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Cambridge NERC Doctoral Training Partnerships

Graduate Research Opportunities

Lead Supervisor: David Al-Attar, Earth Sciences

Co-Supervisor: Sanne Cottaar, Earth Sciences

Brief summary: 
This project will develop new methods for modelling and inverting the Earth's free oscillations, and lead to improved constraints on mantle structure including lateral density variations which remain poorly known.
Importance of the area of research concerned: 
Our understanding of the Earth's internal structure has largely been obtained through analysis and inversion of seismic waves. Most seismic observations are, however, only sensitive to variations in elastic wave speeds, and contain little to no information on density variations. Within the lowermost mantle two large low velocity provinces (the so-called LLVPs) have been conistently imaged within tomographic studies, but their origin and dynamic significance remains unclear. Are they dense or light? What is their relation to mantle plumes? Were they produced through the convection process, or are they a remnant from the Earth's differentiation? It is only at long periods (order one hour and above) that there is an appreciable sensitivity to density, due to the fact that self-gravitational plays an important role in the dynamics. Past studies have investigated the density of LLVPs using free oscillation data, but the results have been inconclusive. This is due to two main factors. First, various approximations have been applied to model free oscillations of questionable accuracy. The second is limitations in the methods used to solve the associated inverse problem.
Project summary : 
This project will develop a new computational approach for normal model coupling calculations in realistic Earth models. The work will build on recent theoretical advances that overcome limitations in past work. In particular, previous studies have accounted for lateral variations in density, boundary topography, and anelasticity only to first-order accuracy, but here an exact treatment will be made. Having developed this code, attention will shift to the inverse problem. Here there will be need for original theoretical work including the derivation of sensitivity kernels using adjoint methods. In constrast to past work the focus will not be on model building, but instead using constrained optimisation to test specific quantitative hypostheis about the Earth. For example, can we determined the average densities of the two LLVPs along with meaningful uncertainties.
What will the student do?: 
Normal mode coupling is based on the expansion of the wavefield in a laterally heterogeneous Earth model in terms of the eigenfunctions of a spherical reference model. In this manner the problem is reduced to infinite-dimensional system of algebraic equations for the expansion coefficients, though this system must be appropriately truncated in numerical work. The tradiational approach to calculating the necessary matrix-elements is based on first-order perturbation theory, and so does not account for the non-linear dependence of certain parameters like density or boundary topography. To overcome this, the student will develop and apply a new pseudo-spectral approach that handles boundary topography exactly via particle relabelling transformations. Adjoint methods will then be applied to obtain sensitivity kernels, and iterative methods used to efficiently solve the optimisation problem within a function-space setting. Uncertainties will be quantified using a constrained-optimisation approch. This will require careful consideration of prior information on the mantle's structure and compostion, and hence engagement with broader aspects of Earth Sciences.
References - references should provide further reading about the project: 
Al-Attar D., Crawford O., Valentine A., Trampert J., 2018. Hamilton's principle and normal mode coupling in an aspherical planet with a fluid core. Geophys. J. Int., 214, 485-507.
Akbarashrafi F., Al-Attar D., Deuss A., Trampert J., Valentine A., 2018. Exact free oscillation spectra, splitting functions and the resolvability of Earth's density structure. Geophys. J. Int., 213, 58-76.
Maitra M., Al-Attar D., 2019. A non-perturbative method for gravitational potential calculations within heterogeneous and aspherical planets. Geophys. J. Int., 219, 1043-1055.
You can find out about applying for this project on the Department of Earth Sciences page.
Dr Sanne Cottaar
Dr David Al-Attar
Department of Earth Sciences Graduate Administrator