Most information about the Earth's sub-surface (e.g., rock stratal geometries, temperatures, pressures, composition, fluid content) comes from either seismic or electromagnetic waves. These propagate through the subsurface and either defract, refract or reflect (echo) back to the surface. There they are recorded and interpreted for sub-surface properties. Traditionally such waves emanate from active energy sources (earthquakes in seismology, or actively-induced seismic or electromagnetic sources in industrial subsurface exploration settings). However, in the past five years a revolutionary new set of methods has developed under the general name of 'Wavefield Interferometry', which have changed the nature of seismology fundamentally. In its most popular form, interferometry allows the energy from passive sources like ocean waves, wind, and anthropogenic activity (previously considered to be background noise) to be used to image the Earth. Interferometry allows this 'noise' field to be converted into signals that look like seismograms from active sources, even though no such sources occurred. The resulting seismograms from such virtual (imagined) sources are used to image the real Earth structure. In only five years this has become a standard technique in surface wave tomography of the Earth's crust and upper mantle, and similar techniques are under development for the exploration industry. Indeed, in the seismological community this has been so successful that signals from earthquakes (the previous data source) are now often ignored - only the background energy field (previously considered to be noise) is used for subsurface imaging. A limiting problem exists with such methods, which has only been fully illuminated over the past two years. Theoretically, interferometry works when the noise field comes equally from all directions. This is never the case on Earth for either passive noise fields, or even when active 'bespoke' fields are used in the industrial setting, principally because the dominant form of energy propagation from sources on or near the Earth's surface is through so-called surface waves, waves that hug the Earth's outermost surface as they travel. Surface waves thus dominate the virtual seismograms to an extent that swamps all body wave information. For industrial exploration it is strictly necessary to use body waves. Since interferometry would open new doors in subsurface exploration, it is highly desirable to be able to alter the interferometric methods to be able to work within biased energy fields. Our research group has recently developed a method, called 'directional balancing', that can be integrated within wavefield interferometric methods to correct biases due to the energy field directionality (provisional patents filed; manuscript submitted for publication). This method promises to reduce approximately-horizontally propagating surface wave energy, while enhancing the more vertically-propagating body wave arrivals to a realistic level. The method requires that energy is recorded on an array of receivers (rather than only by a pair of receivers as in standard interferometry). In industrial seismics, arrays of receivers are always available since they form intrinsic components of the seismic acquisition and processing system. Hence, in principle directional balancing is directly applicable to industrial seismic data, using both passive and active sources of energy. This project will develop the directional balancing method to the point of industrial application, and apply it to real, industrial-scale, seismic data sets provided by the industrial partner. By enhancing the body wave arrivals relative to surface waves, these methods promise to make wavefield interferometry techniques applicable to industrial scale seismics, thereby opening new fields of research, development and creating new and exciting possibilities for subsurface exploration.

Efficient and accurate simulation of wave phenomena is a key enabling technology across science and engineering. Applications span diverse areas, and include the whole of acoustics and noise control, non-destructive testing and ultrasonic and microwave technologies for medical imaging, problems of seismic and radar propagation and imaging, and even quantum scale simulations. But even though the underlying partial differential equations are usually linear and well understood, wave phenomena are complex and hard to simulate whenever the wavelength is small compared to the diameter of the region to be simulated.A main and standard computational tool for simulations of wave problems is the so-called finite element method. The idea of the method is to break up the computational domain into small elements and to approximate the solution on each of them in a simple way, e.g. as a linear variation. However, this gives accurate solutions only if the diameter of each element is small compared to the wavelength. Thus the number of elements needed and the associated computational cost and storage is infeasible if the diameter of the region to be simulated is very large compared to the wavelength, as it is for very many complex problems of wave propagation and scattering, e.g. seismic wave propagation for hydrocarbon exploration.Recently, there has been strong international interest in novel finite element formulations that try to solve this problem by representing the wave field on each element by functions that are themselves waves. This allows much bigger element sizes and so a significant reduction of the computational cost. However, these novel finite element methods are still in their infancy and it is poorly understood how to implement them in an optimal way. For example, one key open problem is the question of which wave functions to use. Another open question is how to achieve numerical stability, i.e. an algorithm whose results are not garbled by effects resulting from the limited accuracy that computers have. These and other questions are particularly unclear for three dimensional problems, although most practical applications are three dimensional.The fellowship addresses this wide open research area. Building upon novel ideas about how to locally model wave phenomena in a stable way it combines fundamental research in diverse areas of applied and computational mathematics in order to develop the next generation of finite element methods for wave problems. These new methods have the potential to be orders of magnitude faster than current methods allowing for numerical simulations of phenomena that are currently out of reach. In close collaboration with partners in science and industry the new methods will be applied to exciting research problems in science and engineering. In particular, a major part of the hydrocarbon exploration business is enabled through the modelling and inversion of large scale 3D seismic and electromagnetic data sets, and Schlumberger Cambridge Research will be a key project partner. Throughout the fellowship annual international workshops on next generation finite element methods for wave problems will be organised, at Reading and Schlumberger. These will bring together leading researchers in the area of numerical wave simulations from academia and industry and will drive this research area forward by intensifying collaborations and developing and exploring application areas for these methods.Numerical wave simulations are an essential technology in science and engineering. Innovations in many areas depend upon the ability to simulate complex wave phenomena. The UK is one of the leading countries for wave-related research. This fellowship will enhance this role by building up an internationally outstanding research group on novel finite element methods for wave problems that will have a strong impact on wave-related research and applications long after the duration of the fellowship.

