New generation finite element methods for seismic forward modelling

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This project is concerned with developing and prototyping new, more effective computer simulation methods for modelling the propagation and reflection of seismic waves in the earth subsurface. The pull for this research comes from the CASE partner, Schlumberger Cambridge Research, who wish to develop more efficient methods for imaging the subsurface below land or marine deposits in order to locate hydrocarbon-bearing rocks. In particular, for accurate imaging in more complex geometries where seismic reflectivity is poor, for instance imaging below basalt or salt structures, the state-of the art is to use imaging methods which proceed iteratively, solving the full mathematical equations describing the wave propagation and reflection at each step. Unfortunately, this full simulation of seismic propagation and reflection by standard numerical methods is hugely expensive in computing resources, because of the complex subsurface geometry and the large 3D region that must be simulated. Large, that is, in diameter in comparison with the wavelengths of the seismic waves, so that a very high resolution is needed to visualise the wave propagation accurately using standard computational methods. These standard methods include so-called 'finite element methods', computer simulation methods in which the earth subsurface is thought of as composed of a large number (e.g. 1,000,000-100,000,000) of small pieces (the 'finite elements') in each of which the seismic wave has a very simple behaviour, e.g. is approximately constant. This project is concerned with exploring the use, for seismic simulations, of a new, more sophisticated class of finite element method. This new class of method differs in using a more sophisticated assumed behaviour in each element, namely a certain standard wave-like behaviour (that of a plane wave or a combination of plane waves). To keep the project to a manageable size, one suitable for proof of concept, and suitable for a PhD student to complete in 3 1/2 years, the modelling will be restricted to two-dimensional simulations (where it is assumed that the geometry is constant in one horizontal direction), and to a simplified acoustic model of the seismic propagation. Main objectives of the project will be: i) To extend previous work of this type, namely the so-called Ultra Weak Variational Formulation, so that the method can deal with spatially varying seismic properties (that is, where the wave speed varies gradually or suddenly with position in the subsurface). This (significant) extension to the current method, which will need both strong mathematical and computing skills, will be essential for the method to be of use for general purpose seismic modelling. ii) To apply a combination of sophisticated mathematics and numerical experiments so as to understand the behaviour of the new algorithm. iii) To test the new method on representative acoustic 2D geological models supplied by Schlumberger, comparing the performance of the new algorithms, as implemented in computer software, with existing methods, based on standard finite elements and so-called finite difference modelling. These standard methods Schlumberger has implemented in existing computer software. In the first 18 months of the PhD the student will receive training, in superb research environments at Reading and Schlumberger, in the mathematics, computing, and knowledge of standard methods for modelling seismic propagation, that will be necessary for completion of the project. In this, the student at Reading will benefit from access to courses forming part of our MSc in Mathematics of Scientific and Industrial Computation, from our membership of an advanced graduate level training consortium in Mathematics (the MAGIC group), and from membership of a large community of academic and research staff and PhD students working on many exciting applications of the mathematics of waves, and the use of waves in imaging.