THEORETICAL GLOBAL SEISMOLOGY I

Credit Hours: 3 · Graduate

The course provides a path through theoretical seismology from a comprehensive analysis perspective. It consists of five parts: (i) The introduction of Earth's elastic-gravitational deformations through the calculus of variations, and the introduction of fluid-solid boundaries involving Earth's core using an action integral. (ii) The variational linearized or weak formulation of Earth's elastic-gravitational deformations. (iii) Energy estimates and well-posedness under appropriate conditions (that, for example, constrain the shapes of the major boundaries) of the system of elastic-gravitational equations describing the oscillations of the earth, and a Volterra equation justifying the extraction of the system describing acousto-elastic waves. (iv) The characterization of the spectrum of the earth, seismic normal modes and the essential spectrum associated with internal or gravity modes and embedded eigenfrequencies. The ``asymptotic’' resolution of the identity or seismic normal mode summation. In radial models such as PREM, a discussion of the Einstein-Brillouin-Keller quantization, trace formula and length spectrum. (v) Incorporation of dynamic ruptures, using rate- and state-dependent friction laws, generating seismic waves through an iterative coupling scheme and viscosity solutions. All parts will be illustrated with computational simulations using numerical formulations closely related to the analysis.