Surface WindSpeed Statistics Modelling: Alternatives to the Weibull Distribution and Performance Evaluation
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Abstract
Windspeed statistics are generally modelled using the Weibull distribution. However, the Weibull distribution is based on empirical rather than physical justification and might display strong limitations for its applications. Here, we derive windspeed distributions analytically with different assumptions on the wind components to model wind anisotropy, wind extremes and multiple wind regimes. We quantitatively confront these distributions with an extensive set of meteorological data (89 stations covering various subclimatic regions in France) to identify distributions that perform best and the reasons for this, and we analyze the sensitivity of the proposed distributions to the diurnal to seasonal variability. We find that local topography, unsteady wind fluctuations as well as persistent wind regimes are determinants for the performances of these distributions, as they induce anisotropy or nonGaussian fluctuations of the wind components. A Rayleigh–Rice distribution is proposed to model the combination of weak isotropic wind and persistent wind regimes. It outperforms all other tested distributions (Weibull, elliptical and nonGaussian) and is the only proposed distribution able to catch accurately the diurnal and seasonal variability.
Keywords
Superstatistics Surface wind Wind anisotropy Wind extremes Wind regimes1 Introduction
To produce e.g. a wind atlas, the best parameter estimates are obtained using the method of moments, which ensures that the energy content of the fitted Weibull distribution equals the energy content of the observed histogram (Troen and Petersen 1989). The method of moments is a method of estimation of distribution parameters introduced by e.g. Pearson (1894, (1902a, (1902b, (1936), and one begins with deriving equations that relate the moments of the distribution (i.e., the expected values of powers) to the parameters of interest. Then a sample is drawn and the moments are estimated from the sample, with the equations then solved for the parameters, using the sample moments. This results in estimates of those parameters. The method of moments ensures the best estimation of windenergy potential but does not ensure the maximum likelihood with the observed histograms. This can lead to large errors when considering only a fraction of the wind distribution, between the cutin and cutout wind speeds of a specific windturbine power curve for instance.
For some design applications including wind loads and structural safety, it is also necessary to have information on the distribution of the complete population of wind speed at a site. Estimation of fatigue damage must account for damage accumulation over a range of extreme winds, the distribution of which is usually fitted with a distribution of the Weibull type (Davenport 1966). In chemistrytransport modelling, the Weibull distribution is used to represent the subgridscale variability of the wind speed. This allows improvement of the simulation of aerosol saltation at the surface and emission fluxes into the atmosphere that are triggered by a threshold in the wind speed (Menut 2008). In such contexts, maximizing the likelihood of fitted distributions to observed windspeed histograms is a major issue.
However, it has been long known that the Weibull distribution is only an approximation and may fit poorly the windspeed statistics, especially in the case of noncircular (i.e. nonisotropic) or nonnormal (i.e. nonGaussian wind components) distributions (Tuller and Brett 1984). The wide use of the Weibull distribution is purely empirical and there is a lack of physical background justifying the use of the Weibull distribution to model wind statistics. Many previous studies have considered the limitations of the Weibull distribution for modelling wind speeds (e.g. Bauer 1996; Erickson and Taylor 1989; Li and Li 2005; Carta et al. 2009; He et al. 2010; Morrissey and Greene 2012). Bauer (1996), Erickson and Taylor (1989) and He et al. (2010) quantified the deviation of the surface windspeed distribution from the Weibull distribution from in situ, remotely sensed and modelled wind speeds. In situ and modelled oceanic surface wind speeds from extratropical latitudes are reasonably well simulated by the Weibull distribution (Bauer 1996). About 30–35 % of modelled surface windspeed frequency distributions are found to be nonWeibull (Erickson and Taylor 1989), and slightly less over the ocean (30–32 %) than over land (30–35 %) with seasonal variations. Conversely, the remotely sensed wind speeds agree poorly with the corresponding empirical distributions. At a more regional scale, He et al. (2010) showed that measured surface windspeed frequency distributions in North America are sensitive to the underlying landsurface types, seasonal and diurnal cycles, and the departure from a Weibull distribution is larger at nighttime.
Possible better suited surface windspeed frequency distributions have been investigated (e.g. Carta et al. 2009, for a review). Some windspeed distributions were based on the use of the bivariate normal distribution with wind speed and direction as variables (Smith 1971; McWilliams et al. 1979; McWilliams and Sprevak 1980; Colin et al. 1987; Weber 1991, 1997), some on more ad hoc distributions based on the fact that wind speed is defined on the positive real line \([0, +\infty [\), which generally have an exponentiallike distribution (Bryukhan and Diab 1993; Li and Li 2005; Morrissey and Greene 2012). The Weibull distribution parameters present a series of advantages with respect to other distributions (e.g. flexibility, dependence on only two parameters, simplicity of the estimation of its parameters) but cannot represent all the wind regimes (e.g. those with high percentages of null wind speeds, bimodal distributions, ...). Therefore, its generalized use cannot be justified. Other distributions based on an expansion of orthogonal polynomials (Morrissey and Greene 2012) or the maximum entropy principle (Li and Li 2005) can produce more accurate estimates of the windspeed distribution than the Weibull function, and can represent a wider range of data types as well. Such distribution functions have been compared to each other (Gamma, Rayleigh, Weibull and Weibull mixture, Beta, lognormal, inverse Gaussian distributions) and the mixture distributions have provided the highest values of coefficient of determination (Carta et al. 2009). In general, the other tested windspeed distributions displayed lower performance.

