1 Introduction
Wind energy plays an important role in many countries’ energy portfolio as a significant renewable source with no major negative environmental impacts (Wiser et al., 2011; Obama, 2017). Earth System Models (ESMs) provide physically consistent projections of wind energy potential, as well as spatially resolved maps in regions with poor observational coverage. However, these models are (more or less accurate) approximations of the actual state of the Earth’s system and the energy assessment is therefore sensitive to changes in the model input. To address this, geoscientists generate a collection (ensemble) of ESMs to assess the sensitivity of the output (including wind) with respect to physical parameters and trajectories of greenhouse gases concentration (forcing scenarios). Recently, the role of the uncertainty due to ESMs’ initial conditions (internal variability) has been identified as a prominent factor for multidecadal projections, hence the importance of quantifying its uncertainty.
The Large ENSemble (LENS) is a collection of 40 runs at the National Center for Atmospheric Research (NCAR) specifically designed to isolate the role of internal variability in the future climate (Kay et al., 2015)
. The LENS required millions of CPU hours on a specialized supercomputer, and very few institutions have the resources and time for such an investigation. Is such an enormous task always necessary for assessing internal variability? While it is absolutely necessary for quantities at the tail of the climate (e.g., temperature extremes), it is not always necessary for simpler indicators such as climate mean and variance. As part of a series of investigations promoted by KAUST on the topic of assessing wind energy in Saudi Arabia,
Jeong et al. (2017) introduced the notion of a stochastic generator (SG), a statistical model that is trained on a small subset of LENS runs. The SG acts as a stochastic approximation of the climate model and hence allows for sampling more surrogate climate runs^{1}^{1}1A brief discussion on the difference between a SG and an emulator is contained in the same work.. In their study, the authors present a SG for the global annual wind and show that only five runs are sufficient to generate synthetic runs visually indistinguishable from the original simulations, and with a similar spatiotemporal local dependence. However, while the SG introduced by the authors is able to approximate annual global data for the Arabian peninsula effectively, an annual scale is not useful for wind energy assessment, and a SG at a finer temporal resolution in the same region is required to provide policyrelevant results.A SG for monthly global wind output requires considerable modeling and computational efforts. From a modeling perspective, data indexed on the sphere and in time require a dependence structure able to incorporate complex nonstationarities across the entire Earth’s system, see Jeong et al. (2017) for a recent review of multiple approaches. For regularly spaced data, as is the case with atmospheric variables in an ESM output, multistep spectrum models are particularly useful as they can provide flexible nonstationary structures for Gaussian processes in the spectral domain while maintaining positive definiteness of the covariance functions (Castruccio and Stein, 2013; Castruccio and Genton, 2014; Castruccio, 2016; Castruccio and Genton, 2016, 2018). Recently, Castruccio and Guinness (2017) and Jeong et al. (2017) introduced a generalization that allows graphical descriptors such as land/ocean indicators and mountain ranges to be incorporated in a spatially varying spectrum.
Besides the modeling complexity, the computational challenges are remarkable, as inference needs to be performed on an extremely large data set. Over the last two decades, the increase in size of spatiotemporal data sets in climate has prompted the development of many new classes of scalable models. Among the many solutions proposed, fixed rank methods (Cressie and Johannesson, 2008), predictive processes (Banerjee et al., 2008), covariance tapering (Furrer et al., 2006), and Gaussian Markov random fields (Rue and Held, 2005) have played a key role in our ability to couple feasibility of the inference while retaining essential information to be communicated to stakeholders; see Sun et al. (2012) for a review. However, even by modern spatiotemporal data set standards, 220 million points is a considerable size, and to perform inference, a methodology that leverages on both the parallel computing and gridded geometry of the data is absolutely necessary. Castruccio and Stein (2013), Castruccio and Genton (2014), Castruccio (2016), Castruccio and Genton (2016), Castruccio and Guinness (2017), Castruccio and Genton (2018), and Jeong et al. (2017) have provided a framework for a fast and parallel methodology for extremely large climate data sets. However, it has so far been limited to Gaussian processes. Whether an extension to nonGaussian models with such a large data set is possible (and how) was an open question.
