Bayesian Inverse Research Articles

A Hadamard fractional total variation-Gaussian (HFTG) prior for Bayesian inverse problems

This paper studies the infinite-dimensional Bayesian inference method with Hadamard fractional total variation-Gaussian (HFTG) prior for solving inverse problems. First, Hadamard fractional Sobolev space is established and proved to be a separable Banach space under some mild conditions. Afterwards, the HFTG prior is constructed in this separable fractional space, and the proposed novel hybrid prior not only captures the texture details of the region and avoids staircase effects, but also provides a complete theoretical analysis in the infinite dimensional Bayesian inversion. Based on the HFTG prior, the well-posedness and finite-dimensional approximation of the posterior measure of the Bayesian inverse problem are given, and samples are extracted from the posterior distribution using the standard preconditioned Crank-Nicolson (pCN) algorithm. Finally, numerical results under different models indicate that the Bayesian inference method with HFTG prior is effective and accurate.

Probabilistic inversion of circular phase spectra: application to two-station phase-velocity dispersion estimation in western Canada

SUMMARY Periodic directional and temporal measurements are common in seismology, and necessitate specific statistical analyses that are appropriate for circular quantities. In this work, we explore the use of a von Mises distribution as a representation of errors on circular seismological observations. Specifically, we automate the estimation of surface-wave phase-velocity dispersion for the teleseismic two-station method, which generally suffers from a 2π phase ambiguity. The use of Bayesian inverse techniques, which aim to rigorously quantify model parameter uncertainty, have become widespread throughout seismology over the last decade. Here, we apply Bayesian inversion to measurements of surface-wave phase spectra in order to estimate 1-D, path-averaged Earth structure between station pairs. The dispersion curve and associated uncertainties are additional results of the inversion, which can then be used as input for subsequent analyses (e.g. tomography). We demonstrate this technique through application to surface-wave recordings from long-running seismic stations throughout western Canada. Our results for over 10 000 station pairs reveal first-order tectonic features consistent with previous studies, which provides confidence in our approach as well as an initial step towards resolving a full 3-D seismic velocity model for the region. This work also presents a foundation for the inversion of surface-wave phase spectra to estimate 3-D Earth structure directly. Finally, the ideas presented in this work are not limited to the inversion of surface-wave phase spectra, but can also be considered for Bayesian geophysical inversion of any circular quantities.

Using Ensemble Meteorological Data Sets to Treat Meteorological Uncertainties in a Bayesian Volcanic Ash Inverse Modeling System: A Case Study, Grímsvötn 2011

AbstractAtmospheric dispersion models are employed in forecasting the atmospheric transport of ash clouds following a volcanic eruption. Errors in input meteorological data can, however, lead to discrepancies between the modeled and observed ash cloud locations. Furthermore, meteorological errors can affect the performance of inversion techniques used to estimate ash emissions. This study explores using ensemble meteorological data sets to overcome issues with meteorological errors in a Bayesian inversion system, Inversion Technique for Emissions Modelling (InTEM) for volcanic ash. Two measures (the evidence value and a cost function) are obtained from the Bayesian framework, and each measure is used to select a “best” meteorological data set from within the ensemble. These two best meteorological data sets are constructed in an iterative manner which is ideally suited for operational use when ash cloud forecasts are updated at regular intervals. The method is applied to the 2011 eruption of the Icelandic volcano Grímsvötn and improvements to ash cloud forecasts using the best meteorological data sets assessed. In this case study, errors in the deterministic meteorological data are significant, resulting in a reduction of ash emissions estimated by InTEM for volcanic ash. We also illustrate how the best meteorological data sets lead to more accurate modeling of the ash cloud.

Petrophysical inversion based on f-s-r amplitude-variation-with-offset linearization and canonical correlation analysis

