Geophys. Suppl. MNRAS 1953 Jones 409 30

7/27/2019 Geophys. Suppl. MNRAS 1953 Jones 409 30

1/24

MONTHLY NOTICES

OF THE

ROYAL ASTRONOMICAL SOCIETYGEOPHYSICAL SUPPLEMENT

Vol. 6 No. 7 1953 June

THE HARMONIC ANALYSIS OF THE EARTHS MAGNETICFIELD, FOR EPOCH 1942

Sir Harold Spencer Jones, F.R.S. and P . J . Melotte

(Received 1952 May I)

SummaryThe results are given of the harmonic analysis of the Admiralty magnetic

charts of declination, horizontal intensity and inclination for the epochWithin the limits of observational error, the Earth's magnetic

field appears to be entirely of internal origin. There is no evidence of adipole field of external origin greater than 0 - 1 per cent of the field of internalorigin. The intensity of the dipole field is at present decreasing at a rateof about 5 per cent per century. The geomagnetic poles have a westerlydrift at a rate of 4"'s per century; the north magnetic dip pole is moving ina direction a little to the west of north, but the south magnetic dip poleappears to be practically stationary. In consequence of the dearth of magneticdata over the oceans since 1929, magnetic charts are becoming less accurateand there is a great need for airborne magnetic surveys of ocean areas.

' 1942.5.

I . In a paper entitled "The Earth's Magnetic Potential", by Dyson andFurner", a harmonic analysis of the Earth's magnetic field was given, based on datafrom the Admiralty Magnetic Charts for

1922. These charts, for magneticdeclination, horizontal intensity and inclination, were compiled at the RoyalObservatory.

2. Declination charts have been prepared at five-year intervals, while chartsfor other components of the magnetic field have been prepared at less frequentintervals. Charts for horizontal intensity and inclination, in addition to declin-ation, were last prepared for the epoch 1942.5. In future, in accordance with arecommendation of the Association of Terrestrial Magnetism of the InternationalUnion for Geodesy and Geophysics, charts for declination will be produced atfive-year intervals, 1955, 60, 65, etc., and charts for horizontal intensity, vertical

intensity, total intensity and inclination at ten-year intervals, 1955, ' 65 , '75, etc.Preparations for the compilation at the Observatory of the complete series ofcharts for the epoch 1955.0 have accordingly been commenced.

*F. Dyson and H. Furner, M.N., Geophys. Suppl., I, 76, 1923.G 30

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4103. Since the unfortunate loss of the non-magnetic ship, the Carnegie, in 1929,

practically no magnetic data for the sea areas have been obtained. Both the valuesof the elements of the field at epoch and their secular change have consequentlybecome increasingly uncertain over the sea areas with lapse of time. It thereforeseemed desirable to make a harmonic analysis of the Earth's magnetic field, asdepicted by the Adm iralty Magnetic Chartsfor 1942, n the expectation that it wouldprovide some indication of the regions in which the charted data were seriously inerror and that comparison between the analysis for the epochs 1922 nd 1942 wouldprovide information about the secular change of the field between those epochs inmuch greater detail than could be obtained from the scanty observations alone.The analysis was completed in 1945, but pressure of work has hitherto prevented.the results from being written up.

4. Values of the declination, horizontal intensity and inclination were readoff from the charts at points of a grid spaced at 10" ntervals in longitude and latitudebetween 80" N. and 80" S. latitudes. From the values so tabulated, the northerly,easterly and vertical components of the field were calculated ; the values of thesecomponents are tabulated in Tables 1-111.

If the magnetic potential is assumed to arise solely from forces situated insidethe Earth, it can be expressed in the usual form :

H . Spencer Jones and P . J . Melotte, The harmonic

V = a 5 e)n+lHnm(X)(gnmos m+ +hnm in 4)),n-0 m-0

where a denotes the radius of the Earth, r the distance from the centre, andgnm ndhnmare numerical coefficients,

( n - m)(n- m - )Hnm(h)= cosm {p"-- - z(zn - ) Pn-m-2

4 is the longitude, A the latitude, and p stands for s ink Th e nomenclature due toC. F. Gauss and used by J. C. Adams (Collected Papers, Vol. 11, "T he Theory ofTerrestrial Magnetism") is here followed. The symbols g and h are known asthe Gauss coefficients.

