MULTIPARAMETER AND POLARIMETRIC METHODS

Perspective

The objective of this section is to demystify polarization measurements of rainfall. As anyone exposed to elementary physics knows, plane electromagnetic waves are characterized by an amplitude, a phase and a polarization. While the detection of quantities related to the amplitude of the waves (such as power) is relatively straightforward, measurements of parameters involving phase and polarization are more technologically demanding. In no small part the recent trend toward using phase and polarization information for rainfall estimation is the consequence of improved microelectronics and faster computer processing capabilities.

It is not surprising then, that the evolution of radar rainfall measurements closely follows that of improvements to radar technology. The measurement of rainfall using radar has turned out to be a much greater challenge than perhaps originally envisioned by the early pioneers in radar meteorology. While the radar reflectivity factor is invaluable for detecting where rainfall is likely to be most intense, we have seen that its utility in measuring rain quantitatively , particularly on the time and space scales of interest to hydrologists, remains to be demonstrated. The wide range of values of instantaneous, point rainfall rates associated with a given Z have been discussed previously; it is only through careful statistical processing and meteorologically intelligent interpretation of Z that it has proven of much quantitative value at all (Atlas et al. 1990; Rosenfeld et al. 1990; Rosenfeld 1994) even on larger space and longer time scales. This has been treated in previous sections. For finer temporal and spatial radar rainfall measurements, greater accuracy requires a more complete use of all the information contained within the waves back scattered to the radar. It was natural and, indeed probably inevitable, that phase and polarization information have become the recent focus for radar rainfall research.

Early Developments

This is not the first time investigators have explored radar polarization and phase measurements of precipitation, however. Atlas (1951) and Atlas et al., (1953) used Gans theory to show theoretically that randomly oriented spheroids could affect intensities back scattered to the radar. Such effects were subsequently observed in the melting layer (Browne and Robinson 1952). Somewhat later at M.I.T. researchers conducted the first systematic polarization measurements in various types of precipitation including rain and snow (Newell and Geotis 1955; Newell et al. 1955; Newell et al. 1957). While the M.I.T. investigators knew that these measurements were telling them something about the shapes of the scatters, Newell later confided, however, that "...we really didn't know what to do with them (the measurements)." Nor was the M.I.T. group the only one exploring the potential application of polarization measurements in precipitation. Kennaugh at Ohio State University pioneered much of the methodology of microwave polarimetry, and while he was primarily funded to study potential military applications, he apparently also gathered measurements in rain in order to characterize its effects as 'clutter' obscuring targets. Simultaneously, in Canada at the National Research Council, polarization measurements were used to study precipitation in order to better understand its effects on communication. In spite of all of this activity, however, it appears that it was not until the late 1960's that a program was directed exclusively toward applying polarization techniques to a meteorological problem, namely hail detection (McCormick, 1968). Even this application, however, was qualitative in nature, and it was really not until Seliga and Bringi (1976) proposed using radar polarization observations to measure rain that the potential quantitative application of polarization measurements became an area of intense research by a group of eccentric radar meteorologists.

Qualitative Background

The motivation behind Seliga and Bringi's insight is illustrated in Fig. 21 for a raindrop having an axis ratio r = a/b. When a microwave interacts with such a drop, energy is scattered in all directions with part of the wave reflecting back to the radar and another part scattering in the forward direction. As the dimensions of the drop suggest, the amount of energy scattered depends upon whether the amplitude of the transmitted wave is parallel to the long axis of the drop (horizontal polarization) or to the short axis (vertical polarization). Since most raindrops fall as shown in Fig.21 with the small axis oriented vertically, one expects intuitively, and indeed it is the case, that the amount of energy back scattered using a horizontally polarized wave exceeds that measured using a wave polarized vertically. In addition, the wave scattered in the forward direction recombines with the forward propagating transmitted wave. When there are a sufficiently large number of drops, the forward scattered wave can actually induce a small but perceptible change in the phase of the net wave (transmitted plus scattered) from what the phase would have been in the absence of rain. Moreover, because of the finite dimension and the geometry of the drops, the magnitude of the phase shift depends upon whether the transmitted wave is vertically or horizontally polarized. Furthermore, this phase shift is carried along and accumulates until the waves finally return to the radar.

