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We present a theoretical framework that links Fermat's principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it yields shortest paths in the considered media through the optimal transport plans. In the discrete setting, our formulation results in physically significant optimal couplings, whose off-diagonal entries identify shortest paths in both directed and undirected graphs. For undirected graphs with positive edge weights, commonly used to parameterize seismic media, our method provides solutions to the Eikonal equation consistent with those from the Dijkstra algorithm. For directed negative-weight graphs, corresponding to transportation cost matrices with negative entries, our approach aligns with the Bellman–Ford algorithm but offers considerable computational advantages. We also highlight potential research directions. These include the use of sparse cost matrices to reduce the number of unknowns and constraints in the considered transportation problem, and solving specific classes of optimal transport problems through the Dijkstra algorithm to enhance computational efficiency. [ABSTRACT FROM AUTHOR]

Concerns raised by Okazaki & Ueda (2022) on the paper by Sambridge et al. (2022) are addressed. Two issues are discussed and some new numerical results presented. The first concerns whether the properties of the Wasserstein time-series misfit introduced in our earlier paper will translate to model space non-uniqueness in a seismic waveform inversion setting. It is argued that this is unlikely, given the special conditions, which must exist between all observed/predicted seismic waveform pairs for non-uniqueness to result. The second issue discussed is the efficacy of using the Sliced Wasserstein algorithm of Bonneel et al. (2015) as an alternate to the marginal Wasserstein algorithm, as proposed by Okazaki & Ueda (2022). It is argued that for optimization-based waveform fitting, the Sliced Wasserstein algorithm is a viable alternate provided care is taken to ensure that conditions arise which do invalidate analytical derivative expressions of the resulting Wasserstein misfit. In practice, this would likely mean recasting the 2D Optimal Transport problem posed in our earlier paper onto unstructured grids. [ABSTRACT FROM AUTHOR]

We propose a new approach to measuring the agreement between two oscillatory time-series, such as seismic waveforms, and demonstrate that it can be used effectively in inverse problems. Our approach is based on Optimal Transport theory and the Wasserstein distance, with a novel transformation of the time-series to ensure that necessary normalization and positivity conditions are met. Our measure is differentiable, and can readily be used within an optimization framework. We demonstrate performance with a variety of synthetic examples, including seismic source inversion, and observe substantially better convergence properties than achieved with conventional L 2 misfits. We also briefly discuss the relationship between Optimal Transport and Bayesian inference. [ABSTRACT FROM AUTHOR]

Monte Carlo methods are widespread in geophysics and have proved to be powerful in non-linear inverse problems. However, they are associated with significant practical challenges, including long calculation times, large output ensembles of Earth models, and difficulties in the appraisal of the results. This paper addresses some of these challenges using generative models, a family of tools that have recently attracted much attention in the machine learning literature. Generative models can, in principle, learn a probability distribution from a set of given samples and also provide a means for rapid generation of new samples which follow that approximated distribution. These two features make them well suited for application to the outputs of Monte Carlo algorithms. In particular, training a generative model on the posterior distribution of a Bayesian inference problem provides two main possibilities. First, the number of parameters in the generative model is much smaller than the number of values stored in the ensemble, leading to large compression rates. Secondly, once trained, the generative model can be used to draw any number of samples, thereby eliminating the dependence on an often large and unwieldy ensemble. These advantages pave new pathways for the use of Monte Carlo ensembles, including improved storage and communication of the results, enhanced calculation of numerical integrals, and the potential for convergence assessment of the Monte Carlo procedure. Here, these concepts are initially demonstrated using a simple synthetic example that scales into higher dimensions. They are then applied to a large ensemble of shear wave velocity models of the core–mantle boundary, recently produced in a Monte Carlo study. These examples demonstrate the effectiveness of using generative models to approximate posterior ensembles, and indicate directions to address various challenges in Monte Carlo inversion. [ABSTRACT FROM AUTHOR]

The 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. Plain Language Summary: The lowermost mantle sits atop the core‐mantle boundary, the most dramatic boundary within our planet, with contrasts in physical properties that exceed those that exist at the surface. Despite significant progress in recent years, this part of the Earth is not well understood, and various tomographic studies on a global scale, along with regional studies that focus on seismic waveform modeling, pave the path towards higher resolution and new understanding. Important questions to answer are on the distribution, shape, size and composition of inhomogeneities in the lowermost mantle, and their critical role in the mantle and core dynamics. While there is a general consensus on the long‐scale length structures inferred from long‐period shear waves, there are notable differences in details of the tomograms of the lowermost mantle, stemming from the use of different imaging methods and datasets. Here, we utilize a large travel time data set of ScS and S waves with a significant addition of new measurements sensitive to the lowermost mantle to perform a probabilistic shear‐wave tomography, and we retrieve a high‐resolution image of the lowermost mantle. The new shear‐wave speed tomogram and comprehensive resolution‐estimations indicate that short and medium scale inhomogeneities are omnipresent features of the lowermost mantle. Key Points: S‐wave tomography of the lowermost mantle using state‐of‐the‐art Bayesian approach with 2D spherical Voronoi cellsThe inversion technique treats the model complexity and the data noise as free parameters and avoids damping and smoothingThis study provides an important bridge between long‐scale features at a global scale and short‐scale features of regional models [ABSTRACT FROM AUTHOR]