## 1. Introduction

The effects of the unresolved clouds in numerical weather prediction (NWP) models are represented in the deep and shallow convection parameterizations. Traditionally, these parameterizations only accounted for the cumulus effects on the temperature and moisture profiles (Stensrud 2007). Thus, the unresolved clouds were transparent to radiative processes. An adverse effect that can be anticipated in these parameterizations is the overestimation of the shortwave radiation that reaches Earth’s surface [i.e., global horizontal irradiance (GHI)]. Indeed, a positive GHI bias in WRF (Skamarock et al. 2008), a state-of-the-science NWP model, has been attributed to an underestimation of cloudiness (Mathiesen and Kleissl 2011; Lara-Fanego et al. 2012; Ruiz-Arias et al. 2016). This hypothesis is reinforced by the relatively good replication of GHI observations by WRF under clear-sky conditions (Ruiz-Arias et al. 2013, 2014; Jiménez et al. 2016).

Recent model developments have incorporated the impact that the unresolved clouds exert on the atmospheric radiative transfer (Alapaty et al. 2012; Berg et al. 2013; Herwehe et al. 2014; Deng et al. 2014). The improved parameterizations produce a reduction in precipitation biases due to the reduced convection intensity (Alapaty et al. 2012; Herwehe et al. 2014) and improved GHI estimations in case studies (Berg et al. 2013). However, a systematic evaluation to quantify the improvements in GHI due to modeling the radiative effects of unresolved clouds has not yet been studied in detail.

The purpose of this work is to quantify the potential improvement in the short-range (0–6 h) predictability of GHI by accounting for the unresolved clouds. This is the time frame in which the performance of NWP models may outperform satellite-based methods (Diagne et al. 2013). We analyze three numerical experiments with a range of complexity in representing the unresolved clouds. This includes the mass-flux parameterization of Deng et al. (2014) recently added to WRF as a set of augmentations in support of solar energy applications (WRF-Solar; Jiménez et al. 2016).

Given the high spatiotemporal variability of clouds, the potential gain in predictability is analyzed using ensembles of WRF-Solar simulations covering the contiguous United States (CONUS) over a 1-yr period. This ensemble approach not only allows us to increase the number of realizations (reducing statistical uncertainty), but also to quantify the uncertainty in the predictions. The ensemble accounts for a portion of the model uncertainty using the stochastic kinetic energy backscatter scheme (SKEBS; Shutts 2005; Berner et al. 2009, 2011, 2015). In SKEBS, kinetic energy from unresolved scales is made available, or backscattered onto, the resolved scales via stochastic perturbations of the streamfunction and potential temperature at selected wavenumbers. The ensemble performance is evaluated against high-quality ground observations from geographically diverse locations in order to accurately quantify the gain in predictability.

## 2. Experimental setup

Three numerical experiments were performed with the WRF-Solar Model based on Advanced Research WRF, version 3.6 (Jiménez et al. 2016). The first experiment (NO-FEEDBACK) does not take into account the effects of the unresolved clouds on the atmospheric radiative transfer. The second experiment (DEEP-FEEDBACK) only activates the radiative feedback of deep convective cumulus clouds as an intermediate step in representing the unresolved clouds. This is accomplished via the deep cumulus feedback to the radiation as implemented in the Grell–Freitas scheme (Grell and Freitas 2014). Although scale aware, at the grid spacing used in this work the scheme represents predominantly the effects of deep convection. Finally, the third experiment (ALL-FEEDBACK) activates the radiative feedbacks of both the deep and shallow cumulus clouds using the mass-flux parameterization implemented in WRF-Solar (Deng et al. 2014). The mass-flux scheme represents the deep and shallow cumulus using a cloud entraining/detraining model to represent updrafts. Two predictive equations for cloud fraction and cloud liquid/ice water content for neutrally buoyant clouds provide the feedback to the radiation scheme.

WRF-Solar was configured to cover the CONUS with 9-km horizontal grid spacing. The model physics is configured to match the High Resolution Rapid Refresh model (HRRR; Benjamin et al. 2011) run operationally by the National Oceanic and Atmospheric Administration (NOAA). The shortwave and longwave radiative processes are parameterized with the Rapid Radiative Transfer Model for Global Models (RRTMG; Iacono et al. 2008). Microphysical processes are accounted for with the Thompson scheme (Thompson et al. 2008) and the planetary boundary layer parameterization is based on Nakanishi and Niino (2009). The land–atmosphere interactions are represented with the Rapid Update Cycle land surface model (Benjamin et al. 2004). Important differences are the activation of the Grell–Freitas and Deng cumulus parameterizations and the increase in the frequency of calls to the radiation scheme from 30 to 15 min. Additionally, the aerosols are represented with a monthly climatological dataset covering North America with 0.05° × 0.05° horizontal grid increments implemented in the WRF-Solar Model (Jiménez et al. 2016).

