1. Introduction

On polarimetric radars that transmit simultaneous horizontal and vertical polarizations three variables must be calibrated. These are reflectivity factor, differential reflectivity, and differential phase. Reflectivity factor remains the most difficult and elusive. Differential phase is not calibrated per se; rather the system differential phase is measured to determine the beginning of the phase unwrapping interval (Zahrai and Zrnic 1993). This can be done by equating the mean phase of ground clutter to the system differential phase (Zrnic et al. 2006). Differential reflectivity Z_DR does need to be calibrated with special care; for example, the rainfall estimator tested by Ryzhkov et al. (2005) has a bias error of 18% if the bias of Z_DR is 0.2 dB.

Hubbert et al. (2003) suggest a way to calibrate differential reflectivity using natural scatterers. The method is well suited for fully polarimetric radars, that is, those that measure all the elements of the backscatter covariance matrix. Its essence is to use sun scan for calibrating the receiving path and cross-polar measurement for calibrating the transmit path including the transmitted powers. Ryzhkov et al. (2005) also relied on the sun scan for calibrating Z_DR in the receiver chain and propose measurement in dry aggregates at high elevation angles for complete transmit–receive calibration.

Soon the national network of Weather Surveillance Radar-1988 Doppler (WSR-88D) radars will be fitted with the polarimetric capability. Calibration of differential reflectivity on the network presents a couple of challenges. First, immediately after retrofitting any of the radars to dual polarization the differential reflectivity should be properly calibrated. Second, calibration should be maintained at all times during operations. This paper suggests how to meet these two calibration requirements on the WSR-88D network. Section 2 describes the systems' hardware pertinent for calibration. The step-by-step procedure is explained in section 3, whereas sections 4 and 5 present measurements in the transmitter and receiver chains of the research WSR-88D radar. Constraints and assumptions are also listed.

2. Calibration ports on the radar system

In developing the procedure we set the following goals. 1) Existing parts and capabilities of the current WSR-88D should be used; this eliminates calibration at vertical incidence, or addition of extra hardware beyond at most one mechanical microwave switch. 2) Calibration must be possible immediately after retrofitting radar; this eliminates reliance on precipitation. 3) Calibration should be over the full dynamic range; this enlists the need for a variable attenuator. 4) There should be no temporal drift in the calibration curve over several volume scans; this demands calibration at the end of each volume scan.

The block diagram in Fig. 1 captures the gist of the research and development radar on which the procedure has been tested. The radar (KOUN) is a modified version of WSR-88D such that it can simultaneously transmit and receive horizontally and vertically polarized waves. It serves as proof of concept for polarimetric development of the WSR-88Ds and has a digital receiver (RVP8 processor; Sigmet 2005) with a 72-MHz sampling rate that will be standard on the operational network. Thus the future dual-polarization WSR-88Ds will be very similar to the KOUN, and the procedure developed herein is directly applicable to radars on the network.

The points labeled 1–4 in Fig. 1 are relevant for the subsequent discussion. Point 1 represents the coupler for transmitter power measurement. Points 2 are two couplers above the elevation rotary joints (EL) for power measurements. The baseline WSR-88D calibration method used these power monitors, but the performance was not satisfactory. The latest versions of the WSR-88D, which are fitted with the Open Radar Data Acquisition System (ORDA), do not use these monitors although the hardware remains in place. The production design for the polarimetric WSR-88D will likely include improved measurement of the transmitted power. Point S is near the antenna plane just outside of the radome and indicates the sun's radiation flux. Points 3 correspond to calibration couplers near the input to the low noise amplifiers (LNAs). In each channel between point 3 and the LNA there is a waveguide filter (bandwidth ∼16 MHz) and receiver protector that are not drawn in Fig. 1. Points 4 represent outputs of the digital receivers, one in the channel for horizontal (H) polarization, and the other in the channel for vertical (V) polarization. These outputs are in-phase and quadrature phase (I, Q) values of H and V components from which various powers, noises, and differential reflectivity are computed.