The prediction of petrophysical properties, such as porosity and rock-fluid volumes, from partially stacked seismic data typically requires a rock-physics model that often is lithology dependent and difficult to calibrate. We adopt canonical correlation analysis (CCA) to infer the underlying relation between petrophysical properties and elastic attributes estimated from seismic data. We develop a two-step inversion approach: first, we predict elastic properties from partially stacked seismic data using a Bayesian linear inverse method based on an amplitude-variation-with-offset (AVO) linearization in terms of fluid, rigidity, and density factors, and then we predict petrophysical properties from the estimated AVO attributes using CCA. The novelty of our approach is the application of CCA to the fluid and rigidity factors, which avoids the calibration of an explicit rock-physics model by automatically deriving a linear relation in the lower dimensional space of the canonical variables. The parameterization of the linearization in terms of fluid, rigidity, and density factors maximizes the correlation with respect to the petrophysical properties of interest. Furthermore, the probabilistic approach is extended to the petrophysical inversion using Bayesian linear theory and the posterior distribution of petrophysical properties conditioned by seismic data is computed by combining the probability distributions obtained from seismic and petrophysical inversion to propagate the uncertainty from the seismic to the petrophysical domain. The inversion is validated on a synthetic case that finds high accuracy of our formulation. A case study with synthetic and real partially stacked seismic data also is presented and compared to a traditional inversion with an explicit rock-physics model.

Solution of physics-based Bayesian inverse problems with deep generative priors

Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is tackling their ill-posed nature. Bayesian inference provides a principled approach for overcoming this by formulating the inverse problem into a statistical framework. However, it is challenging to apply when inferring fields that have discrete representations of large dimensions (the so-called “curse of dimensionality”) and/or when prior information is available only in the form of previously acquired solutions. In this work, we present a novel method for efficient and accurate Bayesian inversion using deep generative models. Specifically, we demonstrate how using the approximate distribution learned by a Generative Adversarial Network (GAN) as a prior in a Bayesian update and reformulating the resulting inference problem in the low-dimensional latent space of the GAN, enables the efficient solution of large-scale Bayesian inverse problems. Our statistical framework preserves the underlying physics and is demonstrated to yield accurate results with reliable uncertainty estimates, even in the absence of information about underlying noise model, which is a significant challenge with many existing methods. We demonstrate the effectiveness of proposed method on a variety of inverse problems which include both synthetic as well as experimentally observed data.

Convergence of spectral likelihood approximation based on q-Hermite polynomials for Bayesian inverse problems

In this paper, q-Gaussian distribution, q-analogy of Gaussian distribution, is introduced to characterize the prior information of unknown parameters for inverse problems. Based on q-Hermite polynomials, we propose a spectral likelihood approximation (SLA) algorithm of Bayesian inversion. Convergence results of the approximated posterior distribution in the sense of Kullback–Leibler divergence are obtained when the likelihood function is replaced with the SLA and the prior density function is truncated to its partial sum. In the end, two numerical examples are displayed, which verify our results.

Bayesian rock-physics inversion with Kumaraswamy prior models

The prediction of rock and fluid volumetric fractions from elastic attributes often is referred to as petrophysical or rock-physics inversion because it requires rock-physics models to map petrophysical properties into geophysical variables, such as velocities and density. Bayesian approaches are suitable for rock-physics inverse problems because the solution, expressed in the form of a probability distribution, can represent the uncertainty of the model predictions due to the errors in the measured data. Bayesian inverse methods often rely on Gaussian prior distributions for their analytical tractability. However, Gaussian distributions are theoretically not applicable to rock and fluid volumetric fractions because, by definition, they are nonzero on the entire set of real numbers, whereas rock and fluid volumetric fractions are bounded between zero and one. The proposed rock physics inversion is based on a Bayesian approach that assumes Kumaraswamy probability density functions for the prior distribution to model double-bounded nonsymmetric continuous random variables between zero and one. The results of the Bayesian inverse problem are the pointwise probability distributions of the rock and fluid volumetric fractions conditioned on the seismic attributes. In the first application, the method is validated using synthetic well-log data for the soft sand and stiff rock-physics models with comparisons with several prior models. In the second application, the method is applied to a 2D real data set to obtain the posterior distribution, the maximum a posteriori, and the confidence intervals of porosity, mineral volumes, and fluid saturations. The most likely model of rock and fluid properties estimated from the posterior distribution assuming a Kumaraswamy prior model finds higher accuracies compared to the corresponding results obtained with a Gaussian prior model.

Fast Bayesian inversion for high dimensional inverse problems

We investigate the use of learning approaches to handle Bayesian inverse problems in a computationally efficient way when the signals to be inverted present a moderately high number of dimensions and are in large number. We propose a tractable inverse regression approach which has the advantage to produce full probability distributions as approximations of the target posterior distributions. In addition to provide confidence indices on the predictions, these distributions allow a better exploration of inverse problems when multiple equivalent solutions exist. We then show how these distributions can be used for further refined predictions using importance sampling, while also providing a way to carry out uncertainty level estimation if necessary. The relevance of the proposed approach is illustrated both on simulated and real data in the context of a physical model inversion in planetary remote sensing. The approach shows interesting capabilities both in terms of computational efficiency and multimodal inference.

Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion

This paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on the combination of several modern techniques such as transport maps and low-rank approximations via a nonintrusive tensor train reconstruction. The central idea is to carry out computations for statistical quantities of interest such as moments based on a convenient representation of a reference density for which accurate numerical methods can be employed. Since the transport from target to reference can usually not be determined exactly, one has to cope with a perturbed reference density due to a numerically approximated transport map. By the introduction of a layered approximation and appropriate coordinate transformations, the problem is split into a set of independent approximations in seperately chosen orthonormal basis functions, combining the notions h- and p-refinement (i.e. “mesh size” and polynomial degree). An efficient low-rank representation of the perturbed reference density is achieved via the Variational Monte Carlo method. This nonintrusive regression technique reconstructs the map in the tensor train format. An a priori convergence analysis with respect to the error terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback–Leibler divergence is derived. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a main motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity and degrees of perturbation of the transport to the reference density. The (superior) convergence is demonstrated in comparison to Monte Carlo and Markov Chain Monte Carlo methods.

Estimating reaction parameters in mechanism-enabled population balance models of nanoparticle size distributions: A Bayesian inverse problem approach.

In order to quantitatively predict nano- as well as other particle-size distributions, one needs to have both a mathematical model and estimates of the parameters that appear in these models. Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth. The Bayesian approach addresses the question of "how well do we know our parameters, along with their uncertainties?." The results reveal that Bayesian inversion statistical analysis on an example, prototype nanoparticle formation system allows one to estimate not just the most likely rate constants and other parameter values, but also their SDs, confidence intervals, and other statistical information. Moreover, knowing the reliability of the mechanistic model's parameters in turn helps inform one about the reliability of the proposed mechanism, as well as the reliability of its predictions. The paper can also be seen as a tutorial with the additional goal of achieving a "Gold Standard" Bayesian inversion ME-PBM benchmark that others can use as a control to check their own use of this methodology for other systems of interest throughout nature. Overall, the results provide strong support for the hypothesis that there is substantial value in using a Bayesian inversion methodology for parameter estimation in particle formation systems.

Uncertainty quantification of quadratically nonlinear local damage size quantification using nonlinear ultrasonics

Backscattered and forward scattered harmonic waves from local damage modeled as quadratically nonlinear material in a 1D bar are used to solve a deterministic inverse problem to find the local damage size by Ghodake (2020). Due to highly nonlinear (|sin(x)/x|) nature of the proposed forward problem, the constrained nonlinear deterministic optimization problem was solved using non-gradient-based algorithms. The forward model is a function of input frequencies, linear wave velocity, and local damage size. In practice, slight uncertainty of these parameters gives a completely different solution. To address this problem, uncertainty quantification of the modified forward problem is carried out along with solving the inverse problem using reliability-based design optimization (RBDO) and Bayesian Inverse (BI) approaches. In RBDO studies, Quantile Monte Carlo (QMC) approach gives more reliable and accurate results in comparison with approaches such as reliability index approach (RIA), performance measure approach (PMA), sequential optimization and reliability assessment (SORA), and inverse first-order reliability method (FORM) integrated with optimization methods such as interior-point (IP) and sequential programming (SQP). Multiple models are solved using the BI approach. Support vector machines regression (SVR), polynomial chaos expansions (PCE), and Kriging surrogate models are used along with RBDO and BI to reduce computational time.

Bayesian Inversion of Soil Hydraulic Properties from Simplified Evaporation Experiments: Use of DREAM(ZS) Algorithm

There is an increasing interest in identifying soil hydraulic properties from simplified evaporation experiments. However, the conventional simplified evaporation method includes a deficit due to using the linear assumption and not accounting for uncertainty in parameters. A suggested alternative method is assessing the parameter uncertainties through inverse modeling. We examined the combination of a Bayesian inverse method, namely, DREAM(ZS), and a numerical simulation model, namely, HYDRUS-1D, for parameter inversion with data in simplified evaporation experiments. The likelihood function could be conditioned only on pressure head observations (single-objective (SO)), or on both pressure head and evaporation rate observations (multi-objective (MO)), with different treatments on the top boundary condition. Three synthetic numerical experiments were generated in terms of the soil types of sand, loam and clay to verify the inverse modeling method. The MO approach performed better than the SO approach and linear assumption when the stage 1 evaporation rate was kept constant. However, the SO inversion was more robust when oscillations existed in the potential evaporation rate. Then, the SO inverse modeling was adopted to investigate two real experiments on loamy-sand soils and compared with the linear assumption. The linear assumption could be reliable for wet conditions with stage 1 evaporation but was not always useable for a relatively dry condition, such as that with stage 2 evaporation. The inverse modeling could be more successful in capturing the whole evaporation process of soils when both stage 1 and stage 2 were involved.

