Introduction

In this paper we shall review the evolution of the half century of research aimed at using radar for the quantitative measurement of rainfall. The importance of the problem is self-evident ranging from urban hydrology through flood warning to the role of precipitation in the global water budget and its influence on climate.

We cannot possibly review all of the steps taken over the last 50 years. Thus, we shall emphasize what seem to us to be those which set the directions. In many cases, seminal developments started to move us in positive directions, but we then reached insurmountable obstacles which required a new approach. Perhaps the most prominent case in point is the work of Marshall and Palmer (1948) which demonstrated the existence of a drop size distribution (DSD) which is a simple function of the rain rate, and therefore led to a correspondingly simple relation between radar reflectivity Z and rain rate R. But by the mid-60s we found that there were many such relations, and we found it impossible to classify them by clear cut physical characteristics except in a gross manner. In this review we shall see that we have evidently overcome this problem. Sad to say, even today we find both scientists and practitioners using the primitive Z-R relations without regard to the nature of the precipitation.

The one thing that we were sure of by about 1970 was that when the radar was calibrated correctly and use was made of the proper radar equation for meteorological targets, that the radar measured Z correctly at short range.. But it was at greater ranges that large discrepancies were found. These were attributed to a long list of factors relating mainly to the manner in which the beam illuminated the targets, the difference in reflectivity within the beam and that at the surface, sharp gradients of Z, and a number of physical effects on the precipitation as it fell to the ground (Zawadzki,1984). In spite of the fact that we attempted to make appropriate corrections and referred to the effective reflectivity, Ze, that measured by the radar instead of that given by the DSD, success still eluded us.

But hope springs eternal. By the early 1980s, the catch phrase became "multi-parameter measurements" - either combinations of radar, microwave and optical extinction, radiometry, or polarimetric radar (Atlas et al.,1984). The essence of such approaches was based upon the realization that the DSD of rain depended upon at least two parameters - size and concentration - and so one needs two independent measurements. Radar polarimetry has indeed produced some remarkable results and we shall discuss these later. But it is based upon characterizing the DSD, and so it is intrinsically limited to the ranges and heights at which the measured rain is similar to that which reaches the surface. And so we have come full circle; we are again depending largely on the DSD and are still limited by many of the range and beam dependent factors discussed above.

We come now to the probabilistic approaches, the area time integral (ATI) or threshold method, and the probability matching method (PMM) which are treated in Section B. These depend upon the existence of well defined probability density function (PDF) of R whose first and second moments are determined by the physical characteristics of the precipitation. The matching of the PDF of R to that of Ze for each precipitation type results in a family of Ze-R relations classified by type. The concept is that one can now relate the rain at the surface to the effective reflectivity aloft on the assumption that the effect of the beam on the measurements and the difference between what is measured aloft and observed at the surface is reproducible on average. Initial results with these methods are promising but we have yet to determine how much further they will take us than all the others. Finally, it is worth considering the combination of statistical and polarimetric methods to obtain effective Z-R relations in real time as has been done by Atlas et al.(l995).

GENESIS - Z-R RELATIONS BASED UPON DROP SIZE DISTRIBUTIONS

Any such review must start with the work of Ryde (1946) who carried out the first studies on the echo intensity from and attenuation by precipitation in order to understand their effects on radar. Indeed, much of this work was done prior to 1941 in anticipation of the development of microwave magnetrons operating at 10 cm wavelength during World War II. The very first thing he did was to compute (by hand!) the normalized radar cross-section of water and ice spheres as a function of the ratio of diameter to wavelength. In order to determine the reflectivity as a function of rain rate R he needed data on the drop size distribution versus R. This he got mainly from the work of Laws and Parsons (1943). With this he was able to compute the quantity NS versus R and wavelength as shown in Fig.1, where N is the particle concentration in cm-3 and S is the radar cross-section divided by 4 p averaged over the size distribution. The caption shows the relation between NS and the reflectivity Z (mm6 m-3).

Few scientists seem to have recognized that this diagram provided the first Z-R relation. Atlas and Ulbrich (1990) showed that the curves between 3 and 10 cm wavelength in Fig.1 correspond to the relation Z=320R1.44. It is astonishing that this is virtually identical to the relation which is widely used around the world, and particularly in the rainfall algorithm used in the present day NEXRAD (WSR-88D) radar systems in the U.S.

