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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 Ϫ1- to 1-dB contour of Z forms roughly are considered to be negligible in radar meteorology. DR a45Њ 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 Ͼ ഠ55 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 Ͼϩ body scattering (Zrnic 1987) because of the theorized dB and ZDR ϽϪ5 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 Ϫ3 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.

᭧ 2000 American Meteorological Society

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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 ␶ and receiver bandwidth but is given a. General here by c␶/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

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Ϫ 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- g⍀TtSGSE i j V ϭ 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 (4␲)(RRrri jij) volume Bm. where Si,j are 2 ϫ 2 bistatic scattering matrices for the Pi,j particles, Gk isa2ϫ 2 bistatic scattering matrix for a ground element, and g represents an overall system 2Rm Ϫ c␶ Յ di ϩ dj ϩ ri ϩ rj Յ 2Rm. (1) gain constant. Here, Et isa1ϫ 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 ϩ rj is constant, and thus the loci of particles in 3D transmit polarization states, while ⍀ 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 ϭ |V |2 . (6) modeled as a cylinder with a constant power across the 3b ͸͸ 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. That is, the cylindrical radar beam is means that the three-body scatter path is bidirectional. subdivided into thin cylinders with a maximum depth The direct backscatter power from the resolution volume of 80 m (a function of radar beam height above the Bm is calculated for a similar density of scatterers as is surface). Scatter from this thin cylindrical cross section used for the three-body scatter calculations: of the radar beam is modeled by a single scatterer lo- g2|⍀TtS E |2 cated at the center of the cylinder. Scatter from the hail P ϭ i . (7) bs ͸ 42 is modeled by the Mie solution for ice spheres and for i (4␲) di water-coated ice spheres. Obviously, summing over a realistic ensemble of hail- Under these approximations the above-mentioned el- stones contained in the main beam of a radar would be lipsoid will have its Pi,j axis parallel to the ground (along computationally impossible. To simplify the problem the line of sight of the radar) and the intersection of the the scatter from a vertical cross section of the radar beam ellipsoid with the ground will de®ne the ellipse is approximated by a single scatterer located at the cen- xy22 h 2 ter of the volume. Since there is a double summation ϩϭ1 Ϫ , (2) over particle pairs in Eq. (6) and only a single sum- 22 2 ab b mation over individual particles in (7), a doubling in where 2a and 2b are the lengths of the major and minor the number of particles will increase the power ratio axis of the ellipse, respectively, and h is the distance P3b/Pbs by 3 dB. The concentration of particles, assumed from the major axis to the ground. The major axis is here to be 1 mϪ3, is accounted for in the model by along the main beam of the radar. The parameters a and increasing the ratio P3b/Pbs by the the amount Pinc: b are found from desired particle number density P ϭ . (8) 2D Ϫ d Ϫ d inc used particle number density a ϭ ijand (3) 2 The number density of hail particles as well as the num- b ϭ [a21/2Ϫ (|d Ϫ d |/2)] . (4) ber density of ground grid points was increased until ij the sum in Eq. (6) converged. The ellipsoid is constrained by the distance from the radar to the back edge of the apparent resolution volume b. Ground model Bm; that is, this distance de®nes the constant distance ri ϩ rj via Eq. (1). Another ellipsoid is de®ned by the The most dif®cult and uncertain part of the analysis distance from the radar to the front edge of Bm or Rm is the modeling of the ground, which may be composed

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FIG. 3. The three-body scattering geometry for a ground element. Incident wave direction is always in the x±z plane. FIG. 4. The backscatter cross section ␥ for the Lommel±Seeliger

model [Eq. (8)] for the ground. Note that ␪i ϭ ␪s in Fig. 3. of trees, shrubs, crops, grasses, roads, buildings, water, etc. Clearly scattering cross sections from such collec- which the copolar H polarization (HH) and copolar V tions vary dramatically. In addition, even though many polarization (VV) bistatic cross sections are equal. The backscatter measurements have been made at S and C advantage of the model is that it is easy to implement, bands, there is a dearth of bistatic measurements. The allows for general quantitative results, and lets ZDR cal- only signi®cant recent measurements that we are aware culation be unbiased from ground effects (i.e., VV ϭ of are reported by Ulaby et al. (1988) for 35 GHz. There HH and VH ϭ HV for ground scatter cross sections). are sophisticated modeling techniques that have ap- The statistical model is actually an algebraic com- peared in the literature (Ulaby et al. 1988; Bahar and bination of two models for rough surfaces: 1) a model Zhang 1996), but they would be dif®cult to implement valid for slightly rough surfaces; and 2) a model valid into our model and would not necessarily yield more for very rough surfaces where slightly rough means that accurate or informative solutions due to the general un- the rms surface height is much less than the wavelength, known and complex nature of the ground. Therefore, while very rough means that the rms surface height is for this general study the computational complexity is much greater than the wavelength. Typical terrain sur- reduced by employing two analytical models for rough faces will be composed of roughnesses of both scales surfaces: 1) an empirical Lommel±Seeliger model and for frequencies considered here and thus the models are 2) a statistical model that treats the surface height above combined with the resulting model yielding scattering a mean planar surface as a random variable. The Lom- cross sections that give good approximation to experi- mel±Seeliger model used (Ruck et al. 1970) has the form mental data (Ulaby and Dobson 1989). Importantly, the VV exceeds the HH backscatter cross section, which is cos␪isϩ cos␪ ␥(␪is, ␪ ) ϭ k , (9) frequently observed experimentally and is necessary cos␪iscos␪ here to obtain signi®cant negative ZDR in hailshafts at where k is a function of the surface properties (but treat- near±ground levels. The statistical model, while giving ed as a constant here), and ␪i and ␪s are the incident a more accurate representation of terrain than the above and scattered angles, respectively. Figure 3 shows the Lommel±Seeliger model, is still analytical, which thus bistatic scattering geometry used for the ground models. allows for fairly simple simulations. Details of the sta- Vertical (V) incident and scattering directions are de- tistical model can be found in the appendix.

