1. Introduction
The estimation of the uncertainty of a model prediction is a critical issue that has become increasingly relevant in the last decades with the use of more complex and sophisticated numerical models. When a variable like temperature is measured with a thermometer at a given location and time, the uncertainty that goes with the measurement can be readily estimated. In the same way, the model predictions at a given grid box and time optimally should come with an uncertainty estimate. The uncertainty of simulated weather and climate variables has been the subject of a significant number of published works. It is inherent to the chaotic nature of the nonlinear equations solved in weather and climate models (Slingo and Palmer 2011) and has important implications for short-term weather forecasts (e.g., Fox and Wikle 2005), future climate change studies (e.g., Salazar et al. 2016; Overland et al. 2016), ocean (e.g., Bitner-Gregersen et al. 2014), and chemistry (e.g., Malmberg et al. 2008) research and intermodel comparison (e.g., Wang et al. 2016). As discussed in Leutbecher and Palmer (2008), there are two main sources of errors in numerical weather prediction models that result in uncertainties: initial condition errors and model errors. The former arises because the data used to initialize the models are not perfect, with some of these errors amplifying during the simulation and resulting in significant forecast errors. The latter is due to deficiencies in the model’s representation of the dynamics and physics of the atmosphere by numerical algorithms. The traditional way to estimate the uncertainty of a model variable is to perform an ensemble of experiments in which one (or more) aspect(s) of the model configuration is varied: for example, different physics and/or dynamics options are selected (e.g., Evans et al. 2012) and the initial and/or boundary conditions are perturbed (e.g., Torn and Hakim 2009). Another commonly followed approach is a multimodel ensemble (e.g., Salazar et al. 2016). The uncertainty sampled with an ensemble of simulations will, of course, depend on how the ensemble is set up. For example, in a multiphysics ensemble, the uncertainty estimate will reflect the sensitivity of the model’s response to the different treatments of a given process (e.g., boundary layer dynamics, cumulus convection, or cloud microphysics processes). Another possibility is to perturb the initial and/or boundary conditions with some measurement of uncertainty associated with them. In this case, it will be an indication of how the uncertainty associated with the forcing data propagates to the model response. Philosophically, such ensemble methods are fundamentally different from measurement error and natural variability of variables in nature. Ensemble uncertainty estimation methods quantify the differences in various researchers’ implementation and interpretation of nature. We point out that such model ensemble error summaries are conceptually different than natural variability plus measurement uncertainty. These methods are not only time consuming but very computationally demanding; therefore, slowing down the progression into higher spatial resolutions is needed for a more realistic simulation of the atmospheric conditions (e.g., Prein et al. 2015). Providing the uncertainty for a time-dependent spatial field like the 2-m temperature is crucial for a majority of weather and climate applications, not least for climate change projections, as it will help guide policy-makers and stakeholders responsible for developing mitigation strategies. There is a need to develop efficient ways of estimating the uncertainty of a model prediction that uses the least possible amount of computational resources. In this paper, such an approach is proposed that makes use of a single model run and a statistical model, the Bayesian hierarchical model (BHM).
The vast majority of the statistical models used in numerical weather research are based on multiple linear regression, such as model output statistics (MOS) techniques Vislocky and Fritsch 1995, 1997) and the Statistical Hurricane Intensity Prediction Scheme (SHIPS; DeMaria and Kaplan 1994; DeMaria et al. 2005). Statistical models based on artificial neural networks (e.g., Kuligowski and Barros 1998; Sideratos and Hatziargyriou 2007), linear stochastic processes (e.g., Toth et al. 2000), and Bayesian model averaging (e.g., Raftery et al. 2005) have also been used. The BHM proposed here is a hierarchical model inferred using the Bayesian approach (i.e., the different layers in the BHM are linked through Bayes’s rule), which allows both observed data and model parameters to be random variables, resulting in a more realistic and accurate estimation of the uncertainty compared to non-Bayesian approaches (e.g., Haining et al. 2007). In addition, flexible prior information can be incorporated into the Bayesian model: as pointed out by Dormann (2007), prior information (i.e., information that accounts for subjective beliefs or expectations regarding the behavior of a given parameter before it is estimated from the data itself) can be very helpful in modeling spatial autocorrelative effects compared to ordinary nonspatial linear effects. These features of the Bayesian model are behind our choice of this approach for this work.