Fig.1 - Reflectivity NSx10 versus wavelength for various values of rainfall rate (after Ryde, 1946) Z = 4(10 l / p ) 4 /K 2 times the ordinate. At l = 10 cm Z = 320 R 1.44

With the exception of the theoretical work of Wexler and Swingle (1947; emanating from a 1945 classified report) on the radar equation for precipitation, the seminal papers on radar rainfall relations must be credited to Marshall et al. (1947) (Fig. 2) and Marshall and Palmer (1948) (Fig.3). The superb correlation between Z and echo power in Fig.2 gave tremendous impetus to many scientists around the world to use radar for rainfall measurements. But note that the radar range was only 8.7 km.

And the remarkably simple exponential drop size distributions found by M-P in Fig.3 with constant intercept No and the relation between slope L and R made it clear that because Z and R were functions of the 6th and roughly 4th moment of the DSD, respectively, that Z was related directly to R by a simple relation such as Z=200R1.6. The exact forms of the M-P equations are presented in Fig. 3. Atlas (1953) later found L =3.67/Do where Do is the median volume diameter of the DSD.

Fig. 2 - Relative echo power at 10.7 cm and reflectivity Z computed from drop size samples at a range of 8.7 km (after Marshall et al., 1947)

While the latter work by Marshall et al inspired the field to get moving, it was not long before a number of us realized that the M-P relation was deceptively simple. Indeed, the M-P DSD relations are based upon highly averaged data that hide the wide variations which exist from moment to moment and from one type of rain to another. Joss and Gori (1978) demonstrated that the shape of the DSD depends upon sample size; as the sample size increases the larger drops are counted more effectively and the drop spectrum more closely approximates the exponential distribution. Moreover, the M-P relation gave the impression that things were so well behaved that there was a deterministic relation between Z and R. But we now know that this is not the case; the relation is a probabilistic one. In any rain No and Do behave as random variables so that both Z and R tend to be lognormally distributed. When this is the case, Z and R are then related by a power law on average (Atlas et al., 1990a). This is elaborated further below.

Fig. 3 - Drop size distribution (solid straight lines) compared with those of Laws and Parsons (broken lines) and those from Ottawa (dotted lines) (after Marshall and Palmer, 1948)

It was Twomey (1953) who first noted that the variations from place to place and from one rain to another are so great as to render estimates of R from Z of limited value. Nevertheless, this did not deter others like Fujiwara (1965) in Japan and Atlas and Chmela (1957) from trying to select the appropriate Z-R relations from the nature of the precipitation itself. It is interesting that the latter efforts and others like them anticipated the modern studies to characterize the Z-R relations by the physical properties of the rain as seen by the radar itself (Rosenfeld et al. 1995a).

In order to relate Z-R to the physical nature of the precipitation Atlas and Chmela (1957) developed the rain parameter diagram for an exponential DSD but with allowance for the variation of No and Do, in lieu of the constant No and the Do-R relation of M-P. A later version of this diagram by Ulbrich and Atlas (1978) is shown as Fig. 4. What this shows is that all integral parameters such as R, Z, W (liquid water content), and S (optical depth) are dependent upon both drop size and concentration. Also included in Fig.4 is an inset diagram showing the relation between differential reflectivity ZDR at horizontal and vertical polarization and D0 (Seliga and Bringi, 1976). Polarimetry is discussed later. Clearly, the relationship between any two of the seven parameters in Fig. 4 determines all other relations.

Fig. 4 - Rain parameter diagram based upon exponential drop size distributions with varying N 0 (thin solid lines), median volume diameter - D 0 (thick solid lines), liquid water content - W (dashed curves), and optical extinction coefficient S (thin dashed lines). Inset diagram shows differential reflectivity Z DR vs D 0. Relation between any two integral parameters provides relations between others (after Ulbrich and Atlas, 1978)

In particular, any single curve which corresponds to a Z-R relation also implies a perfect relation between No and Do. However, actual observations with either airborne or surface disdrometers show that the latter parameters are only poorly correlated. For example, using observations of the DSD from an airborne 2-D precipitation probe during TOGA COARE, Ulbrich and Atlas (1995) report a correlation coefficient of -0.29 between No and Do. This means that there is considerable scatter about the No and Do relation which corresponds to that about the Z-R relation. Again, we see further evidence for the statistical nature of the Z-R relation.

