## 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.

**y**(

**x**,

*t*), that is explicitly dependent on both space,

**x**, and time,

*t*, as

**u**(

**x**,

*t*) is the vector of model inputs. Basic details pertaining to

**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.

**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)

**u**≈

*p*

_{u}(

**u**). In principle, given the underlying pdf, we can sample it to generate ensemble fluxes such that

*j*represents a single replicate from an ensemble of size

*N*and

**u**

_{j}←

*p*

_{u}(

**u**). Since the underlying pdf is generally unknown, we postulate that it can be represented by

*γ*(

_{j}**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,

**C**

_{γ}(

**x**)] is a mean unity, lognormal distribution with covariance

**C**

_{γ}(

**x**) where

**L**is implicit in its formulation. Here

**L**approximates a typical characteristic length for each variable in

**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.

Model states and approximated characteristic length vector, **L**.

### 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.

Maximum number of measurements,

#### 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

**C**

_{yz}is the sample covariance between the prior fluxes and the predicted measurements,

**C**

_{zz}is the sample covariance of the predicted measurements, and

**C**

_{υυ}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,

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,

### 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, **C**_{υυ}, 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, **v**_{j}, is modeled as a random Gaussian process with mean zero noise of covariance **C**_{υυ} 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.

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

Minimum RMSD (across all values of

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

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 *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 ^{−2}) (results not shown), hence there is no substantial difference between the SW flux estimates conditioned on the SRB-LW product over the range

### 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, *σ*_{LW}, and the SW ensemble standard deviation, *σ*_{SW}. 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 ^{−2}) (Li et al. 2005). However, the conditioning procedure utilizing specific products at relatively large values of *σ*_{LW} for *σ*_{SW} for

### 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

Further attempts were made at determining an optimal influence length in the sense that the “optimal”

### 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, **C**_{zz} + **C**_{υυ}, is comparable to the sample covariance between the prior fluxes and the predicted measurements, **C**_{yz}. In addition, the covariance structure found in **C**_{yz} 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 **C**_{yz}, 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).

**Ω**=

**x**

_{r}. In other words,

*t*, as

**C**

_{yz},

**C**

_{zz}, and

**C**

_{υυ}as

*N*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.

_{r}Using the integrated gains computed via Eq. (6), the daily averaged, domain-averaged time series of integrated gains (as a function of

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, **C**_{yz}, 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 **C**_{yz} 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 **C**_{zz} 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 **C**_{zz} relative to **C**_{yz}. As a result, **C**_{yz} tends to increase more than (**C**_{zz} + **C**_{υυ})^{−1}, which has the effect of introducing seasonality into the computed gains. Item 3) occurs because **C**_{yz} and **C**_{zz} become increasingly similar when viewed across increasingly larger regions of space while at the same time **C**_{zz} can increase relative to **C**_{υυ}. Recall that **C**_{yz} is the sample covariance between the prior fluxes and the predicted measurements and that **C**_{zz} 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, **C**_{yz} and **C**_{zz} 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 **C**_{yz} and (**C**_{zz} + **C**_{υυ})^{−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 **C**_{yz} and **C**_{zz} are comparing different types of fluxes, and hence the magnitude of the two covariance matrices can be vastly different. Secondly, the asymptotic behavior of

One last item of note is the high-frequency variations seen in the **C**_{yz}, and hence a larger

### 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

## 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

In general, a reduction in conditioned uncertainty could be achieved, but only when

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

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, **C**_{zz} and its subsequent inversion. For example, Bayesian conditioning of NLDAS-LW requires ~2 times the computational runtime as

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.

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