On-Line Evaluation of Systems with Discrete Observations

When using forward modeling to estimate model parameters, such as the dip, it also is important to estimate the corresponding uncertainty in the model parameters. For gravity data, these uncertainties are dependent on the uncertainty in the Bouguer-corrected data. The uncertainty in the gravity meter reading and the height used in the free-air and Bouguer corrections are among the most important factors influencing the uncertainty in the Bouguer-corrected data. We use two methods for estimating the uncertainty in the Bouguer-corrected data, which give similar answers (0.121 mGal and 0.109 mGal). The uncertainty in the model parameters can be estimated by perturbing the corrected data multiple times by amounts consistent with the estimated uncertainty in the corrected gravity. The standard deviation of the model parameters derived from each perturbed data set gives an estimate of their uncertainty. Using this procedure for Bouguer gravity profiles that cross the Porcupine Destor Fault (a fault that is prospective for gold in the Timmins camp of Ontario, Canada), we find that the uncertainty in the dip is 1° or 2°, assuming a planar or linear fault. If the uncertainty in the corrected data had been 1 mGal (a value typical of regional surveys, instead of 0.1 mGal for a local survey), then the uncertainty in the dip is 41° for the same model. Thus, knowing the uncertainties in the corrected data are very important for estimating the uncertainty in model parameters. Conversely, if a model parameter is known to be required to a specific precision, the survey can be planned so that the corrected gravity has an uncertainty appropriate to achieve that precision.

Data insufficiency of input rock properties is a major issue to analyze the stability of slopes via traditional deterministic and reliability approaches. This data insufficiency in the properties arises due to complexities associated with in-situ and lab testing of rocks. The traditional Bayesian approach overcomes this issue by considering uncertainties in model parameters by combining available prior information neglecting the uncertainty associated with the distribution type/probability model. This study proposes a novel Bayesian multimodel inference approach to incorporate the uncertainties associated with probability models/distribution types along with model parameters for rock properties. The approach first identifies a set of candidate probability models and then employs the Bayesian framework to incorporate the parameter uncertainties for each model. The approach is demonstrated for a rock slope case with the potential of structurally controlled planar failure. It is concluded that the approach effectively treats the statistical uncertainties associated with probability model types and parameters with limited data and provides a more realistic stability assessment than the traditional Bayesian approach. Results show that the uncertainty in probability model parameters affects the stability of rock slope much more significantly than model types.

A stochastic channel embedded in a background facies is conditioned to data observed at wells. The background facies is a fixed rectangular box. The model parameters consist of geometric parameters that describe the shape, size, and location of the channel, and permeability and porosity in the channel and nonchannel facies. We extend methodology previously developed to condition a stochastic channel to well-test pressure data, and well observations of the channel thickness and the depth of the top of the channel. The main objective of this work is to characterize the reduction in uncertainty in channel model parameters and predicted reservoir performance that can be achieved by conditioning to well-test pressure data at one or more wells. Multiple conditional realizations of the geometric parameters and rock properties are generated to evaluate the uncertainty in model parameters. The ensemble of predictions of reservoir performance generated from the suite of realizations provides a Monte Carlo estimate of the uncertainty in future performance predictions. In addition, we provide some insight on how prior variances, data measurement errors, and sensitivity coefficients interact to determine the reduction in model parameters obtained by conditioning to pressure data and examine the value of active and observation well data in resolving model parameters.

