Geomagnetic secular variation over Spain 1970–1988 by means of spherical cap harmonic analysis

Physics of the Earth and Planetary Interiors, 68 (1991) 65—75

65

Elsevier Science Publishers B.V., Amsterdam

Geomagnetic secular variation over Spain 1970—1988 by means of spherical cap harmonic analysis A. Garcia

a

J.M. Torta

b j•j•

Curto b and E. Sanclement C.S.I.C., Madrid, Spain

b

Museo Nacional de Ciencias Naturales,

~ Observatori de L’Ebre, C. S. I. C., Roquetes, Tarragona, Spain (Received 7 June 1990; revision accepted 21 December 1990)

ABSTRACT Garcia, A., Torta, J.M., Curto, J.J. and Sanclement, E., 1991. Geomagnetic secular variation over Spain 1970—1988 by means of spherical cap harmonic analysis. Phys. Earth Planet. Inter., 68: 65—75. A spherical cap harmonic analysis was applied to Spain and neighbouring areas, covering a region with a cap halfangle of 16°.The technique, due to Haines, has allowed us to obtain a model of secular variation (SV) for a time interval of 18 years reduced to the epoch 1987.5. The total number of selected data points was 1581 provided from ten observatories and 60 repeat stations. The SV model has a maximum spatial index of four and temporal degree of two. The coefficients were tested and the statistically non-significant terms removed. Another model covering a smaller region with greater density of data has been developed and both results compared. The average secular variations derived from the definitive geomagnetic reference field (DGRF) and the international geomagnetic reference field (IGRF) were compared with these models and maps plotting the calculated first field derivatives are presented. The integration of the final model will serve to update satellite data in order to produce a regional main field model.

1. Introduction The method most commonly used to describe the spatial distribution of the main geomagnetic field is in terms of an expansion in spherical harmonics. The scales of time variation of this field, which range from approximately one year to a few centuries, form the so-called secular variation which like the field itself originates in the fluid core of the Earth (Barraclough and Clarke, 1988). Likewise, the most useful way of representing them is by means of an expansion in spherical harmonics. The best source of information for developing such representations is the differences between adjoining annual averages provided by the different observatories around the world. Another good database is the magnetic surveys conducted by different countries with locations spatially selected because they are free of 0031-9201/91/$03.50

© 1991

—

Elsevier Science Publishers B.V.

surface anomalies and corrected in order to remove, as far as possible, the effects of the external field during the observation time. The stations are carefully marked during these services in order to be reoccupied afterwards and for this reason they are called repeat stations. A spherical harmonic analysis is specially suitable for modelizing the whole globe because at a first approach it is a sphere, or also if the area in question is defined by a hemisphere (Chapman and Bartels, 1940). Haines (1985a) described an interesting review of the methods developed for modelizing the geomagnetic field or its secular variation over smaller areas. Of all methods, the most widely used at present are the rectangular harmonic analysis of Alldredge (1981) and the spherical cap harmonic analysis (SCHA) developed by Haines. This analysis, to a great extent, solves the problems that arise when trying to

66

A. GARCIA ET AL.

modelize the geomagnetic field over terrestrial regions from the general spherical harmonic analysis, because certain features of the field close to the boundary of the area in question may be distorted. Furthermore, it provides an excellent method for radial representations of the field, which therefore make it especially suitable for analyzing satellite data.

2. Data As stated above, the data input was based on the differences between adjoining annual average values from geomagnetic observatories and on the differences between the field values measured between consecutive occupations of various repeat stations, divided by the corresponding interval of time. The region to be studied had to be defined by a spherical cap, in order to cover the whole of the —30 50

—20 I

.

7

30

-

20

-

0

—

10

~,___._L—~ii~r’

.

50

-

30

-

20

~

~

40.

