On the least-squares fit of small and great circles to spherically projected orientation data |
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| Authors: | Norman H Gray Peter A Geiser James R Geiser |
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| Institution: | 1. Department of Geology and Geophysics, University of Connecticut, 06268, Storrs, Connecticut, USA
2. Department of Mathematics, S.U. N.Y., New Paltz, 12561, New Paltz, New York, USA
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| Abstract: | Given the direction cosines a i ′ = (a 1 i , a 2 i , a 3 i )corresponding to a set of pspherically projected fabric poles, an initial estimate x′ = (x1, x2, x3, x4)for the angular radius x4,and direction cosines of the center of the least-squares small circle which minimizes the sum of the squares of the angular residuals $$r = \sum\limits_p {\left {x_4 - \cos ^{ - 1} \left( {a_1^i x_1 + a_2^i x_2 + a_3^i x_3 } \right)} \right]} ^2 $$ can be iteratively improved by taking xj+1 = xj + Δxwhere xj is the value of xat the jth iteration and $$\Delta x = - H_j^{ - 1} \left {q_j + x_j \left( {x'_j H_j^{ - 1} x_j } \right)\left( {q_j - x'_j H_j^{ - 1} q_j } \right)} \right],$$ where As an initial approximation for xwe have found it convenient to ignore the fact that the data are constrained to lie on the surface of the reference sphere and to use the parameters of a least-squares plane through the given poles. Generalization of this approach to fitting variously constrained great and small circles is easily made. The relative merits of differently constrained fits to the same data can be tested approximately if it is assumed that the errors in the location of the poles are isotropic and normally distributed. It is thus possible to statistically assess the relative significance of conflicting structural models which predict different geometrical patterns of fabric elements. |
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