THEORETICAL GLOBAL SEISMOLOGY
Description: 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 “asympotic’’ resolution of the identity or seismic normal mode summation. In radial models such as PREM, a discussion of the Einstein-Brioullin-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.
Construction and Analysis of Equations
- Calculus of variations and elastic-gravitational deformations
- Linearized variational model of elastic-gravitational deformations
- Analysis of the system of elastic-gravitational equations, well posedness
- Dynamic ruptures generating seismic waves
Spectrum
- Characterization of the spectrum and seismic normal modes
- Resolution of the identity and seismic normal mode summation
Construction and Analysis of Solutions
- WKB eigenfunctions and EBK quantization in SNREI models
- Microlocal analysis of body waves
- Semiclassical analysis of surface waves
Class Resources:
- ‘On the system of elastic-gravitational equations describing the oscillations of the earth’ (2015) with S.F. Holman and H. Pham. View
- ‘Variational formulation of Earth’s elasto-gravitational deformations under low regularity conditions’ (2016) with K. Brazda and G. Hörmann. View
- ‘Semi-classical analysis of elastic surface waves’ (2017) with A. Iantchenko, G. Nakamura and J. Zhai. View
- ‘Analysis of dynamic ruptures generating seismic waves in a self-gravitating planet: an iterative coupling scheme and well-posedness’ (2019) with K. Kumar and R. Ye. View
