Inverse Uncertainty Research Articles

An ensemble neural network atmospheric correction for Sentinel-3 OLCI over coastal waters providing inherent model uncertainty estimation and sensor noise propagation

Accurate atmospheric correction (AC) is a prerequisite for quantitative ocean colour remote sensing and remains a challenge in particular over coastal waters. Commonly AC algorithms are validated by establishing a mean retrieval error from match-up analysis, which compares the satellite-derived surface reflectance with concurrent ground radiometric observations. Pixel-based reflectance uncertainties however, are rarely provided by AC algorithms and those for the operational Ocean and Land Colour Instrument (OLCI) marine reflectance product are not yet recommended for use. AC retrieval errors and uncertainties directly determine the quality with which ocean colour products can be estimated from the marine surface reflectance. Increasingly there is also the need for reflectance uncertainty products to be used as data assimilation inputs into biogeochemical models.This paper describes the development of a new coastal AC algorithm for Sentinel-3 OLCI that provides pixel-based estimation of the inherent model inversion uncertainty and sensor noise propagation. The algorithm is a full-spectral model-based inversion of radiative transfer (RT) simulations in a coupled atmosphere–ocean system using an ensemble of artificial neural networks (ANN) that were initialized differently during the training process, but composed of the same network architecture. The algorithm has been validated against in-situ radiometric observations across a wide range of optical water types, and has been compared with the latest EUMETSAT operational Level 2 processor IPF-OL-2 v7.01.In this analysis we found that the ensemble ANN showed improved performance over the operational Level 2 processor with a band-averaged (412–708 nm) mean absolute percentage error (MAPE) of 16% compared to 37% and a four-times lower band-averaged bias of -0.00045 sr-1.In the ensemble inversion process we account for three uncertainty components: (1) the total model variance that describes the variance of the data from the different ANNs, (2) the prediction variance of the mean, which is based on calculations of the RT simulations and (3) the instrument noise variance of the mean by propagating the OLCI spectral signal-to-noise ratios (SNR). To study algorithm performance and to quantify the contribution of the different uncertainty components to the total uncertainty, we applied the algorithm to an optically complex full resolution (FR) test scene covering coastal waters of the Great Barrier Reef, Australia. The uncertainties associated with the instrument noise variance were found to be two orders of magnitude lower than the uncertainty components of the prediction and total model variances. The overall largest uncertainty component in our uncertainty framework is attributed to the total model inversion error from averaging the responses of the slightly different adapted networks in the ensemble. The algorithm is made publicly available as a Python/C plugin for the Sentinel Application Platform (SNAP).

Bayesian inverse uncertainty quantification of a MOOSE-based melt pool model for additive manufacturing using experimental data

Additive manufacturing (AM) technology is being increasingly adopted in a wide variety of application areas due to its ability to rapidly produce, prototype, and customize designs. AM techniques afford significant opportunities in regard to nuclear materials, including an accelerated fabrication process and reduced cost. High-fidelity modeling and simulation (M&S) of AM processes is being developed in Idaho National Laboratory (INL)’s Multiphysics Object-Oriented Simulation Environment (MOOSE) to support AM process optimization and provide a fundamental understanding of the various physical interactions involved. In this paper, we employ Bayesian inverse uncertainty quantification (UQ) to quantify the input uncertainties in a MOOSE-based melt pool model for AM. Inverse UQ is the process of inversely quantifying the input uncertainties while keeping model predictions consistent with the measurement data. The inverse UQ process takes into account uncertainties from the model, code, and data while simultaneously characterizing the uncertain distributions in the input parameters–rather than merely providing best-fit point estimates. We employ measurement data on melt pool geometry (lengths and depths) to quantify the uncertainties in several melt pool model parameters. Simulation results using the posterior uncertainties have shown improved agreement with experimental data, as compared to those using the prior nominal values. The resulting parameter uncertainties can be used to replace expert opinions in future uncertainty, sensitivity, and validation studies.

Modeling RL Electrical Circuit by Multifactor Uncertain Differential Equation

The symmetry principle of circuit system shows that we can equate a complex structure in the circuit network to a simple circuit. Hence, this paper only considers a simple series RL circuit and first presents an uncertain RL circuit model based on multifactor uncertain differential equation by considering the external noise and internal noise in an actual electrical circuit system. Then, the solution of uncertain RL circuit equation and the inverse uncertainty distribution of solution are derived. Some applications of solution for uncertain RL circuit equation are also investigated. Finally, the method of moments is used to estimate the unknown parameters in uncertain RL circuit equation.

