Notation and model for correlations

The pulse pair processing mode is used for all polarization calculations, except that Z_dr-only processing for the STAR case can be done in either FFT or random phase as well as pulse pair.

As with the standard moments, autocorrelations form the basis for the processing of the polarization variables.

An autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them. It is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal, which has been buried under noise. However, it is also possible to perform a correlation of the S_hh and S_vv signal, which is used to create Z_HV.

The autocorrelations are computed similarly to the standard moments. For example, in pulse pair mode the autocorrelations for the horizontal transmit co-polar channel are:

T_{0 h h} = \frac{1}{M} \sum_{n = 1}^{M} S_{h h}^{n} \times S_{h h}^{n}

R_{0 h h} = \frac{1}{M} \sum_{n = 1}^{M} {s'}_{h h}^{n} \times {s'}_{h h}^{n}

R_{1 h h} = \frac{1}{M - 1} \sum_{n = 1}^{M - 1} {s'}_{h h}^{n} \times {s'}_{h h}^{n + 1}

R_{2 h h} = \frac{1}{M - 2} \sum_{n = 1}^{M - 2} {s'}_{h h}^{n} \times {s'}_{h h}^{n + 2}

For dual polarization systems, these correlations can be applied up to 4 different ways (R_hh, R_vv, R_hv, and R_vh), where R_hv and R_vh are equivalent. The physical model for the channel powers is identical to the model used for the standard moment cases:

Co-channel power

R_{0}^{h h} = \frac{1}{M} \sum_{n = 1}^{M - 1} {s'}_{h h}^{n} \times {s'}_{h h}^{n} = g_{h}^{r} g_{h}^{t} {S'}_{h h} + N_{h}

R_{0}^{v v} = \frac{1}{M} \sum_{n = 1}^{M - 1} {s'}_{v v}^{n} \times {s'}_{v v}^{n} = g_{v}^{r} g_{v}^{t} {S'}_{v v} + N_{v}

Diagonal channel power

R_{0}^{h v} = \frac{1}{M} \sum_{n = 1}^{M - 1} {s'}_{h h}^{n} \times {s'}_{v v}^{n} = \frac{g_{h}^{r} g_{h}^{t} {S'}_{h h} g_{v}^{r} g_{v}^{t} {S'}_{w}}{2}

Here S denotes the actual backscatter average power to the radar. When multiplied by the appropriate transmitter and receiver gains, S yields the actual measured power. Sometimes in comparing powers in 2 channels (for example, Z_dr and LDR), we need to know the relative gains of the 2 channels. However, in many calculations, the relative gains cancel out, and in these cases the algorithms are implemented assuming all the gains are equal to 1.

In the R_hv term, the noise variable is not present because the noise between the horizontal receiver and vertical receiver is random, having a normalized coherency of 0 with an infinite number of samples. A finite number of samples needs to be used, typically between 30 ... 60, in weather radar signal processing. However, due to the noise coherency going to 0, the noise variance also becomes smaller, allowing us to lower the detection thresholds, while having same false alarm rates as the traditional R_hh term. Lowering the detection threshold increases the apparent sensitivity when inserting the R_hv term in the radar equation for reflectivity.

In the following algorithm descriptions, we use the notation common in the literature, for example:

R_{0}^{h h} = \frac{1}{M} \sum_{n = 1}^{M - 1} {s'}_{h h}^{n} \times {s'}_{h h}^{n} = ({| {s'}_{h h} |}^{2}) = S_{h h}

R_{0}^{v v} = \frac{1}{M} \sum_{n = 1}^{M - 1} {s'}_{v v}^{n} \times {s'}_{v v}^{n} = ({| {s'}_{v v} |}^{2}) = S_{v v}

R_{0}^{h v} = \frac{1}{M} \sum_{n = 1}^{M - 1} {s'}_{h h}^{n} \times {s'}_{v v}^{n} = ({| {s'}_{h v} |}^{2}) = S_{h v}