As elaborated on a little later, by alternatively switching between horizontal and vertical polarization, it is possible to deduce something about the average shape of the raindrops using the measurements of back scatter intensity at each polarization. This information can, in turn, be used to say something about the average size (and therefore terminal fall speed) of the drop. Similarly, by using a suitable radar, the polarization phase difference can also be measured. As it turns out this quantity provides a good measure of the concentration of rainwater.

Fig.21 - A schematic diagram of a transmitted microwave (solid line) interacting with a drop having a large (horizontal) dimension b and a small (vertical) dimension a. The drop produces back scattered and forward scattered waves depend upon whether the transmitted wave is horizontally or vertically polarized.

Polarization measurements, therefore, provide information not contained in measurements of just the reflectivity factor alone. As a consequence, they should significantly reduce the error of estimates of the rainfall rate. This is suggested by considering the simple relation

R = 3.6 VmW (12)

where R is the rainfall rate [mm h-1], W is the rainwater content [g cm-3], and Vm is the mass weighted mean fall speed of the drops [m s-1]. That is, all the rain water can be thought of as falling at the velocity Vm. This latter quantity, in turn, is a function of mass weighted mean drop diameter (Dm) as illustrated in Fig.22 (Jameson 1993).

Using (12) the relative error in the rainfall rate is then given by

dR/R =dV/V + dW/W (13)

Polarization measurements help, then, because they tell us something about the size of the drops, thereby reducing dVm/Vm, and they tell us something about the concentration of rainwater, thereby reducing dW/W. In contrast the radar reflectivity factor lumps all the information into one number, a number that can usually be associated with a wide variety of combinations of Vm and W.

Fig.22 - The dependence of the mass weighted mean terminal fall speed (Vm) of drop size distributions on the mass weighted mean diameter (Dm) computed using an ensemble of widely varying drop size distributions (after Jameson 1993 which gives explicit expressions for the curve)

POLARIZATION PARAMETERS AND RAINFALL MEASUREMENT

Before considering explicit polarization rainfall measurement algorithms, it is first necessary to identify what currently appear to be the polarization parameters most useful to radar rainfall measurement. As suggested in the previous section, we, therefore, need to consider those quantities which tell us something about the size of the raindrops and the amount of rain water. Moreover, because some of the most portable and useful radars operate at frequencies which are significantly attenuated by rain, it is convenient to consider quantities most relevant to attenuating as well as so-called 'non-attenuating' frequencies. (In reality, under some conditions and geometries, even radars operating at 'non-attenuating' frequencies may experience significant attenuation (e.g., Ryzhkov and Zrnic, 1995a), but by and large this should not be the case.)

'Non-attenuating' frequencies

This discussion is largely confined to radars transmitting at and less than about 3 GHz. Although radars operating at frequencies as high as 5 GHz often seem to be unaffected by attenuation, experience and calculations show that in the frequency range from 5 to 9 GHz there can be just enough attenuation to obscure a simple quantitative interpretation of many of the radar polarization measurements, but not enough attenuation to be useful (Jameson, 1994a). As mentioned, however, even at 3 GHz attenuation can, at times. be significant. Fortunately, the measurements tell you when caution is required.

The key to the rainfall method proposed by Seliga and Bringi (1976) is the measurement of the so-called differential reflectivity given by

(14)

where z = Z H / Z V , and Z H , Z V are the radar reflectivity factors measured using horizontal and vertical polarization, respectively. It turns out that this quantity is a measure of the radar reflectivity weighted average axis ratio (Jameson 1983) (see Fig.21). However, because the axis ratio is simply related to the drop diameter in the case of quiescent drops (Beard and Pruppacher 1970; Pruppacher and Pitter 1971), this number can be directly converted into an estimate of the reflectivity weighted average drop diameter (Jameson 1983). This, however, is not the drop size most relevant to the rainfall rate. Traditionally, ZDR has been used to estimate what is known as the median drop diameter Do (see Seliga and Bringi 1976; 1978), that drop size splitting the rain water concentration into equal portions with one part consisting of drops having diameters > Do and the other consisting of drops having diameters < Do. Alternatively, (12) suggests that it is more appropriate (and indeed accurate) to use ZDR to estimate Dm, the mass weighted mean drop diameter as illustrated in Fig.23.

