Abstract. Seismically derived depth estimates to the top of the oceanic crust beneath the Hawaiian Islands indicate that the curvature of the deflected lithosphere is much larger than commonly believed. The conservative and model-independent curvature estimates exceed 10-7 m-1 and are comparable in magnitude to curvatures at trenches and outer rise systems. The depth estimates are used to constrain both two-dimensional (2-D) and three-dimensional (3-D) flexural models. The curvature constraints require a 2-D variable elastic thickness that decreases from 35 km in areas away from the volcanic load to 25 km directly beneath the load. In an attempt to understand the nature of the yielding beneath the Hawaiian Islands we introduce two new 3-D models. The first model combines a realistic yield strength based rheology with a new technique for 3-D flexure calculations in which the elastic plate thickness is curvature-dependent. The new variable rigidity model predicts an undeformed (mechanical) plate thickness of 44 km, decreasing to 33 km beneath the big island of Hawaii. The best-fitting mechanical thickness corresponds approximately to the depth to the 600 C isotherm in 90-m.y.-old lithosphere. The second model uses a broken plate, but here the crack is oriented along the weak Molokai fracture zone rather than along the island chain trend. This unconventional flexure model can explain the observed asymmetry in the depth data across the fracture zone without requiring the excessively large elastic thickness of more conventional broken plate models. Both the proposed models imply that modeling with constant thickness plates may underestimate the true mechanical plate thickness by being unduly influenced by the weak zone beneath the seamounts.
The Hawaiian Islands represent some of the largest volcanic loads on the seafloor resulting in regional deformation of the lithosphere beneath them [Vening-Meinesz, 1941]. The nature of the deformation can be used to infer thermomechanical properties of the oceanic lithosphere. Walcott [1970] argued that the Hawaiian Deep and Arch were produced by flexure of an elastic plate fractured along the line of the island chain and estimated the flexural rigidity to be of the order of 1023 Nm. Watts and Cochran [1974], using marine gravity observations to constrain their models, concluded that a flexural rigidity of about 5 1022 Nm was required for a continuous plate model, whereas a somewhat stiffer plate (2 1023 Nm) was needed for a discontinuous (fractured) plate. For a Young's modulus of 100 GPa, the elastic plate thicknesses implied by these rigidity estimates are ~ 18 km and ~ 28 km, respectively. Later two-dimensional (2-D) and three-dimensional (3-D) investigations, using a continuous plate model, have concluded that the elastic plate thickness must lie in the 25- to 40-km range [McNutt and Shure, 1986; Watts, 1978; Watts and ten Brink, 1989].
A two-ship seismic experiment carried out in August-September 1982 provided both multichannel seismic reflection and refraction data, imaging the flexural deformation of the lithosphere caused by the island chain [Watts et al., 1985]. By approximating the digitized depths to the top of the crust with smoothing cubic splines, estimates of curvatures beneath Hawaii that were both model independent (not derived from flexure calculations) and conservative (natural cubic splines minimize the curvature) were presented [Wessel et al., 1989]. The derived curvatures were much higher than previously thought, being comparable in magnitude to values from deep-sea trenches and outer rise systems. If the seismic imaging adequately portrays the flexed surface, then the large curvatures imply considerable yielding beneath the surface load. The large curvatures may require us to reconsider the practice of using constant thickness plate models to approximate the flexural deformation beneath seamounts. Elastic thicknesses determined by conventional methods could be biased toward smaller values by giving more weight to weak zones beneath the loads where the plate effectively is thinner.
While the results from the curvature analysis of Wessel et al. [1989] are intriguing, it is possible that some of the effect is due to the 3-D geometry of the problem. To a first order, the Hawaiian seamount chain may be considered a two-dimensional load, but as we attempt to extract more details a 3-D approach is clearly warranted. The causes of the extensive weakness beneath Oahu may be several, including yielding due to large curvature bending, localized thinning related to magmatic activity, or interactions between bending stresses and remnant thermal stresses [Wessel, 1992]. In the following, we will first document the thickness variations implied by the seismic data using a 2-D approach, and then present a 3-D solution to plate flexure with curvature-dependent rigidity. Finally, we will reconsider the possibility of a broken plate, not cracked along the trend of the chain as has been proposed by numerous investigators [Walcott, 1970; Watts and Cochran, 1974], but instead along the Molokai fracture zone. We will show that in the light of available data, this new flexural model is also a viable alternative to the fixed thickness models.
