Regularisatie Methodes voor Seismische Tomografie (Regularization Methods for Seismic Tomography)

Regularisatie Methodes voor Seismische Tomografie (Regularization Methods for Seismic Tomography)

Caljouw, C.

Van Gijzen, M.B. (mentor)

Electrical Engineering, Mathematics and Computer Science

Numerical Analysis

2015-07-03

In seismic tomography, parts of the Earth are reconstructed by the travel times of seismic waves that are measured by seismometers. These tomographic problems can be formulated as a system of linear equations. Unfortunately, most of the time the system is ill-posed, because of ray bending and noise. This problem can not only be found in geophysics but also in other sciences, like radiology and astrophysics. Regularization methods replaces the ill-posed problem with a "nearby" well-posed problem whose solution is a good approximation of the true solution. The main goal of this report is to understand two regularization methods, Tikhonov and another regularization which we will call the Saunders regularization, and to investigate when these regularization methods are equivalent and which regularization matrices can be used for Saunders regularization. Saunders and Tikhonov regularisation are equivalent when the identity matrix is used. But when an arbitrary matrix is used, the regularization methods are not equivalent. We also examined choices for the regularization matrix for both methods. It was not very clear which matrix one should use for the Saunders regularization, but for Tikhonov regularization method we used a smoothing operator for the solution in both horizontal en vertical direction. For Saunders regularization we also used a smoothing operator but instead for the solution, a smoothing operator for the residuals as regularization matrix.

Regularization Methods
Tikhonov

http://resolver.tudelft.nl/uuid:c46132e4-5bcb-452c-9354-dbd73f7ed82d

Student theses

bachelor thesis

(c) 2015 Caljouw, C.