Bayesian slip inversion with automatic differentiation variational inference

SUMMARYThe Bayesian slip inversion offers a powerful tool for modelling the earthquake source mechanism. It can provide a fully probabilistic result and thus permits us to quantitatively assess the inversion uncertainty. The Bayesian problem is usually solved with Monte Carlo methods, but they are computationally expensive and are inapplicable for high-dimensional and large-scale problems. Variational inference is an alternative solver to the Bayesian problem. It turns Bayesian inference into an optimization task and thus enjoys better computational performances. In this study, we introduce a general variational inference algorithm, automatic differentiation variational inference (ADVI), to the Bayesian slip inversion and compare it with the classic Metropolis–Hastings (MH) sampling method. The synthetic test shows that the two methods generate nearly identical mean slip distributions and standard deviation maps. In the real case study, the two methods produce highly consistent mean slip distributions, but the ADVI-derived standard deviation map differs from that produced by the MH method, possibly because of the limitation of the Gaussian approximation in the ADVI method. In both cases, ADVI can give comparable results to the MH method but with a significantly lower computational cost. Our results show that ADVI is a promising and competitive method for the Bayesian slip inversion.

The use of machine learning for inverse uncertainty quantification in TRACE code based on Marviken experiment

Inverse Uncertainty Quantification (IUQ) methods are gaining recognition in the application of Best Estimate Plus Uncertainty methodologies. Typically, the code input parameters constituted the sources of uncertainties that cannot be estimated other than by user expertise. The IUQ methods aim at providing a more robust and scientific estimation of the distributions of those parameters. In this paper, an approach has been proposed to use machine learning to quantify the uncertainty of critical flow parameters available in the TRACE code.Serving as the means to the ultimate goal of IUQ, the machine learning algorithm has also been utilized in the qualitative assessment of discrepancies between the experiment and calculations. First, a set of Monte Carlo calculations were performed for selected Marviken Critical Flow Tests with permutated values of input parameters of TRACE code. Then, utilizing a supervised machine learning model and a set of features describing the mass flow progression, a Random Forest classifier was used to assign the results to one of the defined classes of prediction accuracy. This analysis has been done using two different training datasets to present the influence of the training data selection on the classification.The machine learning classification resulted in creating a class for which the results with the best fit to the experiment were collected. Derivation of this class created an opportunity to propose ranges and probability distribution functions for CHM12 and CHM22 TRACE internal variables, which is a goal for IUQ methods. Lognormal distribution has been proposed for CHM12 with a mode equal to 0.87, while normal distribution was proposed for CHM22 with a mean equal to 0.9. Variables C1RC2 and C2RC2 proved to be uninfluential on the results. The PDFs for CHM12 and CHM22 have been verified on all 26 Marviken tests. The article proposes a method that can be used in best-estimate analyses to obtain reliable distributions of input variables.

Towards the NASA UQ Challenge 2019: Systematically forward and inverse approaches for uncertainty propagation and quantification

This paper is dedicated to exploring the NASA Langley Challenge on Optimization under Uncertainty by proposing a series of approaches for both forward and inverse treatment of uncertainty propagation and quantification. The primary effort is placed on the categorization of the subproblems as to be forward or inverse procedures, such that dedicated techniques are proposed for the two directions, respectively. The sensitivity analysis and reliability analysis are categorized as forward procedures, while modal calibration & uncertainty reduction, reliability-based optimization, and risk-based design are regarded as inverse procedures. For both directions, the overall approach is based on imprecise probability characterization where both aleatory and epistemic uncertainties are investigated for the inputs, and consequently, the output is described as the probability-box (P-box). Theoretic development is focused on the definition of comprehensive uncertainty quantification criteria from limited and irregular time-domain observations to extract as much as possible uncertainty information, which will be significant for the inverse procedure to refine uncertainty models. Furthermore, a decoupling approach is proposed to investigate the P-box along two directions such that the epistemic and aleatory uncertainties are decoupled, and thus a two-loop procedure is designed to propagate both epistemic and aleatory uncertainties through the systematic model. The key for successfully addressing this challenge is in obtaining on the balance among an appropriate hypothesis of the input uncertainty model, a comprehensive criterion of output uncertainty quantification, and a computational viable approach for both forward and inverse uncertainty treatment.