In addition to the digitized depths to several key reflectors based on multichannel seismic reflection data [Watts and ten Brink, 1989], ship bathymetry and gravity were used to constrain the flexural models presented in this paper. Since both 2-D and 3-D flexural modeling will be considered, the approach taken was to compile 3-D gridded data sets of both ship bathymetry and gravity, and then extract "representative" 2-D profiles across the island chain in the region of seismic coverage. A best-fitting great circle perpendicular to the chain was determined for this purpose.
Fig. 1. Location of each individual depth estimate to the top of the oceanic crust (reflector 2). The data are organized into three transects, marked W, C and E. Double line shows trend of the island chain; the line orthogonal to this trend is the great circle used for presentation of the projected data sets. Heavy dashed line gives approximate location of the Molokai fracture zone.
The Hawaiian seismic experiment (R/V Conrad cruise C2308) provided more than 6000 km of multichannel common depth point (CDP) reflection profiles and 11 expanding spread profiles (ESP) [Watts et al., 1985]. Figure 1 shows the area of interest, the locations of the depth-converted reflection data, and the orientation of the great circle onto which data will be projected. Complete descriptions of acquisition and reduction of the seismic data have been published elsewhere [Lindwall, 1988, 1991; ten Brink and Brocher, 1987, 1988; Watts et al., 1985]. Watts and ten Brink [1989] converted the two-way travel times to depths using interpolations between linear velocity gradients determined for each ESP [ten Brink and Brocher, 1987]. They concluded that the main sources of error were uncertainties in navigation, inaccuracies in digitizing the travel times, and the interpolation between velocity gradients.
Figure 2 presents the depth converted reflections from Reflector 2 (R2, top of the crust), Reflector 3 (R3, Moho), and pieces of Reflector 4 (possibly the base of a subcrustal sill complex). Away from the island of Oahu, the data from the three projected transects (W, C, and E in Figure 1) agree very well on the depth to the respective reflectors. More scatter is observed toward the center of the projected line. This dispersion is partly due to lack of detailed velocity information within the island, but also reflects the fact that the three transects are much further apart, making the three dimensionality of the crustal structure around and beneath Oahu become more apparent. The continuous curves representing R2 and R3 (Figure 2) were constructed by filtering the point data (100 km, full width median filter).
Fig. 2. The crustal structure beneath Oahu as imaged by the multichannel seismic reflections. For Reflector 2 and Reflector 3, 100-km full width median filters were used to estimate smooth trends (double lines). Reflector 4, thought to represent the bottom of a subcrustal sill complex [ten Brink and Brocher, 1987], is also plotted.
All available bathymetry data for the Hawaiian region were compiled and run through a general crossover error detector [Wessel, 1989] to check data consistency and identify notoriously bad cruises. Problematic cruises were removed from the data set prior to gridding. To minimize map distortion and facilitate further analysis, we used an oblique Mercator projection (pole at 68W, 69N) in which the hot spot track (small circle about the pole) projects to a horizontal line. The final gridded bathymetry data set was assembled using a continuous curvature spline in tension algorithm [Smith and Wessel, 1990] with the SYNBABS data set [Van Wyckhouse, 1973] constraining areas of no ship data. Figure 3a presents a contour map of the gridded topography. The data coverage for this region is quite dense: the gray areas indicate bins with data constraints. A gray scale image of the same data set is illustrated in Figure 3b. The marine gravity measurements were treated in a similar manner. The cruises were crossover corrected following Wessel and Watts [1988]; the analysis implied a point precision of approximately 9 mGal.