®ned by the unit vectors ␪i, ␪s, respectively, while hor- Shown in Fig. 5a are backscatter cross sections for izontal (H) incident and scattering directions are de®ned the slightly rough surface and the very rough surface by the unit vectors ␾ i, ␾ s, respectively. The incident models. The slightly rough surface model has the VV vector ␪i is always in the x±z plane so that ␾ i ϭ 0. Note cross section greater than the HH cross section for an- that incident angle de®ned in a standard spherical co- gles greater than about 10Њ. The very rough surface ordinate system is typically taken as ␲ Ϫ ␪i, not ␪i,as model has the VV and HH backscatter cross sections is done here (and in Ruck et al. 1970). A plot of the equal and has larger cross sections for angles less than backscatter cross sections (i.e., ␪i ϭ ␪s, ␾ s ϭ 180Њ) for 10Њ. The composite model shown in Fig. 5b is found the Lommel±Seeliger model is shown in Fig. 4. This by simply adding the backscatter amplitudes from the model gives a reasonable approximation to measured two models. Experimental measurements reported in cross sections of terrain surfaces where dimensions are Ulaby and Dobson (1989) indicate that the composite considerably greater than wavelength for diffuse scat- curves shown in Fig. 5b are quite realistic and, fur- tering but can be invalid for specular scatter (Ruck et thermore, the curves could be easily increased several al. 1970). We use it as a ®rst-order approximation in decibels in magnitude especially at small incidence an-

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FIG. 6. Geometry for dipole ®eld calculation. Dipole is located at height h above x±y plane. Incident wave is along the positive x di-

rection. The lengths ri, riϩ1 de®ne an area between concentric circles over which power from the vertical and horizontal dipole ®elds are summed.

(the ground) coincident with z axis with z ϭ h, where h here refers to the height above the x±y plane. Con- centric circles are shown on the x±y plane de®ned by varying the length of r by 0.15 increments (arbitrary units). The incident wave is in the positive x direction along the line de®ned by z ϭ h, y ϭ 0. The 3D plot of the scattered ®eld intensity for incident V polarization resembles a horizontal ``doughnut'' (e.g., see 3D dipole ®eld pattern in Balanis 1982) with maximum intensity loci in the horizontal plane de®ned by z ϭ h and min- imum intensity (zero) along the z axis. For H-incident FIG. 5. The backscatter cross section (␪i ϭ ␪s ) for (a) the very polarization the maximum intensity is located in the x±z rough and slightly rough surface models and (b) the composite plane and the minimum intensity is along the line de- model. ®ned by x ϭ 0, z ϭ h. Thus a horizontal dipole will have an intensity maximum in the negative z direction, gles. The parameters chosen for the composite model while a vertical dipole will have a minimum (zero). Figure 7 shows the ratio of H to V total dipole power are k 0l ϭ 1.5, k 0h ϭ 0.3, and w ϭ 0.087 49. The di- incident on the area de®ned by the concentric circles r electric constant used for the ground is er ϭ 48.82 ϩ i j15.12 for both S and C band, which was calculated ϭ 0.15i and ri ϩ1 ϭ 0.15(i ϩ 1), (i ϭ 1, 2, 3, . . .) with from a formula given in Ruck et al. [1970, their Eqs. the dipole height h as a parameter. The horizontal axis (9.1)±(41)] The parameters were chosen based more on is ri. As the diameter of the concentric circles becomes the ®nal shape of the resulting composite curve rather than on some physical criterion. The most important result is that the theoretical model used here approxi- mates known experimental measurements.

3. Simulation results a. ZDR signatures on the back side of a hailshaft The two primary factors affecting the nature of three- body ZDR signatures are 1) the difference in the scat- tering characteristics of hail at V and H incident polar- izations and 2) the scattering characteristics of the ground. We ®rst illustrate how scattering characteristics of hail affect ZDR on the back side of high re¯ectivity cores by examining the ®eld patterns of horizontally and vertically oriented dipoles, which are used as simple FIG. 7. The ratio of horizontal to vertical power incident on the models for the scattered ®elds of hailstones. Figure 6 hatched area in Fig. 6 with h as a parameter. The horizontal axis is shows a dipole scatterer located above the x±y plane in arbitrary units of ri and riϩ1 ϭ 0.15 ϩ ri.

