## Abstract

We test the consistency of 108 estimates of skewness (a measure of asymmetry that depends on the orientation of lithospheric magnetization) of magnetic anomaly 32 from the Pacific plate with a model for spreading-rate–dependent anomalous skewness formulated for data in the Arctic, Atlantic, and Indian Oceans. In a prior study, a chron 32 (71.6–73.0 Ma) paleomagnetic pole was determined that best fit these 108 skewness estimates while simultaneously solving for a third parameter, anomalous skewness, assumed to be independent of spreading rate. An analysis of the residuals in skewness was previously used to test for any dependence on spreading rate and indicated an increase in residual skewness with increasing spreading rate, which is opposite in trend to that observed in other ocean basins. In contrast with the prior analysis of residuals, we find the data to be consistent with anomalous skewness increasing with decreasing spreading half rate less than 50 mm yr^{−1}. Thus, the spreading-rate dependence of anomalous skewness in the Pacific is consistent with that found in other ocean basins and with the model for spreading-rate–dependent anomalous skewness. The resulting revised paleomagnetic pole lies only 1.2° from the prior pole. The revised pole, as was the case for the original pole, shows that the Hawaiian hotspot has shifted southward relative to the spin axis by 13° since ca. 72 Ma.

## INTRODUCTION

In a simple magnetization model of oceanic crust with vertical polarity boundaries, the skewness (asymmetry) of an anomaly in total magnetic field intensity depends on the ambient field direction, the remanent magnetization direction, and the strike of the magnetic lineations (Schouten and McCamy, 1972; Blakely and Cox, 1972; Schouten and Cande, 1976). Marine magnetic anomaly skewness is usually quantified as the phase shift that best transforms an observed magnetic anomaly to a shape expected from a simple model of seafloor magnetization (i.e., the rectangular two-dimensional layer 2A prisms with vertical reversal boundaries and vertical magnetization with alternating polarity). This simple first-order representation of the oceanic crustal magnetization model does not account for the details of the source of marine magnetic anomalies. The difference between the skewness that is observed and the skewness that is predicted by the simple magnetization model is called anomalous skewness. Its presence may bias paleomagnetic poles determined from skewness data.

Early studies demonstrated the existence of anomalous skewness (Cande, 1976). The size of anomalous skewness was estimated from the discrepancy between the skewness of anomalies on one plate compared with that of their counterparts on another plate across a mid-ocean ridge. Later works on magnetic anomalies in the Arctic, Atlantic, and Indian Oceans showed that the size of anomalous skewness decreases with increasing spreading half rate less than ∼50 mm yr^{−1}, above which it is negligible (Roest et al., 1992; Dyment et al., 1994).

Several models have been proposed to explain anomalous skewness, including temporal variations of the geomagnetic field intensity within a given chron (Cande, 1978), tectonic rotation of the source layer (Cande, 1978; Verosub and Moores, 1981), acquisition of a secondary magnetization in crustal layer 2A (Raymond and LaBrecque, 1987; Beske-Diehl, 1989), and magnetization of the deep crust and uppermost mantle controlled by the thermal structure of the oceanic lithosphere (Blakely, 1976; Cande, 1976; Kidd, 1977; Harrison, 1987; Arkani-Hamed, 1988, 1989). None of these models, however, accounts for the tendency of anomalous skewness to decrease with increasing spreading rate and to become negligible above spreading half rates of ∼50 mm yr^{−1} (e.g., Roest et al., 1992; Dyment et al., 1994). A model based on spreading-rate–dependent thermo-viscous remanent magnetization of oceanic crustal layer 3 and the uppermost mantle successfully explains these characteristics of anomalous skewness (Dyment and Arkani-Hamed, 1995; Dyment et al., 1997). In this model, the magnetic structure of the oceanic lithosphere depends on spreading rate with two parameters adjusted to fit the observed spreading-rate dependence of anomalous skewness. At a given spreading rate, the anomalous skewness is different for different anomalies. This is related to the effect of unevenly distributed neighboring magnetic sources on the skewness of these anomalies, which Dyment and Arkani-Hamed (1995) refer to as the “sequence effect.”