From this expression, if X, , denote the northerly, easterly and vertical(downwards) components, it follows that at the surface (where r = a ) :

I aV dHnmr ah dh=

- C- g," cos m+ +h n msin m+},= XmHnm ec h {-gnm sin m+ +hnmcos M),=-- 1 avrcosr\ a+avar

= - Z (n + r )Hnm{gnmos m+ +hnm in 4 ) .The first six harmonics were included in the solution.

5. Though the functions H m mhave been extensively used in harmonicanalysis by Laplace, Gauss, Adams and others, Adolf Schmidt introduced neworthogonal functions, Prim(0), defined by

when m = o ,

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3 3 3 353

3 3 3

3 3 3 3 3 3 3 3 2 2 2 2 2 2 2

2 3

6

7

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52

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I 1 1 1 I 1 I 2 2 2 2 2 2 1I01

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05

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.

+

f

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+

+

,

,

,

+ '

+I72

+

+

2 .!!

Y

e

n


4/24

H . Spencer Jones and P . J . Melotte, The harmonic


5/24

TA

I11

2

e

c

F

Umt0

0"

E

10

20

30

4 50

6 7 8 9100

I 0

I20

130

1 150

I 6 1

E

1 1

W

I 6

150

1 130

I20

I

0

I00

9 8 7 6 50

4 30

I0

w2

5 519

5 530

5 5 5 5 5 5 57

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

4 5 5 511

5 5 5 5 5 5 5 6 6 5 5 5 5 5 5 5 5 5 5 6 6 6 6 5 5 5 5 5 5 5

0

5

6

5

4

f4 4 4 4 4 4 5

11

5 5 5 5 5 5 5 5 5 5 5 5 4 4 5 552

5 5 6 6 6 5 5 5 5 5 4 4 $

4 4 4 445

4 4 501

5I

530

5 5 5 5 50

4 4 4 4 4 4 4 4 5

5 5 6 6 5 5 5 4 4 4 45

3 3 3 3 4 4 4 4 4 4 4 4 4 4

3 330

331

3 3 3 4 4 53 4 5 5 5 530

5 4 435

4 3 3

3

+ 2 2 2

2 2 2 310

3 3 3 3 3 3 2 2 2 2 2 2 2 313

3 3 3 4 4

3 4 4

4 445

4 3 3 302

2

+ 0"

0"

-0"

2

3

4

5

6

+ 1

-103

o

1

0

I

+

5

131

1

1

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112

4

9

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3

1

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132

32

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121

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9

-0

1

+ 2

1

I

130

20

I

2

150

3

1

4

1

6

2

8

2

1

2

131

2

1

2

I

2

1

I

+ 0

2

I00

9

-50

-6

-139

-2 2 2 2 2 2 3 3 3 3 3 3 3 3 3

51

330

310

2 2 2 2 2 1 1 152

1 9 6 43 4 6 1 I 1

- 2 2 2 2 3

00

313

3 3 3 43 4 4

4 4

4 4 3 3 3 2 2 2 14 4 3 2 1 115

1 I I I

I I 2

- 2 2 3

01

3 3

3 351

3 4 4 4 4 5 5 5 5 5 5 5 5 4 4 4 3 3 3 3 2

2 2 2 1 I6 I6 1 2 2I

-00

315

330

3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 4 4 4 4 4 3 3 2 2 2 2 2 2 2 2

-3 3 4 4 4

4 4

512

5 5 5 5 535

5 6 6 6 6 6 6 5 5 5 4 4 4 4 3 3 3 313

310

312

323

3

7

-

, 441

4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 5 5 5 5 5 4 4 4 4

4 4 4 4 4

-

5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 55 5 5 5 5 512

5 5 5 55

%

Y

Y

i R


6/24

414 H . Spencer Jones and P . J. Melotte, The harmonic

where P , enotes the associated Legendre function defined by

where = cose,

6 being the co-latitude, i.e.(n/z-A).It is readily shown tha tP , is related toH nm y the relation

( z n- ) !!( n - m ) ! Hnm.

