GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 1.2 91 latitude, longitude, velocities in north, east, up and their standard deviations, and name of the GNSS site is required. Least squares collocation is a minimum variance algorithm based on the mathematical model in Eq. 1: (1) Given a covariance function C(d), for example isotropic, dependent on the squared distance between any two points, the unbiased north and east velocity components at site P are computed by a weighted average of the velocities of all the GNSS sites, withweight dependent on the formal uncertainties of the measured velocities and the relative distance between the computation point and the contributing GNSS site. The velocities of the GNSS sites are assumed uncorrelated, so that the weight matrix W is diagonal. Eq. 1 represents the algorithm used to interpolate the scattered velocities of the GNSS sites to any point P within the network, and estimate the variance of the interpolated velocity, given the variances of the measured velocities (Caporali et al. , 2013). To map the velocities into strain rates it is sufficient to differentiate the first equation in Eq.1 with respect to the north and east components: (2) We then obtain the strain rate matrix in geographical coordinates. The eigenvalues and eigenvectors are finally obtained by matrix diagonalization: (3) We take as positive the extensional strain rate and negative the compressional strain rate. The variances of the contributing velocities can likewise bemapped into variances of the strain rate components. In this way a control of the error propagation can be kept in a consistent manner. As mentioned earlier, we constrain the scale distance d 0 in the covariance function by imposing that the shear strain rate, in the sense of Savage and Simpson (1996) is maximized: