The use of smooth piecewise algebraic approximation in the determination of vertical crustal movements in Eastern Canada

Abstract

The objective of this study is to compile a physically meaningful map of vertical crustal movements (VCM) for Eastern Canada. Average vertical velocities over the past century are determined by repeated precise levelling and monthly mean sea level observations from 17 tide gauges. The spatial vertical velocities may be mathematically expressed in any number of ways. In this study, the uplift rate is calculated using smooth piecewise algebraic polynomial approximation. The mathematical model of the approximation for the geodetic data is given. We show how a vertical velocity surface is approximated using piecewise algebraic polynomials and what conditions should be satisfied to guarantee the smoothness of the surface. First, we divide Eastern Canada into zones. The vertical movement is represented by a different polynomial surface in each zone. The polynomials are joined together at nodal points along the border of adjacent zones in such a way that a certain degree of smoothness (differentiability) of the resulting function is guaranteed. This study shows that piecewise polynomial surfaces can represent the available data in a unified map. The pattern of a northwest to southeast gradient of crustal movements is consistent with the existing Glacial Isostatic Adjustment (GIA) models. Present-day radial displacement predictions due to postglacial rebound over North America computed using VM2 Earth model and ICE-4G adopted ice history show a zero line (hinge line) very similar to ours along the St. Lawrence River. The main advantage of the presented technique is its capability of accommodating in one model, different kinds of information when the re-levelled segments are scattered not only in time but also in space. Piecewise approximations make it easier to get the physically meaningful details of the map, without increasing the degree of polynomials. © Springer-Verlag Berlin Heidelberg 2007.