Abstract. A detailed analysis of the topography and geoid associated with the Hawaii-Emperor seamount chain reveals that while geoid/topography ratios are high over the southeastern part of the chain, they fall off with distance away from the current hot spot location. The topographic expression of the swell has two maxima near the intersections between the seamount chain and the Murray and Molokai fracture zones, respectively. Both geoid and topography rapidly decrease westward of the intersection with the Murray fracture zone. It appears that the region of the Pacific plate between the two fracture zones has been more susceptible to reheating and hot spot penetration than elsewhere along the chain. Alternatively, the observed pattern may be indicative of time variation in the strength of the Hawaiian hot spot.
Of all the volcanic features on the ocean floor, the Hawaiian-Emperor seamount chain has been one of the most studied over the last 50 years (Figure 1). Dietz and Menard  were the first to identify the swell associated with the islands. The swell appears as a broad topographic rise surrounding the youngest part of the chain, where the depths are almost 1500 m shallower than normal seafloor of the same age [Parsons and Sclater, 1977]. Wilson  suggested that the swell was caused by the upwelling of mantle material associated with a deep-seated "hot spot". On the basis of the relationship between gravity and topography from 5 by 5 degree averages, Watts  estimated the depth of compensation to be 100-200 km, and suggested that flow in the upper mantle dynamically supported the swell. Using more accurate data derived from satellite altimetry, later workers have reduced the compensation depth estimate by a factor between 2 and 3, suggesting instead that the cause of the swell is more likely to be uplift due to reheating and thermal expansion within the lower lithosphere [Detrick and Crough, 1978; Marks and Sandwell, 1991; McNutt and Shure, 1986]. However, the normal seismic velocities beneath the swell [Woods et al., 1991] and apparent lack of heat flow [Von Herzen et al., 1989] suggest that a dynamic mechanism also may be viable [Olson, 1990].
Understanding the origin of the Hawaiian swell has
important implications for studies of hot spot-lithosphere
interactions, island chain formation, rheology of the upper
mantle, and marine isostasy in general. It is vital that all
pertinent observations be assembled and analyzed in order to
detect patterns and trends that may put constraints on the many
physical models proposed for swell formation. While heat flow
Herzen et al.,
and seismic velocities
[Woods et al.,
may provide significant clues to the problem, the lack of
widespread coverage of such data limits their usefulness. It is
therefore important to investigate the appropriate data sets with
adequate coverage, which includes shipboard bathymetry and
This paper presents a detailed data analysis of ship bathymetry and satellite-derived geoid over the Hawaiian- Emperor seamount chain. I will examine both the along-chain and across-chain variations of these data and the geoid to topography ratio, and determine to what extent the observations may constrain the various models that have been proposed as mechanisms for the Hawaiian hot spot swell.
Color-shaded relief topography image with artificial illumination from the projection pole. The illumination highlights the horizontal topographic fabric, including fracture zones and short-wavelength artifacts caused by closely spaced, subparallel tracks. Note the infinity shape of the swell bathymetry in the eastern half of the Hawaiian chain. (b) Color-shaded relief image of the residual geoid after a regional geoid has been removed (see text). The geoid mimics the topography in Plate 1a and falls off rapidly west of the Murray fracture zone.
Available ship collected bathymetry data for the Hawaiian region were compiled and merged with the SYNBAPS [Van Wyckhouse, 1973] data set using a continuous curvature splines in tension algorithm [Smith and Wessel, 1990]. By giving the ship data higher weight than the SYNBAPS values, the latter only contributed information in regions not constrained by the ship surveys. Because of the oblique orientation of the Hawaiian seamount chain all data were rotated, using an oblique Mercator projection, about the pole (292E, 69N) that best describes the motion of the Pacific plate over the hot spots. In this projection, hot spot tracks (small circles about the projection pole) plot as horizontal lines which simplifies the subsequent flow line analysis. Figure 2 presents a contour map of the gridded topography. The data coverage for this region is quite dense: the gray areas indicate the ship data constraints. A color-shaded relief image of the same data set is illustrated in Plate 1a. In this image, the Hawaiian Swell appears as two circular regions (shaped like an infinity symbol) of shallow topography centered on Molokai (202.5E, 21.5N) and the Gardner Pinnacles (192E, 24.5N), respectively. This feature is also outlined by the 5 km isobath in Figure 2 (heavy line).
