Inverse Uncertainty Research Articles

An effective one-iteration learning algorithm based on Gaussian mixture expansion for densities

In this study, we utilize Gaussian Mixture Model (GMM) and propose a novel learn algorithm to approximate any density in a fast and simple way. In our previous study, we proposed a idea called GMM expansion which inspired by Fourier expansion. Similar to the base of frequencies in Fourier expansion, GMM expansion assume that normal distributions can be placed evenly along the support as a set of bases to approximate a large set of distribution in good accuracy. In this work, a new algorithm is proposed base on the idea of GMM expansion. A theoretical analysis also given to verify the convergence. Various experiments are carried out to exam the efficacy of proposed method. Experiment result demonstrate the advantages of proposed method and support that this new algorithm perform faster, is more accurate, has better stability, and is easier to use than the Expectation Maximization (EM) algorithm. Furthermore, the benefits of this proposed method helps improve the integration of GMM in neural network. The experiment results show that the neural network with our proposed method significantly improves ability to handle the inverse problem and data uncertainty. Finally, another application, a GMM-based neural network generator, is built. This application shows the potential to utilize distribution random sampling for feature variation control in generative mode.

Flow-dependent observation errors for greenhouse gas inversions in an ensemble Kalman smoother

Abstract. Atmospheric inverse modeling is the process of estimating emissions from atmospheric observations by minimizing a cost function, which includes a term describing the difference between simulated and observed concentrations. The minimization of this difference is typically limited by uncertainties in the atmospheric transport model rather than by uncertainties in the observations. In this study, we showcase how a temporally varying, flow-dependent atmospheric transport uncertainty can enhance the accuracy of emission estimation through idealized experiments using an ensemble Kalman smoother system. We use the estimation of European CH4 emissions from the in situ measurement network as an example, but we also demonstrate the additional benefits for trace gases with more localized sources, such as SF6. The uncertainty in flow-dependent transport is determined using meteorological ensemble simulations that are perturbed by physics and driven at the boundaries by an analysis ensemble from a global meteorology and a CH4 simulation. The impact of direct representation of temporally varying transport uncertainties in atmospheric inversions is then investigated in an observation system simulation experiment framework in various setups and for different flux signals. We show that the uncertainty in the transport model varies significantly in space and time and that it is generally highest during nighttime. We apply inversions using only afternoon observations, as is common practice, but also explore the option of assimilating hourly data irrespective of the hour of day using a filter based on transport uncertainty and taking into account the temporal covariances. Our findings indicate that incorporating flow-dependent uncertainties in inversion techniques leads to more accurate estimates of GHG emissions. Differences between estimated and true emissions could be reduced more effectively by 9 % to 82 %, with generally larger improvements for the SF6 inversion problem and for the more challenging setup with small flux signals.

Some Properties of Lognormal Uncertain Variables

Uncertainty theory, as a mathematical tool to rationally handle degrees of belief of human beings, has been widely applied in science and engineering. As a basic component of uncertainty theory, uncertainty distribution is a carrier to describe the characterization of an uncertain variable. Moreover, inverse uncertainty distribution provides an easy way to calculate the functions of uncertain variables as well as their expected value. This paper mainly provides the formulas to calculate the expected value, variance, moments and entropy of lognormal uncertain variable. Besides, some operational laws of lognormal uncertain variables are proved.

Integrating deep learning and discrete cosine transform for surface waves full-waveform inversion

SUMMARY Accurate estimations of near-surface S-wave velocity (Vs) models hold particular significance in geological and engineering investigations. On the one hand, the popular multichannel analysis of surface waves (MASWs) is limited to the 1-D and the plane wave assumptions. On the other hand, the more advanced and computationally expensive full-waveform inversion (FWI) approach is often solved within a deterministic framework that hampers an accurate uncertainty assessment and makes the final predictions heavily reliant on the starting model. Here we combine deep learning with discrete cosine transform (DCT) to solve the FWI of surface waves and to efficiently estimate the inversion uncertainties. Our neural network approach effectively learns the inverse non-linear mapping between DCT-compressed seismograms and DCT-compressed S-velocity models. The incorporation of DCT into the deep learning framework provides several advantages: it notably reduces parameter space dimensionality and alleviates the ill-conditioning of the problem. Additionally, it decreases the complexity of the network architecture and the computational cost for the training phase compared to training in the full domain. A Monte Carlo simulation is also used to propagate the uncertainties from the data to the model space. We first test the implemented inversion method on synthetic data to showcase the generalization capabilities of the trained network and to explore the implications of incorrect noise assumptions in the recorded seismograms and inaccurate wavelet estimations. Further, we demonstrate the applicability of the implemented method to field data. In this case, available borehole information is used to validate our predictions. In both the synthetic and field applications, the predictions provided by the proposed method are compared with those of a deterministic FWI and the outcomes of a network trained in the full data and model spaces. Our experiments confirm that the implemented deep-learning inversion efficiently and successfully solves the FWI problem and yields more accurate and stable results than a network trained without the DCT compression. This opens the possibility to efficiently train a neural network that provides accurate instantaneous predictions of Vs near-surface models and related uncertainties.

