Abstract: A large number of deconvolution procedures have appeared in the literature during the last three decades, including a number of maximum‐likelihood deconvolution (MLD) procedures. The major advantages of the MLD procedures are (1) no assumption is required about the phase of the wavelet (most of the classical deconvolution techniques assume a minimum‐phase wavelet, an assumption that may not be appropriate for many data sets); (2) MLD procedures can resolve closely spaced events (i.e., they are high‐resolution techniques); and (3) they can efficiently handle modeling and measurement errors, as well as backscatter effects (i.e., reflections from small features). A comparative study of six different MLD procedures for estimating the input of a linear, time‐invariant system from measurements, which have been corrupted by additive noise, was made by using a common framework developed from fundamental optimization theory arguments. To date, only the Kormylo and the Chi‐t algorithms can be recommended.

Abstract: Modeling of Physical systems can be decomposed into four subproblems: representation, measurement, estimation, and validation. This paper shows how this decomposition can be applied to the reflection seismology problems of inversion and deconvolution, and that reflection seismology is a very rich field for practitioners of parameter estimation, system identification, and signal processing.

Abstract: A new algorithm to solve convolution systems of linear equations is described. Modular arithmetic and the Fermat number theoretical transform are used. The algorithm permits elimination of the rounding-off errors of the solution of the deconvolution problem.

Abstract: The presence of random additive noise is the most important degrading factor in the deconvolution of seismic data. Noise-induced distortion of signal phase and amplitude produces severe stack attenuation, makes poststack recovery difficult with spectral enhancement techniques, and leaves the stratigraphic imprint unclear. The random noise component in the data is estimated from trace segments before the first arrivals and at the bottom of the record beyond seismic basement. An autocorrelation of this noise is used to adjust the signal autocorrelation prior to Wiener-Levinson deconvolution filter design. To improve the robustness of the technique, an iterative surface-consistent wavelet solution (common source, receiver, and offset) is used in preference to a single-channel operation. Use of this deconvolution technique is shown by synthetic and case examples to result in correct phase alignment, enhanced stacking fidelity, and extended signal bandwidth even on very noisy data. The improvement, coupled with sensible handling of coherent noise energy, is crucial for the interpretation of subtle stratigraphic plays in many areas.

Abstract: The purpose of this note is to present a straightforward and unified approach for deriving the fixed-point and fixed-interval recursive algorithms of minimum-variance deconvolution. It is different from the approaches derived before.

Abstract: A shift-invariant spatial deconvolution function for single-photon-emission computerized tomography with constant attenuation is presented. Image reconstruction algorithm is similar to conventional convolution-back-projection algorithm except that exponential weight is applied in backprojection process. The deconvolution function was obtained as a solution of a generalized Schifmilch's integral equation. A method to solve the integral equation is described briefly. The present deconvolution function is incorporated with frequency roll-off and image resolution can be preset. At the extreme of ideal image reconstruction, the deconvolution function is identical to that deduced by Kim et al. and its Fourier transform was proved to be identical to the filter deduced by Tretiak and Delaney and Gullburg and Budinger. Variance of the reconstructed image was analyzed and some numerical results were given. The algorithm was tested with computer simulation.

Abstract: Wave theory justifies both pre‐ and post‐NMO deconvolution, but the filters should be estimated simultaneously, not sequentially A linearized theory enables simultaneous estimation of the two filters The theory incorporates spherical divergence independently from statistical weighting Field data test cases show the expected interaction between NMO and deconvolution The tests were not able to establish that simultaneous estimation is superior to sequential estimation The difficulty is ascribed to the inadequacy of NMO as a downward continuation process

Abstract: Deconvolution of data in the Fourier domain is sometimes hampered by the presence of zeros in the kernel. A time domain deconvolution method is proposed that can circumvent such difficulty. Added advantages are: 1) a higher signal-to-noise ratio often attainable in the deconvolved result compared to the Fourier domain solution (even with nonsingular kernels), and 2) a saving in computational speed when the unknown signal is short compared to the kernel. Numerical illustrations and application to motion blur correction are also given.

Abstract: Wiener ‘spiking’ deconvolution of seismic traces in the absence of a known source wavelet relies upon the use of digital filters, which are optimum in a least-squares error sense only if the wavelet to be deconvolved is minimum phase In the marine environment in particular this condition is frequently violated, since bubble pulse oscillations result in source signatures which deviate significantly from minimum phase The degree to which the deconvolution is impaired by such violation is generally difficult to assess, since without a measured source signature there is no optimally deconvolved trace with which the spiked trace may be compared A recently developed near-bottom seismic profiler used in conjunction with a surface air gun source produces traces which contain the far-field source signature as the first arrival Knowledge of this characteristic wavelet permits the design of two-sided Wiener spiking and shaping filters which can be used to accurately deconvolve the remainder of the trace In this paper the performance of such optimum-lag filters is compared with that of the zero-lag (one-sided) operators which can be evaluated from the reflected arrival sequence alone by assuming a minimum phase source wavelet Results indicate that the use of zero-lag operators on traces containing non-minimum phase wavelets introduces significant quantities of noise energy into the seismic record Signal to noise ratios may however be preserved or even increased during deconvolution by the use of optimum-lag spiking or shaping filters A debubbling technique involving matched filtering of the trace with the source wavelet followed by optimum-lag Wiener deconvolution did not give a higher quality result than can be obtained simply by the application of a suitably chosen Wiener shaping filter However, cross correlation of an optimum-lag spike filtered trace with the known ‘actual output’ of the filter when presented with the source signature is found to enhance signal-to-noise ratio whilst maintaining improved resolution

Abstract: Burg's maximum entropy method has been used with success in spectral estimation. This paper is an attempt to generalize the maximum entropy method to the deconvolution of positive signals from a finite set of linear measurements. In this paper, we also investigate the existence of a solution to the deconvolution problem using a geometric approach. By formulating the deconvolution problem as an unconstrained optimization problem, our procedure will provide the maximum entropy deconvolved positive signal when it exists. Otherwise, the algorithm diverges.

