5.4.1 Skewness of Magnetic Anomalies

The measurement of the skewness of magnetic anomalies as described in §5.2.2 provides a method for calculating paleomagnetic poles. The effective inclination /' is easily determined from the geomagnetic field direction at the site (generally calculated using the IGRF, see §1.1.3) and the azimuth A of the source using (5.2.2). Therefore, if the skewness 0 is known, (5.2.1) can be solved for /r . Substituting into (5.2.3) the possible combinations of DR and IR, together with (1.2.9) and (1.2.10), defines a semi-great circle of paleomagnetic pole positions consistent with the observation 0. To further constrain the solution to a single pole position instead of a possible pole path, it is necessary to determine 0 for two sets of contemporaneous anomalies on a plate. The two semi-great circles should intersect to give the paleomagnetic pole for the plate for the age of the anomalies (Fig. 5.21a). In practice the estimated value of 0 in each case can only be determined at a level of confidence A0 of ±5-10°. Therefore, the values 0±A0 define a lune of confidence for lineations of the same age. The intersection of two lunes of confidence for lineations of the same age at two places on a rigid plate then determines the zone of confidence for the paleomagnetic poles (Fig. 5.21b).

Schouten and Cande (1976) referred to the above method for determining paleomagnetic poles as the "theta method". It was first used by Larson and

Chase (1972) on three sets of lineations in the Western Pacific. They found 9 by computing a series of models for their lineations phase-shifted in 10° increments. They then estimated errors by comparing the observed profiles with the models. Although this gives a good estimate of 9 when the profiles are nearly symmetric (9 near 0°), it is not as effective when the profiles are nearly antisymmetric (9 near 90°). Schouten and Cande (1976) proposed that the skewing effect of 9 be removed (deskewed) by inverse phase filtering (Schouten and McCamy, 1972), similar to the method of reduction to the pole of Blakely and Cox (1972). The profiles are then inverse phase-filtered in 5° increments of 9. To calculate 9 and its confidence level A9, suitable individual anomalies are compared with the model profile. This will then give a range of values of 9 corresponding to the best fit for each chosen anomaly. These values of 9 are then averaged to give a mean value for the observed profile and the confidence interval A9 can be calculated.

The presence of anomalous skewness causes a systematic error in the effective remanent inclination, and this limits the accuracy of paleomagnetic poles determined from skewness data. Petronotis et al. (1992) proposed the following procedure to calculate paleomagnetic poles and to allow for the effect of anomalous skewness. The anomalous skewness 9A is given by the difference between the true effective remanent inclination I'K and the apparent effective remanent inclination /AR , according to

If Xs and <j>s are the latitude and longitude of the site, and and <t>p are the corresponding coordinates of the pole position, then /R and DR can be calculated according to (1.2.9) and (1.2.10). Given a trial value for anomalous skewness 9A and a trial pole position (A.p, <j>p), a model apparent effective remanent inclination can be predicted at each site from (5.4.1) and then compared with the observations.

In the Schouten and Cande (1976) approach two distinct observations uniquely define a best fitting paleomagnetic pole (Fig. 5.21) because only two parameters A.p and <|>p were adjusted. To allow for anomalous skewness 9A must also be estimated; therefore, three distinct observations must be used in this case. For three or more observed values of skewness, a pole position will rarely fit all the data perfectly. A pole position is therefore calculated that minimizes the weighted squared differences between the predicted and observed apparent effective remanent inclination at each site.

The relative amplitude factor C also contains information about the remanent magnetization vector from (5.2.4). Since the overall amplitude of the anomalies also depends on the thickness of the source layer and the remanent magnetization, C cannot be resolved from the amplitudes of the observed

Fig. 5.21. Estimating a paleomagnetic pole position from the skewness of magnetic anomalies, (a) Semi-great circles of possible pole positions, determined from unique values of 0 from two sets of magnetic lineations located at =, intersect at the pole position, (b) The zone of confidence for the pole position is determined by the intersection of two lunes of confidence, shown here with A0 = 10° in each case. From Schouten and Cande (1976).

anomalies. It is possible, however, to determine a locus of paleomagnetic poles from a ratio of C for two regions. Assuming that the thickness of the source layer is the same for the two regions and its magnetization only varies as a geocentric axial dipole field, then the ratio of the amplitudes equals the ratio of C times the ratio of the dipole paleointensity. Although the resolution of the ratio of amplitudes is far less than the resolution of 0, the locus of paleomagnetic poles can also be used as an independent source of information (Schouten and Cande, 1976).

Vacquier (1963) showed that if a combined magnetic and bathymetric survey of a seamount is made, it is possible to calculate the direction of magnetization of the seamount under the assumption of uniform magnetization. These directions of magnetization are an equivalent of those determined paleomagnetically by conventional methods and it is possible to calculate a paleomagnetic pole for the seamount. The growth of a small seamount takes place over 10 -105 years. Secular variation will tend to be averaged out in the computation of its average direction of magnetization. Larger seamounts may remain volcanically active over a period of 104—107 years, and the problem arises that several polarity changes may have taken place during this time. The seamount will not then be uniformly magnetized as will be found when the computed anomaly fails to match the observed anomaly.

