Abstract

Temporal differential reflectivity bias variations are investigated using the National Center for Atmospheric Research (NCAR) S-band dual-polarization Doppler radar (S-Pol). Using data from the Multi-Angle Snowflake Camera-Ready (MASCRAD) Experiment, S-Pol measurements over extended periods reveal a significant correlation between the ambient temperature at the radar site and the bias. Using radar scans of the sun and the ratio of cross-polar powers, the components of the radar that cause the variation of the bias are identified. It is postulated that the thermal expansion of the antenna is likely the primary cause of the observed bias variation. The cross-polar power (CP) calibration technique, which is based on the solar and cross-polar power measurements, is applied to data from the Plains Elevated Convection at Night (PECAN) field project. The bias from the CP technique is compared to vertical-pointing bias measurements, and the uncertainty of the bias estimates is given. An algorithm is derived to correct the radar data for the time- and temperature-varying bias. Bragg scatter measurements are used to corroborate the CP technique bias measurements.

1. Introduction

Even though differential reflectivity was first introduced to the radar meteorology community more than 40 years ago by Seliga and Bringi (1976), the calibration of continues to be a topic of research, an issue for most radars, and a quantity whose temporal stability is poorly documented and inadequately understood. To calibrate an instrument well, an unambiguous measurement standard is usually required. For weather radar, a good reference standard would be a radar target that fills the radar resolution volume uniformly and has a guaranteed differential reflectivity value. It is widely thought that such a target is rain when it is viewed vertically by the radar, since in theory such raindrops have intrinsic dB (Seliga et al. 1979; Gorgucci et al. 1999). However, in practice the “guaranteed” criterion is not necessarily met and there are a host of factors that may foil this calibration and bias measurements: the raindrops may exhibit odd oscillations (Beard 1984); the radome, especially a wet radome, or a wet antenna can affect the microwaves; ground clutter from sidelobes can contaminate the radar signals; and the antenna shape, when pointing vertically as compared to pointing horizontally, may be different. Other issues, such as receiver saturation or hardware and signal processing errors, can also bias measurements. Despite all this, most of these possible error sources can be mitigated through careful measurements and by rotating the vertically pointing antenna through several rotations and averaging . Still, an undetected bias can exist that is difficult to assess without another calibration standard. Furthermore, there is little in the literature accounting for possible time-dependent fluctuations of the bias. One such study by Figueras i Ventura et al. (2012) investigates the long-term bias of radars in the Météo-France network via vertical-pointing (VP) measurements and solar “spikes” captured in volume scans (Holleman et al. 2010). The VP measurements were made every 15 min; however, the VP and solar measurements were averaged over a day and thus do not capture bias variations that occur over tens of minutes or less, as is done in this study. A similar study is done by Frech (2013) for the German Weather Service radars. Frech points out the importance of good temperature control inside the radome if bias fluctuations are to be kept to acceptable levels. They do not identify the source of the fluctuations.

There are several other techniques that have been used for the calibration of . An engineering calibration method characterizes the gain of different signal paths of the radar with power meter measurements and solar power measurements (Zrnić et al. 2006; Hubbert et al. 2008b). However, this technique has proven to have unacceptable uncertainty for S-Pol, and for the Next Generation Weather Radar (NEXRAD) polarimetric Weather Surveillance Radar-1988 Doppler (WSR-88DP) radars (Ice et al. 2014). Bragg scatter has been used to estimate the bias, since it is thought to have an intrinsic value of 0 dB (Melnikov et al. 2011). The intrinsic properties of raindrop size distributions or dry aggregates in the ice phase have also been used to estimate bias (Ryzhkov et al. 2005; Bechini et al. 2008). The variability of such precipitation particle distributions can be considerable and the techniques typically require a considerable quantity of data and thus time to execute. Recently, Richardson et al. (2017) examine Bragg scatter via an automated identification algorithm as a means to assess the bias of the NEXRADs. The analysis spans a year, but the Bragg estimates are daily averages.

Another technique uses the principle of radar reciprocity that states that the two cross-polar backscatter cross sections are equal (Saxon 1955; Hubbert et al. 2003). This has been termed the cross-polar power (CP) technique, which uses the integrated ratios of the two CP powers, along with solar measurements, to calculate a system bias. These measurements can also be used to identify which radar components cause the bias to drift in time. The CP technique has been reported previously in Hubbert and Bringi (2003) and Hubbert et al. (2008a).

To analyze the time variability of the bias, measurements over an extended period of time with fine-time resolution (10 min or less) are needed. This is difficult to do using the VP technique, since it requires light rain over the radar. Also, if the rain is too heavy, bias can be introduced due to radome wetting or with dish wetting if no radome is present, and due to receiver saturation. Thus, there are very few studies of bias over extended periods that use VP data. In this respect, the CP technique is advantageous, since it can be executed anytime during daylight hours so that bias variations over time can be observed. Extended period bias variability has been addressed in Hubbert et al. (2008a). Figure 4 in that paper, repeated here as Fig. 1, shows VP data, collected by S-Pol over a 90-min period on 23 May 2007, with an apparent variability and increase in bias of 0.03 dB. The variation is small but a discernible increasing trend can be seen. Figure 6 of that paper, repeated here as Fig. 2, shows a comparison of bias monitoring via test pulse injection above the low-noise amplifiers (LNAs) (purple asterisks) and vertical (V) to horizontal (H) polarization solar power ratios (blue diamonds) from data gathered over 12 h on 21 June 2007 by S-Pol. If the differential gain of the antenna is a constant (since it is an inactive metal device), the two curves should match but they do not. As can be seen, the solar ratio data varies by 0.10 dB over the day—a significant amount for calibration. This discrepancy is explained later in this paper. Figures 1 and 2 indicate that there can be significant variation in bias over a day.

Fig. 1.

Calculation of bias from 90 consecutive vertical-pointing PPI scans from S-Pol data gathered on 23 May 2007 at NCAR’s Marshall field site; taken from Hubbert et al. (2008a). There is one estimated bias per scan. Straight line is a least squares fit. The standard deviation of the data about the fit is given in the figure.

Fig. 1.

Calculation of bias from 90 consecutive vertical-pointing PPI scans from S-Pol data gathered on 23 May 2007 at NCAR’s Marshall field site; taken from Hubbert et al. (2008a). There is one estimated bias per scan. Straight line is a least squares fit. The standard deviation of the data about the fit is given in the figure.

Fig. 2.

Comparison of (purple asterisks) and a pseudo- calculated from test pulses injected above the LNAs (blue diamonds). See  appendix A for the definition of . Data were gathered on 21 June 2007 at NCAR’s Marshall field site. If the differential gain of the antenna is a constant, then the two curves should match. Taken from Hubbert et al. (2008a).

Fig. 2.

Comparison of (purple asterisks) and a pseudo- calculated from test pulses injected above the LNAs (blue diamonds). See  appendix A for the definition of . Data were gathered on 21 June 2007 at NCAR’s Marshall field site. If the differential gain of the antenna is a constant, then the two curves should match. Taken from Hubbert et al. (2008a).

As a practical note, what is the accuracy of required for rainfall estimates? This depends on the rainfall estimator used, the accuracy of the reflectivity estimate, the rainfall rate, and the error required by the user. Bringi and Chandrasekar (2001) show that if rain-rate errors are to be less than about 25%, assuming the reflectivity estimate has less than 1-dB error, then should be estimated to better than 0.1 dB (see their Fig. 8.8). See Ryzhkov et al. (2005) for further discussion on rainfall and error requirements.

Uncertainty

No matter how meticulously any measurement is made, there is a degree of error or uncertainty (Taylor and Kuyatt 1994). In part, this paper addresses and attempts to estimate what the uncertainties are for the CP and VP calibration techniques for data gathered by S-Pol. On the surface, evaluating uncertainty from a measurement perspective seems straightforward: one repeats a measurement many times, holding appropriate variables constant, and then calculates the mean, which represents the systematic error, and the standard deviation σ, which represents the random error. This assumes that there is a calibration measurement reference; that is, we know what the true mean value should be. It is also assumed that the repeated measurements are executed over a period during which the device under test (i.e., the radar) is in a stable state. This means that the state of the radar—that is, the transmit power and the gains—are consistent to within the precision of the desired calibration characterization. Knowledge of the time stability of the bias is thus important in the assessment of the measurement uncertainty. The uncertainty then is a factor k times the standard deviation of the random error, where k defines confidence or “coverage” demanded by the users of the specification. It is widely thought that should be calibrated to “within” 0.1 dB, but the coverage required for this specification is ambiguous and is rarely stated. From our experience, users of S-Pol radar data would like to be “quite confident” that the reported bias is within 0.1 dB of the true bias, and this suggests that a coverage factor of 2 is required at a minimum, which gives a confidence level of 95%.

