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Comparison between the lab observations and DDA computations on the backscattering features of sphere–cone–oblate ice particles
Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
www.elsevier.com/locate/jqsrt
Note
Comparison between the lab observations and DDA computations on the backscattering features of sphere–cone–oblate ice particles Wang Zhenhuia;∗ , Xu Xiaoyonga , Wang Qing-anb , Chao Zengmingc a
Department of Electronic Engineering, Nanjing Institute of Meteorology, Nanjing 210044, China b Department of Atmospheric Science, Nanjing University, Nanjing 210093, China c China National Space Industrial Corporation, Beijing 100854, China Received 16 April 2002; accepted 1 July 2002
Abstract Twenty sphere–cone–oblate ice particles are simulated for their backscattering observations in a microwave lab and calculations are done with the DDA method. The lab experiment and the DDA theory are brie7y presented in this paper. The theoretical results are compared with the lab observations. It is shown that the theoretical results are consistent with the experimental data in general and that the backscattering ability of sphere–cone–oblate ice particles increases as the particle size parameter increases, and 7uctuates in the so-called resonance region. The backscattering cross sections of sphere–cone–oblate ice particles depend not only on their sizes equivolume spherical radii Re but also on their shapes. There is a statistically linear relationship between di:erential re7ectivity ZDR and the shape factor, 2a=h, which is the ratio of the horizontal scale to the vertical scale of a particle. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sphere–cone-oblate; Backscattering; Lab observations; DDA computations
1. Introduction Most of the hailstones are nonspherical. The typical shape is conical during the formation of precipitation and hail [1]. It was reported that 80% of all large hailstones collected in Switzerland contained embryos of conical graupel [2]. Therefore, examining the backscattering cross section ∗
Corresponding author. Fax: +86-25-873-1195. E-mail address: [email protected] (W. Zhenhui).
0022-4073/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 1 6 5 - 6
456
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
behavior of such ice particles in microwave Jeld could possibly lead to new understanding of hydrometeor evolution and new approach to their detection. An initial attempt at solving this complex problem was begun by Aydin et al. [3] who made computations with the T-matrix method (TMM) of backscattering parameters for ice-phase hydrometers having di:erent densities, water contents, water coatings, shapes and sizes. In this work, the backscattering cross sections of a series of sphere–cone–oblate ice particles are experimentally measured within a microwave lab, and are theoretically computed with the discrete dipole approximation (DDA) algorithm [4]. The results from this research are useful not only for the interpretation of signals from hail-detecting radar but also for the assessment of the DDA algorithm and the lab system.
2. Conical model and lab measurement A sphere–cone–oblate spheroidal model was chosen for the lab observations and DDA calculations. Fig. 1 illustrates the features of this model. In the Fig. 1 is the cone angle, h is the overall height, a and b are the semi-major and semi-minor axes of the oblate spheroid section, respectively, and c is the radius of the apex sphere. Table 1 gives the values of the parameters of the sphere–cone–oblate series chosen for this study. Re is the equivolume spherical radius and 2a=h is deJned as the ratio of the horizontal scale to the vertical scale of a particle. The experiment was carried out in the Electromagnetic Scattering Laboratory of China National Space Industrial Corporation [5]. The background radiation from the environment has been taken into account during observational data processing. The radar unit of the system can make frequency scanning. The rotation axis of the target is vertical, perpendicular to the propagation direction of the incident electromagnetic wave. The system measures both the H and V , which are backscattering cross sections of a target for horizontally and vertically polarized incident waves, respectively.
Fig. 1. Geometry of a sphere–cone–oblate particle.
