Geochemistry, Geophysics, Geosystems

Toward a self-consistent, high-resolution absolute plate motion model for the Pacific

Authors


Abstract

The hot spot hypothesis postulates that linear volcanic trails form as lithospheric plates move relative to stationary or slowly moving plumes. Given geometry and ages from several trails, one can reconstruct absolute plate motions (APM) that provide valuable information about past and present tectonism, paleogeography, and volcanism. Most APM models have been designed by fitting small circles to coeval volcanic chain segments and determining stage rotation poles, opening angles, and time intervals. Unlike relative plate motion (RPM) models, such APM models suffer from oversimplicity, self-inconsistencies, inadequate fits to data, and lack of rigorous uncertainty estimates; in addition, they work only for fixed hot spots. Newer methods are now available that overcome many of these limitations. We present a technique that provides high-resolution APM models derived from stationary or moving hot spots (given prescribed paths). The simplest model assumes stationary hot spots, and an example of such a model is presented. Observations of geometry and chronology on the Pacific plate appear well explained by this type of model. Because it is a one-plate model, it does not discriminate between hot spot drift or true polar wander as explanations for inferred paleolatitudes from the Emperor chain. Whether there was significant relative motion within the hot spots under the Pacific plate during the last ∼70 m.y. is difficult to quantify, given the paucity and geological uncertainty of age determinations. Evidence in support of plume drift appears limited to the period before the 47 Ma Hawaii-Emperor Bend and, apart from the direct paleolatitude determinations, may have been somewhat exaggerated.

1. Introduction

Are there plumes in the mantle? If so, are they fixed or do they move? Are linear seamount chains the surface expression of these plumes, and can they be used to infer the motion of the plates over the mantle? These are questions that have been with us since the early days of plate tectonics when Wilson [1963] first suggested that the Hawaiian chain might have formed over a hot spot. Following Morgan's [1971] hypothesis of fixed mantle plumes there was almost unanimous agreement that the Hawaii-Emperor Bend (HEB) was the archetypal example of a large change in absolute plate motion (APM) recorded by plume volcanism. Indeed, almost all modern introductory geology textbooks use this feature, estimated to have formed between 42–44 Ma [Clague and Dalrymple, 1973], to showcase changes in plate tectonic motions (Figure 1). Lately, this interpretation has come under renewed scrutiny. Norton [1995] searched for geological evidence of tectonic activity contemporaneous with the HEB and found none, arguing instead that the HEB was a “nonevent” caused by southward drift of the Hawaiian plume. Fluid dynamic modeling, suggesting southward drift of the Hawaiian plume [Steinberger and O'Connell, 1998], supported that view, which also is in accord with paleomagnetic data [Tarduno and Cottrell, 1996]. Furthermore, African APM models [Müller et al., 1993], projected via the Antarctica plate circuit into the Pacific, fail to reproduce the HEB, in turn suggesting that the HEB perhaps does not reflect APM changes [Cande et al., 1995; Raymond et al., 2000]. Recent studies have argued that it is not possible to fit most Pacific hot spot trails without invoking some drift [Koppers et al., 2001]. Finally, the lack of a major reorganization of the Pacific-Farallon spreading system at HEB time seemed to imply that no large APM change took place [Atwater, 1989]. Collectively, these arguments have placed the fixed hot spot hypothesis in serious question. Indeed, Tarduno et al. [2003] have argued that no plate motion change is needed once hot spot motion is considered. Should these claims prove true, we will soon be witnessing a sea change in global tectonic studies, given that many geologic inferences have been tied to the presumably fixed hot spot system.

Figure 1.

The geometry of the Hawaii-Emperor seamount chain. Stretching westward from the Big Island of Hawaii, the chain changes to a more northerly direction; the point of change is referred to as the Hawaii-Emperor Bend (HEB) and has now been dated at 47–48 Myr (∼Chron 21) by allowing 1–2 Myr for edifice construction. Triangles represent preliminary new dates from Sharp and Clague [2002] except for Nintoku [Tarduno et al., 2003].

While all questions regarding hot spot fixity and APM have yet to be answered, much progress has been made. The available data do seem to suggest that while little or no hot spot motion has occurred since the HEB, drift may indeed be needed prior to that time. However, three salient observations suggesting we exercise caution regarding plume drift have recently surfaced, as outlined below.

1.1. Revised Age of the HEB

Perhaps one of the key discoveries pertaining to the HEB puzzle was the recent realization that rocks dredged from the seamounts closest to the HEB appear to be considerably older than previously reported. Sharp and Clague [1999, 2002] have presented preliminary 40Ar/39Ar dates of ∼47 Ma for Kimmei and Diakakuji seamounts. This is 4 m.y. older than the K/Ar dates of the same samples that were the basis for the traditional 43 Ma age estimate of the HEB [Clague and Dalrymple, 1973]. Since the dated rocks are post-shield transitional to alkalic basalts and trachytes then the earliest eruptions (by analogy with the Quaternary Hawaiian volcanoes) probably took place 1–2 Myr earlier. Sharp and Clague [2002] even suggest an age of ∼50 Ma for the HEB; however, since the rocks are not believed to be from the rejuvenated (post-erosional) stage we suggest a more realistic age of the HEB is around 47–48 Ma, or ∼Chron 21. However, we caution that until the older dates can be confirmed the Chron 21 age remains preliminary.

1.2. “Nonevent” Revisited

Norton [1995] argued that the HEB was a “nonevent” as he was unable to find any significant tectonic events that coincided with it. However, in 1995 the conventional age of the HEB was thought to be ∼43 Ma (∼Chron 19). If the revised age of 47–48 Ma (∼Chron 21) is accepted, then many tectonic events appear to correlate with the HEB. These events include, but are not limited to, (1) major triple junction reorganizations in the South Pacific where Pacific-Antarctic spreading propagated northward to intersect the Pacific-Farallon ridge, which terminated spreading at the Pacific-Aluk ridge at ∼Chron 21, leaving the Henry and Hudson Troughs as conjugate tectonic scars [Cande and Haxby, 1991]; (2) main episode of Cenozoic East-West Antarctic separation and the initiation of seafloor spreading in the Adare basin at the old end of Chron 20 [Cande and Stock, 2005]; (3) earliest arc (forearc) volcanism at 48–51 Ma following the initiation of Izu-Bonin-Mariana subduction in the western Pacific [Cosca et al., 1998]; and perhaps the most significant event and possible “smoking gun,” (4) the collision between India and Eurasia at Chron 22–21 [Patriat and Achache, 1984]. While geologic correlations of these events with the HEB event will always be tenuous due to the uncertainties of dating (for instance, the beginning of East-West Antarctic separation may have begun as early as Chron 27 [Cande and Stock, 2005]), we believe the older age of the HEB would considerably weaken Norton's [1995] original argument and provide a much-improved linkage between APM and RPM models at HEB time.

