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.
Introduction
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 e.g.. No such general expressions
for non-Gaussian multivariate distributions exist. Copula theory
e.g. allows joint distributions to be constructed from
specified marginal distributions. However, which copula model to use for a
given analysis is not generally known a priori and is usually determined
empirically through a statistical model selection exercise.
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 e.g. and in space
e.g. has been considered, relatively
little attention has been paid to developing expressions for the joint
distribution. Previous studies have used copula methods to model horizontal
spatial dependence of wind speeds for wind power applications
and dependence of daily wind speed maxima
. While these earlier analyses have focused on probabilistic
modelling of simultaneous wind speed values at different spatial locations in
the horizontal, dependence structures in the vertical (e.g. for vertical
interpolation of wind speeds) or in time are also of interest. For example,
an analysis in which the need for an explicit parametric form for the joint
distribution of wind speeds at different altitudes has arisen is the hidden
Markov model (HMM) analysis considered in . In an HMM analysis
of continuous variables, it is necessary to specify the parametric form of
the joint distribution within each hidden state. In , the joint
distribution of wind speeds at 10 and 200 m and the potential
temperature difference between these altitudes was modelled as a multivariate
Gaussian distribution, despite the fact that for at least the speeds this
distribution cannot be strictly correct. This pragmatic modelling
approximation was made because of the absence of a more appropriate
parametric distribution for the quantities being considered. The alternative
approach of using the wind components at the two levels (for which the
multivariate Gaussian model may be a better approximation) instead of the
speeds directly has the downside of increasing the dimensionality of the
state vector from three to five, dramatically increasing the number of
parameters to be estimated (with the covariance matrices in particular
increasing from 9 to 25 elements) and reducing the statistical robustness of
the results.
A number of previous studies have constructed univariate speed distributions
starting from models for the joint distribution of the horizontal components
e.g.. One useful benefit of this approach
is that it allows the statistics of the speed and the components to be
related to each other. The specific goal of the present study is to extend
this approach to bivariate distributions, constructing models of the joint
probability distribution of wind speeds that are directly connected to the
joint distributions of the horizontal components of the wind. As the
following results will demonstrate, generalizing this approach to the
bivariate speed distribution results in rather complicated mathematical
expressions. Expressions for multivariate distributions of more than two
speeds will be even more complicated, and may not be analytically tractable.
Through the analysis of the bivariate speed distribution, we will probe how
far the development of closed-form, analytic expressions for parametric speed
distributions based on distributions of components can practically be
extended.
Both Weibull and Rice distributions have been used to model the univariate
wind speed distribution cf., and models of
multivariate distributions with Weibull or Ricean marginals have been
developed e.g.. Much
of this work has been done in the context of wireless communications
: the present study builds upon the results of these
earlier analyses.
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 particularly over the oceans,
e.g., the direct connection of the Rice
distribution to the distribution of the components (which the Weibull
distribution does not have) is useful from a modelling and theoretical
perspective e.g.. The
six-parameter bivariate Rice distribution that we will consider is more
flexible than the five-parameter bivariate Weibull distribution, and able to
model a broader range of dependence structures. Furthermore, it is directly
connected to the univariate distributions and dependence structure of the
wind components. However, the bivariate Weibull distribution is
mathematically much simpler than the bivariate Rice distribution and easier
to use in practice. Other flexible bivariate distributions for non-negative
random variables exist, such as the α-μ distribution discussed in
. Because the Weibull and Rice distributions are common models
for the univariate wind speed distributions, this study will focus
specifically on their bivariate generalizations.
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 , , and
for completeness and because of notational differences between this study and
the earlier ones. In Sect. 3, the ability of these distributions to model
wind speed data from the middle troposphere, and from the near-surface flow
over land and the ocean, is considered. Examples of dependence structures in
both space (horizontally and vertically) and in time are considered. A
discussion and conclusions are presented in Sect. 4.
Empirical models of the bivariate wind speed distribution
As a starting point for developing models of the bivariate wind speed
distribution, we consider the joint distribution of the horizontal wind
vector components ui=(ui,vi), i=1, 2 (where the
subscripts i denote wind vectors at two different locations, two different
points in time, or both). In particular, we assume that
the two orthogonal wind components are marginally Gaussian with isotropic and uncorrelated fluctuations:uivi∼Nu‾iv‾i,σi200σi2i=1,2,
and the cross-correlation matrix of the two vectors is2corr(u1,u2)=corr(u1,u2)corr(u1,v2)corr(v1,u2)corr(v1,v2)=μ1μ2-μ2μ1=ρcosψρsinψ-ρsinψρcosψ,where we have expressed the correlations in both Cartesian and polar
coordinates: (μ1,μ2)=(ρcosψ,ρsinψ), with 0≤ρ=μ12+μ221/2≤1. This assumed
correlation structure implies that the correlation matrix becomes diagonal
when the vector u2 is rotated through the angle -ψ:corr(u1,R(-ψ)u2)=ρ00ρ,whereR(-ψ)u2=cosψsinψ-sinψcosψu2v24=u2cosψ+v2sinψ-u2sinψ+v2cosψ.
The joint distribution of the horizontal components resulting from these
assumptions is
p(u1,u2,v1,v2)=1(2π)2σ12σ22(1-ρ2)×exp-12(1-ρ2)(u1-u‾1)2σ12+(v1-v‾1)2σ12+(u2-u‾2)2σ22+(v2-v‾2)2σ22-2μ1[(u1-u‾1)(u2-u‾2)+(v1-v‾1)(v2-v‾2)]σ1σ25-2μ2u1-u‾1(v2-v‾2)-(v1-v‾1)(u2-u‾2)σ1σ2.
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.
Bivariate Rayleigh distribution
The joint distributions of the speeds wi=ui2+vi2
obtained from the pdf Eq. () when both vector
wind components are mean-zero is the bivariate Rayleigh distribution
e.g.. Transforming variables to wind speed wi and
direction θi=tan-1(vi/ui), the joint distribution (Eq. ) with u‾i=vi‾=0,
i=1,2, becomes
p(w1,w2,θ1,θ2)=w1w2(2π)2σ12σ22(1-ρ2)exp-12(1-ρ2)w12σ12+w22σ226×expρ1-ρ2w1w2σ1σ2cos(θ1-θ2+ψ).
