Bootstrap statistics for paleomagnetic data

The power and utility of paleomagnetic analyses stem largely from the ability to quantify such parameters as the degree of rotation of a rock body, or the orientation of an anisotropy axis. Until recently, estimates for uncertainty in these paleomagnetically determined parameters derived from assumptions concerning the underlying parametric distribution functions of the data. In many geologically important situations, the commonly used parametric distribution functions fail to model the data adequately and the uncertainty estimates so obtained are unreliable. Such essentials as the test for common mean require data sets consistent with a spherically symmetric underlying distribution; their application in inappropriate circumstances can result in flawed interpretations. Moreover, the almost universally used approximation for a cone of 95% confidence for the mean of a sample drawn from a Fisher distribution is quite biased even for moderate dispersions (K = 25). The availablity of inexpensive, powerful computers makes possible the empirical estimation of confidence regions by means of data resampling techniques such as the bootstrap. These resampling schemes replace analytical solutions with repeated simple calculations. We describe a bootstrap approach for the calculation of uncertainties for means or principal directions of paleomagnetic data. The method is tested on means of simulated Fisher distributions with known parameters and is found to be reliable for data sets with more than about 25 elements. Because a Fisher distribution is not assumed, the approach is applicable to a wide range of paleomagnetic data and can be used equally well on directions or associated virtual poles. We also illustrate bootstrap techniques for the discrimination of directions and for the fold test which enable the use of these powerful tests on the wider range of data sets commonly obtained in paleomagnetic investigations.