1.3. Influence of Plate Circuits

Recently, Steinberger et al. [2004] used an alternative plate circuit to that of Cande et al. [1995] to connect Africa and the Pacific via Australia and the Lord Howe Rise, and now the HEB is once again predicted. This sensitivity to the choice of plate circuit suggests that the uncertainty in such circuits, in particular continental deformation in Antarctica, remains a major source of error. Indeed, citing large uncertainties due to continental deformation, Gordon and Horner-Johnson [2004] found no significant motion between Pacific and other hot spots and instead called upon true polar wander to explain the available paleomagnetic data.

In this paper, we outline our new approach to absolute plate motion modeling. We will limit the application of the method and discussion of preliminary results to the Pacific basin, and suggest future modifications that may permit the method to contribute to the two fundamental, yet unresolved aspects of global absolute plate motions: true polar wander and hot spot drift.

2. Modeling of Absolute Plate Motions

While Bullard et al. [1965] were able to fit conjugate Atlantic coastlines back in the 1960s, it was not until the 1980s before a proper misfit criterion for RPM modeling was proposed [Hellinger, 1981], critical insight into the uncertainties of rotations were obtained [Molnar and Stock, 1985], and a rigorous spherical regression technique to solve for rotations and their uncertainties was developed [Chang, 1987, 1988]. Those techniques, now standard within the plate reconstruction community, have had almost 20 years to mature. In contrast, the task of finding APM is arguably more complicated and still under active development [e.g., Andrews and Gordon, 2003; Harada and Wessel, 2003; Kumar et al., 2004; Wessel et al., 2004]. In 1997 we proposed “hot-spotting” [Wessel and Kroenke, 1997] as a novel way to evaluate APM models and determine optimal hot spot locations, while at the same time Harada and Hamano [2000] were developing their Polygon Finite Rotation Method (PFRM). We are now learning how to couple these complementary techniques and determine rigorous uncertainty estimates. As we will show below, we have made much progress toward developing such an approach, and we will present herein a preliminary Pacific APM with estimated covariance matrices reflecting the uncertainties for each rotation.

Recently, O'Neill et al. [2005] presented a new technique for modeling absolute plate motions over fixed or moving hot spots, including the rigorous estimation of the uncertainties in the model rotations. Building on the RPM modeling approach of Chang [1988], they employ a modified Hellinger criterion of fit to the general case of hot spot trail reconstructions and present a revised set of Indo-Atlantic rotations for the last 120 Myr. Our technique shares some similarities with theirs but differs considerably in many aspects; we will compare the two techniques following the development below.

2.1. Absolute Versus Relative Plate Motions

In studies of RPM, the data used are sets of conjugate magnetic isochrons identified in marine magnetic anomalies and orientations of fracture zones [e.g., Cande and Stock, 2004]. The model in question is a finite rotation that relocates points defining (possibly disjointed) isochron segments on plate A onto plate B such that the rotated and fixed sets of points closely approximate great circle segments (Figure 2a). These great circle segments can themselves be expressed in terms of the unknown rotation. The orthogonal distances of both sets of points from each great circle segment express the misfit, and the rotation that minimizes this misfit is sought [e.g., Chang, 1988]. Because the conjugate data can be optimally superimposed using single, total reconstruction rotations, it was natural to define the model in terms of total reconstruction rotations and not stage rotations.

Figure 2.

(a) Modeling of RPM. Identification of a key magnetic isochron and its crossings with fracture zones in ship track measurements yields conjugate points on two plates, A and B. A rotation R is sought so that when points on plate A are rotated to plate B (dashed circles), the fixed and rotated sets closely match great circle segments and their deviation from these segment is minimized. As each great circle segment can be parameterized in terms of R, only three parameters need to be determined [e.g., Chang, 1988; Hellinger, 1981]. (b) Modeling of APM. Seamounts are considered points that must approximately lie along small circles. Given a trial pole location (P′), a revised pole location P is found such that the misfit between seamounts and the local best-fitting small circle is minimized [e.g., Koppers et al., 2001].

In most studies of APM, the data used are surface expressions of assumed hot spot volcanism (seamount and island chains and their measured ages). The traditional approach is to model coeval segments of seamount chains as small circles about stage poles of rotation (Figure 2b); such rotation poles are found by minimizing the distances from each seamount to its locally best-fitting small circle about a candidate pole [e.g., Koppers et al., 2001]. The opening angle is harder to estimate since the ends of each segment can be difficult to determine. Typically, the opening angles are found by trial and error. Given the age range of a particular set of copolar segments, opening rates can be determined. Because the data portray small circles, it was natural to define the model in terms of stage rotations and not total reconstruction rotations.

Although useful as an exploratory technique, the traditional APM modeling approach has many limitations: (1) short segments, possibly reflecting APM changes, are difficult to identify and correlate across several chains; (2) short small-circle segments become indistinguishable from great circles and hence angular distances to poles cannot be accurately determined; (3) without easily identifiable kinks that can be correlated between chain segments, ages are needed to define segments and make the correlation, and these are often lacking; (4) unlike RPM modeling, no rigorous approach for estimating APM uncertainties exists; (5) the method breaks down if hot spots are moving during a stage interval; and, finally, (6) no consistency check for hot spot locations is provided. The last shortcoming was addressed by Wessel and Kroenke [1997] who developed a method to derive optimal hot spot locations from seamount data if the APM is known, whereas Harada and Hamano [2000] made many of the other issues irrelevant by introducing a new technique to determine total reconstruction rotations provided hot spot locations are known. In this paper, we improve the modeling of APM by combining these two complementary methods into a self-consistent hybrid technique. The hybrid technique allows us to (1) determine the best location for hot spots, (2) construct a high-resolution APM model, and (3) introduce estimated covariance matrices for the uncertainty in each rotation. In describing the technique, we will present a preliminary self-consistent Pacific APM with confidence regions for each rotation pole and reconstructed points along the chains. We then discuss how this new technique compares with the technique recently presented by O'Neill et al. [2005]. Finally, the new model is contrasted with traditional Pacific APM models, and the implications of the model for drift within the Pacific hot spot group and the origin of the Hawaii-Emperor bend is addressed.

2.2. Classes of Models

Before describing the hybrid method, it is useful to distinguish between different classes of APM models and determine the types of data available to constrain the various model parameters. With M being the number of hot spot chains considered, for each chain i there are Ni undated seamounts, located at

equation image

where equation imagei,j represents either the geographic (θ, λ) or Cartesian (x, y, z) coordinates of a seamount center. In each chain, there may also be Gi dated seamounts, i.e.,

equation image

where equation imagei,j are the coordinates of a dated seamount center and τi,j its radiometrically determined age in Myr, with individual uncertainty δi,j. Finally, we have specific times when plate reconstructions are desired. These are the prescribed “knot-times” tk, k = 1, K and will normally correspond to times of certain key isochrones used in RPM studies. Such times are preferred because, ultimately, we must use a common temporal reference frame when building global plate circuits and propagating model covariances.

