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Ocean Island Densities and Models of Lithospheric Flexure T. A. Minshull, School of Ocean and Earth Science, University of Southampton, Southampton Oceanography Centre, European Way, Southampton SO14 3ZH, U.K. Ph. Charvis, Unité Mixte de Recherche Géosciences Azur, Institut de Recherche pour le Développement (IRD), BP48, 06235, Villefranche-sur-mer, France. Key words: Gravity anomalies, density, flexure of the lithosphere, volcanic structure, rheology Short title: Ocean island densities Submitted to Geophysical Journal International, December 1999. Revised version, September 2000. Final version, January 2001. ABSTRACT Estimates of the effective elastic thickness (Te) of the oceanic lithosphere based on gravity and bathymetric data from island loads are commonly significantly lower than those based on the wavelength of plate bending at subduction zones. The anomalously low values for ocean islands have been attributed to the finite yield strength of the lithosphere, to erosion of the mechanical boundary layer by mantle plumes, to 1 prexisting thermal stresses, and to overprinting of old volcanic loads by younger ones. A fifth possible contribution to the discrepancy is an incorrect assumption about the density of volcanic loads. We suggest that load densities have been systematically overestimated in studies of lithospheric flexure, potentially resulting in systematic underestimation of effective elastic thicknesses and overestimation of the effects of hotspot volcanism. We illustrate the effect of underestimating load density with synthetic examples and an example from the Marquesas Islands. This effect, combined with the other effects listed above, in many cases may obviate the need to invoke hotspot reheating to explain low apparent elastic thicknesses. INTRODUCTION Our main constraint on the rheology of the oceanic lithosphere comes from its deformation in response to applied stresses. While laboratory studies make an important contribution, empirical relations based on laboratory measurements must be extrapolated over several orders of magnitude in order to apply them to geological strain rates, and such extrapolations must have large uncertainties. The largest stresses applied to the oceanic lithosphere are at plate boundaries and beneath intraplate volcanic loads, and our understanding of its response to stresses on geological timescales comes mainly from gravity and bathymetric studies of these features. It has long been recognised that estimates of effective elastic thickness (Te) from oceanic intraplate volcanoes are significantly lower than estimates of the mechanical thickness of the lithosphere of the same age at subduction zones (McNutt 1984; Fig. 1). Part of this difference arises because the finite yield strength of the oceanic lithosphere can be exceeded due to plate curvature beneath volcanoes (Bodine, Steckler & Watts 1981). 2 Wessel (1992) attributed a further part of the discrepancy to thermal stresses due to lithospheric cooling, which sets up a bending moment which places the lower part of the plate in tension and the upper part in compression. Plate flexure due to volcanic loading releases thermal stresses, so that the plate appears weakened. Wessel (1992) found that a combination of the above effects could partly explain the low Te values from ocean island loading studies, but not completely. The remaining reduction was attributed to "hotspot reheating" - the reduction of lithospheric strength by mechanical injection of heat from the mantle plumes assumed to give rise to ocean islands. This effect has been proposed by many authors (e.g. Detrick & Crough 1978), and was quantified by McNutt (1984), who suggested that the age of the lithosphere was effectively "reset" by plume activity, to a value corresponding to the regional average basement depth. A problem with this suggestion is that hotspot swells exhibit heat flow anomalies which are much smaller than those predicted by the reheating model (Courtney & White 1986; Von Herzen et al. 1989). More recently, anomalously low Te values for some ocean island chains have been attributed to errors in the inferred age of loading where older volcanic loads have been overprinted by younger ones (e.g., McNutt et al. 1997; Gutscher et al. 1999). Here we propose an additional explanation for the low apparent Te of the lithosphere beneath many oceanic volcanoes which comes from the methods of data analysis used rather than from geodynamic processes. 