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

Imaging the internal structure of volcanoes helps highlighting magma pathways and monitoring potential structural weaknesses. We jointly invert gravimetric and muographic data to determine the most precise image of the 3D density structure of the Puy de Dôme volcano (Chaîne des Puys, France) ever obtained. With rock thickness of up to 1,600 m along the muon lines of sight, it is, to our knowledge, the largest volcano ever imaged by combining muography and gravimetry. The inversion of gravimetric data is an ill-posed problem with a non-unique solution and a sensitivity rapidly decreasing with depth. Muography has the potential to constrain the absolute density of the studied structures but the use of the method is limited by the possible number of acquisition view points, by the long acquisition duration and by the noise contained in the data. To take advantage of both types of data in a joint inversion scheme, we develop a robust method adapted to the specificities of both the gravimetric and muographic data. Our method is based on a Bayesian formalism. It includes a smoothing relying on two regularization parameters (an a priori density standard deviation and an isotropic correlation length) which are automatically determined using a leave one out criterion. This smoothing overcomes artifacts linked to the data acquisition geometry of each dataset. A possible constant density offset between both datasets is also determined by least-squares. The potential of the method is shown using the Puy de Dôme volcano as case study as high quality gravimetric and muographic data are both available. Our results show that the dome is dry and permeable. Thanks to the muographic data, we better delineate a trachytic dense core surrounded by a less dense talus.

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Bayesian joint inversion of muographic and gravimetric data for the 3D imaging of volcanoes, case study of the Puy de Dôme

10.5194/egusphere-egu2020-13978 ◽

2020 ◽

Author(s):

Anne Barnoud ◽

Valérie Cayol ◽

Peter Lelièvre ◽

Valentin Niess ◽

Cristina Cârloganu ◽

...

Keyword(s):

Density Distribution ◽

Joint Inversion ◽

A Priori ◽

High Energy ◽

Quality Data ◽

Constant Density ◽

Density Structure ◽

Spatial Covariance ◽

Massif Central ◽

Gravimetric Data

We present a method to jointly invert muographic and gravimetric data to infer the 3D density structure of volcanoes.Muography and gravimetry are two independent methods that are sensitive to the density distribution. The gravimetric inversion allows to reconstruct the 3D density variations but the process is well-known to be ill-posed leading to non unique solutions. Muography provides 2D images of mean densities from the detection of high energy atmospheric muons crossing the volcanic edifice. Several muographic images can be used to reconstruct the 3D density distribution but the number of imagdes is generally limited by instrumentation and field contstraints.The joint inversion of muographic and gravimetric data aims at reconstructing the 3D density structure of an edifice, benefiting from the advantages of both methods. We developed a robust inversion scheme based on a Bayesian formalism. This approach takes into account the data errors and a priori information on the density distribution with a spatial covariance so that smooth models are obtained. The a priori density standard deviation and the spatial correlation length are the two hyperparameters that tune the regularization, hence that control the inversion result. The optimal set of hyperparameters is determined in a systematic way using Leave One Out (LOO) and Cross Validation Sum of Squares (CVSS) criteria (Barnoud et al., GJI 2019). The method also allows to automatically determine a constant density offset between gravimetry and muography to overcome a potential bias in the measurements (Leli&#232;vre et al., GJI 2019).The case of the Puy de D&#244;me volcano (French Massif Central) is studied as proof of principle as high quality data are available for both muography (Le M&#233;n&#233;deu et al., EGU 2016; C&#226;rloganu et al., EGU 2018) and gravimetry (Portal et al., JVGR 2016). We develop and validate the method using synthetic data computed from a model based on the Puy de D&#244;me topography and acquisition geometry, as well as on real data.

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Bayesian joint muographic and gravimetric inversion applied to volcanoes

Geophysical Journal International ◽

10.1093/gji/ggz300 ◽

2019 ◽

Vol 218 (3) ◽

pp. 2179-2194 ◽

Cited By ~ 5

Author(s):

Anne Barnoud ◽

Valérie Cayol ◽

Valentin Niess ◽

Cristina Cârloganu ◽

Peter Lelièvre ◽

...

Keyword(s):

Data Acquisition ◽

Spatial Correlation ◽

Ad Hoc ◽

A Priori ◽

High Energy ◽

Density Model ◽

Test Case ◽

Spatial Correlation Function ◽

Ill Posed ◽

Leave One Out

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.

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An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation

Mathematical Problems in Engineering ◽

10.1155/2020/5865971 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Songshu Liu ◽

Lixin Feng

Keyword(s):

Kernel Method ◽

A Priori ◽

Diffusion Problem ◽

Two Dimensional ◽

A Posteriori ◽

Numerical Examples ◽

Regularization Parameters ◽

Ill Posed ◽

Convergence Estimates ◽

Inverse Diffusion

In this paper, we consider a two-dimensional (2D) time-fractional inverse diffusion problem which is severely ill-posed; i.e., the solution (if it exists) does not depend continuously on the data. A modified kernel method is presented for approximating the solution of this problem, and the convergence estimates are obtained based on both a priori choice and a posteriori choice of regularization parameters. The numerical examples illustrate the behavior of the proposed method.

