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 (Figure 5(a); Table 2). It must be noticed, in addition, that transform faults are,

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; Table 2), so it is relevant to exclude them from statistics on spreading centre topography.

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. [6] was coarser than the definition made here (see Figure 4), the age of the sea-floor determined from Müller et al.’s dataset [6] is about 3 Ma (2.93 Ma after the mean value and 3.04 Ma after the median value both determined on the logarithmic distribution of sea-floor ages (using all spreading centres only); yellow histogram in Figure 5(b)). The age uncertainties calculated by Müller et al. [6] accounts for uncertainties on fitting pairs of isochrones. The mean age uncertainty of 0.5 Ma (median = 0.78 Ma) obtained at ridges using Müller et al.’s dataset [6] therefore most likely underestimate the age error when uncertainties on location are taken into account (Figure 5(c)). Now, uncertainties remain low and this should not fundamentally affect results on the age-depth dependence of the sea-floor (see below).

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 mm²/yr) has been calculated from Euler poles and angles at each spreading centre segment (Figure 5(d)). Knowing every segment lengths, the results have been converted in spreading rates (in mm/yr) and compared with values provided by Müller et al. [6] (Figure 5(e) & Figure 5(f)).

The topographic elevation of ridge inner rifts as function of spreading rates has thus been tested (Figure 6) using both the spreading rates computed by Müller et al. ( [6] ; hereafter termed data. A) and the accretion rates computed here (this study; hereafter termed data. B).

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 Figure 2), and the rise of the inner ridge elevation seems to reach a threshold around 1 mm²/yr (Figure 6(a)) or around 30 mm/yr (Figure 6(b)) whatever the dataset used (i.e. data. A or data. B). Albeit similar relationship using data. A or data. B, the rise of the inner rift elevation is not linear with the spreading rate (Figure 6(c)). The mathematical relationship between elevation and rate is undetermined, although

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; Figure 7(a) & Figure 7(b)). However, the amount of data is rather large, and a good picture of mid-oceanic ridge shape and elevation can be obtained with a dataset limited to 1.333 around ridge axes (Figure 7(c)).

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 Figure 8, and show a nice general shape with the inner rift around −3100 m (μ = −3120.387 m), the ridge crest distant of 0.15 (ca. 17 km) away from the axis with elevation around −2650 m (μ = −2661.473 m), and the ridge flank gently dipping away. In the ridge flank, two parts can be distinguished: the nearest part―below 0.80 in distance from ridge axis (yellow dashed line in Figure 8)―with a steeper slope, and the farthest part―beyond 0.75 (yellow dash-dotted line in Figure 8)―with a shallower slope. The “virtual ridge axis” can therefore have different elevation according to the slope chosen to represent the ridge flank. The “virtual ridge” elevation―the point at which the best linear fit on flank data crosses the ridge axis (Figure 8)―is −2601.615 m using the entire ridge flank, −2559.811 m using near section

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

However, we previously saw (Figure 6) that the rate of sea-floor accretion is an important parameter in the elevation of the mid-oceanic ridge. The ridge dataset (data within 1.333 away from ridge axis) has been therefore divided into sub-datasets according to the accretion rates (bins of 0.5 mm²/yr). Using running averages (50 data sliding window), the relationship of the elevation of the inner rifts to accretion rates becomes clear (Figure 9(a)). In addition, one can see that the ridge crests are also impacted by the accretion rates within a relatively constant segment length ranging from 0.15 to 0.70 - 0.80 from ridge axis, i.e. the nearest section of the ridge flank, hereafter named “near flank”. Beyond 0.80, elevations and slopes are relatively similar; the cause of the observed variations is undetermined but might be due to effects of the geoid and/or the dynamic topography since data per accretion bin are not evenly distributed around the world. Consequently, linear fits have been established on the three sections (inner rift, near flank, far flank) and normalised to the mean value within the far flank section (at 1.03 away from ridge axis) for comparison purpose (Figure 9(b)). Ridge crests is prominent for accretion rates lower than ca. 1.5 mm²/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 Figure 9(c) & Figure 9(d).

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 Figure 4), then the “virtual ridge elevation” may vary quite considerably, and values are summarised in Table 3.

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 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. [23] for solution in inverse problem) and requires using datasets for which uncertainties are probably as large as the problem we want to address here. The Airy-Pratt method for isostasy is therefore regarded here as more adapted to the present concern. The isostatic correction C for sediment load is simply expressed as follow:

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 topographic dataset used for age-depth relationship corresponds to all oceanic data (after ETopo1 [5] , i.e. blue dots in Figure 1) that are not excluded by buffer zones as shown in Figure 2 plotted against ages provided by Müller et al. [6] . In other words, all oceanic data not affected by known main bias were used and corrected from sediment load as described above.