Each experiment consists of 52 WRF-Solar simulations sampling the calendar year 2014 to represent the different synoptic regimes over the CONUS. The first simulation is launched on 1 January 2014 followed by one simulation every week. This configuration reduces the temporal correlations between consecutive forecasts and, at the same time, alleviates the computational cost of obtaining simulations for each day of the year. Each one of the 52 simulations is initialized at 1500 UTC and spans 6 h (no spinup time) in order to capture the middle of the solar day across CONUS. The NOAA Rapid Refresh model (RAP; Weygandt et al. 2011) is used to initialize the simulations and to update the lateral boundary conditions every 3 h. The WRF-Solar output is recorded every 15 min.

The model uncertainty is simulated by running a stochastically generated ensemble with 10 members for each simulation. Thus, there are 520 WRF-Solar runs (52 cases × 10 ensemble members) for each one of the three numerical experiments.

The predictability of GHI is assessed against observations from the Surface Radiation Budget Network (SURFRAD; Augustine et al. 2000, 2005) and the Integrated Surface Irradiance Study (ISIS) network (Hicks et al. 1996). Each network has seven stations available over CONUS (Fig. 1). These are among the highest-quality GHI observations over CONUS. A quality control procedure to ensure consistency between the solar zenith angle and the recorded values was applied to further increase the quality of the data (Roesch et al. 2011). Additionally, the 1-min records from the SURFRAD network were averaged to 3 min to match the records provided by the ISIS network. Data were matched to WRF-Solar’s 15-min output. A 5% error is randomly introduced to the simulated GHI to account for the observational error (Augustine et al. 2000).

## 3. SKEBS configuration and the effects of the effective resolution

Motivated by the tendency of short-range ensembles to be underdispersive (e.g., Raftery et al. 2005), the SKEBS scheme was configured to maximize the spread (the standard deviation of the ensemble members around the ensemble mean) in GHI. First, the amplitude of the streamfunction and temperature perturbations was increased by one order of magnitude (i.e., 10^{−4} m^{2} s^{−3} and 10^{−5} m^{2} s^{−3}, respectively) with respect to the default SKEBS values. Additionally, the decorrelation time (i.e., the memory of the previous perturbation) was reduced from 3 h to 5 min. This change is consistent with the focus on short-range simulations (6 h). Finally, based on theoretical findings of the atmospheric response to the scale of the perturbations using turbulent kinetic energy spectra (Lorenz 1969; Rotunno and Snyder 2008; Durran and Gingrich 2014), only the wavenumber that led to the maximum GHI spread was perturbed.

The wavenumber to perturb was selected based on a sensitivity study of the GHI spread as a function of the perturbed wavenumber. We selectively perturbed different wavenumbers to analyze the standard deviation of the GHI among the 10 members of the ensemble. Figure 2 shows the averaged standard deviation of the GHI computed at each grid point (600 × 354) of the WRF-Solar domain (Fig. 1) as a function of lead time for a day with clouds over CONUS (25 August 2012). As a reference we also show results for perturbing all wavenumbers, which is the default SKEBS configuration (“All” curve). The mean standard deviation increased with increasing wavenumbers until wavenumber 32. Beyond that wavenumber, the spread showed a decay with respect to higher wavenumbers, particularly clear for wavenumber 128 (gray line). This finding suggests that the largest spread for a 6-h forecast of GHI does not result from perturbing the smallest scales of motion.

Average standard deviation of the GHI calculated with the 10 ensemble members for each grid point of the CONUS domain (600 × 354 grid points).

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

Average standard deviation of the GHI calculated with the 10 ensemble members for each grid point of the CONUS domain (600 × 354 grid points).

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

Average standard deviation of the GHI calculated with the 10 ensemble members for each grid point of the CONUS domain (600 × 354 grid points).

