An iterative local updating ensemble smoother for estimation and uncertainty assessment of hydrologic model parameters with multimodal distributions

1. Introduction

• Direct parameter measurements are often difficult or impossible, so hydrologic parameter estimation uses indirect measurements to infer posterior uncertainty.Except in a few simple cases, the posterior has no analytical form and requires Monte Carlo sampling for numerical approximation.Source: §1 · p. 3
• MCMC can struggle with multimodal posteriors and require prohibitive model evaluations for high-dimensional, CPU-demanding systems.Surrogates can reduce cost, but accurate high-dimensional surrogates are difficult to construct except when nonlinearity is low enough for linear approximation.Source: §1 · pp. 3–4
• EnKF and ES are most suitable for Gaussian distributions because they rely on the first two statistical moments, making direct use problematic for multimodal parameters.Strongly nonlinear problems may also require iteration to achieve satisfactory data matches.Source: §1 · p. 5
• Clustering-based ES extensions face uncertain mode counts, stochastic mode identification, and challenges in high-dimensional problems.Other approaches combine EnKF with an inverse method such as particle filtering.Source: §1 · p. 5
• ILUES extends ES to multimodal inverse problems by updating local ensembles without K-means, Gaussian mixture models, or another inverse algorithm.It assimilates measurements multiple times using an iterative ES form to seek satisfactory matches in strongly nonlinear problems.Source: §1 · pp. 5–6

2. Iterative local updating ensemble smoother

• ILUES combines model-response fit and parameter proximity when selecting local ensembles, so updates can distinguish modes while keeping plausible solutions within or near the local ensemble bounds.Response distance filters samples far from the posterior region; parameter distance filters those far from the nearest mode, and either criterion alone can be inadequate.Source: §2 · pp. 7–8
• With α = 1, local updating reduces to original ES and cannot handle multiple posterior modes; α < 1 makes local updating distinct and suitable for multimodal problems.The original ES performs poorly in multimodal settings, whereas local updating uses a subset of the global ensemble.Source: §2 · pp. 8–9
• Overlapping local ensembles can help identify many possible modes and, when multimodality is absent, avoid erroneous mode identification.The method uses Ne seeds, and shared samples can lead the local ensembles to produce updated samples in the same mode.Source: §2 · p. 9
• Repeatedly applying the local update to the latest ensemble assimilates measurements multiple times, with measurement-error covariance inflated by the iteration count.The paper uses this iterative ES scheme for strongly nonlinear problems and repeats it Niter times.Source: §2 · p. 9

3.1. Example 1: A simple case study with infinite number of modes in the posterior

• For the infinite-mode case, ILUES identified the posterior within three iterations; repeated measurement assimilation was needed for convergence in this nonlinear problem.With an ensemble of 400 and α=0.1, the prior-to-posterior average RMSE ratio was 180.17.Source: §3.1 · p. 11

0.4 ) would significantly deteriorate the inversion results. According to our own

• Ensemble size and local-ensemble factor α involve trade-offs: small ensembles can miss posterior regions, while high-dimensional or many-mode problems generally need larger ensembles and computational cost.In the tested infinite-mode case, α=0.1 was recommended; selecting a best-J sample improved data matches in one test but could bias inversion results in another.Source: §0.4 · pp. 12–13
• The tuning study found that when α<0.1, smaller b values could improve data matches, whereas for α≥0.1, relatively larger b values performed better.The authors used b=1 as the default in subsequent examples.Source: §0.4 · pp. 13–14

3.2. Example 2: A 100-dimensional case study with multimodal posterior

• With 1,000 samples, ILUES’ simulated outputs from prior samples could converge to the measurement within five iterations, yielding an average prior-to-posterior RMSE ratio of 20.84, though insufficient to fully characterize the 100-dimensional posterior.The posterior RMSE mean and 95% confidence interval were 0.91 and [0.029, 2.69], close to MCMC’s 0.94 and [0.038, 2.88].Source: §3.2 · p. 15

3.3. Example 3: A rainfall-runoff model with multimodal prior

• For the HYMOD rainfall-runoff case, ILUES estimated parameters within five iterations and produced posterior RMSE statistics close to MCMC using an ensemble of 300.The prior-to-posterior average RMSE ratio was 5.30; the authors warn that very small ensembles risk biased inversion results.Source: §3.3 · p. 16

3.4. Example 4: Contaminant source identification with multimodal posterior

• ILUES identified the bimodal contaminant-source posterior, with posterior RMSE statistics close to DREAM MCMC; multiple-data-assimilation ES did not accurately characterize the bimodality.DREAM used eight parallel chains of length 2,000, totaling 16,000 model evaluations.Source: §3.4 · pp. 18–19

3.5. Example 5: Contaminant source identification with 108 unknown parameters

• With 108 unknown parameters, ILUES identified contaminant-source parameters within seven iterations and estimated the conductivity field; uncertainty quantification remained challenging for both ILUES and MCMC.The prior-to-posterior RMSE ratios were 586.47 for concentration data and 27.44 for head data; the authors note that 108 unknowns challenge accurate uncertainty quantification by both algorithms.Source: §3.5 · pp. 20–21

4. Conclusions and discussions

• ILUES selects each sample’s local ensemble using model-response and parameter distances, then updates it with ES to explore multimodal distributions.It adopts repeated measurement assimilation to achieve satisfactory data matches in nonlinear problems.Source: §4 · pp. 21–22
• Across five numerical cases, ILUES adequately quantified parametric uncertainty in complex systems whether or not multimodality was present.The cases included a posterior with infinitely many modes, a 100-parameter example, and three hydrologic inverse problems.Source: §4 · p. 22
• When model structural error matters, it can be included with measurement error and its distribution parameters estimated alongside the unknown model parameters.The discussion also names data-driven error quantification and Bayesian model averaging as alternative approaches.Source: §4 · p. 22
• The tested multimodality arises from system nonlinearity and scarce measurements, whereas multimodal non-Gaussian parameter fields require additional techniques.Level-set methods and normal-score transforms are cited for discretely distributed fields, which the authors leave for future work.Source: §4 · pp. 22–23

• Tables 1–3 list prior ranges and true values for model parameters in examples three through five; Table 3 covers contaminant-source parameters.Source: Table 1 · p. 29, Table 2 · p. 29, Table 3 · p. 29
• Table 4 is captioned as reporting RMSE between f(mtrue) and measurements, along with means and 95% summaries.Source: Table 4 · p. 29
• Figure 1 presents prior samples and updated samples, while Figures 6–13 include parameter trace plots across examples.Source: Figure 1, Figure 6, Figure 8
• Figures 2–4 show parameter-estimation results under varying α, ensemble size Ne, or both; Figure 4 uses log-transformed RMSE for its comparisons.The plotted configurations also include different α and b values in Figure 5.Source: Tables · p. 30, Figure 2, Figure 3
• Figure 14 presents posterior realizations, a mean estimate, and estimation variance for the log-transformed conductivity field.Source: Tables · p. 31, Figure 14