_{1}

^{*}

The so-called “typical” values for the Earth’s topography are often used in the literature, such as the mean continental altitude (MCA), the Moho depth for “normal” continental crust, or “typical” depth of mid-oceanic ridges. However, the statistical relevance of those values is hardly discussed. Focussed on data for the global topography, this paper presents statistical analyses regarding various environments. It is shown in particular that the definition of the mid-oceanic ridge is not straightforward, and varies considerably according to what is actually considered: the ridge “inner-rift”, the ridge “crest”, or the “virtual ridge” at spreading centre. This definition is also a function of the spreading rate and has strong implications for the rationale on the age-depth relationship of the sea-floor. In addition, the latter relationship is highly dependent on how the topographic data are corrected from sediment load. The correction itself implies numerous aspects that relies on the precision and associated uncertainties of, in particular, the sediment thickness, sediment porosity, and the mantle, water and sediment densities. In this respect, the analysis carried out here favours a plate cooling model (PCM) for the age-depth dependence of the sea-floor. The topographic elevation at trench proves also to be related to the age of the sea-floor through a different PCM equation. Away from the trench, the oceanic lithosphere is affected by flexuration, for which equations can be defined assuming that the end-load position is not located at trench. On the other hand, the elevation of magmatic arc does not appear to be related to sea-floor age or spreading rate. However, the correlation between the arc-trench distance and the topographic elevation of arc for continental crust seems to be an indicator of slab dip and therefore the existence of slab roll-back processes. Along intra-oceanic magmatic arc, a periodicity in topographic elevation suggests a periodicity in the occurrence of magma chambers, and therefore magmatic processes that need to be further studied. At passive margin, the transition between continental and oceanic crust seems to be relatively sharp in average. Subdivision of the datasets according to the age of the continent-ocean boundary (COB) indicates that rift and passive margin shoulders are found within a couple of degree away from the COB and for ages younger than ca . 20 Ma. Finally, the statistical analysis of continental data assumed to be free of thinning or thickening effects suggests that the MCA should rather be considered in terms of “lowlands” and “highlands”. Relying on model of Moho depth, the “normal” crustal depth might be thinner than commonly accepted. In any case, the filtering of reduced topography can help to determine the impact of dynamic topography.

The present-day topography is the sole example of topography that we can access to. It is consequently used as reference when addressing various issues about the topography in the geological past. Therefore, it is of prime importance to properly characterize this topography on the global scale.

A number of studies in Geosciences use and/or discuss data about the topography of the Earth at global scale, most of the time with implications for palaeogeographies (e.g. [

A series of statistical analysis about the present-day topography of the Earth is provided here and discussed for the following environments: mid-oceanic sp- reading centres, depth-age dependence of the oceanic lithosphere, active margin settings and intra-oceanic subduction zone, passive margin settings, and mean continental altitude (MCA).

The aim is to clarify the relationships between physical processes and resulting topography in order to decipher what parameters can be used in assessing topographies in deep time.

A statistical analysis is highly dependent of the dataset used. For this study, statistics were carried out on the following datasets:

・ Global Relief Model Etopo1 [

・ Global sea-floor age model and associated uncertainties [

・ Global sediment thickness map [

Given the uncertainties associated with the datasets used, all statistics were performed assuming a spherical Earth with radius R_{Mean}:

R Mean = ( R E q . ) 2 × R P o l . 3 = ( 6378.137 ) 2 × 6356.752 3 = 6371.001 km (1)

where, R_{eq}_{.} is the equatorial radius and R_{pol}. the polar radius of the ellipsoid of reference (WGS84). In the following, the Earth’s surface is therefore 5.101 × 10^{8} km^{2} and the volume, 1.083 × 10^{12} km^{3}; one arc-degree corresponds to 111.195 km. In addition, because of the various spatial resolution of the aforementioned datasets, resampling with linear interpolation between original grid cells were carried out using a geodetic grid in order not to overestimate polar regions (^{6} data points, which correspond approximately to one data point every 16 km world-wide. The various examined quantities (topographic elevation, sediment thickness, etc.) are therefore taken at the exact same location.

Data are separated between points lying on crust assumed to be continental in nature from those lying on crust assumed to be continental in nature. The definition of the Continent-Ocean Boundaries have been made by hand by following “at best” both prominent features in the first derivative of the map of the Earth gravity field model (Grace GGM02, [

As first put forward by Otto Krümmel [

Separately, statistics show that neither the data points assumed to lie on continental crust (brown dots in _{oceanic} = −4325 m and m_{continental} = +187 m) are more representative or more robust than the mean values, none of them have a correct meaning either and it might be useful to consider the peak of the data distribution (here, peak_{oceanic} = 6.268 × 10^{4} data

in the bin [−4300 m; −4400 m] and peak_{continental} = 1.108 × 10^{5} data in the bin [+0 m; +100 m], respectively). However, oceanic and continental distributions are relatively well-symmetric to first order, and the differences in elevation according to what values are regarded as more representative are relatively small (difference between the mean and median is Δ(μ − m) = 136 m for oceanic dataset and Δ(μ − m) = 30 m for continental dataset;

Full dataset | ETopo1_Full | Oceanic dataset | ETopo1_Oc | Continental dataset | ETopo1_Con |
---|---|---|---|---|---|

Number of values | 2,621,440 | Number of values | 1,576,027 | Number of values | 1,045,413 |

Percentage of the full dataset | 100.000% | Percentage of the full dataset | 60.121% | Percentage of the full dataset | 39.879% |

Sum | −6,374,802,668 | Sum | −6,601,701,529 | Sum | 226,898,861 |

Minimum | −10,726 | Minimum | −10,726 | Minimum | −9848 |

Maximum | 7446 | Maximum | 3969 | Maximum | 7446 |

Range | 18,172 | Range | 14,695 | Range | 17,294 |

Mean | −2431.794 | Mean | −4188.825 | Mean | 217.042 |

Median | −3280 | Median | −4325 | Median | 187 |

D(m-m) | 848.2 | D(m-m) | 136.2 | D(m-m) | 30.0 |

First quartile | −4543 | First quartile | −4991 | First quartile | −42 |

Third quartile | 64 | Third quartile | −3583 | Third quartile | 568 |

Standard error | 1.5009 | Standard error | 0.8828 | Standard error | 1.1089 |

95% confidence interval | 2.9417 | 95% confidence interval | 1.7303 | 95% confidence interval | 2.1734 |

99% confidence interval | 3.8659 | 99% confidence interval | 2.2739 | 99% confidence interval | 2.8562 |

Variance | 5,905,119.368 | Variance | 1,228,217.177 | Variance | 1,285,422.711 |

Average deviation | 2206.273 | Average deviation | 853.877 | Average deviation | 666.635 |

Standard deviation | 2430.045 | Standard deviation | 1108.250 | Standard deviation | 1133.765 |

Coefficient of variation | −0.9993 | Coefficient of variation | −0.2646 | Coefficient of variation | 5.2237 |

Skew | 0.351 | Skew | 0.922 | Skew | −0.090 |

Kurtosis | 1,572,863.520 | Kurtosis | −5470.538 | Kurtosis | −1671.557 |

Kolmogorov-Smirnov stat | 0.156 | Kolmogorov-Smirnov stat | 0.053 | Kolmogorov-Smirnov stat | 0.189 |

Critical K-S stat, alpha = 0.10 | 0.001 | Critical K-S stat, alpha = 0.10 | 0.001 | Critical K-S stat, alpha = 0.10 | 0.001 |

Critical K-S stat, alpha = 0.05 | 0.001 | Critical K-S stat, alpha = 0.05 | 0.001 | Critical K-S stat, alpha = 0.05 | 0.001 |

Critical K-S stat, alpha = 0.01 | 0.001 | Critical K-S stat, alpha = 0.01 | 0.001 | Critical K-S stat, alpha = 0.01 | 0.002 |

tainties based on standard deviation are useless.

