 Article
 Open Access
The second generation of the GFZ Reference Internal Magnetic Model: GRIMM2
 V. Lesur^{1}Email author,
 I. Wardinski^{1},
 M. Hamoudi^{1} and
 M. Rother^{1}
https://doi.org/10.5047/eps.2010.07.007
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2010
 Received: 18 December 2009
 Accepted: 2 July 2010
 Published: 31 December 2010
Abstract
We present the second generation of the GFZ Reference Internal Magnetic Model (GRIMM2), that was derived for the preparation of the GFZ candidate for the 11th generation of the IGRF. The model is built by fitting a vector data set made of CHAMP satellite and observatory data, spanning the period 2001.0 to 2009.5. The data selection technique and the model parametrization are similar to that used for the derivation of the GRIMM model (Lesur et al., 2008). The obtained model is robust over the time span of the data. However, the secular variation above spherical harmonic degree 13 becomes less controlled by the data and is constrained by the applied regularization before 2002 and after 2008.5. At best, only the spherical harmonic degrees 3 to 6 are robustly estimated for the secular acceleration. The problem associated with the first two spherical harmonic degrees of the secular acceleration model arise from the difficulty in separating the core field signal from the external fields and their internally induced counterparts. The regularization technique applied smoothes the magnetic field model in time. This affects all spherical harmonic degrees, but starts to be significant at spherical harmonic degree 5.
Key words
 Geomagnetism
 core field modeling
 IGRF
1. Introduction
The ongoing mission of the CHAMP satellite provides very high quality vector measurements of the Earth’s magnetic field which have led in the recent past to numerous studies about its external and internal sources (Reigber et al., 2005). In 2009, the satellite has been revolving in an orbit of very low altitude (≃320. km). This, combined with the fact that the external magnetic field perturbations were small due to a very long period of low solar activity, serve studies of internal fields. Furthermore, in view of providing the best possible data set for the preparation of the 11th version of the IGRF, fully processed CHAMP vector data have been made available up to 2009.5. This motivate a renewed effort in modeling of the core magnetic field to high spatial and temporal resolution.
In this study we develop the second generation of the GRIMM model, covering the years 2001 to 2009.5. In its prior version (Lesur et al., 2008), the GRIMM model has been built from a data set made of CHAMP satellite data covering years 2001 to 2006 and hourly mean values obtained from 132 geomagnetic observatories. The data selection process was set to optimize the model time resolution. The GRIMM model showed a general agreement in mapping the temporal and spatial characteristics of magnetic field features with the series of CHAOS models (Olsen et al., 2006b, 2009), which were mainly based on satellite data from CHAMP and Ørsted. The data selection criteria are different for both modeling approaches, but they rely on a similar parameterization of the temporal evolution of the magnetic field, and consequently revealed a rapid fluctuation of the secular acceleration. Such detailed and robust description of the secular acceleration of the Earth’s magnetic field allows to investigate processes driving the temporal evolution of the field itself such as changes of the flow inside the liquid outer core. Other core magnetic field models are available (e.g. Maus et al., 2006; Thomson and Lesur, 2007) but with a different time parameterization that does not allow a continuous mapping in time of the secular acceleration.
We use the new large CHAMP data set to derive the second generation of the GRIMM model—GRIMM2, and to derivate our IGRF candidate model. In this work some questions regarding the robustness of secular acceleration estimates are addressed. Although the GRIMM series of models aim to model all aspects of the magnetic field of internal origin, we present here only results concerning the field generated in the Earth’s core. An associated model of the lithospheric field is available, but it has been derived independently from the core field model.
The next section is dedicated to briefly describe the data selection techniques, the model parameterization of GRIMM2, and the model estimation techniques. The modifications introduced for this second generation of the model are highlighted. The third section presents the obtained model which then is discussed in the fourth section.
2. Data Set, Data Selection, Model Parameterization and Model Estimation
2.1 Data set, data selection
The model GRIMM2 is built from CHAMP satellite magnetic vector data, and observatory hourly mean vector data. The most recent version 51 Level2 CHAMP satellite data span the epochs 2001.0 to 2009.58 and include improved time dependent FGMASC orientation corrections (i.e. orientation of the fluxgate magnetometers relative to the reference frame defined by the star cameras). Observatory hourly mean data are only used up to 2009.0.

Positive value of the Zcomponent of the interplanetary magnetic field (IMFB_{ Z }) to minimize possible reconnection of the magnetic field lines with the Interplanetary Magnetic Field (IMF).

20 s minimum between sampling points such that the nonmodeled lithospheric field does not generate correlated errors between data points.

Local time between 23:00 and 05:00, and the sun below the horizon at 100 km above the Earth’s reference radius (a = 6371.2 km), to minimize the contribution from the magnetic field generated in the ionosphere.

Norm of the Vector Magnetic Disturbances (VMD, Thomson and Lesur, 2007) less than 20 nT and norm of its time derivative less than 100 nT/day.

High accuracy of the FGM magnetometer readings (quality flag 1 set to 0) and dual starcamera mode (quality flag 2 set to 3).

