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We propose methods to efficiently explore the generalized nullspace of (non-linear) inverse problems, defined as the set of plausible models that explain observations within some misfit tolerance. Owing to the random nature of observational errors, the generalized nullspace is an inherently probabilistic entity, described by a joint probability density of tolerance values and model parameters. Our exploration methods rest on the construction of artificial Hamiltonian systems, where models are treated as high-dimensional particles moving along a trajectory through model space. In the special case where the distribution of misfit tolerances is Gaussian, the methods are identical to standard Hamiltonian Monte Carlo, revealing that its apparently meaningless momentum variable plays the intuitive role of a directional tolerance. Its direction points from the current towards a new acceptable model, and its magnitude is the corresponding misfit increase. We address the fundamental problem of producing independent plausible models within a high-dimensional generalized nullspace by autotuning the mass matrix of the Hamiltonian system. The approach rests on a factorized and sequentially preconditioned version of the L-BFGS method, which produces local Hessian approximations for use as a near-optimal mass matrix. An adaptive time stepping algorithm for the numerical solution of Hamilton's equations ensures both stability and reasonable acceptance rates of the generalized nullspace sampler. In addition to the basic method, we propose variations of it, where autotuning focuses either on the diagonal elements of the mass matrix or on the macroscopic (long-range) properties of the generalized nullspace distribution. We quantify the performance of our methods in a series of numerical experiments, involving analytical, high-dimensional, multimodal test functions. These are designed to mimic realistic inverse problems, where sensitivity to different model parameters varies widely, and where parameters tend to be correlated. The tests indicate that the effective sample size may increase by orders of magnitude when autotuning is used. Finally, we present a proof of principle of generalized nullspace exploration in viscoelastic full-waveform inversion. In this context, we demonstrate (1) the quantification of inter- and intraparameter trade-offs, (2) the flexibility to change model parametrization a posteriori, for instance, to adapt averaging length scales, (3) the ability to perform dehomogenization to retrieve plausible subwavelength models and (4) the extraction of a manageable number of alternative models, potentially located in distinct local minima of the misfit functional., Geophysical Journal International, 227 (2), ISSN:0956-540X, ISSN:1365-246X

SUMMARY Any search or sampling algorithm for solution of inverse problems needs guidance to be efficient. Many algorithms collect and apply information about the problem on the fly, and much improvement has been made in this way. However, as a consequence of the No-Free-Lunch Theorem, the only way we can ensure a significantly better performance of search and sampling algorithms is to build in as much external information about the problem as possible. In the special case of Markov Chain Monte Carlo (MCMC) sampling we review how this is done through the choice of proposal distribution, and we show how this way of adding more information about the problem can be made particularly efficient when based on an approximate physics model of the problem. A highly non-linear inverse scattering problem with a high-dimensional model space serves as an illustration of the gain of efficiency through this approach.

We present a method to explore the effective nullspace of nonlinear inverse problems without Monte Carlo sampling. This is based on the construction of an artificial Hamiltonian system where a model is treated as a high‐dimensional particle. Depending on its initial momentum and mass matrix, the particle evolves along a trajectory that traverses the effective nullspace, thereby producing a series of alternative models that are consistent with observations and their uncertainties. Variants of the nullspace shuttle enable hypothesis testing, for example, by adding features or by producing smoother or rougher models. Furthermore, the Hamiltonian nullspace shuttle can serve as a tunable hybrid between deterministic and probabilistic inversion methods: Choosing random initial momenta, it resembles Hamiltonian Monte Carlo; requiring misfits to decrease along a trajectory, it transforms into gradient descent. We illustrate the concept with a low‐dimensional toy example and with high‐dimensional nonlinear inversions of seismic traveltimes and magnetic data, respectively., Geophysical Research Letters, 46 (2), ISSN:0094-8276, ISSN:1944-8007

Inverse problems where relationships are linear arise in many fields of science and engineering and, consequently, algorithms for solving them are widespread. However, when the size of the problem increases, the computational challenge becomes huge, hence, unless some simplifying assumptions are yielded, it becomes impossible to solve the problem. Our study addresses these issues for large linear(izable) inverse problems when the uncertainties can be modeled as Gaussian and the forward relationship and covariance matrices can be expressed in terms of Kronecker products. Under these conditions, we illustrate an algorithm capable of addressing very large problems with very limited storage requirements and much faster than the traditional approach. The result is a complete characterization of the posterior distribution in a probabilistic sense in terms of mean and covariance. We extend this method also to nonlinear problems where a Gauss-Newton algorithm is employed. Applications to reflection seismology, magnetic anomaly inversion and image restoration are presented.