The original method proposed by Vacquier (1963) used a linear least squares inversion of the magnetic anomaly constrained by the shape of the seamount. The seamount is assumed to be uniformly magnetized, to be bounded by an upper surface equal to its bathymétrie surface, and to have a known lower surface that is usually flat and assumed to be the same depth as the surrounding ocean floor. This method has been improved or extended using more efficient algorithms (Talwani, 1965; Plouff, 1976). Vacquier (1963) modeled the shape as an aggregate of rectangular blocks, whereas Talwani (1965) used a stack of laminas and Plouff (1976) a stack of layers. Attempts have been made to allow for non-uniform magnetization in seamounts by dividing the topography into discrete blocks. For example, McNutt (1986) divided the topographic shape into two or three blocks, assuming each was uniformly magnetized and used least squares methods to solve for the direction of magnetization within each block.

In order to allow for nonuniform magnetization the boundaries between the uniformly magnetized blocks were determined subjectively, usually by inspection of the residual magnetic anomaly. Parker et al. (1987) proposed an entirely different approach and developed a method using linear inverse theory in a Hilbert space. The method is particularly useful where seamounts have magnetization inhomogeneities. The derived magnetic model maximizes the uniformly magnetized part and minimizes the inhomogeneities. Hildebrand and Parker (1987) applied this method to some seamounts in the Pacific Ocean whose magnetization had previously been determined by the linear least squares method. In most cases the differences between the results obtained by the two methods were small. When modeling seamount magnetization it is assumed that there are no induced or viscous components. This assumption is difficult to verify because few seamount interiors have been sampled. However, it is surprising that there is a predominance of seamounts modeled to have normal polarity magnetization. Gee et al. (1989) sampled an uplifted seamount and found that one-sixth of the magnetization may arise from induced magnetization. Despite this, it seems that as long as the induced and viscous components are less than about 15% (Q > 7), the calculated paleomagnetic pole will be in error by <5° (Sager and Pringle, 1988). A useful review of these methods together with appropriate theory is given by Blakely (1995).

There has been considerable discussion as to the validity of the methods of modeling seamount magnetization using the methods described above. Parker (1988) has shown that the standard least squares method is unsatisfactory because, in the case of nonuniform magnetization, the unfitted field is not due to random contamination as required. He therefore proposed a statistical theory that overcame this problem, showing that the paleomagnetic pole derived in this way can be completely incompatible with that deduced from standard least squares inversion. Also, the magnetization of dredged and drilled samples from the ocean floor shows that both basalts (Lowrie, 1974) and gabbros (Kent et al., 1984) have log-normal distributions of magnetization. Limited sampling of seamounts (Kono, 1980a; Gee et al., 1988, 1989) also shows similar log-normal distributions that vary according to lithology and grain size. This means that the model of uniform magnetization usually assumed is unrealistic. Parker (1991) therefore developed a model for seamount magnetization in which the direction of magnetization is fixed but the intensity of magnetization can vary with no upper limit on magnitude. Application of this model to several seamounts suggests that many published results using the assumption of uniform magnetization may not be as well determined as has previously been thought.

It is useful to calculate some parameter that indicates how well the calculated magnetic anomaly approximates to the observed anomaly. The most widely used parameter is the goodness-of-flt ratio (GFR) defined as

Mean observed magnetic anomaly , . „

Mean residual anomaly where the mean residual anomaly is the mean difference between the observed and calculated anomaly. A GFR <2.0 is usually regarded as an unreliable result (Sager and Pringle, 1988).

Most seamount magnetizations have been determined from the Pacific plate. However, to be useful for paleomagnetic studies the ages of the seamounts must be known. The first attempt at plotting an apparent polar wander path (APWP, see chapter 6) for the Pacific plate was carried out by Francheteau et al. (1970b). They showed that the pole position determined from 17 seamounts in the vicinity of the Hawaiian islands with ages 85-90 Ma lay at 61°N, 16°E with A95 = 8°. The Pacific plate has thus drifted approximately 30° northwards since the Cretaceous. Sager (1987) indicates that there are now more than 90 results with GFR >2.0 from this region. Sager and Pringle (1988) conclude that only 22 of these seamounts can be used for determining the apparent polar wander path for the Pacific jDlate^ There are 17 that have radiometric ages, with all but 3 of these being At/ At total fusion or incremental heating ages, which are considered the most reliable for dating basalts that have undergone submarine alteration. The other 3 ages are high-quality K-Ar ages. Other means, including fossils dredged from the reef cap or magnetostratigraphic arguments (Gordon and Cox, 1980a), have been used to date the remaining 5 seamounts.

5.4.3 Calculating Mean Pole Positions From Oceanic Data

Oceanic plates present the following types of data that provide information about paleomagnetic pole positions.

(i) Paleomagnetic measurements of inclinations (but not declinations) obtained from deep-sea drilling cores (see §3.2.3 for analysis of inclination-only data). This constrains the pole to lie on a small circle, centered on the sampling site, at a distance given by the paleocolatitude (1.2.5).

