GNGTS 2013 - Atti del 32° Convegno Nazionale

(1) where and are the spherical cap harmonic coefficients that determine the model ( INT / EXT stand for internal and external contributions, respectively), are associated Legendre functions that satisfy the appropriate boundary conditions (null potential or co-latitudinal derivative at the border of the cap) and have integer order m and real, but not necessary integer, degree n k ( k is an integer index selected to arrange, in increasing order, the different roots n for a given m ). The number of the coefficients depends on the maximum spatial indexes of expansion K INT and K EXT . This technique was introduced by Haines (1985) for geomagnetism but it has been successfully applied also in gravity and crustal field studies. In this study, we will try to apply this technique to model the crustal field of the European area using only magnetic satellite data. First, we have to model both vector and total intensity data. For this reason, we apply an iterative approach to establish a linear relation between the model coefficients and the intensity data. The geomagnetic field elements are defined as a non-linear function which depends on the model coefficients as: (2) where the vector contains all the model coefficients and is the error which is assumed as Gaussian. To find the optimal set of the model coefficients, we chose the regularized weighted least square inversion applying the Newton-Raphson iterative approach (Gubbins and Bloxham, 1985): (3) where is the matrix of parameters which depends on the spherical cap harmonic functions in space and time (the so-called Frechet matrix). is the data error covariance matrix (the inverse matrix of weights) and is the vector of differences between the input data and modeled data for the i -th iteration. The and matrices are the spatial and temporal regularization matrices, respectively, with damping parameters and . The index i indicates the number of the iteration, which requires a first initial solution . Then, we define a spatial roughness that depends on the norm of the geomagnetic field B 2 in terms of the model coefficients (see Korte and Holme, 2003): (4) where t s and t e are the initial and final epoch respectively, and d is the differential solid angle over the spherical cap at the Earth’s surface (radius a ). The temporal roughness is defined in terms of the second derivative of the geomagnetic field as: (5) Both matrices, and , are diagonal in the global case due to the orthogonality of the basis functions over the sphere. However, for spherical caps the two sets of basis functions involved in the SCHA technique (with indices k – m = odd and k – m = even, respectively) are not orthogonal among themselves (Haines, 1985) and both matrices have non-diagonal elements (see Korte and Holme, 2003 for a review). 165 GNGTS 2013 S essione 3.2