AUTOMATED CALCULATION AND ERROR ANALYSIS OF FABRIC AND STRAIN ELLIPSOIDS FROM MULTIPLE SECTION DATA

Determination of the best-fit fabric ellipsoid from section ellipses is a common problem in structural geology. Numerous methods exist for determining the best-fit section ellipse, however calculation of the best-fit ellipsoid is overdetermined. Two direct non-iterative solutions are those of Robin (2002) and Shan (2008). Robin noted that each section ellipse differs by some error from a section of the solution ellipsoid, and derived a set of linear equations to determine the six independent parameters of the best-fit ellipsoid by minimizing the sum of squared norms of the "error matrices". Shan recognized that if the six ellipsoid parameters are represented as a six-dimensional space, the smallest eigenvector of the data matrix is the optimal solution. Shan's method additionally allows incorporation of lineation data. Robin's and Shan's methods are implemented for non-dilation cases in the free software EllipseFit. The solution ellipsoids are given as radii and directions. For each section the residuals between the data and solution are calculated, allowing identification of outlier section ellipses. Two error analysis techniques are implemented, one randomly perturbs the input data section parameters thousands of times to produce a cloud of solution ellipsoids, from which the confidence regions are determined. This does not use error estimates calculated from the section ellipses, and has similarities to the method used by Mookerjee and Nickleach (2011) for their iterative method. The second method uses bootstrap data files from section ellipses to construct a cloud of solution ellipsoids, from which the confidence regions are determined. This latter procedure has similarities to the method Webber and Klepeis (2018) used for Robin's method. Confidence regions are calculated for the ellipsoid parameters, and displayed on Nadia-Hsu and Flinn-Ramsay plots. Preliminary tests show that Robin and Shan's methods, and the two error analysis methods, give similar results.