Several sets of parameters need to be determined. The actual locations of each hot spot at the present time (t = 0) are

equation image

These are not known, but may be estimated. The positions of hot spots at earlier times tk (i.e., if hot spots are moving over time) are given by

equation image

where Di,kT are finite rotations that rotate the i'th hot spot's present location to its position at time tk. Note we have used the inverse rotation Di,kT since we are rotating from the present hot spot to the past location. Each finite rotation has three parameters: location (θ, λ) of the rotation pole and the opening angle ω; however, since each rotation only applies to a single hot spot, we can require the pole to be 90° away from the hot spot, thus reducing Di,kT to two parameters. We also seek the APM of the plate over the fixed background mantle; it is parameterized by the K rotations Pk = P(tk). Thus the location of a seamount formed by hot spot i at the time tk is theoretically given by

equation image

We will provide initial best guesses for the present hot spot locations, equation imagei (0). It is also understood that the APM opening angles are functions of the knot time, i.e.,

equation image

The inverse function is also useful:

equation image

Below, we distinguish between four cases of APM scenarios considered in the literature.

2.2.1. Case 1: Hot Spots Are Fixed at Known Locations

This is the classic case where Di,kT all equal I (the identity matrix), implying there is no plume drift. Furthermore, if we can assume the hot spot locations to be known, then the PFRM technique of Harada and Hamano [2000] will solve for the optimal APM model. Thus this scenario has already been fully described and solved.

2.2.2. Case 2: Hot Spots Are Fixed at Unknown Locations

Here, we realize that our initial hot spot locations are likely to be incorrect and we thus seek to optimize their locations as part of the APM model. The current approach to this case will be the hybrid technique presented in this paper, in which a hot-spotting step is used to iteratively refine the hot spot locations.

2.2.3. Case 3: Hot Spots Are Moving Along Known Paths

Here, we must consider individual hot spot motions. In this case, the drift Di,kT are prescribed by mantle circulation models [e.g., Steinberger, 2000]. Because Di,kT are then considered known, they are simply used in (4) to give the correct starting points (hot spots) for a given age [O'Neill et al., 2005]. PFRM will yield an APM consistent with the prescribed drift. However, the t(ω) relationship must be modified to account for the drift.

2.2.4. Case 4: Hot Spots Are Moving Along Unknown Paths

In this final case, both hot spot drift and plate motions are considered unknown. Currently, no technique exists that can simultaneously solve for both Di,kT and Pkt. Because the two components enter as a product Pi, kT = PkT · Di,kT, it may be possible to separate along-chain plate motion components from across-chain hot spot motion components geometrically, but ages will clearly be needed to separate along-trail hot spot drift from plate motion.

In this paper we will consider the second case: Stationary hot spots at unknown locations. While, at present, there is paleomagnetic support for some hot spot motion (cases 3–4) prior to the HEB event, we will restrict the scope of this paper in order to present the detailed development and discussion of the simple, fixed-hot spot case, before extending the method to the more general case of moving hot spots and, as a consequence, introduce more unknown parameters.

2.3. Geometric Constraints

Our starting point is the Pacific database of seamounts compiled by Wessel and Lyons [1997] based on the Geosat/ERS-1 altimetry-derived gravity grids of Sandwell and Smith [1997]. We identify all seamounts that may possibly be part of a hot spot chain and assign them a unique chain ID number (Figure 3). For instance, all the seamounts in the data set that appears to belong to the Hawaii-Emperor chain were given an ID of 1. Note that although numerous chains are identified, not all will be included in the APM modeling.

Figure 3.

Seamount data constraints generated from the Wessel and Lyons [1997] seamount database. Unique chain ID numbers have been assigned to each seamount (as color-coded). The PFRM works by brute-force grid search of all possible rotations. For each trial rotation, here exemplified by the rotation pole P at (70°N, 66°W) with an opening angle of ω = 18°, the M hot spot locations (red stars: HI, Hawaii; LV, Louisville; FD, Foundation; PC, Pitcairn; CR, Caroline; and CB, Cobb) are reconstructed (yellow stars). If two or more of these reconstructed points fall on or very close to their respective seamount chain, the rotation is included in the further analysis. The cluster of points near P (red dots) represents the poles of all rotations that passed the test for all opening angles in the 17° ≤ ω ≤ 19° interval. Additional chains (shown but not used in the modeling) are Marquesas (MQ), Samoa (SA), Kodiak (KO), Bowie (BW), Austral-Cook (AC), Society Islands (SO), southern (TS) and northern (TN) Tokelau, southern (SW) and northern (NW) Wake, Tuvalu (TU), Tuamotu (TO), Musicians (MU), and Lilioukalani (SS).

2.4. Acceptable Rotations and the Criterion of Fit

Because of uncertainties in ages and the finite width of seamount chains, the location of a hot spot must be associated with an uncertainty region. The size of this region is likely to be smaller for chains with active or recently active volcanism but may be larger for extinct hot spots or hot spots with very modest output during the last several million years. We implemented these uncertainties by subjectively specifying a radius of confidence for each hot spot (Table 1). While it might ultimately be more correct to assign elliptical confidence regions for each hot spot, here we will use circular regions, as the ensuing calculations are simplified considerably. As Table 1 reveals, we consider the locations of some hot spots (e.g., HI) to be better known than others (e.g., LV).

Table 1. Hot Spot Locations
TagNameθλr, km
HIHawaii19°12′N155°03′W25
LVLouisville52°24′S137°12′W75
FDFoundation37°54′S110°00′W25
PCPitcairn25°54′S129°15′W25
CRCaroline4°48′N163°39′E50
CBCobb45°33′N129°15′W75

Unlike the situation for RPM modeling and traditional APM (Figure 2), our data are not points or lines; rather they are the finite extent of surface volcanism along chains (Figure 4). We have therefore taken the approach that the entire outline of a seamount (i.e., its base contour) should be considered when determining acceptable rotations.

Figure 4.

In our technique, a hot spot (H) is rotated by a trial rotation to yield a reconstructed point (X). If X falls within the base contour of a seamount or within the distance r of the nearest seamount, we consider the rotation viable, provided the same is true for at least one more chain for the same rotation.

