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

Regularization in the reflection angle

Equation 3 can be solved in different domains: poststack image domain (zero subsurface-offset) (Valenciano et al., 2006), prestack subsurface-offset image domain (Valenciano, 2006,2007), or prestack reflection-angle image domain (this paper). Valenciano (2007) shows that a prestack regularization is necessary to reduce the noise in the result without smoothing the image space.

In this paper I discuss the use of the regularization in the reflection-angle domain (Prucha et al., 2000; Kuhl and Sacchi, 2003). The regularization operator is a derivative in the reflection-angle dimension that penalizes the roughness of the image. It works by spreading the image from well illuminated to poorly illuminated reflection angles.

The general fitting goals corresponding to the angle-domain inversion are:

$${\bf H}({\bf x, h};{\bf x',h'}) {\bf S}_{\Theta \rightarrow {\bf h}} \hat{{\bf m}}({\bf x},\Theta)-{\bf m}_{mig}({\bf x},{\bf h}) \approx 0,$$

$$\varepsilon{\bf D}_\Theta \hat{{\bf m}}({\bf x},\Theta) \approx 0,$$ (4)

where ${\bf H}({\bf x, h};{\bf x',h'})$ is the subsurface-offset Hessian (Valenciano, 2006), ${\bf S}_{\Theta \rightarrow {\bf h}}$ is a slant-stack operator that transforms the image from angle to subsurface-offset domain, ${\bf D}_\Theta$ is a derivative operator, ${\bf x}=(z,x,y)$ is a point in the image, ${\bf h}=(h_x,h_y,h_z)$ is the half subsurface-offset, and $\Theta = (\gamma,\theta)$ reflection, and azimuth angle.

In the next section I discuss on the numerical solution of the inversion problem stated in equation 4 applied to the imaging of Sigsbee model. Notice that in this paper I use a 2D example where only the $h_x$ component of the subsurface-offset and the reflection-angle $\gamma$ are used.

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