Calibration of the rock-frame elastic properties is an important step in establishing the rock-physics models for quantitative seismic reservoir characterization. While several empirical and theoretical rock-frame models have been proposed in the literature, their applicability to carbonate rocks is challenged due to their specific textural and varied pore type characteristics. Here, the suitability of an empirical model (Nur’s critical porosity), and a semi-empirical inclusion-based model (Keys-Xu) for a hard rock carbonate reservoir in south-west Iran is examined. Wireline log data in conjunction with deterministic and probabilistic optimization algorithms are employed to calibrate the free parameters in these models, as well as the minerals’ effective elastic moduli and density. The optimum values derived from the very fast simulated annealing algorithm were comparable to the estimated values from a probabilistic calibration workflow (Markov chain Monte Carlo sampling-based method). The uncertainty quantification via the probabilistic approach indicates higher uncertainty is associated with the calibrated clay elastic parameters (10%–13%) compared to the uncertainty associated with the calcite elastic moduli (1%–3%) in both rock-frame models. This could be attributed to the dominance of the calcite mineral and lack of variability in clay volume fraction in the interval of interest. The uncertainty associated with critical porosity and pore aspect ratio are 3% and 7%, respectively. The results of this case study show that simple rock frame models with only one free parameter (critical porosity or pore aspect ratio) could successfully model the elastic logs without resorting to more sophisticated rock-frame models.

We present the theory for and applications of Hamiltonian Monte Carlo (HMC) solutions of linear and nonlinear tomographic problems. HMC rests on the construction of an artificial Hamiltonian system where a model is treated as a high-dimensional particle moving along a trajectory in an extended model space. Using derivatives of the forward equations, HMC is able to make long-distance moves from the current towards a new independent model, thereby promoting model independence, while maintaining high acceptance rates. Following a brief introduction to HMC using common geophysical terminology, we study linear (tomographic) problems. Though these may not be the main target of Monte Carlo methods, they provide valuable insight into the geometry and the tuning of HMC, including the design of suitable mass matrices and the length of Hamiltonian trajectories. This is complemented by a self-contained proof of the HMC algorithm in Appendix A. A series of tomographic/imaging examples is intended to illustrate (i) different variants of HMC, such as constrained and tempered sampling, (ii) the independence of samples produced by the HMC algorithm and (iii) the effects of tuning on the number of samples required to achieve practically useful convergence. Most importantly, we demonstrate the combination of HMC with adjoint techniques. This allows us to solve a fully nonlinear, probabilistic traveltime tomography with several thousand unknowns on a standard laptop computer, without any need for supercomputing resources., Geophysical Journal International, 216 (2), ISSN:0956-540X, ISSN:1365-246X

We present a new methodology for inverting P--to--S receiver function (RF) waveforms directly for mantle temperature and composition. This is achieved by interfacing the geophysical inversion with self-consistent mineral phase equilibria calculations from which rock mineralogy and its elastic properties are predicted as a function of pressure, temperature, and bulk composition. This approach anchors temperatures, composition, seismic properties, and discontinuities that are in mineral physics data, while permitting the simultaneous use of geophysical inverse methods to optimize models of seismic properties to match RF waveforms. Resultant estimates of transition zone (TZ) topography and volumetric seismic velocities are independent of tomographic models usually required for correcting for upper mantle structure. We considered two end-member compositional models: the equilibrated Equilibrium Assemblage (EA) and the disequilibrated Mechanical Mixture (MM) models. Thermal variations were found to influence arrival times of computed RF waveforms, whereas compositional variations affected amplitudes of waves converted at the TZ discontinuities. The robustness of the inversion strategy was tested by performing a set of synthetic inversions in which crustal structure was assumed both fixed and variable. These tests indicate that unaccounted-for crustal structure strongly affects the retrieval of mantle properties, calling for a two-step strategy presented herein to simultaneously recover both crustal and mantle parameters. As a proof of concept, the methodology is applied to data from two stations located in the Siberian and East European continental platforms. ISSN:2169-9313 ISSN:0148-0227 ISSN:2169-9356

