I. 3.
POLARIMETRIC MEASUREMENTS OF RAIN
D. S. Zrnic and A. V. Ryzhkov
Although there are reflectivity factor rain rate relations which can on the average retrieve correct rainfall, quite often instantaneous measurements are erroneous. To a large extend biases occur because the rain rate is specified by a drop size distribution which in its simplest form depends on two parameters. This has motivated Seliga and Bringi (1976) to use reflectivity factors at two linear orthogonal polarization and solve for the exponent and slope of a drop size distribution. Subsequently Seliga and Bringi (1978) suggested that the combination of specific differential phase and differential reflectivity could be used to obtain rainfall estimates independent of absolute calibration.
Humphries (1974) showed that specific differential phase KDP is almost linearly related to rain rate R and that it is not sensitive to the drop size distribution, whereas Sachidananda and Zrnic (1986) obtained an R(KDP) which was much less sensitive to drop size distribution variations than any of the rain rate reflectivity factor relations. Jameson (1985) related KDP to the product of the liquid water content and the departure from unity of the mass-weighted mean axis ratio of drops. Furthermore, Jameson (1991) suggested a two parameter R(Zh/Zv, KDP) relation (Zh, Zv are reflectivity factors for horizontally and vertically polarized waves) and showed its insensitivity to drop size distribution variations.
The purpose of this paper is to further elaborate on the relations between the polarimetric variables and rain amounts, and to discuss pertinent issues of rainfall estimation from specific differential phase.
Although there are several ways to relate the polarimetric variables to rain rate, three have potential for operational applications.
Following the steps of Seliga and Bringi (1976), Sachidananda and Zrnic (1987) show that
R = 6.84 10-3 Zh-3.86 Zv4.86 (1)
is a two parameter relation that approximates the effects of concentration and exponent of the drop size distribution (DSD) on the rain rate (Zh, Zv are in mm6 m-3).
A single parameter relation (Sachidananda and Zrnic 1987)
R = 40.56 KDP0.866, (2)
where R is in mm h-1 and KDP is in deg km-1, has several desirable properties which are discussed in the next section.
The prime contributor to DSD induced errors in R(KDP) is the uncertainty in the median drop diameter Do. Jameson (1991) suggested that the effects of Do could be compensate for with the differential reflectivity. Jameson's (1991) relation cast in a slightly different form (Ryzhkov and Zrnic 1995) is
R = 52.0 KDP0.960 ZDR-0.447 (3)
where R is in mm h-1, KDP is in deg km-1, and ZDR is in dB.
Relations (1), (2), and (3) have been tested by simulations and on radar data. In Fig. 1a are the errors DRDSD (adapted from Ryzhkov and Zrnic 1995) of these three estimators caused by DSD variability. Gamma distributions were simulated with a range of parameters suggested by Ulbrich (1983). It is apparent that the estimator (3) has the lowest error.
In practice it turns out that statistical errors DRS are much larger than the errors induced by the DSD variability. These errors are influenced by the Doppler spread of the weather echo signals, by the correlation between horizontally and vertically polarized echoes _rhv(0)_, by the dwell time, and by the range averaging interval. For typical spectrum widths of 1 to 6 m s-1, correlation of 0.95, dwell times of 0.1 s and range averaging intervals of 4 km the total error
DRtot = (DRDSD2 + DRS2)1/2 (4)
is in Fig. 1b (adapted from Ryzhkov and Zrnic 1995).
Fig. 1- Errors of three polarimetric estimators caused by a) drop size distribution variations, and b) both DSD variations and statistical uncertainty.
As can be seen, in the presence of measurement errors, the estimator R(KDP,ZDR) still exhibits the best performance provided that KDP and ZDR are smoothed over a range of about 4 km. But in practice there are other sources of errors to which specific differential phase is less prone so that R(KDP) may be preferable. Note that radar calibration errors degrade considerably the performance of the R(Zh, Zv) estimator.
Comparison of the three estimators with rain accumulations measured by weighing buckets has been performed for one convective event. The gage network consists of 42 stations in the Little Washita river basin which is situated from 50 to 80 km of the radar (see Atlas et al, 1995).
On June 9, 1993 a squall line passed over the basin during the time that polarimetric radar data were recorded. Volume scans were collected every five minutes from which total rain accumulations at the location of gages were computed from the lowest scan of 0.4 deg; the effective azimuthal beamwidth (Doviak and Zrnic 1993) was 1.8 deg. The number of pulse pairs (HV) was 64. Averaging was over 4 km in range for the total differential phase FDP, 3 km for ZDR, and 1 km for Zh. This four times smaller range averaging interval for Zh did not have adverse effects on our comparisons. The effects of differential attenuation where removed from the ZDR data. We used formula (2) only if KDP > 0.4 deg km-1, i.e., for rain rates larger than about 18 mm h-1, otherwise we used the Marshal-Palmer formula, Z = 200R1.6. This hybrid estimator was chosen because it is known (Sachidananda and Zrnic 1987, and Chandrasekar et al. 1990) that R(KDP) has large random errors at low rain rates.
