Evaluation of candidate geomagnetic field models for IGRF12
 Erwan Thébault^{1}Email author,
 Christopher C. Finlay^{2},
 Patrick Alken^{3, 4},
 Ciaran D. Beggan^{5},
 Elisabeth Canet^{6},
 Arnaud Chulliat^{3, 4},
 Benoit Langlais^{1},
 Vincent Lesur^{7},
 Frank J. Lowes^{8},
 Chandrasekharan Manoj^{3, 4},
 Martin Rother^{7} and
 Reyko Schachtschneider^{7}
Received: 7 April 2015
Accepted: 14 June 2015
Published: 19 July 2015
Abstract
Background
The 12th revision of the International Geomagnetic Reference Field (IGRF) was issued in December 2014 by the International Association of Geomagnetism and Aeronomy (IAGA) Division V Working Group VMOD (http://www.ngdc.noaa.gov/IAGA/vmod/igrf.html). This revision comprises new spherical harmonic main field models for epochs 2010.0 (DGRF2010) and 2015.0 (IGRF2015) and predictive linear secular variation for the interval 2015.02020.0 (SV20102015).
Findings
The models were derived from weighted averages of candidate models submitted by ten international teams. Teams were led by the British Geological Survey (UK), DTU Space (Denmark), ISTerre (France), IZMIRAN (Russia), NOAA/NGDC (USA), GFZ Potsdam (Germany), NASA/GSFC (USA), IPGP (France), LPG Nantes (France), and ETH Zurich (Switzerland). Each candidate model was carefully evaluated and compared to all other models and a mean model using welldefined statistical criteria in the spectral domain and maps in the physical space. These analyses were made to pinpoint both troublesome coefficients and the geographical regions where the candidate models most significantly differ. Some models showed clear deviation from other candidate models. However, a majority of the task force members appointed by IAGA thought that the differences were not sufficient to exclude models that were well documented and based on different techniques.
Conclusions
The task force thus voted for and applied an iterative robust estimation scheme in space. In this paper, we report on the evaluations of the candidate models and provide details of the algorithm that was used to derive the IGRF12 product.
Keywords
Geomagnetism Field modeling IGRFFindings
Introduction
The International Association of Geomagnetism and Aeronomy (IAGA) released the 12th generation of the International Geomagnetic Reference Field (IGRF) in December 2014 (Thébault et al. 2015). The IGRF is a series of standard mathematical models describing the largescale internal part of the Earth’s magnetic field between epochs 1900 A.D. and the present (see for instance Macmillan and Finlay 2011 for a review and Finlay et al. 2010a for the preceding generation). It is used by scientists in a wide variety of studies including longterm dynamics of the Earth’s core field, space weather (e.g., Bilitza and Reinisch 2008), local magnetic anomalies imprinted in the Earth’s crust, or land surveying. It is also used by commercial organizations and individuals as a source of orientation information for drilling or navigation (Meyers and Minor Davis 1990) and has become of increasing interest for space science during the last decade. A task force appointed by IAGA approved the specifications of IGRF12 and issued a call in May 2014. The call requested candidate models for the main field (MF) for the Definitive Geomagnetic Reference Field for epoch 2010 (DGRF2010), for a provisional IGRF model for epoch 2015 (IGRF2015) both to spherical harmonic (SH) degree 13, and for a prediction of its annual rate of change, the secular variation (SV), over the upcoming 5 years (SV20152020) to SH degree 8. The term “definitive” is used because any further substantial improvement of these retrospectively determined models is unlikely. In contrast, the provisional IGRF model will eventually be replaced by a definitive model in a future revision of the IGRF when the community will have more complete knowledge concerning the Earth’s magnetic field for epoch 2015.0.
Candidate models to IGRF12 and participating organizations
Summary of submitted candidate models for IGRF12  

