Dictionary learning algorithms for the downward continuation of the gravitational potential

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.

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.

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.

See also: V. Michel and N. Schneider, A first approach to learning a best basis for gravitational field modelling, arxiv: 1901.04222v2