—10

Iberian Peninsula as well as the islands forming part of the Spanish State, that is, the Balearics and especially the Canary Islands, with a wide enough margin around them so as to prevent, as far as possible, the cap boundary influence. This involved selecting an analysis area with a region containing the most Western part of the European continent, with a great density of data; and a larger area —oceanic zones and North Africa— with an almost non-existent distribution of data. In order to minimize this drawback, a spherical cap was selected (Fig. 1) whose periphery included the observatories of San Miguel (Azores) and Tamanrasset (Algeria) and of course the Santa Cruz de Tenerife observatory in the Canary Islands. The repeat stations were the geomagnetic services of France and Spain. The cap centre was located at a point defined by the coordinates 34°Nand 7°Eand presented a halfangle of 16°, including a total of ten observatories and 60 repeat stations. In order to avoid the influence of the well-

,,

.1 —30

—20

—10

0

10

Fig. 1. Locations of the observatories (stars), repeat stations (dots) and simulated stations (circles), which provide the database for the analysis. The solid line indicates the 16° spherical cap boundary and the broken line the 8° spherical cap boundary.

67

GEOMAGNETIC SECULAR VARIATION OVER SPAIN 1970-1988

known impulse on the secular variation detected towards the year 1969 (Ducruix et al., 1980; Courtillot and Le Mouël, 1984) in a recent analysis similar to this, the data used were recorded after said impulse took place; thus including all the annual observatory averages available and field component measurements in the repeat stations, from 1970 onwards. After observing the residues of these data with regard to the components of the corresponding international geomagnetic reference field (IGRF), a total of four measures distributed between two Spanish repeat stations were rejected, given that they presented a considerable deviation, When these few, presumably erroneous items of data had been dropped out the first derivatives were obtained by simply determining the difference between consecutive values in nanoteslas and dividing by their respective intervals of time in years. These data were then related to an intermediate epoch between those of both measurements. Meanwhile, all the first derivative vectors containing at least one unknown component were dropped out from the database, because the three components were essential for future coordinate conversions. Finally, a total of 1581 first derivatives were computerized so as to adjust the secular variation model. Separately, this corresponds to a total of 351 items of data (117 vectors difference) from the ten observatories and 1230 (410 vectors difference) of the 60 repeat stations. Both sets of data were weighed in the same way in the later least-squares adjustment. In order to enhance the contribution of information in the areas lacking data and thus to facilitate the extrapolation of the model in said areas, a series of simulated repeat stations were added (Newitt and Haines, 1989), the first derivatives being obtained from the recently published DGRF (definitive geomagnetic reference field)— IGRF (Barraclough, 1987). Instead of adjusting cubic polynomials in time to the coefficients for each degree and order during the analysis period as per the Alidredge procedure (1985), it was preferred to only add to the total of the data four values for each of the 18 simulated stations, distributed between the cap boundary and other zones which totally lacked data. These values correspond to intermediate epochs in the intervals of -

-

-

-

.

-

.

-

~

(L~t~18.aN.Lo~=_1O.0E)

— — -

C

-

___________________ “~970

~1975

9~0

_____

~

1990

Fig. 2. Secular variation of the Y component from the spherical cap harmonic model in the position defined by one of the simulated stations (solid line). Such variation may be cornpared with the one that presents the DGRF-IGRF models, which is constant during the 5 year intervals. It can be also compared with the vanation obtained by adjusting cubic polynormals to the coefficients of those models (broken line), following Alldredge’s procedure (1985).

validity of each annual variation published of the international reference field (1972.5, 1977.5, 1982.5 and 1987.5). This decision was taken when it was found that unfortunately the first procedure distorted the model, given that the second degree polynomial variation in the annual change resulting from the Alldredge cubic approximation does not have to be the same with which the model is adjusted for the original data. Furthermore, although this approximation is especially suitable for representing the internal field more realistically than the normal technique of linearly extrapolating from the IGRF values, it is not too useful for representing the secular variation before and after the first and last intermediate epoch respectively, which were used to solve the approximation. Both facts may be seen in the example of Fig. 2. These new simulated data were weighed in the same way as the data from repeat stations and observatories. Later, the first derivative components were converted from geodetic to geocentric and these geocentric coordinates were transformed to a new pole located at 34°N, —7°E, forming the real input database of the secular variation model to be adjusted.

