Chapter 3:

Multiparameter Radar Measurements: Theory and Applications

V. Chandrasekar
Colorado State University, Fort Collins, CO 80523
 

3.1. INTRODUCTION

Conventional radars for meteorological measurements transmit and receive waves of a single fixed polarization. In contrast to the conventional radars, polarization diversity systems provide either for the variation of the transmitted wave polarizations or for dual-channel reception of orthogonally polarized waves or both. Polarization diversity permits the measurement of additional characteristics of meteorological echoes in addition to the basic reflectivity factor. For precipitation these polarization characteristics are related to the mean values and distributions of size, shape, and spatial orientation of hydrometeors, and also to their thermodynamic phase state. Polarization itself is a fundamental descriptor of electromagnetic waves. This paper will discuss the basic description of polarimetric radar and the application of polarimetric radar measurements. The scope of this paper is to provide an overview of polarimetric radar analysis with emphasis on recent developments and will not provide an extensive review of the literature in the area of polarimetric radars. However, reference to some review articles are provided for interested readers.

3.2. POLARIZATION OF AN ELECTROMAGNETIC WAVE

The general form of an electromagnetic wave can be described as an elliptically polarized wave where the electric field vector rotates at a rate of the angular frequency of the wave () in either a clockwise or counter clockwise direction. There are many different notation conventions that one can use to describe polarized electromagnetic waves. In order to avoid confusion one should adopt one convention and then adhere to it. In this discussion, notation will be introduced from first principles.

To begin with, a monochromatic time varying electric field propagating in the positive  direction is described by

where is the angular frequency in radians,  is the wave number and x,y are phase constants. Using the e+jt time convention, the electric field in the z = 0 plane can be written with complex exponentials as
Eq. 3.2
which can be written in vector notation as
Eq. 3.3
It can be shown that as time increases, the tip of electric field vector E traces an ellipse when viewed in a plane of constant z.  Figure 3.1 shows a diagram of this polarization ellipse. To remove any ambiguity, the direction of propagation is taken into the paper as signified by the X at the origin. There are two primary ways to uniquely describe the ellipse: 1) with the phasor descriptors as given in (3); or 2) with the geometric descriptors of  and . The geometric descriptors are defined as
  The angle "epsilon"is called the angle of ellipticity while e is the ellipticity. Also,
= amplitude ratio,        (3.6)
= phase difference,   (3.7)

By examining (3.3)-(3.7) a number of useful relationships can be deduced:
 

Because of the choice of propagation direction in Fig. 3.1, the ellipticity angle is positive for right handed polarizations. The IEEE Standard definition of right handed polarization is that a right handed polarized electric field will rotate clockwise when viewed in a plane of constant z with direction of propagation being away from the viewer. However, according to the IEEE standard is negative for right handed polarization instead of positive as defined here. This deviation from the IEEE standard is done so that for later applications, (involving radars) when viewing the polarization ellipse in Fig. 3.1, the x and y axis can be thought of as the vertical and horizontal axis with  directed into the paper. The phasor descriptors,  and, and the geometric descriptors, and  , have an interesting geometrical interpretation. They uniquely represent a point on a sphere called the Poincare sphere [1].
 

3.3. POLARIMETRIC RADAR

The most general polarization state of a wave is elliptical polarization. However the discussion here will be restricted to linear and circular polarizations for simplicity. When  the polarization is linear;  we have right circular polarization and when we have left circular polarization.

With this brief introduction to polarization we proceed to a discussion of polarimetric radars. A polarimetric radar transmits and receives electromagnetic radiation typically in two orthogonal states of polarization. A fully polarimetric radar transmits alternately two orthogonal polarization states but receives both co-polar and cross-polar states relative to the transmit state simultaneously. Further discussions on polarimetric radar will be restricted to linear polarization states for simplicity.

