Open Access

International Geomagnetic Reference Field: the 12th generation

  • Erwan Thébault1Email author,
  • Christopher C Finlay2,
  • Ciarán D Beggan3,
  • Patrick Alken4, 5,
  • Julien Aubert6,
  • Olivier Barrois7,
  • Francois Bertrand8, 9,
  • Tatiana Bondar10,
  • Axel Boness8, 9,
  • Laura Brocco6,
  • Elisabeth Canet11,
  • Aude Chambodut12,
  • Arnaud Chulliat4, 5,
  • Pierdavide Coïsson6,
  • François Civet1,
  • Aimin Du13,
  • Alexandre Fournier6,
  • Isabelle Fratter14,
  • Nicolas Gillet7,
  • Brian Hamilton3,
  • Mohamed Hamoudi15, 19,
  • Gauthier Hulot6,
  • Thomas Jager8, 9,
  • Monika Korte15,
  • Weijia Kuang16,
  • Xavier Lalanne6,
  • Benoit Langlais1,
  • Jean-Michel Léger8, 9,
  • Vincent Lesur15,
  • Frank J Lowes17,
  • Susan Macmillan3,
  • Mioara Mandea18,
  • Chandrasekharan Manoj4, 5,
  • Stefan Maus4,
  • Nils Olsen2,
  • Valeriy Petrov10,
  • Victoria Ridley3,
  • Martin Rother15,
  • Terence J Sabaka16,
  • Diana Saturnino1,
  • Reyko Schachtschneider15,
  • Olivier Sirol6,
  • Andrew Tangborn20,
  • Alan Thomson3,
  • Lars Tøffner-Clausen2,
  • Pierre Vigneron6,
  • Ingo Wardinski15 and
  • Tatiana Zvereva10
Earth, Planets and Space201567:79

DOI: 10.1186/s40623-015-0228-9

Received: 30 January 2015

Accepted: 13 April 2015

Published: 27 May 2015

Abstract

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.

Keywords

Geomagnetism Field modeling IGRF

Correspondence/Findings

Introduction

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 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.
Table 1

Summary of IGRF generations, their intervals of validity, and related references

Full name

Short name

Valid for

Definitive for

Reference

IGRF 12th generation

IGRF-12

1900.0-2020.0

1945.0-2010.0

Thébault et al., this article

IGRF 11th generation

IGRF-11

1900.0-2015.0

1945.0-2005.0

Finlay et al. (2010a)

IGRF 10th generation

IGRF-10

1900.0-2010.0

1945.0-2000.0

Maus et al. (2005)

IGRF 9th generation

IGRF-9

1900.0-2005.0

1945.0-2000.0

Macmillan et al. (2003)

IGRF 8th generation

IGRF-8

1900.0-2005.0

1945.0-1990.0

Mandea and Macmillan (2000)

IGRF 7th generation

IGRF-7

1900.0-2000.0

1945.0-1990.0

Barton (1997)

IGRF 6th generation

IGRF-6

1945.0-1995.0

1945.0-1985.0

Langel (1992)

IGRF 5th generation

IGRF-5

1945.0-1990.0

1945.0-1980.0

Langel et al. (1988)

IGRF 4th generation

IGRF-4

1945.0-1990.0

1965.0-1980.0

Barraclough (1987)

IGRF 3rd generation

IGRF-3

1965.0-1985.0

1965.0-1975.0

Peddie (1982)

IGRF 2nd generation

IGRF-2

1955.0-1980.0

IAGA (1975)

IGRF 1st generation

IGRF-1

1955.0-1975.0

Zmuda (1971)

Mathematical formulation of the IGRF model

The IGRF is a series of mathematical models of the internal geomagnetic field \(\overrightarrow {B}(r,\theta,\phi,t)\) and its annual rate of change (secular variation). On and above the Earth’s surface, the magnetic field \( \overrightarrow {B}\) is defined in terms of a magnetic scalar potential V by \(\overrightarrow {B}=-\nabla V\) and where in spherical polar co-ordinates V is approximated by the finite series
$$ {\fontsize{9.6pt}{9.6pt}\selectfont{\begin{aligned} {\kern-16.5pt}V(r,\theta,\phi,t)&=a\sum_{n=1}^{N}\sum_{m=0}^{n}\left(\frac{a}{r}\right)^{n+1}\\ &\times\left[ {g_{n}^{m}}(t)\cos \left(m\phi \right) +{h_{n}^{m}}(t)\sin \left(m\phi \right) {P_{n}^{m}}(\cos \theta)\right] \!, \end{aligned}}} $$
(1)

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 \({P_{n}^{m}}(\cos \theta)\) are the Schmidt quasi-normalized associated Legendre functions of degree n and order m. The Gauss coefficients \({g_{n}^{m}}\), \({h_{n}^{m}}\) are functions of time and are conventionally given in units of nanotesla (nT).