In this paper, we propose a SG for monthly winds that is multistep, spectral and can capture a nonGaussian behavior. We adopt a simple yet flexible approach to construct nonGaussian processes in time: the Tukey and autoregressive process (Yan and Genton, 2017), defined as , where is a location parameter, is a scale parameter, is a Gaussian autoregressive process, and is the Tukey and transformation (Tukey, 1977):
(1) 
where controls the skewness and governs the tail behavior. A significant advantage of Tukey and autoregressive processes is that they provide very flexible marginal distributions, allowing skewness and heavy tails to be adjusted. This class of nonGaussian processes is integrated within the multistep spectral scheme to still allow inference for a very large data set. The Tukey and transformation has also been successfully implemented for maxstable processes (Xu and Genton, 2016) and random fields (Xu and Genton, 2017); see Xu and Genton (2015) and references therein for the independent case.
The remainder of the paper is organized as follows. Section 2 describes the wind data set. Section 3 details the statistical framework with the Tukey and autoregressive models and the inferential approach. Section 4 provides a model comparison and Section 5 illustrates how to generate SG runs. The article ends with concluding remarks in Section 6.
2 The Community Earth System Model (CESM) Large ENSemble project (LENS)
We work on global wind data from LENS, which is an ensemble of CESM runs with version 5.2 of the Community Atmosphere Model from NCAR (Kay et al., 2015). The ensemble comprises runs at (latitude longitude) resolution, with each run under the Representative Concentration Pathway (RCP) 8.5 (van Vuuren et al., 2011). Although the full ensemble consists of 40 runs, in our training set we consider only randomly chosen runs for the SG to demonstrate that only a small number of runs is necessary (a full sensitivity analysis for is performed in Jeong et al., 2017). We consider monthly nearsurface wind speed at 10 m above the ground level (U10 variable) from 2006 to 2100. Since our focus is on future wind trends, we analyze the projections for a total of 95 years. We consider all 288 longitudes, and we discard latitudes near the poles to avoid numerical instabilities, due to the very close physical distance of neighboring points and the very different statistical behavior of wind speed in the Arctic and Antarctic regions (McInnes et al., 2011), and consistently with previous works. Therefore, we use 134 bands between S and N, and the full data set comprises approximately 220 million points (). An example is given in Figure 1
(ad) where we show the ensemble mean and standard deviation of the monthly wind speed from the five selected runs, in May and November 2020. We observe that both means and standard deviations show temporal patterns. In particular, between the Northern Tropic and latitude
N, the mean of wind speed over the ocean in November is stronger than that in May.). (c) and (d) are the same as (a) and (b), but those in November 2020. The empirical skewness and kurtosis of wind speed from one ensemble member after removing the trend are reported in (e) and (f), respectively, only for the locations where p
values of a significance test are less than 0.05.For each site, we test the significance of skewness and kurtosis of wind speed over time (Bai and Ng, 2005) after removing the climate. In many spatial locations, the pvalues are smaller than 0.05 as shown in Figure 1
(e) and (f), thus indicating how the first two moments are not sufficient to characterize the temporal behavior of monthly wind in time. Most land points have significant skewness and, consistently with
Bauer (1996), we observe that monthly wind speeds over the ocean are negatively skewed in the tropics and positively skewed outside of that region. The Tropical Indian Ocean and the Western Pacific Ocean, both areas of small wind speeds, represent an exception of positively skewed distribution.3 The SpaceTime Model
3.1 The Statistical Framework
It is known that, after the climate model forgets its initial state, each ensemble member evolves in ‘deterministically chaotic’ patterns (Lorenz, 1963). Climate variables in the atmospheric module have a tendency to forget their initial conditions after a short period, and to evolve randomly while still being attracted by the mean climate. Since ensemble members from the LENS differ only in their initial conditions (Kay et al., 2015), we treat each one as a statistical realization from a common distribution in this work. We define as the spatiotemporal monthly wind speed for realization at the latitude , longitude , and time , where , , , and , and define .
To remove the trend in our model, we consider with . The Gaussian assumption for is not in general valid at monthly resolution (see Figure 1(ef), Figure S1 for a significance test on the skewness and kurtosis, and Figure S2 for Lilliefors and JarqueBera normality tests); we therefore apply the Tukey and transformation (1), so that our model can be written as:
(2) 
where , with
being the vector of the location parameters,
the Kroneker product, the vector of ones of length , , with the vector of scale parameters, , and , with and the vectors of the parameters for the Tukey and transformation at each site. Here represents the elementwise transformation according to (1).We denote by the parameters of the Tukey and transformation and by the vector of covariance parameters, which can be divided into temporal, longitudinal, and latitudinal dependence. The total set of parameters is .
Here is very high dimensional, hence we consider a multistep inference scheme as first introduced by Castruccio and Stein (2013), where the parameters obtained from previous steps are assumed fixed and known:

Step 1. We estimate
and by assuming that there is no crosstemporal dependence in latitude and longitude; 
Step 2. We consider and fixed at their estimated values and estimate by assuming that the latitudinal bands are independent;

Step 3. We consider and fixed at their estimated values and estimate .