Furthermore, mean values of topographic features may be biased by some outliers related to distinct features. For instance, the determination of the “best” sea-floor age-depth relationship may be considered as biased by the presence of oceanic plateaus or seamounts, deep fracture zones, abandoned arcs, … In order to exclude such potential outliers, a buffer zone has been defined (

The often called “mid-oceanic ridges” are important features of the topography of the Earth. They indeed represent a cumulative length of some 65,000 km (e.g. https://en.wikipedia.org/wiki/Mid-ocean_ridge). However, because this feature is fractal (as are length of coast lines for instance), the cumulative length depends on the resolution of the dataset. Using the present resolution and definition (

The definition of what is a spreading centre and what is a transform fault is not as straightforward as one may first think. It is merely considered herein that a segment of ridge with direction within ±45 from the orientation of the motion vector between the two adjacent tectonic plates is a transform fault, whereas others are spreading centres (see sketch in

The use of ETopo1 [

[

The mean value of all spreading centres at the location of inner rifts is μ = −3250.780 m. However, even if there is a maximum of data close to the median value (m= −3231 m), statistics show that the data are not Gaussian distributed (

All data (inner rifts only) | ETopo1 | All segments corresponding to spreading centres | ETopo1 | All segments corresponding to transform faults | ETopo1 | Spreading centres excluding buffer zones | ETopo1 | |
---|---|---|---|---|---|---|---|---|

Number of values | 30,726 | Number of values | 19,278 | Number of values | 11,448 | Number of values | 12,650 | |

Sum | −103,826,726 | Sum | −62,668,546 | Sum | −41,158,180 | Sum | −41,303,926 | |

Minimum | −9758 | Minimum | −9758 | Minimum | −7666 | Minimum | −6590 | |

Maximum | 849 | Maximum | 755 | Maximum | 849 | Maximum | −445 | |

Range | 10,607 | Range | 10,513 | Range | 8515 | Range | 6145 | |

Mean | −3379.116 | Mean | −3250.780 | Mean | −3595.229 | Mean | −3265.132 | |

Median | −3351 | Median | −3231 | Median | −3534 | Median | −3218 | |

D(m-m) | −28.1 | D(m-m) | −19.8 | D(m-m) | −61.2 | D(m-m) | −47.1 | |

First quartile | −4044 | First quartile | −3938 | First quartile | −4266 | First quartile | −3760 | |

Third quartile | −2753 | Third quartile | −2677 | Third quartile | −2965 | Third quartile | −2740 | |

Standard error | 5.8815 | Standard error | 6.9695 | Standard error | 10.2448 | Standard error | 6.7341 | |

95% confidence interval | 11.5274 | 95% confidence interval | 13.6594 | 95% confidence interval | 20.0781 | 95% confidence interval | 13.1978 | |

99% confidence interval | 15.1487 | 99% confidence interval | 17.9501 | 99% confidence interval | 26.3840 | 99% confidence interval | 17.3430 | |

Variance | 1,062,886.233 | Variance | 936,398.946 | Variance | 1,201,540.319 | Variance | 573,656.248 | |

Average deviation | 792.689 | Average deviation | 753.516 | Average deviation | 835.902 | Average deviation | 599.856 | |

Standard deviation | 1030.964 | Standard deviation | 967.677 | Standard deviation | 1096.148 | Standard deviation | 757.401 | |

Coefficient of variation | −0.3051 | Coefficient of variation | −0.2977 | Coefficient of variation | −0.3049 | Coefficient of variation | −0.2320 | |

Skew | 0.213 | Skew | 0.366 | Skew | 0.188 | Skew | −0.173 | |

Kurtosis | 1.257 | Kurtosis | 1.233 | Kurtosis | 1.219 | Kurtosis | 0.569 | |

Kolmogorov-Smirnov stat | 0.044 | Kolmogorov-Smirnov stat | 0.052 | Kolmogorov-Smirnov stat | 0.041 | Kolmogorov-Smirnov stat | 0.044 | |

Critical K-S stat, alpha = 0.10 | 0.007 | Critical K-S stat, alpha = 0.10 | 0.009 | Critical K-S stat, alpha = 0.10 | 0.011 | Critical K-S stat, alpha = 0.10 | 0.011 | |

Critical K-S stat, alpha = 0.05 | 0.008 | Critical K-S stat, alpha = 0.05 | 0.010 | Critical K-S stat, alpha = 0.05 | 0.013 | Critical K-S stat, alpha = 0.05 | 0.012 | |

Critical K-S stat, alpha = 0.01 | 0.009 | Critical K-S stat, alpha = 0.01 | 0.012 | Critical K-S stat, alpha = 0.01 | 0.015 | Critical K-S stat, alpha = 0.01 | 0.014 |

in general, about 300 m - 350 m deeper than inner rifts of spreading centres (344.5 m deeper according to mean values, and 303 m according to median values;

The age of the sea-floor at spreading centres shall be, by definition, 0 million years (Ma). Because the definition of ridges by Müller et al. [

It is commonly accepted that slow spreading ridges have deep inner rifts whereas fast ridges have not. The spreading rate is therefore an important parameter, but as for sea-floor ages, the resolution of the dataset may impact results. The amount of accreted surface of sea-floor per year (in mm^{2}/yr) has been calculated from Euler poles and angles at each spreading centre segment (

The topographic elevation of ridge inner rifts as function of spreading rates has thus been tested (

The mid-oceanic ridges are indeed deeper in average with slow accretion rates, but the dispersion of data is maximum for slow accretion rates (even when potential outliers are excluded using the buffer zones defined in ^{2}/yr (

one may think a power law might rule the system.

Statistics on the elevation of ridges are considered here as more relevant from analysis of ridge crests, ridge flanks and “virtual ridge axes” than from analysis of ridge inner rifts. For such analysis, data were first selected within the area accreted since chron C3 (ca. 6 Ma;

In order to have a clearer picture, the distance to ridge axis has been divided into 0.025 bins. Although the distributions per bin are not Gaussian from a statistical point-of-view, they are relatively symmetric and mean and median values are relatively close (generally below 100 m in difference; Annexe 1 in supplementary data). Mean values per bin are displayed in

of the flank, and −2695.038 m using data of the flank from the farthest section, respectively.

However, we previously saw (^{2}/yr). Using running averages (50 data sliding window), the relationship of the elevation of the inner rifts to accretion rates becomes clear (^{2}/yr and associated with troughs at ridge axis (inner rift), whereas ridge crest are mostly lower (deeper in bathymetry) than ridge axis elevation when rates are higher. Those relationships between topographic elevations, distances to ridge axis and accretion rates are also well perceptible in

If we now consider that the elevation of the ridge axis is not the true elevation affected by processes acting in inner rift but a virtual elevation resulting from the extrapolation of linear fitting on ridge flank (as the red line of inset of

In conclusion, the depth definition of the often called “mid-oceanic ridges” is subject to caution, since values may vary quite considerably according to what features are taken into account. This issue has a particularly strong implication in the definition of an age-depth dependence of the entire sea-floor.

The realization that sea-floor topography (bathymetry) is highest at mid-oceanic ridges and decreases with distance is at least as old as the advent of Plate Tectonics (e.g. [

raphic data and 55 Ma for heat flow data) and their physical meaning remain obscure.