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

The decay in the ensemble spread due to perturbing the smallest scales of motion (higher wavenumbers) can be understood in terms of the “effective resolution” of the model. This is the region affected by implicit numerical diffusion or explicit dissipation techniques needed to remove energy at the smallest scales of motion, and it is around *x* and *y* direction are associated with the same distance. Results of this experiment showed that perturbations under the WRF’s effective resolution,

Based on these findings we configured the SKEBS ensemble to perturb only wavenumber 32 for our CONUS domain because it produced the largest spread for this test case.

## 4. Results

### a. Systematic errors

The benefit of accounting for the effects of unresolved clouds is first indicated by evaluating the GHI bias as a function of the lead time (Fig. 3). The GHI value at the grid point nearest to each of the 14 observational sites is used in the comparison. We use bootstrap resampling to compute 5%–95% confidence intervals and to test statistical significance of the differences. The NO-FEEDBACK experiment showed a positive bias for all lead times with a mean bias of 49 W m^{−2}. Incorporating the feedbacks from deep cumulus clouds, DEEP-FEEDBACK, reduced the bias at all lead times (mean bias of 39 W m^{−2}, ^{−2},

(top) GHI bias as a function of the lead time for the three experiments: NO-FEEDBACK (black), DEEP-FEEDBACK (red), and ALL-FEEDBACK (blue). The bias is calculated using the complete set of SURFRAD and ISIS observations (14 sites) and the entire set of simulations (52 ensembles of 10 members each). The error bars in ALL-FEEDBACK represent 5%–95% confidence intervals calculated with bootstrap resampling. Experiments NO-FEEDBACK and DEEP-FEEDBACK have error bars of similar magnitude. (bottom) Bootstrap confidence intervals of GHI differences: DEEP-FEEDBACK − NO-FEEDBACK (red) and ALL-FEEDBACK − NO-FEEDBACK (blue). Bars above zero indicate a statistically significant bias reduction.

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

(top) GHI bias as a function of the lead time for the three experiments: NO-FEEDBACK (black), DEEP-FEEDBACK (red), and ALL-FEEDBACK (blue). The bias is calculated using the complete set of SURFRAD and ISIS observations (14 sites) and the entire set of simulations (52 ensembles of 10 members each). The error bars in ALL-FEEDBACK represent 5%–95% confidence intervals calculated with bootstrap resampling. Experiments NO-FEEDBACK and DEEP-FEEDBACK have error bars of similar magnitude. (bottom) Bootstrap confidence intervals of GHI differences: DEEP-FEEDBACK − NO-FEEDBACK (red) and ALL-FEEDBACK − NO-FEEDBACK (blue). Bars above zero indicate a statistically significant bias reduction.

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

(top) GHI bias as a function of the lead time for the three experiments: NO-FEEDBACK (black), DEEP-FEEDBACK (red), and ALL-FEEDBACK (blue). The bias is calculated using the complete set of SURFRAD and ISIS observations (14 sites) and the entire set of simulations (52 ensembles of 10 members each). The error bars in ALL-FEEDBACK represent 5%–95% confidence intervals calculated with bootstrap resampling. Experiments NO-FEEDBACK and DEEP-FEEDBACK have error bars of similar magnitude. (bottom) Bootstrap confidence intervals of GHI differences: DEEP-FEEDBACK − NO-FEEDBACK (red) and ALL-FEEDBACK − NO-FEEDBACK (blue). Bars above zero indicate a statistically significant bias reduction.

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

Inspecting the bias at individual locations reinforces the conclusion that it is necessary to parameterize the cumulus feedbacks from both the shallow and deep cumulus clouds (Fig. 4a). Experiments NO-FEEDBACK and DEEP-FEEDBACK displayed a positive bias at all the sites. This systematic error was corrected in the ALL-FEEDBACK experiment that reduced the bias at all sites, and even displayed a small negative bias at one site.

(a) Bias at each SURFRAD and ISIS observational sites and (b) seasonal evolution of the bias calculated with a running mean of the bias from each WRF-Solar simulation. (top) Error bars represent 5%–95% confidence intervals calculated with bootstrap resampling. (bottom) The bootstrap confidence intervals of GHI differences: DEEP-FEEDBACK − NO-FEEDBACK (red) and ALL-FEEDBACK − NO-FEEDBACK (blue). Bars above zero indicate a statistically significant bias reduction.