Given our initial selection of M hot spot locations (HI, LV, FD, PC, CR, CB in Figure 3) we seek to determine all finite rotations that are consistent with the given chain geometries. Figure 3 shows contours of the distance from the nearest seamount for each of the six chains modeled in this paper. A trial rotation is deemed to “fit” if it moves a hot spot location to a reconstructed point that is either on or within a prescribed distance ri from the associated seamount chain (Figure 4). Thus the circular uncertainty in a hot spot location is implicitly treated by allowing rotations to move a hot spot close to, but not necessarily onto, a seamount chain. Because seamount chains may have gaps and sometimes cover different age ranges, we initially will accept all rotations that fit at least Mc chains; typically, Mc ≥ 2. We use a grid search to examine all possible rotation pole coordinates for a wide range of rotation opening angles. Depending on the grid spacing we typically end up with 107–109 trial rotations, most of which will fail the fit criterion. The N combinations of (θj, λj, ωj), j = 1,N that do pass the fit criterion make up the initial population of rotations that we will analyze to determine representative (i.e., average) rotations and their confidence regions. The number of chains fit and the density of rotation poles as functions of pole locations are displayed in Figure 5.

Figure 5.

(a) Location of all rotation poles, color-coded by the maximum number of chains (M) the rotations fit for opening angles in the 0–45° range. Most lie near traditional estimates of poles, which tend to cluster in North America. Note that all the outlying poles to the south occur when the opening angle (ω) approaches zero (for ω = 0° all poles fit perfectly). (b) Density of rotation poles. Hot colors indicate a higher concentration of poles.

Next, we analyze the rotation pole locations as functions of opening angle. Initially, we choose to group rotations by a simple sampling scheme in which we prescribe a desired sampling interval, Δω, and a window half-width, Ω. Then, for each nominal rotation angle ωk = kΔω, k = 1, K, we determine the Nk rotations in the population that have opening angles ωj such that ωk − Ω < ωj< ωk + Ω. Here, we have used Δω = 2° and Ω = 1°. To minimize errors in determining the average rotation, we parameterize each of these rotations using quaternions qj, which is a 4-D Cartesian vector that has a scalar part and a 3-D vector part [e.g., Chang et al., 1990]. To estimate the mean rotation from our set of experimentally determined rotations we calculate the mean components of the quaternions,

equation image

and convert this quaternion back to a mean rotation pole equation imagek = (equation imagek, equation imagek) and mean opening angle equation imagek [e.g., Gramkow, 2001; Moakher, 2002]. By averaging each group of rotations thus (using non-overlapping opening angle windows) we obtain a smooth, representative set of averaged rotations equation imagek = Rot(equation imagek, equation imagek, equation imagek), i = 1, K, which we call the Absolute Plate Path (APP) model. This path is not unique: a more or less smooth APP can be obtained by choosing a different Ω. Our Ω was chosen by examination of the residual misfits to the model, which are strongly related to the average width of the seamount chains (and we return to this point later in the presentation). The APP model is displayed in Figure 6 (blue dots). No time information was used to determine the APP (except in providing the implicit zero-time hot spot locations). However, age constraints will now be required to convert the APP into a true APM model, allowing predictions of an age-progression along each chain.

Figure 6.

(a) All rotation pole longitudes displayed as a function of the opening angle (ω). Superimposed line with blue dots is the smooth APP model obtained by averaging the rotations, using the initial 2° windows. (b) Same for pole latitudes. Note that latitudes appear to be more stable than longitudes and are confined to a relatively narrow band of latitudes (except when ω goes to 0°). Several longitudinal outliers still remain in the population of rotation poles. The number of chains (M) fit for each opening angle is shown by the color bar (see Figure 5a for scale).

Although not shown here, we can display components of all rotations that fit any of the possible combinations of chains. Because not all seamounts used may be part of a hot spot chain, we sometimes find spurious rotations for some angle intervals where the total number of chains fit is small or the chain geometry is ambiguous. For example, the higher scatter in rotation pole longitudes for the interval 17.5° < ω < 27.5° in Figure 6 largely comes from the combination HI/LV/CB, reflecting mostly uncertainties in the Cobb chain. Examining individual combinations may allow us to exclude outliers.

2.5. Hot Spot Locations and Hot-Spotting

While we carefully picked our initial hot spot locations, they are not optimal with respect to the APP. Wessel and Kroenke [1997, 1998] showed that one cannot separate the hot spot locations from the APP model, i.e., they are intimately related. Consider the analogy with fitting the line y = ax + b to many data points: Once the best fit is found we know the slope a and intercept b. This implicitly fixes x0 = −b/a where y = 0; one cannot specify x0 independently after determining a and b. Thus we run hot-spotting on our geometric constraints using the preliminary APP model determined above. In most cases, the optimal location for the hot spot will be slightly offset from our initial location, and we update the hot spot locations accordingly. Given these modified hot spot locations we must repeat the steps above (since the distances between hot spots and seamounts have changed) and obtain a modified APP. We iterate until the solution stabilizes, yielding a final set of hot spot locations that are internally consistent with the APP rotation parameters. For chains that are too short to benefit from hot-spotting we adjust hot spot locations along-track, if this shift is implied by the age-distance relationship (see below).

2.6. Constraints From Observed Radiometric Ages

Age constraints are based on a Pacific basin seamount age compilation [Clouard and Bonneville, 2001], revised to include new ages from the Louisville [Koppers et al., 2004] chain and the new ages for the Hawaii-Emperor bend region [Sharp and Clague, 2002]. Each age determination is assigned a chain ID. Using the APP, we determine opening angles to each dated volcanic center. Since such angular measures (unlike distances in km along chains) are equivalent for all chains and independent of the distance to the rotation pole, we can plot the observed ages versus the estimated opening angles for all chains on a single graph (Figure 7). We then can determine a best-fit linear spline to the distribution of points. This curve relates opening angle and time (i.e., equations (6)(7)), allowing us to calculate opening rates and obtain a true APM model. The gray band approximates the 95% confidence limit for the spline and is used to convert the uncertainty in observed age for a given angle into uncertainty in angle for a given age, such as our preferred Chron ages (see the Chron 5d example in Figure 7). Given these new ranges of opening angles we form new groups of rotations and determine new averages for each group (using equation (8)). Because the age uncertainties translate into a wider range of opening angles for each group, additional rotations with a wider range of opening angles than found in Section 2.5 will be included in any group, and this leads to increased along-track uncertainty in the final set of mean rotations. Furthermore, because some groups of opening angles now overlap our mean rotations are not always independent; this will be discussed later. The new averages prove the basis for our preliminary, fixed-hot spot APM model for the Pacific.

Figure 7.

A t(ω) plot for the six Pacific seamount chains used in APM modeling. The linear spline routine determined that two sections were sufficient back to ∼70 Ma (solid line) and automatically picked the knot point (at ∼25°). Extending the model (heavy dotted line) to the oldest seamounts along Emperor and Louisville (squares) would likely yield a three-segment model. The gray envelope represents 95% confidence level for the spline, which clearly exceeds analytical uncertainties in dating; however, the ω-coordinates are uncertain due to modeling limitations and possible hot spot motion. Rather than pick angles every 2°, we select desired times (e.g., Chron 5d) and determine the corresponding opening angle (∼16.8°) and the 95% uncertainty range (e.g., heavy solid line for ∼14–20°).

