arXiv subjects

Xuebin Zhao

Publications and source records attributed to Xuebin Zhao.

11 recordsLinked to original sources

arXiv2026

Designing Solutions to Geophysical Inverse Problems by Changing Variables

Geoscientists often solve inverse problems to estimate values of parameters of interest given relevant data sets. Bayesian inference solves these problems by combining probability distributions that describe uncertainties in both observations and unknown parameters, and we require that the solution provides unbiased uncertainty estimates in order to inform risk-based decisions. It has been known for over a century that employing different, but equivalent parametrisations of the same information can yield conditional probabilities that are mathematically inconsistent, a property referred to as the BK-inconsistency. Recently this inconsistency was shown to invalidate the solutions to physical problems found using several well-established methods of Bayesian inference. In this study, we explore the extent to which this inconsistency affects solutions to common geophysical problems. We demonstrate that changes in parametrisations result in inconsistent conditional probability densities, even though they represent exactly the same information. We show that this can affect Bayesian posterior solutions dramatically across various geoscientific problems using real and synthetic data. Given that deterministic inversion is often equivalent to finding the maximum a posteriori solution to specific Bayesian problems (the mathematical equations to be solved are identical), the BK-inconsistency also results in inconsistent solutions to deterministic inverse problems. Indeed, we show that solutions can potentially be designed, simply by changing the parametrisation. This study highlights that a careful rethinking of Bayesian inference and deterministic inversion may be required in physical problems: the effects that we demonstrate are likely to affect past and present inverse problem solutions in a variety of different fields of application.

physics.geo-ph↗

arXiv2026

Linearised versus Nonlinear Estimates of Uncertainty in Full Waveform Inversion

Seismic full waveform inversion (FWI) is a powerful technique to generate high resolution images of the Earth's interior. However, significant uncertainty exists in all FWI solutions due to imperfect acquisition geometries, inherent noise in the data, and nonlinearity of the forward problem. Probabilistic Bayesian FWI addresses this non-uniqueness by estimating the entire family of possible model solutions described by the posterior probability density function (pdf). The posterior pdf can be estimated using nonlinear inversion methods to quantify full uncertainties. Alternatively, by linearising the physics relating parameters and observations around the maximum a posteriori solution, the posterior pdf is usually approximated by a Gaussian pdf. This is referred to as the linearised method. In this work, we apply both nonlinear and linearised methods to 2D acoustic Bayesian FWI problems. We use a variational inference algorithm for the nonlinear case, in which a transformed Gaussian is optimised to approximate the posterior pdf. The results can be compared with those from a linearised, locally-Gaussian based method. We also apply Stein variational gradient descent for comparison. The results show that while both the linearised and nonlinear methods recover the posterior mean models accurately, they exhibit significantly different posterior uncertainty structures, especially around layer interfaces, due to the linearisation of wave physics. Linearised uncertainty estimates are shown to be significantly less accurate: they provide far less accurate fits to observed waveform data, and yield biased estimates of inferred meta-properties such as volumes of geological bodies. This work therefore motivates the application of fully nonlinear inversion methods in Bayesian FWI if accurate uncertainty estimates over parameters, or inferred or interpreted meta-properties are important.

physics.geo-ph↗

arXiv2026

Variational and Monte Carlo Methods for Bayesian Inversion of Dynamic Subsurface Flow Simulations Using Seismic and Fluid Pressure Data

In order to predict future performance of subsurface fluid reservoirs under possible operating scenarios, a dynamic, porous-medium flow simulation model must be tuned to include representative properties of the reservoir. Estimating subsurface reservoir properties given remotely sensed or borehole-based observations typically involves finding the solution to a challenging inverse problem. We compare Monte Carlo random sampling to variational inference methods which use optimisation to constrain parametrised uncertainties in nonlinear Bayesian inversions. We use them to estimate the posterior probability distribution of reservoir permeability given fluid pressure and seismic measurements. The methods include automatic differentiation variational inference (ADVI), Stein variational gradient descent (SVGD), and a Monte Carlo method called stochastic SVGD (sSVGD), all of which we benchmark against results from Metropolis-Hastings McMC. We also test an ADVI variant called physically structured variational inference (PSVI): in our implementation this method estimates only spatially-local correlations between model parameters based on the intuition that such correlations are strong in remote sensing problems in which data only inform about spatial-averages of local dynamics. We apply the methods to two- and three-dimensional inverse problems of carbon dioxide storage, inspired by the Endurance field, located in the UK North Sea. Results show that PSVI achieves a good balance between mean-field ADVI and full-rank ADVI in terms of accuracy of the posterior approximation and computational efficiency. SVGD and sSVGD offer more accurate approximations of the target posterior distribution, but at far higher computational cost. Between them, sSVGD outperforms SVGD, exhibiting better computational efficiency and mitigating the problems of mode collapse and spurious correlations.

