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Abstract Due to the inaccessibility of Earth's deep interior, geologists have long attempted to estimate the composition of the continental crust from its seismic properties. Despite numerous sources of error including nonuniqueness in the mapping between composition and seismic properties, the corresponding uncertainties have typically been estimated qualitatively at best. We propose a Bayesian approach that uses mineralogical modeling to combine prior knowledge about the composition of the crust with seismic data to give a posterior distribution of the predicted composition at any location, combined with a Monte Carlo simulation to estimate the average composition of the Earth's crust. Our approach yields an estimated composition of 59.5% silica in the upper crust (90% credible interval 58.9 %–60.1%), 57.9% in the middle crust (90% credible interval 57.2%–58.6%), and 53.6% in the lower crust (90% credible interval 53.0%–54.2%). Our estimate exhibits less compositional stratification over depth and a more intermediate composition in the upper and middle crust than previous estimates. Testing our approach on a simulated crust reveals the importance of prior assumptions in estimating the composition of the crust from its seismic properties, and suggests that future work should focus on quantifying those assumptions. 

Abstract In the problem of spotlight mode airborne synthetic aperture radar (SAR) image formation, it is well-known that data collected over a wide azimuthal angle violate the isotropic scattering property typically assumed. Many techniques have been proposed to account for this issue, including both full-aperture and sub-aperture methods based on filtering, regularized least squares, and Bayesian methods. A full-aperture method that uses a hierarchical Bayesian prior to incorporate appropriate speckle modeling and reduction was recently introduced to produce samples of the posterior density rather than a single image estimate. This uncertainty quantification information is more robust as it can generate a variety of statistics for the scene. As proposed, the method was not well-suited for large problems, however, as the sampling was inefficient. Moreover, the method was not explicitly designed to mitigate the effects of the faulty isotropic scattering assumption. In this work we therefore propose a new sub-aperture SAR imaging method that uses a sparse Bayesian learning-type algorithm to more efficiently produce approximate posterior densities for each sub-aperture window. These estimates may be useful in and of themselves, or when of interest, the statistics from these distributions can be combined to form a composite image. Furthermore, unlike the often-employed ℓ p -regularized least squares methods, no user-defined parameters are required. Application-specific adjustments are made to reduce the typically burdensome runtime and storage requirements so that appropriately large images can be generated. Finally, this paper focuses on incorporating these techniques into SAR image formation process, that is, for the problem starting with SAR phase history data, so that no additional processing errors are incurred. The advantage over existing SAR image formation methods are clearly presented with numerical experiments using real-world data.