Simulation of normal distributed smooth fields by Karhunen-Loéve expansion in combination with kriging

Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.     

Gravity reduction with three-dimensional atmospheric pressure data for precise ground gravity measurements

The redistribution of air masses induces gravity variations (atmospheric pressure effect) up to about 20 μgal. These variations are disturbing signals in gravity records and they must be removed very carefully for detecting weak gravity signals. In the past, different methods have been developed for modelling of the atmospheric pressure effect. These methods use local or two-dimensional (2D) surface atmospheric pressure data and a standard height-dependent air density distribution. The atmospheric pressure effect is consisting of the elastic deformation and attraction term. The deformation term can be well modelled with 2D surface atmospheric pressure data, for instance with the Green's function method. For modelling of the attraction term, three-dimensional (3D) data are required. Results with 2D data are insufficient.From European Centre for Medium-Range Weather Forecasts (ECMWF) 3D atmospheric pressure data are now available. The ECMWF data used here are characterised by a spacing of Δ and Δλ = 0.5°, 60 pressure levels up to a height of 60 km and an interval of 6 h. These data are used for modelling of the atmospheric attraction term. Two attraction models have been developed based on the point mass attraction of air segments and the gravity potential of the air masses. The modelling shows a surface pressure-independent part of gravity variations induced by mass redistribution of the atmosphere in the order of some μgal. This part can only be determined by using 3D atmospheric pressure data. It has been calculated for the Vienna Superconducting Gravimeter site.From this follows that the gravity reduction can be improved by applying the 3D atmospheric attraction model for analysing long-periodic tidal waves including the polar tide. The same improvement is expected for reduction of long-term absolute gravity measurements or comparison of gravity measurements at different seasonal times. By using 3D atmospheric pressure data, the gravity correction can be improved up to some μgal.     

Influence of the zone of weakness on dip angle and shear heating of subducted slabs

We study the importance of the zones of weakness and the pattern of downgoing flow in steady-state models of subducting lithosphere, which interacts mechanically and thermally with the ambient mantle. The non-linear system of governing equations consists of (i) the momentum equation in stream function formulation and (ii) the steady-state heat transfer equation including conduction and advection of heat and dissipation. A finite element method has been applied to this system. We consider the viscosity to be a non-linear function of both the temperature and the stream function. In steady-state two-dimensional (2D) flow, the stream function isolines follow material trajectories. They are used to follow the top of the subducting slab, which because of its possible increase in water content, is assumed to have a lower viscosity. The zone of weakness has been thus obtained in the self-consistent fashion since the stream function as well as the temperature are the output from our modeling and no a priori assumptions about the shape of the bending lithosphere are taken into account. It was shown that several orders decrease of viscosity in the zone of weakness is required to obtain the dip angle of about 45°. If the decrease of viscosity is not sufficient enough, the subducted slab either sinks almost vertically or does not exhibit a plate-like behavior. We have also demonstrated that shear heating can unrealistically increase at the zone of weakness for fast subductions if decrease of viscosity is underestimated.     

Chemical and isotopic equilibrium between CO2 and CH4 in fumarolic gas discharges: Generation of CH4 in arc magmatic-hydrothermal systems

The chemical and isotopic composition of fumarolic gases emitted from Nisyros Volcano, Greece, and of a single gas sample from Vesuvio, Italy, was investigated in order to determine the origin of methane (CH₄) within two subduction-related magmatic-hydrothermal environments.Apparent temperatures derived from carbon isotope partitioning between CH₄ and CO₂ of around 340°C for Nisyros and 470°C for Vesuvio correlate well with aquifer temperatures as measured directly and/or inferred from compositional data using the H₂O-H₂-CO₂-CO-CH₄ geothermometer. Thermodynamic modeling reveals chemical equilibrium between CH₄, CO₂ and H₂O implying that carbon isotope partitioning between CO₂ and CH₄ in both systems is controlled by aquifer temperature.N₂/³He and CH₄/³He ratios of Nisyros fumarolic gases are unusually low for subduction zone gases and correspond to those of midoceanic ridge environments. Accordingly, CH₄ may have been primarily generated through the reduction of CO₂ by H₂ in the absence of any organic matter following a Fischer-Tropsch-type reaction. However, primary occurrence of minor amounts of thermogenic CH₄ and subsequent re-equilibration with co-existing CO₂ cannot be ruled out entirely. CO₂/³He ratios and δ¹³C_CO2 values imply that the evolved CO₂ either derives from a metasomatized mantle or is a mixture between two components, one outgassing from an unaltered mantle and the other released by thermal breakdown of marine carbonates. The latter may contain traces of organic matter possibly decomposing to CH₄ during thermometamorphism.