Development of a discrete numerical approach to the Fredholm Integral Method for evaluating microwave scattering by irregular hydrometeors

A new approach to the implementation of Fredholm Integral Method (FIM) was developed to evaluate microwave scattering by irregular hydrometeors in the melting layer where snowflakes aggregate. These contain air, ice and liquid water and therefore complex to model. In this study, the particles were modelled discretizing their volume, filling it with cubic or spherical cells according to their weighted contents. The FIM presented represents a departure from earlier work where the numerical integration is no longer based on expansion in a set of polynomials but based on direct spatial integration. The strength of our approach is that the computations are performed in the spatial frequency domain. As a result, the angular scattering pattern is strongly connected to the Spatial Fourier Transform of the scatterer; hence, for electrically small particles the angular spectrum is relatively smooth and the number of pivots required for integration is relatively low. The theoretical analysis of the first Born term is comparatively simple. Comparisons show a good agreement between the first Born term using our approach and the exact method by Holt. However, the theory of the second Born term is relatively difficult. The approach taken by Hankel cannot be applied essentially because of the power of p in the integrand being odd. An alternative approach which still involves contour integration method uses quandrantal contour in combination with a conditioning weighting function to control the magnitude of the integrand. The numerical evaluation of the scattering functions are performed and compared. The results suggested similar pattern in comparison with the Mie theory and other established numerical algorithms for homogeneous spherical or ellipsoidal dielectric scatterers. This technique has a good potential to be applied to irregular hydrometeors since only the distribution of the dielectric constants need to be changed.