## Abstract

The statistical predictability of wind speed using Gaussian predictors, relative to the predictability of orthogonal vector wind components, is considered. With the assumption that the vector wind components are Gaussian, analytic expressions for the correlation-based wind speed prediction skill are obtained in terms of the prediction skills of the vector wind components and their statistical moments. It is shown that

at least one of the vector wind components is generally better predicted than the wind speed (often much more so);

wind speed predictions constructed from the predictions of vector wind components are more skillful than direct wind speed predictions; and

the linear predictability of wind speed (relative to that of the vector wind components) decreases as the variability in the vector wind increases relative to the mean.

These idealized model results are shown to be broadly consistent with linear predictive skills assessed using observed sea surface wind from the SeaWinds scatterometer. Biases in the model predictions are shown to be related to the degree to which vector wind variations are non-Gaussian.

## 1. Introduction

Near-surface winds exert a significant influence on surface exchanges of energy, momentum, and mass, and the associated kinetic energy represents a resource of potentially global importance. Recent years have seen a rapid increase in the number of studies examining the predictability of near-surface winds (e.g., Lange and Focken 2005; Costa et al. 2008; Gneiting et al. 2008; Klausner et al. 2009; Ma et al. 2009; Soman et al. 2010; Thorarinsdottir and Gneiting 2010; Giebel et al. 2011; Pinson 2012; Schuhen et al. 2012; Sloughter et al. 2013). Wind prediction models may be entirely empirical, based on statistical relationships, or they may be dynamical, making use of physically based equations of motion. A third class of hybrid models uses statistical postprocessing to correct errors in local wind predictions from dynamical models. Although a primary focus of this research has been wind energy prediction, the prediction of near-surface winds is important for a much broader range of applications (e.g., modeling pollutant transport, assessing the risks of extreme winds to transport or the built environment, or computing surface fluxes).

While a number of studies have addressed direct predictions of wind speed, relatively little attention has been paid to the predictability of speed relative to vector wind components. While the wind speed is of most direct interest for many applications, the vector winds contain information about both speed and direction. As the joint distribution of vector wind components is more closely Gaussian than that of speed and direction, it has the practical advantage (relative to the joint distribution of speed and direction) of being amenable to the use of classical statistical predictive tools. Consideration of the vector wind components also allows for a more direct connection to dynamical models in which the equations of motion are normally expressed in terms of the vector winds rather than the speed and direction.

This study considers relationships between the predictability of vector wind components and that of wind speed. Exact knowledge of the vector wind components of course provides exact knowledge of the speed. What is less clear is the extent to which wind speed variations are predictable when variability in the vector wind components is only imperfectly predicted. In particular, the focus of this study is the statistical prediction of surface vector wind components and wind speed from Gaussian predictors. Such predictors may be individual physical quantities or linear combinations of a number of these (as in a multiple linear regression). For example, in the context of statistical downscaling, a predictor may result from a linear regression model based on a number of large-scale free-atmospheric circulation indices (which may involve physical quantities other than winds) as in Monahan (2012a). The results that will be obtained are insensitive to whether there is a time lag between predictor and predictand (i.e., for a statistical forecast model) or if the predictor and predictand are simultaneous (i.e., for a “nowcast”). All that is assumed about the predictors is that they are Gaussian.

Throughout this study, *u* and *υ* will denote generic orthogonal horizontal vector wind components (*U* and *V* will be used specifically for zonal and meridional components). While the precise orientation of these components will generally be arbitrary, in some instances the components will be aligned along and across the mean vector wind; these instances will be explicitly noted. The wind speed will be denoted by

Note that defining the wind speed in terms of the vector components in this way neglects contributions from the vertical component of the velocity: this is justified by the fact that the focus of this study is eddy-averaged winds. As the components are therefore temporally or spatially filtered quantities, Eq. (1) does not exactly relate the filtered velocity components to the filtered wind speed. In strict terms, *u* and *υ* are the components resulting from reconstructing the wind vector from the eddy-averaged wind speed and direction (the data generally available from meteorological station or satellite-based remotely sensed measurements), rather than the eddy-averaged vector wind components. For notational convenience, however, we will simply refer to these as the vector wind components.

