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For intermediate relative particle size, x of ~20 to ~60, and for even larger particles there are a number of physical optics methods that show promise in calculation of light scattering by nonspherical and/or inhomogeneous particles.
The method of improved geometric optics (IGO) (Zhang Zhibo et al 2004, Yang P and Liou 1996) has been shown to converge to the FDTD method (see also Scattering calculations for nonspherical/inhomogeneous particles: "Exact" methods) at x ~20 for the attenuation efficiency and absorption efficiency (Yang P and Liou 1996) and the p11 element of the phase matrix (Yang P and Liou 1995) assuming the hexagonal ice column as the particle shape.
The physical optics method described by Muinonen K (2003, 1989) could also be applied to ice crystals for 25 < x < 200. The physical optics method combines ray-tracing with diffraction at the particle outline (edge diffraction) which introduces some size dependency. The complete size dependency can be obtained by solving the Kirchhoff integral. However, this is computationally very expensive. More recently, a very efficient edge diffraction method proposed by Hesse E and Ulanowski (2003) has shown good agreement with SVM in terms of the phase function (the p11 element of the scattering phase matrix) and the degree of linear polarization for an x of 50 and 100 in particular orientations (Hesse E and Ulanowski 2003, Hesse E et al 2003).
For large relative particle size, x > ~60, ray-tracing methods have been applied to randomly-shaped particles (Muinonen K et al 1996) as well as to crystalline particles (Macke A et al 1996, Takano and Liou 1995, 1989, Wendling P et al 1979).
An approximation of Mie theory known as the anomalous diffraction approximation (ADA), developed by van de Hulst (1981) for homogeneous spheres, has been extensively used to calculate the attenuation and absorption properties of naturally occuring particles, such as cloud droplets and ice crystals (for example, Sun W and Fu 2001, Ackerman SA and Stephens 1987), as well as phytoplankton cells (for example, Bricaud A and Morel 1986). Note that errors of this approximation may reach 30% in the case of water droplets in air (refractive index, m' ~1.33). ADA may be improved for spheres and circular cylinders by relating the attenuation cross section and absorption cross-section to the internal field inside the particle (Yang P et al 2004). Quirantes A (1997) extended ADA to include the coated sphere.
The processes of wave optical resonance, refraction, and internal reflection have been parameterized and incorporated into a simplified version of ADA based on the effective photon path (proportional to a ratio of the particle mass-to-projected area) with errors relative to Mie theory of less than 10% (Mitchell DL 2000). This modified ADA (MADA) applies to ice particles of any shape having a relative size of 0.5 < x < 1000 by using power law expressions relating particle mass and area to the maximum dimension. It agrees with the FDTD and T-matrix methods to within 15% for all ice particle shapes addressed by these methods (Mitchell DL et al 2006) and agrees with laboratory measurements of the attenuation of light by an ice cloud of hexagonal columns to within 3% on average over a wavelength range of 2-17 µm (Mitchell et al 2006, 2001). MADA treats both single particles and a dispersion, as characterized by the size distribution, analytically. This makes MADA an extremely efficient method computationally.
See also Scattering calculations for nonspherical/inhomogeneous particles: "Exact" methods.
| CITATION: Mitchell D. L., Baran A. J. 2007. Scattering calculations for nonspherical/inhomogeneous particles: Approximate methods (www.tpdsci.com/Tpc/ScaCalcMetNspApx.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 02-Jun-2007 Modified: 02-Jun-2007 Reviewed: PENDING |
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