Search

Surrogate models are widely used to improve the computational efficiency in various geophysical simulation problems by reducing the number of model runs. Conventional one-layer surrogate representations are based on global (e.g. polynomial chaos expansion, PCE) or on local kernels (e.g., Gaussian process emulator, GPE). Global representations omit some details, while local kernels require more model runs. The existing multi-resolution PCE is a promising hybrid: it is a global representation with local refinement. However, it can not (yet) estimate the uncertainty of the resulting surrogate, which techniques like the GPE can do. We propose to join multi-resolution PCE and GPE s into a joint surrogate framework to get the best out of both worlds. By doing so, we correct the surrogate bias and assess the remaining uncertainty of the surrogate itself. The resulting multi-resolution emulator offers a pathway for several active learning strategies to improve the surrogate at acceptable computational costs, compared to the existing PCE-kriging approach it adds the multi-resolution aspect. We analyze the performance of a multi-resolution emulator and a plain GPE using didactic test cases and a CO2 benchmark, that is representative of many alike problems in the geosciences. Both approaches show similar improvements during the active learning, but our multi-resolution emulator leads to much more stable results than the GPE. Overall, our suggested emulator can be seen as a generalization of multi-resolution PCE and GPE concepts that offers the possibility for active learning.

There has been an increasing interest in using multi-model ensembles over the past decade. While it has been shown that ensembles often outperform individual models, there is still a lack of methods that guide the choice of the ensemble members. Previous studies found that model similarity is crucial for this choice. Therefore, we introduce a method that quantifies similarities between models based on so-called energy statistics. This method can also be used to assess the goodness-of-fit to noisy or deterministic measurements. To guide the interpretation of the results, we combine different visualization techniques, which reveal different insights and thereby support the model development. We demonstrate the proposed workflow on a case study of soil–plant-growth modeling, comparing three models from the Expert-N library. Results show that model similarity and goodness-of-fit vary depending on the quantity of interest. This confirms previous studies that found that “there is no single best model” and hence, combining several models into an ensemble can yield more robust results.

Several environmental phenomena require monitoring time-dependent densities in porous media, e.g., clogging of river sediments, mineral dissolution/precipitation, or variably-saturated multiphase flow. Gamma-ray attenuation (GRA) can monitor time-dependent densities without being destructive or invasive under laboratory conditions. GRA sends gamma rays through a material, where they are attenuated by photoelectric absorption and then recorded by a photon detector. The attenuated intensity of the emerging beam relates to the density of the traversed material via Beer–Lambert’s law. An important parameter for designing time-variable GRA is the exposure time, the time the detector takes to gather and count photons before converting the recorded intensity to a density. Large exposure times capture the time evolution poorly (temporal raster error, inaccurate temporal discretization), while small exposure times yield imprecise intensity values (noise-related error, i.e. small signal-to-noise ratio). Together, these two make up the total error of observing time-dependent densities by GRA. Our goal is to provide an optimization framework for time-dependent GRA experiments with respect to exposure time and other key parameters, thus facilitating neater experimental data for improved process understanding. Experimentalists set, or iterate over, several experimental input parameters (e.g., Beer–Lambert parameters) and expectations on the yet unknown dynamics (e.g., mean and amplitude of density and characteristic time of density changes). We model the yet unknown dynamics as a random Gaussian Process to derive expressions for expected errors prior to the experiment as a function of key experimental parameters. Based on this, we provide an optimization framework that allows finding the optimal (minimal-total-error) setup and demonstrate its application on synthetic experiments.

This study evaluates and analyzes creep testing results on salt concrete of type M2. The concrete is a candidate material for long-lasting structures for sealing underground radioactive waste repository sites. Predicting operational lifetime and security aspects for these structures requires specific constitutive equations to describe the material behavior. Thus, we analyze whether a fractional viscoelastic constitutive law is capable of representing the long-term creep and relaxation processes for M2 concrete. We conduct a creep test to identify the parameters of the fractional model. Moreover, we use the Bayesian inversion method to evaluate the identifiability of the model parameters and the suitability of the experimental setup to yield a reliable prediction of the concrete behavior. Particularly, this Bayesian analysis allows to incorporate expert knowledge as prior information, to account for limited experimental precision and finally to rigorously quantify the post-calibration uncertainty.