Proton radiography inversions with source extraction and comparison to mesh methods.

Proton radiography is a central diagnostic technique for measuring electromagnetic (EM) fields in high-energy-density, laser-produced plasmas. In this technique, protons traverse the plasma where they accumulate small EM deflections which lead to variations in the proton fluence pattern on a detector. Path-integrated EM fields can then be extracted from the fluence image through an inversion process. In this work, experiments of laser-driven foils were conducted on the OMEGA laser and magnetic field reconstructions were performed using both "fluence-based" techniques and high-fidelity "mesh-based" methods. We implement nonzero boundary conditions into the inversion and show their importance by comparing against mesh measurements. Good agreement between the methods is found only when nonzero boundary conditions are used. We also introduce an approach to determine the unperturbed proton source profile, which is a required input in fluence reconstruction algorithms. In this approach, a fluence inversion is embedded inside of a mesh region, which provides overconstrained magnetic boundary conditions. A source profile is then iteratively optimized to satisfy the boundary information. This method substantially enhances the accuracy in recovering EM fields. Lastly, we propose a scheme to quantify uncertainty in the final inversion that is introduced through errors in the source retrieval.

A Data Driven Black Box Approach for the Inverse Quantification of Set-Theoretical Uncertainty

Abstract Inverse uncertainty quantification commonly uses the well established Bayesian framework. Recently, alternative interval methodologies have been introduced. However, in their current state of the art implementation, both techniques suffer from a large and usually unpredictable computational effort. Thus, both techniques are not applicable in a real-time context. To achieve a low-cost, real-time solution to this inverse problem, we introduce a deep-learning framework consisting of unsupervised auto-encoders and a shallow neural network. This framework is trained by means of a numerically generated dataset that captures typical relations between the model parameters and selected measured system responses. The performance and efficacy of the technique is illustrated using two distinct case studies. The first case involves the DLR AIRMOD, a benchmark case that has served as reference case for the inverse uncertainty quantification problem. The results demonstrate that the achieved accuracy is on par with the existing interval method found in literature, while requiring only a fraction of its computational resources. The second case study examines a resistance pressure welding process, which is known to require extremely fast monitoring and control due to the high process throughput. Based on the proposed method, and with only a limited selection of simulated responses of the process, it is possible to identify the interval uncertainty of the crucial parameters of the process. The computational cost in this case makes it possible for an inverse uncertainty quantification in a real-time setting.

Inverse uncertainty quantification for stochastic systems by resampling. Applications to modeling of alcohol consumption and infection by HIV

A random differential equation, or stochastic differential equation with parametric uncertainty, is a classical differential equation whose input values (coefficients, initial conditions, etc.) are random variables. Given data, the probability distributions of the input random parameters must be appropriately inferred, before proceeding to simulate the model’s output. This task is called inverse uncertainty quantification. In this paper, the goal is to study the applicability of the Bayesian bootstrap to draw inferences on the posterior distributions of the parameters, by resampling the residuals of the deterministic least-squares optimization with Dirichlet weights. The method is based on repeated deterministic calibrations. Thus, to alleviate the curse of dimensionality, the technique may be combined with the principle of maximum entropy for densities, when there are some parameters that are not optimized deterministically. For illustration of the methodology, two case studies on important health topics are conducted, with stochastic fitting to data. The first one, on past alcohol consumption in Spain, taking social contagion into account. The second one, on HIV evolution considering CD4+ T cells and viral load, with a patient in clinical follow-up. All these applied models are built from a compartmental viewpoint, with a randomized basic reproduction number that controls the long-term behavior of the system.