Integrating over the wind directions to obtain the marginal distribution for
the wind speeds, we obtain
p(w1,w2)=w1w2σ12σ22(1-ρ2)exp-12(1-ρ2)w12σ12+w22σ227Ioρ1-ρ2w1w2σ1σ2,
where we have used the fact that
∫02πeαcosθcoskθdθ=2πIk(α),
where Ik(z) is the modified Bessel function of order k. Note that the
correlation angle ψ drops out after integration over θ1 and
θ2. As a result, for the bivariate Rayleigh distribution,
p(w1,w2) depends only on the three parameters
(σ1,σ2,ρ).
For ρ=0, p(w1,w2) factors as the product of the marginal
distributions of w1 and w2:
p(w1,w2)=w1σ12exp-w122σ12w1σ12exp-w122σ12,
and the wind speeds are independent. As ρ→1, we can use the
asymptotic result
I0(x)∼ex2πx(x≫1)
to show that
p(w1,w2)→w1σ12exp-w122σ12δw1σ1-w2σ2,
where δ(⋅) is the Dirac delta function. In this limit, w1 and
w2 are perfectly correlated and Rayleigh distributed.
Example bivariate Rayleigh distributions p(w1,w2) for ρ=0,0.5, and 0.85, with w1 and w2 scaled respectively by
σ1 and σ2. The upper and left subpanels show the marginal
distributions of w1 and w2 respectively. These marginal
distributions are the same for all three panels.
Moments of the bivariate Rayleigh distribution are given by
12Ew1mw2n=2m/22n/2σ1mσ2nΓ1+m2Γ1+n22F1-m2,-n2,1;ρ2,
where 2F1(α,β,γ;z) is the hypergeometric function
. In particular, we have
13mean(wi)=σiπ2,14var(wi)=2σi21-π4,15corr(w1,w2)=π4-π2F1-12,-12,1;ρ2-1.
Because 2F1(-1/2,-1/2,1;ρ2) is an increasing function of
ρ2 with 2F1(-1/2,-1/2,1;0)=1, corr(w1,w2)
must be non-negative for the bivariate Rayleigh distribution.
Plots of p(w1,w2) for the three values ρ=0, 0.5, and 0.85
are presented in Fig. , along with the marginal
distributions of w1 and w2 (which are the same for all three
panels). The marginal distributions are positively skewed and the contours of
the joint distributions are more tightly concentrated below and to the left
of their peaks than elsewhere. As expected, the distributions become more
tightly concentrated around the 1 : 1 line as the dependence parameter
ρ increases.
Bivariate Rice distribution
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 wi and direction θi, the
joint distribution becomes
p(w1,w2,θ1,θ2)=w1w2(2π)2σ12σ22(1-ρ2)×exp-12(1-ρ2)(w12+u‾12+v‾12)σ12+(w22+u‾22+v‾22)σ22-2μ1(u‾1u‾2+v‾1v‾2)σ1σ2-2μ2(u‾1v‾2-v‾1u‾2)σ1σ2×exp([w1σ1a1cosθ1+w2σ2a2cosθ2+w1σ1b1sinθ1+w2σ2b2sinθ216+11-ρ2w1w2σ1σ2μ1cos(θ1-θ2)+μ2sin(θ1-θ2)]),
where
a1=11-ρ2u‾1σ1-μ1u‾2σ2-μ2v‾2σ217=11-ρ2U‾1σ1cosϕ1-ρU‾2σ2cos(ϕ2-ψ),b1=11-ρ2v‾1σ1-μ1v‾2σ2+μ2u‾2σ218=11-ρ2U1‾σ1sinϕ1-ρU2‾σ2sin(ϕ2-ψ),a2=11-ρ2u‾2σ2-μ1u‾1σ1+μ2v‾1σ119=11-ρ2U‾2σ2cosϕ2-ρU‾1σ1cos(ϕ1+ψ),b2=11-ρ2v‾2σ2-μ1v‾1σ1-μ2u‾1σ120=11-ρ2U2‾σ2sinϕ2-ρU1‾σ1sin(ϕ1+ψ),
and where we have defined the magnitude and direction of the mean vector
wind:
(u‾i,v‾i)=U‾i(cosϕi,sinϕi).
The marginal distribution for the wind speeds is obtained by integrating the
joint distribution over θ1 and θ2. To evaluate this
integral, we make use of the result
1(2π)2∫02π∫02πexpα1cosθ1+α2cosθ2+β1sinθ1+β2sinθ2+γcos(θ1-θ2+ψ)dθ1dθ2=∑k=0∞ϵkcosktan-1β1α1-tan-1β2α2+ψ22×Ikα12+β12Ikα22+β22Ik(γ),
where
ϵk=1k=02k≠0
and it is important that tan-1(b/a) be evaluated as the angle between
the vector (a,b) and the vector (1,0) (that is, as the four-quadrant
inverse tangent). Equation () follows from the
Fourier series
eccosθ=∑k=0∞ϵkIk(c)coskθ
along with repeated use of trigonometric identities and the integral
Eq. ().
Finally, we obtain the expression for the bivariate Rice distribution
p(w1,w2)=w1w2σ12σ22(1-ρ2)×exp-12(1-ρ2)(w12+u‾12+v‾12)σ12+(w22+u‾22+v‾22)σ22-2μ1(u‾1u‾2+v‾1v‾2)σ1σ2-2μ2(u‾1v‾2-v‾1u‾2)σ1σ2×∑k=0∞ϵkcosktan-1b1a1-tan-1b2a2+tan-1μ2μ1×Ikw1σ1a12+b12Ikw2σ2a22+b2225×Ikρ1-ρ2w1w2σ1σ2.
Expressed in terms of the magnitude and direction of the mean
wind vectors,
p(w1,w2)=w1w2σ12σ22(1-ρ2)×exp-12(1-ρ2)w12+U‾12σ12+w22+U‾22σ22-2ρU‾1U‾2cos(ϕ1-ϕ2+ψ)σ1σ2×∑k=0∞ϵkcos(kν)Ikw1σ1a12+b12Ikw2σ2a22+b2226×Ikρ1-ρ2w1w2σ1σ2,
where
27ν=tan-1(1-ρ2)U‾1U‾2σ1σ2sin(ϕ1-ϕ2+ψ)(1+ρ2)U‾1U‾2σ1σ2cos(ϕ1-ϕ2+ψ)-ρU‾12σ12+U‾22σ22
and
28ai2+bi2=11-ρ2U‾i2σi2+ρ2U‾3-i2σ3-i2-ρ2U‾1U‾2σ1σ2cos(ϕ1-ϕ2+ψ)12.
