Velocity-space maps and transforms of tracking observations for orbital trajectory state analysis

The orbital state of a satellite in a central force field can be uniquely described by its velocity hodograph, a circle, rather than the Keplerian conic. Also, its coordinate-frame rotation about the attracting center is definable, without singularity, by the four-parameter set of Euler parameters. A unified state model of orbital trajectory and attitude dynamics has previously been developed by use of state variables of the orbital velocity hodograph and Euler parameters. The dynamical constraint equations of this orbital state model are especially effective in advanced techniques of state estimation, used for orbit determination and prediction. External observations of orbital vehicles, such as provided by optical and radar sensors of tracking systems, are transformable into corresponding velocity state maps, as presented in this paper. These transformations and the consequent state maps are essential for development of the orbit observation matrix used with the unified state matrix, in recursive estimators such as the Kalman filters. Line-of-sight rays and range spheres (or hemispheres) of observations map conformally into orthogonal spherical surfaces in velocity space, as the result of the point-contact transformations. In bispherical coordinates, the field of observation maps for a ground-based tracking system site is shown to be a reduced (or degenerate) form of the general field of observation maps for a satellite-based tracking site. These orbital state maps and transforms are directly useful in development of observation matrices for candidate observation sets, such as range only, angle only, or range plus range-rate tracking schemes. Also, surface coverage patterns can be generated for proposed new tracking systems, in mission analysis and system synthesis studies.     

A canonical theory for short GPS baselines. Part II: the ambiguity precision and correlation

The present contribution is the second of four parts. It considers the precision and correlation of the least-squares estimators of the carrier phase ambiguities. It is shown how the precision and correlation of the double-differenced ambiguities as well as of the widelane ambiguities are effected by the observation weights, by the number of satellites tracked, by the number of observation epochs used, and by the change over time of the relative receiver-satellite geometry. Also the ability of the widelane transformation to decorrelate and to improve the precision is investigated. Received: 16 July 1996 / Accepted: 14 November 1996     

Copper(II) humate mobility in kaolinite soil

An investigation of the influence of humate on the mobility of copper(II) ions in a kaolinite soil using leaching tests and electrokinetic experiments is reported. The data are interpreted in terms of humate–copper–clay interactions and humate electrical charge. Humate is mostly immobile below pH8 but is more mobile in alkaline conditions (sorption to kaolinite reduces its mobility in neutral conditions). Copper humate complexes are mobile in both acidic and alkaline conditions, but not in neutral conditions where they are sorbed. The dissolved copper humate complexes that form in acidic conditions are positively charged. The net effect of humate is to increase cupric ion mobility in kaolinite soil, especially in alkaline conditions.     

Improved Petrological Modal Analyses from X-ray Powder Diffraction Data by use of the Rietveld Method I. Selected Igneous, Volcanic, and Metamorphic Rocks

Mineral modes have been determined for specimens of eight rocktypes from CuK X-ray powder diffraction data using the Rietveldmethod. The samples include two granites, a granodiorite, adamellite,gabbro, basalt, trachyte, and two granulite-facies metamorphicrocks. Up to eight individual mineral components have been measuredin each sample (no glassy phases were observed), with a detectionlimit of {small tilde}1 wt.%, depending on the mineral assemblage.Marked variations in grain size (i.e., granite vs. trachyte)provide no difficulties for the X-ray method. The X-ray resultscompare very favourably with (1) optical modes determined forthe medium–coarse-grained samples by point counting, (2)normative calculations obtained using locally enhanced catanormand mesonorm software, and (3) corresponding Rietveld modesdetermined, for two samples, from neutron powder data. Wheredifferences occur, these are discussed in relation to the limitationsof each of the methods. The improved accuracy of the X-ray method is due primarily tothe incorporation of the full diffraction profile in the Rietveldanalysis calculations, and the elimination of preferred orientationby collecting the data from samples packed in glass capillaries(i.e., Debye–Scherrer mode). The good agreement of theX-ray and neutron modes shows that the usual problems encounteredwith microabsorption, extinction, and sampling are of littleconcern in these rocks. The results highlight one of the majoradvantages provided by Rietveld modal analysis over the moretraditional ‘reference intensity’ X-ray methods,namely, that the crystal chemistry (and thus the calibrationconstants) of the individual phases can be adjusted dynamicallyduring each individual analysis. This not only provides moreaccurate phase abundances, but also gives important supplementaryinformation about the mineralogy of the major components.     

Authigenic iron and titanium oxides in triassic red beds: (St. Bees Sandstone), Cumbria,Northern England

Authigenic iron and titanium oxides occur in three main textural varieties in the St. Bees Sandstone, a Triassic red bed succession in Cumbria. These are: (a) overgrowths of haematite, titaniferous haematite, anatase, and rutile which generally occur on detrital host grains of similar compositions. The overgrowths may occur as syntaxial rim cements or rhombohedral and prismatic projections and are always in optical continuity, (b) discrete crystals of anatase (including octahedrite) and haematite 10-40 μm in size occur in pore spaces, and (c) fine grained ( < 1 μm) pigmentary haematite occurs as grain coatings around detrital grains and in the interstitial matrix. The euhedral nature and pore-filling habit clearly indicate the authigenic origin of these iron- and titanium oxides. All three textural varieties are believed to have been precipitated from ground waters containing dissolved ferrous and titanium ions. These ions were derived from the intrastratal solution of detrital iron and titanium bearing grains including ferromagnesian silicates and iron-titanium oxides. The precipitation of iron oxides from migrating ground waters in the manner described here goes some way to explaining how thick, uniformly red successions can be formed.