At intermediate and low spreading rates, the magnetization polarity boundaries of the lower crust and uppermost mantle are nonvertical and curved, flattening with distance from the ridge axis, and are responsible for the anomalous skewness. The transition occurs at a spreading half rate of ∼50 mm yr^{−1} (above which anomalous skewness is negligible) and is probably controlled by the percolation of hydrothermal fluids, which controls the serpentinization of layer 3 and the uppermost mantle (Harrison, 1987; Dyment et al., 1997). That is, the percolation of hydrothermal fluids to layer 3 and the upper mantle is negligible at half rates above ∼50 mm yr^{−1} and increases with decreasing spreading rate at half rates below ∼50 mm yr^{−1}.

Several Pacific plate paleomagnetic poles have been determined in prior studies in whole or in part from skewness data and have provided strong evidence for northward motion of the Pacific plate and southward motion of Pacific hotspots relative to the spin axis (Gordon, 1982; Petronotis and Gordon, 1989; Acton and Gordon, 1991; Petronotis et al., 1992, 1994; Petronotis and Gordon, 1999; Horner-Johnson and Gordon, 2010). A possible weakness of some of these earlier results is that they do not account for the now-well-established spreading-rate dependence of anomalous skewness (Roest et al., 1992; Dyment et al., 1994). Moreover, in contrast with the results from the other ocean basins, Petronotis and Gordon (1999) inferred an increase in anomalous skewness with increasing spreading rate, which is opposite in trend to that observed by Roest et al. (1992) and Dyment et al. (1994).

Specifically, Petronotis and Gordon (1999) determined the paleomagnetic pole for anomaly 32 (71.6−73.0 Ma for the time scale of Cande and Kent [1995], which is used throughout this paper) that best fit the observed skewness of many crossings of anomaly 32, while simultaneously solving for a third parameter, anomalous skewness, assumed to be independent of spreading rate. They then analyzed the residuals in skewness to test for any dependence on spreading rate and found an increase in residual skewness with increasing spreading rate. If the anomalous skewness of Pacific plate anomaly 32 has an entirely different spreading-rate dependence than the many anomalies in the Arctic, Atlantic, and Indian Oceans investigated by Roest et al. (1992) and Dyment et al. (1994), it would imply different magnetic properties of the oceanic crust in different locations or some difference due to the different locations of these studies relative to the magnetic pole, or possibly question the viability of the model of Dyment and Arkani-Hamed (1995).

There is a weakness, however, in the methods used by Petronotis and Gordon (1999) to analyze the data. They did not correct for the spreading-rate dependence of anomalous skewness for anomaly 32 before inverting their data because corrections were not available for this anomaly. Here, we remedy this deficiency by explicitly correcting the skewness estimates for spreading-rate dependence of anomaly 32 by using predictions determined from the model of Dyment and Arkani-Hamed (1995). Because the anomalous-skewness correction depends on spreading rate, we examine different sets of estimated spreading rates to determine the sensitivity of the estimated pole position to uncertainty in the spreading rates.

In contrast with their conclusions, we find that the data of Petronotis and Gordon (1999) are actually consistent with the predicted spreading-rate dependence of anomalous skewness. Our paleomagnetic pole for the Pacific plate differs little (by 0.2° to 1.2° depending on the set of spreading rates used) from the Petronotis and Gordon (1999) pole, confirming that the Hawaiian hotspot has shifted ∼13° southward relative to the spin axis since ca. 72 Ma (Petronotis and Gordon, 1999).

## METHODS

### Relation between Parameters

*I*, which gives a biased estimate of the true effective remanent inclination,

_{a}^{eff}*I*. The bias is hereinafter called anomalous skewness, θ

_{r}^{eff}_{a}, where

*I*is the ambient effective field inclination (Schouten and Cande, 1976; Petronotis et al., 1992), given by

_{s}^{eff}*A*is the azimuth of the strike of the magnetic lineation at the site, measured 90° clockwise from the direction in which the seafloor becomes younger.

*I*and

*D*are the inclination and declination of the ambient geomagnetic field.

*I*is estimated from a geomagnetic reference field model for a given latitude, longitude, and lineation azimuth.

_{s}^{eff}### Model Estimation

A best-fitting pole and, when desired, anomalous skewness are estimated by weighted least squares from observed apparent effective remanent inclinations (Gordon and Cox, 1980; Gordon, 1982; Petronotis et al., 1992). Ninety-five percent confidence limits for these parameters are determined both from a constant chi-square boundary and by linear propagation of errors.

