International Geomagnetic Reference Field: the 12th generation

The 12th generation of the International Geomagnetic Reference Field (IGRF) was adopted in December 2014 by the Working Group V-MOD appointed by the International Association of Geomagnetism and Aeronomy (IAGA). It updates the previous IGRF generation with a definitive main field model for epoch 2010.0, a main field model for epoch 2015.0, and a linear annual predictive secular variation model for 2015.0-2020.0. Here, we present the equations defining the IGRF model, provide the spherical harmonic coefficients, and provide maps of the magnetic declination, inclination, and total intensity for epoch 2015.0 and their predicted rates of change for 2015.0-2020.0. We also update the magnetic pole positions and discuss briefly the latest changes and possible future trends of the Earth’s magnetic field.

section_content: The International Geomagnetic Reference Field (IGRF) is a series of mathematical models describing the large-scale internal part of the Earth's magnetic field between epochs 1900 A. D. and the present. The IGRF has been maintained and produced by an international team of scientists under the auspices of the International Association of Geomagnetism and Aeronomy (IAGA) since 1965 (Zmuda 1971). It results from a collaborative effort between magnetic field modelers and institutes involved in collecting and disseminating magnetic field data from magnetic observatories (see the Appendix for the list of World Data Centers), ground surveys, and low Earth orbiting (LEO) satellites. The IGRF is used by scientists in a wide variety of studies, for instance, concerning the dynamics of the Earth's core field, space weather, or local magnetic anomalies imprinted in the Earth's crust. It is also used by commercial organizations and individuals as a source of orientation information. 

The IGRF model must be regularly revised in order to follow the continuous temporal changes of the geomagnetic field generated in the Earth's outer core. The period between revisions is however sufficiently short to preserve its utility as a reference model in applications requiring a fixed reference standard. Table 1 provides the nomenclature and a summary of the history of previous generations of the IGRF. At present, each generation consists of three constituent models. One constituent is designated a Definitive Geomagnetic Reference Field (DGRF). The term 'definitive' is used because any further improvement of these retrospectively determined models is unlikely. The second constituent model, referred to as Zmuda (1971) an IGRF model, is non-definitive -it will eventually be replaced by a definitive model in a future revision of the IGRF. The final constituent, referred to as the secular variation (SV), is provided to predict the time variation of the large-scale geomagnetic field for the 5 years following the latest revision of the IGRF. Readers interested in the history of the IGRF should consult Barton (1997), and users can find legacy versions of the IGRF at the online archive located at http://www. ngdc. noaa. gov/IAGA/vmod/igrf_old_models. html. These may prove useful for those wishing to recover data from which a previous generation of the IGRF has been subtracted or who wish to use the latest generation of the IGRF to carry out revised analyses. Here, attention will focus on the most recent 12th-generation IGRF, hereafter referred to as IGRF-12, that provides a DGRF model for epoch 2010. 0, an IGRF model for epoch 2015. 0, and a predictive SV model covering the epochs 2015. 0-2020. 0. IGRF-12 was agreed in December 2014 by a task force of the IAGA Working Group V-MOD. The purpose of this note is to document the release of IGRF-12, to act as a permanent published record of the IGRF-12 set of model coefficients, and to briefly describe some major features of the geomagnetic field at the Earth's surface as revealed by the updated model.
section_name: Introduction

section_name: Table 1 Summary of IGRF generations, their intervals of validity, and related references

section_content: The IGRF is a series of mathematical models of the internal geomagnetic field -→ B (r, θ, φ, t) and its annual rate of change (secular variation). On and above the Earth's surface, the magnetic field -→ B is defined in terms of a magnetic scalar potential V by -→ B = -∇V and where in spherical polar co-ordinates V is approximated by the finite series

with r denoting the radial distance from the center of the Earth, a = 6, 371. 2 km being the geomagnetic conventional Earth's mean reference spherical radius, θ denoting geocentric co-latitude, and φ denoting east longitude. The functions P m n (cos θ) are the Schmidt quasi-normalized associated Legendre functions of degree n and order m. The Gauss coefficients g m n, h m n are functions of time and are conventionally given in units of nanotesla (nT). 

