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Optimal solutions to a linear inverse problem in geophysics.

T H Jordan1, J N Franklin

  • 1California Institute of Technology, Pasadena, Calif. 91109.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1971
PubMed
Summary
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This study optimizes solutions for Earth

Area of Science:

  • Geophysics
  • Inverse Problems
  • Data Analysis

Background:

  • The Backus-Gilbert formulation addresses inverse problems in geophysics.
  • Ill-posed linear systems are common in geophysical data analysis.
  • Stochastic extensions offer methods to handle ill-posed problems.

Purpose of the Study:

  • To apply Franklin's theory of well-posed stochastic extensions to geophysical inverse problems.
  • To develop an optimal solution for linear systems in Earth data analysis.
  • To incorporate prior information about solution smoothness into the model.

Main Methods:

  • Utilizing Franklin's theory for well-posed stochastic extensions.
  • Solving linear systems derived from the Backus-Gilbert formulation.

Related Experiment Videos

  • Implementing a prescribed linear filter to constrain solutions.
  • Estimating statistical variance of noise in geophysical data.

    • Main Results:

    • An optimal solution is derived for the inverse problem under specific constraints.
    • The method allows the integration of prior knowledge on solution smoothness.
    • A preliminary model for Earth's interior density and shear velocity is presented.

    Conclusions:

    • The developed method provides a robust approach to solving ill-posed geophysical inverse problems.
    • Incorporating prior smoothness information enhances the reliability of Earth models.
    • This framework is applicable to various geophysical inverse problems with noisy data.