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Satellite XCO2 retrievals have been widely used in estimating fossil fuel carbon (FFC) emissions at point and urban scales. However, at the regional scale, it remains a significant challenge. Furthermore, current global and regional atmospheric inversions often overlook the uncertainties associated with FFC emissions. To meet the needs of the global carbon stocktake, we developed an inversion method based on Bayesian statistical theory and OCO-2 satellite XCO2 observations to optimize FFC emissions alongside terrestrial ecosystem carbon fluxes (NEE). The methodology’s core is to distinguish the contributions of NEE and FFC to the observed concentrations using their different spatial distributions. We designed an observing system simulation experiment to invert the 2016 FFC emissions. The results showed that posterior FFC emissions were significantly optimized during the non-growing seasons in the regions with high emissions, with the optimization effect diminishing as emissions shrank. Average FFC emissions uncertainty reductions are in the range of 13–82% in the non-growing season for the eight largest emitting regions globally. By assuming the same uncertainty reduction for FFC emissions in both the growing and non-growing seasons, we can optimize annual emissions for high-emission areas. We believe this study provides a new idea for the inversion of FFC emissions at the regional scale, which is important for achieving the goal of carbon neutrality.

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.

This paper presents the implementation of the Bayesian inversion method for the characterization and estimation of different dielectric material properties. The scattering parameters of single and multi-layer materials are measured using a free-space experimental setup using a standard gain horn antenna and a Vector Network Analyzer (VNA) at Ka-band (26–40 GHz). The relative permittivity, material thickness, and material positioning error are defined as model parameters and estimated using the observed (measured) data. The FR4 Epoxy, Rogers RT/Duriod 5880, and Rogers AD600 with different relative permittivities and thicknesses are used in the measurement setup. The results displayed good agreement between model parameters and estimated properties of the presented materials, while the corresponding eigenvectors provided a level of confidence in model parameter values. The results were compared with different reported techniques to showcase the possible use of the presented method in microwave imaging, non-destructive testing, and similar applications.

The chapter contains sections titled: Introduction The Niche for Inverse Dispersion by bLS MO‐bLS in Undisturbed Flow Inverse Dispersion in Strongly Disturbed Flows: 3D‐bLS Applications of MO‐bLS Conclusion

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability – via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

Inconsistencies between observations from long and short period seismic waves and geochemical data mean craton formation and evolution remains enigmatic. Specifically, internal layering and radial anisotropy are poorly constrained. Here, we show that these inconsistencies can be reconciled by inverting cratonic Rayleigh and Love surface wave dispersion curves for shear‐wave velocity and radial anisotropy using a flexible Bayesian scheme. This approach requires no explicit vertical smoothing and only adds anisotropy to layers where required by the data. We show that all cratonic lithospheres are comprised of a positively radially anisotropic upper layer, best explained by Archean underplating, and an isotropic layer beneath, indicative of two‐stage formation. Within the positively radially anisotropic upper layer, we find a variable amplitude low velocity zone within 9 of 12 cratons studied, that is well correlated with observed Mid‐Lithospheric Discontinuities (MLDs). The MLD is best explained by metasomatism after craton formation. Plain Language Summary The ancient cores of the continents, or cratons, are a treasure‐trove of >2.5 billion years of Earth's history. However, scientists disagree on the processes that led to their formation, or whether they have evolved significantly through time. This is because the geological and geophysical methods used to investigate cratons often yield conflicting results. By capitalizing on an up‐to‐date global long‐wavelength seismic data set, we image the cores of 12 cratons using an advanced statistical method, Bayesian inference. The flexible method requires few choices to be made a priori, is driven by the quality of the data itself and measures uncertainties on results. By detecting velocity differences between horizontally and vertically vibrating seismic waves, we show that all cratons likely comprise an upper layer formed during the hot early Earth, by a process that strongly aligns the constituent minerals of the rocks in the horizontal plane. Below this ∼125 km thick upper layer, a lower layer (∼150 km thick) shows no clear alignment of minerals and so was likely formed by a different process, at a later time. Variable slow wavespeed zones exist within the upper layer that match previous results from short‐wavelength seismic data. Key Points Bayesian surface wave inversion reconciles existing differences between seismic images of craton structure from long and short period data Cratons show a shallow low velocity zone (LVZ) within a layer of positive radial anisotropy and a high velocity isotropic layer beneath Cratons are formed in two stages shown by anisotropic structure and are later modified producing a LVZ and mid lithosphere discontinuities

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.

