Reflectivity calibration parameters

For dBZ calculations, the dBZ_o value is required as a calibration constant.

Depending on the polarization case and the technique selected for standard moment calculation, it may also be required to have Z_dr Offset and LDR Offset:

Z_dr Offset

The ratio of the total gains (transmit/receive) of the 2 co-receive channels.
LDR Offset

The ratio of the receiver gains in a dual receiver system.

This is not required for Case 2: Simultaneous Dual Transmit and Receive (STAR) or Case 3: Alternating H/V Transmit: Single Receiver.

RVP10 supports a single calibration reflectivity dBZ₀. In most cases, it is assumed that the dBZ₀ is for the horizontal co-receive (HH) channel. The exception is for fixed vertical polarization, in which the algorithm assumes that the calibration is for the vertical co-receive (VV) channel. LDR Offset and Z_dr Offset are also downloaded and used to adjust the dBZ₀, as required, depending on the user's selection for the standard moments. For example, in STAR mode, if the user selects dBZ to be computed from the VV channel, the dBZ₀ for the HH and a Z_dr Offset adjustment are used to calculate the dBZ in the VV channel.

Case 1H: Fixed Horizontal Transmit, Dual Channel Receive- (HH, VH)
dBZ_o from HH Channel	TTY Setup Question Responses
Calculate T, Z, V, W from:	HXmt	VXmt	CoRcv	CxRcv
HH (co) (Recommended)	—	—	`YES`	`NO`
VH (LDR Offset^-1 weighting)	—	—	`NO`	`YES`
HH+VH (xdr^-1 weighting)	—	—	`YES`	`YES`

HH Channel (Co-Pol)

HH channel (co-pol) is the recommended channel for linear polarization because, for linear polarization, the co-polar channel has the strongest signal. Processing is identical to a conventional radar.

VH Channel (Cross-Pol)

VV channel (cross-pol) is used for circular or elliptic transmit polarization. Since the algorithm assumes that dBZ₀ is from the co-polar channel, xdr is used to adjust the autocorrelations as follows:

T_{0} = {x d r}^{- 1} T_{0}^{v h}

R_{(0)} = {x d r}^{- 1} R_{0}^{v h}

R_{1} = {x d r}^{- 1} R_{1}^{v h}

R_{2} = {x d r}^{- 1} R_{2}^{v h}

N = {x d r}^{- 1} N_{v}

These adjusted autocorrelations are used as input to the standard moment processing for a conventional radar. For example, in reflectivity processing, the radar equation can be written as follows:

Z^{v h} = C S_{v h} r^{2} = [\frac{C r^{2} N_{v}}{g_{v}^{r} g_{h}^{t}}] [\frac{r^{2}}{r_{0}^{2}}] [\frac{T_{0}^{v h} - N_{v}}{N_{v}}]

where $T_{0}^{v h} = g_{v}^{r} g_{h}^{t} S_{v h} - N_{v}$

= [\frac{{C r}_{0}^{2} N_{h}}{g_{h}^{r} g_{h}^{t}}] [\frac{r^{2}}{r_{0}^{2}}] [\frac{T_{0}^{v h} - N_{v}}{N_{h}}]

The third term is 1/XDR and is written as follows:

Z^{v h} = [\frac{{C r}_{0}^{2} N_{h}}{g_{h}^{r} g_{h}^{t}}] [\frac{r^{2}}{r_{0}^{2}}] [\frac{{{x d r}^{- 1} T}_{0}^{v h} - {x d r}^{- 1} N_{v}}{N_{h}}]

The first term is the dBZ₀ for the HH channel. We can use the dBZ₀ for the HH channel to calibrate the cross-channel, if we first adjust the cross-channel noise and power by 1/xdr, and then normalize by N_h. The reflectivity calculation assumes that the calibrated xdr value compensates for any differences in the radar constant between the 2 channels. We do not need separate radar constants for the 2 channels.

HH+VH Channels

HH+VH channels are used for elliptic transmit polarizations that give comparable return signal in both the co- and cross-channels. The approach is to obtain average autocorrelation functions as follows:

T_{0} = \frac{T_{0}^{h h} + {x d r}^{- 1} T_{0}^{v h}}{2}

R_{0} = \frac{R_{0}^{h h} + {x d r}^{- 1} R_{0}^{v h}}{2}

R_{1} = \frac{R_{1}^{h h} + {x d r}^{- 1} R_{1}^{v h}}{2}

R_{2} = \frac{R_{2}^{h h} + {x d r}^{- 1} R_{2}^{v h}}{2}

N = \frac{N_{h} + {x d r}^{- 1} N_{v}}{2}

These adjusted autocorrelations are used as input to the standard moment processing for calibration with respect to the HH channel.

