Autocorrelation for moment estimations

The final spectrum moment calculation (for total power or SNR, mean velocity, and spectrum width) in all processing modes is based on autocorrelation moment estimation techniques.

Typically, the first three lags are calculated, denoted as R₀, R₁ and R₂. There are two ways to calculate these, that is, time domain or frequency domain calculation.

In the PPP mode for dual-polarization, the autocorrelations are computed directly in the time domain.
In DFT mode, they are computed by taking the inverse DFT of the Doppler power spectrum in the frequency domain.

In the DFT case, only the first 3 terms must be calculated.

The time domain and frequency domain techniques are nearly identical, except for that the method of taking the inverse DFT of the power spectrum relies on the assumption that the time series is periodic. Another difference is that for time domain calculation, only a rectangular weighting is used.

In the following table, M is the number of pulses in the time average. Here, s' denotes the clutter-filtered time series, s denotes the original unfiltered time series and the * denotes a complex conjugate. gr and gt represent the transmitter and receiver gains, that is, their product represents the total system gain.

Time domain calculation of autocorrelations and corresponding physical models
Parameter and definition	Physical model
$T_{0} = \frac{1}{M} \sum_{n = 1}^{M} s_{n} * s_{n}$	$g^{r} g^{t} (S + C) + N$
$R_{0} = \frac{1}{M} \sum_{n = 1}^{M} {s'}_{n} * {s'}_{n}$	$g^{r} g^{t} S + N$
$R_{1} = \frac{1}{M - 1} \sum_{n = 1}^{M - 1} {s'}_{n} * {s'}_{n + 1}$	$g^{r} g^{t} S + e^{j π V' - \frac{π^{2} W^{2}}{2}}$
$R_{2} = \frac{1}{M - 2} \sum_{n = 1}^{M - 2} {s'}_{n} * {s'}_{n + 2}$	$g^{r} g^{t} S + e^{j π V' - 2 π^{2} W^{2}}$

Since the RVP10 is a linear receiver, there is a single gain number that relates the measured autocorrelation magnitude to the absolute received power. However, since many of the algorithms do not require absolute calibration of the power, the gain terms are ignored in the discussion of these. T_o for the unfiltered time series is proportional to the sum of the meteorological signal S, the clutter power C and the noise power N. R₀ is equal to the sum of the meteorological signal S and noise power N which is measured directly on the RVP10 by periodic noise sampling. T_o and R₀ are used for calculating the dBZ values - the equivalent radar reflectivity factor which is a calibrated measurement. The physical models for R₀, R₁ and R₂ correspond to a Gaussian weather signal and white noise. W is the spectrum width and V' the mean velocity, both for the normalized Nyquist interval on [-1 to 1].

The autocorrelation lags above and the corresponding physical models have five unknowns: N, S, C, V', W. Because the R₁ and R₂ lags are complex, this yields, effectively, 5 equations in 5 unknowns using the constraint provided by the argument of R₁. This closed system of equations can be solved for the unknowns which is the basis for calculating the moments from the autocorrelations.