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Downloaded 09/29/21 10:53 PM UTC 52 JOURNAL of ATMOSPHERIC and OCEANIC TECHNOLOGY VOLUME 17 JANUARY 2000 HUBBERT AND BRINGI 51 The Effects of Three-Body Scattering on Differential Re¯ectivity Signatures J. C. HUBBERT AND V. N . B RINGI Colorado State University, Fort Collins, Colorado (Manuscript received 18 September 1998, in ®nal form 5 March 1999) ABSTRACT Effects of three-body scattering on re¯ectivity signatures at S and C bands can be seen on the back side of large re¯ectivity storm cores that contain hail. The ®ngerlike protrusions of elevated re¯ectivity have been termed ¯are echoes or ``hail spikes.'' Three-body scattering occurs when radiation from the radar scattered toward the ground is scattered back to hydrometeors, which then scatter some of the radiation back to the radar. Three-body scatter typically causes differential re¯ectivity to be very high at high elevations and to be negative at lower elevations at the rear of the storm core. This paper describes a model that can simulate the essential features of the three- body scattering that has been observed in hailstorms. The model also shows that three-body scatter can signi®cantly affect the polarimetric ZDR (differential re¯ectivity) radar signatures in hailshafts at very low elevation and thus is a possible explanation of the frequently reported negative ZDR signatures in hailshafts near ground. 1. Introduction scatter. The lower panel shows the associated ZDR (dif- ferential re¯ectivity) ®eld. In the area of three-body scat- Under most circumstances multiple scattering effects tering, the 21- to 1-dB contour of Z forms roughly are considered to be negligible in radar meteorology. DR a458 angle with the ground. This contour area separates, However, one situation where the effects of multiple in general, positive Z values (above) from the negative scattering have been observed is in re¯ectivity signa- DR values (below). This pattern in Z is seen quite fre- tures on the back side (away from the radar) of high- DR quently in ¯are echoes in DLR and in Colorado State re¯ectivity cores (Z . ø55 dBZ) that contain hail. h University (CSU)±CHILL (S band, wavelength of about Fingerlike protrusions of elevated re¯ectivity have been observed, which are termed ¯are echoes or ``hail 11 cm) radar data, though it is typically much more pronounced at C band. Note the extrema of Z 9 spikes.'' This type of multiple scattering is termed three- DR .1 body scattering (Zrnic 1987) because of the theorized dB and ZDR ,25 dB, which would be dif®cult to ex- scattering path: transmitted energy is scattered to ground plain microphysically. The expected value of ZDR in this by the illuminated hailstones, the ground then scatters low re¯ectivity area is 0 dB, which is observed in very the energy back toward the main beam where hailstones light rain or in randomly oriented ice particles. again scatter some of the energy back toward the radar. Thus far in the literature, reported three-body scat- Zrnic (1987) modeled three-body scattering via a mod- tering observations has been limited to ¯are echos found i®ed radar equation and was able to predict the decay on the back side of high re¯ectivity cores. In this paper, in intensity of the the ¯are echo with respect to increased the possible effects of three-body scattering contami- range. Shown in Fig. 1 is an example of three-body nating the main signal in storm cores is also considered. scattering from DLR's (German Aerospace Agency) We hypothesize that another possible artifact of three- C-band (wavelength of about 5 cm) radar located at body scattering in ZDR is seen underneath the storm core Oberpfaffenhofen, Germany. The top panel shows re- at ranges from 79 to 88 km, where ZDR is quite negative ¯ectivity with peak values exceeding 65 dBZ. The three- with some values less than 23 dB. Researchers have body ¯are echo is evident on the right side of the panel; explained these negative values microphysically; that is, that is, the direct backscatter for ranges greater than hail of certain size and shape were assumed to be re- about 90 km is very small so that the seen re¯ectivity sponsible (Bringi et al. 1984; Zrnic et al. 1993). Model contours are probably due exclusively to three-body studies at S band show that vertically oriented conical hail less than about 4 cm (Aydin et al. 1984), vertically oriented oblate hail less than about 4 cm, and horizon- tally oriented oblate hail greater than about 4 cm (Aydin Corresponding author address: Dr. John Hubbert, Dept. of Elec- trical Engineering, Colorado State University, Fort Collins, CO and Zhao 1990) can produce such ZDR signatures. How- 80523-1373. ever, it has not been shown that hailstones actually do E-mail: [email protected] fall in such an orientation to cause ZDR