Data assimilation (DA) provides optimal combination of model simulation results with observed values. There are four sources of uncertainty for any DA: 1) inherent uncertainty from limitations of scientific knowledge, 2) data insufficiency, which is insufficient information content in target observations for history matching-constrained parameter selection, 3) observation or measurement error, and 4) model representation error. Null space sensitivity analysis is a technique to examine data sufficiency. Sensitivity is the variation in model solution values due to variability, or uncertainty, in one or more parameter values. Parameters are in the null space when their variation causes minimal change to history matching skill during assimilation. As a result, null space parameters can be set to any value, constrained only by professional judgement, to produce a best fit model. Null space parameters that generate significant changes to important model predictions are however diagnostic of data insufficiency. We present a new null space sensitivity analysis for the iterative ensemble smoother (iES) algorithm, which provides an ensemble method for DA, in PEST++. A fundamental advantage of iES is computational efficiency through efficient and empirical sampling of posterior parameter distributions. Our new method leverages uncertainty analysis post-assimilation rather than robust Monte Carlo sampling, which is computationally expensive, to determine empirical parameter sensitivity and maintain the computational advantages of iES. Sensitivity analysis is generated by an ensemble of models with insensitive parameters varying across the feasible range of parameter values and sensitive parameters fixed to best-fit model values. The case study application of the null space sensitivity analysis identified data insufficiency leading to limited decision support regarding the amount of groundwater storage in the system, and it demonstrated a more than 97% reduction in computational requirements relative to the Null Space Monte Carlo (NSMC) method.

Abstract: Predictive uncertainty analysis focuses on defensible variability in model projected values after estimation of the posterior parameter distribution. Inverse-style parameter estimation selects posterior parameters through history matching where parameters are varied and resulting model simulation values are compared to observations, and parameters are selected balancing goodness-of-fit between simulated and observed values and expert knowledge. When inverse-style parameter estimation approaches are used, parameter sensitivity, which is the change in simulated outputs relative to the change in parameter values, is an important consideration. Variation in null space parameters has a limited impact on history matching skill; however, these parameters become important when they impact predictions. A new null space sensitivity analysis for ensemble methods of data assimilation (DA) using observation error models is developed and implemented for an integrated hydrological model. Empirical parameter sensitivity is estimated by comparing the spreads of prior and posterior parameter distributions. Sensitivity analysis is generated by an ensemble of models with insensitive parameters varying across the prior parameter distribution and sensitive parameters fixed to best-fit model values. The result is identification of insensitive aquifer storage parameters that change storage-related model predictions by as much as two times. This null space analysis describes uncertainty from data insufficiency. Ensemble methods using observation error models also describe predictive uncertainty from noisy measurements and imperfect models.