GNGTS 2013 - Atti del 32° Convegno Nazionale

Finally, to model in time, we use the penalized cubic b-splines (de Bor, 2001). This kind of temporal functions provide more realistic temporal variation than the classical polynomial or sinusoidal functions. Modeling process. In order to know if the present temporal and spatial distribution of the magnetic satellite data allows us to obtain a robust geomagnetic main field model for the European region, we first carried out a test using synthetic data. The synthetic data were obtained from the global model “Comprehensive Model ver. 4”, CM4 (Sabaka et al. , 2004) which was developed by using all the different magnetic sources which characterize the present geomagnetic field, and has a temporal validity from 1960 to 2002. The CM4 model gives information about the internal field (global degree n ≤ 13), crustal field (global degree n > 13) and the external field, and was obtained simultaneously modeling all these contributions at the same time, from this the term “comprehensive” model. As indicated in Eq. (3), we need an initial set of the time-dependent model coefficients. We have fixed the initial values as g 0 0 = -15000 nT and g k m = 0 nT for k >0. The spatial resolution of the model is given by the selection of the degree n k and the size of the spherical cap 0 . In order to test the influence of these two parameters in the modeling process, we have used different values of them. According to the selection area, i.e. the European continent, the cap was centered at 45ºN, 15ºE choosing different half angles of 36º, 43º, 50º and 56º. The degree k in Eq. (1) was fixed between 5 and 8, providing a more or less constant global degree n k of 13 (see Table 1). We have performed different tests changing the spatial parameters but keeping constant the influence of the external field up to degree K EXT = 1. The cubic B-splines control the temporal resolution of the modeling approach. In our work, a set of knot points every 0.5 yr for the temporal interval from 1999.0 to 2005.0 was used. Tab. 1 – Tests with different index K INT and half-angle 0 ( K EXT is maintained equal to 1) and corresponding residuals given as root mean squares (RMS) in nT for the different main field (SCHA and CM4) models; n k is automatically determined by K INT , 0 and the boundary conditions and is comparable with the maximum degree of a typical global model of the main field. K INT K EXT 0 n k RMS X RMS Y RMS Z RMS F RMS XYZF 5 1 36 13.3 58,265 36,033 62,901 75,889 60,018 6 1 43 13.1 6,748 5,055 7,232 8,255 6,920 7 1 50 13 0,713 0,573 0,836 0,996 0,795 8 1 56 13.1 0,271 0,239 0,248 0,334 0,275 After synthesizing all the vector and total intensity data at the same location and time of the real satellite data, we have applied the Eq. (3) to obtain the time-dependent model coefficients. In this case, the synthetic data include both main and crustal fields. The results, not shown here, were extremely good: our models were able to reproduce very well the characteristics of the crustal field provided by the CM4 model in this region. Moreover, these tests also indicated that the best pair of spatial parameters ( K INT , 0 ) was 7 and 50º respectively. These values were selected taking into account the final root mean square (rms) of the models and the different trade-off curves of the spatial and temporal norms of the geomagnetic field given in Eqs. (4) and (5). After the validation of the method, our next step was to apply it to the real satellite data. In this case, we have to pay a special attention for modeling both internal fields, i.e. the main and the crustal field. In order to model the main and crustal fields two different subsequent inversions have been carried out. We first used a K INT = 7 (with 0 = 50º, n k is approximately 13) to model the main field and K EXT =1 to discriminate an eventual external field. After generating the time- dependent model of the main field using eq. 3, we subtracted the model predictions at the real 166 GNGTS 2013 S essione 3.2