Fig. 1. Location map of the Hawaiian-Emperor Seamount Chain. Also shown are major fracture zones taken from Cande et al. . The skewed rectangle represents the outline of the oblique Mercator grid used in the subsequent analysis. All chain topography shallower than 2 km is solid. Shallow bathymetry not created by the Hawaiian hot spot are hachured.
Fig. 2. Contour map of the combined SYNBAPS and ship bathymetry data set. An oblique Mercator projection with the pole at (292E, 69N) is used. Contour interval is 1 km, with the 5 km isobath drawn as a heavy line. The regions constrained by ship measurements and land stations are shown in light gray.
During the Exact Repeat Mission, the Geosat altimeter collected instantaneous sea surface heights along the same tracks as the earlier Seasat mission. However, the geoid profiles derived from the Geosat altimetry are about three times more precise than the Seasat data set [Sandwell and McAdoo, 1988]. The complete geoid (Figure 3) contains long-wavelength information with large amplitudes unrelated to the processes important to this study. Because it is difficult to separate out the geoid effects from large-scale density heterogeneities in the mantle, it is customary to reduce their influence by removing a suitable regional geoid field prior to interpretation [Sandwell and Renkin, 1988]. This study used the GEM-T2 geopotential model [Marsh et al., 1990] to remove these long wavelength components. A cosine taper was used to roll off the spherical harmonic coefficients from degree and order 4 through 18 to avoid Gibbs phenomenon. This regional geoid is similar to that used in earlier studies [Marks and Sandwell, 1991] , except it retains somewhat more of the intermediate wavelength geoid anomalies (2000-3000 km). Although the choice of regional geoid is important [Sleep, 1990] , in this case both the adopted regional and that of Sandwell and Renkin  were almost identical. Residuals were obtained by subtracting the regional geoid from the observed sea-surface; they clearly depict the swell, in particular beneath the south-eastern part of the Hawaiian seamount chain (Plate 1b).
Fig. 3. Contour map of the complete marine geoid from Geosat and Seasat altimetry. Contour interval is 1 m, and the Hawaiian islands are shown as solid areas.
The topographic and geoidal response to reheating and
thinning by a hot spot have been given by Sandwell
terms of Green's functions. Thus, in order to predict topography
and geoid over Hawaii, one would have to know the heat
source distribution, which unfortunately is not available.
Similarly, to predict the topography and geoid over a dynamically
compensated swell would require knowledge of the shape
of the plume-head beneath the plate
Rather than resort to guesses of these parameters, I have
approximated the swell shape by simple exponential functions.
the first to model oceanic swells using a Gaussian function.
Closer inspection of a cross-section of topography (Figure 4)
reveals that the Gaussian curve (dashed line) seems to
overestimate the swell height. A simple way to compensate
for the "peakedness" of the simple Gaussian is to approximate
the swell shape with a "super-Gaussian" instead
where h is the center amplitude and w the half width. The value p = 2 results in the standard Gaussian curve; higher values for p gives flatter, plateau-like shapes. Figure 4 shows examples of best fitting super-Gaussians for various values of p. The curves fit to the data allow for a linear variation in the regional topography while excluding data within 250 km of the swell center. Note that in this region of the Hawaiian seamount chain the volcanic pile is not centered on the swell but offset by about 75 km to the south, a fact also noted by McNutt and Shure .