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three-body power must be greater than the HH three-

body power in order to cause negative observed ZDR obs (Z DR ), with

P bsϩ P 3b Z obs ϭ 10 loghh hh , (10) DR P bsϩ P 3b []vv vv bs bs wherePPhh and vv are the direct backscattered copolar powers at H- and V-incident polarizations (equal for 3b 3b spherical scatterers), andPPhh and vv are the three-body copolar powers for H- and V-incident polarizations, re- bs bs 3b 3b spectively. Suppose that PPPvvϭϭϭ hh vv2P hh; for example, the direct backscatter medium consists of ran-

domly oriented hail (intrinsic ZDR ϭ 0 dB) with the VV three-body power exceeding the HH three-body power FIG. 8. The three-body power ratio HH/VV as a function of dis- obs by 3 dB. In this case the Z DR ϭϪ1.25 dB. If the three- tance in back of the hailshaft. 3b body ZDR is made 3 dB more negative, that is, Pvv ϭ 3b 3b 3b 4PPhhinstead of vvϭ 2P hh, then the three-body VV bs bs larger, the ratio of H to V power becomes smaller. This power can also be 3 dB less, that is, PPvvϭϭ hh 3b obs suggests (to the extent that a dipole ®eld resembles scat- 2PZvv, and these values still yield DR ϭϪ1.25 dB. Thus, ter from hail) that three-body ZDR in back of a hailshaft the three-body VV power can be 3 dB less than the will have a tendency to go from positive to negative as direct backscatter power and still signi®cantly bias the range increases. This type of ZDR pattern is observed in observed ZDR. Fig. 1 at ranges 88±110 km at heights of 3±10 km. Figure 9 shows modeling results using the Lommel± The full three-body scattering model described in sec- Seeliger ground model for panel (a) S band and panel tion 2, which uses the Lommel±Seeliger ground model, (b) C band. There are three sets of curves in each plot is now used to calculate the three-body ZDR signatures corresponding to radar beam heights at 0.01-, 0.1-, and on the back side of a hailshaft. The hailshaft depth, Dp, 0.5-km heights above ground. Obviously, if the center is 3 km and the hail is modeled as 2-cm solid ice spheres. of the radar beam is at 0.01 or 0.1 km, the lower part Shown in Fig. 8 is the ratio of three-body HH power of the beam would be blocked by the earth (assuming to three-body VV power as a function of range from a1Њ beamwidth). The model assumes no beam blockage the back (away from the radar) of the hailshaft (i.e., 0 and is meant to demonstrate the effects of low elevation km corresponds to the back edge of the hailshaft). The angles. The horizontal axis is the diameter of the hail, three curves correspond to 3-, 5-, and 7-km radar beam while the left vertical axis (solid curves) shows the ratio heights above ground. The curves show that the three- of VV three-body power to VV direct backscatter power body ZDR values are very high close to the hailshaft and (in dB), and the right vertical axis (dashed curves) is 3b 3b then decrease monotonically with increasing range and three-body ZDR (PPhh / vv ) (in dB). The dashed curves are become negative. As the height increases, the range at not distinguished since they are similar and show that which the ZDR ®rst becomes negative increases. This is three-body ZDR is typically close to zero (|ZDR| Ͻ 0.7 3b bs similar to what is observed in Fig. 1. Thus, the three- dB) for this model. For S band PPvvϾ vv for D Ͼ 4 3b bs body ZDR signature on the back of hailshafts can be cm at 0.01-km height. At 0.5-km height PPvvϾ vv only attributed to the angular scattering pattern of the hail- for d ϭ 5.5 cm. The model indicates that hail with stones with no preferential scattering from the ground diameters of about 5±6 cm would most likely provide (i.e., VV Ͼ HH ground cross sections) required. suf®cient power in order for the three-body scatter to be large enough to effect the primary backscatter signal. Hail less than about 3.5 cm is much less likely to cause b. Negative Z in hailshafts DR enough three-body power to affect the primary back-

The model is now used to investigate ZDR signatures scatter return. At C band, however, there is a peak at D 3b bs in hailshafts at low elevations. A two-layer model is ϭ 2.75 cm, where PPvvϾ vv at 0.01-, 0.1-, and 0.5-km 3b bs used for the spherical hail: a solid spherical ice core is heights withPPvv / vv ϭ 22.9 dB for h ϭ 0.01 km. Figure covered with a 1-mm liquid water shell, which is a 9 also suggests that three-body scattering effects will typical way to model melting hail (Rassumssen and be more evident at C band than at S band since the Heyms®eld 1987). The ground is ®rst described using occurrence of 2.5±3-cm hail is much more common than the Lommel±Seeliger model given by Eq. (9). There are 5±6-cm hail. Since three-body ZDR (dotted curves) are obs two conditions to be met if three-body scattering is to not negative enough to signi®cantly biasZ DR , we next bias the observed ZDR to negative values: 1) the power use the statistical ground model in order to obtain neg- obs due to three-body scattering must be close in magnitude ativeZ DR . to the power due to direct backscatter; and 2) the VV Figure 10 is similar to Fig. 9 but with the statistical