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
457
Table 1 Sizes of sphere–cone–oblate particles Index
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Re (mm) Deg 2a (mm) b (mm) h (mm) c (mm) 2a=h
0.94 70 2.2 0.5 1.8 0.47 1.22
1.10 90 2.6 0.7 1.8 0.91 1.44
1.78 75 3.9 1.3 3.8 0.55 1.03
1.90 90 4.5 1.2 3.2 1.33 1.41
2.04 69 4.6 1.2 4.4 0.53 1.05
2.64 67 6.0 1.5 5.5 0.95 1.09
2.74 65 5.8 2.0 6.2 0.90 0.94
3.02 69 6.9 1.7 6.4 0.78 1.08
3.65 72 8.5 2.0 7.2 1.40 1.18
3.88 69 8.2 3.0 8.4 1.67 0.98
4.00 68 9.0 2.4 8.8 0.88 1.02
5.65 66 12.4 3.7 12.9 1.24 0.96
5.71 82 13.0 3.8 11.0 2.27 1.18
5.87 59 11.9 4.8 13.5 2.77 0.88
7.66 65 17.2 4.4 16.9 1.97 1.02
7.92 81 18.6 4.7 14.3 4.19 1.30
7.95 61 16.1 6.6 18.8 3.07 0.86
9.60 65 21.2 6.0 20.4 3.82 1.04
9.91 63 20.1 8.4 22.4 4.84 0.90
10.07 87 23.2 6.7 18.0 5.83 1.29
Since ice particles melt at normal temperature, substitutes have been used for the lab experiment. A kind of pasty material was so produced with chemical compounding technique that the dielectric constant after the material is solidiJed is very close to the dielectric constant of ice. The paste was poured into a metal, shelly mould and then was put into an oven for a few hours for solidiJcation. The dielectric constant of the material after the mould was taken o: has been measured in a microwave guide and the result shows that the of the material is 3.14 – 0.004j, very close to ice. 3. Theoretical computation To compute the absorption and scattering cross sections of non-spherical particles, techniques such as discrete dipole approximation, T-matrix method [6–9] and variable separation method [10,11] can be used. In this study, the discrete √ dipole approximation is chosen because it works well for materials with |m − 1| 6 3, where m = is the refractive index [5]. The discrete dipole approximation for computing scattering and absorption of a particle in arbitrary shape was introduced by Purcell and Pennypacker [12] and has undergone a number of theoretical developments for applications since then [13–18,5]. The DDA takes a particle as an array of many small dipoles and the dipole dimension is suOciently small as compared with the wavelength. Every dipole is activated because of both the incident wave and the electric Jeld induced by all other dipoles. The self-consistent solution for the dipole polarizations can be obtained from a set of coupled linear equations. Because the number of the dipoles is often very large, iterative methods rather than direct methods for solving the equations are used. A Jrst guess for the solution is given and then iteratively modiJed until the modiJcation is less than a pre-deJned criterion . In this study, = 10−5 is chosen for the sphere–cone–oblate spheroid and N is determined based on the shape parameters of a particle [16,5]. According to Aydin et al. [3], the di:erential re7ectivity ZDR is deJned as ZDR = 10log( H = V ); dB; where H and V are the backscattering cross sections at horizontal and vertical polarizations, respectively.
458
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
4. Results The lab data and the computational results for sphere–cone–oblate ice particles at wavelengths of 3.2, 5.6 and 10:7 cm are shown in Fig. 2. The backscattering cross sections in the Jgure have been normalized according to qH = H =R2e and qV = V =R2e . In Fig. 2, qH; 3:2; c and qH; 3:2; o stand for the computed and observed results, respectively, at 3:2 cm for horizontal polarization. Other symbols represent the results correspondingly. Fig. 2 indicates that the observed data are generally consistent with the DDA computation. Thereby, the experimental results are reliable and the DDA is an eOcient tool for computing microwave scattering by sphere–cone–oblate ice particles. As is shown in Fig. 2, the backscattering cross sections at both polarizations decrease as the wavelength increases and increase with the equivolume radius for small particles. That is to say, the backscattering ability increases with the particle size parameter deJned by 2Re =, where is the wavelength. As the size parameter increases further, the backscattering cross sections would 7uctuate. For = 3:2 cm, the resonance occurs when the radii of particles are greater than 7:6 mm. This resonance behavior for a sphere–cone–oblate particle is similar to that for a sphere [18]. For longer wavelengths such as 5.6 and 10:7 cm, the resonance appears at larger radius exceeding the largest particle in this study. The di:erence between lab data and theoretical results for large particles is larger than that for small particles. For example, when Re ¿ 7:9 mm, the di:erence between qV; 3; 2; o and qV; 3; 2; c is quite large (see Fig. 2b). This is possibly because of the large errors in both the DDA computation and the lab observation due to the backscattering 7uctuation in the resonance area. The di:erences between the lab data and theoretical results for the 3 cases such as = 3:2 cm and Re = 2:64 