Spherical Harmonic Analyses of the Geomagnetic Field for Eight

Ceophys. J. R. astr. SOC. (1974) 36,497-513

Spherical Harmonic Analyses of the Geomagnetic Field for Eight Epochs between 1600 and 1910

D. R. Barraclough

(Received 1973 October 5)*

Summary

The data for epochs between 1600 and 1910 in the catalogue of Veinberg and Shibaev have been analysed and two sets of spherical harmonic models of the geomagnetic field have been derived. Two methods are used to overcome the shortage of intensity data for the earlier epochs. In the first set of models the ratios of the harmonic coefficients to gIo are determined using both D and I data. The second set uses assumed values of 8,’ for these epochs. In contrast to previous analyses based on this catalogue, the data are weighted according to the number of original observations contributing to the catalogue mean and allowance is made for the oblate- ness of the Earth. The dipolar nature of the field in the Pacific region is investigated and the positions of the geomagnetic poles are derived. The variation of the position of the eccentric dipole with time is studied.

1. Introduction

A valuable source of early geomagnetic data is the catalogue based on B. P. Veinberg’s collection of observations (Veinberg & Shibaev 1969). Selections of these data have been analysed several times using spherical harmonics. Benkova, Adam & Cherevko (1970) analysed the data for 1600, 1650, 1700 and 1750 but published the results only in the form of graphs. Adam, et al. (1970a, b) used the results of these analyses and those of additional analyses for 1550, 1800 and 1850 to investigate the time dependence of various parameters of the geomagnetic field such as the positions of the eccentric dipole, of the geomagnetic poles and of the dip-poles. The values of the spherical harmonic coefficients themselves were not published, however. Braginskii & Kulanin (1971) have analysed the data for 1600, 1650, 1700, 1750 and 1800 using a different method and have published values €or the resulting coefficients and their standard deviations. Braginskii (1972) has used these same data, augmented by data from other sources, to produce a further set of models for these epochs.

Investigations of fluid motions in the Earth’s core in the past few centuries require spherical harmonic models which describe the geomagnetic field as accurately as possible and as far back in time as there are sufficient data available. To this end the data contained in the catalogue for epochs 1600, 1650, 1700, 1750, 1800 and 1850 are here re-analysed and those for 1890 and 1910 are analysed for the first time. These analyses are based on a broader selection of the data than those of Adam, Benkova and their colleagues. Refinements incorporated include the use of an oblate

* Received in original form 1973 August 17.

497

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498 D. R. Barraclough

Table 1 Data used in the anulyses

Epoch

1600 1650 1700 1750 1800 1850 1890 1910

No. of data points D I H

201 0 0 177 0 0 255 5 0 351 100 0 436 206 40 461 448 391 325 298 267 387 412 400

No. of original observations

D Z H 1594 0 0 1199 0 0 3673 8 0 4221 834 0 6551 1115 207

12164 9212 7741 12772 9181 8345 11096 6834 7165

No. of iterations

(Methods C & D)

L 2

spheroid instead of a sphere to approximate the actual shape of the Earth and a more realistic scheme for assigning weights to the data.

2. Data

The catalogue of Veinberg & Shibaev (1969) contains values of the geomagnetic declination (D), inclination (I) and horizontal intensity (H) reduced to 12 epochs between 1550 and 1940, inclusive, and to positions whose latitude and longitude are multiples of lo". The numbers of such data points for each element and for each of the 8 epochs considered here are shown in Table 1 . In general, each of these data points is a mean value based on several original observations. The numbers of such original observations are also given, for each element and epoch, in Table I.

There are a few observations of H for 1800, but for earlier epochs only data for the angular elements D and I are available. Without sufficient intensity data, a complete description of the geomagnetic field in terms of spherical harmonic coefficients is not possible, unless additional assumptions are made. Adam, Benkova and their colleagues used a method which results in an incomplete model whose coefficients are ratios of the values of the second, third, etc. coefficients g,,"', h," to that of the first coefficient g,'. Only the declination data were used to derive these models, which describe the pattern of the geomagnetic field but give no information about the scale. Braginskii & Kulanin produced complete models by using assumed values for the coefficient 8,'.

3. Methods of analysis

3.1 Bauer's method This method, which will be referred to as Method A, is discussed in detail by

Benkova et al. (1970). It was, in fact, first proposed by Bauer (1894) (see also Schmidt 1897, 1935), but has only become practicable since the advent of digital computers. The equations quoted by Benkova el al., which refer to an earth assumed to be spherical, are rewritten here for the case of a spheroidal earth (for further details see Barraclough & Malin 1971).