For Gaussian vector winds with statistical moments,

and the joint probability density function (pdf) of *u* and *υ* is given by

(e.g., Wilks 2005). The marginal distribution of *w* alone is given by expressing the vector wind in terms of magnitude and direction (rather than Cartesian coordinates) and then integrating over direction,

(e.g., Monahan 2006, 2007). The wind speed distribution is not Gaussian. Furthermore, statistical measures of *w* cannot generally be partitioned into separate statistical measures of *u* and *υ* and so closed-form analytic expressions relating the statistical features of wind speed to those of the vector wind components are not generally available. In contrast, the statistics of *w*^{2} separate easily into additive contributions from *u*^{2} and *υ*^{2}. For example, for some variable *Z*, we have

where cov(*x*, *y*) denotes the covariance between the quantities *x* and *y*. Knowledge of the statistical relationship between *Z* and the vector wind components can therefore be used to make statements about the relationship between *Z* and *w*^{2}.

This result is useful in the present context because variations of observed wind speed are highly correlated with those of its square. Correlations between *w* and *w*^{2} from observed winds between 60°S and 60°N were calculated using SeaWinds surface wind data ( appendix B). The resulting distribution of correlations (Fig. 1) demonstrates that these correlations are always greater than 0.9 and generally greater than 0.95. It follows that to a good approximation

for any variable *Z*. For example, estimates of corr(*U*, *w*) and corr(*U*, *w*^{2}) from SeaWinds data are compared in the left panel of Fig. 2; the right panel of this figure similarly compares corr(*V*, *w*) with corr(*V*, *w*^{2}). Clearly, the linear statistical relationships between *w* and the vector wind components are almost indistinguishable from those between *w*^{2} and the components. The high correlation between *w* and *w*^{2} is discussed in detail in Monahan (2012b), in which it is shown that this result follows from the facts that *w* cannot take negative values and the ratio std(*w*)/mean(*w*) is small. A similar result was discussed in Carlin and Haslett (1982), in which it was noted that variations of *w* are highly correlated with those of . The approximation Eq. (10) is central to the results of this study. Together with Eq. (9), it will allow us to obtain analytical expressions for the predictability of *w* in terms of the predictability of *u* and *υ*.

The focus of this study is on deterministic prediction of wind speed and components by single time series, rather than ensemble prediction. Of the various measures that can be used to assess the quality of the deterministic prediction *y _{p}* of the quantity

*y*, the two most common are mean squared error ε

^{2}and correlation coefficient corr(

*y*,

_{p}*y*). These measures are not independent; in fact, it can be shown that

(e.g., Murphy 1988), where the notations mean(*x*) and std(*x*) are used to denote the mean and standard deviation of *x*, respectively. Terms I and II are respectively errors associated with biases in the prediction of the mean and the standard deviation. Term III represents errors associated with unsynchronized variations of the prediction and the predictand, as measured by the correlation.

This study addresses differences in predictability of vector wind components and wind speed inherent to the mathematical relationship between these quantities. As such, we will consider an idealized setting in which all statistical features of the vector winds and wind speeds are assumed to be known. In this case, any prediction biases can be corrected and terms I and II in the mean squared error expansion Eq. (11) can be eliminated by rescaling the mean and standard deviation of the predicted wind speeds to match the known true values. The remaining irreducible errors are associated with imperfect correlations between prediction and predictand. Correlations are a standard skill measures for deterministic predictions and will be the focus of the present analysis. This idealized setting represents a best-case scenario of statistical prediction with Gaussian predictors, in a stationary setting with known statistics.

As the simplest example of the relationship between linear predictive skills of the vector wind components and wind speed, our primary focus will be the prediction of wind speed for the case in which variability in the vector wind components is uncorrelated and isotropic: *σ _{u}* =

*σ*=

_{υ}*σ*and

*r*= 0. This situation is a good first approximation to vector wind variability across much of the World Ocean (e.g., Monahan 2007). In fact, the assumption of uncorrelated, isotropic variability is not necessary; the results of a more general analysis are presented in appendix A.

We will successively consider the following three different prediction scenarios:

The direct prediction of

*w*from a single Gaussian predictor, for which it will be shown that the predictability of*w*is always bounded above by at least one vector wind component (section 2).The prediction of

*w*from predictions of the components*u*and*υ*using a single Gaussian predictor. Although predictions of*w*are shown to improve, we will find that the correlation-based wind speed predictability is still bounded above by that of at least one vector wind component. Furthermore, predictive skill is generally reduced when the variance of the (imperfectly predicted) components is inflated to offset a low variance bias (section 3).The prediction of

*w*from predictions of*u*and*υ*using separate Gaussian predictors. In this case, the predictability of speed is generally (but not always) bounded above by that of the best-predicted vector wind component, and inflation of component variances can in some circumstances improve the quality of the wind speed prediction (section 4).