Based on a geophysical model for elastic loading, the application potential of Global Positioning System (GPS) vertical crustal displacements for inverting terrestrial water storage has been demonstrated using the Tikhonov regularization and the Helmert variance component estimation since 2014. However, the GPS-inferred terrestrial water storage has larger resulting amplitudes than those inferred from satellite gravimetry (i.e., Gravity Recovery and Climate Experiment (GRACE)) and those simulated from hydrological models (e.g., Global Land Data Assimilation System (GLDAS)). We speculate that the enlarged amplitudes should be partly due to irregularly distributed GPS stations and the neglect of the terrain effect. Within southwest China, covering part of southeastern Tibet as a study region, a novel GPS-inferred terrestrial water storage approach is proposed via terrain-corrected GPS and supplementary vertical crustal displacements inferred from GRACE, serving as \"virtual GPS stations\" for constraining the inversion. Compared to the Tikhonov regularization and Helmert variance component estimation, we employ Akaike’s Bayesian Information Criterion as an inverse method to prove the effectiveness of our solution. Our results indicate that the combined application of the terrain-corrected GPS vertical crustal displacements and supplementary GRACE spatial data constraints improves the inversion accuracy of the GPS-inferred terrestrial water storage from the Helmert variance component estimation, Tikhonov regularization, and Akaike’s Bayesian Information Criterion, by 55%, 33%, and 41%, respectively, when compared to that of the GLDAS-modeled terrestrial water storage. The solution inverted with Akaike’s Bayesian Information Criterion exhibits more stability regardless of the constraint conditions, when compared to those of other inferred solutions. The best Akaike’s Bayesian Information Criterion inverted solution agrees well with the GLDAS-modeled one, with a root-mean-square error (RMSE) of 3.75 cm, equivalent to a 15.6% relative error, when compared to 39.4% obtained in previous studies. The remaining discrepancy might be due to the difference between GPS and GRACE in sensing different surface water storage components, the remaining effect of the water storage changes in rivers and reservoirs, and the internal error in the geophysical model for elastic loading.

Water resources underpin human society and economic growth, yet freshwater is unevenly distributed, leaving arid regions severely water-stressed. The Beishan mining district in Inner Mongolia exemplifies this challenge: despite abundant minerals, it lacks surface water and depends almost entirely on groundwater. To improve exploration in such complex settings, we propose a Bayesian joint inversion that leverages the complementary sensitivities of Surface Nuclear Magnetic Resonance (SNMR) and Transient Electromagnetic (TEM) data within a probabilistic framework. Using a transdimensional Markov Chain Monte Carlo (MCMC) algorithm, the method adaptively balances data weighting and model complexity. Tests on synthetic and field datasets show that combining SNMR’s direct sensitivity to water content with TEM’s high-resolution resistivity imaging enhances aquifer detection across depths and enables quantitative uncertainty assessment. Applied in Beishan, the approach delineates promising aquifers, with results confirmed by drilling, offering a robust basis for groundwater exploration and sustainable management in arid regions.

Prior model construction is a fundamental component in geophysical inversion, especially Bayesian inversion. The prior model, usually derived from available geological information, can reduce the uncertainty of model characteristics during the inversion. However, the prior geological data for inferring a prior distribution model are often limited in real cases. Our work presents a novel framework to create 3D geophysical prior models using soil geochemistry and borehole rock sample measurements. We focus on the Bayesian inversion, which enables encoding of knowledge and multiple non-geophysical data into the prior. The new framework developed in our research comprises three main parts, namely correlation analysis, prior model reconstruction, and Bayesian inversion. We investigate the correlations between surface and subsurface geochemical features, as well as the correlation between geochemistry and geophysics, using canonical correlation analysis for the surface and borehole geochemistry. Based on the resulting correlations, we construct the prior susceptibility model. The informed prior model is then tested using geophysical forward modeling and outlier detection methods. In this test, we aim to falsify the prior model, which happens when the model cannot predict the field geophysical observation. To obtain the posterior models, the reliable prior models are incorporated into a Bayesian inversion framework. Using a real case of exploration in the Central African Copperbelt, we illustrate the workflow of constructing the high-resolution 3D stratigraphic model conditioned on soil geochemistry, borehole data, and airborne geophysics.