Abstract: A method for analyzing formations using measurements from a detector in response to energy imparted therein. The measurements have characteristics which exponentially reduce in magnitude with time. The method includes (a) determining, an N-th order integral of the value of each measurement from an initial time to the time of each measurement, wherein N represents a number of exponentially decaying characteristics; b) determining a solution to a system of linear equations relating the measurements to the integrals, the solution representing polynomials of order N related to a decay rate and an initial measurement amplitude for each component; (c) solving the polynomials to determine the decay rate and the initial amplitude for each component; (d) determining if the decay rates and initial amplitudes are within possible limits; and (e) incrementing N and repeating (a) through (d) until the decay rates or the initial amplitudes are not within possible limits.

Abstract: A method for correcting data measured by a well logging instrument for effects of cable yo-yo. The data are first preprocessed to reduce magnitude of spatial frequency components in the data occurring within a bandwidth of axial acceleration of the logging instrument which corresponds to the cable. Then eigenvalues of a matrix are shifted, over depth intervals where the smallest absolute value one of the eigenvalues changes sign, by an amount such that the smallest absolute value eigenvalue does not change sign. The matrix forms part of a system of linear equations by which the measurements made by the instrument are converted to values of a property of interest of earth formations. Artifacts which may remain in the data after the step of preprocessing are substantially removed by the eigenvalue shifting. In one embodiment, the step of preprocessing includes low pass filtering using a cutoff at the axial resolution limit of a sensor on the instrument.

Abstract: A method for correcting data measured by a well logging instrument for effects of cable yo-yo. The data are first preprocessed to reduce magnitude of spatial frequency components in the data occurring within a bandwidth of axial acceleration of the logging instrument which corresponds to the cable. Then eigenvalue, of a matrix are shifted, over depth intervals where the smallest absolute value one of the eigenvalues changes sign, by an amount such that the smallest absolute value eigenvalue does not change sign. The matrix forms part of a system of linear equations by which the measurements made by the instrument are converted to values of a property of interest of earth formations. Artifacts which may remain in the data after the step of preprocessing are substantially removed by the eigenvalue shifting. In one embodiment, the step of preprocessing includes low pass filtering using a cutoff at the axial resolution limit of a sensor on the instrument.

Abstract: A method for inverting a seismic signal, wherein an input seismic signal is represented as an input vector i(t) of length m+n-1, an output seismic signal is represented as an output vector o(t) of length m, and a finite impulse response filter is represented as a solution vector u(t) of length n, satisfying a convolutional equation i(t)Xu(t)=o(t). An m.times.n Toeplitz matrix T(t,.tau.) corresponding to i(t) is calculated, satisfying a matrix equation T(t,.tau.).multidot.u(t)=o(t). Both sides of the matrix equation are transformed to the frequency domain, to generate a transformed equation. Both sides of the transformed equation are spectrally pruned, to generate a pruned equation. The pruned equation is then solved.

Abstract: A method for estimating axial positions of formation layer boundaries from transverse electromagnetic induction signals. A first derivative is calculated with respect to depth of the induction signals. A second derivative of the signals is calculated. The second derivative is muted. Layer boundaries are selected at axial positions where the muted second derivative is non zero, and the first derivative changes sign. The selected boundaries are thickness filtered to eliminate boundaries which have the same axial spacing as the spacing between an induction transmitter and receiver used to measure the induction signals, and to eliminate boundaries having a spacing less than an axial resolution of the induction signals. In a preferred embodiment, the process is repeated using transverse induction measurements made at another alternating current frequency. Layer boundaries selected in both frequencies are determined to be the layer boundaries.

Abstract: A marine seismic signal is transformed from time domain into frequency domain and represented by matrix D. The marine data signal is truncated in time, transformed into the frequency domain and represented by matrix D.sub.T Eigenvalue decomposition D.sub.T=S.LAMBDA.S.sup.-1 of matrix D.sub.T is computed. Matrix product D S is computed and saved in memory. Conjugate transpose [D S]* is computed and saved in memory. Matrix product [D S]*[D S] is computed and saved in memory. Matrix inverse S.sup.-1 is computed and saved in memory. Conjugate transpose (S.sup.-1)* is computed and saved in memory. Matrix product S.sup.-1 (S.sup.-1)* is computed and saved in memory. An initial estimate of the source wavelet w is made. Source wavelet w is optimized by iterating the steps of computing the diagonal matrix [I-w.sup.-1 .LAMBDA.], computing matrix inverse [I-w.sup.-1 .LAMBDA.].sup.-1, computing conjugate transpose [(I-w.sup.-1 .LAMBDA.).sup.-1 ]*, retrieving matrix products [D S]*[D S] and S.sup.-1 (S.sup.

Abstract: A physical signal is represented by a linear system. The linear system is transformed from the time and space domain to the frequency domain, generating a transformed system. The least significant portions in the transformed system are determined. The least significant portions are deleted from the transformed system, generating a smaller pruned system. The pruned system is solved in the frequency domain, generating a solution. The solution is inverse transformed from the frequency domain to the time and space domain, generating an approximation to the physical signal.

Abstract: A method for determining the 2-dimensional distribution of horizontal and vertical electrical conductivities of earth formations surrounding a wellbore using measurements made by a transverse electromagnetic induction well logging instrument. A model is generated of the axial distribution of the horizontal and vertical conductivities, from induction signals acquired by the instrument using two-frequency alternating current. The model is generated by calculating an initial estimate of the conductivity distribution and axially inverting the estimate with respect to the measurements made using the two-frequency alternating current. Shoulder correction is applied to measurements made by the instrument using single-frequency alternating current. An estimate of the radial distribution of the conductivities is generated from the shoulder corrected induction signals acquired using the single-frequency alternating current.

Abstract: A method of compressing digital signals for transmission and/or storage. The method includes lossy compressing the digital signal to generate a lossy compressed signal. The compressed signal is decompressed locally to generate a local lossy decompressed signal. An error signal is calculated as the difference between the original digital signal and the local lossy decompressed signal. The power in the original digital signal and in the error signal is determined. A minimum resolution for quantizing the error signal is determined from a ratio of the power of the original signal with respect to power of the error signal. The error signals is lossy compressed by quantizing to at least the minimum resolution. Both the compressed digital signal and the compressed error signal are transmitted and/or stored. The lossy compressed error signal and the lossy compressed signal are recovered from storage or transmission.

Abstract: A marine seismic signal is transformed from time domain into frequency domain and represented by matrix D. The marine data signal is truncated in time, transformed into the frequency domain and represented by matrix D.sub.T. Eigenvalue decomposition D.sub.T =S.multidot..LAMBDA..multidot.S.sup.-1 of matrix D.sub.T is computed. Matrix product D.multidot.S is computed and saved in memory. Matrix inverse S.sup.-1 is computed and saved in memory. An initial estimate of the source wavelet w is made. Source wavelet w is computed by iterating the steps of computing diagonal matrix [I-w.sup.-1 .LAMBDA.], computing matrix inverse [I-w.sup.-1 .LAMBDA.].sup.-1, retrieving matrix product D.multidot.S and matrix inverse S.sup.-1 from memory, and minimizing the total energy in matrix product [D.multidot.S] [I-w.sup.-1 .LAMBDA.].sup.-1 S.sup.-1. Primary matrix P representing the wavefield free of surface multiples is computed by inserting computed value for w into the expression [D.multidot.S] [I-w.sup.-1 .LAMBDA.].sup.-1 S.