A realistic noise model is essential for trustworthy inversion of geophysical data. Sometimes, as in case of seismic data, quantification of the noise model is non-trivial. To remedy this, a hierarchical Bayes approach can be adopted in which properties of the noise model, such as the amplitude of an assumed uncorrelated Gaussian noise model, can be inferred as part of the inversion. Here we demonstrate how such an approach can lead to substantial overfitting of noise when inverting a 1D reflection seismic NMO data set. We then argue that usually the noise model is correlated, and suggest to infer the amplitude of a correlated Gaussian noise model. This provides better results than assuming an uncorrelated model. In general though, the results suggest that care should be taken using the hierarchical Bayes approach to infer the noise model.

By thoroughly revisiting the classic human action recognition paradigm, we analyzed different training/testing strategies, discovering that standard (cross-validating) testing strategies are not always the suitable validation procedures to assess an algorithm’s performance. As a consequence, we design a novel action recognition architecture, applying a “personalized” strategy to learn how any subject performs any action. We discover that it is advantageous to customize (i.e., personalize) the method to learn the actions carried out by each subject, rather than trying to generalize the actions executions across subjects. Leveraging on that, we propose an action recognition framework consisting of a two-stage classification approach where, given a test action, the subject is first identified before the actual recognition of the action takes place. Despite the basic, off-the-shelf descriptors and standard classifiers adopted, we score a favorable performance with respect to the state-of-the-art as to certify the soundness of our approach.

International audience

Geophysical Journal International, 192 (1), ISSN:0956-540X, ISSN:1365-246X

New aeromagnetic data image the extent and spatial distribution of Cenozoic magmatism and older basement features over the Admiralty Block of the Transantarctic Mountains. Digital enhancement techniques image magmatic and tectonic features spanning in age from the Cambrian to the Neogene. Magnetic lineaments trace major fault zones, including NNW to NNE trending transtensional fault systems that appear to control the emplacement of Neogene age McMurdo volcanics. These faults represent splays from a major NW–SE oriented Cenozoic strike-slip fault belt, which reactivated the inherited early Paleozoic structural architecture. NE–SW oriented magnetic lineaments are also typical of the Admiralty Block and reflect post-Miocene age extensional faults. To re-investigate controversial relationships between strike-slip faulting, rifting, and Cenozoic magmatism, we combined the new aeromagnetic data with previous datasets over the Transantarctic Mountains and Ross Sea Rift. Two key observations can be made from our aeromagnetic compilation: 1) Cenozoic alkaline intrusions along the margin of the Ross Sea Rift lie oblique to the NW–SE strike-slip faults and are not significantly displaced by them; 2) the Southern Cross and the Admiralty Blocks are much more significantly affected by Cenozoic magmatism compared to the adjacent tectonic blocks, thereby indicating major tectono-magmatic segmentation of the Transantarctic Mountains rift flank. We put forward three alternative tectonic models to explain the puzzling observation that major Cenozoic alkaline intrusions emplaced along the Ross Sea Rift margin show no evidence for major strike-slip displacement. Our first model predicts that the alkaline intrusions were emplaced along left-lateral cross-faults, which accommodated distributed right-lateral shearing. In contrast, our second model does not require major distributed strike-slip shearing, and relates the emplacement of Cenozoic alkaline intrusions to sea-floor spreading in the Adare Basin, coupled with intracontinental transfer faulting. The third model is an attempt to reconcile the two opposing hypothesis and relies on a recent inference, which postulates that opening of the Adare Basin relates to fault splaying from the Balleny strike-slip fault zone. A low seismic velocity anomaly in the upper mantle appears to extend from the Ross Sea Rift under the Admiralty and Southern Cross Blocks of the Transantarctic Mountains. Lateral flow of hot upper mantle from the rifted region to the rift flank may explain the observed tectono-magmatic segmentation of the Transantarctic Mountains. We infer that this process caused a regional upwarp of the Curie isotherm under the Admiralty and Southern Cross Blocks of the Transantarctic Mountains, and facilitated extensional faulting, renewed uplift, and volcanism in the Neogene.