When numerically evaluating the bivariate Rice distribution, it is convenient
to transform the infinite series in Eqs. () and
() into an integral which is then
approximated numerically (Appendix A).
Note that p(w1,w2) depends on the relative orientation of the mean
wind vectors and the correlation angle ψ only through the combination
ϕ1-ϕ2+ψ. Because of this symmetry, the quantities
ϕ1-ϕ2 and ψ cannot be determined individually from wind
speed data alone. As a result, p(w1,w2) is determined by six
parameters:
(U‾1,σ1,U‾2,σ2,ρ,ϕ1-ϕ2+ψ). For particular applications, it may be
appropriate to fix either ϕ1-ϕ2 or ψ, allowing the other
angle to be estimated from data. For example, when considering the temporal
dependence structure of winds assumed to have stationary statistics, it can
be assumed that ϕ1-ϕ2=0. The bivariate Rice distribution also
has the discrete symmetry that it is invariant under the transformation
ϕ1-ϕ2+ψ→-(ϕ1-ϕ2+ψ). Note that
these symmetries are in addition to the invariance of the distribution of
components (Eq. ) to the rotation of the
coordinate system θi→θi+Δθ, i=1,2, under which the angles ϕ1-ϕ2 and ψ are individually
invariant.
Integrating over w2 to obtain the marginal distribution for w1 we
obtain the univariate Rice distribution
∫0∞p(w1,w2)dw2=w1σ12exp-w12+U‾122σ12I0w1U‾1σ12,
with mean and variance
30mean(w1)=σ1π21F1-12,1,-U‾122σ1231var(w1)=2σ12+U‾12-σ12π21F12-12,1,-U‾122σ12
(with equivalent expressions for w2 obtained by integrating over
w1), where 1F1(α;β;z) is the confluent
hypergeometric function . Equation () follows
from Eq. () using the integral ∫0∞xe-ax2Ik(bx)Ik(cx)dx=3212aexpb2+c24aIkbc2a,
Neumann's theorem ∑k=0∞ϵkIk(x)Ik(y)coskϕ=I0x2+y2+2xycosϕ,
and the fact that
cosν=34(1+ρ2)U‾1σ1U‾2σ2cos(ϕ1-ϕ2+ψ)-ρU‾12σ12+U‾22σ22(1-ρ2)2(a12+b12)a22+b22.
Note that each wind speed marginal distribution depends only on
the magnitude of the mean wind vector, while the joint distribution also
depends on the angle between the two mean wind vectors. Furthermore, as ρ→0 only the first term contributes to the infinite series in Eq. (), and the joint distribution reduces to the
product of the marginals.
The joint moments of the bivariate Rice distribution can be evaluated using
the Taylor series expansion:
35Ikρ1-ρ2w1w2σ1σ2=∑j=0∞1j!(j+k)!ρ2(1-ρ2)2j+kw1σ12j+kw2σ22j+k,
which allows the double integral defining the moments to factorize as the
products of individual integrals over w1 and w2 that can be
evaluated using
36∫0∞xμe-αx2Ik(βx)dx=βkΓ(k+μ+1)/22k+1α(μ+ν+1)/2Γ(k+1)1F1k+μ+12;k+1;β24α.
The resulting expression for the joint moments is
Ew1mw2n=2m/22n/2σ1mσ2n(1-ρ2)1+m/2+n/2×exp-12(1-ρ2)U‾12σ12+U‾22σ22-2ρU‾1U‾2cos(ϕ1-ϕ2+ψ)σ1σ2×∑l=0∞∑k=0lϵkρ2l-kcos(kν)Γl+m2+1Γl+n2+1l!(l-k)!(k!)2×(1-ρ2)(a12+b12)(a22+b22)2k×1F1l+m2+1;k+1;(1-ρ2)(a12+b12)237×1F1l+n2+1;k+1;(1-ρ2)(a22+b22)2. When the mean vector winds are equal to zero, only
the k=0 terms contribute to this expression and Eq. () is recovered.
Comparison of the correlation coefficient
corr(w1,w2) for bivariate Rice distributed variables
(Eq. ) with the approximate expression
Eq. () for the parameter values
(U‾1/σ1,U‾2/σ2,ϕ1-ϕ2+ψ)=(3,2,0) and (0.5,4,0).
Defining the variables Vi=U‾i/2σi, the correlation coefficient
between w1 and w2 is given by
38corr(w1,w2)=(1-ρ2)2exp-V12+V22-2V1V2cos(ϕ1-ϕ2+ψ)(1-ρ2)G(V1,V2,ρ,ϕ1-ϕ2+ψ)-π41F1-12;1;-V121F1-12;1;-V221+V12-π41F1-12;1;-V12121+V22-π41F1-12;1;-V2212,
where
G(V1,V2,ρ,ϕ1-ϕ2+ψ)=∑l=0∞∑k=0lϵkρ2l-kcos(kν)Γ2l+32l!(l-k)!(k!)2×(1-ρ2)(a12+b12)(a22+b22)2k×1F1l+32;k+1;(1-ρ2)(a12+b12)239×1F1l+32;k+1;(1-ρ2)(a22+b22)2.
The correlation coefficient corr(w1,w2) depends only on the
four quantities (U‾1/σ1,U‾2/σ2,ϕ1-ϕ2+ψ,ρ).
considered the correlation structure of wind speeds using
the approximation corrw1,w2≃corrw12,w22. The assumed covariance
structure of the wind components then results in the approximate expression:
corrw1,w2≃ρ2+u‾1u‾2σ1σ2+v‾1v‾2σ1σ2μ1+u‾1v‾2σ1σ2-v‾1u‾2σ1σ2μ21+u‾12+v‾12σ121+u‾22+v‾22σ2240=ρ2+U‾1U‾2σ1σ2cosϕ1-ϕ2+ψρ1+U‾12σ121+U‾22σ22.
Plots of the correlation coefficient (Eq. ) and
the approximation (Eq. ) as functions of ρ are
shown in Fig. for
(U1‾/σ1,U‾2/σ2,ϕ1-ϕ2+ψ)
= (3,2,0) and (1,5,0). Agreement between the exact and approximate values of
the correlation coefficient is reasonably good in both cases, with the
largest discrepancies generally occurring for larger absolute values of
ρ. Note that negative wind speed correlation values are permitted by the
bivariate Rice distribution.