3 THE DENSITY OF OCEANIC INTRAPLATE VOLCANOES Estimation of Te requires quantification of both the load represented by the volcano and the flexure of the underlying plate. In a few cases the flexure has been quantified by mapping the shapes of the top of the oceanic crust and the Moho by seismic methods (e.g., Watts & ten Brink 1989; Caress et al. 1995; Watts et al. 1997). Even in some of these studies, the shape of the flexed plate beneath the centre of the load is poorly constrained by seismic data, since the sampling of this region by marine shots recorded on land stations is poor, and additional constraints from gravity modelling are needed. The vast majority of the values shown in Fig. 1 come from studies using bathymetric and gravity or geoid data. The limitations of values based on the ETOPO5 global gridded bathymetry and low-resolution satellite gravity data have been discussed elsewhere (e.g., Goodwillie & Watts 1993; Minshull & Brozena 1997). However, even where values have been derived from shipboard data, significant errors can arise because of the trade-off in gravity modelling between the shape of density contrasts and their magnitude. Key parameters in gravity modelling of ocean islands and seamounts are the density of the load and the density of the material filling the flexural depression made by the load. These two quantities are normally set to be equal because the infill material lying beneath the load is likely to have a similar density to the load itself, and allowing the density of the infill to vary laterally introduces significant additional complexity into gravity and flexure calculations. Calculations are further simplified if this density is taken to be equal to that of the underlying oceanic crust, since then if the preloading sediment thickness can be taken to be negligible, only one flexed density contrast (the 4 Moho) needs to be considered. Therefore many authors make this assumption, using typical oceanic crustal density of 2800 kg/m3 (e.g. Watts et al. 1975; Calmant, Francheteau & Cazenave 1990; McNutt et al. 1997). Some support for this value came from drilling studies in the Azores and Bermuda, which led Hyndman et al. (1979) to conclude that "the mean density of the bulk of oceanic volcanic islands and seamounts is about 2.8 g/cm3". Independent constraints on load density and flexural parameters are available in the gravity and bathymetric data themselves. For example, the diameter of the flexural node is a clear indicator of Te, and in some cases this can be defined from bathymetric data (e.g., Watts, 1994). The shape of gravity or geoid anomalies also provides constraints. A least-squares fitting approach where both Te and density are varied may allow an estimate of the density to be made , but commonly there is a strong trade-off between elastic thickness and density which leads to an ill-defined misfit function with large uncertainties (e.g., Watts 1994; Minshull & Brozena 1997). The major contribution to a least-squares fit is the peak amplitude of the anomaly, and this value is highly sensitive to the load density. The trade-off may be avoided by using an alternative optimisation, for example by including the cross-correlation between residual gravity and topography as an additional term in the misfit function (Smith et al. 1989). Constraints on density can also come from wide-angle seismic studies, since for oceanic crustal rocks there is a strong correlation between seismic velocity and density (Carlson & Herrick 1990). Except perhaps for rocks with large fracture porosity, densities can be predicted from seismic velocity measurements with an uncertainty of 5 only a few percent (e.g., Minshull 1996). Using Carlson and Herrick's preferred relation, a density of 2800 kg/m3 corresponds to a mean seismic velocity of 6.0 km/s; these authors suggest a mean density of 2860 ± 30 kg/m3 for normal oceanic crust. However, a number of recent studies suggest that ocean islands and seamounts have mean velocities and therefore mean densities significantly lower than these values (Fig. 2). The mean density depends on the size of the volcanic edifice, since smaller volcanoes are likely to have a greater percentage of low-density extrusive rocks (Hammer et al. 1994), on its age, since erosion and mass wasting cause a general increase in porosity as well as redistributing load and infill material, and on its subaerial extent, since these processes act much more rapidly on a subaerial load. Factors such as the rate of accumulation of extrusive material may also be important. The overall correlation with load size is weak, so unfortunately the load density is not easily estimated from its size. However, in general the density is significantly lower than that of normal oceanic crust; this is not surprising because of the greater proportion of extrusives in ocean islands and seamounts and because of the effects of erosion and mass wasting. EFFECT ON FLEXURE AND ELASTIC THICKNESS ESTIMATES The above evidence suggests that load densities may be systematically overestimated in studies of the flexural strength of the oceanic lithosphere. There are two resulting effects on flexural modelling. Firstly, the vertical stress exerted by the load is overestimated, so for a given Te value the depression of the top of the crust and Moho is overestimated and the corresponding negative gravity anomaly is overestimated. Secondly, the positive gravity anomaly of the load itself is overestimated. The latter 6 effect is always larger, even for Airy isostasy, since