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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]

When working with hydrological data, the ability to quantify the similarity of different datasets is useful. The choice of how to make this quantification has a direct influence on the results, with different measures of similarity emphasising particular sources of error (for example, errors in amplitude as opposed to displacements in time and/or space). The Wasserstein distance considers the similarity of mass distributions through a transport lens. In a hydrological context, it measures the "effort" required to rearrange one distribution of water into the other. While being more broadly applicable, particular interest is paid to hydrographs in this work. The Wasserstein distance is adapted for working with hydrographs in two different ways and tested in a calibration and "averaging" of a hydrograph context. This alternative definition of fit is shown to be successful in accounting for timing errors due to imprecise rainfall measurements. The averaging of an ensemble of hydrographs is shown to be suitable when differences among the members are in peak shape and timing but not in total peak volume, where the traditional mean works well. [ABSTRACT FROM AUTHOR]

When working with hydrological data, the ability to quantify the similarity of different datasets is useful. The choice of how to make this quantification has direct influence on the results, with different measures of similarity emphasising particular sources of error (for example, errors in amplitude as opposed to displacements in time and/or space). The Wasserstein distance considers the similarity of mass distributions through a transport lens. In a hydrological context, it measures the 'effort' required to rearrange one distribution of water into the other. While being more broadly applicable, particular interest is payed to hydrographs in this work. The Wasserstein distance is adapted for working with hydrographs in two different ways, and tested in a calibration and 'averaging' of hydrographs context. This alternate definition of fit is shown successful in accounting for timing errors due to imprecise rainfall measurements. The averaging of an ensemble of hydrographs is shown suitable when differences among the members is in peak shape and timing, but not in total peak volume, where the traditional mean works well. [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]

Author(s): Malcolm Sambridge, Andrew Jackson Author Affiliations: National COVID numbers -- Benford's law looks for errors Given the importance of accurate reporting of COVID-19 cases and deaths to strategies for [...]

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]

We develop a theoretical framework for framing and solving probabilistic linear(ized) inverse problems in function spaces. This is built on the statistical theory of Gaussian Processes, and allows results to be obtained independent of any basis, avoiding any difficulties associated with the fidelity of representation that can be achieved. We show that the results of Backus–Gilbert theory can be fully understood within our framework, although there is not an exact equivalence due to fundamental differences of philosophy between the two approaches. Nevertheless, our work can be seen to unify several strands of linear inverse theory, and connects it to a large body of work in machine learning. We illustrate the application of our theory using a simple example, involving determination of Earth's radial density structure. [ABSTRACT FROM AUTHOR]

By starting from a general framework for probabilistic continuous inversion (developed in Part I) and introducing discrete basis functions, we obtain the well-known algorithms for probabilistic least-squares inversion set out by Tarantola & Valette. In doing so, we establish a direct equivalence between the spatial covariance function that must be specified in continuous inversion, and the combination of basis functions and prior covariance matrix that must be chosen for discretized inversion. We show that the common choice of Tikhonov regularization (⁠|$\mathbf {C_m^{-1}} = \sigma ^2\mathbf {I}$|⁠) arises from a delta-function spatial covariance, and that this lies behind many of the artefacts commonly associated with discretized inversion. We show that other choices of spatial covariance function can be used to generate regularization matrices yielding substantially better results, and permitting localization of features even if global basis functions are used. We are also able to offer a straightforward explanation for the spectral leakage problem identified by Trampert & Snieder. [ABSTRACT FROM AUTHOR]

A method of extracting group and phase velocity dispersions jointly for Love- and Rayleigh-wave observations is presented. This method uses a spectral element representation of a path average Earth model parameterized with density, shear-wave velocity, radial anisotropy, and VP/VS ratio. An initial dispersion curve is automatically estimated using a heuristic approach to prevent misidentification of the phase. A second step then more accurately fits the observed noise correlation function (NCF) between interstation pairs in the frequency domain. For good quality cross correlations with reasonable signal-to-noise ratio, we are able to very accurately fit the spectrum of NCFs and hence obtain reliable estimates of both phase and group velocity jointly for Love and Rayleigh surface waves. In addition, we also show how uncertainties can be estimated with linearized approximations from the Jacobians and subsequently used in tomographic inversions. [ABSTRACT FROM AUTHOR]