For any point on or above the surface of the Earth, the relation

X sin D = Y cos D,

where X and Y are the horizontal components of the geomagnetic field in the directions of geographical north and east, respectively, can be rewritten by expanding X and Y in spherical harmonics. The field is assumed to be derivable from a scalar potential


Spherical harmonic analyses 499

and its sources are all assumed to be internal to the Earth's surface. The resulting relation is

s i n D 5 5 ((+)"" Anm(gnm cos m 4 + hnm sin m4) n = l m = O

n + 2 mPnm n = l m = O sin 8 -COS D 5 2 ((+) (-)(g/ sinqb-h,,"' cosrn4)

where Anm = nXnm cos 6 - (n + 1) Pnm sin 6,

K is the maximum degree of the analysis, ii is the radius of the reference sphere, here taken to be the mean radius of the Earth, 6371.2 km, r is the radial distance from the centre of the Earth, 8 is the geocentric colatitude, 6 is the difference between the geodetic colatitude and the geocentric colatitude, q5 is the longitude measured east from Greenwich, P,"(cos 8) is the associated Legendre function of degree n and order m and nXnm (cos 0) = dPnm(cos 8)/dB. (For simplicity in printing, the dependence of the functions Pnm and Xnm on cos8 is not stated explicitly in equation (1) nor in what follows.)

Using the single index notation of Benkova et al.,

and ($ xj replaces

[A,,"' cosmq5 sin D- (rnP,,"'/sinO) sinrn4 cos D](ii/r)"+' [A,,"' sin rn4 sin D + (mP,"/sin 8) cos m4 cos D](Zj/r)"+'. (2)

aj replaces

Then equation (1) can be rewritten as

where K' = K ( K + 2 ) is the number of coefficients (gnm, h,") to be determined. Since x1 = g 1 O # 0,

K' x1a,+ .c x,aj = 0,

J = 2

or, dividing throughout by xl,

where y j = xj/xl.

form If there are N values of D , a series of N inhomogeneous equations results, of the

K' .c y,aji = -ali ] = a i = l ,2, ..., N , (3)

where aji is the value of aj at the i-th data point. Each equation of condition of the form given by equations (2) and (3) is multiplied

by the square root of the number of original observations contributing to the data point to which the equation refers. Equal weight is thus given to each original observation. The system of equations is then solved using the method of least squares to give the values of y j for j = 2,3, . .., K' or, alternatively, of Gll , HI1, GZ0, . . ., H K K ,



where Grim = g//glo, H," = h,"/gl0. The standard deviations of the coefficients are derived in the usual way from the sum of the squares of the residuals and from the diagonal elements of the inverse of the normal equations matrix.

3 . 2 Bauer's method extended to enable I data to be used When values for the inclination are available at points for whch declination data

also exist, method A can be modified to permit inctusion of these additional data. This method (method B) uses the relation

A 2 cosl = H sin1 = ___ sin I cos D (4)

and was also proposed by Bauer (1894). Proceeding as in method A, and making the same assumptions with regard to

the geomagnetic field, 2 and X are expanded in terms of spherical harmonics. Equation (4) then becomes

sin1 5 2 ((;)n+z A,"(g,,"' cos mq5 + h,"' sin m+)

COSD n = t m = O

Bn"'(gnm cos m4 f h,"' sin m+) K

+cosI c n = l m = O ((f)""

where

Using the single index notation as before,

B," = (n + 1) Pnm cos 6 + nXnm sin 6 .

[A," cos m+ sec D sin I + B,,"' cos m+ cos I](a/r)"+2 [A," sin m+ sec D sin I + B," sin m+ cos I ] ( i i /~ - ) "+~ . (5 )

aj replaces

The number of data points N is now the sum of the number of D values, for each of which there is an equation of condition of the form described by equations (2) and (3), and the number of I values for points at which D values are also available, for each of which there is an equation of condition of the form described by equations ( 5 ) and (3). Each of the equations of condition of the first class is multiplied by the square root of the number of original D observations, as before. Each equation of the second class is multiplied by ( N , . (2jn). arctan (N,/NI))*, where N,, N , are the number of original observations of D, I , respectively, which contribute to the data points to which the equation refers. This function has the required properties that it tends to zero when N , is small and tends to the value of ( N J t when N , is large.

