Comparing Linear and Nonlinear Methods for More Reliable Predictive Uncertainty Quantification and Optimal Design of Experiments

Data acquisition in complex environmental systems is typically expensive. Therefore, experimental designs should be optimized such that most can be learned about the system at least costs. In the past, optimal design (OD) analyses were mainly restricted to linear or linearized problems and methods. Nonlinear OD methods offer more efficient data collection strategies, because they can better handle the non-linearity exhibited by most coupled environmental systems. However, the much higher computational demand restricts their applicability to models with comparatively low run-times. Our goal is to compare the trade-off between computational efficiency and the obtainable design quality between linear and nonlinear OD methods.

In our study, a steady-state model for a section of the river Steinlach (South Germany) was set up and calibrated to measured groundwater head data and on estimated groundwater exchange fluxes. The model involves a Pilot Point parameterization scheme for hydraulic conductivity and six zones with uncertain river bed conductivities.

In the linear OD approach, the initial predictive uncertainty of groundwater exchange fluxes and mean travel times are estimated using the PREDUNC utility (Moore and Doherty 2005) of PEST. The parameter calibration was performed with a non-linear global search. A discrete global search method and PREDUNC was then utilized to identify augmented monitoring strategies (additional n measurement locations and data types) that reduce the predictive uncertainty the most.

For the nonlinear assessment, a conditional ensemble obtained with Markov-chain Monte Carlo represents the initial state of uncertainty and is used as input to a nonlinear OD framework called PreDIA (Leube et al. 2012). PreDIA can consider any kind of uncertainties and non-linear (statistical) dependencies in data, models, parameters and system drivers during the OD process.

The linear and non-linear approaches are compared thoroughly during each step of the model-based OD process. We investigate differences in initial uncertainty levels, estimated relative uncertainty levels and optimized designs. For our case study, we further assess at which step the linear approach starts deviating strongly from the non-linear method, from which point on the further linear analysis becomes unrealistic.