As in Fig. for the bivariate Rice
distribution with
(U‾1,σ1,U‾2,σ2)=(6,4,2,5) and (ρ,ϕ1-ϕ2+ψ)=(0.8,π),(0,0), and
(0.8,0).
Examples of the joint Rice pdf (and the associated marginals) are presented
in Fig. for
(U‾1,σ1,U‾2,σ2)=(6,4,2,5) and (ρ,ϕ1-ϕ2+ψ)=(0.85,π),(0,0), and
(0.85,0). By construction, the marginal distributions are the same in each
panel. The distributions of both w1 and w2 are positively skewed,
and take respective maxima at values of about σ1 and just less than
2σ2. For the different values of the dependence parameter ρ,
the joint distributions have considerably different shapes. The joint
distribution for (ρ,ϕ1-ϕ2+ψ)=(0.8,π) describes (weakly)
negatively correlated variables with a nonlinear dependence structure evident in ridges
of enhanced probability extending to the left and right upward from the
probability maximum. For (ρ,ϕ1-ϕ2+ψ)=(0,0), probability
contours are concentrated towards smaller values of w1 and w2 (as
is the case for the marginal distributions). Finally, w1 and w2 are
evidently positively correlated for (ρ,ϕ1-ϕ2+ψ)=(0.8,0), with a slight curvature in the shape of the distribution indicating
the existence of some nonlinear dependence.
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. )
and the moments (Eq. ) are much more complicated
for the bivariate Rice than the bivariate Rayleigh. Furthermore, while the
univariate Rice distribution is a convenient model for the pdf of wind speed,
observed winds show clear deviation from Ricean behaviour
e.g.. We will therefore consider another
model of the bivariate wind speed distribution with Weibull marginals, which
turns out to result in simpler mathematical expressions (at the cost of a
more artificial derivation than that of the bivariate Rice distribution).
Bivariate Weibull distribution
As in and , we obtain the bivariate Weibull
distribution from the bivariate Rayleigh distribution through separate power
law transformations of w1 and w2. The pdf of a Weibull distributed
variable is
p(x)=baxab-1exp-xab,
with moments
Exm=amΓ1+mb,
where a and b are denoted the scale and shape parameters respectively.
The Rayleigh distribution is a special case of the Weibull distribution with
a=2σ and b=2. Weibull distributed variables remain
Weibull under a power law transformation, with suitably modified scale and
shape parameters: if x is Weibull with scale parameter a and shape
parameter b, xk will be Weibull with scale parameter ak and shape
parameter b/k. A joint wind speed distribution with Weibull marginal
distributions can therefore be constructed from a joint Rayleigh distribution
using the appropriate power law and scale transformations.
If we start with (x1,x2) as bivariate Rayleigh distributed with
σi=1/2, i=1,2, we obtain marginal Weibull
distributions with specified scale and shape parameters through the
transformations
wi=aixi2/bi.
The joint pdfs transform as
p(w1,w2)=∂x1w1∂x2w1∂x1w2∂x2w2-1p(x1,x2)
and so we obtain the bivariate Weibull distribution
p(w1,w2)=11-ρ2b1b2a1a2w1a1b1-1w2a2b2-1×exp-1(1-ρ2)w1a1b1+w2a2b245×I02ρ(1-ρ2)w1a1b1/2w2a2b2/2.
An analogous approach to constructing bivariate Weibull distributions through
nonlinear transformations of a bivariate Gaussian was followed in
; the resulting expressions are considerably more complicated
than those considered here.
Evidently, p(w1,w2) factorizes into the product of the marginal
distributions as ρ→0. As ρ→1, we can use Eq. () to make the approximation
p(w1,w2)≃11-ρ2b1b2a1a2w1a1b1-1w2a2b2-1×exp-1(1-ρ2)w1a1b1+w2a2b2×4πρ(1-ρ2)w1a1b1/2w2a2b2/2-1/2×exp2ρ(1-ρ2)w1a1b1/2w2a2b2/2≃b1a1w1a1b1-1exp-w1a1b1×b22a2w1a1-b1/4w2a23b2/4-1×δw1a1b1/2-w2a2b2/246=p(w1)δw2-a2w1a1b1/b2,
where the last equality follows from the fact that
δf(x)-α=1f′(α)δx-f-1(α).
As expected, w1 and w2 are completely dependent in the limit that
ρ→1 (although they are not perfectly correlated if b1≠b2 as the functional relationship
w2=a2w1a1b1/b2
between the two variables will be nonlinear).
The relatively simple form of the bivariate Weibull distribution permits a
relatively simple expression for the conditional distribution
p(w2|w1)=p(w1,w2)p(w1)=b2a2w2a2b2-1exp-w2a2b2×11-ρ2exp-ρ21-ρ2w1a1b1+w2a2b249×I02ρ(1-ρ2)w1a1b1/2w2a2b2/2.
The factor in the second set of braces characterizes how
conditioning on the value of w1 changes the distribution of w2 from
its marginal distribution and corresponds to a copula density;
e.g.. Note that for w1 sufficiently large we can write
p(w2|w1)≃14πρ(1-ρ2)b2a2w2a23b2/4-1w1a1-b1/450×exp-11-ρ2ρw1a1b1/2-w2a2b2/22.
For ρ not too close to zero, the conditional distribution for large
w1 is concentrated around the nonlinear regression curve
w2a2b2=ρ2w1a1b1.
As in Fig. for the bivariate Weibull
distribution with (a1,b1,a2,b2)=(4,1.5,5,7) and ρ=0,0.7, and 0.95 and the speeds w1,w2 scaled respectively by the
Weibull scale parameters a1, a2.
Computing the moments, we obtain
52Ew1mw2n=a1ma2nΓ1+mb1Γ1+nb22F1-mb1,-nb2,1;ρ2.. The correlation coefficient is then given by
53corr(w1,w2)=Γ1+1b1Γ1+1b22F1-1b1,-1b2,1;ρ2-1Γ1+2b1-Γ21+1b1Γ1+2b2-Γ21+1b2.
