GEOPHYSICAL RESEARCH LETTERS, VOL. 20, NO. 19, PAGES 2135-3137, OCTOBER 8, 1993 THE FOLD TEST IN PALEOMAGNETISM AS A PARAMETER ESTIMATION PROBLEM • GeoffreyS. Watson Department of Mathematics, Princeton University Randolph J. Enkin PacificGeoscience Centre,Geological Survey of Canada Abstract Most proposed fold testformulations use signifi- cance tests to try pre-tilting and post-tiltingremanence hypotheses. We suggest that it is betterto consider the fold testas a parameter estimation problem. Making the usual assumption that the distribution of remanence vectorswas originally roughly parallel, we propose, using a monte carlo simulation technique, to estimate the amountof tectonic tilting at the time of magnetization along with a 95% confidence interval. If, for example, this confidence interval included100% then one could not rule out pre-tilting remanence. In the older terminology, the fold test is positive. The k-ratio test of McElhinny [1964]is often said to be conservative in that if a study passes the k-ratiotest then it certainly passes a more rigorous test. We show with a typical counter-example that this assertion is incorrect. Observational uncertainty of bedding directions is easily included in this formulation. Introduction Determining the relativetiming of magnetic remanence andtectonic deformation is of central importance in paleo- magnetic studies.Paleomagnetists call the analysis of this problem the fold test. (A moregeneral term wouldbe the "tilt test", including mechanisms suchas fault block rota- tions. In this paperwe usethe term "fold test"but under- stand it in thegeneral sense.) A positive foldtest is a strong indication that remanence isprimary, and a negative foldtest shows that the rocks were remagnetized. The fold testwas introduced in Graham's [1949] funda- mental paper. Apart from seeing the future importance of paleomagnetism with remarkable clarity, he recognized that if rockswere stablymagnetized before folding, "restoring the beds to the horizontal causes the magnetizations to move towards parallelism with one another". If, on the contrary, themagnetization was acquired afterfolding, then thein situ remanence directions will be parallel at all sites and unre- lated to the orientation of thebeds. In his day there were no appropriate statistical methods. In an effort to ensure that these observations are not just due to random errors, McElhinny [1964] attemptedto formulate a statistical test of whether the parallelismof •Geological Survey of Canada Contribution 19593. Copyright 1993by the American Geophysical Union. Paper number93GL01901 0094-8534/93/93GL-01901503.00 remanence vectors is significantly stronger after (or before) beddingcorrection. His suggestion was that the Fisher [1953] concentration parameter, K, should be estimated from the data before and after tectoniccorrection,k• and k2 respectively. Thenthe ratioof these estimates is referred to a significance testdevised by Watson [1956] to test the null hypothesis K•= •2. If k2 / k• is above a critical value, dependingon the number of sites, the distribution is accepted to be "significantly" more clustered before tectonic correction. Unfortunately, this is an invalid application of Watson's test, as was pointed out by McFadden and Jones [1981], since k• and k2 are not independent statistical quantities but rather are related through the known bedding attitudes. While a large value of k2/k• is an indication of a positive fold test,there canbe no statistical test capable of evaluating significance sincevariations in the ratio are the result of untilting, not statistical fluctuations. The paleomagnetic community continues to useMcElhinny's formulation since it casts the problem in a simple, although incorrect way. Since it is impossible to test statistically whetherthe remanence is significantly moreclustered after (or before) tectonic correction, McFaddenand Jones [1981] (and most other recent papers on the subject) proposedseparate hypothesis tests on the data before andafter tectonic correc- tion to check whether the data is incompatiblewith remanence acquired when the bedding was in either con- figuration. For example, if the remanence was pre-tilting, onewouldexpect the remanence to be significantly incom- patible before tectonic correction, but not incompatible after untilting. If a hypothesis is tested at the95 % significance level, one expects thatin 5 % of the cases where the null hypothesis is true, the hypothesis will be rejected. These are type I errors, and their probabilityis set by the user. More problematic are typeII errors; times when a null hypothesis is in fact not true but is accepted all the same. In the fold test, this typically occurs whenbeds are not in the correct orientations but dips are small. Unfortunately, the probabil- ity of typeII errors depends on howbig thedeviation of the underlying distributions to the null hypothesis is. In other words,the probability that a post-tilting remanence gives a pre-tilting test statistic can not be known. A furtherdifficultywith a significance testas a fold test is that there are several situations where the remanence is not simply pre- or post-tilting.Several mechanisms exist: if remagnetization occurred during tilting, if a pre-tilting remanence is rotated by grain-scale deformation, if incorrect rotation axes are used to restore the beds to the paleo- horizontal, if the sites do not all record the sameage of 2135