RESIDUAL STATICS ANALYSIS AS A GENERAL LINEAR INVERSE PROBLEM

Abstract

To estimate near‐surface time anomalies, it is commonly assumed that apparent seismic reflection times are comprised of the sum of “surface‐consistent” source and receiver static terms, “subsurface‐consistent” structure and residual normal moveout (RNMO) terms, and indeterminate noise. The model parameters (statics, RNMO, and structural terms) that, in a least‐squares sense, best satisfy traveltime observations in multifold seismic data are solutions to a set of linear simultaneous equations. Because these equations are ill conditioned and their solutions are known to be nonunique, conventional direct methods of solution are not applicable. Problems of this type which have both overdetermined and underconstrained aspects can be analyzed using the general linear inverse methodology. In this approach, observed time deviations are decomposed into linear combinations of orthogonal eigenvectors, each of which determines a related linear combination of model parameters. A property of this decomposition is that the uncertainty (standard deviation) in a model‐parameter eigenvector is functionally related to the uncertainty in its associated observation eigenvector. In particular, statics corrections having spatial wavelengths much shorter than a cable length have smaller uncertainties than do the observations themselves, whereas long‐wavelength corrections have much larger standard deviations and are thus poorly determined. In practice, iterative methods are commonly used to solve the large number of equations encountered for typical seismic profiles. Using the Gauss‐Seidel iterative formalism, we can know in advance how many iterations are required to obtain a given reduction of the original error for any wavelength contribution. Errors in shorter‐wavelength corrections converge rapidly to zero while a heavier price is exacted to compute longer‐wavelength corrections. However, because those longer‐wavelength corrections can be estimated only with large uncertainty, it is desirable to exclude them from the statics solution through judicious choice of the number of iteration cycles.