For b1,b2>1 (the relevant range of shape parameters for
wind speeds), 2F1(-1/b1,-1/b2,1;ρ2) is an increasing
function of ρ2 with 2F1(-1/b1,-1/b2,1;0)=1.
Therefore, the bivariate Weibull distribution is unable to represent
situations in which the wind speeds are negatively correlated.
Examples of the bivariate Weibull distribution for (a1,b1,a2,b2)=(4,1.5,5,7) and values of ρ=0,0.7, and 0.95 are shown in
Fig. . Again, the marginal distributions in all
three cases are the same by construction. The distribution of w1 is
positively skewed with a maximum near a value of w1=0.5a1, while
that of w2 is negatively skewed with a maximum near w2=a2. For
ρ=0 the joint distribution is simply the product of the marginals. As
ρ increases, w1 and w2 become positively correlated - although
the correlation is weak even for ρ=0.7 for this set of parameter
values. At the value of ρ=0.95, while the correlation of the two
variables is only moderate, a strong nonlinear dependence is evident in the
concentration of the distribution around the curve given by Eq. ().
Maximum likelihood parameter estimates for the wind speed data shown
in Figs. , ,
, and .
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 ), with
ϕ1-ϕ2 set to zero. Goodness-of-fit of the distributions was
assessed using a statistical test described in Appendix B. In order to
distinguish how well the parametric joint distributions model the marginal
distributions from how well they represent dependence between variables, the
goodness-of-fit analyses were repeated for each pair of time series with the
values of one of the pair shuffled in time. This shuffling destroys the
dependence structure without affecting the distributions of the marginals.
Use of a bivariate analysis rather than separate univariate goodness-of-fit
tests for the marginals allows direct comparisons of p-values, as exactly
the same test is used for the original and shuffled data.
Joint distributions of 500 hPa DJF 00Z wind speeds at four
different pairs of latitudes along 216∘ W. Wind speeds at the two
latitudes quoted in the column headings are respectively denoted w1 and
w2. (a–d) Scatterplots of wind speed data.
(e–h) Maximum likelihood bivariate Weibull pdfs (white contours)
and kernel density estimates of the observed joint pdf (colours). The
p-value of a goodness-of-fit test with the null hypothesis that the
observed wind speed data are drawn from the corresponding best-fit bivariate
Weibull distribution is given. The values in brackets are the p-values
obtained when the dependence structure is eliminated by shuffling the values
of w2 in time. (i–l) As in the middle row, for the best-fit
bivariate Rice distribution. Values of the best-fit model parameters are
given in Table .
Wind speeds at 500 hPa
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 , subsampled to every second day to minimize
the effect of serial dependence on the goodness-of-fit test
. The wind speed data were computed as the magnitude of
zonal and meridional components.
The joint distributions of wind speeds at four pairs of latitudes along
216∘ W are presented in Fig. . A
moderately strong negative correlation (r=-0.55) is evident between wind
speeds at (39∘ S, 216∘ W) and (54.75∘ S,
216∘ W) (Fig. a). Because it is unable to
model a negative correlation between wind speeds, the best-fit bivariate
Weibull pdf differs substantially from the distribution of the observed winds
(Fig. e). The goodness-of-fit test correspondingly
rejects the null hypothesis that the observations are drawn from this
distribution (p=0). In contrast, the bivariate Rice distribution provides a
reasonable model of the joint distribution of wind speeds at these two
locations (Fig. i) and the goodness-of-fit test
provides no evidence that these data are statistically incompatible with this
distribution (p=0.31). For wind speeds at these two locations, the
bivariate Rice distribution is evidently a better model than the bivariate
Weibull distribution.
In contrast, for wind speeds at (12∘ N, 216∘ W) and
(15.75∘ N, 216∘ W), the null hypotheses of being drawn from
either the bivariate Rice or Weibull distributions are rejected at the 95 %
significance level for both distributions. These wind speeds are weakly
correlated (r=0.37) but show evidence of nonlinear dependence. The joint
pdf of the observed speeds is characterized by two ridges of high probability
extending to the right, upward and downward away from the region of maximum
probability (Fig. b, f, j). These ridges are not
captured by either of the best-fit bivariate Weibull or Rice distributions,
although a hint of this structure is evident in the Rice distribution. While
the fit of one or both of these parametric distributions to these observed
wind speed data may be sufficiently good for practical applications,
nevertheless we can confidently exclude the possibility that these data are
drawn from either distribution.
The wind speeds at (3∘ S, 216∘ W) and (0.75∘ N,
216∘ W) are correlated (r=0.68) and their scatter clusters
around a straight line extending away from the origin
(Fig. c). Both the best-fit bivariate Weibull and
Rice distributions appear to the eye to be good fits to the data (Fig. g, k), and in neither case can the null hypothesis
be rejected that the data are drawn from these distributions. Only small
differences exist between the two best-fit distributions for these data.
Finally, the wind speeds at (15∘ S, 216∘ W) and
(45∘ S, 216∘ W) are uncorrelated (r=0.06) and fit
sufficiently well by both the bivariate Weibull and Rice distributions
(Fig. d, h, l) that in neither case is the null
hypothesis rejected. As in the previous example, the bivariate best-fit Rice
and Weibull distributions are essentially indistinguishable for these data.
The temporal dependence structure of 500 hPa DJF 00Z wind speeds at
(39∘ S, 216∘ W) for lags s= 0.25 to 4 days, with
w1=w(tn) and w2=w(tn+s). (a–c) Scatterplots of
wind speeds separated by 0.5 days (first column), 1.5 days (second column),
and 3 days (third column). (e–g) Maximum likelihood bivariate
Weibull distribution (white contours) and kernel density estimate of the
joint pdf of the lagged data. The p-value of a goodness-of-fit test of the
bivariate Weibull fit is quoted in white. The values in brackets are the
p-values obtained when the dependence structure is eliminated by shuffling
the values of w2 in time. (i–k) As in panels (e–g) for the bivariate Rice
distribution. Panels (d, h, l) show results at a range of different
time lags. (d)p-values of bivariate goodness-of-fit tests for
bivariate Weibull (solid black) and bivariate Rice (solid red). The dashed
lines correspond to the p-values for shuffled w2. The thin black curve
is the 0.95 significance level. (h) Best-fit estimate of the
parameter ρ of the bivariate Weibull distribution. (l) Best-fit
estimates of ρ (black) and cosϕ (blue) for the Rice distribution.