the corresponding density contrast is closer to the observation point. The resulting effect on Te estimates may be quantified by computing the flexure, gravity anomalies and geoid anomalies due to a series of synthetic loads and then estimating the corresponding elastic thickness by least-squares fitting of the anomalies with an incorrect assumed density. The effect varies with the size of the load, so in this study three load sizes are considered (Table 1): a small load, comparable with some seamount chains, a large load comparable to large ocean islands such as Tenerife and La Réunion, and an intermediate load. The result also depends on the shape of the load. The simplest shapes for computational purposes are a cone and an axisymmetric Gaussian bell; most ocean islands and seamounts have their mass more focused close to the volcano axis than a conical shape would imply, so a Gaussian shape is chosen here. For simplicity, the loads were assumed to be entirely submarine. Flexure computations used the Fourier methods of, e.g., Watts (1994), while gravity anomalies were calculated using the approach of Parker (1974), retaining terms up to fifth order in the Taylor series expansion to ensure the gravity anomalies of the steeply-sloping load flanks are well represented. Geoid anomalies were computed by Fourier methods from the corresponding gravity anomalies. Load and infill densities of 2500-2700 kg/m3 were considered, with the density of the water, crust and mantle set to 1030, 2800 and 3330 kg/m3 respectively, and a normal oceanic crustal thickness of 7 km (White, McKenzie & O’Nions 1992). The crustal density used is the most commonly used value, though it is slightly lower than Carlson & Herrick’s (1990) preferred value. For a series of predefined Te values, Te was estimated using an assumed load and infill density of 2800 kg/m3. 7 The results of these computations (Fig. 3) confirm that in all cases Te is underestimated because the magnitude of the plate flexure is overestimated, and the size of the discrepancy increases with decreasing load size. In most cases the discrepancy is comparable to or larger than commonly quoted uncertainties in Te values. The total volume of the volcanic edifice (load plus infill) is also significantly overestimated. The ratio between the inferred infill volume and the true volume deviates little from the ratio for Airy isostasy, whatever the value of Te, because the total compensating mass must be the same in all cases. For a mean edifice density of 2600 kg/m3 this ratio is 1.55, while for a density of 2500 kg/m3 the ratio is 1.89. Estimates of Te are slightly improved if the geoid rather than the gravity anomaly is used, because the geoid anomaly is more sensitive to the shape of the flexed surfaces beneath the load. However, the difference is small and in real applications the geoid may be more strongly influenced by deeper, long-wavelength effects which are not accounted for in the flexural model. For these synthetic examples, the correct value of Te can of course be recovered by simultaneous optimisation of both Te and load density (Fig. 4). The misfit minimum is fairly flat and a small amount of noise due to density variations not considered in the flexural model may shift the minimum away from its true value, but no systematic bias is expected. The uniform density models used for these calculations are somewhat oversimplified; seismic experiments indicate that ocean islands commonly have a high-density intrusive core (Fig. 2). To simulate this type of structure, gravity anomalies were computed for a "large" load with a density of 2500 kg/m3 except in a central core with a radius 80% of the Gaussian decay length and reaching 50% of the height of the load 8 3 above the surrounding ocean floor, which was assigned a density of 2800 kg/m . This structure is loosely based on that of Réunion Island (Fig. 2). In this case the flexure computation is iterative since the overall height of the cylindrical core depends on the amplitude of the flexural depression. The results (Fig. 5) again show that Te is systematically underestimated, and the results from this composite load are very similar to those of a uniform load of the same mean density. REAL EXAMPLE: THE MARQUESAS ISLANDS We further illustrate the problem of assuming a standard oceanic crustal density with a real example from the Marquesas Islands, for which the elastic thickness is constrained independently by seismic determinations of the shape of the flexed oceanic basement and the shape of the flexed Moho beneath the load. Two further effects must be considered here. The first is that of inferred magmatic underplating (Caress et al. 1995). Magmatic underplating places a negative load on the base of the plate which reduces its downward flexure, but also increases the Moho depth and hence the apparent flexure of the base of the plate; the trade-off between these effects depends on the size, shape and density contrast of the underplate. The second is that of the assumed