Bayesian Tomography Research Articles

Bayesian geochemical correlation and tomography

To accurately reconstruct palaeoenvironmental change through time it is important to determine which rock samples were deposited contemporaneously at different sites or transects, as erroneous correlation may lead to incorrectly inferred processes and rates. To correlate samples, current practice interpolates geological age between datable units along each transect, then temporal signatures observed in geochemical logs are matched between transects. Unfortunately spatiotemporally variable and unknown rates of sedimentary deposition create highly nonlinear space-time transforms, significantly altering apparent geochemical signatures. The resulting correlational hypotheses are also untestable against independent transects, because correlations have no spatially-predictive power. Here we use geological process information stored within neural networks to correlate spatially offset logs nonlinearly and geologically. The same method creates tomographic images of geological age and geochemical signature across intervening rock volumes. Posterior tomographic images closely resemble the true depositional age throughout the inter-transect volume, even for scenarios with long hiatuses in preserved geochemical signals. Bayesian probability distributions describe data-consistent variations in the results, showing that centred summary statistics such as mean and variance do not adequately describe correlational uncertainties. Tomographic images demonstrate spatially predictive power away from geochemical transects, creating novel hypotheses attributable to each geochemical correlation which are testable against independent data.

Bayesian tomography using polynomial chaos expansion and deep generative networks

SUMMARY Implementations of Markov chain Monte Carlo (MCMC) methods need to confront two fundamental challenges: accurate representation of prior information and efficient evaluation of likelihood functions. The definition and sampling of the prior distribution can often be facilitated by standard dimensionality-reduction techniques such as Principal Component Analysis (PCA). Additionally, PCA-based decompositions can enable the implementation of accurate surrogate models, for instance, based on polynomial chaos expansion (PCE). However, intricate geological priors with sharp contrasts may demand advanced dimensionality-reduction techniques, such as deep generative models (DGMs). Although suitable for prior sampling, these DGMs pose challenges for surrogate modelling. In this contribution, we present a MCMC strategy that combines the high reconstruction performance of a DGM in the form of a variational autoencoder with the accuracy of PCA–PCE surrogate modelling. Additionally, we introduce a physics-informed PCA decomposition to improve accuracy and reduce the computational burden associated with surrogate modelling. Our methodology is exemplified in the context of Bayesian ground-penetrating radar traveltime tomography using channelized subsurface structures, providing accurate reconstructions and significant speed-ups, particularly when the computation of the full-physics forward model is costly.

Bayesian eikonal tomography using Gaussian processes


 Eikonal tomography has become a popular methodology for deriving phase velocity maps from surface wave phase delay measurements. Its high efficiency makes it popular for handling datasets deriving from large-N arrays, in particular in the ambient-noise tomography setting. However, the results of eikonal tomography are crucially dependent on the way in which phase delay measurements are predicted from data, a point which has not been thoroughly investigated. In this work, I provide a rigorous formulation for eikonal tomography using Gaussian processes (GPs) to smooth observed phase delay measurements, including uncertainties. GPs allow the posterior phase delay gradient to be analytically derived. From the phase delay gradient, an excellent approximate solution for phase velocities can be obtained using the saddlepoint method. The result is a fully Bayesian result for phase velocities of surface waves, incorporating the nonlinear wavefront bending inherent in eikonal tomography, with no sampling required. The results of this analysis imply that the uncertainties reported for eikonal tomography are often underestimated.

Evaluating imaging uncertainty associated with the near surface and added value of vertical arrays using Bayesian seismic refraction tomography

AbstractTraditional tomographic methods do not consider the uncertainties associated with near-surface velocities and static corrections and provide a deterministic solution to the estimation problem. However, these uncertainties significantly affect structural mapping and interpretation of seismic imaging results. On the other hand, Bayesian first-arrival tomography provides multiple near-surface models that fit observed traveltimes equally well and enable the study of potential solution distributions. We demonstrate this approach on a complex synthetic near-surface model, representative of arid environments, to quantify associated velocity and statics uncertainties. We evaluate two different parameterizations for subsurface velocities in the context of near-surface Bayesian tomography: Voronoi tessellation with natural neighbor interpolation and the more conventional Delaunay triangulation with linear interpolation. Our analysis shows that the Voronoi cell parameterization with natural neighbor interpolation is more appropriate for this problem. Finally, the new approach is applied to compare two alternative acquisition geometries comprising conventional surface receivers and surface receivers augmented with vertical receiver arrays. The results demonstrate that adding vertical receiver arrays to conventional surface receivers can significantly reduce the near-surface velocity uncertainty and thus increases the accuracy of the seismic imaging results. Furthermore, the study shows that Bayesian tomography can be used as a tool for evaluating different source and receiver geometries during the acquisition design stage.