Case 1V: Fixed Vertical Transmit and Dual Channel Receive- (VV, HV)
dBZo from VV Channel	TTY Setup Question Responses
Calculate T, Z, V, W from:	`HXmt`	`VXmt`	`CoRcv`	`CxRcv`
VV (co)	—	—	`YES`	`NO`
HV (xdr weighting)	—	—	`NO`	`YES`
VV+HV (xdr weighting)	—	—	`YES`	`YES`

This is the only case for which the calibration constant dBZ₀ for the VV channel should be downloaded to the signal processor.

VV Channel (Co-Pol)

VV channel (co-pol) is the recommended channel for the case of linear polarization. The reason is that for linear polarization, the co-polar channel has the strongest signal. Processing is identical to a conventional radar.

HV Channel (Cross-Pol)

HV channel (cross-pol) is used for circular or elliptic transmit polarization when most of the return is in the cross-pol channel. Since the algorithm assumes that dBZ₀ is from the co-polar channel, xdr is used to adjust the autocorrelations as follows:

T_{0} = x d r T_{0}^{v h}

R_{0} = x d r R_{0}^{h v}

R_{1} = x d r R_{1}^{h v}

R_{2} = x d r R_{2}^{h v}

N = x d r N_{h}

These adjusted autocorrelations are used as input to the standard moment processing with dBZ₀ calibrated with respect to the VV channel.

VV+HV Channels

VV+HV channels are used for elliptic transmit polarizations that give comparable return signal in both the co- and cross-channels. The approach is to obtain average autocorrelation functions as follows:

T_{0} = \frac{T_{0}^{v v} + x d r T_{0}^{h v}}{2}

R_{0} = \frac{R_{0}^{v v} + x d r R_{0}^{h v}}{2}

R_{1} = \frac{R_{1}^{v v} + x d r R_{1}^{h v}}{2}

R_{2} = \frac{R_{2}^{v v} + x d r R_{2}^{h v}}{2}

N = \frac{N_{v} + x d r N_{h}}{2}

These adjusted autocorrelations are used as input to the standard moment processing algorithms with dBZ_o calibrated with respect to the VV channel.

Case 2: Simultaneous Transmit and Receive- STAR (HH, VV) & Case 3: Alternating Transmit Single-Channel Receive (HH, VV)
dBZo from HH Channel	TTY Setup Question Responses
Calculate T, Z, V, W from:	`H-Xmt`	`V-Xmt`	`Co-Rcv`	`Cx-Rcv`
HH	`YES`	`NO`	—	—
VV (Z_dr Offset^-1 weighting)	`NO`	`YES`	—	—
HH+VV (gdr^-1 weighting)	`YES`	`YES`	—	—

A fundamental difference between these 2 cases is that for all standard moment processing choices, the STAR case has double the number of samples compared to the single-channel alternating case. The processing is otherwise identical.

HH Channel

Since the HH channel is directly calibrated, this is the recommended choice. Processing is identical to a conventional radar.

VV Channel

In VV channel, GDR is used to adjust the autocorrelations as follows:

T_{0} = {g d r}^{- 1} T_{0}^{v v}

R_{0} = {g d r}^{- 1} R_{0}^{v v}

R_{1} = {g d r}^{- 1} R_{1}^{v v}

R_{2} = {g d r}^{- 1} R_{2}^{v v}

N = {g d r}^{- 1} N_{v}

These adjusted autocorrelations are used as input to the standard moment processing algorithms with dBZ₀ calibrated with respect to the HH channel.

HH+VV Channels

HH+VV channels approach gives the benefit of doubling the number of samples used for the reflectivity calculation as follow:

T_{0} = \frac{T_{0}^{h h} + {g d r}^{- 1} T_{0}^{v v}}{2}

R_{0} = \frac{R_{0}^{h h} + {g d r}^{- 1} R_{0}^{v v}}{2}

R_{1} = \frac{R_{1}^{h h} + {g d r}^{- 1} R_{1}^{v v}}{2}

R_{2} = \frac{R_{2}^{h h} + {g d r}^{- 1} R_{2}^{v v}}{2}

N = \frac{N_{h} + {g d r}^{- 1} N_{v}}{2}

These adjusted autocorrelations are used as input to the standard moment processing algorithms with dBZ₀ calibrated with respect to the HH channel.

