In this section, we present gravity modeling for profiles 1 and 3 (Figure 1). Profile 1 is taken as an example to estimate the contribution of the different lithospheric features in the observed gravity anomaly. Since the geometry of the other two experiments and the crustal velocity models are quite similar to this one, we assume that the main conclusions can be generalized to the other two transects. Gravity modeling of profile 3 is done for a different purpose; to constrain the Moho geometry for the NW flank of the transect, which is poorly resolved by seismic tomography.
 The goal is to find 2-D density models that can explain the gravity data along the transects and is consistent with the seismic velocity models. Because of the lack of shipboard free-air gravity data, the gravity profiles along both transects have been built using available high-resolution marine gravity data based on satellite altimetry [Sandwell and Smith, 1997]. We consider this data set to be sufficient for our analysis, since we are mainly interested in the long-wavelength crustal features. Our gravity calculation is based on Parker's  spectral method, which allows calculating the gravity anomaly caused by lateral and vertical density variations (see Korenaga et al.  for details).
5.1. Profile 1, Southern Cocos
 A significant part of the free-air gravity is caused by seafloor topography and crustal thickness variations. Therefore prior to any attempt to model the crustal (and mantle) density field, it is necessary to correct the gravity anomaly for these two contributions. The topography contribution can be estimated by calculating the gravity signal of the seafloor-water interface and assuming a crustal layer of uniform thickness (7 km) and density (2800 kg/m3) (Figure 10a). Mantle density is considered to be 3300 kg/m3. The resulting anomaly is the so-called mantle Bouguer anomaly (MBA) (model a in Figure 10e). This anomaly largely overestimates the observed free-air gravity along the profile, reflecting principally the existence of a thick crustal root beneath the Cocos Ridge.
Figure 10. Different density models along profile 1, southern Cocos, corresponding to (a) model a, (b) model b, (c) model c, and (d) model d, described in the text. (e) Observed free-air gravity anomaly (open circles) and calculated gravity anomaly for density models a–d. Shaded zone shows the gravity anomaly uncertainty derived from velocity uncertainty (Figure 6a). RMS misfits are 71, 29, 13, and 5 mGal.
Download figure to PowerPoint
 In the next step, we attempt to correct for the crustal thickness variations (Figure 10b), by incorporating the crust-mantle boundary obtained in the seismic tomography inversion (Figure 5a). The crustal and mantle densities are the same as in model a. The gravity anomaly calculated with this model (model b in Figure 10e) fits better the observed free-air gravity, but there are still differences as large as 40–50 mGal in the central part of the ridge. Once corrected for seafloor and Moho topography, the remaining gravity anomaly misfits must reflect crustal and/or mantle density variations with respect to the reference model (model b).
 In order to test the influence of the crustal density variations, we have built the third simple model (Figure 10c), which includes crudely the main features observed in the velocity model. This model is composed of two layers with different densities. The upper layer (above 7 km depth) corresponds approximately to our layer 2. Its density is 2600 kg/m3. The bottom one (below 7 km depth) represents our layer 3, and includes the denser, crustal root. Its density (3000 kg/m3) is between the density of layer 2 and that of the mantle (3300 kg/m3). The calculated gravity anomaly (model c in Figure 10e) fits the observed gravity very well, and misfit is smaller than 10–15 mGal along most of the transect. The larger differences (around 90 km and 200 km) are directly associated with the bathymetric highs. These bathymetric features correspond to three-dimensional, seamount-like structures, so the observed gravity differences are most likely due to our assumption of 2-D density structure. This simple model accounts therefore for the long-wavelength gravity anomalies without calling for anomalous mantle densities.
 Finally, we have built the fourth model, in which we allowed for lateral and vertical density variations within the crust. The 2-D seismic velocity model (Figure 5a) was directly converted into a crustal density model using empirical velocity-density relations. Because of the lack of seismic reflection data, we employed different conversion rules for each crustal layer only on the basis of the seismic velocities. For velocities lower than 3.2 km/s (approximately sediments) we have used Hamilton's  relation for shale, ρ = 0.917 + 0.747Vp − 0.08Vp2. For velocities between 3.2 and 6.5 km/s (approximately layer 2) we have assumed Carlson and Herrick's  empirical relation, ρ = 3.61 − 6.0/Vp, which is based on Deep Sea Drilling Project (DSDP) and Ocean Drilling Program (ODP) core data. Finally, for velocities higher than 6.5 km/s to the crust-mantle boundary (approximately layer 3), we have used Birch's  law for plagioclase, and diabase-gabbro-eclogite, ρ = 0.375 + 0.375Vp, which is considered to be more adequate to describe the composition of the oceanic lowermost crust. Mantle density is kept constant (3300 kg/m3).