Bayesian Seismic Tomography Based on Velocity-Space Stein Variational Gradient Descent for Physics-Informed Neural Network

Bayesian Seismic Tomography Based on Velocity-Space Stein Variational Gradient Descent for Physics-Informed Neural Network

High-fidelity inference of local impurity profiles in C-2W using Bayesian tomography.

In C-2W (also called "Norman") [1], beam-driven field reversed configuration plasmas embedded in a magnetic mirror are produced and sustained in a steady state. A multi-chord passive Doppler spectroscopy diagnostic provides line-integrated impurity emission measurements near the center plane of the confinement vessel with fast time resolution. The high degree of plasma non-uniformity across optical sightlines can preclude direct fitting of the measured line-integrated spectra. To overcome this challenge, local impurity profiles are inferred using Bayesian tomography, a superior analysis technique based on a complete forward model of the diagnostic. The measured emission of O4+ triplet lines near 278.4nm is modeled assuming two independent populations: thermal and beam ions. Gaussian processes are used to generate and infer local profiles. The inference incorporates details of the geometrical arrangement of the diagnostic, instrument function, intensity calibration, and a noise model. Markov chain Monte Carlo (MCMC) sampling of the posterior distribution of solutions provides high-fidelity uncertainty estimates. The reconstructed O4+ impurity profiles are consistent with data from other diagnostics and show good agreement with expected physics based on previously developed models of biasing circuit and impurity transport.

Tropospheric wet tomography and PPP: Joint estimation from GNSS crowdsourcing data

Water vapor is the dominant greenhouse gas in the atmosphere accounting for 50 to 70% of the total effect, it plays a large role in determining the weather forecasts, and it accounts for up to one fifth of the atmospheric error for GNSS measurements. Hence, measuring the amount of water vapor in the atmosphere is important. Previously, data from GNSS base stations has been used to compute tomographic estimates for the water vapor distribution. Here, we present a technique that enables doing these tomographic estimates with receivers residing at roving positions instead of ones with a pre-determined position. Specifically, we show that combining probabilistic precise point positioning (PPP) with Bayesian tomography with 1 Hz posteriori update rate leads to the convergence of both the receiver position estimates and the wet residual estimates. Moreover, our findings include the fact that even low quality receiver data, such as the one crowdsourced from consumer-level equipment such as smartphones, may be used to improve the water vapor estimate, if the uncertainty models are not too optimistic nor too pessimistic. Simulated GNSS observations with realistic errors for up to one thousand GNSS receivers in Southern France, presented in the antecedent work (De Oliveira Marques et al., 2020), are employed for elaborate testing. We discuss why the proposed method is likely straightforwardly applicable on data coming from moving modern multi-sensor systems, although our testing data is limited to receivers that are not moving. Applications for the proposed technique include benefits in regions that are not covered by high-quality weather measurement instruments, e.g. islands, seas, and other remote areas such as the Arctic.

Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements

Owing in large part to the advent of integrated biphoton frequency combs, recent years have witnessed increased attention to quantum information processing in the frequency domain for its inherent high dimensionality and entanglement compatible with fiber-optic networks. Quantum state tomography of such states, however, has required complex and precise engineering of active frequency mixing operations, which are difficult to scale. To address these limitations, we propose a solution that employs a pulse shaper and electro-optic phase modulator to perform random operations instead of mixing in a prescribed manner. We successfully verify the entanglement and reconstruct the full density matrix of biphoton frequency combs generated from an on-chip Si3N4 microring resonator in up to an 8 × 8-dimensional two-qudit Hilbert space, the highest dimension to date for frequency bins. More generally, our employed Bayesian statistical model can be tailored to a variety of quantum systems with restricted measurement capabilities, forming an opportunistic tomographic framework that utilizes all available data in an optimal way.

