A fast algorithm for 3D azimuthally anisotropic velocity scan

Introduction

Multiazimuth seismic data reveal the Earth's seismic response along different azimuthal directions. Detecting and measuring the anisotropy in such data can be useful for characterizing fractures or stress in the subsurface (Tsvankin and Grechka, 2011). When apparent azimuthal anisotropy is present, conventional single-parameter isotropic velocity scan and normal moveout are usually inadequate. To further flatten the events, a residual anisotropic moveout is necessary. This, however, brings several difficulties in the implementation. First, the computational cost increases dramatically compared with the single-parameter case. If we assume for simplicity that there are $N$ sample points in every dimension of data and model (parameter) domains, then the numerical complexity of a two-parameter velocity scan will be at least $O(N^5)$ ; i.e., summing over $O(N^2)$ data points for each of $O(N^3)$ values (time + two parameters). Second, the simultaneous automatic picking from a high-dimensional semblance volume also poses a challenge (Tao et al., 2012; Adler and Brandwood, 1999; Siliqi et al., 2003; Arnaud et al., 2004).

In this work, we attempt to solve a fundamental problem related to the first difficulty. Specifically, we introduce a fast algorithm to speed up the velocity-scan process. The stacking procedure involved in computing the semblance can be regarded as a generalized Radon transform (Beylkin, 1984). Following our previous work on the hyperbolic Radon transform (Hu et al., 2013,2012), we formulate the time-domain summation as a discrete oscillatory integral in the frequency domain, and apply the 3D version of the FIO (Fourier integral operator) butterfly algorithm (Candès et al., 2009). As a result, complexity of the velocity scan reduces to roughly $O(N^3\log N)$ , where $N$ is representative of the number of points in either dimension of data space or model space. An alternative approach was developed by Burnett and Fomel (2009), but may not be applicable for noisy data.

A fast algorithm for 3D azimuthally anisotropic velocity scan