This conditional stepwise approach implies some degree of error and uncertainty propagation across stages. Castruccio and Guinness (2017) provided some guidelines on how to control for the propagation by using intermediate steps within Step 3. Following the same scheme, we detail the model for each of the three steps and provide a description for the inference (see Sections 3.2, 3.3, and 3.4).
3.2 Step 1: Temporal Dependence and Inference for the Tukey and model
We assume that in (2) evolves according to a Vector AutoRegressive model of order (VAR()) with different parameters for each spatial location:
(3) 
where are diagonal matrices with autoregressive coefficients, and is the diagonal matrix of the standard deviations. Such a model assumes that there is no crosstemporal dependence across locations, and Figure S3(a) in the supplementary material provides diagnostics on this assumption. Hence, at each spatial location, we have a Tukey and autoregressive process of order (Yan and Genton, 2017). The vector of temporal parameters is therefore and .
The vectors and in (2) are estimated simultaneously via a maximum approximated likelihood estimation (MALE, Xu and Genton (2015)), and since the model assumes no crosstemporal dependence, each site can be analyzed independently and it is straightforward to parallelize the inference. For each spatial location, we select a model and compute the optimal order for (3). The results are shown in Figure S3(b). For a substantial share of points (), was selected, hence underscoring the need for a model with temporal dependence, even after differencing the original data from the average across realizations. A map of and is shown in the supplementary material (Figure S5), along with the pvalues (Figure S6).
Estimated values are represented in Figure 2. Here and were estimated with significant nonzero values over many locations (see Figure S4 for the pvalue), and it is apparent how the Gaussian autoregressive model is not suitable for modeling monthly wind speed.
Once all parameters are estimated, the residuals can be calculated as,
(4) 
where , and are the inverse Tukey and transformations at latitude and longitude .
The following sections provide a model for the dependence structure of , i.e., a parametrization of in (3). Specifying a valid model for the entire spherical domain that is able to capture global dependence structures is a nontrivial task. However, the following steps rely on the Gaussianity of , and hence require to specify only the covariance structure.
3.3 Step 2: Longitudinal Structure
Here, we focus on , i.e., we provide a model for the dependence structure at different longitudes but at the same latitude. Since the points are equally spaced and on a circle, the implied covariance matrix is circulant under a stationarity assumption (Davis, 1979), and is more naturally expressed in the spectral domain. The wind behavior on a latitudinal band, however, is not longitudinally stationary. Recently, an evolutionary spectrum approach that allows for changing behavior across largescale geographical descriptors, was successfully implemented for global annual temperature and wind speed ensembles (Castruccio and Guinness, 2017; Jeong et al., 2017; Castruccio and Genton, 2018). Here, we use a similar approach and we model
in the spectral domain via a generalized Fourier transform across longitude. Indeed, if we define
to be the imaginary unit, the wavenumber, then the process can be spectrally represented as(5) 
with being a spectrum evolving across longitude, and the spectral process.
To better account for the statistical behavior of wind speed, we implement a spatially varying model in which ocean, land, and high mountains above 1,000 m (consistently with Jeong et al., 2017) are treated as covariates. Therefore depends on being in a land, ocean, and high mountain domain, with the following expression:
(6) 
where is a smooth function (taper) that allows a transition between the land and the ocean domain. Each of the three components of the spectrum in (6) is parametrized by (Castruccio and Stein, 2013; Poppick and Stein, 2014): , for , where are interpreted as the variance, inverse range, and smoothness parameters, respectively, similarly as for the Matérn spectrum. The parameters are modeled so that their logarithm changes continuously and linearly depends on the altitude, i.e., , and , where , , and is the altitude at location . Similar notation holds for and . Hence, the longitudinal parameters are , and . Since the parameter values for each latitudinal band are independent from other bands, model inference across latitude can be performed independently with distributed computing.
3.4 Step 3: Latitudinal Structure
We now provide a model for latitudinal dependence, and since the model in (5) is independent and identically distributed across and , we take out these two indices for simplicity. While Castruccio and Stein (2013) and later works have proposed an autoregressive model for across (but independent across ), we consider a more general Vector AutoRegressive model of order 1 (VAR(1)) so that is allowed to also depend on neighboring wavenumbers. We define and the latitudinal dependence by , where , and is a matrix of size with coefficients of the autoregressive structure across latitude; encodes the dependence for the innovation. To balance flexibility with computational feasibility, we seek a sufficiently sparse but articulated structure for . We propose a banded diagonally dominant matrix parametrized by for all values (the explicit expression is available in the supplementary material), and , where and for all . Hence, the latitudinal parameters are , .
