The fourth slope: A fundamental new classification of continental margins

Continental margins develop long submarine slopes, linking the shallow shelves along the continental landmasses to the deep abyssal plain. They are the results of a complex interaction between destructive and constructive processes, although by and large they are sites of deposition. There is a great amount of variation between the length, height, smoothness, gradient and variation thereof between the slope profiles; however, there is also recurring similarity in their shape. The similitude has suggested systematic relationships between the shape and the processes forming them, and led to studies on geomorphological categorisation based on curvatures. The potential for prediction of along‐strike variations and connection between morphology and sedimentary process is herein approached through broadening the mathematical functions used, detailed measurement, observation and curve‐fitting of over 150 passive continental margins. Previously, three functions have been used to categorise submarine slopes. The present study finds that four mathematical functions closely match the slopes: Linear, Gaussian, exponential and quadratic (positive and negative/inverse), and reveals that the fourth slope, the quadratic, is by far the most common. While exponential and quadratic slopes are similar there is a crucial difference in the way in which the angle of the slope changes. This study suggests that quadratic slopes represent systematically decreasing sediment deposition with distance, previously attributed to exponential slopes. Exponential slopes meanwhile, represent slope readjustment profiles with upper sediment bypass and lower slope aggradation. Linear slopes, which form the longest low‐angle slopes, form in response to high sediment input. Abrupt shelf‐edges form in shallower water and develop longer slope aprons, suggesting formation from erosional processes. This implies that the quintessential sigmoidal (s‐shaped, Gaussian function) slope, with a smooth rollover, represents the fundamental depositional slope profile.

itude has suggested systematic relationships between the shape and the processes forming them, and led to studies on geomorphological categorisation based on curvatures.The potential for prediction of along-strike variations and connection between morphology and sedimentary process is herein approached through broadening the mathematical functions used, detailed measurement, observation and curve-fitting of over 150 passive continental margins.Previously, three functions have been used to categorise submarine slopes.The present study finds that four mathematical functions closely match the slopes: Linear, Gaussian, exponential and quadratic (positive and negative/inverse), and reveals that the fourth slope, the quadratic, is by far the most common.While exponential and quadratic slopes are similar there is a crucial difference in the way in which the angle of the slope changes.This study suggests that quadratic slopes represent systematically decreasing sediment deposition with distance, previously attributed to exponential slopes.Exponential slopes meanwhile, represent slope readjustment profiles with upper sediment bypass and lower slope aggradation.Linear slopes, which form the longest low-angle slopes, form in response to high sediment input.
Abrupt shelf-edges form in shallower water and develop longer slope aprons, suggesting formation from erosional processes.This implies that the quintessential sigmoidal (s-shaped, Gaussian function) slope, with a smooth rollover, represents the fundamental depositional slope profile.