In the IGRF-12 model, the Gauss coefficients g m n and h m n are provided for the main field (MF) at epochs separated by 5 years between 1900. 0 and 2015. 0 A. D. The time dependence of the Gauss coefficients is assumed to be linear over 5-year intervals and is specified by the following expression

where. g m n respectively

. h m n given in units of nT/year represent the 5-year average first time derivative (the linear secular variation) of the Gauss coefficients. t is the time of interest in units of year and T 0 is the epoch preceding t which is an exact multiple of 5 years, such that T 0 ≤ t < (T 0 + 5. 0). When MF models exist for both T 0 and T 0 + 5. 0, then coefficients ġ m n (T 0 ) can be computed as [ g m n (T 0 + 5. 0)g m n (T 0 )] /5. 0. For the final 5 years of the model validity (between 2015. 0 and 2020. 0 for IGRF-12), 1 and by taking the gradient of V in spherical polar co-ordinates

( 4 )

For some applications, the declination D, the inclination I, the horizontal intensity H, and the total intensity F are required. These components are calculated from X, Y, and Z using the relations,

(5)

In Equation 1, the maximum spherical harmonic degree of the expansion N may vary from one epoch to another. The maximum degree N of the series is equal to 10 up to and including epoch 1995. 0 and the coefficients are quoted to 1-nT precision. For epoch 2000, the coefficients are provided to degree and order 13 and quoted to 0. 1-nT precision, and from epoch 2005 onwards they are quoted to 0. 01-nT precision for the DGRF (and 0. 1 nT for the latest non-definitive IGRF), to take advantage of the higher data quality and good coverage provided by the LEO satellite missions (Finlay et al. 2010a ). The maximum truncation degree N = 13 for epochs after 2000 is defined so as not to include the crustal magnetic field contributions that dominate at higher degrees (see e. g., Langel and Estes 1982). 

The predictive SV coefficients ġ m n (t) and ḣ m n (t) are given to degree and order 8 to 0. 1-nT/year precision. Because of these changes in precision and nomenclature, it is recommended to always use the term 'IGRF-gg, ' where gg represents the generation, in order to keep track of the coefficients that were actually used in applications. This is a simple way to standardize studies carried out at different epochs that makes it apparent whether the results are 'predictive' and therefore less accurate or 'definitive'. For example, one cannot recover the original full-field measurement from an aeromagnetic anomaly map if one does not know which generation of the IGRF was used. This issue has important consequences when comparing magnetic surveys carried out at different epochs (e. g., Hamoudi et al. 2007; Hemant et al. 2007; Maus et al. 2007). 

Equation 1 is expressed in the geocentric system of co-ordinates, but it is sometimes necessary to work in geodetic co-ordinates. When converting between geocentric and geodetic co-ordinates (see for instance Hulot et al. 2007), it is recommended to use the World Geodetic System 1984 (WGS84) datum as present-day satellite magnetic data are often positioned using it. The WGS84 spheroid is defined with major (equatorial) radius A = 6,378. 137 km at the equator and a reciprocal flattening f = 1/298. 257223563 (the polar semi-minor axis is therefore B = A(1-f ) 6,356. 752 km). The number of institutions participating in IGRF-12 was larger than for any previous generation. This reflects the constructive effect of open and unconditional cooperation between scientists involved in modeling the magnetic field, the institutions archiving and disseminating the ground magnetic data, and the national and the European space agencies who actively worked to distribute their expertise, computer programs, and magnetic satellite data with documentation. This latter point was especially important for the MF for epoch 2015. 0 given the short period that elapsed between the launch of the Swarm satellites (in November 2013) and the submission of IGRF candidate models by October 2014. The European Space Agency provided prompt access to the Swarm satellite measurements, including detailed documentation and information on the operational status of the instruments (https://earth. esa. int/web/guest/missions/ esa-operational-eo-missions/swarm). This allowed the teams producing candidate models to rapidly use the Swarm data and helped IGRF-12 to be delivered on time. The collection of ground-based magnetic observatory measurements (see Table 2 ) and the availability of other satellite measurements, from the CHAMP (Reigber et al. 2002), Ørsted (Neubert et al. 2001 ) and SAC-C missions, were also crucial for IGRF-12. 