mm; = 5:6 cm and Re = 2:74 mm, and = 10:7 cm and Re = 3:88 mm are also outstanding. Though the reason for this is still under investigation, the reliability of the lab data for the three points are suspected according to the gradually increasing features of the backscattering cross section as a function of the size parameter. Both qH and qV are in the same order in amplitude, but the di:erence in their variation with Re indicates the e:ect of the particle’s shape on the polarization. Take for example the two particles with radii of 5.65 and 5:71 mm, which have been labeled in Table 1 as 12 and 13, respectively. At all the 3 wavelengths, the qH for 12 is less than that for 13, while the qV for 12 is greater than that for 13. Though the radius of 13 is greater than 12, the cone angle is also greater. Therefore, the horizontal scale increases while the vertical scale decreases. Similar case also happens to the oblate pairs such as 1.78 and 1:9 mm, 7.66 and 7:92 mm, and 9.91 and 10:07 mm. It is indicated that the backscattering cross sections of sphere–cone–oblate ice particles depend not only on their equivolume radii Re but also on their shapes. 5. ZDR –2a=h correlation analysis The ZDR –2a=h correlation analysis is performed to show the relationship between the di:erential re7ectivity ZDR and the particle’s shape factor 2a=h. The results from the ZDR –2a=h correlation analysis, based on the lab-observed data and the DDA-computed results, are shown in Fig. 3.
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
459
Fig. 2. Normalized backscattering cross sections of sphere–cone–oblate ice particles as a function of equivolume radius Re . Panels a and b are for H- and V-polarized incident waves, respectively.
460
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
Fig. 3. Di:erential re7ectivity ZDR of sphere–cone–oblate ice particles at three wavelengths as a function of 2a=h. Panels a and b are for observed and computed results, respectively. “R” stands for linear correlation coeOcient.
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
461
The ZDR depends on many factors, but ZDR statistically increases with 2a=h. For all the three wavelengths, the regression lines of ZDR all cut through the abscissa axis at around 2a=h = 1. It is indicated that ZDR is greater than 0 dB when 2a=h ¿ 1 and ZDR ¡ 0 dB when 2a=h ¡ 1. This feature is the scientiJc basis for detecting the shape of hailstones with a dual-polarized weather radar. Fig. 3b based on the DDA-computed data indicates that the ZDR –2a=h correlation coeOcients, R, are 0.96 and 0.98 for = 5:6 and 10:7 cm, respectively, but R is as small as 0.38 at 3:2 cm. This is again mainly because of the complicated resonance features of backscattering cross sections. Suppose that the four particles, whose 2a=h values are 0.90, 1.02, 1.29 and 1.30, (corresponding Re are 9.91, 7.66, 7.92 and 10:07 mm, respectively) can be discarded from the sample, as shown in Fig. 3b. Then a linear regression analysis between ZDR and 2a=h would show that the R could be as great as 0.96. Fig. 3a is the result from regression analysis on the observed data. It can be seen that for all the three wavelengths, the correlation coeOcients are 0.35, 0.62, and 0.53, respectively, all less than the corresponding numbers shown in Fig. 3b. This implies that the observed data are more discrete than the computed results. 6. Conclusion In this paper, the backscattering features of sphere–cone–oblate ice particles are examined with lab observations and DDA calculations. It has been shown that the theoretical results are generally consistent with the experimental data. Thereby, it can be roughly said that the experimental results are reliable and the DDA is an eOcient tool for computing microwave scattering by sphere–cone–oblate spheroids. Analyses on theoretical and experimental data indicate that the backscattering ability of sphere– cone–oblate ice particles increases as the particle size parameter increases, and 7uctuates in the so-called resonance region. The backscattering cross sections of sphere–cone–oblate ice particles depend not only on their sizes but also on their shapes. Statistically, there is a linear relationship between the di:erential re7ectivity ZDR and the shape factor 2a=h. For all the three wavelengths speciJed, the regression lines of ZDR all intersect the abscissa axis at around 2a=h = 1, implying that ZDR ¿ 0 dB when 2a=h ¿ 1 and ZDR ¡ 0 dB when 2a=h ¡ 1. Acknowledgements Dr. Draine has kindly provided the DDA codes for computation. This study is supported by the National Natural Science Foundation of China under the Grant 49875007. References [1] Mason BJ. The physics of cloud. Oxford: Clarendon Press, 1971. p. 671. [2] List R. Growth and structure of graupel and hailstones. Phys Precipitation Geophys Monogr 1960;5:317. [3] Aydin K, Seliga TA, Bringi VN. Di:erential radar scattering properties of model and mixed-phase hydrometeors. Radio Sci 1984;19(1):58–66.