For 1700 and 1750 all the values of I in the catalogue are coincident with D values so that the total numbers of equations of condition ( N ) are the sums of the numbers of D and I values shown in Table 1. For 1800, N is equal to 636, there being 6 I values which do not have associated D values.

3 . 3 Iterative method The data for 1850, 1890 and 1910 include substantial amounts of horizontal

intensity ( H ) data. They have been analysed using the iterative method which has been widely used in recent years. It has been described by Cain et a1. (1967). A version of this method was used by Braginskii & Kulanin (1971) and by Braginskii (1972) to analyse the data for 1800. The version used here, and referred to as method C, uses a spheroid rather than a sphere to approximate the shape of the Earth. The equations of condition for D, I and H are assigned weights equal to the number of original observations contributing to the particular data point.


Spherical harmonic analyses 50 1

The initial values of the coefficients to be corrected were taken, for epoch 1910, from the model for 1905 which is based on the analyses of the secular variation field performed by Vestine et al. (1947) and quoted in McDonald & Gunst (1967). For the other epochs, the initial values used were those from the next, more recent, epoch (e.g. the initial values for 1890 were the coefficients from the analysis for 1910).

3.4 Iterative method modijed for use when there are insuficient intensity data For epochs before 1850 a modification of method C , similar to that used by

Braginskii & Kulanin (1971) and by Braginskii (1972), has been used and is referred to as method D. Values for the coefficient glo cannot be determined in the analysis unless there is an adequate distribution of intensity data. They are therefore derived from the expression

g lo ( t ) = -31110~3+15~46(t-1914~0), (6) where t is the epoch in years AD. This expression was derived by fitting a straight line to the values of gl0 from 170 spherical harmonic models of the geomagnetic field with epochs between 1829.0 and 1970-0, using the method of least squares. Where there are several models for the same epoch a selection procedure was used to reject any values of glo which differed widely from the mean value for that epoch.

In the subsequent analyses, using the iterative method, the values of glo found from equation (6) were held fixed and the corrections for all the other coefficients were determined. The Earth was assumed to have a spheroidal shape and the same weighting scheme as in method C was used.

Braginskii & Kulanin (1971) and Braginskii (1972) assumed a spherical earth and assigned weights equal to 1.0,0-25 and 1.0, respectively, to the equations of condition for D, I and H. Another feature of their method was that the values of the coefficients g20, 8,' and 8,' were held fixed and equal to zero when no I data were available.

4. Spherical harmonic models

The models produced using the methods just described all have maximum order and degree of K = 4 (K' = 24). The values of the coefficients G,", H," produced by using methods A and B are given in Table 2 together with their standard deviations. Table 3 contains the values of the coefficients g,,"', h,", and their standard deviations, which result from the analyses using methods C and D.

For epochs before 1850 the models in Tables 2 and 3 have been compared. The coefficients g,", h," of Table 3 were divided by gIo and the differences, A, between these values and those in Table 2 for the same epoch were derived. The standard deviation tsA of this difference was computed from the standard deviations of the two coefficients differenced and the quantity lAl/tsA, expressing the absolute value of the difference in units of its standard deviation was calculated. The results are summarized in Table 4. In only one case, G,' for 1600, does the difference exceed 3 standard deviations and in only one other instance, H , 2 for 1600, does it exceed 2 standard deviations. The two methods (classing methods A and B together) thus lead to similar results. The agreement improves for later epochs, reflecting the improvement in the distribution of the data.

It will be noted, in Tables 2 and 3, that the coefficients with the largest standard deviations are those with m = 0. These are also those for which the differences (IAI) between the two sets of models are largest (see Table 4). These standard deviations and differences become smaller for later epochs. This reflects the fact that the zonal harmonics are not well determined by declination data alone.

The two sets of models should, of course, be identical since they are both, in theory, least squares solutions of the same problem using the same data. That they


G,H

n m

G1

1

H1

1

G

20

G

21

H

21

G

22

H

22

G

30

G

31

H

31

G

32

H

32

G

33

H

33

G

40

G

41

H

41

G

42

H

42

G

43

H

43

6

44

H

44

Tabl

e 2

Coe

ficie

nts

Grim

, H,,"

, pro

duce

d by

met

hods

A a

nd B

, and

thei

r st

anda

rd d

evia

tions

16

00

1650

17

00

1750

18

00

0~10

79+

_0~

0069

0.

0946

+_0*

0060

0.

