The Bootstrap for Magnetic Susceptibility Tensors

Constable, C, Tauxe L.  1990.  The Bootstrap for Magnetic Susceptibility Tensors. Journal of Geophysical Research-Solid Earth and Planets. 95:8383-8395.

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In studies of the anisotropy of susceptibility or remanence of paleomagnetic samples it is conventional to specify the anisotropy in terms of the parameters of the anisotropy ellipsoids, namely the directions of the principal axes of the ellipsoid and their associated eigenvalues. Confidence intervals for these parameters have in the past often been estimated by using a linearization scheme to propagate the effect of small changes through the eigenvalue decomposition. The validity of these approximations is explored using a Monte-Carlo simulation from measurements that are presumed normally distributed, showing that there are circumstances in which the linearization scheme gives confidence intervals that are much too small. Q-Q plots indicate that the common assumption that the noise in the measurements is Gaussian does not always hold. Because of these shortcomings in the conventional technique we propose using a bootstrap resampling scheme to find empirically the distribution of uncertainties in the results. Confidence intervals for the eigenvalues are found directly from their empirical distributions. For the principal axes, approximate elliptical regions of confidence on the unit sphere are parameterized in terms of the Kent or FB5 distribution. The number of modes observed in the distribution of eigenvalues obtained by bootstrapping is used to classify the shape of the susceptibility ellipsoid as spherical, oblate, prolate or triaxial. The empirical nature of the bootstrap technique allows the extension of the analysis of uncertainties to parameters derived from the principal susceptibilities, such as percentage anisotropy or shape factor.