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The problem to predict a direction, axis, or orientation (rotation) from corresponding geocoded data is discussed and a general solution by virtue of embedding a sphere/hemisphere in a real vector space is presented. Its explicit justification in terms of mathematical assumptions concerning stationarity/homogeneity and isotropy is included. The data are modelled by a stationary random field, and the spatial correlation is represented by modified multivariate variograms and covariance functions. Various types of isotropy assumptions concerning invariance under translation/rotation of the data locations, the measurements, or a combination of both, can be distinguished and lead to different simplifications of the general cross-covariance function. Beyond spatial prediction a measure of confidence in the estimates is provided. 相似文献
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Computer-aided geometric design of geologic surfaces and bodies 总被引：2，自引：0，他引：2
Bivariate quadratic simplicial B-splines are employed to obtain aC
1-smooth surface from scattered positional or directional geological data given over a two-dimensional domain. Vertices are generated according to the areal distribution of data sites, and polylines are defined along real geological features. The vertices and the polylines provide a constrained Delaunay triangulation of the domain. Note that the vertices do not generally coincide with the data sites. Six linearly independent simplex B-splines are associated with each triangle. Their defining knots and finite supports are automatically deduced from the vertices. Specific knot configurations result in discontinuities of the surface or its directional derivatives. Coefficients of a simplex spline representation are visualized as geometric points controlling the shape of the surface. This approach calls for geologic modeling and interaction of the geologist up front to define vertices and polylines, and to move control points initially given by an algorithm. Thus, simplex splines associated with irregular triangles seem to be well-suited to approximate and allow further geometrically modeling of geologic surfaces, including discontinuities, from scattered data. Applications to mathematical test as well as to real geological data are given as examples. 相似文献
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Dr. H. Schaeben 《International Journal of Earth Sciences》1988,77(2):591-607
Surveying the history of mathematical geology since the times of Lyell it is shown that its characteristic feature is that of interaction between a strongly historically inclined science and rather abstract mathematics/statistics. It is proven that mathematization of geology and experimental geology stimulate one another, and that mathematical geology can be of essential aid in formulating conceptual models and scientific theories to integrate and unify diverse geological phenomena.Examples of the progressing mathematization of geology and the geosciences have been chosen to be most instructive for the purposes of understanding the general law of the development of science in the geosciences, as well as the propagation of mathematical models and their numerical realization. They range from almost conventional application of classical statistics in subdividing the Tertiary, to mathematical analysis of directional and orientation data vital to plate tectonics, deterministic and stochastic approaches for modeling and simulation purposes in characterization and management of natural resources, and the application of bifurcation theory to study differentiated layering as well as research in artificial intelligence and expert systems in exploration. All examples will be briefly presented and discussed, in general terms, avoiding all severe mathematics. Similarities and differences in the lawful development of geology, biology, and physics with respect to their mathematization are mentioned.
Zusammenfassung Die Geschichte der mathematischen Geologie seit Lyell überschauend wird festgestellt, da\ ihr charakteristisches Merkmal das der Interaktion einer historisch orientierten Naturwissenschaft mit auf Abstraktion zielender Mathematik/Statistik ist. Es wird belegt, da\ Mathematisierung der Geologie und experimentelle Geologie sich gegenseitig vorantreiben und da\ mathematische Geologie zur Bildung von konzeptionellen Modellen und wissenschaftlichen Theorien, die unterschiedliche geologische Erscheinungen in Zusammenhang stellen, wesentlich beitragen kann.Die hier vorgestellten Beispiele der fortschreitenden Mathematisierung der Geologie und der Geowissenschaften als Ganzes sind nach dem Kriterium ausgewählt, die allgemeine Gesetzmä\igkeit des Entwicklungsprozesses der Wissenschaften für die geologischen Wissenschaften besonders deutlich darzustellen, und mit der Absicht, mathematische Modelle und ihre numerische Realisierung zu verbreiten. Sie reichen von fast standardmä\iger Anwendung klassischer, statistischer Argumente zur Unterteilung des Tertiärs über die mathematische Analyse von Richtungs- und Orientierungsdaten, welche wesentlich zur Akzeptanz der Plattentektonik beitrug, und deterministische und stochastische Zugänge zur Modellierung und Simulation bei der Charakterisierung und Verwaltung natürlicher Ressourcen bis zur Anwendung der mathematischen Verzweigungstheorie bei der Untersuchung von differenziertem metamorphem Lagenbau und zur Forschung auf dem Gebiet der künstlichen Intelligenz und der Expertensysteme in der Exploration. Die Beispiele werden ohne mathematische Formulierungen diskutiert. Auf ähnlichkeiten und Unterschiede bei der gesetzmä\igen Entwicklung von Geologie, Biologie und Physik in bezug auf ihre Mathematisierung wird hingewiesen.