Research has shown (Jameson 1985; 1994b) that the amount of water is well measured by KDP, the specific differential phase shift (usually expressed as ∞ km-1), i.e., it is the rate at which the difference between phases measured using horizontal and vertical polarization changes with increasing distance from the radar. This is largely because the real component of the forward scattering matrix element is proportional to D3 (Jameson, 1989) so that KDP µ W(1- ) where is the mass weighted mean axis ratio corresponding to Dm [For 'equilibrium shaped' drops (Pruppacher and Beard 1970), =1.03-0.62Dm where Dm is in cm.]. This nearly linear relationship between KDP and W is illustrated in Fig.24 for several different frequencies. It indicates that if Dm is estimated using ZDR, then W can be simply measured using KDP.

Fig.23 - The average relation of ZDR (expressed here in antilog form as z ) to Dm for an ensemble of gamma drop size distribution having the form factors m = 1, 0 or 2 (from Jameson, 1994 a )

The measurement of KDP using alternating horizontal and vertical polarization was first proposed by Mueller (1984) and has since been implemented by several investigators (Jameson and Mueller 1985; Sachidananda and Zrnic 1986; Bringi et al. 1990; Jameson and Caylor 1994). An advantage of measuring KDP is that it is not affected by attenuation unless, of course, the reflected signals fall below the minimum signal to noise of the radar. Consequently, KDP can be used to detect and to estimate the magnitude of attenuation even at 'non-attenuating' frequencies (Holt 1988; Bringi et al. 1990). However, because the attenuation is temperature sensitive so are the techniques to correct for attenuation (Jameson 1992), and sufficiently accurate corrections may not always be possible.

Fig.24 - The relation of KDP to W as a function of Dm for frequencies of 2.8 (3), 5.48 (5) and 9.34 (9), and 13.8 (13) GHz and for an ensemble of widely varying gamma drop size distributions and temperatures. For discussion purposed rainfall rates corresponding to the Sekhon and Srivastava (1971) thunderstorm drop size distributions (Rss) are added to the upper axis. However, the curves themselves in no way require an assumption that the drop size distributions belong to the Sekhon and Srivastava (or any other) family of distributions. (After Jameson and Caylor 1994 which gives explicit expressions for the various curves)

Attenuating frequencies

Most mobile radars which can be readily mounted on vans and aircraft, for example, operate at frequencies above about 9 GHz. At these higher frequencies attenuation by rain is inevitable. While at very high frequencies there can be so much attenuation that penetration of the beam into the rain is too limited to be useful, in the frequency range from around 10-20 GHz, however, attenuation is not usually too large, yet is often large enough to be useful.

In this discussion the most relevant attenuation is polarization differential attenuation, AHV, i.e., the difference between the attenuation using horizontal polarization and that using vertical polarization. This quantity can be estimated using the monotonic decreasing component of ZDR (e.g., Jameson 1994b; Ryzhkov and Zrnic 1995a) where KDP indicates that attenuation is occurring. By combining AHV with KDP it is possible to make quite reasonable estimates of Dm as illustrated in Fig.25 for a frequency of 13.8 GHz. In this sense, AHV/KDP is the surrogate for ZDR at attenuating frequencies.

Fig.25 - The average relation of the ratio of the polarization differential attenuation (AHV) to KDP to the mass weighted mean diameter, Dm for an ensemble of gamma drop size distributions having the form factor m = -1,0, or 2 where X = AHV/KDP in the expression (after Jameson 1994b ). At attenuating frequencies, this ratio is used rather than the differential reflectivity to estimate Dm.

With an estimate of Dm, the measurement of the concentration of rainwater is identical to that illustrated in Fig.24 (since KDP is not affected by attenuation). Consequently, this section shows that for both 'non-attenuating' and attenuating frequencies, polarization measurements yield estimates of the important drop size, Dm, and of W. Using Fig.22 to transform Dm into Vm and Fig.24 to convert KDP into W, it then follows that polarization measurements provide the radar solution to (12) for the rainfall rate. This is considered in the next section where explicit expressions are listed, errors are compared and an example is given.

Polarization algorithms for estimating rainfall rate

The quantities of interest are ZDR, KDP, the radar reflectivity factor measured using horizontal polarization, ZH, and Dm as derived from either ZDR or the combination of AHV and KDP. Beginning with the classical ZDR technique, a list of some of the algorithms is given in Table 1. Note that in all cases ZH is used to account for the concentration of rain water since ZDR is only a function of drop shape (size). The behavior of these algorithms is illustrated in Fig.26 and compared to values computed using an ensemble of three different gamma drop size distributions (Ulbrich 1983). As we shall see, a primary factor limiting the application of these algorithms is the uncertainty in measurements of ZH which should be kept to better than ±0.5 dB or less if this approach is to be competitive with other algorithms.

