reGionAl modelinG of The GeomAGneTic field in euroPe usinG sATelliTe And Ground dATA: GeoloGicAl APPlicATionse. qamili1, f.J. Pavón-carrasco1, A. de santis1,2, m. fedi3, m. milano3

1 Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy2 Università G. D’Annunzio, Chieti, Italy3 Università Federico II, Napoli, Italy

Introduction. The geomagnetic field varies in time and space. In order to describe these spatial and temporal variations, systematic surveys are necessary. Before the satellite era, continuous records at magnetic observatories and repeat station measurements were the principal resources of the geomagnetic field studies. But the Earth’s surface is not uniformly covered by these ground magnetic measurements (especially over the seas). If we want to model properly the geomagnetic field, more data are necessary to cover the area under study and these data should be also distributed at different heights, in order to better sample the entire space of interest. For this reason, satellites play an outstanding role for geomagnetic field measurements and modeling. Different satellite missions (i.e. Magsat, Ørsted, Champ, SAC-C etc.) have produced high quality geomagnetic data with global coverage and high spatial resolution (Matzka et al., 2010). Taking advantages of the high quality of the satellite data, in this work, we want to see if it is possible to model both main and anomaly Earth’s magnetic fields using an appropriate satellite dataset, with a particular objective in mind: to construct a reliable European model of the geomagnetic anomaly field. To do this, we have here selected a set of 6 years (1999-2005) of data (vector and total intensity) from the Ørsted, CHAMP and SAC-C satellites, which is the same dataset used for deriving the CHAOS-4 model (Olsen et al., 2010). In order to properly model the above-mentioned contributions of the geomagnetic field, appropriate selection criteria should be applied to these satellite data. Data selection based on geomagnetic indices and local time minimizes the influence of the external fields. When considering the data selection procedure, it is necessary to take into account also the characteristics of the region under study.

Considering what said above, we considered data that satisfy the following stringent criteria:1. at the European latitudes it is required that the Dst index (that measures the strength of

the magnetospheric ring-current) does not change by more than 2 nT/hr, 2. at non-polar latitudes is required that the geomagnetic activity index should be Kp ≤ 2o, 3. data sampling interval applied is 60 sec, 4. only data from dark regions (sun 10° below horizon) were considered; 5. for the polar region, data are selected satisfying Em < 0.8mV/m.Here we have tried to model the main and crustal fields by means of the Spherical Cap

Harmonic Analysis (SCHA) in space. The crustal field is time invariant so it does not need to be modeled in time, while the changes in time of the main field are modeled by means of the classical penalized cubic B-splines, covering the indicated time interval. We will see below that these techniques provide optimal representation of the internal field over the area of investigation.

Methodology. Modeling the global geomagnetic field at the Earth’s surface and above is usually approached using the well-known Spherical Harmonic Analysis (SHA). However, global techniques do not provide in general higher resolution when the study is carried out in a restricted part of our planet, and then, a regional approach is usually the most plausible mathematical technique. A useful contribution for such a regional analysis is given by the Spherical Cap Harmonic Analysis (SCHA) introduced for the first time by Haines (1985). The SCHA is a powerful analytical technique for modeling the Laplacian potential and the corresponding field components over a spherical cap. The solution of Laplace’s equation in spherical coordinated (r,θ, ) for the magnetic potential V due to internal and external sources over a spherical cap can be written as an expansion of non-integer degree spherical harmonics (Haines, 1985):

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(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 nk (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 KINT and KEXT. 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 B2 in terms of the model coefficients (see Korte and Holme, 2003):

(4)

where ts and te 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).

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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 g0

0 = -15000 nT and gkm = 0 nT for k>0. The spatial resolution

of the model is given by the selection of the degree nk 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 nk 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 KEXT = 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 KINT and half-angle 0 (KEXT 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; nk is automatically determined by KINT, 0 and the boundary conditions and is comparable with the maximum degree of a typical global model of the main field.

KINT KEXT 0nk RMSX RMSY RMSZ RMSF RMSXYZF

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 (KINT, 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 KINT = 7 (with 0= 50º, nk is approximately 13) to model the main field and KEXT =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

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input data to obtain in this way the crustal field. This crustal field represents the magnetization content in the continental and oceanic lithosphere at satellite altitude and its values do not reach absolute values higher than 20 nT.

Then, we used these residual values to model the crustal field using a no-time-dependent model with KINT = 35 (nk is approximately 65). In this model we apply the spatial regularization norm with damping parameter equal to 10-5 nT-2 and the approach is not iterative. The penalized B-cube splines are not used in this part of the work as well as the constraint of the temporal norm of the geomagnetic field.