In addition, the equations for the GDH1 model are of the form:

d ( t ) = d R + f ⋅ t , when t < 20 Ma (i.e. young-aged sea-floor) (2.1)

Bin for accretion rates: | [0.0 - 0.5] | [0.5 - 1.0] | [1.0 - 1.5] | [1.5 - 2.0] | [2.0 - 2.5] | [2.5 - 3.0] | [3.0 - 3.5] | [3.5 - 4.0] | [4.0 - 4.5] | [4.5 - 5.0] | [5.0 - 5.5] | [5.5 - 6.0] | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

From all ridge flank (0.15˚ - 1.333˚): | |||||||||||||

Virtual ridge depth= | −2452.410 | −2562.439 | −2599.913 | −2860.311 | −2711.659 | −2792.679 | −2946.689 | −3000.700 | −2957.560 | −2964.489 | −3074.472 | −2652.467 | |

Slope= | −617.053 | −451.884 | −299.454 | −207.844 | −171.049 | −99.264 | −124.367 | −159.375 | −66.239 | −176.741 | −118.148 | −310.463 | |

Number of data = | 16388 | 18264 | 7237 | 5177 | 4350 | 1777 | 1640 | 922 | 749 | 348 | 150 | 86 | |

R^{2}= | 4.151% | 4.096% | 2.376% | 2.342% | 0.777% | 0.280% | 1.864% | 6.317% | 0.264% | 13.381% | 12.315% | 35.953% | |

From near flank (0.15˚ - 0.80˚): | |||||||||||||

Virtual ridge depth= | −2430.281 | −2488.134 | −2545.865 | −2859.507 | −2709.234 | −2823.927 | −2946.496 | −3003.177 | −2918.789 | −2937.961 | −3076.350 | −2623.863 | |

Slope= | −662.054 | −625.472 | −422.651 | −204.000 | −166.328 | −22.341 | −126.939 | −152.337 | −144.727 | −237.841 | −117.288 | −386.070 | |

Number of data = | 11097 | 10296 | 4037 | 2874 | 2432 | 996 | 912 | 507 | 407 | 196 | 82 | 45 | |

R^{2}= | 1.468% | 2.654% | 1.503% | 0.767% | 0.246% | 0.006% | 0.860% | 2.062% | 0.692% | 8.160% | 3.504% | 15.509% | |

From far flank (0.75˚ - 1.333˚): | |||||||||||||

Virtual ridge depth= | −2641.282 | −2689.400 | −2703.573 | −2897.806 | −2813.491 | −2796.377 | −2914.320 | −3026.533 | −3158.611 | −2908.340 | −3042.853 | −2702.415 | |

Slope= | −434.943 | −325.922 | −196.573 | −172.809 | −76.838 | −100.715 | −153.724 | −136.669 | 121.294 | −221.144 | −147.733 | −266.077 | |

Number of data = | 6024 | 8747 | 3507 | 2532 | 2109 | 858 | 800 | 457 | 369 | 168 | 72 | 46 | |

R^{2}= | 0.778% | 0.496% | 0.238% | 0.361% | 0.035% | 0.057% | 0.513% | 1.066% | 0.144% | 4.701% | 5.311% | 14.616% |

d ( t ) = d R + α ρ M ( T M − T 0 ) Z L 2 ( ρ M − ρ W ) ( 1 − 8 π 2 exp ( − κ π 2 Z L 2 t ) ) , when t ≥ 20 Ma (i.e.

old-aged sea-floor) (2.2)

where,

d(t) is the sea-floor topography (depth in m) as function of age t (in second but usually converted into Ma with a conversion factor τ = 3.15576.10^{+13} s per Ma),

d_{R} is the depth at the mid-oceanic ridge (in m),

f is an ad hoc factor (dimensionless, hereafter shown with the sign Ø),

α is the volume coefficient of thermal expansion (in 1/kelvin or K^{−}^{1}),

κ is the thermal diffusivity (in mm^{2}・s^{−}^{1}),

ρ_{M} and ρ_{W} are the mantle and water densities respectively (or mass per unit volume in kg・m^{−}^{3}),

Z_{L} is the asymptotic thermal plate thickness (i.e. at infinite distance from the ridge, in m),

T_{M} and T_{0} are respectively the basal and the top temperature of the thermal plate (in K).

It must be noticed that both equations are dependent upon d_{R}, which we saw, may vary in definition on the one hand, and varies as function of spreading rates on the other hand.

Furthermore, those equations hold for thermal plates. On oceanic sea-floor however, the crust is covered by sediments, and isostatic calculation shall first be carried out to correct for loading effects. The estimate for sediment thicknesses is subject to caution and the method for correction is not as straightforward as commonly thought.

The isostatic correction for sediment load is crucial to define properly the age- depth dependence of the sea-floor. Although the Veining-Meinesz method may be regarded as a more comprehensive technique for accounting for isostatic correction, the method is complex (e.g. [

C = ( ρ W − ρ S ρ W − ρ M ) ⋅ h S (3)

The correction thus only depends on the mean sea water density ρ_{W}, the mean sediment density ρ_{S}, and the mean mantle density above compensation depth ρ_{M}.

The sea water density can be computed from the International Equation of State of Seawater [

The sea water density ρ_{W} required for the isostatic correction C corresponds to the mean value over the water column, i.e. corresponds to the integration of sea water densities throughout every water columns. The mean sea water density ρ_{W} is shown as function of the height of the sea water column in red in

within bounds in deep yellow.

The density of the sediment cover at global scale is obviously a much more challenging issue. Multiple factors impact the sediment density, the two most important of which are certainly the lithology and the compaction. Winterbourne et al. [_{S} over the sediment pile can be calculated using:

ρ S = 1 h S ∫ 0 h S ( ϕ ρ i g W + ( 1 − ϕ ) ρ g ) d z (4.1)

with,

ϕ = ϕ 0 e ( − z / λ ) after Athy [

where,

h_{S} is the height of the sediment pile or sediment thickness,

ρ_{igW} is the intergranular water density or water density in sediment pores,

ρ_{g} is the density of the solid sediment grains,

φ is the porosity as function of height of sediment,

φ_{0} is the initial porosity,

z is the compaction decay wavelength,

z is the depth within the sediment pile.

Assuming ρ_{igW} = 1013 kg・m^{−}^{3}, ρ_{g} = 2650 kg・m^{−}^{3} and ρ_{M} = 3300 kg・m^{−}^{3}, Winterbourne et al. [_{0} = 0.56 and compaction decay length λ = 4.5 km.

Audet & Fowler [_{W} = 1027.6 kg・m^{−}^{3} (for T = 4 C and S = 34.7‰; P_{atm} = 101,325 Pa), which is higher than the 1013 kg・m^{−}^{3} used by Winterbourne et al. [_{igW} must certainly be even higher.

Now, for sake of simplicity, the present study nonetheless follows the equations of Winterbourne et al. [

φ 0 = 0.5616 ± 0.1003 [Ø]

λ = 4578.6 ± 1112.3 [ m ]

In Winterbourne et al. [_{g} = 2650 ± 250 [kg・m^{−}^{3}]. It is found here (see below, inset in _{g} = 2300 ± 500 [kg・m^{−}^{3}] (“best” value of ρ_{g} = 2292 [kg・m^{−}^{3}]) both improve the correction and is more coherent with the diversity of grains in sediment. The intergranular water density ρ_{igW} is assumed to merely follow the International Equation of State of Seawater (purple curve and associated pink uncertainties in

The mean mantle density above compensation depth is just defined as ρ_{M} = 3150 ± 300 [kg・m^{−}^{3}].

In order to illustrate the amount of correction and associated uncertainties those values correspond to, the sediment density as function of sediment pile and the corresponding correction factor C are defined in _{W} = −4325 m (mean global ocean depth).

The topographic dataset used for age-depth relationship corresponds to all oceanic data (after ETopo1 [

Because the number of data points is large (N = 798,458), age bins of 1 Ma have been used to obtain subdatasets on which statistics have been determined.