Fig. 4. Approximations to the cross section of swell topography (solid dots connected by dashed line) using super-Gaussians. The best fitting conservative curve (p = 5, heavy line) fits both the swell height and flank slopes well while being essentially horizontal under the volcanic load. A traditional Gaussian (heavy dashed line) arbitrarily overestimates the swell amplitude. The thin solid line (p = 20) has flanks that are too steep. Data within gray region were excluded in the estimation.
The main difficulty in estimating the swell shape is the fact that the observed topography is a combination of swell, flexural deformation caused by the volcanoes, and the volcanoes themselves. One cannot separate out one factor without knowing or specifying the other two. McNutt and Shure , assuming the swell was caused by flexure resulting from the buoyant effect of a low density subsurface load, simultaneously modeled the swell shape and the flexural response of the lithosphere from surface loads using linear filters. However, their elegant technique requires the elastic thickness to be a constant, which may not be entirely applicable for the Hawaiian chain [Wessel, 1993]. Instead, I will attempt to determine conservative (minimum) estimates for swell height by fitting super-Gaussians that are essentially flat in the region masked by the volcanic pile. There are no physical reasons why a Gaussian or super- Gaussian should be used for this purpose; they are simply smooth, symmetrical curves that may be used to approximate the swell component of the observed bathymetric signal. Nevertheless, the response of an elastic lithosphere to an impinging plume will basically reflect the shape of the plume head which may be flat-topped, according to some theoretical [Huppert, 1982] and experimental studies [Didden and Maxworthy, 1982]. The conservative value p = 5 was used for this study; larger values only led to minor changes in the swell amplitude and produced unreasonable swell flanks (see thin curve for p = 20 in Figure 4).
To provide better observational constraints on the compensation mechanism for the swell associated with the Hawaiian- Emperor seamount chain, both the topography (Figure 2) and geoid (Figure 3) data sets were analyzed in some detail. I have attempted to extract the height of the swell above the surrounding seafloor as a function of distance from the current hot spot location (Figure 5). The proximity of sizable topographic features not created by the Hawaiian hot spot makes this undertaking somewhat problematic. The median topography (filter width 500 km) along profiles sampled parallel to the chain (horizontal in the oblique Mercator projection maps) in a region away from the volcanic ridge but still on the swell (light gray corridors in Figure 5a) has been estimated. The same was done to more distant data clearly off the swell (dark gray corridors). While the median filtering removes most small-scale features departing from the regional depth [Smith, 1990], problems arise in regions of extensive shallow bathymetry. One such feature is the Hess Rise (Figure 1). In Figure 5b, the two thin lines represent median swell topography to the north and south of the ridge, respectively. The region between these curves has been shaded and gives an indication of the large variability of the data. The heavy solid line tracking between the thinner lines was obtained by filtering the average of the two lines after first "surgically" removing the Hess Rise. Thus, the swell heights in this region (3200-4000 km from hot spot) are based entirely on subjective assumptions of what the seafloor depth would have been without the thickened crust associated with Hess Rise. While not shown here, similar median profiles were constructed parallel to the Emperor seamount chain as well, and combined with the data from the Hawaiian chain. The lowest solid line gives the average regional depth outside the influence of the swell, while the heavy dashed line was computed from crustal ages using the cooling plate model [Parsons and Sclater, 1977]. Note the difference in observed and predicted step in topography between the two fracture zones. The swell height determined from this method varies considerably along the chain, with two prominent maxima at 250 km and 1500 km.
Fig. 5. (a) The region is divided into 20 zones (AT) of equal width (165 km). Median filtered topography was sampled along the gray strips parallel to the island chain in order to estimate swell topography variations. (b) Swell topography as a function of distance from the hotspot. The heavy solid line represents the best estimate of swell bathymetry (from light gray corridors in Figure 5a). Heavy dashed line is predicted depth based on crustal ages and the cooling plate model [Parsons and Sclater,1977]. Solid line beneath is estimate of bathymetry away from the swell (dark gray strips in Figure 5a). Thin dashed line represents predicted depth to a plate rejuvenated to 45 m.y. beneath the hot spot. Swell is largest between the two fracture zones.
