Bayesian Inversion Method Research Articles

A Bayesian inverse approach to proton therapy dose delivery verification

This study presents a proof-of-concept for a novel Bayesian inverse method in a one-dimensional setting, aimed at proton beam therapy treatment verification. Our methodology is predicated on a hypothetical scenario wherein strategically positioned sensors detect prompt- γ s emitted from a proton beam when it interacts with defined layers of tissue. Using these data, we employ a Bayesian framework to estimate the proton beam’s energy deposition profile. We validate our Bayesian inverse estimations against a closed-form approximation of the Bragg Peak in a uniform medium and a layered lung tumour.

Estimating Carbon Dioxide Emissions in Two California Cities Using Bayesian Inversion and Satellite Measurements

AbstractNASA's Orbiting Carbon Observatories (OCO‐2 and OCO‐3) provide measurements of column‐averaged carbon dioxide concentrations (XCO2) with sufficient spatial resolution and precision to constrain bottom‐up estimates of CO2 fluxes at regional scales. We use Bayesian inversion methods assimilating satellite retrievals to improve estimates of CO2 fluxes in the South Coast Air Basin (SoCAB) which surrounds Los Angeles, and in the San Francisco Bay Area Air Basin (SFBA). We study 2020 to understand the impact of the COVID‐19 lockdowns and an active wildfire season. Our results indicated that a 50% (30%) reduction in CO2 emissions relative to 2015 during the COVID‐19 lockdown period was consistent with OCO measurements for SFBA (SoCAB). We find that posterior wildfire emissions differed significantly from the prior at the scale of individual wildfires, though with large uncertainties, and that wildfire emissions in SFBA are significant, attributing 72% of the region's CO2 emissions during August 2020 to wildfires.

Bayesian model averaging and Bayesian inference-based probabilistic inversion method for arch dam zonal material parameters

Inversion analysis based on structural monitoring data is an important part of evaluating the working behaviour of structures. In this paper, a probabilistic inversion method for the elastic modulus of concrete in arch dams based on Bayesian model averaging (BMA) and Bayesian inference is proposed. According to the measured displacement data of an arch dam project, the influence of the separation accuracy of water pressure component, the inversion method and the selection of displacement measuring points on the inversion results of the elastic modulus of the dam body is studied. Example analysis results show that the accuracy of the multi-measurement point displacement and hydraulic pressure component separation model based on BMA-HST (Hydrostatic-seasonal-time) is at least 18.97 % higher than that of the single measurement point, and the relative error of the elastic modulus value of the multi-measurement point zonal inversion based on the Polynomial chaos-Kriging (PCK) proxy model and the Bayesian inference is only 6 % at the maximum, which indicates that the inversion model described in this paper is able to realize the high-precision inversion analysis of the material parameters of the zonal analysis of concrete dams at a relatively low computational cost.

A 650-Myr history of Earth's axial precession frequency and the evolution of the Earth-Moon system derived from cyclostratigraphy.

The preservation of Milankovitch cycles in the stratigraphic record provides independent geological information to study our ancient solar system and can be leveraged to constrain existing theoretical models. Here, we identify 34 high-quality cyclostratigraphic records spanning the past 650 million years and use them to infer the evolution of the Earth-Moon system through a Bayesian inversion method. We reconstruct the time evolution of Earth's axial precession frequency, lunar distance, length of day, and the periods of obliquity and climatic precession cycles. The results indicate an interval of high tidal energy dissipation in the Earth-Moon system at ~300 to 200 million years ago, and are broadly consistent with an independently calculated tidal evolution model. Our results provide an improved determination of the past periods of obliquity and climatic precession for astrochronology applications and yield important constraints on the history of tidal energy dissipation during the Phanerozoic Eon.

