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Antarctica and Greenland hold enough ice to raise sea level by more than 65 m 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.

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]

The predictions of the theory of light propagation in weakly absorbing anisotropic minerals are tested against systematic measurements of the infrared absorbance spectra ofcalcite, olivine, and topaz oriented in both principal and random sections, using both polarized and unpolarized light. We show that if the linear polarized maximum absorbance is smaller than -0.3, or if the ratio of maximum and minimum absorbance is close to unity, then (1) the polarized maximum and minimum absorbances as well as the unpolarized absorbance are, to a good approximation, linearly proportional to thickness, regardless of the direction of the incident light; (2) the angular variation of polarized light absorption is indistinguishable from the theoretical predictions within the uncertainty of the measurements; (3) for any section the unpolarized absorbance is the mean of the polarized maximum and minimum absorbance; and (4) the average unpolarized absorbance of randomly oriented grains is one third of the Total Absorbance (defined as the sure of the three principal absorbances). Therefore, calibrations relating Total Absorbance to absorber concentration in minerals that have been developed from measurements with polarized light parallel to the principal axes may be applied to measurements with unpolarized light on a population of randomly oriented sections. We show that 10 such measurements are sufficient to achieve a petrologically useful accuracy. The method enables water concentrations in nominally anhydrous minerals to be determined from samples where the preparation of oriented specimens is not feasible, such as high-pressure experimental runs and fine-grained mantle xenoliths. The method may also be used for obtaining quantitative measurements on low-symmetry minerals. Keywords: Infrared spectroscopy, absorbance spectroscopy, unpolarized light, olivine, calcite, topaz, nominally anhydrous minerals

A new approach to the use of spectroscopic absorbance measurements for anisotropic crystals allows results to be extracted using unpolarized light incident on random crystal orientations. The theory of light propagation in anisotropic absorbing crystals is developed from Maxwell's equations to devise an expression for the transmittance of linearly polarized light traveling in an arbitrary direction in weakly absorbing media. This theory predicts the distribution of transmittance and absorbance as a function of direction and polarization angle of incident light. It is shown how a previously deduced empirical expression, commonly used in infrared spectroscopy, is a good approximation to the full theory under a wide range of conditions. The new theory shows that principal polarized absorbances correspond to the eigenvalues of an absorbance ellipsoid. An expression is derived for the unpolarized absorbance as a function of the angles describing incident light direction, [A.sub.unpol]([PHI], [PSI]), and the principal polarized absorbances, [A.sub.a], [A.sub.b], [A.sub.c] in an anisotropic crystal [A.sub.unpol]([PHI],[PSI]) = 1/2[[A.sub.a]([cos.sup.2][PHI][cos.sup.2][PSI} + [sin.sup.2][PSI]) + [A.sub.b]([cos.sup.2][PHI][sin.sup.2][PSI] + [cos.sup.2][PSI]) + [A.sub.c][sin.sup.2][PHI]]. Integration of this expression over all incident angles leads to a simple relationship between total measured unpolarized absorbance and the three principal polarized absorbances. Using this theory, a procedure is proposed for estimating both total ([A.sub.a] + [A.sub.b] + [A.sub.c]) and principal absorbances from spectroscopic measurements of absorbance using unpolarized light on a set of randomly oriented crystals. Keywords: Absorption index theory, anisotropic media, unpolarized light, spectroscopy

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]

The application of the geometrical concept of 'natural neighbors' theory to the weighted residual method helps in solving partial differential equations on highly irregular evolving grids. This approach can effectively deal with problems involving large deformation and solid-fluid interactions on a deforming mesh. A two-dimensional example of a sinking elasto-plastic plate in a linear viscosity fluid is used to illustrate the updating of the mesh nodes and the connections between nodes during the calculation.

