For the age distribution of the oceanic crust (hereafter “ocean crust age” represented by τ) we use the data from Müller et al. (2008a; who provide global reconstructions of ocean crust age in 1 Ma intervals for the past 140 Ma. For each reconstructed age in Müller et al. (2008a), ocean crust age, depth to basement, and bathymetry are given. The reconstructed bathymetry based on Müller et al. (2008a) is referred to hereafter as EB08 (EB: EarthByte). The data are in 0.1∘ × 0.1∘ resolution (3601 longitude × 1801 latitude points). For this project, 000 Ma (modern) crustal age reconstruction data are used (Fig. S1 in the Supplement).

We use modern ocean sediment thickness data from Divins (2003) and Whittaker et al. (2013). These data are derived from seismic profiling of the world's ocean basins and other sources. The reported thicknesses are calculated using seismic velocity profiles that yield minimum thicknesses. Data values represent the distance between sea floor and “acoustic basement”. The data are given in 5′×5′ resolution and have been regridded to 0.1∘ × 0.1∘ resolution values (Fig. S2) to match the EB08 grid.

To construct the shelf-slope-rise structures, ETOPO1 modern bathymetry (Amante and Eakins, 2009) is used. We use the “Bedrock” version of ETOPO1, which is available in a 1′×1′ resolution (earthmodels.org), regridded to 0.1∘ × 0.1∘ resolution (Fig. S3) in order to match the EB08 grid (Fig. S1). This version of ETOPO1 includes relief of Earth's surface depicting the bedrock underneath the ice sheets. However, we use only the oceanic points in this data set, so that this has no impact on the reconstructed bathymetry.

Reconstruction of open ocean bathymetry starts with ocean crust age. This information is available only at locations where oceanic crust is preserved or has been reconstructed. The ocean depth to basement is the distance between mean sea level and the top of the basaltic layer of the oceanic crust. Calculation of depth to basement is based on a cooling plate model in which the vertical distance between mean sea level and basement ωτ is expressed as ωτ=ω0+ωd, where the ω0=-2639.8 m is the area-weighted average of mid-oceanic ridge depths from the North Pacific, eastern Atlantic and southeastern Atlantic reported in Crosby et al. (2006), and ωd is the change in depth due to plate cooling. Here we adopt a negative sign to denote depths below mean sea level. The change in depth due to cooling of the oceanic plate ωd is given by (adopted from Eq. 4.211 in Turcotte and Schubert, 2014) ωd=-αρm(Tm-Tw)yL(ρm-ρw)12-4π2∑m=0∞1(1+2m)2exp(-κyL21+2m2π2τ), where α (3×10-5 K-1) is the volumetric coefficient of thermal expansion of the mantle, ρm is (3300 kg m-3) is density of the upper mantle, ρw (1000 kg m-3) is density of sea water, Tm-Tw (1300 K) is the difference between upper mantle and ocean temperature, κ (3.410835×105 m2 s-1) is thermal diffusivity, and yL (2619.7 m) is equilibrium plate thickness, all assumed to have constant values.

The equilibrium depth to basement ωe corresponds to the limit of τ→∞ in Eq. (2), appropriate for the oldest crust: ωe=-αρm(Tm-Tw)yL2(ρm-ρw).

In our reconstruction we use ωe=-5875 m, the mid-point of the range -5750 to -6000 m in the oldest part of the North Pacific (Crosby et al., 2006). We assign an area-weighted average value to the parameter β (Table 1): β=2αρm(Tm-Tw)(ρm-ρw)κπ=329.5ms-12, so that κyL2=βπ2ωe2=4.97×10-2s-1.

In terms of ωe and β, Eq. (2) becomes ωd=ωe124π2∑m=0∞11+2m2exp⁡-βπ2ωe21+2m2π2τ.

We include the first 25 terms in the sum of Eq. (6) to ensure convergence. Lastly, the depth to basement is calculated with Eq. (1).

The addition of sediment and an isostatic correction from sediment loading of the oceanic crust (e.g., Célérier, 1988) is needed to complete the bathymetry. A parameterized multilayer sediment cover, called “OES sediment thickness” (Figs. 2, 3), was isostatically added on top of the depth to basement ωτ (Fig. 4) to complete the open ocean bathymetry (Fig. 5). OES sediment thickness (Fig. 3) was parameterized based on a third-degree polynomial fit between area-corrected global sediment thickness data (Divins, 2003; Whittaker et al., 2013) and age of the underlying oceanic crust τ. Sediment loading was calculated using a multicomponent sediment layer with varying sediment densities given in Table 2 in 100 m increments of the sediment. The variable sediment densities were calculated from a linear extrapolation of sediment densities in Crosby et al. (2006) (Table S1 in the Supplement). For the isostatic correction, in each 100 m sediment layer we calculate an adjusted thickness given by Dz=100(ρm-ρz)(ρm-ρw), where ρz is the density of the zth layer, ρm= 3300 kg m-3 and ρw= 1000 kg m-3. The sediment model has a total of 16 layers in which the basal layer includes all sediment deeper than 1500 m. For a given location we sum Dz to obtain the isostatically adjusted total sediment thickness, which is then added to the depth to basement to obtain the open ocean bathymetry. This loading correction is similar to procedures used by Crough (1983) and Sykes (1996).