K E Y W O R D S
clinoform, continental margin, curvature, slope

| INTRODUCTION
Clinoforms form the building blocks of sedimentary geology and their curiously typical geometry, consisting of a smooth s-shape formed by a near-flat topset, sloping foreset and gentle bottomset, can be recognised in cmscale ripples up to the kilometre-scale slopes connecting the continental shelves to the deep ocean floor (Figure 1).Since ever larger structures require ever larger amounts of sediment to construct, it is natural that the smaller clinoforms building shorelines and subaqueous deltas, feed larger shelf clinoforms, and these in turn contribute to building the continental margin scale clinoform.This conveyor belt of sediment is influenced by periods of regression and transgression and sediment supply, with smaller systems merging with larger systems over longer time-scales (Patruno & Helland-Hansen, 2018;Pellegrini et al., 2020;Pratson et al., 2007).While the processes that build clinoforms and clinothems (denoting the body between clinoforms) on varying scales are different, there are also omnipresent factors that exert control on shape and slope (Anell, 2024 in review).Several studies have addressed the geomorphological characteristics of continental margins and if systematic relationships exist between the shape of the margin and sediment supply, basin morphology, lithology, transport mechanisms, along with the effects of pre-existing physiography and antecedent geology (Adams & Schlager, 2000;Brothers et al., 2013;Cacchione et al., 2002;Goff, 2001;O'Grady et al., 2000;Orton & Reading, 1993).It has been recognised that the slope itself can be categorised developing in three forms: planar, concave and sigmoidal, in general corresponding to linear, exponential and Gaussian functions respectively (Adams et al., 1998;Adams & Kenter, 2014;Adams & Schlager, 2000).The mathematical functions can closely match most submarine slopes forming both on km-scale continental margins as well as smaller shelf and deltaic scaled clinoforms in various depositional environments (Anell et al., 2023;Goff, 2001;Kertznus & Kneller, 2009;Quiquerez & Dromart, 2006).
Continental slopes are sites of complex interaction of construction (sedimentation) and destruction (erosion), but are by and large sites of sediment accumulation, hosting 41% of ocean sediment volume despite covering only 4% of the surface area (Brothers et al., 2013;Kennett, 1983).Clastic sediment enters the marine realm largely through rivers, who lose carrying capacity as the flow changes from confined to unconfined and thus deposit sediment near the river mouth as a delta.Sediment load, grain-size and hydrodynamic factors influence how sediment is partitioned between the shoreline and the commonly developing, subaqueous delta (Figure 1).Deltas can build out across the shelf and eventually supply sediment directly to the slope (Pratson et al., 2007).Fluctuations in sea level can increase the rate of progradation or flood the shelf leading to back-stepping of the shoreline.As clinoforms on different scales require different sediment supply to build outward, smaller scale systems commonly merge with larger ones to create hybrid clinoforms (Figure 1), while significant increases in relative sea-level typically create temporo-spatial accommodation such that deltacontinental margin clinoforms can co-exist in a step-wise compound system.When sediment is trapped along the shoreline the continental margin is starved of sediment and remains stationary, or even retreats through erosion within shelf canyons (Fisher et al., 2021).If waves and currents efficiently transport sediment across the shelf, the shoreline remains largely stationary as the shelf widens and the slope advances (Pratson et al., 2007).Additionally, along-shelf and slope transport, sometimes funnelled downslope by canyons, supplies sediment to the slope and basin floor (Gauchery et al., 2021;Pratson et al., 2007;Weaver et al., 2000).The shape of the continental margin is controlled by sediment distribution, erosion and accretion along-slope.The present study approaches the morphological characterisation of 295 passive continental margins (Figure 2) by broadening the range of mathematical functions applicable in order to narrow categories and refine the potential for prediction of along-strike variations between form and processes.

| METHODS
The study uses bathymetric profiles drawn in Google Earth which extend from the shoreline to the deep abyssal plain.The bathymetric profiles are of relatively low resolution to details, but perfectly adequate resolution for curve-fitting and measuring geometric parameters.F I G U R E 2 Location of the measured profiles grouped into geographic areas, the abbreviated name for each geographic area is written in caps, the n denotes the total number of profiles the numbers within the parenthesis denotes first the number of these which are curve-fitted followed by those on which all other parameters are measured.The profiles are drawn at right angles to the orientation of the shelf-break at the location of the profile.

ANELL
The profiles were selected randomly, with relatively even spacing, generally 200-400 km between, along passive continental margins, and drawn, from visual observation, roughly perpendicular to the shelf break.Lines were occasionally redrawn with minor adjustment of angle or location, when the profile was cross-cut by multiple canyons or the termination towards the sea-floor was obscured by topography (incisional, fault, trench or topographic highs).The study thus excludes profiles in areas across shelf and continental scale clinoform with two rollovers (shelf edge and shelf break; sensu Anell, 2024 in review), such as offshore South-eastern USA and southernmost South America, as well as tectonically complex areas including converging margins.This was done in order to study slope profiles which had clear starting and ending point to consolidate the mathematical validity of the study.In total 295 profiles were selected and various parameters measured (Figure 3, Table 1).The (upper/lower) rollover is the area, or point, between the flat shelf and main slope/ the main slope and basin floor.The upper rollover point on continental margins coincides with the shelf break.The rollover can be a clear break in slope, but often forms a curved segment.The study differentiates between the rollover point, rollover length and the rollover (zone).The upper rollover point is located perpendicular to the intersection of the extrapolation of vertical lines along the shelf and upper slope (Figure 3).The upper rollover is mathematically defined, albeit based on personal interpretation of the surfaces along which to extrapolate.The shelf is not always a flat surface, although it often has a very clear trend to extrapolate, the slope meanwhile, clearly varies in angle.The extrapolation is always along an upper slope segment whose continuation extends above the rollover, not below.The rollover length is defined as the length between the upper inflection and the upper rollover point.This measurement provides information on if the transition from shelf to slope is sudden or gradual.
The lower rollover, is often, but not always, a clear 'kink' marking the transition between slope and continental rise, and was selected based on this demarcation, or when nor clear, based on interpretation, to give a general idea of the length of this 'apron' segment.This point has previously been defined as the point where the inclination drops below the tangent of 0.025 (Heezen et al., 1959).
For curve-fitting, the 'slope' segment needs to be clearly defined.The shelf-break has previously been defined as a marked increase in gradient from the shelf reference line T A B L E Summary table showing the geometric parameters measured including the average for all 295 profiles, the parameters associated with the 174 curve-fitted profiles, and thereafter into groups based on maximum slope, average slope, margin width, rollover length and upper inflection depth.

