fluxes of the same type, but is also witnessed in Figs. 7c and 8a for fluxes of differing type.

One last item of note is the high-frequency variations seen in the matrices. These high-frequency perturbations about the low-frequency seasonal cycle are due to the arrival and subsequent departure of large-scale cloud systems. When SW fluxes are conditioned on a SW product or LW fluxes are conditioned on a LW product, the gain becomes much larger. When SW fluxes are conditioned on a LW product or LW fluxes are conditioned on a SW product, the gain becomes significantly more negative. This behavior is witnessed for all values of including the 1D results. The former arises from a larger covariance between the prior fluxes and the predicted measurements, Cyz, and hence a larger . The latter results from the same effect, but the gains are increasingly negative because of the anticorrelation effect clouds have on downwelling radiative fluxes. An ideal example occurs in early July 2003 when a large storm system entered the domain, causing significant rainfall over several days throughout most of the region (results based on OKMESONET analysis not shown). Figure 8b shows this as a relatively large, negative gain at the largest values of . Coincidentally, Fig. 8d shows a relatively large, positive gain via conditioning on the SW model by the Pinker-SW product. This type of behavior is also found in the other subplots in Figs. 7 and 8, but is not as readily apparent because of the muted response when using the coarser-scale SRB products.

e. Spatial smoothing effect

Since 1D conditioning does not consider error correlations between neighboring measurement pixels, application of the 1D procedure results in a checkerboardlike pattern coincident with the spatial resolution of the coarse-scale measurement product. An example of such an occurrence is shown in Fig. 9c. This is an undesirable outcome when applied to continuous fields such as downwelling radiative flux. The 2D procedure, on the other hand, does not produce these discontinuities along the merged measurement pixel boundaries. Since more than one measurement pixel is utilized during the conditioning procedure at any given measurement location, the resulting update is effectively “smoothed” across space and results in a more continuous field after conditioning. An example of this behavior is shown in Fig. 9d. Not only does 2D conditioning often yield improved estimates in the presence of clouds when an appropriately specified is applied, but the resulting field contains a more continuous representation of radiative flux. This “smoothing” effect demonstrates some of the added utility of applying 2D conditioning to continuous fields.

Fig. 9.
Fig. 9.

Spatial smoothing effect of (c) 1D conditioning relative to (d) 2D conditioning utilizing the SRB-LW product. (a) The prior ensemble shown as the ensemble mean and (b) the measurement product used during conditioning are shown for reference.

Citation: Journal of Hydrometeorology 13, 4; 10.1175/JHM-D-11-073.1

5. Discussion and conclusions

A 2D conditioning procedure was implemented into the original, 1D Bayesian merging scheme outlined in Forman and Margulis (2010b). A series of experiments merging four (4) different readily available downwelling radiation measurement products was conducted and compared against an independent, ground-based radiometer network. A total of 25 experiments (including the 1D experiments) were performed. In general, in the absence of significant cloud systems, consideration of horizontal error correlations adds little value to the conditioning procedure. Rather, as the influence length increases, the posterior ensemble is, in general, degraded relative to the 1D conditioning results. In the presence of significant cloud systems, however, expanding the region of covariance localization improves the conditioned estimate up to a certain influence length. During both clear-sky and cloudy-sky conditions, a threshold influence length is ultimately reached, which is indicative of reduced spatial correlations beyond a particular influence length scale.

The findings suggest significant improvement in downwelling LW can be achieved during certain times of the year when large-scale cloud systems are more prevalent in the SGP, but that limited improvement was witnessed when smaller-scale convective cloud systems are more common such as in June–August (JJA). SW fluxes during both clear- and cloudy-sky conditions, in general, were not improved beyond their 1D counterparts, and in many cases were degraded by the 2D procedure because of the introduction of spurious correlations. In general, application of 2D conditioning offers potential for improved LW flux estimation by simultaneously merging more than one coarse-scale, satellite-derived radiation flux measurement, but only in the presence of large-scale cloud systems. In the absence of such cloud systems, it is best to use the 1D application because of a lack of a sufficient covariance structure for use by the 2D procedure.

