Sparse Bayesian mass-mapping using trans-dimensional MCMC

Upscaling of velocity logs is a common problem in correlating surface seismic and log data and in the generation of coarse scale models from log data to use in modeling or data processing. Backus averaging allows computation of the long wavelength effective elastic moduli of a medium comprised of a stack of thin, isotropic and homogeneous layers. However, it is not obvious how to accurately choose the depth interval over which the average should be applied and where to assign interface depths between upscaled layers. Though running Backus average methods have been proposed, they produce smoothed models, even though it is clear that many settings such as unconformities are best modeled with velocity discontinuities. Here we apply a Markov Chain Monte Carlo (MCMC) method to optimize the depths of interfaces in upscaled log data and to quantify the uncertainty in the resulting models. Application to log data from the North Sea shows that the method provides useful, objective guidelines for assigning model properties. Histograms showing the estimated distribution functions of interface depths indicate both where interfaces should be located and how large is the uncertainty in each location. These results also suggest a simple means for choosing the minimum number of interfaces that should be incorporated into the final model to generate the simplest solution.

Parameter estimation and uncertainty analysis for a watershed model

An increasing number of field development projects include rigorous uncertainty quantification workflows based on parameterized subsurface uncertainties. Reservoir model calibration workflows for reservoir simulation models including historical production data, also called history matching, deliver non-unique solutions and remain technically challenging. In addition, the validation process of the reservoir simulation model often introduces a break of the conceptual connection to the geological model. This raises questions on how to quantify the deviation between the calibrated simulation model and the original geological model. Workflow designs for history matching require scalable and efficient optimization techniques to address project needs. Derivative-free techniques like Markov Chain Monte Carlo (MCMC) are used for optimization and uncertainty quantification. Adjoint techniques derive analytical sensitivities directly from the flow equations. For history matching those sensitivities are efficiently used for property updates on grid block level. Both techniques have different characteristics and support alternative history matching strategies like global vs. local, stochastic vs. deterministic. In this work both techniques will be applied in an integrated workflow design to the Norne field. The Norne field is a North Sea oil-and-gas reservoir with approximately 30 wells, with one third being used for WAG injection for pressure support. Field data was previously released by Statoil and made available for a public benchmark study (NTNU Norway) testing history matching techniques including production and time-lapsed seismic data. We focus on well production data for history matching. MCMC is used for global parameter updates and uncertainty quantification in a Bayesian context. An implementation of an adjoint technique is applied for analytical sensitivity calculations and local parameter adjustments of rock properties. History matching results are presented for field wide and well-by-well production data. Consistency checks between updated and original geological model are presented for rock property distribution maps. Geostatistical measures including spatial correlations are used to quantify deviations between updated and original geological model. In conclusion scalability and performance efficiency of the practical workflow implementation is discussed with a perspective of a consistent feedback loop from history matching to geological modeling.

Parameterization of a process-based tree-growth model: Comparison of optimization, MCMC and Particle Filtering algorithms

The detection of gravitational waves from extreme-mass-ratio inspirals (EMRIs) in space-based antennas like Taiji and Laser Interferometer Space Antenna promises deep insights into strong-field gravity and black hole physics. However, the complex, highly degenerate, and nonconvex likelihood landscapes characteristic of EMRI parameter spaces pose severe challenges for conventional Markov chain Monte Carlo (MCMC) methods. Under realistic instrumental noise and broad priors, these methods demand impractical computational costs but are prone to becoming trapped in local maxima, leading to biased and unreliable parameter estimates. To address these challenges, we introduce flow-matching MCMC (FM-MCMC), a novel Bayesian framework that integrates continuous normalizing flows (CNFs) with parallel tempering MCMC (PTMCMC). By generating high-likelihood regions via CNFs and refining them through PTMCMC, FM-MCMC enables robust exploration of the nontrivial parameter spaces, achieves orders-of-magnitude improvement in computational efficiency, and, more importantly, ensures statistically unbiased inference. By enabling real-time, unbiased parameter inference, FM-MCMC could unlock the full scientific potential of EMRI observations, and would serve as a scalable pipeline for precision gravitational-wave astronomy.

