International Geomagnetic Reference Field — The Eleventh Generation
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The CHAOS3 geomagnetic field model and candidates for the 11th generation IGRF
Earth, Planets and Space volume 62, Article number: 1 (2010)
Abstract
As a part of the 11th generation IGRF defined by IAGA, we propose a candidate model for the DGRF 2005, a candidate model for IGRF 2010 and a candidate model for the mean secular variation between 2010 and 2015. These candidate models, the derivation of which is described in the following, are based on the latest model in the CHAOS model series, called “CHAOS3”. This model is derived from more than 10 years of satellite and ground observatory data. Maximum spherical harmonic degree of the static field is n = 60. The core field time changes are expressed by spherical harmonic expansion coefficients up to n = 20, described by order 6 splines (with 6month knot spacing) spanning the time interval 1997.0–2010.0. The third time derivative of the squared magnetic field intensity is regularized at the coremantle boundary. No spatial regularization is applied.
1. Introduction
The 11th generation IGRF is based on seven candidate models for DGRF 2005 and for IGRF 2010, and eight candidate models for the mean secular variation (SV) for 2010–2015. These candidate models have been submitted to IAGA working group VMOD in October 2009 and evaluated by the IGRF taskforce group (see paper by Finlay et al., 2010, this issue). In the following, we describe the derivation of three of the submitted candidate models, one for each of the above mentioned model groups. Our three candidates are based on the CHAOS3 field model (more specific: CHAOS3α), a new version in the CHAOS model series. Previous versions are CHAOS (Olsen et al., 2006), xCHAOS (Olsen and Mandea, 2008) and CHAOS2 (Olsen et al., 2009).
Compared to its predecessors, CHAOS3 is derived from more recent satellite data and using “revised observatory monthly mean values” (which are corrected for external and induced field contributions). In addition, a different regularization scheme is applied to the time changing part of the model.
During the evaluation procedure of the IGRF candidate models it has been found that the degree1 internal terms of the preliminary model version CHAOS3α—derived in September 2009 and used as parent model for our IGRF candidate models—for epoch 2010.0 are rather different from those of the other candidate models. This is most probably due to the fact that only data until summer 2009 have been used for model version CHAOS3α. It is therefore interesting to investigate the effect of including more recent data on the model behavior near 2010.0. The resulting final model, CHAOS3, allows us to make a further assessment of our model candidates (which are based on CHAOS3α) and also of the final adopted IGRF model for epoch 2010.0.
2. Data
For CHAOS3 we use Ørsted scalar data between March 1999 and December 2009, and vector data between March 1999 and December 2004; CHAMP scalar data between August 2000 and December 2009, and vector data between January 2001 and December 2009; and SACC scalar data between January 2001 and December 2004. The same data selection criteria as for the CHAOS2 model (Olsen et al., 2009) are applied.
To extend the model back in time beyond February 1999 we supplement the satellite data with annual differences of revised observatory monthly means of the North, East and Vertical downward components (X, Y, Z) for the time interval 1997.0–2009.5. However, contrary to CHAOS2, for which we have used monthly means calculated in the traditional way as the arithmetic mean of all data of a given month (e.g., Chulliat and Telali, 2007), we use revised monthly mean values for CHAOS3, to minimize the influence of external fields, which contribute to the traditional monthly means due to the way these are calculated. The revised monthly means are calculated by removing from the observatory hourly mean values the sum of:

a model of the ionospheric (plus induced) field as predicted by the CM4 model (Sabaka et al., 2004), parameterized by the 3monthly means of F_{10.7} solar flux, and