Assessment of the Choked Flow Model of RELAP5 for Application of Inverse Quantification Methods

In the framework of deterministic safety analysis, best estimate plus uncertainty methodologies can provide a more informative detailed analysis of transients than conservative approaches. However, one important step in the application of such methodologies is derivation of uncertainty of the physical models. Probability density functions of the physical model parameters are obtained by applying inverse uncertainty quantification (IUQ) methods to experimental data. The present work deals with evaluation of the experimental database and assessment of the simulation model, which are required steps prior to the application of IUQ methods. The U.S. Nuclear Regulatory Commission system code RELAP5 has been applied for simulation of three experimental databases on choked/critical flow: Sozzi-Sutherland, Super MobyDick, and Marviken Critical Flow Tests. First, the adequacy of the experimental data to the specific target domain is carried out, to then proceed to calibration of the input model parameters by combining the use of Gaussian Process metamodels and optimization schemes. The assessment of the simulation models is performed by evaluating independently accuracy, precision, and consistency through four statistical indicators, including a novel indicator to evaluate the consistency of the results. The final results indicate that the model is capable of reproducing the selected experimental database and overall average accuracy, precision, and consistency below the 10% threshold and can be used for application of IUQ methods.

Crustal stress near the Yakutat microplate collision from probabilistic earthquake focal mechanisms

Earthquake source characteristics provide a valuable constraint on crustal stress and regional plate tectonics. Earthquake source studies in Yukon and northern British Columbia have been limited by sparse seismic network coverage. In this work, we leverage recent seismic network improvements to estimate focal mechanisms for small- and moderate-magnitude ( M > 2.0) earthquakes from P-wave first-motion polarity data. We invert these data within a probabilistic framework to rigorously quantify mechanism uncertainties. Subsequently, probabilistic earthquake focal mechanisms are used as input for inversions for the orientation of principal stress axes, and stress shape ratio, for spatial windows throughout the region. We implement a novel, data-driven approach to propagate focal mechanism uncertainty in stress inversion. Our results improve the spatial coverage of existing earthquake focal mechanisms and enable an orogenic-scale study of crustal stress near the Yakutat–North America collision. Overall, the region is characterized by a transpressive stress regime. Maximum horizontal compressive stress orientations exhibit a pattern orthogonal to the Yakutat collision syntaxis. Stress inversion results also reveal an abrupt change in orientation east-to-west, which we interpret as a change in stress regime across the Fairweather–Connector–Totschunda fault system that is likely related to coupling between the subducting Yakutat slab and North America. This work improves our understanding of potential earthquake hazards in southwestern Yukon and the surrounding region.

Uncertainty reduction in MASW inversion and ground response analysis using a-priori information

Multi-channel analysis of surface waves (MASW) method is commonly used for estimating the shear wave velocity (Vs) profile of soil and subsequent ground response analysis (GRA). However, the uncertainties from inversion non-uniqueness in MASW testing and the absence of soil type information in GRA can undermine their accuracy and affect seismic designs. Incorporating a-priori information from geotechnical investigations can be a potential solution, but its effectiveness in MASW testing is not well understood. This study aims to quantify the uncertainty reduction achieved by incorporating a-priori information in MASW testing and GRA. Extensive numerical simulations of MASW and GRA for nine idealized soil profiles were carried out. Subsequently, the findings were validated by carrying out field testing with and without a-priori information. The a-priori information included number and thickness of soil layers, soil type and index properties. The findings demonstrate that incorporating a-priori information significantly reduces inversion uncertainty in MASW and GRA, enhancing the reliability and accuracy of the analyses. This work provides a quantitative assessment of uncertainty reduction using a-priori information for different possible site and earthquake scenarios. The findings have important implications for design of earthquake-resistant structures and for improving the safety of people and infrastructure in earthquake-prone areas.