depth of the top of pre-existing oceanic crust, which defines the size of the load. Both effects are explored below. A comprehensive gravity and bathymetric dataset for the Marquesas Islands (Fig. 6) was published by Clouard et al. (2000). The depth to pre-existing oceanic basement has been determined by multichannel seismic reflection profiling and by coincident sonobuoy wide-angle seismic data (Wolfe et al. 1994), and an elastic thickness of 18 9 km has been determined. Gravity anomalies were computed for a region extending one degree in all directions beyond the edges of the region shown in Figure 6, to avoid edge effects. Models assumed an oceanic crustal thickness of 6 km, consistent with sonobuoy refraction data (Wolfe et al. 1994; Caress et al. 1995), an oceanic crustal density of 2800 kg/m3, and a range of elastic thicknesses and load and infill densities. The root mean square misfit was evaluated for the region shown. Our purpose here is not to place new constraints on the rheology of the lithosphere in this region, but rather to illustrate the bias that may be introduced by an incorrect choice of load density. The island chain forms a well-defined bathymetric feature, though there is some interference from the Marquesas Fracture Zone in the south-east corner of the region. If a horizontal base level is used for the load, there is an uncertainty of a few hundred metres in what this base level should be. The smallest root-mean-square misfit was found for a base level of 4200 m (Fig. 7a) and this corresponds to a Te of 19 km, similar to the seismically constrained value, and a density of 2550 kg/m3, slightly smaller than the 2650 kg/m3 used by Wolfe et al. (1994). The fit between observed and calculated gravity is good, with the misfit only exceeding 20 mGal locally around some islands and seamounts, where density variations within the volcanic edifices make a significant contribution (Clouard et al. 2000). A slightly shallower base level of 4000 m results in a slightly larger misfit, but no significant difference in Te or the best-fitting load density (Fig. 7b). An alternative possibility is that part of the seabed relief is due to a long-wavelength swell. Inclusion of the swell in the load could result in a biased Te value. An objective method for separating a regional bathymetric signal such as a hotspot swell from a 10 residual due volcanic loading by using a median filter which maximises the residual volume above a particular contour was suggested by Wessel (1998). The method places a clear lower limit on the optimal filter length, but unfortunately the residual volume changes slowly at large filter lengths, so that small changes due to the choice of area included in the analysis can change significantly the filter length corresponding to the maximum, and an upper limit is less clearly defined. To illustrate the effect of accounting for a swell, we analysed a series of models using a residual load after removing a regional defined by a 500 km median filter, which gives a peak swell amplitude of about 500 m relative to the abyssal plain at the edge of the area considered. This regional relates to the central part of the swell only, since the swell extends well beyond the area considered here (Sichoix et al. 1998). In this case the best-fitting Te is slightly larger, while the best-fitting load density is unchanged (Fig. 7c). Wolfe et al. (1994) found that the observed basement deflection could be matched by the addition of a basal load equal to 25% of the low-pass filtered top load, with a filter cut-off at 300 km, and interpreted as due to magmatic underplating. McNutt & Bonneville (2000) have suggested that such underplating could be the origin of the long-wavelength swell. Including such a basal load also has little effect on the optimal Te and load density (Fig. 7d). These analyses show that the optimum load density required to fit the gravity data is always significantly less than 2800 kg/m3, and that if a load density of 2800 kg/m3 were assumed for the Marquesas, the elastic thickness would be underestimated by ~25-40%. Calmant et al. (1990) made just such an assumption, and found a Te value of 14±2 km, significantly less than the seismically constrained value. This bias in elastic thickness is much greater than the bias which 11 would arise due to reasonable errors in the base level or due to the effect of underplating. DISCUSSION The above results show that great care should be taken in interpreting estimates of Te, and hence of the amplitude of lithospheric flexure and the volume of volcanic edifices, from gravity or geoid-based seamount loading studies of oceanic lithosphere unless there are independent seismic constraints, or at least the load density has been optimised to match the gravity data. Values are likely to be underestimated by large amounts, particularly for small loads, if a density of 2800 kg/m3 has been assumed. Most recent studies have indeed taken a more rigorous approach, with the use of seismic constraints (e.g. Watts et al. 