Curvature Rollover Length
Note: The overall slope angle is calculated based on the length of the shelf and the depth of the inflection while the average shelf dip is the average between the shoreline and inflection depth.Cont., Continental.'Apron' is the length of the continental rise within the measured slope (the value is based on interpretation and is therefore more of a guideline).The black box highlights the data-column which was used for the calculations and the trend thus bears no significance beyond providing average values.The red is used to highlight the upper value, the blue for the lower value.(Vanney & Stanley, 1983).Herein the continental slope is defined as the segment from where the shelf dips (consistently) above 1% -a point termed the upper inflection, to a point where the basin floor dips (consistently) below 1% (the lower inflection).While the upper inflection is often very clear, the lower can be harder to define, when difficult, the lower inflection was selected at the point where the average dip of the slope is consistently below 1%.Around 174 of the 295 bathymetric profiles were digitised and exported to excel for curve-fitting.Digitised curves were placed at equal starting values for x (100 m) and equal finishing values for f(x) (100 m).A best-fit least square regression (R 2 ) method was used to find the best fitting curve.Curves with ambiguous results, that is, showing a close fit with two or more types of curves, were tested at other placements within the quadrant to ensure best fit.As quadratic equations without limitations can mimic linear slopes with short terminal angle reductions towards the sea-floor, limitations were placed on fitting quadratic equations.Quadratic best-fit curves required the vertex of the curve to be within less than twice the total distance of the curve and less than twice the height of the curve.This applies to both positive and negative indefinite quadratic equations.

