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ESTIMATION OF DISCONTINUOUS COEFFICIENTS AND BOUNDARY PARAMETERS FOR HYPERBOLIC SYSTEMS

IP.com Disclosure Number: IPCOM000149109D
Original Publication Date: 1899-Dec-30
Included in the Prior Art Database: 2007-Apr-12
Document File: 41 page(s) / 2M

Publishing Venue

Software Patent Institute

Related People

Lamm, Patricia K.: AUTHOR [+3]

Abstract

ESTIMATION OF DISCONTINUOUS COEFFICIENTS AND BOUNDARY PARAMETERS FOR HYPERBOLIC SYSTEMS Patricia K. Lamm Southern Methodist University Katherine A. Murphy Brown UniversFty ABSTRACT We consider the problem of estimating discontinuous coefficients, including locations of discontinuities, that occur in second order hyperbolic systems typical of those arising in 1-D surface seismic problems. In addition, we treat the problem of identifying unknown parameters that appear in boundary conditions for the system. A spline-based approximation theory is presented, together with related convergence findings and representative numerical examples. Research supported in part by NSF grant DMS-8200883 (P.K.L.), and NSF grants DMS-8205355, DMS-8504316 and AFOSR contract 84-0398 (K.A.M.). Parts of the research were carried out while both authors were visitors at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA which is operated under NASA contract Nos. NAS1- 17070 and NAS1-18107.

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ESTIMATION OF DISCONTINUOUS COEFFICIENTS
AND BOUNDARY PARAMETERS FOR

HYPERBOLIC SYSTEMS

     Patricia K. Lamm
Southern Methodist University

Katherine A. Murphy
Brown UniversFty

ABSTRACT

    We consider the problem of estimating discontinuous coefficients,
including locations of discontinuities, that occur in second order hyperbolic
systems typical of those arising in 1-D surface seismic problems. In
addition, we treat the problem of identifying unknown parameters that appear
in boundary conditions for the system. A spline-based approximation theory is
presented, together with related convergence findings and representative
numerical examples.

Research supported in part by NSF grant DMS-8200883 (P.K.L.), and NSF grants
DMS-8205355, DMS-8504316 and AFOSR contract 84-0398 (K.A.M.). Parts of the
research were carried out while both authors were visitors at the Institute
for Computer Applications in Science and Engineering (ICASE), NASA Langley
Research Center, Hampton, VA which is operated under NASA contract Nos. NAS1-
17070 and NAS1-18107.

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1. Introduction.

In this paper we consider a one-

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                         imensional seismic inverse problem: Our goal is to construct a parameter stimation scheme for the hyperbolic

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model equations treated in [7] extend4ng these ideas to allow the estimation of discontinuous coefficients (incluJ ng the location of discontinuities). The approach taken here contrasts wit4 that taken in [7] i n that we consider a different decomposition of the wave equation, yielding a different operator and state space for theoretical argum nts. We combine these ideas with a

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variation of the approximation schemein [I51 which was developed to treat the problem of estimating discontinuous c efficients in parabolic systems.

4

   The underlying theoretical appro ch to the identification problem follows
the outline of other related papers ( g. [3], 1141, [5], [6], [7], [I51 in that we define the (infinite di mensional ) i de tification problem and construct associated a~proximate identification problems; nder compactness assumptions on our parameter space, we prove convergence of parame er estimates and approximating state

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variables. Our arguments here are baded on an application of the Trotter-Kato Theorem. In Section 5 we describe t h numerical algorithm and conclude with some prel imi nary computational exampl

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