Fig. 5 is a simplified version of Fig. 4 in which we display only the isopleths of No and Do. The shaded area covers the range of the 69 Z-R relations tabulated by Battan (1973). This shows that a measure of Z alone cannot determine R to better than about ±6 dBR at Z=100 and ±8 dBR at Z=105 mm6 m-3, where dBR=10LogR. The dotted curve represents the Z-R relation of Joss and Waldvogel (1970) and is close to that used widely today. The large variability of the coefficient A and exponent b in the Z-R relations was also found by Fujiwara (1965). He found only a weak relationship between A and b such that larger A corresponds to smaller b, but no clear cut distinction with precipitation type. He also reported that A ranges from about 60 to 1100 and b from 1 to 2 .

In spite of the overlap in Z-R relations, both Joss et al (1970) and Austin (1987) suggested using relations with coefficients A increasing from about 100 for drizzle, 200 - 300 for widespread rain, and 400 for thunderstorms. The measurements of Joss et al (1970) were based upon the new Joss and Waldvogel (1967) momentum impact drop size instrument, the disdrometer, which revolutionized our ability to measure rain drop size spectra.

Recently, however, Tokay and Short (1996) have found that Z-R relations computed from DSD measurements at the ground in the tropics result in larger Z by a factor of about 2 in stratiform rain compared to that in convective rain of the same R for R £ 10 mm/hr . This discrepancy from conventional wisdom has not yet been adequately resolved. The most straightforward explanation is as follows. The difference from DSDs in thunderstorm rain such as that found by Jones (1956) and Foote (1966), both of which report larger drops and Zs than in temperate latitude stratiform rain, is probably due to the fact that the more vigorous thunderstorms of higher latitudes contain stronger updrafts and produce larger graupel and hail, and thus larger drops, than are found in convective showers in the tropics. In other words, while tropical stratiform rains have larger drops and Zs than do their convective counterparts (at R<10 mm/hr) the more intense temperate latitude thunderstorms produce still larger drops than do tropical stratiform rains. A more detailed description of the tropical precipitation process and its affects on the particle size distributions is presented in the section "EVOLUTION OF THE Ze - R COMPONENTS IN A CONVECTIVE COMPLEX".

Fig. 5 - A simplified plot of the rain parameter diagram showing only isopleths of total drop concentration NT (m 3 ) and median volume diameter, D0. The dotted line corresponds to the Z-R relation of Joss and Waldvogel (1970). The shaded area covers the range of the 69 Z-R relations surveyed y Battan (1973) after Ulbrich and Atlas (1978)

The work of Waldvogel et al. (1995) is also pertinent. They show that the intensity of the bright band relative to the reflectivity below, decreases with the degree of riming of the ice crystals above the melting layer. In such cases the clumping or aggregation of crystals into large snowflakes within and above the melting layer is reduced and fewer large raindrops are found below the BB. Since riming in the sub-zero temperature zone is associated with updrafts and convection, the result is a smaller number of large drops, a larger No, and a smaller Z in convective than in stratiform rain of the same intensity. This behavior also appears to be responsible for the earlier findings of Waldvogel (1974) on the jump in No in DSDs.

RADAR RAINFALL MEASUREMENTS - PRACTICAL LIMITATIONS

Radar rainfall studies were also made extensively at the MIT Weather Radar Project (Austin and Williams, 1951; Austin and Geotis, 1960; Austin, 1987). In the early work, like that elsewhere, they found that the radar underestimated the rainfall by a factor of 3 to 7 dB. Austin and Geotis emphasized the importance of measuring both the effective gain and beam width of the radar. But it was Probert-Jones (1962) who clarified the picture substantially with his paper on "The Radar Equation in Meteorology", and thus accounted for much of the discrepancy. Still, this did not explain many of the problems in actual radar measurements which were subsequently found.