Seven candidate MF models for the DGRF epoch 2010. 0 and ten candidate MF models for the IGRF epoch 2015. 0 were submitted. In addition, nine SV models were submitted for the predictive part covering epochs 2015. 0-2020. 0. Team A was from BGS, UK (Hamilton et al. 2015) ; team B was from DTU Space, Denmark (Finlay et al. 2015) ; team C was led by ISTerre, France, with input from DTU Space (Gillet et al. 2015) ; team D was from IZMIRAN, Russia; team E was from NGDC/NOAA (Alken et al. 2015) ; team F was from GFZ, Germany (Lesur et al. 2015) ; team G was led by GSFC-NASA, USA, in collaboration with UMBC; team H was from IPGP (Fournier et al. 2015; Vigneron et al. 2015), France, in collaboration with the CEA-Léti (Léger et al. 2015) and with input from LPG Nantes and CNES, France; team I was led by LPG Nantes, France (Saturnino et al. 2015) with input from CNES; team J was from ETH Zurich, Switzerland. These teams contributed to all or parts of the three model constituents of IGRF. Following the IGRF specifications, the MF candidate models had a maximum spherical harmonic degree N = 13 and the predictive SV model had a maximum spherical harmonic degree N = 8. 

The final IGRF-12 MF models for epochs 2010. 0 and 2015. 0 as well as the predictive SV model for 2015. 0-2020. 0 were calculated using a new weighting scheme of the candidate models. For the previous generation of IGRF, fixed weights were assigned to each candidate model based on information gleaned from the evaluations (see Finlay et al. 2010b, for instance) and most weight was given to those models showing the smallest scatter about the arithmetic mean of the candidate models. For IGRF-12, the evidence for significant systematic errors in one or more models was not thought to be sufficient to reject any of the models. A robust weighting scheme was instead applied to the candidate models in space, as agreed by a vote of the IGRF-12 task force. The specification of the candidate models and details of the evaluations and weighting scheme are described in a dedicated paper in this special issue (Thébault et al. 2015).
section_name: Mathematical formulation of the IGRF model

section_content: Table 3 lists the Schmidt semi-normalized spherical harmonic coefficients defining IGRF-12. In IGRF-12, only coefficients after epoch 2005. 0 are modified, but all coefficients are included to serve as a complete record of the model since 1900. This should help to avoid any confusion with previous generations of IGRF, particularly with their provisional parts. The coefficients are given in units of nT -31543-31464-31354-31212-31060-30926-30805-30715-30654-30594-30554-30500-30421-30334-30220-30100-29992-29873-29775-29692-29619. 4-29554. 63-29496. 57-29442. 0 10. 3 g 1 -2298 -2298 -2297 -2306 -2317 -2318 -2316 -2306 -2292 -2285 -2250 -2215 -2169 -2119 -2068 -2013 -1956 -1905 -1848 -1784 -1728. 2 -1669. 05 -1586. 42 -1501. 0 -1469 -1494 -1524 -1559 -1600 -1645 -1692 -1740 -1790 -1834 -1889 -1944 -1992 -2038 -2091 -2144 -2180 -2208 -2239 -2267 -2288. 0 -2305. 83 -2326. 54 -2352. 3 -5. 5 h 3 -330 -357 -389 -421 -445 -462 -480 -494 -499 -499 -476 -462 -414 -404 -366 -333 -336 -310 -284 -262 -227. 6 -198. 86 -160. 40 -115 -90 -109 -134 -163 -195 -226 -252 -276 -278 -274 -278 -269 -266 -265 -257 -249 -240 -236 -231. 9 -225. 23 -211. 03 -188 -93 -102 -111 -119 -125 -131 -136 -141 -142 -147 -152 -156 -157 -160 -159 -162 -164 -165 -166 -168. 6 -168. 05 -163. 17 -157. 5 1. 4 h 5 -124 -125 -126 -126 -125 -122 -118 -115 -113 -119 -122 -121 -114