There are two primary polarization-dependent constituents in a polarimetric radar namely: a) the polarization-dependent backscatter and b) polarization-dependent transmission mode and receiver of the radar system. Consider a dual-polarized radar that transmits and receives electromagnetic radiation polarized in both the vertical and horizontal directions. Then the received wave amplitudes at the radar, , in the horizontal and vertical directions respectively from a single scatterer can be written as

where [P] represents the propagation matrix , which is the scattering matrix of the ith scatterer, and  are the transmission wave amplitudes at the antenna input. Shh, Svv are the co-polar terms and Shv is the cross-polar term in the backscatter matrix. The terms of the propagation matrix can be approximated as
which indicates there is a differential attenuation and phase between waves polarized in the horizontal and the vertical direction, given by the real and imaginary parts respectively of  The backscatter from an ensemble of particles filling a radar resolution volume as can be obtained as Rh = Rhi, Rv = Rvi.

A typical fully polarimetric radar system consists of orthogonal dual polarized transmission mode and a dual channel receive mode that are set to be co-polar and cross-polar to the transmit state of polarization. The transmit polarization state is switched every pulse repetition time between two orthogonal states. Figure 3.2 shows the schematic of transmission and reception scheme of a fully polarimetric radar. XX and YY are the copolar received signals and XY and YX are the cross-polar received signals where X and Y are the two orthogonal transmission states. Neglecting the effect of the propagation, the properties of the backscatter medium can be studied by looking at the covariance matrix of the fully polarimetric radar signals. The covariance matrix, in terms of the copolar returns RHH, RVV and the cross-polar return RHV is given by

The elements of the covariance matrix can be estimated from the measurements of a fully polarimetric radar. The diagonal terms involve power measurements, but estimating the off diagonal terms require coherent radars (capability to measure phase also). All existing polarimetric radars do not measure all elements of the covariance matrix, but only a subset of them. The ZDR radars typically measure only a subset of  given by
where DP is the differential propagation phase between the two polarization states H and V.

If we exclude the effects of propagation medium then the covariance matrix can be shown to represent

where SHH, SVV and SHV represent the scattering matrix elements of the scatterers in the resolution volume. In the presence of propagation medium the subset represented by (3.11) is equivalent to
The elements of the covariance matrix are directly related to the physical properties of the medium.
 
 

3.4. MEASUREMENT TECHNIQUES

 
Figure 3.2 describes the pulsing scheme of a fully polarimetric radar alternating between two polarization states H and V. In this section we will study measurement procedures to obtain some of the elements (or subset) of the covariance matrix.
 

Let HH2n VV2n+1 represent the co-polar return due to the transmit pulse at H and V polarization state, respectively. Let HV2n represent the cross-polar return due to the transmit pulse at H polarization state. Then the mean powers at H and V polarization states (PHH, PVV) and the differential power and phase between two polarization states (ZDR and DP respectively) can be estimated as,

where

The specific differential propagation phase constant KDP can be computed from the estimates of  at two ranges r1 and r2 as


 

The cross correlation estimate HV which is the cross covariance <SHHSVV>  normalized with respect to <|SHH|2>  and <|SVV|2> can be obtained as
 

where Ts is the pulse repetition time and

and (Ts) is the autocorrelation estimate of the H/V polarized signals. This estimate is obtained by assuming a Gaussian Doppler spectrum for which  Note that (2Ts) is obtained directly from the H/V polarized returns. The cross polar power can be obtained as
using which we can obtain the normalized power in the crosspolar channel with respect to co-polar channel, called the Linear Depolarization Ratio (LDR), as
Similarly the cross covariance between (HV) returns and (HH) returns can be obtained as
Using (3.23) we can estimate the cross correlation between (HH) and (HV) signals as
Thus expressions (3.14) through (3.24) show the algorithms to obtain the terms of the covariance matrix from the signals of a fully polarimetric radar. In the following section we will study the relation between the covariance estimates and the properties of the back scatter medium.

