References

1. Tensor calculation: From displacement vectors on the grid, we calculate the strain rate tensor inside each square K. For each grid we compute the variation in length for the four sides and the two diagonals, we write the following equations : The calculation of tensor components uses the finite element method (Pagarete and al, 1990), for infinitesimal deformations the relative variations of each side between two epochs is written (Kasser and Thom, 1995; Welsch, 1983): Where: is tensor component in X, is tensor component in Y and is shearing component Respectively are the azimut, the length and the variation of length of the side k In this equation we have three unknowns His matrix coefficients following the equation (2) e is components vector U vector of the sex relative variation lengths of each segment of the mesh K Where: A new approach of mapping a strain rate field by tensors for crustal deformations studies EISSA Leila1,3,4, KASSER Michel² Email : leila.eissa@ensg.eu, michel.kasser@ensg.eu (1)LAREG (Laboratoire de recherche En Géodésie), (2)ENSG, 6 – 8 av Blaise Pascal, 77455 Marne la Vallée cedex 2, France, (3) universitéParis-Est, Cité Descartes 5, bd Descartes, Champs sur Marne 77454 MARNE LA VALLEE, (4)Tishreen University, Lattaquia Syria. • Introduction: • Thespacegeodesy measurements quantifiesthedeformation of the earth crust with an accuracy of level of mm per year. • Our aim is : • to assess the deformations for a geologically active site : tectonic zone, seismic zone, landslide etc … • to represent the strain rate by tensors evenly spread (independent ofreferenceframeandof location ofmeasurement sites). • to representthedegreeof significanceoftheobtaineddeformations. • To makethelinkbetweenthegeodeticmeasuresandthegeophysicalinterpretation. Computation and representation of errors: The variance analysis is particularly complex, given the unstability of the representation of the tensor form around the isotropic case, we therefore adopt a method of Monte-Carlo. We have two sets of available measurements on a given geodetic network with the resulting variances of the two compensations. We then synthesize for the two periods random sets of coordinates for each measuring point. Thus we calculate the corresponding tensors. We repeat this operation 100 times. Therefore as we have 100 datasets we compute 100 times the interpolation grid. At the end we will have 100 tensors components. From these datasets we estimate uncertainties of normal and shear components as well as uncertainty of maximum shear direction. Now we can represent the strain rate tensor components and their uncertainties on the same map. • Data sources: • Data sources to assess and represent the deformation of the crust are: • Displacementfields derivedfromspacegeodetic techniques such as (GPS, SLR,VLBI, DORIS). • Displacementfieldsfromcorrelationbetweentwoortho-images The determination of these three unknown requires at least three equations, but we have six equations (one for each side) so we solve by least square adjustment. We build the following matrix system: Example : We apply this methodology on the well known fault area of San Andreas fault in California. GPS EUREF Ortho-images correlation Conclusion: The tool in development will make the “connection” between the expertise of geodesists and that of geophysicists in order to acquire all available data, without any omission or artificially added ones. This will greatly facilitate the interpretation of phenomena, directly from a velocity field measured by space geodesy or any other techniques of measurement. This software can be integrated with GMT or at least will have an output GMT. Interpolation: In order to represent the rate tensors of region of interest, we must interpolate a grid covering the study area. Many interpolation methods have been tested such as: linear, bicubic, nearest neighbours and spline interpolations. We have conducted a test on a fictive site. Geographic situation Interpolation grid Tensor classical representation We note that it is difficult to interpret this kind of representation, furthermore we have no idea about the uncertainties of obtained tensors. So the heart of our work begins here : Aim : To represent the isotropic deformation and shearing part on the same map in an intuitive mapping. We want to represent these deformations with colour themes, and to combine them so as to ease the interpretation. Choice : A colour scale to represent the isotropic part (applied over the entire mesh) with maximum saturation and a segment of variable width to represent the shearing part, its value and its direction. The variation of the saturation will be used for the error analysis display. Combined solution with nearest neighbours and spline interpolations References J.Pagarete, m.Kasser, J.-C. Ruegg Évaluation et représentation des erreurs sur les déformations d’un réseau géodésique: utilisation de la méthode de Monte Carlo. Bull. Géod. 64 (1990) pp. 63-72 Hans-Gert Kahle, Marc Cocard, Yannick Peter, Alain Geiger, Robert Reilinger, Aykut Barka, and George Veis GPS-derived strain rate field within the boundary zones of the Eurasian, African, and Arabian Plates. J Geophys Res , VOL.105, NO.B10, PAGES 23,353-23,370, oct 10, 2000 Chung-Pai Chang, Tsui-Yu Chang, Jaques Angelier, Honn Kao, Jian-cheng Lee, Shui-Beih Yu strain and stress field in Taiwan oblique convergent system: constraints from GPS observation and tectonic data. Earth and planetary Science Letters 214 (2003) 115-127 A.Caporali, S.Martin, M.Massironi and S.Baccini state of strain in the Italian crust from geodetic data. Sébastien Leprince, Sylvain Barbot, François Ayoub, and Jean-Philippe Avouac IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 6, JUNE 2007 Spline interpolation Bicubic interpolation bicubic interpolation may give an unexpected discontinuity spline interpolation works well but it diverges when onemoves away from sites ofmeasurements Compromise:use spline to interpolate (insidetheconvex hull of measurement sites) and nearestneighbours to extrapolate (outsidethis convex hull) What is a strain rate tensor? Any deformation deforms an infinitesimal circle into an ellipse. Its axes are greater or smaller than the radius of initial circle. This can characterize the deformation. So a strain tensor is generally represented either by axes of the ellipse either by the triplet isotropic part, shearing part and direction of maximum shear. a classical strain rate tensor A