We discuss a backward Monte-Carlo technique for muon transport problem, with emphasis on its application in muography. Backward Monte-Carlo allows exclusive sampling of a final state by reversing the simulation flow. In practice it can be made analogous to an adjoint Monte-Carlo, though it is more versatile for muon transport. A backward Monte-Carlo was implemented as a dedicated muon transport library: PUMAS. It is shown for case studies relevant for muography imaging that the implementations of forward and backward Monte-Carlo schemes agree to better than 1 %.

Sea level has been routinely measured by satellite altimetry for nearly three decades, while, for about 15 years, Argo floats and GRACE (Gravity Recovery and Climate Experiment; Tapley et al., 2019) and GRACE Follow-On (GRACE-FO; Landerer et al., 2020) allow quantifications of the steric effect and ocean mass change respectively, the two main components of the global mean sea level (GMSL) budget. In recent years, numerous studies have been devoted to assess the GMSL budget, that is, comparing the altimetry-based GMSL time series with the sum of components (e.g.,

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.

This study analyzes the interannual variability of the water mass transport measured by satellite gravity missions in regard to eight major climate modes known to influence the Earth’s climate from regional to global scales. Using sparsity promoting techniques (i.e., LASSO), we automatically select the most relevant predictors of the climate variability among the eight candidates considered. The El Niño–Southern Oscillation, Southern Annular Mode and Arctic Oscillation are shown to account for a large part the interannual variability of the water mass transport observed in extratropical ocean basins (up to 40%) and shallow seas (up to 70%). A combination of three Pacific and one Atlantic modes is needed to account for most (up to 60%) of the interannual variability of the terrestrial water storage observed in the North Amazon, Parana and Zambezi basins. With our technique, the impact of climate modes on water mass changes can be tracked across distinct water reservoirs (oceans, continents and ice-covered regions) and we show that a combination of climate modes is necessary to explain at best the natural variability in water mass transport. The climate modes predictions based on LASSO inversions can be used to reduce the inter-annual variability in satellite gravity measurements and detect processes unrelated with the natural variability of climate but with similar spatio-temporal signatures. However, significant residuals in the satellite gravity measurements remain unexplained at inter-annual time scales and more complex models solving the water mass balance should be employed to better predict the variability of water mass distributions.

Abstract. The Earth energy imbalance (EEI) at the top of the atmosphere is responsible for the accumulation of heat in the climate system. Monitoring the EEI is therefore necessary to better understand the Earth's warming climate. Measuring the EEI is challenging as it is a globally integrated variable whose variations are small (0.5–1 W m−2) compared to the amount of energy entering and leaving the climate system (∼340 W m−2). Since the ocean absorbs more than 90 % of the excess energy stored by the Earth system, estimating the ocean heat content (OHC) change provides an accurate proxy of the EEI. This study provides a space geodetic estimation of the OHC changes at global and regional scales based on the combination of space altimetry and space gravimetry measurements. From this estimate, the global variations in the EEI are derived with realistic estimates of its uncertainty. The mean EEI value is estimated at +0.74±0.22 W m−2 (90 % confidence level) between August 2002 and August 2016. Comparisons against estimates based on Argo data and on CERES measurements show good agreement within the error bars of the global mean and the time variations in EEI. Further improvements are needed to reduce uncertainties and to improve the time series, especially at interannual timescales. The space geodetic OHC-EEI product (version 2.1) is freely available at https://doi.org/10.24400/527896/a01-2020.003 (Magellium/LEGOS, 2020).

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is a relatively new geophysical imaging method that uses muons to provide estimates of average densities along particular lines of sight. Muography can only see above the horizontal elevation of the detector and it is therefore attractive to attempt a joint inversion of muography data with gravity data, which is also responsive to density but generally requires combination with another geophysical data set to overcome issues related to non-uniqueness and poor depth resolution. Some previous work has investigated this joint inverse problem and 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. Particularly important is how to account for the fact that even though the two data types are sensitive to the same physical quantity, density, they respond through different response functions. Moreover, the two measurements are affected by different systematic uncertainties that are difficult to model. In this work, we considered an approximation where the two density quantities, inferred from the two data types, can be related by an unknown scalar offset. We considered various existing and new joint inversion methods that might solve this problem and we applied them to a synthetic volcano imaging scenario based on the Puy de Dôme volcano in the Central Massif region of France. We used unstructured meshes in our modelling to adequately honour the significant topography in that scenario. Our experiments indicated that the most successful joint inversion method for this type of geological scenario was one in which the data misfit function is reformulated to automatically determine the best-fitting offset following a least-squares minimization argument. However, other approaches showed merit and we suggest several of the investigated methods be applied and compared for any specific joint inversion scenario.

TURTLE is a C library providing utilities allowing to navigate through a topography described by a Digital Elevation Model (DEM). The library has been primarily designed for the Monte Carlo transport of particles scattering over medium to long ranges, e.g. atmospheric muons. But, it can also efficiently handle ray tracing problems with very large DEMs (10 9 nodes or more), e.g. for neutrino simulations. The TURTLE library was built on an optimistic ray tracing algorithm, detailed in the present paper. This algorithm proceeds by trials and errors, approximating the topography within the modelling uncertainties of the DEM data. This allows to traverse a topography in constant time, i.e. independently of the number of grid nodes, and with no added memory. Detailed performance studies are provided by comparison to other ray tracing algorithms and as an application to muon transport in a Monte Carlo simulation.