31 research outputs found

    Kinematic artifacts in prestack depth migration.

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    Strong refraction of waves in the migration velocity model introduces kinematic artifacts¿coherent events not corresponding to actual reflectors¿into the image volumes produced by prestack depth migration applied to individual data bins. Because individual bins are migrated independently, the migration has no access to the bin component of slowness. This loss of slowness information permits events to migrate along multiple incident-reflected ray pairs, thus introducing spurious coherent events into the image volume. This pathology occurs for all common binning strategies, including common-source, common-offset, and common-scattering angle. Since the artifacts move out with bin parameter, their effect on the final stacked image is minimal, provided that the migration velocity model is kinematically correct. However, common-image gathers may exhibit energetic primary events with substantial residual moveout, even with the kinematically accurate migration velocity model

    A Differential Semblance Criterion for Inversion of Multioffset Seismic Reflection Data

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    Mean-square error leading to least-squares inversion of multioffset reflection seismograms is insensitive to velocity trend information except in the immediate vicinity of a kinematically correct model. In contrast, differential semblance retains sensitivity to velocity trend changes over a wide range of models. The differential semblance criterion combines mean-square error with the mean-square differences of inverted models from datasets at neighboring shot positions (or offsets, or slownesses, ...). Differential semblance compares model estimates at nearby acquisition parameters which are similar even when the model velocity trends are incorrect. Because the method inverts the data, so that the estimated model amplitudes are meaningful, simple differences between the (unstacked) model estimates give a reliable measure of velocity error. A mathematical investigation indicates that the differential semblance criterion is smooth and convex over a large range of velocity models. Numerical simulation using synthetic data sets verifies this contention

    Iso-spectral deformations of general matrix and their reductions on Lie algebras

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    We study an iso-spectral deformation of general matrix which is a natural generalization of the Toda lattice equation. We prove the integrability of the deformation, and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szeg\"{o}. Based on the root spaces for simple Lie algebras, we consider several reductions of the hierarchy. These include not only the integrable systems studied by Bogoyavlensky and Kostant, but also their generalizations which were not known to be integrable before. The behaviors of the solutions are also studied. Generically, there are two types of solutions, having either sorting property or blowing up to infinity in finite time.Comment: 25 pages, AMSLaTe

    The Plane-Wave Detection Problem

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    The plane-wave detection problem is: to estimate the incidence angle and waveform of a transient plane traveling wave, from samples recorded at a linear array of receivers. This simple problem shares several important mathematical features with other inverse problems of wave propagation, and is of interest in its own right as a model problem in ocean acoustic signal analysis. Straightforward formulation as a nonlinear least squares problem yields a nonconvex objective for which the minima are not stably dependent on the data. In contrast, an infeasible point formulation, in which the signal at each receiver is explained to some extent independently, proves to yield a smooth convex optimization problem with stable optima. Numerical experiments illustrate the theoretical results about the infeasible point approach, differential semblance optimization

    Hilbert class library: A library of abstract C++ classes for optimization and inversion

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    AbstractAccording to the Object-Oriented Programming paradigm, a computer program should be organized around the fundamental objects it manipulates. In the C++ programming languages, these objects are embodied in classes. The Hilbert Class Library (HCL) is a collection of C++ classes designed for implementing numerical optimization algorithms in the context of Hilbert spaces. HCL includes base classes for defining vectors, linear operators, nonlinear operators and functionals, and related mathematical objects. Using these base classes, algorithms can be coded in a natural style that does not refer to application-specific details; nonetheless, the code can be applied to arbitrarily complex applications. Thus, HCL is intended to provide a way to bridge the often large gap between sophisticated numerical optimization routines and complicated simulation-based applications

    Domain Decomposition Algorithms for Linear Hyperbolic Equations

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    The use of parallel computers for solving partial differential equations is important in areas such as fluid dynamics, reservoir simulation, and structural analysis, where many of the problems of interest cannot be solved without the use of supercomputers. One technique for applying parallel computers to the solution of these problems is known as domain decomposition where the domain of interest is subdivided into several smaller subdomains and the task of solving the partial differential equation on each subdomain problem is assigned to a different processor. The global solution is then synthesized from the solutions computed on the individual subdomains. Much of the current work in the application of domain decomposition techniques has been in the area of elliptic partial differential equations, with very little attention being given to hyperbolic equations. We propose to use the methods of domain decomposition for the solution of linear hyperbolic equations. The idea of using overlapping domains is introduced in the context of linear hyperbolic equations to develop a domain decomposition algorithm which is shown to be well suited for parallel processors. The issues of communication costs and load balancing are addressed and a simple strategy for assigning jobs to processors to achieve load balancing is presented

    Deflated Krylov Subspace Methods for Nearly Singular Linear Systems

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    This paper concerns the use of Krylov subspace methods for the solution of nearly singular nonsymmetric linear systems. We show that the Incomplete Orthogonalization Methods (IOM) in conjunction with certain deflation techniques of Stewart and Chan can be used to solve large nonsymmetric linear systems which are nearly singular
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