One obstacle, however, hinders the use of a Bayesian model: the approximations step. The normalizing constant used in Eq. (3), relevant for model-selection problems but not important for parameter estimation, is typically intractable with the analytical derivation replaced by numerical approximations that are often very costly. The Markov chain Monte Carlo (MCMC; Steinsland and Jensen 2010) method is a key step to make the inference. The basic idea is to draw a sequence of dependent samples from a target density of interest. As the number of draws tends to infinity, the resulting output behaves as if it were a sample from the density of interest. However, as the data become larger and their inherent structure more complex, this procedure becomes very computationally expensive. What is more, for hierarchical models, such as the BHM used here, the convergence of the MCMC is very slow and unpredictable (De Smedt et al. 2015). The estimation of hyperparameters is also challenging for the MCMC. An alternative to the MCMC method for which faster approximations of posterior distributions even for more complex models are possible is the integrated nested Laplace approximation (INLA; Rue et al. 2009) method. The INLA method uses a combination of analytical approximations and numerical integrations to speed up the estimation of the posterior distribution instead of generating a long sequence of dependent samples such as in the MCMC. Methods such as INLA make the Bayesian model increasingly more popular in several applications such as for spatial or spatiotemporal data in imaging analysis and geographical analysis (Cressie and Wikle 2011; Blangiardo et al. 2013). Even with these new approximation methods, the implementation of a Bayesian model still requires large amounts of computational resources [in particular, random-access memory (RAM)] compared to non-Bayesian approaches. In particular, it takes about 50 min and roughly 12 gigabyte (GB) RAM on a computer with an Intel Xeon central processing unit (CPU) E3-1270, version 3, at 3.5 GHz to implement a single BHM model. Hence, it has to be applied on a high-performance server that can handle big data with a complex dependent structure.
The Weather Research and Forecasting (WRF; Skamarock et al. 2008) Model used in this work is a fully compressible, nonhydrostatic model that uses a terrain-following hydrostatic pressure-based coordinate in the vertical and the Arakawa C grid staggering for horizontal discretization. It has been used in a wide variety of applications including polar meteorology studies (e.g., Hines and Bromwich 2008) and boundary layer research (e.g., Banks et al. 2016). In this work, the uncertainty of the daily mean 2-m temperature is estimated from one WRF run using the BHM. To assess how realistic the WRF-BHM uncertainty estimate is, it will be compared to that obtained from an ensemble of WRF simulations in which different planetary boundary layer (PBL) schemes are employed. The uncertainty that will be targeted in this work is that associated with different treatments of the boundary layer dynamics and representation of the near-surface flow, which play a crucial role in the simulation of the daily mean near-surface air temperature. It is important to stress that WRF-BHM is not capable of generating the full set of products that can be obtained from an ensemble of experiments. An example is the most likely event, such as for precipitation type. Knowing which precipitation type (e.g., rain, snow, ice pellets, or freezing rain) is more likely can be achieved by checking which is predicted by the majority of the ensemble members with precipitation but cannot be obtained from the BHM. As a result, the BHM cannot be considered as a true alternative to ensemble forecasting. Having said that, ensemble products such as the probability of exceedance, box-and-whisker diagrams, background-error covariance for ensemble Kalman filter, and hybrid variational data and uncertainty estimation, the focus of this work, can be obtained in a computationally cheaper and faster way with the BHM. In this paper, the uncertainty estimation is considered for the daily-mean 2-m temperature, but the methodology developed can be easily extended to other fields.
2. Experimental setup
In this study, WRF, version 3.7.1, is forced with the National Centers for Environmental Prediction (NCEP) Climate Forecast System Reanalysis (CFSR; Saha et al. 2010) 6-hourly data (horizontal resolution of 0.5° × 0.5°) and is run for one month in the boreal summer (July 2016) and winter (January 2017) seasons. To minimize the accumulation of integration errors, and following Lo et al. (2008), each month’s run is broken into three overlapping 11–12-day periods (e.g., for July 2016, the model is run from 0000 UTC 30 June to 0000 UTC 11 July, from 0000 UTC 10 July to 0000 UTC 21 July, and from 0000 UTC 20 July to 0000 UTC 1 August) with the first day regarded as model spinup.