Team  Model DGRF2010  Model IGRF2015  SV20152020  Organization  Publication 
A  YES  YES  YES  BGS  Hamilton et al. 2015 
B  YES  YES  YES  DTU Space  Finlay et al. 2015 
C  YES  YES  YES  ISTerre  Gillet at al. 2015 
D  YES  YES  YES  IZMIRAN   
E  YES  YES  YES  NGDCNOAA  Alken et al. 2015 
F  YES  YES  YES  GFZ  Lesur et al. 2015 
G  YES  NO  YES  NASA/GSCF   
H  NO  YES  YES  IPGP/CEA/LPG Nantes  
I  NO  YES  YES  LPG Nantes  Saturnino et al. 2015 
J  NO  YES  NO  ETH Zürich   
The first section of this paper summarizes the statistical criteria used by the task force members and the evaluators for the testing and the intercomparison of the candidate models to IGRF12. The results of this analysis that served as the basis for internal discussion are then detailed. The task force chair prepared a ballot paper containing a selection of weighting options that was voted on by the task force in December 2014. The last section gives some details of adopted weighting scheme. The resulting IGRF12 coefficients were prepared and checked before being made available to the public through the IAGA Div V, WG VMOD webpage http://www.ngdc.noaa.gov/IAGA/vmod/igrf.html before 1 January 2015.
Mathematical definitions and criteria used in evaluations

Difference between models

Weighted mean model

Spherical harmonic power spectrum and total root mean square vector field

Azimuthal power spectrum

Sensitivity matrix

Spherical harmonic correlation
which is, following the standard definition of the Pearson’s correlation coefficient, the covariance between two models divided by the product of their standard deviation. It gives 1 for a full positive correlation, 0 for no correlation, and −1 for a full anticorrelation degree per degree.
Evaluation of main field candidate models
Analysis of DGRF2010 candidate models
Summary of DGRF2010 candidate models submitted to IGRF12
DGRF candidate models for main field epoch 2010  

Team  Model  Organization  Data  Comments (parent model etc.) 
A  DGRF2010A  BGS  Ørsted; CHAMP; Swarm A, B, C;  Based on parent model using order 6 Bsplines 
Observatory hourly means  with 1 year knot spacing  
B  DGRF2010B  DTU Space  Ørsted; CHAMP; SACC; Swarm A, B, C;  Based on CHAOS5 
Observatory monthly means  using order 6 splines with 6 months spacing  
C  DGRF2010  ISTerre  Ørsted; SACC; CHAMP; Swarm B  Based on COVOBS.x1 
observatory monthly mean  using order 4 Bsplines with 2 year spacing  
D  DGRF2010D  IZMIRAN  CHAMP 2009.02010.75  Spherical Harmonics for each day 
no data selection but numerical filtering  then linear regression centered on 2010.0  
E  DGRF2010E  NGDCNOAA  CHAMP  Based on parent model using quadratic expansion 
F  DGRF2010F  GFZ  CHAMP from 2009.0 to 2011.0  Based on parent model using order 6 Bsplines 
USTHB/EOST  observatory hourly means  with 6 month spacing  
G  DGRF2010G  NASA/GSFC  Ørsted; CHAMP; SACC  Based on CM5 
observatory hourly means  using order 5 Bsplines with 6 month spacing 
RMS vector field differences _{ i,j } R in units nT between DGRF2010 candidate models and also between candidates and the arithmetic mean reference models M and median reference model M _{med} in the rightmost columns. The bottom two rows are simple arithmetic means \({~}_{i}\!\,\overline {R}\) of the _{ i,j } R where the means include all candidates
_{ i,j } R / nT  A  B  C  D  E  F  G  M  M _{med} 