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]

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]

This paper develops a probabilistic Bayesian approach to the problem of inferring the spatiotemporal evolution of earthquake rupture on a fault surface from seismic data with rigorous uncertainty estimation. To date, uncertainties of rupture parameters are poorly understood, and the effect of choices such as fault discretization on uncertainties has not been studied. We show that model choice is fundamentally linked to uncertainty estimation and can have profound effects on results. The approach developed here is based on a trans-dimensional self-parametrization of the fault, avoids regularization constraints and provides rigorous uncertainty estimation that accounts for model-selection ambiguity associated with the fault discretization. In particular, the fault is parametrized using self-adapting irregular grids which intrinsically match the local resolving power of the data and provide parsimonious solutions requiring few parameters to capture complex rupture characteristics. Rupture causality is ensured by parametrizing rupture-onset time by a rupture-velocity field and obtaining first rupture times from the eikonal equation. The Bayesian sampling of the parameter space is implemented on a computer cluster with a highly efficient parallel tempering algorithm.The inversion is applied to simulated and observed W-phase waveforms from the 2010 Maule (Chile) earthquake. Simulation results show that our approach avoids both over- and underparametrization to ensure unbiased inversion results with uncertainty estimates that are consistent with data information. The simulation results also show the ability of W-phase data to resolve the spatial variability of slip magnitude and rake angles. In addition, sensitivity to spatially dependent rupture velocities exists for kinematic slip models.Application to the observed data indicates that residual errors are highly correlated and likely dominated by theory error, necessitating the iterative estimation of a non-stationary data covariance matrix. The moment magnitude for the Maule earthquake is estimated to be ∼8.9, with slip concentrated in two zones updip of and north and south of the hypocentre, respectively. While this aspect of the slip distribution is similar to previous studies, we show that the slip maximum in the southern zone is poorly resolved compared to the northern zone. Both slip maxima are higher than reported in previous studies, which we speculate may be due to the lack of bias caused by the regularization used in other studies. [ABSTRACT FROM AUTHOR]

Non-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, known as Parallel Tempering, which originated in the field of computational statistics but is finding increasing numbers of applications in fields ranging from Chemical Physics to Astronomy. To date, experience in use of Parallel Tempering within earth sciences problems is very limited. In this paper, we describe Parallel Tempering and compare it to related methods of Simulated Annealing and Simulated Tempering for optimization and sampling, respectively. A key feature of Parallel Tempering is that it satisfies the detailed balance condition required for convergence of Markov chain Monte Carlo (McMC) algorithms while improving the efficiency of probabilistic sampling. Numerical results are presented on use of Parallel Tempering for trans-dimensional inversion of synthetic seismic receiver functions and also the simultaneous fitting of multiple receiver functions using global optimization. These suggest that its use can significantly accelerate sampling algorithms and improve exploration of parameter space in optimization. Parallel Tempering is a meta-algorithm which may be used together with many existing McMC sampling and direct search optimization techniques. It's generality and demonstrated performance suggests that there is significant potential for applications to both sampling and optimization problems in the geosciences. [ABSTRACT FROM AUTHOR]

Coda waves arise from scattering to form the later arriving components of a seismogram. Coda-wave interferometry (CWI) is an emerging tool for constraining earthquake source properties from the interference pattern of coda waves between nearby events. A new earthquake location algorithm is derived which relies on coda-wave-based probabilistic estimates of earthquake separation. The algorithm can be used with coda waves alone or in tandem with arrival-time data. Synthetic examples (2D and 3D) and real earthquakes on the Calaveras fault, California, are used to demonstrate the potential of coda waves for locating poorly recorded earthquakes. It is demonstrated that CWI: (a) outperforms traditional earthquake location techniques when the number of stations is small; (b) is self-consistent across a broad range of station situations; and (c) can be used with a single station to locate earthquakes. [ABSTRACT FROM AUTHOR]

Geodynamical models and seismic observations suggest that the Earth's solid inner core rotates at a different rate than the mantle. However, discrepancies exist in rotation rate estimates based on seismic waves produced by earthquakes. Here we investigate the inherent assumption of a constant rotation rate using earthquake doublets-repeating earthquakes that produce similar waveforms. We detect that the rotation rate of the Earth's inner core with respect to the mantle varies with time. We perform an inverse analysis of 7 doublets observed at the College station, Alaska, as well as 17 previously reported doublets, and reconstruct a history of differential inner-core rotation between 1961 and 2007. We find that the observed doublets are consistent with a model of an inner core with an average differential rotation rate of 0.25-0.48° yr−1 and decadal fluctuations of the order of 1° yr−1 around the mean. The decadal fluctuations explain discrepancies between previous core rotation models and are in concordance with recent geodynamical simulations. [ABSTRACT FROM AUTHOR]