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 (Figure 12). Actually, most distributions are closer to Laplace distributions implying that the mean values μ are relevant and representative of the distributions whereas the standard deviations σ simply overestimate the dispersions.

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 Figure 13. The sediment load correction appears satisfactory when compared with the average raw topographic elevation from ETopo1 since the old part exhibit a quasi-flat relationship as expected.

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 (Figure 13). Another method―not carried out here―to define the best parameters for isostatic correction (namely ρ_igW, ρ_g, ρ_M, φ₀, 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 Figure 12). Because the uncertainty associated with this value would need more proper quantification, the density of solid grain in sediment is, here, just arbitrarily chosen to be ρ_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 Figure 13) has the following equation:

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

with coefficient of determination R² = 88.3%.

As emphasized by previous authors, the fit is much better when applied on data for the first 70 Ma (e.g. [18] , although those authors finally chose 20 Ma as limit in their GDH1 model) but the physical meaning of splitting the data between young-aged and old-aged data is unresolved. Using a fit for data younger than 70 Ma only, the HSCM equation (purple in Figure 13) becomes:

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

with coefficient of determination R² = 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 Figure 13) is:

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

with coefficient of determination R² = 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. [6] . PCM equations of the form d ( t ) = A + B ⋅ exp ( − C ⋅ t ) were thus fitted to every sub-datasets and Figure 14 shows the variation of coefficient A, B and C as function of spreading rates (see also Table 4). Polynomial fit of degree 3 (and associated 95% confidence intervals) are merely shown to highlight the first order variation of each coefficient. Beyond 100 - 120 mm・yr⁻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 ( [18] ; see Equation (2)), where d_R corresponds to the “virtual” ridge axis (see analysis of the mid-oceanic ridges above;

Section 3.2) and is given as d_R = 2600 m.

The definition of d_R from the coefficients determined in Figure 14 and Table 4 is simply d R = A + B × ( π 2 / 8 ) . For the other parameters however, the determination is not trivial. Stein & Stein [18] , in particular, noticed the difficulty to quantify uncertainties.

To do so, the genuine equation for Plate Cooling Model (PCM; [14] [22] [28] ) was used and is written as follow:

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 [14] , the equation can be re-written as:

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 [29] [30] in another context was used here to quantify the unknown parameters, namely ρ_M, α, T_M (T₀ is assumed to be the temperature defined at the sea-floor as in Figure 10), κ, and Z_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 mm²・s⁻¹ or × 10⁻⁶ m²・s⁻¹) ; Thermal diffusivity, which value is commonly chosen around 1 mm².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, Figure 15 and Figure 16 shows the outcome for all the data together, i.e. not separated by bin of spreading rates.

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 (Figure 15) than the range around the mean. In order to avoid being limited and biased by minimal values of 2σ and δ_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 (Figure 16) are neither Gaussian distributed nor clearly symmetrical. It seems therefore that many possibilities exist to combine the values for those parameters and no “best” solution satisfactorily stands out. The conclusion is the same when the dataset is divided per bin of spreading rates.

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 Figure 13, and smoothed using polynomial fits) and using more “realistic” ranges for the parameters:

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 mm²・s⁻¹); Thermal diffusivity.

The number of iteration has to reach 2,728,362 to obtain only 100 accepted curves (Figure 17(a)).

Similarly, the outcome for each parameter does not allow clearly defining favoured values (Figure 17). The following mean μ (parameter) and median m (parameter) values are therefore provided for information only:

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 [18] noticed that their estimates of the different parameters with one standard deviation (1σ) are lower than Parsons & Sclater’s [22] estimate. Although the values given just here are closer to those of the latter authors, one can see that the range of possible values is large and the use of standard deviation has no real meaning.

Trenches are defined in Figure 1 and Figure 2 as the deepest topographic points where one tectonic plate subducts beneath another. Elevation, however, varies considerably (Figure 19(a)). Most of the sea-floor entering subduction is rather young in age (age < 50 Ma; Figure 19(b)) and the amount of sediment at trenches (deepest point) is not very large (Figure 19(c)).

Old ages of the sea-floor (age > 180 Ma) exists in Müller et al.’s [6] dataset and correspond to the sea-floor of the East Mediterranean Sea. Although many arguments support Palaeozoic ages in this area (e.g. [31] [32] [33] [34] [35] ), the precise age is questionable and the isochrones provided by Müller et al. [6] are highly speculative. The sediment thickness in this region is equally subject to discussion. Consequently, data older than 150 Ma at trenches have merely been discarded herein.

