Strain rate analysis over the Central Apennines from GPS velocities: the development of a new free software

Displacements or velocities obtained by GPS data processing over repeated surveys can provide useful information on tensional states of terrestrial crust, in those areas in which many stations well spatially distributed are present. In particular, the strain (or strain rate) can be computed over the nodes of a regular grid with suitable size to define a high density deformation field. A new method was deployed to generate easily and quickly the deformation pattern from GPS velocities and to evaluate the significance: values, related to an assigned grid point, can be truly considered only if the GPS stations are well distributed around it. The approach validation was performed by means of synthetic data derived from the theoretical displacement field generated by a Mogi model source. A complete analysis on the velocity pattern of the CaGeoNet network (Central Apennine chain, Italy) was performed providing strain rates and showing both extensional and compressive behaviour at the same values, along the Appennine chain axis. Key-words: GPS, CAGeoNet, velocity, strain, continuous. Introduction The GPS data are collected and processed in order to provide station coordinates within a few millimeters’ accuracy. The networks characterized by a high density point distribution, rarely allows the installation of permanent stations for a continuous monitoring on all vertices, but can provide a very interesting sub-regional and detailed study of the area by means of repeated surveys. In a previous study, Anzidei et al. (2005) estimated for the first time a significant compressive strain along the Central Apennine chain, from the CaGeoNet network data. The network extends from Tyrrhenian to Adriatic coasts, and the distribution of stations in the middle is very dense, with a mean relative distance of a few kilometers. In order to provide consistent results, independent from the choice of the reference frame, a strain rate estimation was performed through the development of a new software (the grid_strain MatlabTM program). It allows the definition of a deformation pattern with a chosen spatial distribution in the whole area over a regular grid. Several routines are applied during computation to accept or exclude local estimates at nodes around which GPS data points can be poorly distributed from the geometrical point of view. In the next paragraphs a complete and detailed application is described and results are analyzed to provide detailed kinematics of the investigated area. Theoretical aspect and approach validation Within a continuous material it is possible to define mathematical relationships between displacement (or velocities) and the acting deformation (or deformation rate). Introducing the 2D spatial displacement gradient L (Lij ) and free to choose a convenient reference system, planimetric movements around a point are given, at the first order as: , where U i j ij i U dx L u + = i are constants and dxj the infinitesimal increment along x and y axes. The gradient tensor ( ) can be separated in two parts j i ij u L , = Ω E L + = ( ij ij ij L ω e + = ), representing the strain and rotation effect respectively. The first matrix ( 2 / ) ( ji ij ij u u + = e ), symmetric, is the internal deformation while the second one ( 2 / ) ( ji ij ij u u − = ω ) is a whole-body rigid motion. The strain tensor diagonalization led to eigenvectors and eigenvalues which satisfy the relation k k k v v λ = E (k = 1, 2), providing the diagonal matrix Ed , in which k kk λ = Ed , by means of the transformation (Arfken, 1985; Wolfram, 2006): the columns of V are the two eigenvectors, that is EV V Ed 1 − = [ 2 1 v v = V ]. In the next the eigenvalues will be named emax and emin following the usual geophysical notation. In this new reference system no shear deformation is present and variations occur along the principal axes only. Taking into account the time dependence, using velocity data instead of displacements and the velocity gradient tensor instead of the displacement one, the same formula allow to define the strain rate (that is the strain for unit time). Going from continuous to discrete and from theory to application, using sparse velocity vectors estimated at each station of a GPS network, the equations become i n j ij n i U x L u + ∆ = ) ( ) ( , where ( ) 0 ) ( ) ( j n j n j x x x − = ∆ , xj0 is the coordinate of a reference point (for example the centre of mass) and n means one of the available N velocity data. Under the hypothesis of uniform strain field condition, and solving the redundant equation system with a linear least square inversion, the velocity gradient components are estimated together with their errors; the latter, as known, are obtained from the residuals between estimated values and observed ones. This method allows the determination of only one single value for horizontal deformation tensor in the whole investigated area. The approach previously described can be applied also over a dense distributed data set to attempt to define a continuous deformation pattern. Considering a regular grid, created by taking into account the points distribution (to define its dimension and characteristics), the estimation for a local strain can be performed over each node but using a suitable weighting strategy to automatically lower the contribution of stations distant from the node. All available data are involved in computation but errors are rescaled using an appropriate function, which increases with distance and is not divergent within a chosen area around the point. Following Shen et al. (2000), who used a similar method to fit a model of continuous deformation, we adopted the weight function , where d is the node-point distance and d0 is a parameter defined for rescaling (see below). Figure 1 was made to become familiar with the use of this smoothing function: it shows a family of curves in a 100 km range, for different d0, and it also suggests that a convenient choice for determining a local strain estimation (selecting observables near the point) can be to relate d0 with the mean inter-distance between stations. d0 d e / − Fig.1 Weight functions family for different d0: values go to zero, increasing the distance between point and node. In particular, at the distance d = 0.7d0 the error is doubled. Fig.2 The weight function decreases with d/d0 dimensionless distance: in the d0 around of each node the scaling factor ranges from 1