The results are plotted in Figs. 1 and 2. The Fig. 1 shows two different profiles at 450 km of altitude in the north-south and west-east directions centered at 45ºN, 43ºE. We have chosen this value of altitude because is approximately the mean altitude of the different satellite orbits. For comparison we also show the model prediction of the CHAOS-4 model (Olsen et al., 2010). This global model was developed by using the same set of satellite data including ground data

Fig. 1 – Model prediction (red) at 450 km from Earth’s surface at 45°N (a) and 15°E (b) compared with CHAOS-4 model (in blue).

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and covering the time interval 1997 – 2011. As we can see, our regional model presents a higher spatial var-iability than the global CHAOS-4 model due to the regional character-istic of our model. This characteristic is presented for all the geomagnetic field components and for the total field intensity.

In order to see the spatial predic-tion of the model, we plot different crustal field maps in Fig. 2. The crus-tal field model was also calculated at 450 km of altitude and was compared with the prediction given by the CHAOS-4 model. Our results show a clear agreement with the global mod-el, but present more details in terms of the spatial wavelength.

Upward continuation transforms anomalies measured at one surface into those that would have been measured at a higher altitude sur-face. In the simplest case, i.e., lev-el-to-level, upward continuation is merely a convolution of the original data, performed in either the space or Fourier domain. We made a lev-el-to-level continuation of the Earth’s surface CM4 data to 450 km. We first transformed the data coordinates from spherical to Cartesian, then used a maximum entropy extrapola-tor to realize a square map and final-ly extended the grid by 10% with a periodic extrapolator. The results are shown in Fig. 3.

Fig. 2 – Maps of the X, Y, Z and F (from top to bottom) crustal field at 450 km as modeled by a set of real satellite data (right) and the crustal field calculated with CHAOS-4 model (left).

Fig. 3 – Level-to-level upward continuation of total intensity F. The CM4 data at ground level were continued to 450 km (a) with a level-to-level convolution filter. Only little differences (c) among CM4 and continued data occur in a wide central area (within 1 nT), while major differences appear at the edges, i.e., where the edge-effect and the errors for not having taken into account the Earth’s curvature are stronger. However, by changing border extrapolator (not shown here) we observed that the edge-effect changes led the continued field to be improved in some areas and to get worse in others. This should demonstrate that most of the error is due to the edge-effect.

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Conclusions. The aim of this work was to find a useful technique in order to construct a European model of the main and crustal magnetic fields from real geomagnetic satellite data.

Here, we analyzed different sets of magnetic data, first a set of synthetic data from CM4 global model in order to validate the analysis procedure, and then a real satellite data set covering a period from 1999 to 2005.

The regional model we present here shows a clear agreement with other global models like CM4 and CHAOS-4, but presents more details in terms of the spatial wavelength. We found that both the datasets allow us to model very well the crustal field, at 450 km height when comparing with the global CHAOS-4 model in the European region. Also, a level-to-level upward continuation to 450 km of the CM4 data at ground level works reasonably well, proving the efficiency of present level-to-level techniques in a satellite-altitude framework. references de Boor, C. 2001. A Practical Guide to Splines. Springer, New York, p. 368. Gubbins, D., and Bloxham J. 1985. Geomagnetic field analysis - III. Magnetic fields on the core-mantle boundary,

Geophysical Journal of the Royal Astronomical Society 80, 695-713.Haines, G. V. 1985. Spherical cap harmonic analysis, Journal of Geophysical Research, 90 (B3), 258-2591.Korte, M., and Holme, R. 2003. Regularization of spherical cap harmonics, Geophysical Journal International 153, 253–

262.Matzka, J., Chulliat, A., Mandea, M., Finlay, C., Qamili, E., 2010. Geomagnetic Observations for Main Field Studies: from

Ground to Space. Space Science Reviews 155 (1-4), 29-64.Olsen, N., Luehr, H., Sabaka, T. J., Michaelis, I., Rauberg, J., Tøffner-Clausen, L. 2010. CHAOS-4: A high-resolution

geomagnetic field model derived from low-altitude CHAMP data, AGU Fall Meeting.Sabaka, T.J., Olsen, N., and Purucker, M.E. 2004. Extending comprehensive models of the Earth’s magnetic field with

Ørsted and CHAMP data. Geophysical Journal International 159, 521–547, 2004.

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