    Asymptotic pseudounitary stacking operators  Next: THEORETICAL DEFINITION OF A Up: Asymptotic pseudounitary stacking operators Previous: Asymptotic pseudounitary stacking operators

# Introduction

Integral (stacking) operators play an important role in seismic imaging and seismic data processing. The most common applications are common midpoint stacking, Kirchhoff migration, and dip moveout. Other examples include (listed in random order) Kirchhoff datuming, back-projection tomography, slant stack, velocity transform, offset continuation, and azimuth moveout. The use of the integral methods increases in prestack three-dimensional processing because of their flexibility with respect to irregularities in the data geometry.

An integral operator often is used to represent the forward modeling problem, and we invert it to solve for the model. In this paper, I consider two different approaches to inversion. The first is least-squares inversion, which requires constructing the adjoint counterpart of the modeling operator. The second approach is asymptotic inversion, which aims at reconstructing the high-frequency (discontinuous) parts of the model. I compare the two approaches and introduce the notion of asymptotic pseudo-unitary operator pair that ties them together.

In practice, least squares inversion is often applied as an iterative process (Ronen and Liner, 2000). The advantage of connecting it with the asymptotic inverse theory is the ability to speed up the iteration. This approach was used, in the context of seismic migration, by Jin et al. (1992) and Lambaré et al. (1992). Asymptotic pseudo-unitary operators, introduced in this paper, provide a more universal theoretical tool. One can use them to construct an appropriate preconditioning operator for accelerating the convergence of the least-squares methods.

The first part of this paper contains a formal definition of a stacking operator and reviews the theory of asymptotic inversion, following the fundamental results of Beylkin (1985) and Goldin (1990,1988). According to this theory, the high-frequency asymptotic inverse of a stacking operator is also a stacking operator. To connect this theory with the theory of adjoint operators, I show that the adjoint of a stacking operator can also be included in the class of stacking operators. The adjoint operator has the same summation path as the asymptotic inverse but a different weighting function. These two results combine together to form the definition of asymptotic pseudo-unitary integral operators. I apply such operators to define a general preconditioning operator for least-squares inversion. While one can apply Beylkin's theory directly for constructing an appropriate asymptotic preconditioner, pseudo-unitary operators accomplish the job in a more straightforward and computationally attractive way.

The second part of the paper addresses such examples of commonly used stacking operators as wave-equation datuming, migration, velocity transform, and offset continuation. The theory is specified for these particular applications and accompanied by numerical examples. The examples demonstrate the practical advantages of asymptotic pseudo-unitary operators.    Asymptotic pseudounitary stacking operators  Next: THEORETICAL DEFINITION OF A Up: Asymptotic pseudounitary stacking operators Previous: Asymptotic pseudounitary stacking operators

2013-03-03