Ill-conditioning and pre-whitening in seismic deconvolution

Abstract

S U M M A R Y Treitel & Wang (1976) observed that autocorrelation matrices, as used for time-domain design of digital deconvolution filters, are ill-conditioned in certain cases. They present an example, where the solution of such a system of linear equations results in significantly different filter points, when the solution is performed on different computers. This paper presents a survey of the causes of such problems from a mathematical point of view. Later, the effect of pre-whitening on numerical stability is examined.For the purposes of discussion here, deconvolution isDeconvolution attempts to produce a function g(s) E G given f(t -s) E F and y ( t ) E Y, where G, F, and Y are the respective function spaces to which the functions belong. Such a problem would be said to be correctly posed if (i) for every function y ( t ) there corresponds a solution g(s) to the problem, (ii) the solution g(s) is unique for a given y ( t ) , and (iii) the solution g(s) is continuous with respect to y ( t ) .Equation (1) is a special case of a Fredholm integral equation of the first kind: b y ( t ) = 1 K(t, s)g(s) ds, c 5 t I d.a As stated by Tihonov (1963), it is not true that a solution g(s) may be produced for any given y ( t ) for equations of this type. So there may be no function, g(s), which, when convolved with a given filter, f, will yield a desired output y ( t ) , which means that deconvolution is incorrectly posed. If the left-hand side of equation (1) is only known to a finite accuracy, the different numerical methods to solve equation (1) lead to quite erratic results. Phillips (1962) presents some interesting numerical examples, and attributes this phenomenon to the fact that the integral operator with kernel K(t, s) generally has no bounded inverse. Franklin (1970) noted these effects, and discussed the use of stochastic processes to provide information about ill-posed linear problems.

Analysis and classification of tissue with scatterer structure templates

“…The advantage of this normalization is that the effects of spectral shape due to the transducer and overlaying tissue are removed along with energy variation from segment to segment. The usual ill-conditioned problems associated with deconvolution are limited using this form of normalization [19]. Whether the scaled off-diagonal components result from frequencies in a high or low signal-to-noise (SNR) region of the spectrum, their magnitudes are all scaled to 1.…”

The Wiener-Levinson algorithm and ill-conditioned normal equations

“…n3 versus n2 operations to solve a system of order n) than the Wiener-Levinson algorithm, and this cost may become prohibitive for large-order systems. Pre-whitening has been noted by Treitel & Wang (1976) and O'Dowd (1990) to be a procedure which results in less ill-conditioned normal equations.…”

“…An example of an ill-conditioned autocorrelation matrix is described by Treitel & Wang (1976). A survey of causes of ill-conditioned autocorrelation matrices is given by O'Dowd (1990).…”

“…First lo00 ms window of the deconvolved outputs. RE C 0 G N I Z I N <; I L L -C 0 N D IT I 0 NI N G Causes of ill-conditioned autocorrelation matrices have been discussed byO'Dowd (1990). Rather than u:ing those properties to identify tests of whether the autocorrelation matrix is ill-conditioned, this section focuses on the question: can intermediate results of the Wiener-Levinson algorithm be employed to provide an indication of ill-conditioning?The Wiener-Levinson algorithm solves a linear equation of the formRf = g,where R is a symmetric, Toeplitz matrix:…”