Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering

Abstract

A linear digital filtering algorithm is presented for rapid and accurate numerical evaluation of Hankel transform integrals of orders 0 and 1 containing related complex kernel functions. The kernel for Hankel transforms is defined as the non‐Bessel function factor of the integrand. Related transforms are defined as transforms, of either order 0 or 1, whose kernel functions are related to one another by simple algebraic relationships. Previously saved kernel evaluations are used in the algorithm to obtain rapidly either order transform following an initial convolution operation. Each order filter is designed with identical abscissas over a large range so that an adaptive convolution procedure can be applied to a large class of kernels. Different order Hankel transforms with related kernels are often found in electromagnetic (EM) applications. Because of the general nature of this algorithm, the need to design new filters should not be necessary for most applications. Accuracy of the filters is comparable to that of single‐precision numerical quadrature methods, provided well‐behaved kernels and moderate values of the transform argument are used. Filtering errors of less than 0.005 percent are demonstrated numerically using known analytical Hankel transform pairs. The digital filter accuracy is also illustrated by comparison with other published filters for computing the apparent resistivity for a Schlumberger array over a horizontally layered earth model. The algorithm is written in Fortran IV and is listed in the Appendix along with a test driver program. Detailed comments are included to define sufficiently all calling parameter requirements.