For calibrating reflectivity and instantaneous gains and phases of the automatic gain control the WSR-88D has a special circuit (Fig. 1, left part with the RF continuous wave and noise generators, and the variable attenuator), which, as will be demonstrated here, is crucial for maintaining calibrated Z_DR. The built-in variable attenuator (Fig. 1) can be changed in steps of 1 dB over a 103-dB range. Under computer control it connects to one of the four internal sources (three are shown in Fig. 1); the continuous wave (CW) generator and noise generator are pertinent for calibrating Z_DR. After the attenuator the signal is directed by a switch to either the input of the LNA (points 3 in Fig. 1) or the input to the first mixer (not shown), or it is disabled.

The gains and attenuations in the H and V channels differ and cause bias of differential reflectivity. The essence behind the procedure is to partition calibration into parts that measure constant system biases (such as the ones caused by waveguides and power splitter) and parts that measure time-varying biases (such as in active circuits). The time-invariant parts are measured once to establish the constant bias. The slowly varying bias is tracked over the full dynamic range from volume scan to volume scan analogous to calibration of the automatic gain control (AGC) circuit on the WSR-88Ds. Initially, the sun scan is used to determine the two-way bias between the elevation rotary joints and the space outside the radome (Melnikov et al. 2003).

The basic assumption for calibration is that the time constant of the slow drift is considerably longer than the duration of volume scan. This is needed to ensure that correction for bias from volume scan to volume scan can be made. Further, measurements on various contributors of constant bias are sequential and some involve common paths, so we assume that the bias does not change between such measurements. This assumption must be checked by immediately repeating the measurements that have a common path; that is what we did and recommend to others. Second, the sun's radiation is randomly polarized so that the horizontal and vertical components have equal mean powers. We have verified that this holds by making the measurements over a long period of time.

Bias between any two points is denoted with the symbol Δ_ij, where the first subscript (i) is the input point and the second (j) is the output. Although Δ_ij = Δ_ji, some of the biases can be easily measured in one direction but not in the opposite because of the placement of couplers and signal flow direction. In general the signal flow direction both during measurement and actual operation is from the port indicated by the first index. Unless stated otherwise, the bias is expressed in decibels.

3. Bias measurement procedure

The calibration consists of separate measurements of the constant bias Δ_C and the time-varying bias Δ₃₄, which are summed to obtain the total bias. At the end of each volume scan the time-varying part should be updated. The constant bias must be measured initially and should be checked occasionally after changes on the system have been made, or during periodic maintenance. On the KOUN we do not automatically update the time-varying part but make a manual correction at the start of data collection. Nonetheless, we present herein the reasons for automatic update and demonstrate, via repetitive measurements over several hours, that it is sound.

The fixed bias from transmission and back to the calibration ports (3, Fig. 1) is

View Expanded

where Δ_S2 is the bias between the EL joint (at ports 2 in Fig. 1) and the antenna output referred to outside of radome (Fig. 1). The factor 2 accounts for the two-way path (transmit and receive). The imbalance in transmitted powers at the elevation joints is denoted by Δ₁₂, and Δ₂₃ is the bias between the elevation joints and the input to the LNA. Constant bias Δ_C is time invariant because it is caused by fixed passive components. We have determined that it is independent of the angular positions of the rotary joints (i.e., changes are smaller than 0.01 dB). The remaining bias Δ₃₄ from calibration ports (3 in Fig. 1) to output of digital receiver is caused by the two active receiver paths (input to the LNA, mixers, and amplifiers). It varies with time and might depend on the received power. Thus Δ_C is added to the variable bias to obtain the total bias.

Here Δ₁₂ is directly measured, whereas Δ_S2 and Δ₂₃ are measured indirectly as

View Expanded

View Expanded

where Δ₃₄ is measured using an external signal generator as described in section 5c. Bias Δ₃₄ is measured using the internal generator (Fig. 1), which is under computer control; hence the measurement is fast and nonintrusive. But, because the differential gains might change in time, measurement of Δ₂₃ consists of three steps. First, we measure Δ₃₄, then Δ₂₃, and then again Δ₃₄; if the Δ₃₄ does not change, the measurement is valid, otherwise it is repeated until a change smaller than a tolerable error is obtained. Similarly, any measurement from points other than 3 to 4 (output of the digital receiver) must be bracketed by a measurement from points 3 to 4. This type of differential measurement has another advantage in that only the input point of measurement changes while the output is always the digital receiver, so the intrusion into the system is minimal.

