The joint probability distribution of wind speeds at two separate locations in space or points in time completely characterizes the statistical dependence of these two quantities, providing more information than linear measures such as correlation. In this study, we consider two models of the joint distribution of wind speeds obtained from idealized models of the dependence structure of the horizontal wind velocity components. The bivariate Rice distribution follows from assuming that the wind components have Gaussian and isotropic fluctuations. The bivariate Weibull distribution arises from power law transformations of wind speeds corresponding to vector components with Gaussian, isotropic, mean-zero variability. Maximum likelihood estimates of these distributions are compared using wind speed data from the mid-troposphere, from different altitudes at the Cabauw tower in the Netherlands, and from scatterometer observations over the sea surface. While the bivariate Rice distribution is more flexible and can represent a broader class of dependence structures, the bivariate Weibull distribution is mathematically simpler and may be more convenient in many applications. The complexity of the mathematical expressions obtained for the joint distributions suggests that the development of explicit functional forms for multivariate speed distributions from distributions of the components will not be practical for more complicated dependence structure or more than two speed variables.

A fundamental issue in the characterization of atmospheric variability is
that of dependence: how the state of one atmospheric variable is related to
that of another at a different position in space, or point in time. The
simplest measure of statistical dependence, the correlation coefficient, is a
natural measure for Gaussian-distributed quantities but does not fully
characterize dependence for non-Gaussian variables. The most general
representation of dependence between two or more quantities is their joint
probability distribution. The joint probability distribution for a
multivariate Gaussian is well known, and expressed in terms of the mean and
covariance matrix

The present study considers the bivariate joint probability distribution of
wind speeds. As these are quantities which are by definition bounded below by
zero, the joint distribution and the marginal distributions are non-Gaussian.
While the correlation structure (equivalently, the power spectrum) of wind
speeds in time

A number of previous studies have constructed univariate speed distributions
starting from models for the joint distribution of the horizontal components

Both Weibull and Rice distributions have been used to model the univariate
wind speed distribution

The two probability density functions (pdfs) we will consider, the bivariate
Rice and Weibull distributions, both start with simple assumptions regarding
the distributions of the wind components. The bivariate Rice distribution
follows directly from the assumption of Gaussian components with isotropic
variance, but nonzero mean. In contrast, the bivariate Weibull distribution
is obtained from nonlinear transformations of the magnitudes of Gaussian,
isotropic, mean-zero components. While the univariate Weibull distribution
has been found to generally be a better fit to observed wind speeds than the
univariate Rice distribution

The bivariate Rice and Weibull distributions are developed in Sect. 2,
starting from discussion of the bivariate Rayleigh distribution (which is a
limiting case of both of the other models). In this section, we repeat some
of the formulae obtained by

As a starting point for developing models of the bivariate wind speed
distribution, we consider the joint distribution of the horizontal wind
vector components

the two orthogonal wind components are marginally Gaussian with isotropic and uncorrelated fluctuations:

and the cross-correlation matrix of the two vectors is

The joint distribution of the horizontal components resulting from these
assumptions is

Note that only considering the horizontal components of the wind vector implicitly restricts the resulting distributions to timescales sufficiently long that the vertical component of the wind contributes negligibly to the speed.

The joint distributions of the speeds

For

Example bivariate Rayleigh distributions

Moments of the bivariate Rayleigh distribution are given by

Plots of

The assumptions leading to the bivariate Rayleigh distribution are too
restrictive to model observed wind speeds in most circumstances. A more
general distribution results from assuming that the wind components are
Gaussian, isotropic, and uncorrelated, but with nonzero mean (Eq.

Changing variables to wind speed

The marginal distribution for the wind speeds is obtained by integrating the
joint distribution over

Finally, we obtain the expression for the bivariate Rice distribution

Note that

Integrating over

The joint moments of the bivariate Rice distribution can be evaluated using
the Taylor series expansion:

Comparison of the correlation coefficient

Defining the variables

As in Fig.

Examples of the joint Rice pdf (and the associated marginals) are presented
in Fig.

Although the bivariate Rice distribution differs from the bivariate Rayleigh
distribution only by allowing for nonzero mean wind vector components, the
resulting expressions for the joint pdf (Eq.

As in

If we start with

Evidently,

The relatively simple form of the bivariate Weibull distribution permits a
relatively simple expression for the conditional distribution

As in Fig.

Computing the moments, we obtain

Examples of the bivariate Weibull distribution for

Maximum likelihood parameter estimates for the wind speed data shown
in Figs.