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Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
[4] Draine BT. The discrete dipole approximation for light scattering by irregular targets. In: Mishchenko MI, Hovenier JW, Travis LD, editors. Light scattering by nonspherical particles: theory, measurements, and applications. New York: Academic Press, 1999. p. 131–45. [5] Wang Qing’an, Ouyang Zixiang, Liu Liping, Chao Zengming. A comparative study on the backscattering ability of raindrops and ice particles (hail). Contrib Atmos Phys 1998;71(4):377–86. [6] Barber P, Yeh C. Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies. Appl Opt 1975;14: 2864–72. [7] Barber P, Hill SC. Light scattering by particles: computational methods. Singapore: World ScientiJc Publishing, 1990. p. 79 –98. [8] Mishchenko MI. Light scattering by randomly oriented axially symmetric particles. J Opt Soc Am A 1991;8: 871–82. [9] Mishchenko MI, Travis LD. Capabilities and limitations of a current Fortran implementation of the T-matrix method for randomly oriented rotationally symmetric scatterers. JQSRT 1998;60(3):309–24. [10] Asano S, Yamamoto. Light scattering by a spheroidal particle. Appl Opt 1975;14:29 – 49. [11] Asano S. Light scattering properties of spheroidal particles. Appl Opt 1979;18:712–23. [12] Purcell EM, Pennypacker CR. Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 1973;186:705–14. [13] Draine BT. The discrete-dipole approximation and its application to interstellar graphite grains. Astrophys J 1988;333:848–72. [14] Goodman JJ, Draine BT, Flatau PJ. Application of fast-Fourier-transform techniques to the discrete dipole approximation. Opt Lett 1991;16:1198–200. [15] Draine BT, Goodman J. Beyond Clausius–Mossotti wave propagation on a polarizable point lattice and the discrete dipole approximation. Astrophys J 1993;405:685–93. [16] Draine BT, Flatau PJ. Discrete-dipole approximation for scattering calculations. J Opt Soc Am A 1994;11:1491–9. [17] Hovenier JW, Lumme K, Mishchenko MI. Computations of scattering matrices of four types of non-spherical particles using diverse methods. JQSRT 1996;l55(6):695–705. [18] Liu Chunlei, Illingworth AJ. Error analysis of backscattering from discrete dipole approximation for di:erent ice particle shapes. Atmos Res 1997;44:231–41.
www.elsevier.com/locate/jqsrt
Note
Comparison between the lab observations and DDA computations on the backscattering features of sphere–cone–oblate ice particles Wang Zhenhuia;∗ , Xu Xiaoyonga , Wang Qing-anb , Chao Zengmingc a
Department of Electronic Engineering, Nanjing Institute of Meteorology, Nanjing 210044, China b Department of Atmospheric Science, Nanjing University, Nanjing 210093, China c China National Space Industrial Corporation, Beijing 100854, China Received 16 April 2002; accepted 1 July 2002
Abstract Twenty sphere–cone–oblate ice particles are simulated for their backscattering observations in a microwave lab and calculations are done with the DDA method. The lab experiment and the DDA theory are brie7y presented in this paper. The theoretical results are compared with the lab observations. It is shown that the theoretical results are consistent with the experimental data in general and that the backscattering ability of sphere–cone–oblate ice particles increases as the particle size parameter increases, and 7uctuates in the so-called resonance region. The backscattering cross sections of sphere–cone–oblate ice particles depend not only on their sizes equivolume spherical radii Re but also on their shapes. There is a statistically linear relationship between di:erential re7ectivity ZDR and the shape factor, 2a=h, which is the ratio of the horizontal scale to the vertical scale of a particle. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sphere–cone-oblate; Backscattering; Lab observations; DDA computations
1. Introduction Most of the hailstones are nonspherical. The typical shape is conical during the formation of precipitation and hail [1]. It was reported that 80% of all large hailstones collected in Switzerland contained embryos of conical graupel [2]. Therefore, examining the backscattering cross section ∗
Corresponding author. Fax: +86-25-873-1195. E-mail address: [email protected] (W. Zhenhui).