There can be subtle systematic bias present due to data processing techniques and other radar system factors that are not revealed by repeated measurements. For example, one reason why the engineering calibration technique produces unacceptably large uncertainties is that the external power measurement introduces systematic errors (Hubbert et al. 2008a). Also, it has been assumed that VP data integrated over several 360° rotations of the antenna will yield 0-dB , but this is seldom corroborated with other calibration techniques.

In this paper, we use the CP technique for investigating the variability of bias and then use other bias estimators to corroborate the results. The CP technique uses two measurements: 1) the V-to-H solar power ratio and 2) the ratio of transmit-V receive-H polarization power () to transmit-H receive-V polarization power (), or . Like the VP technique, no external signal generators or power meters are required that introduce an associated uncertainty to the calibration. An advantage of the CP technique is that both the solar scans and the CP ratio (CPR) can be measured at nearly anytime during daylight hours. Heavy precipitation can cause differential attenuation in the solar measurements thus biasing results; however, we avoid such events. Such extended, fine-time (10 min or less) resolution calibration measurements are rare in the literature. Of particular interest, we find that the radar antenna system contributes significantly to the variation of bias in response to changes of radar site ambient temperature. This has not been shown before. The CP calibration results are corroborated with VP data and with Bragg scatter measurements. Uncertainty analyses are given for the CP and VP calibration techniques.

Experimental data from NCAR’s S-Pol radar are used to illustrate the theory. Of particular interest to the scientific community, we use data from the recent Plains Elevated Convection at Night (PECAN) (Geerts et al. 2017) field campaign to demonstrate the CP technique.

This paper is organized as follows. Section 2 gives a description of S-Pol and theory for the CP technique. In section 3, the temperature dependence of bias is demonstrated with S-Pol data. In section 4, the CP bias measurements are compared to VP bias measurements. Section 5 gives an uncertainty analysis for the CP and VP bias estimations. Section 6 gives bias estimates from Bragg scatter measurements. Section 7 discusses the thermal expansion of the antenna as a possible source of the observed temperature-sensitive bias, and section 8 gives a summary and conclusions. In the appendix, the CP calibration technique is developed for simultaneous H and V transmit (SHV) radars.

2. The CP calibration technique for S-Pol

The CP method has been successfully applied to the Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) radar and S-Pol data to calibrate (Hubbert et al. 2003, 2008a). The technique uses the property of radar reciprocity, which means that the off-diagonal terms of the radar-scattering matrix , are equal (Bringi and Chandrasekar 2001). Practically, this means that the two cross-polar powers incident on the antenna should be equal if the H and V transmit powers are equal. If the radar-measured cross-polar powers are not equal, then this indicates a radar system bias.

Shown in Fig. 3 is a simplified block diagram of S-Pol that captures the essential radar components. The term represents the H and V transmit powers, represents the losses associated with the circulators on transmission, represents the losses associated with the circulators on reception, represents waveguide losses, represents the antenna gains, represents the receiver gains, represents the LNA gains, represents the insertion losses associated with various paths through the intermediate frequency (IF) switch, and represents the received powers. The dashed vertical line on the right represents the reference plane where both the transmit power is monitored and test signals are injected. Physically, this reference plane is located in a sea container that houses both the transmitter and receive electronics. Thus, all radar components to the left of the reference plane are situated in a temperature-controlled sea container, which is dimensionally 8 ft × 8 ft × 20 ft. To the right of the reference plane, the waveguides (H and V channels) exit the sea container to the outside and go to the antenna, which has no radome.

Fig. 3.

S-Pol block diagram. Reference plane (dashed line) is located in a sea container that contains everything to the left of the of the reference plane. To the right of the reference plane, there are several feet of waveguide that then exit the sea container to the outside-exposed environment and go to the antenna, which has no radome. The sea container is temperature controlled, but the temperature naturally oscillates a small amount with the cycling of the air conditioners.

Fig. 3.

S-Pol block diagram. Reference plane (dashed line) is located in a sea container that contains everything to the left of the of the reference plane. To the right of the reference plane, there are several feet of waveguide that then exit the sea container to the outside-exposed environment and go to the antenna, which has no radome. The sea container is temperature controlled, but the temperature naturally oscillates a small amount with the cycling of the air conditioners.

S-Pol uses a single transmitter and a fast mechanical polarization switch to transmit alternate pulses of H and V polarization (termed FHV mode). Because of the IF transfer switch, there are four possible paths through the receiver chain. S-Pol uses the IF switch so that the copolar (and cross polar) signals always use the same receiver path from the IF switch to the in-phase and quadrature (I and Q) samples; that is, S-Pol uses copolar and cross-polar receivers instead of H and V receivers. This is done so that errors in are reduced.

The following equations use variables that are in linear form, whereas the data in this paper are always shown in log form. Details of the derivation of the equations below are given in  appendix A. The essential CP technique equations for our analysis are the V-to-H solar power ratio ,

 
formula

the CPR,

 
formula

and the calibration equation,

 
formula

where is the measured and is calibrated . Thus, the bias correction factor is defined as

 
formula

with the bias defined as the negative of the correction factor (dB). A receiver bias can also be defined as

 
formula

where is the ratio of the V and H transmit powers at the reference plane. Again, more details are given in  appendix A. Note that is contained in so that

 
formula

This will be useful in identifying the sources of temporal bias variability. The CP technique is not limited to radars that operate in FHV mode and can be applied to SHV radars. This is discussed in  appendix B.

a. Cross-polar power technique: Data collection

Execution of the CP technique requires two measurements: 1) solar scans and 2) cross-polar power measurements. Solar scan data have been described and shown before in Hubbert et al. (2010).

1) Solar measurement

Solar radiation at S band is assumed unpolarized and thus the H and V powers are equal. During high sunspot activity, there can be circularly polarized radiation (Tapping 2001). However, circularly polarized radiation will split equally into H and V polarized components and thus we assume that such solar activity will not bias our measurements significantly. The sun is scanned within a box that is approximately in azimuth angle by in elevation angle with 0.1° elevation steps at a scan rate of s-1. The radar scanning software tracks the sun and automatically changes the scanning limits so that the center of the sun is maintained in the center of the scanning box. A total of 128 H and V samples are integrated per point, and the power samples are corrected for background noise. The noise can be calculated while the radar is pointing away from the sun during the solar box scan. Upon completion of the box scan, the data are interpolated to a × rectangular resolution grid. To avoid ground clutter contamination, only data from beyond the 112-km range are used so that approximately 25 600 I and Q samples are integrated per grid point. The center of the solar scan is established (i.e., the maximum power point) and then the ratio of V to H powers is integrated over a solid angle centered on the high power point [see Hubbert et al. (2010) for example solar plots]. It is noted that the diameter of solar disc is approximately 0.53° and these solar plots are a convolution of the S-Pol antenna pattern with the solar disk.

2) CPR measurement

For FHV operations the CP power ratio can be calculated from either weather or clutter targets and is typically an average of hundreds or thousands of measured cross-polar power ratios from a PPI scan or an entire volume scan. The CP technique takes advantage of the principle of radar reciprocity, which applies to the entire radar antenna pattern so that power transmitted and received through the antenna sidelobes does not bias the measurement. For the following datasets, S-Pol was in FHV mode with a pulse repetition time (PRT) of 1 ms. Thus, a cross-polar power pair, separated by 1 ms, comes from nearly the identical resolution volume of scatterers, since neither they nor the antenna moves appreciably in 1 ms. To ensure good data quality, several thresholds are used for the CP powers: 20 dB dB (signal-to-noise ratio) and , where the clutter phase alignment (CPA) is a clutter identification metric (Hubbert et al. 2009). The CP ratios that pass this criterion are then averaged over the input dataset. For FHV operations, a single low-level PPI scan can yield sufficient data for an accurate (uncertainty less than 0.05 dB) estimate of CPR.