A key result of this analysis is that the predictability of *w* relative to the vector wind components is determined by the local wind climate through the means and standard deviations of the components. In general, the predictability of *w* is reduced as the variability of the vector winds becomes large relative to the mean. This study's conclusions are presented in section 5. Generalizations of the results of sections 2–4 to correlated, anisotropic winds are presented in appendix A. A discussion of the dataset considered is given in appendix B.

## 2. Direct prediction of *w* with a single Gaussian predictor

We first consider the prediction of wind speed directly by the Gaussian predictor, *x*, such that

are the (correlation based) predictive skills of *x* for *u* and *υ*, respectively. Without loss of generality, we can take mean(*x*) = 0 and std(*x*) = 1. Note that *x* itself may be a composite of a number of different predictors (e.g., *x* may result from a multiple linear regression); we require only that *x* be Gaussian and possess the specified correlations with the vector wind components.

To obtain an analytic expression for the correlation-based predictive skill of *w* in terms of that of the components, we will adopt the following strategy: knowing the statistics of *u* and *υ* (, and *σ*), and their predictive skills by *x* (*ρ _{u}* and

*ρ*), we can compute the covariance of

_{υ}*x*with

*u*

^{2}and with

*υ*

^{2}and therefore the covariance of

*x*with

*w*

^{2}. Then, using the joint distribution of

*u*and

*υ*, we can compute the mean and variance of

*w*

^{2}. Combining var(

*w*

^{2}) and cov(

*x*,

*w*

^{2}), we obtain an expression for corr(

*x*,

*w*

^{2}). Finally, we make use of the approximation Eq. (10) to obtain the desired analytic expression for the correlation-based predictive skill of wind speed, corr(

*x*,

*w*).

Note that, while we can always define the basis used to define the vector wind components so that *ρ _{u}* ≥ 0,

*ρ*≥ 0, there is not complete freedom in specifying the parameters

_{υ}*ρ*and

_{u}*ρ*. The fact that the determinant the correlation matrix of

_{υ}*x*,

*u*, and

*υ*must be nonnegative implies that

Empirical studies of vector wind predictability have demonstrated that it can be highly anisotropic (e.g., van der Kamp et al. 2012; Culver and Monahan 2013; Sun and Monahan 2013), so the values of *ρ _{u}* and

*ρ*can be quite different. We will denote by

_{υ}*ρ*

_{max}the correlation-based predictive skill of the best-predicted vector wind component in any direction.

As noted above, we first compute

Defining the standardized anomaly *u*′ as

the first term in Eq. (15) is given by

It is at this step that we have made use of the fact that *x* and *u* are Gaussian: For (*x*, *u*) bivariate Gaussian, *x* and *u*′^{2} are uncorrelated. For non-Gaussian (*x*, *u*), no such general statement can be made. Although *x* and *u*′^{2} are uncorrelated, nonzero correlations between *x* and *u*^{2} arise when because variations in *u*^{2} include contributions from *u*′ and *u*′^{2}. A similar calculation gives

so

As the final step in calculating corr(*x*, *w*^{2}) we compute the variance of *w*^{2},

so, making use of the approximation Eq. (10),

Aligning the coordinate system so that *u* is the wind component oriented along the climatological-mean vector wind, so that , we have

It follows that the linear predictability of *w* by the Gaussian predictor *x* is always less than that of the along-mean wind component by a factor determined the statistics of the local wind climate (in particular, the relative sizes of the vector wind mean and standard deviation). Note that, if , *x* carries no direct linear predictive information regarding *w*, irrespective of how good a predictor it is of *u*. Similarly, the predictive skill of vector wind variations in the cross-mean wind direction *ρ _{υ}* has no bearing on the linear predictability of speed. These results generalize to the prediction of

*w*for correlated, anisotropic vector wind components ( appendix A).

## 3. Prediction of *w* from components with a single Gaussian predictor

The previous section considered the direct prediction of wind speed from a Gaussian predictor. As wind speed is a non-Gaussian quantity, the potentially poor performance of Gaussian predictors is not surprising. In contrast, the vector wind components have been assumed to be Gaussian, so perhaps *w* is better predicted by first predicting the components *u* and *υ* by *x* and then combining these into a wind speed prediction. The following analysis will demonstrate that this is fact the case.