physics.geo-ph↗

arXiv2025

Detecting Model Misspecification in Bayesian Inverse Problems via Variational Gradient Descent

Bayesian inference is optimal when the statistical model is well-specified, while outside this setting Bayesian inference can catastrophically fail; accordingly a wealth of post-Bayesian methodologies have been proposed. Predictively oriented (PrO) approaches lift the statistical model $P_\theta$ to an (infinite) mixture model $\int P_\theta \; \mathrm{d}Q(\theta)$ and fit this predictive distribution via minimising an entropy-regularised objective functional. In the well-specified setting one expects the mixing distribution $Q$ to concentrate around the true data-generating parameter in the large data limit, while such singular concentration will typically not be observed if the model is misspecified. Our contribution is to demonstrate that one can empirically detect model misspecification by comparing the standard Bayesian posterior to the PrO `posterior' $Q$, providing a novel and widely-applicable diagnostic tool for the standard Bayesian workflow. To operationalise this, we present an efficient numerical algorithm based on variational gradient descent. A simulation study, and a more detailed case study involving a Bayesian inverse problem in seismology, confirm that model misspecification can be automatically detected using this framework.

stat.ME↗

arXiv2025

A Semi-analytic but Biased Uncertainty Assessment Method using Sample Extensions, Analysed for Nonlinear Travel Time Tomography

Many geophysical problems can be cast as inverse problems that estimate a set of parameter values from observed data. Within a Bayesian framework, solutions to such problems are described probabilistically by the so-called posterior probability distribution functions (pdf's). To obtain robust inference results often requires millions of model parameter value samples to be drawn, and their simulation to be performed; this is a computationally expensive procedure. We investigate the concept of sample extensions as a means to improve efficiency when solving fully nonlinear inverse problems. A sample's extension is defined as the set of models or parameter values whose forward function values are directly accessible from a sample for which the forward function has already been evaluated, obviating the need for additional forward function evaluations. In a specific case of first-arrival travel time calculations used in seismic travel time tomography, we apply sample extensions to obtain a continuous region with non-zero hypervolume within parameter space, across all of which the forward function values are known given only a single forward simulation. We devise a deterministic sampling technique that identifies the most informative extensions by solving an optimisation problem. In an illustrative tomographic example that involves a single travel time datum, we find 51 optimal samples, and use them to construct an analytic approximation to the Bayesian posterior pdf. Additionally, we propose an extensions-based algorithm for real-world tomography scenarios and apply it to a synthetic 2D example. This study highlights two fundamental problems that make the method inefficient: (1) limited hypervolumes of extensions and (2) neglecting parameter correlations to simplify analytic calculations. Finding solutions to these problems defines possible directions for future research.

physics.geo-ph↗

arXiv2025

Direct-3D Variational Bayesian Surface Wave Inversion and Its Application to Ambient Noise Tomography beneath Great Britain

We present a new, variational, fully nonlinear, probabilistic ambient noise tomography method, which estimates subsurface structure and quantifies the corresponding uncertainties directly in three dimensions (3D) from inter-receiver seismic surface wave dispersion data. We use the method to invert for high resolution 3D seismic velocity models of the upper crust beneath Great Britain using seismic ambient noise data recorded around the region - a task that proved too high-dimensional and hence computationally demanding for Monte Carlo sampling to converge to a stable solution. We compare the inversion results from the new method to those obtained from two standard, indirect inversion methods, in which 2D (geographical) surface wave velocity maps and 1D (depth) shear velocity profiles are estimated in two separate, consecutive steps. The results show that the direct-3D scheme preserves better lateral continuity and produces better data fit than the two-step methods, and provides information about lateral correlations that is absent from the two-step solutions. The inversion results are consistent with large-scale geology of Great Britain, and for the first time provide seismologically-imaged evidence of the Great Glen Fault and other major tectonic faults. We therefore propose that direct-3D inversion schemes should be used where possible for surface wave inversion as they provide improved results at little additional computational cost.

physics.geo-ph↗

arXiv2024

Efficient Bayesian Full Waveform Inversion and Analysis of Prior Hypotheses in 3D

