Mathematical and Numerical Problems in Seismic Imaging

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GENERAL FIELDThis project is in general area of the numerical analysis ofwave-propagation problems. More specifically, the project focuses on seismic imaging. The context is as follows: Aspart of the seismic exploration process, waves are emitted from a source into the earth and then the reflections ofthese waves from the subsurface of the earth are measured by sensors. For seismic exploration in a marineenvironment, the source and measurement sensors are towed behind boats. The seismic waves can be described bypartial differential equations (PDEs), formulated in the time or frequency domain. Given the sensor measurements,the seismic properties of the subsurface may be inferred. This process is termed Full Waveform Inversion (FWI). FWIrequires the numerical solution of the PDE, to generate predicted data, and an optimisation step, in which thedifference between the predicted and measured data is minimised to obtain an improved modelof the subsurface.EARLY TRAININGAs a student in the CDT SAMBa, Shaunagh will undertake the SAMBa training programme for first-year PhDstudents, including participating in an Integrative Think Tank, and taking units relevant to her project area. Inparticular she will take a reading course on the wave equation and will do a relevant Level 4 course on large-scaleoptimisation. Further details of the training she will undergo are given in the section "Training Requirements" below.FINAL AIMS AND METHODOLOGYThis project will consider different mathematical and numericalproblems in seismic imaging, in particular related to the process of FWI. The mathematical tools used will bemethods from numerical analysis, optimisation, and analysis of PDEs. One initial problem we seek to investigate,which is of great practical interest in petroleum prospecting, and is not considered in the standard approach to FWI,is the optimal positioning of the sensors in order to obtain the best return from the seismic exploration process. Thekey question in this problem would be "given prior information about the likely make-up of the subsurface (in the formof a class of generic models), can one optimise the location of the sensors to retrieve sufficient information about thesubsurface?''