"After MeElhinny et al. (1996). N is the number of sites. A95 is the radius of the circle of 95% confidence about the mean.

"After MeElhinny et al. (1996). N is the number of sites. A95 is the radius of the circle of 95% confidence about the mean.

When all the data are used (sedimentary and igneous rocks), the means are disceraibly different at the 95% confidence level (Fig. 6.3a), whereas they are not when only igneous rocks are considered (Fig. 6.3b). Note that the mean poles always fall to the right of the line joining the zero common-site longitude to the geographic pole (particularly for the reverse polarity data). Wilson (1971, 1972) referred to this as the right-handed effect. Egbert (1992) has shown that, for simple statistically homogeneous models of secular variation, the distribution of VGP longitudes peaks 90° away from the sampling locality. The bias is not large, but it may contribute to the right-handed effect.

Wilson (1970, 1971) modeled the far-sided effect as originating from an axial dipole source that is displaced northward along the axis of rotation rather than being geocentric but was unable to explain the right-handed effect satisfactorily.

(a) All Data (b) Igneous Rocks Only

Fig. 6.3. Common-site longitude representation of the normal and reverse data shown in Fig. 6.2 and listed in Table 6.3 with circles of 95% confidence about each mean. Polar stereographic projection at 80°N. After MeElhinny et al. (1996).

However, such modeling of the sources of the geomagnetic field is nonunique and there are many possible sources that could equally satisfy the data. It is thus always more appropriate to use spherical harmonics as suggested by (6.3.1). Following James and Winch (1967), the magnetic potential V at the Earth's surface of a dipole of strength m displaced a distance x along the axis of rotation is given by

Ana Ana

If this is related to the expansion of (6.3.1), namely a o o, a of3cos20-1

For a small offset, therefore, Wilson's offset-dipole model is (nonuniquely) equivalent to a geocentric axial dipole ( ) plus a geocentric axial quadrupole ( g2 ), and the displacement x of the offset dipole can be expressed in terms of the zonal coefficients (Wilson, 1970) as x = -4ra . (6-3.5)

McElhinny et al. (1996) examined in some detail what second-order terms, beyond the geocentric axial dipole term, can realistically be deduced from paleomagnetic data for the past 5 Myr (see also Merrill et al., 1996). Their analysis suggests that there are no discernible nonzonal terms present and that only a geocentric axial quadrupole term can be distinguished with the present data set. This conclusion has been confirmed independently by Quidelleur and Courtillot (1996). However, some authors (Johnson and Constable, 1995; 1997; Gubbins and Kelly, 1993; Kelly and Gubbins, 1997) have attempted a full spherical harmonic analysis of the data and claim that significant nonzonal terms are present. The second-order terms are small in relation to the geocentric axial dipole term and their determination is sensitive to a variety of artifacts of the data that can be shown to be present. A significant problem seems to be the incomplete magnetic cleaning of Brunhes-age overprints in reversely magnetized rocks (Merrill and McElhinny, 1983; McElhinny et al., 1996). Until a significant data set becomes available with the demagnetization procedures and directional analyses represented by DC = 4 (Table 6.2), it seems unlikely that further progress can be made. In the absence of such a data set, it seems likely that the apparent differences between the normal and reverse data given in Table 6.3 and shown in Figs. 6.2 and 6.3 arise as a consequence of the current data being inappropriate for the resolution of such second-order problems.

For the time interval 0-5 Ma the best estimate of the geocentric axial quadrupole term is given by the ratio ^S\ = 0.038±0.012 (McElhinny et al., 1996). That is, the geocentric axial quadrupole present in the time-averaged paleomagnetic field is less than 4% of the geocentric axial dipole and has the same value for the normal and reverse fields. The presence of such a geocentric axial quadrupole would mean that paleomagnetic poles would be in error by no more than 3-4° if only a geocentric axial dipole field is assumed. This is much less than the typical 95% confidence limits determined for paleomagnetic pole positions. Therefore, the GAD is an acceptable model for paleomagnetic data in the range 0-5 Ma.

Attempts have been made to measure the ratio g\/g\ for epochs older than 5 Ma, but these analyses suffer even more from data artifact problems than the analysis of data for the time interval 0-5 Ma discussed in §6.3.1. It is reasonable to assume that continental drift has been small over the past few million years and that the relation of present continents to the axis of rotation has remained unchanged. For older times it is necessary to reconstruct the continents using sea-floor spreading data and then analyze the paleomagnetic data for each configuration. For Cretaceous and younger epochs these analyses suggest that it is unlikely that the geocentric axial quadrupole term has ever been more than a few percent of the geocentric axial dipole term (Coupland and Van der Voo, 1980; Livermore et al., 1984).