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FIG. 10. As in Fig. 9, except the statistical ground model is used. FIG. 9. The ratio of three-body scatter power to direct backscatter power (left axis) and three-body ZDR as a function of hail diameter. Spherical hail is modeled as an ice core with a liquid water coat (1 mm). The Lommel±Seeliger ground model is used at (a) S band and 4. Model modi®cations (b) C band. a. Three-body power as a function of height As shown in Figs. 9 and 10 the ratio of three-body 3b bs ground model used in place of the Lommel±Seeliger power to direct backscatter power (PR ϭ PPvv/)isa vv ground model. The solid curves are similar in shape and strong function of height and will vary signi®cantly magnitude to the solid curves of Fig. 9 and similar con- across the vertical dimension of the radar beam. To ac- clusions can be drawn. However, the dotted curves, showing three-body ZDR, are now quite negative espe- cially in the resonant regions of interest, that is, D Ͼ 4 cm for S band and 2.5 cm Ͻ D Ͻ 3 cm for C band. Thus, the more realistic statistical ground model pro- obs vides for negative three-body ZDR that can causeZ DR to be negative. Again, VV are frequently greater than HH ground cross sections for many different terrain types (Ulaby and Dobson 1989). Use of the water-coated ice model for melting hail is 3b bs not necessary to obtainPPvv / vv Ͼ 1.0. To illustrate this, the hail is modeled as ``spongy ice'' using various ice/ water ratios (Bohren and Batten 1982) at S band. Shown in Fig. 11 is the ratio of three-body power to direct backscatter power for hail modeled as 80% ice, 20% water; 90% ice, 10% water; 95%, ice 5% water; solid ice; and the two-layer (water-coated) model used in Fig. 10. All curves exhibit peaks above 0 dB for D Ͼ 4cm FIG. 11. The ratio of three-body scatter power to direct backscatter except the curve for 95% ice, 5% water. Thus a variety power for various mixtures of ice and water. Two-layer denotes water- 3b bs of dielectric constants for hail will produce PPvvϾ vv. coated ice spheres.

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count for this the PR is integrated numerically across disperse distributions. Using Eq. (12) along with Eq. (11), the beamwidth as a function of height. The cylindrical the total three-body power can be estimated. The total radar beam is divided into eight sections of equal height direct backscatter power is found from the simple (inco- with PR at the center of the sections found by inter- herent) addition of the backscatter powers of the individual polation from the previous calculated PR at heights of size classes. This estimation method increases the particle 0.01, 0.01, and 0.5 km. These power ratios (in linear density by a factor of 2 (for a size distribution with two scale) are, approximately, proportional to hϪ1. The radar distinct classes) and this is accounted for by an adjustment beam is taken to be 0.5 km in diameter (i.e., a 1Њ beam- of the particle density factor by 0.5 for this case. It is a width at about 30-km range) and is centered at 0.25 km simple matter to expand this method to an arbitrary number above ground (e.g., a 0.5Њ elevation angle). The inte- of hail sizes. In this way the number density of the total gration at S band for D ϭ 5.5 cm and at C band for D population of particles is kept at 1 mϪ3 with each size class ϭ 2.75 cm and for the statistical ground model gives having equal number densities. For S band, particle sizes

PR as 5 and 9.5 dB, respectively. In comparison, using from 4 to 5.5 cm are integrated for hail modeled as water- the previous model that places hail only at the center coated ice spheres. Using the statistical ground model, a of the beam (at 0.25-km height) gives PR as 2.5 and 7.3 beamwidth of 0.5-km diameter, a beam height 0.25 km, dB, respectively. If the center of the beam is at 0.5-km and powers derived above by integrating across the radar height (ഠ1Њ elevation angle), the beamwidth-integrated beam, the resulting power ratio is PR ϭϪ9.4 dB. A similar PR become Ϫ0.2 and 4.6 dB, respectively, as compared integration at C band but for particles sizes of 2.5, 2.75, to Ϫ0.5 and 4.2 dB, respectively, if the hail is located and 3.0 cm yields PR ϭϪ11.8 dB. This low value may only at the center of the radar beam. Thus, the integra- be unexpected since PR is quite high (about 11 dB for h tion across the beam of the radar only changes the power ϭ 0.1 km; see Fig. 10b) for D ϭ 2.75 cm at C band. This ratio by no more that 2.5 dB for these two cases but large value, 11 dB, is more a result of the reduction of does increase the PR in all cases. Note that because PR direct backscatter power than a large increase in three- is proportional to hϪ1, the closer the center of the radar body power. For these particular size distributions, the beam is to the ground, the more likely the three-body power ratios PR need to be increased about 6±8 dB to power will affect the observed radar signals. bring them to the Ϫ3-dB level, where three-body scattering