Although none of the distributions per bin is properly Gaussian from statistical point-of-view, all are relatively well-symmetrical, “bell-shaped” with median values close to the mean (

Consequently, the mean topographic elevation μ per 1 Ma bin with associated 2σ error bars are plotted against the age of the sea-floor in

It must be noticed that topographic elevations accounting for uncertainties (i.e. uncertainties on the raw datasets augmented by uncertainties on sediment load correction as described above) mainly increase the dispersion around the mean values defined per 1 Ma bin, but the mean values (μ− and μ+) are not themselves drastically shifted up or down (_{igW}, ρ_{g}, ρ_{M}, φ_{0}, λ) might therefore consist in minimizing the dispersion around the mean in every 1 Ma bin. Nevertheless, the difference between μ− and μ+ has been used to search for the “best” value ρ_{g}, the density of solid grain in sediment, which is required in Equation (4.1). The cumulative difference between μ− and μ+ is minimal for ρ_{g} = 2292 kg・m^{−}^{3} (Inset in _{g} = 2300 ± 500

kg・m^{−}^{3}.

A square-root mathematical equation (HSCM) can fit the relationship between depth (μ) and age within 2σ error bars―and even more easily within bounds when uncertainties are accounted for; i.e. (μ+) + 2σ and (μ−) − 2σ―but the mean (or median) values prove to be badly fitted. The “best” HSCM (shown in pink in

d ( t ) = X + Y ⋅ t = − 3425.880 − 186.594 × t , (5.1)

with coefficient of determination R^{2} = 88.3%.

As emphasized by previous authors, the fit is much better when applied on data for the first 70 Ma (e.g. [

d ( t ) = X + Y ⋅ t = − 2758.431 − 30173 × t , (5.2)

with coefficient of determination R^{2} = 99.5% (on data younger than 70 Ma).

The fit using an exponential equation (PCM) is much better. Excluding data younger than 3 Ma because of the presence of the mid-oceanic inner rift (see Section 3.1), the PCM equation (green in

d ( t ) = A + B ⋅ exp ( − C ⋅ t ) = − 5632.290 + 2527.251 × exp ( − 0.02554 × t ) , (6)

with coefficient of determination R^{2} = 98.9%.

Instead of dividing the topographic elevation data according to their age (age bins), the dataset can be divided relative to the spreading rates (rate bins) as defined by Müller et al. [^{−}^{1}, however, the number of data per sub-datasets becomes small, and the increase in coefficient B and C suggested by polynomial fit is likely an artefact linked to the natural shape of 3^{rd} degrees equations. It is suspected that the value of the different coefficients stabilize within the 95% confidence level as spreading rates keep on growing.

Besides the coefficients correspond to:

A = d R + α ρ M ( T M − T 0 ) Z L 2 ( ρ M − ρ W ) ; B = − α ρ M ( T M − T 0 ) Z L 2 ( ρ M − ρ W ) . 8 π 2 ; C = − κ π 2 Z L 2 (7)

in the equation provided by Stein & Stein ( [_{R} corresponds to the “virtual” ridge axis (see analysis of the mid-oceanic ridges above;

Rate Bin | N | A | B | C | R^{2} | d_{R} |
---|---|---|---|---|---|---|

1 | 3527 | −5316.959 | 1014.853 | 0.04367 | 9.25% | −4064.93 |

3 | 2799 | −6118.764 | 2096.280 | 0.02049 | 16.94% | −3532.58 |

5 | 3799 | −5830.703 | 2098.546 | 0.02135 | 22.53% | −3241.73 |

7 | 5751 | −5833.222 | 2556.115 | 0.02132 | 36.87% | −2679.74 |

9 | 10,362 | −5870.105 | 2897.157 | 0.01991 | 49.20% | −2295.88 |

11 | 16,505 | −5562.654 | 3003.276 | 0.02619 | 52.52% | −1857.51 |

13 | 22,792 | −5393.061 | 2744.569 | 0.02934 | 53.67% | −2007.09 |

15 | 28,311 | −5608.740 | 2628.115 | 0.02523 | 53.63% | −2366.43 |

17 | 30,469 | −5635.213 | 2714.170 | 0.02776 | 53.88% | −2286.74 |

19 | 35,283 | −5749.779 | 2866.548 | 0.02406 | 65.58% | −2213.32 |

21 | 36,449 | −5640.174 | 2819.417 | 0.02715 | 67.57% | −2161.86 |

23 | 35,138 | −5462.335 | 2684.412 | 0.03125 | 60.57% | −2150.57 |

25 | 34,115 | −5533.051 | 2660.040 | 0.02858 | 56.93% | −2251.36 |

27 | 33,860 | −5610.924 | 2708.482 | 0.02904 | 60.64% | −2269.47 |

29 | 37,618 | −5753.818 | 2840.822 | 0.02665 | 71.16% | −2249.09 |

31 | 35,682 | −6007.830 | 2798.322 | 0.02097 | 71.67% | −2555.54 |

33 | 35,172 | −6231.765 | 2976.717 | 0.01797 | 74.56% | −2559.39 |

35 | 29,119 | −5568.833 | 2396.410 | 0.02650 | 66.47% | −2612.38 |

37 | 30,151 | −5616.449 | 2407.637 | 0.02669 | 66.67% | −2646.15 |

39 | 20,484 | −5594.930 | 2547.755 | 0.03056 | 62.45% | −2451.76 |

41 | 17,367 | −5633.495 | 2588.690 | 0.02702 | 61.77% | −2439.83 |

43 | 15,321 | −5716.029 | 2502.104 | 0.02152 | 57.65% | −2629.18 |

45 | 12,485 | −5496.043 | 2454.802 | 0.02850 | 60.65% | −2467.55 |

47 | 14,440 | −5468.659 | 2443.662 | 0.03038 | 69.44% | −2453.91 |

49 | 9419 | −5530.076 | 2440.006 | 0.02710 | 65.79% | −2519.84 |

51 | 9062 | −5599.072 | 2390.794 | 0.02496 | 59.18% | −2649.55 |

53 | 9017 | −5732.973 | 2520.546 | 0.02193 | 58.80% | −2623.37 |

55 | 7304 | −5623.763 | 2452.729 | 0.02434 | 59.71% | −2597.83 |

57 | 9727 | −5561.822 | 2564.116 | 0.02969 | 67.02% | −2398.47 |

59 | 9794 | −5566.580 | 2523.694 | 0.02870 | 67.83% | −2453.10 |

61 | 8415 | −5599.637 | 2488.689 | 0.02678 | 69.24% | −2529.34 |

63 | 9820 | −5690.045 | 2476.506 | 0.02301 | 71.59% | −2634.78 |

65 | 12,668 | −5619.003 | 2484.961 | 0.02605 | 71.52% | −2553.31 |

67 | 11,906 | −5701.874 | 2616.598 | 0.02391 | 73.41% | −2473.78 |

69 | 11,138 | −5820.804 | 2655.983 | 0.01847 | 69.81% | −2544.12 |

71 | 9995 | −5654.646 | 2522.371 | 0.02209 | 67.79% | −2542.80 |

73 | 13,152 | −5678.048 | 2616.927 | 0.02565 | 74.55% | −2449.54 |

75 | 8241 | −5724.068 | 2500.552 | 0.01958 | 69.99% | −2639.13 |

77 | 9206 | −5716.711 | 2454.034 | 0.01872 | 66.83% | −2689.17 |

79 | 5995 | −5551.384 | 2347.610 | 0.02410 | 65.78% | −2655.14 |

81 | 6182 | −5548.405 | 2441.323 | 0.02693 | 70.66% | −2536.54 |

83 | 5717 | −5594.010 | 2440.536 | 0.02602 | 70.64% | −2583.12 |
---|---|---|---|---|---|---|