3.5. RADAR RESOLUTION VOLUME PROPERTIES

Starting from the radar equation [2] it is easy to show that the reflectivity factor at horizontal polarization ZHH is proportional to the received power :

where C is the radar constant and r is the distance from the radar. However the reflectivity factor at horizontal and vertical polarizations can be expressed based on the radar cross section of the scatterers as

where HH, VV, are the radar cross sections at H and V polarization states respectively; the wavelength is , while K = (r-1)/(r+2), r being the complex permittivity of the particle. The integrals are over N(D) which is the hydrometeor number distribution over diameter D in the resolution volume. The differential reflectivity of the radar resolution volume is defined as
The cross polar reflectivity ZHV is defined as
where HV(D) is the cross-polar cross section. Using ZHV and ZHH we can define the linear depolarization ratio (LDR) as
The co-polar correlation coefficients of the resolution volume can be obtained in terms of the scattering matrix as
where

where  is the mean differential phase shift on backscatter. Similarly the correlation between co-polar and cross-polar returns can be obtained as

If fHH(D) and fVV(D) are the forward scatter amplitudes of the scatterers, then the specific differential propagation phase KDP can be obtained as
where Ko is the free space propagation constant. The two-way differential propagation phase between ranges r1 and r2 can be written as
Thus equations (3.14-24) provide algorithms for the estimation of various elements of the covariance matrix whereas equations (3.25-34) provide the corresponding relationship to the scattering properties of the observation medium.
 
 

3.6. MICROPHYSICAL PROPERTIES OF HYDROMETEORS AND POLARIMETRIC RADAR MEASUREMENTS

In the previous two sections we have defined the polarimetric properties of a radar resolution volume based on the properties of the constituent scatterers and the possible measurables that can be obtained from a polarimetric radar. In the following we will see the dependence of polarimetric radar signatures on the microphysical properties of precipitation.

3.6.1 Rain Medium

Microphysical properties of the rain medium that are important for radar observations are the raindrop size, shape, and orientation distributions. The size distribution N(D) may be modeled as a gamma distribution of the form
where N(D) is the number of raindrops per unit diameter per unit size interval D to D + D. Note that D refers to the equivalent spherical diameter, whereas Do is the median volume diameter. Drops larger than Do contribute to half the total rain-water content. Together with No and µ the gamma distribution is characterized by three parameters.

The equilibrium shape of raindrops at terminal fall speed is that shape for which the forces due to surface tension, hydrostatic pressure and aerodynamic pressure (due to airflow around the drop) are in balance. A good approximation for the shape of large raindrops is an oblate spheroid with axis ratio (ratio of minor to major axis, a/b) decreasing uniformly with increasing D, as [3]

a/b = (1.03 - 0.62D)                                           (3.36)

The orientation of raindrops has been studied theoretically to show that the mean canting angle, that is, their deviation from a purely vertical orientation, should be close to zero with standard deviation of the order of a few degrees. These deductions are in good agreement with the observations of the canting-angle distributions of raindrops using linear and circular polarization measurements. In this section we assume that the raindrops are equi-oriented with the minor axis of the oblate spheroid being vertical.

It is well known that in the Rayleigh limit ZHH is related to the sixth moment of the raindrop size distribution (RSD), while ZDR is related to the reflectivity-weighted mean axis ratio of the raindrops filling the radar resolution volume. Because the mean axis ratio can be related to a mean size, ZDR is a monotonic function of the reflectivity-weighted mean drop size. In addition it can be shown from theoretical considerations that |HV| > 0.99 at S-band even in intense rain. In the Rayleigh limit, = 0o at S-band, however, at higher frequencies (Mie region) can be quite significant. At long wavelengths KDP is related to the mass-weighted (i.e. D3 weighted) mean axis ratio of the raindrops. Equivalently, KDP is nearly related to the D4th moment of the RSD since axis ratio and D are related. Note that ZHH, ZDR, HV(0) and  depend on the backscatter properties of the rain medium whereas KDP is related to the forward scatter property of the rain medium.