Magnetotelluric (MT) techniques, which employ ultra low frequency electromagnetic waves in order to reveal subsurface geological structures are becoming increasingly important tools for geological exploration, particularly under conditions where standard seismic methods are expensive to implement or produce ambiguous results. A major drawback of present MT methods is a reliance on sporadically occurring natural sources of such waves or highly localised controlled sources in the form of a small transmitter towed along by an ocean going survey vessel. Over the past three decades techniques have been developed which allow ultra low frequency electromagnetic waves to be artificially excited in the Earth's upper atmosphere by irradiating it with modulated high power, high frequency radio waves from ground-based transmitters. This programme will explore the use of such experiments, and the forecasting of naturally-occurring waves, in order to improve the efficiency and cost-effectiveness of current MT survey techniques.

The computation of wave phenomena is widely needed in many application areas, for example models of radar and telecommunications devices require the computation of electromagnetic waves while the implementations of seismic and medical imaging algorithms use acoustic, elastic, and electromagnetic waves to obtain information about the earth's subsurface and the human body respectively. Computer models of the propagation of waves arise naturally in the design and implementation of these technologies. Medical imaging technicians use computer models of how the material composition of the human body scatters incoming electromagnetic waves in order to solve the "inverse problem'' of reconstructing the internal makeup of a human being from an observed scattered wave field. Similarly, seismologists use computer models of how the material properties of the earth's subsurface affects the transmission of elastic waves in order to reconstruct the earth's subsurface properties from observed echoes of elastic waves This technology is hugely useful, for example in the medical context it means we can often diagnose health problems without a need for more invasive techniques. In the seismology case it makes something seemingly impossible become possible - since it is never physically possible to explore all of the earth's subsurface properties by simply boring holes. However the fast and accurate computer modelling of such wave phenomena is complicated and costly (in terms of computer time), principally (but not solely) because of the highly oscillatory nature of the waves and the complicated media through which they pass. Thus there is a strong need for new methods that speed up such models and that task is a principal focus of this research. This project will devise and mathematically justify new families of fast methods for implementing these computer wave models, and will make the new methods available through two software platforms which are accessible to a wide range of scientists as well as in an additional specialist high performance computing library. As well as devising new methods for modelling (which work well on today's multiprocessor computers), the project will also involve direct collaboration with two companies - Schlumberger (a Project Partner, interested in seismology) and ABB (interested in electromagnetic computations) - as well as two academic groups, one in geosciences and one in electromagnetics.

We propose a 9-month project as an extension of an existing joint project between British Antarctic Survey and Schlumberger Gould Research. Schlumberger's vessels conduct seismic surveys of the seabed and underlying geography at many locations around the globe to assess offshore oil and gas reserves. The vessels tow behind them an array about 1 km in width of a dozen cables known as "streamers", typically 6-10 km long (though only 5 cm in diameter). The streamers hold many thousands of sensors (usually hydrophones) to detect the reflected signal from a seismic source. The streamers also hold a number of steering devices, which can be used to actively modify the position of the streamer through the water. In this project we do not use the seismic data recorded by the hydrophones but instead the physical information of the streamers, such as position, tension and steering forces from which ocean current models can be derived (the seismic data of ocean density may however prove useful for follow-on studies to improve our understanding of submesoscale processes). To-date a forward model has been developed and tested to estimate the motion of streamers as they are towed through an ocean current field. Here an inverse model will be developed to take recorded information related to the streamers, in particular the tension at the head end and the velocities and angles along the cable, and to deduce the horizontal, divergence-free currents along the cable, thus providing velocity information at the submesoscale (approximately 1-10 km in the horizontal). The project will refine, validate against independent observations and implement this inverse model. Components of the project will directly build off NERC-funded work conduced as part of a number of past and present projects. The results will be of immediate benefit to Schlumberger as they will provide information for expected currents in the area with which to plan repeat surveys and they will also feed into a number of NERC research projects investigating the role of submesoscale processes in the ocean.