Many wind datasets from different locations are available, and it is
impracticable to consider joint distributions of wind speeds from even a small fraction of
these. In this section, we will consider examples of the joint distribution
of wind speeds using data from a representative range of settings. Bivariate
distributions of wind speeds at both different locations in space and
different points in time will be considered. The sampling of the wind speeds
considered will be temporal (that is, individual samples will correspond to a
specific time for spatial joint pdfs and a specific pair of times for
temporal joint pdfs). Best-fit values of the parameters of the bivariate
Weibull and Rice distributions we present were obtained numerically as
maximum likelihood estimates (Table

Joint distributions of 500 hPa DJF 00Z wind speeds at four
different pairs of latitudes along 216

We first consider the joint distribution of 00Z December, January, and
February 500 hPa wind speeds from 1979 to 2014. These data were taken from
the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim
Reanalysis

The joint distributions of wind speeds at four pairs of latitudes along
216

In contrast, for wind speeds at (12

The wind speeds at (3

Finally, the wind speeds at (15

The temporal dependence structure of 500 hPa DJF 00Z wind speeds at
(39

Considering the spatial correlation structure of these 500 hPa winds, we
find cases in which one distribution (the Rice) is evidently a better fit to
the data than the other (the Weibull), in which neither distribution provides
a statistically significant fit to the data, and in which both distributions
fit the data equally well.

The temporal dependence structure of the wind speed at (39

Joint distribution of JAS wind speeds at 10 and 200 m measured at
Cabauw, NL.

We next consider wind speeds at altitudes of 10 and 200

Maximum likelihood estimates of the bivariate Weibull and Rice distributions
for the full nighttime data show evident disagreement between the scatter of
the data and the best-fit distributions
(Fig.

Joint distribution of DJF QuikSCAT wind speeds at (6.5

Twice-daily December, January, and February level 3.0 gridded SeaWinds
scatterometer equivalent neutral 10

Joint distributions of wind speed at (6.5

The large variations in best-fit estimates of

An indication of why increases in

This study has considered two idealized probability models for the joint distribution of wind speeds, both derived from models for the joint distribution of the horizontal wind components. The first, the bivariate Rice distribution, follows from assuming that the wind vector components are bivariate Gaussian with an idealized covariance structure. The second, the bivariate Weibull distribution, arises from nonlinear transformations of variables with a bivariate Rice distribution in the limit that the mean vector winds vanish (the bivariate Rayleigh distribution). While the bivariate Rice distribution has the advantage of being more flexible and naturally related to a simplified model for the joint distribution of the wind components, the bivariate Weibull distribution is mathematically much simpler and easier to work with. Through consideration of a range of joint distributions of observed wind speeds (over land and over the ocean; at the surface and aloft; in space and in time) the bivariate Rice distribution was shown to generally model the observations better than the bivariate Weibull distribution. However, in many circumstances the differences between the two distributions are small and the convenience of the bivariate Weibull distribution relative to the bivariate Rice distribution is a factor which may motivate its use.

The fact that the bivariate Rice distribution is easier to work with, but
less flexible, than the bivariate Weibull distribution is evident from
inspection of their analytic forms and the relative number of parameters to
fit (five vs. six). If the bivariate Weibull distribution was generically
appropriate for modelling the bivariate wind speed distribution, there would
be no need to consider more complicated models such as the bivariate Rice.
This study provides an empirical assessment of the relative practical utility
of the two models, trading off the ability to model more general dependence
structures (e.g. negatively correlated speeds) against model simplicity.
Neither the univariate nor the bivariate Weibull or Rice distributions are
expected to represent the true distributions of wind speeds

Many of the assumptions that have been made regarding the distribution of the
wind components are known not to hold in various settings. For example, the
vector wind components are generally not Gaussian, either aloft or at the
surface

While it is possible to relax the assumption of Gaussian components for
univariate speed distribution

Ultimately, it would be best for models of the joint distribution of wind
speeds to arise from physically based (if still idealized) models, as has
been done for the univariate case in

The ERA-Interim 500 hPa zonal and meridional wind
components were obtained from the European Centre for Medium-Range Weather
Forecasts at

Equation (

Goodness-of-fit of the bivariate distributions considered was assessed as
follows. For the speed dataset

After computation of

A second goodness-of-fit test proposed by

The author declares that he has no conflict of interest.

This work was supported by the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the provision of the 500 hPa data by the ECMWF, the tower data by the Cabauw Experimental Site for Atmospheric Research (CESAR), and the sea surface wind data by the NASA Jet Propulsion Laboratory Physical Oceanography Distributed Active Archive Center. The author acknowledges helpful comments on the manuscript from Carsten Abraham, Arlan Dirkson, Nikolaos Sagias, and two anonymous reviewers. Edited by: Stefano Pierini Reviewed by: Nikolaos Sagias and two anonymous referees