Fig. 3. (a) Contour map of the gridded topography data set. An oblique Mercator projection (pole at 68W, 69N) was used. Contour interval is 1 km. The regions constrained by ship measurements and land stations are shown in light gray. (b) Gray shaded image of the gridded topography. The ""-shaped outline of the swell is clearly seen.
Based on the gridded bathymetry and gravity data, representative profiles for each data set were extracted to constrain the 2-D modeling. The grids were sampled along several tracks parallel to the main great circle shown in Figure 1. This step gave 21 profiles spaced every 10 km over a 200-km-wide corridor. The profiles were averaged using a median stacking routine, creating the representative profiles presented in Figure 4 (heavy solid lines). Apart from giving estimates of the uncertainties involved, the stacking procedure highlights the deviations from the 2-D assumption (gray envelopes). The stack deviations (1s) are very low, especially on the north side of Oahu, implying that the 2-D assumption is reasonably well justified for this region of the Hawaiian seamount chain.
Fig. 4 (a) Twenty one subparallel bathymetry profiles (thin dashed curves) and the median stacked representative profile (thick curve). The 1 s error envelope is plotted in gray. Median stacking is preferred since outliers (e.g., seamounts) have less influence on the final solution. Other curves show approximations to the swell topography using super-Gaussians. Best-fitting curve (p = 5, double line) fits both the swell height and flank slopes well. A traditional Gaussian (p = 2, dashed curve) severely overestimates the swell amplitude, while p = 20 (thin curve) results in flanks that are too steep. (b) Same for the gravity profile. The double line is the signal due to the swell. Dashed curve is again the traditional Gaussian fit.
It is obvious from Figure 4 that the observed topography and gravity is "contaminated" by a long-wavelength signal associated with the hot spot swell. Without complete knowledge of the swell signal we cannot remove it. McNutt and Shure [1986] were able to estimate the swell shape and the flexural deformation caused by the islands simultaneously using linear filters. However, their spectral method requires the elastic thickness, Te, to be constant, while we will investigate variable thickness models. We will therefore only approximate the effect of the swell and remove it from the observations. Crough [1978] suggested the use of Gaussians to fit oceanic swells. The heavy dashed curve in Figure 4 is such a Gaussian, but appears to overestimate the swell height. Since the volcanic load masks the swell shape, Wessel [1993] sought conservative swell shape estimates by fitting super-Gaussians, defined as
where h0 is the swell amplitude, w, its standard deviation, and p controls the shape of the curve (p = 2 gives a Gaussian distribution). For p > 2 the curves flatten, while p < 2 lead to sharper curves. A value of p = 5 provided a satisfactory fit to the swell flanks, and connected them without introducing an unconstrained central high (double line).
The R2 data set only extends from -300 km to +400 km (0 corresponds to the axis of the chain), which is within the window where plate flexure is important. The data set was extended by assuming that R2 is subparallel to the seafloor beyond the range of the seismic data set. The short wavelength noise in the seismic data was suppressed with a median filter (5 km full width). Another concern with the reflection data is the degree of scatter directly beneath the islands, were the three transects are far apart. While the large scatter in the depths directly beneath the islands makes detailed 2-D modeling there infeasible, the seismically determined depths are quite consistent from transect to transect away from the central region. Therefore only the points away from the inconsistent central area were modeled. Figure 5a presents the modified data set R2 (solid squares) after filtering and extending the range. The depths clearly reflect the swell topography as well as the flexural deformation. Using the swell shape determined previously, the swell effect was then removed from the depth estimates. In addition, points inside the 100-km-wide inconsistent central area were removed. The corrected data set (open squares) represents a better data set for 2-D flexural modeling. The flexural bulges are now less outstanding (the bulge from a 2-D line load is theoretically ~4 % of the maximum deflection beneath the load). Note that a Gaussian curve (heavy dashed curve in Figure 4a) would have removed the bulges completely. The assumption that the elastic plate must be continuous across the island chain implies that the two branches of depth estimates on either side of the data gap must lie on a continuous curve. Our goal is to determine the minimum curvature required for a continuous line to pass through both data branches. A smoothing natural cubic spline was used to approximate the trend suggested by the squares in Figure 5a. The knot points were chosen to be 100 km apart. To ensure that the exact location of the knot points did not bias the solution, all knots were shifted in increments of 10 km and the best-fitting smoothing spline for all intermediate knot configurations were determined. The similarity of the solutions (Figure 5a) indicates that the curves are insensitive to the knot location. The curvatures implied by the smooth curves (Figure 5b) exceed 10-7 m-1, thus being comparable to curvatures at trenches and outer rises [Judge and McNutt, 1991]. The curvatures exceed 10-7 m-1 even if no swell correction is applied, and increases only slightly for swells with p 2. It is evident that a significant amount of yielding must have taken place beneath the load. The estimates of curvature presented here are both model independent and conservative, since a natural cubic spline seeks to minimize the curvature. These estimates therefore differ from curvatures derived from flexure calculations which strongly depend on the model assumed. The analysis suggests that curvatures beneath seamounts, at least the larger ones, may be much larger than previously expected and warrant the use of variable rigidity models. Most of the deformation at seamounts occurs directly beneath the load. Hence fixed Te models that minimize the fit to the deformation will tend to find a thinner plate, as we shall see shortly.