3. Model adjustment First, a secular variation analysis was applied to a cap halfangle 16° with a computer program

68

A. GARCIA ET AL.

written in FORTRAN 77 adapted from the program published by Haines (1988). Initially, we chose a maximum spatial index K = 4 and a maximum degree Q in the temporal polynomial of two, the time variable being reduced to epoch 1987.5. The adjustment by least squares is based on a stepwise regression which must be provided with the F levels defining the required degree of significance (Draper and Smith, 1981). In fact, the algorithm described by Efroymson (1960) was used, which has the facility of adding and eliminating variables according to the significant contribution level they provide. Consequently, an F level of four was selected, given that it is a common value in this type of procedure; as a result, only 24 statistically significant coefficients appeared from the 75 coefficients present in expansion in spherical harmonics with K 4 and Q 2. The average square error of the estimation or standard error (calculated as the positive square root of the quotient between the sum of the squares of the residues and the degrees of freedom) in this first adjustment was 15.9 nT year~. In order to improve the goodness of fit as far as possible, the input database was cleaned so that any data that were separated by more than twice the recorded standard error were dropped out, The remaining data were used for a new adjustment, providing a total of 32 statistically significant coefficients and a standard error of 11.1 nT year1. The coefficients of this new model served to plot different secular variation maps in epochs within the 1970—1990 period. These were compared with those corresponding to the same epochs, but obtained with the DGRF-IGRF. Despite the fact that there was no excessive deviation of the values calculated with the model with regard to the original data, considerable discrepancies were noted in the spatial distribution of the isopors with regard to that corresponding to the international reference field, especially in the X and Y components. Various adjustments were then tested with new combinations of K and Q without any essential improvements being observed. The fact that these discrepancies appeared stronger in some components than in others led us to believe that the deviations that appeared in one =

=

component might negatively influence the obtention of the rest, given that the adjustment was realized jointly for the three. In order to remedy this disadvantage, it was decided to adjust two different models, one for the X and Y components and another for the Z. This involved rerunning the computer program for each model, because the sum of squares and products matrix or least square matrix have to be sized differently since the same number of non null elements does not appear in each expansion and therefore a different number of coefficients is produced. The short delay in the adjustment of the coefficients and the later obtention of the values calculated with the same, which involves having to work on each model independently, is more than compensated by an improved data extrapolation, shown by a much more real spatial distribution of the isopors. Moreover, there is no need either to talk of (k + 1)2(Q + 1) coefficients for each cornponent, given that when differentiation is not necessary with regard to colatitude, only a series of basic functions are required: those with k m even (Haines, 1988); and so only around half the total number of coefficients are needed for the corresponding component. Furthermore, a given component may be adjusted with a lower maximum spatial index than the remaining ones and this also effects a smaller overall number of coefficients. Consequently, both models were adjusted separately using the values corresponding to the north and east components on the one hand and the vertical component on the other, of the initial input database formed by the 527 first derivative vectors and the 72 corresponding to the fictitious stations. The process was started again with K 4 and Q 2 and with only one series of basic functions for the vertical component. After separately examining the statistically significant coefficients which appeared and the secular variation maps which were plotted, it was seen that it is sufficient to define a maximum K 3 index for the vertical component model. Once this analysis had been repeated with new parameters for the second model and in accordance with each standard error recorded in each of the two input databases, those separated by more than twice the corresponding —

=

=

=

=

69

GEOMAGNETiC SECULAR VARIATION OVER SPAIN 1970-1988

r.m.s. deviation were dropped out. The two final adjustments provided 19 statistically significant coefficients from a total of 72 for the joint model of the X and Y components, and 13 from a total of 30 for the Z component (see Table 1). The corresponding standard errors were 10.8 nT year~ and 8.1 nT year~ respectively, meaning an overall r.m.s. deviation of 9.6 nT year~. Figures 3 and 4 show the temporal variations of each model, in the positions defined by three different observatories and repeat stations. The goodness of the temporal adjustment may be seen. In order to check the correctness of the data extrapolation, taking into account the low density of the same in a large part of the cap, another analysis was applied to a smaller cap covering an area with a greater density of data (see Fig. 1). This new cap has its pole at 41°N, —3°Eand a halfangle of 8°.The data from observatories and repeat stations it includes were completed with a total of 20 first derivatives obtained as before from five simulated stations throughout the peripheral zone of the cap lacking information. The result of this analysis allowed for the plotting of secular variation maps (Fig. 5) which may be compared with the corresponding areas of those