Another set of algorithms uses KDP directly, usually in the form of a power law as listed in Table 2. These algorithms have the advantage that they do not require measurements of ZH. On the other hand they essentially apply to only one form of the drop size distribution as illustrated in Fig.31. Consequently, the errors of the estimates depend upon how significantly the drop size distributions vary from this form. There is, then, a potential for some significant errors at times since measurements seem to indicate that µ can often vary between -1 and 3 (Ulbrich and Chilson 1994).

Fig. 26 - The average relation (solid line) of ZDR (expressed here in antilog form as Z to the ration of the radar reflectivity factor measured using horizontal polarization (Z3H) at 2.8 GHz to the rainfall rate for an ensemble of gamma drop size distributions (circles) having the form factor m = 1,0, or 2 (from Jameson 1994b ). For comparison relations derived by other investigators (see Table 1) are plotted as well.

On the other hand, KDP can be combined with ZDR or AHV (vis à vis Dm) to minimize such problems. In addition these combinations significantly reduce the need to assume a particular drop shape-drop size relation (Jameson 1991). The drops, for example, need not have an average equilibrium shape because factors which increase (decrease) drop distortion (and hence ZDR and AHV) will also increase (decrease) KDP as well so that departures from equilibrium tend to cancel when Dm (from ZDR or AHV) and KDP are combined to estimate R. This approach, which is equivalent to the radar solution to (12), is illustrated in Fig.28. Coefficients for the fits may be found in Table 3 (Jameson 1994a).

Table1: Relations for estimating the rainfall rate R using the differential reflectivity, ZDR and radar reflectivity factor measured using horizontal polarization, ZH [mm6 m-3].

Table2: Coefficients and exponents used in power law relations of the form R=CKDPp where R is the rainfall rate [mm h-1] and KDP is the specific differential phase shift ( km-1.)

With this plethora of algorithms, it is reasonable to wonder which one is 'best'. In reality the selection of a particular algorithm will depend on the hardware and computational capabilities of the available radar as well as application objectives. Nevertheless, it would be nice to have some basis for comparing the different algorithms. Since adequate experimental data are not yet readily available, we resort to a numerical comparison. While less than satisfactory, it is better than nothing, and it at least provides some guidance until real data become available.

Fig.27 - Plots of least square error power laws of the form R=CKPDP corresponding to gamma drop size distributions having form factors m = -1,0 or 2. For comparison power laws derived by investigators (see Table 2) are plotted as well and generally all seem to correspond to a m = 1. This figure also shows that the errors of such power laws may increase when a drop size distribution has a m considerable different from unity.

Table3: The coefficients (C ) of least square error polynomial fits between the mass weighted mean drop diameter (Dm) and the ratio KDP /R= ∑ Ci Dmi. (From Jameson 1994a)

ALGORITHM COMPARISONS

Since statistical fluctuations can be minimized with sufficient averaging, the initial concern is with bias errors. While instruments can introduce biases as well, these generally add to fundamental physical biases so that the first concern here is those biases which arise from the physics of the measurement. In particular we want to know how sensitive the various algorithms are to variations in the size distributions of the drops. As is well known and as will be shown in a subsequent example, drop size distributions can vary widely. Algorithms which automatically track these variations are preferred to ones which are locked into place by assuming a particular form of the drop size distribution since these latter algorithms may introduce significant bias errors.

Fig.28 - The relation of KDP to R as a function o Dm for frequencies of 2.8(3), 5.48 (5), 9.34 (9), and 13.8 (13) GHz and for an ensemble of widely varying gamma drop size distributions and temperatures. For discussion purposes rainfall rates corresponding to the Sekhon and Srivastava (1971) thunderstorm drop size distribution (Rss) are added to the upper axis. However the curves themselves in no way require an assumption that the drop size distribution belong to the Sekhon and Srivastava (or any other) family of distributions (after Jameson, 1994 a). These curves (see Table 3) are essentially the solution to (12) in the text.

