Statistical downscaling of historical monthly mean winds over a coastal region of complex terrain. I. Predicting wind speed

Abstract

Surface wind speed is a key climatic variable of interest in many applications, including assessments of storm-related infrastructure damage and feasibility studies of wind power generation. In this work and a companion paper (van der Kamp et al. 2011), the relationship between local surface wind and large-scale climate variables was studied using multiple regression analysis. The analysis was performed using monthly mean station data from British Columbia, Canada and large-scale climate variables (predictors) from the NCEP-2 reanalysis over the period 1979–2006. Two regression-based methodologies were compared. The first relates the annual cycle of station wind speed to that of the large-scale predictors at the closest grid box to the station. It is shown that the relatively high correlation coefficients obtained with this method are attributable to the dominant influence of region-wide seasonality, and thus contain minimal information about local wind behaviour at the stations. The second method uses interannually varying data for individual months, aggregated into seasons, and is demonstrated to contain intrinsically local information about the surface winds. The dependence of local wind speed upon large-scale predictors over a much larger region surrounding the station was also explored, resulting in 2D maps of spatial correlations. The cross-validated explained variance using the interannual method was highest in autumn and winter, ranging from 30 to 70% at about a dozen stations in the region. Reasons for the limited predictive skill of the regressions and directions for future progress are reviewed.

Appendix

Beginning with the simplest case of univariate linear regression of a monthly mean predictor variable P against a predictand y, the correlation coefficient is

$$ r^{2} = \frac{{\text{cov}^{2} (P,y)}}{{\sigma_{{P_{j} }}^{2} \sigma_{y}^{2} }},\quad \text{cov} (P,y) = M^{ - 1} \sum\limits_{m = 1}^{{M_{k} }} {y_{m} P_{m} } ,\quad \sigma_{y}^{2} = M^{ - 1} \sum\limits_{m = 1}^{{M_{k} }} {y_{m}^{2} } ,\quad \sigma_{P}^{2} = M^{ - 1} \sum\limits_{m = 1}^{{M_{k} }} {P_{m}^{2} } $$

where cov(P, y) is the covariance of P and y over the M = 12 month period of interest and the σ’s are the corresponding variances. The above assumes that both predictor and predictand are expressed as monthly anomalies from their respective annual means, i.e., $\bar{y} = \bar{P} = 0$. For an exactly sinusoidal predictand with amplitude α and zero phase and a predictor with amplitude β and phase θ, we have

$$ y_{m} = \alpha \sin \gamma_{m} ,\quad P_{m} = \beta \sin (\gamma_{m} + \theta ), $$

where γ _m  = 2πm/12. Evaluating r ², one finds that the amplitudes in the quotient cancel, leaving:

$$ {r^{2} = \cos^{2} }\theta . $$

That is, r ² differs from unity only inasmuch as the predictor-predictand phase difference differs from zero.

For two or more predictors P _j , with P _j  = β _j  sin (γ_m + θ_j), the coefficient of determination is given by the appropriate generalization of the above, i.e.,

$$ R^{2} = \, \sum\limits_{j = 1}^{N} {b_{j} } \frac{{\text{cov} (P_{j} ,y)}}{{\sigma_{y}^{2} }},\quad \text{cov} (P_{j} ,y) = M^{ - 1} \sum\limits_{m = 1}^{{M_{k} }} {y_{m} P_{j,m} } , $$

where b _j are the coefficients of the best-fit multilinear function to the predictand y, determined by minimization of the least squares error:

$$ \chi^{2} = \, \sum\limits_{m = 1}^{M} {\left[ {y_{m} - \left( {a + \sum\limits_{m = 1}^{N} {b_{j} P_{j,m} } } \right)} \right]}^{2} . $$