In the IGRF-12 model, the Gauss coefficients \({g_{n}^{m}}\) and \({h_{n}^{m}}\) 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
$$ \begin{aligned} {g_{n}^{m}}(t)& ={g_{n}^{m}}(T_{0})+\overset{.}{g}_{n}^{m}(T_{0}).(t-T_{0}), \end{aligned} $$
(2)
$$ \begin{aligned} {h_{n}^{m}}(t)& ={h_{n}^{m}}(T_{0})+\overset{.}{h}_{n}^{m}(T_{0}).(t-T_{0}), \end{aligned} $$
(3)
where \(\overset {.}{g}_{n}^{m} \left (\text {respectively}\; \overset {.}{h}_{n}^{m}\right)\) 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 0t<(T 0+5.0). When MF models exist for both T 0 and T 0+5.0, then coefficients \(\dot { {g_{n}^{m}}}(T_{0})\) can be computed as \( [{g_{n}^{m}}(T_{0}+5.0)-{g_{n}^{m}}(T_{0})]/5.0\). For the final 5 years of the model validity (between 2015.0 and 2020.0 for IGRF-12), the coefficients \( \dot {{g_{n}^{m}}}(t)\) and \(\dot {{h_{n}^{m}}}(t)\) of the predictive average SV are explicitly provided. The geocentric components of the geomagnetic field in the northward, eastward, and radially inwards directions (X, Y and Z) are obtained from the model coefficients using Equation 1 and by taking the gradient of V in spherical polar co-ordinates
$$ X={\frac{1}{r}}{\frac{\partial V}{\partial \theta }},\quad Y=-{\frac{1}{ r\sin \theta }}{\frac{\partial V}{\partial \phi }},\quad Z={\frac{\partial V }{\partial r}}. $$
(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,
$$ \begin{aligned} {}H&=\sqrt{X^{2}+Y^{2}},\quad F=\sqrt{X^{2}+Y^{2}+Z^{2}},\quad \\ D&=\arctan {(Y/X)},\quad I=\arctan {(Z/H)}. \end{aligned} $$
(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 \(\dot {{g_{n}^{m}}}(t)\) and \(\dot {{h_{n}^{m}}}(t)\) 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 12th-generation IGRF

IGRF-12, the 12th generation of IGRF, is derived from candidate models prepared by international teams who answered a call issued by the IGRF-12 task force in May 2014. This call requested candidates for the Definitive Geomagnetic Reference Field (DGRF) for epoch 2010, for a provisional IGRF model for epoch 2015, and for a predictive SV model for the interval 2015.0-2020.0. The IGRF-12 model coefficients remain unchanged for epoch 2005 and earlier.

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.
Table 2

Magnetic observatories contributing data used in the construction of IGRF-12

Supporting Agencies

Country

Observatory IAGA code

Centre de Recherche en Astronomie, Astrophysique et Geophysique

Algeria

TAM

Universidad Nacional de la Plata

Argentina

TRW

Servicio Meteorologico Nacional

Argentina

ORC

Geoscience Australia

Australia

ASP, CKI, CNB, CSY, CTA, DVS, GNA, GNG, KDU, LRM, MAW, MCQ

Zentralanstalt für Meteorologie und Geodynamik

Austria

WIK

Institut Royal Météorologique

Belgium

DOU, MAB

CNPq-Observatorio Nacional

Brazil

VSS

Academy of Sciences

Bulgaria

PAG

Geological Survey of Canada

Canada

ALE, BLC, CBB, FCC, IQA, MEA, OTT, PBQ, RES, STJ, VIC,YKC

Centro Meteorológico Regional Pacifico

Chile

IPM

Academy of Sciences

China

BMT

China Earthquake Adminstration

China

CDP, GLM, GZH, KSH, LZH, MZL, QGZ, QIX, QZH, SSH, THJ, WHN

Academy of Sciences

Czech Republic

BDV

Danish Technical University-Space

Denmark

TDC

Addis Ababa University

Ethiopia

AAE

Finnish Meteorological Institute

Finland

NUR

Geophysical Observatory

Finland

SOD

Institut de Physique du Globe de Paris

France

AAE, BOX, CLF, KOU, IPM, LZH, MBO, PHU, QSB, PPT, TAM

Ecole et Observatoire des Sciences de la Terre

France

AMS, CZT, DMC, DRV, PAF, TAN

Institut Français de Recherche Scientifique pour le Développement

France

BNG, MBO

Academy of Sciences

Georgia

TFS

Universität München

Germany

FUR

Alfred-Wegener-Institute for Polar Marine Research

Germany

VNA

GFZ Hemholtz Centre Potsdam

Germany

NGK, TDC, WNG

Universität Stuttgart

Germany

BFO

Institute of Geology and Mineral Exploration

Greece

PEG

Academy of Sciences

Hungary

NCK

Eötvös Loránd Geophysical Institute

Hungary

THY

University of Iceland

Iceland

LRV

Indian Institute of Geomagnetism

India

ABG, PND, SIL, TIR, VSK

National Geophysical Research Institute

India

HYB

Meteorological and Geophysical Agency

Indonesia

KPG, PLR, TND

Meteorological Service

Ireland

VAL

Survey of Israel

Israel

AMT, BGY, ELT

Instituto Nazionale di Geofisica e Vulcanologia

Italy

AQU, DMC

Japan Coast Guard

Japan

HTY

Japan Meteorological Agency

Japan

CBI, KAK, KNY, MMB

Geographical Survey Institute

Japan

ESA, KNZ, MIZ

Institute of the Ionosphere

Kazakhstan

AAA

National Centre for Geophysical Research

Lebanon

QSB

Université d’Antananarivo

Madagascar

TAN

Gan Meteorological Office/ETH Zurich

Maldives/Switzerland

GAN

Universidad Nacional Autonoma de México

Mexico

TEO

Institute of Geological and Nuclear Sciences

New Zealand

API, EYR, SBA

University of Tromsø

Norway

BJN, DOB, TRO

Instituto Geofisico del Peru

Peru

HUA

Academy of Sciences

Poland

BEL, HLP, HRN

Directorate General of Telecommunications

Republic of China

LNP

Instituto Nacional de Geologia

República de Moçambique

LMM

Geological Survey of Romania

Romania

SUA

Academy of Sciences

Russia

ARS, BOX, LVV, MOS, NVS

Institute of Solar-Terrestrial Physics

Russia

IRT

Dept. of Agriculture, Forestry, Fisheries & Meteorology

Samoa

API

Geomagnetic College Grocka

Serbia and Montenegro

GCK

Slovenska Akademia Vied

Slovakia

HRB

National Research Foundation

South Africa

HBK, HER, KMH, TSU

Observatori de l’Ebre

Spain

EBR, LIV

Real Instituto y Observatorio de la Armada

Spain

SFS

Instituto Geográfico Nacional

Spain

GUI, SPT

Sveriges Geologiska Undersökning

Sweden

ABK, LOV, LYC, UPS

Swedish Institute of Space Physics

Sweden

KIR

Bŏgaziçi University

Turkey

IZN

Academy of Sciences

Ukraine

AIA

British Geological Survey

United Kingdom

ASC, ESK, HAD, JCO, KEP, LER, PST, SBL

US Geological Survey

United States

BRW, BOU, BSL, CMO, DLR, FRD, FRN, GUA

  

HON, NEW, SIT, SJG, SHU, TUC

Academy of Science and Technology

Vietnam

PHU

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).