3. bias as a function of ambient temperature

In this section a bias analysis is presented from two experiments, Multi-Angle Snow Camera and Radar (MASCRAD) and PECAN. MASCRAD took place from December 2014 to March 2015 (Kennedy et al. 2015). There were large ambient temperature swings (C) at the MASCRAD S-Pol radar site that caused the bias to vary about 0.4 dB. PECAN took place from 1 June to 15 July 2015. Similar observations are made during PECAN, providing a second dataset with which to evaluate and compare various bias corrections.

a. The MASCRAD dataset

During December 2014 through March 2015, S-Pol was located at its Front Range Observational Network Testbed (FRONT) field site near Firestone, Colorado, and gathered data for MASCRAD. For calibration purposes, many solar scans were made over a wide range of temperatures that proved advantageous in detecting a relationship between and temperature. The ambient temperature at S-Pol was obtained from a National Oceanic and Atmospheric Administration (NOAA) Meteorological Assimilation Data Ingest System (MADIS) weather station located relatively close to S-Pol (about 4 km to the west of S-Pol). While this temperature will be somewhat different than the actual ambient temperature at S-Pol, it is a reasonable approximation and does track the trends of temperature in the Firestone site vicinity.

Figure 4 shows an example S-Pol time series of , CPR, and the resulting [Eq. (4)] from 15 January 2015. Colorado time is UTC h. As can be seen, there is a significant increase of about 0.15 dB in and 0.2 dB in bias over the 7.5-h measurement period. The CPR increases about 0.05 dB. Changes of 0.2 dB in bias are significant and need to be understood and corrected for accurate quantitative use of . What is causing this change in ? Figure 5 shows and the approximate ambient temperature versus time at the Firestone site and suggests that temperature could be a factor. In fact, it is very likely that thermal expansion of S-Pol’s antenna is the primary cause of the increase in and to substantiate this, other possible sources of the change in bias will be eliminated below. The possible sources of bias variability are 1) the antenna (from the reference plane through the antenna) 2) the receiver path (from the reference plane through the receivers), and 3) the transmit power as measured at the reference plane. The receiver path can be broken into two parts: 1) from the reference plane to the IF switch and 2) from the IF switch through the copolar and cross-polar receivers. The analysis below will examine each of these possible error sources.

Fig. 4.

, CPR +1, and for 15 Jan 2015.

Fig. 4.

, CPR +1, and for 15 Jan 2015.

Fig. 5.

and temperature at the S-Pol Firestone site for 15 Jan 2015.

Fig. 5.

and temperature at the S-Pol Firestone site for 15 Jan 2015.

Examine in Eq. (1). The variability of could be caused by either 1) the antenna and waveguide terms, , or 2) the remaining terms in Eq. (1), which are part of the receiving network. Various solar, test pulse, and cross-polar power ratios are compared in order to identify which of these radar components are most likely to be causing the bias to vary over time.

Figure 6 shows power ratios of the copolar receiver to the cross-polar receiver for both H and V received signals. The data are from 15 January 2015. From Fig. 3 the power ratios are

 
formula
 
formula

These equations are valid for V and H signals resulting from either the sun or test pulse signals injected at the reference plane. Test pulse signals are injected routinely at the end of the maximum unambiguous range for a particular PRT. S-Pol typically operates with a PRT of 1 ms so that the injected test pulse signal appears at the 149-km range ring. These ratios are only a function of the gains in the receive path from the IF switch through the copolar and cross-polar receivers. The curves for both the test pulse and solar powers are theoretically identical. There is a negligible difference of about 0.01 dB on average between these two signals (the bottom two curves in Fig. 6). More importantly, the and signals in Fig. 6 are nearly flat during the 7.5-h measurement period and show no ambient temperature dependence. The signals do show a 0.1-dB offset from the signals. This is apparently due to the IF switch and surrounding connections but is not important, since the CP calibration procedure will account for this. Also, the offset is unimportant in terms of this analysis of the temperature dependence of bias. It can be concluded that the differential gain of the corresponding hardware components (i.e., the IF switch and the copolar and cross receivers) was very stable on 15 January 2015 and is not the source of the observed bias temporal variation. The same can be concluded for all S-Pol data from December 2014 through February 2015. To be clear, the measurement on S-Pol uses only the copolar receiver because the IF switch always directs the copolar signal to . Because of this does not appear in the equations for , CPR, nor . Nevertheless, it is important to show the stability of the IF switch and the receivers so that they are eliminated from possible sources of bias variations shown in this article.

Fig. 6.

V-copolar to V-crosspolar, and H-copolar to H-crosspolar power ratios for solar data and for test pulse data. Test pulse signals are injected at the test plane of Fig. 3.

Fig. 6.

V-copolar to V-crosspolar, and H-copolar to H-crosspolar power ratios for solar data and for test pulse data. Test pulse signals are injected at the test plane of Fig. 3.

Other possible sources of bias variability are the transmit power ratio [Eq. (A5)] and . Thus, it is instructive to compare to CPR, since is contained in the CPR of Eq. (2); that is, variations in should be seen in CPR measurements. Figure 7 shows that and CPR curves correlate (i.e., similar trends vs time) quite well. Thus, it is apparent that the increasing trend of is the dominant factor in the increase of CPR. The radar components most likely affecting are the transmit power divider (fast mechanical switch) and the circulators on transmit (). Figure 8 shows CPR and the ratio [Eq. (5)], that is, the measured transmit power ratio is subtracted (dB) from CPR. From above, since the differential gains vary little (based on data in Fig. 6), it can be concluded that the variability in is attributable to the LNAs and circulators on receive (i.e., ). This variability is about 0.02–0.03 dB over the 7.5 h, and there is little trend over the measurement period. If the circulators behave similarly on receive as they do on transmit, then the increasing trend in CPR for 15 January is likely due to the transmit power divider hardware. This increase is about 0.04 dB over the 7.5-h measurement period on 15 January.

Fig. 7.

CPR (solid line) and corresponding to Fig. 6.

Fig. 7.

CPR (solid line) and corresponding to Fig. 6.

Fig. 8.

CPR (solid line), and difference of the CPR and V/H transmit power ratio [RG, Eq. (5)] (dashed line) in decibels.

Fig. 8.

CPR (solid line), and difference of the CPR and V/H transmit power ratio [RG, Eq. (5)] (dashed line) in decibels.

Thus, it has now been shown that the radar components in the complete receive path (from the reference plane through the receivers) cause relatively little variability. Importantly, the components in are not responsible for the observed increase in bias on 15 January. Equation (1) shows that is contained in . Thus, the increasing trend in seen on 15 January is due to the antenna and possibly the waveguide from the sea container up to the S-Pol feed horn. However, since the H and V waveguides that travel in parallel up to the antenna are made of the same material and the waveguide attenuation characteristic are minimally dependent on temperature, it is very likely that the waveguide is not the source of the seen bias dependence on temperature.

The relationship between the ambient temperature and for S-Pol at the Firestone site was consistent throughout MASCRAD. Solar scan data were gathered on 24 and 26 December 2014; 9, 10, 11, 12, and 15 January 2015; and 22, 25, 26, and 27 February 2015. There are 328 analyzed solar scans. The transmitter was on during the solar scans so that both CPR and could be estimated from the solar scans. The solar powers were estimated from gates beyond 112-km range so that sidelobe ground clutter is eliminated. CPR was estimated from gates close to the radar where sidelobes yielded significant cross-polar ground clutter power. Note that it is necessary that the transmitter be on during solar scans or else a bias will be incurred. It was found that the circulators’ H-to-V differential gain changes when there is no high-power pulse passed through them. Shown in Fig. 9 is a scatterplot of the 328 estimates versus ambient temperature at the S-Pol site. Clearly there is a trend in the data with increasing approximately 0.4 dB over the temperature range of −20° to 20°C. Figure 9 also shows the linear least squares regression fit,

 
formula

where T is the temperature (°C). The Pearson linear coefficient of correlation is 0.902. The conclusion is that the variation in as a function of ambient temperature during MASCRAD is primarily from the antenna rather than from the receive electronics. However, the bias is a function of both and CPR, and CPR is a function of the transmit power ratio. Figure 4 shows that on 15 January 2015, the increase in the correction factor is partly due to (about 75%) and CPR (about 25%). The reason for the increasing trend of the transmit power ratio for the 15 January data is unknown and since the transmitter electronics are in a temperature-controlled environment, this increase in power ratio is not directly related to the outside temperature. There was no consistent relationship between the transmitter power difference and ambient temperature during MASCRAD.