Bayesian tomography with prior-knowledge-based parametrization and surrogate modelling

SUMMARYWe present a Bayesian tomography framework operating with prior-knowledge-based parametrization that is accelerated by surrogate models. Standard high-fidelity forward solvers (e.g. finite-difference time-domain schemes) solve wave equations with natural spatial parametrizations based on fine discretization. Similar parametrizations, typically involving tens of thousand of variables, are usually employed to parametrize the subsurface in tomography applications. When the data do not allow to resolve details at such finely parametrized scales, it is often beneficial to instead rely on a prior-knowledge-based parametrization defined on a lower dimension domain (or manifold). Due to the increased identifiability in the reduced domain, the concomitant inversion is better constrained and generally faster. We illustrate the potential of a prior-knowledge-based approach by considering ground penetrating radar (GPR) traveltime tomography in a crosshole configuration with synthetic data. An effective parametrization of the input (i.e. the permittivity distributions determining the slowness field) and compression in the output (i.e. the traveltime gathers) spaces are achieved via data-driven principal component decomposition based on random realizations of the prior Gaussian-process model with a truncation determined by the performances of the standard solver on the full and reduced model domains. To accelerate the inversion process, we employ a high-fidelity polynomial chaos expansion (PCE) surrogate model. We investigate the impact of the size of the training set on the performance of the PCE and show that a few hundreds design data sets is sufficient to provide reliable Markov chain Monte Carlo inversion at a fraction of the cost associated with a standard approach involving a fine discretization and physics-based forward solvers. Appropriate uncertainty quantification is achieved by reintroducing the truncated higher order principle components in the original model space after inversion on the manifold and by adapting a likelihood function that accounts for the fact that the truncated higher order components are not completely located in the null space.

Fast Bayesian Tomography of a Two-Qubit Gate Set in Silicon

Benchmarking and characterising quantum states and logic gates is essential in the development of devices for quantum computing. We introduce a Bayesian approach to self-consistent process tomography, called fast Bayesian tomography (FBT), and experimentally demonstrate its performance in characterising a two-qubit gate set on a silicon-based spin qubit device. FBT is built on an adaptive self-consistent linearisation that is robust to model approximation errors. Our method offers several advantages over other self-consistent tomographic methods. Most notably, FBT can leverage prior information from randomised benchmarking (or other characterisation measurements), and can be performed in real time, providing continuously updated estimates of full process matrices while data is acquired.

Polynomial surrogates for Bayesian traveltime tomography

This paper tackles the issue of the computational load encountered in seismic imaging by Bayesian traveltime inversion. In Bayesian inference, the exploration of the posterior distribution of the velocity model parameters requires extensive sampling. The computational cost of this sampling step can be prohibitive when the first arrival traveltime prediction involves the resolution of an expensive number of forward models based on the eikonal equation. We propose to rely on polynomial chaos surrogates of the traveltimes between sources and receivers to alleviate the computational burden of solving the eikonal equation during the sampling stage. In an offline stage, the approach builds a functional approximation of the traveltimes from a set of solutions of the eikonal equation corresponding to a few values of the velocity model parameters selected in their prior range. These surrogates then substitute the eikonal-based predictions in the posterior evaluation, enabling very efficient extensive sampling of Bayesian posterior, for instance, by a Markov Chain Monte Carlo algorithm. We demonstrate the potential of the approach using numerical experiments on the inference of two-dimensional domains with layered velocity models and different acquisition geometries (microseismic and seismic refraction contexts). The results show that, in our experiments, the number of eikonal model evaluations required to construct accurate surrogates of the traveltimes is low. Further, an accurate and complete characterization of the posterior distribution of the velocity model is possible, thanks to the generation of large sample sets at a low cost. Finally, we discuss the extension of the current approach to more realistic velocity models and operational situations.

Lowermost Mantle Shear‐Velocity Structure From Hierarchical Trans‐Dimensional Bayesian Tomography

AbstractThe core‐mantle boundary (CMB) is the most extreme boundary within the Earth where the liquid, iron‐rich outer core interacts with the rocky, silicate mantle. The nature of the lowermost mantle atop the CMB, and its role in mantle dynamics, is not completely understood. Various regional studies have documented significant heterogeneities at different spatial scales. While there is a consensus on the long scale length structure of the inferred S‐wave speed tomograms, there are also notable differences stemming from different imaging methods and datasets. Here we aim to overcome over‐smoothing and avoid over‐fitting data for the case where the spatial coverage is sparse and the inverse problem ill‐posed. We present an S‐wave tomography model at a global scale for the Lowermost Mantle (LM) using the Hierarchical Trans‐Dimensional Bayesian Inversion (HTDBI) framework, LM‐HTDBI. Our LM‐HTDBI analysis of ScS‐S travel times includes uncertainty, and the complexity of the model is deduced from the data itself through an implicit parameterization of the model space. Our comprehensive resolution estimates indicate that short‐scale anomalies are significant and resolvable features of the lowermost mantle regardless of the chosen mantle‐model reference to correct the travel times above the D’’ layer. The recovered morphology of the Large‐Low‐Shear‐wave Velocity Provinces (LLSVPs) is complex, featuring small high‐velocity patches among low‐velocity domains. Instead of two large, unified, and smooth LLSVPs, the newly obtained images suggest that their margins are not uniformly flat.