We consider ten sequential subsamples of 95 years (10 years except for the last partition) to reduce the computation burden. The parameter estimates of and from 10 subsamples are similar, as shown in Figure 3 (other estimates of longitudinal dependence parameters show similar patterns). Since there is no evidence of a change in latitudinal dependence over time, we consider the average of parameter estimates. Such value is used for combining multiple latitudinal bands and generating surrogates in Section 5. and are also shown in Figure S7.
4 Model Comparison
To validate our proposed model based on the Tukey and autoregressive (TGHAR) process, we compare it with both a Gaussian autoregressive (GAR) process, and two models with special cases of spatial dependence structure from steps 2 and 3 detailed in Sections 3.3 and 3.4.
4.1 Comparison with a Gaussian temporal autoregressive process
In our first comparison, we notice that the GAR process can be obtained from (2) by assuming , , , and ; therefore a formal model selection can be performed. Figure 4 represents the Bayesian Information Criterion (BIC) between the two models at each site from one ensemble member. Positive and negative values indicate better and worse model fit of TGHAR compared to GAR, respectively. TGHAR outperforms GAR in more than of spatial locations, with a considerable improvement in the BIC score (the map scale is in the order of ). The fit for land sites is overall considerably better for TGHAR, with peaks in the North Africa area near Tunisia, and in and around Saudi Arabia, in the region of study in Section 5. The tropical Atlantic also shows large gains.
4.2 Comparison with submodels of global dependence
The TGHAR model is also compared with one of no altitude dependence, i.e., where
for all in (6). The model still assumes an evolutionary spectrum with changing behavior across land/ocean (Castruccio and Guinness, 2017), and is denoted by LAO. We further compare TGHAR with a model having an autoregressive dependence across latitude, i.e., a model in which in the parametrization of in Section 3.4, which we denote as ARL.
Since both LAO and ARL are special cases of the TGHAR, a formal comparison of their model selection metrics can be performed (see Table 1). There is evidence of a considerable improvement from LAO to ARL, hence the need to incorporate the altitude while modeling the covariance structure. The additional, smaller (although nonnegligible, as the BIC improvement is approximately ) improvement from ARL to TGHAR underscores the necessity of a flexible model that is able to account for dependence across both wavenumbers and latitude.
Model  LAO  ARL  TGHAR 

# of parameters  
loglik  
BIC ( 
All three models can also be compared via local contrasts, as the residuals in (4) are approximately Gaussian. We focus on the contrast variances to assess the goodness of fit of the model in terms of its ability to reproduce the local dependence (Jun and Stein, 2008):
(7) 
where and denote the eastwest and northsouth contrast variances, respectively.
Metric  Region  th  th  th 

ocean  
land  
mountain  
ocean  
land  
mountain 
We compare ARL with TGHAR, and compute the squared distances between the empirical and fitted variances. We find that the TGHAR shows a better model fit in the case of the northsouth contrast variance but that there is no noticeable difference between the two models in the case of the eastwest variances. A representation of these differences for the small region of interest near South Africa ( and ) is given in Figure S8. Positive values are obtained when TGHAR is a better model fit than the ARL; negative values are obtained when ARL is the better model fit. Figure S8(a) and (b) show that dark red colors are more widely spread over mountains, and that no clear difference is shown over the ocean. Results presented in Table 2 are consistent with the visual inspection and the two metrics, in particular, over mountain areas, show larger values than those obtained for the ocean areas. In a global mean or median of the metrics, there is no significant difference between the two models.
5 Generation of Stochastic Surrogates
Once the model is properly defined and validated, we apply it to produce surrogate runs and train the SG with climate runs. A comprehensive sensitivity analysis on the number of elements in the training set can be found in Jeong et al. (2017). We use the SG to obtain an assessment of the uncertainty of monthly wind power density, and compare it with the results of the full extent of the LENS runs.