IGRF-12 model coefficients and maps

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 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.
Table 3

12th Generation International Geomagnetic Reference Field

 

Degree

Order

IGRF

IGRF

IGRF

IGRF

IGRF

IGRF

IGRF

IGRF

IGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

DGRF

IGRF

SV

g/h

n

m

1900.0

1905.0

1910.0

1915.0

1920.0

1925.0

1930.0

1935.0

1940.0

1945.0

1950.0

1955.0

1960.0

1965.0

1970.0

1975.0

1980.0

1985.0

1990.0

1995.0

2000.0

2005.0

2010.0

2015

2015-20

g

1

0

-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

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

18.1

h

1

1

5922

5909

5898

5875

5845

5817

5808

5812

5821

5810

5815

5820

5791

5776

5737

5675

5604

5500

5406

5306

5186.1

5077.99

4944.26

4797.1

-26.6

g

2

0

-677

-728

-769

-802

-839

-893

-951

-1018

-1106

-1244

-1341

-1440

-1555

-1662

-1781

-1902

-1997

-2072

-2131

-2200

-2267.7

-2337.24

-2396.06

-2445.1

-8.7

g

2

1

2905

2928

2948

2956

2959

2969

2980

2984

2981

2990

2998

3003

3002

2997

3000

3010

3027

3044

3059

3070

3068.4

3047.69

3026.34

3012.9

-3.3

h

2

1

-1061

-1086

-1128

-1191

-1259

-1334

-1424

-1520

-1614

-1702

-1810

-1898

-1967

-2016

-2047

-2067

-2129

-2197

-2279

-2366

-2481.6

-2594.50

-2708.54

-2845.6

-27.4

g

2

2

924

1041

1176

1309

1407

1471

1517

1550

1566

1578

1576

1581

1590

1594

1611

1632

1663

1687

1686

1681

1670.9

1657.76

1668.17

1676.7

2.1

h

2

2

1121

1065

1000

917

823

728

644

586

528

477

381

291

206

114

25

-68

-200

-306

-373

-413

-458.0

-515.43

-575.73

-641.9

-14.1

g

3

0

1022

1037

1058

1084

1111

1140

1172

1206

1240

1282

1297

1302

1302

1297

1287

1276

1281

1296

1314

1335

1339.6

1336.30

1339.85

1350.7

3.4

g

3

1

-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

1

-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

-115.3

8.2

g

3

2

1256

1239

1223

1212

1205

1202

1205

1215

1232

1255

1274

1288

1289

1292

1278

1260

1251

1247

1248

1249

1252.1

1246.39

1232.10

1225.6

-0.7

h

3

2

3

34

62

84

103

119

133

146

163

186

206

216

224

240

251

262

271

284

293

302

293.4

269.72

251.75

244.9

-0.4

g

3

3

572

635

705

778

839

881

907

918

916

913

896

882

878

856

838

830

833

829

802

759

714.5

672.51

633.73

582.0

-10.1

h

3

3

523

480

425

360

293

229

166

101

43

-11

-46

-83

-130

-165

-196

-223

-252

-297

-352

-427

-491.1

-524.72

-537.03

-538.4

1.8

g

4

0

876

880

884

887

889

891

896

903

914

944

954

958

957

957

952

946

938

936

939

940

932.3

920.55

912.66

907.6

-0.7

g

4

1

628

643

660

678

695

711

727

744

762

776

792

796

800

804

800

791

782

780

780

780

786.8

797.96

808.97

813.7

0.2

h

4

1

195

203

211

218

220

216

205

188

169

144

136

133

135

148

167

191

212

232

247

262

272.6

282.07

286.48

283.3

-1.3

g

4

2

660

653

644

631

616

601

584

565

550

544

528

510

504

479

461

438

398

361

325

290

250.0

210.65

166.58

120.4

-9.1

h

4

2

-69

-77

-90

-109

-134

-163

-195

-226

-252

-276

-278

-274

-278

-269

-266

-265

-257

-249

-240

-236

-231.9

-225.23

-211.03

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5.3

g

4

3

-361

-380

-400

-416

-424

-426

-422

-415

-405

-421

-408

-397

-394

-390

-395

-405

-419

-424

-423

-418

-403.0

-379.86

-356.83

-334.9

4.1

h

4

3

-210

-201

-189

-173

-153

-130

-109

-90

-72

-55

-37

-23

3

13

26

39

53

69

84

97

119.8

145.15

164.46

180.9

2.9

g

4

4

134

146

160

178

199

217

234

249

265

304

303

290

269

252

234

216

199

170

141

122

111.3

100.00

89.40

70.4

-4.3

h

4

4

-75

-65

-55

-51

-57

-70

-90

-114

-141

-178

-210

-230

-255

-269

-279

-288

-297

-297

-299

-306

-303.8

-305.36

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-329.5

-5.2

g

5

0

-184

-192

-201

-211

-221

-230

-237

-241

-241

-253

-240

-229

-222

-219

-216

-218

-218

-214

-214

-214

-218.8

-227.00

-230.87

-232.6

-0.2

g

5

1

328

328

327

327

326

326

327

329

334

346

349

360

362

358

359

356

357

355

353

352

351.4

354.41

357.29

360.1

0.5

h

5

1

-210

-193

-172

-148

-122

-96

-72

-51

-33

-12

3

15

16

19

26

31

46

47

46

46

43.8

42.72

44.58

47.3

0.6

g

5

2

264

259

253

245

236

226

218

211

208

194

211

230

242

254

262

264

261

253

245

235

222.3

208.95

200.26

192.4

-1.3

h

5

2

53

56

57

58

58

58

60

64

71

95

103

110

125

128

139

148

150

150

154

165

171.9

180.25

189.01

197.0

1.7

g

5

3

5

-1

-9

-16

-23

-28

-32

-33

-33

-20

-20

-23

-26

-31

-42

-59

-74

-93

-109

-118

-130.4

-136.54

-141.05

-140.9

-0.1

h

5

3

-33

-32

-33

-34

-38

-44

-53

-64

-75

-67

-87