Surface‐Wave Tomography of the Northern Canadian Cordillera Using Earthquake Rayleigh Wave Group Velocities

AbstractWe measure fundamental‐mode Rayleigh wave group velocity dispersion from regional earthquakes and employ transdimensional Bayesian tomography to invert for azimuthally anisotropic group velocity maps at 10–60 s periods. Seismic azimuthal anisotropy within the crust and uppermost mantle shows fast‐axis alignment with principal northern Canadian Cordillera (NCC) tectonic structures suggesting large‐scale and coherent deformation across the crust and upper mantle. We extract group velocity dispersion curves at each grid point of the isotropic group velocity maps and invert them independently for VS as a function of depth in a probabilistic framework. At mid‐to‐lower crustal depths, our pseudo 3‐D VS model reveals an anomalously low VS structure (<3.8 km/s) across the NCC, which likely reflects elevated crustal temperatures that buoyantly support regional high elevations. Two high‐velocity regions (≥4.5 km/s) at uppermost mantle depths underpin the edges of the arcuate Mackenzie fold and thrust belt. We observe variations in the dip direction of the high‐velocity craton edge in the upper mantle, from subvertical in the northern NCC to westward in the southern NCC. We interpret a low‐velocity region (≤4.3 km/s) between the Tintina and Denali faults at uppermost mantle depths to reflect upwelling asthenosphere, which likely postdates 430 km of previously estimated Eocene lithosphere‐scale horizontal displacement along the Tintina fault. Our surface‐wave model resolves a high‐velocity region (≥4.4 km/s) dipping north‐northeast below the Wrangell volcanic field, which we interpret as a down‐going segment of the Pacific slab and may contribute to the observed volcanism in the area.

Bayesian seismic tomography using normalizing flows

SUMMARYWe test a fully non-linear method to solve Bayesian seismic tomographic problems using data consisting of observed traveltimes of first-arriving waves. Rather than using Monte Carlo methods to sample the posterior probability distribution that embodies the solution of the tomographic inverse problem, we use variational inference. Variational methods solve the Bayesian inference problem under an optimization framework by seeking the best approximation to the posterior distribution from a family of distributions, while still providing fully probabilistic results. We introduce a new variational method for geophysics—normalizing flows. The method models the posterior distribution by using a series of invertible and differentiable transforms—the flows. By optimizing the parameters of these transforms the flows are designed to convert a simple and analytically known probability distribution into a good approximation of the posterior distribution. Numerical examples show that normalizing flows can provide an accurate tomographic result including full uncertainty information while significantly decreasing the computational cost compared to Monte Carlo and other variational methods. In addition, this method provides analytic solutions for the posterior distribution rather than an ensemble of posterior samples. This opens the possibility that subsequent calculations that use the posterior distribution might be performed analytically.

Bayesian Random Tomography of Particle Systems.

Random tomography is a common problem in imaging science and refers to the task of reconstructing a three-dimensional volume from two-dimensional projection images acquired in unknown random directions. We present a Bayesian approach to random tomography. At the center of our approach is a meshless representation of the unknown volume as a mixture of spherical Gaussians. Each Gaussian can be interpreted as a particle such that the unknown volume is represented by a particle cloud. The particle representation allows us to speed up the computation of projection images and to represent a large variety of structures accurately and efficiently. We develop Markov chain Monte Carlo algorithms to infer the particle positions as well as the unknown orientations. Posterior sampling is challenging due to the high dimensionality and multimodality of the posterior distribution. We tackle these challenges by using Hamiltonian Monte Carlo and a global rotational sampling strategy. We test the approach on various simulated and real datasets.