compute alternative distributions analytically with different assumptions on the wind components: (1) normal distributions with different variances to model wind anisotropy; (2) nonGaussian distributions, from the superstatistics theory, to better model wind extremes; (3) a mixture of normal distributions to model multiple wind regimes.

perform indepth verification of these distributions against observations covering various subclimatic regions in France to identify those distributions that perform optimally, and the reasons for this, such as terrain complexity and dominant weather regimes.

analyze the sensitivity of the proposed distributions to the diurnal to seasonal variability.
2 Observations and Methodology
2.1 Wind Measurements
2.2 Wind Regimes at the Studied Stations
In the northern and western regions, all wind directions are experienced, even though this part of France is located in the storm track so that strong winds are often from the west, from the Atlantic Ocean (Vautard 1990; Plaut and Vautard 1994; Simonnet and Plaut 2001). Indeed, over the northwest Atlantic cyclones originate, travel eastwards and affect the European continent. In the southern region, frequent channelled flow can persist for several days. The strongest and most frequent channelled valley flow is the mistral, which derives from the north/northwest in the Rhône valley (Drobinski et al. 2005), and occurring when a synoptic northerly flow impinges on the Alpine range. As the flow experiences channeling, it is substantially accelerated and can extend offshore over horizontal ranges exceeding few hundreds of kilometres (Salameh et al. 2007; Lebeaupin Brossier and Drobinski 2009). The mistral occurs all year long but exhibits a small seasonal variability either in speed and direction, or in its spatial distribution (Orieux and Pouget 1984; Guénard et al. 2005, 2006). The mistral shares its occurrence with a northerly land breeze and southerly sea breeze (e.g. Bastin and Drobinski 2005, 2006; Drobinski et al. 2006, 2007), which can also be channelled in the nearby valleys (Bastin et al. 2005b) or interact with the mistral (Bastin et al. 2005a, 2006). In such a region, accounting for such persistent wind systems for modelling the windspeed statistics is thus mandatory.
We will often refer to three stations as examples among the 89 stations: Nantes is in flat terrain, such as most stations in northwestern France; Pau is in a more complex topographic region, close to the Pyrénées mountains, and Orange is in the Rhône valley and influenced by the mistral channelled flow.
2.3 Goodnessoffit of the Distributions
We introduce several windspeed distributions and analyze how they fit the observational data. The comparison is made on the cumulative density function (CDF) for wind speed, and we compute a distance score between the CDF of the tested distribution (F) and the observed empirical CDF (\(\hat{F}_n\)). It must be noted that the Weibull distribution leads to an analytic expression for the CDF, but not so for the other distributions that will be derived hereafter. Therefore the CDF (F) are numerically computed from their probability density function (PDF) expressions. To be consistent among distributions, even the Weibull CDF is numerically computed from its PDF.
Goodnessoffit statistics
Name  \(\varDelta _n^2\)  \(\omega (x)\) 

Cramer–von Mises (CvM)  \(W_n^2\)  1 
Anderson–Darling (AD)  \(A_n^2\)  \([F(x) (1 F(x))]^{1}\) 
Righttail AD (ADR)  \(R_n^2\)  \([1 F(x)]^{1} \) 
Righttail AD of second degree (AD2R)  \(r_n^2\)  \([1 F(x)]^{2}\) 
In the case of the Cramer–von Mises (CvM) statistic (\(W_n^2\)), the weight is constant (\(\omega (x)=1\)) so that the centre of the distribution actually dominates the equation. Here, the centre of the distribution is not one single point but the region around the median, mean or maximum of the distribution (where the PDF is above, say, 0.1). The Anderson–Darling score (\(A_n^2\)) puts weight on the tails of the distributions, where the tail corresponds to the part of the distribution that exceeds the 90th centile. In order to analyze the upper tail corresponding to strong winds, we can use the modified righttail Anderson–Darling (ADR) statistic (\(R_n^2\)) (Sinclair et al. 1990) or, for even greater weight placed on the tail, the modified righttail ADR of second degree (AD2R) statistic (\(r_n^2\)) as defined by Luceño (2006).
We will use the \(W_n^2\) and \(r_n^2\) scores to assess the goodnessoffit on the centre and tail of the distributions. As in any goodnessoffit test, there are thresholds for rejection of the null hypothesis (i.e. the hypothesis that the observed data are drawn from F) for different significance levels. But these thresholds depend on the distribution F, on the autocorrelation in the data and could only be estimated by simulations (e.g. Ahmad et al. 1988). Therefore the question of limit values for the metrics is very complex and beyond the scope of the present work. Rather than defining thresholds we will simply compare the scores of different fits at each station.
3 Performance of Weibull Distribution
In the flat terrain of northern France, as at Nantes (Fig. 3a), the Weibull distribution describes well the centre of the distribution but it tends to underestimate the tails of the distributions. In wind energy, the tail of the wind distribution is not important for estimating the wind resource but it is of high importance when addressing wind loading and damage fatigue, pollutant transport or the impact of wind storms. In more complex terrain, the fit to the Weibull distribution is less accurate. For example, at Pau in the Pyrénées, the distribution is more peaked than for the Weibull approximation (Fig. 3b), and is worse in the southern valleys such as at Orange (Fig. 3c). The windspeed distribution exhibits a peak at low wind speeds (about \(2~\hbox {m}~\hbox {s}^{1}\)) followed by a “shoulder”, with a more concave shape between 6 and \(12~\hbox {m}~\hbox {s}^{1}\) due to the channelled flow. Figure 3c shows that the Weibull distribution cannot fit such a complex shape, and we estimate that, in these particular cases, using a Weibull distribution leads to errors exceeding 10 % on occasions regarding the wind energy.