-6 -6. 2 -6. 72 -7. 14 -6. 8 - for the MF models and of nT/year for the predictive SV model. The coefficients are also available at http://www. ngdc. noaa. gov/IAGA/vmod/igrf. html, together with software to compute the magnetic field components at times and locations of interest, in both geodetic and geocentric reference frames. IGRF-12 is also available from the World Data Centers listed at the end of this paper. We display in Figure 1 maps of the declination D, inclination I, and total intensity F in 2015. 0 on the Earth's reference sphere (r = a) in a Mercator projection that is well suited to navigation. The green lines are the zero contours; in the declination map, the line shows the agonic line where true geographic and magnetic north/south as predicted by the model coincide on the Earth's surface. The general features shown by the maps in 2015 are well known (e. g., Finlay et al. 2010a ) and have slowly evolved through the 115 years covered by IGRF-12. In particular, the minimum of magnetic intensity (see Figure 1 The positions of the geomagnetic poles and the magnetic dip poles in the northern and southern hemispheres, tabulated in Table 4, are presented in Figure 3 on the Earth's reference sphere. We recall that the geomagnetic poles are the points of intersection between the tilted axis of a central inclined magnetic dipole and the sphere of radius a = 6, 371. 2 km. Their positions, expressed in the geocentric co-ordinate system, are antipodal and can be determined from only the three dipole (n = 1) Gauss coefficients. The magnetic dip poles are defined as the points on the Earth's surface where the magnetic field inclination, as determined from the entire field model to degree n = N, is vertical. They are referred to the north and south magnetic poles and are given in Table 4 for the field as observed in the geodetic WGS84 co-ordinate system. The comparison between the locations of the geomagnetic poles and the dip poles is of interest as, seen in the spherical frame, they would coincide if the Earth's magnetic field was perfectly dipolar. However, this is not the case. The comparison also illustrates the comparatively slower drift in time of the Earth's geomagnetic dipole compared to other contributions of the magnetic field. Interestingly, the movements of the north and south magnetic poles have not been erratic and have constantly moved northward since 1900. The tilt between the geomagnetic and the geographic axes is at present reducing with time; it is about 9. 7°in 2015. 0 and projected to be 9. 4°in 2020. The north magnetic pole appeared to be accelerating rather smoothly over the last century (Figure 4 ) from about 5 to about 50 km/year with an increased acceleration around 1990 (Chulliat et al. 2010). The peculiar acceleration of the north and south magnetic poles between 1945 and 1955 as calculated by IGRF should be regarded with caution; see Xu (2000) for a discussion. Perhaps the most striking feature of IGRF-12 is that the north magnetic pole appears to have started a phase of deceleration with a velocity of about 53. 2 km/year in 2015 and a projected velocity of 42. 6 km/year in 2020. Note however that the later estimate relies on the predictive (SV) part of IGRF-12 for epoch 2015. 0 to 2020. 0 and that retrospective analysis has shown that errors could be significant (e. g., Finlay et al. 2010b ). The locations computed from models are also intrinsically approximate due to the limited spatial resolution of the IGRF-12 models. For further details on the limitations of the IGRF for various applications and on difficulties in estimating its accuracy, readers should refer to Lowes (2000) or consult the IGRF 'Health Warning' found at http://www. ngdc. noaa. gov/IAGA/vmod/igrfhw. html.
section_name: IGRF-12 model coefficients and maps

section_content: Further general information about the IGRF: http://www. ngdc. noaa. gov/IAGA/vmod/igrf. html. The coefficients of IGRF-12 in various file formats: http://www. ngdc. noaa. gov/IAGA/vmod/igrf12coeffs. txt Fortran software for synthesizing the field from the coef-
section_name: IGRF-12 online data products