-98

-117

-126

-139

-152

-151

-154

-153

-143

-133.1

-123.45

-118.06

-119.3

-1.2

g

5

4

-86

-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

4

-124

-125

-126

-126

-125

-122

-118

-115

-113

-119

-122

-121

-114

-97

-91

-83

-78

-75

-69

-55

-39.3

-19.57

-0.01

16.0

3.4

g

5

5

-16

-26

-38

-51

-62

-69

-74

-76

-76

-82

-76

-69

-63

-62

-56

-49

-48

-46

-36

-17

-12.9

-13.55

-8.03

4.1

3.9

h

5

5

3

11

21

32

43

51

58

64

69

82

80

78

81

81

83

88

92

95

97

107

106.3

103.85

101.04

100.2

0.0

g

6

0

63

62

62

61

61

61

60

59

57

59

54

47

46

45

43

45

48

53

61

68

72.3

73.60

72.78

70.0

-0.3

g

6

1

61

60

58

57

55

54

53

53

54

57

57

57

58

61

64

66

66

65

65

67

68.2

69.56

68.69

67.7

-0.1

h

6

1

-9

-7

-5

-2

0

3

4

4

4

6

-1

-9

-10

-11

-12

-13

-15

-16

-16

-17

-17.4

-20.33

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-20.8

0.0

g

6

2

-11

-11

-11

-10

-10

-9

-9

-8

-7

6

4

3

1

8

15

28

42

51

59

68

74.2

76.74

75.92

72.7

-0.7

h

6

2

83

86

89

93

96

99

102

104

105

100

99

96

99

100

100

99

93

88

82

72

63.7

54.75

44.18

33.2

-2.1

g

6

3

-217

-221

-224

-228

-233

-238

-242

-246

-249

-246

-247

-247

-237

-228

-212

-198

-192

-185

-178

-170

-160.9

-151.34

-141.40

-129.9

2.1

h

6

3

2

4

5

8

11

14

19

25

33

16

33

48

60

68

72

75

71

69

69

67

65.1

63.63

61.54

58.9

-0.7

g

6

4

-58

-57

-54

-51

-46

-40

-32

-25

-18

-25

-16

-8

-1

4

2

1

4

4

3

-1

-5.9

-14.58

-22.83

-28.9

-1.2

h

6

4

-35

-32

-29

-26

-22

-18

-16

-15

-15

-9

-12

-16

-20

-32

-37

-41

-43

-48

-52

-58

-61.2

-63.53

-66.26

-66.7

0.2

g

6

5

59

57

54

49

44

39

32

25

18

21

12

7

-2

1

3

6

14

16

18

19

16.9

14.58

13.10

13.2

0.3

h

6

5

36

32

28

23

18

13

8

4

0

-16

-12

-12

-11

-8

-6

-4

-2

-1

1

1

0.7

0.24

3.02

7.3

0.9

g

6

6

-90

-92

-95

-98

-101

-103

-104

-106

-107

-104

-105

-107

-113

-111

-112

-111

-108

-102

-96

-93

-90.4

-86.36

-78.09

-70.9

1.6

h

6

6

-69

-67

-65

-62

-57

-52

-46

-40

-33

-39

-30

-24

-17

-7

1

11

17

21

24

36

43.8

50.94

55.40

62.6

1.0

g

7

0

70

70

71

72

73

73

74

74

74

70

65

65

67

75

72

71

72

74

77

77

79.0

79.88

80.44

81.6

0.3

g

7

1

-55

-54

-54

-54

-54

-54

-54

-53

-53

-40

-55

-56

-56

-57

-57

-56

-59

-62

-64

-72

-74.0

-74.46

-75.00

-76.1

-0.2

h

7

1

-45

-46

-47

-48

-49

-50

-51

-52

-52

-45

-35

-50

-55

-61

-70

-77

-82

-83

-80

-69

-64.6

-61.14

-57.80

-54.1

0.8

g

7

2

0

0

1

2

2

3

4

4

4

0

2

2

5

4

1

1

2

3

2

1

0.0

-1.65

-4.55

-6.8

-0.5

h

7

2

-13

-14

-14

-14

-14

-14

-15

-17

-18

-18

-17

-24

-28

-27

-27

-26

-27

-27

-26

-25

-24.2

-22.57

-21.20

-19.5

0.4

g

7

3

34

33

32

31

29

27

25

23

20

0

1

10

15

13

14

16

21

24

26

28

33.3

38.73

45.24

51.8

1.3

h

7

3

-10

-11

-12

-12

-13

-14

-14

-14

-14

2

0

-4

-6

-2

-4

-5

-5

-2

0

4

6.2

6.82

6.54

5.7

-0.2

g

7

4

-41

-41

-40

-38

-37

-35

-34

-33

-31

-29

-40

-32

-32

-26

-22

-14

-12

-6

-1

5

9.1

12.30

14.00

15.0

0.1

h

7

4

-1

0

1

2

4

5

6

7

7

6

10

8

7

6

8

10

16

20

21

24

24.0

25.35

24.96

24.4

-0.3

g

7

5

-21

-20

-19

-18

-16

-14

-12

-11

-9

-10

-7

-11

-7

-6

-2

0

1

4

5

4

6.9

9.37

10.46

9.4

-0.6

h

7

5

28

28

28

28

28

29

29

29

29

28

36

28

23

26

23

22

18

17

17

17

14.8

10.93

7.03

3.4

-0.6

g

7

6

18

18

18

19

19

19

18

18

17

15

5

9

17

13

13

12

11

10

9

8

7.3

5.42

1.64

-2.8

-0.8

h

7

6

-12

-12

-13

-15

-16

-17

-18

-19

-20

-17

-18

-20

-18

-23

-23

-23

-23

-23

-23

-24

-25.4

-26.32

-27.61

-27.4

0.1

g

7

7

6

6

6

6

6

6

6

6

5

29

19

18

8

1

-2

-5

-2

0

0

-2

-1.2

1.94

4.92

6.8

0.2

h

7

7

-22

-22

-22

-22

-22

-21

-20

-19

-19

-22

-16

-18

-17

-12

-11

-12

-10

-7

-4

-6

-5.8

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-2.2

-0.2

g

8

0

11

11

11

11

11

11

11

11

11

13

22

11

15

13

14

14

18

21

23

25

24.4

24.80

24.41

24.2

0.2

g

8

1

8

8

8

8

7

7

7

7

7

7

15

9

6

5

6

6

6

6

5

6

6.6

7.62

8.21

8.8

0.0

h

8

1

8

8

8

8

8

8

8

8

8

12

5

10

11

7

7

6

7

8

10

11

11.9

11.20

10.84

10.1

-0.3

g

8

2

-4

-4

-4

-4

-3

-3

-3

-3

-3

-8

-4

-6

-4

-4

-2

-1

0

0

-1

-6

-9.2

-11.73

-14.50

-16.9

-0.6

h

8

2

-14

-15

-15

-15

-15

-15

-15

-15

-14

-21

-22

-15

-14

-12

-15

-16

-18

-19

-19

-21

-21.5

-20.88

-20.03

-18.3

0.3

g

8

3

-9

-9

-9

-9

-9

-9

-9

-9

-10

-5

-1

-14

-11

-14

-13

-12

-11

-11

-10

-9

-7.9

-6.88

-5.59

-3.2

0.5

h

8

3

7

7

6

6

6

6

5

5

5

-12

0

5

7

9

6

4

4

5

6

8

8.5

9.83