85 | 6173 | −5922.604 | 2730.984 | 0.02146 | 79.31% | −2553.39 |

87 | 3919 | −5608.449 | 2300.972 | 0.02052 | 67.81% | −2769.74 |

89 | 3991 | −5762.094 | 2449.970 | 0.01872 | 71.81% | −2739.56 |

91 | 3660 | −5627.699 | 2293.941 | 0.02137 | 69.75% | −2797.66 |

93 | 2690 | −5772.420 | 2389.686 | 0.01667 | 66.05% | −2824.26 |

95 | 3188 | −5656.486 | 2350.696 | 0.02074 | 68.86% | −2756.43 |

97 | 3282 | −5573.957 | 2350.762 | 0.02497 | 65.33% | −2673.82 |

99 | 3381 | −5638.498 | 2356.601 | 0.02186 | 65.76% | −2731.16 |

102.5 | 8454 | −5643.653 | 2316.986 | 0.02414 | 69.91% | −2785.19 |

107.5 | 7956 | −6004.234 | 2539.286 | 0.01313 | 71.22% | −2871.52 |

112.5 | 5410 | −5372.904 | 2162.869 | 0.02590 | 58.40% | −2704.57 |

117.5 | 5829 | −5445.186 | 2185.277 | 0.02328 | 76.59% | −2749.21 |

122.5 | 4829 | −5459.457 | 2088.414 | 0.02172 | 73.45% | −2882.98 |

127.5 | 4489 | −5253.976 | 2380.026 | 0.04202 | 74.21% | −2317.74 |

132.5 | 2823 | −5680.783 | 2177.532 | 0.01629 | 62.66% | −2994.36 |

137.5 | 3121 | −5386.019 | 2775.200 | 0.04021 | 75.09% | −1962.25 |

142.5 | 1898 | −5420.062 | 2795.848 | 0.03431 | 69.04% | −1970.82 |

147.5 | 1272 | −5752.568 | 2713.227 | 0.02527 | 73.08% | −2405.26 |

Section 3.2) and is given as d_{R} = 2600 m.

The definition of d_{R} from the coefficients determined in

To do so, the genuine equation for Plate Cooling Model (PCM; [

d ( t ) = ρ M α ( T M − T 0 ) Z L 2 ( ρ M − ρ W ) ( 1 − 4 π ∫ 0 1 ∑ n = 1 ∞ 1 n exp ( − κ n 2 π 2 Z L 2 t ) sin ( n π Z ) d Z ) (8.1)

After evaluation of the integral [

d ( t ) = ρ M α ( T M − T 0 ) Z L 2 ( ρ M − ρ W ) ( 1 2 − 4 π 2 ∑ m = 0 ∞ 1 ( 1 + 2 m ) 2 exp ( − κ n 2 π 2 Z L 2 t ) ) (8.2)

The Bayesian-Markov chain Monte-Carlo inversion method employed by Scholer [_{M}, α, T_{M} (T_{0} is assumed to be the temperature defined at the sea-floor as in _{L} (ρ_{W} being equally defined as above).

This stochastic inversion simply consists in randomly choosing values for each parameter within given bounds, and in only accepting resulting curves from Equation (8.2) that fit within defined limits.

Here, the parameters were chosen with rather large and conservative ranges as follow:

ρ M ∈ [ 0 ; 5000 ] (in kg・m^{−3}) ; mean density of the mantle above compensation depth, which value is commonly chosen around 3300 kg・m^{−3} in the literature.

α ∈ [ 0 ; 10 × 10 − 5 ] (in K^{−1}) ; volume coefficient of thermal expansion, which value is commonly chosen around 3 × 10^{−5} K^{−1} in the literature.

T M ∈ [ 0 ; 5000 ] (in K); Temperature at compensation depth, which value is commonly chosen around 1625 K in the literature, so that ( T M − T 0 ) ≃ 1350 ˚ C .

Z L ∈ [ 0 ; 200000 ] (in m); Thermal plate thickness, which value is commonly chosen around 125 km in the literature.

κ ∈ [ 0 ; 10 ] (in mm^{2}・s^{−1} or × 10^{−6} m^{2}・s^{−1}) ; Thermal diffusivity, which value is commonly chosen around 1 mm^{2}.s^{-1} in the literature.

Note that in Equation (8.2), the sum is carried out up to m = 10,000.

For sake of clarity,

The acceptance of the resulting curves was defined in two ways: 1) curves that fit within two standard errors around the mean values per bin of age (μ ± 2σ), and 2) curves that fit within 95% of the number of data between the minimum and maximum values for every bin that are the closest to the median (m ± δ_{95%}). The second method has the advantage to discard the main outliers (5% of the subdatasets) and to be independent of the data distribution. However the range of acceptable curves around the median is larger (_{95%} respectively, bounds have been smoothed using polynomial functions. The simulation stops after 5000 curves fit within bounds. Because the possibility for simulated curves to fit within bounds is smaller around the mean than around the median, the number of iterations was much larger to obtain the 5000 curves using (μ ± 2σ) than using (m ± δ_{95%}) (N_{iter} = 184,603 and N_{iter} =

26,888 respectively).

The distributions of potential values for the aforementioned parameters (

In order to see the effect of limiting the range of possibilities, the same computation has been carried out using the upper and lower bounds corresponding to the average mean topographic data accounting for uncertainties (μ+ and μ− as in

ρ M = 3150 ± 300 , i.e. ρ M ∈ [ 2850 ; 3450 ] (in kg・m^{−3}); Mean density of the mantle above compensation depth,

α = 3 × 10 − 5 ± 4 × 10 − 5 , i.e. α ∈ [ 0 ; 7 × 10 − 5 ] (in K^{−1}); Volume coefficient of thermal expansion,

Δ T = 1350 ± 350 , i.e. Δ T ∈ [ 1000 ; 1700 ] (in K); Difference in temperature between the base and the top of the thermal plate,

Z L = 125000 ± 50000 , i.e. Z L ∈ [ 75000 ; 175000 ] (in m); Thermal plate thickness,

κ = 1 ± 5 , i.e. κ ∈ [ 0 ; 6 ] (in mm^{2}・s^{−}^{1}); Thermal diffusivity.

The number of iteration has to reach 2,728,362 to obtain only 100 accepted curves (

Similarly, the outcome for each parameter does not allow clearly defining favoured values (

μ ( ρ M ) = 3144.930 kg ⋅ m − 3 and m ( ρ M ) = 3154.485 kg ⋅ m − 3 ,

μ ( α ) = 3.569 × 10 − 5 K − 1 and m ( α ) = 3.351 × 10 − 5 K − 1 ,

μ ( Δ T ) = 1359.687 K and m ( Δ T ) = 1367.602 K ,

μ ( Z P l a t e ) = 130786.628 m and m ( Z P l a t e ) = 133353.594 m ,

μ ( κ ) = 1.079 × 10 − 6 m 2 ⋅ s − 1 and m ( κ ) = 1.074 × 10 − 6 m 2 ⋅ s − 1 ,

μ ( Z R i d g e ) = 2839.611 m and m ( Z R i d g e ) = 2834.526 m .

Stein & Stein [

Data related to subduction zones are shown as blue hatched zones in

Trenches are defined in

Old ages of the sea-floor (age > 180 Ma) exists in Müller et al.’s [

After correction from sediment load (as detailed above), and when data are cleared from main disturbing features (use of the exclusion buffer zones other than subduction zone buffer themseleves, as shown in _{T} with age (t) are:

d T ( t ) = A × t + B = − 22.219 × t − 5615.355 ; R 2 = 58.43 % (9.1)

d T ( t ) = A + B ⋅ exp ( − C ⋅ t ) = − 8619.514 + 4079.148 × exp ( − 0.01916 × t ) ; R 2 = 67.63 % (9.2)

On the contrary, any relationship between sea-floor depth and the rate at which the lower tectonic plate subducts beneath the upper one cannot be determined (^{−}^{1}) and the age of the sea-floor entering subduction (in Ma).