The physics parameterizations used include the Goddard 6-class microphysics scheme (Tao et al. 1989), the Rapid Radiative Transfer Model for Global Circulation Models (RRTMG) models for both shortwave and longwave radiation (Iacono et al. 2008), the Mellor–Yamada Nakanishi and Niino (MYNN), level 2.5 (Nakanishi and Niino 2006); with Monin–Obukhov surface layer parameterization (Monin and Obukhov 1954); the 4-layer Noah land surface model (Chen and Dudhia 2001); and for cumulus convection the Betts–Miller–Janjić scheme (Janjić 1994) coupled with the precipitating convective cloud scheme described in Koh and Fonseca (2016). A climatological aerosol distribution based on Tegen et al. (1997) is used in the radiation scheme. An interactive prognostic scheme for the sea surface skin temperature (SSKT) based on Zeng and Beljaars (2005) is also added to the model. The lower boundary condition to the SSKT scheme is the 6-hourly SSTs from CFSR, linearly interpolated in time in order to have a continuously varying forcing on the skin layer. The fractional sea ice coverage and sea ice depth are also read from CFSR every 6 h with the sea ice albedo a function of air temperature, skin temperature, and snow (Mills 2011). Gravitational settling of cloud drops in the atmosphere is parameterized as in Duynkerke (1991) and Nakanishi (2000), while cloud water (fog) deposition onto the surface because of turbulent exchange and gravitational settling is treated using the simple fog deposition estimation (FogDES) scheme (Katata et al. 2011).
In addition to the control experiment (model configuration stated above, with the MYNN, level 2.5, PBL scheme), nine additional simulations are performed using different PBL schemes. As the focus of this work is on the 2-m temperature, it makes sense to perturb the PBL scheme as the near-surface fields are highly sensitive to how the boundary layer dynamics are represented. This is particularly true for Scandinavia, which has a very large seasonal contrast: snow-covered terrain, sea ice in the Gulf of Bothnia, and a very short daylight period in winter, contrasting with open water everywhere, snow-free land, and long daylight hours in summer. These sensitivity runs are conducted over a 10-day period at the beginning of January 2017 and July 2016 with a 1-day spinup before. The winter period selected comprises both milder days, when baroclinic systems from the Atlantic prevail, and colder days, when areas of high pressure are in control and rather cold temperatures, dropping below −30°C in some places, occur. The summer period started with a southerly flow and rather warm temperatures, but the weather conditions became more unstable in the middle and latter parts of the period. Both are representative of the different atmospheric conditions experienced in Scandinavia in the cold and warm seasons. The PBL schemes considered are the Yonsei University (YSU; Hong et al. 2006); asymmetric convective model, version 2 (ACM2; Pleim 2007); quasi-normal scale elimination (QNSE; Sukoriansky et al. 2005); Mellor–Yamada–Janjić (MYJ; Janjić 1994); Bougeault–Lacarrère (BouLac; Bougeault and Lacarrère 1989); the University of Washington (UW) scheme (Bretherton and Park 2009), Grenier–Bretherton–McCaa (GBM; Grenier and Bretherton 2001); Shing–Hong (SH) scheme (Shin and Hong 2015); and MYNN, level 3 (MYNN3; Nakanishi and Niino 2006). Further information about these schemes can be found in Banks et al. (2016) and Cohen et al. (2015).
WRF is run in a nested configuration with the domains shown in Fig. 1a. The outermost grid, at a horizontal resolution of 15 km, comprises the entire Scandinavian Peninsula. The 3-km grid includes the full Botnia–Atlantica region that consists of the Nordic counties of Nordland, Norway; Västerbotten, Sweden; and Pohjanmaa, Finland. The cumulus scheme is switched off in the second domain, while in the outermost grid analysis, nudging toward CFSR is employed. The nudging regions are given in Fig. 1b: the water vapor mixing ratio is nudged in the mid- and upper troposphere, whereas the horizontal wind components and potential temperature perturbation are nudged in the upper troposphere and lower stratosphere, with the nudging time scale set to 1 h for all variables. To prevent the reflection of waves near the model top, a sponge layer is included (Skamarock et al. 2008). Figure 1b also shows the model vertical levels: 60 levels, concentrated in the PBL with about 20 levels in the lowest 200 m. The model output of domain 1 is discarded, and that of domain 2 is stored every 1 h and used for analysis.
(a) Spatial extent of the 15- (blue) and 3-km (red) WRF grids and (b) model (eta) levels with the nudging regions and sponge layer highlighted in red and green rectangles, respectively. The vertical coordinate in (b) is in logarithmic scale.
