Results for “Track reconstruction” — Page 11

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Argon Isotopic Constraints on Planetary Degassing and the Evolution of Earth's Atmosphere: Insights from K-bearing Phases

An enduring question in planetary science is the nature of volatile degassing from rocky planets during their evolution. The reactive nature of planetary volatiles (e.g. H2O, CO2, N, H) make them difficult to directly track through time. Noble gases, like argon, can act as passive tracers of planetary processes, with isotopic ratios providing clues on planetary volatile inventories and fluxes. Previous studies [e.g. 1, 2] use Earth’s argon degassing history to constrain the growth of continental crust. However, that record is limited to the modern 40Ar/36Ar ratio of 298.56 [3] and three time slices recorded in hydrothermal quartz of the Archean Dresser formation [1]: from ~143 (3.5 Ga) to ~212 (2.7 Ga). Due to the paucity of data, partly a result of using nominally K-free minerals, the temporal evolution of atmospheric 40Ar/36Ar remains poorly constrained. K-bearing phases from igneous rocks, forming at high temperature and upper mantle or crustal conditions, are rarely considered viable for recording ancient atmospheric 40Ar/36Ar values. We show that this possible and a potential path to more precise records of ancient atmosphere by leveraging the radiogenic endmember of isotope correlation diagrams. An isochron of coeval biotite grains from the Bushveld Complex [4], with little indication of alteration or excess 40Ar, yield an initial 40Ar/36Ar value that matches the atmospheric 40Ar/36Ar value of ~228 at 2055 Ma in the model of [2]. In addition, new multi-mineral data from the Palisade Sill also match the modelled 40Ar/36Ar value of ~293 at 202 Ma, providing another tie point in the late Triassic. Combined, the Bushveld and Palisades samples provide two new constraints on the ancient atmosphere and help define the history of planetary degassing. These results demonstrate the possibility of accurately and precisely obtaining atmospheric information from igneous rocks, not exclusively K-free phases, thus providing a surplus of ways to reconstruct planetary atmospheres and processes. A robust atmospheric 40Ar/36Ar history also has practical uses and we explore the systematic offsets that arise from using a modern 40Ar/36Ar value in 40Ar/39Ar age calculations of old samples.

J. B. Setera↗

NASA NTRS2015

Rao-Blackwellization for Adaptive Gaussian Sum Nonlinear Model Propagation

When dealing with imperfect data and general models of dynamic systems, the best estimate is always sought in the presence of uncertainty or unknown parameters. In many cases, as the first attempt, the Extended Kalman filter (EKF) provides sufficient solutions to handling issues arising from nonlinear and non-Gaussian estimation problems. But these issues may lead unacceptable performance and even divergence. In order to accurately capture the nonlinearities of most real-world dynamic systems, advanced filtering methods have been created to reduce filter divergence while enhancing performance. Approaches, such as Gaussian sum filtering, grid based Bayesian methods and particle filters are well-known examples of advanced methods used to represent and recursively reproduce an approximation to the state probability density function (pdf). Some of these filtering methods were conceptually developed years before their widespread uses were realized. Advanced nonlinear filtering methods currently benefit from the computing advancements in computational speeds, memory, and parallel processing. Grid based methods, multiple-model approaches and Gaussian sum filtering are numerical solutions that take advantage of different state coordinates or multiple-model methods that reduced the amount of approximations used. Choosing an efficient grid is very difficult for multi-dimensional state spaces, and oftentimes expensive computations must be done at each point. For the original Gaussian sum filter, a weighted sum of Gaussian density functions approximates the pdf but suffers at the update step for the individual component weight selections. In order to improve upon the original Gaussian sum filter, Ref. [2] introduces a weight update approach at the filter propagation stage instead of the measurement update stage. This weight update is performed by minimizing the integral square difference between the true forecast pdf and its Gaussian sum approximation. By adaptively updating each component weight during the nonlinear propagation stage an approximation of the true pdf can be successfully reconstructed. Particle filtering (PF) methods have gained popularity recently for solving nonlinear estimation problems due to their straightforward approach and the processing capabilities mentioned above. The basic concept behind PF is to represent any pdf as a set of random samples. As the number of samples increases, they will theoretically converge to the exact, equivalent representation of the desired pdf. When the estimated qth moment is needed, the samples are used for its construction allowing further analysis of the pdf characteristics. However, filter performance deteriorates as the dimension of the state vector increases. To overcome this problem Ref. [5] applies a marginalization technique for PF methods, decreasing complexity of the system to one linear and another nonlinear state estimation problem. The marginalization theory was originally developed by Rao and Blackwell independently. According to Ref. [6] it improves any given estimator under every convex loss function. The improvement comes from calculating a conditional expected value, often involving integrating out a supportive statistic. In other words, Rao-Blackwellization allows for smaller but separate computations to be carried out while reaching the main objective of the estimator. In the case of improving an estimator's variance, any supporting statistic can be removed and its variance determined. Next, any other information that dependents on the supporting statistic is found along with its respective variance. A new approach is developed here by utilizing the strengths of the adaptive Gaussian sum propagation in Ref. [2] and a marginalization approach used for PF methods found in Ref. [7]. In the following sections a modified filtering approach is presented based on a special state-space model within nonlinear systems to reduce the dimensionality of the optimization problem in Ref. [2]. First, the adaptive Gaussian sum propagation is explained and then the new marginalized adaptive Gaussian sum propagation is derived. Finally, an example simulation is presented.

state estimation↗