To model the shelf-slope-rise structure, profiles from various modern shelf-slope-rise structures at active and passive margin regions from ETOPO1 were examined, along with their corresponding sediment thicknesses taken from Divins (2003). As a representative active margin, the west coast of South America was chosen (Fig. 6). For passive margins, the Atlantic Ocean (north, south and central) and part of the Southern Ocean were chosen as representative because their complete rifting history is preserved (Figs. 7, S4).

Profiles from these representative regions were used to parameterize the widths of the continental shelf, slope and rise as follows. The basic parameters of the shelf-slope-rise structure (Fig. 8a) include continental shelf width lsh, continental slope width lsl and continental rise width lr. The location of the maximum extent of oceanic crust according to EB08 is labeled M, and another anchor point labeled P marks the boundary between the shelf-slope-rise structure and the open ocean. These are derived as lsh+lsl=M,lsh+lsl+lr=P,lr=-0.290lsl+437.2,lsl+lr=-8.28×10-3lsh2+5.486lsh, where M and P are the distances of the coastline from points M and P, respectively.

The numerical coefficients in Eq. (8a) and (8d) were obtained from fits to ETOPO1 profiles (Figs. 6, 7, S4). In Fig. 8b we plot the width of the slope + rise versus the width of the shelf from a set of passive margin regions that span a range of shelf widths. We then fit a parabola to this data, constraining the parabola to pass through the origin in order to model the structure at active continental margins. We apply this parabolic fit to active margins and to passive margins where the shelf width is less than the parabola maximum, approximately 350 km. Shelves having widths greater than this maximum are treated individually as special cases.

To determine the corresponding depths, we work outward from the coast. First we apply a uniform gradient of 3.2∘ in depth over the width of the shelf. This value of the shelf gradient was obtained from analysis of 17 ETOPO1 transects (Fig. S4). For the depth distribution along the slope and rise, we assume another uniform gradient, as illustrated in Fig. 8a, joining the depth at the shelf break with the depth calculated for the open ocean at point P.

This methodology works for all shelf-slope-rise regions except where the shelf is anomalously extended, for example, north of Siberia, the Falkland Islands region, and the complex regions in Southeast Asia. If the M point is too far from the coastline, so that lsh+lsl>800 km, or too close to the coastline, so that lsh+lsl<100 km, then the relationship among the three widths no longer holds. For these regions we assume that P=M (Fig. 1c). To complete the reconstruction, these regions were filled by interpolation from neighboring regions.

ETOPO1 bathymetry reveals that active margins lack extensive shelves (Fig. 6) and their slope gradient is anomalously large. Likewise, sediment thickness profiles show that active margins have little sediment cover near or far from the coast. In particular, sediment thickness on the shelves of active margins rarely exceeds 250 m and gradually thins out beyond the subduction zone towards the open ocean.

In contrast to active margins, passive margins are characterized by significant shelf-slope-rise regions. Three out of the 16 passive margin cross sections studied are shown in Fig. 7. The extent of the shelf region varies substantially along passive margin coastlines, which accounts for the scatter among the profiles in Fig. 7. For example, in the profile between the southern tips of Africa and South America, the South American side has a very wide, platform-like shelf region that extends for more than 500 km, whereas on the African side the shelf is at most 100 km wide.

The bathymetric gradients at passive continental margin slopes in Fig. S5 vary significantly, from -0.004 to -0.018. Compared to active margins, passive margins are characterized by greater thickness of sediments and more lateral variability. The greater sediment thickness on passive margins and its greater lateral variability are evident in the 13 passive margin transects shown in Fig. S4.

Figure 8 shows the relationship between the widths of the shelves and the widths of the adjacent slope-rise structure. A transect east–northeast of Newfoundland in the northern part of the Atlantic Ocean (Fig. S4, Set 3, center panel) includes 300 km of continental shelf and nearly 900 km of continental slope-plus-rise structure. The presence of the widely extended Gulf of St. Lawrence may contribute to this anomaly.

Our depth-to-basement reconstruction is shown in Fig. 4. The isostatically adjusted, sediment-loaded model bathymetry of the open ocean is shown in Fig. 5, for which only ocean basin areas with ocean crust ages have an assigned bathymetry. The gap between the coastline and open ocean bathymetry is reconstructed with the shelf-slope-rise model described in Sect. 3.3.

The mid-oceanic ridge systems in our open ocean bathymetry in Fig. 4 have an average depth of approximately -2675 m. Away from the mid-ocean ridges, ocean depth increases systematically, and reaches a maximum depth of approximately -5575 m at old crustal ages. In Fig. 5, the open ocean bathymetry is shown with the modeled sediment cover from Fig. 3 isostatically loaded on to it. With this sediment cover added, the bathymetry ranges between -2675 and -4900 m in the open ocean regions and the maximum depth of the reconstructed bathymetry is approximately -6500 m. The depth range between -4900 and -6500 m is associated with old ocean crust (crustal age in the range of τ= 100–120 Ma) along the flanks of the Atlantic, Pacific, Southern and Indian oceans, and the Bay of Bengal.

