Earthquakes and their corresponding tsunamis pose significant hazard to popu- lated regions around the world. Therefore, it is critically important that we seek to more fully understand the physics of the combined earthquake-tsunami system. One way to address this goal is through numerical modeling. The work discussed herein focuses on combining dynamic earthquake and tsunami models through the use of the Finite Element Method (FEM) and the Finite Difference Method (FDM). Dynamic earthquake models ac- count for the force that the entire fault system exerts on each individual element of the model for each time step, so that earthquake rupture takes a path based on the physics of the model; dynamic tsunami models can incorporate water height variations to produce water wave formation, propagation, and inundation. Chapter 1 provides an introduction to some important concepts and equations of elastodynamics and fluid dynamics as well as a brief example of the FEM. In Chapter 2, we investigate the 3-D effects of realistic fault dynamics on slip, free surface deformation, and resulting tsunami formation from an Mw 9 megathrust earthquake offshore Southern Alaska. Corresponding tsunami models, which use a FDM to solve linear long-wave equations, match sea floor deformation, in time, to the free surface deformation from the rupture simulations. Tsunamis generated in this region could have large adverse effects on Pacific coasts. In Chapter 3, we construct a 3-D dynamic rupture model of an earthquake on a reverse fault structure offshore Southern California to model the resulting tsunami, with a goal of elucidating the seismic and tsunami hazard in this area. The corresponding tsunami model uses final seafloor displacements from the rupture model as initial conditions to compute local propagation and inundation, resulting in large peak tsunami amplitudes northward and eastward due to site and path effects. In Chapter 4, we begin to evaluate 2-D earthquake source parameters from characteristics of the Rayleigh-wave field by running a suite of 2-D dynamic rupture models on thrust/reverse faults that vary in dip angle and fault curvature, and with equivalent prestress conditions such as constant traction across the fault or variable prestress distributions. We compare traveling Rayleigh-wave breakout amplitudes with fault slip distribution. Such Rayleigh- wave analysis has implications for early estimation of far-field tsunami amplitude, since source parameters are directly related to tsunami generation and propagation.

The functional form of the friction law employed in dynamic earthquake simulations is directly related to the energy partitioning of rupture and slip (e.g., Kanamori and Rivera, 2006). Earthquake ruptures can propagate along geometrically complex faults by bending and jumping along fault segments (Wesnousky, 1988), however, this is dependent on how much energy is available to further rupture propagation and how much energy is needed. For planar fault stepovers with homogeneous stress, rupture can jump segments or it can arrest at the edge of a segment. Since friction laws, along with stress regimes, implicitly specify the energy budget, different laws can lead to different rupture dynamics. Using realistic, laboratory-derived friction laws to simulate rupture is a constant ambition in seismology. The rate-state friction formulation includes intuitive variables such as time, slip speed history, and normal stress history as well as exhibits reasonable properties such as stick-slip sliding, creep, and healing (Dieterich, 1978b, 1979; Linker and Dieterich, 1992). Particularly, including the change in friction coefficient with normal stress history is imperative for earthquakes that produce large dynamic fluctuations in normal stress, including both stepovers and dip-slip faults that penetrate the free surface.

In this study we use FaultMod to run dynamic earthquake simulations on fault stepovers using the various friction formulas listed above. Firstly, we investigate effects of the functional form of the friction laws on jumping rupture at stepovers. The functional forms are associated with unique energy budgets that can determine how far rupture can jump within a given amount of time. We find that making the energy budgets similar between the friction parameterizations makes the rupture processes more similar at the stepover region (e.g., similar maximum jump distances). We also find that for larger jumps the rupture speeds increase to supershear speeds, even though the initial stress conditions would preclude such a result on a planar strike-slip fault (Andrews, 1976b). Secondly, we add to the stepover simulations by introducing the Linker-Dieterich formulation, in which friction coefficient depends on normal stress. It is well known that stepovers exhibit perturbations in normal stress along the offset region during rupture, with the sign being opposite for compressional and dilational stepovers. Adding the normal-stress-dependent state variable to the friction formulation decreases the maximum rupture jump for both types of stepover, however.

Additionally, we apply the Linker-Dieterich formulation to normal and reverse dip-slip faults, both of which are known to exhibit dynamic fluctuations in normal stress due to the free surface (Brune, 1996; Nielsen, 1998; Oglesby et al., 1998). Differences in shear stress direction between normal and reverse faults produce asymmetric normal stress perturbations along the fault during rupture, causing the motion from a reverse fault to be larger than that of an otherwise equivalent normal fault. Adding a normal-stress-dependent state variable serves to mitigate this effect. We also find that decreasing the initial shear stress inhibits rupture more on normal faults than on reverse faults, given that the faults intersect the free surface.