Uncertain Independent Increment Processes are Contour Processes

Uncertain processes are used to model dynamic indeterminate systems associated with human uncertainty, and uncertain independent increment processes are a type of uncertain processes with independent uncertain increments. This paper mainly verifies a basic property about the sample paths of uncertain independent increment processes, which states that uncertain independent increment processes defined on a continuous uncertainty space are contour processes, a type of uncertain processes with a spectrum of sample paths as the skeletons. Based on this property, the extreme values and the time integral of an uncertain independent increment process are investigated, and their inverse uncertainty distributions are obtained.

Uncertain Circuit Equation

Differential equation is a powerful tool for investigating the transient and steady-state solutions of electrical circuit in the time domain. By considering the noise in actual circuit system, this paper first presents an uncertain circuit equation, which is a type of differential equation driven by Liu process. Then the solution of uncertain circuit equation and the inverse uncertainty distribution of solution are derived. Following that, two applications of solution are provided as well. Based on the observations, the method of moments is used to estimate the unknown parameters in uncertain circuit equation. In addition, a paradox for stochastic circuit equation is also given.

Towards improving the predictive capability of computer simulations by integrating inverse Uncertainty Quantification and quantitative validation with Bayesian hypothesis testing

The Best Estimate plus Uncertainty (BEPU) approach for nuclear systems modeling and simulation requires that the prediction uncertainty must be quantified in order to prove that the investigated design stays within acceptance criteria. A rigorous Uncertainty Quantification (UQ) process should simultaneously consider multiple sources of quantifiable uncertainties: (1) parameter uncertainty due to randomness or lack of knowledge; (2) experimental uncertainty due to measurement noise; (3) model uncertainty caused by missing/incomplete physics and numerical approximation errors, and (4) code uncertainty when surrogate models are used. In this paper, we propose a comprehensive framework to integrate results from inverse UQ and quantitative validation to provide robust predictions so that all these sources of uncertainties can be taken into consideration.Inverse UQ quantifies the parameter uncertainties based on experimental data while taking into account uncertainties from model, code and measurement. In the validation step, we use a quantitative validation metric based on Bayesian hypothesis testing. The resulting metric, called the Bayes factor, is then used to form weighting factors to combine the prior and posterior knowledge of the parameter uncertainties in a Bayesian model averaging process. In this way, model predictions will be able to integrate the results from inverse UQ and validation to account for all available sources of uncertainties. This framework is a step towards addressing the ANS Nuclear Grand Challenge on “Simulation/Experimentation” by bridging the gap between models and data.

Three-dimensional uncertain heat equation

Heat equation is a partial differential equation describing the temperature change of an object with time. In the traditional heat equation, the strength of heat source is assumed to be certain. However, in practical application, the heat source is usually influenced by noise. To describe the noise, some researchers tried to employ a tool called Winner process. Unfortunately, it is unreasonable to apply Winner process in probability theory to modeling noise in heat equation because the change rate of temperature will tend to infinity. Thus, we employ Liu process in uncertainty theory to characterize the noise. By modeling the noise via Liu process, the one-dimensional uncertain heat equation was constructed. Since the real world is a three-dimensional space, the paper extends the one-dimensional uncertain heat equation to a three-dimensional uncertain heat equation. Later, the solution of the three-dimensional uncertain heat equation and the inverse uncertainty distribution of the solution are given. At last, a paradox of stochastic heat equation is introduced.

Complex uncertain differential equations with application to time integral

In this paper, we propose complex uncertain differential equations (CUDEs) based on uncertainty theory. In order to describe the evolution of complex uncertain phenomenon related to belief degrees, we apply the complex Liu process to CUDEs. Firstly, we pose a concept of a linear CUDE and prove that homogeneous linear CUDE and general linear CUDE have solutions. Then, we prove existence and uniqueness theorem of a special CUDE. Further, we design a numerical algorithm to obtain inverse uncertainty distribution of the solution. Finally, as an application, we analyse the inverse uncertainty distributions of time integral of CUDEs and design numerical algorithms to obtain inverse uncertainty distributions of time integral.

Inverse uncertainty quantification for imprecise structure based on evidence theory and similar system analysis

Inverse uncertainty quantification for imprecise structure based on evidence theory and similar system analysis

