Tractable Analytic Expressions for the Wind Speed Probability Density Functions Using Expansions of Orthogonal Polynomials

The use of the two-parameter Weibull function as an estimator of the wind speed probability density function (PDF) is known to be problematic when a high accuracy of fit is required, such as in the computation of the wind power density function. Various types of nonparametric kernels can provide excellent fits to wind speed histograms but cannot provide tractable analytical expressions. Analytic expressions for the wind speed PDF are needed for many applications, particularly in the downscaling of model or satellite wind speed estimates to the regional or point scale. It is demonstrated that the judicious use of an expansion of orthogonal polynomials can produce more accurate estimates of the wind speed PDF than relatively simply parametric functions, such as the commonly used Weibull function. This study examines four such expansions applied to two different surface wind speed datasets in Oklahoma. The results indicate that the accuracy of fit of a given expansion is strongly related to how close the basis weight function in an expansion resembles the wind speed histogram. It is shown that this basis function, which is the first term in the expansion, acts as a first “best guess” to the true wind speed PDF and that the additional terms act to “adjust” the fit to converge on the true density function. The results indicate that appropriately chosen orthogonal polynomials can provide an excellent fit and are quite tractable.