0022-4073/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 2 ) 0 0 1 6 5 - 6
456
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
behavior of such ice particles in microwave Jeld could possibly lead to new understanding of hydrometeor evolution and new approach to their detection. An initial attempt at solving this complex problem was begun by Aydin et al. [3] who made computations with the T-matrix method (TMM) of backscattering parameters for ice-phase hydrometers having di:erent densities, water contents, water coatings, shapes and sizes. In this work, the backscattering cross sections of a series of sphere–cone–oblate ice particles are experimentally measured within a microwave lab, and are theoretically computed with the discrete dipole approximation (DDA) algorithm [4]. The results from this research are useful not only for the interpretation of signals from hail-detecting radar but also for the assessment of the DDA algorithm and the lab system.
2. Conical model and lab measurement A sphere–cone–oblate spheroidal model was chosen for the lab observations and DDA calculations. Fig. 1 illustrates the features of this model. In the Fig. 1 is the cone angle, h is the overall height, a and b are the semi-major and semi-minor axes of the oblate spheroid section, respectively, and c is the radius of the apex sphere. Table 1 gives the values of the parameters of the sphere–cone–oblate series chosen for this study. Re is the equivolume spherical radius and 2a=h is deJned as the ratio of the horizontal scale to the vertical scale of a particle. The experiment was carried out in the Electromagnetic Scattering Laboratory of China National Space Industrial Corporation [5]. The background radiation from the environment has been taken into account during observational data processing. The radar unit of the system can make frequency scanning. The rotation axis of the target is vertical, perpendicular to the propagation direction of the incident electromagnetic wave. The system measures both the H and V , which are backscattering cross sections of a target for horizontally and vertically polarized incident waves, respectively.
Fig. 1. Geometry of a sphere–cone–oblate particle.
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
457
Table 1 Sizes of sphere–cone–oblate particles Index
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Re (mm) Deg 2a (mm) b (mm) h (mm) c (mm) 2a=h
0.94 70 2.2 0.5 1.8 0.47 1.22
1.10 90 2.6 0.7 1.8 0.91 1.44
1.78 75 3.9 1.3 3.8 0.55 1.03
1.90 90 4.5 1.2 3.2 1.33 1.41
2.04 69 4.6 1.2 4.4 0.53 1.05
2.64 67 6.0 1.5 5.5 0.95 1.09
2.74 65 5.8 2.0 6.2 0.90 0.94
3.02 69 6.9 1.7 6.4 0.78 1.08
3.65 72 8.5 2.0 7.2 1.40 1.18
3.88 69 8.2 3.0 8.4 1.67 0.98
4.00 68 9.0 2.4 8.8 0.88 1.02
5.65 66 12.4 3.7 12.9 1.24 0.96
5.71 82 13.0 3.8 11.0 2.27 1.18
5.87 59 11.9 4.8 13.5 2.77 0.88
7.66 65 17.2 4.4 16.9 1.97 1.02
7.92 81 18.6 4.7 14.3 4.19 1.30
7.95 61 16.1 6.6 18.8 3.07 0.86
9.60 65 21.2 6.0 20.4 3.82 1.04
9.91 63 20.1 8.4 22.4 4.84 0.90
10.07 87 23.2 6.7 18.0 5.83 1.29
Since ice particles melt at normal temperature, substitutes have been used for the lab experiment. A kind of pasty material was so produced with chemical compounding technique that the dielectric constant after the material is solidiJed is very close to the dielectric constant of ice. The paste was poured into a metal, shelly mould and then was put into an oven for a few hours for solidiJcation. The dielectric constant of the material after the mould was taken o: has been measured in a microwave guide and the result shows that the of the material is 3.14 – 0.004j, very close to ice. 3. Theoretical computation To compute the absorption and scattering cross sections of non-spherical particles, techniques such as discrete dipole approximation, T-matrix method [6–9] and variable separation method [10,11] can be used. In this study, the discrete √ dipole approximation is chosen because it works well for materials with |m − 1| 6 3, where m = is the refractive index [5]. The discrete dipole approximation for computing scattering and absorption of a particle in arbitrary shape was introduced by Purcell and Pennypacker [12] and has undergone a number of theoretical developments for applications since then [13–18,5]. The DDA takes a particle as an array of many small dipoles and the