1025

+0*0

036

0*10

43+_

0.00

32

0-09

97+_

0-00

22

-0.0

706

0.00

59

-0.0

858

0.00

52

-0.1

105

0.00

30

-0.1

403

0.00

22

-0.1

610

0.00

20

-0.2

914

0.04

46

0.07

58

0.04

86

-0.0

213

0.02

10

-0.0

202

0.00

44

-0.m

3

0.00

33

-0.1

010

0.01

20

-0.0

569

0.01

12

-0.0

817

0.00

71

-0.0

757

0.00

36

-0.0

660

0.00

26

-0.0

125

0.00

92

-0.0

586

0.00

71

-0.0

326

0.00

57

-0.0

282

0.00

26

-0.0

229

0.00

25

0.05

74

0.00

55

0.05

93

0.00

45

0.05

60

0.00

28

0.03

58

0.00

19

0.01

75

0-00

14

0.0908

0.00

45

0.05

70 0.0048

0.02

57

0.00

20

-0.0

095

0.00

18

-0.0

346

0.00

16

0.10

90

0.03

28

0.03

37

0.02

92

0.01

61

0.01

81

-0.0

311

0.00

46

-0.0

278

0.00

33

0.10

04

0.01

48

0.02

36

0.01

35

0.03

57

0.00

87

-0.0

042

0.00

31

0.03

15

0.00

25

-0.0

138

0.01

16

-0.0

052

0.01

08

0.00

47

0.00

82

0.02

12

0,00

27

0.01

81

0.00

26

-0.0

371

0.00

75

-0.0

096

0.00

46

-0.0

138

0.00

36

-0.0

326

0.00

22

-0.0

406

0.00

15

-0.0

529

0.00

92

0.01

28

0.00

63

-0.0

094

0.00

30

-0.0

055

0.00

18

-0.0

109

0.0017

0.00

45

0.00

33

0.00

28

0.00

35

0.00

67

0-00

15

0.01

26

0*00

14

-0.0

016

0.00

12

0.00

16

0.00

35

-0.0

073

0*00

30

-0.0

148

0.00

15

-0.0

255

0.00

14

-0.0

286

0.00

12

-0.0

067

0.01

37

-0.0

476

0.01

19

-0.0

450

0.00

92

-0.0

244

0.00

%

-0.0

221

0.00

24

-0.0

596

0.00

94

-0.0

198

0.00

99

-0.0

620

0.00

87

-0.0

265

0.00

31

-0.0

390

0.00

21

0.02

68

0.00

79

0.02

22

0.00

62

0.01

48

0.00

23

0.00

10

0.00

23

0.01

24

0-00

73

0.05

47

0.00

66

0.03

84

0.00

56

0.03

12

0.00

32

0.02

30

0.00

18

0.02

04

0.00

17

O.ooo6

0.00

13

0.00

29

0.00

11

0.00

27

0.00

34

-0.0

065

0.00

23

-0.0

039

0.00

18

0.01

34

0.00

38

0.00

84

0.00

21

0.01

16

0.00

16

0.00

73

0.00

13

0-01

11

0-00

12

-0.0

080

0.00

21

-0.0

108

0.00

22

0.00

07

0.00

11

0.00

35

0.00

11

0.00

54

0.00

10

-0.0

021

0.00

22

0.00

92

0.00

20

0.00

95

0.00

11

0.00

72

0.00

10

0.00

60

0~00

11

-0.0

035

0.00

51

-0.0

159

0.00

41

-0.0

144

0.00

32

0.00

21

0.00

18

-0.0

177

0*00

15

p P


nm

1

0

11

1

1

20

2

1

21

2

2

22

3

0

31

3

1

32

3

2

33

3

3

40

4

1

41

4

2

42

4

3

43

4

4

44

Tab

le 3

C

oefic

ient

s g,,"', h

,", p

rodu

ced

by m

etho

ds C

and

D, a

nd th

eir

stan

dard

dev

iatio

ns.

