In geoscientific applications, such as rainfall-runoff prediction, space-time estimation of groundwater levels in data-scarce regions, or geostatistical inverse modeling, a frequent task is to quantify the uncertainty of a modeled/estimated/predicted function of time, space, or state variables. Unfortunately, such functions and their uncertainties are often non-stationary (e.g., dependent on space, system states, or local configurations) and exhibit more complex space-time structures than the well-researched linear and multi-Gaussian case of standard geostatistical or Gaussian process theory.
As two application examples, we will use groundwater table estimation and the prediction of side effects from shallow geothermal utilization. On these examples, we will develop methods to model non-stationary marginal distributions of these random space functions. This will allow transforming the random function to have a spatially stationary marginal; yet it will still have a non-Gaussian mutual dependence structure. Then we will explore approaches to describe this remaining dependence, so that structural inference, random generation, and conditional simulation are possible. Candidates are spatial copulas, multipoint geostatistics, or diffusion models. The overall model complexity will include the stationary (trans-)multi-Gaussian case as a special case.