The larger amount of improvement when applying the 2D procedure to LW fluxes relative to SW fluxes has less to do with the actual application of the 2D procedure itself and more to do with the quality of the prior SW estimate. That is, Forman and Margulis (2010b) showed the prior SW model performed quite well in the SGP and that even without application of a conditioning procedure the prior estimates contained a comparable (or even smaller) amount of error relative to the SRB-SW and Pinker-SW products. Therefore, when using 2D conditioning on SW fluxes there is effectively less room from improvement within the SGP. Application of the SW prior outside of the SGP in more variable terrain (e.g., mountainous regions) may yield a less accurate prior estimate; hence, there may be more room for improvement using the conditioning procedure. Granted, even though the conditioned SW estimates from the Bayesian merging framework in the SGP did not see as large of an improvement as for the LW fluxes, the conditioned estimates contain less error and uncertainty than the prior estimate or the measurements alone. This finding suggests an increased level of confidence within the posterior ensemble as a result of the conditioning procedure.

A concerted attempt was made to define an optimal as a function of CCF (as well as for the cloud states listed in Table 1) for use in 2D conditioning, but a clear-cut determination could not be found. Because of the discontinuous nature of clouds (i.e., clear patches next to cloudy patches), a comparison between the cloud length scale and the corresponding optimal was difficult to make and contained a significant amount of variability for a given CCF or individual cloud state. Preliminary results show promise for a rigorous method of defining a cloud length scale at a ~ daily time scale, but this investigation is well beyond the current scope of work and should be considered in future studies.

In general, a reduction in conditioned uncertainty could be achieved, but only when is less than or equal to the approximate characteristic lengths for the states that dominate the modulation of the radiative flux processes. The findings suggest that not only is the cloud scale important in determining the optimal , but so is the actual cloud structure itself. That is, the 2D procedure behaves differently in optically thin clouds versus optically thick clouds even though the shape and length scales of the two different cloud systems may be identical.

Regarding the question of reduced error as a function of influence length scale or as a function of spatial resolution, the results are somewhat inconclusive. The integrated gain analysis shown in Figs. 7 and 8 clearly show the covariance structure is larger in magnitude when merging the finescale measurements for a given value. For example, for in Fig. 7a versus Fig. 7b, the gain is more than twice as large in the NLDAS-LW product than in the SRB-LW product. However, with regard to the issue of reduced error, sometimes the conditioned ensemble is improved at the longer values (e.g., Fig. 3), whereas other times the ensemble is degraded at the longer values compared to its 1D counterpart. Therefore, the most meaningful conclusion that may be derived is that the finer-scale products used during Bayesian conditioning yields the best results because of their ability to capture more finescale detail within the radiative flux measurements, and that the successful application of 2D conditioning is largely dependent on the presence of a meaningful covariance structure between the prior fluxes and the predicted measurements if any beneficial information transfer from the measurements into the conditioned estimates is to take place, which was also shown to be the case during 1D condition (Forman and Margulis 2010b).

Despite the error reduction in LW and SW fluxes during cloudy-sky conditions, the application of 2D conditioning comes at increased computational expense. As the size of the measurement vector, , increases so does the computational burden associated with computing Czz and its subsequent inversion. For example, Bayesian conditioning of NLDAS-LW requires ~2 times the computational runtime as and increases significantly with an ever-increasing . When 2D conditioning utilizes ~10 or more measurements during the update, computational performance is no longer constrained by read–write operations but rather is constrained by CPU speed. Therefore, a trade-off exists between the amount of improvement in radiative flux estimation and computational expense when merging the finest scale (i.e., NLDAS-LW and Pinker-SW) measurement products.

One final point to make involves consideration of multiscale processes and flow-dependent behavior in clear-sky versus cloudy-sky fluxes. Over large areas of space, radiative fluxes are often found in both clear-sky and cloudy-sky regions. Differences between clear- and cloudy-sky conditions can introduce differences in the scales of the processes affecting downwelling radiative fluxes. Similarly, scale differences within a cloud system (e.g., regions of low optical thickness adjacent to regions of high optical thickness) can often introduce multiscale processes that modulate downwelling radiative flux. The presence of this multiscale behavior lends itself to the potential application of an ensemble multiscale filter (Pan and Wood 2009a; Zhou et al. 2008; Willsky 2002). Application of a multiscale procedure is beyond the scope of this current study, but should be considered in future applications.