Uni

ts: g

amm

as

1600

-342

1 f 3

20

- 35

965

3055

29

3

2026

75

3 15

42

479

-193

36

18

-198

0 23

0 -3

282

184

-266

4 16

63

-232

0 79

8 13

81

553

512

381

185

570

-226

14

2 -4

3 14

4 13

17

655

2088

41

9

583

235

-208

9 30

9 -60

144

-559

16

0 28

9 85

-1

25

105

-120

7 38

1

1650

-

3519

2 -3

396f

22

8 29

36

205

614

2570

27

99

564

1902

29

7 -2

053

167

-214

7 17

2 -2

500

1332

-1

335

630

-29

451

655

202

-134

30

4 -1

92

121

318

118

680

469

1803

41

1 -4

12

330

355

164

-144

4 22

7 16

6 88

416

86

-298

77

-256

70

1700

-

3441

9 - 3

497 5

126

3761

10

8 10

94

807

2668

26

4 11

51

210

-206

0 10

1 -8

72

69

251

676

-755

33

0 -5

96

302

656

133

312

110

524

51

807

334

1497

31

2 -3

77

220

376

112

-939

11

6 16

5 64

-413

57

-3

4 39

-3

42

39

-236

55

1750

-3

3646

-3

6225

110

4763

76

49

0 14

8 24

52

123

1075

90

294

64

696

151

171

112

-568

90

11

51

75

140

66

-448

49

87

2 51

80

0 84

10

87

102

-117

8 65

-540

I8

-6

9 63

-9

33

61

-1

45

-254

46

-1

29

38

-222

36

1800

-333

9k

74

5282

67

15

0 10

2 23

09

89

717

82

1108

53

73

2 10

8

- 32

873

-522

47

-111

1 81

-5

76

84

1332

52

36

3 57

83

39

90440

724

75

1332

67

86

76

58

1 49

-6

64

54

-106

38

-3

39

41

-214

34

-2

00

36

1850

- 32

222 5

94

-282

3 48

57

87

49

2791

48

154

37

1472

37

11

30

36

-192

41

-193

48

-930

38

-2

74

38

1466

37

28

35

24

8 32

80

2 33

73

4 32

78

4 31

10

30

37

5 32

-1

93

32

-247

29

-2

92

30

44

29

-247

29

1890

-3

1718

f99

-266

4 67

57

64

68

-244

79

29

72

68

-834

67

70

4 62

13

03

66

700

70

-140

8 58

-2

39

53

1471

58

13

61

41

4 57

55

1 54

97

5 39

10

47

39

-73

38

501

38

-99

38

-422

38

-1

92

38

149

39

-273

38

1910

- 3

1246

f 65

-2

367

49

5932

48

2945

49

1237

38

10

03 40

1008

38

-658

41

-102

6 45

-154

3 34

-3

46

34

1141

37

20

39

62

1 32

39

8 32

87

3 30

91

3 28

72

28

66

6 30

18

29

-5

50

29

-211

29

16

7 27

-7

6 27

v\

0

w



Table 4 Comparison of models produced by different methods

Epoch I4 /.a (value)(coeff s .) RMS Maximum I A( /ua

1600 1 . 8 6.5 G2' 1650 0 .7 1 .6 G4' 1700 0 . 7 1.6 G40 1750 0 . 8 1.9 G42 1800 0 .6 1.1 Gz', GJ', H4'

are not identical is possibly caused by the use of a small number of iterations in Methods C and D (see Table 1).

A similar comparison has been made between the models of Table 3 and the fourth order models of Braginskii & Kulanin (1971) for 1600, 1650, 1700, 1750 and 1800. Table 5 gives, for each epoch, the root mean square values of lAI/oA, its maximum value and the coefficient to which this maximum value applies. The coefficients 8,' and, for 1600, 1650 and 1700, gZo, g3*, 8,' were not included. The agreement is quite good for all epochs. There are 3 cases where lAl/oA is greater than 3.0 and another 4 where it is greater than 2.0 but less than 3.0. It is interesting to note that for four of the five epochs the coefficient g,' has the maximum value of

There are three reasons why the models derived here differ from those of Braginskii and Kulanin (1 97 1). Firstly, different weighting schemes have been used, the weights assigned here taking better account of the quality of the data. Secondly, the methods described above all take account of the non-spherical shape of the Earth whereas Braginskii and Kulanin assume the Earth to be spherical. Finally, the expressions used to derive values of gIo for the earlier epochs differ somewhat, that used in the present work being based on all published spherical harmonic models with epochs of 1829.0 and later.

