Wave-equation inversion is a powerful technique able to build clean images with balanced amplitudes in complex subsurface areas relative to migration alone. This paper illustrates how to perform wave-equation inversion in image space without making any velocity model or acquisition geometry approximations. The method explicitly computes the least-squares Hessian matrix, defined from the modeling/migration operators, and uses an iterative solver to find the solution of the resulting system of equations. This technique can handle 3-D data in a target-oriented fashion. The inversion in the presence of a complex overburden leads to an ill-conditioned system of equations that needs to be regularized to obtain a stable numerical solution. Regularization can be implemented in the poststack image-domain (zero subsurface offset), where the options for a regularization operator are limited to a customary damping, or in the prestack image-domain (subsurface offset), where a physically-inspired regularization operator (differential semblance) can be applied. Though the prestack imagedomain inversion is more expensive than the poststack imagedomain inversion, it can improve the reflectors continuity into the shadow zones with an enhanced signal-to-noise ratio. We demonstrate the utility of both these methods by improving the subsalt-sediment images of a 3-D Gulf of Mexico field data set.
Seismic imaging (modeling/migration) operators are nonunitary (Claerbout, 1992, 1985, 2001), meaning that if L is a modeling operator, and L'' is its adjoint (migration), their product L''L ? I, where I is the identity matrix. As a result, images of the subsurface are blurred when produced by a migration operator. Even for a simple subsurface, the bandlimited characteristic of the seismic data in the time and space domain would prevent L and L'' from being unitary (Claerbout, 1985; Chavent and Plessix, 1999; Plessix and Mulder, 2004).Moreover, where the subsurface is complex, and the seismic acquisition geometry is limited and irregular, the non-unitary characteristic of the imaging operators causes migration to produce images with artifacts and biased amplitudes.
One way to improve on the use of an adjoint operator such as that for migration is to use inversion (Tarantola, 1987). The "relative reflectivity" can be inverted from the seismic data to obtain images that better represent the subsurface properties and geometry than do those from migration (Nemeth et al., 1999; Prucha and Biondi, 2002; Kuhl and Sacchi, 2003; Clapp, 2005). The assumption behind the reflectivity inversion (as for migration) is that the long-wavelength model (migration velocity) is known.
Solution of the Ill-posed inversion problem
Solving linear systems generally is not difficult, but the high dimensionality and the ill-posedness of the seismic-imaging inverse problem in complex areas, makes its solution challenging. There are two main physical reasons for this. The first is the limited and irregular acquisition geometry of the seismic experiment (Nemeth et al., 1999; Duquet and Marfurt, 1999; Ronen and Liner, 2000). The second is the complexity of the overburden (Rickett, 2003; Guitton, 2004; Clapp, 2005). These two effects combined produce incomplete and irregular illumination of the subsurface. In the worst cases "shadow zones" can be created.
Image segmentation offers a means of automatically delineating salt bodies in seismic images, an otherwise human-intensive and time-consuming task. Current segmentation algorithms successfully pick salt boundaries; a logical extension of such work is to apply these methods to the task of building and updating seismic velocity models. The method presented here successfully applies image segmentation tools in conjunction with sediment- and saltflood migration techniques to identify the top and base of a salt body. Furthermore, previously existing velocity models may be updated based on the results of segmentation and automated boundary picking. In the latter case, the prior model acts as a priori information for the picking algorithm in areas of relative uncertainty, producing an "optimized" boundary path across an image. For both synthetic and real seismic data, migrations with velocity models derived from this method produce greatly improved images.
The purpose of image segmentation is to automatically divide an image into sections based on specific attributes. Because of its global optimizaton properties, one algorithm with a variety of potential applications to seismic interpretation is Normalized Cuts Image Segmentation (NCIS) (Shi and Malik,2000). The NCIS method was first applied to seismic data by Hale and Emanuel (2002;2003), who used it to paint 3D atomic meshes of seismic images. Recent work by Lomask and others (Lomask,2007; Lomask et al.,2007) presents an image segmentation algorithm for automatic picking of salt boundaries. Such a scheme offers many potential benefits for the seismic velocity model building process, which we explore in this paper.
Lomask''s algorithm divides a seismic image into two segments; the boundary separating them is the salt interface. The segmentation is based on a specific seismic attribute, in most cases instantaneous amplitude, that can clearly differentiate between a salt body and the surrounding sediments. The most basic step is to create a weight matrix Wthat relates each pixel in a migrated seismic image to a random collection of neighboring pixels. Low weights are assigned to pixel pairs most likely to be separated by a salt boundary. A path across the image which minimizes the sum of the weights through which it passes is the salt boundary.
Extension to velocity estimation
In this paper, we adapt the methods outlined above for use in iterative velocity model construction and updating. A reasonably accurate velocity model is an essential component of the seismic imaging process. Much of today''s seismic data is acquired in regions characterized by complex salt bodies; in such cases, clearly delineating salt interfaces is often one of the most human-intensive, time consuming and inexact aspects of velocity estimation. Correct salt interface interpretation becomes especially important when the imaging target is located sub-salt, as is often the case for modern surveys. The method we propose here is designed to function as a tool for either velocity model construction, or updating. We show our method to be highly effective when combined with widely used sediment- and salt-flooding migration techniques to make original salt interface interpretations.