We used the method of Petronotis et al. (1992), but first corrected for spreading-rate–dependent anomalous skewness. Thus, unbiased estimates of the effective remanent inclinations can presumably be directly calculated from the skewness data (Eq. 1). The spreading-rate–dependent values of anomalous skewness for anomaly 32 were predicted from the model of Dyment and Arkani-Hamed (1995) (Table 1; Fig. 1).

## MAGNETIC PROFILES AND SPREADING RATES

Anomalous-skewness corrections were applied to 108 skewness estimates of anomaly 32 (Fig. 2): 19 recording Pacific-Kula spreading, 55 Pacific-Farallon spreading, three Pacific-Aluk spreading, 11 Pacific-Bellingshausen spreading, and 20 Pacific-Antarctica spreading (Petronotis and Gordon, 1999). The profiles collected over seafloor produced by Pacific-Farallon spreading were further subdivided into three groups: northern (36 profiles north of 22°N), central (15 profiles between the equator and 22°N), and southern (four profiles south of 5°S).

Using the Cande and Kent (1992) time scale, Petronotis and Gordon (1999) estimated the spreading rate appropriate for each crossing of anomaly 32 by comparing the observed sequence of anomalies 30–33 with synthetic magnetic-anomaly profiles. When no rate estimate was available for an individual crossing, they used an average for the subregion (Fig. 2). We have revised their rate estimates to be consistent with the time scale of Cande and Kent (1995), which results in rates that are ∼15% lower than before (Table 2; Fig. 3).

The rate estimates of Petronotis and Gordon (1999) did not follow a sine curve with angular distance from a pole of rotation (Figs. 4A and 4B), which would be expected for spreading between rigid plates. These departures from a sine curve are unlikely to be caused by plate nonrigidity, however. Investigations of geologically current plate motion also result in spreading-rate estimates that do not perfectly follow a sine curve, and they have standard deviations in the rate estimates along the East Pacific Rise of ∼3 mm yr^{−1} (DeMets et al., 2010). The dispersion of the rate estimates of Petronotis and Gordon (1999) appears to be a little larger than this (Figs. 4A and 4B). This is unsurprising because the rates of Petronotis and Gordon (1999) were estimated from anomalies from only one side of a mid-ocean ridge, because their counterparts have been subducted. Thus, locally asymmetric spreading may have increased the dispersion in the rates.

Therefore, rates determined by fitting many data simultaneously while assuming that plates are rigid may be more accurate than rate estimates from individual crossings from only one side of the ridge. In particular, such estimates will give a more accurate estimate of the relative rates. Additionally, Petronotis and Gordon (1999) were not able to estimate the spreading rate for every individual crossing of anomaly 32, and an average for the subregion was used where no rate estimate was available. Thus, we also estimated some spreading rates from published stage poles and angles (Table 3) using updated ages from the time scale of Cande and Kent (1995) (Table 4). However, a possible weakness of using published stage poles and angles is that they average plate motion over longer time intervals than do the rates of Petronotis and Gordon (1999) and thus may not be representative of spreading during chron 32.

The rates of Petronotis and Gordon (1999) for Pacific-Farallon spreading are similar to those calculated from the anomaly 34 to 25 stage pole and angle of Engebretson et al. (1984), and also to those calculated from the anomaly 32a to 30/31 stage pole and angle of Rosa and Molnar (1988) (Fig. 4A), and we suspect that the stage poles provide a more accurate set of rates than the individual crossings, just as we think that the best-fitting angular velocities for geologically current plate motion determined by DeMets et al. (2010) can be used to estimate rates more accurately than can be done from a single magnetic anomaly crossing of a mid-ocean-ridge segment.

