Computational analysis of extended full waveform inversion

Introduction

Seismic velocity analysis methods can be divided into two major groups. First, there are techniques that aim to minimize the misfit in the data domain, e.g., full waveform inversion (FWI) (Pratt, 1999; Luo and Schuster, 1991; Tarantola, 1984). Second, there are other techniques (Biondi and Sava, 1999; Symes and Carazzone, 1991; Shen, 2004; Zhang et al., 2012), that aim to improve the quality in the image domain, such as migration velocity analysis (MVA). These approaches try to measure some quality of the image and then invert for the estimated image perturbation using a linearized wave-equation operator.

There are significant advantages to minimizing the residual in the image space: global convergence, increased signal-to-noise ratio, and decreased complexity of the data (Tang et al., 2008). However, a common drawback to doing velocity analysis in the image domain is that only the transmission effects of the velocity are used. This results in incomplete vertical resolution in the estimated model updates. On the other hand, FWI does not have that problem, since it utilizes the information from both the forward-scattered and back-scattered wavefields. This results in higher resolution model estimates. Moreover, the data misfit is computed in the data space directly without the need to go to another domain or to separate the data into several components. This direct computation results in a relatively simple relationship between the data residuals and the model updates. However, FWI has the disadvantage that its objective function is far from being smooth and convex; it requires the starting model to be very close to the true model to avoid convergence to local minima.

Biondi (2012) presents a generalized framework for full waveform inversion that avoids the cycle-skipping problem while utilizing all the components of seismic data to invert for the medium parameters. This is achieved in two steps: first by extending the velocity model through an additional degree of freedom, and second by imposing a regularization to constrain this added degree of freedom.

In this paper, I compare the cost of conventional full waveform inversion to extended inversion in model space that uses subsurface offset and time lags. I also compare the cost of the extended inversion to linearized inversion by scale separation (Almomin and Biondi, 2012). Next, I propose extending full waveform inversion through a data space axis, such as source location or source ray parameter, instead of model space axes. Finally, I test the source ray parameter extension on the Marmousi model.

Computational analysis of extended full waveform inversion