Numerical integration of related Hankel transforms by quadrature and continued fraction expansion

Abstract

An algorithm is presented for the accurate evaluation of Hankel (or Bessel) transforms of algebraically related kernel functions, defined here as the non‐Bessel function portion of the integrand, that is more widely applicable than the standard digital filter methods without enormous increases in computational burden. The algorithm performs the automatic integration of the product of the kernel and Bessel functions between the asymptotic zero crossings of the latter and sums the series of partial integrations using a continued fraction expansion, equivalent to an analytic continuation of the series. The integrands may be saved to allow the rapid computation of related transforms without recalculating the kernel or Bessel functions, and the integration steps use interlacing quadrature formulas so that no function evaluations are wasted when it is necessary to increase the order of the quadrature rule. The continued fraction algorithm allows very slowly divergent or even formally divergent integrals to be computed quite easily. The local error is controlled at each step in the algorithm, and accuracy is limited largely by machine resolution. The algorithm is written in Fortran and is listed in an Appendix along with a driver program that illustrates its features. The driver program and subroutine are available from the SEG Business Office.