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Book Chapter

Exercises

Published:
January 01, 1997

Abstract

  • Derive the Wave Equation

    The two most important physical laws underlying the wave equation are Newton’s second law of motion and Hooke’s Law. Use these to derive the wave equation for sound waves in a thin rod (1D Wave Equation).

  • Derivation of Derivatives, Finite differencing of the Wave Equation. Second derivatives are the essence of the wave equation. In order to compute these second derivatives we need to relate the data to the derivatives. The Taylor Series does this for us. Apply the Taylor Series and develop the expressions for f(h) and f(−h) to compute the f″(0).

  • Grid Dispersion

    The rule of thumb for avoiding grid dispersion using second order methods is to use 10 grid points per wavelength.

  • Transparent Reflectors

    A nifty idea for suppressing multiples in reverse-time migration is to make the reflectors transparent. The variable density wave equation is used because the density can be manipulated to make the reflection coefficient zero or very small.

    For those more mathematically inclined, the iteration matrix represented by the finite difference operator has eigenvalues. Violation of the stability criteria causes an eigen-value greater than one, and any number greater than one raised to a large power quickly goes to infinity.

  • Model Definition Sketch in the box below a rather simple geological model. Assume you are going to generate a set of synthetic shot records. Place the model dimensions in the upper right hand corner and the lower left hand corner. Pencil in the interval velocity values for each layer.

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Contents

Society of Exploration Geophysicists Course Notes

Seismic Modeling and Imaging with the Complete Wave Equation

Ralph Phillip Bording
Ralph Phillip Bording
Institute for Geophysics University of Texas at Austin
Austin, Texas
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Larry R. Lines
Larry R. Lines
Department of Geology and Geophysics University of Calgary
Calgary, Alberta
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Society of Exploration Geophysicists
Volume
8
ISBN electronic:
9781560801870
Publication date:
January 01, 1997

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