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GEOREF RECORD

Velocity-stack and slant-stack stochastic inversion

Jeffrey R. Thorson and Jon F. Claerbout
Velocity-stack and slant-stack stochastic inversion
Geophysics (December 1985) 50 (12): 2727-2741

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 L (super T) (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 L (super T) 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 L (super -1) , 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.


ISSN: 0016-8033
EISSN: 1942-2156
Coden: GPYSA7
Serial Title: Geophysics
Serial Volume: 50
Serial Issue: 12
Title: Velocity-stack and slant-stack stochastic inversion
Affiliation: Sierra Geophys., Redmond, WA, United States
Pages: 2727-2741
Published: 198512
Text Language: English
Publisher: Society of Exploration Geophysicists, Tulsa, OK, United States
References: 20
Accession Number: 1986-040489
Categories: Applied geophysics
Document Type: Serial
Bibliographic Level: Analytic
Illustration Description: illus.
Secondary Affiliation: Stanford Univ., Dep. Geophys., USA, United States
Country of Publication: United States
Secondary Affiliation: GeoRef, Copyright 2017, American Geosciences Institute. Reference includes data supplied by Society of Exploration Geophysicists, Tulsa, OK, United States
Update Code: 1986
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