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 Figure 2), a relationship between sea-floor depth and age at trenches arises (Figure 20). Within two standard errors around the mean (μ ± 2σ), a linear mathematical relationship cannot be formally ruled out but an exponential function (PCM) may be viewed as more coherent. Numerically, the equations relating the depth at trench d_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 (Figure 21). As found by other authors (e.g. [36] [37] ), there is no apparent relationship between the subduction rate (i.e. the sum of the upper plate motion and lower plate motion defined from Euler poles at every point location at trenches; in mm・yr⁻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 Figure 2, but discarding data belonging to exclusion buffers), the lithospheric plate clearly shows a flexuration (Figure 22). The distance of the bulge―the most elevated point before trench― is located at 1.09 and 1.00 from trench using polynomial smoothing on median and mean values respectively.

The flexuration can be modelled using equations for the bending of an elastic lithosphere (e.g. [14] ; see also [38] ). The deflection of the plate w [m] as function

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 w₀ 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 ; x₀ 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 Figure 23(b). The match between data (mean values μ per bin) and model is quite poor.

In theory, if the end load is applied at trench, the maximal depression w₀ corresponds to the difference between the elevation at trench and the elevation far from the trench, and w₀ can be estimated from Equation (6) and Equation (9.2).

As shown in Figure 19(b), the age of the sea-floor at trenches varies considerably. Assuming however, these ages roughly follow an exponential decay (inset in Figure 23(a)), the mean age defined from natural log distribution (Figure 23(a)) is about 25 Ma. The PCM equation (Equation (6)) provides an estimation of the depth of the sea-floor of d(25) = −4297.689 m. The depth at trench is given by Equation (9.2) and corresponds to d_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 Figure 23(b). However, such calculation does not fit the data at all.

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 Figure 23(b)); 2) It does not seem correct to regard the trench as the position of the end load.

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 Figure 23(b)) much better fit the data (R² = 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 (Figure 24) to try to determine the “best” parameters of Equation (10) (Equations (10.1) to (10.5)) within bounds defined as the two standard errors around the mean (μ ± 2σ) and 95% of the data between the minimum and maximum values per bin around the median (m ± δ_95%).

The parameters were chosen within the following ranges:

ρ_W is defined from the International Equation of State of Seawater [24] as above.

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 ×10³ m) ; the elastic thickness of oceanic lithosphere is commonly chosen around 50 × 10³ m in the literature (see in particular [39] [40] ).

E = 10 ± 5 i.e. E ∈ [ 5 ; 15 ] (in ×10¹⁰ Pa) ; the Young’s modulus value is commonly chosen around 7 × 10¹⁰ Pa in the literature.

The outcome for the parameters used in Equation (10) does not allow clearly defining favoured values (Figure 25). The following mean μ (parameter) and median m (parameter) values are therefore provided for information only:

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 (Figure 26(a)) and the variation of coefficients A, B, C & D as function of age is shown in Figure 26(b) (see also Table 5). Polynomial fit of degree 3 (and associated 95% confidence intervals) are merely shown to highlight the first order variation of each coefficient. Concerning the coefficients B, C & D, the variation of the 3^rd order polynomial fits does not highly depart from linear fits with equations:

B = 542.191 × ( Age ) + 1055.494 ; R² = 41.0% (relative to data); R² = 97.4% (relative to 3^rd polynomial fit);

C = − 0.00401 × ( Age ) + 1.466 ; R² = 7.25% (relative to data); R² = 90.0% (relative to 3^rd polynomial fit);

D = − 0.01276 × ( Age ) − 0.506 ; R² = 8.52% (relative to data); R² = 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 Figure 23(b)) with equation:

A = − 5035.361 + 2410.361 × exp ( − 0.04050 × ( Age ) ) ; R² = 85.6% (relative to data); R² = 95.8% (relative to 3^rd polynomial fit).

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, Figure 27(a); Table 6). There is no relationship between this distance

and the age of the sea-floor entering subduction (Figure 27(b); the linear fit is near flat with equation y = 0.00159 × x + 1.89714 and mean y value = 1.958 ).

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 (Figure 28(a); Table 6). The distribution of arc elevation in continental crust is sharper and lesser symmetrical so that the mean and median values differs significantly. In addition, a number of data corresponding to the region of the Altiplano in South America marks a cluster of elevation between 4000 and 5000 m (Figure 28(a) & Figure 28(b)) and shift the mean value.

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 Figure 28(b)). For active margins however, arcs closer to their trenches seem to predominantly exhibit lower elevation than arcs located farther away. The reason seems to be straightforward: When the arc is close to the trench, the subducting slab is dipping with high angle usually in agreement with slab roll- back processes. The upper plate undergoes extension leading to the lowering of the elevation of the arc. On the contrary, under compression, the upper plate is squeezed and the arc uplifted.

