A physically meaningful model of vertical crust movements in Canada using smooth piecewise algebraic approximation: Constraints for glacial isostatic adjustment models

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Different kinds of analytical models of crustal deformation have both advantages and limitations, and are appropriate to different deformation patterns. However, in wide areas where multiple geophysical phenomenon are responsible for the deformation, such as in Canada, it would be a challenge to infer a physically meaningful model that accommodates different kinds of scattered geodetic data, while offering the optimum approximation to them. We develop an efficient method to automatically compute a smooth approximation of large functional scattered re-levelling data and historical tide gauge records given over Canada and northern US to thereby compile a unified map of Vertical Crustal Movements (VCM). The area of study is divided into patches and piecewise algebraic surfaces are fitted to 2D observation points and tilt between them, where constraints are enforced between the parameters of the surfaces. When the surfaces are fitted to the data, the set of constraints is imposed in such a way that rather than the surfaces being fitted sequentially, they are fitted simultaneously. The VCM model accomplished in this research is computationally demanding and numerically manageable. Enforcing the continuity and smoothness in the first derivatives throughout the surfaces, the VCM model highlights the long wavelength spatial variations of the crust in Canada, mainly due to Post Glacial Rebound (PGR). The rate of changes of orthometric height obtained from the map of VCM (H& ) is compared with the map of rate of gravity changes (ġ) in Canada (Pagiatakis and Salib, 2003). The PGR hinge line follows the same pattern in both maps and the close correlation between the map of VCM and ġ map is easily traceable and is in a fairly good agreement with theoretical model of Jachens (1978) in different areas. The VCM is also compared to geodetic height changes based on GPS solutions in Canadian Base Network (CBN) stations (Henton et al., 2006). This investigation shows disagreement with the GPS solution in Canadian prairies. In this study, some of theprobable causes of such inconsistencies are explored. VCM is also collated to theoretical predictions based on the published ICE-3G and ICE-4G loading history and on a model of Earth rheology characterized by stratified viscosity variations (Tushingham and Peltier, 1991; Peltier, 1994). In this study, a map of ratio between gravity changes to height changes is compiled and physically interpreted. Using VCM and geodetic height changes from CBN-GPS solution, a map of rate of geoidal height changes is also compiled and interpreted in some areas in Canada. The VCM constraints on Glacial Isostatic Adjustment (GIA) model parameters are investigated by varying, one at a time, two key parameters: 1) viscosity in different layers, and 2) the thickness of Laurentide ice over individual ice disks in Eastern Canada, and the Prairies, to obtain better fits to the VCM. In Eastern Canada, the VCM is consistent with an increase in the upper mantle viscosity. In The Great Lakes, the VCM has a better agreement with the predictions of GIA computed considering a lower viscosity for different layers of mantle. This study shows also that near the centre of rebound at Churchill, present day vertical crustal movement is most sensitive to the viscosity in the shallow part of the lower mantle and the transition zone (UM2). The VCM is consistent with a thinning of the Laurentide ice-sheet over the Prairies relative to both standard ice models. These analyses leads to better understanding of the trade-offs between Earth rheology and ice sheet history and hence some suggestions are made to improve postglacial rebound model.