Denoting the respective vector wind component predictions from *x* as *u _{p}* and

*υ*, we have

_{p}As is natural for predictions of Gaussian predictands using a Gaussian predictor, the expressions for *u _{p}* and

*υ*take the form of linear regressions of

_{p}*u*and

*υ*on

*x*, respectively. From the predictions of the vector wind components, we construct the wind speed prediction,

Note that the variances of *u _{p}* and

*υ*are generally biased low relative to

_{p}*u*and

*υ*[std(

*u*) =

_{p}*ρ*, std(

_{u}σ*υ*) =

_{p}*ρ*] because the square of the correlation coefficient (bounded above by one) corresponds to the fraction of variance accounted for by the prediction (e.g., Wilks 2005).

_{υ}σAs in the previous section, we will develop an analytic expression for the correlation-based predictability of *w* by *w _{p}* by first computing an expression for , combining this with expressions for and var(

*w*

^{2}), and finally making use of the approximation

The covariance between and *w*^{2} is given by

Computing the first of these terms,

where we have made use of the fact that, for *x* and *y* Gaussian, with corr(*x*, *y*) = *r*, then corr(*x*^{2}, *y*^{2}) = *r*^{2}. Note that this result does not hold in general for non-Gaussian *x* and *y*. For example, for *x* that is Gaussian with mean zero and unit variance and *y* = |*x*|, we have corr(*x*, *y*) = 0 but corr(*x*^{2}, *y*^{2}) = 1.

The remaining terms in the expression for follow similarly,

so

Calculating the variance of the predicted squared wind speed,

from which it follows that

[having made use of Eq. (20) for var(*w*^{2})]. As was the case with the expression for corr(*x*, *w*) in the case of direct predictions of wind speed, the predictive skill of wind speeds constructed from predictions of the vector components is a function of the skills of the component predictions (*ρ _{u}* and

*ρ*) and the local wind climate (, and

_{υ}*σ*).

Considering Eq. (33) in the along- and across-mean wind basis (for which ) and making use of the inequality Eq. (14), we find that the predictability of *w* is again bounded above by that of the best-predicted vector wind component,

Although the predictive skills of *w* by *x* and *w _{p}* are both bounded above by that of the vector wind, they are not equivalent: specifically, the prediction of

*w*by

*w*is always superior to that directly by

_{p}*x*,

[where we have made use of Eq. (21) and the fact that by construction std(*x*) = 1]. Furthermore, in contrast to corr(*x*, *w*), corr(*w _{p}*,

*w*) does not vanish in the limit that .

We see that, from the perspective of correlation-based predictive skill, for a single Gaussian predictor it is always better to first predict *u* and *υ* and then compute *w*, rather than predict *w* directly (insofar as the vector winds are Gaussian). For correlated, anisotropic vector winds, we also find that wind speed predictions are improved by first predicting the components. Interestingly, in this more general situation there is a small parameter range over which speeds are slightly better predicted than the components ( appendix A), but it must be emphasized that this range is very small.

As discussed above, for imperfect predictions of *u* and *υ* (*ρ _{u}*,

*ρ*< 1), the Gaussian predictions of

_{υ}*u*and

*υ*have a low variance bias that will result in biases in both the mean and standard deviation (as well as the higher-order moments) of constructed wind speed

*w*. Assuming that the mean and variance of

_{p}*w*are known,

*w*can be shifted and rescaled to match these statistics (the process of “double bias correction”; e.g., Lange and Focken 2005). Alternatively, the bias correction can be applied to

_{p}*u*and

_{p}*υ*before these are used to construct

_{p}*w*. As we will now demonstrate, doing so actually degrades the wind speed linear predictive skill corr(

_{p}*w*,

_{p}*w*). Compensating for the variance biases of the predicted vector wind components, we take our predictions of

*u*and

*υ*to be

and define

In this case, we have

and

Taking (without loss of generality), we define

From the perspective of the linear predictability of wind speed, there is no benefit to removing the variance bias of the vector wind components. This result generalizes to the case of correlated, anisotropic winds ( appendix A) and demonstrates that the appropriate strategy is to apply double bias correction to the mean and standard deviation of *w _{p}*, which does not affect the correlation skill corr(

*w*,

_{p}*w*). Note that, while this strategy can eliminate biases in the mean and standard deviation of the predicted wind speed, it will not generally eliminate the biases in higher-order moments or percentiles.

