Evaluating Lower Computational Burden Approaches for Calibration of Large Environmental Models

Abstract

Realistic environmental models used for decision making typically require a highly parameterized approach. Calibration of such models is computationally intensive because widely used parameter estimation approaches require individual forward runs for each parameter adjusted. These runs construct a parameter-to-observation sensitivity, or Jacobian, matrix used to develop candidate parameter upgrades. Parameter estimation algorithms are also commonly adversely affected by numerical noise in the calculated sensitivities within the Jacobian matrix, which can result in unnecessary parameter estimation iterations and less model-to-measurement fit. Ideally, approaches to reduce the computational burden of parameter estimation will also increase the signal-to-noise ratio related to observations influential to the parameter estimation even as the number of forward runs decrease. In this work a simultaneous increments, an iterative ensemble smoother (IES), and a randomized Jacobian approach were compared to a traditional approach that uses a full Jacobian matrix. All approaches were applied to the same model developed for decision making in the Mississippi Alluvial Plain, USA. Both the IES and randomized Jacobian approach achieved a desirable fit and similar parameter fields in many fewer forward runs than the traditional approach; in both cases the fit was obtained in fewer runs than the number of adjustable parameters. The simultaneous increments approach did not perform as well as the other methods due to inability to overcome suboptimal dropping of parameter sensitivities. This work indicates that use of highly efficient algorithms can greatly speed parameter estimation, which in turn increases calibration vetting and utility of realistic models used for decision making.

Figure 1

Plot of change in model outputs (y‐axes) to small increments of change in one model parameter (vertical hydraulic conductivity, Kv, on the x‐axes) for two different observations. Each dot represents one model run; the straight line is the best fit through the dots. Because the true parameter sensitivity derivative is approximated using a 1% parameter perturbation, sequential 1% perturbations are expected to provide a coherent change (e.g., the monotonically changing line shown in (b)). Poor derivatives calculated by perturbation (a) can confound derivative‐based parameter estimation methods; tighter solver closure as shown in (b) provides more coherent derivatives. An influence statistic (Cook's D) for the two observations is also listed, where higher values represent more influence on the regression (modified from Feinstein et al. ; Anderson et al. 2015).

Figure 2

Extents and surface‐water system in Clark and Hart (2009) and the updated model used in this study. MERAS, Mississippi Embayment Regional Aquifer Study; SFR, stream flow routing.

Figure 3

Layer 2 extents and location of hydraulic conductivity and specific yield pilot points.

Figure 4

Farfield and nearfield (inset) calibration target locations. MERAS, Mississippi Embayment Regional Aquifer Study.

Figure 5

Change in model fit for the four approaches by PE iteration. The blue shaded rectangle represents a “desirable” range of fit, one reflecting adequate history matching and reasonable parameter fields. Larger symbols indicate iteration used for comparison; dashed lines reflect overfitting. IES, iterative ensemble smoother; phi, measurement objective function; SVD, singular value decomposition.

Figure 6

Improvement of fit versus number of forward model runs of the MODFLOW‐NWT model (>6 h per run). The blue shaded rectangle represents a “desirable” range of fit, one reflecting adequate history matching and reasonable parameter fields. Larger symbols indicate iteration used for comparison; dashed lines reflect overfitting. IES, iterative ensemble smoother; phi, measurement objective function; SVD, singular value decomposition.

Figure 7

Optimal parameters for the nearfield area of interest at the desirable level of fit, where the IES results reflect a single parameter field from the PEST++ reported “base” realization. The location of the nearfield is shown in Figures 3 and 4. IES = iterative ensemble smoother; Kh = horizontal hydraulic conductivity; specific yield [dimensionless]; SVD = singular value decomposition.