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

Along the Marianas, for example, the topographic elevation varies as in Figure 29. At first glance, one may see an irregular longwave variation, in the order of 1000 km (outlined with polynomial fit in green in Figure 29(a)), and a shorter, with variation of the order of 200 km (outlined with a sinusoidal function in purple in Figure 29(a)).

Using periodograms (created with Past [41] ), the power versus frequency diagram (Figure 30) shows several significant peaks. In particular, the multiple-taper spectral analysis [42] highlights peaks at −2.3336, −1.8720, and −1.6581 on the log10 of the frequency.

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.

The thickness of the crust has been investigated by Mooney et al. [50] with the

Crust.5.1 model (5 × 5 model) now updated to the Crust.1.0 model ( [8] ; 1 × 1 model available online: http://igppweb.ucsd.edu/~gabi/crust1.html). The distribution of the Moho depth for data assumed to lie on continental crust mainly exhibits two peaks (Figure 32(b)). The first is located at a depth below 40 km and the second below 30 km (−37.00 km and −29.36 km after multiple terms Gaussian fit, respectively). The linear relationship between the Moho depth d [km] and the topographic elevation e [m] is given by the equation:

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

(red line in Figure 32(c)).

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 (Figure 32(b)), two main peaks stand out: the most prominent at a depth of 36.998 km and the second, at 29.359 km. From the linear relationship (Equation (13)), and even from the polynomial fits of higher degrees, the corresponding two main topographic elevations should be +725.047 m and −505.971 m (+741.088 m & −491.721 m with a polynomial fit of degree 2; +642.537 m and −422.962 m with degree 6), respectively. Those results do not compare with the histogram of the topographic elevation (Figure 32(a)).

It can be inferred from this that the Moho depth defined by Laske et al. [8] is too crude to be compared with the ETopo1 model [5] , because a large number of data for the Moho does not properly correspond to the histogram of the topographic elevation. However, the general trend between topography and Moho depth is clear and Equation (13) can be regarded as a good first approximation.

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 R² = 56.35%) or degree 6 (for which R² = 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 (Figure 33(a); same as per Figure 32(a)), the mean altitude of continent is μ_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; Table 7). This observation probably explains why the standard value of +250 m is commonly adopted.

Now, the shape of the data distribution (Figure 33(a)) clearly indicates that continental crust with topographic elevation lower than 0 m to −1000 m are altered by thinning effects. In parallel, a bulge in data distribution around ca. +1000 m suggests the effects of crustal thickening. The range of values for “normal crust” certainly lies between those bounds but cannot be precisely determined from this dataset only. Consequently, the bounds for “normal crust” (i.e. crust considered to be not significantly affected by thinning or thickening processes) have been arbitrarily chosen to be −333 m and +777 m. The corresponding distribution within those limits is shown in Figure 33(b), and the spatial distribution is shown in Figure 33(c) (blue dots below -333 m, yellow and orange dots comprised between −333 m and +777 m, and red dots above +777 m).

Using the limited subdataset (i.e. [−333; +777]), Figure 33(b)―which actually corresponds to a zoom on the peak of Figure 33(a)―discloses the irregular shape of the data distribution. A single term Gaussian fit (dark blue curve in Figure 33(b)) proves the mismatch; the mean value for “normal crust” is μ_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 Figure 33(b)), the signification of the location and shape parameters of each term is obscure. At first, nevertheless, the data distribution displays two main peaks. A Gaussian fit with two terms (blue curve in Figure 33(b)) can bring out those peaks for which the location and shape parameters are: μ_nC.1 = +24.35 m (± 97.02 m; 1 σ₁) and μ_nC.2 = +310.2 m (±160.3 m;

1 σ₂). 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

to cratonic areas (Figure 33(c)).

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% (Table 7) of continental crust in the world at present-day.

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. [51] described the dynamic topography as “the long-wavelength part of non-isostatic topography [which] is supposed to be generated by mantle dynamics and is up to now not well studied”. Those authors produced a global map of dynamic topography from gravity field data.

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 Figure 34(a), and Figure 34(b) and −34.c show the correction that the two end-members (degree 1 and degree 10) imply. In other words, Figure 34(b) and Figure 34(c) show how the topographic elevation would reduce if the continental crust was in general constant. Spatially, the topographic elevation is shown in Figure 35(a) and Figure 35(c) after reduction from polynomial fit of degree 1 (linear) and degree 10 respectively. The use of a Gaussian filter (Figure 35(b) & Figure 35(d)) consequently highlights area where the dynamic topography presumably acts by uplift (red) or subsidence (blue). The amplitude of the obtained dynamic topography is relative because highly dependent on the filter applied. Now, most of the amplitude topography is comprised with ± 1000 m, although stronger negative values are obtained close to subduction zones.