Fig.29a is a plot of bias errors computed by Jameson (1994a) for various algorithms using an ensemble of gamma drop size distributions having a form parameter µ of -1, 0, and 2 and quiescent drops ranging in size from 0.01 to 0.6 cm diameter. In this figure eB is the relative bias error (D R/R) due to drop size variation. The combinations (AHV,KDP) at 13.8 GHz and (ZDR,KDP) at 2.8 GHz correspond to the algorithms in Fig.8, while KDP alone refer to power law algorithms, ZH refers to the standard Z-R relation and (ZH,ZDR) refers to the classical ZDR algorithm (as given by Jameson in Table 1). According to the figure the (ZH,ZDR), (AHV,KDP) and (ZDR,KDP) algorithms all have bias errors near zero in contrast to the KDP power law algorithms and especially to the Z-R relation computed from a least square error fit to the entire ensemble of raindrops.

Fig.29 (a) - The algorithm bias errors eb = D R/R computed for an ensemble of widely varying gamma drop size distributions having form factors of m = -1,0, or 2. Measurement errors are excluded but biases in estimates Dm are included when appropriate. KDP denotes biases associated with R-KDP power laws, ZH denotes those for a least square error Z-R relation derived for this ensemble of drop size distribution, while ZH , ZDR denotes those corresponding to the classic ZDR technique (as given by Jameson in Fig. 6). AHV, KDP and ZDR, KDP denote the techniques that use the two parameters first to estimate Dm and then use KDP to estimate R (Fig. 28). (b) The total error of the estimates of R including measurement errors given in the text. For discussion purposed Rss is added to the top of the figures.

However, the situation changes somewhat after adding in reasonable measurement errors (Fig.29b). (The assumed errors are ±10% in KDP at 2.8 GHz, ±8% in AHV at 9 GHz, ±0.1 dB in ZDR, and ±0.5 dB in Z. If desired, see Jameson (1994a) for further discussion particularly with regard to frequency scaling of the errors.) In relative order, the most desirable algorithms over a wide range of rainfall rates are then (AHV,KDP) at 13.8 GHz, (ZDR,KDP) at 2.8 GHz, KDP at 13.8 GHz, (ZDR,ZH) at 2.8 GHz, KDP at 2.8 GHz and, in last place, the Z-R relation. Using a 3 GHz radar, recent experimental measurements (Ryzhkov and Zrnic 1995b) are in very good agreement with these theoretical results. It is worth noting, however, that once the rainfall rate becomes very light (less than around 1 mm h-1), the Z-R relation does just about as well because there are so few distorted drops that polarization measurements provide little additional information.

Using (ZDR,KDP) to Determine Z-R: An Example

Because there are only a few radars equipped to make the necessary polarization measurements even at 3 GHz much less at other frequencies, a complete set of sample results is not available. Rather the intent here is just to illustrate that such measurements are possible and can yield useful and interesting results. Before proceeding, however, we must mention that unlike Z and ZDR which can be measured directly by the radar, KDP must be computed as the derivative with increasing distance from the radar of the measured accumulated propagation differential phase shift (DP ) (Bringi et al.,1990; Jameson, 1994b; Jameson and Caylor, 1994). While somewhat more computationally intensive, KDP can be calculated in real time even by modest computers.

The data in this example were collected in 1991 during the CaPE (Convection and Precipitation Experiment) project centered on the NASA Kennedy Space Center in Florida. The NCAR (National Center for Atmospheric Research) CP-2 radar was especially equipped by Colorado State University (CSU) under NASA support for this project to gather propagation differential phase measurements at 3 GHz. In addition, this radar system also measures ZDR at 3 GHz.

The basic radar measurements in a rather average, mature Florida summertime shower (Byers and Braham, 1949) are shown in Fig.30a. Ancillary data indicate that the height of the 0˚C level is at approximately 5 km so that some of this shower is at temperatures above freezing. Most of the larger KDP occur where there is liquid water below this altitude. Significant KDP are also coincident with the

Fig.30 - Line plots of KDP (∞Km-1) deduced at different altitudes (elevation angles) as a function of radar range with superimposed contour plots of the radar reflectivity factor (dBZ) at 3 GHz and horizontal polarization (Z3H) and the 3 GHz differential reflectivity (db) measured in a summertime in Florida tropical rainstorm as part of a CaPE project. The height of 0∞ C is at about 5 km. (b) A plot of the height structure of Dm (mm) deduced from ZDR and of the rainfall rate is still air (mm h -1) deduced using KDP and Dm (i.e., ZDR) shown in Fig. 30 a. Note that the largest rainfall rates lie above the maximum reflectivity factors at this time, while the largest drops are found below about 3 km. (From Jameson 1994a).

'core' of the shower defined by the larger reflectivity factors at 3 GHz, Z3H. However, significant KDP also occur at locations removed from where Z3H is largest.

