Target-oriented wave-equation inversion: regularization in the reflection angle

Introduction

Conventional imaging techniques such as migration cannot provide an accurate picture of poorly illuminated areas (Clapp, 2005). In such areas, migration artifacts or multiples can easily obscure the small amount of signal that exists, making difficult to obtain correct positioned reflectors with useful amplitudes. One reason that makes the structural image and the amplitudes unreliable in this areas is the different amount of energy illuminating the target reflectors at different angles. This is a consequence of the complexity of the subsurface and the limited acquisition geometry of the seismic experiment.

One way to improve the estimates of subsurface-acoustic properties is to use inversion (Tarantola, 1987). A linear version linking the reflectivity to the data has being applied to solve imaging problems (Nemeth et al., 1999; Clapp, 2005; Kuhl and Sacchi, 2003). This procedure computes an image by convolving the migration result with the inverse of the Hessian matrix. When the dimensions of the problem get large, the explicit calculation of the Hessian matrix and its inverse becomes unfeasible. That is why Valenciano and Biondi (2004) and Valenciano et al. (2006) proposed the following approximations: (1) to compute the one-way wave equation Green functions from the surface to the target (or vice versa); (2) to compute an approximate Hessian, exploiting its sparse structure; and (3) to compute the inverse image following an iterative inversion scheme. The last item renders unnecessary an explicit computation of the inverse of the Hessian matrix.

The wave-equation inversion problem has a big null space. That is why a model regularization needs to be added. Two different regularization schemes for wave-equation inversion have been discussed in the literature. First, a geophysical regularization which penalizes the roughness of the image in the offset-ray-parameter dimension (which is equivalent the reflection-angle dimension) (Prucha et al., 2000; Kuhl and Sacchi, 2003). Second, a differential semblance operator to penalize the energy in the image not focused at zero subsurface-offset (Shen et al., 2003; Valenciano, 2006,2007).

In this paper I study the regularization in the reflection angle of the target-oriented wave-equation inversion. That choice for the regularization forces the inverse image to be smooth with the reflection angle. It works by spreading the image from well illuminated to poorly illuminated reflection angles. In order to impose this smoothness constraint I implemented a chain of the subsurface-offset Hessian and a slant-stack (reflection-angle to subsurface-offset) operator. I used the Sigsbee synthetic model to validate the methodology, showing that the inversion reduces the effect of the uneven illumination in the angle gathers and in the angle stack.

Target-oriented wave-equation inversion: regularization in the reflection angle