Estimators (2) and (3) compare well with the gages (Figs. 2a,b) whereas the estimator (1) fared poorly because of contamination by hail. Exclusion of three gages over which polarimetric data indicated hail produced the following rms and percentage differences between radar and gages: for R(Zh Zv) - 8.7 mm or 38%, for R(KDP) - 5.3 mm or 28%, and for R(KDP,ZDR) - 4.5 mm or 22 %.
Use of the R(Zh, Zv) relation (1) to estimate rainfall results in the largest overall difference between radar and gage estimates; this is attributed to two reasons. First is the hail contamination which could not be completely accounted for and which reduced the observed ZDR values. The second reason is the presence of melting hail aloft in this type of storm which results in the excess of large drops, possibly, supported by ice cores that leads to increase in ZDR.
This analysis of the raingage comparisons shows a modest improvement of the R(KDP,ZDR) estimator over the alternate R(KDP) estimator. Consideration of factors other than random errors points out several practical advantages of the R(KDP) estimators which are discussed next.
Fig. 2 - Scattergram of cumulative rainfall versus rain gage totals for a) R(KDP), and b) R(KDP,ZDR). The storm occurred on June 9, 1993.
Frequency and phase measurements are preferred to amplitude measurements because of better immunity to additive noise and much lower sensitivity to distortions of amplitudes, to wit the crispy clear sounds of commercial FM radio stations. Because specific differential phase is obtained from phase measurements it retains several advantages over power measurements. These are: 1) It is independent of receiver and transmitter calibrations; 2) it is not affected by attenuation; 3) it is immune to beam blockage, 4) it is not biased by ground clutter cancelers; 5) it is insensitive to variations in drop size distribution; 6) it is little biased by the presence of hail; 7) it can be used to detect anomalous propagation. Advantages 1 through 4 are all rooted in the relative independence of phase measurements from amplitude of the signals. This holds as long as there is enough signal strength to allow an accurate estimate of phase. The remaining benefits are tied to physical principles.
With the decrease of signal to noise ratio (SNR) the phase measurements have an increased uncertainty but no bias. Eventually there is an abrupt transition as a function of SNR beyond which the estimates are not reliable. This transition occurs at signal to noise ratios somewhat less than one. It is caused by the dependence of phase error which is primarily a sum of linear and quadratic terms in SNR-1 for low values of SNR (Doviak and Zrnic 1993). Thus at SNR < 1 the quadratic term takes over and errors become excessive but, up to that point the phase measurements excel over power measurements.
Measurements of rainfall from sites with significant blockages are compromised because a) observations are at heights where precipitation might be partially or totally frozen, and b) in many directions the beam might be partially blocked and complicated corrections need to be made if a Z-R relation is to be used. Reflectivity - or power corrections require precise survey in azimuth, and knowledge of the vertical axis orientation of the antenna (Harju and Puhakka, 1980); furthermore the correction scheme is highly sensitive to small differences between indicated radar azimuth and correct azimuth of obstacles. Experience has shown that simple correction of rainfall field is possible if less than 60% of the beam is blocked and the obstacle is flat (Harrold et al. 1974).
If differential phase is used, it is possible to make accurate measurements in regions of partial beam blockage even if the horizon is rugged. Therefore it might be worth lowering the elevation scan by even a few tenths of a degree to where the beam is partially blocked so that the effective height of the truncated beam is closer to the ground. Scanning at low elevation angles is important to minimize the influence of a) changes of precipitation with height, and b) the horizontal drift of hydrometeors as they fall from the resolution volume to the ground.
Survey of terrain from the south east to south west of the Cimarron radar indicates an almost uniform height blockage at 0.2 deg in elevation. If the antenna is pointing at 0 deg in elevation the power loss due to this blockage is 6.4 dB. Measurements were made that confirm this loss.
On May 25th 1994 an isolated storm developed in south-central Oklahoma. The storm split into a right and left moving cells and about two hours of data were collected on these storms. At the beginning of data acquisition the right moving storm was at 120 km south from the radar and by the end of acquisition it was at 180 km. Although these are considerable distances, general polarimetric features were evident throughout data collection. We submit that this is important because use of all polarimetric variables in a classification/quantification scheme should produce better estimates of water accumulation than possible with any single parameter.
The integral of the rain rate over the storm area (Fig. 3) indicates that rain estimates from KDP at 0 and 0.5 deg agree with estimates from Z (200 R1.6 was used) at 0.5 deg in the early time of the record. At nearly the same time using the 0 deg elevation data the rainfall R(Z) is underestimated by about 5 dB.