Fig. 5. (a) A 5-km median filtered version of Reflector 2 which have been extended assuming that the seafloor is subparallel to the top of the crust away from the island (solid squares). Open squares show Reflector 2 after correcting for the swell shape (determined in Figure 4a) and excluding data points from the central region. Ten smoothing cubic splines were fitted to the adjusted data sets to obtain smooth, representative profiles of the deformed surface. Estimates of curvature based on cubic splines are conservative since a cubic spline always tries to minimize the curvature between the knots. (b) The curvatures determined in Figure 5a. Large curvatures are implied directly beneath the load. The thin line is a smooth approximation to the derived curvatures (heavy line).
Flexure of an elastic beam overlying a weak fluid substratum is given by
where w is the deflection, r is the density contrast between the material below and above the plate, g is normal gravity, q is the variable load, and D is the flexural rigidity related to the elastic thickness Te by
where E is Young's modulus and n is Poisson's ratio. The 2-D nature of the flexural deformation beneath Oahu has been investigated using two conventional models for plate flexure. The first model assumed a fixed value for Te across the seamount chain, as proposed by Watts and Cochran [1974]. The 2-D load shape was estimated from the average topography profile by removing the swell shape and a base level of 4200 m. The model minimizing the rms misfit to the seismically determined depths had an elastic thickness of 25 km. This constant thickness model is displayed as a dotted curve in Figure 6a. While achieving a satisfactory fit to the northern branch, the model underestimates the flexure on the southern side of the island chain. However, the gravity anomaly predicted by the 25-km plate model provides quite a good fit to the data (dotted curve in Figure 6b), a result also noted by many earlier workers [Watts, 1978; Watts and Cochran, 1974].
Fig. 6. (a) Best-fitting flexural models using conventional, two-dimensional techniques. The dotted curve is the response of a 25-km-thick elastic plate while the solid curve is the response of a variable thickness plate where the thickness ranges from 35 km away from the load to 25 km beneath the load. (b) Gravity predictions based on the models in Figure 6a. Within the uncertainties of the data, both models provide reasonable fits, although the constant thickness model fits slightly better.
To investigate the amount of yielding implied by the curvature estimates, an analytical expression relating the effective elastic thickness to plate curvature was derived (Appendix A). This technique was used with a smooth approximation to the curvature estimates in Figure 5b (thin line) to design a variable thickness model. The result was a plate model in which the ratio of effective elastic thickness to mechanical (undeformed) thickness varied from 1 (away from the load) to 0.7 (beneath the load), reflecting the smooth shape of the curvature variations. A search for the best mechanical plate thickness gave 35 km which means that the effective elastic thickness decreases to 25 km beneath the load. The flexural response of this model is drawn as a thick curve in Figure 6a. As one would expect, the model with more free parameters provides an improved fit to the data. While the small improvement would normally not justify the use of a more complicated model, the a priori information of curvature estimates provides the best argument why a variable rigidity solution is more appropriate than a fixed thickness plate. Curiously, the constant thickness model seems to fit the gravity data better (Figure 6b). This conflict is most likely due to the shortcomings of the 2-D model but could also be explained ad hoc with density or crustal thickness variations across the chain, in particular since the Molokai fracture zone cuts across in this area. Furthermore, the material imaged above R4 may act as a buoyant buried load and modify the flexural response somewhat. Unfortunately, the along strike extent of this hypothetical sill complex is not constrained, in fact its existence is still controversial [Lindwall, 1988, 1991].