Accurate heat flux estimation in continuous casting Molds via MH-MCMC Bayesian Inverse Method

The present work focuses on accurately estimating heat flux variations on the hot faces of the billet and slab molds during continuous casting using the Metropolis Hastings–Markov Chain Monte Carlo (MH-MCMC) Bayesian inverse method and experimental data. Accurate estimation of heat flux variation is crucial for operational efficiency and ensuring high-quality steel products. The variation in the hot face heat flux (qn(y)) of the billet mold is estimated using the proposed MH-MCMC Bayesian inverse method and experimental data for various casting speeds. The study extends this methodology to the slab mold for estimating the variations in the heat fluxes (qnr(y), qw(x,y)) on the hot narrow and wide faces of the slab mold. The estimated heat fluxes for both molds are used in respective forward models to obtain temperatures. The temperatures obtained excellently match experimental temperatures and show superior accuracy in estimating the heat fluxes. From the present methodology, the errors between these simulated and experimental temperatures are much less compared with the literature. The predicted heat flux variations on the hot faces of the billet and slab molds are also compared with the heat flux obtained from the literature.This study provides valuable insights into accurately estimating heat fluxes in continuous casting molds (billet and slab) using the MH-MCMC Bayesian inverse method. The accurately estimated heat fluxes of the molds are essential for ensuring the quality, efficiency, and innovation of continuous casting processes.

Characterizing multifarious hydroclimatic patterns using geodetic measurements in the Australian mainland

Benefiting from the tremendous advance of modern geodetic technologies, accurate measurements provide us with novel means to weigh terrestrial water storage (TWS) and shed critical light on hydroclimatic interaction mechanisms. In this study, GNSS and GRACE observations are jointly used to investigate spatiotemporal TWS fluctuations and their connections with large-scale climate patterns in the Australian mainland. A variational Bayesian principal component analysis-based inversion method is designed to handle random data missing in GNSS and long-term data gaps in GRACE. The jointly inferred TWS estimates manifest more spatial details compared to the GRACE Mascon solutions and good resistance to non-hydrological mechanisms in GNSS data. The joint inversion results are compared with other water-related storage/flux estimates (i.e., GLDAS, GRACE, and precipitation), which have considerable coherence with TWS at both temporal and spatial scales. The results indicate obvious annual variations in TWS changes in north Australia in contrast to inappreciable seasonal hydrological signatures in other vast arid zones. Hydrological wet/dry conditions are quantified with drought severity index (DSI) datasets derived from multiple means and the joint-based DSI datasets temporally agree with the other datasets and correspond well with precipitation anomalies in most watersheds. In addition, the jointly inverted TWS estimates are used to declassify multifarious hydroclimatic features, and multi-source hydrometeorological datasets are used to attribute interannual TWS changes. Our study contributes to better observing the water cycles for the hydrological community and illuminates more application prospects in the climate sciences.

Bayesian Inversion with Neural Operator (BINO) for modeling subdiffusion: Forward and inverse problems

Fractional diffusion equations have been an effective tool for modeling anomalous diffusion in complicated systems. However, traditional numerical methods require expensive computation cost and storage resources because of the memory effect brought by the convolution integral of time fractional derivative. We propose a Bayesian Inversion with Neural Operator (BINO) to overcome the difficulty in traditional methods as follows. We employ a deep operator network to learn the solution operators for the fractional diffusion equations, allowing us to swiftly and precisely solve a forward problem for given inputs (including fractional order, diffusion coefficient, source terms, etc.). In addition, we integrate the deep operator network with a Bayesian inversion method for modeling a problem by subdiffusion process and solving inverse subdiffusion problems, which reduces the time costs (without suffering from overwhelm storage resources) significantly. A large number of numerical experiments demonstrate that the operator learning method proposed in this work can efficiently solve the forward problems and Bayesian inverse problems of the subdiffusion equation.

Earth-Moon dynamics from cyclostratigraphy reveals possible ocean tide resonance in the Mesoproterozoic era.

Cyclostratigraphy is an important observational window into the history of the Earth-Moon system. However, there is limited information from the Mesoproterozoic era (1.0 to 1.6 billion years ago); accordingly, only weak constraints on Earth-Moon separation and tidal dissipation are available for this time. To close this knowledge gap, we analyze cyclostratigraphy from the Yemahe Formation (~1.2 billion years ago), Wumishan Formation (~1.5 billion years ago), and Chuanlinggou Formation (~1.6 billion years ago) in China. We use a Bayesian inversion method to analyze the three cyclostratigraphic sections. We combine previous results with these three estimates to construct an updated Earth-Moon system evolution and tidal dissipation history after 2.5 billion years ago. The results show a tidal dissipation peak that is consistent with the model predictions within the error range but also that there may be an additional resonance fluctuation in the Mesoproterozoic era.

