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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.

Discrete optimal transportation problems arise in various contexts in engineering, the sciences, and the social sciences. Often the underlying cost criterion is unknown, or only partly known, and the observed optimal solutions are corrupted by noise. In this paper we propose a systematic approach to infer unknown costs from noisy observations of optimal transportation plans. The algorithm requires only the ability to solve the forward optimal transport problem, which is a linear program, and to generate random numbers. It has a Bayesian interpretation and may also be viewed as a form of stochastic optimization. We illustrate the developed methodologies using the example of international migration flows. Reported migration flow data captures (noisily) the number of individuals moving from one country to another in a given period of time. It can be interpreted as a noisy observation of an optimal transportation map, with costs related to the geographical position of countries. We use a graph-based formulation of the problem, with countries at the nodes of graphs and nonzero weighted adjacencies only on edges between countries which share a border. We use the proposed algorithm to estimate the weights, which represent cost of transition, and to quantify uncertainty in these weights.

The geologic record of Milankovitch climate cycles provides a rich conceptual and temporal framework for evaluating Earth system evolution, bestowing a sharp lens through which to view our planet’s history. However, the utility of these cycles for constraining the early Earth system is hindered by seemingly insurmountable uncertainties in our knowledge of solar system behavior (including Earth–Moon history), and poor temporal control for validation of cycle periods (e.g., from radioisotopic dates). Here we address these problems using a Bayesian inversion approach to quantitatively link astronomical theory with geologic observation, allowing a reconstruction of Proterozoic astronomical cycles, fundamental frequencies of the solar system, the precession constant, and the underlying geologic timescale, directly from stratigraphic data. Application of the approach to 1.4-billion-year-old rhythmites indicates a precession constant of 85.79 ± 2.72 arcsec/year (2σ), an Earth–Moon distance of 340,900 ± 2,600 km (2σ), and length of day of 18.68 ± 0.25 hours (2σ), with dominant climatic precession cycles of ∼14 ky and eccentricity cycles of ∼131 ky. The results confirm reduced tidal dissipation in the Proterozoic. A complementary analysis of Eocene rhythmites (∼55 Ma) illustrates how the approach offers a means to map out ancient solar system behavior and Earth–Moon history using the geologic archive. The method also provides robust quantitative uncertainties on the eccentricity and climatic precession periods, and derived astronomical timescales. As a consequence, the temporal resolution of ancient Earth system processes is enhanced, and our knowledge of early solar system dynamics is greatly improved.

In this paper we develop a framework for parameter estimation in macroscopic pedestrian models using individual trajectories—microscopic data. We consider a unidirectional flow of pedestrians in a corridor and assume that the velocity decreases with the average density according to the fundamental diagram. Our model is formed from a coupling between a density dependent stochastic differential equation and a nonlinear partial differential equation for the density, and is hence of McKean-Vlasov type. We discuss identifiability of the parameters appearing in the fundamental diagram from trajectories of individuals, and we introduce optimization and Bayesian methods to perform the identification. We analyze the performance of the developed methodologies in various situations, such as for different in- and outflow conditions, for varying numbers of individual trajectories, and for differing channel geometries.

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

Quantifying 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. Plain Language Summary Improving our knowledge of the locations, magnitudes, and types of methane sources is important for implementing effective emissions mitigation technologies and regulations. Methane emissions are often challenging to quantify because a wide variety of sources can emit methane, and these disparate sources are often intermingled. We demonstrate how a dynamic linear model can use multi‐month time series of two tracer gases, ethane and ammonia, to effectively separate methane emissions from the energy and agriculture sectors. Incorporating these data into a Bayesian inverse analysis refines the magnitude and distribution of methane fluxes from each sector. Our analysis reveals that methane from agriculture is several times higher than inventory estimates. While this is in part due to the spatial distribution of sources, more monitoring is needed to improve agriculture emissions factors. Energy sector emissions factors optimized in this work are consistent with other regional studies of energy sector methane emissions. A synthesis of these works demonstrates a regional decline in energy sector emissions despite a concomitant increase in oil and gas extraction; however, current emissions are similar to 2008 estimates. Key Points A dynamic linear model apportions energy and agriculture methane emissions from multi‐month trace gas measurements in Northern Colorado An estimated 0.4 ± 0.2 kg CH4 are emitted per barrel of oil equivalent produced, yielding a Wattenberg Field emission rate of 15 Mg CH4/hr Optimized agriculture methane emissions are higher than inventory predictions, in part due to mislocated fluxes in the inventory

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.

