Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances

Quantities like tropospheric zenith delays or station coordinates are repeatedly measured at permanent VLBI or GPS stations so that time series for the quantities at each station are obtained. The covariances of these quantities can be estimated in a multivariate linear model. The covariances are needed for computing uncertainties of results derived from these quantities. The covariance matrix for many permanent stations becomes large, the need for simplifying it may therefore arise under the condition that the uncertainties of derived results still agree. This is accomplished by assuming that the different time series of a quantity like the station height for each permanent station can be combined to obtain one time series. The covariance matrix then follows from the estimates of the auto- and cross-covariance functions of the combined time series. A further approximation is found, if compactly supported covariance functions are fitted to an estimated autocovariance function in order to obtain a covariance matrix which is representative of different kinds of measurements. The simplification of a covariance matrix estimated in a multivariate model is investigated here for the coordinates of points of a grid measured repeatedly by a laserscanner. The approximations are checked by determining the uncertainty of the sum of distances to the points of the grid. To obtain a realistic value for this uncertainty, the covariances of the measured coordinates have to be considered. Three different setups of measurements are analyzed and a covariance matrix is found which is representative for all three setups. Covariance matrices for the measurements of laserscanners can therefore be determined in advance without estimating them for each application.     

Regional gravity field modeling using free-positioned point masses

A two-step free-positioned point mass method is used for regional gravity field modeling together with the remove-compute-restore (RCR) technique. The Quasi-Newton algorithm (L-BFGS-B) is implemented to solve the nonlinear problem with bound constraints in the first step, while in the second step the magnitudes of the point masses are re-adjusted with known positions in the least-squares sense. In order to reach a good representation of the gravity field, a number of parameter sets have to be defined carefully before the computations. The effects of four important parameter sets (depth limits, number of point masses, original/reduced basis functions and optimization directions) are investigated for regional gravity field modeling based on two numerical test cases with synthetic and real data. The results show that the selection of the initial depth and depth limits is of most importance. The number of point masses for obtaining a good fit is affected by the data distribution, while a dependency on the data variability (signal variation) is negligible. Long-wavelength errors in the predicted height anomalies can be reduced significantly by using reduced basis functions, and the radial-direction optimization proves to be stable and reliable for regular and irregular data scenarios. If the parameter sets are defined properly, the solutions are similar to the ones computed by least-squares collocation (LSC), but require fewer unknowns than LSC.     

Archaean high-K granitoids produced by remelting of earlier Tonalite–Trondhjemite–Granodiorite (TTG) in the Sangmelima region of the Ntem complex of the Congo craton,southern Cameroon

We present a geochemical and isotopic study that, consistent with observed field relations, suggest Sangmelima late Archaean high-K granite was derived by partial melting of older Archaean TTG. The TTG formations are sodic-trondhjemitic, showing calcic and calc-alkalic trends and are metaluminous to peraluminous. High-K granites in contrast show a potassic calc-alkaline affinity that spans the calcic, calc-alkalic, alkali-calcic and alkalic compositions. The two rock groups (TTG and high-K granites) on the other hand are both ferroan and magnesian. They have a similar degree of fractionation for LREE but a different one for HREE. Nd model ages and Sr/Y ratios define Mesoarchaean and slab-mantle derived magma compositions respectively, with Nb and Ti anomalies indicating a subduction setting for the TTG. Major and trace element in addition to Sr and Nd isotopic compositions support field observations that indicate the derivation of the high-K granitic group from the partial melting of the older TTG equivalent at depth. Geochemical characteristics of the high-K granitic group are therefore inherited features from the TTG protolith and cannot be used for determining their tectonic setting. The heat budget required for TTG partial melting is ascribed to the upwelling of the mantle marked by a doleritic event of identical age as the generated high-K granite melts. The cause of this upwelling is related to linear delamination along mega-shear zones in an intracontinental setting.     

Access and Assimilation: Pivotal environmental information challenges

The world-wide environmental exchange of information is hampered by difficulties bridging terminologies and languages so that trans-cultural, cross-sectoral, and multi-dimensional information can be made available in a usable form at low cost. The alternative concepts and elements described are planned to contribute to HEMIS, UNEP-HEMs world-wide information and reference system. The potential content includes: environmentally relevant organizations, activities, systems, methods, data sources, qualities, and access methods. The key design element is a multi-lingual thesaurus which has harmonization effect resulting from the structuring and strategic access to information. Documentation and retrieval is another concern beside the provision of tools to ease identification, dissemination, and validation of information. The maintenance of validated source data and documents will reduce the volumes of requested information and thereby counteract the present information overkill.     