Obtaining slip distributions for earthquakes results in an ill-posed inverse problem. While this implies that only limited and uncertain information can be recovered from the data, inferences are typically made based only on a single regularized model. Here, we develop an inversion approach that can quantify uncertainties in a Bayesian probabilistic framework for the finite fault inversion (FFI) problem. The approach is suitably efficient for rapid source characterization and includes positivity constraints for model parameters, a common practice in FFI, via coordinate transformation to logarithmic space. The resulting inverse problem is nonlinear and the most probable solution can be obtained by iterative linearization. In addition, model uncertainties are quantified by approximating the posterior probability distribution by a Gaussian distribution in logarithmic space. This procedure is straightforward since an analytic expression for the Hessian of the objective function is obtained. In addition to positivity, we apply smoothness regularization to the model in logarithmic space. Simulations based on surface wave data show that smoothing in logarithmic space penalizes abrupt slip changes less than smoothing in linear space. Even so, the main slip features of models that are smooth in linear space are recovered well with logarithmic smoothing. Our synthetic experiments also show that, for the data set we consider, uncertainty is low at the shallow portion of the fault and increases with depth. In addition, a simulation with a large station azimuthal gap of 180° significantly increases the slip uncertainties. Further, the marginal posterior probabilities obtained from our approximate method are compared with numerical Markov Chain Monte Carlo sampling. We conclude that the Gaussian approximation is reasonable and meaningful inferences can be obtained from it. Finally, we apply the new approach to observed surface wave records from the great Illapel earthquake (Chile, 2015, M w = 8.3). The location and amplitude of our inferred peak slip is consistent with other published solutions but the spatial slip distribution is more compact, likely because of the logarithmic regularization. We also find a minor slip patch downdip, mainly in an oblique direction, which is poorly resolved compared to the main slip patch and may be an artefact. We conclude that quantifying uncertainties of finite slip models is crucial for their meaningful interpretation, and therefore rapid uncertainty quantification can be critical if such models are to be used for emergency response. [ABSTRACT FROM AUTHOR]

Following the linearized attenuation tomography from our previous study (Pejić et al., 2017, https://doi.org/10.1002/2016JB013692), we perform hierarchical transdimensional Bayesian tomography of the upper ≈400 km of the inner core, using 398 globally distributed t∗ estimates. The results are in good agreement with the ones obtained through linearized tomography: they show more complex attenuation pattern than the purely hemispherical one, and the noise estimated from the hierarchical inversion is in good agreement with estimates obtained from the Discrepancy Principle in the previous study. The attenuation pattern we observe gives more weight to the geodynamical models that couple the thermal anomalies of the lowermost mantle to the inner core boundary. Key Points: Transdimensional sampling of Voronoi cells on a spherical surface applied to a problem in global tomographyNo hemispherical pattern is observed in attenuation structure of the upper inner coreThe study provides a strong seismological hint for a strongly attenuating inner core [ABSTRACT FROM AUTHOR]

The use of Bayesian trans‐dimensional sampling in 2‐D and 3‐D imaging problems has recently become widespread in geophysical inversion. Its benefits include its spatial adaptability to the level of information present in the data and the ability to produce uncertainty estimates. The most used parameterization in Bayesian trans‐dimensional inversions is Voronoi cells. Here we introduce a general software, TransTessellate2D, that allows 2‐D trans‐dimensional inference with Voronoi cells and two alternative underlying parameterizations, Delaunay triangulation with linear interpolation and Clough‐Tocher interpolation, which utilize the same algorithm but result in either C0 or C1 continuity. We demonstrate that these alternatives are better suited to the recovery of smooth models, and show that the posterior probability solution is less susceptible to multimodalities which can complicate the interpretation of model parameter uncertainties. Key Points: We present a new software for trans‐dimensional surface reconstruction incorporating hierarchical error estimation, Hamiltonian Monte Carlo, and parallel temperingWe propose two alternative parameterizations to the ubiquitous Voronoi cellsThese alternate parameterizations may open up the application of trans‐dimensional inversion to a wider variety of geophysical problems [ABSTRACT FROM AUTHOR]