Citation: Journal of Applied Meteorology and Climatology 58, 3; 10.1175/JAMC-D-18-0018.1
3. Statistical model
The INLA (Rue et al. 2009) package in the software R is used to implement the BHM as described above. The R code is quite straightforward; it codes the BHM model as presented above directly into the R language, and is freely available online.1
4. Methods
In this section, the setup of the BHM and the diagnostics used to evaluate the performance of the WRF-BHM are presented.
a. WRF data
The focus is on the daily mean 2-m temperature from the 3-km WRF grid, Fig. 1a, which contains 301 × 381 grid points. The BHM is set up in a 20-day period in the winter (1–20 January 2017) and summer (1–20 July 2016) seasons. For each month, the BHM was first fit to the WRF data using different length periods multiple of 5 days and up to 30 days. It was found that there was very little sensitivity in the parameters for a period longer than 20 days. Hence, and for computational reasons, a 20-day period for each season is selected to develop the BHM. The variance 
Method I: The 10 000 grid points (N = 10 000) are randomly selected from the available 301 × 381 in the model domain. The assumption is that the variance
Method II: The 10 000 grid points (N = 10 000) around coastlines and inland sites randomly are selected. A grid point is considered to be a coastline point if one of its nearest spatial neighbors is totally (or partially) covered by an ocean or sea. It is considered to be an inland point if its nearest coastal point is beyond its third-order neighbors. Here, it is assumed that spatial heterogeneity in
Method III: The 10 000 grid points (N = 10 000) at two different elevations (namely, ≤673.2 and >673.2 m) are randomly selected. A threshold of 673.2 m is chosen as to have at least 10 000 grid points in each category. The assumption is that spatial heterogeneity in
Method IV: The domain is divided into nine regions (shown in Fig. A1) each containing a minimum of 10 000 grid points (N = 10 000). The purpose of this approach is to have the largest possible spatial heterogeneity in
b. BHM parameter setting
1) Covariates
While the 2-m temperature is a function of many covariates, for simplicity only one, the elevation, is considered (i.e., 
2) Prior distributions for parameters
How to build the prior distribution (informative or noninformative approach) of a given parameter is a challenging task in Bayesian analysis and will depend on whether there is preexisting information about the parameter being modeled. Both approaches have their strengths and weaknesses (e.g., Berger 2006). When one has limited information about the parameter, weakly informative prior is generally a good choice (Simpson et al. 2015). In this study, the default specifications in INLA for the distributions of the parameters are used, which correspond to noninformative prior for 
The noninformative prior assigned to 
c. BHM evaluation
5. Results and discussion
a. Winter experiment (January 2017)
Table 1 shows the summary statistics of the posterior distributions of the BHM parameters for this season. Only parameters statistically significantly different from zero at the 95% confidence interval 
Summary statistics (including mean μ, standard deviation σ, and quantiles Q) of the posterior distributions of the BMH parameters for the winter experiment. For method IV, only the minimum and maximum values are given, full data is provided in Table A1.
Regarding method I, 
Comparable values are obtained for methods II–IV. For method II, a lower 
Having fully set up the BHM, the results for the four diagnostics will now be presented. Figure 2a shows the diagnostic D1 for the winter period. Over water bodies and southwestern Finland, the RMSE values are rather small, generally less than 0.5 K. Conversely, and mostly over the northeastern part of the domain in northern Sweden and Finland, they are generally in excess of 0.75 K. Here, rather cold surface temperatures associated with stably stratified boundary layers are common in winter (e.g., Pepin et al. 2009). The WRF Model is known to underperform in such environments (e.g., García-Díez et al. 2013) as some of the dominant physical processes are not well modeled (e.g., Steeneveld 2014). By using different PBL schemes that represent the boundary layer dynamics in different ways (e.g., Cohen et al. 2015; Banks et al. 2016), a wide range of values for the 2-m temperature is obtained, resulting in a larger discrepancy between the two approaches. An inspection of the temperature data revealed that for a given day in the 10-member ensemble experiment, a temperature range in excess of 10 K is obtained in the Arctic Scandinavia, whereas farther south the range is generally smaller, typically less than 2 K (not shown). Larger RMSE values are also seen in valleys and low-elevation regions, where cold-air pooling is a common occurrence in winter, and over the high terrain, where the WRF surface-temperature forecasts are known to be sensitive to the choice of the PBL scheme (e.g., Zhang et al. 2013). Elsewhere, the RMSE values are mostly in the range 0.5–0.75 K. The results in Fig. 2a show that the mean 2-m temperature estimates using the two approaches are generally similar (RMSE differences mostly within 0.75 K). Figure 3 shows the results for 
Diagnostic D1, an indication of how similar the WRF-BHM’s mean 2-m temperature is to that estimated from an ensemble of WRF simulations with different PBL schemes, for the (a) winter period (1–10 Jan 2017) and (b) summer period (1–10 Jun 2016). The optimal value for diagnostic D1 is zero.