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

(a) Bias at each SURFRAD and ISIS observational sites and (b) seasonal evolution of the bias calculated with a running mean of the bias from each WRF-Solar simulation. (top) Error bars represent 5%–95% confidence intervals calculated with bootstrap resampling. (bottom) The bootstrap confidence intervals of GHI differences: DEEP-FEEDBACK − NO-FEEDBACK (red) and ALL-FEEDBACK − NO-FEEDBACK (blue). Bars above zero indicate a statistically significant bias reduction.

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

(a) Bias at each SURFRAD and ISIS observational sites and (b) seasonal evolution of the bias calculated with a running mean of the bias from each WRF-Solar simulation. (top) Error bars represent 5%–95% confidence intervals calculated with bootstrap resampling. (bottom) The bootstrap confidence intervals of GHI differences: DEEP-FEEDBACK − NO-FEEDBACK (red) and ALL-FEEDBACK − NO-FEEDBACK (blue). Bars above zero indicate a statistically significant bias reduction.

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

The seasonal evolution of the bias further supports the benefits of parameterizing the radiative effects of the unresolved clouds (Fig. 4b). Summer and spring (winter and autumn) months showed the largest (smallest) positive bias in NO-FEEDBACK. Activating the effects of the unresolved deep cumulus in experiment DEEP-FEEDBACK revealed a positive feedback during spring and summer. The summer and spring behavior was further improved when the effects of the shallow cumulus were activated in experiment ALL-FEEDBACK. That experiment also had the largest GHI bias reduction during autumn. Small impacts associated with radiative effects of unresolved clouds were obvious for the winter.

### b. Error characterization and prediction

The root-mean-square error (RMSE) and the spread versus the lead time are plotted in a dispersion diagram in Fig. 5a. A perfect statistically consistent ensemble would have identical RMSE and ensemble spread (i.e., perfect error prediction) whereas an underdispersive/overdispersive ensemble would have lower/higher spread than RMSE (i.e., the error is underestimated/overestimated). The three experiments all had similar RMSE (solid lines) at the different lead times. Experiment ALL-FEEDBACK produced slightly lower RMSE values than the other experiments, but this reduction is within the 5%–95% confidence intervals. In addition, although experiments DEEP-FEEDBACK and ALL-FEEDBACK produced an increase in the ensemble spread (dashed lines), no experiment displayed matching RMSE and spread.

Dispersion diagrams using (a) the nearest grid point to SURFRAD and ISIS sites and (b) including the effects of the model’s effective resolution. (top) Error bars represent 5%–95% confidence intervals calculated with bootstrap resampling. (bottom) The confidence intervals of the ratio between the spread and RMSE calculated with bootstrap resampling. Perfect consistency occurs if the bars overlap with the

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

Dispersion diagrams using (a) the nearest grid point to SURFRAD and ISIS sites and (b) including the effects of the model’s effective resolution. (top) Error bars represent 5%–95% confidence intervals calculated with bootstrap resampling. (bottom) The confidence intervals of the ratio between the spread and RMSE calculated with bootstrap resampling. Perfect consistency occurs if the bars overlap with the

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

Dispersion diagrams using (a) the nearest grid point to SURFRAD and ISIS sites and (b) including the effects of the model’s effective resolution. (top) Error bars represent 5%–95% confidence intervals calculated with bootstrap resampling. (bottom) The confidence intervals of the ratio between the spread and RMSE calculated with bootstrap resampling. Perfect consistency occurs if the bars overlap with the

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

The rank histograms (Anderson 1996) together with the missing rate error [MRE; i.e., the percentage of measurements failing outside the highest/lowest values of the ensemble members; Eckel and Mass (2005)] provide further evidence of the underdispersive nature of the ensembles (Figs. 6a,b). A larger positive (negative) MRE reveals a more underdispersive (overdispersive) ensemble. Experiment NO-FEEDBACK exhibited a relatively flat histogram, but it is underdispersive (MRE = 16.90%) due to the tendency for all ensemble members to overestimate GHI (Fig. 6a). Activating the effects of the unresolved clouds in experiment ALL-FEEDBACK reduced the tendency to overestimate GHI (Fig. 6b). However, all members underestimated GHI more frequently exhibiting similar underdispersion (MRE = 17.20%). Experiment DEEP-FEEDBACK showed similar results (not shown).

