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 paleom agnetic data in plate tectonics, require estimates of the angles betwe en directions and confidence intervals for these angles. To give a set of interrelated simple methods, we approximate all estimated directio ns by appropriate concentrated Fisher distributions. In preliminary nu merical experiments we verify previous results for the distribution of the means of samples drawn from Fisher distributions and study the di stributions 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 me ans''). We find an approximate expression for the geometric bias in th is 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 tha t it works well for theta > 2 theta(crit). (When, for example, Rk = 25 0 for both samples, theta(crit) = 5.1 degrees.) We then show that the bias-corrected estimator of theta, theta = theta - B(theta*), is norm ally distributed with mean equal to the true angle and variance sigma( theta)(2) = theta(crit)(2). Thus one can construct a 95% confidence in terval for this angle with the formula (theta - 1.96 sigma(theta), th eta + 1.96 sigma(theta)). We show by extensive simulation that covera ge probabilities for this confidence interval are slightly conservativ e for theta > 2 theta(crit) and good for theta > 3 theta(crit). We der ive related results for angles between various combinations of fixed d irections and Fisher means. We find that the estimator alpha of the an gle between two great circles containing a known fixed direction and t wo different Fisher means is unbiased and normally distributed with va riance sigma(alpha)(2) = 1/(R(1)k(1) sin(2) theta(1)) + 1/(R(2)k(2) si n(2) theta(2)) where the theta(i) are the angular distances between th e Fisher means and the known fixed direction. Thus the confidence inte rval on this rotation angle is (alpha - 1.96 sigma(alpha), B + 1.96 si gma(alpha)). We give examples of applying these techniques to paleomag netic data analysis, especially for determining terrane motions. Our m ethods provide confidence intervals for poleward displacement. For rot ation, simulation suggests that they provide results at least as accur ate as previous methods [McWilliams, 1984; Demarest, 1983] over a wide range of relevant parameter values, while being simpler or more flexi ble. We apply these methods to estimating rates of apparent polar wand er and point out an additional bias due to dating errors.