Confidence limits and bias correction for estimating angles between directions with applications to paleomagnetism

Many problems in the earth sciences, particularly in the use of paleomagnetic data in plate tectonics, require estimates of the angles between directions and confidence intervals for these angles. To give a set of interrelated simple methods, we approximate all estimated directions by appropriate concentrated Fisher distributions. In preliminary numerical experiments we verify previous results for the distribution of the means of samples drawn from Fisher distributions and study the distributions of six estimators of the Fisher concentration parameter re. We selected Fisher's original estimator for kappa, k = (n - 1)/(n - R), for use in our subsequent simulations. We then study the bias in, and confidence intervals around, the estimator ti of the angle between the means of samples drawn from two Fisher distributions (''Fisher means''). We find an approximate expression for the geometric bias in this angle, B(theta) = - tan theta + [tan(2) theta + theta(crit)(2)](1/2) where theta(crit) = [1/(R(1)k(1)) + 1/(R(2)k(2))](1/2), and show that it works well for theta > 2 theta(crit). (When, for example, Rk = 250 for both samples, theta(crit) = 5.1 degrees.) We then show that the bias-corrected estimator of theta, theta* = theta - B(theta*), is normally distributed with mean equal to the true angle and variance sigma(theta)(2) = theta(crit)(2). Thus one can construct a 95% confidence interval for this angle with the formula (theta* - 1.96 sigma(theta), theta* + 1.96 sigma(theta)). We show by extensive simulation that coverage probabilities for this confidence interval are slightly conservative for theta > 2 theta(crit) and good for theta > 3 theta(crit). We derive related results for angles between various combinations of fixed directions and Fisher means. We find that the estimator alpha of the angle between two great circles containing a known fixed direction and two different Fisher means is unbiased and normally distributed with variance sigma(alpha)(2) = 1/(R(1)k(1) sin(2) theta(1)) + 1/(R(2)k(2) sin(2) theta(2)) where the theta(i) are the angular distances between the Fisher means and the known fixed direction. Thus the confidence interval on this rotation angle is (alpha - 1.96 sigma(alpha), B + 1.96 sigma(alpha)). We give examples of applying these techniques to paleomagnetic data analysis, especially for determining terrane motions. Our methods provide confidence intervals for poleward displacement. For rotation, simulation suggests that they provide results at least as accurate as previous methods [McWilliams, 1984; Demarest, 1983] over a wide range of relevant parameter values, while being simpler or more flexible. We apply these methods to estimating rates of apparent polar wander and point out an additional bias due to dating errors.