2.7. Confidence Regions on Rotation Poles and Reconstructed Points

Since, for each opening angle group k, we find the entire population of rotations compatible with the geometric data, our analysis of this population determines not just the mean rotation equation imagek but also provides an estimate of the variations about these means. Following Chang et al. [1990] we consider the population of Nk rotations Rj as small rotations ΔRj followed by the average rotation equation imagek. In other words,

equation image

We can then solve for the small rotations

equation image

That is, each compatible rotation is presented as a small perturbation to the mean rotation. We use pseudo-vectors rj to parameterize these small rotations. Following Chang et al. [1990] we must use a pseudo-vector to calculate Ck = Cov (equation imagek) instead of Cov (equation imagek) as the latter is undefined. Thus we calculate the covariance matrix from the distribution of the pseudo-vectors rj, i.e.,

equation image

where Xk is the matrix of all the Nk pseudo-vectors and equation imagek is the mean pseudo-vector in Xk. Note it is customary to publish Cov(rk) as a list of the 8 values equation imagek and akgk which are related to Ck via the relation

equation image

typically with gk = 10−5; here, equation imagek are all assumed to be 1 since we are simply describing the variance of the population. The covariance matrix can be used to define an ellipsoidal confidence region for the rotations r satisfying

equation image

The radial projection of this ellipsoid onto the surface of the Earth gives a confidence region S(θ, λ) for the pole of rotation, and the upper and lower limits on the opening angle may be contoured within this boundary (Figure 8); Appendices A and B describe how these items are derived. The covariance matrices are also used to estimate the uncertainties in projected points along the chain (see Appendix C for details). The predictions of our preliminary APM model for Pacific hot spot chains, with confidence regions on reconstructed points, are displayed over the Pacific bathymetry grid in Figure 9. The parameters of the APM model are listed in Table 2.

Figure 8.

Rotation poles and their confidence regions for the Pacific APM model shown in Figure 9. The large ellipses for the youngest three rotation poles are not shown. Star indicates pole locations, and their colors relate to angle or age. (a) Lower surface of opening angles (as deviation Δω from the mean angle). (b) Same, but for upper surface of opening angles.

Figure 9.

Pacific APM model fit with 95% confidence ellipses. Solid lines indicate predictions for the six chains used in the modeling (with red, crossed circles marking the corresponding hot spots), while open lines with dashed ellipses are used for predictions for other chains (with white, crossed circles marking the hot spots), all for ages younger than ∼70 Ma. Older APM predictions (double-dashed lines) are from Euler stage poles based on manual trial and error [Kroenke et al., 2004]. Orange arrows highlight seafloor evidence for short-lived, northward plate motion around ∼23–27 Ma and again around ∼13–17 Ma. The locations of the HI hot spot and both the HEB and the LV bends (yellow arrows) were determined and assigned 95% confidence ellipses (yellow), allowing for the determination of a bend separation of 72.61±1.10° (orange dashed lines). Heavy, yellow line near LV with dashed, copolar lines on either side indicates the implied distance (with 95% uncertainty) the LV should be from HI. Sector of white annulus around the youngest Louisville seamount represents likely region for the hot spot. Optimal pick for LV agrees well with both location constraints. Orange [Raymond et al., 2000] and green [Wessel and Kroenke, 1997] squares show previous estimates of the Louisville hot spot. White arrows indicate possible HEB-contemporary bends, but new dates do not support this speculation [Koppers and Staudigel, 2005]. See caption of Figure 3 for chain abbreviations.

Table 2. APM Finite Rotations and Their Covariance Matricesa
θλtωabcdef
  • a

    Opening angle ω is in degrees, and t is in Myr. The covariance matrix for each rotation R is defined in equation (12), with af given in radians, g = 10−5, and equation image set to 1.

49°11′N49°49′W0.780.911.681−0.9573.1722.487−3.4757.114
52°55′N55°18′W2.581.572.064−1.1956.6733.992−8.50517.35
58°54′N66°50′W5.895.153.779−0.6825.8225.566−8.40124.45
62°56′N69°00′W8.868.653.2160.7536.4843.496−10.0631.95
67°17′N61°53′W12.2911.47.5062.8303.43014.2574.29940.27
69°43′N65°38′W17.4717.917.533.9944.624−3.177−2.69411.70
69°16′N68°48′W24.0620.822.51−9.45732.4120.78−51.87102.1
67°40′N68°12′W28.2825.211.12−5.62721.238.456−16.2636.47
66°21′N66°06′W33.5428.013.34−8.86721.1810.56−11.6920.57
65°21′N63°12′W40.131.312.06−2.7006.4215.663−10.8924.19
63°40′N65°46′W47.9134.316.3911.9845.34−7.609−15.5113.47
60°45′N69°27′W53.3536.016.0720.71100.5−6.286−18.6910.02
55°10′N73°50′W61.139.224.9642.25139.5−10.43−23.9410.51
52°24′N75°50′W67.6741.013.6316.0378.01−5.967−12.859.576
50°27′N78°47′W7644.3
47°07′N81°11′W8448.2
46°21′N79°55′W9655.9
51°13′N79°20′W11059.6
53°27′N82°47′W12571.2
59°05′N98°21′W14075.5

2.8. Comparison With Other Techniques

Since both our technique and that of O'Neill et al. [2005] (herein called the OMS method) overlap in scope, we will discuss some of their similarities and differences. However, because the approaches taken are so different and have been applied to different ocean basins a direct comparison of model parameters is not possible.

Unlike classical APM modeling techniques, both techniques find total reconstruction rotations and provide rigorous covariance matrices that describe the uncertainties in the model rotations. Furthermore, the OMS method works equally well with both fixed and moving hot spots (using prescribed past hot spot locations as a function of time), whereas our method currently requires past hot spot locations as a function of opening angle; iterations will be required to find a compatible angle-time relationship (equation (6)). The current locations of hot spots are input parameters for both methods. However, for fixed hot spots our method will adjust these locations for an optional fit using the hot-spotting test; this is not done in OMS. The OMS method needs to linearly interpolate between dated samples to determine points along the chain for the times of interest. The large uncertainty in ages therefore propagates directly into the locations chosen to determine the rotation. Our method does not need to use uncertain ages to determine points to be used in determining rotations; we first use all points to determine an accurate motion path and only then analyze the stacked ages to determine the history. The OMS technique must also make the approximation that the data can be considered great circle segments; however this is only likely to introduce minor errors. Finally, while our method produces a smooth APM model by virtue of averaging, the OMS technique has no mechanism for smoothing and consequently can generate a non-smooth age progression along the hot spot track [O'Neill et al., 2005].

3. Discussion

Although there are numerous implications of this APM model that we would like to address, we will discuss only some aspects here. We note that our APM provides a very good fit to the six chains that provide data, as well as reasonable predictions for other Pacific chains not included in the analysis (Figure 9). In addition to the large change at Chron 21 seen in the Hawaii-Emperor chain, we note apparent geometric support for similar changes in other, poorly dated chains (white arrows). However, several of these are not supported by newly acquired age data [Koppers and Staudigel, 2005], underscoring the need to use both ages and geometry when studying plate motions.

