Physics of the Earth and Planetary Interiors, 74(1992)209—217 Elsevier Science Publishers B.V., Amsterdam
209
New representation of geomagnetic secular variation over restricted regions by means of spherical cap harmonic analysis: application to the case of Spain J.M. Torta
a A. Garcia b ~•j• Curto a and A. De Santis Observatori de l’Ebre, C.S.I.C., Roquetes (Tarragona), Spain b Museo Nacional de Ciencias Naturales, C.S.I.C., Madrid, Spain Istituto Nazionale di Geofisica, Roma, Italy a
(Received 22 July 1991; revision accepted 15 May 1992)
ABSTRACT Torta, J.M., Garcia, A., Curto, J.J. and De Santis, A., 1992. New representation of geomagnetic secular variation over restricted regions by means of spherical cap harmonic analysis: application to the case of Spain. Phys. Earth Planet. Inter., 74: 209—217. A model of the secular variation (SV) of the geomagnetic field over Spain and adjacent regions, using a particular application of spherical cap harmonic analysis (SCHA), is given. SCHA was directly applied as presented in a previous paper; however, some drawbacks were detected, making evident the limitations of the technique for fields with low spatial harmonic content, as in the case for SV over relatively small regions. After trying some alternative approaches to the conventional procedure, it has been discovered that the solution lies in increasing the cap half-angle (but the data continue to be restricted within the original cap) to obtain more suitable harmonic contributions. It is shown that inspection of the spatial power spectra of the SV field on different spherical caps is useful in selecting the cap half-angle.
1. Introduction
ysis (SCHA). By applying the boundary conditions for the cap, it follows that the most general
When observations of the geomagnetic field (or its temporal variations) are only available for a small portion of the Earth’s surface, or the analysis is only required over such area, the associated Legendre functions (with integer degree) in colatitude, valid for the case of global analyses, are no longer suitable basis functions to satisfy the new boundary conditions. Haines (1985a) showed that it is possible to find orthogonal basis functions over a region bounded by a spherical cap which provide a uniformly convergent expan-
solution to Laplace’s equation for a magnetic potential V is
sion for the potential and its derivatives. The technique is termed spherical cap harmonic anal-
K~
V=
a \nk(m)+1 r)
k
k=0 m==0
~.
Pnk(m)(cos
0)
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X ~ q +
a
cos(m4) +h~ sin(mcb)~t~
[g~
r I k~1 m=O ~ a) k
EXT
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~
—
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(cos 0)
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[g~~~ 1e
Correspondence to: J.M. Torta, Observatori de l’Ebre, C.S.I.C., Roquetes (Tarragona), Spain. 0031-9201/92/$06.00 © 1992
—
q=o
Elsevier Science Publishers B.V. All rights reserved
cos(m~)+h7~”,~ sin(m~)]t~
210
J.M. TORTA ET AL.
where r is the radial distance, 0 the colatitude and 4 the longitude, expressed in the spherical cap coordinate system; a is a reference radius, generally the average radius of the Earth. The constants g~q and hTq are known as the spheriK1~ represents the maximum spatial index for cal cap harmonic coefficients. The parameter the expansion that accounts for the internal sources, and KEXT the maximum index for the expansion corresponding to the external sources. As may be seen in (1), a temporal dependence has been included, with t a time or epoch, and the parameters q and Q defined as the order and degree of the polynomial in t, respectively. The expansion in spherical harmonics over the cap is then similar to that described for the conventional global spherical harmonic analysis, with the exception that now the degrees of the associated Legendre functions of the first kind, nk(m), are no longer integer parameters. If 0~is the colatitude of the cap boundary, the values of equations (k is an integer index selected to arnk(m) are found as the roots of the following range, in increasing order, the different roots n for a given m): 0P~”’(m)(COS0~)=
P:(m)(cosoo)=o
0
for k
—
m
=
even
(2)
p,~~s
I
(a) — —
—
—
\
—
\
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~‘~V°~°~ (b)
\
k=0
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—
1.0
.