H nm s identical with the function which Schm idt denotes byPapm.It follows tha t

( n - m ) ! 112 (zn - ) !! p n , ,n= { z m } ( n - m ) !

= 5 p n .m (SaY).T h e valuesof 4 = Pnrn/Pnnmor n = I to 6, m = o to 6 have been tabulated by

T h e use of the functionsPnm n geophysical investigations in preference t o theTheir properties are

Schmidt (Zoc. cit. )has tabulated the values of the functionPnm cos0 ) and the

Schmidt.*

functions Hnm has been recommended byS. Chapman.described inGeomagnetism,Vol. 11, Chap. X V II, by Chapman and Bartels.

associated functionsXn m dPnm COS e ) /nd o ; Ynm mPnm cose ) / n in 0

fo r values ofm and n up to 6 and for values of0 (the co-latitude) at intervalsof 5"from 0' to go'. His tables can therefore be used t o obtain the values ofdHnm/dA,mHnm ec A and (n+ ) Hnm n the expressions forX, , bove.

6. Vestinet carried out a harmonic analysisof the U.S. magnetic charts forepoch1945. H e expresses th e magnetic potential fro m internal forces in the fo rm:

* + I 1

n= aCX

()- Pnm(Anmosm+

+Bnm in m+)

and, at the surface,X = CXnm{Anm os m+ +Bnm in 4},Y X Ynm{ Anm in 4 +Bnm osm+},

n + I

n= C-

nn&A n m osm+ +Bnm in m+),whereXnm, nm, nm re as tabulated by Schm idt.

I n the Gaussian notation

5Xn" = - 'Hnm/dA Y n m= - mH,m secA); Prim= H,".n n

* A . Schmidt, Tafeln der Normierten Kugelfunctionen, Gotha, 1935, p. 20.t E. H. Vestine and others, The Geomagnetic Field, Its Description and Analysis, Camegie

Institution, Washington. Publication 580, 1947.


7/24

analysis of the Earth's rnagneticJield, fo r epoch 1942 415The values of the Gaussian coefficientsgnm,hnm are therefore related t o th e

coefficientsA,,", Bnm sed by Vestine by the formulae

These form ulae can be used fo r comparing the results of th e analysis by Vestineof the American charts for epoch 1945 with those of the present analysis of theBritish charts for epoch1942.

It should be noted that Vestine, in his TableIX, tabulates the first eightGaussian coefficients of the Earth's magnetic potential from a series of analysesgoing back to the earliest, by Gauss,1835. Th ese coefficients aregnm,hnmup tothe value of2 fo r m and n . T h e tabulated values are not, however, those of thecoefficients as defined by Gauss, but those of the coefficientsAnm, Bnm. It isadvisable tha t a distinction should be m ade in th e nom enclature for, and thedesignation of, the coefficients according to whether the expressions used by Gauss

or by Schm idt are adopted.7. T h e values ofX, , , iven in T ablesI, I1 and 111, were analysed to givetheg and h coefficients, the different latitude zones being com bined together withthe relative weights used by Dyson and Fu rne r, viz.:-

Lat. W t. Lat. Wt. Lat. W t.oo I 0 f 30" 8 &60 3

& 1 O 0 I0 f 4 0 " 7 f70" 2+ z o o g f 50" 5 *soo I

These preliminary solutions having been made, the weighted residuals foreach latitude zone fo r each of t he solutions were tabulated, and the mean residualsobtained fo rX, , eparately. T h e mean residuals corresponding to unit weightfor each latitude zone and for each component were then obtained. ForX nd Ythe zonal values ran smoothly, but in Z th e scatter was large, particularly in southernlatitudes. Fo r each component the mean residual was plotted against latitude anda smoo th curve was drawn through the plotted points. T h e values readoff fromthis curve were adopted as the mean residual for each zone of latitude and for eachcomponent. From these values, relative weights were obtained, which representmore accurately the uncertainties in the chart values than the provisional arbitrary

weights originally used. T h e relative weightsso derived, which were used innew solutions to obtain the definitive values of t h e g andh coefficients,are given inTable IV.