We further note the presence of two northward jogs (orange arrows) in several chains during the ∼13–17 Ma and ∼23–27 Ma intervals (clearly visible in the Hawaii and Cobb chains, and possibly represented by a small offset in the Louisville chain near 161.5°W as first suggested by Lonsdale [1988]). We have previously correlated these jogs with major circum-Pacific tectonic events [Sterling et al., 2000]. Here, we suggest that these are real manifestations of changes in APM and that our technique should have high enough resolution to reveal such second-order perturbations that appear related to plate boundary forces [e.g., Lithgow-Bertelloni and Richards, 1998]. It is reasonable that a high-resolution APM should reflect these perturbations; certainly RPM models reflect rapid changes of plate motion [e.g., Cande et al., 1995]. These two jogs are a robust feature of many of our APP modeling results, but are not well resolved in our APM.

We explore this in more detail in Figure 10, which zooms in on the Gulf of Alaska. Here, we show both an earlier APP [Wessel et al., 2004] with its geometric confidence ellipses (i.e., uncertainty in a projected point given an opening angle in the APP) in red and the present APM with its complete confidence ellipses (i.e., uncertainty in a projected point for a given age in the APM) in black. Clearly, the APM is much smoother than the APP, and there are two reasons why this is so: (1) The uncertainties in observed ages are mapped into wider opening angle intervals for averaging (Figure 7), which results in overlapping windows and acts as a low-pass filter, and (2) unlike the earlier APP, the present modeling explicitly assigns a confidence region to each hot spot. This also has the effect of blurring the APP further.

Figure 10.

Close-up of the predicted APP (red) and APM (black) tracks for the Cobb seamount chain, with corresponding confidence regions. Yellow circle indicates uncertainty assigned to the Cobb hot spot location. Dashed lines and ellipses are predictions for the Kodiak-Bowie chain (not used in the modeling). Whereas the postulated northward jogs at ∼23–27 Ma and 13–17 Ma are resolved in the APP, they are not resolved at that level of precision in the APM due to the higher age scatter in this range (Figure 7) and the ambiguity of geometry (see longitude pole scatter around ω ≈ 20° in Figure 6).

While the present APM is conservative and more realistically handles the uncertainties, it is intriguing to note that where those two aforementioned jogs occur and the APP changes appear to approach the magnitude of the angular change at the HEB, we see numerous linear, volcanic ridges perpendicular to the APP direction at the time, suggesting they might have formed at leaky transforms and fracture zones in relatively young lithosphere in response to extensional stresses caused by such rapid APM changes. Note, however, that while these jogs are well resolved in our APP models [Kroenke et al., 2004; Wessel et al., 2004], the confidence regions on projected points of a specified time in the APM are too large to confirm this hypothesis at present. Also, the strongest evidence for the jogs comes from the Cobb chain whose geometry is poorly determined (exemplified by the large scatter of rotation pole longitudes in Figure 6). More and better age determinations are needed to reduce the ambiguities for this interval of plate motions in the Pacific and hopefully bring the APP and APM into closer agreement.

Many models of Pacific APM have included the Easter-Line Islands (Tuamotu) chain in order to have three seamount chain constraints during the Emperor stage [e.g., Duncan and Clague, 1985; Morgan, 1971]. In fact, when Harada and Hamano [2000] first published their PFRM method they included this chain. Here, we have excluded it for two reasons: (1) It is not clear if the southern Line Islands form a continuation of the Tuamoto Plateau/chain beyond an apparently HEB-coeval bend (Figure 9), and (2) the age support along the chain is somewhat inconsistent. We note, as a consequence of excluding it, our rotation poles in general match Harada and Hamano's [2000] poles in latitude, but longitudes are offset by a few degrees to the east.

The locations of both the HEB and the LV bends were determined by inspection of the bathymetry grid and empirically assigned 95% confidence ellipses (Figure 9; see caption for details). From these parameters we determined the great circle separation between the bend points to be 72.61 ± 1.10°. If plates are rigid and the hot spots have not drifted significantly since HEB time (which is supported by new paleomagnetic analyses along the Hawaiian chain [Sager et al., 2005]), then the present locations of HI and LV hot spots (red circles) should be separated by the same distance. The thin, dashed orange line from HI to LV terminates at a heavy yellow line near LV, with dashed, copolar yellow lines on either side, which indicates the implied distance (with 95% uncertainty) that LV should be separated from HI. Furthermore, the white sector of an annulus around the youngest (1.112 Myr) Louisville seamount [Koppers et al., 2004] is our estimate of possible location for the hot spot solely based on reasonable perturbations of age and likely plate velocities. Our optimal pick for the LV hot spot location (red circle) is within the confidence zones implied both by geometry and ages. However, it is farther east than most conventional choices such as the hotpot location preferred by Raymond et al. [2000] (orange square) but not as far south as the Hollister Ridge location (green square) originally suggested by Wessel and Kroenke [1997]. While both of these earlier estimates are slightly outside our confidence half-annulus (Raymond et al. [2000]'s location is slightly “upstream” of the 1.112 Myr old seamount whereas the Wessel and Kroenke [1997] location is slightly too far from Hawaii), we cannot say if this is statistically significant. The general paucity of Louisville chain volcanism since ∼12 Ma makes it difficult to assess the true location of the hot spot.

The misfit between the observed ages and the modeled t(ω) relationship can be converted to spatial misfits by back-tracking dated seamounts to their presumed origin. We can then rotate all these relocated points, i.e., the cluster for each hot spot, to a common reference point. We can also do the same for the undated seamounts by using the determined opening angles as proxies for age. The resulting combined scatterplot is displayed in Figure 11. As expected, the scatterplot for undated seamounts shows very low along-track scatter (inconsequential, but serves as a consistency check) but the across-track scatter represents the geometric misfit, reflecting both the finite widths of seamount chains as well as inadequacies in the geometric modeling. The 1-σ standard deviation for across-track scatter is ∼50 km. While a comparison between the scatter of the two populations is tempered by the uncertainties in the t(ω) relationship, it appears that the geometry of trails is generally better modeled than the age progressions along them.

Figure 11.

Overall scatterplot for back-tracked dated (red) and undated (green) seamounts for all chains, referred to a common reference frame. The ∼50 km across-track r.m.s. for undated seamounts reflects the level of APP fit, while the much higher along-track r.m.s. for the dated seamounts reflects the APM fit. We therefore use 2° angle windows for averaging rotations as this reflects the track width scatter.