The resulting matrix equation,

$$ \begin{gathered} {\mathbf{Y}} = {\mathbf{A}} \cdot {\mathbf{B}}, \hfill \\ {\mathbf{Y}} = \left( {\begin{array}{*{20}c} {\sum\limits_{m} {y_{m} } } \\ {\sum\limits_{m} {P_{1m} y_{m} } } \\ {\sum\limits_{m} {P_{2m} y_{m} } } \\ \vdots \\ {\sum\limits_{m} {P_{Nm} y_{m} } } \\ \end{array} } \right),\quad {\mathbf{B}} = \left( {\begin{array}{*{20}c} a \\ {b_{1} } \\ {b_{2} } \\ \vdots \\ {b_{N} } \\ \end{array} } \right),\quad {\mathbf{A}} = \left( {\begin{array}{*{20}c} M & {\sum\limits_{m} {P_{1m} } } & {\sum\limits_{m} {P_{2m} } } & \cdots & {\sum\limits_{m} {P_{Nm} } } \\ {\sum\limits_{m} {P_{1m} } } & {\sum\limits_{m} {P_{1m}^{2} } } & {\sum\limits_{m} {P_{1m} } P_{2m} } & \cdots & {\sum\limits_{m} {P_{1m} } P_{Nm} } \\ {\sum\limits_{m} {P_{2m} } } & {\sum\limits_{m} {P_{2m} } P_{1m} } & {\sum\limits_{m} {P_{2m}^{2} } } & \cdots & \vdots \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\sum\limits_{m} {P_{Nm} } } & {\sum\limits_{m} {P_{Nm} } P_{1m} } & \cdots & \cdots & {\sum\limits_{m} {P_{Nm}^{2} } } \\ \end{array} } \right), \hfill \\ \end{gathered} $$

may be solved for the b _j as

$$ b_{j} = \frac{{|{\mathbf{A^{\prime}}}(j + 1)|}}{{|{\mathbf{A}}|}}, $$

where A′(j) is equal to A with the jth column replaced by y.

As will become clear, performing the calculation for N = 2 is sufficient to establish a general result that holds for any number of predictors. After some straightforward algebra and noting that sums of odd periodic functions (e.g., sin [γ_i + θ], etc.) over a complete period vanish, one finds:

$$ \begin{gathered} b_{1} = \, \frac{{|{\mathbf{A^{\prime}}}(2)|}}{{|{\mathbf{A}}|}} \, = \frac{\alpha }{{\beta_{1} }} \, \frac{{\cos \theta_{1} - \cos \theta_{2} \cos (\theta_{1} - \theta_{2} )}}{{\sin^{2} (\theta_{1} - \theta_{2} )}} \, \hfill \\ b_{2} = \, \frac{{|{\mathbf{A^{\prime}}}(3)|}}{{|{\mathbf{A}}|}} \, = \frac{\alpha }{{\beta_{2} }} \, \frac{{\cos \theta_{2} - \cos \theta_{1} \cos (\theta_{1} - \theta_{2} )}}{{\sin^{2} (\theta_{1} - \theta_{2} )}} \, \hfill \\ \end{gathered} $$

$$ \sigma_{y}^{2} = 6\alpha^{2} M^{ - 1} ,\quad \text{cov} (P_{1} ,y) = 6\alpha \beta_{1} M^{ - 1} \cos \theta_{1} ,\quad \text{cov} (P_{2} ,y) = 6\alpha \beta_{2} M^{ - 1} \cos \theta_{2} \, . $$

Finally, constructing R ², the amplitudes again cancel, leaving

$$ R^{2} = \frac{{\cos^{2} \theta_{1} + \cos^{2} \theta_{2} - 2\cos \theta_{1} \cos \theta_{2} \cos (\theta_{1} - \theta_{2} )}}{{\sin^{2} (\theta_{1} - \theta_{2} )}} = 1, $$

where the latter follows by using trigometric identities. This establishes the general result that only two sinusoidal predictors are needed to give perfect correlation with a sinusoidal predictand. It mirrors the well known result that any sinusoidal function with a given frequency and phase can be expressed as a linear combination of sines and cosines of the same frequency but zero phase, provided that certain definite relations hold between the various coefficients and phases (Davis 1963). While such an exact relation cannot obtain here due to the independence of predictand/predictor amplitudes and phases, it is still the case that complete correlation is obtained on the basis of the additional degrees of freedom provided by multiple predictors.