## 4. Prediction of wind speed from components: Separate predictions of *u* and *υ*

In the previous analyses, we assumed that a single predictor *x* was available for predictions of *u*, *υ*, and *w*. In fact, if *u* and *υ* are to be separately predicted through, for example, statistical downscaling, then distinct predictors will be obtained for each of these. The resulting predictive skills of *u* and *υ* will therefore be better than (or at least no worse than) those obtained with the single predictor *x*, but how much better will the predictions of *w* constructed from these component predictions be? We will now demonstrate that, although for the majority of parameter values the predictive skill of *w* is bounded above by that of the components, this is not true in all circumstances. Furthermore, there are situations in which predictions of *w* are improved by correcting for the variance biases in *u _{p}* and

*υ*before constructing

_{p}*w*, although these are still a minority of cases and these predictive skills are still generally bounded above by those of the vector wind components. Finally, predictions of observed sea surface winds will be used to demonstrate that, despite the strong assumptions that have been made about the distribution of vector winds, the results of this analysis are in good agreement with empirically determined, correlation-based wind speed predictive skills.

_{p}We define the new vector wind predictions *u _{p}* and

*υ*as Gaussian quantities such that

_{p}Again, we may think of *u _{p}* and

*υ*as resulting from linear regression predictions based on, for example, large-scale free-tropospheric predictors. Although the predictions

_{p}*u*and

_{p}*υ*are distinct and potentially constructed individually, in general they will be correlated. Without loss of generality, we can orient the wind component axes so that

_{p}*ρ*

_{11}> 0 and

*ρ*

_{22}≥ 0. Also, we can assume that

*ρ*

_{11}> |

*ρ*

_{21}| and

*ρ*

_{22}> |

*ρ*

_{21}|; that is, we can assume that

*υ*is not a better prediction of

_{p}*u*than

*u*is (and similarly for

_{p}*u*and

_{p}*υ*). The requirement that the correlation matrix of

*u*,

_{p}*υ*,

_{p}*u*, and

*υ*have a nonnegative determinant implies the following constraint on the correlation coefficients,

We recover the situation considered in the previous section in the limit that *P* = 1, *ρ*_{11} = *ρ*_{21} = *ρ _{u}*, and

*ρ*

_{12}=

*ρ*

_{22}=

*ρ*, in which case

_{υ}*u*and

_{p}*υ*both become equivalent to predictions from the single predictor

_{p}*x*.

To compute corr(*w _{p}*,

*w*), we follow the same procedure as in section 3. We first calculate the covariance between squared wind speed and its prediction,

The variance of the predicted squared wind speed is

The general expression for corr(*w _{p}*,

*w*) is rather complicated; in the along- and across-mean wind basis, it simplifies to

In contrast to the situation when a single predictor *x* is used for both components, for the present setting in which *u* and *υ* are predicted separately there are parameter sets for which the linear predictability of wind speed exceeds that of the better predicted of the along- or across-wind components. Defining the ratio

we find that Γ is not bounded above by unity for all sets of parameter values. To demonstrate this, the ratio Γ was sampled uniformly over the following parameter ranges: 0 ≤ *ρ*_{11}, *ρ*_{22} ≤ 1, , and [subject to the constraints imposed by inequality (47)]. For the majority of parameter sets, the (correlation based) wind speed predictability is lower than that of the best-predicted vector wind component (Fig. 3). Only 1.9% of the parameter sets yield a value of Γ exceeding 1. Most of the parameter sets yielding Γ > 1 are found for one (or both) of *ρ*_{12} and *ρ*_{21} < 0; when these cross correlations are constrained to be nonnegative, Γ > 1 occurs for only 0.03% of the parameter sets [Fig. 3; while the expression Eq. (50) is insensitive to the sign of the cross correlations, this is not true of the inequality Eq. (47)]. Note that this analysis does not rule out the possibility of a vector wind projection between the along- and across-wind directions being better predicted than either of these, such that corr(*w _{p}*,

*w*) ≤

*ρ*

_{max}.

In the previous section, we found that the linear predictability of *w* was generally reduced by correcting the variance biases of *u _{p}* and

*υ*obtained from a single predictor. A similar analysis will now be carried out for the case of

_{p}*u*and

_{p}*υ*predicted separately. Again denoting the variance-corrected vector wind component predictions as and , we find

_{p}and

Once again, we define the ratio

The ratio *A* was sampled uniformly over the parameter ranges used earlier to sample Γ (Fig. 4, left). We see that there are parameter values for which *A* ≰ 1: for these (about 11% of the parameter value sets), the wind speed prediction is improved by correcting for the variance bias of *u _{p}* and

*υ*. The prediction skill is unchanged or degraded by this strategy for the remaining 89% of the parameter sets. Although correcting the variance bias increases wind speed predictability for some parameter sets, it generally remains below that of the best-predicted vector wind component. The ratio

_{p}sampled over the same parameter range (Fig. 4, right) shows little difference from the ratio Γ associated with the non-bias-corrected wind speed predictions. Thus, while there are circumstances in which it is helpful to correct for the variance bias in the vector wind components, the predictability of wind speeds is still in the great majority of situations lower than that of the along- and across-wind components.

