Advanced Geostatistics for the Bayesian Modelling of Terrain Uncertainty – GAMBIT

Geostatistics aims to provide a range of techniques to model, estimate, simulate regionalized variables using random field theory. From its mining origin, geostatistics was initially interested in small data sets, which led it to develop parsimonious estimation tools that happen to be optimal in the stationary Gaussian case.

This framework has had to evolve in order to grasp the applications for which the Gaussian and / or stationary assumptions were inadequate, using respectively Gaussian anamorphosis and / or local variogram modelling

In recent years, it has been necessary to have new geostatistical tools to model, estimate and simulate massive regionalized variables using non-Gaussian and non-stationary random fields . In response to this request, at least for the Gaussian case, F. Lindgren and H. Rue have introduced in 2011 a novel approach (Stochastic Partial Differential Equation, known by its acronym SPDE). To the knowledge of the authors of this proposal, this approach has not yet been studied in France.

The first objective of GAMBIT is to allow a French research team, creator of geostatistics and internationally recognized in the field, to assess to what extent the SPDE approach responds to these new needs. In particular, IC3i and the geostatistical team of Mines ParisTech will review the not fully solved technical problems in the article F.Lindgren and H. Rue and in the many subsequent communications, such as those of D. Bolin, which sought to extend the applications domain.
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The GAMBIT methodology, to be studied in the project, will use the SPDE modelling (inference, simulation) in a Bayesian framework. The usefulness of GAMBIT will be assessed in two completely different dual-nature applications. .

The first application concerns the modelling of terrain uncertainty. It addresses the need to provide both military and civil operational users of Digital Elevation Models (DEM), with a probabilistic knowledge of their altimetric error. This error is modeled as the solution of a SPDE model with a non-Gaussian second member and whose parameters are functions (splines) of a set of predictors, which can be computed from the DEM itself or from other available data. The study will investigate the introduction of constraints in the SPDE model (such as the consistency with hydrography) and the fusion of different DEMs, each with its own uncertainty model.

The second application concerns the spatio-temporal modelling of the average monthly atmospheric concentration of carbon dioxide since 2010, estimated from measurements of the AIRS (Atmospheric InfraRed Sounder) sensor onboard the NASA Aqua satellite. This problem, fundamental to the study of climate change, is based on a space-time SPDE model defined on the surface of the Earth, assumed to be spherical.