Evidence for a reducing Archean ambient mantle and its effects on the carbon cycle

Chemical reduction-oxidation mechanisms within mantle rocks link to the terrestrial carbon cycle by influencing the depth at which magmas can form, their composition, and ultimately the chemistry of gases released into the atmosphere. The oxidation state of the uppermost mantle has been widely accepted to be unchanged over the past 3800 m.y., based on the abundance of redox-sensitive elements in greenstone belt–associated samples of different ages. However, the redox signal in those rocks may have been obscured by their complex origins and emplacement on continental margins. In contrast, the source and processes occurring during decompression melting at spreading ridges are relatively well constrained. We retrieve primary redox conditions from metamorphosed mid-oceanic ridge basalts (MORBs) and picrites of various ages (ca. 3000–550 Ma), using V/Sc as a broad redox proxy. Average V/Sc values for Proterozoic suites (7.0 ± 1.4, 2s, n = 6) are similar to those of modern MORB (6.8 ± 1.6), whereas Archean suites have lower V/Sc (5.2 ± 0.4, n = 5). The lower Archean V/Sc is interpreted to reflect both deeper melt extraction from the uppermost mantle, which becomes more reduced with depth, and an intrinsically lower redox state. The pressure-corrected oxygen fugacity (expressed relative to the fayalite-magnetite-quartz buffer, DFMQ, at 1 GPa) of Archean sample suites (DFMQ –1.19 ± 0.33, 2s) is significantly lower than that of post-Archean sample suites, including MORB (DFMQ –0.26 ± 0.44). Our results imply that the reducing Archean atmosphere was in equilibrium with Earth’s mantle, and further suggest that magmatic gases crossed the threshold that allowed a build-up in atmospheric O2 levels ca. 3000 Ma, accompanied by the first “whiffs” of oxygen in sediments of that age

A multivariate piecing-together approach with an application to operational loss data

The univariate piecing-together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. We propose a multivariate extension. First it is shown that an arbitrary copula is in the domain of attraction of a multivariate extreme value distribution if and only if its upper tail can be approximated by the upper tail of a multivariate GPD with uniform margins. The multivariate PT then consists of two steps: The upper tail of a given copula $C$ is cut off and substituted by a multivariate GPD copula in a continuous manner. The result is again a copula. The other step consists of the transformation of each margin of this new copula by a given univariate distribution function. This provides, altogether, a multivariate distribution function with prescribed margins whose copula coincides in its central part with $C$ and in its upper tail with a GPD copula. When applied to data, this approach also enables the evaluation of a wide range of rational scenarios for the upper tail of the underlying distribution function in the multivariate case. We apply this approach to operational loss data in order to evaluate the range of operational risk.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ343 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

An offspring of multivariate extreme value theory: the max-characteristic function

This paper introduces max-characteristic functions (max-CFs), which are an offspring of multivariate extreme-value theory. A max-CF characterizes the distribution of a random vector in Rd, whose components are nonnegative and have finite expectation. Pointwise convergence of max-CFs is shown to be equivalent to convergence with respect to the Wasserstein metric. The space of max-CFs is not closed in the sense of pointwise convergence. An inversion formula for max-CFs is established

The multivariate Piecing-Together approach revisited

The univariate Piecing-Together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. A multivariate extension was established by Aulbach et al. (2012a): The upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula that coincides in its central part with C. In addition to Aulbach et al. (2012a), we achieve in the present paper an exact representation of the PT-copula's upper tail, giving further insight into the multivariate PT approach. A variant based on the empirical copula is also added. Furthermore our findings enable us to establish a functional PT version as well.Comment: 12 pages, 1 figure. To appear in the Journal of Multivariate Analysi

Focusing through random media: eigenchannel participation number and intensity correlation

Using random matrix calculations, we show that, the contrast between maximally focused intensity through random media and the background of the transmitted speckle pattern for diffusive waves is, \mu_N =1 +N_{eff}, where N eff is the eigenchannel participation number for the transmission matrix. For diffusive waves, N_{eff} is the inverse of the degree of intensity correlation, \kappa. The profile of the focused beam relative to the ensemble average intensity is expressed in terms of the square of the normalized spatial field correlation function, F(\Delta r), and \kappa. These results are demonstrated in microwaves experiments and provide the parameters for optimal focusing and the limits of imaging