Résumé Si on résume l'histoire de la géologie mathématique depuis Lyell, on constate que son trait caractéristique réside dans l'interaction d'une science (naturelle) à orientation historique avec une mathématique-statistique tendant à l'abstraction. Il est de fait que la mathématisation de la géologie et la géologie expérimentale se stimulent mutuellement et que la géologie mathématique peut aider de manière significative à l'élaboration de modèles conceptuels et de théories scientifiques qui tendent à intégrer et à unifier les divers phénomènes géologiques.Les exemples présentés ici de ce processus de mathématisation progressive de la géologie et des sciences de la Terre ont été choisis pour Être les plus significatifs possible, afin d'en éclairer le développement et de justifier en mÊme temps le dessein de propager l'usage des modèles mathématiques et leur réalisation numérique. Le premier exemple concerne l'application, presque conventionnelle, de la statistique classique à la subdivision du Tertiaire. On poursuit par l'analyse mathématique des données de direction et d'orientation, vitales dans l'étude du modèle de la tectonique des plaques. Viennent ensuite les approches déterministes et stochastiques de l'élaboration et de la simulation en vue de caractériser et de gérer les ressources naturelles. Les exemples se poursuivent par l'application de la théorie mathématique de la bifurcation à l'étude du rubanement métamorphique et à la recherche dans le domaine de l'intelligence artificielle et des systèmes d'experts appliqués à l'exploration. Tous les exemples sont brièvement présentés et discutés, en termes généraux, à l'exclusion de formulation mathématique. On souligne les ressemblances et les différences entre les lois du développement de la géologie, de la biologie et de la physique en regard de leur mathématisation.
Lyell'a , . , .. . . , . .: , . . , . .相似文献
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Spherical geometry of quaternions is employed to characterize the Bingham distribution on the 3-dimensional sphere
as being uniquely composed of a bipolar, a circular and a spherical component. A new parametrization of its dispersion parameters provides a classification of patterns of crystallographic preferred orientations (CPO, or textures). It is shown that the Bingham distribution can represent most types of ideal CPO patterns; in particular single component, fiber and surface textures are represented by rotationally invariant bipolar, circular and spherical distributions, respectively. Pole figures of given crystal directions are derived by the spherical Radon transform of the Bingham probability density function of rotations, which are displayed for general and special cases. 相似文献
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Heinrich Siemes Helmut Schaeben Carlos A. Rosire Horst Quade 《Journal of Structural Geology》2000,22(11-12)
The crystallographic preferred orientation of hematite in banded iron ores and the orientation of both the measured and the calculated principal susceptibility axes are strongly related. The maximum susceptibility is aligned with the lineation and the pole of the foliation coincides with the minimum susceptibility, although there are often distinct differences between the measured and calculated values of the susceptibilities. A wide variety of configurations of c-axis pole figures modeled by varying the parameters of the Bingham distribution and Bingham–Mardia-distribution reveal that quite different c-axis patterns of hematite ores may have the same anisotropy of the magnetic susceptibility (AMS) parameters. Large deviations between calculated and experimental AMS-data should initiate further investigations to resolve a probably unnoticed heterogeneity of the fabric. The present investigations show that the structural analysis of the preferred orientation of hematite ores by means of the rather inexpensive and fast magnetic method must be accompanied by the more expensive but unambiguous determination of preferred orientation by x-ray and neutron diffraction experiments in order to accomplish a complete and sound interpretation. 相似文献
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H. Schaeben 《Mathematical Geology》1984,16(2):139-153
An algorithm to classify data points on the sphere in distinct cluster groups is defined. The characteristics of the cluster groups and the rule for assigning data to the groups are related to a continuous differentiable density estimation. The modes of the estimated density are assumed to be representative of the groups; data points are then assigned to the mode reached by the steepest ascent. The major advantage of this procedure is its sensitivity in detecting cluster groups independently of their geometry and configuration. As a consequence, the procedure is capable of handling orientation data that may be arranged in girdles. 相似文献