For a more quantitative analysis, we use the Cramer–von Mises (\(W_n^2\)) and AD2R scores (\(r_n^2\)) to quantify the distance between the empirical and the predicted CDFs. The quantities \(W_n^2\) and \(r_n^2\) measure the distance between the empirical CDF (\(\hat{F}_n\)) and the fitted distribution’s CDF, with a weight on the values in the centre or at the right tail of the distribution, respectively. For example at Nantes, Pau and Orange, the Cramer–von Mises (\(W_n^2\)) scores for the Weibull distribution are 2, 44 and 24 respectively, and the AD2R scores (\(r_n^2\)) are 840, 1560 and 1460. We need to be careful when comparing stations, for example we see a lower \(W_n^2\) score at Orange than at Pau whereas the Weibull fit appears poorer. Indeed the score values depend on the function \(\hat{F}_n\), different at each station and dependent on the number of observations n. So we cannot compare one station to another but we can compare the scores of several fits at a unique station (see Sect. 5). Nevertheless, it is important to provide an estimate of the fit quality. Based on our observations of all fits, we consider that a Cramer–von Mises score (\(W_n^2) < 2\) indicates a good fit for the centre of the distribution, and an AD2R score (\(r_n^2) <100\) indicates a good fit for the tail of the distribution. This is consistent with that we observed at the three stations where only the fit at the centre of the distribution at Nantes is excellent.
4 Alternative WindSpeed Statistics Models
4.1 Elliptical Distribution
4.2 NonGaussian Distribution
With the previous elliptical distribution, we assumed the wind components to follow a Gaussian distribution. However, this assumption is not always valid, since Gaussian curves sometimes fail to describe the histograms—see Fig. 5. We can evaluate the departure of each component u and v from a Gaussian shape by computing the ADR score for the two components, i.e. \(A_n^2(u)\) and \(A_n^2(v)\); Fig. 6b shows the sum \(A_n^2(u) + A_n^2(v)\). Theoretically, the strict Gaussianity is reached when the sum equals zero, however, from visual inspection, a value around 20 can still be considered as reasonably Gaussian. In the flat terrain of northwestern France, the scores are not too high, indicating a good fit to a Gaussian. This is however not the case in southern and eastern France. In the following, we use superstatistics defined by Beck and Cohen (2003) to address such a deviation from Gaussianity. This approach consists in representing the longterm stationary state by a superposition of different states that are weighted with a certain probability density.
4.3 The Rayleigh–Rice Distribution
 (i)random flow: the wind components have zero means and similar variances. This windspeed statistic is well described by a Rayleigh distribution: equal variances \(\sigma _u^2=\sigma _v^2=\sigma _1^2\) and zero means \(\mu _u = \mu _v = 0\). The Rayleigh distribution is a particular case of a Weibull distribution with shape parameter \(k=2\), and is a particular case of the previously introduced elliptical distribution (Eq. 6 with equal variances: \(b=0\) so \(I_0(b\,M^2)=1\)),$$\begin{aligned} P_\mathrm{Rayleigh}(M;\sigma _1^2) = \frac{M}{\sigma _1^2} \exp \left( \frac{M^2}{2\sigma _1^2} \right) \end{aligned}$$(14)
 (ii)channelled flow: the wind components have different means. The Rice distribution describes well this windspeed statistic: equal variances \(\sigma _u^2=\sigma _v^2=\sigma _2^2\) and non zero means \(\mu _u \ne \mu _v\),where \(\mu = \sqrt{\mu _u^2+\mu _v^2}\) and \(I_0\) is the modified Bessel function of the first kind of order zero.$$\begin{aligned} P_\mathrm{Rice}(M;\mu ,\sigma _2^2) = \frac{M}{\sigma _2^2} \exp \left( \frac{M^2+\mu ^2}{2\sigma _2^2} \right) I_0 \left( \frac{M\mu }{\sigma _2^2} \right) , \end{aligned}$$(15)
5 Performance of the Alternative Distributions
The three distributions introduced in the previous section are fitted to the observations, using a minimization algorithm on the ADR statistic, such as decided with the Weibull distribution in Sect. 3. This means that the two or four parameters of each distribution are adjusted in order to minimize the \(R_n^2\) distance between the CDF and the observed empirical distribution. For the elliptical distribution, the fit determines the best parameters \(\sigma _u^2\) and \(\sigma _v^2\) of Eq. 6. For the nonGaussian distribution the PDF is more complex due to the hypergeometric term in Eq. 11, making it more difficult to fit than a Weibull distribution. The Rayleigh–Rice distribution for wind speed in Eq. 16 is more difficult to fit because it has four parameters, especially because of the nonlinear effect of the \(\alpha \) parameter that modulates the respective weights of the Rayleigh and Rice distributions. To overcome this difficulty, we first fit the distribution for only three parameters and a fixed value of \(\alpha \), repeat this for a series of different \(\alpha \) values, and choose the best of all fits. This best fit is then used as a first estimate to fit with four parameters and it rapidly converges.