Rank histograms for GHI at the nearest grid point to SURFRAD and ISIS sites from the (a) NO-FEEDBACK and (b) ALL-FEEDBACK experiments. The rank histograms as a result of including the model’s effective resolution for the (c) NO-FEEDBACK and (d) ALL-FEEDBACK experiments are also shown. The panels also show the MRE defined as the fraction of observations lower (higher) than the lowest (highest) ranked prediction above or below the expected missing rate of

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

Rank histograms for GHI at the nearest grid point to SURFRAD and ISIS sites from the (a) NO-FEEDBACK and (b) ALL-FEEDBACK experiments. The rank histograms as a result of including the model’s effective resolution for the (c) NO-FEEDBACK and (d) ALL-FEEDBACK experiments are also shown. The panels also show the MRE defined as the fraction of observations lower (higher) than the lowest (highest) ranked prediction above or below the expected missing rate of

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

Rank histograms for GHI at the nearest grid point to SURFRAD and ISIS sites from the (a) NO-FEEDBACK and (b) ALL-FEEDBACK experiments. The rank histograms as a result of including the model’s effective resolution for the (c) NO-FEEDBACK and (d) ALL-FEEDBACK experiments are also shown. The panels also show the MRE defined as the fraction of observations lower (higher) than the lowest (highest) ranked prediction above or below the expected missing rate of

Citation: Monthly Weather Review 144, 9; 10.1175/MWR-D-16-0104.1

### c. Impact of considering effective resolution

The underdispersion can be largely ameliorated by considering the effects of the effective resolution. The

The positive impacts on the ensemble statistical consistency as a result of accounting for the effective resolution are illustrated by the rank histograms shown in Figs. 6c and 6d. The frequency that all members of the ensemble overestimated GHI (Figs. 6a,c) was decreased from experiment NO-FEEDBACK, although it was still underdispersive (MRE = 7.26%, Fig. 6c). Hence, the effects of effective resolution on the resolved clouds only partially improved the underdispersion. The combination of both, accounting for the effective resolution and including the effects of the unresolved deep and shallow cumulus, produces an ensemble with better statistical consistency (MRE = 5.85%, 20% of improvement, Fig. 6d).

Including the impacts of the effective resolution also produces positive effects in the dispersion diagram (Fig. 5b). The spread and the RMSE are now in better agreement, particularly for the ALL-FEEDBACK experiment after lead times beyond 2 h. The match of the RMSE–spread lines indicates that accounting for both the effects of unresolved clouds and the effective resolution of the model provides a meaningful quantification of the simulated uncertainty. These results could be considered encouraging taking into account that it has been shown (e.g., Buizza et al. 2003) that is difficult is to reach a good statistical consistency at these short lead times without any postprocessing being applied.

## 5. Conclusions

The role of the unresolved clouds on short-range GHI predictability has been systematically analyzed using observations and ensemble simulations spanning a 1-yr period. Including the radiative effects of unresolved cumulus, both deep and shallow, is necessary to reduce a systematic positive bias in GHI. The bias improvements are consistent across all sites analyzed. Summer months reveal nearly unbiased GHI simulations whereas winter months show small impact to the radiative effects of unresolved clouds, which suggests that any errors are mostly due to other cloud processes that may not be properly represented in the model. Although slight improvements were observed in the RMSE, the error reduction is not statistically significant. Accounting for the unresolved clouds, therefore, produces a better cloudiness field from the standpoint of GHI prediction, but the timing of the modeled clouds does not necessarily match the observations. This finding stresses the difficulties of simulating small atmospheric scales of motion, which are inherently uncertain, and necessitates the ensemble approach of this assessment to quantify uncertainties in the estimations.

A meaningful quantification of the prediction uncertainties (consistency between the RMSE and the spread of the ensemble) can be demonstrated with proper representation of the model’s effective resolution. This includes both a selective perturbation of the atmospheric scale of motion, which should not exceed the model’s effective resolution, and representation of the variability of grid points within the effective resolution. Thus, one should consider the model’s effective resolution, as well as the inherent uncertainty in modeling the specific location of convective clouds, when assessing GHI predictions that include the radiative effects of unresolved clouds.

## Acknowledgments

This work was largely supported by NSF-based funds. It was also partially supported by DOE Project DE-EE0006016. Special thanks are offered to Chris A. Gueymard for providing us with the aerosol dataset used in the WRF-Solar simulations. The simulations used resources of the National Energy Research Scientific Computing Center as well as NCAR’s Computational and Information Systems Laboratory (ark:/85065/d7wd3xhc). Discussion with Josh Hacker and Jimy Dudhia were helpful during this work. Two anonymous reviewers are also acknowledged for their constructive comments.

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