This paper applies parallel tempering within a Bayesian formulation for strongly nonlinear geoacoustic inverse problems. Bayesian geoacoustic inversion consists of sampling the posterior probability density (PPD) of seabed parameters to estimate integral properties, such as marginal probability distributions, based on ocean acoustic data and prior information. This sampling is usually carried out using the Markov-chain Monte Carlo method of Metropolis-Hastings sampling. However, standard sampling methods can be very inefficient for strongly nonlinear problems involving multi-modal PPDs with the potential to miss important regions of the parameter space and to significantly underestimate parameter uncertainties. Parallel tempering achieves efficient/effective sampling of challenging parameter spaces with the ability to transition freely between multiple PPD modes by running parallel Markov chains at a series of increasing sampling temperatures with probabilistic interchanges between chains. The approach is illustrated for inversion of (simulated) acoustic reverberation data for which the PPD is highly multi-modal. While Metropolis-Hastings sampling gives poor results even with very large sample sizes, parallel tempering provides efficient, convergent sampling of the PPD. Methods to enhance the efficiency of parallel tempering are also considered. [ABSTRACT FROM AUTHOR]

SUMMARY A meaningful interpretation of seismic measurements requires a rigorous quantification of the uncertainty. In an inverse problem, the data noise determines how accurately observations should be fit, and ultimately the level of detail contained in the recovered model. A common problem in seismic tomography is the difficulty in quantifying data uncertainties, and thus the required level of data fit. Traditionally, the complexity of the solution model (defined by both the number of basis functions and the regularization) is defined arbitrarily by the user prior to inversion with only limited use of data errors. In the context of multiscale problems, dealing with multiple data sets that are characterized by different noise variances and that span the Earth at different scales is a major challenge. Practitioners are usually required to arbitrarily weigh the contribution of each data type into the final solution. Furthermore, the basis functions are usually spatially uniform across the velocity field and regularization procedures are global, which prevents the solution model from accounting for the uneven spatial distribution of information. In this work we propose to address these issues with a Hierarchical Bayesian inversion. The new algorithm represents an extension of the transdimensional tomography to account for uncertainties in data noise. This approach has the advantage of treating the level of noise in each data set, as well as the number of model parameters, as unknowns in the inversion. It provides a parsimonious solution that fully represents the degree of knowledge one has about seismic structure (i.e. constraints, resolution and trade-offs). Rather than being forced to make decisions on parametrization, level of data fit and weights between data types in advance, as is often the case in an optimization framework, these choices are relaxed and instead constrained by the data themselves. The new methodology is presented in a synthetic example where both the data density and the underlying structure contain multiple length scales. Three ambient noise data sets that span the Australian continent at different scales are then simultaneously inverted to infer a multiscale tomographic image of Rayleigh wave group velocity for the Australian continent. The procedure turns out to be particularly useful when dealing with multiple data types with different unknown levels of noise as the algorithm is able to naturally adjust the fit to the different data sets and provide a velocity map with a spatial resolution adapted to the spatially variable information present in the data. [ABSTRACT FROM AUTHOR]

There is significant seismic activity in the region around Australia, largely due to the plate boundaries to the north and to the east of the mainland. This activity results in serious seismic and tsunami hazard in the coastal areas of Australia. Hence seismicity is and will be monitored in real time by Geoscience Australia (GA), which uses a network of permanent broadband seismometers. Seismic moment tensor (MT) solutions are currently determined using 1-D, radially symmetric models of Earth and this requires augmentation by recording stations located outside of Australia. A 3-D model of the Australian continent developed recently using full waveform tomography now offers the opportunity to significantly improve the determination of MT solutions of earthquakes from tectonically active regions. A complete-waveform, time-domain MT inversion method has been developed using a point-source approximation. A series of synthetic tests using first a 1-D and then a 3-D structural model has been performed. The feasibility of deploying 3-D versus 1-D Earth structure for the inversion of seismic data has been studied and the advantages of using the 3-D structural model were illustrated with examples. The 3-D model is superior to the 1-D model, as a number of sensitivity tests show. The ultimate goal of this work is an automated MT inversion system in Australia relying on GA and other international stations, although more work remains to be done before the full implementation of such a scheme in real time. [ABSTRACT FROM AUTHOR]