8. It will be seen tha t in general the weight for each com ponent fo r a southernlatitude belt is appreciably less than the weight for the corresponding northernlatitude belt. This is because the land areas are more extensive in the northe rnhemisphere than in the southern; over many of the land areas the magnetic dataare reasonably well known f rom land magnetic surveys, whereas there have beenvery few magnetic observations in the sea areas since 1929. An additional reasonis tha t the re are m any more permanent m agnetic observatories and secular changerepeat stations in the n orthern hemisphere than in the southern, in consequence ofwhich the secular change data are more reliable for the northe rn hem isphere thanfor the southern.

The weights in TableN have been formed on a uniform basis for eachcomponent. It will be noted that the weights in X a n d Y ar e of the same order, the


8/24

416 H. Spencer Jones and P. J. Melotte, The harmotzic

weights in Y eing mostly greater than those inX n n orthern latitudes, but some-what smaller in southern latitudes. T h eX and Y alues are formed from th eobserved values of declination and of horizontal intensity; for most of the regionof th e Earth's surface between60" N. and 60" S. latitude, t he declination does notexceed30". A given error inH will therefore produce a larger error i nX han in Y,while a given error inD will produce a larger error inY han in X. I n th esouthern Indian Ocean the declination is known to be very uncertain, as in thisregion the secular change in declinationis large, while the rate of change ofdeclination with latitude is also large.

TABLE V

Relative Weightsfor each Latitude Belt and fo r each Component ( X , Y , Z)

Lat.+80

7060

504030

+I 02 0

0

- 020

3040

506070

- 0

XI a612 -63

4'35. 6.95

8.6210'2

I I '913.2I 3 '913.511'1

8 206.17

3 '792-280 '740.36

Y2 a06

3 '055 0 0

I I '4015'2

19-220.819.215'2

10.9

7-58

7 255 '00

3 '471 '79o .800 42

2

0.350.460 630.87I '251.51

1 '541-39I '24

0.76

0.530'350.180.15

0.130.13

1 - 0 1

T h e weights of the vertical intensity are relatively low. T hi s componentis formed from the charted values of horizontal intensity and inclination. I t seemsprobable that there are appreciable errors in the inclination: over mu ch of thesurface of the globe, moreover, the inclination exceeds45", o that the errors ininclination are much enhanced in the vertical intensity.

9. If a portion of the Earth's magnetic field is of external origin, the magneticpotential will be expressible in the form

+u X (i)"H," (g-nmcosm$ +hWnmin m$)].T h e analysis, in the fo rm explained above, by which the coefficients of the te rmsdepending on the longitude are derived will determine

for X, g,m+g-,m,

nfor z g,"- ---g-,m.The coefficients determined from the separate analysesof the northerly and

easterly componentsof the magnetic field should therefore agree, apart fro m theeffect of accidental errors. T h e values of the coefficients obtained from the


9/24

analysis of the Earths mag net icfie ld, fo r epoch 1942 417

analysis of these two components, with the weights first assumed, were in fact i nclose agreement. A new solution was accordingly made by combining th e datafor these two components and using the definitive weights given in Table IV.The derived values of the coefficients are given in the second column of Table V.The vertical component was analysed separately, using the definitive weights givenin Table IV , and the derived values of the coefficients are given in the thirdcolumn of Table V.

10. If there is an external component of the total field the differencebetween the values of g,, derived separately from (X+ ) nd 2 s given by( z n + ~ ) g - , m / ( n + ~ ) .n comparing the values given in columns z and 3 ofTable V it should be noted that the values in column z have a weight that isseven or eight times the weight of the values in column 3 , the combined weightbeing given in column 5 , and that the probable error corresponding to unitweight is about +o.0035. From an examination of columns z and 3, in con-junction with the relative weights and probable errors, no evidence of the existenceof an external component of the field is found. I n relation to the probable errorsof the various quantities, there are no significant differences between the corre-sponding values in columns z and 3 .