Bioremediation experiments and dynamic model of petroleum hydrocarbon contaminated soil

Clarifying the occurrence and morphological characteristics of petroleum hydrocarbons (PHs) in soil can facilitate a comprehensive understanding of their migration and transformation patterns in soil/sediment. Additionally, by establishing the dynamic transformation process of each occurrence state, the ecological impact and environmental risk associated with PHs in soil/sediment can be assessed more precisely. The adsorption experiments and closed static incubation experiments was carried out to explore the PHs degradation and fraction distribution in aged contaminated soil under two remediation scenarios of natural attenuation (NA) and bioaugmentation (BA) by exogenous bacteria through a new sequential extraction method based on Tenax-TA, Hydroxypropyl-β-cyclodextrin and Rhamnolipid (HPCD/RL), accelerated solvent extractor (ASE) unit and alkaline hydrolysis extraction. The adsorption experiment results illustrated that bioaugmentation could promote the desorption of PHs in the adsorption phase, and the soil-water partition coefficient Kd decreased from 0.153 L/g to 0.092 L/g. The incubation experiment results showed that compared with natural attenuation, bioaugmentation could improve the utilization of PHs in aged soil and promote the generation of non-extractable hydrocarbons. On the 90th day of the experiment, the concentrations of weakly adsorbed hydrocarbons in the natural attenuation and bioaugmentation experimental groups decreased by 46.44% and 87.07%, respectively, while the concentrations of strongly adsorbed hydrocarbons and non-extractable hydrocarbons increased by 77.93%, 182.14%, and 80.91%, and 501.19%, respectively, compared their initial values. We developed a novel dynamic model and inverted the kinetic parameters of the model by the parameter scanning function and the Markov Chain Monte Carlo (MCMC) method based on the Bayesian approach in COMSOL Multiphysics® finite element software combined with experimental data. There was a good linear relationship between experimental interpolation data and model prediction data. The R2 for the concentrations of weakly adsorbed hydrocarbons ranged from 0.9953 to 0.9974, for strongly adsorbed hydrocarbons from 0.9063 to 0.9756, and for non-extractable hydrocarbons from 0.9931 to 0.9982. These extremely high correlation coefficients demonstrate the high accuracy of the parameters calculated using the Bayesian inversion method.

Comparison of Machine-learning and Bayesian Inferences for the Interior of Rocky Exoplanets with Large Compositional Diversity

In previous work, we demonstrated that machine-learning techniques based on mixture density networks (MDNs) are successful in inferring the interior structure of rocky exoplanets with large compositional diversity. In this study, we compare the performance of a well-trained MDN model with the conventional Bayesian inversion method based on the Markov chain Monte Carlo (MCMC) method, under the same observable constraints. Considering that MCMC inversion is generally performed with the prior knowledge of planetary mass, radius, and bulk molar ratios of Fe/Mg and Si/Mg, we regenerate a substantial data set of interior structure data for rocky exoplanets and train a new MDN model with inputs of planetary mass, radius, Fe/Mg, and Si/Mg. It has been found that the well-trained MDN model has comparable performance to that of the MCMC method but requires significantly less computation time. The MDN model presents a practical alternative to the traditional MCMC method, surpassing the latter with minimal requirements for specialized knowledge, faster prediction, and greater adaptability. The developed MDN model is made publicly available on GitHub for the broader scientific community’s utilization. With the advent of the James Webb Space Telescope, we are ushering in a new epoch in exoplanetary explorations. In this evolving landscape, the MDN model stands out as a valuable asset, particularly for its ability to rapidly assimilate and interpret new data, thereby substantially advancing our understanding of the interior and habitability of exoplanetary systems.