—
i /
—
—
—
05
(3)
To visualize the appearance of these boundary conditions more clearly we can compare the case of the cap with the case of a hemisphere (0~= 90°),where the solution is the same as that for conventional harmonic analysis, i.e. for the whole sphere (0~= 180°).In this case (Fig. 1(a); adapted from De Santis, 1991), the basis functions are characterized by the associated Legendre functions with integer degree, n, forming two subsets which at the cap boundary (the equator) have either zero derivative with respect to colatitude or zero value, depending on whether n m is even or odd. If we now pass to a cap with 00 < 90°,we expect to find functions similar to those shown in Fig. 1(a), whose value or derivative is zero at the cap boundary (Fig. 1(b)). Since the derivative of a polynomial is again a polynomial, the formula described for the poten-
-
—
12°
for k—m=odd
-
0 I 0
6•
~°
-~
—
or —
—
—
(k —
m) m)
= =
even
~
— —
odd
~ = dO P,~(O=O~)= 0
Fig. 1. Associated Legendre functions with m = 0 for (a) the hemisphere and (b) a cap of 16°.For n — m = even (dashed lines), the functions are constrained to have their derivatives equal to zero at the boundary (equator); for n — m odd (solid lines), the functions must be zero. Case (b) is the local extension of (a); in this case the index k replaces n as it does in the summation of the potential expansion (adapted from De Santis, 1991).
tial of a field is valid not only for the main geomagnetic field (whose components are obtamed directly from (1) by taking the negative
211
REPRESENTATION OF GEOMAGNETIC SECULAR VARIATION
gradient of the potential), but also for any of its temporal derivatives, such as those which represent the secular variation (SV). In this way Haines (1985b) successfully developed an SV spherical cap harmonic model for Canada with a spherical cap of half-angIe 30°,and used it, with a previous integration, to update old main field data in order to get a reference field model (Haines and Newitt, 1986; Newitt and Haines, 1989). However, when the cap region is relatively small compared with the wavelengths involved in the phenomenon, the direct application of SQ-IA to model SV fields (or fields that vary smoothly over the Earth’s surface, i.e. with long wavelength content) is not so straightforward. (Haines (1985b, p. 12573) in his conclusions wrote about SV application over ‘large’ portions of the Earth’s surface.) By studying the application of SCHA to the simple case of a dipole field we will analyse the reasons for this disadvantage and propose an alternative solution. Finally, an improved Spanish SV model is obtained,
2. Attempts at modelling geomagnetic SV in Spain Garcia et al. (1991) modelled geomagnetic secular variation in Spain and neighbouring areas over a roughly 30 year time span using SCHA with K1~ 4, QINT 2, KEXT QEXT 0, t =
=
=
=
from the IGRF for the same epochs. A symptom of this problem was the presence of a significant value of the monopole term. Garcia et al. (1991) attributed this incongruity to a probable deficiency of the technique when required to handle all three components together in this particular case: SV data over a small cap. In fact when Haines (1985b) applied SCHA to SV in Canada he used a 30°cap, with better-distributed data. A significant improvement was found by modelling the Z (vertical) and X + V horizontal components separately. The spherical cap harmonic expansions included maximum spatial indices K 3 (Z model) and K 4 (X + Y model), with maximum temporal degree Q 2. The resulting r.m.s. values did not change too much, but the new SV models were more in agreement with the IGRF SV models. However, the final models did still show some unrealistic shapes, especially in the X and Y components. Of course, carrying out two separate analyses raises some questions about the application of SCHA, because it is made at the expense of partial violation of Laplace’s equation. In any case the work drew attention to the need for care in using SCHA, especially for SV models extending over small caps. Another explanation could be incomplete coverage of the harmonic contents in the spherical cap harmonic models used. =
=
=
year-1987.5. A total of 527 three-component observatory and repeat station data spanning the time period 1970—1988 were used; because of the sparseness of the observations, 72 simulated data were used to fill gaps in the region. These simulated data were computed from the SV values of the international geomagnetic reference field (IGRF) (e.g. Barraclough, 1987) at the epochs: 1972.5, 1977.5, 1982.5 and 1987.5. The model was based on data within a cap of half-angle 16°centred at 34°N,7°W.An attempt was made to model the data by means of the conventional procedure, i.e. modelling the potential including all the three components X, Y, Z. Even though the fits to the data were reasonable, the models themselves showed some gross instabilities. This strange behaviour was more evident when compared with the global models derived