A  0.00  2.76  6.61  4.01  2.40  2.55  4.92  1.97  1.70 
B  2.76  0.00  7.06  4.91  2.04  2.37  5.38  2.50  1.96 
C  6.61  7.06  0.00  7.28  6.72  7.66  5.81  5.45  5.99 
D  4.01  4.91  7.28  0.00  4.27  4.52  5.66  3.53  3.72 
E  2.40  2.04  6.72  4.27  0.00  2.42  4.81  1.92  1.51 
F  2.55  2.37  7.66  4.52  2.42  0.00  5.47  2.68  2.18 
G  4.92  5.38  5.81  5.66  4.81  5.47  0.00  3.69  4.19 
Mean Diff  3.88  4.09  6.86  5.11  3.78  4.17  5.34  3.10  3.04 
Analysis of IGRF candidate models for epoch 2015
Summary of IGRF2015 candidate models submitted to IGRF12
IGRF candidate models for main field epoch 2015  

Team  Model  Organization  Data  Comments (parent model, propagation to 2015) 
A  IGRF2015A  BGS  Ørsted; CHAMP; Swarm A, B, C;  Based on parent model evaluated in 2015.0 
Observatory hourly means  extrapolation from steady core flow hypothesis  
B  IGRF2015B  DTU Space  Ørsted; CHAMP; SACC; Swarm A, B, C;  Parent CHAOS5 evaluated in 2015.0 
Observatory monthly means  linear extrapolation from 2014.75  
C  IGRF2015C  ISTerre  Ørsted; SACC; CHAMP; Swarm B  Parent COVOBS.x1 model evaluated in 2015 
observatory monthly mean  using forward integration of a stochastic model  
D  IGRF2015D  IZMIRAN  Swarm A, B, and C vector data  Parent model evaluated in 2015.0 
Nov2013 to Sep2014, no data selection  linear extrapolation  
E  IGRF2015E  NGDCNOAA  Ørsted; Swarm A, B, C  Parent model evaluated in 2015.0 
linear extrapolation from 2014.3  
F  IGRF2015F  GFZ  Swarm A, B, C;  Parent model evaluated in 2015.0 
USTHB/EOST  observatory hourly means  lineral extrapolation from 2014.5  
H  IGRF2015H  IPGP  Swarm A, B, C Nov2013 to Sep2014  Parent model evaluated in 2015.0 
CEA/CNES  only ASM experimental vector data  
I  IGRF2015I  LPG Nantes  Swarm A and C  Parent model evaluated in 2015.0 
CNES  Nov2013 to Sep2014  linear extrapolation from 2014.3  
J  IGRF2015J  ETH Zurich  Swarm C;  Parent model evaluated in 2015.0 
GFZ  Dec2013 to Sep2014  linear extrapolation 
RMS vector field differences _{ i,j } R in units of nT between IGRF2015 candidates and also between them and the arithmetic mean of all candidates M and the median M _{med}. The bottom row displays the mean of the RMS vector field differences between each candidate model and all other candidate models \({~}_{i}\overline {R}\) from Eq. 8 labelled “Mean Diff”
_{ i,j } R / nT  A  B  C  D  E  F  H  I  J  M  M _{med} 

A  0.0  6.8  12.1  14.1  7.3  6.3  9.1  10.3  16.2  6.2  5.8 
B  6.8  0.0  9.8  13.3  4.8  5.1  5.4  9.3  15.3  3.8  3.2 
C  12.1  9.8  0.0  17.0  12.5  10.1  10.9  11.8  15.4  8.8  8.9 
D  14.1  13.3  17.0  0.0  14.3  12.9  16.1  14.5  18.4  11.8  12.6 
E  7.3  4.8  12.5  14.3  0.0  6.5  7.0  9.9  16.3  5.8  5.2 
F  6.3  5.1  10.1  12.9  6.5  0.0  7.6  9.2  15.0  4.1  3.5 
H  9.1  5.4  10.9  16.1  7.0  7.6  0.0  11.8  17.3  7.0  6.4 
I  10.3  9.3  11.8  14.5  9.9  9.2  11.8  0.0  14.9  7.4  7.8 
J  16.2  15.3  15.4  18.4  16.3  15.0  17.3  14.9  0.0  12.9  13.8 
Mean Diff  10.3  8.7  12.4  15.1  9.8  9.1  10.6  11.5  16.1  7.5  7.5 
Analysis of IGRF12 SV20102015 candidate models
Summary of SV20152020 candidate models submitted to IGRF12
Predictive SV candidate models for epoch 20152020  

