Modeling, migration, and inversion in the generalized source and receiver domain

introduction

Shot-profile migration is an accurate imaging technique. The computation is performed in each shot gather; thus it closely mimics the actual physical experiment. However, migrating shot by shot is expensive, since the number of shot gathers is usually very big for a typical seismic survey. To reduce the cost, Whitmore (1995), Zhang et al. (2005) and Duquet and Lailly (2006) develop plane-wave source or delayed-shot migration, which migrates a small number of synthesized shots made by linear combination of the original shot gathers after linear time delays. In fact, plane-wave source or delayed-shot migration is only a special case of a more general class of migration technique, phase-encoding migration (Romero et al., 2000; Liu et al., 2006), where the source encoding functions can be any type of phase functions, such as linear phase functions, random phase functions, etc.. Though the methods mentioned above involve encoding only the original sources, there is no reason that the receivers could not be encoded. For example, besides assuming tilted line sources (plane-wave source or delayed-shot migration), we can also assume the data are recorded by tilted line receivers, or assume both line sources and line receivers at the same time. Such ideas have been explored by Stoffa et al. (2006), who develop the ray-based asymptotic theory for plane-wave source migration, plane-wave receiver migration and plane-wave source and receiver migration. In this paper, I extend those ideas from the plane-wave domain to more general cases and unify them under the generalized source and receiver domain.

Another important aspect of imaging is how to preserve the amplitude information of the reflectors. It is widely known that because of the non-unitary nature of the Born forward modeling operator, its adjoint, the migration operator, can only preserve the kinematic information of the reflectors (Lailly, 1983). To better preserve the amplitude, the inversion should be used. Bleistein (2007) derives closed-form asymptotic inversion formulas based on the synthesized shot gathers (e.g. plane-wave sources) under the assumption that the acquisition geometry is infinite. However, we never have infinite acquisition geometry in practice. In fact, the limited acquisition geometry is an important factor distorting the amplitude of the reflectors, especially in complex geologies; hence it should not be neglected.

In this paper, I also extend the target-oriented inversion theory (Valenciano, 2008) to the generalized source and receiver domain for limited acquisition geometry. The effect of limited acquisition geometry is then taken into account in the least-squares sense and corrected by the pseudo-inverse of the target-oriented Hessian, the second derivative of the least-squares misfit functional with respect to the model parameters (Tang, 2008; Plessix and Mulder, 2004; Valenciano, 2008). I demonstrate that in the generalized source and receiver domain, the target-oriented Hessian can be more efficiently computed without storing the Green's functions, which is a major obstacle for the Hessian computation in the original shot-profile domain. I also show that the Hessian obtained in the generalized source and receiver domain is essentially the same as the phase-encoded Hessian (Tang, 2008), the physics of which, however, was not carefully discussed in Tang (2008). Therefore, from this perspective, this paper completes the discussion of the phase-encoded Hessian from the physical point of view. The modeling, migration and target-oriented Hessian formulas are all derived in terms of Green's functions, so that any type of Green's functions can be used under this framework, such as ray-based asymptotic Green's functions, Green's functions obtained by solving one-way wave equations, and Green's functions obtained by solving two-way wave equations. Anisotropy can also be taken into account, provided that the Green's functions are properly modeled.

This paper is organized as follows: I first briefly review the theory of Born modeling, migration and the target-oriented Hessian in the original shot-profile domain. Then I extend the theory to the encoded source domain, the encoded receiver domain, and the encoded source and receiver domain. Finally, I introduce a new phase-encoding scheme, which mixes both random and plane-wave phase encoding, to compute the Hessian operator. The new scheme combines advantages of both random phase encoding and plane-wave phase encoding. Finally, I apply the mixed phase-encoding scheme to a simple synthetic model.

Modeling, migration, and inversion in the generalized source and receiver domain