Introduction

Migration is the adjoint of a linear forward modeling operator rather than the inverse [e.g. Claerbout (1995)]. This means that, although migration treats kinematics correctly, the amplitudes of migrated images do not accurately represent seismic reflectivity.

Geophysical inverse theory provides a rigorous framework for estimating earth models that are consistent with some observed data. Typically the matrices involved in industrial-scale geophysical inverse problems are too large to invert directly, and we depend on iterative gradient-based linear solvers to estimate solutions. However, operators such as prestack depth migration are so expensive to apply that we can only afford to iterate a handful of times, at best.

In this paper I compute diagonal weighting functions that can be applied directly to migrated images to compensate for the inadequacies of the adjoint with respect to seismic amplitudes. Furthermore, these weighting functions can be applied as preconditioning operators that speed the convergence of iterative linear solvers, facilitating least-squares recursive depth migration.

As well as looking at model-space weights, I also consider data-space weighting functions derived from the operator ${\bf A}\, {\bf A}'$(where ${\bf A}$ is our linear forward modeling operator), and develop a framework for computing and applying both model and data-space weights simultaneously.