Attempts have been made to derive models, using methods A and D, from the 41 values of D for 1550 included in the catalogue, but without success. In all cases the standard deviations of the resulting coefficients were at least of the same order of magnitude as the coefficients themselves and in some cases they were many times larger. It was suspected that this was caused by the normal equations matrix being ill conditioned. This would be a reasonable surmise since the data for 1550 are sparsely and unevenly distributed. There are no data South of latitude 40" S or in longitudes between 80" E and 310" E. It is possible to derive upper and lower bounds for the condition number P of a matrix quite simply (see, for example, Golden (1965), pp. 232-233). For 1550 P was found to be greater than 3 x lo6 showing that the normal equations matrix was indeed very ill conditioned. The matrices for epochs later than 1700 were all well conditioned ( P < 100). For 1650 and 1700 P was of the order of 1000 and the matrices were reasonably well conditioned, but the conditioning was poorer for 1600, P being greater than 5000.

I Alb*.

Table 5 Comparison with models of Braginskii and Kulanin

RMS Maximum I A1 /aa Epoch I 4 /.a (value)(coeffs.) 1600 1 - 1 2 .2 g41,g44 1650 0.9 1 . 7 g4' 1 700 1 . 0 2.2 h43 1750 1.4 3 . 1 h43, g4' 1800 1 .3 3-4 841



FIG. 1. Chart of the declination computed from the model for epoch 1600 (Dashed curves indicate negative or westward declination).







Spherical harmonic analym 507





In Figs 1-6 are shown contour charts of the geomagnetic declination as synthesized from the models derived by methods C and D for 1600, 1650, 1700, 1750, 1800 and 1850, respectively. When these charts are compared with Halley's famous world chart for 1700, with the charts for 1600 and 1700 drawn by Hansteen (1819) and with those of van Bemmelen (1 899) for 1600, 1650 and 1700, a reasonable degree of agreement is found.

5. The geomagnetic field in the region of the Pacific

Doell & Cox (1971) have shown that the geomagnetic field in the region of the Pacific Ocean has, at the present time, a more nearly dipolar character than the field over the rest of the Earth. They also present evidence which suggests that this has been true in the past.

The geomagnetic field in the Pacific for the epochs covered by the set of models derived using methods C and D has been investigated. The first three coefficients of each model were used to compute the declination, inclination and horizontal intensity for the centred dipole field at points where catalogue data were available. The residuals between these synthesized values and the data were derived and root mean square residuals were computed for the points situated in the Pacific area and for those in the remaining parts of the world. Each residual for the declination ( A D ) was multiplied by the value of the horizontal intensity ( H ) synthesized at the data point.

The root mean square values of AH and of H A D are lower, for all epochs, in the Pacific than elsewhere, though there are indications that the differences between the two regions are smaller for the earlier epochs. These results therefore support the findings of Doell & Cox (1971). The root mean square residuals for the inclination (AZ) are smaller in the Pacific than elsewhere for 1850 and for later epochs but they are larger for epochs before 1850. A possible reason for this could be that the whole of the Pacific was included whereas Doell & Cox (1971) base their conclusions largely on data for Hawaii. Their findings thus refer to the centre of the Pacific region which is where the present minimum of the non-dipole field occurs.

6. Positions of the geomagnetic pole and of the eccentric dipole

The positions of the geomagnetic poles (which are the axis poles and the dip-poles of the centred dipole field) can be derived from the three first degree coefficients of a spherical harmonic model. The position of the eccentric dipole can be computed from the 8 coefficients of the first and second degrees (see Bartels 1936).

These positions have been computed from the coefficients of the models produced using methods C and D and the results are shown in Table 6. In this Table & and & are the latitude and longitude, respectively, of the northern geomagnetic pole in degrees (the southern geomagnetic pole is antipodal). The position of the eccentric dipole is described by re, its radial distance (in kilometres) from the centre of the Earth, 8, its colatitude and 4, its east longitude and also by its Cartesian coordinates (in kilometres) x,, ye , z, referred to the centre of the Earth as origin. The x-axis of this system points towards the north geographic pole, the y-axis towards the point (0 = 90°, 4 = 00) and the z-axis towards the point (8 = 90°, 4 = 900).

The two sets of models (that produced using Methods A and B and that produced using Methods C and D) were found to give similar positions for the geomagnetic poles except that for 1600 the longitudes of the pole differ quite considerably. The pole positions show a steady movement over the period 1650 to 1850 of 0.03"yr-' southwards and 0-1 1" yr-I westwards. After 1850 the southward drift becomes very small.


Tabl

e 6

Dip

ole

para

met

ers

Epo

ch

1600

.0

1 650

.0

1700

.0

1750

.0

1800

.0

1850

.0

1890

.0

1910

.0

1942

.5

1962

.5

82.7

" k 0

.5

82.7

0.

