Obtaining a clean, coherent seismic image in areas of complex subsurface can be difficult. This is particularly true when the subsurface lends itself to shadow zones, such as those under the edges of salt bodies. In these areas, little of the seismic energy gets to the reflectors, and even less energy makes it back to the surface Muerdter et al. (1996); Prucha et al. (1998). In addition to the shadow zones, proper imaging is made difficult by multipathing. Multipathing occurs when energy that follows different paths through the subsurface arrives at the receiver at the same time. Solving these issues is no simple matter.
It has been suggested that it is possible to eliminate artifacts caused by multipathing by imaging in the reflection angle domain via angle-domain Kirchhoff migration Xu et al. (1998, 2001). However, Kirchhoff methods are not necessarily optimal for complex subsurfaces SEG Workshop (2001). Additionally, Stolk and Symes (2002) have shown that angle-domain Kirchhoff migration may not eliminate artifacts in strongly refracting media, even when using the correct velocity and all arrivals. To address these issues, angle-domain wave-equation migration methods have been developed Fomel and Prucha (1999); Prucha et al. (1999a,b); Sava et al. (2001). Stolk and De Hoop (2001) have shown that these methods do not suffer from image artifacts when migrated at the correct velocity.
Unfortunately, migration alone does not necessarily provide the best image. A better image in complex areas can often be obtained using least-squares inversion Duquet and Marfurt (1999); Nemeth et al. (1999); Ronen and Liner (2000). However, inversion often cannot improve the image in shadow zones without many iterations. With many iterations, noise in the null space caused by poor illumination can cause the inversion process to blow up.
To prevent these blow-ups, since we often have some idea of what the image in the shadow zone should look like, we can impose some sort of regularization on the inversion carried out in the angle domain Kuehl and Sacchi (2001); Prucha et al. (2000, 2001). In this paper, we will use a regularization that tends to create dips in the image from a priori selected reflectors and therefore can be applied by the use of steering filters Clapp et al. (1997); Clapp (2001). The inversion uses these steering filters to strengthen existing events to help fill in shadow zones. This regularization may even improve amplitude behavior Prucha and Biondi (2000).
The inversion process we use in this paper lends itself to two different regularization schemes. The ``1-D approach'' tends to create flat events along the reflection angle axis. This helps to fill in holes in different angle ranges caused by poor illumination. The ``2-D approach'' cascades the 1-D approach with an attempt to create dips along picked reflectors in the common midpoint (CMP) - depth plane. This allows the user to test possible interpretations within shadow zones. If the picked reflector doesn't cause the regularization to tend to create dips that interfere with the data, the model will contain those ``new'' dips. If the picked reflectors are incompatible with the data, the inversion will reject the dips.
In this paper, we will first review the theory and implementation of our inversion, then show the results of this regularized inversion scheme on a fairly complex synthetic model. We will then study the impact of the regularization operator more closely. Finally, we will discuss some of our future plans.