3. The dipole case In attempts to verify this question, we started with a simple model, a centred dipole. We synthesized values at the Earth’s surface on a 1°x 1° grid from a magnetic dipole with field components X= C~sin 0, Z 2 C• cos 0, Y= 0, 0 is the geocentric colatitude and C is a constant chosen so that values computed at the reference surface are comparable with those of SV data, or anomaly data; here we chose C 38.5 nT. Cornputing different spherical cap harmonic models from these sample data showed some incongruities: all the models that included external terms fit the data better than the SCHA models that included internal terms only. For example, the model with KINT 4 and KEXT 0 showed a =
=
=
=
212
J.M. TORTA ET AL
r.m.s. deviation of 3.72 nT and eight statistically significant coefficients with F levels 4; by using both KINT 4 and KEXT 4, the r.m.s. deviation decreased to 0.10 nT. Hence two questions arose: why do the external terms improve the fit so significantly?, and is it a problem with SCHA itself? The inclusion of external terms in the expansion practically means a doubling of the number of harmonic coefficients. This drives the expansion to adjust itself better (its terms) to fit the long wavelengths included by the original data; no physical considerations could be given to justify this fact (the dipole is definitely inside the cap). Separate modelling is an alternative to the previous method which can be explained by the following: experimental data include wavelengths with (little) different coverage among the magnetic components; with separate modelling some spherical (cap) harmonic functions can better approximate the behaviour of the horizontal components without the constraint of also fitting the vertical component, and vice versa. Other partial answers (G. Haines and F.J. Lowes, personal communications, 1991) to these questions could be: the non-orthogonality of the basis functions (SCHA involves the use of two different sets of complete basis functions, formed by Legendre generalized functions with non-integer harmonic degree; functions in one set are non-orthogonal to functions in the other), a need to weigh the horizontal and vertical components differently, and the inadequate harmonic spatial degree coyerage. The last point emphasizes the problem one meets when applying SCHA to low-degree fields (especially SV fields because they are more influenced by lower harmonics than the anomaly field). After some trials with the case of the dipole, we realized that the main reason for the lack of fit is that it is difficult to fit low harmonics with a finite series of terms of higher harmonics. With a spherical cap of half-angle 16°,the boundary conditions (see Fig. 1(b)) involve the use of Legendre functions with (apart from the monopole term): a first harmonic of n 6.1481, second harmonic n 8.1068, third harmonic n 10.5214, and so on. However, we have not to forget that we are =
=
=
=
=
=
trying to fit a field that we know has the contribution of n 1 only. With an infinite series, convergence would be reached (Haines, 1985a), but in practice this is not the case. We have to find a way to include lower-degree harmonics in the model. To achieve this we simply enhanced the size of cap (the new boundary conditions involve ~k’ with lower values than before) but using the same dipolar data set, i.e. that over the 16°cap alone (all models were deduced by least-squares regression). It quickly became evident the improvements to the fit resulted. By using the sameset of grid values, but with a spherical cap with a half-angle of 45°,the new spherical cap harmonic model gave an r.m.s. of 0.10797 nT and only seven coefficients. This is because with a larger cap the model includes lower harmonics. If we increase the half-angle to 60°,we obtain an r.m.s. of 0.03372 nT. Finally, if we reach the hemisphere, the r.m.s. becomes negligible (0.00039 nT), and, of course, the coefficients obtained are only those corresponding to n 1, with m 0 and m 1. =
=
=
=
4. Attempts at modelling IGRF SV by SCHA Since our final objective was to obtain an adequate SV spherical cap harmonic model from the observed SV over Spain and neighbouring areas, we decided to apply the above-mentioned procedure to SV data. To verify the validity of such a procedure, it is logical to start with uniformly distributed data uncontaminated by the errors involved in the measurement processes. Thus, first of all we considered a 1°X 1° grid synthesized by IGRF SV values (1975—1980) over the same 16°cap. Then, leaving the other parameters (KINT 4, QINT 2, KEXT QEXT 0) fixed, we determined by means of a least-squares procedure various spherical cap harmonic models considering different cap angles: 16°,and from 20 to 90°(i.e. apart from the 16°model, a spherical cap harmonic model every 5°).We expected some trouble in computing the models by least squares because the data did not completely cover all the cap used each time, so that each set of basis functions is orthogonal within an interval greater =
=
=
=
213
REPRESENTATION OF GEOMAGNETIC SECULAR VARIATION
~
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interested tion with IGRF (the values 16° cap); onlythe in the behaviour area weofwere the o
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Cap half—angle (deg.)