The most substantial ZDR in Fig.30a occur below an altitude of 3 km and largely coincide with the greatest Z3H. Since ZDR is a measure of the reflectivity factor weighted mean raindrop axis ratio (Jameson, 1983), the bigger the ZDR the larger the raindrops (Fig.23). More specifically estimates of Dm computed from ZDR are plotted in Fig.30b. The spatial coincidence of the largest Dm and greatest Z3H strongly suggests that the reflectivity factor in these locations is significantly enhanced by the presence of the larger drops.

However, these Z3H are also undoubtedly increased by a higher concentration or rain water as well. The combination of the larger drops with larger water contents should coincide with greater rainfall rates. In fact if we were to estimate R using a typical Z-R relation, the rainfall rates would be maximum precisely where the reflectivity factors are greatest. On the other hand, estimates of R using KDP and ZDR reveal an entirely different structure (Fig.30b) with the largest R occurring at altitudes of 3-4 km, considerably above the height of the maximum Z3H. One plausible interpretation of these observations is that the largest, fastest falling raindrops producing the largest values of ZDR and Z3H at lower levels are descending ahead of the bulk of the rainfall higher up in the storm.

This example demonstrates two points, namely that size sorting can be significant at times and that KDP provides a method to decouple estimates of R from Z. By estimating R using (ZDR,KDP) and comparing to Z, it is possible to correlate Z with R over identical sampling volumes permitting the development of Z-R relations useful for certain applications by conventional radars. This is illustrated in Fig.31a at some sample heights through the storm shown in Fig.30. In the Z-R relations, the greater the power of R, the larger the drops (Jameson 1991) and consequently the smaller the coefficient needed to yield the same radar reflectivity factor. Thus, at the lowest level where the drops are largest, the Z-R relation is characterized by a small coefficient and large power of R, while at the top where the drops are smaller (Fig.30b), the coefficient is much larger and the power is near unity. Consequently, the effect of size sorting present in Fig.30b is clearly evident in the behavior of the Z-R relations with height. Moreover, when all of the data from these levels are combined, we get yet another relation as shown in Fig.31b. Interestingly, this expression is

Fig. 31 - (a) Plots of the Z-R relations computed using the R deduced from the combination of Dm from ZDR and KDP from (Figs. 23 and 28) for 3 GHz and the observations of ZDR and KDP from Fig. 30. Note that the variability among the relations is most likely due to drop size sorting with altitude. (b) The least square error Z-R relation deduced by combining all the data used in (a). Note the remarkable similarity of the total Z-R relation to that for Sekhon and Srivastava (1971) thunderstorm drop size distributions (Z=300R1.35).

remarkable close to the Z-R relation of Sekhon and Srivastava (1971) for thunderstorms (Z=300R1.35). This example also clearly shows the dangers inherent in using a Z-R relation derived from a large sample to estimate instantaneous, point values of R using Z alone. As enticing as this example is, however, these data are limited and only represent some of the first simultaneous measurements of these variables. Consequently, they illustrate just the beginning of what can and must be done in the future as discussed briefly below.

Some Encouraging Results

While there is an undeniable need for extensive experiments, it is unlikely that any single test will provide the 'final' answer. It is much more realistic to expect that each experiment will be one more contribution toward the long process of amassing several different and independent checks which, while incomplete separately, make a strong case when considered together.

In spite of the known limitations of rain gages for the validation of radar rainfall estimates they remain the 'standard' means of comparison and have been used in the following recent experiments. For example, over a range of rates from about 5 to 20 mm hr-1, Ryzhkov and Zrnic (1995b) report that algorithms using KDP in combination with ZDR perform best, with r.m.s. errors between gages and radar of 18-25%. The use of KDP alone resulted in r.m.s. errors of 28-32%. Significantly greater errors, ranging from 38-55%, occurred even when using the combination of Z and ZDR, presumably due in part to the difficulty of measuring Z with sufficient accuracy. These numbers are in good agreement with the expectations for 'instantaneous, point' estimates discussed in Jameson and Caylor (1994).

In another study, Bolen et al. (1995) report that the algorithm using KDP alone yielded an average r.m.s. error of only 21% at rates between about 30 and 75 mm hr-1; this is about 7 to 10% lower than the values reported by Ryzhkov and Zrnic (1995b). The difference may be due in part to the larger rain rates. Moreover, these estimates apparently were not significantly affected by radar beam blockage.