Citation: Journal of Applied Meteorology and Climatology 58, 3; 10.1175/JAMC-D-18-0018.1
Diagnostic D2, an indication of how similar the WRF-BMH’s σe is to the uncertainty estimated from an ensemble of WRF simulations with different PBL schemes, obtained from (a) method I, (b) method II, (c) method III, and (d) method IV for the period 1–10 Jan 2017. The optimal value for diagnostic D2 is zero.
Citation: Journal of Applied Meteorology and Climatology 58, 3; 10.1175/JAMC-D-18-0018.1
To illustrate the closeness between the WRF-BHM and ensemble temperature distributions, Figs. 4a and 4c show diagnostics D3, the K-S statistic, and D4, the K-L divergence, respectively. Both metrics are nonnegative with the optimal value being zero. The K-S values are mostly within 0.8, being less than 0.5 over the water bodies and in parts of southern Sweden and southern and western Finland. Here, the WRF-BHM and ensemble 2-m temperature CDFs are expected to be more similar given the smaller RMSE values shown in Figs. 2a and 3. Diagnostic D4 gives a similar picture with the two PDFs being more divergent in the northeastern part of the domain where their means and standard deviations are farther apart. Quantitatively speaking, the scores for the K-S and K-L diagnostics shown in Fig. 4 are not very good. As shown in Fig. 5, in which an explicit comparison of the two temperature distributions at several locations in the model domain is performed, and as discussed in more detail below, the lower K-L (and K-S) scores arise from a more significant deviation of the ensemble temperature distribution from a Gaussian shape.
Diagnostic D3, K-S statistic, for the (a) winter period (1–10 Jan 2017) and (b) summer period (1–10 Jun 2016). (c),(d) As in (a) and (b), but for the diagnostic D4, K-L divergence. Diagnostics D3 and D4 are nonnegative with the optimal value being zero.
Citation: Journal of Applied Meteorology and Climatology 58, 3; 10.1175/JAMC-D-18-0018.1
The 2-m temperature distribution at four grid points for which the diagnostics D1, D2, and D4 indicate good and not-so-good WRF-BHM performance for the first day of the winter (1 Jan 2017) and summer (1 Jul 2016) periods. The blue curve is the WRF-BHM temperature distribution, and the red curve is that obtained from the multiphysics ensemble experiment. For a given grid point, the ensemble temperature distribution is calculated using the kernel density estimation method with the 10 ensemble members. See the title of each plot for information regarding the location of the grid points and the scores.
Citation: Journal of Applied Meteorology and Climatology 58, 3; 10.1175/JAMC-D-18-0018.1
As to allow for a more direct evaluation in the left panels of Fig. 5 and at a given day and for two grid points over land and water for which higher and lower D1, D2, and D4 (similar results for D3) scores are obtained, the WRF-BHM and ensemble temperature distributions for the winter season are plotted. It is important to note that similar results are obtained at other grid points (not shown), and hence these curves are representative of those obtained elsewhere. Regarding the shape of the distribution, and despite the presence of a secondary maximum, the ensemble temperature distribution generally resembles a Gaussian distribution, supporting the assumption of a Gaussian shape made in section 3 for the BHM. What is more, and as expected, the agreement between the curves is better if the D1, D2, and D4 scores are higher, which also confirms that the use of these verification diagnostics for this work is justified. For all the winter results, the mean value of the WRF-BHM temperature distribution is within 1 K of that obtained with the ensemble of simulations. Because of the presence of the referred secondary maximum, possibly arising from the small number (10) of ensemble members considered, there are some differences in the spread of the two distributions. In any case, and when taken as a whole, the results given in Fig. 5 highlight that the simpler and computationally cheaper WRF-BHM approach can be used to estimate the ensemble-generated temperature distribution at least for this region and season.
b. Summer experiment (July 2016)
Table 2 is as Table 1, but for the summer experiment. For this season, 
Summary statistics (including mean μ, standard deviation σ, and quantiles Q) of the posterior distributions of the BMH parameters for the summer experiment. For method IV, only the minimum and maximum values are given, full data is provided in Table A2.
