Current migration methods are done mostly in the offset domain and the shot domain, which are prone to multipathing ten Kroode et al. (1999) in complex areas. A fairly new method that is being used to image complex areas is migration in the reflection angle domain (RAD). This can be done both by Kirchhoff methods Xu et al. (1998) and wave-equation methods Prucha et al. (1999). The RAD avoids the problem of multipathing, and therefore contains fewer artifacts than the more commonly used domains. Unfortunately, even with fewer multipathing artifacts, in complex areas migration may not be not enough. Since we are interested in complex areas, we can reformulate our imaging problem as an inversion problem Chemingui (1999).
Although imaging by inversion can give better results than migration, an inversion problem can be unstable Claerbout (1991). A trick used to constrain inversion problems to a reasonable result is regularization Fomel (1997); Harlan (1986). Theory states that for a particular point in the subsurface, the reflectivity as a function of reflection angle should vary smoothly Richter (1941). Therefore, the obvious choice for a regularization operator in the reflection angle domain is one that smooths along the reflection angles. To speed the convergence, we can reformulate the regularization problem as a preconditioned problem Fomel et al. (1997). We intend to show that applying this method in the reflection angle domain will improve the common image gathers (CIGs), making the events more continuous, reducing artifacts, and attenuating multiples.
In this paper, we will first explain how to image in the reflection angle domain and how to apply regularization and preconditioning. Then we will show the results of applying regularization to a RAD inversion problem on two different synthetic datasets.