Now we discuss the performances of the three distributions, first at stations Nantes, Pau and Orange and afterwards in a more systematic fashion at all 89 stations.
5.1 Examples of Fits at Three Stations
Goodnessoffit scores of the distributions in Fig. 8. The quantities \(W_n^2\) (Cramer–von Mises) and \(r_n^2\) (righttail ADR of second degree) measure the distance between the empirical and fitted distributions with focus on the centre and right tail of the distribution, respectively. A lower value indicates a better fit
Nantes  Pau  Orange  

\(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  
Weibull  2.0  839  43.9  1563  24.2  1458 
Elliptical  1.0  392  21.9  4067  43.6  486 
NonGaussian  0.9  82  11.5  292  91.0  3076 
Rayleigh–Rice  0.9  57  7.4  89  1.1  61 
5.2 Diurnal and Seasonal Variability
Same as Table 2 for daytime (observations only at 1200 UTC) and nighttime (observations only at 0000 UTC)
NIGHT (0000 UTC)  DAY (1200 UTC)  

Nantes  Pau  Orange  Nantes  Pau  Orange  
\(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  
Weibull  0.4  156  1.9  320  2.0  75  0.1  60  1.5  47  0.8  62 
Elliptical  0.2  171  1.1  713  5.8  31  2.0  29  0.7  62  0.8  35 
NonGaussian  0.1  16  0.9  15  3.0  156  2.0  30  0.5  12  2.8  119 
Rayleigh–Rice  0.1  49  0.5  3  0.1  25  0.1  3  0.3  11  0.1  10 
Same as Table 2 for extended winter (October–March) and summer (April–September)
Winter  Summer  

Nantes  Pau  Orange  Nantes  Pau  Orange  
\(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  \(W_n^2\)  \(r_n^2\)  
Weibull  1.0  974  30.7  1508  20.9  953  1.5  49  15.7  251  7.3  604 
Elliptical  0.5  498  15.9  6234  40.6  355  0.9  16  6.9  344  8.9  270 
NonGaussian  0.5  132  8.5  144  64.1  1854  0.9  16  6.8  180  32.4  1272 
Rayleigh–Rice  0.2  25  1.6  141  0.4  29  0.9  79  3.2  21  0.7  94 
The impact of the seasonal cycle on the windspeed statistics has also been investigated. Erickson and Taylor (1989) showed that overland 35 % of the windspeed distributions are judged to be nonWeibull in January versus 30 % in July. Table 4 gives the values of the \(W_n^2\) and \(r_n^2\) scores for the fits of the Weibull, elliptical, nonGaussian and Rayleigh–Rice distributions at the three stations, Nantes, Pau and Orange in winter (October–March) and summer (April to September) (the scores should be multiplied by 2 to be comparable to the scores of Table 2. For “perfect fit” the values of \(W_n^2\) and \(r_n^2\) scores should be lower than about 1 and 50, respectively). Table 4 shows a less clear behaviour. At Nantes, all distribution fits give larger \(W_n^2\) score in summer than in winter suggesting a better fit of the centre of the distribution in summer. It is however the reverse for the tail of the distribution, except for the Rayleigh–Rice distribution. At the other stations, the Weibull distribution, as well as the elliptical and nonGaussian distributions, better fit the observations in summer for both the centre and the tail. The Rayleigh–Rice distribution displays in general the opposite behaviour, generally performing better during winter. This can easily be explained by the higher probability of persistent strong winds over France, which produce the secondary peak or shoulder at higher wind speeds, enabling a more accurate and reliable fit of the Rayleigh–Rice distribution. However, in any case, in absolute value the Rayleigh–Rice distribution generally outperforms the other distributions.
5.3 Systematic Quantification of Performances
We now generalize the findings from the three example stations and make a systematic comparison of the performances of each distribution against the Weibull. At each station, we compute the \(W_n^ 2\) and \(r_n^ 2\) scores of each distribution, such as we did for the Weibull (Fig. 4). Then the comparison of the \(W_n^ 2\) (respectively \(r_n^ 2\)) scores indicates which distribution performs best on the centre (respectively the tail) of the distribution. We consider that two distributions are similar when the difference in \(W_n^ 2\) scores is \(<\)2 (\(<\)100 for the \(r_n^ 2\) scores).
Figure 11 summarizes the performances of the Rayleigh–Rice distribution. Figure 11b shows that it is doing similar or better than the Weibull on the centre of distribution at all stations, and better or similar on the tail at 73 over 89 stations. It does not only outperform the Weibull but Fig. 11a also shows that the fits are very good: \(W_n^ 2 \approx 0\) at almost all stations. The Pau site, where the peak is not well fitted by the Rayleigh–Rice, is actually an exception since it is among the five worst \(W_n^ 2\) scores. The Rayleigh–Rice distribution is designed for regions of flows channelled in valleys, or where sustained wind field prevails. Indeed, it performs very well in the southern region at stations where channelled flows create shouldered distributions, such as at Orange. The Rayleigh–Rice is capable of fitting the two peaks, so it improves the representation of the windspeed statistics in these complex areas. Surprisingly, even in other areas without bimodal distribution, the Rayleigh–Rice distribution brings some improvement, so superposing two regimes enables to better represent the shape of the wind speed statistics. This is consistent with the review of Carta et al. (2009) regarding mixture distributions involving the Weibull distribution.