Knowledge of the Earth's interior relies on indirect information collected at or near the surface. Typically, data do not uniquely constrain the subsurface properties and are contaminated by noise, and therefore the solution of ill-posed inverse problems is required. Geophysicists have traditionally addressed such problems through deterministic approaches, seeking a single 'best-fitting' model defined by optimality criteria that reflect our understanding of the problem at hand. In recent decades, Bayesian approaches have become increasingly common, as they characterise the posterior probability distribution of the model conditioned on the observations, thereby quantifying uncertainty.In this context, Markov chain Monte Carlo (MCMC) methods have emerged as a crucial tool as they allow sampling from posterior distributions of arbitrary complexity. At the core of many MCMC algorithms lies the Metropolis-Hastings scheme. This combines a proposal distribution with a probabilistic acceptance criterion to construct a Markov chain that has the desired target distribution as its stationary distribution. The algorithm is versatile as it rests on mild technical conditions on the proposal, and is thus widely adopted across a broad range of geoscientific inference problems. Yet when the parameter space is large or the forward models are computationally expensive---both common scenarios in geophysical applications---it can become inefficient, resulting in poor chain mixing and slow convergence to the target (posterior) distribution. These challenges underscore the importance of effective proposal mechanisms.In this presentation, we introduce a novel approach to designing Metropolis-Hastings proposals based on adaptive transport maps. The framework is inspired by recent developments from the field of Applied Mathematics linking Bayesian inference with Optimal Transport theory. The idea is to find a monotone, nonlinear transformation to recast a (complex) target probability distribution into a (simpler) reference distribution that is more amenable to standard MCMC steps. Our key contribution is to parameterise these transformations using invertible neural networks, ensuring monotonicity while gaining the flexibility and expressiveness that neural architectures afford.The proposed method progresses iteratively. We begin with a standard sampling strategy (e.g., a random-walk Metropolis) to obtain initial draws from the target distribution. These samples inform the training of an invertible neural network that learns to map from the target to a simpler reference distribution, specifically a standard (multivariate) Gaussian. Subsequent proposals are then generated in the reference space, either as global independence moves or local perturbations, and are accepted or rejected following a suitably modified Metropolis-Hastings criterion. As more samples accumulate, the network's parameters are updated, improving overall sampling efficiency. Ultimately, the approach yields not only an ensemble of samples representative of the desired target distribution---just as in standard MCMC---but also a compact, learned representation of it in the network's weights.We illustrate the proposed paradigm both theoretically and through examples. Preliminary results indicate that transport-map-enhanced MCMC has the potential to significantly accelerate Bayesian sampling across a range of applications.

Markov chain Monte Carlo (MCMC) methods are widely used in the solution of parameter estimation problems arising in acoustics and other applications. The use of MCMC to estimate the parameters of a single model is well established. However, in many applications, there is not a single model for the data but rather a number of competing models. A common method of dealing with multiple models is to use MCMC to compute the posterior probability and estimate the parameter values of each model in turn. However, for problems with many models, it is more efficient to combine the parameter spaces of all models into a single space and use MCMC to perform across-model sampling of the joint space. Although the development of an MCMC algorithm of this sort is sufficiently difficult so as to be unprofitable for the non-specialist, the acoustician wishing to solve their multi model parameter estimation problem using MCMC can still do so using an existing algorithm. This presentation gives an overview and brief tutorial of MCMC for parameter estimation and then discusses and gives an example of using the open source computer program BayeSys [Skilling, 2004] to determine the model order of a simple atomic model.

PurposeThe application of statistical inversion theory provides a powerful approach for solving estimation problems including the ability for uncertainty quantification (UQ) by means of Markov chain Monte Carlo (MCMC) methods and Monte Carlo integration. This paper aims to analyze the application of a state reduction technique within different MCMC techniques to improve the computational efficiency and the tuning process of these algorithms.Design/methodology/approachA reduced state representation is constructed from a general prior distribution. For sampling the Metropolis Hastings (MH) Algorithm and the Gibbs sampler are used. Efficient proposal generation techniques and techniques for conditional sampling are proposed and evaluated for an exemplary inverse problem.FindingsFor the MH-algorithm, high acceptance rates can be obtained with a simple proposal kernel. For the Gibbs sampler, an efficient technique for conditional sampling was found. The state reduction scheme stabilizes the ill-posed inverse problem, allowing a solution without a dedicated prior distribution. The state reduction is suitable to represent general material distributions.Practical implicationsThe state reduction scheme and the MCMC techniques can be applied in different imaging problems. The stabilizing nature of the state reduction improves the solution of ill-posed problems. The tuning of the MCMC methods is simplified.Originality/valueThe paper presents a method to improve the solution process of inverse problems within the Bayesian framework. The stabilization of the inverse problem due to the state reduction improves the solution. The approach simplifies the tuning of MCMC methods.

Recent progress using geometry in the design of efficient Markov chain Monte Carlo (MCMC) algorithms have shown the effectiveness of the Fisher Riemannian structure. Furthermore, the theory of the underlying geometry of spaces of statistical models has made an important breakthrough by extending the classical theory on exponential families to their closures, the so-called extended exponential families. This paper looks at the underlying geometry of the Fisher information, in particular its limiting behaviour near boundaries, which illuminates the excellent behaviour of the corresponding geometric MCMC algorithms. Further, the paper shows how Fisher geodesics in extended exponential families smoothly attach the boundaries of extended exponential families to their relative interior. We conjecture that this behaviour could be exploited for trans-dimensional MCMC algorithms.

