Mathematical and Numerical Problems in Seismic Imaging

GENERAL FIELD
This project is in general area of the numerical analysis of
wave-propagation problems. More specifically, the project focuses on seismic imaging. The context is as follows: As
part of the seismic exploration process, waves are emitted from a source into the earth and then the reflections of
these waves from the subsurface of the earth are measured by sensors. For seismic exploration in a marine
environment, the source and measurement sensors are towed behind boats. The seismic waves can be described by
partial 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). FWI
requires the numerical solution of the PDE, to generate predicted data, and an optimisation step, in which the
difference between the predicted and measured data is minimised to obtain an improved model
of the subsurface.
EARLY TRAINING
As a student in the CDT SAMBa, Shaunagh will undertake the SAMBa training programme for first-year PhD
students, including participating in an Integrative Think Tank, and taking units relevant to her project area. In
particular she will take a reading course on the wave equation and will do a relevant Level 4 course on large-scale
optimisation. Further details of the training she will undergo are given in the section "Training Requirements" below.
FINAL AIMS AND METHODOLOGY
This project will consider different mathematical and numerical
problems in seismic imaging, in particular related to the process of FWI. The mathematical tools used will be
methods 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. The
key question in this problem would be "given prior information about the likely make-up of the subsurface (in the form
of a class of generic models), can one optimise the location of the sensors to retrieve sufficient information about the
subsurface?''

The impact of the SAMBa CDT will occur principally through the following two pathways:

1. Direct engagement with industrial partners, leading to PhD projects that are collaborative with industry, and that are focussed on topics with direct industrial impact.

2. The production of PhD graduates with
(a) the mathematical, statistical and computational technical skill sets that have been identified as in crucial demand both by EPSRC and by our industrial partners, coupled to
(b) extensive experience of industrial collaboration.

The underlying opportunity that SAMBa provides is to train graduates to have the ability to combine complex models with 'big data'. Such people will be uniquely equipped to deliver impact: whether they continue with academic careers or move directly to posts in industry, through quantitative modelling, they will provide the information that gives UK businesses competitive advantages. Our industrial partners make it clear to us that competitiveness in the energy, manufacturing, service, retail and financial sectors is increasingly dependent on who can best and most quickly analyse the huge datasets made available by the present information revolution.

During their training as part of SAMBa, these students will have already gained experience of industrial collaboration, through their PhD projects and/or the Integrated Think Tanks (ITTs) that we propose, that will give all SAMBa students opportunities to develop these transferable skills. PhD projects that involve industrial collaboration, whether arising from ITTs or not, will themselves deliver economic and social benefits to UK through the private companies and public sector organisations with which SAMBa will collaborate.

We emphasise that Bath is at the forefront of knowledge transfer (KT) activities of the kind needed to translate our research into impact. Our KT agenda has recently been supported by KT Accounts and Impact Acceleration Accounts from EPSRC (£4.9M in total) and a current HEFCE HEIF allocation of £2.4M. Bath is at the forefront of UK activity in KTPs, having completed 150 and currently holding 16 KTP contracts worth around £2.5M.

The SAMBa ITTs are an exciting new mechanism through which we will actively look for opportunities to turn industrial links into research partnerships, supported in the design of these projects by the substantial experience available across the University.

More widely, we envisage impact stemming from a range of other activities within SAMBa:

- We will look to feed the results of projects involving ecological or epidemiological data directly into environmental and public health policy. We have done this successfully many times and have three REF Case Studies describing work of this nature.

- Students will be encouraged to make statistical tools available as open source software. This will promote dissemination of their research results, particularly beyond academia. There is plenty of recent evidence that such packages are taken up and used.

- Students will discuss how to use new media to promote the public understanding of science, for example contributing to projects such as Wikipedia.

- Students will be encouraged to engage in at least one outreach activity. Bath is well known for its varied, and EPSRC-supported, public engagement activities that include Royal Institution Masterclasses, coaching the UK Mathematics Olympiad team, and reaching 50 000 people in ten days with an exhibit at the Royal Society's 350th Anniversary Summer Exhibition in 2010.