Fig. 3 - Areal integral of rain rates at elevation angles of 0 and 0.5 deg for a storm cell that occurred on May 25, 1994. R(KDP) refers to the estimates obtained from the specific differential phase whereas R(Z) refers to estimates from the reflectivity factor.
The rain area integral obtained from R(Z) after 70 min is most likely in error because hail started at that time and produced a 10 km hail swath by the end of data collection. Presence of hail was deduced from a sharp increase of reflectivity accompanied by a decrease of differential reflectivity to less than 1 dB in the main core.
Although there were no rain gages in the path of this storm, the example clearly illustrates the potential of combined polarimetric measurements.
A rainfall for which raingage data is available occurred on Feb 19, 1994. An intense but shallow squall line moved over the Little Washita river basin where 43 closely spaced raingages are located (about one gage per 35 km-2). Total accumulation (sum of rain depths from all gages) was 438 mm. Because the gage network is located close to both the Cimarron polarimetric radar and the WSR-88D operational radar, two independent estimates based on the same Z-R relation are possible. The total for Cimarron was 153 mm and for WSR-88D it was 185 mm. The two radars are about 50 km apart and the higher estimates by the WSR-88D might be due to a more favorable aspect angle (less attenuation). We used the differential phase to correct attenuated values of the Cimarron reflectivities and the KDP,Z scattergram to derive a more representative Z=67R1.62 relation. With this new relation we obtained a total of 390 mm, which is in reasonable agreement with the gages. The hybrid R(KDP) estimator (eq. 2 for KDP>0.4 deg km-2, and Z=67R1.62 otherwise) produces the best result, 416 mm. If the Marshall-Palmer relation 200R1.6 is used in the hybrid algorithm the result is 364 mm.
Normally the Z=67R1.62 is not expected in a squall line. But this line occurred in Feb and had a melting level at about 2 km. A detailed analysis of the vertical structure indicates that a narrow distribution of graupel was present above 2 km. It is likely that the melting graupel did not have enough time to evolve in an equilibrium MP type drop size distribution. Modeling of DSDs indicates consistency of the Z and KDP data if nearly monodispersed DSD is assumed with a large number of small (but still nonspherical) drops.
The fields of rain accumulations in the area of the gages are in Fig. 4. The field from rain gages (Fig. 4a) is obtained by interpolation. The R(Z) fields were obtained by applying the Marshall-Palmer formula to the WSR-88D data (Fig. 4b) and the Cimarron data (Fig. 4d); although they are similar the underestimation is evident. The hybrid field agrees well with the gages. The differences between radar and gage fields (other than the bias for the two derived from Z) are caused by the structure of convective elements. Radar fields show streaks caused by advecting small convective cells. Evidently, even this dense gage network does not resolve the fine structure of precipitation.
Fig. 4 - Fields of rainfall accumulations: a) From the rain gages; b) From the MP relation applied to the WSR-88D data; c) From the MP relation applied to the Cimarron data; c) From the hybrid R(KDP) algorithm.
Remaining issues that concern specific differential phase deal with the range averaging interval, and data artifacts. In light rain this interval should be longer so that there is enough distance for differential phase to accumulate. That is adaptive filter length for convective and stratiform precipitation would be advantageous. Sometimes, even after averaging, KDP has negative values; these could be due to sidelobes and/or partial beam filling which create decreases of FDP. Other data (including Z and ZDR) can be examined to recognize and correct these artifacts.
Because of relative insensitivity to DSD variation it is unlikely that there would be a plethora of KDP-R relations, at least for moderate to very large rain rates. The few proposed ones are all within 12% of each other (Aydin et al. 1995).
Three estimators of rain rate based on polarimetric measurements have been reviewed. These estimators were evaluated using simulated gamma drop size distributions, from which radar observable (valid for 11 cm wavelength) were derived. Simulation reveals that a relation R(KDP,ZDR) similar to the one proposed by Jameson (1991) produces the lowest standard error of the rain rate estimate. Thus, in theory this estimator is virtually independent of the DSD variations found in natural rains. Furthermore it is least affected with processing errors.
In practice, the high accuracy of R(KDP,ZDR) estimates cannot be achieved because of possible bias errors in ZDR. Limitations in the present radar system cause total errors of both R(KDP,ZDR) and R(KDP) estimators to be larger than theoretical predictions. Comparison between cumulative rain amounts obtained with the two polarimetric estimators and a dense network of rain gages reveals similar results. We show that the R(KDP) estimator is preferred because it is immune to several sources of bias that plague the other estimators.
A case is made to capitalize to the fullest extent possible on the information contained in the polarimetric variables. With this information the rain estimation becomes a two step process. First precipitation is classified into types and then appropriate optimum polarimetric relation can be used to determine the amount.
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