can signi®cantly effect observed ZDR. The needed addi- tional power could be obtained by increasing the scattering b. Hail size distribution cross sections of the ground. Ulaby and Dobson (1989) To study the effect of integration over a size distri- show backscatter cross sections for grasses at S band, bution, the three-body power from a distribution con- which have the 95% occurrence level curve at about 16 sisting of just two sizes with equal number of particles dB for vertical incidence backscatter with the similar curve in each class is considered ®rst. The three-body power for C band at 13 dB. In other words, 5% of the experi- is expressed as mental observations exceeded these levels. In comparison, P ϭ ␣ G ␣ ϩ ␤ G ␤ the statistical ground model parameters assumed here gave 3bikjikj͸͸ ͸͸ i,j k i,j k backscatter cross sections of about 5 dB at vertical inci- dence. Furthermore, the Ulaby and Dobson VV curves ϩ 2 ␣ G ␤ , (11) ͸͸ ikj typically lie above the HH curves especially at small in- i,j k cident angles and more so at C band than at S band. The where ␣ and ␤ are the scattering amplitudes for the two statistical model parameters used here gave equal VV and size classes. The computation time required for a dis- HH cross sections for small (Ͻ10Њ) incidence angles (see tribution of particles sizes would be unfeasible (unless Fig. 5). Additional three-body power can be gained by the number of particles sizes is kept very small), and increasing the size of the radar resolution volume. The therefore an estimation of the power from the interaction modeled uses a radar resolution volume based on a 1Њ of the two classes is used. The ®rst two summations are beamwidth at 30-km range. If the range is doubled, the known from previous monodisperse calculations and the size of resolution volume is increased by a factor of 4, third summation, which accounts for the interaction be- and this increase in volume (assuming the hail density tween the two classes, is estimated from Ϫ3 remains at 1 m ) will increase PR by 6 dB due to the 1/2 double summation (over particle pairs) in Eq. (6). Thus 6 dB of additional power is gained by simply increasing the ␣ G ␤ ϭ ␣ G ␣␤G ␤ . ͸͸ikj ͸͸ ikj ͸͸ ikj range of the hailshaft to 60 km; however, similarly P is i,j k΂΃Ά·Ά· i,j k i,j k R reduced by 6 dB by decreasing the range to the hailshaft (12) to 15 km. In any event, it is quite possible that conditions This approximation assumes that three-body power from for strong three-body scattering do occur according to the the monodisperse distributions bound the sum representing models used and the experimental observations. This, the three-body power from the interaction of the two clas- however, does not preclude other factors that can cause ses and can be estimated by Eq. (12), which is the geo- negative ZDR via direct backscatter due to shape and ori- metric mean of the three-body powers for the two mono- entation effects. Negative ZDR could also be caused by

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differential attenuation (ADP) due to by rain along the prop- investigate the effects of three-body scatter on ZDR sig- agation path, high re¯ectivity gradients across the antenna natures. Scattering from spherical hailstones was mod- beam and sidelobes, direct backscatter from the ground, eled using Mie theory, while a general Lommel±Seeliger or a combination of these factors. In fact, ADP and three- and a statistical model were used to represent scattering body scattering can produce similar ZDR signatures on the from the ground. The Lommel±Seeliger model shows back side of high-re¯ectivity areas at low-elevation angles. VV and HH cross sections to be equal (as well as VH obs The ADP lowers the observed ZDR according to ZDR ϭ ZDR and HV). This model was suf®cient to explain the ZDR Ϫ (2ADP)r, where ZDR represents the intrinsic ZDR of the signature seen on the back side of the hail core shown medium r is the range (in km) and ADP is the one-way in Fig. 1. This model was also used to investigate the differential attenuation (in dB kmϪ1) for an aligned me- possible effects of three-body scatter at low-elevation dium. To distinguish between these two mechanisms, the angles within hail cores. At very low elevations large LDR (linear depolarization ratio) as a function of range hail produced enough three-body power to affect the can be examined if available. If the LDRh (LDRh is primary backscatter signal. However, the model also

VH/HH, while LDR␷ is HV/VV) is very high, say, greater showed that three-body ZDR was about 0 dB so that the than Ϫ15 dB, and the re¯ectivity is low (Z Ͻ 35 dBZ), intrinsic ZDR would not be signi®cantly biased negative three-body scattering is very likely causing the negative (or positive). A more accurate statistical-based ground

ZDR since intrinsic scatterers that would cause the low model was then employed that provided for VV ground re¯ectivity would likely have much lower LDR (e.g., light cross sections to exceed HH ground cross sections, rain, ice crystals, dry graupel). Also, the ADP can be es- which is frequently observed experimentally (Ulaby and timated from the copolar speci®c differential phase (KDP), Dobson 1989). Using this ground model caused three- which is the slope of the range pro®le of copolar differ- body ZDR to be quite negative, and thus three-body scat- ential phase (␾ DP) (Bringi et al. 1990) by ADP ϭ ␤KDP tering was shown to be a possible explanation for the with ␤ ϭ 0.003 67 at S band and ␤ ϭ 0.0157 at C band. frequently observed negative ZDR close to ground in However, recent experimental evidence (Ryzhkov and hailshafts. The hail size that gave the greatest three-