The reliability sensitivity index based on the safety/failure classification of model predictions is a valuable tool to measure how uncertain parameters affect the failure of engineering systems. The key to estimate this index is to estimate the failure-conditional probability density function (PDF) of parameters, which can be regarded as a posterior PDF from the perspective of Bayesian inference. Due to the binary property of the failure indicator function, it is difficult to directly sample the failure-conditional PDF. In this work, a new efficient sampling method is proposed to estimate the failure-conditional PDF and the reliability sensitivity index through a two-stage Markov chain Monte Carlo (MCMC) simulation. In the first stage, a different criterion based on the distance to the failure domain is adopted to update the chain and finally get a first sample in the failure domain. Then, starting with this sample, a normal Markov chain is run in the second stage to simulate the failure-conditional PDF and estimate the reliability sensitivity index. The preconditioned Crank-Nicolson algorithm (a special MCMC method) is adopted to deal with high-dimensional problems (with many uncertain parameters), and adaptive parameter tuning is used to enhance the performance of the proposed two-stage MCMC method. Several test examples show the high efficiency of the proposed two-stage MCMC method compared to subset simulation. The method is currently designed for problems with a single failure domain, yet it could be extended.

Given a set of competing models of some phenomenon together with measurement data, Bayesian model selection (BMS) is a process of finding the model that is the best candidate for being the true dat...

Global sensitivity analysis is a widely used tool for uncertainty apportionment and is very useful for decision making, risk assessment, model simplification, optimal design of experiments, etc. Density-based sensitivity analysis and regional sensitivity analysis are two widely used approaches. Both of them can work with a given sample set of model input-output pairs. One significant difference between them is that density-based sensitivity analysis analyzes output distributions conditional on input values (forward), while regional sensitivity analysis analyzes input distributions conditional on output values (reverse). In this paper, we study the relationship between these two approaches and show that regional sensitivity analysis (reverse), when focusing on probability density functions of input, converges towards density-based sensitivity analysis (forward) as the number of classes for conditioning model outputs in the reverse method increases. Similar to the existing general form of forward sensitivity indices, we derive a general form of the reverse sensitivity indices and provide the corresponding reverse given-data method. Due to the shown equivalence, the reverse given-data method provides an efficient way to approximate density-based sensitivity indices. Two test examples are used to verify this connection and compare the results. Finally, we use the reverse given-data method to perform sensitivity analysis in a carbon dioxide storage benchmark problem with multiple outputs, where forward analysis of density-based indices would be impossible due to the high-dimensionality of its model outputs.

Bayesian mechanistic modeling often suffers from overconfident and biased posterior distributions for parameters and predictions. This phenomenon arises because the fundamental assumption of Bayesian Model Analysis is violated: the underlying model is assumed to be true, but in fact, it is a simplification of reality with structural errors that show at least during some periods of the modeled time span. As a result, a compromise solution in parameter space is identified that can formally fit the full data set best, but this parameter set will not be representative of the true system state. Neither will it be representative of the “compensation mode” in which the model is whenever structural error kicks in. As a logical consequence, predictions will be biased and their intervals too narrow. The longer the data set used for calibration, the stronger the misleading effect. Typical sources of severe structural deficits that produce dynamically occurring errors are missing or misspecified processes in the model. We propose a formal time-windowed Bayesian analysis to overcome this general problem. When performing Bayesian updating on shorter time windows, the assumption of a (quasi-) true model becomes more plausible, and by sliding this window through the calibration time series, we let the model adjust its posterior parameter distributions according to the current strength of error. These time-shifting parameter distributions allow us to (1) identify periods of statistically significant model error occurrence via measuring time-varying Bayesian model evidence, (2) diagnose potential sources of model error by understanding the time-varying parameter compensation mechanisms, and (3) predict with more realistic uncertainty intervals by distribution averaging. We demonstrate the proposed method on a set of synthetic and real-world scenarios of soil moisture modeling. With this example, we also highlight its usefulness to analyze dynamic systems in a wide range of disciplines, such as water quality modeling, decision support, and risk assessment. Results show that the time sequence of posterior parameter distributions (and dependent model mechanisms such as water retention curves and unsaturated hydraulic conductivity functions) provides valuable insights into the model’s weaknesses, and it also provides guidance for model improvement.

Mechanistic models facilitate understanding complex biogeochemical interactions and process chains in soil. However, biogeochemical soil models often have weakly constraint parameters and show sloppiness. That means the dimensionality of parameter spaces is overly large and parameters often cannot be inferred based on available experimental data. Thus, equifinality arises, i.e. many different parameter combinations lead to very similar or identical model predictions.Expert knowledge represents a synthesis of existing knowledge on processes in soil systems that can be used to find viable parameter regions such that models give plausible predictions in line with evidence-based expectations. Here, we present an approach to leverage expert knowledge. This is achieved by formulating expert knowledge in terms of parameter and process constraints that must be fulfilled. Viable parameter sets are then identified by model conditioning using a novel Bayesian constraint-based parameter search algorithm that extends a previously published iterative constraint-based parameter search method. The algorithm successively applies stricter conditions by increasing the minimum acceptable number of process constraints to be satisfied in each iteration. We present the concept of the algorithm and demonstrate a successful application to a complex model simulating biodegradation of the herbicide Atrazine that has a high-dimensional parameter space. The presented approach can be widely applied to other soil biogeochemical models and provides a powerful tool to leverage expert knowledge for constructing robust prior parameter distributions for model sensitivity analysis or calibration.

