Daniel Erdal

Pumping and tracer tests are site-investigation techniques frequently used to determine hydraulic conductivity. Tomographic test layouts, in which multiple tests with different combinations of injection and observation wells are performed, gain a better insight into spatial variability. While hydraulic tomography has repeatedly been applied in the field, tracer tomography lags behind. In a previous publication, we presented a synthetic study to investigate whether the ensemble Kalman Filter (EnKF) or the Kalman Ensemble Generator (KEG) performs better in inverting hydraulic- and tracer-tomographic data. In this work, we develop an experimental method for solute-tracer tomography using fluorescein as a conservative tracer. We performed hydraulic- and tracer-tomographic tests at the Lauswiesen site in Germany. We analyzed transient drawdown and concentration data with the EnKF and steady-state hydraulic heads and mean tracer arrival times with the KEG, obtaining more stable results with the KEG at lower computational costs. The spatial distribution of the estimated hydraulic conductivity field agreed with earlier descriptions of the aquifer at the site. This study narrows the gap between numerical studies and field applications for aquifer characterization at high resolution, showing the potential of combining ensemble-Kalman filter based methods with data collected from hydraulic and solute-tracer tomographic experiments.

In global sensitivity analysis and ensemble-based model calibration, it is essential to create a large enough sample of model simulations with different parameters that all yield plausible model results. This can be difficult if a priori plausible parameter combinations frequently yield non-behavioral model results. In a previous study (Erdal and Cirpka, 2019), we developed and tested a parameter-sampling scheme based on active-subspace decomposition. While in principle this scheme worked well, it still implied testing a substantial fraction of parameter combinations that ultimately had to be discarded because of implausible model results. This technical note presents an improved sampling scheme and illustrates its simplicity and efficiency by a small test case. The new sampling scheme can be tuned to either outperform the original implementation by improving the sampling efficiency while maintaining the accuracy of the result or by improving the accuracy of the result while maintaining the sampling efficiency.

Ensemble-based uncertainty quantification and global sensitivity analysis of environmental models requires generating large ensembles of parameter-sets. This can already be difficult when analyzing moderately complex models based on partial differential equations because many parameter combinations cause an implausible model behavior even though the individual parameters are within plausible ranges. In this work, we apply Gaussian Process Emulators (GPE) as surrogate models in a sampling scheme. In an active-training phase of the surrogate model, we target the behavioral boundary of the parameter space before sampling this behavioral part of the parameter space more evenly by passive sampling. Active learning increases the subsequent sampling efficiency, but its additional costs pay off only for a sufficiently large sample size. We exemplify our idea with a catchment-scale subsurface flow model with uncertain material properties, boundary conditions, and geometric descriptors of the geological structure. We then perform a global-sensitivity analysis of the resulting behavioral dataset using the active-subspace method, which requires approximating the local sensitivities of the target quantity with respect to all parameters at all sampled locations in parameter space. The Gaussian Process Emulator implicitly provides an analytical expression for this gradient, thus improving the accuracy of the active-subspace construction. When applying the GPE-based preselection, 70–90% of the samples were confirmed to be behavioral by running the full model, whereas only 0.5% of the samples were behavioral in standard Monte-Carlo sampling without preselection. The GPE method also provided local sensitivities at minimal additional costs.

We compare two ensemble Kalman-based methods to estimate the hydraulic conductivity field of an aquifer from data of hydraulic and tracer tomographic experiments: (i) the Ensemble Kalman Filter (EnKF) and (ii) the Kalman Ensemble Generator (KEG). We generated synthetic drawdown and tracer data by simulating two pumping tests, each followed by a tracer test. Parameter updating with the EnKF is performed using the full transient signal. For hydraulic data, we use the standard update scheme of the EnKF with damping, whereas for concentration data, we apply a restart scheme, in which solute transport is resimulated from time zero to the next measurement time after each parameter update. In the KEG, we iteratively assimilate all observations simultaneously, here inverting steady-state heads and mean tracer arrival times. The inversion with the dampened EnKF worked well for the transient pumping-tests, but less for the tracer tests. The KEG produced similar estimates of hydraulic conductivity but at significantly lower costs. We conclude that parameter estimation in well-defined hydraulic tests can be done very efficiently by iterative ensemble Kalman methods, and ambiguity between state and parameter updates can be completely avoided by assimilating temporal moments of concentration data rather than the time series themselves.