1997), or simultaneous optimisation of load density and elastic thickness (e.g., Smith et al. 1989). In some of these studies the crustal density has been fixed equal to the load density (e.g., Filmer et al. 1993; Lyons et al. 2000). Such an approach simplifies gravity calculations, but its effect on the estimation of Te is not yet quantified. Even ignoring such problems, the published global dataset for seamount loading (as compiled by, e.g., Wessel 1992) is reduced to very few points (the filled circles and squares of Fig. 1) if values using assumed densities and values derived from ETOPO5 bathymetry and/or low-resolution satellite gravity are excluded. The less well-constrained values of Te (open symbols of Fig. 1) are too scattered to resolve statistically a bias toward lower values by comparison with the betterconstrained values (filled symbols). In some cases the bias due to using an assumed 12 density may be counteracted by an upward bias due, for example, to the use of a twodimensional analysis (Lyons et al. 2000). However, it is clear from the above analysis that such a bias is likely to be frequently present. Most published values for ocean islands could be considerably improved by a reanalysis using shipboard bathymetry and a combination of shipboard, land, and high-resolution satellite gravity/geoid measurements and using an approach which maintains the load density as a free variable or constrains it from seismic data. Without such reanalysis, and given the variety of other possible explanations for reduced elastic thickness beneath ocean islands and seamounts, suggestions that the mechanical thickness of the lithosphere is eroded significantly by plumes must be treated with some caution. CONCLUSION From our study we draw the following conclusions: 1. Ocean islands and seamounts commonly have a mean density which is significantly lower than that of the oceanic crust beneath. 3 2. If a load density of 2800 kg/m is assumed in gravity studies of flexure due to seamount loading, effective elastic thicknesses are significantly underestimated, by an amount which can be up to a factor of three or four for smaller loads. The infill volume can be overestimated by a factor of at least 50%, or more in the presence of underplating. 13 3. For the Marquesas Islands, where the elastic thickness is independently constrained by seismic data, the bias in elastic thickness which would be introduced by assuming a load density of 2800 kg/m3 is much larger than biases which would be introduced by reasonable variations in the chosen base of the load or by failing to allow for the contribution of underplating with a volume consistent with seismic data. 4. Few flexural studies of ocean islands and seamounts have used data of sufficient resolution and with sufficient constraint on load densities to give an accurate estimate of the effective elastic thickness; the paucity of reliable values severely limits their use in constraining models of plume-lithosphere interaction. ACKNOWLEDGEMENTS TAM is supported by a Royal Society University Research Fellowship. We thank A. Watts and I. Grevemeyer for supplying digital data used in Fig. 2, and P. Wessel and M. McNutt for constructive reviews. The GMT package of Wessel & Smith (1998) was used extensively in this study. This research was initiated during visits by both authors to the Instituto de Ciencias de la Tierra (Jaume Almera), Barcelona, Spain. UMR Géosciences Azur contribution 356. 14 REFERENCES Bonneville, A., Barriot, J.P. & Bayer, R., 1988. Evidence from geoid data of a hot spot origin for the southern Mascarene Plateau and Mascarene Islands (Indian Ocean), J. Geophys. 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B., Cochran, J. R., & Selzer, G., 1975. Gravity anomalies and flexure of the lithosphere: a three-dimensional study of the Great Meteor seamount, northeast Atlantic, , J. Geophys. Res., 80, 1391-1398. Watts, A.B., Peirce, C., Collier, J., Dalwood, R., Canales, J.P. & Henstock, T.J., 1997. A seismic study of lithospheric flexure in the vicinity of Tenerife Canary Islands, Earth Planet. Sci. Lett., 146, 431-447. Weigel, W., & Grevemeyer, I., 1999. The Great Meteor seamount: seismic structure of a submerged intraplate volcano, J. Geodynamics, 28, 27-40.. Wessel, P., 1992. Thermal stresses and the bimodal distribution of elastic thickness estimates of the oceanic lithosphere, J. Geophys. Res., 97, 14177-14193. Wessel, P., 1993. A reexamination of the flexural deformation beneath the Hawaiian islands, J. Geophys. Res., 98, 12177-12190. Wessel, P., 1998. An empirical method for optimal robust regional-residual separation of geophysical data, Math. Geol., 30, 391-408. Wessel, P. & Smith, W. H. F., 1998. New, Improved Version of Generic Mapping Tools Released, EOS Trans. AGU, 79 (47), 579. 20 White, R. S., McKenzie, D. & O’Nions, R. K., 1992. Oceanic crustal thickness from seismic measurements and rare earth element inversions, J. Geophys. Res., 97, 1968319715. Wolfe, C. J., McNutt, M. K., & Detrick, R. S., 1994. The Marquesas archipelagic apron: seismic stratigraphy and implications for volcano growth, mass wasting, and crustal underplating, J. Geophys. Res., 99, 13591-13608. 