Application of Bayesian tomography method to the visible spectroscopic diagnostic on HL-2A tokamak

As the commission of a visible spectroscopic diagnostic to the HL-2A tokamak for the measurements of both bremsstrahlung continuum radiation and impurity line emission, a Bayesian tomography method has been developed to this diagnostic for the reconstruction of a 2D emission distribution from line-integrated data. In this method, Gaussian process prior is adopted as an effective way to regularize the smoothness of the emission field, which can be truncated to enforce non-negative constraint of the emission values. For the reliability and quality assessment, uncertainty of the reconstruction can be examined through its confidence interval derived from a posterior probability. Performance of this Gaussian processtomography method is exemplified by the reconstructions of the bremsstrahlung continuum emission, which coincide with the equilibrium magnetic flux surfaces with respect to position and shape. In another case the reconstruction of CVI line emission in the visible range (n = 8→7, 529.05 nm) disclose the existence of a bright radiation belt due to the plasma–wall interaction.

A workflow for seismic imaging with quantified uncertainty

The interpretation of seismic images faces challenges due to the presence of several uncertainty sources. Uncertainties exist in data measurements, source positioning, and subsurface geophysical properties. Understanding uncertainties’ role and how they influence the outcome is fundamental in the earth sciences and essential in the oil and gas industry decision-making process. Geophysical imaging is time-consuming. When we add uncertainty quantification, it becomes both time and data-intensive. In this work, we propose a workflow for seismic imaging with quantified uncertainty. We build the workflow upon Bayesian tomography, reverse time migration, and image interpretation based on statistical information. The workflow explores an efficient hybrid parallel computational strategy to decrease the reverse time migration execution time. High levels of data compression are applied to reduce data transfer among workflow activities and data storage. We capture and analyze provenance data at runtime to improve workflow execution, monitoring, and debugging with negligible overhead. Numerical experiments on the Marmousi2 Velocity Model Benchmark demonstrate the workflow capabilities. We observe excellent weak and strong scalability, and results suggest that lossy data compression does not hamper the seismic imaging uncertainty quantification. We explore the propagation of the velocity uncertainties to the seismic images by confidence maps and probability density functions in control points. The multi-modal character revealed by such high-order statistics reinforces the need for the machinery we have put together in our workflow.

Fully Arbitrary Control of Frequency-Bin Qubits.

Accurate control of two-level systems is a longstanding problem in quantum mechanics. One such quantum system is the frequency-bin qubit: a single photon existing in superposition of two discrete frequency modes. In this Letter, we demonstrate fully arbitrary control of frequency-bin qubits in a quantum frequency processor for the first time. We numerically establish optimal settings for multiple configurations of electro-optic phase modulators and pulse shapers, experimentally confirming near-unity mode-transformation fidelity for all fundamental rotations. Performance at the single-photon level is validated through the rotation of a single frequency-bin qubit to 41 points spread over the entire Bloch sphere, as well as tracking of the state path followed by the output of a tunable frequency beam splitter, with Bayesian tomography confirming state fidelities F_{ρ}>0.98 for all cases. Such high-fidelity transformations expand the practical potential of frequency encoding in quantum communications, offering exceptional precision and low noise in general qubit manipulation.

A practical and efficient approach for Bayesian quantum state estimation

Bayesian inference is a powerful paradigm for quantum state tomography, treating uncertainty in meaningful and informative ways. Yet the numerical challenges associated with sampling from complex probability distributions hampers Bayesian tomography in practical settings. In this article, we introduce an improved, self-contained approach for Bayesian quantum state estimation. Leveraging advances in machine learning and statistics, our formulation relies on highly efficient preconditioned Crank–Nicolson sampling and a pseudo-likelihood. We theoretically analyze the computational cost, and provide explicit examples of inference for both actual and simulated datasets, illustrating improved performance with respect to existing approaches.

Exponential families for Bayesian quantum process tomography

A Bayesian approach to quantum process tomography has yet to be fully developed due to the lack of appropriate probability distributions on the space of quantum channels. Here, by associating the Choi matrix form of a completely positive, trace preserving (CPTP) map with a particular space of matrices with orthonormal columns, called a Stiefel manifold, we present two parametric probability distributions on the space of CPTP maps that enable Bayesian analysis of process tomography. The first is a probability distribution that has an average Choi matrix as a sufficient statistic. The second is a distribution resulting from binomial likelihood data that enables a simple connection to data gathered through process tomography experiments. These are examples of continuous, nonunitary random CPTP maps that capture meaningful prior information for use in Bayesian estimation. We show how these distributions can be used for point estimation using either maximum a posteriori estimates or expected a posteriori estimates, as well as full Bayesian tomography resulting in posterior credibility intervals. This approach will enable the full power of Bayesian analysis in all forms of quantum characterization, verification, and validation.