Acknowledgments

Funding provided by the NASA Earth System Science Fellowship (Contract NNX07AN64H) and NASA Grants NNG04GO74G and NNG05GE58G. Helpful comments by three anonymous reviewers are gratefully acknowledged, especially reviewer 1 for a detailed and constructive critique.

REFERENCES

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
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    • Search Google Scholar
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    • Search Google Scholar
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Save
  • Brock, F., Crawford K. , Elliott R. , Cuperus G. , Stadler S. , Johnson H. , and Eilts M. , 1995: The Oklahoma mesonet: A technical overview. J. Atmos. Oceanic Technol., 12, 519.

    • Search Google Scholar
    • Export Citation
  • Cosgrove, B., and Coauthors, 2003: Real-time and retrospective forcing in the North American Land Data Assimilation System (NLDAS) project. J. Geophys. Res., 108, 8842, doi:10.1029/2002JD003118.

    • Search Google Scholar
    • Export Citation
  • De Lannoy, G., Reichle R. , Houser P. , Arsenault K. , Verhoest N. , and Pauwels V. , 2010: Satellite-scale snow water equivalent assimilation into a high-resolution land surface model. J. Hydrometeor., 11, 352369.

    • Search Google Scholar
    • Export Citation
  • Forman, B., and Margulis S. , 2009: High-resolution satellite-based cloud-coupled estimates of total downwelling surface radiation for hydrologic modelling applications. Hydrol. Earth Syst. Sci., 13, 969986.

    • Search Google Scholar
    • Export Citation
  • Forman, B., and Margulis S. , 2010a: Assimilation of multiresolution radiation products into a downwelling surface radiation model: 1. Prior ensemble implementation. J. Geophys. Res., 115, D22115, doi:10.1029/2010JD013920.

    • Search Google Scholar
    • Export Citation
  • Forman, B., and Margulis S. , 2010b: Assimilation of multiresolution radiation products into a downwelling surface radiation model: 2. Posterior ensemble implementation. J. Geophys. Res., 115, D22116, doi:10.1029/2010JD013950.

    • Search Google Scholar
    • Export Citation
  • Garbrecht, J., Van Liew M. , and Brown G. , 2004: Trends in precipitation, streamflow, and evapotranspiration in the Great Plains of the United States. J. Hydrol. Eng., 9, 360367.

    • Search Google Scholar
    • Export Citation
  • Gupta, S., 1989: A parameterization for longwave surface radiation from sun-synchronous satellite data. J. Climate, 2, 305320.

  • Gupta, S., Darnell W. , and Wilber A. , 1992: A parameterization for longwave surface radiation from satellite data: Recent improvements. J. Appl. Meteor., 31, 13611367.

    • Search Google Scholar
    • Export Citation
  • Janjić, T., Nerger L. , Albertella A. , Schröter J. , and Skachko S. , 2011: On domain localization in ensemble-based Kalman filter algorithms. Mon. Wea. Rev., 139, 20462060.

    • Search Google Scholar
    • Export Citation
  • Li, Z., Cribb M. , Chang F.-L. , Trishchenko A. , and Luo Y. , 2005: Natural variability and sampling errors in solar radiation measurements for model validation over the Atmospheric Radiation Measurement Southern Great Plains region. J. Geophys. Res., 110, D15S19, doi:10.1029/2004JD005028.

    • Search Google Scholar
    • Export Citation
  • Minnis, P., and Coauthors, 1995: Clouds and the Earth’s Radiant Energy System (CERES) algorithm theoretical basis document. NASA Langley Research Center Tech. Rep., NASA Reference Publ. 1376, Vol. III, 70 pp.

  • Minnis, P., and Coauthors, 2008: Cloud detection in nonpolar regions for CERES using TRMM VIRS and Terra and Aqua MODIS data. IEEE Trans. Geosci. Remote Sens., 46, 38573884.