Most linear inverse problems require regularization to ensure that robust and meaningful solutions can be found. Typically, Tikhonov-style regularization is used, whereby a preference is expressed for models that are somehow 'small' and/or 'smooth'. The strength of such preferences is expressed through one or more damping parameters, which control the character of the solution, and which must be set by the user. However, identifying appropriate values is often regarded as a matter of art, guided by various heuristics. As a result, such choices have often been the source of controversy and concern. By treating these as hyperparameters within a hierarchical Bayesian framework, we are able to obtain solutions that encompass the range of permissible regularization parameters. Furthermore, we show that these solutions are often well-approximated by those obtained via standard analysis using certain regularization choices which are—in a certain sense—optimal. We obtain algorithms for determining these optimal values in various cases of common interest, and show that they generate solutions with a number of attractive properties. A reference implementation of these algorithms, written in Python, accompanies this paper. [ABSTRACT FROM AUTHOR]

Antarctica and Greenland hold enough ice to raise sea level by more than 65m if both ice sheets were to melt completely. Predicting future ice sheet mass balance depends on our ability to model these ice sheets, which is limited by our current understanding of several key physical processes, such as iceberg calving. Large-scale ice flow models either ignore this process or represent it crudely. To model fractured zones, an important component of many calving models, continuum damage mechanics as well as linear fracture mechanics are commonly used. However, these methods have a large number of uncertainties when applied across the entire Antarctic continent because the models were typically tuned to match processes seen on particular ice shelves. Here we present an alternative, statistics-based method to model the most probable zones of the location of fractures and demonstrate our approach on all main ice shelf regions in Antarctica, including the Antarctic Peninsula. We can predict the location of observed fractures with an average success rate of 84% for grounded ice and 61% for floating ice and a mean overestimation error rate of 26% and 20 %, respectively. We found that Antarctic ice shelves can be classified into groups based on the factors that control fracture location. [ABSTRACT FROM AUTHOR]

We address the problem of unpolarized light spectroscopy of geological materials. Using infrared radiation, the aim of this technique is to learn about the absorbing species, such as hydroxyl. The use of unoriented samples leads to the need to perform a rigorous statistical analysis, so that the three principal absorbances of the crystal can be retrieved. We present here such an analysis based on a derivation of the probability density function for a single random measurement. Previous methods for retrieval of the absorbances are shown to be suboptimal, producing biased results that are sometimes even unphysical (e.g., negative estimates for an inherently positive quantity). The mathematical structure of the problem is developed to use the maximum likelihood estimation method, and we show how to optimize for the three absorbance parameters. This leads to good parameter retrieval on both synthetic and real data sets. [ABSTRACT FROM AUTHOR]

A method of extracting group and phase velocity dispersions jointly for Love- and Rayleigh-wave observations is presented. This method uses a spectral element representation of a path average Earth model parameterized with density, shear-wave velocity, radial anisotropy, and VP/VS ratio. An initial dispersion curve is automatically estimated using a heuristic approach to prevent misidentification of the phase. A second step then more accurately fits the observed noise correlation function (NCF) between interstation pairs in the frequency domain. For good quality cross correlations with reasonable signal-to-noise ratio, we are able to very accurately fit the spectrum of NCFs and hence obtain reliable estimates of both phase and group velocity jointly for Love and Rayleigh surface waves. In addition, we also show how uncertainties can be estimated with linearized approximations from the Jacobians and subsequently used in tomographic inversions. [ABSTRACT FROM AUTHOR]