11.83

13.3

0.1

g

8

4

1

1

1

2

2

2

2

1

1

9

11

6

2

0

-3

-8

-7

-9

-12

-14

-16.6

-18.11

-19.34

-20.6

-0.2

h

8

4

-13

-13

-13

-13

-14

-14

-14

-15

-15

-7

-21

-23

-18

-16

-17

-19

-22

-23

-22

-23

-21.5

-19.71

-17.41

-14.6

0.5

g

8

5

2

2

2

3

4

4

5

6

6

7

15

10

10

8

5

4

4

4

3

9

9.1

10.17

11.61

13.4

0.4

h

8

5

5

5

5

5

5

5

5

5

5

2

-8

3

4

4

6

6

9

11

12

15

15.5

16.22

16.71

16.2

-0.2

g

8

6

-9

-8

-8

-8

-7

-7

-6

-6

-5

-10

-13

-7

-5

-1

0

0

3

4

4

6

7.0

9.36

10.85

11.7

0.1

h

8

6

16

16

16

16

17

17

18

18

19

18

17

23

23

24

21

18

16

14

12

11

8.9

7.61

6.96

5.7

-0.3

g

8

7

5

5

5

6

6

7

8

8

9

7

5

6

10

11

11

10

6

4

2

-5

-7.9

-11.25

-14.05

-15.9

-0.4

h

8

7

-5

-5

-5

-5

-5

-5

-5

-5

-5

3

-4

-4

1

-3

-6

-10

-13

-15

-16

-16

-14.9

-12.76

-10.74

-9.1

0.3

g

8

8

8

8

8

8

8

8

8

7

7

2

-1

9

8

4

3

1

-1

-4

-6

-7

-7.0

-4.87

-3.54

-2.0

0.3

h

8

8

-18

-18

-18

-18

-19

-19

-19

-19

-19

-11

-17

-13

-20

-17

-16

-17

-15

-11

-10

-4

-2.1

-0.06

1.64

2.1

0.0

g

9

0

8

8

8

8

8

8

8

8

8

5

3

4

4

8

8

7

5

5

4

4

5.0

5.58

5.50

5.4

-

g

9

1

10

10

10

10

10

10

10

10

10

-21

-7

9

6

10

10

10

10

10

9

9

9.4

9.76

9.45

8.8

-

h

9

1

-20

-20

-20

-20

-20

-20

-20

-20

-21

-27

-24

-11

-18

-22

-21

-21

-21

-21

-20

-20

-19.7

-20.11

-20.54

-21.6

-

g

9

2

1

1

1

1

1

1

1

1

1

1

-1

-4

0

2

2

2

1

1

1

3

3.0

3.58

3.45

3.1

-

h

9

2

14

14

14

14

14

14

14

15

15

17

19

12

12

15

16

16

16

15

15

15

13.4

12.69

11.51

10.8

-

g

9

3

-11

-11

-11

-11

-11

-11

-12

-12

-12

-11

-25

-5

-9

-13

-12

-12

-12

-12

-12

-10

-8.4

-6.94

-5.27

-3.3

-

h

9

3

5

5

5

5

5

5

5

5

5

29

12

7

2

7

6

7

9

9

11

12

12.5

12.67

12.75

11.8

-

g

9

4

12

12

12

12

12

12

12

11

11

3

10

2

1

10

10

10

9

9

9

8

6.3

5.01

3.13

0.7

-

h

9

4

-3

-3

-3

-3

-3

-3

-3

-3

-3

-9

2

6

0

-4

-4

-4

-5

-6

-7

-6

-6.2

-6.72

-7.14

-6.8

-

g

9

5

1

1

1

1

1

1

1

1

1

16

5

4

4

-1

-1

-1

-3

-3

-4

-8

-8.9

-10.76

-12.38

-13.3

-

h

9

5

-2

-2

-2

-2

-2

-2

-2

-3

-3

4

2

-2

-3

-5

-5

-5

-6

-6

-7

-8

-8.4

-8.16

-7.42

-6.9

-

g

9

6

-2

-2

-2

-2

-2

-2

-2

-2

-2

-3

-5

1

-1

-1

0

-1

-1

-1

-2

-1

-1.5

-1.25

-0.76

-0.1

-

h

9

6

8

8

8

8

9

9

9

9

9

9

8

10

9

10

10

10

9

9

9

8

8.4

8.10

7.97

7.8

-

g

9

7

2

2

2

2

2

2

3

3

3

-4

-2

2

-2

5

3

4

7

7

7

10

9.3

8.76

8.43

8.7

-

h

9

7

10

10

10

10

10

10

10

11

11

6

8

7

8

10

11

11

10

9

8

5

3.8

2.92

2.14

1.0

-

g

9

8

-1

0

0

0

0

0

0

0

1

-3

3

2

3

1

1

1

2

1

1

-2

-4.3

-6.66

-8.42

-9.1

-

h

9

8

-2

-2

-2

-2

-2

-2

-2

-2

-2

1

-11

-6

0

-4

-2

-3

-6

-7

-7

-8

-8.2

-7.73

-6.08

-4.0

-

g

9

9

-1

-1

-1

-1

-1

-1

-2

-2

-2

-4

8

5

-1

-2

-1

-2

-5

-5

-6

-8

-8.2

-9.22

-10.08

-10.5

-

h

9

9

2

2

2

2

2

2

2

2

2

8

-7

5

5

1

1

1

2

2

2

3

4.8

6.01

7.01

8.4

-

g

10

0

-3

-3

-3

-3

-3

-3

-3

-3

-3

-3

-8

-3

1

-2

-3

-3

-4

-4

-3

-3

-2.6

-2.17

-1.94

-1.9

-

g

10

1

-4

-4

-4

-4

-4

-4

-4

-4

-4

11

4

-5

-3

-3

-3

-3

-4

-4

-4

-6

-6.0

-6.12

-6.24

-6.3

-

h

10

1

2

2

2

2

2

2

2

2

2

5

13

-4

4

2

1

1

1

1

2

1

1.7

2.19

2.73

3.2

-

g

10

2

2

2

2

2

2

2

2

2

2

1

-1

-1

4

2

2

2

2

3

2

2

1.7

1.42

0.89

0.1

-

h

10

2

1

1

1

1

1

1

1

1

1

1

-2

0

1

1

1

1

0

0

1

0

0.0

0.10

-0.10

-0.4

-

g

10

3

-5

-5

-5

-5

-5

-5

-5

-5

-5

2

13

2

0

-5

-5

-5

-5

-5

-5

-4

-3.1

-2.35

-1.07

0.5

-

h

10

3

2

2

2

2

2

2

2

2

2

-20

-10

-8

0

2

3

3

3

3

3

4

4.0

4.46

4.71

4.6

-

g

10

4

-2

-2

-2

-2

-2

-2

-2

-2

-2

-5

-4

-3

-1

-2

-1

-2

-2

-2

-2

-1

-0.5

-0.15

-0.16

-0.5

-

h

10

4

6

6

6

6

6

6

6

6

6

-1

2

-2

2

6

4

4

6

6

6

5

4.9

4.76

4.44

4.4

-

g

10

5

6

6

6

6

6

6

6

6

6

-1

4

7

4

4

6

5

5

5

4

4

3.7

3.06

2.45

1.8

-

h

10

5

-4

-4

-4

-4

-4

-4

-4

-4

-4

-6

-3

-4

-5

-4

-4

-4

-4

-4

-4

-5

-5.9

-6.58

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-7.9

-

g

10

6

4

4

4

4

4

4

4

4

4

8

12

4

6

4

4

4

3

3

3

2

1.0

0.29

-0.33

-0.7

-

h

10

6

0

0

0

0

0

0

0

0

0

6

6

1

1

0

0

-1

0

0

0

-1

-1.2

-1.01

-0.96

-0.6

-

g

10

7

0

0

0

0

0

0

0

0

0

-1

3

-2

1

0

1

1

1

1

1

2

2.0

2.06

2.13

2.1

-

h

10

7

-2

-2

-2

-2

-2

-2

-2

-1

-1

-4

-3

-3

-1

-2

-1

-1

-1

-1

-2

-2

-2.9

-3.47

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-4.2

-

g

10

8

2

2

2

1

1

1

1

2

2

-3

2

6

-1

2

0

0

2

2