Zrnic 1995; Carey et al. 1997) suggests that the results in body scatter power to direct backscatter power ratio (PR) Bringi et al. (1990) may underestimate, in some cases, the was around 2.75 cm for C band and 5.5 cm at S band amount of differential attenuation per degree of differential for hail modeled as water-coated ice spheres. Hail was phase by a factor of 2. For the C-band case shown in Fig. also modeled as ice spheres with various dielectric con-

1, the total ␾ DP through the main precipitation shaft was stants, and the results showed that such hailstones also about 40Њ, so that the amount of differential attenuation yielded PR values similar to values for water-coated ice could be 1.26 dB if ␤ in the relationship, as given by so that a wide variety of hailstone types will yield sim- Bringi et al. (1990), is doubled. This still cannot account ilar three-body powers. The effects of three-body scat- for the Ϫ2 dB and smaller ZDR values observed close to tering at low elevations would most likely be observed ground level. In addition, LDR is high, ranging from about on the back edge of hailshafts where the three-body

Ϫ16 to Ϫ6 dB, and the ␳h␷ (copolar correlation coef®cient) scattering would remain strong but where the direct scat- is quite low, ranging from about 0.1 to 0.4 in this area, ter from hail has decreased due to decreased concen- which also suggests three-body scatter. In areas where tration of hail. Also, due to the bidirectional nature of three-body scattering clearly dominates (e.g., in the ¯are three-body scatter, the larger the radar resolution vol- echo regions) LDR is typically Ϫ5 dB or higher and ␳h␷ ume, the larger the power ratio PR becomes since back- is low, that is, less than 0.8 and more typically 0.5 or less. scatter power increases by n (number of hydrometeors), These signatures are due to the scattering characteristics while three-body power increases by n 2. This means of the ground. If the negative ZDR was due to vertically that three-body signatures become stronger with in- aligned oblate or prolate hail, then such low LDR and ␳h␷ creased range, all other factors being equal. It is also would be unlikely. In addition, differential attenuation will possible that if HH ground cross sections were to exceed cause LDRh to increase, while LDR␷ will decrease and VV ground cross sections, ZDR would be biased positive. thus this can be another check for the presence of differ- To distinguish negative ZDR caused by three-body ential attenuation. scattering from negative ZDR caused by differential at- tenuation (ADP) due to rain, especially on the back side of storm cores, the differential phase (␾ ), linear de- 5. Conclusions DP polarization ratio (LDR), and copolar correlation co-

The three-body re¯ectivity signatures known as ¯are ef®cient (␳h␷ ) signatures can be examined. The amount echoes or ``hail spikes'' are routinely seen on the back of ADP due to rain can be estimated from the ␾ DP so that side of high-re¯ectivity cores that contain hail. How- the amount of negative ZDR due to ADP can be also es- ever, the effects of three-body scattering on differential timated. Recent studies (Ryzhkov and Zrnic 1995; Ca- re¯ectivity (ZDR) are not as well known. It is important rey et al. 1997) have shown that more ADP can occur to understand the possible origin of these signatures, (about a factor of 2) than what is predicted from ␾ DP especially for future operational systems (e.g., Zahrai according to relationships given in Bringi et al. (1990). and Zrnic 1997). This paper used a numerical model to If the observed ZDR is more negative than that predicted

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by ADP, then other possible causes need to be considered, a. Slightly rough surface model including three-body scatter. In ¯are echo regions where three-body scattering clearly dominates, the observed The solution given here for a slightly rough surface LDR was extremely high, Ϫ10 to 0 dB, and ␳h␷ was was formulated by Rice (1965) and derived by Peake very low at less than 0.8 and typically around 0.5 or (1959a,b) employing a perturbation technique and can less. Thus, if the LDR is higher and the ␳h␷ lower than be found in Ruck et al. (1970). It is important to note can be explained by the type of hydrometeors present, that the present application is valid for polarization- then three-body scatter could be an alternate explana- dependent bistatic scatter. tion. In contrast, differential attenuation does not effect The average (incoherent) scattering amplitude per ␳h␷ and only increases LDR by ADPr, where r is the unit surface area is range along the rain ®lled propagation path (assuming that the propagation matrix is diagonal). Unfortunately, 4 2 spq ϭ kh0 cos␪ispqcos␪␣ I, (A1) LDR can be high (and ␳h␷ can be low) due to direct ␲ backscatter from hail but typically not as high (or low) as that seen for three-body scattering. Note that other where p, q represent the scattered and incident polari- factors, such as steep re¯ectivity gradients (along with zation states, respectively; h is the square root of the mismatched copolar antenna patterns; e.g., see Hubbert mean-square roughness height; k is the wavenumber; et al. 1998) and clutter, can also have similar effects. 0 ␣pq (given below) is directly proportional to the scat- The end result is that it can be very dif®cult to identify tering matrix element. The quantity I is given here for and especially separate these various effects. a Gaussian surface-height correlation coef®cient (Ruck Acknowledgments. The authors acknowledge support et al. 1970): from the National Science Foundation and the U.S. Ϫkl22(␰ 2ϩ ␰ 2) Weather Research Program via ATM-9612519. The I ϭ ␲l exp0 xy , (A2) DLR C-band radar data were made available through []2 Dr. Peter Meischner. where APPENDIX ␰xissϭ sin␪ Ϫ sin␪ cos␾ , (A3) Statistical Rough Surface Model ␰yisϭ sin␪ sin␪ , (A4) The two statistical rough surface models are given. They are combined to make the composite rough surface and the quantity l is the correlation length. The bistatic model used in the study. scattering elements, ␣pq, for H and V polarizations are