Case 4: Alternating Dual-Channel (HH, VH, VV, HV)
dBZo from HH Channel	TTY Setup Question Responses
Calculate T, Z, V, W from:	`HXmt`	`VXmt`	`CoRcv`	`CxRcv`
HH	`YES`	`NO`	`YES`	`NO`
VH (xdr¹ weighting)	`YES`	`NO`	`NO`	`YES`
VV (gdr^-1 weighting)	`NO`	`YES`	`YES`	`NO`
HV (xdr/gdr weighting)	`NO`	`YES`	`NO`	`YES`
HH+VV (gdr^-1 weighting)	`YES`	`YES`	`YES`	`NO`
HV+VH (xdr & gdr weighting)	`YES`	`YES`	`NO`	`YES`

HH Channel

Since the HH channel is directly calibrated, this is the recommended choice. Processing is identical to a conventional radar.

VH Channel

Processing is identical to Case 1: Fixed Transmit: Dual-Channel Receiver.

VV Channel

Processing is identical to Case 2: Simultaneous Dual Transmit and Receive (STAR)) and Case 3: Alternating H/V Transmit: Single Receiver.

HV Channel

The weighting in HV channel uses both xdr and gdr as follows:

T_{0} = \frac{x d r}{g d r} T_{0}^{h v}

R_{0} = \frac{x d r}{g d r} R_{0}^{h v}

R_{1} = \frac{x d r}{g d r} R_{1}^{h v}

R_{2} = \frac{x d r}{g d r} R_{2}^{h v}

N = \frac{x d r}{g d r} N_{h}

These adjusted autocorrelations are used as input to the standard moment processing algorithms with dBZ₀ calibrated with respect to the HH channel.

HH+VV Channels

Processing is identical to Case 2: Simultaneous Dual Transmit and Receive (STAR)) and Case 3: Alternating H/V Transmit: Single Receiver.

HV+VH Channels

The weighting in HV+VH processing must correct for both transmitter and receiver effects in order to use the HH channel dBZ₀ as follows:

T_{0} = \frac{\frac{x d r}{g d r} T_{0}^{h v} + {x d r}^{- 1} T_{0}^{v h}}{2}

R_{0} = \frac{\frac{x d r}{g d r} R_{0}^{h v} + {x d r}^{- 1} R_{0}^{v h}}{2}

R_{1} = \frac{\frac{x d r}{g d r} R_{1}^{h v} + {x d r}^{- 1} R_{1}^{v h}}{2}

R_{2} = \frac{\frac{x d r}{g d r} R_{2}^{h v} + {x d r}^{- 1} R_{2}^{v h}}{2}

N = \frac{\frac{x d r}{g d r} N_{h} + {x d r}^{- 1} N_{v}}{2}

These adjusted autocorrelations are used as input to the standard moment processing algorithms with dBZ₀ calibrated with respect to the HH channel.

Suppose that we want to compute the average of the reflectivities for the VH and HV channels. An example this weighted averaging is as follows:

Z^{h v + v h} = {C r}^{2} \frac{{S_{h v} + S}_{v h}}{2}

= {C r}^{2} \frac{\frac{T_{0}^{h v} - N_{h}}{g_{h}^{r} + g_{h}^{t}} + \frac{T_{0}^{v h} - N_{v}}{g_{v}^{r} + g_{h}^{t}}}{2} = \frac{{C r}^{2}}{g_{h}^{r} g_{h}^{t}} \frac{(T_{0}^{h v} - N_{h}) \frac{g_{h}^{t}}{g_{v}^{t}} + (T_{0}^{v h} - N_{v}) \frac{g_{h}^{r}}{g_{v}^{r}}}{2}

but since $x d r = \frac{g_{h}^{r}}{g_{v}^{r}}$ and $g d r = \frac{g_{v}^{r} g_{v}^{t}}{g_{h}^{r} g_{h}^{t}}$

Z^{v h + h v} = \frac{{C r}^{2}}{g_{h}^{r} g_{h}^{t}} [\frac{\frac{x d r}{g d r} T_{0}^{h v} + {x d r}^{- 1} T_{0}^{v h}}{2} - \frac{\frac{x d r}{g d r} N_{h} + {x d r}^{- 1} N_{v}}{2}]

Z^{v h + h v} = \frac{{C r}^{2}}{g_{h}^{r} g_{h}^{t}} [T_{0} - N] = [\frac{{C r}^{2} N_{h}}{g_{h}^{r} g_{h}^{t}}] [\frac{r^{2}}{r_{0}^{2}}] [\frac{T_{0} - N}{N_{h}}]

The first term in brackets is precisely dBZ₀ for the HH channel. If we average the correlations using the appropriate gdr and xdr weighting, the average reflectivity is obtained by using conventional processing with the HH channel dBZ₀.