Papers by Author: K.Gerald van den Boogaart

Abstract: A long time has past since the introduction of the harmonic method for the reconstruction of the ODF from polefigure measurements, and it has been replaced by discrete methods of inversion, because of its incapability with respect to ghost effects. The harmonic method is still not in its best possible state: it disregards the high order harmonics; it disregards measurement errors and therefore gives suboptimal results; it does not provide standard errors, neither for the C-coefficients nor for the ODF; and there are the ghost effects. However, the harmonic method is a well established inversion method and it can improved at these points. Statistical considerations based on geostatistics and a model of the unknown ODF as a random function in a Baysian approach yields an inversion method, which can be characterized as a smoothing spline method. This new method is statistically optimal among all linear methods and resembles favorable features of the harmonic method in an improved way. It provides an optimal linear reconstruction of the even part of the ODF. It does not truncate the harmonic series expansion at a fixed level, but accounts for every even harmonic space in an optimal way with respect to its signal to noise ratio in the polefigure measurements. The method applies for irregularly sampled and incomplete pole figures. The method accomplishes standard errors for the ODF and the C-coefficients. Discrete inversion methods, explicitly or not, reconstruct the odd harmonic part of the function based on the principle of maximum entropy. Based on the theory of exponential families a continuous odd part (and the truncated even part) can be computed based on the entropy principle and the C-coefficients estimated by the spline method.

Abstract: The determination of an ODF, C-coefficients, property tensors and portions of texture components from EBSD orientation measurements is afflicted with statistical errors introduced by incomplete sampling of the grains. Since the measurements are highly spatially correlated and stochastically dependent, classical sampling theory does not apply. A general statistical method for error estimation in the presence of stochastically dependent observations has been developed and applied to the most important quantities of texture analysis. The method is based on the assumption of a finite range of dependence between different measurements and on the estimation of the covariance in the observed set of orientation. The methods allows the computation of standard measurement errors and confidence limits for the mentioned texture quantities. It can be used for an objective decision whether two textures are statistically equal or not, based on the comparison of estimated ODFs. Further we can decide statistically whether the ODF obeys certain types of symmetry (e.g. whether it is a girdle textures or whether it is symmetric about the shear plane observed in the field).