3

5

4.2

3.77

3.09

2.4

-

h

10

8

4

4

4

4

4

4

4

4

4

-2

6

7

6

3

3

3

4

4

3

1

0.2

-0.86

-1.99

-2.8

-

g

10

9

2

2

2

2

3

3

3

3

3

5

10

-2

2

2

3

3

3

3

3

1

0.3

-0.21

-1.03

-1.8

-

h

10

9

0

0

0

0

0

0

0

0

0

0

11

-1

0

0

1

1

0

0

-1

-2

-2.2

-2.31

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-1.2

-

g

10

10

0

0

0

0

0

0

0

0

0

-2

3

0

0

0

-1

-1

0

0

0

0

-1.1

-2.09

-2.80

-3.6

-

h

10

10

-6

-6

-6

-6

-6

-6

-6

-6

-6

-2

8

-3

-7

-6

-4

-5

-6

-6

-6

-7

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-8.7

-

g

11

0

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

2.7

2.95

3.05

3.1

-

g

11

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-1.7

-1.60

-1.48

-1.5

-

h

11

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.1

0.26

0.13

-0.1

-

g

11

2

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-1.9

-1.88

-2.03

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-

h

11

2

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1.3

1.44

1.67

2.0

-

g

11

3

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1.5

1.44

1.65

2.0

-

h

11

3

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.9

-0.77

-0.66

-0.7

-

g

11

4

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.1

-0.31

-0.51

-0.8

-

h

11

4

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-2.6

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-1.1

-

g

11

5

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.1

0.29

0.54

0.6

-

h

11

5

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.9

0.90

0.85

0.8

-

g

11

6

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.7

-0.79

-0.79

-0.7

-

h

11

6

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.7

-0.58

-0.39

-0.2

-

g

11

7

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.7

0.53

0.37

0.2

-

h

11

7

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-2.8

-2.69

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-

g

11

8

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1.7

1.80

1.79

1.7

-

h

11

8

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.9

-1.08

-1.27

-1.4

-

g

11

9

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.1

0.16

0.12

-0.2

-

h

11

9

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-1.2

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-

g

11

10

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1.2

0.96

0.75

0.4

-

h

11

10

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-1.9

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-

g

11

11

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

4.0

3.99

3.75

3.5

-

h

11

11

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.9

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-

g

12

0

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

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-

g

12

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

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-0.21

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-

h

12

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

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-

g

12

2

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.2

0.21

0.30

0.4

-

h

12

2

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.23

0.27

0.4

-

g

12

3

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.9

0.89

1.04

1.2

-

h

12

3

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

2.5

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1.9

-

g

12

4

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.2

-0.38

-0.63

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-

h

12

4

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-2.6

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-

g

12

5

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.9

0.96

0.95

0.9

-

h

12

5

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.7

0.61

0.49

0.3

-

g

12

6

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.5

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0.1

-

h

12

6

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.40

0.59

0.7

-

g

12

7

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.46

0.52

0.5

-

h

12

7

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.0

0.01

0.00

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-

g

12

8

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

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-

h

12

8

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.0

0.02

0.13

0.3

-

g

12

9

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

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-

h

12

9

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.28

0.27

0.2

-

g

12

10