The addition of the shelf-slope-rise model completes the OESbathy (Fig. 9), though excluded are ocean islands, seamounts, trenches, plateaus and other localized anomalies plus the underlying dynamical topography. Below we evaluate the modeled OESbathy with respect to ETOPO1 and EB08.

Examples of the global ocean sediment thickness data of Divins (2003) are displayed as cross-sectional profiles from the coastline to the abyssal ocean in Figs. 6, 7 and S4. In these profiles, the sediment thickness contribution is shown separately from ETOPO1. These profiles highlight the fact that the greatest sediment accumulations occur in the shelf-slope-rise regions, whereas open ocean regions accumulate far less. Active margins as in Fig. 6 have thin sediment cover, whereas passive margins as in Figs. 7 and S4 have much thicker sediment cover. On the passive margins, lateral heterogeneity in sediment thickness reflects a complex buried topography of the seafloor on which the sediment accumulated. This topography consists of rifted, stretched and sagged lithosphere in kilometer-scale relief, first in-filled by syn-rift sediment and then buried by post-rift sediment (e.g., Watts et al., 2009; Davison and Underhill, 2012). The thickness profiles of the Atlantic margins reflect subsurface graben structures related to the Jurassic–Cretaceous rifting of Pangea (Peron-Pinvidic et al., 2013; Franke, 2013).

The shelf-slope-rise model in Figs. 1 and 8 is based on modern-day bathymetry with three well-defined gradient changes from the coast to the open (deep) ocean. There is no accounting in the model for the complex types of internal architecture in shelf-slope-rise structures just described.

For paleo-ocean reconstructions, extrapolation back through time will produce proportionate narrowing of shelf-slope-rise geometry at passive margins. Highly variable internal structures strongly suggest that simple backward extrapolation may not accurately produce paleo-shelf-slope-rise bathymetries, especially for the oldest paleo-oceans. Rifting depends on local lithospheric strength, mantle dynamics, and global tectonics, all contributing to the evolution of a passive margin in ways that are not easy to parameterize (Ziegler and Cloetingh, 2003; Corti et al., 2004). Thus, additional data such as from seismic profiling and ocean margin drill cores must be consulted before applying these types of corrections for deep time reconstructions.

Lastly, we point out that our shelf-slope-rise formulation constitutes a marked improvement over simple bathymetric interpolation between the coastline and oldest oceanic crust. Bathymetric interpolation would not resolve the extreme differences in slope between shelf and rise, nor would it faithfully represent the heterogeneity in shelf lengths found in the modern ocean.

The Divins sediment thickness (Fig. S2) may be isostatically subtracted from ETOPO1 (Fig. S3) to yield a sediment-stripped bathymetry that should be in isostatic equilibrium with the mantle (Fig. 12a). To detect deviations in this bathymetry from isostatic equilibrium, the OESbathy-modeled depth to basement (Fig. 4), which is in isostatic equilibrium with the mantle (Eqs. 2, 3), is subtracted from the sediment-stripped bathymetry. This residual bathymetry (Fig. 12b) is comparable to the residual basement maps of Müller et al. (2008a; their Fig. 11), with differences attributable to the isostatic corrections applied to sediment removal; and the predicted crustal (depth-to-basement) models OESbathy subjected to the same treatment as ETOPO1 provide a secondary check of our methodology (Fig. 13a). Removing sediments, including their loading, results in a difference map with deeper values than from ETOPO1 with the same sediment correction applied (compare Figs. 12a and 13a). This difference also appears in the residual OESbathy (Fig. 13b), which shows slightly negative mid-ocean ridges, mostly positive coastlines, and very negative terrigenous sediment fans.

It remains unclear whether the differences between true and reconstructed bathymetry produce qualitatively important impacts on climate. One fundamental process for which bathymetry is potentially important is ocean tidal amplitude, which depends sensitively on basin resonances (which in turn depend sensitively on the ocean depth affecting the speed of gravity waves; Arbic et al., 2009). As noted above, both lateral (Krupitsky et al., 1996) and vertical (Sijp and England, 2005) ocean circulation have also been hypothesized to be sensitive to the details of bathymetry. Work to evaluate these sensitivities in modern models will be a future focus of research.

Another key issue concerns reconstructed paleobathymetry with simple vertical ocean margins, i.e., no realistic shelf-slope-rise structures, which if applied to paleo-oceans could result in substantially inaccurate paleoclimate simulations. The shelf-slope-rise structure is known to present-day ocean models but not for paleo-ocean models; the “modular” aspect of the OESbathy reconstruction provides a convenient means to test the effect of shelf-slope-rise structures on modern climate simulations. Obviously, such a test could be undertaken by simply removing the actual shelf-slope-rise structures from ETOPO1, but to our knowledge this has never been done.