dipole dimension is suOciently small as compared with the wavelength. Every dipole is activated because of both the incident wave and the electric Jeld induced by all other dipoles. The self-consistent solution for the dipole polarizations can be obtained from a set of coupled linear equations. Because the number of the dipoles is often very large, iterative methods rather than direct methods for solving the equations are used. A Jrst guess for the solution is given and then iteratively modiJed until the modiJcation is less than a pre-deJned criterion . In this study, = 10−5 is chosen for the sphere–cone–oblate spheroid and N is determined based on the shape parameters of a particle [16,5]. According to Aydin et al. [3], the di:erential re7ectivity ZDR is deJned as ZDR = 10log( H = V ); dB; where H and V are the backscattering cross sections at horizontal and vertical polarizations, respectively.
458
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
4. Results The lab data and the computational results for sphere–cone–oblate ice particles at wavelengths of 3.2, 5.6 and 10:7 cm are shown in Fig. 2. The backscattering cross sections in the Jgure have been normalized according to qH = H =R2e and qV = V =R2e . In Fig. 2, qH; 3:2; c and qH; 3:2; o stand for the computed and observed results, respectively, at 3:2 cm for horizontal polarization. Other symbols represent the results correspondingly. Fig. 2 indicates that the observed data are generally consistent with the DDA computation. Thereby, the experimental results are reliable and the DDA is an eOcient tool for computing microwave scattering by sphere–cone–oblate ice particles. As is shown in Fig. 2, the backscattering cross sections at both polarizations decrease as the wavelength increases and increase with the equivolume radius for small particles. That is to say, the backscattering ability increases with the particle size parameter deJned by 2Re =, where is the wavelength. As the size parameter increases further, the backscattering cross sections would 7uctuate. For = 3:2 cm, the resonance occurs when the radii of particles are greater than 7:6 mm. This resonance behavior for a sphere–cone–oblate particle is similar to that for a sphere [18]. For longer wavelengths such as 5.6 and 10:7 cm, the resonance appears at larger radius exceeding the largest particle in this study. The di:erence between lab data and theoretical results for large particles is larger than that for small particles. For example, when Re ¿ 7:9 mm, the di:erence between qV; 3; 2; o and qV; 3; 2; c is quite large (see Fig. 2b). This is possibly because of the large errors in both the DDA computation and the lab observation due to the backscattering 7uctuation in the resonance area. The di:erences between the lab data and theoretical results for the 3 cases such as = 3:2 cm and Re = 2:64 mm; = 5:6 cm and Re = 2:74 mm, and = 10:7 cm and Re = 3:88 mm are also outstanding. Though the reason for this is still under investigation, the reliability of the lab data for the three points are suspected according to the gradually increasing features of the backscattering cross section as a function of the size parameter. Both qH and qV are in the same order in amplitude, but the di:erence in their variation with Re indicates the e:ect of the particle’s shape on the polarization. Take for example the two particles with radii of 5.65 and 5:71 mm, which have been labeled in Table 1 as 12 and 13, respectively. At all the 3 wavelengths, the qH for 12 is less than that for 13, while the qV for 12 is greater than that for 13. Though the radius of 13 is greater than 12, the cone angle is also greater. Therefore, the horizontal scale increases while the vertical scale decreases. Similar case also happens to the oblate pairs such as 1.78 and 1:9 mm, 7.66 and 7:92 mm, and 9.91 and 10:07 mm. It is indicated that the backscattering cross sections of sphere–cone–oblate ice particles depend not only on their equivolume radii Re but also on their shapes. 5. ZDR –2a=h correlation analysis The ZDR –2a=h correlation analysis is performed to show the relationship between the di:erential re7ectivity ZDR and the particle’s shape factor 2a=h. The results from the ZDR –2a=h correlation analysis, based on the lab-observed data and the DDA-computed results, are shown in Fig. 3.
