Learning geospatial data with Gaussian Random Field models

Understanding the world’s water cycle and its geographical change is critical in a world short of water. When water is added or moved on a continental landmass, the Earth’s surface is displaced [RD01-1]. The displacement of the Earth’s surface can be detected using the more than 10 thousand permanent GNSS stations around the world.

Objectives: to develop robust state-of-the-art computational tools for use in the geophysical sciences allowing us to generate spatio-temporal models from data with probabilistic estimates of uncertainty (stemming from poor spatial resolution of GNSS data). We can link the change in the surface displacement to a change in distribution of surface mass using elasticity models.

Methodology. A recent breakthrough was the link made in [RD01-2] between Matern-class Gaussian random fields that can be used and the solution of stochastic PDEs. These equations can be solved using state-of-the-art multigrid and finite element techniques on HPC architecture, giving new algorithms for Bayesian uncertainty quantification applied to geophysics problems with millions of unknown parameters [RD01-3,4].

[RD01-1] van Dam, T., Colilieux, X., Wuite, J., Altamimi, Z. and Ray, J. 2012, Nontidal ocean loading: amplitudes and potential effects in GPS height series, J. of Geodesy, 86, 1043-1057, DOI 10.1007/s00190-012-0564-5.

[RD01-2] Lindgren, F., Rue, H. and Lindström, J. 2011, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73, 423–498.

[RD01-3] Hauseux P, Hale JS, Bordas SPA, 2017, Accelerating Monte-Carlo estimation with derivatives of high-level finite element models, Computer Methods in Applied Mechanics and Engineering 318, 917-936.

[RD01-4] Isaac T, Petra N, Stadler G, Ghattas O, 2015, Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet, Journal of Computational Physics 296, 348-368.