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Aluminosilicate glasses and melts are of paramount importance for geo- and materials sciences. They include most magmas, and are used to produce a wide variety of everyday materials, from windows to smartphone displays. Despite this importance, no general model exists with which to predict the atomic structure, thermodynamic and viscous properties of aluminosilicate melts. To address this, we introduce a deep learning framework, 'i-Melt', which combines a deep artificial neural network with thermodynamic equations. It is trained to predict 18 different latent and observed properties of melts and glasses in the K 2 O-Na 2 O-Al 2 O 3 -SiO 2 system, including configurational entropy, viscosity, optical refractive index, density, and Raman signals. Viscosity can be predicted in the 100–1015 log 10 Pa·s range using five different theoretical frameworks (Adam-Gibbs, Free Volume, MYEGA, VFT, Avramov-Milchev), with a precision equal to, or better than, 0.4 log 10 Pa·s on unseen data. Density and optical refractive index (through the Sellmeier equation) can be predicted with errors equal or lower than 0.02 and 0.006, respectively. Raman spectra for K 2 O-Na 2 O-Al 2 O 3 -SiO 2 glasses are also predicted, with a relatively high mean error of ∼25% due to the limited data set available for training. Latent variables can also be predicted with good precisions. For example, the glass transition temperature, T g , can be predicted to within 19 K, while the melt configurational entropy at the glass transition, Sconf(T g) , can be predicted to within 0.8 J mol−1 K−1. Applied to rhyolite compositions, i-Melt shows that the rheological threshold separating explosive and effusive eruptions correlates with an increase in the fraction of non-bridging oxygens in rhyolite melts as their alkali/Al ratio becomes larger than 1. Exploring further the effect of the K/(K + Na) ratio on the properties of alkali aluminosilicate melts with compositions varying along a simplified alkali magmatic series trend, we observe that K-rich melts have systematically different structures and higher viscosities compared to Na-rich melts. Combined with the effects of the K/(K + Na) ratio on other parameters, such as the solubility, solution mechanisms and speciation of volatile elements, this could ultimately influence the eruptive dynamics of volcanic systems emitting Na-rich or K-rich alkali magmas. [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]

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]

Glacial isostatic adjustment (GIA) modeling is not only useful for understanding past relative sea‐level change but also for projecting future sea‐level change due to ongoing land deformation. However, GIA model predictions are subject to a range of uncertainties, most notably due to uncertainty in the input ice history. An effective way to reduce this uncertainty is to perform data‐model comparisons over a large ensemble of possible ice histories, but this is often impossible due to computational limitations. Here we address this problem by building a deep‐learning‐based GIA emulator that can mimic the behavior of a physics‐based GIA model while being computationally cheap to evaluate. Assuming a single 1‐D Earth rheology, our emulator shows 0.54 m mean absolute error on 150 out‐of‐sample testing data with <0.5 s emulation time. Using this emulator, two illustrative applications related to the calculation of barystatic sea level are provided for use by the sea‐level community. Plain Language Summary: Piecing together the history of ice sheet change during past glacial cycles is not only important for understanding past sea‐level change but also for predicting how ongoing glacial rebound contributes to future sea‐level change. Traditionally, a physics‐based "sea‐level model" is used to predict the sea‐level change associated with a particular reconstruction of past ice sheet change and compare the results with geological records of past sea level. However, a fundamental limitation of this approach is the need to compute sea‐level change for a large number of plausible ice histories, which is often prohibited by the computational resources required to repeatedly solve the complex physical equations. In this paper, we describe a machine‐learning‐based statistical model that can mimic the behavior of a physics‐based sea‐level model. This statistical model is computationally cheap and we demonstrate that it is able to accurately predict global sea‐level change for a suite of 150 "unseen" ice histories. Our statistical model predicts sea‐level change 100–1,000 times faster than a physics‐based model, making it an ideal tool for investigating and improving our understanding of global ice sheet change. Key Points: The first attempt to build a deep‐learning based Glacial isostatic adjustment (GIA) emulator that can accurately predict global sea‐level change based on a given ice modelThis emulator (GEORGIA) can predict global sea‐level change history within 0.5 s with minor emulation errorThis GIA emulator along with two illustrative applications are available for use by the wider sea‐level community [ABSTRACT FROM AUTHOR]

