This proposal initiates a new collaboration aimed at improving methods of waveform inversion by including topography in seismic wave modeling and reducing the reliance of these methods on data at extremely low-frequency. The Principal Investigators propose to adapt and improve methods developed for simulating scattering from a diffraction grating. Their approach has extensions to linear elasticity and provides general improvements for solving the inverse problem. Significant mathematical advances are required to develop robust and efficient techniques for this application. The Boundary Perturbation Method extends the Field Expansion approach to an arbitrary number of layers with independent topographies and allows for rapid and accurate simulation of acoustic wave propagation in two dimensions. Extensions to three dimensions and to the general equations of elasticity are required. Moreover, the extension of frequency-independent discretizations based on Geometric Acoustics to multilayered elastic models is a significant and necessary mathematical advance. Additionally, advances in the inverse problem are also necessary to make the technique truly applicable to the seismic imaging problem. The standard method of "full-waveform inversion" requires data at frequencies which are too low to record in practice. The PIs' approach casts the forward problem as the application of a sequence of topography-dependent operators where the interface shapes appear rather explicitly. A number of iteration schemes are proposed for the recovery of these shapes, using re-arrangements of the compositions coupled to standard regularizing techniques from the theory of ill-posed problems.
The propagation properties of seismic waves in layers of sediment are crucial in many technologies including the determination of inner earth properties and structure, earthquake detection and prediction, and hydrocarbon (oil and gas) exploration. In light of its many important applications, it is not surprising that a vast array of numerical and experimental techniques have been brought to bear upon this problem. However, several gaps in understanding and capability still exist. The PIs' address some of these questions through sophisticated numerical simulations which will be validated against both laboratory experiments and field measurements from the Tibetan plateau.