Bayesian joint muographic and gravimetric inversion applied to volcanoes

Abstract

SUMMARY Gravimetry is a technique widely used to image the structure of the Earth. However, inversions are ill-posed and the imaging power of the technique rapidly decreases with depth. To overcome this limitation, muography, a new imaging technique relying on high energy atmospheric muons, has recently been developed. Because muography only provides integrated densities above the detector from a limited number of observation points, inversions are also ill-posed. Previous studies have shown that joint muographic and gravimetric inversions better reconstruct the 3-D density structure of volcanic edifices than independent density inversions. These studies address the ill-posedness of the joint problem by regularizing the solution with respect to a prior density model. However, the obtained solutions depend on some hyperparameters, which are either determined relative to a single test case or rely on ad-hoc parameters. This can lead to inaccurate retrieved models, sometimes associated with artefacts linked to the muon data acquisition. In this study, we use a synthetic example based on the Puy de Dôme volcano to determine a robust method to obtain the resulting model closest to the synthetic model and devoid of acquisition artefacts. We choose a Bayesian approach to include an a priori density model and a smoothing by a Gaussian spatial correlation function relying on two hyperparameters: an a priori density standard deviation and an isotropic spatial correlation length. This approach has the advantage to provide a posteriori standard deviations on the resulting densities. Using our synthetic volcano, we investigate the most reliable criterion to determine the hyperparameters. Our results suggest that k-fold Cross-Validation Sum of Squares and the Leave One Out methods are more robust criteria than the classically used L-curves. The determined hyperparameters allow to overcome the artefacts linked to the data acquisition geometry, even when only a limited number of muon telescopes is available. We also illustrate the behaviour of the inversion in case of offsets in the a priori density or in the data and show that they lead to recognizable structures that help identify them.

Joint inversion methods with relative density offset correction for muon tomography and gravity data, with application to volcano imaging

“…Some previous work has investigated joint inversion of those two data types through either synthetic (Davis & Oldenburg 2012;Jourde et al 2015;Barnoud et al 2019) or both synthetic and real data studies (Nishiyama et al 2014(Nishiyama et al , 2017Rosas-Carbajal et al 2017). Those studies demonstrated the potential improvements to be gained by jointly inverting muography and gravity data.…”

“…However, there has yet to be a thorough investigation of different numerical approaches for formulating the joint inverse problem to account for the fact that the two data sets respond to density through different mathematical response functions, which results in some generally unknown offset. The work of Davis & Oldenburg (2012), Jourde et al (2015), Nishiyama et al (2017) and Barnoud et al (2019) considered joint inversion methods that assumed both data sets were responsive to the same density quantity, which requires an accurate manual relative shifting of the two data sets prior to inversion, for example, as guided by prior information. Instead of relying on such prior information, Rosas-Carbajal et al (2017) included an additional scalar model parameter in the inversion representing the unknown relative density offset.…”

Robust Bayesian Joint Inversion of Gravimetric and Muographic Data for the Density Imaging of the Puy de Dôme Volcano (France)

“…Using this approach, the resulting 3D density models are free of artifacts linked to the acquisition geometry, even with a limited number of muographic view points. Besides, comparing the inversions of synthetic data with one and three muographic view points, Barnoud et al (2019) show that the resolution of the resulting density model is improved when using multiple points of view. Another difficulty in such a joint inversion is that the density estimated by muography is often lower than the density estimated by gravimetry due to the detection of 1) non-ballistic low-energy muons scattered by the volcanic edifice (Nishiyama et al, 2014a(Nishiyama et al, , 2016Gómez et al, 2017;Rosas-Carbajal et al, 2017), 2) muons coming from the backward direction (Jourde et al, 2013) and 3) other charged particles (Oláh and Varga, 2017;Saracino et al, 2017).…”

“…However, so far inversions have shown artifacts related to the muography acquisition geometry and to the limited number of muon detectors. Based on synthetic data, Barnoud et al (2019) designed a Bayesian inversion scheme where two regularization parameters, an a priori density standard deviation and a correlation length, can be determined in a robust way using a Cross-Validation Sum of Squares criterion, such as the Leave One Out (LOO). Using this approach, the resulting 3D density models are free of artifacts linked to the acquisition geometry, even with a limited number of muographic view points.…”

Muography and Its Potential Applications to Mining and Rock Engineering

“…Up till now muography has been applied to or tested in many fields. For example, muography has been used in the imaging of volcanoes (Nagamine et al 1995;Tanaka et al 2007Tanaka et al , 2009Tanaka et al , 2014Okubo and Tanaka 2012;Lesparre et al 2012;Marteau et al 2012Marteau et al , 2015Shinohara and Tanaka 2012;Carlôganu et al 2013;Tanaka and Yokoyama 2013;Nishiyama et al 2014;Ambrosino et al 2015b;Jourde et al 2016;Tioukov et al 2017;Noli et al 2017;Kaiser 2019;D'Alessandro et al 2019;Oláh et al 2019;Tanaka 2019;Barnoud et al 2019;Lelièvre et al 2019), in mining exploration (Schouten 2019), in the imaging of underground structures (Bonneville et al 2019;Saracino et al 2019), in archaeology and tunnel detection (Basset et al 2006;Menichelli et al 2007;Levy et al 1988;Celmins 1990;Caffau et al1997;Morishima et al 2017), in the monitoring of carbon capture storage sites (Kudryavtsev et al 2012;Jiang et al 2013;Klinger et al 2015;Gluyas et al 2019), in scanning old mining sites to detect the possible presence of unknown cavities (Baccani et al 2019;Mitrica et al 2019), in investigation of mineral deposits and rock density measurements…”