Star camera outputs checked and corrected (Flag digit describing the attitude processing technique larger than 1).

Data are selected at all local time, and independently of the sun position.

Data sampled in singlecamera mode are used.
The residual mean (M) and standard deviation (SD) for all data types in nT, where Sat. and Obs. stand for satellite and observatory, respectively. HL corresponds to high latitudes and SM to the SM coordinate system.
Data types  Camera mode  Number of data  First run  GRIMM2  

SD  M  SD  
Sat. X (SM)  Dual  568361  3.7  0.04  2.69 
Sat. Y (SM)  Dual  568361  3.9  −0.43  3.15 
Sat. X (HL)  Dual  1100140  48.2  1.37  44.25 
Sat. Y (HL)  Dual  1100140  53.8  0.44  49.51 
Sat. Z (HL)  Dual  1100140  20.2  −0.91  17.98 
Sat. X (HL)  Single  332643  62.8  −6.05  59.81 
Sat. Y (HL)  Single  332643  73.3  −0.35  69.81 
Sat. Z (HL)  Single  332643  26.7  −1.51  24.64 
Obs. X (SM)  —  345446  3.3  0.06  3.32 
Obs. Y (SM)  —  345446  3.5  0.02  3.45 
Obs. X (HL)  —  102695  19.7  −1.38  19.13 
Obs. Y (HL)  —  102695  11.3  0.05  11.14 
Obs. Z (HL)  —  102695  17.3  0.22  16.98 
The same selection criteria as for mid and lowlatitudes satellite data are applied to hourly mean data of geomagnetic observatories (downloaded from the Word Data Center (British Geological Survey—BGS, Edinburgh)). From the 148 observatories with available data for the epochs of interest, the data of 18 observatories were rejected, because of a strong contamination with instrumental noise, baseline jumps and drifts. We point out that we are using hourly mean data, without further processing.
The data selection criteria used here are very similar to those used in the derivation of the first generation of GRIMM and have proven to lead to robust and accurate core field models.
2.2 Model parameterization
The model parameterization has been simplified compared to the GRIMM model. We do not attempt here to estimate the toroidal magnetic field generated by the field aligned currents, nor the field generated in the ionosphere at high latitudes. It has been found during the derivation of GRIMM that modeling these contributions only marginally improve the fit to the data, mainly because the temporal parameterization does not provide a useful description of the variations caused by processes in high latitude ionosphere and field aligned currents. Furthermore, coestimating these magnetic fields and the core magnetic field carries the risk that part of the core field is explained in terms of these external fields and vice versa (Lesur et al., 2008). Also, the lithospheric field is only coestimated up to Spherical Harmonic (SH) degree 30 together with the other components of the field. Apart from the lithospheric field, the model includes the core field, a representation of the large scale external fields and their associated internally induced counterparts. The crustal offsets at observatory locations are also estimated.
2.3 Model estimation
In order to obtain a robust core field model for the entire data time span, constraints have to be applied on the model parameters. As we will see below, the spatial complexity of the obtained SV model is not robust for SH degrees higher than 13. Normally constraints should be applied to control the spatial complexity in order to have an acceptable model at high SH degrees. Because of the fact that the modeled SV at high SH degrees shows reasonable behavior in some regions, i.e. mid latitudinal regions, we abandoned the idea of using a spatial constraint in the derivation of GRIMM2.
The model estimation is made in three successive steps. After computing data densities on a quasiregular triangular mesh, a first rough model is obtained by a leastsquare fit, where a satellite datum is weighted depending on the density of the triangular cell it belongs to. The resulting fit to the data is given in Table 1. In the second step, the leastsquare fit to the data is made using as weights the inverse of the error variances estimated in the first run. We did not introduced a specific treatment to handle the anisotropic error of singlecamera mode data because their errors are dominated by the unmodeled contributions of the magnetic field generated by field aligned currents. As the problem is linear, a large range of damping parameter can be investigated rapidly. Once acceptable values of the damping parameters are set, the model is further improved by five runs done using a reweighted leastsquares algorithm with an L_{1} measure of the misfit. These runs are computationally demanding because the set of normal equations has to be recalculated each time. The starting model used is the output of the second step. By doing only five iterations, the iterative process is not fully converged, but we verified that the part of the model associated with the core field does not vary significantly if further iterations are made.
3. Results
4. Discussion
The GRIMM2 model was presented in the previous sections. We now turn to the problem of estimating the robustness of the model. In particular it is compared with CHAOS2s (Olsen et al., 2009), a recent model covering nearly the same time span.
The effect of the constraints is seen on the SA spectrum from SH degree 7. At SH degrees higher than 8 the SA of the unconstrained model is unrealistically large and the obtained SV model oscillates in time around the average value given by the constrained model.
