Velocity‐stack and slant‐stack stochastic inversion

Abstract

Normal moveout (NMO) and stacking, an important step in analysis of reflection seismic data, involves summation of seismic data over paths represented by a family of hyperbolic curves. This summation process is a linear transformation and maps the data into what might be called a velocity space: a two‐dimensional set of points indexed by time and velocity. Examination of data in velocity space is used for analysis of subsurface velocities and filtering of undesired coherent events (e.g., multiples), but the filtering step is useful only if an approximate inverse to the NMO and stack operation is available. One way to effect velocity filtering is to use the operator [Formula: see text] (defined as NMO and stacking) and its adjoint L as a transform pair, but this leads to unacceptable filtered output. Designing a better estimated inverse to L than [Formula: see text] is a generalization of the inversion problem of computerized tomography: deconvolving out the point‐spread function after back projection. The inversion process is complicated by missing data, because surface seismic data are recorded only within a finite spatial aperture on the Earth’s surface. Our approach to solving the problem of an ill‐conditioned or nonunique inverse [Formula: see text], brought on by missing data, is to design a stochastic inverse to L. Starting from a maximum a posteriori (MAP) estimator, a system of equations can be set up in which a priori information is incorporated into a sparseness measure: the output of the stochastic inverse is forced to be locally focused, in order to obtain the best possible resolution in velocity space. The size of the resulting nonlinear system of equations is immense, but using a few iterations with a gradient descent algorithm is adequate to obtain a reasonable solution. This theory may also be applied to other large, sparse linear operators. The stochastic inverse of the slant‐stack operator (a particular form of the Radon transform), can be developed in a parallel manner, and will yield an accurate slant‐stack inverse pair.