An efficient discrete Hankel transform based on convolutional Gaussian fast Fourier transform

Abstract The numerical evaluation of Hankel transforms is fundamental to electromagnetic (EM) geophysics. Typically, this problem is addressed via two distinct paradigms: the digital linear filter (DLF) method, prioritized for its computational speed, and direct numerical integration methods, valued for their rigorous error control. However, the accuracy and efficiency of the DLF method depend critically on the optimal selection of filter coefficients, which are intrinsically tied to the wavenumber in EM forward modeling. Moreover, since the filter coefficients are sensitive to the analytical form of the input function, the versatility of the DLF method is inherently limited. On the other hand, direct integration methods such as quadrature-with-extrapolation (QWE) can achieve higher accuracy, but at the cost of significantly increased computation time. This paper presents a new discrete Hankel transform method based on a convolution-type Gaussian fast Fourier transform (Conv-Gauss-FFT) algorithm that is both efficient and highly accurate. By recognizing that the Hankel transform in EM forward problems can be expressed as a continuous convolution-type integral, we first approximate the integral using a weighted sum of shifted discrete convolutions. The weights and the shifted sampling points are optimized via Gauss–Legendre quadrature rules. Each discrete convolution is structured as a Toeplitz matrix and solved efficiently using circulant embedding techniques in combination with the FFT algorithm. Additionally, a segmented sampling strategy is introduced to mitigate the accuracy degradation associated with exponential sampling at large source-receiver offset. A series of EM geophysical case studies demonstrate that the proposed algorithm achieves computational speed superior to both the DLF and QWE methods while maintaining comparable accuracy. Although DLF and QWE retain their respective advantages in simplicity and analytical precision, the algorithm proposed in this study offers an efficient computational alternative. Its notable performance advantages make it particularly suitable for large-scale controlled-source electromagnetic (CSEM) forward modeling. Graphical abstract