Improving inversions of regional outer core surface flow models through the application of spherical slepian function

There are multiple applications of the Earth's magnetic field to industry and science (for example navigation and satellite operations) so it is critical to understand how the magnetic field is generated and to accurately forecast magnetic field change. The main field generated by the movement of conductive liquid in the outer core, in a process that is often referred to as the geodynamo. The main field is the most dominant contribution ($>90\%$) to the Earth's magnetic field measured at the Earth's surface and changes on monthly and annual timescales. However, the main field change is difficult to predict due to unpredictable changes in acceleration in the fluid movement and our poor understanding of how the liquid is moving at the core surface. This is best illustrated by the re-release of the 2015-2020 World Magnetic Model which failed to meet the success criteria (errors smaller than 1$^{\circ}$ RMS at sea level worldwide) after only 3.5 years. By assuming that changes in the magnetic field in the Earth's outer core are advection-dominated on short timescales, models of the core surface flow can be deduced. This assumption allows us to state that magnetic field lines are `frozen' into packets of liquid at the Earth's outer core surface, so the change in magnetic field strength (Secular Variation, SV) can be related to motion of liquid iron at the core surface. Despite having an inversion methodology to study flow at the Earth's Core-Mantle Boundary (CMB), core surface flow models are known to be under-determined and thus require other assumptions to produce feasible flows. Also, we do not have good understanding of how regional features at the base of the mantle affect the flow within the core. Core flow and magnetic field models at the CMB tend to be described by spherical harmonics, which are not suitable for separation into individual regions due to large leakage being generated during the separation. Spherical Slepian functions are one way to investigate regional separations of spherical harmonic data, such as geodetic, gravity or crustal magnetic fields. Spherical Slepian functions can spatially and spectrally separate bandlimited potential fields by transforming the spherical harmonic coefficients into the Slepian basis and sorting the functions by contribution to the patch. The unitary matrix that translates between the spherical harmonic and Slepian basis is constructed by optimisation over the region of interest and the sphere, so the first functions concentrate their signal to be within the region of interest, and the last functions in the matrix corresponds to the region complement. This compromise means there is some leakage outside the region of interest but with a better concentration of the signal within the region. Altitude-cognizant Slepian functions have also been created to solve for a regional separation at a different height from the data collection with an additional downward continuation factor. The motivation behind the work is two-fold. Firstly, we wish to understand the best technique for the application of spherical Slepian functions to core surface flows. For an optimal decomposition, the result of summing the basis functions for the `in' and `out' regions should be identical to a global spherical harmonic function and the `in' region will be fully recreated with no signal outside the region of interest. Secondly, we wish to make geophysical interpretations of the impact of the Large Low Velocity Provinces (LLVPs) on the core surface flow over time. LLVPs are two antipodal regions of anomalously low seismic velocity covering $\sim 25\%$ of the CMB surface. Long-lived features in the Earth's magnetic field have been speculated to be linked to the LLVP structures as evidence for top-down control on the geodynamo. Whether these features apply a thermal forcing, a chemical exchange, dynamic topography or other effect to the core remains to be understood. We begin with the simplest application of spherical Slepian functions for investigating regional core surface flow, by using scalar spherical Slepian functions on prior core flow models. Flow is a vector quantity and, therefore, we had to decompose the flow potentials as opposed to the flow itself. We conclude that it is possible to produce core flow separations with scalar Slepian functions but it will usually generate large leakage at the region boundary. The cause of this leakage is predominantly due to the bandlimiting for the maximum spherical harmonic degree and large gradients along the region boundary due to constraining the potentials in the global model to be zero within the complementary region. The leakage generated during the Slepian decomposition was found to be due to the splitting of spherical harmonic input coefficients into a larger magnitude value for inside the LLVPs and an oppositely-signed coefficient (of similar magnitude) for outside the LLVPs. We have successfully incorporated spherical Slepian functions into regional inversions for SV and core surface flow modelling but the resulting separations require careful consideration of the parameters used in the decomposition. Decompositions of SV models at the Earth's surface produce good separations but increasing the downward continuation of SV models from Earth surface height to CMB produces un-Earth-like values, which cannot be easily corrected for. We tested regularisation methods that were dependent on the number of eigenfunctions and the surface area of the region of interest but our `optimal' solutions are chosen by trial and error adjustments of the division between `in' and `out', and the unconstrained degree-dependent damping parameter. The altitude-cognizant spherical Slepian functions generally improve decompositions due to the downward continuation factor acting as an additional damping at large degrees. Decomposing surface core flow was not possible due to unconstrained parameters and large un-Earth-like flows generated and instead we study the SV, which is linearly related to flow. Finally, we inverted for regional SV for satellite data inversions 2000--2021 and 150 years of COV-OBS.x2 SV model coefficients to investigate how LLVPs may be affecting core surface flow over time. We have assumed that the `optimal' methodology is suitable for all timesteps and the variation in the SV decompositions over time are due to changes in the flow structure. We identify that the proportion of SV energy underneath LLVPs is incrementally changing over time and there is good correlation between periods of known acceleration change and inflection points in the spectra at $l=2$ and $l=4$. Inversions of satellite energy within the LLVPs have been relatively constant over the last 20 years and is roughly proportional to the surface area of the LLVPs but the longer time series shows a reduction in spectral energy within the LLVP over time. A longer time-series is needed to see if LLVP energy varies in a constant or predictable manner