Bottom Multi-Parameter Bayesian Inversion Based on an Acoustic Backscattering Model

The geoacoustic and physical properties of the bottom, as well as spatial distribution, are crucial factors in analyzing the underwater acoustic field structure and establishing a geoacoustic model. Acoustic inversion has been widely used as an economical and effective method to obtain multi-parameters of the bottom. Compared with traditional inversion methods based on acoustic propagation models, acoustic backscattering models are more suitable for multi-parameter inversion, because they contain more bottom information. In this study, a Bayesian inversion method based on an acoustic backscattering model is proposed to obtain bottom multi-parameters, including geoacoustic parameters (the sound speed and loss parameter), partial physical parameters of the sediment, and statistical parameters of the seafloor roughness and sediment heterogeneity. The bottom was viewed as a kind of fluid medium. A high-frequency backscattering model based on fluid theory was adopted as the forward model to fit the scattering strength between the model prediction and the measured data. The Bayesian inversion method was used to obtain the posterior probability density (PPD) of the inversion parameters. Parameter estimation, uncertainty, and correlation were acquired by calculating the maximum a posterior (MAP), the mean values, the one-dimensional marginal distributions of the PPD, and the covariance matrix. Finally, the high-frequency bottom backscattering strength from the Quinault Range site was employed for inversion tests. The estimated values and uncertainties of various bottom parameters are presented and compared with the directly measured bottom parameters. The comparison results demonstrate that the method proposed herein can be used to estimate the sediment/water sound speed ratio, the sediment/water density ratio, and the spectral exponent of the roughness spectrum effectively and reliably.

PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments

This paper describes PyOED, a highly extensible scientific package that enables developing and testing model-constrained optimal experimental design (OED) for inverse problems. Specifically, PyOED aims to be a comprehensive Python toolkit for model-constrained OED . The package targets scientists and researchers interested in understanding the details of OED formulations and approaches. It is also meant to enable researchers to experiment with standard and innovative OED technologies with a wide range of test problems (e.g., simulation models). OED, inverse problems (e.g., Bayesian inversion), and data assimilation (DA) are closely related research fields, and their formulations overlap significantly. Thus, PyOED is continuously being expanded with a plethora of Bayesian inversion, DA, and OED methods as well as new scientific simulation models, observation error models, and observation operators. These pieces are added such that they can be permuted to enable testing OED methods in various settings of varying complexities. The PyOED core is completely written in Python and utilizes the inherent object-oriented capabilities; however, the current version of PyOED is meant to be extensible rather than scalable. Specifically, PyOED is developed to “enable rapid development and benchmarking of OED methods with minimal coding effort and to maximize code reutilization.” This paper provides a brief description of the PyOED layout and philosophy and provides a set of exemplary test cases and tutorials to demonstrate the potential of the package.

Excess Ground Ice Profiles in Continuous Permafrost Mapped From InSAR Subsidence

AbstractExcess ground ice formation and melt drive surface heave and settlement, and are critical components of the water balance in Arctic soils. Despite the importance of excess ice for the geomorphology, hydrology and biogeochemistry of permafrost landscapes, we lack fine‐scale estimates of excess ice profiles. Here, we introduce a Bayesian inversion method based on remotely sensed subsidence. It retrieves near‐surface excess ice profiles by probing the ice content at increasing depths as the thaw front deepens over summer. Ice profiles estimated from Sentinel‐1 interferometric synthetic aperture radar (InSAR) subsidence observations at 80 m resolution were spatially associated with the surficial geology in two Alaskan regions. In most geological units, the estimated profiles were ice poor in the central and, to a lesser extent, the upper active layer. In a warm summer, units with ice‐rich permafrost had elevated inferred ice contents at the base of the active layer and the (previous years') upper permafrost. The posterior uncertainty and accuracy varied with depth. In simulations, they were best (≲0.1) in the central active layer, deteriorating (≳0.2) toward the surface and permafrost. At two sites in the Brooks Foothills, Alaska, the estimates compared favorably to coring‐derived profiles down to 35 cm, while the increase in excess ice below the long‐term active layer thickness of 40 cm was only reproduced in a warm year. Pan‐Arctic InSAR observations enable novel observational constraints on the susceptibility of permafrost landscapes to terrain instability and on the controls, drivers and consequences of ground ice formation and loss.

