R0, R1 width algorithm

Given samples of the Doppler autocorrelation function, numerous estimates of spectral variance can be computed (Passarelli & Siggia, 1983). The particular estimator used by the RVP10 employs the magnitudes of R₀ and R₁ and assumes that the Doppler spectrum is Gaussian and that the signal-to-noise ratio is large.

Specifically we have (similar to Srivastava, et al 1979):

V a r i a n c e = 2 \ln [\frac{R_{0}}{| R_{1} |}] = - 2 \ln [S Q I]

where ln represents the natural logarithm. This can be compared to the expression in the preceding section for SQI to illustrate that this expression for the variance is only valid when:

\frac{S N R}{S N R + 1} \approx 1

which occurs when the SNR is large.

This variance estimator is normalized to the Nyquist interval in units of [-π, π]. For example, a variance of Π²/25 would be obtained from a Gaussian spectrum having a standard deviation equal to one fifth of the total width of the plotted spectral distribution. For scientific purposes, the spectrum width (standard deviation) is more physically meaningful than the variance, since it scales linearly with the severity of wind shear and turbulence. For these reasons, the width W is output by RVP10:

W = \frac{\sqrt{V a r i a n c e}}{π}

For efficient packing in 8-bits, width is normalized to the Nyquist interval [-1, 1 ]. For the example given above, the output width W would be (1/5). To obtain the width in m/s, multiply the output width by V_u.