Figure 2. Spatial Distribution ISOPORIC MODELLING OF THE GEOMAGNETIC FIELD COMPONENTS AT THE EPOCH PENINSULA AND BALEARIC ISLANDS. V.M. Marin(1), F.J. PAVON CARRASCO(1, 2) and I. Socias(1) (1) Servicio de Geomagnetismo. Instituto Geografico Nacional. C/General Ibanez de Ibero, Madrid. Spain (2) Dpto. Fisica de la Tierra I: Geofisica y Meteorologia. UCM Madrid. Spain The Earth's magnetic field is a potential field resulting of the contribution of different fields: ones of internal origin, as is the main field (whose origin is the external core), the local field that causes the magnetic anomalies and the induced fields in the crust. Others of external origin, mainly due to the Earth-Sun interaction. These fields suffer temporary variations of different periods and amplitudes. The long period variations are related to the internal origin field and receive the name of Secular Variation (SV). This variation is also spatial, so if we want to study this phenomenon is necessary to determine its characteristics in the zone from the data obtain at the repeat stations. There exist many methods for modelling SV: in a global way (data of the whole Earth) or in case we have data distributed over a region, we can use regional adjustments. These generally, give a model of better resolution than the global ones for the area of study, because of high density data in the area itself. Here we are going to analyze two regional methods, the polynomial method, that adjusts the SV to a polynomial of second degree of the repeat stations coordinates respect to mean values, and a second method that applies the spherical cap harmonic analysis (SCHA) to a cap that sufficiently covers the zone. The SV can be in a first approximation separated in a temporal and a spatial part, that can be analyzed separately. The temporary part in a station and during a short period of time adjusts to a polynomial of second degree, of the form: E= E 0 + A·t + B· t 2 where E 0 is the geomagnetic component value at time origin and A and B coefficients are calculated by mean squares adjustment of the equation, being these related with secular variation. The geomagnetic field values have been obtained from the repeat station network that the IGN has over the Peninsula and Balearic islands and from San Pablo Toledo observatory. Table 1. Temporal Distribution 3-REGIONAL MODELLING METHODS. APPLICATION TO IBERIA Polynomial Method The method fits the geomagnetic field components to a second degree polynomial distribution as a function of the repeat stations coordinates. This method has already been applied in Iberia (Ardizone,1998) If E is the component of the geomagnetic field or their SV, the equation is (Barraclough and Clarke, 1988):  E i = A 0 + A 1 (  i -  0 ) + A 2 ( i - 0 ) + A 3 (  i -  0 )( i - 0 ) + + A 4 (  i -  0 ) 2 + A 5 ( i - 0 ) 2 Fitting this expression to the observed data we determine the model coefficients for each component (table 2) as well as their statistical parameters (figure 3). The method has two main limitations, the first is that the Laplace equation is not fulfilled over the studied region and the second is that it is not possible to take into account the repeat stations altitude. Figure 3- Standard deviation against number of data SCHA Method The global geomagnetic field can be represented by means of spherical harmonic analysis, for it is supposed that on the regions there are no electrical currents nor magnetic sources. Then the Laplace equation is fulfilled The solutions are the spherical harmonic (orthogonal functions on the sphere depending on colatitude and longitude). This is useful for global models, but if the study is restricted to a given area, we can use the spherical harmonics over a cap (Haines, 1985). If we consider only the contribution of the main field, the solution is very similar to the global one being g and h the SCHA coefficients that define the model, obtained by mean square fitting (Haines, 1985). This method has already been applied on Iberia (Torta,1990) Figure 4- Standard deviation against number of data Figure 5-Isoporic maps of Iberia for Left polynomial model, right SCHA model coefficients ∆X(nT/year)∆ Y(nT/year)∆ Z(nT/year) A0A A1A A2A A3A A4A A5A kmn k (m ) ∆g m k (nT/year)∆h m k (nT/year) CONCLUSSIONS rms error Polynomial (nT/year) SCHA (nT/year) rms X rms Y rms Z rms XYZ Table 4- Rms error of both models for individual and combined components. In the polynomial model, the coefficients (A 0, A 1, A 2, A 3, A 4 and A 5 ) are independent for each component X,Y and Z that define the Geomagnetic Field, so the global polynomial model is defined by a total of 18 coefficients (6 for each component). The SCHA technique allows a joined model of the SV of the three components through 9 coefficients (the number of coefficients in a spherical harmonic development is a function of K: n=(K+1) 2, as in this case K=2 then N=9). Table 4 shows the comparison of rms errors (root mean square, sum of the standard deviation divided by the degrees of freedom of the model) of the isoporic fittings obtain with the two different models, the polynomial and the SCHA. The error of each component (rms x, rms Y, and rms Z ) is slightly smaller in the polynomial fitting, that is due to the independent fitting of each magnetic component in that model. However, the global rms error (rms xyz ) of the three components together is smaller in the SCHA model than in the polynomial, because the number of coefficients in the first is 9, while in the polynomial it goes up to 18. A study of standard deviation in both models a bigger concentration of error close to cero in the SCHA theory than in the polynomial fitting, as can be seen in figures 3 and 4, the error distributions are characteristics of a gaussian distribution. The coefficients of the polynomial technique have no geophysical meaning, while in the SCHA technique, he coefficients SCH (gmk and hmk) are related with the coefficients of the spherical harmonic global development (SHA, Spherical Harmonic Analysis), necessary in the geomagnetic field spectral study and of the contributions of the different magnetic fields to the terrestrial magnetism. The geomagnetic field secular variation modeling allows the elaboration of the geomagnetic maps of Spanish mainland and Balearic Islands, transporting the magnetic components data from earlier years ( since for X, Y and Z components and since for magnetic declinations) up to , obtaining finally the geomagnetic maps for Spanish mainland and Balearic Islands for the epoch (IGN, 2006). 1-SECULAR VARIATION 2- SECULAR VARIATION CALCULATION 5- REFERENCES Ardizone, J.A. (1998): "Análisis de datos aeromagnéticos. Metodologías y aplicación al levantamiento aeromagnético de España peninsular". Instituto Geográfico Nacional, Publicación técnica nº 32. Barraclough D.R and E.Clarke, (1988):”Statistical analysis of geomagnetic variations” BGS Tec. Rep. WM/88/15C, 62pp. De Miguel, L. (1980): "Geomagnetismo". Instituto Geográfico Nacional. Haines, G.V. (1985): "Spherical cap harmonic analysis". J. Geophys. Res., 90, Haines, G.V. (1988): "Computer programs for spherical cap harmonic analysis of potential and general fields". Computers & Geosciences., 14 No.4, IGN, Servicio de Geomagnetismo: "Mapa Geomagnético de España, ".. Torta, J.M. (1990): "Modelización regional del campo geomagnético sobre España: campo anómalo, variación secular y campo de referencia". Publicaciones del Observatorio ∆ ∆ ∆ ∆ ∆ y scha ∆ z scha