If paleomagnetic poles for a given geological epoch (e.g., the Permian of Europe or North America) are consistent for rocks sampled over a large geologically stable region such as a craton of continental extent, this provides compelling evidence that the dipole assumption used in calculating the poles is essentially correct. However, this does not in itself provide any information about whether the dipole is axial. Testing the axial nature of the field requires some independent measure of paleolatitude as will be described in §6.3.4.

Evans (1976) suggested a test of the dipolar nature of the geomagnetic field throughout the past. For any given magnetic field a definite probability distribution of magnetic inclination, I, exists for measurements made at geographical sites uniformly distributed around the globe. This is easily obtained by simply estimating the surface area of the globe corresponding to any set of |7| classes. For example, the dipole field is horizontal at the equator and has |7| = 10°

at latitude 5.0°. At the poles the field is vertical and has |/| = 80° at latitude 70.6°. The surface areas of these two zones imply that if sampling is sufficient and geographically random the 0° < |/| < 10° band would make up 8.8% of results and the 80° < |/| < 90° band would make up only 5.7% of results.

The present uneven distribution of land on the Earth's surface and the existence of areas of relatively intense study means that in present-day terms a random geographical sampling of paleomagnetic data has not been undertaken from this uniform distribution. However, over the past 600 Myr or more, considerable movement of continents relative to the pole has taken place and it might be assumed that this has been sufficient to render the paleomagnetic sampling random in a paleogeographical sense. Evans (1976) therefore compared the observed frequency distribution of |/| from paleomagnetic data over the past 600 Myr with that expected for the first four axial multipole fields using 1271 observations. This analysis was later updated by Piper and Grant (1989) using 4787 observations covering the past 3000 Myr. Kent and Smethurst (1998) analyzed 6419 mean inclinations for data covering the past 3500 Ma. The dependence of |/| on colatitude 0 for axial multipoles can be obtained from the relationship tan/; = ~(1 + °P/ , (6.3.6)

where P/ is the Legendre polynomial of degree / and /, is the inclination arising from an axial multipole of degree I. The expected frequency distributions of |7| for the first three multipoles are illustrated in Fig. 6.4a.

In their analysis of global paleomagnetic data Kent and Smethurst (1998) used the data listed in the Global Paleomagnetic Database (§6.2.3). They excluded only those results described by the authors in each case as representing secondary magnetization. This rather liberal acceptance criterion allowed a large data inventory, but Kent and Smethurst (1998) argued that any systematic errors introduced through the selection of data that have been remagnetized can be modeled and evaluated statistically. To minimize any bias from the overconcentration of data from any region, the observed values of |/| were averaged within 10° x 10° latitude/longitude areas for eleven geological periods in the Phanerozoic (Neogene, Paleogene, Cretaceous,..., Ordovician, and Cambrian) and in 50-Myr intervals in the Precambrian. This technique results in fewer but more evenly distributed data points and is similar to that used by Evans (1976). These data were then grouped to construct frequency distributions of |/| for the Cenozoic (253 points), Mesozoic (342 points), Paleozoic (352 points), and Precambrian (531 points), a total of 1478 binned data points, or about one-quarter of the 6419 discrete inclinations. For the Phanerozoic the 5142

Fig. 6.4. (a) Comparison of the probabilities of observing magnetic inclination |7| within 10° latitude bands for the first three axial multipole fields, (b) The observed frequency distribution of |7| for the Cenozoic, Mesozoic, Paleozoic and Precambrian compared with that expected for the GAD. The numbers in brackets are the numbers of 10° * 10° latitude/longitude areas averaged in each case. After Kent and Smethurst (1998), with permission from Elsevier Science.

Fig. 6.4. (a) Comparison of the probabilities of observing magnetic inclination |7| within 10° latitude bands for the first three axial multipole fields, (b) The observed frequency distribution of |7| for the Cenozoic, Mesozoic, Paleozoic and Precambrian compared with that expected for the GAD. The numbers in brackets are the numbers of 10° * 10° latitude/longitude areas averaged in each case. After Kent and Smethurst (1998), with permission from Elsevier Science.

inclination values analyzed by Kent and Smethurst (1998) are represented by 947 bins, compared with the 1271 values analyzed by Evans (1976) in 430 bins.