Framework for the correct treatment of model input parameters for Bayesian updating problems in nuclear engineering

Bayesian updating (BU) tools are increasingly used in nuclear engineering for inverse uncertainty quantification (IUQ) and calibration combined with uncertainty reduction. Their goals are quantifying or reducing the uncertainty of some of the uncertain model input parameters. Since it is often the case that available experimental data only allows for updating the most influential input parameters, researchers often ignore the less important ones during BU. This paper explores the consequences of neglecting the uncertainties of uncalibrated model input parameters (UMIP). It also proposes how to include them properly and which BU algorithms are the best choices for various types of inverse problems. The analysis is based on exploring two toy problems and one in nuclear engineering concerning multiplication factor calculations. The results clearly show that the improper treatment of UMIP during BU often leads to underestimating posterior uncertainties — either of the calibrated input parameters or the simulated integral parameters, depending on how the BU was conducted. The proposed methods of correct UMIP treatment will improve the rigorousness of the BU processes and boost confidence in the resulting posterior distributions.

Full-field experiment-aided virtual modelling framework for inverse-based stochastic prediction of structures with elastoplasticity

Forecasting of stochastic ductile failures in fabrication and service stages are challenging tasks of advanced structures in practical engineering. Due to the prohibitive costs of repetitive experimental tests to quantify optimal failure-related parameters, numerous studies have turned to simulation-based uncertainty quantification. However, the credibility of these approaches is frequently doubted by the function-generated distribution of random data involved. Thus, an accurate assessment of the material parameters ascertains the appropriate estimation of uncertain parameters, which is essential for the risk/safety assessment of structures in different stages. In this paper, a full-field experiment-aided virtual modelling framework for inverse-based prediction of stochastic elastoplastic failure (EVMF-ISP) is proposed to deliver precise insights regarding the effective distribution range of mechanical parameters. To this end, all available experiment observations serve as the reference for assessing the prior and posterior probability density of the unknown parameters through the real-time inverse uncertainty quantification (UQ) module. The framework can be divided into three parts, where initially a pre-virtual model (PRVM) is formulated, and Bayesian inference is implemented to propagate the experiment observations backwards to ascertain the uncertain parameters. Then, an advanced multidimensional slice sampling method is developed to deal with the derived complex posterior probability density function (PDF) of mechanical parameters. In the end, a reliable stochastic elastoplastic analysis can be conducted with the revised uncertain samples and finalised with post-virtual models (POVMs) for the concerned structures. Such that, accurate and efficient determination of nonlinear response of structures can be directly predicted. The EVMF-ISP framework is logically presented and thoroughly illustrated with practical applications.

Average Waiting Time of Customers in Single-Server Uncertain Queueing System

Uncertain queueing system is an important research object in the field of operations research, and widely exists in logistics, production, insurance and other fields. In order to further analyze the performance of the single-service uncertain queueing system, this paper derives the generalized inverse uncertainty distributions of the waiting times of customers, and provides the calculation formulas for the average waiting times of customers on this basis. Finally, this paper also provides two examples to illustrate the generalized inverse uncertainty distributions of waiting times and the calculation formulas of the average waiting times under specific uncertainty distributions.

Negative cumulative extropy of uncertain variables with applications

Recently, a measure of uncertainty termed by negative cumulative extropy (NCEX) is proposed by Tahmasebi and Toomaj (2022). It is a flexible extension of extropy measure which was proposed by Lad, Sanfilippo, and Agro (2015). To measure the uncertainty of uncertain variables, in this article, we propose the concept of NCEX through uncertainty distributions. Some properties of NCEX are derived, and some practical examples of uncertain variables are given. Several basic theorems are proposed. A formula of cross-extropy is defined via inverse uncertainty distributions. We also delve into a pattern recognition problem to highlight the importance of cross-extropy. Finally, applications to portfolio selection and signal processing are presented.

Development of higher order deterministic approaches of forward and inverse uncertainty quantification for nonlinear engineering systems