Abstract. The contribution of rainfall forcing errors relative to model (structural and parameter) uncertainty in the prediction of soil moisture is investigated by integrating the NASA Catchment Land Surface Model (CLSM), forced with hydro-meteorological data, in the Oklahoma region. Rainfall-forcing uncertainty is introduced using a stochastic error model that generates ensemble rainfall fields from satellite rainfall products. The ensemble satellite rain fields are propagated through CLSM to produce soil moisture ensembles. Errors in CLSM are modeled with two different approaches: either by perturbing model parameters (representing model parameter uncertainty) or by adding randomly generated noise (representing model structure and parameter uncertainty) to the model prognostic variables. Our findings highlight that the method currently used in the NASA GEOS-5 Land Data Assimilation System to perturb CLSM variables poorly describes the uncertainty in the predicted soil moisture, even when combined with rainfall model perturbations. On the other hand, by adding model parameter perturbations to rainfall forcing perturbations, a better characterization of uncertainty in soil moisture simulations is observed. Specifically, an analysis of the rank histograms shows that the most consistent ensemble of soil moisture is obtained by combining rainfall and model parameter perturbations. When rainfall forcing and model prognostic perturbations are added, the rank histogram shows a U-shape at the domain average scale, which corresponds to a lack of variability in the forecast ensemble. The more accurate estimation of the soil moisture prediction uncertainty obtained by combining rainfall and parameter perturbations is encouraging for the application of this approach in ensemble data assimilation systems.

In operational weather models, the effects of turbulence in the atmospheric boundary layer (ABL) on the resolved flow are modeled using turbulence parameterizations. These parameterizations typically use a predetermined set of model parameters that are tuned to limited data from canonical flows. Using these fixed parameters results in deterministic predictions that neglect uncertainty in the unresolved turbulence processes. In this study, we perform a machine learning-accelerated Bayesian inversion of a single-column model of the ABL. This approach is used to calibrate and quantify uncertainty in model parameters of Reynolds-averaged Navier–Stokes turbulence models. To verify the data-driven uncertainty quantification methodology, we test in an idealized setup in which a prescribed but unobserved set of parameters is learned from noisy approximations of the model output. Following this verification, we learn the parameters and their uncertainties in two different turbulence models conditioned on scale-resolving large-eddy simulation data over a range of ABL stabilities. We show how Bayesian inversion of a numerical model improves flow predictions by investigating the underlying mean momentum budgets. Further, we show that uncertainty quantification based on neutral ABL surface layer data recovers the relationships between parameters that have been predicted using theoretical modeling, but that learning the parameters based on stable ABL data or data from outside the surface layer can lead to different parameter relationships than neutral surface layer theory. Efforts to systematically reduce parameter uncertainty reveal that (1) sampling wind speed up to the ABL height can reduce uncertainty in key model parameters by up to $$84\%$$ , and (2) assimilating fluid flow quantities beyond first-order moment statistics can further reduce uncertainty in ways that baseline wind speed assimilation alone cannot achieve. The parameters learned using Bayesian uncertainty quantification generally yield lower error than standard deterministic parameters in out-of-sample tests and also provide uncertainty intervals on predictions.

The remaining useful life (RUL) estimation is an important metric that helps in condition-based maintenance. Damage data obtained from the diagnostics techniques are often noisy, and the RUL estimated from the data is less reliable. Estimating the probabilistic RUL by quantifying the uncertainty in the predictive model parameters using the noisy data increases confidence in the predicted values. In this work, Bayesian inference is used to inversely quantify the uncertainty in the model parameters of the predictive damage model in adhesively bonded joints, and quantified parameter uncertainty is represented by a probability distribution function (PDF). It is difficult to obtain a closed-form solution for the RUL from the PDF of model parameters due to the nonlinearity of the damage models. Instead, statistical simulations are used to generate the model parameter samples from the PDF, and the RUL is estimated from the samples. In the statistical simulations, the predictive damage model is evaluated for each parameter sample. Using a physics-based model as the predictive model increases computational time, which makes the statistical techniques to be computationally expensive. It is essential to reduce the computational time to estimate the RUL in a feasible time. In this work, real-time probabilistic RUL estimation is demonstrated in adhesively bonded joints using the Sequential Monte Carlo (SMC) sampling method and cloud-based computations. The SMC sampling method is an alternative to traditional Markov Chain Monte Carlo (MCMC) methods, which enables generating statistical parameter samples in parallel. The parallel computational capabilities of the SMC methods are exploited by running the SMC simulation on multiple cloud calls. This approach is demonstrated by estimating fatigue RUL in the adhesively bonded joint. The accuracy of probabilistic RUL estimated by SMC is validated by comparing it with RUL estimated by the MCMC and the experimental values. The SMC simulation is run on the cloud and the computational speedup of the SMC is demonstrated.

