¹Daniela Rommel, ¹Arne Grumpe, ¹Marian Patrik Felder, ¹Christian Wöhler, ²Urs Mall,
³Andreas Kronz
Icarus (in Press) Link to Article [http://dx.doi.org/10.1016/j.icarus.2016.10.029]
¹Image Analysis Group, TU Dortmund University, Otto-Hahn-Str. 4, D–44227 Dortmund, Germany
²Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, D–37077 Göttingen, Germany
³Geowissenschaftliches Zentrum Göttingen, Goldschmidtstr. 1, D–37077 Göttingen, Germany
Copyright Elsevier

While the interpretation of spectral reflectance data has been widely applied to detect the presence of minerals, determining and quantifying the abundances of minerals contained by planetary surfaces is still an open problem. With this paper we address one of the two main questions arising from the spectral unmixing problem. While the mathematical mixture model has been extensively researched, considerably less work has been committed to the selection of endmembers from a possibly huge database or catalog of potential endmembers. To solve the endmember selection problem we define a new spectral similarity measure that is not purely based on the reconstruction error, i.e. the squared difference between the modeled and the measured reflectance spectrum. To select reasonable endmembers, we extend the similarity measure by adding information extracted from the spectral absorption bands. This will allow for a better separation of spectrally similar minerals. Evaluating all possible subsets of a possibly very large catalog that contain at least one endmember leads to an exponential increase in computational complexity, rendering catalogs of 20–30 endmembers impractical. To overcome this computational limitation, we propose the usage of a genetic algorithm that, while initially starting with random subsets, forms new subsets by combining the best subsets and, to some extent, does a local search around the best subsets by randomly adding a few endmembers. A Monte-Carlo simulation based on synthetic mixtures and a catalog size varying from three to eight endmembers demonstrates that the genetic algorithm is expected to require less combinations to be evaluated than an exhaustive search if the catalog comprises 10 or more endmembers. Since the genetic algorithm evaluates some combinations multiple times, we propose a simple modification and store previously evaluated endmember combinations. The resulting algorithm is shown to never require more function evaluations than a full exhaustive search and the number of required function evaluations appears to grow less than exponentially. It thus requires considerably less time than an exhaustive search because the number of function evaluations is a hardware independent measure of the computational complexity. To evaluate the spectral similarity measure, we created a spectral reflectance catalog of selected lunar analog minerals. Based on precisely prepared mixtures of two to three components, we show that the proposed spectral similarity measure selects less false endmembers from the catalog than a similarity measure that is purely based on the reconstruction error.

¹Olivier Verhoeven, ¹Pierre Vacher
Planetary and Space Science (in Press) Link to Article [http://dx.doi.org/10.1016/j.pss.2016.10.005]
¹Laboratoire de Planétologie et Géodynamique, UMR-CNRS 6112, Université de Nantes, 2 rue de la Houssinière 44322 Nantes, France

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^1,2,3Rebecca A. Fischer, ¹Andrew J. Campbell, ¹Fred J. Ciesla
Earth and Planetary Science Letters (in Press) Link to Article [http://dx.doi.org/10.1016/j.epsl.2016.10.025]
¹University of Chicago, Department of the Geophysical Sciences, 5734 S Ellis Ave, Chicago, IL 60637, USA
²National Museum of Natural History, Smithsonian Institution, PO Box 37012, MRC 119, Washington, DC 20013-7012, USA
³University of California Santa Cruz, Department of Earth and Planetary Sciences, 1156 High St, Santa Cruz, CA 95064, USA
Copyright Elsevier

The Earth and other terrestrial planets formed through the accretion of smaller bodies, with their core and mantle compositions primarily set by metal–silicate interactions during accretion. The conditions of these interactions are poorly understood, but could provide insight into the mechanisms of planetary core formation and the composition of Earth’s core. Here we present modeling of Earth’s core formation, combining results of 100 N-body accretion simulations with high pressure–temperature metal–silicate partitioning experiments. We explored how various aspects of accretion and core formation influence the resulting core and mantle chemistry: depth of equilibration, amounts of metal and silicate that equilibrate, initial distribution of oxidation states in the disk, temperature distribution in the planet, and target:impactor ratio of equilibrating silicate. Virtually all sets of model parameters that are able to reproduce the Earth’s mantle composition result in at least several weight percent of both silicon and oxygen in the core, with more silicon than oxygen. This implies that the core’s light element budget may be dominated by these elements, and is consistent with ≤1–2 wt% of other light elements. Reproducing geochemical and geophysical constraints requires that Earth formed from reduced materials that equilibrated at temperatures near or slightly above the mantle liquidus during accretion. The results indicate a strong tradeoff between the compositional effects of the depth of equilibration and the amounts of metal and silicate that equilibrate, so these aspects should be targeted in future studies aiming to better understand core formation conditions. Over the range of allowed parameter space, core and mantle compositions are most sensitive to these factors as well as stochastic variations in what the planet accreted as a function of time, so tighter constraints on these parameters will lead to an improved understanding of Earth’s core composition.