The raindrop model can now be used to study the radar parameters defined in (3.26-33) at S- and C-bands. For each RSD (triplet of parameters (No, Do, m)) we calculate the various radar observables are calculated using the previously defined shape and orientation assumptions. The parameters of the gamma RSD are then varied over the range of natural rainfall so that a scatter plot is generated, the scatter being representative of natural variability in rain. Figures 3.3-3.6 show the scatterplot of the polarimetric radar measurement of the corresponding frequency band being denoted by S or C attached to the radar observables. Figs. 3a and b show ZDR against ZHH at S- and C-band, respectively. At ZHH(S) = 50 dBz, ZDR(S) varies between 1.5 and 3 dB, while at C-band ZDR(C) varies between 1.5 and 4 dB. These ZDR variations at a fixed reflectivity level reflect the physical variability of the RSD.

Figures 3.4a and b show KDP against ZHH at S- and C-band, respectively. At ZHH = 50 dBz, KDP(S) varies between 1.5 and 2km-1 whereas KDP(C) varies between 2 and 4km-1. Differential propagation phase causes depolarization of circularly polarized waves even at long wavelengths. With intense rain rates and the possibility of long propagation paths, the use of circular polarization for radars with only crosspolar reception (i.e. transmit RHC, receive LHC) should be avoided.

Figure 3.5 shows HV(0) against ZHH(S) and HV(0) against ZDR(C). Several factors contribute to HV(0) deviating from unity. Principal among these are non-zero values of  and/or mixed phase precipitation, i.e. rain mixed with partially melting ice or hail. At C-band large raindrops contribute substantially to  as shown in Fig. 3.6.

3.6.2 Ice Medium

The radar parameters defined earlier are dependent on the distributions of size, shape, orientation, and dielectric constant of particles filling the radar resolution volume. For raindrops these distributions are well known. However, for a large variety of ice particles they are highly variable and thus assumptions must be made, particularly with respect to shape, orientation and composition of the ice particles. The melting of ice particles into raindrops can be modelled using a melting model. The general shapes used for ice particles are either conical or spheroidal, with extreme oblate-prolate shapes being used for columnar, needle or plate like crystals. The fall behavior can be quite complex but is often modelled as Gaussian or simple harmonic with the symmetry axes aligned in a mean sense along the vertical or horizontal directions. In summary the polarimetric radar studies of ice medium are still approached empherically with theory and observations providing feedback to each other to advance the microphysical understanding of ice phase.

3.7. POLARIMETRIC RADAR DATA

Data collected from polarimetric radar is presented in this section for comparison with theoretical results presented so far. The measurements were obtained from the Colorado State University CHILL radar and the NCAR CP-2 radar. CP-2 radar can collect a subset of the full polarimetric measurements using an advanced signal processor connected to the radar [4]. Among the several parameters discussed here the CP-2 radar's new advanced processor calculates ZHH, ZDR, DP, HV, LDR and HH,HV. Among these LDR and HH,HV are obtained from the radar's X-band system. The CSU-CHILL radar system located at Colorado State University is currently capable of measuring ZHH, ZDR, DP and HV in real time. By Summer 1994 the radar system will be upgraded to measure LDR and HH,HV also. Table 3.1 lists the characteristics of the CSU-CHILL radar. The characteristics of the CP-2 radar can be found in [6].
 

3.7.1 Intercomparison of Polarimetric Signatures in Rain

A scatterplot of ZDR(S) against ZHH(S) was constructed from a fairly large database and is shown in Fig. 3.7. This plot can be compared with Fig. 3.3a. For convenience, the two dashed curves shown in Fig. 3.7 were obtained from Fig. 3.3a as the upper and lower bounds in ZDR for a given ZHH. It is clear that the measurements fall in the general region given by the rainfall model. Figure 8 shows a plot of HV versus ZDR in rain. The vertical bar lines are the 95% confidence interval of the estimated mean of HV(0). The mean HV(0) from data decreases with increasing ZDR in agreement with the rain model. These values of HV(0) in rain are limited by the antenna system. The new antenna at the CSU-CHILL facility has been shown to give HV(0) of 0.999 close to the theoretical limit.