~ (b)
TA
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0 •
0
•
•
.
•
_________ ___________________
iii 7~I~i 111111 I 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
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Fig. 2. (a) The left hand axis shows r.m.s. deviation (solid line) of the spherical cap harmonic fits (with K = 6,0) versus the cap half-angle. The data were provided from the IGRF sv model for the time interval 1975—1980, calculated on a 1°x1° rectangular (geographical latitude—longitude) grid inside a spherical cap of half-angle 16°centred at 34°N,7°W.The right hand side axis shows the number of statistically significant coefficients (with F level = 4) obtained for each of the models (star symbols). (b) The same as (a) with spherical cap harmonic fits (with K = 4,0; Q 2) to real secular variation data over the cap of half-angle 16°,for a time interval from 1970 to 1988 reduced to the epoch 1987.5.
model outside this region is not relevant, and therefore we did not fill with IGRF data outside the original cap. The enlargement of the cap over which we imposed the Haines’ boundary conditions is a trick to include the suitable wavelengths belonging to the original observations in the basis functions. This is realized at the expense of the orthogonality of the functions within the same set (which are orthogonal on the large cap boundary, but not on the data interval), but the gain is better To visualize spatial spectral the problem coverage. the low-degree spectrum better, we decided toofcompute a spatial power spectrum from the model coefficients in order to examine the itspatial harmonic of each model and how converges. It is content important to emphasize that when developing the spatial power spectra of fields on a spherical cap, we must have in mind that such fields are represented by functions that are periodic on an interval larger (the whole sphere) than that on which the functions are known. Haines (1991) has demonstrated that the power spectra can be evaluated in a similar way to that used for conventional spherical harmonic analysis, but first global coefficients gg~and hg~ have to be computed from the spherical cap harmonic coefficients g~
than that covered by the data. In fact, we found that the r.m.s. taken over all three components changed for different caps, reaching lower values in the range 25—75° (Fig. 2(a)).
150
ICC -~
5. Application to actual geomagnetic SV data in Spain
1
~ Encouraged by the good results mentioned above, we applied the same simple technique to the actual SV data (Fig. 2(b)). There is a good improvement from the case of the smallest cap (16°) to larger caps (especially above 25°) (of course the r.m.s. values, for real SV data, are not as small as in the case of the IGRF values). As we did with the previous SV model (Garcia et al., 1991), we filled the gaps in data distribu-
1
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I 0
0
10
20
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Fig. 3. Sub-periodic spatial power spectrum for the 16°spherical cap harmonic fit to the real SV data, up to degree n = 36.
214
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J.M. TORTA ET AL.
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215
REPRESENTATION OF GEOMAGNETIC SECULAR VARIATION
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the north, east and vertical components, in nanoTesla per year, plotted from the IGRF the interval 1975—1980. (b) Isoporic maps for 1977.5 from the spherical cap harmonic model to SV data (after removing all data that differed by more than twice the standard deviation of a preliminary fit), using a cap of half-angle 16°.(c) The same as (b), but using a cap of half-angle 25°.(d) The same as (b) and (c), but using a cap of half-angle 45°. (a) Annual variation maps of
coefficients
for
216
J.M. TORTA ET AL
and h~(eqn. (33) in Haines, 1991). Once this is done, for the spectrum of the potential, the power at each degree of the Earth’s surface is found from 2 Sn (2n + 1) asin2( 0~/2) =
x
~
[(ggD)2
(4)
+ (hgrn)2]
m=0 =
TABLE 1 Maximum (l1max) and minimum (flmin) degrees of spherical cap harmonic expansions (with same KINT = 4) developed at different half-angles, together with their associated wavelengths over the Earth’s surface (wmjn and ~ respectively)
___________________________________________ ~min
0min
(km) 16° 20° 25° 30° 35° 40° 45°
24.6277 19.6044
15.5864 12.9083 10.9958 9.5619 8.4471