Thus, the process of testing polarimetric methods for rainfall estimation is well underway. However, we need many more such experiments and different approaches to determine the conditions under which the various methods perform best. With perseverance the ultimate hope is to reach sufficiently definitive conclusions to convince even a hardened skeptic that polarization measurements offer a real and significant improvement.

OVERVIEW OF POLARIMETRY

It should be emphasized, however, that polarization measurements alone do not assure accurate rain estimation all the time, everywhere. As with radar reflectivity factor measurements, they are subject to the same deleterious effects of beam width, clutter, noise and transformations in the rain before it reaches the ground. In particular, polarization techniques will work only where there are polarization signals, i.e. when there are a sufficient number of drops larger than about 1 mm diameter to induce polarization dependent differences in the measurements. Thus, in lighter rain there is no particular advantage, and, in fact, there may be a definite disadvantage to using polarization algorithms to estimate rainfall. But since polarization radars measure Z anyway, in light rain it should be trivial to switch to reflectivity based algorithms as the polarization signatures become too small. What is important to remember is that the polarization techniques work best in the more intense, and presumably, often the more important rain.

Moreover, polarization measurements are neither more uncertain nor more inaccurate than are measurements of the reflectivity factor. In fact an important advantage of (most) polarization techniques is that they do not require an absolute calibration of the radar, often a laborious and unsatisfying experience which usually leaves a dB or more uncertainty anyway. Polarization measurements are less affected by partial beam filling by a cloud, and measurements of the cross-correlation function can be used to detect data affected by ground clutter. Furthermore, since the fundamental parameters are more physically related to the rainfall rate than is Z, polarization techniques are less subject to physical biases than are Z-R relations, for example. After all, it only makes common sense to measure first those quantities which are most directly related to the rainfall rate.

On the other hand polarization measurements do require more expensive hardware including excellent antennas and receivers. The radars should be coherent, and the phase and amplitude balances must be carefully maintained. In addition, more than one frequency is required to achieve maximum performance over as wide a dynamic range of rainfall rates as possible. In short, polarization measurements are certainly going to be more expensive than simple measurements of the reflectivity factor. Consequently, it is unlikely that they will become commonplace for years to come. Therefore, it is going to continue to be important to use the radar reflectivity factor in the most effective and optimal manner.

CONCLUDING REMARKS

The physical processes responsible for the production of rain lead to a wide spectrum of spatial and temporal scales from meters to hundreds and more kilometers. Moreover, we have come to appreciate that the physics of rain is not described only in terms of the characteristics of the drops themselves and how they scatter microwaves, but, as importantly, by the spatial and temporal statistics of the rain as well. This realization has, quite literally, revolutionized our understanding of radar rainfall measurement in recent years.

From the first measurements of the radar reflectivity factor in rain, Z was used to describe rainfall quantitatively throughout the entire spectrum of scales. While sometimes apparently quite successful, especially at larger scales, inaccuracies and failures increased markedly at finer and finer resolutions. Much of the subsequent research, therefore, focused on the microphysics of microwave scattering by rain in an attempt to understand the small scale ineffectiveness of Z. This has produced a number of important new techniques for measuring quantities much more physically related to the rain than is Z. While still awaiting extensive experimental verification, techniques such as those using polarization appear poised to improve considerably rainfall estimation at the finer scales . Simultaneously, while we were improving our understanding of why Z fails at the smaller scales of the spectrum, there were advances in understanding why Z succeeds at the larger ones. In particular, we now know that even without such improved measurement techniques and by using Z alone, the powerful tool of rainfall statistics provides useful quantitative estimates of rain even on scales far exceeding those normally associated with radar observations. Moreover, aside from these largest scales, recent research shows that with intelligence, the reflectivity factor alone can be applied successfully to finer spatial and temporal resolutions than previously thought possible. Presumably, replacing Z with quantities more physically based on the rainfall will likely extend these statistical approaches to even finer resolutions. Thus, over the rainfall spectrum there seems to be a continuum from physically based algorithms at the finest resolutions to those which are statistically based at courser resolution.

In summary, it appears likely that in the near future it will be possible for someone with specific resolution requirements to choose an optimum approach limited only by their financial and hardware resources. Although it has taken nearly half a century, we have finally learned that while there is no silver bullet applicable to the entire rainfall spectrum, there is at least a golden arsenal of techniques for the radar measurement of rainfall. It is time to take advantage of these new insights.