From the results presented in Figure 6, it appears that the lower values for Te (~ 25 km) previously reported using simple, elastic flexure models must represent the weaker, highly deformed region beneath the load rather than the stronger, less deformed flanks. This result also implies that rigidity estimates can be quite model dependent, and that additional constraints like multichannel seismic imaging of the deformed surface may be required, at least beneath large surface loads like the Hawaiian seamount chain, in order to reliably estimate Te.
To probe beyond what can be learned from 2-D modeling, a 3-D load model was constructed from the gridded topography. We will contrast three different flexural models: One with constant thickness, one with variable thickness, and finally a new broken plate model. No longer confined to a projected line, the seismically derived depth estimates are now used as constraints at their original locations (Figure 1). A density of 2700 kg m-3 was assigned to the island load.
Constant Plate Thickness
The 3-D constant thickness problem was also considered by McNutt and Shure [1986] and Watts and ten Brink [1989]. A parameter search for the combination of plate thickness and infill density that provides the best least squares fit to the depth estimates was performed. Figure 7 shows a contour plot of the normalized rms misfit. The best fit was achieved by Te = 25 km with an infill density of 2550 kg m-3 and is thus comparable to the constant thickness 2-D results. The Te = 25 km solution also agrees well with that of McNutt and Shure [1986] (Te = 24 km) but is much less than the 40 km reported by Watts and ten Brink [1989] using a similar data set. The difference may be attributed to the swell which was not removed in the latter study. The S and N mark the locations of the best fit when only the data south or north of the Molokai fracture zone were used as constraints, respectively. The best estimate is, in a sense, a weighted average of these two end members. The implied asymmetry may indicate that the two plate segments have different thermomechanical properties and respond differently to the volcanic load.
Fig. 7. Contour plot of normalized rms misfit as a function of elastic thickness and infill density for the 3-D constant thickness plate model. The best fit corresponds to Te = 25 km and ri = 2550 kg m-3. When only the data south (or north) of Molokai fracture zone is used to calculate the rms misfit, the location of the best parameters are marked by S (or N).
Curvature-Dependent Plate Thickness
The curvature analysis, 2-D modeling, and 3-D constant rigidity modeling strongly suggest that a single constant thickness plate is not entirely applicable beneath the Hawaiian Islands; hence a variable 3-D solution must be sought. In the 2-D case, we simply used the curvature data to design an ad hoc variable thickness plate; for the 3-D case this is no longer practical. Instead, we will use a technique in which the variations in plate thickness is caused entirely by the brittle and ductile failure implied by the bending itself. An iterative approach is followed in which the previous step's solution of vertical deflections is used to estimate the variations in the effective elastic plate thickness, Te (x,y). These refined thicknesses are then used to complete the next deflection solution. We used the relation between Te and plate curvature as derived in Appendix A. For the solution to the 3-D variable rigidity flexure problem, we employ a new iterative spectral method, presented in Appendix B. A similar parameter search for infill density and undeformed (mechanical) plate thickness was then carried out, giving the results in Figure 8. As expected, the best fit requires a stronger plate away from the load (Te = 45 km), while the infill density remains more or less the same (2530 kg m-3). Again, the north and south plate segments give diverging estimates. Figure 9a presents the deflections predicted by the best-fitting variable rigidity model, with the implied variations in plate thickness shown in Figure 9b. The modeling predicts that the yielding has reduced the effective elastic thickness to 33 km beneath the big island; unfortunately, no data constraints exist in this area.