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.1

0.08

0.21

0.2

-

h

12

10

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.9

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-

g

12

11

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.2

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-

h

12

11

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

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-0.23

-0.1

-

g

12

12

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

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0.04

0.0

-

h

12

12

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.8

0.88

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0.7

-

g

13

0

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.2

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0.0

-

g

13

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.9

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-

h

13

1

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.9

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-

g

13

2

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.30

0.31

0.4

-

h

13

2

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.2

0.33

0.30

0.4

-

g

13

3

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.1

0.28

0.42

0.5

-

h

13

3

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1.8

1.72

1.66

1.6

-

g

13

4

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

-0.43

-0.45

-0.5

-

h

13

4

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

-0.54

-0.59

-0.5

-

g

13

5

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

1.3

1.18

1.08

1.0

-

h

13

5

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-1.0

-1.07

-1.14

-1.2

-

g

13

6

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

-0.37

-0.31

-0.2

-

h

13

6

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.1

-0.04

-0.07

-0.1

-

g

13

7

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.7

0.75

0.78

0.8

-

h

13

7

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.7

0.63

0.54

0.4

-

g

13

8

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.4

-0.26

-0.18

-0.1

-

h

13

8

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.21

0.10

-0.1

-

g

13

9

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.35

0.38

0.3

-

h

13

9

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.6

0.53

0.49

0.4

-

g

13

10

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.1

-0.05

0.02

0.1

-

h

13

10

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.3

0.38

0.44

0.5

-

g

13

11

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.4

0.41

0.42

0.5

-

h

13

11

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.2

-0.22

-0.25

-0.3

-

g

13

12

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.0

-0.10

-0.26

-0.4

-

h

13

12

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.5

-0.57

-0.53

-0.4

-

g

13

13

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

0.1

-0.18

-0.26

-0.3

-

h

13

13

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.9

-0.82

-0.79

-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.

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 bottom), also known as the South Atlantic Anomaly, has continuously drifted westward and decreased since 1900. The point of minimum intensity at the Earth’s surface is currently over Southern Paraguay and is expected to cross the political boundary with Argentina during the second half of 2016. Maps of the predictive annual rate of change for D, I, and F between 2015 and 2020 at the Earth’s surface are shown in Figure 2. They are consistent with the continuation of the long-established westward drift and deepening of the South Atlantic Anomaly.
Figure 1

Maps of the magnetic declination D (top, units are degrees), inclination I (middle, units are degrees), and total intensity F (bottom, units are nT) at the Earth’s mean radius r=a in 2015; the red dot indicates the minimum intensity. Projection is Mercator.

Figure 2

Maps of the predicted rate of change per year in the declination D (top, units are degrees/year), the inclination I (middle, units are degrees/year), and total intensity F (bottom, units are nT/year) at the Earth’s mean radius r=a for the interval 2015.0 to 2020.0. Projection is Mercator.

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.
Figure 3

Motion of the magnetic dip pole (red) and geomagnetic pole (blue) since 1900 from IGRF-12 in the northern hemisphere (left) and the southern hemisphere (right). Stereographic projection is employed. The scale bar gives an indication of distance on the WGS84 ellipsoid that is correct along lines of constant longitude and also along the middle lines of latitude shown.

Figure 4

The northward velocity of the geomagnetic dip poles in the northern (purple dots) and southern (orange crosses) hemisphere as estimated by IGRF-12 on the WGS84 spheroid.