Our analysis up to this point has focused on theoretical results for wind speed predictability, based on the assumption of Gaussian predictors and isotropic, uncorrelated, and Gaussian vector winds. To assess the relevance of these results to predicting real winds, we constructed synthetic predictions of the SeaWinds data at each spatial location in the dataset (between 60°S and 60°N). This was done by taking the observed time series of along- and across-mean wind components *u* and *υ* and corrupting these with synthetic random noise. Specifically, starting from the vector wind component standardized anomalies *u*′ and *υ*′, we constructed the prediction standardized anomalies and ,

The random processes ε_{u} and ε_{υ} are mutually uncorrelated series of independent, Gaussian random variables of unit variance and zero mean. The parameter *ρ* determines the correlation-based predictive skill of *u* and *υ* by *u _{p}* and

*υ*: for simplicity, we assumed equally predictable components for which

_{p}*ρ*

_{11}=

*ρ*

_{22}=

*ρ*. From these anomalies, the predictions

were computed using the vector wind means and standard deviations taken from observations [defining ]. The vector wind predictions *u _{p}* and

*υ*were computed for

_{p}*ρ*= 0.25, 0.5, and 0.75. From these, the correlations corr(

*w*,

_{p}*w*) between observed and predicted wind speed were computed and compared to the theoretical correlation values computed from Eq. (50). Agreement is generally good between the observed and theoretically derived wind correlations corr(

*w*,

_{p}*w*) (Fig. 5). As expected from the model results, it is observed that corr(

*w*,

_{p}*w*) <

*ρ*for the most part; note that sampling variability will contribute to the small number of locations for which this inequality is not satisfied. Also, the spatial joint distribution of the theoretically derived and observed values of corr(

*w*,

_{p}*w*) approximately parallels the 1:1 line.

The results of the theoretical model are not perfect: in particular, it predicts larger values of corr(*w _{p}*,

*w*) than are observed. A potential contributor to this bias is the fact that, in contradiction to the assumptions underlying the model, variability in the vector wind components is not Gaussian (e.g., Monahan 2004, 2006). Furthermore, as

*u*and

_{p}*υ*are themselves constructed from the non-Gaussian

_{p}*u*and

*υ*, these predictions themselves will be non-Gaussian. That the non-Gaussianity of the vector winds is an important factor in biasing the theoretical value of corr(

*w*,

_{p}*w*) is demonstrated by consideration of the spatial joint distribution of this bias with the skewness of the along-mean wind component (Fig. 6). To a first approximation, this bias varies linearly with the vector wind skewness. Note that, for

*ρ*= 0.5 and particularly for

*ρ*= 0.75, there are nonzero offsets in the bias of corr(

*w*,

_{p}*w*) for skew(

*u*) ≃ 0. These offsets demonstrate that the model bias is affected by factors other than non-Gaussianity of the vector winds.

More extensive empirical investigations of the predictability of vector wind components relative to wind speed predicted directly are presented for land surface winds over Canada in Culver and Monahan (2013) and for sea surface winds in Sun and Monahan (2013). Linear regression models with large-scale free-tropospheric predictors were used in these studies to predict variations in the statistics of surface winds from surface meteorological stations and buoys. The fully cross-validated empirical wind speed predictive skills found in these studies are in good agreement with the theoretical results presented here.

## 5. Conclusions

This study has considered the Gaussian statistical predictability of wind speed variations relative to that of the vector wind components. Analytic expressions for the correlation-based linear prediction skill of wind speed have been obtained, based on the assumption that vector wind variations can be approximated as bivariate Gaussian. The following main results have been obtained:

For any Gaussian predictor

*x*, there will be at least one vector wind component that is better predicted than the wind speed. The predictability of the wind speed relative to the best-predicted component decreases as vector wind variations become much larger than the mean vector wind. In the limit that the mean vector wind vanishes,*x*carries no direct linear predictive information about*w*irrespective of how well the vector wind components are predicted.Predictions of