Nuclear data target accuracy requirements for advanced reactors: The ALFRED case

Uncertainties estimation is essential for a sound assessment and a clear definition of safety margins, which in turn are crucial for licensing approval of new nuclear reactors by safety authorities. Moreover, uncertainties are also matter for target accuracy requirements (TARs), whose achievement could disclose new or enhanced opportunities. The concept of target accuracy requirements has been over the years one of the returning focuses of Massimo Salvatores, who just recently relaunched an update exercise within the OECD/NEA Working Party on International Nuclear Data Evaluation Co-operation (WPEC) aiming at providing new estimates having as reference the advanced systems meant for tomorrow.This work presents an evaluation of the current accuracies on the main core integral parameters of the Advanced Lead-cooled Fast Reactor European Demonstrator (ALFRED), as a starting point to discuss which would be the target values that might introduce beneficial improvements in the design. Despite the use of one of the most recent nuclear data libraries (ENDF/B-VIII.0), and the broad margins set in the design of ALFRED according to its mission as demonstrator, room for improvement is found on the multiplication factor. Then, moving from the target accuracies on the integral parameters, an inverse uncertainty problem is solved to derive the corresponding target accuracies on nuclear data, meeting which would guarantee the achievement for the former ones. The inverse problem is solved introducing three different sets of cost parameters, reflecting possible challenges related to the execution of the differential experiments requested to achieve the aimed reductions of the accuracies. By this, the resulting target accuracies on nuclear data here presented are deemed useful to obtain information for the high-priority lists driving the selection of isotopes and reactions for future refinement experiments.

Complex Source Behaviors and Spatiotemporal Evolution of Seismicity During the 2015–2016 Earthquake Sequence in Cushing, Oklahoma

AbstractIn September and October 2015, three M4+ earthquakes occurred as a sequence along a fault northwest of the Cushing city, Oklahoma, followed by another M5 earthquake in November 2016. While previous studies have shown that moderate‐size earthquakes in Oklahoma are likely induced by wastewater injections, it is still not clear what controls the rupture process and spatiotemporal evolutions of seismicity during individual sequences. In this study, we investigated the rupture process of these four M4‐5 events in 2015–2016 with finite fault model (FFM) inversions, and computed the static stress changes during this sequence. We found that the rupture processes of four M4‐5 earthquakes were very complex, and each of them had several subevents with different rupture directivities. The 2016 M5 earthquake started near the region where three M4+ events initiated, but the majority of the slip occurred a few kilometers away in the northeast direction. In comparison, the 2015 M4.3 event mainly ruptured toward the southwest direction. Due to data limitation and inversion uncertainties, we were unable to constrain the rupture directivities for the other two M4+ events. The foreshocks 3 days before the first M4+ earthquake in 2015 occurred in a region of positive shear stress changes caused by previous earthquakes in 2014–2015 on unmapped faults several kilometers to the south. Our results suggest small‐scale heterogeneity in controlling complex seismicity and rupture patterns in the 2015–2016 Cushing sequence, and possible triggering of this sequence by a small stress perturbation on order of a few kilopascals.

Identification of point source emission in river pollution incidents based on Bayesian inference and genetic algorithm: Inverse modeling, sensitivity, and uncertainty analysis

Identification of pollution point source in rivers is strenuous due to accidental chemical spills or unmanaged wastewater discharges. It is crucial to take physical characteristics into account in the estimation of pollution sources. In this study, an integrated inverse modeling framework is developed to identify a point source of accidental water pollution based on the contaminant concentrations observed at monitoring sites in time series. The modeling approach includes a Markov chain Monte Carlo method based on Bayesian inference (Bayesian-MCMC) inverse model and a genetic algorithm (GA) inverse model. Both inverse models can estimate the pollution sources, including the emission mass quantity, release time, and release position in an accidental river pollution event. The developed model is first tested for a hypothetical case with field river conditions. The results show that the source parameters identified by the Bayesian-MCMC inverse model are very close to the true values with relative errors of 0.02% or less; the GA inverse model also works with relative errors in the range of 2%–7%. Additionally, the uncertainties associated with model parameters are analyzed based on global sensitive analysis (GSA) in this study. It is also found that the emission mass of pollution source positively correlates with the dispersion coefficient and the river cross-sectional area, whereas the flow velocity significantly affects release position and release time. A real case study in the Fen River is further conducted to test the applicability of the developed inverse modeling approach. Results confirm that the Bayesian-MCMC model performs better than the GA model in terms of accuracy and stability for the field application. The findings of this study would support decision-making during emergency responses to river pollution incidents.