On the other hand, the Pacific-Kula rates of Petronotis and Gordon (1999) are higher than the rates inferred from the anomaly 32b to 31 stage pole and angle of Engebretson et al. (1984), and they are considerably higher than the rates inferred from the anomaly 32a to 30/31 stage pole and angle of Rosa and Molnar (1988) (Fig. 4B). These differences in estimated rates can cause differences of up to almost 15° in estimated anomalous-skewness correction (Figs. 1 and 5). Because Pacific-Kula spreading slowed from 72 Ma to 56 Ma (Engebretson et al., 1984), the stage poles and angles from Engebretson et al. (1984) and from Rosa and Molnar (1988) give spreading rates that may be biased toward lower than appropriate values for anomaly 32. Thus, the true uncertainty in anomalous-skewness correction may be smaller than implied by the large range of spreading rates that we used. However, as will be shown later herein, the skewness estimates for Pacific-Kula spreading only account for a few percent of the information used to estimate the pole position. Thus, use of the stage pole and angle for Pacific-Kula spreading rates, while not ideal, has little effect on the estimated paleomagnetic pole.

Because published stage poles are only available for Pacific-Farallon and Pacific-Kula profiles, rate estimates of Petronotis and Gordon (1999) were used for other regions in all cases.

## RESULTS

If no correction is made for anomalous skewness, the sum-squared normalized misfit, *r*, is 165.2 (Petronotis and Gordon, 1999). With the spreading-rate–dependent anomalous-skewness corrections derived from the spreading rates of Petronotis and Gordon (1999) (Table 2; Fig. 3), the best-fitting pole is located at 71.8°N, 23.0°E (95% confidence ellipse: 4.2° major semi-axis oriented 96° clockwise from north and 1.5° minor semi-axis) with *r* = 137.7 (Fig. 6). This reduction in misfit, without the introduction of additional adjustable parameters, is generally supportive of the model for spreading-rate–dependent skewness, although the improvement in fit is not statistically significant (*F* = 1.20 with 106 vs. 106 degrees of freedom; the probability, *p*, of finding a value of *F* this large or larger, if the two distributions have identical variances, is 17%.). If anomalous skewness is allowed to adjust in the inversion (after the spreading-rate–dependent anomalous-skewness correction has already been applied), the best-fitting pole is located at 71.7°N, 22.9°E, and the best-fitting value of additional anomalous skewness is –0.6° ± 3.7° (95% confidence limits).

When anomalous-skewness corrections determined from the rates of Engebretson et al. (1984) are applied, the chron 32 paleomagnetic pole is located at 71.8°N, 24.1°E (95% confidence ellipse: 4.1° major semi-axis oriented 97° clockwise from north and 1.5° minor semi-axis) with r = 126.9 (Fig. 6). If anomalous skewness is allowed to adjust in the inversion, the best-fitting pole is located at 71.5°N, 23.7°E, and the best-fitting value of additional anomalous skewness is –1.6° ± 3.7° (95% confidence limits).

For the rates of Rosa and Molnar (1988), the pole position is 72.2°N, 27.3°E (95% confidence ellipse: 4.0° major semi-axis oriented 98° clockwise from north and 1.7° minor semi-axis) with *r* = 123.6 (Fig. 6). If anomalous skewness is allowed to adjust in the inversion, the best-fitting pole is located at 71.6°N, 26.5°E, and the best-fitting value of additional anomalous skewness is –3.0° ± 3.7° (95% confidence limits).

Thus, in all three cases, if anomalous skewness is allowed to adjust after the spreading-rate–dependent correction is applied, no significant additional anomalous skewness is found, thus indicating consistency with the spreading-rate–dependent model of Dyment and Arkani-Hamed (1995). Also, in all three cases, use of the spreading-rate–dependent anomalous-skewness correction results in reductions of the misfit of 17% to 25% without the introduction of any new adjustable parameters. The three new poles lie very near one another and the pole of Petronotis and Gordon (1999) (Fig. 6).

The corrections derived from the rates from available stage poles reduce the misfit more than does the use of the spreading rates of Petronotis and Gordon (1999). None of the spreading-rate–dependent corrections reduces the misfit as much, however, as when anomalous skewness is treated solely as a spreading-rate–independent adjustable parameter (*r* = 108.7; Petronotis and Gordon, 1999). Because the latter involves an additional adjustable parameter not used in the spreading-rate–dependent corrections, it is not surprising that the misfit is smaller. It does not necessarily indicate that the spreading-rate–independent results should be preferred.