Spatially 3-dimensional seismic full waveform inversion (3D FWI) is a highly nonlinear and computationally demanding inverse problem that constructs 3D subsurface seismic velocity structures using seismic waveform data. To characterise non-uniqueness in the solutions, we demonstrate Bayesian 3D FWI using an efficient method called physically structured variational inference, and apply it to 3D acoustic Bayesian FWI. The results provide reasonable posterior uncertainty estimates, at a computational cost that is only an order of magnitude greater than that of standard, deterministic FWI. Furthermore, we deploy variational prior replacement to calculate Bayesian solutions corresponding to different classes of prior information at low additional cost. The results obtained using prior information that models should be smooth show loop-like high uncertainty structures that are consistent with fully nonlinear inversion results presented previously. These structures disappear when smoothing is not imposed, so we conclude that they may be caused by smoothness constraints in tomographic problems. We further analyse a variety of prior hypotheses by constructing Bayesian L-curves, which reveal the sensitivity of the inversion process to different prior assumptions. To our knowledge, this is the first study that allows such prior hypotheses to be compared in probabilistic 3D FWI at feasible computational cost. This work shows that fully probabilistic 3D FWI can be performed and can be used to test different prior hypotheses, at a cost that may be practical, at least in small problems.

physics.geo-ph↗

arXiv2024

Variational Prior Replacement in Bayesian Inference and Inversion

Many scientific investigations require that the values of a set of model parameters are estimated using recorded data. In Bayesian inference, information from both observed data and prior knowledge is combined to update model parameters probabilistically by calculating the posterior probability distribution function. Prior information is often described by a prior probability distribution. Situations arise in which we wish to change prior information during the course of a scientific project. However, estimating the solution to any single Bayesian inference problem is often computationally costly, as it typically requires many model samples to be drawn, and the data set that would have been recorded if each sample was true must be simulated. Recalculating the Bayesian inference solution every time prior information changes can therefore be extremely expensive. We develop a mathematical formulation that allows the prior information that is embedded within a solution, to be changed using variational methods, without recalculating the original Bayesian inference. In this method, existing prior information is removed from a previously obtained posterior distribution and is replaced by new prior information. We therefore call the methodology variational prior replacement (VPR). We demonstrate VPR using a 2D seismic full waveform inversion example, in which VPR provides similar posterior solutions to those obtained by solving independent inference problems using different prior distributions. The former can be completed within minutes on a laptop computer, whereas the latter requires days of computations using high-performance computing resources. We demonstrate the value of the method by comparing the posterior solutions obtained using three different types of prior information: uniform, smoothing and geological prior distributions.

stat.ME↗

arXiv2023

Bayesian Inversion, Uncertainty Analysis and Interrogation using Boosting Variational Inference

Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non-linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so-called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease with which they can be optimised. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods of solution. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate probabilistic solutions to fully non-linear, high-dimensional Bayesian full waveform inversion problems.

physics.geo-ph↗

arXiv2022

An introduction to variational inference in Geophysical inverse problems

In a variety of scientific applications we wish to characterize a physical system using measurements or observations. This often requires us to solve an inverse problem, which usually has non-unique solutions so uncertainty must be quantified in order to define the family of all possible solutions. Bayesian inference provides a powerful theoretical framework which defines the set of solutions to inverse problems, and variational inference is a method to solve Bayesian inference problems using optimization while still producing fully probabilistic solutions. This chapter provides an introduction to variational inference, and reviews its applications to a range of geophysical problems, including petrophysical inversion, travel time tomography and full-waveform inversion. We demonstrate that variational inference is an efficient and scalable method which can be deployed in many practical scenarios.

physics.geo-ph↗

arXiv2022

Error-based Knockoffs Inference for Controlled Feature Selection

Recently, the scheme of model-X knockoffs was proposed as a promising solution to address controlled feature selection under high-dimensional finite-sample settings. However, the procedure of model-X knockoffs depends heavily on the coefficient-based feature importance and only concerns the control of false discovery rate (FDR). To further improve its adaptivity and flexibility, in this paper, we propose an error-based knockoff inference method by integrating the knockoff features, the error-based feature importance statistics, and the stepdown procedure together. The proposed inference procedure does not require specifying a regression model and can handle feature selection with theoretical guarantees on controlling false discovery proportion (FDP), FDR, or k-familywise error rate (k-FWER). Empirical evaluations demonstrate the competitive performance of our approach on both simulated and real data.

stat.ME↗