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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.

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.

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.

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.

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

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.

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.

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.

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.

The mean land-surface temperature represents an important boundary condition for many geothermal studies. This boundary is particularly important to help constrain the models made to analyse resource systems, many of which are shallow in nature and observe relatively small thermal gradients. Consequently, a mean land-surface temperature map of the Australian continent has been produced from 13 years of MODIS satellite imagery, for the period 2003–2015. The map shows good agreement with independent methods of estimating mean land-surface temperature, including borehole surface-temperature extrapolation and long-term, near-surface ground measurements. In comparison to previously used methods of estimating mean land-surface temperature, our new estimates are up to 12 °C warmer. The MODIS-based method presented in this study provides spatially continuous estimates of land-surface temperature that can be incorporated as the surface thermal boundary condition in geothermal studies. The method is also able to provide a quantification of the uncertainties expected in the application of these estimates for the purposes of thermal modelling.

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.

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 trans-dimensional 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.

It has been proposed that volcanic hotspots and the reconstructed eruption sites of large igneous provinces (LIPs) are preferentially located above the margins of two deep mantle large low shear-wave velocity provinces (LLSVPs), beneath the African continent and the Pacific Ocean. This spatial correlation has been interpreted to imply that LLSVPs represent long-lived, dense, stable thermo-chemical piles, which preferentially trigger mantle plumes at their edges and exert a strong influence on lower-mantle dynamics. Here, we re-analyse this spatial correlation, demonstrating that it is not global: it is strong for the African LLSVP, but weak for the Pacific. Moreover, Monte Carlo based statistical analyses indicate that the observed distribution of African and Pacific hotspots/reconstructed LIPs is consistent with the hypothesis that they are drawn from a sample that is uniformly distributed across the entire areal extent of each LLSVP: the stronger spatial correlation with the margin of the African LLSVP is expected as a simple consequence of its elongated geometry, where more than 75% of the LLSVP interior lies within 10° of its margin. Our results imply that the geographical distribution of hotspots and reconstructed LIPs does not indicate the extent to which chemical heterogeneity influences lower-mantle dynamics.

Seismic tomography is a powerful tool for illuminating Earth structure across a range of scales, but the usefulness of any image that is generated by this method is dependent on our ability to quantify its uncertainty. This uncertainty arises from the ill-posed nature of the tomographic inverse problem, which means that multiple models are capable of satisfying the data. The goal of this review is to provide an overview of the current state of the art in the assessment of uncertainty in seismic tomography, and issue a timely reminder that compared to the rapid advances made in many other areas of Earth imaging, uncertainty assessment remains underdeveloped and is often ignored or given minimal treatment in published studies. After providing a historical perspective that dates back to the pioneering work of the early 1970s, the factors that control solution nonuniqueness are discussed, which include data coverage, data noise, choice of parameterization, method used for data prediction and formulation of the inverse problem. This is followed by a description of common methods used to assess solution uncertainty and a commentary on their strengths and weaknesses. The final section of the review presents four case studies involving data sets from Australia and Europe that use different methods to assess uncertainty. The descriptive nature of this review, which does not contain detailed mathematical derivations, means that it is suitable for the many nonspecialists who make use of seismic tomography results but may not have a full appreciation of their reliability.

The major cause of sea-level change during ice ages is the exchange of water between ice and ocean and the planet's dynamic response to the changing surface load. Inversion of ∼1,000 observations for the past 35,000 y from localities far from former ice

Over the last 25 years, several studies have tested for a link between geomagnetic field intensity and reversal frequency. However, despite a large increase in the number of absolute paleointensity determinations, and improved methods for obtaining such data, two competing models have arisen. Here we employ a new tool for objectively analyzing paleomagnetic time series to investigate the possibility of a link between reversal frequency and paleointensity. Transdimensional Markov chain Monte Carlo techniques are applied to a quality-filtered version of the global paleointensity (PINT) database for the last 202 Myr to model long-term paleointensity behavior. A large ensemble of models is sampled, from which a final representative mean model is extracted. The resulting paleointensity model confirms published conclusions that the single-silicate crystal method gives significantly different results from more conventional whole rock paleointensity methods; this makes it difficult to jointly model the two data types in the same analysis. When the much larger whole rock data set is considered, a stable paleointensity of 5.46±0.28×1022 A/m2 for the last 202 Myr is consistent with the 95% confidence interval of the paleointensity model. Statistical tests indicate no significant correlation between reversal frequency and field intensity at the 0.05 level. However, this result is likely due to the characteristics of the PINT database rather than being a genuine, physically representative conclusion. Given the paucity of data and general state of the global paleointensity database, concerted efforts to increase the number of high-quality, well-dated paleointensity data are required before conclusions about a link between geomagnetic field intensity and reversal frequency can be confidently drawn. Key Points We investigate a link between geomagnetic field intensity and reversal frequency The global absolute paleointensity database is analyzed using Bayesian methods No evidence to

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.