The problem of determining the mean Doppler velocity, assuming there is only
one velocity component in the scattering area, can be solved using a
very simple approach. The radar interpreter produces a complex
autocovariance function *A* given by:

where is the amplitude at a given lag, *k* the lag index and
the phase at a given lag.
The phase of the complex value as a function of lag is
determined by the Doppler velocity spectrum. For a system with a single
velocity component, the phase will change linearly with lag time. The phase
increases for positive velocities (toward the radar) and decreases for
negative velocities (away from the radar). Therefore a simple least squares
fit of the phase of the ACF as a function of time will result in a straight
line which passes through 0 at lag 0. The slope of this line corresponds
to the mean Doppler frequency , which can be used to determine the
Doppler velocity . The problem with this method is that the
measurements can be used to calculate the phase
only within the range to (the *residual phase*,
),
but for large velocities the phase change over the entire ACF can easily exceed .
There exists a unique and correct solution which can be found
through various methods.
Using a ``brute-force'' method, one can minimize the error function

where is the measured residual phase at lag *k*, is an integer calculated for each lag and
is, as before, the Doppler shift frequency. This calculation can be
performed for a
number of different values until a minimum is determined. The
corresponding to this minimum is taken to be the desired Doppler
frequency.
The amount of processing required for the brute-force solution is relatively
high and a different technique is
employed in the actual program. It involves calculating a first guess
using one or more close pairs of residual phases and then
adjusting until a minimum error is found. The Doppler frequency
(which can be negative) corresponds to the mean
Doppler velocity and can be converted to the mean line-of-sight component of
the plasma drift velocity using

where is the radar transmission frequency and the assumption is made that .

The backscattered power and the spectral width are parameters that can
be determined from the ACF once the Doppler velocity is known. The amplitude
variation of the ACF is independent of the phase variation. If the scattered
signal has a spectrum *S* which is Lorentzian in shape, that is

where is the maximum of the associated autocovariance function and a constant, then the amplitude variation of the ACF is well represented by an exponential decay. This is usually the case [HVG 93]. There are two exceptions to this rule. Very narrow spectra observed in the E-region often appear to be more characteristic of a Gaussian shape [Han91], while spectra in the cusp region are broad and typified by multiple Gaussian components. Disregarding these exceptions, the amplitude variation of the ACF can be written as the exponential form:

where *C* is a maximum backscattered power, becomes a decay constant,
*k* is the lag index and the basic lag separation time.
A linear least-squares fit to
the measured values of can be used to determine
numerical values for *C* and . Since the autocovariance function and
the spectral density function constitute a Fourier transform pair,
we can determine the relationship of these values
to the half-power spectral width of the power spectrum and arrive at a
formula for the Doppler velocity width:

This follows because the full width at half maximum is .

The method described here is computationally not expensive and is well adapted to a situation where the Doppler power spectrum is dominated by a single component. Although it is used for all calculations on all radars, it has a number of shortcomings [Bak90], as follows.

- The method cannot handle multiple Doppler components in general. It does not give any indication of the presence of multiple Doppler velocities in the target volume and will return wrong results for this case.
- The progressive-fit method employed for the actual (real-time) calculation of the velocity can produce wrong results if a local minimum in the error function is found rather than the global minimum.
- The power and width estimates are based on the assumption of an exponential amplitude decay; this assumption is wrong in certain cases.

The validity of the current method has been checked in several studies
involving the comparison between different techniques.
Grant *et al.* [GMR 95] arrive at a variational difference of in direction and % in speed with no systematic difference,
when comparing SuperDARN drift velocities with measurements from the
Canadian Advanced Digital Ionosonde (CADI).

Wed Oct 9 10:05:17 CST 1996