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Patterns of crystallographic preferred orientation are referred to as texture. The specific subject of texture analysis is
the experimental determination and interpretation of the statistical distribution of orientations of crystals within a specimen
of polycrystalline material, which could be metals or rocks. The objective is to relate an observed pattern of preferred orientation
to its generating processes and vice versa. In geosciences, texture of minerals in rocks is used to infer constraints on their
tectono-metamorphic history. Since most physical properties of crystals, such as elastic moduli, the coefficients of thermal
expansion, or chemical resistance to etching depends on crystal symmetry and orientation, the presence of texture imparts
directional properties to the polycrystalline material.
A major issue of mathematical texture analysis is the resolution of the inverse problem to determine a reasonable orientation
density function on SO(3) from measured pole intensities on
, which relates to the inverse of the totally geodesic Radon transform. This communication introduces a wavelet approach into
mathematical texture analysis. Wavelets on the two-dimensional sphere
and on the rotational group SO(3) are discussed, and an algorithms for a wavelet decomposition on both domains following the
ideas of Ta-Hsin Li is given. The relationship of these wavelets on both domains with respect to the totally geodesic Radon
transform is investigated. In particular, it is shown that the Radon transform of these wavelets on SO(3) are again wavelets
on
. A novel algorithm for the inversion of experimental pole intensities to an orientation density function based on this relationship
is developed. 相似文献
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Mathematical Geosciences - The majority of popular methods of prospectivity modeling refer to the raster mode of digital two-dimensional map images or three-dimensional geomodels, thus requiring a... 相似文献
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Comparison of Mathematical Methods of Potential Modeling 总被引：1，自引：0，他引：1
Helmut Schaeben 《Mathematical Geosciences》2012,44(1):101-129
Various attempts are known to turn the “catalogue” of mineral deposit models compiled by Cox and Singer (1986) operational, and have initiated activities called “potential mapping”, “potential modeling”, or “targeting”. The common
ultimate objective is to estimate the probability for a given location that a mineralization of a given type occurred. The
mathematics range from “weights of evidence” and others featuring a Bayesian approach to logistic regression by maximum likelihood,
and include other realizations by means of fuzzy methods, genetic programming, and artificial neural nets. Once developed
and coded, applications are not restricted to mineral prospection and exploration but include any kind of occurrences and
their estimated probabilities, e.g., risk assessment of land slides and many others. 相似文献
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The probability density function of orientations of crystals generally cannot be measured directly without destruction of the specimen. Therefore it is usual practice to sample pole density functions of several crystal forms in diffraction experiments with a texture goniometer. Determining a reasonable orientation density function from experimental pole density functions is then the crucial prerequisite of quantitative texture analysis. This mathematical problem may be addressed as a tomographic inversion problem specified by the crystal and statistical specimen symmetries and the properties of the diffraction experiment. Its solution with maximum entropy preferred orientation portion and maximum uniform portion is proposed because it yields the most conservative orientation density function with systematically reduced correlation effects, thus avoiding artificial texture ghost≓ components caused by the specific properties of the diffraction experiment. 相似文献
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Helmut Schaeben 《Mathematical Geosciences》2014,46(7):887-893
Despite a missing definition of equivalence of mathematical models or methods by Zhang et al. (Math Geosci, 2013), an “equivalence” (Zhang et al., Math Geosci, 2013, p. 6,7,8,14) of modified weights-of-evidence (Agterberg, Nat Resour Res 20:95–101, 2011) and logistic regression does not generally exist. Its alleged proof is based on a previously conjectured linear relationship between weights of evidence and logistic regression parameters (Deng, Nat Resour Res 18:249–258, 2009), which does not generally exist either (Schaeben and van den Boogaart, Nat Resour Res 20:401–406, 2011). In fact, an extremely simple linear relationship exists only if the predictor variables are conditionally independent given the target variable, in which case the contrasts, i.e., the differences of the weights, are equal to the logistic regression parameters. Thus, weights-of-evidence is the special case of logistic regression if the predictor variables are binary and conditionally independent given the target variable. 相似文献
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While crystallography conventionally presumes that a single crystal carries a unique crystallographic orientation, modern
experimental techniques reveal that a single crystal may exhibit an orientation distribution. However, this distribution is
largely concentrated; it is extremely concentrated when compared with orientation distributions of polycrystalline specimen.
A case study of a deformation experiment with a single hematite crystal is presented, where the experimental deformation induced
twining, which in turn changed a largely concentrated unimodal “parent” orientation distribution into a multimodal orientation
distribution with a major mode resembling the parent mode and three minor modes corresponding to the progressive twining.
The free and open source software MTEX for texture analysis was used to compute and visualize orientations density functions
from both integral orientation measurements, i.e. neutron diffraction pole intensity data, and individual orientation measurements,
i.e. electron back scatter diffraction data. Thus it is exemplified that MTEX is capable of analysing orientation data from
largely concentrated orientation distributions. 相似文献
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H. Schaeben 《Mathematical Geology》1982,14(3):205-216
Density estimation on the unit sphere by kernel methods may be conceived as a process of approximation by singular integrals. This concept aids in the solution of the main problems concerning the contouring of fabric diagrams. The optimal size of the counting element in Schmidt's method with respect to the mean integrated square error (MISE)of the density estimation is given. It proved that the optimal size is not only a function of the sample size but seriously depends on the smoothness of the density of directions on the sphere. In the light of approximation theory the Schmidt method of contouring is qualified as a moving average process; an example of a more refined density estimator is given. 相似文献