4 81

.5

0.2

79.9

0.

2 79

.2

0.1

78

.7

0.1

78.7

0.

1 78

.4

0.1

78.6

0.

1 78

.6

0.1

bC

318.

2"+

3.8

319.

2 2.

8 31

2-9

1.3

307.

2 0.

9 30

2.3

0-7

296.

0 0.

4 29

4.8

0.6

291.

8 0.

4 29

2.2

0.4

290.

4 0

-4

.ye

(km)

15+4

33

-60

314

-107

10

0 -5

2 19

-2

2 13

1

6

-1

11

39

6 78

5

117

6

Ye

-214

& I9

-2

80

60

-256

28

-2

37

13

-222

10

-2

79

6 -3

12

8 -3

25

6 -3

53

4 -3

64

4

(km)

(2,

-106

k52

-162

34

-9

9 23

-1

07

10

-84

9 5

5

67

8 89

5

159

5 21

2 5

re

(km)

239 &

19

329

I9

295

45

265

13

239

10

279

6 31

9 8

339

6 39

5 4

438

5

86"k

103

10

0 54

11

1 18

10

1 4

95

3 9

01

9

02

83

1

79

1 14

1

206"

& 1

4 21

0 7

201

5 20

4 2

201

2 17

9 1

168

1 16

5 1

156

1 15

0 I



900

I loo

I D E 160" 170° 180" 190" 2000 2100 0,

FIG. 7. Position of eccentric dipole: colatitude 0, plotted against longitude +#.

4 FIG. 8. Position of eccentric dipole: radial distance re plotted against colatitude (&).



\

.h. 1890 -0.

I --$- 180

& PE)

Fro. 9. Position of eccentric dipole: d(= re sin 0,) plotted against longitude (&).

Neglecting the value for 1600 which appears to be very inaccurate, the distance (re) of the eccentric dipole from the centre of the Earth is found to have decreased at a rate of 0.8 km yr-I from 1650 to 1800 and then to have increased by 0.9 km yr-*.

Figs 7, 8 and 9 depict some other aspects of the motion of the eccentric dipole. The values for 1942.5 and for 1962.5 are taken from the results of Malin (1969). They are representative of the values given by several other models for these or neighbouring epochs. In Fig. 7 the positions of the eccentric dipole are projected onto a sphere (e.g. the surface of the Earth), in Fig. 8 they are projected onto a (variable) meridian plane (polar plot of I , against d,) and in Fig. 9 they are projected onto the equatorial plane (polar plot of 4, against d = I , sinee). The uncertainties in the values plotted are indicated by lines one standard deviation long. In Figs 7 and 8 these lines would extend beyond the edges of the diagram for Oe in 1600 and 1650. They have therefore been omitted in these two instances. The dominant feature in each of these diagrams is the large amount of scatter for epochs earlier than 1800. In most cases the scatter is within the error limits indicated and is a reflection of the inaccuracies of the coefficients g,,"', h,"' (n = 1,2) for these epochs. Because of this scatter it is not possible to confirm the conclusion of Adam et al. (1970a) that the position of the eccentric dipole has followed an approximately elliptical path over the past 400 years.

7. Conclusions

The models whose coefficients are presented here represent the geomagnetic field for the 8 epochs from 1600 to 1910 as well as the distributions of the data in the catalogue of Veinberg & Shibaev (1969) allow. The linear extrapolation of the values of the coefficient gIo, which was used to overcome the shortage of early intensity data, leaves something to be desired but it has the merits of being simple to use and of being not too implausible. The other methods (A and B) used for the earlier epochs give similar results and do not involve any additional assumptions. They are suitable for studies where knowledge of the scale of the geomagnetic field is not important.


5 12 D. R. Barraclough

On the basis of these models, the dipolar character of the field in the Pacific has been studied. In general the results agree with Doell & Cox (1971) though there are some inlcations that the differences between this region and the rest of the world may not have been as great 300 years ago as they are now.

The northern geomagnetic pole is found to have drifted at a rate of 0.1 1" yr-' westwards and 0.03" yr-' southwards between 1650 and 1850. The eccentricdipole is found to have moved inwards at a rate of 0.8 km yr-' before 1800. Subsequently the movement has been outwards at about 0.9 km yr-'. The derived positions for epochs before 1750 show a considerable amount of scatter and it has not been possible to confirm that the eccentric dipole moves along an elliptical path.

