Wave-equation tomography by beam focusing

Introduction

Tomographic velocity estimation based on wave-equation operators can improve seismic imaging in areas where wavefield-continuation migration is needed. However, it is well-known that the straightforward application of waveform inversion to estimate migration velocity fails to converge to an accurate model when the starting model is too far from the correct one. This failure to converge is caused by the non-linear relationship between data amplitudes and velocity. To avoid this failure the velocity-estimation problem can be formulated in the image domain as the maximization (or minimization) of objective functions that are more sensitive to the data kinematics than to the data amplitudes. Two important examples of this approach are the Wave-Equation Migration Velocity Analysis (WEMVA) method (Sava and Biondi, 2004b,a; Sava, 2004; Biondi and Sava, 1999) and the Differential Semblance Optimization (DSO) (Shen et al., 2005; Symes and Carazzone, 1991; Shen, 2004). Luo and Schuster (1991) introduced a method based on a kinematic objective function to solve the problem of transmission tomography. Both their method and the WEMVA method suffer from the drawback that they require the picking of kinematic parameters: correlation lag in one case (Luo and Schuster, 1991), and a residual migration parameter for WEMVA.

In this paper I develop a framework to update migration velocity by maximizing an objective function defined in the image domain. The objective function is defined as a function of moveout parameters but velocity updating can be performed without explicit picking of the residual moveout parameters. Therefore, it overcomes one of the main difficulties of the WEMVA methods. The methodology is general and can be thus used to optimize the image as a function of arbitrary residual moveout parameters, and possibly of residual migration parameters.

I also introduce a new objective function that overcomes limitations of known methods. This novel objective function has two components: the first term measures the power of the stack over local subarrays (beams) as a function of the moveout curvature. The second term measures the power of the stack across the beams as a function of a bulk-shift of each beam. I then apply the general theory to the computation of the gradient of the proposed objective function with respect to velocity perturbations.

I develop the theory and show the results of numerical tests for a transmission tomography problem because transmission tomography is simpler than reflection tomography, and therefore better suited to the illustration of the basic concepts. I leave to future reports the application to reflection tomography of the method developed for transmission tomography in this report.

Wave-equation tomography by beam focusing