Looking at data according to their distance to the trench (maximum distance of 9 corresponding to data within blue buffer in

The flexuration can be modelled using equations for the bending of an elastic lithosphere (e.g. [

of the distance (x) [m] from a loading point is written as follows:

w ( x ) = w b 2 ⋅ exp ( π 4 ) . exp ( − π 4 ( x − x 0 x b − x 0 ) ) ⋅ sin ( π 4 ( x − x 0 x b − x 0 ) ) (10.1)

where,

Equation (10.2)― w b = − w 0 ⋅ exp ( − 3 π 4 ) ⋅ cos ( 3 π 4 ) ; w_{b} is the relative elevation

of the flexural bulge and w_{0} is the maximum depression (in m) relative to the plate elevation far from the end load (x → ∞),

Equation (10.3)― x 0 = π 2 φ ; x_{0} is the distance at which the bended plate

crosses the elevation of the plate far from the end load (x → ∞),

Equation (10.4)― x b = 3 π 4 φ ; x_{b} is the distance of the bulge to the end load, i.e.

the location of the highest point due to flexuration,

Equation (10.5)― φ = ( 1 1000 ) × ( 4 E h e 3 12 ( 1 − ν 2 ) / ( ρ M − ρ W ) ⋅ g ) 1 4 ; φ is the flexure

parameter, with:

E: the young modulus in Pascal [Pa].

h_{e}: the elastic thickness of the plate in metre [m].

ν: the Poisson’s coefficient [Ø].

g: the gravitational acceleration [m・s^{−}^{2}].

ρ_{M} and ρ_{W}: the density of the mantle and the density of the water column respectively [kg・m^{−}^{3}].

The equation expressing the depth due to flexuration as function of distance to an end load is thus of the form:

z ( x ) = A + B × exp ( − C × x − ( π / 2 ) ) × sin ( C × x − ( π / 2 ) ) = A − B × exp ( − C × x − ( π / 2 ) ) × cos ( C × x ) (11)

Seeking the “best” fit, the parameters found are:

A = − 4528.251 ; B = 5078.168 ; C = 2.98073 ;

And the resulting curve is shown in green in

In theory, if the end load is applied at trench, the maximal depression w_{0} corresponds to the difference between the elevation at trench and the elevation far from the trench, and w_{0} can be estimated from Equation (6) and Equation (9.2).

As shown in _{T}(25) = −6092.880 m. The maximum depression would therefore be w 0 = d ( 25 ) − d T ( 25 ) = 1795.190 m . The flexural bulge would have a relative elevation of w_{b} = 120.313 m. Assuming:

A = d ( 25 ) = − 4297.689 ; B = w b ⋅ ( 2 ) 1 / 2 . exp ( π / 4 ) = 373.183 ; C = 2.98073 as before;

The flexuration model is shown as pink curve in

The reasons are: 1) the peak in natural log distribution of age is pk ≈ 3.55 (instead of the value of 3.184 with the mean), which corresponds to a sea-floor age of roughly 35 Ma and sea-floor depth from PCM equation of d(35) = −4598.500 m. This value is much more consistent with the mean value of d = −4599.630 m

determined from a linear fit on data located between 4 and 9 away from trench (deep yellow line in

Assuming the position of the force responsible for the bending of the elastic plate is not located at trench (i.e. end load position ≠ trench position), the best fitting parameters can be found with eq.11 re-written as:

z ( x ) = A + B × exp ( − C × ( x − D ) − ( π / 2 ) ) × sin ( C × ( x − D ) − ( π / 2 ) ) , (12)

with:

A = − 4583.442 ; B = 25358.096 ; C = 1.12803 ; D = 0.9503.

Using Equation (12), the flexural model (blue in ^{2} = 80.26%). The end load is located −0.95 or 105.671 km in the back of the trench (i.e. in the direction of the upper plate). The elevation of the bulge is w_{b} = 353.290 m above the plate elevation far from the end load (x → ∞) found to be A = −4583.442 m. The bulge location is defined at x_{b} = 1.138 from the trench.

The Bayesian-Markov chain Monte-Carlo inversion method has been used here as well (_{95%}).

The parameters were chosen within the following ranges:

ρ_{W} is defined from the International Equation of State of Seawater [

ρ M = 3150 ± 300 i.e. ρ M ∈ [ 2850 ; 3450 ] (in kg・m^{−}^{3}); mean density of the mantle above compensation depth, which value is commonly chosen around

3300 kg・m^{−}^{3} in the literature.

g is chosen constant and g = 9.80665 m・s^{−}^{2}.

v = 0.25 ± 0.15 i.e. v ∈ [ 0.1 ; 0.4 ] (Ø) ; the Poisson’s coefficient value is commonly chosen around 0.25 in the literature.

h e = 50 ± 50 i.e. h e ∈ [ 0 ; 100 ] (in ×10^{3} m) ; the elastic thickness of oceanic lithosphere is commonly chosen around 50 × 10^{3} m in the literature (see in particular [

E = 10 ± 5 i.e. E ∈ [ 5 ; 15 ] (in ×10^{10} Pa) ; the Young’s modulus value is commonly chosen around 7 × 10^{10} Pa in the literature.

The outcome for the parameters used in Equation (10) does not allow clearly defining favoured values (

μ ( ρ M ) = 3152.340 kg ⋅ m − 3 and m ( ρ M ) = 3150.950 kg ⋅ m − 3 ,

μ ( ν ) = 0.25070 and m ( ν ) = 0.24874 ,

μ ( h e ) = 40938 m a n d m ( h e ) = 44255 m ,

μ ( E ) = 9.88249 × 10 10 Pa and m ( E ) = 9.88255 × 10 10 Pa ,

μ ( w 0 ) = − 1289.030 m and m ( w 0 ) = − 2590.540 m ,

μ ( E n d L d ) = 2.484 ∘ and m ( E n d L d ) = 2.5113 ∘ .

The set of data that belong to the subduction zones (i.e. within 9 from the trench) have been divided according to the age of the sea-floor (per bins of 10 Ma). For every subdataset, a curve using Equation (12) was fitted (^{rd} order polynomial fits does not highly depart from linear fits with equations:

B = 542.191 × ( Age ) + 1055.494 ; R^{2} = 41.0% (relative to data); R^{2} = 97.4% (relative to 3^{rd} polynomial fit);

C = − 0.00401 × ( Age ) + 1.466 ; R^{2} = 7.25% (relative to data); R^{2} = 90.0% (relative to 3^{rd} polynomial fit);

Bin | Age | Param. A | Param. B | Param. C | Param. D | N | R^{2} |
---|---|---|---|---|---|---|---|

[000 - 010] | 5 | −3197.3978 | 4969.6705 | 1.152340 | −0.151244 | 1377 | 7.32% |

[010 - 020] | 15 | −3541.4213 | 8465.7563 | 2.074623 | −0.120540 | 1461 | 5.11% |

[020 - 030] | 25 | −4078.8420 | 18,441.8052 | 1.395312 | −0.687011 | 1767 | 5.37% |

[030 - 040] | 35 | −4419.5271 | 19,641.1181 | 1.191842 | −0.760313 | 1340 | 4.78% |

[040 - 050] | 45 | −4694.6329 | 30,350.5987 | 0.971203 | −1.116648 | 1361 | 7.01% |

[050 - 060] | 55 | −4808.0226 | 17,324.1453 | 1.521831 | −0.701405 | 1030 | 0.99% |

[060 - 070] | 65 | −4733.6068 | −1692.6036 | 1.719051 | 1.096102 | 873 | 1.71% |

[070 - 080] | 75 | −5067.2774 | 60,462.6872 | 0.449038 | −3.569603 | 723 | 9.14% |

[080 - 090] | 85 | −5614.0248 | 97,310.5232 | 0.276939 | −6.716695 | 705 | 12.75% |

[090 - 100] | 95 | −5203.1923 | 68,367.6805 | 0.447062 | −3.094557 | 798 | 10.43% |

[100 - 110] | 105 | −4716.5434 | 10,027.9472 | 2.667755 | 0.096645 | 954 | 6.07% |

[110 - 120] | 115 | −4818.7717 | 32,588.8659 | 1.277478 | −0.555772 | 815 | 6.26% |

[120 - 130] | 125 | −5018.8923 | 98,387.5296 | 0.739402 | −1.827786 | 924 | 17.94% |

[130 - 140] | 135 | −4985.7572 | 115,984.5423 | 0.406327 | −3.160042 | 697 | 22.72% |

[140 - 150] | 145 | −4729.0305 | 45,166.7531 | 1.184919 | −0.671680 | 516 | 8.21% |

D = − 0.01276 × ( Age ) − 0.506 ; R^{2} = 8.52% (relative to data); R^{2} = 33.8% (relative to 3^{rd} polynomial fit).