(ii) Effective inclinations, calculated from the skewness of marine magnetic anomalies, which constrain the pole to lie on a specific half great circle

(iii) Ratio of relative amplitude factors from two sets of magnetic lineations of the same age but created at different locations on the same plate. This constrains the pole to lie on a locus defined by the intensity structure of the geocentric axial dipole field (§5.4.1). Naturally these provide only a low-resolution result.

(iv) Paleomagnetic poles determined from magnetic anomalies over seamounts

(v) Paleoequators determined from geological analysis of marine sediment cores (van Andel, 1974). In this case the equatorial zone of upwelling is also a region of high biological productivity so that this zone is characterized by high sedimentation rate and higher biogenic content than at other latitudes. It is possible in some deep-sea cores to identify the age at which the drill site crossed the equator. A datum from a single site can thus be considered to be a paleocolatitude of 90°, defining a great circle of possible pole positions. Gordon and Cox (1980b) presented a method to yield a best fit pole position and confidence limits for a combination of data of the types listed above. The two fundamental assumptions in their method are that (a) each of the observations arises from a common paleomagnetic pole, and (b) the errors from each datum are random and mutually independent. It is then a question of identifying the pole position that is most compatible with all the observations. Gordon and Cox (1980b) chose maximum likelihood as the appropriate measure of compatibility. That is, given the probability distribution for each observation, the most appropriate pole is the one that gives the maximum joint probability (likelihood) for the actual observations. They further assumed that for small errors the error distribution for each datum is Gaussian. With this assumption, maximum likelihood reduces to a weighted least squares estimation. Hence, as their test parameter they used

where the summation is over all observations, a, is the error in the i"1 observation, and 8, is the deviation of the observation from the calculated value for the current test position of the pole.

For example, for inclination-only data, the datum is considered to be the equivalent paleocolatitude. Thus, for these data

where p°bs is the observed paleocolatitude and p,model is the colatitude of the site for the current test pole position. Seamount poles constrain two degrees of freedom (the latitude and longitude of the pole) in the analysis. The same constraint can be achieved by having two great circles intersecting at right angles at the seamount pole position. Therefore, seamount data with elliptical confidence limits from one site are computationally equivalent to paleoequator data from two sites with different errors. However, paleoequators are in effect just a paleocolatitude with a nominal value of 90°. Therefore, the analyses of

Fig. 5.22. The 72-Ma paleomagnetic pole for the Pacific plate and the data used in its calculation. The solid dot is the location of the pole and the ellipse is its 95% confidence region. The stars indicate paleomagnetic poles from seamounts, solid lines give the locus of possible pole positions determined from paleocolatitudes, dotted lines show the half great circle locus of paleomagnetic poles determined from the skewness of magnetic anomalies, and the dashed line represents the locus of possible pole positions determined from the relative amplitude factor. From Sager and Pringle (1988).

Fig. 5.22. The 72-Ma paleomagnetic pole for the Pacific plate and the data used in its calculation. The solid dot is the location of the pole and the ellipse is its 95% confidence region. The stars indicate paleomagnetic poles from seamounts, solid lines give the locus of possible pole positions determined from paleocolatitudes, dotted lines show the half great circle locus of paleomagnetic poles determined from the skewness of magnetic anomalies, and the dashed line represents the locus of possible pole positions determined from the relative amplitude factor. From Sager and Pringle (1988).

seamount poles, inclination-only data, and paleoequator data are all reduced to the analysis of paleocolatitudes from a computational viewpoint. For anomaly skewness the datum is the effective inclination and so the difference between observed and model effective inclination is used for 8; in (5.4.4). The best estimate of the pole position is that which gives the minimum value of % .

Confidence limits in the estimated pole position are obtained by standard linear propagation of errors. With these errors it is possible to calculate the importance of each datum, bearing in mind that a datum such as a seamount pole provides two constraints and therefore two importances, which should be summed. Data importances are a measure of the contribution that any particular datum makes in constraining the final estimate. If the importance of one datum is substantially greater than that of the others, then there are three possibilities that should be considered: (a) The uncertainty in that datum may have been grossly underestimated, (b) the remainder of the data genuinely do not provide a useful constraint on the result, and (c) the datum in question may be incompatible with the rest of the data. Naturally, (c) is the most serious because it suggests that the datum with the highest "importance" is in error either because of incorrect measurement or because the age of the datum has been incorrectly assigned. Bearing in mind the problems discussed in §5.4.2 regarding the modeling of seamount magnetization, it is critical that seamount poles be accurately (as distinct from precisely) determined. It is inevitable that seamount poles will have the greatest importance since they constrain two dimensions. This is especially the case if the great circles from the other data are subparallel because then the result will be controlled entirely by the seamount data.

An example of the application of this method to a 72-Ma paleomagnetic pole determined for the Pacific plate by Sager and Pringle (1988) is illustrated in Fig. 5.22. As listed in Table 5.3 the analysis combines data from paleocolatitudes

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