Probability uniformization and application to statistical palaeomagnetic field models and directional data

SUMMARY Weintroduceandapplytheconceptof2-Dprobabilityuniformizationtopalaeomagneticdirectional data. 2-D uniformization belongs to a very general class of probability transformations thatmapmultivariateprobabilitydistributionsintomultivariateuniformdistributions.Ourgoal is to produce joint tests of directional data sets assumed generated by a common statistical model, but with different sampling distributions. This situation is encountered when testing so-called Giant Gaussian Process (GGP) models of the Earth’s magnetic field against palaeomagnetic directional data collected from different geographical sites, the predicted sampling distributions being site-dependent. To introduce the concept, we first consider 2-D Gaussian distributions in the plane R 2 , before turning to Angular Gaussian and more general 2-D distributions on the unit sphere S 2 . We detail the approach when applied to the 2-D distributions expected for palaeomagnetic directional data, if these are to be consistent with a GGP model while affected by some Fisherian error. We finally provide some example applications to real palaeomagnetic data. In particular, we show how subtle inhomogeneities in the distribution of the data, such as the so-called right-handed effect in palaeomagnetism, can be detected. This effect, whether of geomagnetic origin or not, affects the Brunhes data in such a way that they cannot easily be reconciled with GGP models originally built with the help of these data. 2-D probability uniformization is a powerful tool which, we argue, could be used to build and test better GGP models of the mean palaeomagnetic field and palaeosecular variation. The software designed in the course of this study is available upon request from the authors. It can also be downloaded from http://geomag.ipgp.fr/download/PSVT.tgz.