*w*are always improved, relative to predicting*w*from*x*directly, by first predicting the components*u*and*υ*from*x*and then constructing the predicted speed. In this approach, the predictive skill of*w*does not vanish in the limit that the mean vector wind goes to zero. For uncorrelated, isotropic vector wind variations, there will always be some vector wind component that is better predicted than wind speed (considerably so, in general). For correlated or anisotropic vector winds, the wind speed is better predicted than any vector component over a very small parameter region. Rescaling the predicted vector wind components to correct for the low variance bias always reduces the prediction skill of wind speeds.Predicting the vector wind components separately rather than using a single predictor always results in an improved wind speed prediction. For isotropic and uncorrelated vector wind variations, it is possible through the use of separate predictions of

*u*and*υ*to obtain wind speed predictions that are better than those of either the along- or across-mean vector wind component (for about 2% of the parameter sets). Furthermore, in this limit, it is possible to improve wind speed predictions by correcting for the variance biases of the predicted vector wind components, but these improvements do not result in an increase in the parameter range over which wind speeds are better predicted than the vector winds. For the great majority of parameter sets, it is the case that wind speed predictions are less skillful than those of vector winds.The results of the theoretical analysis were broadly supported by predictions of sea surface vector winds in low and midlatitudes. Using vector wind predictions produced by corrupting observed vector wind time series with synthetic noise, it was found that wind speed prediction skills are generally less than those of the vector winds. The theoretical model predicts wind speed correlation skills that are somewhat larger than those that are observed. These biases were shown to be closely related to the skewness of the vector winds.

It may seem self-evident that the predictability of vector wind components should limit that of wind speed. However, one can imagine a situation in which the wind speed is perfectly predictable, while the wind direction has no predictability. In such a case, predictions of wind speed would be expected to be considerably better than those of any vector wind component. Such a scenario is ruled out by the results of the present study, for Gaussian vector winds and Gaussian predictors.

The analysis of observed winds in this study has indicated that skewness of the vector winds results in a small but systematic overestimate of wind speed predictability by the idealized model. Other non-Gaussian features of the vector wind component joint distribution, such as multimodality (e.g., Zhang et al. 2013), would also affect the accuracy of the model. A natural next step would be to extend the present analysis to account for the observed non-Gaussianity in *u* and *υ*; such a step is facilitated by the fact that the skewness of sea surface vector winds is a feature that is well understood (e.g., Monahan 2004, 2006). How best to specify a non-Gaussian distribution with given moments remains an open question (e.g., Monahan 2007). A possible approach would be to use a bivariate generalization of the Gram–Charlier expansion (e.g., Longuet-Higgins 1964; Lokas 1998); the resulting expressions for corr(*w _{p}*,

*w*) would be considerably more complicated than those obtained in the present study. It is important to emphasize that the bias in the modeled value of corr(

*w*,

_{p}*w*) is systematically positive; that is, the idealized Gaussian model suggests higher values of wind speed predictability than are observed over the oceans. Thus, the upper bounds presented here for the Gaussian predictability of wind speeds relative to vector wind components are conservative.

A crucial step in the theoretical characterization of the predictability of wind speed was the recognition that variations in speed are highly correlated with those of its square, so that for any variable *x* we can take corr(*x*, *w*) ≃ corr(*x*, *w*^{2}) as an excellent approximation. This result will hold for any power of *w*, so long as the truncated Taylor series approximation

is sufficiently accurate. In particular, the predictability of both *w*^{1/2} and *w*^{3} should be similar to that of *w*^{2}, to which these quantities are strongly correlated (Fig. 7). The first of these quantities is interesting because square root transforms have been used to produce wind speed distributions that are approximately Gaussian and therefore more amenable to prediction by the techniques of classical time series analysis (e.g., Carlin and Haslett 1982; Brown et al. 1984). The second of these quantities is proportional to the wind power density. The present analysis indicates that to a first approximation these quantities should be subject to similar predictability constraints (with Gaussian predictors) as wind speed.

The results of this study also relate to the prediction of any quantity which is the square root of the sum of squared Gaussians. In particular, if and vary with time (e.g., if they represent monthly-mean vector winds), then the present results indicate that the predictability of the mean vector wind magnitude will be bounded above by that of the mean vector wind components. This result has important consequences for the predictability of time-mean wind speeds (e.g., Culver and Monahan 2013; Sun and Monahan 2013).

The assumption that the bias terms I and II in Eq. (11) can be eliminated using known mean(*w*) and std(*w*) is a substantial simplification. In real prediction applications, nonstationarities or slow variations in these wind speed statistics will result in nonzero values for these bias terms. While these biases could potentially be minimized by determining mean(*w*) and std(*w*) over the recent past (rather than the whole record), weighted by a memory kernel (as in Pinson 2012), it cannot be expected that these biases will be exactly zero. By focusing on correlation-based measures of deterministic predictability, we have demonstrated the existence of limitations to the predictability of wind speed (relative to the vector wind components) even in the ideal limit when other biases can be neglected. An important direction of future study is the extension of this analysis to include these biases.