    • Search Google Scholar
    • Export Citation
  • Mlynczak, P., Smith G. , Stackhouse P. , and Gupta S. , 2006: Diurnal cycles of the Surface Radiation Budget Data Set. Proc. 18th Conf. on Climate Variability and Change, Atlanta, GA, Amer. Meteor. Soc., P1.6. [Available online at http://ams.confex.com/ams/Annual2006/techprogram/paper_101147.htm.]

  • Pan, M., and Wood E. , 2009a: A multiscale ensemble filtering system for hydrologic data assimilation. Part I: Implementation and synthetic experiment. J. Hydrometeor., 10, 794806.

    • Search Google Scholar
    • Export Citation
  • Pan, M., and Wood E. , 2009b: A multiscale ensemble filtering system for hydrologic data assimilation. Part II: Application to land surface modeling with satellite rainfall forcing. J. Hydrometeor., 10, 14931506.

    • Search Google Scholar
    • Export Citation
  • Pinker, R., and Laszlo I. , 1992: Modeling surface solar irradiance for satellite applications on a global scale. J. Appl. Meteor., 31, 194211.

    • Search Google Scholar
    • Export Citation
  • Pinker, R., and Coauthors, 2003: Surface radiation budgets in support of the GEWEX Continental-Scale International Project (GCIP) and the GEWEX Americas Prediction Project (GAPP), including the North American Land Data Assimilation System (NLDAS) project. J. Geophys. Res., 108, 8844, doi:10.1029/2002JD003301.

    • Search Google Scholar
    • Export Citation
  • Reichle, R., and Koster R. , 2003: Assessing the impact of horizontal error correlations in background fields on soil moisture estimation. J. Hydrometeor., 4, 12291242.

    • Search Google Scholar
    • Export Citation
  • Willsky, A., 2002: Multiresolution Markov models for signal and image processing. Proc. IEEE, 90, 13961458.

  • Zhou, Y., McLaughlin D. , Entekhabi D. , and Ng G.-H. C. , 2008: An ensemble multiscale filter for large nonlinear data assimilation problems. Mon. Wea. Rev., 136, 678698.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Examples of (left) 1D and (middle),(right) 2D measurement setups for , , and merging measurements spaced on a 1° × 1° grid. The size of the influence length (black ellipse) dictates which measurements are included (black × symbols) and which are excluded (gray × symbols) for the local region of conditioning (black square).

  • Fig. 2.

    Map showing the SGP topographical domain. Markers represent ARM-SGP SIRS (•) and Oklahoma Mesonet (▿) observation stations.

  • Fig. 3.

    Computed statistics of (top) RMSD and (bottom) ρ for longwave and shortwave fluxes during clear- and cloudy-sky conditions for SON.

  • Fig. 4.

    Domain-averaged, temporal-averaged ensemble standard deviation of (top) LW and (bottom) SW radiative fluxes as a function of .

  • Fig. 5.

    LW RMSD examples of 1D (0.125°) and 2D (0.50°) conditioning of NLDAS-LW during November 2004. Cloud covered percent is shown as the thick line. Dates are labeled as month/day/year.

  • Fig. 6.

    Influence length achieving a minimum RMSD (left-hand axis) as a function of time in late August 2004. The corresponding cloud cover percentage (thin dashed line) and cloud characteristic length scale (thick dashed line) are shown along the right-hand axis. Dates are labeled as month/day/year.

  • Fig. 7.

    Daily-averaged, domain-averaged integrated gain for (a),(b) LW flux, , and (c),(d) SW flux, , as a function of influence length, , and time for (a),(c) SRB-LW and (b),(d) NLDAS-LW. Dates are labeled as month/day/year.

  • Fig. 8.

    As in Fig. 7 except for (a),(c) SRB-SW and (b),(d) Pinker-SW.

  • Fig. 9.

    Spatial smoothing effect of (c) 1D conditioning relative to (d) 2D conditioning utilizing the SRB-LW product. (a) The prior ensemble shown as the ensemble mean and (b) the measurement product used during conditioning are shown for reference.

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