The main portion of the field is represented by the dipole field of a uniformlymagnetized sphere (Section 16). The intensity of this field, H o, the co-latitude,do, and the longitude, rJo, of the northern pole of this field, as derived separatelyfrom the north and east and from the vertical components of the field, are asfollows :-

HO 6 0 4 0

From X and Y 0.3096 +o*ooo I I O - I - 8O.4From 2 0.310 4 + 0 - 0 0 1 0 IIo.O - 9O.4It can be concluded from these results that there is no evidence of a dipole

field of external origin which exceeds one-tenth of one per cent of the field ofinternal origin.

It has therefore been assumed that there is no portion of the Earths magneticfield that is of external origin. The equations for the (X+ Y) and the 2 com-ponents were therefore combined together with appropriate weights and solved togive the values listed in column 4 of Table V , he relative weights being given in

column 5.11. It is of some interest to compare the results of the present analysis with

the results obtained by Vestine. The charts from which the data for the twoanalyses were read off differ in epoch by only three years. Though the Britishand American charts were prepared quite independently, they are based essentiallyon the same data. But over many areas recent observations are completelylacking and both series of charts consequently have been based in some areason extrapolation, involving the assessment of the secular change of the Earthsmagnetic field and on the rate of change of this secular change.

The comparison between the Gaussian coefficients and the first four harmonics

from (X+ Y), and from 2, as derived by Vestine and in the present investigation,is given in Table VI.

The agreement between the coefficients derived from the north and eastcomponents is satisfactory. The discordances between those derived from th evertical component are appreciably greater, which is to be expected from the


10/24

H . Spencer Jones and P . J . Me lotte, The harmonicTmm V

Values of C o e f i c k t s

FromX a n d Y

f . 3 0 3 8+a017 8- 025 5- 039 7+ 030 3--021 0

+ . O Z I g- - 0 5 1 2+ '052 4- 040 5- 030 2- 007 5

- 055 4+ 026 6+ 019 4- or4 I+a006 I- 002 3-*013 3-so23 9- 023 6- 013 5

--004 2- 003 4+a008 5- 00 2 4- 0189- 007 5+ 008 9+ 00 2 3+.om 7

' 0 0 0 0

+ ~ O I

+'OOO 7+moo 8

+ 003 3+ a 0 2 I

+.OOI 0

+*003 4+moo 8

+ * o m 5--OOO 4- - ' 0 0 0 5+ 00 0 5+-OOO 6+ . 0 0 0 2

--oar 0

- 002 0

FromZ

+ 304 7+ ' 0 1 5 9- 0 2 45- 039 7+.OZI I--021 9+ '020 8- 047 6+'044 3- 032 9- 042 6- - 0 1 1 7

- -055 4+ '022 4+.or6 2- - 0 1 0 I- - 0 0 1 2

-*003 0

--015 9--or9 4- 027 9-Om7 3+.- 5- 006 8- m o o 4

+ 007 o- 036 5- 006 6+-5 3+.OIZ 9+.or1 I

- - 0 0 1 7+ * 0 0 5 0

+ 00 0 7+ 0 10 0+ a 0 0 0 I

-moo3 8

+.oo1 9+.003 I

+-OOO2

--'ooo- 0 0 0 6+ 0 0 0 3+ . 0 0 0 2

-*OOI 6

+.003 9

+ ' 0 0 0 9

' 0 0 0 0

Fromr, Yand(spherica

Earth)

+ 3039+a0176- 0255- 0398+ 0293- a 0 2 11

+.0218- 0509+ ' 0 5 1 5

-0397- 0329

- 0073

- 0555+ 0260+.0190

+ 0057- 0026--or35- 0236--0238--or30- 0007- 0044- 0033+ 0076--0018- 0204

+so087+.OZIO

+*001g

+a0018--0018+ 0034+ 0010+ 0032+*-9

+ 0005- om4--om4+ o o q+a006+ 'OOO2

- 0139

- 0074

f '003 I

- OOOI

+ 0009

+ 0017

Relativeweights

975928

5

I9394

I56

I9394

156

53211532

35321 I532

3983I3931

8

98313931

8

152316830

152316830

2 891962189

19628352835

1 2

2

38

2

38

2

I 0

I 0

Fromr, Yand;spheroidd

Earth)