Application of Bayesian method for mining-induced tremors: A case study of the Xinjulong coal mine in China

Mining-induced tremor is a dynamic phenomenon occurring during the fracturing or slipping of rock and coal mass in coal mines, and the tremor mechanism is one of the key problems in the field of rockburst. The Bayesian inversion method can more accurately analyse the source rupture type under the premise of considering waveform data and model errors. First, we added different noise levels to the waveform and determined filtering parameters based on the spectrum. The rationality of the Bayesian inversion was verified by observing the focal mechanism solution. Finally, a total of fourteen mining-induced tremors were selected, and the parameters of the moment tensor (MT), type of rupture, and fault plane solutions were obtained. The results demonstrate that (1) the tremor events in the roof are mainly tensile/shear failure, and compression failure mainly occurs in the coal seam. (2) Events on the floor are closely related to gob and mining activities. (3) The rupture form of the events near the tectonic region is more complex, and the fault plane solutions (strike, dip, and slip) were obtained by MT conversion. (4) The relation between rockbursts and mining-induced tremors is discussed by analysing events 1# and 14# that occurred in the Xinjulong coal mine. These research findings can provide a reference for the analysis of source parameters and rupture forms of mining-induced tremors.

Apportionment and Inventory Optimization of Agriculture and Energy Sector Methane Emissions Using Multi‐Month Trace Gas Measurements in Northern Colorado

AbstractQuantifying sector‐resolved methane fluxes in complex emissions environments is challenging yet necessary to improve emissions inventories and guide policy. Here, we separate energy and agriculture sector emissions using a dynamic linear model analysis of methane, ethane, and ammonia data measured at a Northern Colorado site from November 2021 to January 2022. By combining these sector‐apportioned observations with spatially resolved inventories and Bayesian inverse methods, energy and agriculture methane fluxes are optimized across the study's ∼850 km2 sensitivity area. Energy sector fluxes are synthesized with previous literature to evaluate trends in energy sector methane emissions. Optimized agriculture fluxes in the study area were 3.5× larger than inventory estimates; we demonstrate this discrepancy is consistent with differences in the modeled versus real‐world spatial distribution of agricultural sources. These results highlight how sector‐apportioned methane observations can yield multi‐sector inventory optimizations in complex environments.

The Bayesian Inversion Method With a Surrogate Modeling Based on Neural Network for GATEM Data

The Bayesian Inversion Method With a Surrogate Modeling Based on Neural Network for GATEM Data

Oilfield Reservoir Parameter Inversion Based on 2D Ground Deformation Measurements Acquired by a Time-Series MSBAS-InSAR Method

Time-series ground deformation monitoring and reservoir parameter inversion are crucial for the dynamic assessment of oilfield resources and sustainable exploitation in oilfields. As some of the regions with the richest oil reserves in China, the oilfield areas in the western Qaidam Basin were selected as a typical study area. Firstly, we used SAR images collected by the Sentinel-1A satellite from January 2021 to December 2022 and applied the multidimensional small baseline subset (MSBAS) method to obtain vertical and east–west deformation measurements. On this basis, a nonlinear Bayesian inversion method was applied to model the shallow reservoir in a series of complex deformation areas, based on a single-source model and a multi-source model, respectively. As a result, the ground deformation monitoring results obtained by long time-series InSAR clearly reflect the uneven ground deformation caused by the oil extraction and water injection operation processes. There was slight subsidence in the Huatugou oilfield, while significant uplift deformation occurred in the Ganchaigou oilfield and the Youshashan oilfield, with a maximum uplift rate of 48 mm/year. Further analysis indicated that the introduction of the 2D deformation field helps to improve the robustness of oilfield reservoir parameter inversion. Moreover, the dual-source model is more suitable than the single-source model for inverting reservoir parameters of complex deformation. This study not only fills the gap of InSAR deformation monitoring for the oilfields in the western Qaidam Basin but also provides a theoretical reference for the model and method selection of reservoir parameter inversion in other oilfields.