Fig. 8. Contour plot of normalized rms misfit as a function of undeformed (mechanical) thickness and infill density for the 3-D variable thickness plate model. Best fit was achieved for h = 44 km and ri = 2530 kg m-3. If the rms were computed separately for each side of the fracture zone, the minima would occur at the locations marked by S and N.
Fig. 9. (a) Depth to the top of the oceanic crust in kilometers as predicted by the 3-D flexure model with curvature-dependent rigidity. The flexural bulge to the SW of the island of Hawaii is larger than that to the NE. This effect reflects the deviation from the linear island trend at its SE end. (b) Effective elastic thickness variations in kilometers implied by the model in Figure 9a. As expected, the largest yielding is found beneath the largest load (the island of Hawaii).
Broken Plate Revisited
With no reliable data constraints directly beneath the islands is clear that a broken plate may produce reasonable fit to the available data [Watts and ten Brink, 1989]. Indeed, it has been argued that the bending strain directly beneath the load may be so large that a broken plate may be a more appropriate model [Walcott, 1970]. Both Watts and Cochran [1974] and Watts and ten Brink [1989] investigated flexural models in which the plate was cracked along the length of the island chain. In general, the broken plate model requires a much thicker elastic plate to fit the same data as the continuous model since it predicts more deformation at the plate edge. However, it is difficult to reconcile the large elastic (not mechanical) thicknesses (40-60 km [Watts and ten Brink, 1989]) with the thermal properties of 90-m.y.-old oceanic lithosphere. A 60-km plate would indicate that the elastic plate thickness is controlled by the depth to the 850 C isotherm, which is considerably higher than the 450 C estimate determined elsewhere [Calmant and Cazenave, 1986]. Instead, we suggest that a broken plate model may be applicable if the crack is aligned along the Molokai fracture zone system. The Molokai fracture zone represents a large age offset (~16 m.y.) in this region but is associated with subdued topography, gravity, and geoid anomalies. In this regard it is anomalous with respect to the other large offset Pacific fracture zones and thought to be weak [Bonneville and McNutt, 1992; Wessel and Haxby, 1990]. We have determined the 3-D flexural deformation of a broken plate under a two-dimensional load trending at 30 to the fracture and crossing it. Figure 10a illustrates the geometry and shows a contour plot of the deflections caused by the load (not shown). The solution was arrived at by first applying the load to a continuous plate (using (B6)), and then splitting the load and deformation grids along the Molokai fracture zone. The separate south and north segments were then iterated on further using a simple finite difference algorithm, but this time a free boundary condition was used along the fracture zone. Because the two plates are decoupled, they respond independently to the load. The response is two dimensional away from the crack but changes character closer to the fracture. The lines orthogonal to the trend of the load in Figure 10a indicate the approximate location of the projected multichannel data. Because of the uncertainty in the exact location of the Molokai fracture zone beneath the volcanic load, three cross sections at increasing distance from the point of symmetry (i.e., the right-most track) are presented. The deflection profiles along these tracks are plotted in Figure 10b. Note that the flexure curves take on intermediate shapes between the fully broken and continuous plate models, a trend in harmony with the data. This new model for plate flexure beneath Hawaii is capable of explaining the "broken" and asymmetrical character of the data without resorting to unrealistically high elastic plate thicknesses. In fact, the predictions displayed in Figure 10 were based on a fixed elastic thickness of 25 km. The more complicated calculations for a three-dimensional, variable rigidity plate with a crack was not attempted but could arguably lead to an improved fit. It is clear that the presence of the obliquely cross-cutting Molokai fracture zone in the only region with high-quality seismic observations hampers our ability to determine uniquely the most appropriate flexural model for the Hawaiian seamount chain.
Fig. 10. (a) Results from a new flexural model where the plate is broken along the weak Molokai fracture zone. The deformation is caused by a two-dimensional load with the cross-sectional shape of the Hawaiian ridge (determined from Figure 4a). The deformation is sampled along three straight lines normal to the strike of the ridge. (b) Flexural profiles normal to the ridge. The deformation takes on a shape in between that of a continuous and broken plate. Various degrees of asymmetry occur depending on the distance from the point of crossing between the ridge and the crack.