+-3022+.or76- 0292- 0385+a0182- -0142

+a0218- 0507+ 0508- 0384- 0342- 0054--0553+ 0259+'OIg4-'Or43+ 0053--001g--or35- 0235

--or23+ 0 0 0 2- 0044- 0033+ 0076--m17- 0206- 0074+ 0087+ 0033+*ozog

- 0234

- 0001

+*0019

+'0017+ 'ooog--0018

+.0018

+.OoIO

+ 0032+ .ooog

+ 0005- 0004- 0004+ om5+ 0006+ '0002

+ 0033


11/24

analysis of the Earth's magnetic jie ld , for epoch 1942 419

relatively low weight of 2 rom the British charts, as shown by Table IV. Theagreement between the coefficients derived from the elements ( X + Y ) and 2is better from the American charts than from the British ; the American chartsof vertical intensity are the better.

TABLE IGaussian c o e f i c k t s derived by Vestine and by Jones and Melotte

x+ Y

Vestine

+ 305 7+*019 o- 028 7--wo

+.02I 0

--051 2

+ '051 9- * w 3 0

--058 I+ 028 7+.or5 9--007 8

--014 I- 023 4- - 0 2 2 5

--*- - w 3 5+.OIO 8

- 006 9f *007

--OOO 4+-0017

- 0 0 2 2

+*OOO 9

Jones andMelotte

+*303 8+.017 8- -025 5- 039 7

f . 0 2 1 9--051 2

+ 0 5 2 4- 040 5

- 055 4+ 026 6+*OI9 4--014 I

- or3 3- 023 9--oz3 6

-a04 2- 003 4+a008 5

- 007 5+*008 9

' 0 0 0 0

f - 0 0 1 6

--'002

+.oo1 0

Difference

Vestine

+ 305 7- o26 7- 043 7+ 0 22 7- - 0 g 1 0

+-053 8--'041 5

--057 9

+'Or7 9

+*029 7+ 01 4 3--.006 4

--014 4- 023 3-a023 2

-0004 I--004 0+*or1 4

--.007 3+ 008 5'OOO 0

f . 0 0 2 2

--'002 9

+.OOo 9

z

Jones andMelotte

+ 304 7+.or5 9--ow- 039 7+ -020 8- 047 6+w4 3- 032 9

-- '055 4+ 02 2 4+-016- -010 I

- 015 9- 019 4- 027 9

-*006 8-- 'ooo 4--OOI 6

-so06 63

- a 0 1

7+.005 0

+.OOO I

+.OOO 9

Difference

+ 0010+ 20

2 2

40

+ 1934

+ 95- 86

25+ 73

I9+ 37

+ 1539+ 47+ 27- 36

-

-

-

-

-

-

+ 1307

+ 32

+ I7- 2830

-

-

0

Vestine has not given the weights of his coefficients; he used the weightsthat were adopted by Dyson and Furner for combining the data for differentlatitudes, which give the southern latitudes too great a weight relative to thenorthern. On the basis of the assumption, which is not strictly correct, thatthe weights of each coefficient deduced from the X and Y harts are equal in thetwo investigations, it was found, from a comparison of all the coefficients for thefirst six harmonics, that the probable error corresponding to unit weight was&0.0054. The value obtained in the present investigation was k 0.0035.

The concordance of these two values is sufficiently close, in view of the fact


12/24

420 H . Spencer Jones and P . J . Melotte, The harmonicthat the basic assumption about the weights is not strictly accurate, to suggestthat the weights of the two determinations are not greatly different and that thediscordances between the individual coefficients are mainly accidental.