Exhumation plays a crucial role in shaping the evolution and distribution of resource systems in sedimentary basins, affecting mineral and energy resource exploration. Accurate exhumation estimates, derived primarily from empirical equations based on compaction and thermal datasets, are essential but are often compromised by data errors and unquantified uncertainties in model parameters. For instance, model parameters are usually assumed not to be affected by uncertainties despite varying within measurable ranges. Uncertainties from such variation can propagate and compromise the accuracy of exhumation estimates.This study introduces a novel and refined approach to exhumation estimation using Markov Chain Monte Carlo (MCMC) methods to quantify and address uncertainties in data and model parameters. Using this approach, we developed a workflow for quantifying exhumation magnitudes and their associated uncertainties and applied it to sonic log datasets from the Canning and Bonaparte Basins. The impact of uncertainty propagation on exhumation results was assessed by examining four scenarios: assuming no uncertainty in the model or data, considering data noise without model uncertainty, considering model uncertainty without data noise, and considering model uncertainties and data noise together.Our study yielded robust exhumation estimates in the Canning and Bonaparte Basins. Comparison with previous studies shows similarities and differences in exhumation estimates for multiple episodes, with discrepancies potentially arising from variations in exhumation models, data quality and coverage. Uncertainty propagation analysis reveals that considering data-related and model uncertainties together produces variable distributions of exhumation estimates with wider uncertainty ranges. Overall, data quality and coverage proved more critical for the accuracy and precision of exhumation estimates than model refinement. Our models can be integrated into basin evolution studies, help refine fluid migration models, and improve understanding of sedimentation and ore preservation to optimise resource exploration in sedimentary basins.

In this paper, we focus on the synthesis of a hybrid observer for a class of nonlinear switched systems where the evolution of the discrete state is governed by a Petri Net. The switched systems, herein, considered are characterized by switching laws that can depend on both the continuous states and external events. A new observer composed of a continuous and a discrete observer in interaction is designed. The continuous observer is based on the second order sliding mode approach to reconstruct the continuous state without using any information about the discrete state after the transformation of the original system in a canonical form. From these estimated continuous states, some discrete modes are determined thanks to the depending of states switching laws. The discrete modes depending on external events cannot be estimated from the knowledge of the continuous states. The estimated modes from the continuous states are considered as observables modes by the discrete observer in order to give the complete discrete states estimation. An illustrative example ends the paper and shows the efficiency of the proposed approach.

Hydrologists have applied inverse techniques to obtain estimates of subsurface permeability and porosity variations and their associated uncertainties. Although inverse methods are now well established in hydrology, important aspects of inverse theory, the analysis of resolution, and the trade‐off between model parameter resolution and model parameter uncertainty have not been utilized. In this paper the concept of model parameter resolution is incorporated into the analysis of hydrological experiments. Model parameter resolution is a measure of the spatial averaging implicit in estimates of a distributed hydrological property such as permeability. There are two important uses of resolution and uncertainty estimates in hydrology. The first use is to plan a hydrologic testing program. Resolution matrices can be developed for proposed well tests in a variety of synthetic media. Then the effectiveness of the test design can be evaluated in terms of model parameter resolution and uncertainty. Secondly, when real data are available and used in an inversion determining the distribution of hydrologic parameters, estimates of model parameter resolution and uncertainty analysis can indicate the reliability of the solution. For synthetic tests in which the hydraulic conductivity varies and porosity does not, it is found that tracer data can provide better spatial resolution of subsurface hydraulic conductivity variations than transient pressure data. Pressure data are most sensitive to hydraulic conductivity variations immediately surrounding the well. Both pressure and tracer data better determine barriers to flow rather than channels to flow. The methodology is applied to a set of transient pressure data gathered at the Grimsel Rock Laboratory of the Swiss National Cooperative for the Storage of Radioactive Waste. In the fracture under study a low hydraulic conductivity region appears to partition the fracture plane into two distinct zones.

