Improvement of the geoid in local areas by satellite-to-satellite tracking

Summary The concept of satellite-to-satellite tracking measuring the relative velocity of two orbiting satellites spaced some hundreds kilometers on a close orbit, provides now possibilities for the investigation of the Earth’s gravity field. In the paper only medium and short wave length effects affecting the measured relative velocity have been considered. Collocation is used in such an analysis of local geoid improvement, because this method allows to combine heterogeneous data in a consistent way. Covariance functions relevant for the particular case of a circular equatorial orbit are given. Two kinds of observation equations have been formulated. The choice of observation equation with regard to satellites configuration is discussed. It is found that it is sufficient to have a limited number of satellite-to-satellite observations in a 7^o×7^o area around the estimation point with distances between profiles of about 1^o.5 and between the two satellites forming the pair of 200+350 km; the altitude of satellite-to-satellite observations should be as low as possible. The accuracy of the geoid determination strongly depends on the degree and order of the reference field used. An accuracy of about ±1 m can be achieved with an assumed reference field of (40,40). The influence of measuring errors is discussed and it is shown that only satellite-to-satellite observations with accuracy better then 0.1 mm/sec will give an improvement of the geoid. Finally, some results on the combination of low-low satellite-to-satellite tracking and terrestrial gravity data are given. The proposed method seems to be especially interesting for unsurveyed areas. Furthermore, it has the practical advantage that only a local coverage data is needed.     

Spectral analysis of oscillations with a time-variable frequency

Summary The paper presents a method of evaluating oscillations with a time-variable frequency using a computer. The given function is first interpolated at points which are not distributed equidistantly in time, but the digitizing step varies with time according to a known optional regularity. The spectrum of the obtained function is computed for various of these interpolations and tests are run to determine when the interpolation best compensates the time variation of the frequency. The initial and terminal frequency in the given sample is then determined. The usability of this method with respect to various types of oscillations with a variable frequency and its accuracy in comparison to sonagrams are discussed.     

Geomorphological models as a tool for environmental studies

Relief is understood as a fundamental component of the environmental system. The structure of geomorphodynamic processes and of the relief sphere as the main energy transformation surface is a determining factor for environmental conditions of both sites and areas. Modelling is the instrument of the applied and systems analytic approach to geomorphology. Geomorphological tools are appropriate means of determining, structuring, assessing and predicting the geoecological potential of the environment. This is demonstrated by quantitative and multivariate models to describe spatial patterns of the environment (fluvial network analysis) and to evaluate and present environmentally relevant geomorphological disposition (slope stability analysis).     

Earthquake strong motion simulated by stress drop models

Shallow strike slip earthquakes on vertical faults are modelled as two-dimensional antiplane strain ruptures in a uniformly prestressed homogeneous halfspace. Behind the rupture front, which is specified, the stress drops to a lower value. The elastodynamic boundary value problem is solved with a finite difference method. Several cases are studied, which include symmetric and one-directional rupture propagation, surface faulting, multiple events, variable rupture velocity, sticking and rebreaking of the fault plane. The time function of displacement, velocity and acceleration are interpreted as signals generated by events in the focus, namely starting, stopping and breaking through the surface of the rupture. The model explains peak velocity and peak acceleration in the near field of M5.5–6 earthquakes; which are typically about 0.2 m/s and 5 m/s² at 10 km epicentral distance, if the rupture velocity is close to the shear wave velocity. Sticking of the fault does not alter the accelerograms significantly, but it increases the seismic moment in simple events and decreases it in multiple events.Contribution No. 226, Geophysical Institute, University of Karlsruhe.     

Mineral isotopic age relationships in the polymetamorphic Amîtsoq gneisses,Godthaab district,West Greenland