Figure 12 gives a visual summary of the comparison of the four distributions. In the left panel (Fig. 12a), we compare only the Weibull, elliptical and nonGaussian distributions which all depend on two parameters only, whereas the right panel (Fig. 12b) also includes the Rayleigh–Rice fourparameter distribution. The white dots in Fig. 12a correspond to stations in northwestern France where the wind components are close to Gaussian shape and without too much anisotropy (see Fig. 6), such as Nantes. The fits of all four distributions are quite accurate and very close, except on the tail where the Weibull and elliptical distributions tend to underestimate the probability of strong winds. In the other areas, we saw that the distributions are less accurate and either one of the three is the best fit, depending on the wind characteristics.
Finally, Fig. 12b shows the benefit of a mixed distribution such as the Rayleigh–Rice to model windspeed statistics for a wide range of environments. This new distribution outperforms the other three almost everywhere. One can note that the Rayleigh–Rice distribution performs best even where the anisotropy ratio is much larger than 1 and/or where the wind components are not Gaussian. This could be seen as contradictory with the fact that the Rayleigh–Rice distribution is the mixture of two normal distributions. This suggests that the nonGaussianity of the observed windspeed distribution, which can be partly reproduced by our nonGaussian distribution, is probably dominated by the bimodal nature of the distribution. Regarding anisotropy, the good behavior of the Rayleigh–Rice distribution suggests that the anisotropic nature of the windspeed distribution is most probably carried by the existence of a sustained prevailing flow rather than different windcomponent variances as proposed in McWilliams et al. (1979), McWilliams and Sprevak (1980) and Weber (1997). It also explains why the elliptical distribution performs worse than the Rayleigh–Rice distribution.
Other quantitative comprehensive evaluations could be used. A global performance index could be for instance the power of the distribution, namely the third moment of the distribution, which is maybe of more practical importance and allows comparisons between stations. We did not use this indicator since it does not ensure the best good fit of the distributions to the observations, which is a key aspect of this study. However the analysis of such index (not shown) confirms the analysis using the CvM and AD2R scores. The Rayleigh–Rice outperforms the other distributions with on average less than 2 % relative error with respect to the observations. The nonGaussian is the less efficient with relative error often exceeding 20 % (half of the stations). The Weibull and elliptical distributions display similar performance (slightly better for the elliptical distribution) with relative errors ranging between 5 and 20 % in most stations.
6 Conclusion
The use of the Weibull distribution for wind statistics modelling is a convenient and powerful approach. It is however based on empirical rather than physical justification and might display at times strong limitations for its application. Based on wind measurements collected at 89 locations throughout France, for a wide range of environments, from flat to complex orography with different weather regimes, we compared the Weibull distribution and two other twoparameter probability density functions for the wind speed, here called elliptical and nonGaussian. We therefore provide greater physical insight into the validity domain of the Weibull distribution, depending on the wind characteristics, mainly the fluctuations and anisotropy. The elliptical distribution assumes a Gaussian shape for the wind components but takes into account the anisotropy by assuming different variances for each component. The nonGaussian distribution is based on the recently developed superstatistics theory. It assumes fluctuating variances of the two wind components, which are eventually modelled by a Gaussian distribution over a short time interval. But for analytic calculation purpose, the proposed windspeed distribution does not take into account the anisotropy. Where the wind components are close to Gaussian shape and without too much anisotropy, such as at most stations in northwestern France, all fits are quite close and rather accurate, except on the tail of the distribution where only the nonGaussian distribution does not underestimate the strong wind probability. In more complex regions, close to the mountains, in southern and eastern France, the wind field can present anisotropy and/or departure from Gaussian shape, and either the elliptical or the nonGaussian distribution can be better suited than the Weibull to represent the wind statistics. We also introduced a Rayleigh–Rice fourparameter distribution as a combination of a Rayleigh distribution to model the isotropic wind field and a Rice distribution to model persistent wind regimes. This gives excellent results, especially for the weather stations located in the Rhône or Aude valleys (where the mistral and tramontane channelled flows accur, respectively) where the Weibull or other twoparameter distributions, are not able to reproduce the observed shouldered distributions. Combining Rayleigh and Rice distributions is another way of applying the superstatistics theory, which models the wind system as the superposition of local dynamics at different intervals with different mean wind speeds.
Finally, this study points out the limits of using a unique analytic expression to model the wind statistics, since the wind field and its statistical distribution can greatly vary spatially. The more sophisticated distributions obviously fit more complex wind regimes better but with less simple estimation of their parameters. This is the case for our Rayleigh–Rice distribution that by far outperforms the other distributions at most stations. One use of parametric distributions, especially the Weibull distribution, is the statistical downscaling of nearsurface wind speed to produce regional windspeed climatologies (Pryor et al. 2005). We showed that a number of analytical distributions can represent wind speed distributions. Knowing properties such as surrounding topography, anisotropy, existence of persistent wind regimes can help in determining which distribution performs optimally. However, we also advocate nonparametric statistical methods, based on the windspeed cumulative distribution function, or percentiles that would not be sensitive to the complexity of the observed windspeed distribution (e.g. Michelangeli et al. 2009; Salameh et al. 2009; Lavaysse et al. 2012; Vrac et al. 2012).
Footnotes
 1.
The data were recorded in knots and not in \(\hbox {m}~\hbox {s}^{1}\).