But before we get to the practical problems let us review a classical experiment by Joss et al. (1970). They made measurements directly overhead at a height of only 200 m with a 4.6 cm radar. They compared the reflectivity-deduced rain rate to the average measured by four rain gages, all within 20 m of each other, and to that from a disdrometer. Fig.6 shows one of the results of this experiment. Here there is superb agreement between the Zs measured by the radar and that deduced from the disdrometer, thus showing that the radar is in fact measuring correctly. There is also good agreement between the R(DSD) and R=(Z/300)2/3 except for a 30 min period starting at 0425. Inspection of the N0 record reveals that these deviations result mainly from the large variability of N0 ( greater than a factor of 10) from the value implied by the coefficient 300 in their choice of Z-R relation. But the most important conclusion of this work is that under ideal conditions (i.e. well calibrated radar, short ranges and small pulse volumes such that the rain at the surface is virtually identical to that aloft) the radar is performing according to theory.

At this point we note that henceforth all measurements of the reflectivity factor by disdrometer will be designated by Z, while those measured by radar will be denoted by Ze, where the subscript e refers to the "effective" Z which includes all the measurement problems. When it is obvious that we are dealing with measured reflectivity, we drop the subscripte.

Fig. 6 - 80 minute record of the rainfall measured deduced by radar at vertical incidence at height of 200 m and by disdrometer at the surface, July 17, 1968, Locamo-Monti, Switzerland. Note excellent agreement between radar and disdrometer-based Z values (after Joss et al., 1970)

Unfortunately, most of our observations are not made under such ideal conditions as those of Joss et al.(1970). In particular they are generally made at longer ranges where: 1) the beam is above the surface, 2) the pulse volume is large and intercepts reflectivities which are either less than or greater than that at the surface, 3) where the volume is not filled uniformly, 4) sharp gradients of Z cause erroneous measurements in the averaging process, 5) large Zes are detected on the beam side lobes to extend the apparent area of the rain, and 6) drafts and size sorting by drafts and windshear affect the rain rates such that the Ze-R relations are meaningless. Add to these the effects of evaporation, growth, attenuation by the rain and a wet radome, and one wonders how we can expect any useful results except at short ranges. These factors have been discussed by a variety of investigators (Browning, 1978, Zawadzki, 1984).

Fig.7 from Joss and Waldvogel (1990) shows the effects of the vertical profiles of reflectivity (black areas) within a 1 degree beam as a function of range on measured reflectivities in convective (top) widespread (center) and snow with low level rain (bottom). The number in each figure gives the percentage, referred to the true, melted water at the ground, in rain rate deduced from the maximum reflectivity aloft. Only in the convective rains do the measurements come close to the true values (although factors other than the vertical profile have been neglected). With the bright band (BB) of widespread rain, the surface rain is overestimated at short range and underestimated at long range.

Fig. 7 - Vertical profiles seen by the radar at various ranges in convective (top), stratiform (center), and low level rain or snow. The numbers along each profile are the percentages of the true melted water at the ground in rain rate deduced from the maximum aloft (after Joss and Waldvogel, 1990)

And low level rain is severely underestimated because the beam is actually measuring the greatly reduced Zs aloft

Rosenfeld et al. (1992) have also demonstrated the averaging down of the peak reflectivity and the broadening of the actual storm reflectivity pattern by the beam such that the average rain rate over the storm area is overestimated. It is also not uncommon for a strong storm to be detected on the radar side lobes when there is little or no rain over a gage (Austin, 1987).

The various factors discussed above result in measurements such as those shown in Fig.8 (Austin, 1987). Here the ordinate represents the total gage accumulations in 18 gages on June 18,1977 over a period of about 3 hours, while the abscissa represents the radar deduced accumulations using various Z-R relations. The total accumulation of all gages was 239 mm. We see that the most conventional Z-R relation (Z=230R1.4, upper left) grossly underestimates the gage totals everywhere. In order to achieve the optimum agreement with all 18 gages, it was necessary to use the relation Z=230R1.1 (lower right). This exponent is exceedingly low and unrealistic. On the other hand the best agreement with the sum of the accumulations at all gages is shown at the upper right with Z=100R1.4. But this agreement comes as a result of compensating overestimates in some gages by underestimates at others. This illustrates that it is an exercise in futility to apply any single Z-R relation to a point in the space - time domain. During a three hour period the rain varies in type from convective to stratiform, with varying effects, and one needs to adapt the Ze-R to the precipitation type. Accordingly, it is also clear that totaling gages are grossly inadequate to validate Ze-R relations.