which will be shown next and which are the true isoporic maps plotted by our final analysis. By means of this comparison it can be observed that the extrapolation of the data to oceanic or terrestrial zones lacking information is acceptable. Therefore it will also enable us to carry out both this study and further studies which will follow it, of a whole area which covers zones of great interest. The data extrapolation, moreover, has no cxcessive influence on the spatial distribution of the isopors in the zone with a high density of information, and the deviations observed are mainly caused by the negative influence in this last analysis of the nearby cap boundary.

4. Isoporic maps The spherical harmonic model of our cap, or rather the two of them, allow for maps to be plotted of the variation of the geomagnetic field components, or isoporic maps, throughout the interval under study, or also for extrapolation to future epochs. Figure 6 shows the isoporic maps of the north, east and vertical components respectively, plotted

TABLE I Statistically significant coefficients obtained for each of the models

xk

Y m

nk(m)

g~’

0

1

6.1481

11.344

2 2 2 3 3

0 1 2 0 1

13.2304 13.2304 10.5214 19.2652 18.6075

—1.458 —2.806 —0.829 0.763 1.238

3

2

0.306

4 4

1 2

17.9248 24.6277 23.5484

k

m

nk(m)

0 1

0 1

2 2

0 2

3

1

0.000 0.000

h~”’0

g~”1

—23.112

0000

8.757 —4.180

0.000 0.156 0.000

h~ 0.709

g~°2 0.032

h~’2 0.071

—3.731

0.000

0.000

0.000 0.000 0.000 0.000 0.000

1.267 0.872 —0.681

0.000 0.000 0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000 —0.426

0.000

—0.010 —0.020 0.000

z 0.0000 6.1481 13.2304 10.5214 18.6075

g~’0 31.087 5.967 —0.737 0.000

0.000

h~0

g,”1

h~’1

g~”2

—0.167

—0.080 0.006

—1.245 —8.778

0.000

—0.688 0.289

—0.030 —0.031 0.039

h~’2

0.000

0.000 0.000

0.000

0.000 0.000

0.000 0.000

70

A. GARCIA ET AL.

8

ever, our model detected an inflection in the isopors over the Iberian Peninsula for this component in the last years, an inflection which does not appear in the international reference field map. This may be due both to the fact that the annual variation plotted from the last IGRF is always a prediction and as such, liable to have to be cor-

CHAMBON-LA-FORET

.

o

8

.

I

I

1970

1975

1980

1985

1990

-

E8RO o

.~

—

a

-

—

-

—

-

-

~

©

rected; it could also be the result of the negative influence of the measurements in some Spanish repeat stations, because they have a high residue. This negative influence is also seen in the Y cornponent isoporic maps; although the Y is generally obtained from the measurement of the horizontal component H and the declination D (Y H =

0 0

T1970

8

__________________________________ 1975

1980

1985

1990

8

BURGOS

ALEERIA

‘1970

1970

1975

1980

1985

1975

1980

1985

1990

1990

Fig. 3. Secular variation of the X (long broken line), Y (short broken line) and Z (solid line) components from the spherical

2

cap harmonic model, in the positions defined by some of the observatories. The symbols are the differences between adjoin. ing annual mean values from these observatories (circles, triangles and squares; respectively)

8

CIUDAD RODRIGO —. — —

—~ •

—

—

— —

—

• •

°

~

.

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—

-

0

I 1970

from the corresponding coefficients obtained for each model. They refer to intermediate epochs of the five year intervals from 1970 to 1990. This allows them to be compared with the annual vanation maps plotted from the fourth generation DGRF-IGRF coefficients (Barraclough, 1987) shown in Fig. 7. As is known, said annual variation is constant throughout each of these intervals and represents the average variation during the interval and so it is for the intermediate epoch when it is closest to the real annual variation, The companson is excellent for the Z component in the four epochs descnbed and quite good for the X component in the three first ones. How-

1975

2

1980

1985

1990

MALACA .