Notes
Acknowledgments
This research has received funding from the French Environment and Energy Management Agency (ADEME) through the MODEOL project (contract 1205C01467). The authors are very grateful to Samuel Humeau and Jerry Szustakowski at Ecole Polytechnique for fruitful discussion. They are also grateful to Paul Poncet (GDF Suez), Robert Bellini (ADEME) and Nicolas Girard (Maïa Eolis) for their feedback and advice. Bénédicte Jourdier is funded by ADEME and GDF Suez. Wind data and information on the Integrated Surface Database are available at http://www.ncdc.noaa.gov/isd.
References
 Ahmad M, Sinclair C, Spurr B (1988) Assessment of flood frequency models using empirical distribution function statistics. Water Resour Res 24:1323–1328CrossRefGoogle Scholar
 Baïle R, Muzy JF, Poggi P (2011) An MRice wind speed frequency distribution. Wind Energy 14(6):735–748. doi: 10.1002/we.454 CrossRefGoogle Scholar
 Bastin S, Drobinski P (2005) Temperature and wind velocity oscillations along a gentle slope during seabreeze events. BoundaryLayer Meteorol 114(3):573–594. doi: 10.1007/s1054600412376 CrossRefGoogle Scholar
 Bastin S, Drobinski P (2006) Seabreezeinduced mass transport over complex terrain in southeastern France: a casestudy. Q J R Meteorol Soc 132(615):405–423. doi: 10.1256/qj.04.111 CrossRefGoogle Scholar
 Bastin S, Champollion C, Bock O, Drobinski P, Masson F (2005a) On the use of gps tomography to investigate water vapor variability during a mistral/sea breeze event in southeastern france. Geophys Res Let 32(L05808). doi: 10.1029/2004GL021907
 Bastin S, Drobinski P, Dabas A, Delville P, Reitebuch O, Werner C (2005b) Impact of the Rhône and Durance valleys on seabreeze circulation in the Marseille area. Atmos Res 74(1–4):303–328. doi: 10.1016/j.atmosres.2004.04.014 CrossRefGoogle Scholar
 Bastin S, Drobinski P, Guénard V, Caccia JL, Campistron B, Dabas AM, Delville P, Reitebuch O, Werner C (2006) On the interaction between sea breeze and summer mistral at the exit of the Rhône valley. Mon Weather Rev 134(6):1647–1668. doi: 10.1175/MWR3116.1 CrossRefGoogle Scholar
 Bauer E (1996) Characteristic frequency distributions of remotely sensed in situ and modelled wind speeds. Int J Climatol 16:1087–1102CrossRefGoogle Scholar
 Beck C, Cohen EGD (2003) Superstatistics. Physica A Stat Mech Appl 322:267–275. doi: 10.1016/S03784371(03)000190 CrossRefGoogle Scholar
 Bernardin F, Bossy M, Chauvin C, Drobinski P, Rousseau A, Salameh T (2009) Stochastic downscaling method: application to wind refinement. Stoch Environ Res Risk Assess 23(6):851–859. doi: 10.1007/s0047700802769 CrossRefGoogle Scholar
 Bryukhan F, Diab R (1993) Decomposition of empirical wind speed distributions by laguerre polynomials. Wind Eng 17:147–151Google Scholar
 Carta J, Ramírez P (2007) Analysis of twocomponent mixture weibull statistics for estimation of wind speed distributions. Renew Energy 32:518–531CrossRefGoogle Scholar
 Carta J, Ramírez P, Bueno C (2008) Considerations of the effects of winds on the drift of oil slicks at sea: statistical and temporal aspects of wind velocity, direction and persistence. Energy Convers Manag 49:1309–1320CrossRefGoogle Scholar
 Carta JA, Ramírez P, Velázquez S (2009) A review of wind speed probability distributions used in wind energy analysis: case studies in the canary islands. Renew Sustain Energy Rev 13(5):933–955. doi: 10.1016/j.rser.2008.05.005 CrossRefGoogle Scholar
 Chew V, Boyce R (1962) Distribution of radial error in the bivariate elliptical normal distribution. Technometrics 4:138–139. doi: 10.2307/1266181 Google Scholar
 Colin B, Coupal B, Frayce D (1987) Considerations of the effects of winds on the drift of oil slicks at sea: statistical and temporal aspects of wind velocity, direction and persistence. Wind Eng 11:51–65Google Scholar
 Cook NJ (2001) “Discussion on modern estimation of the parameters of the Weibull wind speed distribution for wind speed energy analysis” by J.V. seguro, T.W. lambert. J Wind Eng Ind Aerodyn 89(10):867–869. doi: 10.1016/S01676105(00)00088X CrossRefGoogle Scholar
 Crutcher HL, Baer L (1962) Computations from elliptical wind distribution statistics. J Appl Meteorol 1(4):522–530. doi: 10.1175/15200450(1962)001<0522:CFEWDS>2.0.CO;2
 Davenport AG (1966) The treatment of wind loading on tall buildings. In Proceedings of the symposium on tall buildings. University of Southampton, Pergamon Press, LondonGoogle Scholar
 Drobinski P (2012) Wind and solar renewable energy potential resources estimation. AddisonWesley, Reading, MAGoogle Scholar
 Drobinski P, Bastin S, Guénard V, Caccia JL, Dabas A, Delville P, Protat A, Reitebuch O, Werner C (2005) Summer mistral at the exit of the Rhône valley. Q J R Meteorol Soc 131(605):353–375. doi: 10.1256/qj.04.63 CrossRefGoogle Scholar