Another way of estimating the swell height was introduced in the previous section. Instead of analyzing topography profiles parallel to the seamount chain at various distances from the ridge one can investigate profiles across the chain at various locations. Figure 5a shows the region subdivided into 20 zones of equal width (~165 km). For each zone, 10 profiles were extracted and stacked using a median stacking routine to produce a single, representative profile. The best fitting super- Gaussians (p = 5) for these average topography profiles were then determined. The five parameters (height, width, center location, regional depth, and regional slope) were solved for simultaneously using a standard nonlinear procedure (Levenberg-Marquardt). Figure 6 presents the estimated swell shapes for the 18 zones where the method converged. In fitting the swell shape, data within 250 km of the swell center were excluded (removing the influence of the volcanic ridge), and in several instances bathymetric features unrelated to the swell and situated beyond the swell flanks (e.g., the Mid-Pacific Mountains and Hess Rise) were "surgically" removed. For most of the eastern profiles it was fairly obvious which features to exclude (e.g., zone E), while farther west much of the data were so contaminated that accurate estimates were difficult to achieve (e.g., zone Q). In general, the formal errors in swell height (determined from the covariance matrix) were on the order of 200-500 m (1 s). The swell heights are plotted in Figure 7a together with the difference between swell topography and regional topography as determined in Figure 5b. It appears that the swell heights determined by fitting the super- Gaussians are consistently higher than the other estimates. The most likely reason for this mismatch is that the ridge- parallel profiles (light gray in Figure 5a) sometimes sampled parts of the swell flank rather than the swell top, thus underestimating swell height. In that regard the circles better represent the conservative height of the Hawaiian swell. A maximum swell height of ~1300 m is determined, somewhat less than most previous studies and almost a factor of 2 less than the best fitting swell height found by McNutt and Shure  in their inversion for swell parameters and compensation depth. It should be noted that their inversion is very sensitive to the size of the geoid anomaly which is difficult to isolate completely [Sleep, 1990]. The uncertainties in the geoid anomaly directly affect the swell height estimates. For this and other reasons discussed below, I prefer the conservative estimates of swell height derived here to those of McNutt and Shure . However, my estimates should be treated as minimum values for swell height.
Fig. 6. The median stacked topography profile and best fitting swell shape for each zone were determined following the technique in Figure 4. For several zones the fitting routine needed subjective guidance to ignore features not related to the Hawaiian chain. The swell amplitudes and their formal 1 s errors are plotted in Figure 7.
Studies of swell compensation have benefited from the uniform coverage of the marine geoid obtained by satellite altimetry. Because geoid anomalies over isostatically compensated features are non-zero and depend on the compensation mechanism [Haxby and Turcotte, 1978] , the relationship between geoid and depth anomalies has traditionally been considered a sensitive indicator of the state of compensation beneath oceanic swells and plateaus [Crough, 1978]. For instance, Sandwell and MacKenzie  and Marks and Sandwell  recently demonstrated that while many oceanic plateaus are compensated by thickened crust, swells associated with hot spots attained higher geoid to topography ratios (GTR) and are better explained by a combination of thermal reheating/thinning and dynamic uplift. In particular, the Hawaiian swell exhibits high GTR of 4-5 m/km [Marks and Sandwell, 1991]. These findings notwithstanding, the interpretation of the GTR values in terms of an Airy or Pratt-type compensation mechanism is somewhat tenuous. Studying the effect of a shallow low-viscosity zone on the apparent compensation of mid-plate swells, Robinson et al.  showed that the estimated compensation depth can become arbitrarily small for reasonable viscosity contrasts. In deed, Ceuleneer et al.  found that the observed GTR values world wide could be explained by simple thickening of the lithosphere at the expense of a low-viscosity layer only 50 times less viscous than the mantle beneath, provided the depth to the base of the low-viscosity zone is fixed. In most GTR studies, band-passed topography and geoid from regions surrounding the swells were regressed to determine a single ratio. Consequently, the GTR reveals nothing about the variations along the swell. While the interpretation of the GTR clearly may give non-unique compensation depths, it is still possible that variation in GTR along the chain could provide new constraints on the compensation mechanism.