to be negative. q 2000 American Meteorological Society Unauthenticated | Downloaded 09/29/21 10:53 PM UTC 52 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 17 FIG. 1. An example of three-body scattering in a hailstorm from DLR's (German Aerospace Agency) C-band radar located at Oberpfaf- fenhofen, Germany. The three-body signature is easily seen as the protruding re¯ectivity area on the right-hand side in the top panel. The bottom panel shows the associated ZDR (differential re¯ectivity). Since such negative ZDR signatures are frequently seen scattering associated with ¯are echoes. His model in- in hailshafts, it seems unlikely that they can always be cluded several approximations and simpli®cations, attributed to microphysics. As an alternate explanation, which are not appropriate for simulating the effects on the three-body scatter model calculations described in ZDR. In contrast, we employ a numerical technique, this paper show that for larger hailstones close to ground which sums the scattering contributions from all particle level, three-body scattering can bias ZDR negative if pairs that contribute to a particular radar resolution vol- ground scatter cross sections are large. The three-body ume, to simulate total power from three-body scatter. scattering effects are more pronounced at C band than The geometry is shown in Fig. 2. The terms Pi and Pj at S band, which agrees with experimental data. represent two hailstones in the main beam of the radar, This paper then, explores the effects of three-body while ri,j represent the distance from Pi,j to an incre- scattering on ZDR signatures at S and C bands. Two storm mental ground area and d (not shown) represent the regions are considered: 1) the ¯are echo region (i.e., the i,j distance from Pi,j to the radar. The direct scatter comes region in back of high-re¯ectivity cores) and 2) near from the radar resolution volume B at a distance R ground level in hailshafts. m m from the radar, where Rm terminates at the back edge of the precipitation medium of depth of D. The depth 2. Model description of the resolution volume is controlled by the radar trans- mit pulse width t and receiver bandwidth but is given a. General here by ct/2, where c is the speed of light. Pairs of Zrnic (1987) derived a closed-form solution to predict scatterers in the radar beam will contribute to the ob- the re¯ectivity and velocity signatures of three-body served echo corresponding to Bm if Unauthenticated | Downloaded 09/29/21 10:53 PM UTC JANUARY 2000 HUBBERT AND BRINGI 53 2 150 m for this study. In this way, two concentric, 3D ellipsoids are de®ned. The intersection of the ground and the concentric ellipsoids is represented by the hatched elliptical shell in Fig. 2. The total power due to three-body scattering corresponding to the Bm reso- lution volume is found by integrating over all pairs of scatterers that lie in the radar beam and integrating over the corresponding ground areas. Mathematically, for the ith and jth particles, FIG. 2. A schematic of three-body scattering. Signal path: radar → particle P → ground → particle P → radar. The hatched area rep- gVTtSGSE i j V 5 jki , (5) resents the area on the ground where the three-body path has the 3b 2 same time delay as the direct path from the radar to the resolution (4p)(RRrri jij) volume Bm. where Si,j are 2 3 2 bistatic scattering matrices for the Pi,j particles, Gk isa23 2 bistatic scattering matrix for a ground element, and g represents an overall system 2Rm 2 ct # di 1 dj 1 ri 1 rj # 2Rm. (1) gain constant. Here, Et isa13 2 transmit vector with For a given range, Rm and particle pair Pi,j, the sum ri [1 0]T and [0 1]T representing horizontal and vertical 1 rj is constant, and thus the loci of particles in 3D transmit polarization states, while V is the receive po- space that can contribute to three-body scattering cor- larization vector of the radar. The scattering matrices responding to range Rm is described by a 3D ellipsoid for the hail are found from Mie theory and are functions (prolate spheroid) with the locations of Pi,j as the foci. of incident and scattered directions. The total power is To simplify the calculations, the following assump- found by summing over all particle pairs and over all tions/simpli®cations are made: 1) the radar beam is con- the corresponding ground areas: sidered to be parallel to ground, 2) the radar beam is P 5 |V |2 . (6) modeled as a cylinder with a constant power across the 3b OO 3b beamwidth, and 3) scatter from a particular incremental i,j k volume along the radar beam is approximated by a sin- It is important to note that the left summation is a double gle scatterer located at the center of the volume along summation over all particle pairs, which physically the beam axis.
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