Acknowledgments

1 am grateful to Mr B. R. Leaton and to Dr S . R. C. Malin for helpful discussions during the course of this work and to Mrs F. I. Penfold and Mr M. Fisher for help in producing the figures. Ths work was undertaken as part of the research programme of the Institute of Geological Sciences and acknowledgment is made to the Director for permission to publish this report.

Geomagnetism Unit, Institute of Geological Sciences,

Herstmonceux Castle, Hailsham, Sussex.

References

Adam, N. V., Baranova, T. N., Benkova, N. P. & Cherevko, T. N., 1970a. Geo- magnetic field variation, according to magnetic declination data for 1550-1960, Geomag. Aeron., U.S.S.R., 10(6), 1068-1074, (854-859 in English translation).

Adam, N. V., Baranova, T. N., Benkova, N. P. & Cherevko, T. N., 1970b. Spherical harmonic analysis of declination and secular geomagnetic variation 1550-1960, Earth Planet. Sci. Lett., 9, 61-67.

Barraclough, D. R. & Malin, S. R. C., 1971. Synthesis of International Geomagnetic Reference Field values, Inst. Geol. Sci., Rept. No. 71/1.

Bartels, J., 1936. The eccentric dipole approximating the earth's magnetic field, Terr. Magn. atmos. Elect., 41, 225-250.

Bauer, L. A., 1894. An extension of the Gaussian potential theory of terrestrial magnetism, Proc. Amer. Ass. Adv. Sci., 43, 55-58.

Benkova, N. P., Adam, N. V. & Cherevko, T. N., 1970. Application of spherical harmonic analysis to magnetic declination data, Geomag. Aeron., U.S.S.R., 10(4), 673-680, (527-532 in English translation),

Spherical analyses of the main geomagnetic field, 1550-1800, Geomag. Aeron., U.S.S. R. , 12(3), 524429 (464-468 in English translation).

Spherical analysis of the geomagnetic field from angular data and the extrapolated g l o value. 11, Geomag. Aeron., U.S.S.R., 11(5), 931-933 (786-788 in English translation).

Cain, J. C., Hendricks, S. J., Langel, R. A. & Hudson, W. V., 1967. A proposed model for the International Geomagnetic Reference Field-1965, J. geomagn. geoelect., Kyoto, 19(4), 335-355.

Doell, R. R. & Cox, A., 1971. Pacific geomagnetic secular variation, Science, 171,

Golden, J. T., 1965. FORTRAN ZV Programming and Computing, Prentice-Hall, Inc., New Jersey.

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248-254.



Hansteen, C., 1819. Untersuchungen iiber den Magnetismus der Erde, Lehmann & Grondahl, Christiania.

McDonald, K. L. & Gunst, R. H., 1967. An analysis of the earth’s magnetic field from 1835 to 1965, ESSA Tech. Rept. IER 46-IES 1, Boulder, Colorado.

Malin, S. R. C., 1969. Geomagnetic secular variation and its changes, 1942.5 to 1962.5, Geophys. J. R. astr. SOC., 17, 415-441.

Schmidt, A., 1897. Met. Zeits., 14, 39-42. Translation by L. A. Bauer: The magnetic condition of the earth expressed as a function of time by V. Carlheim- Gyllenskold, Terr. Magn. atmos. Elect. 2, 150-155 (1897).

Tafeln der normierten Kugelfunktionen und ihrer Ableitungen nebst den Logarithmen dieser Zahlen sowie Formeln zur Entwicklung nach Kugel- funktionen, Engelhard-Reyher Verlag, Gotha. Translation: NASA TT F-888 (1 964).

van Bemmelen, W., 1899. Die Abweichung der Magnetnadel. Beobachtungen, Sacular-Variation, Wert- und Isogonensysteme bis zur Mitte des XVIIIten Jahrhunderts, Obsns. magii. met. Obs. Batavia, 21 (Supp.)

Catalogue. The results of magnetic determinctions at equidistant points and epochs, 1500-1940, IZMIRAN, Moscow. Translation: No. 0031 by Canadian Department of the Secretary of State, Translation Bureau (1970).

Vestine, E. H., Laporte, L., Lange, I. & Scott, W. E., 1947. The geomagnetic field its description and analysis, Publ. Carneg. Znst. No. 580.

Schmidt, A., 1935.

Veinberg, B. P. & Shibaev, V. P. (Editor-in-Chief A. N. Pushkov), 1969.


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Spherical Harmonic Analyses of the Geomagnetic Field for Eight