Subsalt imaging by target-oriented wavefield least-squares migration: A 3-D field-data example

introduction

Successful geological interpretation requires accurate reflectivity images of the subsurface. Reflectivity images obtained by prestack depth migration, however, are often distorted by uneven subsurface illumination. This is because the migration operator is only the adjoint of the forward Born modeling operator (Lailly, 1983), which is non-unitary due to the limited acquisition geometry, complex overburden and band-limited wavefields. The distorted image, as exemplified by biased amplitudes and the shadow zone effect, presents significant difficulties for accurate geological interpretation.

To correct the effects of uneven illumination, the reflectivity imaging problem can be posed as a linear inverse problem, which, instead of using the adjoint operator, uses the pseudo-inverse of the Born modeling operator to optimally reconstruct the reflectivity. This inversion-based imaging method is also widely known as least-squares migration (Clapp, 2005; Nemeth et al., 1999; Valenciano, 2008; Kühl and Sacchi, 2003).

Least-squares migration can be implemented in either the data domain or the image domain. In this paper, we focus on the image-domain inversion scheme because it can be implemented in a target-oriented fashion and hence is more suitable for large-scale 3-D applications. The target-oriented image-domain formulation allows us to invert only areas of particular interest, enabling accurate imaging at the reservoir level.

As shown by Valenciano (2008), image-domain least-squares migration contains three main steps: (1) compute the migrated image for a chosen target area, (2) compute the Hessian, the normal operator of the Born modeling operator, for the same target area, and (3) deblur the migrated image using the Hessian with an iterative solver. Among the three steps, the explicit computation of the Hessian is the most computationally intensive part, because it requires either storing a huge number of Green's functions for reuse (Valenciano, 2008), or performing a large number of wavefield propagations. (Each receiver-side Green's function has to be recomputed for each shot, if the Green's functions are not stored) (Tang and Lee, 2010). Fortunately, the computational cost can be drastically reduced by using the phase-encoding method, which does not require storing any Green's functions and significantly reduces the required number of wavefield propagations (Tang, 2009). In this paper, we extend phase-encoding theory to 3-D and show how the cost of Hessian can be drastically reduced by using a simultaneous phase-encoding scheme in the 3-D conical-wave domain.

Regularization is a crucial component of solving an ill-posed inverse problem. One important advantage of the image-domain inversion scheme is that solving the linear inversion problem (step 3) is very fast, which involves only sparse-matrix and vector multiplications. Therefore, different regularization parameters or schemes can be tried at very low cost. The high efficiency of this method also makes interpretation-driven interactive reflectivity imaging possible, where we can repeat the inversion with regularizations that incorporate different geological scenarios and obtain the results in almost real time. In this paper, we precondition the inversion problem with non-stationary dip filters (Claerbout, 2008; Hale, 2007; Clapp, 2003), which impose smoothness on the reflectivities along given dip directions. We show that the dip filter naturally incorporates prior knowledge based on interpreter's geological interpretation into the inversion, and it helps the inversion converge to a geologically meaningful result.

This paper is organized as follows: we first review the theory of image-domain least-squares migration. Then we show how the phase-encoded Hessian can be efficiently computed in the 3-D conical-wave domain. Following that, we discuss how to precondition the inversion with dip filters. Finally, we apply the method to a 3-D field data set acquired from the Gulf of Mexico (GOM), where we invert subsalt reflectivities in a target-oriented fashion.

Subsalt imaging by target-oriented wavefield least-squares migration: A 3-D field-data example