Mapping landforms on the Moon is of great interest and importance for future human settlements and resources exploration. One of the first steps is to map the topography and investigate their shape and geometry in great detail and resolution, which would provide the first conditions for assessing their suitability for future on-site analysis. However, data from the Lunar Orbiter Laser Altimeter (LOLA) provide low resolution elevation maps in comparison to the size of detailed geological features. To improve resolution, we developed an inverse method to upscale topographic maps to a higher resolution using photographic data from the Lunar Reconnaissance Orbiter Camera (LROC). The method, which exploits the relation between topographic gradients and degrees of shading of incoming sunlight, shows an improvement from ~60 metres per pixel to 0.9 metres per pixel, bringing it to the same resolution as the optical images from LROC. Our method can detect craters as small as ~3 metres of diameter and, if illumination from several angles are available, is potentially a way to remove shades from complex features such as caves. It is also possible to estimate the error of the model due to uncertainties in the albedo.

Markov Chain Monte Carlo (MCMC) sampling of solutions to large-scale inverse problems is, by many, regarded as being unfeasible due to the large number of model parameters. This statement, however, is only true if arbitrary, local proposal distributions are used. If we instead use a global proposal, informed by the physics of the problem, we can dramatically improve the performance of MCMC and even solve highly nonlinear inverse problems with vast model spaces. This is illustrated by a seismic full-waveform inverse problem in the acoustic approximation, involving close to 10^6 parameters.

Recent estimates of the crustal thickness of Mars show a bimodal result of either ∼20 km or ∼40 km beneath the InSight lander. We propose an approach based on random matrix theory applied to receiver functions to further constrain the subsurface structure. Assuming a spiked covariance model for our data, we first use the phase transition properties of the singular value spectrum of random matrices to detect coherent arrivals in the waveforms. Examples from terrestrial data show how the method works in different scenarios. We identify three new converted arrivals in the InSight data, including the second multiply reflected phase from a deeper third interface. We then use this information to jointly invert receiver functions with the absolute S-wave velocity information in the polarization of body waves. Results show a crustal thickness of 43±5 km beneath the lander with two mid-crustal interfaces at depths of 8.5±1.5 km and 22±3 km.

Since the ’60s of the last century, the calculation of the magnetic anomaly caused by 2D uniformly-polarized bodies with polygonal cross-section was mainly performed using the popular algorithm of Talwani & Heirtzler (1962, 1964). Recently, Kravchinsky et al. (2019) claimed errors in the above algorithm formulation, proposing new corrective formulas and questioning the effectiveness of almost sixty years of magnetic calculations. Here we showed, demonstrating analytical equivalence of the two approaches, that Kravchinsky et al.’s formulas simply represent an algebraic variant of those of Talwani & Heirtzler. Moreover, we performed intensive numerical analysis generating a large amount of random magnetic scenarios, involving both changing-shape polygons and a realistic geological model, showing a complete agreement among the magnetic responses of the two discussed algorithms and the one proposed by Won & Bevis (1987). Additionally, we released the source code of the algorithms in Julia and Python languages.

The estimation of crustal structure and thickness is instrumental in understanding the formation and evolution of terrestrial planets. Initial planetary missions with seismic instrumentation on board face the additional challenge of dealing with seismic activity levels that are only poorly constrained a priori. For example, the lack of plate tectonics on Mars leads to low seismicity which could in turn hinder the application of many terrestrial data analysis techniques. Here we propose using a joint inversion of receiver functions and apparent incidence angles, which contain information on absolute S-wave velocities of the subsurface. Since receiver function inversions suffer from a velocity depth trade-off, we in addition exploit a simple relation which defines apparent S-wave velocity as a function of observed apparent P-wave incidence angles to constrain the parameter space. We then use the Neighbourhood Algorithm for the inversion of a suitable joint objective function. The resulting ensemble of models is then used to derive uncertainty estimates for each model parameter. In preparation for analysis of data from the InSight mission, we show the application of our proposed method on Mars synthetics and sparse terrestrial data sets from different geological settings using both single and multiple events. We use information theoretic statistical tests as a model selection criteria and discuss their relevance and implications in a seismological framework.

In an ideal application of sequential simulation, parameters are simulated one at a time conditioned to all previously simulated parameters. This requires that marginal distributions of all dimensions (used to derive the conditional distributions) from the random field can be extracted and used for the simulation. However, in practice, only incomplete information from limited-size marginals is used for sequential simulation due to, e.g., computational unwieldiness or to ensure adequate pattern statistics. In this paper, we start out by addressing the problem of how to reconstruct an unknown random field that is consistent with known limited-size marginals. This problem turns out to be highly underdetermined (i.e., infinitely many solutions exist). Therefore, we describe possible additional constraints to supplement the marginals in order to reconstruct well-defined random fields. Secondly, we investigate which random field (out of infinitely many) that is sampled by sequential simulation algorithms using limited-size marginals. We show that sample distributions of such algorithms may depend on the sampling sequence and, sometimes, are inconsistent with the known marginals. We reviewed a formulation of a Markov random field that provides a well-defined solution to the underdetermined problem. Finally, we investigate the relation between marginal-size and information content of reconstructed random fields.

We developed a method for quantification of uncertainties on spatial data acquired at points with uncertain location coordinates. We applied it to well-log data from wells with uncertain trajectories. The method is fully probabilistic and requires few a priori assumptions on the Earth model and the wellbore geometry. We performed simulations of well trace and well-logging showing how the total uncertainty of well-log data vary according to the variability of subsurface properties, and how the reliability of well-log interpretations close to layer boundaries may be compromised.

Inversion of seismic data using information from horizontal wells is often is hampered by cumulative well- location errors. These errors can be significant and propagate to the well-log measurements in the borehole. To achieve a proper data integration and arrive at correct uncertainty estimates, we formulate the problem in a fully probabilistic framework and present a numerical approach for improving subsurface imaging using uncertain well-log data and their uncertain locations, as well as uncertain seismic data. The result is improved model error quantification in the seismic inversion process.