 Drobinski P, Bastin S, Dabas A, Delville P, Reitebuch O (2006) Variability of threedimensional sea breeze structure in southern France: observations and evaluation of empirical scaling laws. Ann Geophys 24(7):1783–1799CrossRefGoogle Scholar
 Drobinski P, Saïd F, Ancellet G, Arteta J, Augustin P, Bastin S, Brut A, Caccia JL, Campistron B, Cautenet S, Colette A, Coll I, Corsmeier U, Cros B, Dabas A, Delbarre H, Dufour A, Durand P, Guénard V, Hasel M, Kalthoff N, Kottmeier C, Lasry F, Lemonsu A, Lohou F, Masson V, Menut L, Moppert C, Peuch VH, Puygrenier V, Reitebuch O, Vautard R (2007) Regional transport and dilution during highpollution episodes in southern france: summary of findings from the field experiment to constraint models of atmospheric pollution and emissions transport (ESCOMPTE). J Geophys Res 112:D13105. doi: 10.1029/2006JD007494
 Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG, Bateman H (1953) Higher transcendental functions, vol 1. McGrawHill, New York, 302 ppGoogle Scholar
 Erickson D, Taylor J (1989) Nonweibull behavior observed in a modelgenerated global surface wind field frequency distribution. J Geophys Res 94:12,693–12,698CrossRefGoogle Scholar
 Fisher R (1912) On an absolute criterion for fitting frequency curves. Messenger Math 41:155–160Google Scholar
 Fisher R (1922) On the mathematical foundations of theoretical statistics. Philos Trans R Soc Lond Ser A 222:309–368CrossRefGoogle Scholar
 Gryning SE, Batchvarova E, Floors R, Peña A, Brümmer B, Hahmann AN, Mikkelsen T (2014) Longterm profiles of wind and weibull distribution parameters up to 600 m in a rural coastal and an inland suburban area. BoundaryLayer Meteorol 150:167–184. doi: 10.1007/s1054601398573 CrossRefGoogle Scholar
 Guénard V, Drobinski P, Caccia JL, Campistron B, Bench B (2005) An observational study of the mesoscale mistral dynamics. BoundaryLayer Meteorol 115(2):263–288. doi: 10.1007/s105460043406z CrossRefGoogle Scholar
 Guénard V, Drobinski P, Caccia JL, Tedeschi G, Currier P (2006) Dynamics of the MAP IOP 15 severe mistral event: observations and highresolution numerical simulations. Q J R Meteorol Soc 132(616):757–777. doi: 10.1256/qj.05.59 CrossRefGoogle Scholar
 He Y, Monahan A, Jones C, Dai A, Biner S, Caya D, Winger K (2010) Land surface wind speed probability distributions in North America: observations, theory, and regional climate model simulations. J Geophys Res 115(D04):103. doi: 10.1029/2008JD010708 Google Scholar
 Justus CG, Hargraves WR, Yalcin A (1976) Nationwide assessment of potential output from windpowered generators. J Appl Meteorol 15(7):673–678. doi: 10.1175/15200450(1976)015<0673:NAOPOF>2.0.CO;2
 Justus CG, Hargraves WR, Mikhail A, Graber D (1978) Methods for estimating wind speed frequency distributions. J Appl Meteorol 17(3):350–353. doi: 10.1175/15200450(1978)017<0350:MFEWSF>2.0.CO;2
 Lavaysse C, Vrac M, Drobinski P, Lengaigne M, Vischel T (2012) Statistical downscaling of the French Mediterranean climate: assessment for present and projection in an anthropogenic scenario. Nat Hazards Earth Syst Sci 12(3):651–670. doi: 10.5194/nhess126512012 CrossRefGoogle Scholar
 Lebeaupin Brossier C, Drobinski P (2009) Numerical highresolution air–sea coupling over the Gulf of Lions during two tramontane/mistral events. J Geophys Res 114:D10110. doi: 10.1029/2008JD011601
 Li M, Li X (2005) Meptype distribution function: a better alternative to weibull function for wind speed distributions. Renew Energy 30:1221–1240CrossRefGoogle Scholar
 Luceño A (2006) Fitting the generalized Pareto distribution to data using maximum goodnessoffit estimators. Comput Stat Data Anal 51(2):904–917. doi: 10.1016/j.csda.2005.09.011 CrossRefGoogle Scholar
 McWilliams B, Sprevak D (1980) The estimation of the parameters of the distribution of wind speed and direction. Wind Eng 4:227–238Google Scholar
 McWilliams B, Newmann M, Sprevak D (1979) The probability distribution of wind velocity and direction. Wind Eng 3:269–273Google Scholar
 Menut L (2008) Sensitivity of hourly Saharan dust emissions to NCEP and ECMWF modeled wind speed. J Geophys Res 113:D16201. doi: 10.1029/2007JD009522
 Michelangeli PA, Vrac M, Loukos H (2009) Probabilistic downscaling approaches: application to wind cumulative distribution functions. Geophys Res Lett 36:L11708. doi: 10.1029/2009GL038401
 Morrissey M, Greene J (2012) Tractable analytic expressions for the wind speed probability density functions using expansions of orthogonal polynomials. J Appl Meteorol Clim 51:1310–1320CrossRefGoogle Scholar
 Orieux A, Pouget E (1984) Le mistral: contribution à l’étude de ses aspects synoptiques et régionaux. Monographie 5, Direction de la MétéorologieGoogle Scholar