This paper introduces an innovative numerical scheme that may accurately quantify the parameter and response distributions with minimal computational costs, with specific applications to the numerical computation of multiphase flows under various fluid conditions. The probabilistic approach of uncertainty quantification, e.g., the Monte Carlo simulation, although known to be effective in estimating the parameter/response distribution for highly nonlinear problems, is usually not applicable to multi-physics and multi-scale integrated systems due to its high computational cost. The objective of this study is first, to reduce the computing demand for complex physical system calculations and, at the same time, to accurately predict actual probability distributions by developing a higher order deterministic approach to uncertainty quantification. To achieve this, instead of executing simulation codes multiple times to estimate the probability distribution, the first to fourth moments of the probability density function are derived directly by employing a second order Taylor expansion form of the responses along with Bayesian statistics. This characterizes the parameter and response distributions computed by inverse and forward uncertainty quantification based on their mean value, mode, covariance, skewness, and kurtosis. The parameter distribution is estimated by the numerically calculated higher moments of the parameters with a posteriori probability density function being derived by Bayesian inference. The parameter distribution is then mapped into the second order approximated response function to compute the characteristics of the response distribution through both analytical calculation and numerical approximation of the moments of the responses. The proposed approach is numerically demonstrated through simulations of two benchmark problems: MIT pressurizer and FEBA tests. The prediction of the responses is improved by inverse uncertainty quantification as numerical calculations with the a posteriori model parameters are in better agreement with the measured data over a wide range of transient periods. Additionally, the response distribution estimated by the proposed approach of forward uncertainty quantification accurately predicts the distribution computed by the Monte Carlo simulation.

Aerosol‐Calibrated Matched Filter Method for Retrievals of Methane Point Source Emissions Over the Los Angeles Basin

AbstractMethane, with a global warming potential roughly 86 times greater than carbon dioxide over a 20‐year timeframe, plays a crucial role in global warming. Remote sensing retrieval is a pivotal methodology for identifying methane emission sources, with accuracy influenced largely by surface and atmospheric properties, including aerosols. In this study, we propose an Aerosol‐Calibrated Matched Filter (ACMF) algorithm to improve the traditional Matched Filter (MF) method. Our new approach incorporates an aerosol scattering correction factor to reduce the aerosol‐induced bias on methane retrievals. Validating our algorithm through simulated spectra, we demonstrate that considering the aerosol scattering effect significantly reduces retrieval errors compared to MF methods by an average of approximately 90%. We apply our newly developed algorithm to hyperspectral data obtained from the Airborne Visible/Infrared Imaging Spectrometer—Next Generation in the Los Angeles Basin and focus on 11 plumes identified through case studies. Our results reveal that ACMF estimates of emission rates and inversion uncertainties exhibit an average reduction of approximately 4% compared to corresponding MF results, with deviation increasing with aerosol optical depth (AOD).

Global Sensitivity Analysis for Segmented Inverse Uncertainty Quantification in the Safety Analysis of Nuclear Power Plants

Within the Best Estimate Plus Uncertainty framework for the safety analysis of Nuclear Power Plants, the quantification of the uncertainties affecting the Thermal-Hydraulics (T-H) codes used is crucial. For this, Inverse Uncertainty Quantification (IUQ) methodologies are being developed for determining the probability density functions of relevant T-H codes input parameters, based on experimental data from Separate Effect Tests (SETs) experimental facilities. In practice, IUQ is challenged by the large range of variability of the experimental data in terms of Initial and Boundary Conditions (ICs & BCs), because the experimental campaigns are designed to cover the widest possible domain of conditions with the smallest number of experiments, so that same or similar ICs and BCs are seldomly repeated. To address this issue, we propose to use global sensitivity analysis, to tailor the IUQ on specific sub-regions described by segmented ICs & BCs domains. The methodology proposed is exemplified on two SETs, namely Sozzi-Sutherland and Super Moby Dick, whose experimental databases have been made available in the ATRIUM (Application Tests for Realization of Inverse Uncertainty quantification and validation Methodologies in thermal hydraulics) project promoted by the OECD/NEA/CSNI. The results obtained are superior to those of traditional IUQ methodologies for models highly sensitive to ICs & BCs.

Predicting multiphase flow behavior of methane in shallow unconfined aquifers using conditional deep convolutional generative adversarial network

