Fitting smooth histories to rotation data

Consider two tectonic plates diverging at a mid-ocean ridge. Geophysicists are able to estimate the rotation of one plate relative to the other at a d iscrete sequence of times in the earth's history. also they usually have in formation as to the likely errors in these rotation estimates. We address t he problem of fitting a smooth history to such rotation data. We employ a m odification of the method used by Jupp and Kent in their 1987 article deali ng with fitting a smooth history to time-labeled points on the surface of t he unit sphere in three-dimensional space. They use parallel translation to "unroll" data from the surface of the sphere to a plane. We replace unroll ing via parallel translation by unrolling via left group multiplication, us ing the group structure of SO(3). We explain why our understanding of the e rrors in tectonic plate reconstructions dictates that left group multiplica tion is preferable both to parallel translation and to right group multipli cation. To choose the smoothing parameter we use the discrepancy method; fo r the Central Atlantic data set which we consider this method gives conside rably better results than cross-validation. (C) 2000 Academic Press.