A probabilistic fatigue life prediction for adhesively bonded joints via ANNs-based hybrid model

The paper is aimed at developing an efficient and robust probabilistic fatigue life prediction framework for adhesively bonded joints. This framework calibrates the fatigue life model by quantifying uncertainty in the fatigue damage evolution relation using a set of experimental fatigue life data. Probabilistic assessment of fatigue life is simulated through damage evolution along the bondline and Bayesian inference via the Markov chain Monte Carlo (MCMC) sampling method for inverse uncertainty quantification (UQ). To expedite the fatigue life simulation, a hybrid model composed of physics-based fatigue damage evolution relation and a data-driven artificial neural networks (ANNs) model is employed. The degradation of the adhesive is evaluated by the fatigue damage evolution relation which is then mapped to the strain redistribution along the bondline using the ANNs model. Once the mapping is learned by the ANNs, through data from FEA simulations, the probabilistic fatigue life prediction framework involves three successive modules: (I) fatigue damage growth (FDG) simulator, (II) uncertainty quantification (UQ), and (III) confidence bounds for fatigue life prediction. The FDG simulator can be used for simulating fatigue degradation rapidly for a given geometric configuration under any arbitrary fatigue loading spectra. The quantified uncertainties from the framework correspond to the intrinsic statistical material properties that can be used for probabilistic fatigue life prediction in any joint configuration with the same adhesive material. The probabilistic framework is verified using a single lap joint (SLJ) by quantifying uncertainties which are then used for probabilistic fatigue life prediction in laminated doublers in the bending (LDB) joint, that uses the same adhesive material as SLJ, and successfully compared with experimental data. The framework is also tested and validated by estimating probabilistic fatigue life in other joint configurations under constant and variable amplitude fatigue loading spectra.

Elastic Properties Estimation From Prestack Seismic Data Using GGCNNs and Application on Tight Sandstone Reservoir Characterization

Traditional optimization algorithms are usually applied to estimate the elastic parameters of the subsurface by using field seismic data. However, these optimization algorithms highly depend on prior knowledge (e.g., the initial model setup and sparsity), leading to serious inversion uncertainties. Nowadays, with the rapid development of neural networks, convolutional neural networks (CNNs) have been widely imposed on estimating elastic parameters from field data. However, the deficiency of labeled seismic data impedes the CNNs application in seismic inversion. Moreover, both the size and diversity of labeled datasets are also critical factors influencing the accuracy and resolution of predicted parameters when using the CNNs-based inversion techniques. In this work, taking the unconventional tight sandstone formation as an example, we develop a geological and geophysical model driven CNNs (GGCNNs), named as GGCNNs. The proposed GGCNNs allow us to take advantage of both the prior geological information and basic geophysical model from the generated synthetic labeled prestack seismic datasets, representing essential characteristics of the subsurface. Moreover, under the consideration of data diversity, the GGCNNs model enables us to make a tradeoff between the inversion accuracy and labeled data size. Applications on both synthetic and field data clearly demonstrate the effectiveness of the proposed GGCNNs model for predicting elastic parameters by using prestack seismic data, i.e., its predicted results are with high accuracy in the vertical profile and continuity and smooth in the horizon slice.

Error-lumped inverse uncertainty quantification of automotive heat exchangers (HEXs) using large-scale database from system level tests

Error-lumped inverse uncertainty quantification of automotive heat exchangers (HEXs) using large-scale database from system level tests

Solutions of linear uncertain fractional order neutral differential equations

Uncertain fractional order neutral differential equation is an important model to describe the evolution process of uncertain dynamical system. This paper devotes to studying linear uncertain fractional order neutral differential equations. After providing the analytic solutions for linear uncertain fractional order neutral differential equations by the Mittag-Leffler function, the author investigates the inverse uncertainty distribution of the solution to linear uncertain fractional order neutral differential equation by the α-path and has a discussion of the dependence of the solution on initial function based on the generalized Gronwall inequality.

Data-driven inverse uncertainty quantification in the transverse tensile response of carbon fiber reinforced composites

Uncertainty quantification is critical for the full exploitation of composite materials’ potential. Inverse methods offer the possibility of indirectly characterizing the uncertainty of microscopic parameters by employing data sets from standard structural tests in higher scales. Two crucial requirements though, are the efficient modeling especially for the nonlinear prediction, and the measurement error availability from the tests which affects the updated scatter. This study employs effective stiffness and strength experimental data in order to quantify uncertainties of a carbon fiber UD composite in the microscale. A polynomial chaos surrogate model is trained from finite element simulations, able to efficiently predict the homogenized stiffness and strength for the uncertainty quantification procedure. The random parameters which are influential enough to be updated, are identified via a variance-based global sensitivity analysis. The inverse problem is solved with the Bayesian inference method, which updates any prior estimation of the probability models of the input parameters, based on output observations from the tests. Results show significant uncertainty reduction in comparison with typically used variance values in the literature and can be used to enrich the composite material databases. The proposed methodology is applied for the transverse tensile load case, although its non-intrusive nature allows applications for more load cases and various setups.