Bayesian Geophysical Inversion Using Invertible Neural Networks

AbstractConstraining geophysical models with observed data usually involves solving nonlinear and nonunique inverse problems. Neural mixture density networks (MDNs) provide an efficient way to estimate Bayesian posterior marginal probability density functions (pdf's) that represent the nonunique solution. However, it is difficult to infer correlations between parameters using MDNs, and in turn to draw samples from the posterior pdf. We introduce an alternative to resolve these issues: invertible neural networks (INNs). These are simultaneously trained to represent uncertain forward functions and to solve Bayesian inverse problems. In its usual form, the method does not account for uncertainty caused by data noise and becomes less effective in high dimensionality. To overcome these issues, in this study, we include data uncertainties as additional model parameters, and train the network by maximizing the likelihood of the data used for training. We apply the method to two types of imaging problems: One‐dimensional surface wave dispersion inversion and two‐dimensional travel time tomography, and we compare the results to those obtained using Monte Carlo and MDNs. Results show that INNs provide comparable posterior pdfs to those obtained using Monte Carlo, including correlations between parameters, and provide more accurate marginal distributions than MDNs. After training, INNs estimate posterior pdfs in seconds on a typical desktop computer. Hence they can be used to provide efficient solutions for repeated inverse problems using different data sets. Also even accounting for training time, our results show that INNs can be more efficient than Monte Carlo methods for solving inverse problems only once.

Variational Bayesian inversion for the reaction coefficient in space-time nonlocal diffusion equations

In the paper, a variational Bayesian method is used to identify the reaction coefficient for space-time nonlocal diffusion equations using nonlocal averaged flux data. To show the posterior measure to be well-defined, we rigorously prove that the forward operator is continuous with respect to the unknown reaction field. Then, gradient-based prior information is proposed to explore oscillation features in the reaction coefficient. Moreover, the Bayesian inverse problem is shown to be well-posed in Hellinger distance. To accurately characterize the posterior density using uncorrelated samples, an efficient variational Bayesian method is used to estimate the reaction coefficient in the nonlocal models. A few numerical results are presented to illustrate the efficacy of the proposed approach and confirm some theoretic discoveries.

HIPPYlib

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitively expensive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYlib is that the solution of the inverse problem is computed at a cost, measured in linearized forward PDE solves, that is independent of the parameter dimension. The mean of the posterior is approximated by the MAP point, which is found by minimizing the negative log-posterior with an inexact matrix-free Newton-CG method. The posterior covariance is approximated by the inverse of the Hessian of the negative log posterior evaluated at the MAP point. The construction of the posterior covariance is made tractable by invoking a low-rank approximation of the Hessian of the log-likelihood. Scalable tools for sample generation are also discussed. hIPPYlib makes all of these advanced algorithms easily accessible to domain scientists and provides an environment that expedites the development of new algorithms.

Bayesian inversion for electromyography using low-rank tensor formats

The reconstruction of the structure of biological tissue using electromyographic (EMG) data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, EMG data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue. Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach. Solving the discretized Bayesian inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a Metropolis–Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes’ theorem proves the well-posedness of the Bayesian inverse problem. The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm. Numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters. The Metropolis–Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods.

Two-dimensional Bayesian inversion of magnetotelluric data using trans-dimensional Gaussian processes.

