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ABSTRACT: Singular value decomposition (SVD) is applied to the identification of seismic reflections by using two different models: the impulse response model, where a seismic trace is assumed to consist of a known signal pulse convolved with a reflection coefficient series plus noise, and the delayed pulse model, where the seismic signal is assumed to consist of a small number of delayed pulses of known shape and with unknown amplitudes and arrival times.SVD clearly shows how least-squares estimation of the reflection coefficients may become unstable, since a division by the singular values is required. Two methods for stabilizing this procedure are investigated. The inverse of the singular values may be replaced by zeros when they are less than a given threshold. This is called the SVD cut-off method. Alternatively, we may use ridge regression which in filter design corresponds to assuming white noise. Statistical methods are used to compute an optimal SVD cut-off level and also to compute an optimal weighting parameter in ridge regression. Numerical studies indicate that the use of SVD cut-off or ridge regression stabilizes the least-squares procedure, but that the results are inferior to maximum-likelihood estimation where the noise is assumed to be filtered white noise.For the delayed pulse model, we use a linearization procedure to iteratively update the estimates of both the reflection amplitudes and the arrival times. In each step, the optimal SVD cut-off method is used. Confidence regions for the estimated reflection amplitudes and arrival times are also computed. Synthetic data examples demonstrate the effectiveness of this method. In a real data example, the maximum-likelihood method assuming an impulse response model is first used to obtain initial estimates of the number of reflections and their amplitudes and traveltimes. Then the iterative procedure is used to obtain improved estimates of the reflection amplitudes and traveltimes.
Geophysical Prospecting 04/2006; 33(6):773 - 799. · 1.32 Impact Factor
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ABSTRACT: ABSTRACTA seismic trace is assumed to consist of a known signal pulse convolved with a reflection coefficient series plus a moving average noise process (colored noise). Multiple reflections and reverberations are assumed to be removed from the trace by conventional means. The method of maximum likelihood (ML) is used to estimate the reflection coefficients and the unknown noise parameters. If the reflection coefficients are known from well logs, the seismic pulse and the noise parameters can be estimated.The maximum likelihood estimation problem is reduced to a nonlinear least-squares problem. When the further assumption is made that the noise is white, the method of maximum likelihood is equivalent to the method of least squares (LS). In that case the sampling rate should be chosen approximately equal to the Nyquist rate of the trace. Statistical and numerical properties of the ML- and the LS-estimates are discussed briefly. Synthetic data examples demonstrate that the ML-method gives better resolution and improved numerical stability compared to the LS-method.A real data example shows the ML- and LS-method applied to stacked seismic data. The results are compared with reflection coefficients obtained from well log data.
Geophysical Prospecting 04/2006; 33(2):233 - 251. · 1.32 Impact Factor
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B. URSIN
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ABSTRACT: In the mathematical theory of seismic signal detection and parameter estimation given, the seismic measurements are assumed to consist of a sum of signals corrupted by additive Gaussian white noise uncorrelated to the signals. Each signal is assumed to consist of a signal pulse multiplied by a space-dependent amplitude function and with a space-dependent arrival time. The signal pulse, amplitude, and arrival time are estimated by the method of maximum likelihood.For this signal-and-noise model, the maximum likelihood method is equivalent to the method of least squares which will be shown to correspond to using the signal energy as coherency measure. The semblance coefficient is equal to the signal energy divided by the measurement energy. For this signal model we get a more general form of the semblance coefficient which reduces to the usual expression in the case of a constant signal amplitude function.The signal pulse, amplitude, and arrival time can be estimated by a simple iterative algorithm. The effectiveness of the algorithm on seismic field data is demonstrated.
Geophysical Prospecting 04/2006; 27(1):1 - 15. · 1.32 Impact Factor
