Fast Computation of Partial Fourier Transforms

TL;DR

This paper presents two efficient algorithms for partial Fourier transforms in 1D and 2D, leveraging multiscale decomposition to achieve near-linear complexity, with applications in reflection seismology.

Contribution

The paper introduces novel multiscale algorithms for partial Fourier transforms that are faster and more efficient than existing methods, especially in 2D.

Findings

1D algorithm is exact with $O(N ext{log}^2 N)$ complexity.

2D algorithm is approximate but accurate with $O(N^2 ext{log}^2 N)$ complexity.

Algorithms are effective for wave extrapolation in seismology.

Abstract

We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to decompose the summation domain of into simpler components in a multiscale way. Existing fast algorithms are then applied to each component to obtain optimal complexity. The algorithm in 1D is exact and takes $O(N\log^2 N)$ steps. Our solution in 2D is an approximate but accurate algorithm that takes $O(N^2 \log^2 N)$ steps. In both cases, the complexities are almost linear in terms of the degree of freedom. We provide numerical results on several test examples.