Wind-profiler radar return signals are Doppler shifted according
to the radial component of the wind, i.e. that along the beam
pointing direction; the transverse component of the wind has no
effect. The radar return signals for a vertically directed beam
might therefore be expected to be influenced only by the vertical
component of the wind. However, owing to the fact that radar beams
have finite widths, the horizontal wind, v_{H}, actually
has a small effect. As can be seen from the diagram, it is only
for the very centre of the beam that the horizontal wind has only
a transverse component, v_{HT}; the magnitude of the
radial component, v_{HR} = v_{H}sinθ,
increases with increasing angle from zenith, θ, and the
sign depends on whether it is the side of the beam facing into or
away from the horizontal wind. This leads to the radar return
signal power being spread across a range of Doppler velocities, an
effect known as beam-broadening, with the mean Doppler shift
corresponding to the mean vertical velocity. Additional broadening
can occur if the vertical velocity fluctuates as a result of
turbulent motions. In principle, therefore, the intensity of the
turbulence can be derived from the radar return spectral widths
after correcting for the effects of beam-broadening.

Since the (two-way) polar diagram of a vertically directed radar beam has a Gaussian form:

where the standard deviation σ_{0} represents
the beam-width, the spread of Doppler velocities, v_{D},
as a result of beam-broadening will also have a Gaussian form
centred about the mean vertical velocity:

where the standard deviation &sigma_{BEAM} is given
by:

The spread of velocities caused by turbulent activity will also
have a Gaussian distribution about the mean vertical velocity,
with a standard deviation σ_{TURB} which gives a
measure of the turbulence intensity, and so the standard deviation
of the combined distributions, i.e. the spectral width of the
radar return signal, σ_{SW}, is given by:

Note that the spectral widths in the old format files of NERC MST Radar data are defined in a non-standard way; the values must therefore be multiplied by a scaling factor of 1.25 (explained in the file format page) before the following correction can be applied. The variance of vertical velocity variations caused by turbulence can therefore (in principle) be extracted from the observed spectral widths through the relationship:

The diagram below shows a scatter plot of observed vertical
beam spectral widths as a function of horizontal wind speed for 24
hour's worth of observations made by the NERC MST radar. The red
line corresponds to v_{H}sinσ_{0},
i.e. σ_{BEAM}, where
sin^{2}σ_{0} = 2.47 × 10^{-4},
which corresponds to a one-way half-power half-width of
1.5°. It should be noted that beam broadening corrected values
of spectral can only be extracted for points lying above the
line. Moreover, attention should confined to those points which
lie significantly above the line. Since the beam broadening
contribution to the observed spectral width increases with
increasing wind speed, the liklihood of being able to extract
turbulence information decreases with increasing wind speed.