## The Self-Similar Rayleigh-Gans Approximation

The Self-Similar Rayleigh-Gans Approximation (SSRGA) provides a means to compute the scattering properties of aggregated ice particles and snowflakes in the microwave and millimetre parts of the spectrum. In this regime the soft sphere/spheroid approximation tends to significantly underestimate scattering, while the Discrete Dipole Approximation (DDA) is very computationally costly. The SSRGA is a fast method that computes the ensemble average scattering properties of aggregate particles and which matches DDA calculations well in the case of unrimed particles. It combines two assumptions:

• Rayleigh-Gans Approximation: the electric field within the particle can be approximated by the incident field, valid if the effective dielectric constant on the scale of the wavelength is close to one. This is also known as the Born approximation.
• Self-Similar Approximation: the process of aggregation leads to the particle having an internal structure that is fractal in nature and can be described by a power law, found to be valid for ice aggregates simulated by two completely different aggregation models. For particles larger than the wavelength, it turns out that the internal structure of ice particles is crucial for determining their scattering properties.

The theory has been developed in two papers:

• Hogan and Westbrook (2014) derived the SSRGA equation for radar backscatter and fitted the three SSRGA parameters to aggregates generated by Westbrook's aggregation model. (Hogan, R. J., and C. D. Westbrook, 2014: Equation for the microwave backscatter cross section of aggregate snowflakes using the Self-Similar Rayleigh-Gans Approximation. J. Atmos. Sci., 71, 3292-3301)
• Hogan et al. (2017) extended the method to derive the scattering and absorption cross-sections and the full phase function. They showed how the Rayleigh-Gans Approximation should be modified to account for the enhanced backscatter and absorption by non-spherical monomers, and tested the revised method using DDA calculations performed on aggregates simulated by two different models. (Hogan, R. J., R. Honeyager, J. Tyynela and S. Kneifel, 2017: Calculating the millimetre-wave scattering phase function of snowflakes using the Self-Similar Rayleigh-Gans Approximation. Q. J. R. Meteorol. Soc., in press)

Example of a simulated aggregate of bullet rosettes from the model of Westbrook et al. (2004), shown in the plane of its longest (x) dimension and shortest (z) dimension, with the intensity of the shading proportional to the amount of ice in the third (y) dimension. Simulations such as these were used to demonstrate the self-similar nature of the struture of aggregates. Taken from Hogan and Westbrook (2014).

### Download

• scatter-1.0.tar.gz (15 Dec 2016): A program to generate particle scattering properties (in the form of NetCDF files) for use in cloud, aerosol and precipitation retrieval algorithms from active and passive sensors. It includes the SSRGA algorithm as well as Mie theory and the T-matrix method. It is written a mixture of C and Fortran and is released under the terms of the Apache License, Version 2. If you are just interested in the SSRGA algorithm then you can use the ssrga.c and ssrga.h source files.
• ssrga_matlab-2.0.tar.gz (15 Dec 2016): Matlab demonstration of SSRGA, including deriving the SSRGA parameters from simulated aggregate structures. See the README file in the package for further details. This code is in the Public Domain: no copyright is asserted. Basically this means you can do what you like with it.
• ssrga_matlab-1.0.tar.gz (12 July 2014): Older version based solely on the Hogan and Westbrook (2014) paper. Running the script compare_ice_scattering_models.m produces the following images:

Comparison of the Self-Similar Rayleigh-Gans Approximation (SSRG) with several other models for the 94-GHz backscatter cross-section of aggregated ice particles and snowflakes. All models assume the Brown and Francis (1995) mass-size relationship, and two of them assume the particles to be horizontally aligned with an aspect ratio of 0.6. The SSRGA model assumes the parameters for aggregates of bullet rosettes reported by Hogan and Westbrook (2014) of κ=0.19, β=0.23 and γ=5/3. The panels are (a) backscatter cross section of ensembles of individual particles, all of the same size but with different realizations of their internal structure, and (b) the relationship between radar reflectivity factor and ice water content for the Field et al. (2005) size distributions.

### Doesn't this contradict what you said in Hogan et al. (2012)?

Hogan et al. (2012) claimed that homogeneous oblate spheroids ("soft spheroids") are a good model for scattering by irregular ice particles at millimetre wavelengths. While this is true for particles whose dimension in the direction of propagation of radiation is smaller than the wavelength, it is not true for larger particles. Note that the other results of Hogan et al. (2012) are unaffected:

• The Brown & Francis (1995) mass-size relationship produces works well in radar applications provided it is applied to the mean of the maximum and minimum particle dimensions, rather than the maximum particle dimension.
• Ice aggregates tend to have an aspect ratio of around 0.6.
• Soft spheroids work well for predicting radar parameters (particularly reflectivity factor and differential reflectivity) when the wavelength is longer than the size of the particle in the direction of propagation.

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