Insights From Layered Anisotropy Beneath Southern New England: From Ancient Tectonism to Present‐Day Mantle Flow

AbstractSeismic anisotropy beneath eastern North America, as expressed in shear wave splitting observations, has been attributed to plate motion‐parallel shear in the asthenosphere, resulting in fast axes aligned with the plate motion. However, deviations of fast axes from plate motion directions are observed near major tectonic boundaries of the Appalachians, indicating contributions from lithospheric anisotropy associated with past tectonic processes. In this study, we conduct anisotropic receiver function (RF) analysis using data from a dense seismic array traversing the New England Appalachians in Connecticut to examine anisotropic layers in the crust and upper mantle and correlate them with past tectonic processes as well as present‐day mantle flow. We use the harmonic decomposition method to separate directionally‐dependent variations of RFs and focus on features with the same harmonic signals observed across multiple stations. Within the crust, there are multiple features that may be correlated with stratification in the Hartford Basin, faults in the Taconic thrust belt, shear zones formed during Salinic/Acadian terrane accretion events, and orogen‐parallel crustal flow in the Acadian orogenic plateau. We apply a Bayesian inversion method to obtain quantitative constraints on the direction and strength of intra‐crustal anisotropy beneath the Hartford Basin. In the upper mantle, we identify a fossil shear zone possibly formed during oblique subduction of Rheic Ocean lithosphere. We also find evidence for a plate motion‐parallel flow zone in the asthenosphere that is likely disturbed by mantle upwelling near the southern margin of the Northern Appalachian Anomaly in the eastern part of the study area.

Sensitivity analysis on mechanical parameters of continuum models for simulating in-plane tests on existing masonry panels

Understanding and modeling the structural behavior of existing masonry buildings is a fundamental task in view of their management and conservation, especially in seismic-prone areas. Among the different modeling approaches for the computational analysis of unreinforced masonry structures, this work investigates the ability of two material continuum models, the modified masonry-like material and Mazars’ material, to simulate real in-plane failure modes of masonry panels, considering the influence of the variability of mechanical parameters on the results. As the first part of a broader benchmark, this work focuses on the study of two recurring test patterns for masonry panels subject to both horizontal and vertical loads, given the availability of experimental data. To account for the parameter variability that typically affects masonry materials, uncertainty propagation is performed in a probabilistic framework, and a gPCE-based proxy model makes it possible to turn time-consuming numerical runs into significantly faster analytical evaluations. Sensitivity analyses are carried out in terms of Sobol’ indices to evaluate the influence of input variability on the ultimate horizontal load. The most relevant masonry parameters for each model are thus highlighted and the issue of model calibration is finally addressed by relying on Bayesian inverse methods. This work also underlines the pros and cons of the different hypotheses, which underlie the two nonlinear constitutive laws.

Proxy-based Bayesian inversion of strain tensor data measured during well tests

Recent instrument developments have made it possible to measure the strain tensor caused by injecting or pumping fluid from aquifers or reservoirs, but the full value of these data is limited because the long runtimes of poroelastic forward models makes it impractical to use many inversion schemes. This limits the interpretation of strain data for managing the recovery of resources or storage of wastes in the subsurface. This paper describes a method of inverting deformation data using a poroelastic numerical simulator so the results can be used to manage reservoirs or aquifers. We developed a workflow designed to reduce the number of simulations sufficiently to make it feasible to use DREAMzs, an advanced Bayesian inversion method that translates the uncertainties from different sources into unbiased posterior parameter distributions and uncertainty envelopes around the field data. Using a KNN proxy model for the poroelastic simulator is key to reducing the overall computations, and the workflow includes a strategy for ensuring the proxy model results converge on the results from the simulator. The workflow is tested using an idealized example that verifies the ability to correctly identify parameters and characterize noise used to perturb the data. Field data from an injection test at an oil reservoir near Tulsa, Oklahoma, are also used to evaluate the efficacy of the workflow with a real dataset. The workflow identified 265 history matching solutions out of 1240 total simulation runs (21% acceptance ratio), where the results were used to characterize posterior parameter distribution and evaluate the prediction uncertainty. This workflow is significant because it enables strain tensor, or other geomechanical measurements to be interpreted to guide decision-making during energy and environmental processes in the subsurface.