 Pearson K (1894) Contribution to the mathematical theory of evolution. Philos Trans R Soc Lond Ser A 195:71–110CrossRefGoogle Scholar
 Pearson K (1902a) On the systematic fitting of curves to observations and measurements, part I. Biometrika 1:265–303CrossRefGoogle Scholar
 Pearson K (1902b) On the systematic fitting of curves to observations and measurements, part II. Biometrika 2:1–23Google Scholar
 Pearson K (1936) Method of moments and method of maximum likelihood. Biometrika 28:34–59CrossRefGoogle Scholar
 Plaut G, Vautard R (1994) Spells of lowfrequency oscillations and weather regimes in the northern hemisphere. J Atmos Sci 51(2):210–236. doi: 10.1175/15200469(1994)051<0210:SOLFOA>2.0.CO;2
 Pryor SC, Schoof JT, Barthelmie RJ (2005) Empirical downscaling of wind speed probability distributions. J Geophys Res 110:D19109. doi: 10.1029/2005JD005899
 Ramírez P, Carta JA (2005) Influence of the data sampling interval in the estimation of the parameters of the Weibull wind speed probability density distribution: a case study. Energy Convers Manag 46(15–16):2419–2438. doi: 10.1016/j.enconman.2004.11.004 CrossRefGoogle Scholar
 Rizzo S, Rapisarda A (2005) Application of superstatistics to atmospheric turbulence. In: Beck C, Benedek G, Rapisarda A, Tsallis C (eds) Complexity, metastability and nonextensivity: proceedings of the 31st workshop of the international school of solid state physics. World Scientific Publishing Company, Incorporated, pp 246–254Google Scholar
 Salameh T, Drobinski P, Menut L, Bessagnet B, Flamant C, Hodzic A, Vautard R (2007) Aerosol distribution over the western Mediterranean basin during a Tramontane/Mistral event. Ann Geophys 25:2271–2291CrossRefGoogle Scholar
 Salameh T, Drobinski P, Vrac M, Naveau P (2009) Statistical downscaling of nearsurface wind over complex terrain in southern france. Meteorol Atmos Phys 103(1–4):253–265. doi: 10.1007/s0070300803307 CrossRefGoogle Scholar
 Seguro JV, Lambert TW (2000) Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis. J Wind Eng Ind Aerodyn 85(1):75–84. doi: 10.1016/S01676105(99)001221 CrossRefGoogle Scholar
 Simonnet E, Plaut G (2001) Spacetime analysis of geopotential height and SLP, intraseasonal oscillations, weather regimes, and local climates over the North Atlantic and Europe. Clim Res 17(3):325–342. doi: 10.3354/cr017325 CrossRefGoogle Scholar
 Sinclair C, Spurr B, Ahmad M (1990) Modified Anderson Darling test. Commun Stat Theory Methods 19(10):3677–3686. doi: 10.1080/03610929008830405 CrossRefGoogle Scholar
 Smith A, Lott N, Vose R (2011) The integrated surface database: recent developments and partnerships. Bull Am Meteorol Soc 92(6):704–708. doi: 10.1175/2011BAMS3015.1 CrossRefGoogle Scholar
 Smith O (1971) An application of distributions derived from the bivariate normal density function. In: Proceedings of the international symposium on probability and statistics in the atmospheric sciences, pp 162–168Google Scholar
 Sura P, Gille ST (2003) Interpreting winddriven southern ocean variability in a stochastic framework. J Mar Res 61(3):313–334. doi: 10.1357/002224003322201214 CrossRefGoogle Scholar
 Takle ES, Brown JM (1978) Note on the use of Weibull statistics to characterize windspeed data. J Appl Meteorol 17(4):556–559. doi: 10.1175/15200450(1978)017<0556:NOTUOW>2.0.CO;2
 Troen I, Petersen EL (1989) European Wind Atlas. Risø National Laboratory, RoskildeGoogle Scholar
 Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52:479–487. doi: 10.1007/BF01016429 CrossRefGoogle Scholar
 Tuller SE, Brett AC (1984) The characteristics of wind velocity that favor the fitting of a Weibull distribution in wind speed analysis. J Appl Meteorol Clim 23(1):124–134. doi: 10.1175/15200450(1984)023<0124:TCOWVT>2.0.CO;2
 Vautard R (1990) Multiple weather regimes over the North Atlantic: analysis of precursors and successors. Mon Weather Rev 118(10):2056–2081. doi: 10.1175/15200493(1990)118<2056:MWROTN>2.0.CO;2
 Vrac M, Drobinski P, Merlo A, Herrmann M, Lavaysse C, Li L, Somot S (2012) Dynamical and statistical downscaling of the French Mediterranean climate: uncertainty assessment. Nat Hazards Earth Syst Sci 12(9):2769–2784. doi: 10.5194/nhess1227692012 CrossRefGoogle Scholar
 Weber R (1991) Estimator for the standard deviation of wind direction based on moments of the Cartesian components. J Appl Meteorol 30:1341–1352CrossRefGoogle Scholar
 Weber R (1997) Estimators for the standard deviation of horizontal wind direction. J Appl Meteorol 36:1407–1415CrossRefGoogle Scholar
 Weisser D (2003) A wind energy analysis of Grenada: an estimation using the ‘Weibull’ density function. Renew Energy 28(11):1803–1812. doi: 10.1016/S09601481(03)000168 CrossRefGoogle Scholar
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