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Other title
Numerical solution
Geophysical modelling
Inverse problem
Fast Fourier transform
Type of item
Degree grantor
University of Alberta
Author or creator
Zhang, Yile
Supervisor and department
Wong, Yau Shu (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Han, Bin (Department of Mathematical and Statistical Sciences)
Yu, Xinwei (Department of Mathematical and Statistical Sciences)
Wong, Yau Shu (Department of Mathematical and Statistical Sciences)
Minev, Peter (Department of Mathematical and Statistical Sciences)
Anton, Cristina (Department of Mathematical and Statistical Sciences)
Sivaloganathan, Siv (Department of Applied Department of Applied Mathematics, University of Waterloo)
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
Doctor of Philosophy
Degree level
Mathematical model and numerical computation play a pivotal role in modern geophysical exploration. By applying computational algorithms to the observed field data, the underground structure can be inferred. This process is generally referred as a geophysical inversion problem. However, due to the model complexity, numerical stability and computing time, solving a geophysical inversion problem is a very challenging task. A typical inversion problem may involve several million of unknowns, and this frequently requires considerable amount of computing time even by using a super-workstation. This thesis focuses on modelling and developing fast and efficient numerical algorithms for geophysical exploration. By recognizing a Block-Toeplitz Toeplitz-Block (BTTB) structure in a potential field inversion problem and combining the conjugate gradient method with the BTTB structure, a class of efficient numerical schemes are proposed. From the simulation results applied to synthetic and field data, we conclude that the proposed schemes significantly improve the stability and accuracy of a downward continuation problem, and they are more superior to the existing methods. Since a regularization process inherently induces distortion in the inversion solution, we construct a novel non-regularized inversion scheme based on a multigrid (MG) technique. The MG based scheme not only preserves the stability of a regularization method, but it also induces less distortion in the reconstructed magnetization solution. We expand our 2D results to a 3D gravity field inversion by proposing a 2D multi-layer model to approximate the density distribution. Based on the multi-layer model, an efficient 3D inversion scheme is proposed, in which all formulation including the regularization, preconditioning and inversion are conducted under a BTTB-based framework. Mathematical analysis for convergence and consistency are presented, and a multi-resolution simulation confirms the efficiency and accuracy of the proposed numerical scheme. As an indispensable tool in high precision exploration, electromagnetic (EM) method is frequently applied to reconstruct the conductivity distribution. We propose an implicit ADI-FDTD scheme to model the diffusion behavior of the EM wave. The time and space grids in our proposed scheme can be much larger than that used in the conventional Du-Fort-Frankel method, while more accurate numerical solution is obtained. Numerical analysis and computational simulation are presented to demonstrate the effectiveness of the proposed scheme.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
Y. Zhang and Y. S. Wong. BTTB-based numerical schemes for threedimensional gravity field inversion. Geophysical Journal International, 203(1): 243 - 256, 2015.Y. Zhang, Y. S. Wong, J. Deng, S. Lei, and J. Lambert. Numerical inversion schemes for magnetization using aeromagnetic data. International Journal of Numerical Analysis & Modeling, 12(4): 684 - 703, 2015.Y. Zhang, Y. S. Wong, and Y. Lin. BTTB-RRCG method for downward continuation of potential field data. Journal of Applied Geophysics, 126: 74 - 86, 2016.W. Li, Y. Zhang, Y. S. Wong, and D. Liang. ADI-FDTD method for twodimensional transient electromagnetic problems. Communications in Computational Physics, 19(1): 94 - 123., 2016.

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