Fig. 9.

Scatterplot of vs temperature for S-Pol data from December 2014 through February 2015 at the Firestone site. Each of the 328 points represents one solar box scan.

Fig. 9.

Scatterplot of vs temperature for S-Pol data from December 2014 through February 2015 at the Firestone site. Each of the 328 points represents one solar box scan.

To visually demonstrate how temperature affects the bias in terms of the antenna patterns, Fig. 10 shows two antenna patterns gathered at 1512 UTC (Fig. 10a) and 2246 UTC (Fig. 10b) 15 January 2015. For the antenna pattern, is calculated for each azimuth and elevation angle. The two white circles are the and 2° solid angles, respectively. The color scale is in 0.2-dB increments. For calibration purposes, is integrated over the solid angle (which yields nearly the same number as when integrating over the solid angle) and is 0.732 dB (Fig. 10a) and 0.891 dB (Fig. 10b). The difference between the two antenna patterns is easily seen visually.

Fig. 10.

” antenna patterns from 15 Jan 2015 at (a) 1512 and (b) 2246 UTC with integrated being (a) 0.732 and (b) 0.891 dB. Here and (see  appendix A for definitions) are calculated for each azimuth and elevation angle of the given patterns. White crosshairs show the estimated center of the antenna pattern, whereas the blue 0° center is derived from the known position of the sun and the S-Pol pointing angle measurements. Difference between the white and blue crosshairs is an estimate of the S-Pol’s pointing angle error.

Fig. 10.

” antenna patterns from 15 Jan 2015 at (a) 1512 and (b) 2246 UTC with integrated being (a) 0.732 and (b) 0.891 dB. Here and (see  appendix A for definitions) are calculated for each azimuth and elevation angle of the given patterns. White crosshairs show the estimated center of the antenna pattern, whereas the blue 0° center is derived from the known position of the sun and the S-Pol pointing angle measurements. Difference between the white and blue crosshairs is an estimate of the S-Pol’s pointing angle error.

b. Measurements during PECAN

From 1 June to 15 July 2015, S-Pol collected data for the field campaign PECAN, which was centered at Hays, Kansas. S-Pol was located about 26 mi southeast of Hays, close to McCracken, Kansas. During the initial part of PECAN, S-Pol suffered several mechanical failures, such as the rotary joint, transmitter, and air conditioners. The S-Pol system did not achieve stability until 16 June and thus the data given below are restricted to 16 June–16 July.

As found during MASCRAD, the bias of S-Pol data during PECAN is also a function of the ambient temperature at the S-Pol site. Figure 11 shows 130 data points of versus the temperature of the S-Pol antenna. The regression least squares fit is

 
formula

where T is in degrees Celsius. The Pearson linear coefficient of correlation is and the standard deviation is 0.02 dB. This standard deviation represents the uncertainty of estimating when using the regression fit curve. During PECAN the temperature of the S-Pol antenna was monitored by five temperature probes mounted to the metal structure of the antenna. The temperature of the antenna in Fig. 11 is the average of those measurements made at 10-min intervals. Again, is highly correlated to the temperature of the S-Pol antenna in Kansas, but interestingly the slope of the line is negative 0.0077 dB °C-1,whereas the slope of the line in Fig. 9 is positive 0.0082 dB °. The reason for this is not well understood. Note that the temperature ranges during MASCRAD and PECAN do not overlap and thus the measurements could be compatible. This will be further investigated with future measurements. The process of dismantling the antenna for shipping and then erecting it again could affect measurements. The same operating frequency, 2809 MHz, was used at both sites.

Fig. 11.

Scatterplot of vs temperature for S-Pol data from 19 Jun to 16 Jul 2015 with the least squares regression curve. Each of the 130 points represents one solar box scan. Correlation coefficient is .

Fig. 11.

Scatterplot of vs temperature for S-Pol data from 19 Jun to 16 Jul 2015 with the least squares regression curve. Each of the 130 points represents one solar box scan. Correlation coefficient is .

To illustrate the long-term nature of the abovementioned temperature dependence during PECAN, Fig. 12 shows , CPR, , and the temperature of the S-Pol antenna for 21 June to 17 July 2015. According to Eq. (5), the variability of is due to the LNAs and the circulators (neglecting ), though it does include all the electronics from the reference plane through the IF switch. As can be seen, the curve varies only about 0.03 dB over the 27-day period. It follows that the variation of versus temperature is primarily due to the differential antenna gain varying with antenna temperature.

Fig. 12.

Time series of CPR (crosspolar power ratio), (transmit power ratio) [log scale; showing the power variations due to the receiver; Eq. (5)], and the temperature of the S-Pol antenna.

Fig. 12.

Time series of CPR (crosspolar power ratio), (transmit power ratio) [log scale; showing the power variations due to the receiver; Eq. (5)], and the temperature of the S-Pol antenna.

Figure 13 shows a scatterplot of low-pass filtered time series of CPR versus low-pass filtered time series of from 21 June to 16 July that demonstrates the correlation of these two measurements. The filter is used to minimize measurement noise and to better show the correlation of the mean trends of the two variables. This again shows that the variability of CPR is primarily a function of . Also seen from Fig. 12 is the negative correlation between temperature and : in general when the temperature is higher, is lower. Even though the transmitter electronics were in a temperature-controlled environment, it is likely that the air conditioning of the sea container affected the transmit network. Comparing the temperature and profiles in Fig. 12, when the temperatures are lower, varies less. The air conditioning of the S-Pol sea container is being improved to better regulate the temperature. This example clearly illustrates the ability of CPR to track and thereby compensate the factor for these variations.

Fig. 13.

Scatterplot of CPR vs for S-Pol data from 21 Jun to 16 Jul 2015. Times series of CPR and are smoothed with a low-pass filter.

Fig. 13.

Scatterplot of CPR vs for S-Pol data from 21 Jun to 16 Jul 2015. Times series of CPR and are smoothed with a low-pass filter.

Since the state of S-Pol was quite stable from 16 June to 16 July, the regression curve from Fig. 11 can be used to estimate with a 2σ uncertainty of 0.04 dB. The temperature of S-Pol’s antenna is known (measured in 10-min intervals). CPR is estimated from both PPI and RHI scans over the entire measurement period and this provides about 6-min temporal resolution for CPR. Under these conditions, data from PECAN can be accurately calibrated to account for the ambient temperature variations by using Eq. (10) for an estimate in Eq. (4). The resulting estimates are plotted in Fig. 14 for 21 June to 16 July. In this way, can be calibrated even at night using the CP technique and the regression analysis of against antenna temperature. Note that the ambient temperature at S-Pol followed quite closely the antenna temperature so that if only the ambient temperature was available, then could still be well calibrated. The differences between the ambient and antenna temperatures were less than about 3°C and this translates to a correction difference of 0.023 dB. To corroborate the CP estimates, they are next compared to VP bias estimates during PECAN.

Fig. 14.

Estimates of the correction factor (negative of the bias) from 21 Jun to 16 Jul 2015 during the PECAN experiment using the CP technique, where is estimated from the regression curve given in Eq. (10).

Fig. 14.

Estimates of the correction factor (negative of the bias) from 21 Jun to 16 Jul 2015 during the PECAN experiment using the CP technique, where is estimated from the regression curve given in Eq. (10).