The flexural models that best account for the curvatures implied by the multichannel seismic reflection data feature a broad zone of weakness beneath the load. There are several mechanisms that could weaken the lithosphere in such a way. First, it is possible that reheating associated with hot spot volcanism could contribute to some of the observed changes in rigidity [McNutt, 1984]. The yield strength is very temperature dependent, and the extra heat would reduce the depth to the isotherm that controls ductile yielding. The main difficulty with this explanation is the apparent lack of significant heat flow [Von Herzen et al., 1989] and the conclusions of Woods et al. [1991] that seismic velocities beneath the swell appear to be similar to those of old oceanic lithosphere elsewhere. Second, the Molokai fracture zone system crosses the region where the reduced strength is implied. As can be seen from the topography image in Figure 3b, the Molokai fracture zone splits into several smaller fracture zone strands over a wider area, which could explain the weakening of the lithospheric plate. The 3-D modeling results indicate that the sheer size of the volcanic load could also drive the plate into brittle and ductile failure, and this deformation would necessarily be concentrated beneath the island chain. Such yielding could be increased by interactions between flexural and residual thermal stresses in regions where the curvature is convex up (i.e., beneath the load) [Wessel, 1992].
Based on 3-D, fixed thickness flexure modeling, Watts and ten Brink [1989] suggested an elastic thickness approaching 40 km. The differences in elastic thickness values for the continuous plate model (25 km versus 40 km) may relate to difference in how the 3-D loads were determined. However, Watts and ten Brink [1989] did not correct the depth-converted multichannel reflection data for the shape of the swell (Figure 4a). The corrected data (Figure 5a) exhibits more deformation, which requires a thinner plate for the same load geometry.
The wide range in elastic thickness solutions simply demonstrates that such estimates strongly depend on the model being used, in particular the difference between the 2-D and 3-D results. Given the magnitudes of the inferred curvatures, it seems only reasonable to use a self-consistent, curvature-dependent plate thickness model instead of a constant thickness plate. While the limited coverage and quality of the seismically determined depths make it difficult to draw firm conclusions, it is clear that the new models presented here fit the data equally well or slightly better than earlier models. It is also intriguing that the flexural model in which the plate is broken along the Molokai fracture zone may explain the character of the data better than more conventional models. It should be noted that the mechanical thicknesses were, for the most part, constrained by the seismically determined depths only. Further 3-D modeling, including gravity data constraints, will undoubtedly give revised, perhaps lower, estimates.
The elastic thickness of the oceanic lithosphere has been shown to increase with the age of the lithosphere at the time of loading [Watts, 1978]. Most Te determinations have come from studies of flexure beneath seamounts, with the anomalous gravity field as data constraints. The elastic plate thickness seems to be controlled by the depth to the 450 C isotherm in the cooling plate model, except beneath seamounts in the anomalous French Polynesia area [Calmant and Cazenave, 1987]. Modeling of plate flexure at trenches [Caldwell and Turcotte, 1979; McAdoo et al., 1985] have suggested that somewhat hotter isotherms (600-700C) may control the plate thickness, especially after the effect of yielding has been taken into account [McNutt, 1984]. The 35- to 44-km mechanical plate thicknesses determined by the variable, curvature-dependent flexure model imply isotherms in the latter range. This study suggests that curvatures beneath seamounts may be much larger than previously anticipated and require the use of variable rigidity models, since fixed Te models tend to underestimate the undeformed plate thicknesses. In light of our results it seems likely that much of this discrepancy in isotherms would disappear if mechanical rather than elastic thicknesses were reported for seamount as well as trench studies. Hopefully, the 3-D flexural modeling approach presented here will be considered in future investigations of seamount isostasy.
Acknowledgments. Advice from Tony Watts and Bill Haxby were useful during the early stages of this study. Comments by two anonymous reviewers and the associate editor led to dramatic improvements in the text. This work was supported by a School of Ocean and Earth Science and Technology (SOEST) postdoctoral fellowship. SOEST contribution 3198.
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