section_name: Appendix: World Data Centers

section_content: WORLD DATA SERVICE FOR GEOPHYSICS, BOULDER NOAA National Centers for Environmental Information, NOAA, 325 Broadway, E/GC, Boulder, CO 80305-3328 UNITED STATES OF AMERICA INTERNET: http://www. ngdc. noaa. govWORLD DATA CENTRE FOR GEOMAGNETISM, COPENHAGEN DTU Space, Diplomvej, Building 327, DK 2800, Kgs. Lynbgy, DENMARK TEL: +45 4525 9713 FAX: +45 353 62475 EMAIL: cfinlay@space. dtu. dkINTERNET: http://www. space. dtu. dk/English/Research/Scientific_data_and_models WORLD DATA CENTRE FOR GEOMAGNETISM, EDINBURGH British Geological Survey Murchison House, West Mains Road Edinburgh, EH9 3LA UNITED KINGDOM TEL: +44 131 650 0234 FAX: +44 131 668 4368 EMAIL: wdcgeomag@bgs. ac. ukINTERNET: http://www. wdc. bgs. ac. uk/WORLD DATA CENTRE FOR GEOMAGNETISM, KYOTO

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) program. 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. ' W. Kuang and A. Tangborn were funded by NASA and the NSF. This work was partly supported by the French 'Agence Nationale de la Recherche ' under the grant ANR-11-BS56-011 and by the Région Pays de Loire, France. I. Wardinski was supported by the DFG through SPP 1488. The IGRF-12 task force finally wishes to express their gratitude to C. Manoj and A. Woods for maintaining the IGRF web pages at NGDC. This is IPGP contribution no. 3625.
section_name: Appendix: World Data Centers

section_content: ficients: http://www. ngdc. noaa. gov/IAGA/vmod/igrf12. f C software for synthesizing the field from the coefficients (Linux): http://www. ngdc. noaa. gov/IAGA/vmod/geomag70_ linux. tar. gz C software for synthesizing the field from the coefficients (Windows): http://www. ngdc. noaa. gov/IAGA/vmod/geomag70_ windows. zip Online computation of field components from the IGRF-12 model: http://www. ngdc. noaa. gov/geomag-web/?model=igrf http://www. geomag. bgs. ac. uk/data_service/models_ compass/igrf_form. shtml http://wdc. kugi. kyoto-u. ac. jp/igrf/point/index. html Archive of legacy versions of the IGRF model: http://www. ngdc. noaa. gov/IAGA/vmod/igrf_old_ models. html

section_name: Competing interests

section_content: . 9 -0. 7 7 -0. 6 6 -0. 7 -

. 9 0. 9 0 0. 8 5 0. 8 -

. 9 0. 9 6 0. 9 5 0. 9 -

. 9 -0. 8 2 -0. 7 9 -0. 8 -

Here, Schmidt semi-normalized spherical harmonic coefficients are provided. Coefficients for degrees n=1-13 in units of nanotesla are listed for IGRF and definitive DGRF main-field models. Coefficients for degrees n=1-8 in units of nanotesla/year are listed for the predictive secular variation. Undefined coefficients are marked with '-'; these should be set to 0. 0 in numerical calculations as is the case in the coefficient files available online.

section_content: The authors declare that they have no competing interests.
section_name: Competing interests

section_content: ET and CCF coordinated the work with full support from the IGRF-12 task force members. CDB generated Figure 3 and verified with independent software the values given in Table 4. All authors participated to the construction of magnetic field candidate models referenced in the manuscript. All authors analyzed and discussed the final IGRF-12 model and approved the final version of the manuscript.
section_name: Authors' contributions

section_content: ET, CCF, CDB, PA, AD, GH, WK, VL, FJL, SM, NO, VP, and TJS are members of the IGRF-12 task force.
section_name: Authors' information