4. bias from vertical-pointing data

To support the CP calibration estimates, they are first compared to VP bias estimates, which are typically regarded as the most reliable, accurate technique for bias estimation. During PECAN there were six VP measurement episodes. Table 1 shows the comparison between CP and VP bias estimates, and the antenna temperature. The largest difference between CP and VP estimates is 0.033 dB on 14 July. There is an entry from 14 June reported in Table 1 and this time is outside the S-Pol stable period (16 June–16 July) so that the regression curve of Eq. (10) is not optimal. However, a regression curve can be calculated from the eight solar scans executed on 13 and 14 June over a C range, which represents more accurately the calibration state of S-Pol for that period. These data and the resulting regression fit ( dB) are shown in Fig. 15. The correlation is −0.86 and the standard deviation is 0.018 dB. The CP bias reported in Table 1 is calculated using this regression curve, and the VP and CP bias estimates are within 0.02 dB of each other. If the regression curve from Fig. 11 is used instead, then the CP bias estimate is dB, which is 0.069 dB higher than the VP bias of −0.142 dB. This again shows that if a temperature-to- relationship can be established, then accurate bias estimates can be made while the radar remains in the stable state during which the solar measurements were made. On 15 June several repairs were made (trigger amplifier was replaced a couple of days prior and required adjustments) to S-Pol that caused the differential receiver gain of S-Pol to change by about 0.1 dB.

Table 1.

Comparison of bias estimates from VP data and from the CP technique as a function of antenna temperature.

Comparison of  bias estimates from VP data and from the CP technique as a function of antenna temperature.
Comparison of  bias estimates from VP data and from the CP technique as a function of antenna temperature.
Fig. 15.

Scatterplot of vs temperature for S-Pol data from 13 to 14 Jun 2015 with the linear regression curve. Resulting correlation is and the standard deviation is 0.018 dB.

Fig. 15.

Scatterplot of vs temperature for S-Pol data from 13 to 14 Jun 2015 with the linear regression curve. Resulting correlation is and the standard deviation is 0.018 dB.

The 2 July VP measurements present an interesting 50-min-long case, during which there was a 4.7°C drop. Figure 16 shows a comparison of VP-measured and CP-estimated bias along with antenna temperature. Normally, the VP bias is calculated from multiple revolutions of the antenna but here the VP bias is shown for single revolutions (50 estimates). The bias from the CP method is calculated using the regression curve from Fig. 11 and a CPR of −0.760 dB. CPR was not estimated during the VP collection period but CPR was nearly unchanged (0.760 dB at 1401 UTC and 0.770 dB at 1453 UTC) before and after the VP data collection period. There is good agreement between the VP- and CP-calculated biases. There is a small 0.02-dB difference between the CP and average VP for times later than 14.6 UTC. There is, however, a significantly larger variation in the VP-calculated biases from 14 to 14.5 h than is typically seen with S-Pol data, which is exemplified from 14.5 to 15 h in Fig. 16. If the bias change is due to thermal expansion of the antenna, then the transition of the shape of the antenna from 14 to 14.5 UTC may not be symmetric, thus causing the variations seen for the VP-calculated bias. In any case, the VP and CP bias estimates show similar trends. Figure 16 also demonstrates the complexities and uncertainty of bias estimation.

Fig. 16.

Comparison of the bias for 2 Jul 2015 calculated from VP data and from the CP method. VP biases are calculated from single revolutions of the S-Pol antenna. Markers on the VP bias curve indicate the data points. Antenna dish temperature (°C) is shown (dotted curve).

Fig. 16.

Comparison of the bias for 2 Jul 2015 calculated from VP data and from the CP method. VP biases are calculated from single revolutions of the S-Pol antenna. Markers on the VP bias curve indicate the data points. Antenna dish temperature (°C) is shown (dotted curve).

5. Uncertainty analysis

a. CP technique

To estimate the uncertainty of the CP technique bias estimate, the uncertainty of both the estimate and the CPR estimate must be determined. To determine the uncertainty of the estimate, the experimental data are used. For MASCRAD, the data from 15 January (Fig. 5) are low-pass filtered as shown in Fig. 17. The low-pass filtered data are assumed to represent the true unbiased trend in the data so that the measurement error, or uncertainty, can be estimated from the differences of the filtered and unfiltered data. Term σ is 0.0084 dB, which is considered the uncertainty of the estimate at the 1σ confidence level. Similar measurement errors were determined from other days during MASCRAD and PECAN.

Fig. 17.

unfiltered and low-pass filtered. Filtered data are an estimate of the true mean trend of the data collected at the Firestone site on 15 Jan 2015. Uncertainty of the estimate is calculated as the standard deviation of the differences between the two curves.

Fig. 17.

unfiltered and low-pass filtered. Filtered data are an estimate of the true mean trend of the data collected at the Firestone site on 15 Jan 2015. Uncertainty of the estimate is calculated as the standard deviation of the differences between the two curves.

Next, the uncertainty of CPR is addressed. Figure 18 shows a scatterplot of the cross-polar powers versus for data collected at 0007 UTC 26 June 2015 from one volume scan. The thresholds applied are 4 km range km, , where is the correlation coefficient between the two cross-polar time series (i.e., the I and Q samples) for a resolution volume. Since reciprocity states that the two cross-polar backscatter cross sections are equal, the correlation of the two cross-polar time series of is a good threshold to eliminate data points that are potential outliers. The data line up well along the 1-to-1 line; however, it is difficult to see variations on the order of 0.10 dB and less on the scale of the plot. To view the small variations of CPR, the data in Fig. 18 are binned in 2-dB increments of the average . The binned averages are shown in Fig. 19. An SNR of 12 dB corresponds to about dBm on the x axis. For cross-polar power from to dBm, the mean trend in CPR is about constant at an average of approximately dB. This demonstrates that the CPR is not dependent on the magnitude of the cross-polar power data from this range and that this is a good SNR range over which to calculate CPR. The deviations in CPR at each end of the horizontal axis are caused by low SNR (left end) or receiver saturation (right end).

Fig. 18.

Scatterplot of FHV vs (crosspolar powers) for data gathered at 0007 UTC 26 Jun 2015; and pairs are separated by 1 ms.

Fig. 18.

Scatterplot of FHV vs (crosspolar powers) for data gathered at 0007 UTC 26 Jun 2015; and pairs are separated by 1 ms.

Fig. 19.

The average CPR in 2-dB bin increments vs corresponding to the data in Fig. 18.

Fig. 19.

The average CPR in 2-dB bin increments vs corresponding to the data in Fig. 18.

To estimate the uncertainty of the mean CPR from the abovementioned volume of data, bootstrap resampling is employed (Efron and Tibshirani 1998). There are 12 303 CPR values, and this dataset is resampled with replacements to create a dataset with 10 000 values. The mean of the resampled dataset is then calculated. This process is repeated 10 000 times and the mean of the means is dB with a standard deviation of 0.003 48 dB, which is the uncertainty of the mean CPR estimate from the original dataset (−0.7576 dB).

The uncertainty of the CPR estimate can also be found from the time series of CPR calculated from RHI and PPI volume data collected during PECAN. Figure 20 shows those CPR estimates from 21 June to 16 July (one estimate per volume scan). The red curve is a low-pass filtered version of the raw black curve and it represents the mean trend of CPR. The diurnal oscillations are primarily due to fluctuations in the transmit power ratio, which in turn are likely due to temperature oscillations inside the transmitter sea container. The difference of the two curves yields a standard deviation of 0.004 09 dB, which is an estimate of the measurement uncertainty. This compares very well with the uncertainty estimate of 0.003 48 dB calculated from bootstrap resampling mentioned above. Below the uncertainty from the bootstrap technique is used.

Fig. 20.

CPR calculated from RHI and PPI scan data from PECAN. Low-pass filter version (red) of the raw data (black curve). Mean trend of CPR primarily due to fluctuation in the transmit power ratio (red curve). Difference between the two curves yields a standard deviation of 0.004 09 dB, which is an estimate of the measurement uncertainty.

Fig. 20.

CPR calculated from RHI and PPI scan data from PECAN. Low-pass filter version (red) of the raw data (black curve). Mean trend of CPR primarily due to fluctuation in the transmit power ratio (red curve). Difference between the two curves yields a standard deviation of 0.004 09 dB, which is an estimate of the measurement uncertainty.