—

.~-

~•

~-~—-._

T

~

—

—.

• —

-=

-.

0

8 ~

197519801985

1990

Fig. 4. Secular variation of the X (long broken line), Y (short broken hne) and Z (solid line) components from the spherical cap harmonic model, in the positions defined by some of the repeat stations. The symbols are the differences between adjoining annual mean values from these Stations (circles, triangles and squares; respectively)

71

GEOMAGNETIC SECULAR VARIATION OVER SPAIN 1970-1955

X’ —10 —8

1987.5

—6 —4

~ ~ -c(

—2

X

0

2

4

10 -8

45 ‘“c~ ç

~

~

~

—10 —8

—6

43

—4

—2

2

—10 —8

~60fl~ 7’ —6

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2

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4

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—10 —8

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1987.5 —4

1987.5 —4

2

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43

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l~l

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i

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—2

i

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2

4

I

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1/ ~ f_~

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0

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—6

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i

N

2

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35 —10 —8

1 4~I••~.. J. . I —6 —4 —2 0

I 2

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Fig. 5. Comparison between some of the secular variation maps obtained from a second developed analysis that covers a smaller cap with greater density of data (left), and the corresponding areas of those which were obtained from our final analysis (right).

72

A. GARCiA CT AL.

X ~

1972.5

V

1~V\,ç

40

~

~

40

40

1972.5

7 45

~sJ~

1972.5

~

45

40

II~~

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1977.5 ~9

-4

~

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7

1977.5 -19

____________

N

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-( ~

40 T~

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116

~35

V

1987.5

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1982.5

Z

1982.5

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1982.5

~I9~5

413

~‘64

~

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~~ X

~19

413

1977.5

-H

1987.5

_____________

7 E

1987.5

_____________

Fag. 6. Isoporic maps of the north, east and vertical components, in nT year’, from the SCI-I model.

.~\

40

73

GEOMAGNETIC SECULAR VARIATION OVER SPAIN i970-I988

DGRF 1970—1975

X —19 45

—14

—9 —4 \~P~j\

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:~

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6 ~l

—19

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A 25 6

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_______

/

25 /i~~~ —19 —14

DGRF 1975—1980 —4

45.

V

1

6

~

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49

—14

-

25 ____________________ —19 —14 —9 —4 1

X

45

~35

/J)

301

—19 ~

DGRF 1970—1975

Z

,~)

~

~

DGRF 1970—1975

V

—19 45

45.

40

40

30 25 —9

—4

1

6

N

30 25

—19

DGRF 1975—1980

—14

—9

—4

1

~. O\iN’~~~(~\ i~k

Z 6

—~

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ij~

30 25

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—14

—9

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OGRE 1975—1980

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5

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25 —19.,...I —14 .1/ J~’—9

X —19 45

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—14

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25—19 I —14 ‘1

y

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—19 ~5

—14

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40

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35

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30

30

-/

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—9

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—4

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—4

—19

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—19

25—19

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~

—14

—9

—4

I

6

DORE 1980—1985 —9

45

45

40

4~

35

35

25

______ 25—19 \ —14 1 ~ —9

—4

1

6

I

L ~5

40

~

Y

‘~“

—14

1

6

GRE 1985—1990 —9

—4

1

Z 5

—19

—14

—4 ~ I

—~

513Q 25

GRE 1985—1990 —9

—4

1 Il,

6 45

40

~-~:

45

1

45 35

~

~

)\~30 —4

30

DGRF 1980—1985

______ —14 —9 —4

6125

‘~r~’

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‘‘~~~N,t 1 5 25

30 35

64

V

—9

/ —4 I /

30 ~

:

~

GRE 1985—1990

—14

‘p

4° -\ ~.,

-\

~

30 ~

—19

25

6 \

~ \

~

30 35

OGRF 19801985

—14

.\

25 —19

)

\I\

-

~

413

—4

[~~‘~-

30 ~

6

25

25—19___________ —14 —9

4°

~N~s\~ ,~c40

0

030

_____

—4

1

6

25

25—19

—14

—9

—4

1

6 25

Fig. 7. Annual variation maps, in nT year~, plotted from the four generation DGRF-IGRF coefficients (Barraclough, 1987).