The errorbars shown in Fig. 7 are known to be underestimated (Lowes and Olsen, 2004) and furthermore the model inversion process is regularized. However, they show that is much less constrained than coefficients at higher SH degrees. By a close inspection, one can see that its formal standard deviation estimates have an annual periodicity due to external field contributions to the magnetic data. It is not clear if the observed rapid variation of the is due to a signal coming from the core or if it originates in the external fields or its induced counterpart. As seen in Fig. 3, these rapid oscillations of the contributes significantly to the temporal variation of the SA energy, but they do not affect significantly the patterns mapped in Fig. 5.
The first temporal derivatives of the other Gauss coefficients present clear and sharp changes in their temporal evolution, generally around 2006 and 2007. These changes are well above the noise level as it could be estimated from the errorbars. We particularly point out the change of slope of the just before 2008 that set some challenges for the prediction of the SV over the coming years, and the strong increase of absolute value of coefficient. This increase from 2006 is the main contribution to the increase in SV energy already observed in Fig. 3.
5. Conclusion
We have presented the second generation of the GFZ Reference Internal Magnetic Model (GRIMM2). As for the first generation, the model has been derived to provide an accurate description of the core field, its temporal behavior and in particular of the secular acceleration. The data set used covers the years from 2001.0 to 2009.5. The core field model is reasonably accurate over the data time span. The Bspline functions used allow the calculation of field values from 2000 to 2011, however, users should be particularly careful when extrapolating the model outside the time interval [2001:2009.5] . We have seen that the secular variation model is controlled by the regularization applied for the first and last year of the model, i.e. it should be used with caution outside the time interval [2002:2008.5]. Above SH degrees 12 or 13 the power spectra of the secular variation model diverges and regularization is required. As an example, a regularization can follow the tapering approach applied in Wardinski et al. (2008) to obtain estimates of spatially controlled small scale SV.
Regarding the secular acceleration we see that it evolves rapidly and reaches absolute values as large as 25 nT/y^{2}. There is no evidence of repetitive patterns, strengthening or weakening of the acceleration. However, the regularization applied affects the acceleration as early as SH degree 4 or 5, and some work is still required before one can downward continue the acceleration model at the core mantle boundary.
This model has been used as the parent model for the GFZ candidate to the 11th version of the IGRF. The model can be downloaded together with some FORTRAN 95 softwares at http://www.gfzpotsdam.de/magmodels.
Declarations
Acknowledgments
We would like to acknowledge the work of CHAMP satellite processing team and of the scientist working in magnetic observatories. We particularly thanks R. Holme and an anonymous reviewer for their constructive comments. IW was supported by the European commission under contract No. 026670 (EC research project MAGFLOTOM).
Authors’ Affiliations
References
 Lesur, V., I. Wardinski, M. Rother, and M. Mandea, GRIMM—The GFZ Reference Internal Magnetic Model based on vector satellite and observatory data, Geophys. J. Int., 173, doi:10.1111/j.1365246X.2008.03724.x, 2008.Google Scholar
 Lowes, F J. and N. Olsen, A more realistic estimate of the variances and systematic errors in spherical harmonic geomagnetic field models, Geophys. J. Int., 157, 1027–1044, 2004.View ArticleGoogle Scholar
 Maus, S., M. Rother, C. Stolle, W. Mai, S.C. Choi, H. Lühr, D. Cooke, and C. Roth, Third generation of the Potsdam Magnetic Model of the Earth (POMME), Geochem. Geophys. Geosyst., 7, Q07008, doi:10.1029GC001,269, 2006.View ArticleGoogle Scholar
 Olsen, N., R. Haagmans, T. J. Sabaka, A. Kuvshinov, S. Maus, M. E. Purucker, M. Rother, V. Lesur, and M. Mandea, The Swarm EndtoEnd mission simulator study: Separation of the various contributions to the Earth’s magnetic field using synthetic data, Earth Planets Space, 58, 359–370, 2006a.View ArticleGoogle Scholar
 Olsen, N., H. Lühr, T. Sabaka, M. Mandea, M. Rother, L. TøffnerClausen, and S. Choi, CHAOS—A model of the Earth’s magnetic field derived from CHAMP, Oersted, and SACC magnetic satellite data, Geophys. J. Int., 166(1), 67–75, doi:10.1111/j.1365246X.2006.02959.x, 2006b.View ArticleGoogle Scholar
 Olsen, N., M. Mandea, T. J. Sabaka, and L. TØffnerClausen, CHAOS2 —a geomagnetic field model derived from one decade of continuous satellite data, Geophys. J. Int., 179(3), 1477–1487, doi:10.1111/j.1365246X.2009.04386.x, 2009.View ArticleGoogle Scholar
 Reigber, C, H. Lühr, P. Schwintzer, and J. Wickert (Eds.), Earth Observation with CHAMP Results from Three Years in Orbit, SpringerVerlag, 2005.Google Scholar
 Thomson, A. and V. Lesur, An improved geomagnetic data selection algorithm for global geomagnetic field modelling, Geophys. J. Int., 169, 951–963, doi: 10.1111/j.1365246X.2007.03354.x, 2007.View ArticleGoogle Scholar
 Wardinski, I., R. Holme, S. Asari, and M. Mandea, The 2003 geomagnetic jerk and its relation to the core surface flows, Earth Planet. Sci. Lett., 267, 468–481, doi:10.1016/j.epsl2007.12.008, 2008.View ArticleGoogle Scholar