Abstract: Cross correlation of seismograms provides new information on the Earth both through the exploitation of ambient noise and specific components of earthquake records. Here we cross‐correlate recordings of large earthquakes on a planetary scale and identify a range of hitherto unobserved seismic phases in Earth's correlation wavefield. We show that both arrivals with the timing expected for the regular seismic wavefield and previously unexplained phases are produced by interference between seismic paths having the same ray parameter but with only a subset of propagation legs in common. This insight explains the origin and generation mechanism of the features of Earth's correlation wavefield and opens up new ways of addressing issues in global seismology. Strong similarity between observed and synthesized correlation wavefields indicates that the Earth's radial structure is remarkably well constrained in the intermediate period range. [ABSTRACT FROM AUTHOR]

This paper presents the application of a novel trans-dimensional sampling approach to a time domain airborne electromagnetic (AEM) inverse problem to solve for plausible conductivities of the subsurface. Geophysical inverse field problems, such as time domain AEM, are well known to have a large degree of non-uniqueness. Common least-squares optimisation approaches fail to take this into account and provide a single solution with linearised estimates of uncertainty that can result in overly optimistic appraisal of the conductivity of the subsurface. In this new non-linear approach, the spatial complexity of a 2D profile is controlled directly by the data. By examining an ensemble of proposed conductivity profiles it accommodates non-uniqueness and provides more robust estimates of uncertainties. [ABSTRACT FROM AUTHOR]

In geophysical inversion, inferences of Earth's properties from sparse data involve a trade-off between model complexity and the spatial resolving power. A recent Markov chain Monte Carlo (McMC) technique formalized by Green, the so-called trans-dimensional samplers, allows us to sample between these trade-offs and to parsimoniously arbitrate between the varying complexity of candidate models. Here we present a novel framework using transdimensional sampling over tree structures. This new class of McMC sampler can be applied to 1-D, 2-D and 3-D Cartesian and spherical geometries. In addition, the basis functions used by the algorithm are flexible and can include more advanced parametrizations such as wavelets, both in Cartesian and Spherical geometries, to permit Bayesian multiscale analysis. This new framework offers greater flexibility, performance and efficiency for geophysical imaging problems than previous sampling algorithms. Thereby increasing the range of applications and in particular allowing extension to trans-dimensional imaging in 3-D. Examples are presented of its application to 2-D seismic and 3-D teleseismic tomography including estimation of uncertainty. [ABSTRACT FROM AUTHOR]

This paper outlines a methodology for the estimation of the environmental noise equivalent reflectance in aquatic remote sensing imagery using an object-based segmentation approach. Noise characteristics of remote sensing imagery directly influence the accuracy of estimated environmental variables and provide a framework for a range of sensitivity, sensor specification, and algorithm design studies. The proposed method enables estimation of the noise equivalent reflectance covariance of remote sensing imagery through homogeneity characterization using image segmentation. The method is first tested on a synthetic data set with known noise characteristics and is successful in estimating the noise equivalent reflectance under a range of segmentation structures. Testing on a Portable Hyperspectral Imager for Low-Light Spectroscopy (PHILLS) hyperspectral image in a coral reef environment shows the method to produce comparable noise equivalent reflectance estimates in an optically shallow water environment to those previously derived in optically deep water. This method is of benefit in aquatic studies where homogenous regions of optically deep water were previously required for image noise estimation. The ability of the method to characterize the covariance of an image is of significant benefit when developing probabilistic inversion techniques for remote sensing. [ABSTRACT FROM PUBLISHER]