Global sensitivity analysis and uncertainty quantification of nonlinear models may be performed using ensembles of model runs. However, already in moderately complex models many combinations of parameters, which appear reasonable by prior knowledge, can lead to unrealistic model outcomes, like perennial rivers that fall dry in the model or simulated severe floodings that have not been observed in the real system. We denote these parameter combinations with implausible outcome as “non-behavior”. Creating a sufficiently large ensemble of behavioral model realizations can be computationally prohibitive, if the individual model runs are expensive and only a small fraction of the parameter space is behavioral. In this work, we design a stochastic, sequential sampling engine that utilizes fast and simple surrogate models trained on past realizations of the original, complex model. Our engine uses the surrogate model to estimate whether a candidate realization will turn out to be behavioral or not. Only parameter sets that with a reasonable certainty of being behavioral (as predicted by the surrogate model) are simulated using the original, complex model. For a subsurface flow model of a small south-western German catchment, we can show high accuracy in the surrogate model predictions regarding the behavioral status of the parameter sets. This increases the fraction of behavioral model runs (actually computed with the original, complex model) over total complex-models runs to 20-90%, compared to 0.1% without our method (e.g., using brute-force Monte Carlo sampling).  This notable performance increase depends on the choice of surrogate modeling technique. Towards this end, we consider both Gaussian Process Emulation (GPE) and models based on polynomials of active variables determined by Active Subspace decomposition as surrogate models. For the GPE-based surrogate model, we also compare random search and active learning strategies for the training of the surrogate model.

In global sensitivity analysis and ensemble-based model calibration it is essential to create a large enough sample of model simulations with different parameters, which all yield plausible model results. This can be difficult if a-priori plausible parameter combinations frequently yield non-behavioral model results. In a previous study (Erdal and Cirpka, 2019), we developed and tested a parameter-sampling scheme based on active subspace decomposition. While in principle this scheme worked well, it still implied testing a substantial fraction of parameter combinations that ultimately had to be discarded because of implausible model results. This technical note presents an improved sampling scheme and illustrates its simplicity and efficiency by a small test case. The new sampling scheme can be tuned to either outperform the original implementation by improving the sampling efficiency while maintaining the accuracy of the result, or by improving the accuracy of the result while maintaining the sampling efficiency.

Integrated hydrological modeling of domains with complex subsurface features requires many highly uncertain parameters. Performing a global uncertainty analysis using an ensemble of model runs can help bring clarity as to which of these parameters really influence system behavior and for which high parameter uncertainty does not result in similarly high uncertainty of model predictions. However, already creating a sufficiently large ensemble of model simulation for the global sensitivity analysis can be challenging, as many combinations of model parameters can lead to unrealistic model behavior. In this work we use the method of active subspaces to perform a global sensitivity analysis. While building up the ensemble, we use the already-existing ensemble members to construct low-order meta-models based on the first two active-subspace dimensions. The meta-models are used to pre-determine whether a random parameter combination in the stochastic sampling is likely to result in unrealistic behavior so that such a parameter combination is excluded without running the computationally expensive full model. An important reason for choosing the active-subspace method is that both the activity score of the global sensitivity analysis and the meta-models can easily be understood and visualized. We test the approach on a subsurface-flow model including uncertain hydraulic parameters, uncertain boundary conditions and uncertain geological structure. We show that sufficiently detailed active subspaces exist for most observations of interest. The pre-selection by the meta-model significantly reduces the number of full-model runs that must be rejected due to unrealistic behavior. An essential but difficult part in active-subspace sampling using complex models is approximating the gradient of the simulated observation with respect to all parameters. We show that this can effectively and meaningfully be done with second-order polynomials.

Integrated hydrological modelling of domains with complex subsurface features requires many highly uncertain parameters. Performing a global uncertainty analysis using an ensemble of model runs can help bring clarity which of these parameters really influence system behavior, and for which high parameter uncertainty does not result in similarly high uncertainty of model predictions. However, already creating a sufficiently large ensemble of model simulation for the global sensitivity analysis can be challenging, as many combinations of model parameters can lead to unrealistic model behavior. In this work we use the method of active subspaces to perform a global sensitivity analysis. While building-up the ensemble, we use the already existing ensemble members to construct low-order meta-models based on the first two active subspace dimensions. The meta-models are used to pre-determine whether a random parameter combination in the stochastic sampling is likely to result in unrealistic behavior, so that such a parameter combination is excluded without running the computationally expensive full model. An important reason for choosing the active subspace method is that both the activity score of the global sensitivity analysis and the meta-models can easily be understood and visualized. We test the approach on a subsurface flow model including uncertain hydraulic parameter, uncertain boundary conditions, and uncertain geological structure. We show that sufficiently detailed active subspaces exist for most observations of interest. The pre-selection by the meta-model significantly reduces the number of full model runs that must be rejected due to unrealistic behavior. An essential but difficult part in active subspace sampling using complex models is approximating the gradient of the simulated observation with respect to all parameters. We show that this can effectively and meaningful be done with second-order polynomials.