Aside from the general conclusions in this section we shall collect some of the key points from the body of the paper which we believe deserve to be emphasized. Some of these may be obvious to the well informed reader, but we include them to provide the perspective which comes from intensive study of radar rainfall methods over many years.

• The exponential drop size distribution (DSD) of Marshall and Palmer (1948) and its dependence upon rainfall rate is a long term average; short term samples tend to show narrower DSDs. It is also characteristic of stratiform rain in temperate latitudes.

• In temperate latitudes, DSDs tend to have more larger drops in thunderstorms than in stratiform rains. In the tropics, the converse occurs at rain rates less than about 10 mm/hr.

• Z-R relations are intrinsically statistical in nature. Both Z and R tend to be lognormally distributed. The PDF of R has a standard deviation which is between 1.7 and 2.2 times its mean.

• The power law relation between Z and R of the form Z=ARb is a natural consequence of the lognormality of their PDFs.

• There is a weak tendency for the coefficient A to increase and the exponent b to decrease with the intensity of the rain in temperate latitudes. The converse is true in the tropics at R<10 mm/hr.

• The PDFs of Z and R and the associated Z-R relations can be characterized by the physical properties of the rain as determined by the three dimensional radar observations and the radar range. The PDF of Z is dependent upon the radar, the range, and the distribution of precipitation reflectivity.

• Measured values of Ze are corrupted by a variety of effects such as side lobes, beam filling, sharp reflectivity gradients, attenuation, etc. Ze-R relations (where R is a surface measurement) are further corrupted by the difference between the reflectivity in the pulse volume and that near the surface as well as changes in the DSD during the fall of the precipitation to the surface. The combination and variability of all these factors makes it virtually impossible to establish a deterministic Ze-R relation for any specific rain. These effects speak strongly for the use of statistically based Ze-R relations.

• The use of surface rain gages to correct for anomalous gage/radar rain (G/R) ratios is useful only when the rain is uniform over distances comparable to the gage spacing. Otherwise, one requires so many gages that the radar itself is no longer needed.

• Accumulating rain gages are of limited value for validating Ze-R relations because of the space-time dependence of the precipitation types and corresponding relations. Rain rate gages are preferable.

• The Area Time Integral (ATI) method of estimating lifetime storm rainfall or area-wide rainfall is a useful approach because it is not subject to the above mentioned factors which corrupt point measurements.

• ATI depends upon the existence of a representative PDF of R and the ability to characterize that PDF on the basis of the radar (or other available) characteristics of the precipitation, and the selection of an optimum rain rate threshold. An error in the threshold produces a corresponding bias in the rainfall estimate.

• The Window Probability Matching Method (WPMM) permits both the classification of the precipitation type and the development of a matched Ze-R relation appropriate to that type and radar range.

• Initial tests demonstrate that WPMM with classification of the precipitation type enhances the accuracy of rainfall measurements relative to that of conventional Ze-R relations. The development of a library of Ze-R relations corresponding to the precipitation types in advance should permit the use of WPMM with classification in real time. While further trials are necessary, this appears to be the method of choice, especially when one does not have access to polarimetric radar.

• Polarimetric radar techniques provide the most physically based direct radar measurement of rainfall and are particularly well suited to measurements on the finest temporal and spatial scales and for understanding the evolution of rain within clouds.

• Polarization estimates improve as the rainfall rate increases because of enhanced signals caused by the increasing abundance of larger, more distorted drops.

• To encompass the largest possible dynamic range of rainfall rates, such measurements need to be made at two frequencies such as 3 and 14 GHz.

• Algorithms based upon KDP and estimates of Dm are the least affected by any potential ignorance of the correct relation between drop sizes and shapes as might happen if the raindrops are oscillating.

• While polarimetry is less sensitive than Z to some effects such as partial beam filling, they are sensitive to other problems such as differences between the radiation patterns at different polarization.

ACKNOWLEDGEMENTS:

We are most appreciative to all of our colleagues in our home institutions and those with whom we have been collaborating under the Tropical Rainfall Measuring Mission (TRMM). Most of the support for this work has been provided either directly or indirectly through NASA grants under the auspices of TRMM. We also acknowledge stimulating discussions with Prof. Carlton Ulbrich of Clemson University and Frank Marks and Paul Willis of the Hurricane Research Division, NOAA Atlantic Oceanographic and Atmospheric Laboratory.

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