Abstract: We consider the problem of estimating the filter generating a non-Gaussian linear process and the decon-voiution of that process when the spectral density of the process has zeros. Without using a minimum phase assumption we show that often if there are only finitely many zeros there are procedures to effect such an estimation and deconvolution.

Abstract: Within the scope of the Fresnel diffraction theory the threedimensional transfer function OTF of the microscope was obtained. Inverse filtering with OTF-1 can be used for the reconstruction of 3-D light-microscopic images. Comparison with Stokseth's approximation for the optical transfer function used in other works on 3-D reconstruction shows a rather large relative error for this approximation. Transforming OTF-1 back into the spatial domain a closed form for the inverse point spread function IPSF (with respect to convolution of the point spread function PSF) was obtained. Direct deconvolution of 3-D images in the spatial domain proofs to be advantageous compared to deconvolution in the frequency domain. Closer examination of the reconstruction procedure shows that the spatial resolution defined by the Rayleigh criterion can be improved by a factor 1.8 perpendicular and by 1.5 parallel to the optical axis.

Abstract: Restoration of an image distorted by a linear shift-invariant system is a 2-D deconvolution problem which is treated here in a Bayesian framework to stabilize the solution. The usual introduction of dynamics into the state equation to reduce the problem dimensions requires an artificial causality assumption. Thus we propose state-space models where the state is constant and equal to the entire object to be restored, and where dynamics appear only in the observation equation which may be either vectorial or scalar. When the image is scanned row by row, the shift properties of the convolution summation allow derivation of a fast optimal Kalman filter through factorization techniques. When the image is scanned pixel by pixel, the computational requirement can be further reduced at the expense of an extra assumption. Finally, a sub-optimal asymptotic filter with a reduced update is derived.

Abstract: The method of recursive least squares, which has been used extensively in the field of system identification will be developed for adaptive parametric spectral estimation of digital signals. The method will be applied for adaptive pulse estimation and deconvolution of seismic data. Unlike the Levinson-type deconvolution method, the adaptive least-squares method does not need to estimate a priori the autocorrelation function and it avoids the windowing problem of a time-limited signal. Because of the recursive nature of the method, it is suitable for adaptive estimation and removal of time-varying pulses, and it is computationally simple and suitable for implementation on small computers with less memory requirements. The method has been implemented on different simulated examples, and the results are given and discussed.

Abstract: We propose a method to model a signal as the output of an all-pole linear system, with noncausal impulse response, driven by a mixed signal (sparse impulses plus noise sequence). The filter parameters and the excitation sequences are obtained by separating the output of a forward-backward predictor into two signals. The mean absolute value of the first one is minimized and the absolute values of the second one are hounded. Its features and advantages with respect to other techniques are pointed out.

Abstract: This paper is concerned with the estimation of the target magnitude response given the non-coherent receiver measurements and the complex information of the transmitted signal. This is a signal reconstruction problem from partial information which involves phase retrieval as well as deconvolution. Nonlinear least squares and error reduction algorithm are formulated for solving this complex non-coherent deconvolution problem. Some preliminary experimental results are included.

Abstract: : Deconvolution in the presence of additive noise is a well known problem for which there exists a Wiener filter which simultaneously spectrally whitens while suppressing noise. A simple variant of this standard Wiener filter incorporates a parameter, P say, which is intended to allow further weight to be given to noise suppression. We shall call such a filter a modified Wiener filter. To design such a filter it is required to know precisely the frequency response of the spread function or wavelet, plus the spectra of the input and additive noise. In practice some response function is taken to be appropriate, and the modified Wiener filter designed from it. If the design response function is thought of as one chosen from a set of allowable response functions - a realistic practical viewpoint - then it is shown how the selection of the design response, the chosen value of the parameter P and the noise/input power spectral ratio effectively determine the characteristics of this set of possible wavelet response functions. This is demonstrated for two different error criteria - (i) the minimization of the average mean-squared error, and (ii) the minimization of the maximum mean-squared error. It is shown how to calculate deconvolution filters which solve sub-optimal versions of (i) and (ii), but which are robust to uncertainty in the wavelet's frequency response.

Abstract: Background and Overview. 1. Stochastic Processes and Models. 2. Wiener Filters. 3. Linear Prediction. 4. Method of Steepest Descent. 5. Least-Mean-Square Adaptive Filters. 6. Normalized Least-Mean-Square Adaptive Filters. 7. Transform-Domain and Sub-Band Adaptive Filters. 8. Method of Least Squares. 9. Recursive Least-Square Adaptive Filters. 10. Kalman Filters as the Unifying Bases for RLS Filters. 11. Square-Root Adaptive Filters. 12. Order-Recursive Adaptive Filters. 13. Finite-Precision Effects. 14. Tracking of Time-Varying Systems. 15. Adaptive Filters Using Infinite-Duration Impulse Response Structures. 16. Blind Deconvolution. 17. Back-Propagation Learning. Epilogue. Appendix A. Complex Variables. Appendix B. Differentiation with Respect to a Vector. Appendix C. Method of Lagrange Multipliers. Appendix D. Estimation Theory. Appendix E. Eigenanalysis. Appendix F. Rotations and Reflections. Appendix G. Complex Wishart Distribution. Glossary. Abbreviations. Principal Symbols. Bibliography. Index.