The and the CPR uncertainty estimates are considered independent so that they can be combined using the root-mean-square average (Taylor 1997). The uncertainty of the CP calibration technique then is dB at the level and 0.019 dB at the level (i.e., 95% confidence). However, if the bias is to be estimated from from the regression curve of Fig. 11, then the uncertainty of that estimate must be used (from above, dB). Then the uncertainty of the bias estimate becomes dB at the 1σ level and 0.0408 dB at the 2σ level.

b. VP technique

To estimate the uncertainty of the VP bias estimate, the mean trend of extended period experimental VP datasets is first estimated. This was shown in Fig. 1, where the regression fit curve is assumed to track the unbiased mean trend of the VP data. As shown, the standard deviation of the residuals is 0.0071 dB so that the uncertainty is 0.0142 dB. This is calculated from bias estimates for single revolutions of the antenna. Normally, the bias would be calculated from an average of several revolutions of the antenna. This would reduce the uncertainty of the bias estimate.

To further illustrate the uncertainty of the VP bias estimate, the PECAN VP data gathered from about 1403 to 1453 UTC 2 July 2015 in Fig. 16 are examined. To help establish the data quality of VP datasets, one usually visually examines histograms of the values. The data thresholds used are 3 km range 15 km, 13 dB SNR 60 dB, and . If the histogram is fairly narrow (i.e., a standard deviation of a couple of tenths of a decibel) and is visually symmetric about the mean, then the data are considered well behaved. Instead of showing 50 histograms here, Fig. 21 shows the mean, median, mode, and standard deviation of the 1-min VP datasets of Fig. 16. It is well known that for symmetrically distributed histograms, the mean, median, and mode are equal. In Fig. 21, the mean and mode are nearly equal, while the median does show some differences on the order of several hundredths of a decibel; however, as compared to the standard deviation of the data (about 0.10 dB and greater), the median is considered to be in reasonable agreement with the mean and mode. Thus, the given statistics indicate that the VP histograms are quite symmetric with small standard deviations. The standard deviations of all the bias histograms from a single revolution of the antenna are less than 0.17 dB (light blue curve is the standard deviation divided by 3). Visual inspection of the histograms also indicates that the VP data are not anomalous, and it follows that the given bias estimates should be indicative of the true bias of S-Pol. The number of samples for each histogram from 14.1 to 14.5 UTC is greater than 25 000 and from 14.5 to 14.85 UTC it is greater than 10 000. The green curve is a five-point running average of the mean estimates (red curve) and this represents a typical amount of data used for a bias estimate for S-Pol. The dashed magenta curve is a low-pass filtered version of the mean estimates, and it is assumed to represent the true unbiased trend of the bias. Then the uncertainty of the VP bias estimate can be determined from the standard deviation of the differences of the magenta curve and the red or green curves. Even though the red curve is not quite as smooth as the green curve, both have a standard deviation of about 0.015 dB or a uncertainty of 0.03 dB. If the uncertainty is estimated for times later than 14.5 h, then the uncertainty drops to 0.0128 dB, which is in close agreement with the previous dataset.

Fig. 21.

Bias of for 2 Jul 2015 calculated from VP data. Mean as shown in Fig. 16 (red curve). Also shown are the median (black), mode (blue), and five-point average of the mean (green). Low-pass filtered version of the green curve (magenta dashed curve), and it is considered to represent the true mean trend of the data. For visual purposes, the standard deviation, divided by 3, of the estimated biases at each point (light blue curve).

Fig. 21.

Bias of for 2 Jul 2015 calculated from VP data. Mean as shown in Fig. 16 (red curve). Also shown are the median (black), mode (blue), and five-point average of the mean (green). Low-pass filtered version of the green curve (magenta dashed curve), and it is considered to represent the true mean trend of the data. For visual purposes, the standard deviation, divided by 3, of the estimated biases at each point (light blue curve).

These uncertainty estimates assume that there are no undetected systematic biases. For example, if the shape of the antenna were to change from when it points horizontally to when it points vertically, then the vertical-pointing bias estimate could cause a systematic error in bias for data gathered when the antenna points horizontally. However, since the VP and CP bias estimates are in good agreement for PECAN data, it is very likely that any undetected systematic bias is small. The 2 July data do illustrate the problem with limited VP bias estimates. For example, using 5 min of VP data (five 360° antenna revolutions) at about 14.15 h, the bias estimate would be about . If this value were used as the bias estimate, then for data gathered at 14.5 h (just 20 min later), would be underestimated by about 0.06 dB. If only a single 360° scan is used to estimate the bias, then greater errors could occur.

6. bias estimation from Bragg scatter

In this section the bias estimated from Bragg scatter detected during PECAN is used to further corroborate the CP-calculated biases. Bragg scatter has been well discussed and reviewed in the literature, for example see Gossard (1977, 1990), Doviak and Zrnić (1993), and Wilson et al. (1994). More recently observations of Bragg scatter with a WSR-88DP have been addressed by Melnikov et al. (2011, 2013). Bragg scatter occurs in the atmosphere when turbulence mixes air masses of different refractive indices on the scale of half the radar wavelength. It is thought that these half-wavelength scattering centers are distributed isotropically throughout the resolution volume so that the observed scattered power is independent of the observation direction and therefore should be 0 dB. Polarimetric radar provides two excellent parameters to distinguish biota echo from Bragg scatter: insects and birds typically are manifest by high and low (copolar correlation coefficient, the correlation of the H and V copolar signals; Bringi and Chandrasekar 2001) while Bragg scatter is characterized by near-zero (dB) and high () while Melnikov et al. (2011) reports Bragg scatter between and 0.06 dB using WSR-88DP data, the WSR-88DP’s have proven difficult to calibrate (Ice et al. 2014). Since S-Pol is very well calibrated for PECAN, this dataset provides an opportunity to evaluate from Bragg scatter and to further corroborate the CP bias estimates.

To identify Bragg scatter, the following thresholds are used:

  • the resolution volume is identified as “cloud drops” by the particle identification (PID)1 algorithm

  • 4 km range 30 km

  • reflectivity 0 dBZ

  • 3 dB SNR 50 dB

  • 0.98 (no noise correction)

  • radial velocity 1.5 m s-1

A estimate from identified Bragg scatter is made for each PPI volume and RHI volume (the RHI volumes consist of 12 vertical cuts at multiples of in azimuth) provided that at least 400 valid range gates with Bragg scatter per volume are found. The resulting estimates from 20 June through 3 July are shown in Fig. 22. The bias correction has been applied using the CP calibration technique (data from Fig. 14). Figure 22 illustrates the variability of from identified Bragg scatter resolution volumes, at least with our present identification algorithm. Nearly all of the outliers (i.e., ) can be attributed to either relatively small or thin Bragg scatter layers or to precipitation echo identified as Bragg scatter. Still, a large majority of the values in Fig. 22 are between dB. Thus, this dataset shows the potential of Bragg scatter to estimate bias; however, Fig. 22 also demonstrates the uncertainty. To better compare and corroborate the CP technique bias estimates, a subset of the Bragg scatter is examined next.

Fig. 22.

Estimates of from identified Bragg scatter. CP bias correction has been applied to the data. Gaps in the data indicate insufficient Bragg scatter was identified. Numbers in the plot are the average contained in the oval areas.

Fig. 22.

Estimates of from identified Bragg scatter. CP bias correction has been applied to the data. Gaps in the data indicate insufficient Bragg scatter was identified. Numbers in the plot are the average contained in the oval areas.

Stronger Bragg scatter events are manually identified that are large in spatial extent and continuity in the high and near-zero signatures. Using the radar data and using the east-, west-, north-, and south-facing cameras that were located at S-Pol during PECAN, possible biasing clouds and precipitation can be identified and avoided. The times when there are strong Bragg scatter echoes, with likely minimal contamination, are marked in Fig. 22 by the ovals. The averages of the data in the ovals are shown in the figure. The estimated values are now confined largely to dB except for the case around 21 June 00 UTC, when the average is −0.069 dB. From 19 to 21 June several solar scans were made and a scatterplot of versus temperature is shown in Fig. 23 along with the resulting regression curve. The regression curve from Fig. 11 (dashed line) is included as well, and it shows that the 19–21 June data points lie above that regression curve and indicate that the calibration state of S-Pol was about 0.04 dB higher on average for that period. From above, the 2σ uncertainty of the is less than 0.02 dB, and since two points in Fig. 23 are greater than 0.02 dB from the dashed curve and the seven other points are greater than about 0.04 dB above the dashed curve, the probability that the nine from 19 to 21 June represent a different calibration state for this time frame is nearly 1.

