Graduate and Postdoctoral Studies
Robust computation of seismic normal modes with Rayleigh-Ritz method in a spherically symmetric earth
Wednesday, April 5, 2017
to 3:00 PM
303 Sewall Hall
People have been dedicating huge amount of efforts to study the seismic normal modes for decades and the theory has been well established. However, the numerical computation is still a long way off, even in a spherically symmetric non-rotating Earth. The most popular software Mineos suffers from the clustering eigenvalue problem and also fails to calculate some special modes accurately. Hence our motivation is to correct these issues and furthermore achieve higher accuracy. In this work we extend Wiggins' and Buland's work, and reformulate the Sturm-Liouville problem to a generalized eigenvalue problem with Rayleigh-Ritz Galerkin method. Our new scheme is absolutely absent from the artificial degeneracies, and we can compute the aforementioned special modes accurately, even for Stoneley modes with exponentially decaying behavior across the solid-fluid boundary.
The weak variational form is utilized to perform the Rayleigh-Ritz procedure, which contributes to preserving the high accuracy across the solid-fluid boundary. Moreover, it avoids the artificial singularity at the center of the Earth, and therefore abandons the unstable shooting method. To project out the non-seismic modes concentrated in the fluid outer core, we propose a remarkable modified curl-free formula starting from Helmholtz decomposition. Based on Lehoucq’s shift-invert strategy, we employ a sparse, direct solver provided by ARPACK to speed up the algorithm.
Our work has numerous applications. Now we can confidently rely on the purely orthogonal eigenfunctions to study the modes classification, and furthermore perform the mode coupling to cast light on the deep Earth's structure. Additionally, we can provide the benchmark for our ongoing parallel package of three-dimensional non-symmetric normal modes computation.