Brighton Webs Ltd
statistics for energy and the environment
The distribution takes its name from the English scientist John Strutt, 3rd Baron Rayleigh. Rayleigh used the
distribution during his work on sound and the propogation of waves (source needed). One of its application is as
a model for wind speed under certain circumstances. Rayleigh's work on the scattering of light by particles much
smaller than the radiation's wavelength is also applied to models of solar irradiance.
The range is determined by the scale parameter and the skewness and kurtosis
are constants. It is equivalent to a Weibull
distribution with these parameters:
||A float with a value greate than zero
The range of values of a random variable with a Rayleigh distribution is from
zero to +ve infinity.
As the scale factor is also the mode, the equations can
also be written in terms of the mode.
The Rayleigh distribution is used as a model for daily average wind speed.
The model describes the distribution of wind speed over the period of a year.
The two graphics below show the distribution of daily average wind speed, one
onshore and the other at an exposed offshore location, in both cases the quality
of the fit is good. This type of analysis can be used for
estimating the energy recovery from a wind turbine.
The following pages use the Rayleigh distribution in
a sample calculation to estimate energy recovery from the wind:
During the curve fitting for this example, an number of
points emerged. First, the quality of the fit was dependent on the
range of wind speed at a given location. The quality of the fit was
lower for locations where the range was from say, 5 knots to 35 knots as
would be expected in an offshore location, than where it was from zero to
25 knots, as in the onshore example above. The effect of this was
that the model would underestimate the energy available to a wind turbine.
This is apparent in the offshore example, where the model overestimates the
number of calm days and under estimates the number of days with typical
Where the Rayleigh distribution is a poor model for a
given location, it may be appropriate to fit the data to a Weibull
distribution. This is effectively changing the shape factor from an
independent variable to a dependent one (the Rayleigh distribution is a
Weibull one with the shape factor set to two).
The height of the observations is also important.
The standard height for meteorological observations is 10m, whilst wind turbines
are often mounted on towers which are 30 - 80 metres high to gain advantage
of reduced friction with the earth's surface.
Finally, the best fit (i.e. lowest chi squared score)
was derived using the matching moment equation for the scale factor (the
mode), suggesting that the average is a reasonable estimator of the modal
Whilst the distribution of wind speed over a year is
useful for determining the total energy available, it may also be necessary
to take account of the variation over the year. The graphic below is
a box and whisker plot showing the monthly variation for the Offshore
Parameter Estimation - Matching Moments
The matching moment equation for the scale factor is
derived by rearranging the equation for the mean:
An attractive feature of the Rayleigh distribution is
that the mode can be estimated from the mean.
Random Number Generation
Random number generation (referred to as r) for a
Rayleigh distribution can be performed by transforming a continuous
uniform variable in the range (0,1) i.e. greater than zero and less than
one (referred to as u) with the
distribution's inverse probability function:
Using Basic style code, the function would be similar to:
r = sqr(-2 * scale * scale * log(u))