Combining forward-scattered and back-scattered wavefields in velocity analysis

Introduction

Seismic velocity-analysis methods can be divided into two major groups. First, there are techniques that aim at minimizing the misfit in the data domain, such as full-waveform inversion (Luo and Schuster, 1991; Tarantola, 1984). Second, there are other techniques that aim at improving the quality in the image domain, such as migration velocity analysis (MVA) (Biondi and Sava, 1999; Symes and Carazzone, 1991; Biondi, 2010; Shen, 2004). These techniques try to measure the quality of the image in several ways and then invert the estimated image perturbation using a linearized wave-equation operator.

There are several advantages to minimizing the residual in the image-space, such as increasing the signal-to-noise ratio and decreasing the complexity of the data (Tang et al., 2008). However, a common drawback in doing velocity analysis in the image domain is that only forward-scattered wavefields are used. This results in a loss of vertical resolution in the estimated model updates.

On the other hand, full-waveform inversion (FWI) does not have that problem, since it utilizes the information from both the forward-scattered and back-scattered wavefields, i.e. both the kinematics and the dynamics. This results in higher resolution in the model estimates. However, FWI has the disadvantage of being highly nonlinear, which requires the starting model to be very close to the true model to avoid converging to local minima.

A straightforward solution is to first invert for the velocity model using MVA techniques and then use the output as the initial model for FWI. However, this practice might not work if the results of MVA are not accurate enough for FWI to start. This could be a result of the larger null space that forward-scattered wavefields do not constrain. Moreover, the convergence rate of the MVA techniques is going to be sub-optimal, since they do not use all of the information in the data.

In this paper, I try to improve the MVA results and convergence rate by supplying the back-scattered information to the gradient. First, I decompose the FWI gradient into two components: the forward-scattering gradient and the back-scattering gradient. The decomposition is done in Fourier domain based on the direction of wave propagation. Second, I develop a new gradient that combines the MVA gradients with the back-scattered component of the FWI gradient using the proper weighting function. This weighting function aims to emphasize the components of the back scattered FWI gradients that overlap with the MVA gradient. Finally, I apply the combined gradient in a synthetic example.

Combining forward-scattered and back-scattered wavefields in velocity analysis