The results of mantle convection simulations are fully determined by the input parameters and boundary conditions used. These input parameters can be for initialisation, such as initial mantle temperature, or can be constant values, such as viscosity exponents. However, knowledge of Earth-like values for many input parameters are very poorly constrained, introducing large uncertainties into the simulation of mantle flow. Convection is highly non-linear, therefore linearised inversion methods cannot be used to recover past configurations over more than very short periods of time, which makes finding both initial and constant simulation input parameters very difficult. In this paper, we demonstrate a new method for making inferences about simulation input parameters from observations of the mantle temperature field after billions of years of convection. The method is fully probabilistic. We use prior sampling to construct probability density functions for convection simulation input parameters, which are represented using neural networks. Assuming smoothness, we need relatively few samples to make inferences, making this approach much more computationally tractable than other probabilistic inversion methods. As a proof of concept, we show that our method can invert the amplitude spectra of temperature fields from 2D convection simulations, to constrain yield stress, surface reference viscosity and the initial thickness of primordial material at the CMB, for our synthetic test cases. The best constrained parameter is yield stress. The reference viscosity and initial thickness of primordial material can also be inferred reasonably well after several billion years of convection. [ABSTRACT FROM AUTHOR]

Owing to the increasing availability of computational resources, in recent years the probabilistic solution of non-linear, geophysical inverse problems by means of sampling methods has become increasingly feasible. Nevertheless, we still face situations in which a Monte Carlo approach is not practical. This is particularly true in cases where the evaluation of the forward problem is computationally intensive or where inversions have to be carried out repeatedly or in a timely manner, as in natural hazards monitoring tasks such as earthquake early warning. Here, we present an alternative to Monte Carlo sampling, in which inferences are entirely based on a set of prior samples--that is, samples that have been obtained independent of a particular observed datum. This has the advantage that the computationally expensive sampling stage becomes separated from the inversion stage, and the set of prior samples--once obtained--can be reused for repeated evaluations of the inverse mapping without additional computational effort. This property is useful if the problem is such that repeated inversions of independent data have to be carried out. We formulate the inverse problem in a Bayesian framework and present a practical way to make posterior inferences based on a set of prior samples. We compare the prior sampling based approach to a Markov Chain Monte Carlo approach that samples from the posterior probability distribution. We show results for both a toy example, and a realistic seismological source parameter estimation problem. We find that the posterior uncertainty estimates obtained based on prior sampling can be considered conservative estimates of the uncertainties obtained by directly sampling from the posterior distribution. [ABSTRACT FROM AUTHOR]

Whenever a geophysical image is to be constructed, a variety of choices must be made. Some, such as those governing data selection and processing, or model parametrization, are somewhat arbitrary: there may be little reason to prefer one choice over another. Others, such as defining the theoretical framework within which the data are to be explained, may be more straightforward: typically, an 'exact' theory exists, but various approximations may need to be adopted in order to make the imaging problem computationally tractable. Differences between any two images of the same system can be explained in terms of differences between these choices. Understanding the impact of each particular decision is essential if images are to be interpreted properly—but little progress has been made towards a quantitative treatment of this effect. In this paper, we consider a general linearized inverse problem, applicable to a wide range of imaging situations. We write down an expression for the difference between two images produced using similar inversion strategies, but where different choices have been made. This provides a framework within which inversion algorithms may be analysed, and allows us to consider how image effects may arise. In this paper, we take a general view, and do not specialize our discussion to any specific imaging problem or setup (beyond the restrictions implied by the use of linearized inversion techniques). In particular, we look at the concept of 'hybrid inversion', in which highly accurate synthetic data (typically the result of an expensive numerical simulation) is combined with an inverse operator constructed based on theoretical approximations. It is generally supposed that this offers the benefits of using the more complete theory, without the full computational costs.We argue that the inverse operator is as important as the forward calculation in determining the accuracy of results. We illustrate this using a simple example, based on imaging the density structure of a vibrating string. [ABSTRACT FROM AUTHOR]

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

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

We apply Hamilton's principle to obtain the exact equations of motion for an elastic planet that is rotating, self-gravitating and comprises both fluid and solid regions. This variational problem is complicated by the occurrence of tangential slip at fluid--solid boundaries, but we show how this can be accommodated both directly and using the method of Lagrange multipliers. A novelty of our approach is that the planet's motion is described relative to an arbitrary reference configuration, with this generality offering advantages for numerical calculations. In particular, aspherical topography on the free surface or internal boundaries of the planet's equilibrium configuration can be converted exactly into effective volumetric heterogeneities within a geometrically spherical reference body by applying a suitable particle relabelling transformation. The theory is then specialized to consider the linearized motion of a planet about a steadily rotating equilibrium configuration, with these results having applications to normal mode coupling calculations used within studies of long-period seismology, tidal deformation and related fields. In particular, we explain how our new theory will, for the first time, allow aspherical boundary topography to be incorporated exactly within such coupling calculations. [ABSTRACT FROM AUTHOR]