Team  Model  Organization  Data  Comments (parent model, propagation to 2015) 
A  SV20152020A  BGS  Ørsted; CHAMP; Swarm A, B, C;  Based on core flow parent model evaluated 
Observatory hourly means  and averaged SV from 2015.0 to 2020.0  
B  SV20152020B  DTU Space  Ørsted; CHAMP; SACC; Swarm A, B, C;  Based on parent CHAOS5 model 
Observatory monthly means  evaluated from splines at 2014.0  
C  SV20152020C  ISTerre  Ørsted; SACC; CHAMP; Swarm B  Based on parent ensemble COVOBS.x1 model 
observatory monthly mean  evaluated and averaged SV from 2015.0 to 2020.0  
D  SV20152020D  IZMIRAN  Swarm A, B, C  Natural orthogonal components (NOCs) 
Nov2013 to Sep2014, no data selection  estimated at 2014.7 (sept2014)  
E  SV20152020E  NGDCNOAA  Ørsted; Swarm A, B, C  From parent model 
firstorder Taylor series with slope at 2015.0  
F  SV20152020F  GFZ  Swarm A, B, C;  From parent model 
USTHB/EOST  observatory hourly means  evaluated and averaged SV from 2013.5 to 2014.5  
G  SV20152020G  NASA  Geodynamo simulation and assimilation from CALS3K.2,  
UMBC  gufm1, CM4, CHAOS4+; average SV from 2015.0 to 2020.0  
H  SV20152020H  IPGP  Swarm A, B, C  Geodynamo simulation and assimilation from Swarm 
LPG Nantes  evaluated and averaged SV from 2015.0 to 2020.0  
I  SV20152020I  LPG Nantes  Swarm A and C  From parent model 
CNES  Nov2013 to Sep2014  firstorder Taylor series with slope at 2014.3 
The simple mathematical extrapolations are often poor predictors as was illustrated by Finlay et al. (2010b, their figure thirteen). For this reason, models B, D, E, F, and I propose candidate models to the SV centered on an epoch close to the available data that does not exceed 2015.0. Forecasting the field to more distant epochs is numerically easier with physically based models although they also have a limited horizon of predictability (for instance Lhuillier et al. 2011). The teams who favored the physically based approach therefore submitted candidates averaged over the upcoming 5 years and centered them on epoch 2017.5. These two families of candidate models, hereafter referred to as the mathematical and physical models therefore follow distinct philosophies. The mathematical models aim to better predict the main field changes for the next 1–2 years simply assuming that its rate of change will be statistically identical to the one observed in recent epochs. On the contrary, physical models hope to predict the field better on average over the full 5year interval.
RMS vector field differences _{ i,j } R in units nT/yr between SV20152020 candidate models and also between these and the mean model M and the median model M _{med}. The final row labelled “Mean Diff” is the mean \({~}_{i}\!\,\overline {R}\) of the _{ i,j } R for each candidate or mean model
_{ i,j } R in nT/yr  A  B  C  D  E  F  G  H  I  M  M _{med} 