The correlation 
The results for diagnostics D1 and D2 for the summer period are given in Figs. 2b and 6, respectively. The RMSE for the mean temperature is smaller than that for the winter season with values generally less than 0.5 K everywhere in the domain. RMSE values in excess of 0.5 K are seen mostly in coastal regions. As stated before, land–sea breeze circulations and associated temperature variations are a common occurrence in the summer along coastlines, and their representation is sensitive to the PBL option chosen (e.g., Steele et al. 2013, 2015). Hence, and at these sites, a larger difference between the predicted 2-m mean temperatures using the two methods is likely to occur. An inspection of the temperatures for the ensemble experiment reveals that in coastal regions the predicted 2-m temperatures can differ by as much as 3 K, whereas at inland sites they are mostly within 1 K (not shown). Larger RMSE values are also seen over the high terrain where the model temperature forecasts are known to be sensitive to the PBL scheme used (e.g., Zhang et al. 2013). As is the case for the diagnostic D1, the RMSE of the standard deviation estimates, given in Fig. 6, is also generally smaller than that obtained in the winter season. As for the summer period, 
As in Fig. 3, but for the summer period (1–10 Jul 2016).
Citation: Journal of Applied Meteorology and Climatology 58, 3; 10.1175/JAMC-D-18-0018.1
Figures 4b and 4d show the K-S and K-L divergence diagnostics for the summer season. The magnitude and spatial pattern of the two scores are similar to those obtained in the winter season with lower values mostly over the Gulf of Bothnia and Atlantic Ocean and higher values in some inland sites. As opposed to the winter period, the lowest scores are not seen in the northeastern (Arctic) part of the domain but in some coastal and high-terrain regions. Here, as shown in Figs. 2b and 6, the mean and standard deviation of the 2-m temperature distributions obtained using the WRF-BHM and multiphysics ensemble differ the most, so the correspondent CDFs and PDFs are expected to be farther apart. These lower scores can be attributed to a more significant deviation of the ensemble temperature distribution from a Gaussian shape as seen in Fig. 5.
The right panels in Fig. 5 show the two temperature distributions at four grid points for the summer season (similar results are obtained at other locations in the domain, not shown). As for the winter case, the WRF-BHM distribution generally resembles that obtained with an ensemble of simulations with the respective means being within 1 K of each other. The presence of a secondary maximum in the ensemble temperature distribution may be an artifact of the limited number of samples used. The results presented here highlight the success of the proposed WRF-BHM formulation in simulating the daily mean 2-m temperature distribution as given by the multiphysics ensemble.
6. Summary and conclusions
When running a numerical model, the gridbox-averaged value is a standard output, but the uncertainty that goes with that prediction is generally not provided. A common way to estimate the uncertainty associated with weather and climate projections is to perform an ensemble of model simulations. This can be achieved in different ways, for example, by changing the model physics and/or dynamics configuration, perturbing the initial and/or boundary conditions, or even by considering different models. In this paper, we propose a novel approach that only requires one simulation: fitting a BMH to the model data and from the estimated distribution extracting the uncertainty of a given variable (here, the 2-m temperature, a crucial field for climate projections, is considered). It is important to note that WRF-BHM is not a true alternative to ensemble forecasting in the sense that it cannot generate the same set of metrics. For example, the most likely event (e.g., for precipitation type) cannot be obtained with the BHM. However, and for commonly used ensemble products such as the uncertainty estimation, probability of exceedance, and background-error covariance for ensemble Kalman filter and hybrid variational data assimilation schemes, it provides an easy to apply, computationally cheap, and faster alternative to ensemble forecasting. We would also like to stress that, although not discussed here, the BHM can also be run in prediction mode. In particular, it can analyze data possessing spatiotemporal dependence and make use of the estimated dependence to predict the values of a given field at different locations and time periods. However, it can be argued that using the BHM for an understanding of the spatiotemporal dependency inherent to a dataset is more relevant than using it for prediction purposes.