Figure  3.9 shows the mean KDP versus ZHH for the central Florida and Colorado datasets gathered with CP-2 and CSU-CHILL respectively. The points marked 'Florida' and 'Colorado' come from the corresponding geographical regions. The solid line is the mean relationship of KDP against ZH at S-band corresponding to Fig. 3.4a of the rain model. The data are in good agreement with the rain model discussed in the previous section. The very high KDP values (>5/km) observed in Colorado were from an intense storm which occurred on 24 June 1992 Colorado which caused serious flooding in Fort Collins and the Colorado State University campus.

3.7.2 Vertical Profile of ZHH and ZDR

Figure 3.10 shows the vertical cross sections through an intense part of a squall line observed on 24 August 1991 in Central Florida, USA. The data shows contours of ZDR overlaid on ZHH. Darker shades represent higher values of ZDR. ZH contours start at 10 dBz and are incremented by 10 dBz. The leading edge of the squall line is at 18 km range. Note the positive ZDR column at 22 km, with the 2 dB contour reaching 6.5 km agl indicative of growth region. The peak ZDR is 4.5 dB at 5 km agl. Also we note that in general ZDR goes to 0 dB above 5 km, indicative of ice. Thus ZDR can be very useful in rain/ice discrimination. LDR signal can also be very effective in detecting ice, especially the rain water transition.

3.7.3 ZDR Hail Detection

A severe storm developed near Fort Collins on 24 June 1992 and moved towards the CSU-CHILL radar, located nearby Greeley, Colorado, USA. It dumped nearly 3" of rain between 15:15 ~ 15:54 MDT at CSU's main campus and in Fort Collins. Observers at CSU noted that precipitation started with a few big drops followed by intense rain and then mixed with hailstones of 0.75" diameter.

Figure 3.11 shows range profiles at an elevation angle of 1.58 taken at 15:36 through two convective cores located at ranges of 22 and 42 km. Both cores show large KDP ([bottom], dashed line) and hence large rainfall rates of 100 and 145 mm/h, respectively. The corresponding profile differential phase data (lower frame, dashed line) are shown in Fig. 3.11. KDP in Figure 3.11 is computed using finite differences based on DP curve. Beyond the second KDP peak (at 42 km) ZDR decreases which indicates hail. At 42 km, ZH = 62 dBz, KDP = 4.7/km, ZDR = 0 dB. At 45.5 km, ZH = 62 dBz, KDP = 1.8/km, ZDR = -1 dB. ZDR value dropping to zero or near negative value indicates the presence of hail. However if ZDR is nearly zero dB, but KDP is positive that signature indicates rain mixed with hail. Thus, at 42 km, rainfall dominates the mixture, and between 42 ~ 45 km, the hail and rain contribution become nearly similar.

Figure 3.12 shows a low elevation angle PPI (Plan Position Indication) showing contours of reflectivity with overlaid half tone ZDR. Figure 3.12 clearly depicts the two precipitation cores. The ZDR hail signature (Figure 3.12) can be clearly identified in the core located at the x-y coordinates (-39, 16) km observing the ZDR hole. A large region of positive ZDR is evident in Fig. 3.12 SE of the core located at x-y coordinates (-20, 7) km. This is adjacent to the leading edge of the complex where the surface level flow was from the SE.

3.7.4 Rainfall Estimation

Historically linear polarimetric radar research was initiated for rainfall estimation applications. However, this research has found great success in the area of microphysical observations. Currently there are two schemes to obtain rainfall estimates. One is using ZHH and ZDR and the other using KDP. Each technique has its advantages and disadvantages and the problem of rainfall estimation using dual-polarization radar is a continuing topic of active research. Therefore some of the rainfall algorithms provided in this chapter may be improved. Again here the final algorithms will be provided without extensive review of the literature.

There are two types of ZHH,ZDR based algorithms currently in use. They are of the form

The coefficients C1, C2, 1, 2, 1, 2 depend on the frequency band of operation. At low rainrates, the first algorithm (3.37) is noisy and, it is preferable to use (3.38). The second algorithm is more recent in the literature. The coefficients can be evaluated theoretically studying the radar backscatter properties of raindrop size distributions. Extensive discussion of such procedures are provided in Chandrasekar and Bringi [5].