We set out to achieve an overall error smaller than 0.1 dB; therefore (assuming independent errors), a desirable value for the error in one measurement is about 0.04 dB so that the cumulative rms error of the four measurements is about 0.08 dB.

Note that the expression for Δ_S2 consists of four terms that could be reduced to two but are not. The reason is to avoid the deleterious effects of time variations. That is, measure Δ₃₄, then Δ_S4, and then Δ₃₄; if the two measurements of Δ₃₄ agree within 0.03 dB, we accept the Δ_S4. Therefore, Δ_S4 − Δ₃₄ is measured in three steps and so is Δ₂₄ − Δ₃₄.

4. Measurement of constant bias in transmission chain

Waveguides were opened at the elevation rotary joints (points 2, Fig. 1). Power was injected at the coupler (point 1, Fig. 1) just above the transmitter. The difference (H power – V power) at the output of the two waveguides was about 0.04 dB (i.e., the bias Δ₁₂ = −0.04 dB). This value is also the smallest that we can discern on the power meter; hence the error in measurement could be at most 0.04 dB. We measured and recorded the coupling losses; these agreed exactly with the values stamped by the manufacturer. Therefore, on similar high-quality radars measurement can be done at the couplers' outputs as explained in the following paragraph.

The waveguides were properly connected, the transmitter was turned on, and measurements were made at the couplers above the elevation rotary joints. For safety it may be wise to reduce the transmitter power by 10 or 20 dB, which we have not done on the KOUN. Because the measurements were made sequentially, we have recorded the transmitter power during each measurement. This is highly recommended. The difference of losses between the H and V channels was 0.06 dB (H had higher loss), in excellent agreement with the measurement on open waveguides. Thus we record Δ₁₂ = −0.06 dB.

The combined specified loss (not measured by us) of the coupler above the elevation joints and the path to the feed horn for the vertical and horizontal channels is 0.2 dB. Therefore, the bias between the two channels must be much smaller than 0.2 dB and is deemed excellent. Nonetheless, there could be a differential gain of the antenna at two polarizations and minute differential loss through the radome. Measurements from sun scan are needed to estimate the total effect.

5. Measurements of bias in the receiving chain

Herein described are measurements of the variable bias, the fixed bias, and the receiver noise in the two channels. For Z_DR calibration no absolute value of power is needed, but the changes in relative values of power at the digital receiver output must be known. Measurements are differential in nature and all the comparisons are done by using digital powers in arbitrary units at the output of the digital receiver.

b. Variable bias (Δ₃₄)

As stated in section 3, the total bias Δ consists of two parts, long-term stable Δ_C and the slowly changing part Δ₃₄, mainly due to drifts in the amplifiers of the two channels. At different powers of the digital receiver output the bias can have different values; therefore, we explicitly introduce this power P_hk (in arbitrary units) to the argument of the bias. The subscript “h” indicates horizontal polarization and k indicates power levels (i.e., setting of the attenuator). Thus,

View Expanded

In the argument of Δ₃₄, P_hk is given by

View Expanded

that is, the noise is subtracted from the power at the output of the digital receiver. The units of powers in (5) are internal to the RVP8.

The variable bias needs to be measured over the full dynamic range of the receiver (i.e., for the full range of H powers P_hk measured at the digital receiver output) at the end of each volume scan. This has not been done on the KOUN although stepping of the attenuation is automated and time series data are recorded from which the bias can be computed. Further, we have semiautomated the procedure to step automatically through all settings of the attenuator.

The bias curve Δ₃₄(P_hk) in Fig. 2 is computed as

View Expanded

where the index k refers to the sequence of output power changes (corresponding to 2-dB attenuation steps), h and v refer to the polarization channels in the digital receiver, and the signal and noise powers are estimates from time series data.

In Fig. 2 two curves are shown, the noise-corrected one [as in (6)], which supposedly should be applicable, and the curve for which the noise has not been subtracted on the right-hand side of Eq. (6); in both cases power P_h in the abscissa has no noise. Rather than straightening the bias cure, the noise correction at low SNRs (∼0 dB) is in the wrong direction. We have determined that the cause is coherent leakage in the internal CW frequency generator, which is present only when this generator is connected to make the calibration; it affects the output at high attenuator settings for which the SNR < 30 dB.