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3-D Joint Inversion of Magnetic and Gravimetric Data With A Priori Information

Geophysical Journal International ◽

10.1111/j.1365-246x.1993.tb01452.x ◽

1993 ◽

Vol 112 (2) ◽

pp. 244-256 ◽

Cited By ~ 28

Author(s):

Hermann Zeyen ◽

Jaume Pous

Keyword(s):

Joint Inversion ◽

A Priori ◽

A Priori Information ◽

Gravimetric Data ◽

Priori Information

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Simultaneous Joint Inversion of Gravity and Self-Potential Data Measured Along Profile: Theory, Numerical Examples, and a Case Study From Mineral Exploration With Cross Validation From Electromagnetic Data

IEEE Transactions on Geoscience and Remote Sensing ◽

10.1109/tgrs.2021.3071973 ◽

2021 ◽

pp. 1-20

Author(s):

Salah A. Mehanee

Keyword(s):

Cross Validation ◽

Joint Inversion ◽

Mineral Exploration ◽

Numerical Examples ◽

Electromagnetic Data ◽

Self Potential

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Complex wavefront sensing based on coherent diffraction imaging using vortex modulation

Scientific Reports ◽

10.1038/s41598-021-88523-x ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Rujia Li ◽

Liangcai Cao

Keyword(s):

Phase Retrieval ◽

Dynamic Range ◽

A Priori ◽

Measured Intensity ◽

Physical Constraints ◽

Coherent Diffraction Imaging ◽

Effective Dynamic ◽

Ill Posed ◽

Retrieval Problem ◽

Diffraction Imaging

AbstractPhase retrieval seeks to reconstruct the phase from the measured intensity, which is an ill-posed problem. A phase retrieval problem can be solved with physical constraints by modulating the investigated complex wavefront. Orbital angular momentum has been recently employed as a type of reliable modulation. The topological charge l is robust during propagation when there is atmospheric turbulence. In this work, topological modulation is used to solve the phase retrieval problem. Topological modulation offers an effective dynamic range of intensity constraints for reconstruction. The maximum intensity value of the spectrum is reduced by a factor of 173 under topological modulation when l is 50. The phase is iteratively reconstructed without a priori knowledge. The stagnation problem during the iteration can be avoided using multiple topological modulations.

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Is There a Justification for Differential a Priori Weighting in Coding Sequences? A Case Study from rbcL and Apocynaceae s.l.

Systematic Biology ◽

10.1080/10635150050207410 ◽

2000 ◽

Vol 49 (1) ◽

pp. 101-113 ◽

Cited By ~ 35

Author(s):

Bengt Sennblad ◽

Birgitta Bremer

Keyword(s):

A Priori ◽

Coding Sequences

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Regularization of backward time-fractional parabolic equations by Sobolev-type equations

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0062 ◽

2020 ◽

Vol 28 (5) ◽

pp. 659-676

Author(s):

Dinh Nho Hào ◽

Nguyen Van Duc ◽

Nguyen Van Thang ◽

Nguyen Trung Thành

Keyword(s):

Error Estimates ◽

Parabolic Equations ◽

Regularization Parameter ◽

A Priori ◽

Sobolev Type ◽

A Posteriori ◽

Parameter Choice ◽

Numerical Tests ◽

Ill Posed ◽

Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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Magnetotelluric static shift: Estimation and removal using the cokriging method

Geophysics ◽

10.1190/1.2400625 ◽

2007 ◽

Vol 72 (1) ◽

pp. F25-F34 ◽

Cited By ~ 11

Author(s):

Benoit Tournerie ◽

Michel Chouteau ◽

Denis Marcotte

Keyword(s):

Apparent Resistivity ◽

A Priori ◽

Shift Factor ◽

Static Shift ◽

Geostatistical Model ◽

Data Set ◽

Resistivity Sounding ◽

The Difference ◽

Cross Variogram

We present and test a new method to correct for the static shift affecting magnetotelluric (MT) apparent resistivity sounding curves. We use geostatistical analysis of apparent resistivity and phase data for selected periods. For each period, we first estimate and model the experimental variograms and cross variogram between phase and apparent resistivity. We then use the geostatistical model to estimate, by cokriging, the corrected apparent resistivities using the measured phases and apparent resistivities. The static shift factor is obtained as the difference between the logarithm of the corrected and measured apparent resistivities. We retain as final static shift estimates the ones for the period displaying the best correlation with the estimates at all periods. We present a 3D synthetic case study showing that the static shift is retrieved quite precisely when the static shift factors are uniformly distributed around zero. If the static shift distribution has a nonzero mean, we obtained best results when an apparent resistivity data subset can be identified a priori as unaffected by static shift and cokriging is done using only this subset. The method has been successfully tested on the synthetic COPROD-2S2 2D MT data set and on a 3D-survey data set from Las Cañadas Caldera (Tenerife, Canary Islands) severely affected by static shift.

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