(⑀rsϪ 1) cos␾ ␣hh ϭϪ , (A5) 22 cos␪irϩ ͙⑀ Ϫ sin ␪ isr)(cos␪ ϩ ͙⑀ Ϫ sin ␪ s)

2 sin␾sr(⑀ Ϫ 1)͙⑀ rϪ sin ␪ s ␣␷h ϭϪ , (A6) 22 cos␪irϩ ͙⑀ Ϫ sin ␪ irsrs)(⑀ cos␪ ϩ ͙⑀ Ϫ sin ␪ )

2 sin␾sr(⑀ Ϫ 1)͙⑀ rϪ sin ␪ i ␣h␷ ϭ , and (A7) 22 ⑀ricos␪ ϩ ͙⑀ rϪ sin ␪ i)(cos␪ sϩ ͙⑀ rϪ sin ␪ s)

22 (⑀rϪ 1)[⑀ rsssin␪ sin␪ Ϫ cos␾ sr͙⑀ Ϫ sin ␪ ir͙⑀ Ϫ sin ␪ s] ␣␷␷ ϭ . (A8) 22 ⑀ricos␪ ϩ ͙⑀ rϪ sin ␪ irsr)(⑀ cos␪ ϩ ͙⑀ Ϫ sin ␪ s)

The dielectric constant, ⑀␶ used for both S and C bands weight present in the vegetation, which is taken as 60% is 48.8152 ϩ 15.12i, which was calculated from (Peake here. Since hail will typically be falling on ground that 1959a; Ruck et al. 1970) is wet from accompanying rain, this is reasonable as- sumption even if the vegetation was previously dry. ⑀ ϭ 2.5(1 Ϫ f) ϩ f⑀ , (A9) ␶ ␶w Wetting of the ground will, in general, increase the mag- where ⑀␶w is the dielectric constant for water taken here nitude of the cross sections of the ground (Ulaby and as 77.90 ϩ i13.24 and f is the fraction of water by Dobson 1989).