At the high end of dynamic range the departure from the constant value is due to saturation effects. Otherwise, the curve is flat down to about −55 dB. From there on the departure above the constant value is caused by the coherent leakage. This we have established by measurements with an external generator (Fig. 3), with the internal noise generator (Fig. 4), and by spectral analysis. Maximum variation in the flat part is about 0.02 dB. The fact that the curve is flat indicates that the gain characteristics of the two receivers are well matched and the gains are offset by a fixed amount. It will shortly be shown that the good match extends to low signals as well; hence it suffices to extrapolate the bias to the low end of dynamic range.

c. Difference in receiver chain from EL couplers to the calibration port (Δ₂₃)

Two measurements are needed to obtain Δ₂₃. These are measurement of Δ₂₄ and measurement of Δ₃₄ at the same output signal power. Then

View Expanded

is the difference bias. On the KOUN we recorded time series data to obtain the values.

Output power from a CW signal (HP generator) was split and injected at the couplers above the elevation rotary joints (ports 2 in Fig. 1) to obtain the bias Δ₂₄(P_h) in Fig. 3. Here P_h is the power (horizontal channel) at the output of digital receiver in arbitrary units internal to the processor. The SNR_h = 0 dB corresponds to –113 dBm, referred to the LNA input. In P_h on the abscissa the noise power N_h has been subtracted; that is, P_h = P_htotal – N_h. Two bias curves are shown; in one the noises N_h and N_v have been subtracted from P_h and P_v to compute the bias; in the other they have not. This is to illustrate the match between the two channels and to determine the value at which receivers' noises begin to influence the measurement. Definitely the noise-corrected bias is constant and reliable down to –5 dB of SNR; beyond, the bias correction is futile because estimates of differential reflectivity would be dominated by uncertainty in measurements.

In Fig. 3 Δ₂₄(P_h) covers a much larger dynamic range than needed for determining this constant bias. It suffices to make measurements at few values in the middle of the dynamic range (anywhere between −40 and −60 dB in Fig. 3) and average the result.

d. Difference from the outside of radome to EL couplers (Δ_S2)

This measurement also requires two steps and use of previously determined value of Δ₂₃. The equation is

View Expanded

Note that Δ_S4 is measured with the sun scan, and Δ₃₄(N_g) must be measured with the internal noise generator. To distinguish the Δ₃₄ measured with internal noise generator we have included (N_g) in parentheses. There are two reasons why the noise generator is used. 1) The sun and noise generator are broad band as is the weather signal, whereas the signal generator is narrow band. The two receivers might not be identical: there is the obvious difference of gains that is calibrated with the narrowband CW generator, but the small difference of integrated powers (over all frequencies) of broadband signals could be of the order of 0.1 dB (no such difference was found on the KOUN). 2) On the KOUN there is a coherent leakage signal if the internal CW generator is connected; it affects measurements at SNR < 30 dB. No coherent leakage signal is present if the internal noise generator is connected.

To measure Δ₃₄(N_g) we use the internal noise generator and built-in attenuator in the range from 0 to 20 dB; this results in an output signal (from the noise generator) to receiver noise ratio of 22 to 2 dB. In the same round we repeated the noise measurements N_h and N_v by increasing the attenuation to a very large value (>80 dB).

The Δ₃₄(Ng_hk) measured on the KOUN is plotted in Fig. 4. For the noise-corrected curve the receiver noise powers were subtracted before computation, but in either case the noise generator power on the abscissa N_g = P_htotal − N_h, where P_htotal is the estimate at the output of digital receiver and N_h is the receiver noise power estimate (also at output of digital receiver).

The noise-corrected bias (Fig. 4) is fairly uniform down to the SNR_h = 0 dB. The change thereafter might be due to small coherent leakage and noise. Because the power from the sun is typically between −75 and −80 units, the measurement (on the KOUN) is not sensitive to these effects.

To determine total receiver bias we scanned the antenna over a small sector at an elevation close to the elevation of the sun. We let the sun drift through the scanning plane and record Δ_S4 (that is Z_DR) and P_h (see Fig. 5). The peak of P_h indicates that the sun is centered on the beam. There will be several peaks (Fig. 5) as the sun enters the plane of the scan. The highest of the peaks is identified and differential reflectivity is computed (after subtracting N_h and N_v, which are estimated at the end of sun scan) for that point and all the points up to 2 dB below it. The average of these values is the differential reflectivity bias Δ_S4.

