Publication Date

8-2017

Date of Final Oral Examination (Defense)

5-24-2017

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Supervisory Committee Chair

Jodi L. Mead, Ph.D.

Supervisory Committee Member

John Bradford, Ph.D.

Supervisory Committee Member

Grady Wright, Ph.D.

Abstract

The first mention of joint inversion came in [22], where the authors used the singular value decomposition to determine the degree of ill-conditioning in inverse problems. The authors demonstrated in several examples that combining two models in a joint inversion, and effectively stacking discrete linear models, improved the conditioning of the problem. This thesis extends the notion of using the singular value decomposition to determine the conditioning of discrete joint inversion to using the singular value expansion to determine the well-posedness of joint linear operators. We focus on compact linear operators related to geophysical, electromagnetic subsurface imaging.

The operators are based on combining Green’s function solutions to differential equations representing different types of data. Joint operators are formed by extend- ing the concept of stacking matrices to one of combining operators. We propose that the effectiveness of joint inversion can be evaluated by comparing the decay rate of the singular values of the joint operator to those from the individual operators. The joint singular values are approximated by an extension of the Galerkin method given in [9, 18]. The approach is illustrated on a one-dimensional ordinary differential equation where slight improvement is observed when naively combining differential equations. Since this approach relies primarily on the differential equations representing data, it provides a mathematical framework for determining the effectiveness of joint inversion methods. It can be extended to more realistic differential equations in order to better inform the design of field experiments.

DOI

https://doi.org/10.18122/B28H89

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