Fig. 23.

Scatterplot of vs temperature for S-Pol data from 19 Jun to 21 Jun 2015. Least squares regression fit (solid line) is on the plot along with the regression curve from Fig. 11 (dashed line). Slopes are nearly identical.

Fig. 23.

Scatterplot of vs temperature for S-Pol data from 19 Jun to 21 Jun 2015. Least squares regression fit (solid line) is on the plot along with the regression curve from Fig. 11 (dashed line). Slopes are nearly identical.

Figure 24 shows a comparison of the 0000 UTC 21 June Bragg data with curve A calibrated using the regression curve from Fig. 23 and curve B calibrated using the regression curve from Fig. 11. Since Bragg scatter should have 0-dB , curve A is judged to have been better calibrated. Again, since the regression curve in Fig. 23 is limited to solar scan data gathered around 21 June, those values represent more accurate estimates of differential gain of the antenna for the 21 June Bragg data.

Fig. 24.

for Bragg scatter on 20–21 Jun 2015. Curve A is calibrated using the regression curve from Fig. 23, while curve B uses the regression curve from Fig. 11.

Fig. 24.

for Bragg scatter on 20–21 Jun 2015. Curve A is calibrated using the regression curve from Fig. 23, while curve B uses the regression curve from Fig. 11.

7. Possible cause of the bias temperature sensitivity

The cause of the temperature-dependent estimates is likely thermal expansion of the antenna and that is supported next. To make a first-order estimate of the amount of antenna dish movement required for a significant change in bias, first consider the phase difference incident at the vertex of the parabolic antenna for a 1-MHz change in operating frequency. This is relevant, since it has been observed that the antenna pattern can indeed change significantly for a 1-MHz change in receive frequency (S-Pol operates at 2809 MHz) and this is demonstrated in Fig. 25. A change from 2808 to 2809 MHz causes more than a 0.1-dB change in . The premise is that this small increment in frequency causes the phases of the radio frequency wave from the feed horn to the parabolic reflector to change, which in turn changes the H and V far-field patterns. Using 2800 and 2801 MHz, this gives 35.5727 and 35.560 wavelengths from the feed horn to the antenna vertex, respectively, which is 150 in away, for a difference of 0.0127 wavelengths. This translates to a phase difference of 4.57 and a distance of 1.37 mm. It is assumed that this amount of phase change of the incident wave onto the antenna dish is indicative of the source of the variation in the antenna patterns. The question is whether the thermal expansion of the antenna is sufficient to make such a change in dimension. A typical thermal expansion coefficient for aluminum alloy (6061) is about 23.6 μcm−1 cm−1 (°C)−1. The S-Pol support struts are about 18 ft (243.8 cm) long, so for a change of 10°C this translates to an expansion of 1.30 mm, which is of the same order of magnitude as 1.37 mm. While this is not proof that the thermal expansion of the S-Pol antenna is the cause of the calibration variance as a function of temperature, it does show that it is physically possible. However, the field of radio astronomy has long been aware of the problem of temperature dependency of antenna structures (Nothnagel 2009) for very long-baseline interferometry (VLBI) data analysis. In VLBI the issue is the movement of the VLBI reference point due to thermal expansion. For bias it is the movement of the phase center of the feed horn and the expansion/contraction of the parabolic reflector that are likely the issues. The exact cause of this temperature‐dependent bias is a topic of future investigation.

Fig. 25.

vs operating frequency from S-Pol data gathered on 7 Sep 2016 at the Marshall field site. A 1-MHz change in operating frequency can significantly change and thus the bias. The cause of this is not well understood.

Fig. 25.

vs operating frequency from S-Pol data gathered on 7 Sep 2016 at the Marshall field site. A 1-MHz change in operating frequency can significantly change and thus the bias. The cause of this is not well understood.

8. Summary and conclusions

This paper has presented a detailed analysis of calibration for S-Pol, first via an extended period solar scan and cross-polar power ratio (CPR) measurements that are required for the CP calibration technique. Solar scan data, made over many days during the MASCRAD and PECAN experiments, revealed that the solar power ratio , which is a function of the antenna’s V-to-H differential gain, had a strong correlation with the ambient temperature at the S-Pol sites. Measurements of CPR and the transmit power ratio provided additional information so that it could be concluded that the temperature’s effects on the antenna’s differential gain was the primary cause of the variability of and, consequently, the bias. For MASCRAD, changed 0.0082 dB C-1 and for PECAN data dB C-1 as determine from regression fits of onto temperature. The reason for the change in sign is not known; however, it would seem that it is a characteristic of the thermal expansion of the antenna. Note that the range of temperatures in MASCRAD was nonoverlapping with the temperatures in PECAN so that both datasets can be compatible. For PECAN, the temperature of S-Pol’s antenna was monitored so that could be estimated in 10-min intervals using the regression curve of on to temperature. CPR was estimated regularly from the RHI and PPI data during PECAN so that a temperature-dependent correction factor could be calculated via Eq. (4). This is the first time that such a temperature-dependent correction factor has been shown.

To corroborate the CP bias estimates, they were compared to VP bias measurements during PECAN. Of the seven cases compared over five separate days, the CP and VP bias estimates were within 0.033 dB of each other. This lends validity to both techniques and our estimation algorithms. Since Bragg scatter is characterized by 0-dB , regions of Bragg scatter from PECAN with no discernible contamination from insects or precipitation were manually identified. The Bragg scatter (over 200 data points), corrected with the CP technique, was contained to dB with a mean of dB. This again validates our CP calibration results and confirms the notion that in Bragg scatter layers is very close to 0 dB if uncontaminated by biota or precipitation. Finding usable Bragg scatter is difficult however. For example, Bragg scatter is most frequently found at the top of the boundary layer and is typically bordered by biological scatter (e.g., insects), which typically has very high ; 10-dB is not uncommon. If 10-dB insects contaminate the Bragg scatter and are 20 dB below the Bragg scatter power, then the incurred bias is 0.039 dB.

The uncertainty of the CP calibration technique was assessed, and it was 0.02 dB at the 2σ (95% confidence) level. Such low uncertainty is contingent upon the algorithm used, the data quality of the radar, the stability of the radar, and the diligence of the radar staff. This uncertainty can be considered valid for a snapshot measurement of the calibration state of the radar and may be good for only a few tens of minutes as it was for S-Pol when the ambient temperature was rapidly changing. For S-Pol, it was possible to derive a calibration equation that took into account the effect of temperature on the differential gain of the antenna so that could be calibrated as a function of temperature. The uncertainty of the estimate of the temperature-dependent bias then increases to about 0.041 dB at the 2σ level.

It is shown in  appendix B, using experimental data, that the CP technique can be successfully applied to SHV radars. The mean CPR estimates from FHV data and SHV data were nearly identical. A radar transmit circuit was illustrated that would support CP calibrations for SHV radars.

Next, the data from Fig. 2, which in part motivated the research for this paper, are revisited. The data are replotted in Fig. 26 along with the ambient temperature at the S-Pol site added in red. The temperature increase from 12.5 to 17.5 h correlates well with the increase in and is approximately 0.008 dB °C-1. This is nearly identical to the slope of the regression fit for for MASCRAD data in Fig. 9 of 0.0082 dB °C-1. Thus, it is apparent that S-Pol’s temperature dependence has always been there. Why has this not been seen previously? There is a dearth of calibrations that extends over a sufficient period with sufficient granularity and accuracy (uncertainty) so that variations on the order of 0.10 dB and less could be positively identified. Also, prior to this, conventional wisdom said that the antenna was an inert device whose gain was a constant. It appears now that the antenna differential gain is a significant function of antenna temperature.

Fig. 26.

data in Fig. 2 replotted here along with the ambient temperature at the S-Pol site (red). Temperature increase correlates well with the increase in . Increase in is approximately 0.008 dB °C-1, which is nearly identical to the slope of the regression fit for for MASCRAD data in Fig. 9 of 0.0082 dB °.

Fig. 26.

data in Fig. 2 replotted here along with the ambient temperature at the S-Pol site (red). Temperature increase correlates well with the increase in . Increase in is approximately 0.008 dB °C-1, which is nearly identical to the slope of the regression fit for for MASCRAD data in Fig. 9 of 0.0082 dB °.