To illustrate geometric fit, we can use APM models to project seafloor bathymetry in such a way as to depict what Pacific hot spot chains would look like if there was only one pole of rotation (at the North pole) and each hot spot was located at the Equator (Figure 12). With our preliminary APM model, almost all seamounts in the Hawaii-Emperor and Louisville chains project within a narrow, 150-km-wide corridor (WHK05 in Figure 12). For comparison, the influential APM model of Koppers et al. [2001] is shown, together with the classic model of Duncan and Clague [1985] and our previous, manually determined model [Wessel and Kroenke, 1997]. A misfit in geometry can systematically affect the calculation of opening angle (ω) and therefore degrade the t(ω) relationship and falsely imply hot spot motion, even if hot spots are fixed. Thus, in order to assess true hot spot motions, it is imperative to have an optimally fit model.

Figure 12.

(a) Seafloor near the Hawaii-Emperor chain projected to Equator using the Pacific APM models of Koppers et al. [2001] (KMMS01), Wessel and Kroenke [1997] (WK97), Duncan and Clague [1985] (DC85), and our APM derived in this paper (WHK05). Thin horizontal lines are 150 km apart. Because of model inconsistencies, particularly in KMMS01, significant parts of the chain are outside this corridor. (b) Same projection for the seafloor near the Louisville chain. Here, for both KMMS01 and DC85, significant parts of the chain are outside this corridor. The youngest seamount (“Lonsdale's seamount”) at ω = 1° is inside for most models.

Koppers et al. [2001] determined their APM model for the Pacific plate and used it to conclude that “relative motion between hot spots may be required to reconcile the observed age progressions with the predicted plate velocities from their modelled Euler poles.” Figure 12 shows that the model presented by Koppers et al. [2001] is limited in its ability to fit the observed geometry, but in fairness to them their main concern was to address age progressions all the way back to 140 Ma. However, because they conclude that inter-Pacific hot spot motion may be required, we need to examine their model in some detail. First, we ran hot-spotting, which resulted in the cumulative volcano amplitude (CVA) image in Figure 13. The maximum CVA location represents the optimal location for the Hawaiian hot spot, which their model places on Maui, more than 200 km northwest of active volcanism at Loihi/Kilauea and their own assigned hot spot location (A. A. Koppers, personal communication, 2003). Accordingly, their assigned hot spot location for Hawaii is not consistent with their APM model, further demonstrating that the two cannot be determined independently.

Figure 13.

Hot spotting using both the published list of seamounts and the Pacific APM rotations from Koppers et al [2001]. The optimal hot spot location for this APM is on Maui, more than 200 km northwest of their assumed hot spot location (bull's-eye) and active volcanism (green triangles). The APM model is therefore not self-consistent.

One key concept in establishing hot spot motion is the “goodness of fit”. Analytical precision of laboratory-determined radiometric ages is now very high [e.g., Koppers et al., 2004], yet uncertainty enters in the form of not knowing exactly what has been dated, given the longevity of volcanism at some hot spot seamounts [e.g., Pringle et al., 1991]. Thus, although possible, it is as yet unresolved if drift between Hawaii and Louisville hot spots is truly required by the age data (i.e., separate from being required by other data, such as paleolatitudes [Tarduno et al., 2003]). In other words, how tight must the 95% confidence band on t(ω) be in order to represent a significant fitΔ

As demonstrated in this paper, our preliminary modeling of the Pacific APM based on both the hot-spotting [Wessel and Kroenke, 1997] and Polygon Finite Rotation Methods [Harada and Hamano, 2000] does allow numerous Pacific chains to be fit simultaneously, apparently without requiring significant plume drift. Our model does not address the possibility that the entire Pacific hot spot group has moved together in a copolar fashion (which would satisfy the Emperor paleolatitudes). O'Neill et al. [2005] found no significant difference between a fixed and moving Indo-Atlantic hot spot reference frame for the last 80 Ma (the duration of our model), but concluded that drift of Pacific plumes is required to reconcile their Indo-Atlantic model with the Pacific. However, to satisfy paleolatitude estimates [Tarduno et al., 2003] they had to use an unrealistically low conduit viscosity (1016 Pa) for Hawaii. We will assess the key issue of the APM of Africa using our technique in a separate manuscript (Y. Harada and P. Wessel, manuscript in preparation, 2006) since it has been suggested that some, if not all, of the mismatch between the Indo-Atlantic and Pacific hot spot reference frames might be related to true polar wander [e.g., Andrews et al., 2004; e.g., Gordon and Horner-Johnson, 2004; Harada and Wessel, 2003]. Continued refinement of the plate circuits will hopefully reduce the sensitivity of projected results to the choice of plate circuit [Steinberger et al., 2004].

While the hybrid modeling offers a new way to study absolute plate motions, there are several aspects of it that can be improved: (1) Our experience so far has indicated that the exact locations of hot spots do influence the rotation parameters we obtain, hence it is important to optimize the hot spot locations. Unfortunately, hot-spotting can assist in this task only for chains with at least one significant kink, such as for Hawaii and Louisville. Other, younger chains must have their hot spot locations adjusted on the basis of trial and error, possibly assisted by a grid search. (2) The averaging of rotations (equation (8)) is sensitive to outliers as it is a least squares operation. While care is taken to exclude groups of poles that clearly are outliers, the whole process could be made more robust by using statistics less sensitive to outliers, such as spherical medians [e.g., Fisher, 1985] or, ultimately, a technique based on the least median of squares criterion [Rousseeuw and Leroy, 1987] to flag outliers and exclude them from the rotations to be averaged. (3) While we have attempted to quantify the 95% confidence limit on the t(ω) relationship, it is not a formal determination. It is possible that a better parameterization using smoothing splines could allow proper spline confidence bands to be estimated [e.g., Wahba, 1990]. The merits of an alternative spline basis (e.g., quadratic instead of linear) warrant further exploration as well. (4) We would like to extend the hybrid modeling further back in time, but because of apparent chain overlap (e.g., particularly in the Ratak-Gilbert-Ellice chain), we will need to modify our procedures so that a seamount can simultaneously “belong” to more than one hot spot chain. Finally, (5) while our method allows for moving hot spots (given prescribed hot spot paths), it presently requires such paths to be given as functions of opening angle and not time. Because time and opening angles are coupled (Figure 7) we must iterate to obtain a revised ω(t) curve since the prescribed hot spot drift will affect the regular t(ω) relationship. Further investigation is needed to ensure proper convergence to an optimal fit.

Because the knot-times tk were taken to match well-known Chrons used in RPM studies, several of our APM finite rotations are not statistically independent since there is overlap in the ranges of opening angles used in the averaging. If a minimal but truly independent set of rotations is desired, one could simply determine a new set of knot-times spaced so that no angle overlap takes place. However, obtaining a rotation that could be added to a RPM rotation for a given Chron would then have to be obtained by interpolation, bringing us back to our starting point. We will continue to explore better ways to extract the maximum number of independent rotations; for now we note Table 2 is a useful starting point.

