Methods for estimating the seismic reflection response

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The convolutional model of the seismic trace consists of a seismic pulse convolved with a reflectivity series plus measurement noise. The seismic deconvolution problem is to estimate the reflectivity series, given the data and an estimate of the seismic pulse. The classical solution to this problem is a weighted least‐squares estimate of the reflectivity series, which is optimal when the noise covariance matrix is known and there are no errors in the pulse. The seismic convolutional model has been reformulated, taking into account errors in the pulse and measurement noise, which is taken to be white noise filtered by a finite‐impulse‐response filter. All variables are assumed to be Gaussian, with known a priori mean values and covariance matrices. The unknown parameters may be the reflectivity series, the noise‐filter coefficients, and the white noise variance or, when the noise covariance matrix is known, just the reflectivity series. This results in maximum a posteriori (MAP) and maximum likelihood (ML) estimates of the reflectivity series that take into account uncertainty in the seismic pulse and colored noise. These estimates generally can be computed by solving a nonlinear minimization problem. The constrained total least‐squares (CTLS) estimate of the reflectivity series is found by minimizing a function that contains one less term than does the function that gives the ML estimate. When there is no uncertainty in the pulse and the noise covariance matrix is known all estimates are linear functions of the data corresponding to weighted least squares. The stabilized least‐squares (SLS) estimate of the reflectivity series is a special case of the MAP estimate with a simple statistical model. The problem of estimating the seismic pulse, given seismic data and an estimate of the reflectivity series, is identical to the problem of estimating the reflectivity series, except that the initial conditions in the convolutional model are slightly different.