The dominant contribution to the uncertainty in internal dose assessment can often be explained by the uncertainty in the biokinetic model structure and parameters. The International Commission on Radiological Protection (ICRP) is currently updating its biokinetic models, including the Human Respiratory Tract Model (HRTM). Gregoratto et al. (2010) proposed a physiologically-based particle transport model that simplifies significantly the representation of particle clearance from the alveolar interstitial region. Bayesian inference using the Weighted Likelihood Monte-Carlo Sampling (WeLMoS) method is applied to the bioassay and autopsy data from the U.S. Transuranium and Uranium Registries' (USTUR) tissue donors 0202 and 0407 exposed to "high fired," refractory PuO2 aerosols in order to examine the applicability of the revised model and to estimate the uncertainties in model parameters and the lung doses as expressed by the posterior probability distributions. It is demonstrated that, with appropriate adjustments, the Gregoratto et al. particle transport model can describe situations involving exposure to highly insoluble particles. Significant differences are observed in particle clearance pattern characteristics to these two individuals' respiratory systems. The respiratory tract of registrant 0202 was most likely compromised by his prior occupational exposure to coal dust, smoking habit, and chronic obstructive pulmonary disease, while donor 0407 was a non-smoker and had no prior history of lung disorder. However, the central values of the particle transport parameter posterior distributions for both cases are found to be still within the 68% probability range for the inter-subject variability derived by Gregoratto et al. PuO2 particles produced by the plutonium fire were extremely insoluble, with about 99% absorbed into blood at a rate of approximately 4.8 × 10 d (Case 0202) and 5.1 × 10 d (Case 0202). When considering this type of plutonium material, doses to other body organs are small in comparison to those to tissues of the respiratory tract. More than 95% of the total committed weighted equivalent dose is contributed by the lungs.

The last few decades have seen considerable progress in the quantification of environmental model uncertainty. Initially the emphasis has been on uncertainty in model parameters. A more recent trend has been to consider uncertainties in both model structure and parameters, most commonly by analyzing jointly predictions generated by several alternative models of the environment. This has been motivated by a growing recognition that the open and complex nature of environmental systems renders them suitable to multiple conceptualizations and mathematical descriptions. Predictions generated by a single model are prone to statistical bias (by reliance on an invalid model) and underestimation of uncertainty (by under-sampling the relevant model space) (Neuman 2003; Neuman and Wierenga 2003). Some multimodel approaches blend or average statistical results generated by a set of alternative models. A common approach to model averaging is to (1) postulate several alternative models for a site, (a) associate each model with a weight or probability, and (c) generate weighted average predictions and statistics of all the models. Ways to accomplish this have varied; some are included in a public-domain code (Multimodel Analysis or MMA by Poeter and Hill 2007) recently reviewed by Ye (2010). This special issue of SERRA focuses on such and other emerging methods of model and parameter uncertainty quantification. The special issue contains seven papers devoted to model averaging. Diks and Vrugt compare model averaging methods that weigh models in different ways, without always requiring that the weights sum up to unity. The methods are applied to two sites and compared in term of their predictive performance measured by out-of-sample root mean squared prediction error. Sain and Furrer estimate weights based on variation and correlation of alternative hierarchical models. They use a Bayesian hierarchical model to estimate correlation between models and the impact of parameter estimation uncertainty on the weights. Ajami and Gu use the Bayesian Model Averaging (BMA) approach of Raftery et al. (2005) to assess uncertainty in a suite of biogeochemical models having various levels of complexity to simulate the fate and transport of nitrate at a field site in California. Their results demonstrate that whereas single models, regardless of their complexity levels, are incapable of representing all active processes at the site, the 95% uncertainty bounds of BMA bracket 90% to 100% of the observations. Tsai use a variance-window (Tsai and Li 2008) version of Maximum Likelihood (ML) BMA (MLBMA; Neuman 2003; Ye et al. 2004) to quantify model uncertainty in managing groundwater within a thick sandy aquifer in Louisiana where saltwater intrusion is of concern. Alternative models are postulated to reflect uncertainty in conceptualizing hydraulic head boundaries and geostatistical parameterization through variogram models. The results M. Ye (&) Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA e-mail: mye@fsu.edu

