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matrix computations, eigenvalues, QR algorithmEach iteration of the multishift QR algorithm of Bai and Demmel requires the computation of a "shift vector" defined by m shifts of the origin of the spectrum, which control the convergence of the process. A common choice of shifts consists of the eigenvalues of the trailing principal submatrix of order m, and current practice includes the computation of these eigenvalues in the determination of the shift vector. In this paper, we describe an algorithm based on the evaluation of the characteristic polynomial of a Hessenberg matrix, which directly produces the shift vector without computing eigenvalues. This algorithm is stable, more accurate, faster, and simpler than the current alternative. It also allows for a consistent shift strategy with dynamic adjustment of the number of shifts. ABSTRACTEach iteration of the multishift QR algorithm of Bai and Demmel requires the computation of a "shift vector" defined by m shifts of the origin of the spectrum, which control the convergence of the process. A common choice of shifts consists of the eigenvalues of the trailing principal submatrix of order m, and current practice includes the computation of these eigenvalues in the determination of the shift vector. In this paper, we describe an algorithm based on the evaluation of the characteristic polynomial of a Hessenberg matrix, which directly produces the shift vector without computing eigenvalues. This algorithm is stable, more accurate, faster, and simpler than the current alternative. It also allows for a consistent shift strategy with dynamic adjustment of the number of shifts.

This paper describes schemes for the frequency‐domain migration of constant‐offset sections in homogeneous and horizontally layered media. Their main advantage is that individual constant‐offset sections can be processed in isolation. The basis for these schemes is a generalization of the following plane‐wave analysis of migration for the zero‐offset case. Migration can be viewed in the time domain as a process whereby the plane waves echoed from a source to a geophone by a reflecting point are shifted in time to a common intersection. The shift applicable to each plane wave can be derived from the knowledge of the diffraction curve for the reflecting point in a neighborhood of the geophone abscissa. This time shift in turn determines the phase shift applicable in the frequency domain to the associated Fourier component of the seismic section. The central part of our methods is the construction of those elements of the diffraction curve pertinent to the shifts from the velocity map of the time section. The result is a table of shifts for an appropriate sample of plane‐wave directions, to be used for interpolation in the migration process. The effectiveness of the method is illustrated by computer experiments with synthetic data.

The phase shift method (Gazdag, 1978) is based on the solution, in the frequency domain, of an approximation (Claerbout, 1976) to the one‐way wave equation with initial conditions defined by a zero‐offset seismic section. Wave velocity is assumed to be constant within each layer of the section grid and is allowed to vary from layer to layer. Under these conditions, the equation written in the frequency domain reduces to a system of independent ordinary differential equations with initial values that can be solved analytically for each layer. The integration process simply amounts to multiplying the initial values by a complex number of unit modulus. The main advantages of this method are simplicity, stability, and high accuracy, since the precision of the Fourier approximation is limited only by the granularity of the seismic section. From a practical viewpoint of computer implementation, the phase shift method offers a great deal of flexibility. Some accuracy can be traded for speed, as needed, by excluding the waves traveling at an angle that exceeds a specified limit, or by ignoring frequencies at the higher end of the spectrum. The migration of a group of reflections can be excluded as well, since the phase shift angles can be accumulated for application to the frequency representation of the section only within a specified depth interval. The most interesting feature of the method may well reside in the fact that the Fourier coefficients of a seismic section can be treated independently of one another. This is of particular importance if the method is implemented for an array processor. In this case, cross‐sections of the frequency data can be sent to the array processor one‐by‐one (or in groups, as may be suitable) for phase shift operations, at minimal storage cost. A formulation of the method for an array processor is outlined here, which takes advantage of the features mentioned above. It is described in a step‐wise algorithmic fashion, using natural language and standard mathematical notations. Functions for discrete Fourier transformations, evaluation of square roots and circular functions (or fast table look‐up or interpolation), and vector reversal are assumed to exist in the array processor, in addition to elementary vector operations. This formulation can also be used for efficient implementation on conventional computers.

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