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Normal ACF analysis using a Phase Fit

  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 tex2html_wrap_inline5060 is the amplitude at a given lag, k the lag index and tex2html_wrap_inline5064 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 tex2html_wrap_inline5066 , which can be used to determine the Doppler velocity tex2html_wrap_inline5068 . The problem with this method is that the measurements can be used to calculate the phase tex2html_wrap_inline5064 only within the range tex2html_wrap_inline5072 to tex2html_wrap_inline5074 (the residual phase, tex2html_wrap_inline5076 ), but for large velocities the phase change over the entire ACF can easily exceed tex2html_wrap_inline5078 . 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 tex2html_wrap_inline5076 is the measured residual phase at lag k, tex2html_wrap_inline5084 is an integer calculated for each lag and tex2html_wrap_inline5066 is, as before, the Doppler shift frequency. This calculation can be performed for a number of different tex2html_wrap_inline5066 values until a minimum is determined. The tex2html_wrap_inline5066 corresponding to this minimum is taken to be the desired Doppler frequency. The amount of processing required for the brute-force solution is relatively highgif and a different technique is employed in the actual program. It involves calculating a first guess tex2html_wrap_inline5092 using one or more close pairs of residual phases and then adjusting tex2html_wrap_inline5066 until a minimum error is found. The Doppler frequency tex2html_wrap_inline5066 (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 tex2html_wrap_inline5098 is the radar transmission frequency and the assumption is made that tex2html_wrap_inline5100 .

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 tex2html_wrap_inline5104 is the maximum of the associated autocovariance function and tex2html_wrap_inline5106 a constant, then the amplitude variation of the ACF is well represented by an exponential decay. This is usually the case [HVG tex2html_wrap_inline4466 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, tex2html_wrap_inline5106 becomes a decay constant, k is the lag index and tex2html_wrap_inline4464 the basic lag separation time. A linear least-squares fit to the measured values of tex2html_wrap_inline5116 can be used to determine numerical values for C and tex2html_wrap_inline5106 . 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 tex2html_wrap_inline5122 .

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 validity of the current method has been checked in several studies involving the comparison between different techniques. Grant et al. [GMR tex2html_wrap_inline4466 95] arrive at a variational difference of tex2html_wrap_inline5124 in direction and tex2html_wrap_inline5126 % in speed with no systematic difference, when comparing SuperDARN drift velocities with measurements from the Canadian Advanced Digital Ionosonde (CADI)gif.

next up previous
Next: Power Spectrum Analysis using Up: Analysis of the Data Previous: Analysis of the Data

Andreas Schiffler
Wed Oct 9 10:05:17 CST 1996