We present a globally distributed data set of ∼400 000 frequency-dependent SH-wave traveltimes. An automated technique is used to measure teleseismic S, ScS and SS traveltimes at several periods ranging from 10 to 51 s. The targeted seismic phases are first extracted from the observed and synthetic seismograms using an automated time window algorithm. Traveltimes are then measured at several periods, by cross-correlation between the selected observed and synthetic filtered waveforms. Frequency-dependent effects due to crustal reverberations beneath each receiver are handled by incorporating crustal phases into WKBJ synthetic waveforms. After correction for physical dispersion due to intrinsic anelastic processes, we observe a residual traveltime dispersion on the order of 1–2 s in the period range of analysis. This dispersion occurs differently for S, ScS and SS, which is presumably related to their differing paths through the Earth. We find that: (1) Wavefront-healing phenomenon is observed for S and to a lesser extent SS waves having passed through very low velocity anomalies. (2) A preferred sampling of high velocity scatterers located at the CMB may explain our observation that ScS waves travel faster at low-frequency than at high-frequency. (3) A frequency-dependent attenuation , with , is compatible with the globally averaged dispersion observed for S waves. [ABSTRACT FROM AUTHOR]

The reversible jump algorithm is a statistical method for Bayesian inference with a variable number of unknowns. Here, we apply this method to the seismic tomography problem. The approach lets us consider the issue of model parametrization (i.e. the way of discretizing the velocity field) as part of the inversion process. The model is parametrized using Voronoi cells with mobile geometry and number. The size, position and shape of the cells defining the velocity model are directly determined by the data. The inverse problem is tackled within a Bayesian framework and explicit regularization of model parameters is not required. The mobile position and number of cells means that global damping procedures, controlled by an optimal regularization parameter, are avoided. Many velocity models with variable numbers of cells are generated via a transdimensional Markov chain and information is extracted from the ensemble as a whole. As an aid to interpretation we visualize the expected earth model that is obtained via Monte Carlo integration in a straightforward manner. The procedure is particularly adept at imaging rapid changes or discontinuities in wave speed. While each velocity model in the final ensemble consists of many discontinuities at cell boundaries, these are smoothed out in the averaged ensemble solution while those required by the data are reinforced. The ensemble of models can also be used to produce uncertainty estimates and experiments with synthetic data suggest that they represent actual uncertainty surprisingly well. We use the fast marching method in order to iteratively update the ray geometry and account for the non-linearity of the problem. The method is tested here with synthetic data in a 2-D application and compared with a subspace method that is a more standard matrix-based inversion scheme. Preliminary results illustrate the advantages of the reversible jump algorithm. A real data example is also shown where a tomographic image of Rayleigh wave group velocity for the Australian continent is constructed together with uncertainty estimates. [ABSTRACT FROM AUTHOR]

The capillary pressure in a reservoir determines the saturation distribution, and hence the total in situ volumes of fluids (oil/water/gas). The accurate knowledge of the capillary pressure distribution is one of the primary factors that may be decisive in the reliable estimation of hydrocarbon reserves. The centrifuge procedure provides laboratory data, which can be inverted to derive capillary pressure curves in laboratory conditions. The derived laboratory capillary pressure curves are then scaled up for full-field simulation of petroleum reservoirs. The inversion procedure is uncertain and gives nonunique capillary pressure curves. The standard industrial practice, however, has been to derive a single capillary pressure curve, and ignoring the uncertainty. Since the capillary pressure is uncertain, estimates of important reservoir parameters dependent on capillary pressure distribution are uncertain. This paper shows how the uncertainty in centrifuge capillary pressure can be quantified. It also shows how this uncertainty propagates through the scale-up process and impacts on the estimate of the oil recovery potential. We illustrate using a simple, two-phase (oil/water) synthetic reservoir model. [ABSTRACT FROM AUTHOR]