Bayesian inversion of electromagnetic data produces crucial uncertainty information on inferred subsurface resistivity. Due to their high computational cost, however, Bayesian inverse methods have largely been restricted to computationally expedient 1-D resistivity models. In this study, we successfully demonstrate, for the first time, a fully 2-D, trans-dimensional Bayesian inversion of magnetotelluric (MT) data. We render this problem tractable from a computational standpoint by using a stochastic interpolation algorithm known as a Gaussian process (GP) to achieve a parsimonious parametrization of the model vis-a-vis the dense parameter grids used in numerical forward modelling codes. The GP links a trans-dimensional, parallel tempered Markov chain Monte Carlo sampler, which explores the parsimonious model space, to MARE2DEM, an adaptive finite element forward solver. MARE2DEM computes the model response using a dense parameter mesh with resistivity assigned via the GP model. We demonstrate the new trans-dimensional GP sampler by inverting both synthetic and field MT data for 2-D models of electrical resistivity, with the field data example converging within 10d on 148 cores, a non-negligible but tractable computational cost. For a field data inversion, our algorithm achieves a parameter reduction of over 32× compared to the fixed parameter grid used for the MARE2DEM regularized inversion. Resistivity probability distributions computed from the ensemble of models produced by the inversion yield credible intervals and interquartile plots that quantitatively show the non-linear 2-D uncertainty in model structure. This uncertainty could then be propagated to other physical properties that impact resistivity including bulk composition, porosity and pore-fluid content.

Stein Variational Reduced Basis Bayesian Inversion

We propose and analyze a Stein variational reduced basis method (SVRB) to solve large-scale PDE-constrained Bayesian inverse problems. To address the computational challenge of drawing numerous samples requiring expensive PDE solves from the posterior distribution, we integrate an adaptive and goal-oriented model reduction technique with an optimization-based Stein variational gradient descent method. We present detailed analyses for the reduced basis approximation errors of the potential and its gradient, the induced errors of the posterior distribution measured by Kullback--Leibler divergence, as well as the induced errors of the Stein variational samples. To demonstrate the computational accuracy and efficiency of SVRB, we report results of numerical experiments on a Bayesian inverse problem governed by a diffusion PDE with random parameters with both uniform and Gaussian prior distributions. Over 100X speedups can be achieved while the accuracy of the approximation of the potential and its gradient is preserved.

Quantifying methane emissions from Queensland's coal seam gas producing Surat Basin using inventory data and a regional Bayesian inversion

Abstract. Methane (CH4) is a potent greenhouse gas and a key precursor of tropospheric ozone, itself a powerful greenhouse gas and air pollutant. Methane emissions across Queensland's Surat Basin, Australia, result from a mix of activities, including the production and processing of coal seam gas (CSG). We measured methane concentrations over 1.5 years from two monitoring stations established 80 km apart on either side of the main CSG belt located within a study area of 350 km × 350 km. Using an inverse modelling approach coupled with a bottom-up inventory, we quantify methane emissions from this area. The inventory suggests that the total emission is 173.2 × 106 kg CH4 yr−1, with grazing cattle contributing about half of that, cattle feedlots ∼ 25 %, and CSG processing ∼ 8 %. Using the inventory emissions in a forward regional transport model indicates that the above sources are significant contributors to methane at both monitors. However, the model underestimates approximately the highest 15 % of the observed methane concentrations, suggesting underestimated or missing emissions. An efficient regional Bayesian inverse model is developed, incorporating an hourly source–receptor relationship based on a backward-in-time configuration of the forward regional transport model, a posterior sampling scheme, and the hourly methane observations and a derived methane background. The inferred emissions obtained from one of the inverse model setups that uses a Gaussian prior whose averages are identical to the gridded bottom-up inventory emissions across the domain with an uncertainty of 3 % of the averages best describes the observed methane. Having only two stations is not adequate at sampling distant source areas of the study domain, and this necessitates a small prior uncertainty. This inverse setup yields a total emission of (165.8 ± 8.5) × 106 kg CH4 yr−1, slightly smaller than the inventory total. However, in a subdomain covering the CSG development areas, the inferred emissions are (63.6 ± 4.7) × 106 kg CH4 yr−1, 33 % larger than those from the inventory. We also infer seasonal variation of methane emissions and examine its correlation with climatological rainfall in the area.

Non-stationary multi-layered Gaussian priors for Bayesian inversion

In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. The aim of the multi-layered hierarchical prior is to provide enough complexity structure to allow for both smoothing and edge-preserving properties at the same time. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We then build the computational inference method using a finite-dimensional Galerkin method. We show that the proposed approximation has a convergence-in-probability property to the solution of the original multi-layered model. We then carry out Bayesian inference using the preconditioned Crank–Nicolson algorithm which is modified to work with multi-layered Gaussian fields. We show via numerical experiments in signal deconvolution and computerized x-ray tomography problems that the proposed method can offer both smoothing and edge preservation at the same time.