WRF is run at 3-km spatial resolution over the Botnia–Atlantica region for one month in the winter (January 2017) and summer (July 2016) seasons. In addition, an ensemble of simulations is generated with 10 different PBL schemes for the first 10 days of January 2017 and July 2016. The uncertainty sampled in this work is that associated with the way the boundary layer dynamics and near-surface flow are represented in the model and their impact on the daily mean 2-m temperature predictions. A fundamental assumption of comparing the BHM to the multiphysics ensemble is that the error is primarily associated with PBL physics uncertainties and that the multiphysics ensemble provides a meaningful estimate of that.
The BHM uses nearby information in both space and time to estimate statistical properties of our WRF variable of interest at each grid location. The 2-m temperature, at a given time and spatial location, is assumed to follow a normal distribution with mean 
The BHM parameters obtained with WRF data are consistent with the expected values. Given the differences in some of them between the two seasons, and at least for this region, it is not advisable to consider just one BHM for the full year. Studies that wish to extend this work for a complete annual cycle should fit a BHM to at least each season separately, as the regional and local-scale dynamics are very different throughout the year. For both summer and winter seasons, the mean and uncertainty estimated using the WRF-BHM are found to be generally comparable to those obtained from an ensemble of simulations with different PBL schemes. The exceptions are (i) Arctic Scandinavia and low-level regions in winter, where cold and stably stratified boundary layers are ubiquitous; (ii) coastal regions in the summer, where land–sea breeze circulations are more significant; and (iii) over the high terrain in both seasons. Here, the WRF-predicted 2-m temperature has been reported to be particularly sensitive to the choice of the PBL scheme (e.g., García-Díez et al. 2013; Steele et al. 2015; Zhang et al. 2013) resulting in larger differences in the simulated temperature distributions using the two approaches. A direct inspection of the WRF-BHM and ensemble temperature distributions revealed the presence of a secondary maximum in the latter, with this stronger deviation from a Gaussian shape probably explaining the lower scores. This feature possibly arises from the limited number of ensemble members considered. Despite differences in the spread, the means of the two distributions are very similar, within 1 K of each other. Hence, the WRF-BHM approach can be used to estimate the uncertainty and, more generally, the distribution of the 2-m temperature as given by the multiphysics ensemble. Since these results are comfortably interpreted based on our general understanding of the weather variables for the Scandinavian Peninsula, we expect the application of the BHM for estimation of modeled weather variable uncertainty to perform similarly in other temporal and spatial domains. What is more, if the ensemble WRF distributions displayed in Fig. 5 were associated with actual surface observations, and we used an appropriate inference method to assess the equality of the distribution pairs, the approach followed in this work then allows a statistical assessment on the equality of the model results and observations. Such inferential statistical analyses have important applications for geospatial research, including climate change.
In conclusion, the main contributions of this work are as follows: (i) we apply the BHM method to estimate the uncertainty of a WRF output variable (2-m temperature) using space- and time-dependent data; (ii) we prove the consistency of the method by demonstrating that the uncertainty estimate correctly summarizes the aggregated space/time data; (iii) we show that the temperature distribution obtained with WRF-BHM is similar to that computed from an ensemble of model simulations with perturbed physics. The methodology developed here is fully general and can easily be applied to any other variable and numerical model. Possible future extensions of this work include (i) employing more covariates in the process model; (ii) developing a data-driven 
Acknowledgments
We acknowledge Botnia–Atlantica, an EU program financing cross-border cooperation projects in Sweden, Finland, and Norway, for their support of this work through the WindCoE project. We thank the High Performance Computing Center North (HPC2N) for providing the computer resources needed to perform the numerical experiments presented in this paper. We thank two anonymous reviewers for their detailed and insightful comments and suggestions that helped to improve the quality of the paper.
APPENDIX
BHM Parameters for Method IV
In this appendix, the values of the parameters for each of the nine regions (Fig. A1) of method IV are given in Tables A1 and A2 for the winter and summer experiments, respectively.
Nine spatial regions used in method IV.
Citation: Journal of Applied Meteorology and Climatology 58, 3; 10.1175/JAMC-D-18-0018.1
Summary statistics (including mean μ, standard deviation σ, and quantiles Q) of the posterior distributions of the BMH parameters for each of the nine regions considered in method IV for the winter experiment.
Summary statistics (including mean μ, standard deviation σ, and quantiles Q) of the posterior distributions of the BMH parameters for each of the nine regions considered in method IV for the summer experiment.
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