 All the v-ρ relationships for sediments, layer 2, and layer 3 are calculated at some reference temperature, T, and pressure, P (e.g., 1 GPa, 25°C), and they can be used to calculate the densities from seismic velocities at in situ conditions using estimates of the pressure and temperature partial derivatives [e.g., Korenaga et al., 2001]. The in situ crustal temperatures are calculated using the heat conduction equation (1), and the lithostatic pressure is given by equation (2):
where T0 and P0 are the temperature and pressure at the surface (assumed to be 0 MPa, 0°C), Qs is the surficial heat flow, κ is the conductivity (∼2.1 W/m °K for gabbro basalt), ρ is the mean density (∼2850 kg/m3 for gabbro basalt), H is the heat production rate, and y is depth.
 In oceanic crust, H is very small, and its contribution to the temperature regime is negligible. Qs is not accurately known at young ocean basins, but it can be inferred using Parsons and Sclater's  empirical law, in which heat flow is applied as a function of the age of the plate, t, Qs = 473.02/t1/2 mW/m2, for 0 < t < 120 Ma. Hydrothermal circulation is known to be a significant cooling mechanism for a young oceanic crust [e.g., Morton and Sleep, 1985], though it is difficult to quantify its influence. One option is to perform a sensitivity test by comparing velocity corrections for 100% conductive heat loss (no hydrothermal cooling) and for 50% conductive heat loss (Figure 11). Larger P-T velocity corrections (up to 0.10–0.15 km/s) correspond to the case without hydrothermal cooling, so we took this value as an upper bound for correction of the in situ crustal velocities.
Figure 11. P-T velocity corrections from in situ conditions to the reference state (400°C, 600 MPa) obtained along profile 1, southern Cocos. Solid line corresponds at 100% conductive heat loss and dashed line corresponds at 50% conductive heat loss.
Download figure to PowerPoint
 The 2-D density model along profile 1 obtained after correcting for P-T conditions is shown in Figure 10d. The uncertainty of the velocity model (Figure 6a) is propagated to that of the calculated gravity anomaly on the basis of the 100 Monte Carlo ensembles, though errors associated with the uncertainties of the conversion rules are not considered here (Figure 10e). We can see that the gravity anomaly is fully recovered with misfits smaller than 5–10 mGal along the entire profile, with the only exception at the bathymetric highs. Therefore we can conclude that the seismic velocity model obtained along profile 1 (Figure 5a) is also consistent with the gravity data.
5.2. Profile 3, Malpelo Ridge
 As we described earlier, the Moho geometry in the NW flank of profile 3 (0–75 km) is not constrained by PmP reflections. Therefore the main purpose of the gravity analysis along this profile is to determine the Moho geometry that best fits the observed gravity anomalies in this part of the transect. To do that, we fixed seismic velocities along the whole profile and the Moho geometry between 75 km and 245 km (Figure 5c), and the model was converted into densities following the same approach as in profile 1 (section 5.1). We used the same empirical conversion laws for the different crustal levels, mantle densities were fixed (3300 kg/m3), and P-T corrections were calculated. Then, we varied the Moho geometry between 0 and 75 km in order to obtain a good fit between observed and calculated gravity anomaly. Finally, uncertainties of the velocity model (Figure 6e) were propagated into that of the gravity anomaly. The final density model is represented in Figure 12a, and the observed and calculated free-air gravity anomalies are shown in Figure 12b. To account for the gravity anomaly, it is necessary to consider a dramatic crustal thinning toward the NW flank of the ridge (Figure 12a). In this part, the crust is only ∼6.0 km thick, which is consistent with the shallow mantle-like velocity body in seismic tomography (Figure 5c). Layer 2 is also thinner than in the rest of the transect (∼2 km).
Figure 12. (a) Velocity-derived density model along profile 3, Malpelo Ridge. Densities are in g/cm3. (b) Observed free-air gravity anomaly (open circles) and calculated gravity anomaly for model a. Shaded zone shows the gravity anomaly uncertainty. RMS misfit is 3.9 mGal.
Download figure to PowerPoint