Fig. 7. (a) Estimated swell heights (circles) from Figure 6 with 1 s error bars. Swell height (solid line) is obtained by subtracting swell topography and regional topography as determined in Figure 5. The latter systematically underestimates the swell height, probably by sampling the swell flank rather than the swell top. Maximum swell height is ~1300 m. (b) Geoid/topography ratios estimated from bandpass-filtered geoid and topography (see text). The method used to determine the ratios is illustrated in Figure 8. The gray rectangle represents the range of values obtained by an earlier study [Marks and Sandwell, 1991] .
I have attempted to estimate the GTR for each of the zones introduced in Figure 5a. The procedure involved bandpass- filtering the topography and geoid following Marks and Sandwell , determining stacked profiles for each zone, and investigating how the GTR varies across and along the swell. The GTR for each zone are plotted in Figure 7b, while Figure 8 illustrates how these values were obtained for profile B. (Note that the topography here has been filtered and therefore differs from the data displayed in Figure 6.) The GTR varies across the swell since only part of the data contains the desired information about the swell compensation. In terms of the bottom panel in Figure 8, the pertinent information will be contained in the two areas 2-3 and 5-6. The middle region (3-5) is dominated by the attraction and shape of the volcanic ridge and associated flexural deformation, and gives predictably lower ratios than the flanking areas. The top panel shows the best fitting L1 regression lines to data over the swell flanks. The average slope is reported and their difference used as an error estimate. It is clear that fitting a single line to all these points will produce an average of many effects and lead to much scatter. While zone B permitted a stable GTR to be calculated, complications arose farther west where off-axis features and their geoidal response overprint the desired signal. For this reason, only the GTRs from the first 13 zones could be determined (Figure 7b). The average GTR agrees well with that of Marks and Sandwell . However, the ratios only remain high until the intersection with the Murray fracture zone where they appear to fall off rapidly, reflecting the corresponding drop in swell topography (Figure 7a) and geoid (Plate 1b). It is clear that the fall-off cannot simply be explained by the increase in lithospheric age since the low-viscosity layer model predicts an increase rather than a decrease in GTR with plate age [Ceuleneer et al., 1988]. Thus, the variation is likely to reflect a change in the nature of compensation away from the hot spot.
Fig. 8. Example of bandpass-filtered topography (bottom panel), geoid (middle panel), and geoid versus topography (solid circles, top panel). The variation in GTR (e.g., local slope defined by the solid circles in the top panel) can be followed across the ridge by relating the circled numbers to their positions in the bottom panel. While short-wavelength information has been attenuated by the filtering, it is still clear that some of that signal remains. The GTR over the swell flank (regions between points 2Ð3 and points 5Ð6) are different from those over the islands (regions between points 3Ð4 and points 4Ð5). A line fit using all the solid circles in the top panel will therefore underestimate the true GTR associated with the swell. Instead, regression lines were fit to data from the two flanks separately. The average of these two slopes is the reported GTR, while their difference is used as the uncertaintly estimate. This plot illustrates the difficulties in obtaining reliable estimates of the GTR.
The analysis of bathymetry and geoid over the Hawaiian swell points to some interaction between fracture zones and the hot spot, a subject also discussed by Epp  and McNutt et al. . In addition to the larger amounts of extrusive volcanism associated with these areas [Clague and Dalrymple, 1987] , the underlying swell (on the basis of its cross-sectional area and height) also seems to have been affected (Figure 6). It appears that the younger plate segment between the Molokai and Murray fracture zones has been more susceptible to reheating and magmatism. The excess elevation of this segment cannot be explained solely from the age contrast with adjoining plate segments. The thin dashed line in Figure 5 represents the depth that would be expected if the plate had been thermally rejuvenated by 40 m.y., making the thermal age of the lithosphere above the hot spot ~45 m.y. Clearly, simple adjustments to the thermal age of the plate will not provide a comprehensive explanation for the sudden drop-off in swell height after the Murray fracture zone intersection.