The coefficients in (3.35) and (3.36) at S and C band are as follows:
 

S band:
C1 = 3 x 10-3  1 = 0.96 1 = - 1.59
C2 = 10.1 x 10-3 2 = 0.92  1 = - 0.4
 
 
C band:
C1 = 3.61 x 10-3  1 = 0.95 1 = - 0.28
C2 = 7.6 x 10-3 2 = 0.93  1 = - 0.281
 
 
The rainfall algorithm based on KDP takes the form
The KDP algorithms are very noisy for low rainrate but work very well for high rainrates. Again the coefficients C3 and depend on the frequency of operation. These coefficients for typical S and C band are as follows:
 
S band: C3 = 40.5 = 0.85
C band: C3 = 19.8 = 1

3.7.5 Winter Storm Observations

Recently polarimetric radar observations have been shown to provide useful signatures in winter storms also. The pristine ice crystals combining to form aggregates can be seen easily using polarimetric radars.

Figures 3.13a,b and c show vertical cuts of reflectivity, ZDR and copolar correlation coefficient over a winter storm in Colorado, USA. The data was collected by the CSU-CHILL radar on 26th January 1994 through a shallow upslope snow storm in northern Colorado foothills. We can see that in contrast to some of the summer storm data the storm top is only 2.5 km with max reflectivity levels of 12 dBZ. The ZDR picture of Fig. 3.13b shows enhanced ZDR aloft in the low reflectivity region. The contour overlay in Fig. 3.13b is reflectivity factor. Aircraft penetrations through this storm at various levels indicated pristine dendrites aloft and aggregates at the lower altitudes. Thus we can see that high values of ZDR at higher altitudes indicate aligned pristine crystals. In winter storms the aligned pristine crystals provide the maximum variation between horizontal and vertically polarized returns. When the crystals aggregate, they provide nearly identical returns at horizontal and vertical polarizations. Therefore, the pristine ice crystal region is low in co-polar correlation coefficient. Fig. 3.13c shows a vertical 'cut' of HV through the same winter storm. We can see from Fig. 3.13c that the pristine crystal region identified through high ZDR is also region of low correlation coefficient.

Figures 14a,b and c show measurements of reflectivity, ZDR and HV respecetively through a winter storm that exhibited a bright band close to the surface. The data was collected by the CSU-CHILL radar on 28 February 1994 in the northern Colorado front range (USA). We can see very enhanced signatures of ZDR and HV in the bright band. The value of HV drops significantly in the bright band due to water/ice mixtures. Again in this storm we can see region of high ZDR aloft indicative of pristine crystals.

3.8 ENGINEERING CONSIDERATIONS

There are several fairly strict engineering requirements that are responsible for successful measurements using a polarimetric radar for weather applications. This section will discuss the engineering aspect very briefly and interested readers are referred to the review article by Bringi and Hendry [6] for details. Most precipitation particles depart only moderately from spherical shapes and therefore accurate determination of the polarization diversity signatures requires careful measurement of small variations. For example the dynamic range of differential reflectivity is not more that 10 dB whereas the horizontal and vertical reflectivities can vary over a 70 dB scale. Measurements of cross polar measurements such as LDR require measurements of a weak signal response, which is several times smaller than (15~20 dB below) the main channel signal level. Therefore the antenna microwave circuitry and all parts of transmission and reception circuitry should meet fairly stringent requirements to be useful for meteorological applications.

Two components are selected here for detailed discussion, owing to their importance in the measurement process namely a) the polarization switch and b) the antenna system.

3.8.1 Polarization Switch

Many polarimetric weather radars use ferrite switches whose polarization state can be switched on a pulse to pulse basis. Typically these single circulator switches have isolations of the order of 20-25 dB depending upon parameters such as power level and operating temperature. It is also to be noted that the ferrite switches are non reciprocal devices and they have different isolations on transmit and receive states. Typically the receive isolation is better than transmit isolation. Some radars use mechanical switches and pin diode switches for polarization control. It is to be noted that the switching speed of mechanical switches is slow and the operating power levels of pin diode switches are small. Though the isolation of the individual switches may be small, three such switches can be connected together to double the isolation levels (see [6] for details).