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.

Methodological advances in seismic tomography are often driven by the quality of data sets. The dense and homogeneous spatial coverage of the AlpArray seismic network, including hundreds of permanent and temporary broadband stations, has motivated a series of methodological developments of ambient-noise-based tomography of the lithosphere across the entire Alps-Apennines regions, which have been published and are reviewed here. To take full advantage of the ocean-bottom seismometers (OBS) in the Ligurian-Provence basin, reconstructed Rayleigh wave signals between OBS have been improved by second-order correlations with onland stations. A Bayesian or fully transdimensional formalism has been introduced in both steps of isotropic ambient noise tomography. The three-dimensional S-wave velocity models have been further improved by wave-equation based inversions accounting for the physics of seismic wave propagation, including elastic–acoustic coupling at the sea bottom. A beamforming approach has been developed to avoid systematic errors in the measurement of azimuthal anisotropy from seismic noise. Probabilistic inversions for depth variations of azimuthal and radial anisotropy have provided robust estimates of anisotropic parameters in the crust and upper mantle that differ significantly from earlier surface-wave tomography studies. These methodological improvements have taken the full benefit of the quality of available seismic data to significantly improve knowledge of the seismic structure of the crust and shallow mantle beneath the Alps-Apennines system. Our findings include detailed mapping of strong and abrupt Moho depth changes under the Western Alps, contrasting orientations of fast velocity directions between the upper and lower Alpine crust, and the absence of significant radial anisotropy everywhere in the European crust and shallow upper mantle, except in the Apenninic lower crust. These methods can be applied to similar dense arrays with equivalent potential benefits.

The phenomenon of differential rotation of the Earth's inner core relative to the mantle is a subject of interest in geodynamo modeling that has been validated by seismological observations, mainly via the earthquake‐doublets method. Although recent studies converge on the time‐varying differential rotation of the inner core relative to the mantle, favoring a decadal variation, the inferred models significantly differ. Here, considering the data selection, the observed data structure, and the subjective model parameterizations, which we avoid by employing a Bayesian transdimensional approach, we show that the recent best‐fit model by Yang and Song (2023, https://doi.org/10.1038/s41561‐022‐01112‐z) featuring the 70‐year decadal variation is not obtained when all available data are considered. Namely, including only a small number of discarded earthquake doublets (<10%) changes the period of the inner‐core differential rotation fluctuation to 20–30 years. More earthquake‐doublet data are required to address the non‐uniqueness of the inversion problem. Plain Language Summary Seismologists use repetitive earthquakes (earthquake doublets) and their waves' fixed paths through the Earth's inner core to measure the rate of the inner core spinning relative to the mantle. A recent study in Nature Geoscience (Yang & Song, 2023, https://doi.org/10.1038/s41561‐022‐01112‐z) based on 31 selected earthquake doublets derived a 70‐year decadal cycle of the inner core spinning rate relative to the mantle. We show that the made selective choices change the data structure and possibly miss the inner core 20–30 year periodicity in the earlier times. Key Points We revisit the problem of the differential rotation of the inner core relative to the mantle from the existing earthquake‐doublet data The data selection, the data structure, and the data‐model relationship are critical in inverting the inner core differential rotation Consequently, the model derived by Yang and Song (2023, https://doi.org/10.1038/s41561‐022‐01112‐z) is not a unique model of the inner core's differential rotation