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
459
Fig. 2. Normalized backscattering cross sections of sphere–cone–oblate ice particles as a function of equivolume radius Re . Panels a and b are for H- and V-polarized incident waves, respectively.
460
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
Fig. 3. Di:erential re7ectivity ZDR of sphere–cone–oblate ice particles at three wavelengths as a function of 2a=h. Panels a and b are for observed and computed results, respectively. “R” stands for linear correlation coeOcient.
Wang Zhenhui et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 77 (2003) 455 – 462
461
The ZDR depends on many factors, but ZDR statistically increases with 2a=h. For all the three wavelengths, the regression lines of ZDR all cut through the abscissa axis at around 2a=h = 1. It is indicated that ZDR is greater than 0 dB when 2a=h ¿ 1 and ZDR ¡ 0 dB when 2a=h ¡ 1. This feature is the scientiJc basis for detecting the shape of hailstones with a dual-polarized weather radar. Fig. 3b based on the DDA-computed data indicates that the ZDR –2a=h correlation coeOcients, R, are 0.96 and 0.98 for = 5:6 and 10:7 cm, respectively, but R is as small as 0.38 at 3:2 cm. This is again mainly because of the complicated resonance features of backscattering cross sections. Suppose that the four particles, whose 2a=h values are 0.90, 1.02, 1.29 and 1.30, (corresponding Re are 9.91, 7.66, 7.92 and 10:07 mm, respectively) can be discarded from the sample, as shown in Fig. 3b. Then a linear regression analysis between ZDR and 2a=h would show that the R could be as great as 0.96. Fig. 3a is the result from regression analysis on the observed data. It can be seen that for all the three wavelengths, the correlation coeOcients are 0.35, 0.62, and 0.53, respectively, all less than the corresponding numbers shown in Fig. 3b. This implies that the observed data are more discrete than the computed results. 6. Conclusion In this paper, the backscattering features of sphere–cone–oblate ice particles are examined with lab observations and DDA calculations. It has been shown that the theoretical results are generally consistent with the experimental data. Thereby, it can be roughly said that the experimental results are reliable and the DDA is an eOcient tool for computing microwave scattering by sphere–cone–oblate spheroids. Analyses on theoretical and experimental data indicate that the backscattering ability of sphere– cone–oblate ice particles increases as the particle size parameter increases, and 7uctuates in the so-called resonance region. The backscattering cross sections of sphere–cone–oblate ice particles depend not only on their sizes but also on their shapes. Statistically, there is a linear relationship between the di:erential re7ectivity ZDR and the shape factor 2a=h. For all the three wavelengths speciJed, the regression lines of ZDR all intersect the abscissa axis at around 2a=h = 1, implying that ZDR ¿ 0 dB when 2a=h ¿ 1 and ZDR ¡ 0 dB when 2a=h ¡ 1. Acknowledgements Dr. Draine has kindly provided the DDA codes for computation. This study is supported by the National Natural Science Foundation of China under the Grant 49875007. References [1] Mason BJ. The physics of cloud. Oxford: Clarendon Press, 1971. p. 671. [2] List R. Growth and structure of graupel and hailstones. Phys Precipitation Geophys Monogr 1960;5:317. [3] Aydin K, Seliga TA, Bringi VN. Di:erential radar scattering properties of model and mixed-phase hydrometeors. Radio Sci 1984;19(1):58–66.
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