a model of the magnetospheric (plus induced) field as predicted by the external field part of CHAOS2, parameterized by the E_{st} and I_{st} indices.
After subtracting these field corrections from the observed hourly mean values we calculate the robust mean (using Huber weights with tuning constant of 1.5) of all hourly mean values of a given month, each observatory, and each of the three elements X, Y and Z. Finally, we take annual differences of the resulting revised monthly means (annual difference value at time t is obtained by taking the difference between those at t+6 months and t−6 months, thereby eliminating a remaining annual variation in the data). This yields 16,493 values of the first time derivative of the vector components, dX/dt, dY/dt, dZ/dt, for 137 observatories. The location of these observatories are shown in Fig. 1.
A preliminary version of CHAOS3, named CHAOS3 α, has been used as parent model for our IGRF candidates. These candidates were derived and submitted at the end of September 2009, when the very recent satellite and observatory data were not yet available. Therefore, this preliminary model (CHAOS3 α) is based on Ørsted and CHAMP satellite data until August 2009, respectively July 2009, and observatory data until 2009.0. Model parameterization is identical though to that of the final model CHAOS3, apart from the fact that the extended timespan of the final model leads to slightly more model parameters for describing the timebinned magnetospheric field and the timevarying CHAMP Euler angles, as explained in the following section.
3. Model Parameterization and Regularization
The time dependence of core field coefficients up to spherical harmonic degree n = 20 is described by order 6 Bsplines with a 6month knot separation and fivefold knots at the endpoints, t = 1997.0 and t = 2010.0. This yields 27 interior knots (at 1997.5, 1998.0,…, 2009.5) and 6 exterior knots at each endpoint, 1997.0 and 2010.0, resulting in 31 basic Bspline functions, M_{ l }(t). Internal field coefficients for degrees n = 21–60 are static. Timedependent terms (for degrees n = 1–20) and static terms (for n = 21– 60) together results in a total of 16,920 internal Gauss coefficients.
Largescale external (magnetospheric) sources are parameterized in a manner similar to the CHAOS2 model, with an expansion of the remote magnetospheric sources (magnetotail and magnetopause) in Geocentric Solar Magnetospheric (GSM) coordinates (up to n = 2) and of near magnetospheric sources (magnetospheric ring current) in the Solar Magnetic (SM) coordinate system (also up to n = 2). The time dependence of degree1 magnetospheric terms in SM coordinates is parameterized by the E_{st} and I_{st} indices (Maus and Weidelt, 2004; Olsen et al., 2005). In addition, we solve for largescale timevarying degree1 external coefficients in bins of 12 hours length (for ), or 5 days length (for ), similar as for the CHAOS2 model. This gives a total of 6,411 external coefficients (6,151 for CHAOS3 α, due to the slightly shorter time span).
As part of the field modeling we also perform an inflight instrument calibration and solve for the Euler angles of the rotation between the coordinate systems of the vector magnetometer and of the star sensor providing attitude information. For the Ørsted data, this yields two sets of Euler angles, while for CHAMP we solve for Euler angles in bins of 10 days (i.e. 213 sets of angles). This yields additional 3 × (2 + 213) = 639 model parameters (618 for CHAOS3α). The total number of model parameters is 16,920 + 6,411 + 639 = 23,970 (23,689 for CHAOS3 α).
These model parameters are estimated by means of a regularized Iteratively Reweighted LeastSquares approach using Huber weights, minimizing the cost function
where m is the model vector, the residuals vector e = d_{obs} − d_{mod} is the difference between observation d_{obs} and model prediction d_{mod}, and is the data covariance matrix. Details can be found in Olsen et al. (2009). and are block diagonal regularization matrices which constrain the third, respectively second, order time derivatives of the core field. minimizes the mean squared magnitude of , integrated over the core surface dΩ (radius c = 3485 km) and averaged over time:
Regularization of the third time derivative alone leads to highly oscillating field behavior. To avoid this we also minimize at the core surface at the model endpoints t = 1997.0 and 2010.0. This is implemented via the regularization matrix . Note that only acts on 12 (the first and last six) of the 31 spline basis functions. The parameters λ_{3} and λ_{2} control the strength of the regularization. We considered several values for these two parameters and finally selected λ_{3} = 1 (nT/yr^{3} )^{−2} and λ_{2} = 10 (nT/yr^{2} )^{−2}.
This regularization is different from that used for CHAOS2, for which the time average of the second time derivative, , is minimized at the core surface. However, we also derive a model using the same model regularization as CHAOS2s, but applied to the extended data set of CHAOS3. In the following sections, this model is referred to as the “extended CHAOS2s” model.
4. Results and Discussion
The total number of data points, residual means and root mean squared (rms) values of the two model versions are listed in Table 1. Means and rms are the weighted values calculated from the model residuals e = d _{obs} − d_{mod} using the Huber weights w obtained in the last iteration.
The CHAOS3 rms misfits for the satellite data are slightly lower than those of the CHAOS3 α model, and lower than those of CHAOS2 (cf. table 1 of Olsen et al. (2009). Compared to CHAOS2, most significant is the decrease of the observatory misfit by a factor 2 for the horizontal components and and by about 30% for , which is probably due to the use of revised monthly mean values compared to traditional monthly means. In addition to the lower rms misfit, the nonzero means of the observatory and found in CHAOS2 are no longer present in CHAOS3, indicating external field contributions in the traditional monthly means that were used for CHAOS2. The revised monthly means are obviously less contaminated by external field contributions.
Figure 2 shows power spectra of the first time derivative (secular variation, circles) and of the second time derivative (secular acceleration, asterisks). CHAOS3 has considerably higher secular acceleration power at degrees n > 6 compared to the two versions of CHAOS2s, due to the fact that the third time derivative of the field (and not its second time derivative) is regularized. In addition, CHAOS3 shows a secular acceleration power rather similar in 2005 and 2010. The rather different secular acceleration power of the two versions of CHAOS2s at epoch 2010 is due to the fact that this epoch is beyond the data span used for CHAOS2s (only data until March 2009), while data until December 2009 were used for the updated version CHAOS2s.
Following the approach described in Mandea and Olsen (2006) and Olsen and Mandea (2007) we use CHAMP satellite data to calculate “virtual observatory” monthly mean values from January 2001 to December 2009, from which we derive time series of Gauss coefficients and . The CHAMP satellite covers all local times within 132 days, which is roughly 4 months. To minimize the effect of external currents that depend on local time, we have applied a 4month running mean to the Gauss coefficients. This clearly reduces the scatter of the coefficients. Figures 3 and 4 (updates of figures 3 and 4 from Olsen and Mandea (2007)) show time series of the first time derivative, and , of the internal Gauss coefficients for n = 1–6. The symbols present annual differences of the coefficients while the curves show model values from CHAOS2s (red), CHAOS3 α (green) and CHAOS3 (blue). The magenta curve is for the “extended CHAOS2s” model. The scatter of the individual monthly solutions (black dots) is largest for the lowerorder, and especially for the zonal coefficients, indicating the influence of the ionospheric field contributions in the polar ionosphere. The semiannual variations in some degree1 coefficients (e.g., ) is probably also caused by contributions from polar ionospheric currents. While all four models are in good agreement before 2008, there are considerable differences towards 2010.0. The largest differences are observed for the coefficients and , as shown in Table 2, for which the values given by CHAOS3 are in a much better agreement to the final IGRF11, compared with CHAOS3α.
The time change of the sectorial coefficients which represent lowlatitude field changes, are better resolved, as indicated by the lower scatter. The temporal variations for the lower degrees are generally well described by the spline models (and especially by CHAOS3). However, higher degree sectorial terms (e.g., ) show rapid field fluctuations that are less well described by the models because of the applied temporal regularization, which increases with degree n. Obviously smallscale lowlatitude rapid fluctuations exist but are not captured by the present spline models.
A comparison of the annual differences of the observatory monthly means and model estimations are shown in Figs. 5 and 6 for the 16 selected observatories indicated by red symbols in Fig. 1. These observatories are arranged according to their geographical latitude from North to South. The black symbols are the annual differences of the revised monthly means, used both for CHAOS3 and CHAOS3α, while the green symbols represent revised monthly mean values calculated after determination of CHAOS3α. These values have only been used for estimating CHAOS3, but not for CHAOS3α.
For comparison, annual differences of the traditional monthly means (calculated as arithmetic mean of all values of the month) are shown with light blue symbols. Compared to the traditional monthly means these revised monthly means have considerably less scatter; a reduction by a factor of three is not unusual (Olsen, 2009). Their use allows cleaner extraction of the core field signal. It also allows rapid field changes like geomagnetic jerks to be studied, not only in the Eastcomponent Y (as is typically done) but also in the other two components X and Z (which are more contaminated by magnetospheric sources and therefore require a careful removal of external fields.) However, despite our attempt to remove ionospheric and magnetospheric fields when calculating revised monthly means there is probably still some contamination from external sources. The peculiar behavior of around 2003 (a year of exceptionally high geomagnetic activity) at the observatories MMB, KAK, PHU, KOU, HUA and HER is an indication of this. From the parent model CHAOS3 α our three candidate models for IGRF11 have been derived in the following way:

our candidate model for DGRF2005 is the degree n = 1–13 part of CHAOS3 α computed for the epoch t = 2005.0;

our candidate model for IGRF2010 is the degree n = 1–13 part of CHAOS3 α computed for the epoch t = 2010.0 (note that 2010 is the last spline knot of this model, but since only data until August 2009 have been used to determine the parent model CHAOS3 α, an extrapolation in time beyond the data span is performed);

our candidate model for an average secular variation from 2010.0 to 2015.0 is the degree n = 1–8 part of the first time derivative of CHAOS3 α computed for the epoch t = 2010.0.
Using satellite and observatory data (selected and processed as indicated before), high quality models of the recent geomagnetic field have been developed. They provide a detailed picture of the internal field, of the core field secular variation and secular acceleration at Earth’s surface. Moreover, over the last decade, these models have dramatically improved the core field description at the coremantle boundary and brought new insights into core dynamics.
Coefficients and data sets for the CHAOS3 α and CHAOS3 model versions are available at www.space.dtu.dk/files/magneticmodels/CHAOS3/.
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Acknowledgments
The Ørsted Project was made possible by extensive support from the Danish Government, NASA, ESA, CNES, DARA and the Thomas B. Thriges Foundation. The support of the CHAMP mission by the German Aerospace Center (DLR) and the Federal Ministry of Education and Research is gratefully acknowledged. We would like to thank the staff of the geomagnetic observatories and INTERMAGNET for supplying highquality observatory data, and Susan MacMillan for providing us with preliminary observatory hourly mean values for 2009. The work by MM is considered as IPGP contribution 3071.
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Olsen, N., Mandea, M., Sabaka, T.J. et al. The CHAOS3 geomagnetic field model and candidates for the 11th generation IGRF. Earth Planet Sp 62, 1 (2010). https://doi.org/10.5047/eps.2010.07.003
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DOI: https://doi.org/10.5047/eps.2010.07.003
Key words
 Geomagnetic reference model
 IGRF/DGRF
 spherical harmonic analysis