An Innovative Slepian Approach to Invert GRACE KBRR for Localized Hydrological Information at the Sub-Basin Scale

GRACE spherical harmonics are well-adapted for representation of hydrological signals in river drainage basins of large size such as the Amazon or Mississippi basins. However, when one needs to study smaller drainage basins, one comes up against the low spatial resolution of the solutions in spherical harmonics. To overcome this limitation, we propose a new approach based on Slepian functions which can reduce the energy loss by integrating information in the spatial, spectral and time domains. Another advantage of these regionally-defined functions is the reduction of the problem dimensions compared to the spherical harmonic parameters. This also induces a drastic reduction of the computational time. These Slepian functions are used to invert the GRACE satellite data to restore the water mass fluxes of different hydro-climatologic environments in Africa. We apply them to two African drainage basins chosen for their size of medium scale and their geometric specificities: the Congo river basin with a quasi-isotropic shape and the Nile river basin with an anisotropic and more complex shape. Time series of Slepian coefficients have been estimated from real along-track GRACE geopotential differences for about ten years, and these coefficients are in agreement with both the spherical harmonic solutions provided by the official centers CSR, GFZ, JPL and the GLDAS model used for validation. The Slepian function analysis highlights the water mass variations at sub-basin scales in both basins.

Application of spherical Slepian functions to the inversion of virtual observatory satellite magnetic data into localised regions of flow on the core-mantle boundary

<p>Spherical Slepian functions (or ‘Slepian functions’) are mathematical functions which can be used to decompose potential fields, as represented by spherical harmonics, into smaller regions covering part of a spherical surface. This allows a spatio-spectral trade-off between aliasing of the signal at the boundary edges while constraining it within a region of interest. While Slepian functions have previously been applied to geodetic and crustal magnetic data, this work further applies Slepian functions to flows on the core-mantle boundary. There are two main reasons for restricting flow models to certain parts of the core surface. Firstly, we have reason to believe that different dynamics operate in different parts of the core (such as under LLSVPs) while, secondly, the modelled flow is ambiguous over certain parts of the surface (when applying flow assumptions). Spherical Slepian functions retain many of the advantages of our usual flow description, concerning for example the boundary conditions it must satisfy, and allowing easy calculation of the power spectrum, although greater initial computational effort is required.</p><p><br>In this work, we apply Slepian functions to core flow models by directly inverting from satellite virtual observatory magnetic data into regions of interest. We successfully demonstrate the technique and current short comings by showing whole core surface flow models, flow within a chosen region, and its corresponding complement. Unwanted spatial leakage is generated at the region edges in the separated flows but to less of an extent than when using spherical Slepian functions on existing flow models. The limited spectral content we can infer for core flows is responsible for most, if not all, of this leakage. Therefore, we present ongoing investigations into the cause of this leakage, and to highlight considerations when applying Slepian functions to core surface flow modelling.</p>

Dictionary learning algorithms for the downward continuation of the gravitational potential

<p>A fundamental problem in the geosciences is the downward continuation of the gravitational potential. It enables us to learn more about the system Earth and, in particular, the climate change.</p><p>Mathematically, we can model a (downward continued) signal in a 'best basis' consisting of local and global trial functions from a dictionary. In practice, our dictionaries include spherical harmonics, Slepian functions and radial basis functions. The expansion in dictionary elements is obtained by one of the Inverse Problem Matching Pursuit (IPMP) algorithms.</p><p>However, it remains to discuss the choice of the dictionary. For this, we further developed the IPMP algorithms by introducing a learning technique. With this approach, they automatically select a finite number of optimized dictionary elements from infinitely many possible ones. We present the details of our method and give numerical examples.</p><p>See also: V. Michel and N. Schneider, <em>A first approach to learning a best basis for gravitational field modelling,</em> arxiv: 1901.04222v2</p>