Several aspects of absolute plate motions were not addressed in this paper. For example, considerable debate is ongoing with regards to plate circuits and whether there has been motion between the Pacific and Atlantic groups of hot spots [e.g., Andrews and Gordon, 2003; Harada and Wessel, 2003; Raymond et al., 2000; Steinberger et al., 2004]. As this paper has focused on method development with application to a single tectonic plate (the Pacific), we are unable to address these important issues here. However, future work will extend our hybrid modeling technique to other plates and hopefully allow us to engage in that important debate. For the same reasons, we have not attempted to model directly the latitudinal shift that paleomagnetic evidence implies [Tarduno et al., 2003], since our modeling of a single plate provides no constraints on such motions. Our preliminary work [Harada and Wessel, 2003] leads us to believe that no significant motion between the two groups of hot spots has taken place, a conclusion also reached by others [Andrews et al., 2004]. However, such determinations are perhaps premature as we believe the extension of the present modeling technique to multiple plates will be required in order to rigorously asses global absolute plate motions, hot spot motions, and true polar wander.

Appendix A:: Confidence Region for a Rotation

To determine the projection of the ellipsoidal confidence region given by equation (13), we partition C = V · Λ · VT which gives us the matrix V with the eigenvectors or principal axes u, v, and w, of the ellipsoid, and the matrix

equation image

which contains three eigen-values that each equals the square of the length of one of the ellipsoid axes [e.g., Chang et al., 1990]. In this local coordinate system (u, v, w) the ellipsoid is defined by

equation image

for some choice of confidence level, α. The radial projection of this ellipsoid onto the Earth's surface is equivalent to finding the set of vectors rt that are tangents to the ellipsoid (Figure A1). Collectively, these vectors define a path P(u, v, w) on the surface of the ellipsoid and this path projects radially to S(θ, λ) on the surface of the Earth; this is the “plunging blimp” in the modeling literature. Here, we seek a simpler formulation that allows the graphical representation of these confidence regions than that first presented by Hanna and Chang [1990]. First, we consider P'(u, v), the projection of P(u, v, w) onto the uv plane and parameterize its (uP, vP) coordinates by

equation image

where η = η(β) is the in-plane radial length scales we must determine for all angles β (0–2π) in the uv plane. Since the tangent point must lie on the ellipsoid and satisfy (A2), we obtain two candidate coordinates

equation image

At the true tangent point, the dot product between the tangent vector from the Earth's center, rt, and the outward normal vector to the ellipsoid, n, must be zero. The outward non-normalized normal vector to the ellipsoid at points (u, v, ±w) is given (in local coordinates) by

equation image

which, when using (A2), yields

equation image

To evaluate the dot product we must express the vector rt in the (u, v, w) coordinate system. First, in the Earth's coordinate system we have rt = r0 + Δr, with r0 the coordinates at the center of the ellipsoid, and converted to local coordinates we find

equation image

which evaluates to

equation image

Now, the equation n · rt′ = 0 can be expressed as

equation image

which can be reduced and rearranged to yield

equation image

or, in general,

equation image

This expression can be further rearranged and squared to yield an ordinary quadratic equation whose solutions are

equation image

Two situations may arise: (1) The origin u = v = 0 lies inside P'(u, v). Here, (A9) will give two real solutions of opposite sign. We simply select all positive η(β) as negative solutions are simply −∣η(β+π)∣. (2) The origin lies outside P′(u, v). Here we recognize that no real solution to (A9) may exist for some range of β, manifested by complex roots. The remaining real solutions will come in pairs of the same sign. We only need the positive pairs where the upper values η(β) define the outer branch and the lower values η(β) represent the inner branch, both given in counterclockwise order. Reversing the order of the inner branch and appending it to the outer branch yields the complete solution which now are double-valued for any β. Using η(β) and equations (A3)(A4), we calculate the vectors to the two potential tangent points, at (uP, vP, ±wP), form the normals to the ellipsoid at those two points, and retain the vector that satisfies n · rt = 0. The resulting (uP, vP, wP) coordinates, for all β, once converted back to Earth coordinates, uniquely determine S(θ, λ).

Figure A1.

Projection of a 3-D confidence ellipsoid for a finite rotation represented by its pseudo-vector, r0. We seek to project the outline of this ellipsoid onto the Earth's surface, yielding a confidence region S(θ, λ) for the pole of rotation. This is achieved by tracking the path P(u, v, w) of the tangent vector rt along the ellipsoid. See text for details.

Appendix B:: Confidence Region for the Opening Angles

For rotation poles within S(θ, λ), the opening angle ω can take on a range of values. We wish to determine L(θ, λ) and U(θ, λ) which are surfaces of minimum and maximum opening angle, respectively, inside the region defined by S(θ, λ). Using (A2) we can generate any point r on the surface of the ellipsoid, and hence a unit vector equation image in S(equation image, λ) (Figure A1). Given such a vector, let r = ω · equation image. Then, the difference vector Δr relative to r0 is ω · equation imager0. For some value(s) ω this difference vector will lie on the ellipsoid, but in order to use (A2) we must first project the difference vector into local coordinates using Δr′ = VT · Δr, which evaluates to

equation image

The three components of any Δr′ are thus

equation image

Inserting these coordinates into (A2) yields

equation image

We rewrite this equation as

equation image

which is of the standard quadratic form. Its two solutions (ωl, ωu) represent the lower and upper rotation angles (in radians), respectively, at the (equation image, λ) point that corresponds to equation image. Thus the collections of all (θ, λ, ωu) and (θ, λ, ωl) define the two surfaces of interest.

Appendix C:: Uncertainties in a Reconstructed Point

Consider a total reconstruction rotation R, a hot spot location h, and the covariance of the rotation, C = Cov (r). The reconstructed point somewhere along the hot spot chain is then y = RT·h, where we again use the inverse rotation RT since we are rotating from the present (hot spot) to the past (seamount). That also means we first need to determine Ct, the covariance matrix of RT. Following Kirkwood et al. [1999], this quantity is given by

equation image

Then, the covariance Cov (y) is given by

equation image

where

equation image

By substituting in equation (C1) we find

equation image

This covariance matrix has y as its 3rd eigenvector with zero eigenvalue σw2; hence the matrix describes an elliptical confidence region around y in the local plane normal to y. To find the axes lengths and azimuths of this ellipse, we use the unit vector equation image = (0, 1, 1) to find the local east/north components (Δx, Δy) of the major axis in the tangent plane at y:

equation image

from which we can determine the azimuth ζ (in degrees) and the lengths (in km) of the major σ1 and minor σ2 axes:

equation image

where Re is the radius (in km) of the (spherical) Earth.

Acknowledgments

This work was supported by the National Science Foundation grant OCE 99-06773. Comments by Dietmar Müller, Joann Stock, Doug Wilson, and Associate Editor Bob Duncan led to improvements in the manuscript. SOEST contribution 6723.

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