The observed frequency distributions of |/| for the Cenozoic, Mesozoic, Paleozoic, and Precambrian calculated by Kent and Smethurst (1998) are shown in Fig. 6.4b. Data for the Cenozoic and Mesozoic closely resemble the GAD model and a %2 test confirms that there is no statistical reason to reject the hypothesis that these data conform with the predicted GAD distributions. However, the data for the Paleozoic and Precambrian are decidedly skewed toward lower values and a %2 test confirms that these distributions differ significantly from the expectations of the GAD model. Kent and Smethurst (1998) argue that if the bias to low inclination values for the Paleozoic and

Precambrian is due to contamination by younger magnetizations then one would expect such overprinting to produce frequency distributions more like the GADlike patterns of the Mesozoic and Cenozoic. They therefore conclude that the paleomagnetic field in the Paleozoic and Precambrian can best be explained by a geomagnetic source model which includes a relatively modest (-25%) contribution to the geocentric axial dipole from a zonal octupole field and an arbitrary zonal quadrupole contribution. An alternative, and perhaps more likely, explanation is that the underlying assumption of random sampling of the globe through continental drift during the Paleozoic and Precambrian is invalid. In this case, if the GAD model is assumed, the results may reflect a tendency for continental lithosphere to have been continuously cycled into the equatorial belt.

The intensity of the Earth's magnetic field varies from the equator to the pole and for a geocentric axial dipole field has a latitude variation given by (1.2.7). This variation may be rewritten as

4n a where F0 is the equatorial value of the intensity of the geocentric axial dipole field, m is the Earth's dipole moment and a is the radius of the Earth (see §1.2.4). If the GAD model is valid in the past, then worldwide paleointensity measurements should show an equator to pole variation given by (6.3.7), in which the intensity at the pole is twice the equatorial value. For more details on the determination of paleointensities in the geological past, readers are referred to Merrill et al. (1996).

The determination of paleointensities is a much more difficult problem than the determination of paleomagnetic field directions. This is because the method that is generally considered most reliable requires that the sample be heated and cooled in successively higher temperatures up to the Curie temperature and this can result in chemical changes taking place. Thus often many measurements are made that produce no reliable result (e.g. Kosterov et al., 1998), and the number of determinations that have been made globally is relatively few. Also, it has been found that paleointensities for any epoch can vary quite widely with standard deviations of -50% of the mean value to be expected (McElhinny and Senanayake, 1982; McFadden and McElhinny, 1982; Kono and Tanaka, 1995; Perrin and Shcherbakov, 1997). Therefore, in order to determine reasonably well-defined mean values for any latitude, it is necessary to make a large number of measurements.

Paleomagnetic Latitude

Fig. 6.5. Global paleointensities plotted as a function of paleomagnetic latitude. Paleointensities are averaged over 20° latitude bands. Vertical bars show the 95% confidence limits. The numbers of units used in averaging are indicated. Crosses with dashed vertical error bars are for 0-10 Ma, and solid circles with solid vertical error bars are for 0-400 Ma. The curves represent the best fits for a geocentric axial dipole field; dashed curve for 0-10 Ma, solid curve for 0-400 Ma. From Perrin and Shcherbakov (1997).

Fig. 6.5. Global paleointensities plotted as a function of paleomagnetic latitude. Paleointensities are averaged over 20° latitude bands. Vertical bars show the 95% confidence limits. The numbers of units used in averaging are indicated. Crosses with dashed vertical error bars are for 0-10 Ma, and solid circles with solid vertical error bars are for 0-400 Ma. The curves represent the best fits for a geocentric axial dipole field; dashed curve for 0-10 Ma, solid curve for 0-400 Ma. From Perrin and Shcherbakov (1997).

The most recent summaries of the latitude variation of paleointensities in the past are given by Tanaka et al. (1995) for the time interval 0-10 Ma, and by Perrin and Shcherbakov (1997) for 0^100 Ma. Mean values averaged over 20° latitude bands given by these authors are summarized in Fig. 6.5 and the best fit curves to each data set for a geocentric axial dipole field are drawn through the data. Although all the mean values for each band are shown in Fig. 6.5, it should be noted that Perrin and Shcherbakov (1997) take the view that when the number of units being averaged is less than 20 then the calculated mean value cannot be considered reliable. A chi-square test indicates that the data are consistent with the latitude variation to be expected from a geocentric axial dipole field, so the current data set gives no statistical reason to reject the GAD model. This provides useful confirmation of the acceptability of the GAD model for the past 400 Myr. A subset of the global data set for the time of the Mesozoic dipole low (120-260 Ma) is also consistent with the GAD model (Perrin and Shcherbakov, 1997). Therefore, it appears that the GAD remains a reasonable first-order model irrespective of the variation in the Earth's mean dipole moment. The best fitting curves shown in Fig. 6.5 give the mean geocentric axial dipole moment as

7.8 x 1022 Am2 for 0-10 Ma and 6.5 x 1022 Am2 for O^tOO Ma. These may be

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