Optimized formulas for the gravitational field of a tesseroid

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid’s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.     

Rapid U‐Pb Geochronology by Laser Ablation Multi‐Collector ICP‐MS

Detrital zircon (DZ) U‐Pb laser ablation‐inductively coupled plasma‐mass spectrometry (LA‐ICP‐MS) has revolutionised the way geologists approach many Earth science questions. Although recent research has focused on rapid sample throughput, acquisition rates are limited to 100–300 analyses h^?1. We present a method to acquire zircon U‐Pb dates at rates of 120, 300, 600 and 1200 analyses h^?1 (30, 12, 6 and 3 s per analysis) by multi‐collector LA‐ICP‐MS. We demonstrate the efficacy of this method by analysing twelve zircon reference materials with dates from ~ 3465 to ~ 28 Ma. Mean offset from high‐precision dates increases with faster rates from 0.9% to 1.1%; mean random 1s uncertainty increases from 0.6% to 1.3%. We tested this new method on a sandstone sample previously characterised by large‐n DZ geochronology. Quantitative comparison shows increased correspondence among age distributions comprising > 300 dates. This new method holds promise for DZ geochronology because (a) it requires no major changes to hardware, but rather modifications to software; (b) it yields robust age distributions well‐suited for quantitative analysis and maximum depositional age calculations; (c) there is only a minor sacrifice of accuracy and measurement uncertainty; and (d) there is less burden to researchers in terms of time investment and analytical cost.     

Mantle fragments in the Scandinavian caledonides

Mantle fragments of ultramafic composition are widespread in the Scandinavian Caledonides (SC). Lenses and boudins of Alpine-type peridotites in the Scandinavian Caledonides represent parts of dismembered ophiolite sequences and fragments of sub-continental upper mantle. Metaperidotites of nappes in internal positions are generally isofacial with the metamorphic envelope, usually Caledonian metasediments but in places also Precambrian metagranitoids forming the basement cores of the nappes. Caledonian metamorphism strongly modified the texture and mineralogy of the peridotites and resulted in a systematic metamorphic pattern which is consistent with the pattern observed in the envelope.

Metaperidotites of the external massifs display at least a two-stage metamorphic history: an early Caledonian high-pressure high-temperature phase related to early crustal stacking and a late Caledonian regional metamorphic overprint which produced a regular Barrovian-type metamorphic pattern of in-situ metamorphism.

Metaperidotites from nappes in intermediate positions (Iapetus Ocean ophiolites and ultramafic rocks from island arc environments) show strongly diverging histories. Metaperidotites from internal ophiolites (oceanic ophiolites, Köli) lack any evidence of subduction metamorphism, are serpentinized to various degrees, show abundant primary mantle relic mineralogies and the Caledonian metamorphic overprint is low. Metaperidotites from external (island arc) ophiolites and other associations (Seve) often show relic high-pressure metamorphism related to the Finnmarkian phase of the Caledonian orogeny. The Seve metaperidotites are occasionally associated with eclogites and show a weak overprint of late Caledonian regional metamorphism. Alpine-type peridotites are absent in the foreland of the Baltic Shield and in the innermost nappes (Lofoten).

The metamorphic characteristics and evolution recorded by the metaperidotites in the Scandinavian Caledonides allow a general reconstruction of the dynamics of collision belt formation.     

Das Pamir-System

Ohne Zusammenfassung     

Ergebnisse neuer geologischer Forschung im Tian-Schan

Ohne Zusammenfassung     

Meereskundlich-chemische Untersuchungen mit Hilfe des Pulfrich-Photometers von Zeiss

Zusammenfassung Die bisherigen Methoden zur Bestimmung des Chlorophylls im Meerwasser benötigen zu ihrer Durchführung Wassermengen von einem bis zu einigen Litern. Sie sind infolgedessen zur Ausfühung serienmäßiger ozeanischer Messungen noch nicht voll geeignet. Durch Ausnutzung der intensiven Rotfluoreszenz des Chlorophylls und Benutzung kleinerer und engerer Kolorimeterröhren läßt es sich ermöglichen, den Wasserbedarf bis auf 100 ccm und weniger je Probe herabzudrücken und somit diesen Mangel zu beheben. Die für diesen Zweck geschaffene gleiche Arbeitsapparatur ermöglicht weiterhin, im Anschluß an die Chlorophyllbestimmung ohne wesentlich erhöhte Arbeitsbelastung die Bestimmung der Eigenfluoreszenz gleichfalls auf dem Wege der Mikromethodik an nur 1 ccm Meerwasser durchzuführen.

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Investigations in chemical oceanography carried out with the aid of the Pulfrich photometer

Summary The old method of determining the chlorophyll content of sea water required a comparatively great quantity of sea water (one or severel litres) and, hence, was not suitable in case of serial measurements. The utilization of the intensive red fluorescence of chlorophyll and the use of small and narrow colorimeter tubes render it possible to minimize the required quantity of water to 100 ccm and less per sample. The apparatus which was especially constructed for this purpose being fit for the determination of both the chlorophyll content and the proper fluorescence, the latter can be easily determined immediately after the determination of the chlorophyll content by means of the micro-method from only 1 ccm of sea water.

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Investigations chimiques dans le domaine océanographique à l'aide du photomètre de PulfrichVII^e communication. Les micro déterminations de la teneur en chlorophylle et de la fluorescence propre de l'eau de mer

Résumé La quantité minimale nécessaire pour déterminer la teneur en chlorophylle selon la méthode en usage jusqu' à présent se montait d'un à plusieurs litres; à cause de cela cette méthode n'était pas convenable pour des mensurations en séries. En se servant de l'intensive fluorescence rouge de la chlorophylle et en employant un petit tube étroit à colorimètre il est possible de limiter la quantité d'eau à 100 ccm et moins par échantillon. En outre, on peut se servir du même appareil pour déterminer la fluorescence propre selon la nouvelle méthode. Pour cette détermination qui peut être effectué sans peine immédiatement après celle de la chlorophylle on a besoin de seulement 1 ccm d'eau de mer.

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Die sechs vorhergehenden Mitteilungen sind in den Annalen der Hydrographie und maritimen Meteorologie erschienen, und zwar I:59, 1931, 313; II:61, 1933, 124; III:62, 1934, 65; IV:63, 1935, 58: V:63, 1935, 195; VI:65, 1937, 276.     

Die wirkliche Solarkonstante

Zusammenfassung Die Solarkonstante 2.4 wird begründet.

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Summary The true solar constant 2.4 is proved.