Table 4

Magnetic pole positions since 1900 as determined from IGRF-12 in WGS84 geodetic latitude

 

North dip pole

South dip pole

North geomagnetic pole

South geomagnetic pole

Epoch

Latitude

Longitude

Latitude

Longitude

Latitude

Longitude

Latitude

Longitude

1900.0

70.46

-96.19

-71.72

148.32

78.68

-68.79

-78.68

111.21

1905.0

70.66

-96.48

-71.46

148.55

78.68

-68.75

-78.68

111.25

1910.0

70.79

-96.72

-71.15

148.64

78.66

-68.72

-78.66

111.28

1915.0

71.03

-97.03

-70.80

148.54

78.64

-68.57

-78.64

111.43

1920.0

71.34

-97.39

-70.41

148.20

78.63

-68.38

-78.63

111.62

1925.0

71.79

-98.00

-69.99

147.63

78.62

-68.27

-78.62

111.73

1930.0

72.27

-98.69

-69.52

146.79

78.60

-68.26

-78.60

111.74

1935.0

72.80

-99.34

-69.06

145.77

78.57

-68.36

-78.57

111.64

1940.0

73.30

-99.87

-68.57

144.60

78.55

-68.51

-78.55

111.49

1945.0

73.93

-100.24

-68.15

144.44

78.55

-68.53

-78.55

111.47

1950.0

74.64

-100.86

-67.89

143.55

78.55

-68.85

-78.55

111.15

1955.0

75.18

-101.41

-67.19

141.50

78.54

-69.16

-78.54

110.84

1960.0

75.30

-101.03

-66.70

140.23

78.58

-69.47

-78.58

110.53

1965.0

75.63

-101.34

-66.33

139.53

78.60

-69.85

-78.60

110.15

1970.0

75.88

-100.98

-66.02

139.40

78.66

-70.18

-78.66

109.82

1975.0

76.15

-100.64

-65.74

139.52

78.76

-70.47

-78.76

109.53

1980.0

76.91

-101.68

-65.42

139.34

78.88

-70.76

-78.88

109.24

1985.0

77.40

-102.61

-65.13

139.18

79.04

-70.90

-79.04

109.10

1990.0

78.09

-103.68

-64.91

138.90

79.21

-71.13

-79.21

108.87

1995.0

79.09

-105.42

-64.79

138.76

79.39

-71.42

-79.39

108.58

2000.0

80.97

-109.64

-64.66

138.30

79.61

-71.57

-79.61

108.43

2005.0

83.19

-118.24

-64.55

137.85

79.82

-71.81

-79.82

108.19

2010.0

85.02

-132.84

-64.43

137.32

80.09

-72.21

-80.09

107.78

2015.0

86.29

-160.06

-64.28

136.59

80.37

-72.63

-80.37

107.37

2020.0

86.39

169.80

-64.11

135.76

80.65

-73.17

-80.65

106.83

IGRF-12 online data products

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 coefficients: 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

Appendix: World Data Centers

WORLD DATA SERVICE FOR GEOPHYSICS, BOULDERNOAA National Centers for Environmental Information, NOAA, 325 Broadway, E/GC, Boulder, CO 80305-3328UNITED STATES OF AMERICAINTERNET: http://www.ngdc.noaa.govWORLD DATA CENTRE FOR GEOMAGNETISM, COPENHAGENDTU Space, Diplomvej, Building 327, DK 2800, Kgs. Lynbgy, DENMARKTEL: +45 4525 9713FAX: +45 353 62475EMAIL: cfinlay@space.dtu.dkINTERNET: http://www.space.dtu.dk/English/Research/Scientific_data_and_models

WORLD DATA CENTRE FOR GEOMAGNETISM, EDINBURGHBritish Geological SurveyMurchison House, West Mains Road Edinburgh, EH9 3LAUNITED KINGDOM TEL: +44 131 650 0234FAX: +44 131 668 4368EMAIL: wdcgeomag@bgs.ac.ukINTERNET: http://www.wdc.bgs.ac.uk/

WORLD DATA CENTRE FOR GEOMAGNETISM, KYOTOData Analysis Center for Geomagnetism and SpaceMagnetism Graduate School of Science, Kyoto UniversityKitashirakawa-Oiwake Cho, Sakyo-kuKyoto, 606-8502, JAPANTEL: +81 75 753 3929FAX: +81 75 722 7884EMAIL: iyemori@kugi.kyoto-u.ac.jpINTERNET: http://wdc.kugi.kyoto-u.ac.jp

WORLD DATA CENTRE FOR GEOMAGNETISM, MUMBAIIndian Institute of GeomagnetismColaba, Mumbai, 400 005, INDIATEL: +91 22 215 0293FAX: +91 22 218 9568EMAIL: abh@iigs.iigm.res.inINTERNET: http://iigm.res.in

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) 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.

Authors’ Affiliations

(1)
Laboratoire de Planétologie et Géodynamique de Nantes, University of Nantes
(2)
DTU Space, National Space Institute, Technical University of Denmark
(3)
British Geological Survey, Murchison House
(4)
Cooperative Institute for Research in Environmental Sciences, University of Colorado
(5)
NOAA National Centers for Environmental Information (NCEI)
(6)
Institut de Physique du Globe de Paris, Sorbonne Paris Cité, Univ. Paris Diderot
(7)
University Grenoble Alpes
(8)
Université Grenoble Alpes
(9)
CEA, LETI, MINATEC Campus
(10)
Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, IZMIRAN
(11)
ETH Zürich Institut für Geophysik, Earth and Planetary Magnetism Group
(12)
Institut de Physique du Globe de Strasbourg, UMR 7516-CNRS, Université de Strasbourg/EOST
(13)
Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics
(14)
Centre National d’Etudes Spatiales
(15)
GFZ German Research Centre for Geosciences
(16)
Planetary Geodynamics Laboratory, NASA GSFC
(17)
School of Chemistry, University of Newcastle upon Tyne
(18)
CNES, Centre National d’Etudes Spatiales
(19)
Départment de Géophysique USTHB, University of Algiers
(20)
Joint Center for Earth Systems Technology, UMBC

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Copyright

© Thébault et al. 2015