Values of the best-fit model parameters are given in
Table .
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. p-values from the analyses with temporally
shuffled data, quoted in parentheses in Fig. , show
that for none of the four pairs of wind speeds considered can the null
hypotheses of univariate Weibull or univariate Rice distributions for
the pair of
marginals be rejected. This result indicates that for these data the
rejection of the full bivariate distributions occurs because of a failure to
adequately represent the dependence structure between the two random
variables.
The temporal dependence structure of the wind speed at (39∘ S,
216∘ W) is illustrated in Fig. . As
with the previous calculations, the pairs of lagged wind speeds
(w(tn),w(tn+s)) were subsampled to 2-day resolution to minimize the
effect of serial dependence on the results of the goodness-of-fit tests. As
the lag increases, the value of the dependence parameter ρ decreases as
expected for both the Weibull and Rice distributions
(Fig. h, l). For most lags, the null hypothesis
of the Weibull distribution as a model for the joint distribution is rejected
(p<0.05; Fig. d). The rejection of the null
hypothesis of a bivariate Weibull distribution is most robust for lags
shorter than 3 days. In contrast, the null hypothesis of a bivariate Rice
distribution is rejected less often than it is not – although p<0.05 for
more than one-third of the lags (Fig. d).
Inspection of the example distributions shown demonstrates that the bivariate
Weibull distributions are broader around their principal axis for small to
intermediate wind speeds in a way that is not consistent with observations
(Fig. e–g). Such structures are not seen in
the best-fit Rice distributions (Fig. i–k).
Note that for lags of 0.5 and 1.5 days the observed distributions suggest a
flaring out of the joint distribution for large wind speeds that is accounted
for by neither the bivariate Weibull nor Rice distributions. There is good
evidence that these data were not drawn from a bivariate Weibull
distribution, and the evidence that they are drawn from a bivariate Rice
distribution is not strong. p-values of fits obtained after shuffling of
the lagged wind speed time series (in brackets in
Fig. e–g, i–k and dashed lines in
Fig. d) are generally larger than those of the
unshuffled data: with a few exceptions (e.g. a lag of 1.75 days), the
rejection of the null hypothesis of the full dataset being drawn from either
of the parametric distributions considered is not associated with a failure
to fit the marginals.
Joint distribution of JAS wind speeds at 10 and 200 m measured at
Cabauw, NL. (a–d) Wind speed scatterplots. (e–h) Kernel
density estimate of the joint pdf of wind speeds (colour) and maximum
likelihood bivariate Weibull distribution (contours). The p-value of a
goodness-of-fit test of the bivariate Weibull distribution is quoted in white. The
values in brackets are the p-values obtained when the dependence structure
is eliminated by shuffling the values of w200 in time.
(i–l) As in the middle row, but for the best-fit bivariate Rice
distribution (contours). (a, e, i) All nighttime data
(20:00–05:00 UTC). (b, f, j) Nighttime data conditioned on being
in regime R2 (very stable boundary layer). (c, g, k) Nighttime
data conditioned on being in regime R1 (weakly stable boundary layer).
(d, h, l) All daytime data (08:00–16:00 UTC). Values of the
best-fit model parameters are given in
Table .
Wind speeds over land at 10 m and 200 m
We next consider wind speeds at altitudes of 10 and 200 m measured
from a 213 m tower in Cabauw, Netherlands (51.971∘ N,
4.927∘ E), maintained by the Cabauw Experimental Site for
Atmospheric Research CESAR; with 10 min
resolution from 1 January 2001 through 31 December 2012. We will focus on
data from July, August, and September (JAS), separated into daytime
(08:00–16:00 UTC) and nighttime (20:00–05:00 UTC) periods. These data
were subsampled in time to account for serial dependence. Only every 50th
point was used in the following analysis. A small number of zero wind speed
values were removed from the dataset.
demonstrated the existence of two distinct regimes of
the nocturnal boundary layer in these data, corresponding to the very and
weakly stratified boundary layers vSBL and wSBL; e.g..
These regimes, denoted respectively R1 and R2, were separated in
using a two-state HMM. Conditioning the data on the HMM state,
the scatterplot of wind speeds at 10 and 200 m separates into two distinct
populations (Fig. a–c). The scatter of wind
speeds at these two altitudes shows no evident regime structure during the
day (Fig. d). In all cases, the wind speeds at the
two altitudes are highly correlated.
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. e, i). Goodness-of-fit tests for both
distributions were rejected (with p=0 in both cases). When conditioned on
being in either regime R1 or R2, both the bivariate Rice and
Weibull distributions result in much better representations of the nighttime
data (Fig. f, g, j, k). In all cases the p-values
exceed 0.05, so the fits cannot be rejected at the 95 % significance level.
While neither the bivariate Rice nor Weibull distributions is a good
probability model for the full joint distribution of wind speeds at these two
altitudes, both are reasonable models for the distributions conditioned by
regime occupation. Finally, the fits of neither the bivariate Rice
nor
Weibull distributions to the daytime data are statistically significant at
the 95 % significance level. In particular, the joint Rice distribution is
too broad (relative to observations) for small values of w10 and
w200 and neither distribution is broad enough at higher wind speed
values. p-values of fits with w200 shuffled in time are all larger
than those of fits to the original data; only for the Rice fit to the full
nighttime data is the shuffled p-value below 0.05. As seen in the previous
data considered, there is no systematic evidence that the failure of the
joint Weibull or Rice distributions to model the joint distributions of the
Cabauw data results from a failure to model the marginals.
Joint distribution of DJF QuikSCAT wind speeds at (6.5∘ S,
135∘ W), denoted w1, and westward along a zonal transect to
165∘ W (denoted w2). (a–c) Scatterplots of wind speed
at (6.5∘ S, 135∘ W) and at the points (6.5∘ S,
162∘ W), (6.5∘ S, 152∘ W), and (6.5∘ S,
142∘ W). (e–g) Kernel density estimate of the joint pdf
(colour) as well as the maximum likelihood bivariate Weibull distribution
(white contours). The p-value of a goodness-of-fit test of the bivariate
Weibull distribution is quoted in white. The values in brackets are the
p-values obtained when the dependence structure is eliminated by shuffling
the values of w2 in time. (i–k) As in panels (e–g) but for the
bivariate Rice distribution. (d)p-values of the bivariate
goodness-of-fit tests for the wind speeds along the transect, for the
bivariate Weibull distribution (solid black) and for the bivariate Rice
distribution (solid red). The dashed lines correspond to the p-values for
shuffled w2. (h) Estimate of the parameter ρ from the
best-fit bivariate Weibull distribution along the transect.