Sleep  argued that the rapid motion of the Pacific plate in the hot spot reference frame may make it possible to separate the effects of uplift caused by hot asthenosphere from that of reheated lithosphere. Due to the change in plate motion at 43 Ma, part of the Hawaiian-Emperor chain will have been less affected by hot asthenosphere, because asthenospheric material moves slower than the lithospheric plate. Figure 9, modified from Sleep , illustrates the case. Hot material injected into the lithosphere moves with the plate while hot material discharged in the asthenosphere only moves at, say, half the speed. Thus, while the heat continuously discharged between the time of the bend (B) and now (H) will have affected the entire lithosphere between B and H, only the section E-H will be underlain by hotter asthenosphere. Before the change in plate motion a similar argument would hold for the section A-B, where heat added between the times A and B would only affect the asthenosphere under B-C. After the plate motion change, the hot asthenosphere under B-C was sheared by the plate motion and smeared over the region BCDE. Thus, Sleep's  model qualitatively explains the observed geoid and topography fall-off by arguing that the eastern part of the chain may be underlain by hotter than normal asthenosphere, while the western segment overlies normal asthenosphere. Of course, the model does not address the coincidence of the fracture zones and the observed maxima. Furthermore, no bathymetric evidence of extra heat in the hachured area in Figure 9 can be observed.
Fig. 9. The change in plate motion over the Hawaiian hot spot may have made it possible to distinguish various causes of uplift. While hot lithospheric material moves with the plate, hot asthenospheric material moves at a slower velocity (say half the plate speed). Heat supplied to the lithosphere between 43 Ma and present will have affected the plate between B and H, while hot material discharged into the asthenosphere during the same time period will today only reside beneath the eastern section (E-H). This implies that the western part of the Hawaiian chain and the southern Emperor chain is underlain by normal asthenosphere while the eastern Hawaiian chain is situated over hotter asthenosphere. The hot asthenosphere beneath the section B-C at the time of the bend has since been sheared latterally and would today underlie the hachured region.
The initial rapid rise in swell height over a distance of less than 1000 km may be an indication that a dynamic mechanism is partly responsible for the swell topography [Olson, 1990]. While a rapid rise is easier to understand in terms of dynamic uplift, both thermal and dynamic models give similar topographical predictions "down-stream" from the hot spot, which makes it difficult to separate out their individual contributions. The GTR shown in Figure 7b seem to indicate that the ratio is decreasing with distance from the swell, which also is predicted by both thermal and dynamic models [Olson, 1990]. The rapid fall-off in topography and geoid west of the Murray fracture zone is more difficult to explain, and will be a strong constraint on any model attempting to predict the swell.
Davies  has also recently investigated the Hawaiian Swell by studying its bathymetric expression along the chain. He argues that the seismic and heat flow evidence indicate that very little thinning of the lithosphere has taken place. Davies  sees the along-axis undulations in the bathymetry as evidence of past variation in the buoyancy flux of the Hawaiian plume, and concludes that the flux doubled 25-30 m.y. ago, corresponding approximately to the location of the Murray fracture zone crossing. While it is likely that the plume buoyancy flux has fluctuated over geologic time, this explanation renders the correlation between geoid and topography fall-off and fracture zone crossings coincidental. It thus appears reasonable to propose that the rapid transition at the Murray fracture zone is most likely caused by a difference in the response of the lithosphere to the hot spot. The response may have been modified to some degree by the transfer from normal to hot asthenosphere (Figure 9), and perhaps variable hot spot activity during the last 20-30 m.y. A better understanding of hot spot-lithosphere interactions will be required in order to separate out the lithospheric and asthenospheric contributions and assess the long term variability in plume buoyancy flux.
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