Conceptual uncertainty is considered one of the major sources of uncertainty in groundwater flow modelling. In this regard, hypothesis testing is essential to increase system understanding by refuting alternative conceptual models. Often a stepwise approach, with respect to complexity, is promoted but hypothesis testing of simple groundwater models is rarely applied. We present an approach to model-based Bayesian hypothesis testing in a simple groundwater balance model, which involves optimization of a model in function of both parameter values and conceptual model through trans-dimensional sampling. We apply the methodology to the Wildman River area, Northern Territory, Australia, where we set up 32 different conceptual models. A factorial approach to conceptual model development allows for direct attribution of differences in performance to individual uncertain components of the conceptual model. The method provides a screening tool for prioritizing research efforts while also giving more confidence to the predicted water balance compared to a deterministic water balance solution. We show that the testing of alternative conceptual models can be done efficiently with a simple additive and linear groundwater balance model and is best done relatively early in the groundwater modelling workflow. [ABSTRACT FROM AUTHOR]

Working with a large temporal dataset spanning several decades often represents a challenging task, especially when the record is heterogeneous and incomplete. The use of statistical laws could potentially overcome these problems. Here we apply Benford's Law (also called the 'First-Digit Law') to the traveled distances of tropical cyclones since 1842. The record of tropical cyclones has been extensively impacted by improvements in detection capabilities over the past decades. We have found that, while the first-digit distribution for the entire record follows Benford's Law prediction, specific changes such as satellite detection have had serious impacts on the dataset. The least-square misfit measure is used as a proxy to observe temporal variations, allowing us to assess data quality and homogeneity over the entire record, and at the same time over specific periods. Such information is crucial when running climatic models and Benford's Law could potentially be used to overcome and correct for data heterogeneity and/or to select the most appropriate part of the record for detailed studies. [ABSTRACT FROM AUTHOR]

• A quantitative method resolves the fine structure of the D″ layer. • Small-scale heterogeneities exist at CMB beneath eastern Alaska & northern Pacific. • There is a lack of correlation between D″ topography and shear velocity variation. • The D″ layer may be a mixed boundary layer comprised of thermo-chemical anomalies. • Inferred accumulated slab material suggests different mantle convection styles. The D″ layer, regarded as a thermal boundary layer and a chemically distinct region above the core-mantle boundary (CMB), has been associated with the phase transition from bridgmanite (Bm) to post-perovskite (pPv) in the lowermost mantle. However, the composition of the lowermost mantle and thermal conditions where Bm-pPv phase-transition occurs is still debatable. The methods typically used to study the fine-scale seismic features in the D″ layer has provided important information. However, trial-and-error seismic waveform modelling cannot uniquely quantify D″-layer properties because of subjective model-parameterization choices and inherent non-uniqueness of solutions and the waveform inversion method has a limited resolution of the velocity gradient and depth of the D″ discontinuity. We develop a grid-search scheme to constrain the detailed 1-D shear-wave velocity structure in the lowermost mantle beneath Alaska and the northern Pacific, accompanied with quantitative assessment of the uncertainty of 1D models. Our results show strong lateral variations of the D″ discontinuity from west to east beneath Alaska, along with the existence of smaller-scale heterogeneities in the east. We find a broad velocity increase, as thick as 240 km, at the top of D″ that indicates this region may involve a composite of downwelling thermo-chemical anomalies at the CMB. There are even smaller scale heterogeneities of approximately 120 km × 120 km in size with larger lateral variations in the lowermost mantle beneath northern Pacific. Both the magnitude and gradient of the velocity at the top of the D″ layer vastly change in adjacent regions, with an increase from 2.8% to 4.5% in magnitude and from 0.08% to 1.2% in gradient, but with a relatively consistent depth of the D″ discontinuity at ∼340 km above the CMB. The weak correlation between D″ topography and velocity variations indicate chemical heterogeneities must be present beneath the northern Pacific, which might come from north-westward subducted Pacific oceanic lithosphere. Our characterisation of the spatial pattern of small-scale heterogeneities in the lowermost mantle supports a hybrid thermo-chemical boundary layer (TCBL) model beneath Alaska and northern Pacific. [ABSTRACT FROM AUTHOR]