12. With the adopted values of the coefficients, given in column 4 f Table V,the values of X, , 2 t each point of the 10" network were computed and com-pared with the values derived from the charts. The differences between thechart data and the computed values are given for the three elements respectivelyin Tables VI I, VI II and IX. Th e quantities in these tables necessarily run onthe whole smoothly, because the chart data are smoothed to eliminate as far aspossible the effects of local magnetic anomalies, while the computed valuesnecessarily run smoothly. Comparison between these tables and the corre-sponding tables in Dyson and Furner's paper (loc. cit. pp. 85-87) is of interest.In Tables VII and VIII there are several regions in which the residuals aresystematic and of appreciable magnitude. The residuals in these tables arelarger on the average than those based on the 1922 charts, an indication that the1942 harts are of lower accuracy than the 1922 harts, as would be expected fromthe paucity of ocean data since 1929. There is, moreover, little correlationbetween the corresponding residuals from the 1922 nd 1942 harts ; if the chartswere of high accuracy and the same regions stood out i n the same direction atdifferent epochs, it would indicate that the general magnetic field was notrepresented with adequate accuracy by the six harmonics. The comparisonbetween the 1922 and I942 residuals suggests that there must be appreciableerrors in the secular change field that was adopted for bringing forward observa-tions to the epoch 1942.5 of the later charts, while the large systematic runs of

residuals suggest that there are areas where the charts are appreciably in error.Comparison between the tables clearly shows that the vertical force dataare of much lower reliability than those for the north and east components. Thelargest residuals are found in regions where the errors in horizontal intensityand in inclination are additive and where, in addition, the tangent of the dip hasa large value. In middle latitudes (between 50"N. and 50" S.) the residualstend to be larger than those obtained from the 1922 charts, but in the polar capsthe very large residuals obtained from the 1922 charts have been reduced.

13. The charts on which this analysis was based were charts of declination,horizontal intensity and inclination. From th e values of X, and 2 computed

from the coefficients listed in Table V, he values of these three components werederived. Tables X, XI , XI 1 give the differences between the chart values andthe computed values for the three elements declination, horizontal intensity andinclination. Table X is of special interest because of the use of the charts ofdeclination for navigational purposes. To o much regard need not be paidto the large residuals in high north and south latitudes, because of the proximityto the magnetic and geographical poles, which are singular points in the systemof isogonals. In latitudes 50" S. and 60" S. there are some considerable residualsin the southern Indian Ocean, which suggest that in places in this area thecharts m,ay be at least 5" in error. The southern Indian Ocean has been recognizedfor many years as a difficult region, for not only are the isogonals more closelyspaced there than elsewhere, but in addition it is a region where the secularchange of declination is both large and ill-determined. It is clear from Table Xthat the charts in declination are not of the accuracy that is needed for certainmodern developments in navigational aids.


13/24

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6 7 8 91 I

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17/24

analysis of the Earth's magnetic fie ld , fo r epoch 1942 425

" 0 e m : 2 d m ~ m N O 0 0 -N m m N N O W t - W v) N v)m 2 kI l l l l + + + + l l l l l l l l l + + + + + + + + + + + + I l l l

v) n n - W e m m n

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a ~ m m ~ ~ ~ m ~ m ~ ~ ~ t - ~ ~ ~ ~ ~ ~ m

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19/24

analysis of the Earth's magnetic field, for epoch 1942 427

14. t seemed desirableto redetermine the coefficients, taking into accountthe spheroidal figure of the Earth. T h e formulae required for this analysis aregiven by J. C. Adams in Section VI, PartI1 of his Collected Papers, Vol. 11.They can be summarized as follows:-

H n m sinm8'Gnm,Xm m a n + 2 [ ( n - m ) inm+lB'G,m+l-msinm-l8' co~e'G,m]/r"+~,

y ,m = an+2mSinm-1B'G ' m/rn+2,Z ,m = a n 2(n+ I) sinm8' G,m/rn+2.

T h e values of ln (a n/ rn ) for values ofn from I to 12; of lncos8', ofIn sinm8' for values ofm from I to 10 and the values of G," are tabulated inSection 11, Part 11. r denotes the radius of the Earth and8' the geocentriclatitude.

From the values ofXnm, Ynm, Znm re computed the primed quantitiesdefined by

X n m Xnm osy5+ZNm iny5,Y," = Y,",Z',m = - , m sin y5+Z ,m cos4,

y5deno tes the angle of th e vertical; the values of Incosy5 an d lnsiny5 aretabulated by Adams. Un prim ed quantities relate to axes along and at rightangles to the radius vector; primed quantities to the normal and tangentialcomponents.