Because the spreading rates from stage poles indicate rates that are too low for chron 32 Pacific-Kula spreading, we consider the pole obtained using Petronotis and Gordon's (1999) rates to be the new preferred pole (Fig. 6). In Figure 7, each observed remanent effective inclination (calculated from the phase shifts using the spreading-rate–dependent anomalous-skewness correction) is compared with its corresponding model effective inclination (calculated from the new preferred pole position). Both are plotted against site paleolatitude. Near the paleo-equator, effective inclination changes rapidly with paleolatitude and gives the best constraints for determining the pole. The largest information contribution thus unsurprisingly comes from the profiles near the paleo-equator (Fig. 8).

## DISCUSSION

The first main issue addressed by this paper is a test of the consistency of the spreading-rate–dependent anomalous-skewness model of Dyment and Arkani-Hamed (1995) with the set of Pacific plate anomaly 32 skewness estimates of Petronotis and Gordon (1999). In contrast with the results of the analysis of residuals by Petronotis and Gordon (1999), we find that the data are consistent with the model. Use of the model corrections reduces the sum-squared–normalized misfit by 17% to 25% (depending on the set of spreading rates used to estimate the anomalous-skewness corrections) relative to an inversion with no correction for anomalous skewness. Moreover, no significant additional spreading-rate–independent anomalous skewness is indicated by the data if first corrected for spreading-rate–dependent anomalous skewness. Thus, Pacific plate skewness data for anomaly 32 are consistent with the model of Dyment and Arkani-Hamed (1995), and one less adjustable parameter is required for analysis of paleomagnetic poles.

The second main issue addressed herein is the extent to which, if at all, spreading-rate–dependent anomalous skewness alters the chron 32 paleomagnetic pole obtained by Petronotis and Gordon (1999) and potentially alters other poles determined using spreading-rate–independent adjustments for anomalous skewness. The results indicate that the pole position obtained from the Pacific plate anomaly 32 skewness data of Petronotis and Gordon (1999) is robust with respect to the treatment of anomalous skewness, as the revised pole lies merely 0.2° to 1.2° from the prior pole, depending on the set of spreading rates used. This robustness is largely a consequence of the insensitivity of the pole position to errors in apparent effective inclinations when determined from crossings of low-paleolatitude, nearly north-south–paleostriking magnetic lineations (Acton and Gordon, 1991). Because of this insensitivity, the poles determined assuming the presence of anomalous skewness (e.g., the diamond and stars in Fig. 6) differ by only a few degrees from a pole determined while ignoring the presence of anomalous skewness (e.g., the red-filled circle in Fig. 6). The poles for which the anomalous-skewness correction is broadly similar, while differing in detail, result in even smaller differences between poles (e.g., between any two of the diamonds and three stars in Fig. 6).

The same may not be true for other data sets in different plate geometries, however. Given that the spreading-rate dependence of anomalous skewness seems well established from analysis of conjugate profiles in the Atlantic and Indian Ocean basins, we think that it is important to apply this correction in future work estimating paleomagnetic poles from skewness data. It seems especially important to apply the correction when the data set does not include many crossings at low paleolatitudes across lineations that strike close to paleo–north-south. Use of the spreading-rate–dependent correction has the further advantage of decreasing the number of adjustable parameters in the paleomagnetic pole determination from three to two.

## CONCLUSIONS

The model for spreading-rate–dependent anomalous skewness of Dyment and Arkani-Hamed (1995) is consistent with the anomalous skewness present in the Pacific plate anomaly 32 skewness data set. The largest uncertainties in anomalous-skewness correction are due to uncertainties in spreading rates. Paleomagnetic poles estimated from the anomaly 32 data set of Petronotis and Gordon (1999) differ little, however, when the method of estimating anomalous skewness or the set of estimated spreading rates is changed. The paleomagnetic results of Petronotis and Gordon (1999), in particular the inferred 13° ± 3° (95% confidence limits) southward shift of the Hawaiian hotspot since ca. 72 Ma, are supported by the revised pole determined using a spreading-rate–dependent correction for anomalous skewness.

We thank Katerina Petronotis for supplying us with the set of spreading rates she estimated for the Pacific plate crossings of anomaly 32 used in Petronotis and Gordon (1999). Koivisto's and Gordon's efforts were supported by National Science Foundation grants OCE-0527375, OCE-0928961, and OCE-1061222. Arkani-Hamed was supported by the National Sciences and Engineering Research Council of Canada (NSERC). This is Institut de Physique du Globe de Paris contribution 3224.