Furthermore, this study considers deterministic rather than ensemble predictions. The statistical postprocessing of ensemble predictions of surface vector winds and wind speeds is an emerging area of research which has seen considerable activity in recent years (e.g., Gneiting et al. 2008; Pinson 2012; Schuhen et al. 2012; Sloughter et al. 2013). Another important direction of future study would be to extend the present analysis to an ensemble prediction setting. In such a setting, correlation-based measures of predictive skill are not adequate, and predictability metrics appropriate to probabilistic forecasting must be used (e.g., Gneiting et al. 2008).

This study has focused on the statistical prediction of vector wind components and wind speed using Gaussian predictors, such as those characteristic of the large-scale flow used for statistical downscaling (e.g., Monahan 2012a). The limits to wind speed predictability that have been found could in principle be avoided through the use of non-Gaussian predictors. In particular, the results obtained in this study do not exclude the possibility that wind speed predictability can be increased through the use of appropriate nonlinear statistical tools, such as nonlinear regression, neural networks (e.g., Kretzschmar et al. 2004), or analog methods (e.g., Carter and Keislar 2000; Klausner et al. 2009). Furthermore, in principle excellent (non-Gaussian) predictors of *u*^{2} and *υ*^{2} could yield an excellent prediction of *w*^{2} without carrying any (linear) predictive information regarding *u* or *υ*. The limits to wind speed predictability obtained in this study apply only to the extent that the predictors are Gaussian.

## Acknowledgments

The author gratefully acknowledges helpful comments on the manuscript from Charles Curry, Aaron Culver, and Cangjie Sun. The manuscript was also greatly improved by the thoughtful comments of three anonymous reviewers. This research was supported by the Natural Sciences and Engineering Research Council of Canada.

### APPENDIX A

#### Prediction of Wind Speeds for Correlated, Anisotropic Vector Winds

For correlated, anisotropic variations (*r* ≠ 0, *σ _{u}* ≠

*σ*), the results of sections 2–4 generalize as follows: We consider first the direct prediction of

_{υ}*w*from a single Gaussian predictor

*x*, for which the general form of the inequality Eq. (14) is

We find that

In the along- and across-mean wind basis, we have

and so the predictability of *w* by *x* is also bounded above by that of the along-mean vector wind component in this general case.

For prediction of *w* from predictions of the components by the single predictor *x*, we now define

from which we then compute

As in the case of isotropic, uncorrelated variations, prediction of *w* by *w _{p}* is always superior to that by

*x*directly,

The general expression for corr(*w _{p}*,

*w*) in the along- and across-wind basis [Eq. (33) with ] is not bounded above by the predictability of either vector wind component. This can be demonstrated by sampling the ratio

uniformly over the parameter ranges 0 ≤ *ρ _{u}*,

*ρ*≤ 1, −1 ≤

_{υ}*r*≤ 1, , and −1 ≤ log

_{10}(

*σ*/

_{υ}*σ*) ≤ 1 (Fig. A1). While the great majority of parameter sets result in values of Γ < 1, this ratio exceeds unity for a small fraction (0.05%) of the parameter sets.

_{u}As was the case for isotropic, uncorrelated winds, compensating for the variance bias of the vector wind predictions by defining

generally degrades the wind speed prediction skill,

Considering finally the prediction of *w* from wind component predictions using distinct predictors, the general form of inequality (47) is

We further compute the covariance between and *w*^{2},

and the variance of the predicted wind speed,

The resulting general expression for is complicated but can be computed using these expressions.

### APPENDIX B

#### Description of the SeaWinds Data

The surface wind dataset considered in this study consists of level 3.0 gridded SeaWinds scatterometer equivalent neutral 10-m zonal and meridional winds between 60°S and 60°N from the National Aeronautics and Space Administration (NASA) Quick Scatterometer (QuikSCAT) satellite (Perry 2001), available twice daily at a resolution of 0.25° × 0.25° from 19 July 1999 to 23 November 2009. These data are available for download from the NASA Jet Propulsion Laboratory (JPL) Distributed Active Archive Center (http://podaac-www.jpl.nasa.gov/dataset/QSCAT_LEVEL_3). Those data points flagged as having been possibly corrupted by rain were excluded from the analysis. No further processing of the dataset was carried out.

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