Plotted in Fig. 5 are P_h and P_v (in linear units at the output of digital receiver) and differential reflectivity bias Δ_S4 (dotted curve). The noises N_h and N_v have not been subtracted in any of these plots, but have been accounted for in the listed results. One can see that the bias is near −0.6 dB. At the time of about 310 s there is a sharp increase of differential reflectivity and drop in signal powers. This is when we began receiver noise measurement. Clearly, the difference in noise powers between the two receivers causes discontinuity in differential reflectivity. In the KOUN case the SNR from the sun scan is typically 10 dB.

To give readers a feel of the stability of the measured bias using the sun (noise corrected), we list (Table 1) values obtained over about 1 h of time on 11 March 2005 (same day as for data in Fig. 5). Noises have been subtracted to compute Δ_S4. The rms deviation about the mean (∼−0.62 dB) of these measurements is 0.025 dB.

The estimated values of Δ_S4 and Δ₃₄(N_g = P_sun) are substituted into (8) to obtain the bias Δ_S2. Because the Δ₃₄(N_g) is available in discrete increments, one might need to interpolate to obtain the Δ₃₄(N_g = P_sun) at the power corresponding to the power from the sun.

Next, in Fig. 6 we present Δ_S3 = Δ_S4 – Δ₃₄ obtained over a 5-month period from March until July of 2005. The sun scan measurement was preceded with the “before” and followed by “after” measurement of Δ₃₄ using the noise generator. The time between two measurements with the noise generator was about 10 min. Note the remarkable stability of the difference Δ_S4 – Δ₃₄. The mean value is Δ_S3 = −0.30 dB; the standard deviation = 0.028 dB might be caused by variations in the polarization of sun's radiation or in the position of the rotary joints. In all but two cases there were no changes in the “before” and “after” values of Δ₃₄.

6. Conclusions

A procedure for calibrating differential reflectivity has been developed and tested on the KOUN polarimetric radar. At its essence is separation of the bias into a constant part and time-varying part (due to active receiver components). It proceeds as follows.

• Constant part is measured at the time of radar installation and at times when pertinent components are replaced; it is due to the transmitter path and part of the receiver path.

• Transmitter path is measured at the couplers above the elevation rotary joints using the transmitted signal.

• Receiver path consists of two pairs of measurements. One pair uses internal noise generator, sun scan, and again the noise generator to establish the difference between the outside of radome and the LNAs inputs. The other pair uses the internal CW generator, external signal generator, and again the internal CW generator to measure the difference between the elevation rotary joints and the LNAs inputs.

• From the differences in the receiver path one computes the part between the elevation rotary joints and the outside of radome.

• The time-varying part must be updated at the end of each volume scan.

• The total bias is the sum of constant and time-varying parts.

Overall there are four sets of measurements, one in the transmission path and three in the receiving path. We have shown that each of the sets has an rms error smaller than 0.04 dB; hence the total is less than 0.08 dB.

The recommended procedure can achieve errors within 0.1 dB, if caution and care is exercised such that 1) all differences in coupling losses are accounted for, 2) noises are correctly estimated and subtracted from the H and V powers, 3) some measurements are repeated or bracketed by a companion measurement, and 4) high-quality equipment is used.

Assuming that the quality of upgraded WSR-88D is equal or better than that of the KOUN, the procedure tested herein is directly transferable. Nonetheless, there may be residual problems of a practical nature in perfecting the technique on any radar network. Most measurements can be automated under computer control. This applies to those with internal CW and noise generators, and to sun scans. Measurement with the external generator attached to couplers above the EL rotary joints can be semiautomated so that the results from the output of the digital receiver are readily available.

Because absolute calibration of reflectivity within 1 dB is still hard to achieve it will appear to casual readers that our claim of 0.1-dB accuracy is exaggerated. The principal reason that this is not so is in the relative comparisons of the two channels. Because the transmitted power is split, on transmit the relative difference is constant; similarly, the relative difference in much of the receiver path is constant and what remains is to continuously track the relative difference in the receiver gains.