Applications of the conjugate gradient fast Fourier Hankel transfer method with an improved fast Hankel transform algorithm

Abstract

The conjugate gradient fast Fourier‐Hankel transforms (CG‐FFHT) method was recently proposed to solve the problems of electromagnetic wave propagation and scattering in axisymmetric inhomogeneous media. This new technique uses the CG method together with the FFHT to solve the wave equation iteratively. Each iteration of the CG method requires O(N log2 N) complex multiplications (N is the number of unknowns). For the application of low‐frequency induction logging, the number of iterations is very small (less than eight). Furthermore, the CG‐FFHT method only requires the storage of several vectors of dimension N. In this paper we present an improved fast Hankel transform (FHT) algorithm as well as some applications of the CG‐FFHT method. It is shown that the improved FHT algorithm results in better accuracy and is more efficient than the other FHT algorithms. Moreover, with this FHT algorithm there is no need to pad the function to be transformed with zeros. Several numerical examples will be shown to illustrate the use of the improved FHT algorithm as well as the applications of the CG‐FFHT method.