Some Advances in Computational Geophysics: Seismic Wave and Inverse Geodynamic Modeling

In this thesis, I develop computational methods that link theory with geophysical observations, with one part devoted to the development of forward models of seismic wave propagation through the mantle and core, and a second part devoted to the inversion of viscous flow in the mantle. First order seismic structure of the earth has been well described radially since the PREM model was introduced. With the help of seismic tomography methods, many large-scale heterogeneous structures have become well imaged. Based on this progress, the information in seismic waveforms, which provides extra constraints, is becoming more important in determination of the detailed structure within the earth's interior. However, 3-D modeling of seismic wave propagation remains computationally expensive, especially at high frequency, because the computing cost scales with fourth power of frequency. Thus 2-D modeling is often used, and in many cases is sufficient for the problem. To use 2-D modeling in global seismology, several issues need to be considered: how to handle the differences in geometric spreading between 2-D and 3-D modeling, how to incorporate earthquake sources into a 2-D code, and how to handle the spherical geometry of the earth. In the first part of my thesis, we solve all three problems, using a 2-D staggered finite difference method with a post-processing step. The post-processing automatically corrects the geometric spreading difference between 2-D and 3-D wave propagation; the earthquake sources are added to the 2-D finite difference simulation using a momentum source and transparent box approaches; the earth-flattening is discussed, especially for the density transformation. Benchmarks of the new method against with 1-D and 3-D code demonstrates the the accuracy of the method. We then use the new code in a study of the interface between outer and inner core. Inner Core Boundary (ICB) is thought to be crucial in estimating the energy released in generating the geomagnetic field. One direct constraint on ICB properties is using reflected P wave from ICB, the PKiKP phase. Due to its small amplitude, near distance PKiKP is seldom observed. However, we find several events beneath Central American as having good set of PKiKP recordings from the USArray seismic network, as well as other core phases like P wave reflection with Core Mantle Boundary (CMB). The amplitude of the phases display large scatters across the stations, which are potentially caused by many factors, including site effects of the stations, upper mantle inhomogeneity, or a bumpy structure along either the CMB or ICB. After comparing amplitude ratio of between PKiKP and PcP phase, analyzing how this ratio changes for different nearby events, and computing forward models using our new method that investigate different factors influence the PKiKP phases, we attribute a stacked amplitude pattern as caused by ICB structure, in which PKiKP phase amplitude rapidly changes within a small range. Finally, we model this observed seismic pattern as a small dome-like anomalies above ICB, where the material changes from that of the outer core to that of inner core gradually. The final part of my thesis is on a geodynamic inversion problem for mantle convection. Mantle convection is an important process that determines plate motions and subduction. Numerous forward models indicate that the constitute relation (viscosity law) is of key importance for mantle convection. Despite substantial effort attempting to determine the viscosity structure of the mantle, either through forward and inverse geophysical models or through laboratory work, many first order questions remain. We assume the realistic viscosity structure, which is temperature and strain-rate dependent, can be parameterized using a set of scalar parameters. Given this set of viscosity parameters and an initial temperature, the mantle evolves following a set of partial differential equations (PDEs). Our goal with the inverse problem is to recover the viscosity parameters and initial temperature by fitting the observational data, which here includes plate motion history and the present day temperature distribution of the mantle. We formulate this inversion problem following a PDE constrained optimization framework. We first define the cost function we want to minimize; then, the derivative of the cost function with respect to viscosity parameters and initial temperature is calculated following the discrete adjoint equations; finally, a gradient-based optimization method, limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) approach is used to find the minimum. To accelerate the optimization process, we modified the traditional LBFGS by adding a preconditioner, and achieve a more rapid convergence. To test our method, we use two synthetic cases: a sinking cylinder within a viscous layer and a realistic subduction model. We find that in the initial temperature-only inversion, the initial temperature can be recovered well; in the joint inversion of initial temperature and viscosity parameters, the temperature, as well as effective viscosity, can also be recovered reasonably, but there are trade offs between viscosity parameters. Presumably, the trade off in viscosity parameters is related to the ill-posedness of the problem.