LINES, L.R. and TREITEL, S . 1984, Tutorial: A Review of Least-Squares Inversion and its Application to Geophysical Problems, Geophysical Prospecting 32, 159-186.Geophysical inversion involves the estimation of the parameters of a postulated earth model from a set of observations. Since the associated model responses can be nonlinear functions of the model parameters, nonlinear least-squares techniques prove to be useful for performing the inversion. A common type of inversion applies iterative damped linear least squares through use of the Marquardt-Levenberg method. Traditionally, this method has been implemented by solving the associated normal equations in conventional ways. However, Singular Value Decomposition (SVD) produces significant improvements in computational precision when applied to the same system of normal equations. Iterative leastsquares modeling finds application in a wide variety of geophysical problems. Two examples illustrate the approach: (1) seismic wavelet deconvolution, and (2) the location of a buried wedge from surface gravity data. More generally, nonlinear least-squares inversion can be used to estimate earth models for any set of geophysical observations for which an appropriate mathematical description is available. . G E O P H Y S I C A L INVERSE PROBLEMSGeophysical inversion may be viewed as an attempt to fit the response of an idealized subsurface earth model to a finite set of actual observations. We distinguish between a model, its model parameters, and the associated model response. A model consists of a set of relations representing a particular mathematical abstraction of an observed process. These equations in turn depend on a certain number, say p , of model parameters which we desire t o estimate from the actual data. The model response consists of the synthetic data produced by a particular realization of * A model response can be either a linear or a nonlinear function of the model parameters. Thus for the linear convolutional model y, = a, * b,, where * denotes convolution, the data y, are a linear function of the model parameters a, and b, , butif, say, y, = a:, then y, is a nonlinear function of a,. A comprehensive discussion of the relation between nonlinear and linear least-squares inversion methods can be found in the book by Draper and Smith (1981, chapter 10). A remark on terminology is in order: Draper and Smith are statisticians and use the word "regression"where we use "inversion ". To complicate matters further, electrical engineers frequently refer to these methods as "systems identification" or "parameter identification". Other terms are also in use.

Geophysical inversion by iterative modeling involves fitting observations by adjusting model parameters. Both seismic and potential‐field model responses can be influenced by the adjustment of the parameters of the rock properties. The objective of this “cooperative inversion” is to obtain a model which is consistent with all available surface and borehole geophysical data. Although inversion of geophysical data is generally non‐unique and ambiguous, we can lessen the ambiguities by inverting all available surface and borehole data. This paper illustrates this concept with a case history in which surface seismic data, sonic logs, surface gravity data, and borehole gravity meter (BHGM) data are adequately modeled by using least‐squares inversion and a series of forward modeling steps.

Seismic source wavelet deconvolution can be treated within the framework of the Backus‐Gilbert (BG) inverse theory. A time shift‐invariant version of this theory leads to the Wiener shaping filter, which has enjoyed widespread use for source wavelet deconvolution in exploration seismology. The model of the BG theory is the ground impulse response, the BG mapping kernel is the source wavelet, and the BG resolving kernel is the convolution between the source wavelet and the Wiener shaping filter. BG inversion involves the minimization of an optimality criterion under a set of constraints. The application of the BG “filter energy” or “noise output power” constraint to Wiener filter design leads to the familiar prewhitening parameter that stabilizes the filter on the one hand, but degrades resolution on the other. The BG “unimodular” constraint produces an unbiased estimate of the model, or ground impulse response. These constraints provide novel insights into the performance of deconvolution filters.

Reliable seismic depth migrations require an accurate input velocity model. Inaccurate velocity estimates will distort point diffractors into smiles or frowns on a depth section. For both poststack and prestack migrated sections, high velocities cause deep smiles while low velocities cause shallow frowns on migrated gathers. However, for prestack images in the offset domain, high velocities cause deep frowns while low velocities cause shallow smiles. If the velocity is correct, there will be no variation in the depth migration as a function of offset and no smiles or frowns in the offset domain. We explain migration responses both mathematically and graphically and thereby provide the basis for depth migration velocity analysis.

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