It is likely that all weather radars with center-fed parabolic reflector antennas experience the temperature-dependent bias fluctuations shown in this paper for S-Pol. Additionally, it was shown that the bias depends on the operating frequency. Many weather radars use magnetron transmitters that are not as frequency stable as klystron transmitters, especially if the transmit frequency is synchronized with a global positioning system (GPS) timing server such as is done with S-Pol. Magnetron transmitter frequency can vary with power, duty cycle, and operating temperature, and thus this can be an additional bias factor. Shorter wavelength antennas are smaller, but smaller dimensional change is required to produce similar effects as shown here. The change in bias is about dB °C-1 for S-Pol operating at 2809 MHz, and the nature of this temperature dependence is likely to change for different operating frequencies. Depending upon the application, this amount of bias may be acceptable for some users of radar data. However, if it is desired to realistically calibrate a weather radar to within 0.10 dB with a confidence level of at least 95%, such temperature effects should be measured and corrected. Since for S-Pol the bias had a near-linear relationship to the antenna temperature (though nonlinear relationships with high-order regression fits would also be effective), it was possible to accurately compensate the bias as a function of temperature. This study does show the importance of monitoring the bias over extended periods of time with good time resolution. The CP calibration technique, based on solar scans and cross-polar power measurements, provides such extended period analyses to reveal time-dependent fluctuations.

Acknowledgments

This work was supported in part by the Radar Operations Center (ROC) of Norman, Oklahoma (EOL-2017-0711). The author would like to acknowledge the EOL/RSF technical staff for its time, effort, and interest in the collection of the experimental data used in this paper. In particular, Dr. Michael Dixon designed and wrote the solar scan and CPR analysis software and provided valuable technical discussions. The author also acknowledges Richard Ice, who recently retired from the ROC, and Frank Pratte, a former engineer at NCAR, both of whom have provided many helpful technical discussions and insights over the years. The helpful comments of three anonymous reviewers, which greatly improved the manuscript, are appreciated. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Science Foundation.

APPENDIX A

Derivation of the CP Technique for Calibration

The following equations are in linear form (i.e., not dB form). From Fig. 3 it follows that

 
formula

and

 
formula

where is the measured differential reflectivity, is the measured linear depolarization ratio (for transmit H polarization), LNA is low-noise amplifier, and where , , , and represent the insertion losses associated with various paths through the switch. For example, represents the path from the V IF switch input through the switch to the copolar receiver. Note that is the intrinsic value we wish to isolate with angle brackets () denoting time average. The ratio of the cross-polar powers becomes

 
formula

where and are the transmit V polarization receive H, and the transmit H polarization receive V cross-polar powers. By reciprocity and these terms cancel, which gives

 
formula

Thus, the cross-polar power ratio captures two sources of bias variations: 1) in the transmit power ratio, defined at the reference plane,

 
formula

and 2) the components in the receiver path from the reference plane to the output of the IF switch (plane A, Fig. 3),

 
formula

Since the transmit V/H power ratio [Eq. (A5)] is experimentally measured at the reference plane, it can be subtracted (dB) from the measured CPR in Eq. (A3). In this way can be experimentally estimated, and this is useful for identifying sources of bias drift.

Corresponding to the four receiver paths, the following are four solar calibration measurements that characterize the system:

 
formula
 
formula
 
formula
 
formula

where the superscript “s” denotes a solar measurement. To calibrate and ( is included for completeness), the three following ratios are needed:

 
formula
 
formula
 
formula

The ratio is

 
formula

Then for the S-Pol configuration, and can be calibrated using

 
formula
 
formula

where is calibrated . Thus, the bias correction factor is defined as

 
formula

In this derivation any cross coupling effects are neglected. S-Pol’s cross coupling is assessed from examining in light precipitation regions. The system limit is estimated to be about −37 dB. Such a small cross coupling level can be neglected for this study. Additionally, the principle of radar reciprocity still holds regardless of the level of cross coupling.

APPENDIX B

Using the CP Technique for Calibrating SHV Radars

Most weather radars, including NEXRAD, use simultaneous H and V (SHV) transmission and reception to achieve dual polarization, and it is shown next how the CP technique can still be applied to such radars. The SHV technology is based on the premise that the H and V transmitted waves do not cross couple in a rain medium, and therefore the resulting H and V radar returns yield accurate estimates of the H–V basis dual-polarization variables (Doviak et al. 2000). Figure 3 can be used to derive the CP calibration equation by eliminating the IF switch and using H and V receivers instead of co- and cross receivers. This was essentially done in Hubbert and Bringi (2003) and shown in their Eq. (9). The SHV calibration equation is

 
formula

where

 
formula

where are the H and V receivers that replace the co- and cross receivers. The issue is that near-simultaneous samples of H and V cross-polar returns are not available for the calculation of CPR for SHV radars. However, if the high-power front end of SHV radars is designed correctly, then the CP correction technique can still be used. Shown in Fig. B1 is a block diagram of S-Pol’s high-power front end (transmitter and switching circuitry). Waveguide switch A sends the transmit pulses either to the fast switch for fast-alternating operations or to the Magic-T (a 3-dB power divider with good isolation) for SHV operations. The important parts of the design are the two waveguide switches B and C. They either send the H(V) power to the antenna or to a dummy load that absorbs the power. These two switches provide SHV operation, or transmit only-H or only-V operations where cross-polar power data can be collected for the calculation of CPR. Note the H-only and V-only transmit paths are identical to the SHV operation H and V paths, respectively. This is very important for accurate CP calibration. The other critical point is that the two cross-polar powers and now will not be gathered in close time proximity but will be separated by tens of seconds if not minutes. For the cross-polar power to be equal (principle of radar reciprocity), the H and V transmit pulses must illuminate the same scatterers. Thus, stationary ground clutter targets must be used and a very accurate radar antenna position controller must be used so that the radar can point to the identical ground clutter scatterers for the measurement of the two cross-polar powers. S-Pol’s pointing angle accuracy is about 0.01°.

Fig. B1.

Block diagram of S-Pol’s high-power front end (transmitter and switching circuitry).

Fig. B1.

Block diagram of S-Pol’s high-power front end (transmitter and switching circuitry).

One technique for the evaluation of CPR is to alternate between only-H and only-V transmission on a PPI-to-PPI basis. If the beams are indexed, then cross-polar powers from the same resolution volumes (but from different PPI scans) can be paired and used for the CP calibration. Another viable technique is to simply point the radar along a radial where there are good clutter targets. The slow waveguide switch can alternate H and V transmit polarizations that illuminate the same clutter targets.

The question to be addressed is, can the CPR be calculated accurately from H and V samples that are separated in time on the order of a minute? Such SHV data can be simulated from FHV data. The H-polarization data from one FHV PPI scan can be compared to the V-polarization data from the next FHV PPI scan; the SHV CPR can be calculated and then compared to the equivalent FHV CPR from the same data.

On 19 April 2011, a consecutive series of thirty-four elevation angle PPIs were made by S-Pol in the FHV mode in clear weather at the Marshall Field site close to Boulder, Colorado. To minimize possible nonstationary clutter targets, a threshold of was used where CPA is a clutter detection metric (Hubbert et al. 2009). The CPR for FHV pairs of CPs, separated by 1 ms, is 0.043 dB. A scatterplot of the two CPs and a regression curve fit can be found in Hubbert et al. (2012). The correlation coefficient of the data is 0.9998 and the standard deviation is 0.204 dB. The CPR can also be calculated from cross-polar power pairs selected from consecutive PPI scans so that they are separated in time by about 1 min. The CPR for these pairs is 0.022 dB, which is in good agreement with the abovementioned 0.043-dB estimate. A scatterplot of the cross-polar powers again can be found in Hubbert et al. (2012) with a correlation coefficient of 0.989 and a standard deviation of 1.29 dB, which is significantly larger than the 0.204 dB mentioned above. A confidence interval for this mean estimate can be calculated by dividing the standard deviation by the square root of the number of samples, 12 898, which yields 1 dB. These results show that the CP calibration approach is amenable to SHV radars. The cost of the addition of two mechanical waveguide switches and two dummy loads seems worth the benefit of accurate, reliable calibration.

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