Deformation analysis of the Earth crust based on manifold intrinsic geometry Case Study: Deformation analysis of the geodynamic network of Iran within 1999 - 2005

Unlike the classical deformation analysis of the Earth crust, which derives the planar and vertical strains separately, in this study, we have offered a method for 3-D deformation study based on intrinsic geometry of the manifolds on the topographic surface of the Earth. In this way, our method would be based on the 2-D metric tensor of horizontal deformation and 2-D curvature tensor of vertical deformation of the topographic surface of the Earth, which solve the problem of classical 2 and 1-D deformation study separately, while does not have the interpretation problem of extrinsic deformation study in 3-D space which results in 3-D strain tensors. From the derived metric tensor in our method two invariant deformations measures, i.e. dilatation (changes in the scale), maximum shear, and from curvature tensor two other invariant deformation measures, i.e. mean curvature and Gaussian curvature can be obtained. Our method algorithmically can be defined through its main computational steps as follows: (i) Computation of 3-D displacement vectors from repeated geodetic observations. (ii) Computation of the covariant and contravariant components of the displacement vector in the Gaussian moving frame. (iii) Discretization of the domain (Earth crust) in to finite surface elements. (iv) Computation of the strain and curvature tensors within the finite surface elements. As the case study, using repeated GNSS observations of the geodynamic network of Iran, crustal deformation within the coverage of the network is computed. The results show that the crust in most parts of the mentioned area is under contraction with the maximum value at South-West of the region. The maximum shear strain has also occurred in the southern part of the geodynamic network. The result of the vertical strain reveals uplift of the crust with maximum values at the South and South-East of the region. The result of the computation and the evaluations by comparison with the seismic map of the region show the success and usefulness of the presented method for deformation study of the curst.