U-Th-Pb, Pb-Pb, Rb-Sr and K-Ar radiometric relationships in the minerals from six selected Amîtsoq gneiss samples reveal a complicated history of variable mineral response to polymetamorphism.K-Ar dates on biotite range from 2170 to 3220 m.y. (excess argon present), on hornblende from 2340 to 2510, and on a single muscovite at 1670 m.y.Rb-Sr whole rock results give an apparent isochron of at least 4065 m.y., but this result is likely fortuitous from a small sample selection since Pb-Pb whole rock analyses give ~ 3600 m.y. and the zircons in these rocks yield a concordia-discordia intersection at 3600 m.y. Rb-Sr mineral analyses generally give a confusing and variable pattern of isotopic relationships; but hornblende, K-feldspar, apatite, allanite and sphene appear to have last responded to metamorphism at 2200–2600 m.y. Rb-Sr in biotite, epidote and, in part, plagioclase have been affected by an event at ~ 1550 m.y.U-Th-Pb data from sphene, apatite and allanite give almost concordant dates at 2500–2600 m.y. $\text{soul}{}^{207}\text{Pb}{}^{204}\text{Pb}$ vs $\text{soul}{}^{206}\text{Pb}{}^{204}\text{Pb}$ plots yield two separate lines for apatite (slope age 2435 m.y.) and for sphene + allanite (slope age 2530 m.y.), indicating apatite to have a different (less-radiogenic) ‘initial’ Pb than that for sphene and allanite. A similar pattern is found for the $\text{soul}{}^{208}\text{Pb}{}^{204}\text{Pb}$ vs $\text{soul}{}^{207}\text{Pb}{}^{204}\text{Pb}$ plot for sphene and apatite. The Pb-isotopic composition of the feldspars is very homogeneous and the least-radiogenic of all components, pointing towards a homogeneous parent material for the now lithologically diverse Amîtsoq gneisses. Using a₀ = 9.307, b₀ = 10.294, C₀ = 29.476, t₀ = 4.56 b.y., ω = 6.9 and $\text{soul}{}^{232}\text{Th}{}^{204}\text{Pb = 27.1}$; the feldspars give a model Pb age of 3500–3600 m.y. by either U-derived or Th-derived Pb. The segregation of the present Amîtsoq gneisses from the homogeneous parent material was apparently accompanied by a U and Th loss with preservation or enrichment of Pb at ~ 3600 m.y. ago. No consistent treatment of the present U-Th-Pb data will produce viable data indicating an age > 3600 m.y. for the parent materials of the Amîtsoq gneiss.Petrographie observations generally concur with radiometric results and permit the postulation of the reaction: Hbl + K-feld→ biotite + epidote + sodic plag, to account for some of the effects of the latest metamorphism.The total internal radiometric evidence indicates three major metamorphic events affected the Amîtsoq gneisses close to 3600, 2500 and 1550 m.y.     

Improvement of the geoid in local areas by satellite gradiometry

Summary Satellite gradiometry is studied as a means to improve the geoid in local areas from a limited data coverage. Least-squares collocation is used for this purpose because it allows to combine heterogeneous data in a consistent way and to estimate the integrated effect of the attenuated spectrum. In this way accuracy studies can be performed in a general and reliable manner. It is shown that only three second-order gradients contribute significantly to the estimation of the geoidal undulations and that it is sufficient to have gradiometer data in a 5°×5° area around the estimation point. The accuracy of the geoid determination is strongly dependent on the degree and order of the reference field used. An accuracy of about ±1 m can be achieved with a reference field of (12, 12). There is an optimal satellite altitude for each reference field and this altitude may be higher than 300 km for a field of low degree and order. The influence of measuring errors is discussed and it is shown that only gradiometer data with accuracies better than ±0.05 E will give a significant improvement of the geoid. Finally, some results on the combination of satellite gradiometry and terrestrial gravity measurements are given. The proposed method seems to be well suited for local geoid determinations down to the meter range. It is especially interesting for unsurveyed and difficult areas because no terrestrial measurements are necessary. Furthermore, it has the practical advantage that only a local data coverage is needed.     

Multiscale approach to geo‐composite cellular structures subjected to rock impacts

Geo‐composite cellular structures are an efficient technological solution for various applications in civil engineering. This type of structure is particularly well adapted to resisting rockfalls and can act as a defensive structure. However, the design of such structures is for the most part empirically based; this lack of research‐based design stagnates optimization and advanced development. In this paper, the mechanical behaviour of a geo‐composite cellular structure is investigated using a multi‐scale approach, from the individual cell made up of an assembly of rocky particles contained in a wire netting cage to the entire structure composed of a regular array of cells. Based on discrete modelling of both the cell and structure scales, a computational tool has been developed for design purposes. Copyright © 2007 John Wiley & Sons, Ltd.     

Discrete modelling of time‐dependent rockfill behaviour

This modelling study deals with the time‐dependent behaviour of rockfill media, which is of particular interest during the life of rockfill dams. Breakage of rock blocks and crack propagation are the main processes responsible for rockfill creep and collapse. The modelling procedure presented here is performed on two scales: on the rock block scale, where the grain is taken to be an assembly of rigid particles initially endowed with cohesive bonds, and on the rockfill scale, which is taken to involve a set of breakable grains interacting via contact and friction processes. The grain breakage process is described in term of a thermodynamically consistent damage interface model, where the damage is a gradual delayed process. This model was implemented in a non‐smooth contact dynamics code. The effects of the main parameters involved were analysed by performing numerical studies. The ability of the model to predict the creep behaviour of rockfill media is confirmed by presenting several simulations. Copyright © 2008 John Wiley & Sons, Ltd.