Boundedness of Turbulent Temperature Probability Distributions,and their Relation to the Vertical Profile in the Convective Boundary Layer

Higher-order moments, minima and maxima of turbulent temperature and water vapour mixing ratio probability density functions measured with an eddy-covariance system near the ground were related to each other and to vertical boundary-layer profiles of the same scalars obtained through airborne soundings. The dependence of kurtosis on squared skewness showed a kurtosis intercept below the Gaussian expectation, suggesting a compression of the probability density function by the presence of natural boundaries. This hypothesis was corroborated by comparing actual minima and maxima of turbulent fluctuations to estimates obtained from the first four sample moments by fitting a four-parameter beta distribution. The most sharply defined boundaries were found for the minima of temperature datasets during the day, indicating that negative temperature fluctuations at the sensor are limited by the availability of lower temperatures in the boundary layer. By comparison to vertical profiles, it could be verified that the turbulent minimum of temperature near the ground is close to the minimum of potential temperature in the boundary layer. The turbulent minimum of water vapour mixing ratio was found to be equal to the mixing ratio at a height above the minimum of the temperature profile. This height roughly agrees with the top of the non-local unstable domain according to bulk Richardson number profiles. We conclude that turbulence statistics measured near the surface cannot be solely explained by local effects, but contain information about the whole boundary layer including the entrainment zone.     

The role of the Kupferschiefer in the formation of hydrothermal base metal mineralization in the Spessart ore district, Germany: insight from detailed sulfur isotope studies

The Spessart district (SW Germany), located at the southwestern margin of the Permian Kupferschiefer basin in Central Europe, hosts abundant stratabound and structurally controlled base metal mineralization. The mineralization styles identified are (1) stratabound Cu-Pb-Zn-(Ag) ores in Zechstein sedimentary rocks, (2) structurally controlled Cu-As-(Ag) ores in Zechstein sedimentary rocks, (3) crosscutting Co-Ni-(Bi)-As and Cu-Fe-As veins, (4) stratabound metasomatic Fe-Mn carbonate ores in Zechstein dolomite, (5) barren barite veins, and (6) Fe-Mn-As veins in Permian rhyolites. Building on previous work that involved mineralogical, textural, and chemical characterization of the major mineralization types, we have performed a comprehensive sulfur isotope study that applied both conventional and novel laser-ablation multi-collector inductively coupled plasma mass spectrometry techniques. The δ³⁴S values of sulfide minerals from the different ore types are consistently negative and highly variable, in the range between −44.5‰ and −3.9‰, whereas the δ³⁴S values of barite are all positive in the range between 4.7‰ and 18.9‰. Remarkably, stratabound and structurally controlled mineralization in Zechstein sedimentary rocks has the least negative δ³⁴S values, whereas vein-type deposits have consistently more negative δ³⁴S values. The observed pattern of sulfide δ³⁴S values can be best interpreted in terms of fluid mixing at the basement-cover interface. Hydrothermal fluids originating from the crystalline basement migrated upward along subvertical fault zones and were periodically injected into groundwaters that were flowing in the post-Variscan sedimentary cover. These groundwaters had interacted with the Zechstein sedimentary rocks, resulting in fluids characterized by elevated concentrations of reduced sulfur (with negative δ³⁴S values) and alkaline pH. Repeated mixing between both chemically contrasting fluids caused rapid and efficient precipitation of sulfide ore minerals in hydrothermal veins with highly variable but distinctly negative δ³⁴S values.     

Present state and future developments of the European geoid

Regional geoid resp. quasigeoid determinations are nowadays required with an accuracy of ±1 to 10 cm over distances from 100 to some 1000 km in order to meet the demands of geodesy, geophysics, oceanography and engineering. Especially the combination of GPS heighting with classical leveling is one of the primary drivers for precise geoid computations. As a consequence, the IAG International Geoid Commission recognized at its meeting in Milano, 1990, that there is an urgent need for a new European geoid computation. This solution should be significantly improved in spatial resolution and accuracy as compared to presently available models. This led to the decision to form a Subcommission for the Geoid in Europe, and the Institut für Erdmessung (IfE) was asked to serve as a computing center in this project.In the first part of this paper early geoid/quasigeoid computations for the area of Europe as well as more recent results obtained at IfE are summarized. The latter solutions include a gravimetric and an astrogravimetric quasigeoid, which have a spatial resolution of about 20 km and a relative accuracy of some dm. Then the possibilities for an improved European quasigeoid calculation are outlined, considering the availability of new and better global and regional data sets. An overview is given on the procedures currently under study at IfE and on the work performed at IfE since 1990. This work includes the collection and screening of new point gravity and terrain data, some investigations on the use of topographic information available at present, and the calculation of a preliminary quasigeoid solution for central, northern and western Europe including a GPS/leveling control. The paper closes with a survey on future activities at IfE within the European geoid project.