T he equationsI;X,mg,m=x',,

r, yrnmgnm = ', ,I;z,mg,m = z',,

with similar equations inh,m, are then used to determine th e coefficients, the right-hand members being derived from the tabulated chart data for the north, eastand vertical components,as in the solutions for the spherical Earth. T h ecombining weights for the different latitude zones, given in Table IV, were

again used, anda single solution combining the data from all three componentsof the field was made.T h e values of th e coefficients from th e solution for the spheroidal Ea rth are

given in the last column of Table V. T h e firstsix coefficients(gn0) re changedslightly, but the remaining coefficients are substantially unaltered. T h e solutiondoes not add anythingof special interest to the solution fo r the spherical Ea rthand it does not appear that th e additional labour involved in th e solution for thespheroidal Ear th is justified in fu ture harmonic analyses.

15. Comparison between the principal coefficients and those determinedfrom other spherical harmonic analyses, includ ing two determinations of slightlylater epoch, are given in Table X II I. T h e coefficientsglo, g20,g22 nd h,l appearto change linearly with the time,h21 having the most rapid rate of change. T h echange ing l l appears not to be linear and is slower now than a century ago. T h evalues ofg21 and hll show a considerable scatter bu t no appreciable change withtime is apparent.

G 31*


20/24

428 H . Spencer Jones and P. J. Melotte, The harmonic

The values of the other coefficients have been compared with their valuesas found by Adams for the epochs 1845 and 1880,by Dyson and Furner for 1922,and by Vestine and others for 1945. The coefficients of a number of the higherharmonic terms show appreciable changes in the course of a century; gso, g31,

g41, hg1,g42, S2, S2 , he2are those that show the largest and most systematic changes.The fact that many of the coefficients change so rapidly is an indication of thecomplicated structure of the secular change field.

TABLE I11Values o principal coe f i ch t s

&o gao g1l gal gasErman-Petersen 1829 +3 201 + 1 2 +284 -445 + 12Gauss I835 3235 - 76 3 1 1 50 5 -I- 2Adams I845 3 219 - I 3 278 491 - 4Adams 1880 3 168 + 73 243 514 53Schmidt 1885 3 168 75 222 481 56Fritsche 1885 3 164 53 7.41 495 59Dyson&Furner 1922 3 095 133 226 518 125Jones&Melotte 1 9 4 2 3 039 176 218 509 135Afanasieva I945 3 032 187 229 498 130Vestine&Lange 1945 +3057 +190 +210 -512 -141

hl '

625578603

-601

595

591592555590

-581

haa- 27

136116129129123734442

- 46

16. The first-order harmonic, which corresponds to the dipole field, is givenby glo in h +(g: cos +hll sin 4)cos A .Writing glo = H , cos 0, ; g,l= H , sin 0, cos do; hI1= H , sin 0, sin 4,)H , gives the intensity of the dipole field ; 0,, d oare the colatitude and the longitudeof the northern pole of the dipole field, or the northern geomagnetic pole. Thesouthern geomagnetic pole is the antipodal point to the northern : these poles aret o be distinguished from the magnetic dip poles.

The positions of thetwo poles from the present analysis are 78O.9 N., 68O.5 W., and 78". S., 111O.5 E.

TABLE IVThe Dipole Field

The values of H,, , and do are given in Table XIV.

Epoch HO 00 dErman-PetersenGaussAdamsAdamsSchmidtFritscheDyson & FurnerJones & MelotteAfanasievaVestine & Lange

I 829I835I845I880I885I8851922I942I 9 4 5

I945

'326 9'330 9.328 2'323 4'323 1-322

-315 9'309 7'309 7'311 9

0

11.712'1

11'2

11.611'3

I I '411.611'1

11.111.4

a

-64.763 '564 3

69 '568.0

67 869.168 568-870 0

Th e values of H,,, ,) can be represented by the formulae:

Ho = 0.318 7 - .017

Documents

Geophys. Suppl. MNRAS 1953 Jones 409 30