Coefficient A is most probably related to age with a PCM equation (light blue dashed curve in

A = − 5035.361 + 2410.361 × exp ( − 0.04050 × ( Age ) ) ; R^{2} = 85.6% (relative to data); R^{2} = 95.8% (relative to 3^{rd} polynomial fit).

Lallemand et al. [

Dealing with topographic elevation only, the results shown herein are therefore a global overview. Further analysis combining at least the aforementioned components would be necessary to decipher the relationship between those processes and better understand the observed topographic result. Nevertheless, the statistical outcomes provided below are not reported in the literature.

The distance between arc and trench is broadly distributed around a mean value of ca. 215 km (1.937 ± 1.949; μ ± 2σ being aware that the distribution is not Gaussian,

and the age of the sea-floor entering subduction (

The arc elevation obviously differs as function of the nature of the upper plate. The arc related to intra-oceanic subduction zones are largely under-water and the distribution of elevation is quasi-Gaussian with a mean value close to -1300 m (

No relationship can be determined between the arc elevation and the arc ? trench distance for intra-oceanic subduction zones (quasi-flat blue linear regression in

The topographic elevation along arcs does not appear to be randomly distributed.

Percentage of the full dataset | 100.000% | Percentage of the full dataset | 26.745% | Percentage of the full dataset | 73.255% |
---|---|---|---|---|---|

Sum | 114,267.1144 | Sum | −10,556,692 | Sum | 19,083,708 |

Minimum | 0 | Minimum | −4762 | Minimum | −3827 |

Maximum | 5.826 | Maximum | 2343 | Maximum | 5709 |

Range | 5.826 | Range | 7105 | Range | 9536 |

Mean | 1.937 | Mean | −1299.605 | Mean | 857.733 |

Median | 1.864 | Median | −1298 | Median | 460 |

D(m-m) | 0.073 | D(m-m) | −1.6 | D(m-m) | 397.7 |

First quartile | 1.184 | First quartile | −1993 | First quartile | 3 |

Third quartile | 2.589 | Third quartile | −511 | Third quartile | 1443 |

Standard error | 0.0041 | Standard error | 11.7867 | Standard error | 9.8970 |

95% confidence interval | 0.0080 | 95% confidence interval | 23.0991 | 95% confidence interval | 19.3972 |

99% confidence interval | 0.0105 | 99% confidence interval | 30.3527 | 99% confidence interval | 25.4903 |

Variance | 0.989 | Variance | 1,128,500.828 | Variance | 2,179,293.473 |

Average deviation | 0.813 | Average deviation | 851.548 | Average deviation | 1106.897 |

Standard deviation | 0.995 | Standard deviation | 1062.309 | Standard deviation | 1476.243 |

Coefficient of variation | 0.5134 | Coefficient of variation | −0.8174 | Coefficient of variation | 1.7211 |

Skew | 0.394 | Skew | −0.165 | Skew | 0.870 |

Kurtosis | 12.121 | Kurtosis | −0.137 | Kurtosis | 0.944 |

Kolmogorov-Smirnov stat | 0.034 | Kolmogorov-Smirnov stat | 0.027 | Kolmogorov-Smirnov stat | 0.120 |

Critical K-S stat, alpha = 0.10 | 0.005 | Critical K-S stat, alpha = 0.10 | 0.014 | Critical K-S stat, alpha = 0.10 | 0.008 |

Critical K-S stat, alpha = 0.05 | 0.006 | Critical K-S stat, alpha = 0.05 | 0.015 | Critical K-S stat, alpha = 0.05 | 0.009 |

Critical K-S stat, alpha = 0.01 | 0.007 | Critical K-S stat, alpha = 0.01 | 0.018 | Critical K-S stat, alpha = 0.01 | 0.011 |

Along the Marianas, for example, the topographic elevation varies as in

Using periodograms (created with Past [

While the longwave variations might be caused by numerous factors, one can speculate that the short-wave variations are linked to volcanoes, and therefore to a rather periodic distribution of magma chambers beneath arcs. Adjusting Gaussian fit, the first peak of the periodogram corresponds to a phase of 215.278 km (±4.5 km at the 95% confidence level).

Using all data for arcs together (including data from cordillera and from island arcs), Fourier transforms, periodograms and other techniques such as

wavelets analysis fail to provide clear picture of periodic signal. However, it might be useful to carry out further analysis with refined datasets in order to decipher typical signal of magma-crust interaction versus modified signal related to other processes.

Passive margins are no plate limit but encompass the transition between continental crust and oceanic crust. The definition of continent-ocean boundaries (COBs) is however not straightforward for three main reasons: 1) COBs are buried under a large amount of sediment and/or under volcanic material, which make the COB difficult to identify accurately even with powerful geophysical tools; 2) fragments of tilted block of continental nature (or “extensional allochthons”) may be left apart and separated from the main continent (e.g. [

Because the thickness of the crust is highly varying and therefore subject to high uncertainty, the thickness of sediment is equally subject to caution, the presence, thickness and extension of magmatic rocks is difficult to assess, and the nature of the underlying mantle itself may be problematic, no correction for sediment load was attempted for data from passive margins.

Using all topographic data within a distance of ±9 away from COBs (as defined in section.2;

If the general shape resembles (R^{2} = 99.45%) a hyperbolic tangent function of

equation f(x) = 2236.076 × tanh(−0.77391 × x) − 1936.493 (purple curve in

Data have then been divided by bins of 1 relative to the distance to COBs and plotted against the age of the COB (age at which the two continental domains are separated whatever the nature of the rocks at sea-floor; ^{th} order polynomial curve (

Although largely neglected, the topographic elevation of continents has major implications for many processes of the Earth evolution, including eustatism, climate, erosion-sedimentation-sediment fluxes, etc. Flament [

The global statistics for topographic elevation of crust of continental nature are provided in

The thickness of the crust has been investigated by Mooney et al. [

Crust.5.1 model (5 × 5 model) now updated to the Crust.1.0 model ( [

d = − 0.006206095756 × e − 32.49861243 ; R 2 = 56.27 % (13)

(red line in

Note that, in fact, eq.13 does not significantly depart from the linear fit obtained from data corrected from ice loading-post-glacial rebound-geoid since the equation is:

d = − 0.006218679627 × e − 32.14407886 ; R 2 = 56.78 % . (14)

From the histogram of the Moho depth (

It can be inferred from this that the Moho depth defined by Laske et al. [

According to Equation (13), the continental crust thickness is null at a depth of −4509.878 m. Assuming a mantle density ρ_{M} = 3150 ± 300 [kg・m^{−}^{3}] as above and using this linear relationship (Equation (13)), the mean crustal density ρ_{C} shall correspond to the following equation:

ρ C = ( ρ M × d ′ ) / h C , (15)

where d' is the Moho depth below −4509.878 m, and h_{C} is the thickness of the crust (i.e. (e + d)).