21 TABLES Table 1: Synthetic Gaussian load sizes. "Scale length" is the radius at which the height decrease to 1/e of its maximum. All calculations use a square grid of 256 by 256 nodes. Load Volume (km3) Height Scale Length Grid Interval (km) (km) Small 4 15 2827 2 Medium 5 25 9817 4 Large 6 45 30159 8 (km) 22 FIGURE CAPTIONS Figure 1. Circles mark estimates of the effective elastic thickness of the oceanic lithosphere as a function of age, with filled symbols where the estimate is based on gravity or geoid modelling with simultaneous optimisation of load density, or on seismic data, open symbols for other estimates, where in most cases a load density of 2800 kg/m3 has been assumed, and smaller symbols where the value is based on lowresolution satellite altimetry and/or ETOPO5 bathymetry. Squares mark values from French Polynesia, with the same convention. Triangles mark estimates of the mechanical thickness of the lithosphere from subduction zones. Solid isotherms are from the plate model of Parsons & Sclater (1977) and dashed isotherms are from the model of Stein and Stein (1992). Data are from the compilation of Wessel (1992) and references therein, with additional values from Bonneville et al. (1988), Wolfe et al. (1994), Goodwillie & Watts (1993), Harris & Chapman (1994), Watts et al. (1997), Minshull & Brozena (1997), Kruse et al. (1997) and McNutt et al. (1997). Where there is more than one published value for the same feature, only the most recent has been retained, except in a few cases where a value based on shipboard data is followed later by a value based on low-resolution satellite data. The large, filled symbols are considered the most reliable (see text). Figure 2. Density models derived from seismic velocity models for ocean islands and seamounts (a) Tenerife (Watts et al. 1997). (b) A model for La Réunion based on seismic velocities (Gallart et al., 1999; Charvis et al., 1999) and the velocity-density 23 relation of Carlson and Herrick (1990). (c) Great Meteor seamount (Weigel & Grevemeyer 1999). Note that the horizontal scale varies from one plot to another. Figure 3. (a) True elastic thickness vs estimated elastic thickness from gravity anomalies for the “large” Gaussian load of Table 1, if a load density of 2800 kg/m3 is assumed. Curves are annotated with the true density in kg/m3. The behaviour at very low true values of Te (less than 5 km) is complex and not fully sampled. (b) Results for three different load sizes if a load density of 2800 kg/m3 is assumed and the true load density is 2600 kg/m3. Here the solid curves result from fitting gravity anomalies and the dashed curves from fitting geoid anomalies. Figure 4. Contoured root mean square gravity misfit (in mGal) for a series of models with different assumed load densities compared with the “medium” Gaussian load of Table 1 with a load density of 2600 kg/m3. The misfit is evaluated for a 308 km by 308 km box at the centre of the 508 km box for which the gravity calculations are done. Note the elongation of the misfit contours in the direction of lower elastic thicknesses and higher densities. Figure 5. True elastic thickness vs estimated elastic thickness for a Gaussian load of density 2500 kg/m3 with a high-density cylindrical core of density 2800 kg/m3. The solid curve represents results from fitting gravity anomalies and the dashed curve represents results from fitting geoid anomalies. 24 Figure 6. (a) Bathymetry around the Marquesas Islands (Clouard et al. 2000). Land areas are shaded in grey. Contour interval is 500 m (b) Free air gravity anomalies, contoured at 50 mGal intervals. (c) Residual gravity for best-fitting flexural model (Fig. 7a), contoured at 20 mGal intervals. (d) Free air gravity anomalies of best-fitting flexural model contoured at 20 mGal intervals. Figure 7. Contoured root mean square gravity misfit (in mGal) for a series of flexural models of the Marquesas Islands with different elastic thicknesses (incremented in 1 km intervals) and load densities (incremented in 50 kg/m3 intervals). Circles mark best-fitting model in each case, and triangles mark the elastic thickness which would be inferred if a load density of 2800 kg/m3 were assumed. (a) All material above 4200 m depth is included in the load. (b) All material above 4000 m depth is included in the load. (c) The load is considered to overly a hotspot swell defined by the application of a 500 km median filter to the bathymetric data. (d) The top loading is as in (a), but in addition there is a basal load with an amplitude equal to the amplitude of the top load for wavelengths larger than 350 km, of zero amplitude for wavelengths less than 250 km, and tapering between these wavelengths, and with a density contrast equal to 25% of the density contrast of the top load. This basal load approximates the underplate inferred by Wolfe et al. (1994). 25