3.8.2 Antenna System

Polarization purity of the antenna system is critical to polarization diversity measurements. Since the weather radar measurements are made over distributed targets, the polarization characteristics over the complete main beam is important (not just boresight). An ideal antenna for polarization diversity measurements has zero side lobes, perfectly matched mainlobe at all polarization states that it is used for, and zero cross polarization pattern [7]. The ideal antenna specification is impossible to satisfy, however many antenna systems have attempted to approximate the specifications as best as possible under practical restrictions. The recently installed CSU-CHILL antenna system has some excellent characteristics making it suitable for measurements of full polarization scattering matrix.

The CSU-CHILL antenna system is a fully steerable prime focus parabolic reflector 8.5m in diameter. The antenna's characteristics are as follows:

a) 3dB beam width 1.0 degree,
b) Directivity 45 dB,
c) Maximum side lobe level at any () plane: <-27 dB,
d) Maximum cross pol level at any plane: < -30 dB, for polarizations radiated/received: horizontal or vertical.
Fig. 3.15 shows the antenna patterns of the CSU-CHILL system. The four curves are copolar and cross polar patterns along the 135o diagonal, at the following four states namely: a) transmit horizontal/receive horizontal b) transmit vertical/receive vertical, c) transmit horizontal/receive vertical, d) transmit vertical/receive horizontal. We can see from Fig. 3.15 that the copolar patterns at horizontal and vertical polarizations are matched well in the main beam, and the first side lobe. We can also see that the cross polar return is maximum away from boresight down by 30 dB approximately. We need to note here that the cross polar pattern is worst on the 45/135 degree plane and they typically have nulls on boresight. The cross polar patterns at all other planes are lower than those shown in Fig. 3.15. It is advantageous to have the peaks of copolar and cross polar patterns displaced. Thus the pattern plots of Fig. 3.15 shows a fairly well designed dual polarization antenna system for weather radar applications.

3.9. SUMMARY AND CONCLUDING REMARKS

This paper provides an overview of polarimetric radar theory and analysis in the context of weather observations. Emphasis was placed on discussion of recent developments in the field. The contents of the paper was provided as an overview, without formal developments of each section. Significant portions of the paper dealt with development of polarimetric radar measurables, and their relation to storm microphysics. Subsequently several examples were provided to illustrate the polarimetric radar measurement features under various microphysical conditions. Rainfall estimation is an important application of polarimetric radars and algorithms to estimate rainfall based on ZDR and KDP measurements were discussed. The application of polarimetric radar techniques to the remote sensing of storm microphysics has recently reached a level of maturity that places it into the mainstream of radar meteorology.

A good account of the early history of dual-polarization radar as applied to meteorology can be found in [8]. While the early pioneering work by McCormick, Hendry and co-workers at the National Research Council of Canada was based on circular polarization techniques [9], the use of linear polarizations and in particular the differential reflectivity technique by Seliga and Bringi [10] and Hall et al [11] set the stage for accelerated research in polarimetric methods for the next two decades including the developoment of differential phase measurement [12-13]. Currently the parameters reflectivity, ZDR, KDP, LDR and HV(0) are actively used by researchers as a multiparameter set to understand the microphysical evolution of storms.
 

The field of polarimetric radar measurements of storms has been evolving rapidly in the past two decades and is a topic of active current research. Since the field is relatively young as well as actively evolving, there are not many review articles. Nevertheless, some recent reviews have appeared, for example, the articles by Bringi and Hendry [6] and Jameson and Johnson [4]. Polarimetric radars are likely to play a major role in the understanding of storm microphysics in the next few decades.

ACKNOWLEDGEMENTS

The author acknowledges helpful discussions with Professor Bringi at Colorado State University. Drs. Liu Li and John Hubbert helped with some figures. Several results presented in this paper were obtained from research effort supported by the United States National Science Foundation (ATM-9019596 and ATM-9200761).
 
 

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