Numerical modeling is an essential tool for geoscience applications involving multiphase flow behavior. Performing numerical simulation is, however, computationally intensive due to multi-scale, non-linear, and multi-physics nature of mechanisms controlling multiphase flow dynamics. Additionally, the computational challenges associated with traditional numerical simulations limit their applicability for repetitive simulations required in inverse modeling, uncertainty analysis, and optimization tasks. To overcome these limitations, surrogate modeling has recently emerged as a viable solution. A surrogate model, which functions as a regression model, is trained using numerical simulation data. It approximates the input–output relationships within a dynamic fluid environment. We design surrogate models, based on conditional deep convolutional generative adversarial network (cDC-GAN), to map the cross-domain between input and output pairs in a multiphase multicomponent system, focusing on methane (CH4) migration in shallow unconfined aquifers. The cDC-GAN technique is selected due to its ability to generate high-fidelity surrogate models that effectively capture complex spatial distributions and heterogeneities. This technique excels in handling high-dimensional data, learning the intricate relationships between inputs and outputs, and producing realistic representations of subsurface phenomena. The trained cDC-GANs consider capillary entry pressure heterogeneity as input because this geologic attribute places first-order control on CH4 migration in subsurface. The outputs include CH4 saturation, water saturation, residual saturation of CH4, CH4 mole fraction, and CH4 molality. For each output, a separate cDC-GAN model is trained. Once trained, cDC-GANs can at any given time determine CH4 distribution in a shallow unconfined aquifer, both qualitatively and quantitatively. The cDC-GAN approach can be considered as a valuable complement to numerical simulations for predicting multiphase flow behavior in other geoscience-, energy-, and environmental applications such as contaminant transport, hydrocarbon production, and geological carbon dioxide and hydrogen storage.

Reduction of the model prediction error in vibration-based SHM applications

The accuracy of an SHM scheme is partly controlled by the accuracy of the predictive models involved within the design process of the detector that associates sensor data with a corresponding health state. On the other hand, the accuracy of the predictive model, which is quantified by the error between predictions and observations, is partly controlled by its parameter magnitudes. This work provides a unified framework that allows for reducing the model prediction error by updating the model parameters in the light of experimental observations. The problem is cast in a probabilistic setting and tackled through inferential statistics. Forward and inverse uncertainty quantification approaches are employed in a sequential manner. The framework is able to yield point estimates in accordance to a Maximum Likelihood Estimator (MLE) and is also able to construct credible intervals through computational Bayesian updating. The former is treated by genetic algorithms whereas the later through Markov Chain Monte Carlo computational statistics. The framework is showcased on a vibration problem of the simplified benchmark case named as “Jim Beam”. The first ten modal parameters are considered as the discriminant features of interest . A surrogate of the finite element model of the considered specimen serves as the predictive model and the eigenfrequencies are extracted from a linear modal analysis. The parameters of inferential interest are the Young’s modulus and an artificial stiffness related parameter that is introduced so as to model the flexural rigidity of the specimen’s bolted connections. Experimental Modal Analysis and Operational Modal Analysis are the two approaches that were employed for experimentally identifying the modes and their corresponding frequencies. Results are provided from both the MLE and the Bayesian updating process and the effectiveness of each approach is discussed.

Uncertainty analysis for forest height inversion using L / P band PolInSAR datasets and RVoG model over kryclan forest site

Forest height, as a measure of the quantity and quality of forest resources, plays a significant role in the study of the ecological functions performed by forests. Although the polarimetric synthetic aperture radar interferometry (PolInSAR) technique has evolved as a potent method for forest height inversion, uncertainties still exist in the process of estimating forest height, and the uncertainties in predicted forest height directly lead into the uncertainty of terrestrial carbon stock calculation results. In this study, we took the Random Volume over Ground (RVoG) model as likelihood function and constructed a hierarchical Bayesian framework to calculate and reduce the uncertainty of forest height inversion using L / P band PolInSAR airborne data via RVoG model. Uncertainties resulted from five canopy types and three forest densities were analyzed, respectively. The results showed that among the five different canopy types, L band has the highest prediction accuracy in pure coniferous canopy with Acc. = 0.90. The uncertainty is extremely low for pure forest, with the ratio of uncertainty values of 0.09 for L band and 0.15 for the P band in pure coniferous canopy, and uncertainty values of 0.16 for L band and 0.11 for P band in pure broadleaf canopy, respectively. Furthermore, when the forest density is between 300 and 600 stems/ha, the ratio of uncertainties for the L band is 0.27, whereas the P band is 0.24. As forest density increases, the uncertainty in forest height estimates decreases for both bands. The changes in canopy types and forest density affect forest height estimation uncertainties obviously, the effects are different at each frequency. The forest height inversion accuracy of the L band in pure coniferous canopy surpasses that in other canopy types, with the lowest uncertainty. P band performed well in broadleaf canopy forest height inversion. The inversion uncertainties at both frequencies decrease with increase of forest densities.