A  0.0  9.7  14.2  16.6  11.0  10.9  11.6  10.7  14.1  8.4  8.8 
B  9.7  0.0  9.0  13.7  5.2  6.4  12.2  9.9  10.4  4.2  3.4 
C  14.2  9.0  0.0  15.6  8.9  10.1  19.0  12.3  13.3  9.3  8.4 
D  16.6  13.7  15.6  0.0  14.1  12.1  20.0  15.0  12.3  11.6  12.3 
E  11.0  5.2  8.9  14.1  0.0  7.5  13.6  10.8  11.3  5.6  4.8 
F  10.9  6.4  10.1  12.1  7.5  0.0  14.1  9.1  10.7  5.1  5.2 
G  11.6  12.2  19.0  20.0  13.6  14.1  0.0  14.4  15.6  11.8  12.1 
H  10.7  9.9  12.3  15.0  10.8  9.1  14.4  0.0  9.9  7.2  7.8 
I  14.1  10.4  13.3  12.3  11.3  10.7  15.6  9.9  0.0  8.1  8.9 
Mean diff  12.3  9.6  12.8  14.9  10.3  10.1  15.1  11.5  12.2  7.9  8.0 
Weighting scheme applied to derive the IGRF12 models
_{ i } ω  A  B  C  D  E  F  G  H  I  J 