Summary The joint inversion of radio magnetotelluric and electrical resistivity tomography data has the potential to reduce the uncertainties in the subsurface conductivity model. This is particularly beneficial when the datasets offer complementary information about the subsurface. However, the traditional gradient-based inversion methods pose challenges in quantifying uncertainty, as they yield a single model with limited appraisal of parameter uncertainty. The Bayesian inversion approach stands out for its capacity to provide quantitative assessments of uncertainty in the inverted model parameters. This is accomplished by generating an ensemble of models, leading to a posterior distribution that encapsulates both prior information concerning model parameters and the dataset information. We have implemented a transdimensional Markov chain Monte Carlo algorithm to perform the joint inversion of radio magnetotelluric and electrical resistivity tomography data. Through synthetic data studies, we illustrate how the inclusion of two complementary datasets can effectively reduce uncertainties in model parameters and how the model parameter uncertainties can be quantified. Subsequently, the developed algorithm is tested using exemplary field data from a waste site near Roorkee, India. Intensive prior geoelectric and transient electromagnetic as well as radio magnetotelluric studies investigated possible waste water seepage with a potential to contaminate the shallow aquifers. The derived subsurface structure from our transdimensional Bayesian results compare well with the deterministic results for the exemplary profile, but in addition provide comprehensive uncertainty estimates.

Whereas quality is an increasingly important aspect of peach fruit [ Prunus persica (L.) Batsch] production at this time, it is still not adequately addressed in crop models. Our objective was to develop a model to assess an essential trait of peach fruit quality (the refractometric index at harvest) to include it in existing crop models and to address the issue of quality in programs dealing with the improvement of crop management. The model predicts the fruit refractometric index, an indicator of sugar content (the most decisive parameter in consumer satisfaction) commonly used by the fruit industry. The model was simple enough so that it could be easily linked to carbon-based crop models. It was calibrated and tested using several independent data sets representing many growing conditions. To account for the effect of uncertainty in input and model parameters, the output of the model was qualified by a prediction interval. Results indicated that the model accurately predicted refractometric indices under 12% (relative root mean squared error values of 0.09 and 0.12 for two data sets), which corresponds to the fruit industry's range of interest. Prediction intervals revealed that the uncertainty in model parameters has moderate effects, whereas the uncertainty of the model input has important effects.

Parameterized fragility-based uncertainty influence quantification and sensitivity analysis methodology: Concept, formulation, and application

To solve the problem of ship automatic berthing control due to unknown time-varying disturbance and dynamic uncertainty of model parameters, an automatic berthing control law based on predefined performance time function is proposed. First, a predefined performance time function is designed and coupled with tracking error to achieve the predetermined performance of tracking error. Secondly, radial basis function neural network is used to approach the dynamic uncertainty of ship model parameters, and the complex uncertainty of model parameters and unknown time-varying disturbance is represented by linearized parameter form with single virtual parameter, which makes the calculation simple and easy to implement in engineering. On this basis, the reverse step control law is designed. Thirdly, the stability of the system is proved based on Lyapunov stability theory. Finally, the simulation results show that the control law can make the ship reach the desired position and heading angle, and realize the automatic berthing of the ship. The control law and berthing controller designed in this paper have good applicability and robustness, which provides a theoretical basis for the subsequent control research of surface intelligent ships.