Dimension reduction for integrating data series in Bayesian inversion of geostatistical models

This study explores methods with which multidimensional data, e.g. time series, can be effectively incorporated into a Bayesian framework for inferring geostatistical parameters. Such series are difficult to use directly in the likelihood estimation procedure due to their high dimensionality; thus, a dimension reduction approach is taken to utilize these measurements in the inference. Two synthetic scenarios from hydrology are explored in which pumping drawdown and concentration breakthrough curves are used to infer the global mean of a log-normally distributed hydraulic conductivity field. Both cases pursue the use of a parametric model to represent the shape of the observed time series with physically-interpretable parameters (e.g. the time and magnitude of a concentration peak), which is compared to subsets of the observations with similar dimensionality. The results from both scenarios highlight the effectiveness for the shape-matching models to reduce dimensionality from 100+ dimensions down to less than five. The models outperform the alternative subset method, especially when the observations are noisy. This approach to incorporating time series observations in the Bayesian framework for inferring geostatistical parameters allows for high-dimensional observations to be faithfully represented in lower-dimensional space for the non-parametric likelihood estimation procedure, which increases the applicability of the framework to more observation types. Although the scenarios are both from hydrogeology, the methodology is general in that no assumptions are made about the subject domain. Any application that requires the inference of geostatistical parameters using series in either time of space can use the approach described in this paper.