DGRF2010  1  1  0  0  1  1  0       
IGRF2015  0  1  0  0  1  1    0  0  0 
SV20152020  0  1  0  0  1  1  0  0  0   
For some of the candidate models submitted for the IGRF12, systematic deviation to the weighted arithmetic mean could be understood in the light of the model descriptions as coming from the scientific choices made during their construction. As a result, a majority of the task force thought that the internal discrepancies between different groups of models were not sufficient to reject any of the models.
This provided more significant weights to the coefficients of each candidate models in agreement with residuals observed between the mean and the candidate models in both the spectral and physical domains. However, as already mentioned by Finlay et al. (2010b), this form of IRLS treats each spherical harmonic coefficient \({g_{n}^{m}}\) (or \({h_{n}^{m}}\)) as independent and thus neglects possible correlation between Gauss coefficients of a single candidate model. It was argued that an application of the Huber weighting in space would be more appropriate since the IGRF is mainly used for mapping purposes.
Discussion and conclusion
In the previous sections, we have described some of the statistical tests carried out by the IGRF12 task force in order to evaluate candidate models for DGRF2010, IGRF2015, and SV20152020. Evaluation results clearly illustrate that some models agree better amongst themselves than others. Investigating whether a specific model is flawed, however, is no trivial matter, and selfconsistency between some models was not always thought sufficient to exclude candidate models that are well documented and based on solid but different scientific choices. Most of the differences between models result from different choices of data selection, the removal of (or correction for) the disturbing fields of other sources, choice of analytical method and weighting, and physical hypotheses on the nature of the sources. We faced the situation where in general there was little uncertainty about the parameterization of the candidate models. ESA’s Swarm satellite mission promises further insights concerning the leakage and contamination from different source fields; in particular regarding models describing the Earth’s internal main field. Models of the various magnetic field sources will be derived throughout the Swarm mission’s lifetime by the Swarm Satellite Constellation Application and Research Facility (SCARF) using both comprehensive and dedicated sequential approaches (Olsen et al. 2013). The different source fields derived from the comprehensive description of the Earth’s magnetic field (Sabaka et al. 2013) will then be compared to dedicated models for the main (Rother et al. 2013), magnetospheric (Hamilton 2013), ionospheric (Chulliat et al. 2013), and lithospheric (Thébault et al. 2013) fields using a common global dataset. The results of these intercomparisons will help the community to better identify those structures in the spectral and physical domains that are the most robust (Beggan et al. 2013).
For the previous generation of the IGRF model (Finlay et al. 2010b), the weighting approach for the DGRF and IGRF models involved giving most weight to the group of models that show smallest scatter about an appropriate mean, such as the simple arithmetic mean, and allocating zero weight to the others. For this generation of IGRF, applying the same philosophy would have led to the rejection of more than half of the candidate models, including all of the physically based candidate models to the predictive SV20152020 (see the possible set of weights summarized Table 8 that were discussed by the task force). However, we know from past experience that the secular variation is not constant in time and can change rapidly on a timescale of perhaps only 1 or 2 years as a result of rapid (e.g., Olsen et al. 2006; Lesur et al. 2008) or strong acceleration (e.g., Chulliat et al. 2010) thus making extrapolation shortsighted and a poor predictor at the end of the 5year interval. Rejecting all magnetic field models incorporating recent advances in modeling capability, including predictive SV or for the internal induction parts, could lead to biased solutions estimated from candidate models merely relying on similar approaches.
The IGRF12 task force voted in favor (but not unanimously) to allocate Huber weights in space and to compute the Gauss coefficients using a robust iterative weighted least squares algorithm. This allowed the inclusion of all candidate models in the calculation of IGRF12 but to downweight the most dissimilar aspects of certain models in space. This was thought to be a compromise solution that would both encourage modeling improvements and keep IGRF activities at the forefront of magnetic field modeling. The robust weighting scheme in space is based on a welldefined distribution of misfit and associated statistical measures in order to obtain the final weights. For epochs 2010.0, 2015.0, and 20152020, we see that models that were statistically different from the simple arithmetic mean still receive full weight in many geographical regions, mostly at midlatitudes. Conversely, none of the candidate models that compared well to the arithmetic mean model are allocated full weight for all three components. The regions where all models are in good agreement are highlighted by the Huber weighing scheme in space, thus providing an interesting indicator of where each of the candidate models agreed. These regions show where the IGRF12 constituent models are best constrained in space by all candidate models.
The Huber weighting in space is however not a technique without drawbacks. First, all models must be treated as a whole and the procedure cannot be applied when certain coefficients of a candidate model are wrong or cannot be easily explained from a carefully evaluated scientific compromise. In such clearly identified situations, the manually defined fixed weighting scheme on the coefficients might be more appropriate and defensible. Secondly, this is a purely statistical approach that allows little control on the weights assigned numerically to the candidate models. It is therefore important to test the output against better controlled techniques. To do this, we verified that the models computed using the fixed weights (as was discussed among the task force, see Table 8) and the Huber weighted estimates were not far apart using all of the above defined criteria. This analysis can be summarized by the estimates of the RMS of their difference, which were found to be respectively 1.5 nT for epoch 2010.0, 2.8 nT for epoch 2015.0, and 3.0 nT/yr for the predictive part. These values can be compared, for instance, to the rounding error of IGRF2015 (and SV20152020) that is equal to about 1.5 nT (or nT/yr) for a precision of 0.1 nT on each Gauss coefficient (see Eq. 9). This difference between the two approaches is therefore relatively small and is always less than the RMS differences between any of the candidate models (see Tables 3, 5, and 7) and much less than the uncertainty suggested for each part of the IGRF model in the “Health Warning” (http://www.ngdc.noaa.gov/IAGA/vmod/igrfhw.html). The model coefficients for IGRF12 can be found in electronic (http://www.ngdc.noaa.gov/IAGA/vmod/igrf.html) or print (Thébault et al. 2015) forms.
Availability and requirements
Project name: International Geomagnetic Reference Field, the twelfth generationProject home page: http://www.ngdc.noaa.gov/IAGA/vmod/igrf.html Operating system(s): Platform and browser independentProgramming language: C, Fortran, Matlab Other requirements: noneLicense: noneAny restrictions to use by nonacademics: none
Declarations
Acknowledgements
The institutes that support magnetic observatories together with INTERMAGNET are thanked for promoting high standards of observatory practice and prompt reporting. The support of the CHAMP mission by the German Aerospace Center (DLR) and the Federal Ministry of Education and Research is gratefully acknowledged. The Ørsted Project was made possible by extensive support from the Danish Government, NASA, ESA, CNES, DARA, and the Thomas B. Thriges Foundation. The authors also acknowledge ESA for providing access to the Swarm L1b data. E. Canet acknowledges the support of ESA through the Support to Science Element (STSE) programme. This work was partly funded by the Centre National des Etudes Spatiales (CNES) within the context of the project of the “Travaux préparatoires et exploitation de la mission Swarm.” We would like to thank R. Home and an anonymous reviewer for their useful comments.
Authors’ Affiliations
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