Geophys. J. Inf. zyxwvutsrqpon ( zyxwvutsrqponmlkjihgfe 1996) 126,495-504 zyxwvutsrqp Statistical analysis of palaeomagnetic inclination data Randolph J. Enkin' and Geoffrey S. Watson' Geological Suriley zyxwvutsrqpo of Canada-Pacijic, PO Box 6000, Sidney. BC, Canada. V8L 4B2, Canada Department zyxwvutsrqponm of Mathematics, Fine Hall, Princeton University, Princeton, NJ, 08544-1000, zyxwvu USA Accepted 1996 March 12. Received 1996 February 28; in original form 1996 May 11 SUMMARY Palaeomagnetic studies on bore core or on tectonically disturbed localities often lose declination information, but the inclination still offers important palaeogeographic information. While the arithmetic mean of inclinations, f, is a biased estimator, the bias is negligible with shallow data. Using co-inclination 6 = 90" - and precision z K* = llvariance, we find that the arithmetic mean and associated 95 per cent confidence interval are acceptable estimates when 6fi > 400". When inclination is steep and /or precision low, numerical methods must be applied. We develop the likelihood function for f3 and K and offer an efficient method to find its maximum, (8, R), and to calculate the confidence interval. When O& < 200°, the confidence interval is asymmetric about the mean. When sites are collected from several rigid blocks, the relative declinations within each block can be useful. Using 'block-rotation Fisher analysis', better inclination estimates with tighter confidence intervals can be made, even on very steep data. We describe how to apply these methods to an inclination-only fold test. The techniques are illustrated on real data and are tested extensively using numerical simulations. Key words: numerical techniques, palaeomagnetism. INTRODUCTION The time-averaged geomagnetic field is well approximated by a geocentric axial dipole, so that the mean inclination (I) is a simple function of the latitude of the measuring site. The goal of many palaeomagnetic studies is to determine the palaeo- latitude zyxwvutsrq (A) of a site at the time the studied rock sequence was magnetized. In the ideal case, there is sufficient geological information to restore samples to the orientations they had when they became magnetized. Sometimes these restorations cannot be made. Bore core, for example, is seldom azimuthally oriented, and, in orogenic belts, unknown vertical axis rotations can make it impossible to bring different sections into their original orientations. But so long as the attitude of the palaeohorizontal plane can be determined, rotation about the strike (i.e. simple horizontal-axis rotation) brings the inclination to its correct pre-tilt value, whatever the full structural correction might be; the measured palaeomagnetic declination may not be meaning- ful, but the inclination is well defined and may be used for tectonic reconstructions and other purposes. Because of secular variation of the geomagnetic field and measuring uncertainties, palaeomagnetic directions must be correctly averaged together to produce a geologically signifi- cant interpretation. When measurements of both inclination and declination are available ('full data'), one applies the techniques introduced by Fisher (1953). However, if only inclination is available ('inclination-only data') it is necessary to consider its statistical distribution so that estimates of the mean inclination and its confidence interval can be made. Since palaeomagnetic observations are distributed on a sphere, the arithmetic average of measured inclinations is biased towards shallow directions; that is, in any rotationally symmetrical distribution, there will be more inclinations shallower than the mean (nearer the equator) than steeper (nearer the pole) (Fig. la). The problem is aggravated when the mean is near vertical (high palaeolatitude) or when dis- persion is large, because some directions 'overshoot' the pole, biasing downwards the calculated average (Fig. lb). As pointed out by Briden & Ward (1966), if the true inclination is 90", all measured directions will have shallower inclinations and thus the arithmetic average will be less than 90". The averaging of inclination-only data has been dealt with before (Briden & Ward 1966 Kono 1980 McFadden & Reid 1982; Clark & Morrison 1983; Clark 1983, 1988; Cox & Gordon 1984). These workers assume, as we do, that palaeo- magnetic directions are distributed according to the Fisher (1953) distribution. Explicit estimates of the mean are not possible, and these workers obtain estimates using various analytical approximations and numerical solutions. These pro- cedures differ considerably, but most yield similar estimates of mean inclination and its confidence interval. All authors agree that estimates are very good, except for near-vertical or highly dispersed data. Cya 1996 RAS 495 Downloaded from https://academic.oup.com/gji/article/126/2/495/624038 by guest on 24 March 2022