As a consequence, the mean density of the continental crust is ρ_{C} = 2712.870 ± 258.369 [kg・m^{−}^{3}].

However, the right relationship between topographic elevation and Moho depth is undetermined. When polynomial fits of degree 2 (for which R^{2} = 56.35%) or degree 6 (for which R^{2} = 58.86%) are used for instance, the mean densities of continental crust does not directly correspond to a single value (as per eq.15), but can be inferred from the data distribution. Hence, the peak values are ρ_{C} = 2678 ± 254.0 [kg・m^{−}^{3}] and ρ_{C} = 2691 ± 256.3 [kg・m^{−}^{3}] for polynomial fits of degree 2 and degree 6, respectively.

The literature often refers to a “normal” or “typical” continental crust, id est a crust that has not been affected by thinning or thickening processes. It is therefore commonly accepted to use the value of ca. 250 m for the topographic elevation and 35 - 40 km for the Moho depth.

Equation.13 places the Moho at a depth of 34.050 km if the elevation is 250 m (33.699 km using Equation (14)), and conversely, the topography elevation shall rise to an elevation comprised between +403.053 m and +1208.712 m if the “typical” Moho depth is considered. Note that the Moho is even set at a depth of 33.917 km and 34.376 km if the polynomial fits of degree 2 and degree 6 are respectively considered. As previously noticed, those values are not quite in agreement with those generally accepted.

Focussing herein on topography, the definition of a “normal” continental crust will be tempted from the present-day global topography, corrected from ice, post-glacial rebound, and geoid.

From the distribution of the global topography (_{C} = 273.607 m. Although the distribution is clearly not Gaussian, the peak is well-marked and the distribution is broadly symmetrical, so that the median value is relatively close to the mean (m = 235.208 m; Δ(μ − m) = 38.399 m;

Now, the shape of the data distribution (

Using the limited subdataset (i.e. [−333; +777]), _{nC} = +185.6 m (± 408.1 m; 1σ) and is not statistically representative. Although a Gaussian fit with multiple terms obviously better fit the distribution (example with a six term fit in light blue in _{nC}_{.1} = +24.35 m (± 97.02 m; 1 σ_{1}) and μ_{nC}_{.2} = +310.2 m (±160.3 m;

1 σ_{2}). The boundary between the two peaks can be located at ca. +270 m. It seems to separate what can be termed the “lowlands” (with μ_{nC}_{.1} = +24.35 m) from the “highlands” (with μ_{nC}_{.2} = +310.2 m). And the latter largely corresponds

ETopo1 continental topography | Altitude [m] | Corrected continental topography | Altitude [m] | Topography of “normal crust” | Altitude [m] | ||
---|---|---|---|---|---|---|---|

Number of values | 1,045,101 | Number of values | 1,045,101 | Number of values | 686,241 | ||

Percentage of the full dataset | 100.000% | Percentage of the full dataset | 100.000% | Percentage of the full dataset | 65.663% | ||

Sum | 227,067,466 | Sum | 285,947,084.6 | Sum | 149,069,223.4 | ||

Minimum | −9848.000 | Minimum | −9775.862 | Minimum | −332.997 | ||

Maximum | 7446.000 | Maximum | 7018.265 | Maximum | 777.000 | ||

Range | 17,294.000 | Range | 16,794.127 | Range | 1109.997 | ||

Mean | 217.268 | Mean | 273.607 | Mean | 217.226 | ||

Median | 187.000 | Median | 235.208 | Median | 182.450 | ||

D(m-m) | 30.268 | D(m-m) | 38.399 | D(m-m) | 34.776 | ||

First quartile | −41.000 | First quartile | −20.748 | First quartile | 19.081 | ||

Third quartile | 569.000 | Third quartile | 668.625 | Third quartile | 404.273 | ||

Standard error | 1.1090 | Standard error | 1.1121 | Standard error | 0.3053 | ||

95% confidence interval | 2.1736 | 95% confidence interval | 2.1797 | 95% confidence interval | 0.5984 | ||

99% confidence interval | 2.8564 | 99% confidence interval | 2.8645 | 99% confidence interval | 0.7864 | ||

Variance | 1,285,253.581 | Variance | 1,292,539.971 | Variance | 63,969.185 | ||

Average deviation | 666.556 | Average deviation | 680.079 | Average deviation | 211.622 | ||

Standard deviation | 1133.690 | Standard deviation | 1136.899 | Standard deviation | 252.921 | ||

Coefficient of variation | 5.2179 | Coefficient of variation | 4.1552 | Coefficient of variation | 1.1643 | ||

Skew | −0.090 | Skew | −0.192 | Skew | 0.288 | ||

Kurtosis | −2502.852 | Kurtosis | −2502.852 | Kurtosis | 927.544 | ||

Kolmogorov-Smirnov stat | 0.189 | Kolmogorov-Smirnov stat | 0.193 | Kolmogorov-Smirnov stat | 0.060 | ||

Critical K-S stat, alpha = 0.10 | 0.001 | Critical K-S stat, alpha = 0.10 | 0.001 | Critical K-S stat, alpha = 0.10 | 0.001 | ||

Critical K-S stat, alpha = 0.05 | 0.001 | Critical K-S stat, alpha = 0.05 | 0.001 | Critical K-S stat, alpha = 0.05 | 0.002 | ||

Critical K-S stat, alpha = 0.01 | 0.002 | Critical K-S stat, alpha = 0.01 | 0.002 | Critical K-S stat, alpha = 0.01 | 0.002 | ||

to cratonic areas (

Interestingly however, those peak values (μ_{nC}_{.1} = 24.35 m & μ_{nC}_{.2} = 310.2 m) are quite far from the standard of ca. +250 m. And, according to the definition taken herein, “normal” or “typical” continental crust represents ca. 65.66% (

Besides, albeit the misgivings set out above (Section 8.1), the corresponding Moho depth shall be―to first order (Equation (13))―of 32.650 km under lowlands, and 34.424 km under highlands.

Kaban et al. [

It can also be considered that if continental crust was of same thickness everywhere, the long-wave length variation of topography, once corrected from ice loading-post-glacial rebound-geoid, would be due to dynamic topography.

Considering the trend between topographic elevation and Moho depth as manifest, an attempt is made here to correct the effect of that trend in order to highlight the dynamic topography component. It is not determined, however, how the trend should be underlined. Polynomial fits of various degrees (from degree 1 (i.e. linear) to degree 10) are presented in

The statistics on the Earth’s topography presented herein shows that most of the values commonly chosen in the literature as “typical” are subject to caution. The so-called “typical” MCA of +250 m, “typical” Moho depth of ca. 35 km - 40 km,

or “typical” depth of mid-oceanic ridges of −2600 m are values that largely mismatch the values obtained here.

In general, the use of mean values (μ) is inappropriate. Depending on the precision required, the median value (m) might be more relevant because it is more robust even if the data distribution is relatively symmetrical. In addition, the use of standard deviation (μ) is statistically incorrect because a large part of the data considered herein is not Gaussian distributed.

As the present-day topography is the sole example of topography that we have, more caution should be taken regarding the use of statistical values of the topography, in particular for palaeotopography issues. Indeed, many physical processes are at work behind those “mean values”, and it is important to further study them in order to better understand what parameters are predominant in these processes.

Now, much more should be done using the statistics of the Earth’s topography and also using other datasets (heat flux, magnetics, gravimetry, etc.) in order to decipher the role of the various processes. I hope this paper will pave the way for further studies in this direction.

Vérard, C. (2017) Statistics of the Earth’s Topography. Open Access Library Journal, 4: e3398. https://doi.org/10.4236/oalib.1103398

Annexe 1. Statistics of elevation data (ETopo1) for every 0.025˚ bin of distance to ridge axis