(l) Estimate of the parameters ρ (black) and cosψ (blue)
from the best-fit bivariate Rice distribution along the transect. Values of
the best-fit model parameters are given in
Table .
Sea surface wind speeds
Twice-daily December, January, and February level 3.0 gridded SeaWinds
scatterometer equivalent neutral 10 m wind speeds between
60∘ S and 60∘ N at a resolution of
0.25∘× 0.25∘ from the National Aeronautics and
Space Administration (NASA) Quick Scatterometer QuickSCAT;
are available from December 1999 through February 2008. Data flagged as
having possibly been corrupted by rain were excluded from the following
analysis. Although the data are nominally twice-daily, it is often the case
that data for either the ascending or descending pass of the satellite are
missing. The maximum likelihood parameter estimates of bivariate wind speed
distributions and goodness-of-fit tests were carried out using every third
non-missing data point in order to minimize the effect of serial dependence.
For the goodness-of-fit tests M=10 quantiles were used because of the
relatively small sample sizes. Because of the near-polar orbit of the
satellite results, observations of wind speed at different locations are not
simultaneous. The joint distributions we consider therefore combine
dependence in both space and time.
Joint distributions of wind speed at (6.5∘ S, 135∘ W) with
speeds at points along a zonal transect to 165∘ W (in increments of
1∘) were estimated. As the distance between the two positions
increases, there is a decreasing trend in the best-fit bivariate Weibull
dependence parameter ρ (Fig. h) with some small
fluctuations likely due to sampling variability. The same is not true for the
best-fit bivariate Rice dependence parameter ρ, which fluctuates wildly
(Fig. l). Large fluctuations also seen in the
best-fit value of cosψ are clearly correlated with those of ρ:
where one parameter is anomalously large (relative to the spatial trend), the
other is anomalously small. Of the 30 pairs of points considered, the null
hypothesis of a bivariate Rice distribution is rejected (at the 95 %
significance level) at only one (Fig. d). In
contrast, p<0.05 for the bivariate Weibull distribution at several
longitudes, particularly close to the base point at (6.5∘ S,
135∘ W). Inspection of the best-fit bivariate Weibull pdfs
(Fig. e–g) shows that they are broader for smaller
wind speeds than for larger values, a feature not evident in the best-fit
bivariate Rice distributions (Fig. i–k) or the
scatter of data (Fig. a–c). From these results, we
find only equivocal evidence that the pairs of wind speed data along this
zonal transect are drawn from a bivariate Weibull distribution and no strong
evidence to reject the null hypothesis that they are drawn from a bivariate
Rice distribution. Again, the p-values of fits to temporally shuffled wind
speeds are generally similar to or larger than those of fits to the full
distribution. The bivariate Rice distribution with the wind speed at
142∘ W (Fig. k) illustrates a rare
example in which the parametric fit to the original data passed the
goodness-of-fit test at the 5 % significance level while the fit to the
shuffled data did not; it is evident from Fig. d that
this situation is not common. The surface wind vector components in the
tropics are known to be non-Gaussian e.g., so we have a
priori reasons to believe the joint distribution should not be Ricean. The
fact that the data do not generally allow for a rejection of the null
hypothesis that the winds are bivariate Rice (for either the original or
shuffled datasets) is likely a consequence of the relatively small sample
size.
(a–c) Estimates of ρ and cosψ from
50 realizations of bivariate Ricean variables with
(U‾1,σ1,U‾2,σ2,ρ,ϕ1-ϕ2+ψ)=(7.3,1.5,6.9,1.5,0.4,0) for each of the sample sizes N=250,1250,
and 9000. The open circles correspond to estimates with
cos(ϕ1-ϕ2+ψ)≥0.99, while the stars are estimates with
cos(ϕ1-ϕ2+ψ)<0.99. (d–f) Contours of bivariate
Rice pdfs corresponding to 10 of the 50 best-fit parameter estimates
(randomly chosen). The contour values are the same for all pdfs within each
subplot. (g–i) Scatterplot of the sample correlation coefficient
between the two Ricean variables and the correlation coefficient given by the
approximate expression Eq. (). The 1 : 1 line is
given in solid black. The open circles and stars are as in the upper row.
The large variations in best-fit estimates of ρ and cosψ for the
bivariate Rice distribution result from the fact that for some parameter
values the distribution is only weakly sensitive to simultaneous changes in
these parameters: increases in ρ can be offset by decreases in
cosψ with only small changes to the joint distribution. To demonstrate
this weak sensitivity, 50 realizations of bivariate Ricean variables with
(U‾1,σ1,U‾2,σ2,ρ,ϕ1-ϕ2+ψ)=(7.3,1.5,6.9,1.5,0.5,0) were generated for each of the sample sizes of
N=250, 1500, and 9000. Maximum likelihood estimates of these
parameters obtained from these realizations demonstrate that for the smaller
samples ρ and cosψ show strong and correlated sampling
variability, with large increases in ρ combined with large decreases in
cosψ (Fig. a). As expected, these
sampling fluctuations become smaller as the sample size increases(Fig. b, c). Despite
the large variation of ρ and cosψ for small to intermediate sample
sizes, there is relatively little variation in the structure of the
corresponding bivariate Rice distributions
(Fig. d–f).
An indication of why increases in ρ should counterbalance decreases in
cosψ with only small effects on the joint distribution is given by the
approximate expression for the correlation coefficient, Eq. (). The value of this approximation is unaffected by
changes in ρ and ψ that leave the numerator invariant. The
compensation between sampling variations in ρ and cosψ is evident
the fact that corr(w1,w2) given by Eq. () is an excellent approximation to the sample
correlation coefficient even for estimates of ρ and cosϕ which are
far away from the population values (Fig. g–i). Note that there is no evident
relationship between sampling fluctuations in corr(w1,w2) and
those of ρ and ψ: the range of sample correlation values for
(ρ,cosψ) near the population values of (0.5,1) (open circles) is the
same as that for values of (ρ,cosψ) far from these values (stars).
In these parameter ranges, the dependence between w1 and w2
constrains ρ and ψ not individually, but together – over large
ranges of values for sufficiently small sample sizes.