degree and wavelength for several caps with K 1~ 4; the better behaviour of the 45° model at lower harmonics is evident). Figure 2(a) is important because it shows the particular behaviour of spherical cap models with change of half-angles as derived from IGRF over the investigated area; best r.m.s. occurs at 55°,but 40 and 45°are also quite good. Figure 4h shows that the first four harmonics of the IGRF SV spectrum are in decreasing order of magnitude. From Fig. 4(a)—(e) it is evident that the first time n 1 spectral contribution becomes the highest value at 45°. This spherical cap model is, in our opinion, the best compromise for including both proper long and short wavelengths; moreover, the 45°model contains the smallest number of coefficients (together with 50 and 65°models; stars in Fig. 2(b)) with respect to the others. In summary, we found that the major problem in modelling SV (or any low harmonic field) data in relatively small regions comes from the poor spectral coverage in low harmonics when we use Legendre non-integer degree functions which start with higher values of n. The best compromise is to enlarge the size of the cap in order to cover those harmonics which are probably more significant, especially in SV fields (for the choice of the most suitable half-angle, the IGRF SV =
We first computed (omit2) up the to power degreespectrum n 36 for the ting the term a potential associated with the 16° spherical cap harmonic model (Fig. 3); it is evident that the most important harmonic contribution does not come from the first harmonics, and the slow convergence is clear. Similar spectra were found and drawn for the 25—55°spherical cap harmonic models (Fig. 4(a)—(g)), verifying both the increasing weight of the lower degrees and the rate of convergence. At the end we drew the IGRF spectrum (Fig. 4(h)) with the appropriate behaviour at the lower harmonics. Finally we compared the IGRF model with three spherical cap harmonic models characterized by 16, 25 and 45°half-angles (Fig. 5). In the progression from 16 to 25° spherical cap harmonic model, the improvement in the SV modelling is significant, not only in terms of the r.m.s. fit (already seen in Fig. 2(b) but also in terms of the ‘shape’ of the isopores. All the models shown in Fig. 5 were computed after rejecting data that differed by more than twice the standard deviation from a preliminary model. Final r.m.s. values were: 11.1 (16°) and 9.6 (both 25 and 45°) nT
~ma6
year~. It was finally decided to choose the 45° SV model because it showed the best agreement with the corresponding IGRF SV model. The choice of 45°cap model moreover comes from a careful look at Fig. 2(a), (b) and from considerations of spectral content, also presented in Table 1 (it contains minimum and maximum
1625 2042 2568 3101 3641 4187 4738
~~max
(Ion) 6.1481 4.8432 3.8056 3.1196 2.6347 2.2754 2.0000
6511 8265 10519 12832 15194 17593 20016
___________________________________________
=
maps can give a good orientation). This fact does not prejudice the Laplace constraints, instead of fitting the vertical and horizontal components separately. In any case, the latter way could be investigated further. At the moment, we do not go further because of the (apparent) solution to the problem by means of the application of SCHA on larger caps. 6. Discussion The problem of the appropriate application of SCHA on sparsely distributed data, with a strong
217
REPRESENTATION OF GEOMAGNETIC SECULAR VARIATION
presence of low degree spatial harmonics, especially in case of SV modelling, has been carefully investigated in order to determine a model for the SV field as observed in a region including the Iberian Peninsula and the Spanish islands. We would like to summarize some important points that emerged during the work: at the very first stage it is important to detect the harmonic contribution associated with the kind of field that has to be analysed; the removal of IGRF (or of its SV) does not always take into account the appropriate low harmonic degree contribution, especially when the global model, during its original construction, did not consider many measurements belonging to the region of interest. Moreover, IGRF gives an unrealistically constant SV for 5 year intervals. Thus, during the modelling of SV data, the representation of the smooth (long wavelength) part of the data by means of SCHA is a rather hard task. We found a quite simple solution by applying SCHA with boundary conditions on a larger cap than the cap which contains the analysed data; this implies the use of longer wavelengths, useful to model those typical of secular variation values. Acknowledgements The authors wish to thank Gerry Haines and Frank Lowes for attending to some questions and
providing some appreciated suggestions. The paper was improved after the remarks made by D.J. Kerridge and an anonymous referee. This work is part of research project No. PB87-0390 supported by the D.G.I.C.Y.T., Ministerio de Educación y Ciencia, Spain.
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