74

A. GARCIA ET AL.

1990.0

V

1990.0

::jI~~: X

:~~1L_~::

~

~-H-9-4I6~

~4-4J

1992.5

_____

V

1992.5

Z

1990.0

7

1992.5

_____

~19-H-9-4i5

~t1~W~f~ ~JU 2525

Fig. 8. Predictive isoporic maps for 1990.0 and 1992.5, in nT year

sin( D)), and this has a value of almost zero in the whole area and interval under study, so a small error in its measurement is displayed by an important error in the Y value, which results in greater uncertainty in the set of data provided for this component. Likewise, and taking into account the fact that the extrapolation in time is given on the assumption that the behaviour of the field over the last 18 years will remain the same during the next years (Haines, 1985b), predictive isoporic maps were plotted for each component (Fig. 8). More specifically, those given correspond to 1990.0 and 1992.5, which we will soon be able to contrast with the annual variation charts for the 1990—1995 period, to be published from the new IGRF in 1990. Once again, the prediction seems perfectly reasonable for the secular variation of the Z component, but not so much for the X component, which continues to display, and even to increase, an inflec-

~,

from the SCH model.

tion in the isopors for latitudes corresponding to the Iberian Peninsula; but rather for the Ycomponent variation, which tends to present values which seem too small towards the north of the map and which cause the isopors to steadily move closer, from north to south throughout the map. Given these circumstances and considering the abovementioned assumption rapidly loses validity as one moves away from the period analysed, no maps have been plotted for epochs after 1992.5. In order to do so, the database of this analysis would need to be supplemented with the latest data from observatories and repeat stations. The extrapolation of-the secular variation must serve (and this is one of the main objectives of this analysis) to reduce data of the main field components measured during the analysis interval, to an epoch which will be selected approximately 2 years after the last item of secular variation data used for this analysis. This will enable regional main

GEOMAGNETIC SECULAR VARIATION OVER SPAIN i970-1988

field models to be obtained over the same cap and will involve the integration of the secular variation models which have been presented.

5. Conclusions A spherical cap harmonic analysis was applied to Spain and neighbouring areas in order to model the secular variation of the components of the

geomagnetic field of said area. This type of analysis is one of the most appropriate modern techniques for mathematically describing the main field or its derivatives, over terrestrial regions of less than a hemisphere which may approach a spherical cap. The inclusion of temporal dependence offers the possibility of extrapolations and updating measurements usable in future analyses. As has been described, it allows for one model to be obtained for the north and east components and a separate model for the vertical component, a different statistical analysis being realized for each of them. This involves a greater overall number of statistically significant coefficients and a longer running time, but this is compensated by the more realistic data adjustments that are at-

tamed. Our final models comprise 19 statistically significant coefficients for the secular variation of the X and Y components and 13 for the Z component. The average square deviation of the values calculated by these models with respect to the original data is 9.6 nT year_i. The smallest contribution to said error is made by the secular variation of the Z component.

75

viding data and opportune suggestions which were very helpful. This work was partially supported by the Direcciôn General de Investigaciôn CientIfica y Técnica, under the research project No. PB87-0390 at the Observatori de 1’Ebre.

References

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Alldredge, L.R., 1981. Cubic approximations of definitive geo. magnetic reference field models. J. Geophys. Res., 90: 8719—8728. Barraclough, DR., 1987. International Geomagnetic Reference Field: the fourth generation. Phys. Earth Planet. Inter., 48:

279-292.

Barraclough, DR. and Clarke, E., 1988. Statistical analysis of geomagnetic variations. Brit. Geol. Surv. Geomagn. Res.

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Acknowledgements

We would like to thank Dr. D.R. Barraclough, Dr. G.V. Haines and Dr. H. Nevanlinna for pro-

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