The Bayesian approach to inverse problems, in which the posterior probability distribution on an unknown field is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computationally feasible for partial differential equation (PDE) inverse problems. Balancing the sources of error arising from finite-dimensional approximation of the unknown field, the PDE forward solution map and the sampling of the probability space under the posterior distribution are essential for the design of efficient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diffusion coefficient. We provide complexity analyses of several Markov chain Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data δ. Particular attention is given to bounds on the overall work required to achieve a prescribed error level ε. Specifically, we first bound the computational complexity of ‘plain’ MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) ‘surrogate’ representation of the forward response map of the PDE over the entire parameter space, and, second, a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the forward PDE. For both of these strategies, we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular, we provide sufficient conditions on the regularity of the unknown coefficients of the PDE and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over ‘plain’ MCMC algorithms for the Bayesian inversion of PDEs.

Evaluating uncertainties in electrochemical impedance spectra of solid oxide fuel cells

Seismic full waveform inversion (FWI) is a powerful tool for monitoring subsurface changes during carbon capture and storage (CCS) operations, but its ill-posed nature makes uncertainty quantification (UQ) essential for reliable interpretation. Sampling-based Bayesian methods such as Markov chain Monte Carlo (McMC) provide rigorous UQ but are computationally demanding. In conventional FWI, elastic properties are assigned to densely discretized space-filling cells, resulting in a high-dimensional parameterization that makes large-scale elastic FWI computationally infeasible. To address this challenge, we propose a rock-physics-guided parameter-reduction strategy that compactly represents the CO2 plume geometry using cubic splines controlled by a limited number of nodes. This parsimonious parameterization not only significantly reduces the number of model parameters and the forward simulations required for an effective UQ using sampling methods but also has the potential to improve the practicality and efficiency of other types of UQ methods. Numerical experiments on a cross-well synthetic scenario and a field-scale case based on the Aquistore storage site in Saskatchewan, Canada, demonstrate that the method efficiently reconstructs the plume shape and its extent and that it converges to consistent posterior distributions across multiple Markov chains.

Sensitivity analysis of uncertain nonlinear systems plays an important role in analyzing chemical engineering process models. Even though a variety of different techniques exists for analyzing nonlinear systems, these methods often do not update the parameter distribution using available measurement data. This forms the motivation behind this work as a sensitivity analysis procedure is introduced that deals with nonlinear systems and uncertainty in model parameters, and also incorporates the use of available data. The technique computes two covariance matrices: a covariance matrix that describes the effect that changes of the parameters values, sampled from a prior distribution, have on the outputs, and a covariance matrix that determines the effect of available data on the posterior distribution of the parameters using a Markov chain Monte Carlo method. The information contained in these two covariance matrices is then simultaneously evaluated by balancing the two matrices. The results returned by this technique are the directions in parameter space that have the largest impact on the system behavior according to the given parameter uncertainty as well as the contribution of individual parameters to these important directions in parameter space. The presented technique is applied to three examples.

This thesis is concerned with the calibration of disease models in order to inform decisions about public health problems. These models are used to make predictions about the likely outcomes of disease spread (e.g. morbidity and mortality) in a population, given different interventions (e.g. vaccination, screening and treatment). These models necessitate simpli cations of the underlying mechanisms of disease progression, but have become essential tools in areas such as cost-effectiveness research. Although there are many types of models that can be used, all of them involve parameters, which affect the model's outputs, for example, the age of onset of illness, duration of particular health states or disease specifc mortality. Identifying areas of the model's parameter space that are consistent with data is referred to as model calibration. One of the areas of concern when searching a model's parameter space to get a prediction from the model (that best agrees with observed data) is the propagation of uncertainty. If one has a prediction about an outcome, the question of how uncertain we are about this prediction should be raised. In model calibration, the uncertainty we are interested in is uncertainty in parameters[5]. There are many approaches to model calibration with little consensus on the best practice. Additionally, within the current approaches, some signifi cant problems, that one could face when attempting calibration, merit further examination. We highlight this with a literature search. We believe, that many popular approaches have not yet addressed fundamental issues and more research may be needed to tackle these issues. The model calibrations in this thesis are carried out in a Bayesian framework, using Markov chain Monte Carlo (MCMC) techniques. Although the ultimate goal of calibration should be predictions about some quantities of interest, the emphasis of this work is on the techniques and methodology of model calibration in general. Many possible difficulties of using MCMC and some other sampling approaches are presented and discussed. Avenues for additional research are identified.

An Algebraic Structure of Orthogonal Wavelet Space