Unauthenticated | Downloaded 09/29/21 10:53 PM UTC JANUARY 2000 HUBBERT AND BRINGI 61 b. Very rough surface model The composite rough surface model is a simple addition of the above two models. As the surface roughness increases with respect to wavelength, the scattered ®eld becomes more incoher- ent. A specular-point model is employed in which it is REFERENCES assumed that scattered ®eld results from areas that spec- ularly re¯ect the incident wave. This is also referred to Aydin, K., and Y. Zhao, 1990: A computational study of polarimetric as an optic approach (Ruck et al. 1970). The bistatic radar observables in hail. IEEE Trans. Geosci. Remote Sens., scattering amplitudes are 28, 412±421. , A. Seliga, and V. N. Bringi, 1984: Differential radar scattering S ϭ ␤ J, (A10) properties of model hail and mixed phase hydrometeors. Radio pq pq Sci., 19, 58±66. where p, q refer to polarization states, ␤ is proportional Bahar, E., and Y. Zhang, 1996: A new uni®ed full wave approach to to the scattering matrix element, and J is de®ned as evaluate the scatter cross sections of composite rough surfaces. IEEE Trans. Geosci. Remote Sens., 34, 973±980. 2 ␰ 22ϩ ␰ Balanis, C. A., 1982: Antenna Theory: Analysis and Design. Harper J ϭ exp Ϫ xy, (A11) and Row, 790 pp. w␰ 2w␰ 2 Bohren, C. F., and L. J. Batten, 1982: Radar backscattering of mi- []z crowaves by spongy ice spheres. J. Atmos. Sci., 39, 2623±2628. where w 2 ϭ 4h 2/l2, h 2 is the mean-square roughness Bringi, V. N., T. A. Seliga, and K. Aydin, 1984: Hail detection using height and l is the surface correlation length. The terms a differential re¯ectivity radar. Science, 225, 1145±1147. ␰ are de®ned in Eqs. (A3) and (A4), respectively, and , V. Chandrasekar, N. Balakrishnan, and D. S. ZrnicÂ, 1990: An x,y examination of propagation effects in rainfall on radar mea- ␰ ϭϪcos␪ Ϫ cos␪ . (A12) surements at microwave frequencies. J. Atmos. Oceanic Tech- z i s nol., 7, 829±840. The scattering matrix elements for the H±V basis are Carey, L. D., S. A. Rutledge, and T. D. Keenan, 1997: Correction of attenuation and differential attenuation at S- and C-band using 2 differential propagation phase. Preprints, 28th Int. Conf. on Ra- aaR(␫) ϩ sin␪ sin␪ sin ␾ R⊥(␫) 23 ࿣ is s dar Meteorology, Austin, TX., Amer. Meteor. Soc., 25±26. ␤␷␷ ϭ (A13) aa14 Hubbert, J., V. N. Bringi, L. D. Carey, and S. Bolen, 1998: CSU± CHILL polarimetric radar measurements in a severe hailstorm Ϫ sin␪i aR3 ࿣(␫) ϩ sin␪saR2 ⊥(␫) in eastern Colorado. J. Appl. Meteor., 37, 749±775. ␤h␷ ϭ sin␾s (A14) Peake, W. H., 1959a: The interaction of electromagnetic waves with aa14 some natural surfaces. Tech. Rep. 898-2, Antenna Laboratory, Ohio State University, 110 pp. [Available from Electroscience sin␪saR2 ࿣(␫) Ϫ sin␪i aR3 ⊥(␫) Laboratory, Ohio State University, 1320 Kinnear Rd., Columbus, ␤ ϭ sin␾ (A15) ␷hs aa OH 43212.] 14 , 1959b: Theory of radar return from terrain. IRE National Con- vention Record 7, Part 1, 27±41. Ϫ sin␪ sin␪ sin␾ R (␫) Ϫ aaR⊥(␫) is s࿣ 23 Rasmussen, R. M., and A. J. Heyms®eld, 1987: Melting and shedding ␤hh ϭ , (A16) aa14 of graupel and hail. Part I: Model physics. J. Atmos. Sci., 44, 2754±2763. where R⊥(␫), R࿣(␫) are Fresnel re¯ection coef®cients Rice, S. O., 1965: Re¯ection of electromagnectic waves by a slightly rough surface. The Theory of Electromagnetic Waves, M. Kline, 2 ⑀rrcos␫ Ϫ ͙⑀ Ϫ sin ␫ Ed., Dover, 351±378. R࿣(␫) ϭ (A17) Ruck, G., D. Barrick, W. Stuart, and C. Krichbaum, 1970: Radar 2 ⑀rrcos␫ ϩ ͙⑀ Ϫ sin ␫ Cross Section Handbook. Vol. 2. Plenum Press, 949 pp. Ryzhkov, A. V., and D. S. ZrnicÂ, 1995: Precipitation and attenuation 2 cos␫ Ϫ ͙⑀r Ϫ sin ␫ measurenents at a 10-cm wavelength. J. Appl. Meteor., 34, 2121± R⊥(␫) ϭ , (A18) 2134. 2 cos␫ ϩ ͙⑀r Ϫ sin ␫ Ulaby, F. T., and M. C. Dobson, 1989: Handbook of Radar Scattering Statistics for Terrain. Artech House, 357 pp. with ⑀␶ being the relative permittivity (the relative per- , and J. J. van Zyl, 1990: Scattering matrix representation for meability has been assumed to be unity). The angles of simple targets. Radar Polarimetry for Geoscience Applications, incidence argument ␫ for the Fresnel coef®cients and F. T. Ulaby and C. Elachi, Eds., Artech House, 1±52. the other quantities are de®ned as , T. E. Van Deventer, J. R. East, T. F.Haddock, and M. E. Coluzzi, 1988: Millimeter-wave bistatic scattering from the ground and 1 vegetation targets. IEEE Trans. Geosci. Remote Sens., 26, 229± cos␫ ϭ ͙1 Ϫ sin␪issin␪ cos␾ sϩ cos␪ icos␪ s(A19) 243. ͙2 Zahrai, A., and D. ZrnicÂ, 1997: Implementation of polarimetric ca- pability for the WSR-88D (NEXRAD) radar. Preprints, 28th In- a1 ϭ 1 ϩ sin␪issin␪ cos␾ sϪ cos␪ icos␪ s (A20) ter. Conf. on Radar Meteorology, Austin, TX, Amer. Meteor. Soc., 284±285. a2 ϭ cos␪issin␪ ϩ sin␪ icos␪ scos␾ s (A21) ZrnicÂ, D. S., 1987: Three-body scattering produces precipitation sig- nature of special diagnostic value. Radio Sci., 22, 76±86. a3 ϭ sin␪iscos␪ ϩ cos␪ isssin␪ cos␾ (A22) , V. N. Bringi, N. Balakrishnan, K. Aydin, V. Chandrasekar, and J. Hubbert, 1993: Polarimetric measurements in severe hail- a4 ϭ cos␪isϩ cos␪ . (A23) storm. Mon. Wea. Rev., 121, 2223±2238.

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