Deterministic hydrological models with uncertain, but inferred‐to‐be‐time‐invariant parameters typically show time‐dependent model errors. Such errors can occur if a hydrological process is active in certain time periods in nature, but is not resolved by the model or by its input. Such missing processes could become visible during calibration as time‐dependent best‐fit values of model parameters. We propose a formal time‐windowed Bayesian analysis to diagnose this type of model error, formalizing the question “In which period of the calibration time‐series does the model statistically disqualify itself as quasi‐true?” Using Bayesian model evidence (BME) as model performance metric, we determine how much the data in time windows of the calibration time‐series support or refute the model. Then, we track BME over sliding time windows to obtain a dynamic, time‐windowed BME (tBME) and search for sudden decreases that indicate an onset of model error. tBME also allows us to perform a formal, sliding likelihood‐ratio test of the model against the data. Our proposed approach is designed to detect error occurrence on various temporal scales, which is especially useful in hydrological modeling. We illustrate this by applying our proposed method to soil moisture modeling. We test tBME as model error indicator on several synthetic and real‐world test cases that we designed to vary in error sources (structure and input) and error time scales. Results prove the successful detection errors in dynamic models. Moreover, the time sequence of posterior parameter distributions helps to investigate the reasons for model error and provide guidance for model improvement., Key Points: We propose a data‐driven method for model‐structural error detection. Our method rests on a statistically rigorous Bayesian framework without prior assumptions about error sources or patterns. We confirm successful error detection on various temporal scales in synthetic test cases and present insights from a real‐world case study., German Research Foundation (DFG), Cluster of Excellence, University of Stuttgart, https://doi.org/10.18419/darus-1836

In conventional Bayesian geostatistical inversion, specific values of hyperparameters characterizing the prior distribution of random fields are required. However, these hyperparameters are typically very uncertain in practice. Thus, it is more appropriate to consider the uncertainty of hyperparameters as well. The preconditioned Crank-Nicolson Markov chain Monte Carlo with parallel tempering (pCN-PT) has been used to efficiently solve the conventional Bayesian inversion of high-dimensional multi-Gaussian random fields. In this study, we extend pCN-PT to Bayesian inversion with uncertain hyperparameters of multi-Gaussian fields. To utilize the dimension robustness of the preconditioned Crank-Nicolson algorithm, we reconstruct the problem by decomposing the random field into hyperparameters and white noise. Then, we apply pCN-PT with a Gibbs split to this “new” problem to obtain the posterior samples of hyperparameters and white noise, and further recover the posterior samples of spatially distributed model parameters. Finally, we apply the extended pCN-PT method for estimating a finely resolved multi-Gaussian log-hydraulic conductivity field from direct data and from head data to show its effectiveness. Results indicate that the estimation of hyperparameters with hydraulic head data is very challenging and the posterior distributions of hyperparameters are only slightly narrower than the prior distributions. Direct measurements of hydraulic conductivity are needed to narrow more the posterior distribution of hyperparameters. To the best of our knowledge, this is a first accurate and fully linearization free solution to Bayesian multi-Gaussian geostatistical inversion with uncertain hyperparameters.

To provide flexibility in power systems, hydropower frequently operates in strong hydropeaking schemes (i.e. resulting in severe flow flashiness) that harm the downstream river ecology. Energy storage systems are becoming an accessible source of flexibility, and when combined with new transmission infrastructure, could alleviate hydropeaking. Here, we explore how these technologies can mitigate severe hydropeaking with an emphasis on the temporal and spatial river flashiness. More precisely, in a case study on Chile for a fully renewable system (planned with a multi-objective framework, LEELO), we illustrate the costs of mitigating hydropeaking, select efficient solutions, and discuss the overall resulting storage needs. Our findings are relevant for decision makers related to river ecology. From the found trade-offs, we see how system-wide flashiness can be mitigated for little extra cost: 5–9% by relying more strongly on other storage technologies (depending on the level of flashiness), and only 1–4% if transmission expansion is considered. When focusing on the main reservoirs of each basin and the seasonality of their flashiness index, we observe that to avoid sacrifice zones the prescribed system-wide hydropeaking index needs to be very small. Minimizing hydropeaking requires more power capacity but less energy capacity from other storage technologies.