Conclusions
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
e.g.. The results of this analysis characterize the
practical utility of these models, rather than making a claim to their
“truth”. It is noteworthy that for the data considered in this study, the
failure of either the bivariate Rice or Weibull distributions to adequately
fit the joint distribution of wind speeds (at a significance level of
5 %) is not generally associated with a corresponding failure of the
parametric distribution to model the marginals. An interesting direction of
future study would be consideration of other parametric models for the joint
distribution of non-negative quantities, such as the α-μ
distribution , copula-based models e.g., or
distributions obtained through nonlinear transformations of multivariate
Gaussians e.g..
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 e.g., and fluctuations will not
generally be isotropic especially over land; cf..
Furthermore, when used to model temporal dependence the assumed correlation
structure cannot account for the anisotropy in autocorrelation of orthogonal
components in either space e.g. or time
e.g.. Relaxing the assumptions regarding isotropy of
correlation structure results in expressions for the joint speed
distributions involving integrals over angle which are not analytically tractable.
While it is possible to relax the assumption of Gaussian components for
univariate speed distribution e.g., extending
this analysis to the bivariate case would involve specifying a non-Gaussian
dependence structure for the components. At present, there is no physically
based model for such dependence. Without any such physical justification, the
only option is an empirical investigation of the ability of different copula
models to represent observed joint wind speed distributions. It is unlikely
that a copula-based model for dependence of components will admit
analytically tractable expressions for joint speed distributions. A
copula-based analysis of either the components or the speeds directly is also
likely necessary for modelling extreme wind speeds (either large percentiles,
peaks over threshold, or block maxima), as the tails of the bivariate Rice
and Weibull distributions may not be adequate for this task. Finally,
extending the approach used in this study to obtain explicit closed-form
results for the bivariate wind speed distribution to a higher-dimensional
multivariate setting – of wind speeds alone, or of a mixture of wind speeds
and other meteorological quantities – will be analytically intractable for
any except the simplest (and likely unrealistic) covariance structures. It
may not be practical to extend the program of obtaining closed-form
expressions for joint speed distributions from models of the component
distributions much further beyond the bivariate Rice and Weibull speed
distributions considered in this study.
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 and .
The development of such models represents an interesting direction of future
study.
Data availability
The ERA-Interim 500 hPa zonal and meridional wind
components were obtained from the European Centre for Medium-Range Weather
Forecasts at
http://www.ecmwf.int/en/research/climate-reanalysis/era-interim (last
access: 17 November 2015). The Cabauw tower data were downloaded from the
Cabauw Experimental Site for Atmospheric Research (CESAR) at
http://www.cesar-database.nl/ (last access: 13 June 2014). The
Level 3.0 QuickSCAT data were downloaded from the NASA Jet Propulsion
Laboratory Physical Oceanography Distributed Active Archive Center
(http://podaac.jpl.nasa.gov/dataset/QSCAT_LEVEL_3_V2, last access:
18 February 2010).
Numerical computation of bivariate Rice pdf
Equation () is difficult to evaluate numerically
when the arguments of the Bessel functions become large. We have found that a
computationally more stable result is obtained when this equation is
expressed in the form
p(w1,w2)=w1w22πσ12σ22(1-ρ2)∫02πexpf(w1,w2,λ)dλ,
where
f(w1,w2,ϕ)=-12(1-ρ2)w12+U‾12σ12+w22+U‾22σ22-2ρU‾1U‾2cos(ϕ1-ϕ2+ψ)σ1σ2+lnI0A2+B2+2ABcosλA2+Ccosνcosλ+lncosh(Csinνsinλ)dλ,
with
A3A=w1a12+b12σ1,A4B=w2a22+b22σ2,A5C=ρ1-ρ2w1w2σ1σ2,
and the integral is evaluated numerically. Equation () is obtained from Eq. () using the fact that
Ik(x)Ik(y)=12π∫02πI0(x2+y2+2xycosϕ)coskϕdϕ
(which follows from Eq. ) and use of Eq. (). When wi/σi becomes large,
numerical evaluations of the Bessel function in Eq. () become unreliable. For the
present computations using Matlab, values of Inf occur in such cases. This
problem was not solved by using the approximation Eq. () when the argument of the Bessel function is
large, as this approximation is not sufficiently accurate.
Bivariate goodness-of-fit test
Goodness-of-fit of the bivariate distributions considered was assessed as
follows. For the speed dataset wj,n, j=1,2 and n=1,…,N, evenly
spaced quantiles qj,i=i/M, i=0,…,M for the marginals are
estimated. The quantiles qj,0=0 and qj,M=1 are estimated
respectively as 0.9 times the smallest observed value and 1.1 times the
largest observed value. The number of pairs of observations falling
simultaneously into all pairs of quantiles are computed:
B1fkl=∑n=1N1w1,n∈(q1,k,q1,(k+1)]∩w2,n∈(q2,l,q2,(l+1)],
where 1(⋅) is the indicator function. The pdf with
maximum likelihood parameters θ is then integrated to obtain the
expected number of observations in these intervals
gkl=N∫q1,kq1,(k+1)∫q2,lq2,(l+1)p(w1,w2;θ)dw1dw2
and the test statistic A is computed:
A=1M2∑k,l=1M|fkl-gkl|.
Any elements gkl which take the value of Inf (because of numerical
difficulties in evaluating the Bessel function for large arguments; cf.
Appendix A) are excluded from the calculation of the test statistic A.
After computation of A from the observations, an ensemble of B random
realizations of length N from p(w1,w2;θ) is generated and the
corresponding Ãk, k=1,…,B values of the test statistic
are generated in the manner described above. The p-value of the null
hypothesis that the observations are drawn from the specified distribution is
finally computed as the fraction of Ãk values falling above A.
Throughout this analysis, we use M=20 and B=250 (unless otherwise
noted). This goodness-of-fit test assumes independence of the random draws
(w1n,w2n),(w1m,w2m), m≠n. To minimize the effect of
serial dependence in data, in this study we subsample the datasets considered
with a sampling interval sufficiently large to balance reducing serial
dependence with maintaining sample size.
A second goodness-of-fit test proposed by was also
considered, but the statistical power was found to be lower than the test
described above for the distributions considered in this study.
Competing interests
The author declares that he has no conflict of interest.
Acknowledgements
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
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