The mean structure of the model is obtained by smoothing the ensemble mean , but such estimate is highly variable. For each latitude and longitude (i.e., each and ), we fit a spline which minimizes the following function (Castruccio and Guinness, 2017; Jeong et al., 2017): , being the discrete Laplacian. We impose
to give significant weight to the spline interpolant in order to reflect the varying patterns of monthly wind fields over the next century. For each spatial location, harmonic regression of a time series may also be used to estimate the mean structure, but for the sake of simplicity, we opt for a nonparametric description. Once
is estimated from the training set, surrogate runs can be easily generated by the Algorithm 1.We generate forty SG runs with the model presented in this work and compared them with the original forty LENS runs. As clearly shown in Figures 1(a) and S9(a), the ensemble means from the training set and the SG runs are visually indistinguishable.
We also evaluate both models in terms of structural similarity index; to that end, we compare local patterns of pixel intensities that have been standardized for luminance and contrast (Figure S10) (Wang et al., 2004; Castruccio et al., 2017). We observe that the SG runs from the Tukey and case produce maps that are visually more similar to the original LENS runs than those in the Gaussian case (see also Figure S12 for the measures of skewness and kurtosis).
We further compare LENS and SG in terms of nearfuture trend (20132046), a reference metric for the LENS (Kay et al., 2015) that was used to illustrate the influence of the internal variability on global warming trends. We compute nearfuture wind speed trends near the Indian ocean for each of the SG and LENS runs. Results are shown in Figure 5(a) and (b). One can clearly see that the mean nearfuture wind trends by the SG runs are very similar to those from the training set of LENS runs.
We subsequently provide an assessment of the wind energy potential. The wind power density (WPD) (in ) evaluates the wind energy resource available at the site for conversion by a wind turbine. WPD can be calculated as , where is the air density ( kgm in this study), is the wind speed at a certain height , is the known wind speed at a reference height , and (Peterson and Hennessey Jr, 1978; Newman and Klein, 2013). We focus our analysis on the Gulf of Aden (E and N), a narrow channel connecting the Red Sea to the Indian Ocean with high wind regimes (Yip et al., 2017), and we choose to work on WDP at 80 meters, a standard height for wind turbines (Holt and Wang, 2012; Yip et al., 2017), in the year 2020.
Results for March and September are represented in Figure 5(c) and (d), with the histogram representing the SG runs, a superimposed estimated nonparametric density in red, and the LENS runs on top with an asterisk marker. Both histograms have rightly skewed distributional shapes, as also reflected by the distribution of the entire LENS. It is clear that the distribution resulting from the SG runs is more informative than the five LENS runs in the training set (see red cross markers on top), and matches the uncertainty generated by the forty LENS runs. Figure S10 also reports the time series of five LENS runs against five SG runs.
In Figure 6
, we report the boxplots of the distribution of WPD in 2020 for the LENS against the SG runs across all months. The point estimates and ranges of the WPD values from the LENS runs are wellmatched by those from the SG, with a slight misfit in April and November. The importance of such results cannot be understated: the SG is able to capture the interannual WPD patterns, as well as its internal variability in a region of critical importance for wind farming. The internal variability in the months of high wind activity such as July is such that the WPD can be classified from fair to very high according to standard wind energy categories
(Archer and Jacobson, 2003), and the SG can reproduce the same range with as little as five runs in the training set.6 Discussion and Conclusion
In this work, we propose a nonGaussian, multistep spectral model for a global spacetime data set of more than 220 million points. Motivated by the need for approximating a computer output with a faster surrogate, we provide a fast, parallelizable and scalable methodology to perform inference on a massive data set and to assess the uncertainty of global monthly wind energy.
Our proposed model relies on a transGaussian process, the Tukey and, which allows controlling skewness and tail behavior with two distinct parameters. This class of models is embedded in a multistep approach to allow inference for a nonstationary global model while also capturing sitespecific temporal dependence, and it clearly outperforms currently available Gaussian models.
Our model has been applied as a Stochastic Generator (SG), a new class of stochastic approximations that assesses more efficiently the internal variability for wind energy resources in developing countries with poor observational data coverage, using global models. Our results suggest that the uncertainty produced by the SG with a training set of five runs is very similar to that from the forty LENS runs in regions of critical interest for wind farming. Therefore, our model can be used as an efficient surrogate to assess the variability of wind energy at the monthly level, a clear improvement from the annual results presented by Jeong et al. (2017), and an important step forward towards the use of SGs at policyrelevant time scales.
Given the ubiquity of big data in complex structures evolving in space and time, it is natural to incorporate scalable algorithms from other disciplines into our modeling framework. To handle extremely large covariance matrices, we can consider the integration of highperformance computing in exact likelihood inference and predictions (Abdulah et al., 2017) and approximations through parallel Cholesky decompositions of matrices (Hackbusch, 1999, 2015) on different architectures (Litvinenko et al., 2017).
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