| Slope curvature
Continental slopes can develop into five visually different forms: even slopes (constant angle/planar), decreasing (from steep to gentle/concave), increasing (from gentle to steep/convex), tripart (gentle-steep-gentle/sigmoidal) and stepped/complex.Decreasing and tripart slopes form by far the most common types accounting for around 83% of the observed slopes.Meanwhile, because the slope angle distribution and length of these segments varies significantly, the best fitting mathematical function does not simply correspond to the appearance of the geometric form.
The results indicate that there are five fundamentally different basic mathematical functions that can define passive continental margin slope curvature.One showing a dominantly increasing gradient (quadratic negative indefinite), one showing an even (linear) slope, two showing overall decreasing slope gradient (exponential and quadratic positive definite/indefinite), and one displaying the quintessential s-shaped gentle-steep-gentle (or normal distribution) geometry (Gaussian/sigmoidal) (Figures 4 and 5).Although the slopes can at times be closely matched by two, or more, mathematical functions, the best-fitting functions are generally strongly characteristic, with other functions visibly deviating from the bestfitting function (Figure 5).
The average continental margin, based on all 295 measured segments, slopes on average 2.3°, with a maximum average of 8°.The slope is nearly 3 km high and 100 km, long starting around 100 km from the shoreline (Table 1).The average distance from the inflection to the rollover point is 10 km, with ca 35 km forming the main slope segment and 55 km forming the continental rise (slope apron).The shelf dips on average 0.25°, or 0.1° if measured as a depth-width angle (Figure 3).
In earlier works studies on submarine curvature have used exponential, linear and Gaussian functions to fit bathymetric profiles (Adams & Schlager, 2000;Covault et al., 2011;Gerber et al., 2009;Schlager & Adams, 2001).This study indicates that the most common type of slope matches quadratic (positive) equations, comprising 46% of the curve-fitted slopes (Table 1).Values for these slopes tend to fall within the averages and not the extremes, although they rarely form particularly high slopes, and typically have short rollover lengths and relatively long 'apron' segments (Figure 3).
Exponential slopes, which were previously used to define most concave slopes (Adams et al., 2001;Adams & Schlager, 2000;Goff, 2001), are not as widely common as previously thought.These slopes have very abrupt shelf-edges, more steeply dipping maximum angles than quadratic slopes, and long slope aprons leading to low average slope values.Exponential slopes are commonly associated with wide margins, but conversely with shallow inflection depth and gently dipping shelves.Wide margins typically have inflection points in deeper water as a result of the length of the dipping shelf, therefore the shallow inflection is anomalous.The average width is an effect of five margins over 200 km wide within the category, without which the average is much lower.(Table 1).
Linear slopes tend to develop along narrow margins forming the longest, lowest-angle slopes.The rollover is typically short and angular.Wholly linear slopes, as compared to linear segments, are relatively rare, meanwhile, they are found along all studied margins, but with a higher density around Africa.
Gaussian (sigmoidal) and inverse quadratic slopes share many similarities as they are often steep, both max and average, have short slopes and owe their mathematical function to a much longer rollover length than other forms (Table 1).The inverse quadratic slopes have the highest maximum slope angles, while the sigmoidal have the highest average slope angles.Inverse quadratic slopes are characterised by extremely long rollover lengths and close to no formation of an apron (continental rise), they are often associated with the tallest slopes averaging almost 4 km.Gaussian (sigmoidal) slopes are the second most common curvature measured.These slopes are typified as short and steep, with a more even, albeit basinward skewed, distribution between rollover and apron compared to other slopes (Table 1).
Normalised height and length profiles of some selected slope examples clearly show the disparity between the five curvature types, exponential and inverse quadratic defining clear outliers, linear forming a mid-line divide between the two, and the most common types, quadratic and Gaussian developing profiles between the outliers (Figure 5).

| Slope parameters
The steepest slopes, both maximum and average, are typically associated with narrow margins and steeply dipping | 9 of 15 EAGE ANELL shelves.These slopes are typically short and high, and the curve-fitted profiles with the steepest segments (>20°) all correspond to Gaussian (sigmoidal) functions.As these narrower margins tend to have steeper dipping shelves, the inflection points can be in deeper water than wider shelves.This also depends on the total width of the shelf, the deepest inflection points are overall associated with the widest shelves (Table 1).As shelf width increases so often does the length of the rollover and the total length of the slopes.The longest rollover lengths are therefore also more commonly found forming at deep inflection points.The longest rollover lengths (>3 km) are found on very high and long slopes with wide margins, meanwhile for rollovers covering a wide span from 0 to 30 km, fairly similar values for studied parameters are observed.

| DISCUSSION
Mathematical curve-fitting of continental margins indicates five prominent characteristics which separate the slopes from each other: the formation of long gentle rollover lengths (Gaussian and inverse quadratic), abrupt sea-floor termination/lack of continental rise (inverse quadratic), equal sediment distribution along entire slope (linear) and formation of abrupt shelf-breaks followed by two different decreasing slope declivities (exponential and quadratic; Figure 3).

| Rounded or abrupt shelf-breaks
The shelf edge/break marks an important transition from dominantly wave and current (advective) induced transport to dominantly gravity-driven (diffusive) slope transport (Pratson et al., 2007), and additionally alongslope contour current deposits (Faugères & Mulder, 2011;Hernández-Molina et al., 2008;Masson et al., 2002).The sigmoidal shape has been attributed to reworking of original exponential profiles with rounding associated with significant base-level fluctuations (Adams & Schlager, 2000).Schlager and Adams (2001) suggest that most of the Gaussian (sigmoidal) profiles on passive continental margins are associated with Holocene transgression, and rounding of the shelf edge during Quaternary sea-level cycles.They propose that most of these margins are presently relatively sediment starved.Headward canyon growth has also been proposed as modifying exponential slopes towards a Gaussian shaped profile (Goff, 2001).On the other hand, it has also been suggested that the sigmoidal profiles might rather be the unmodified slope, while exponential slopes result from incision and erosion (Kertznus & Kneller, 2009).
In deltas abrupt transitions between topset and foreset are associated with coarser sediment and a more sudden change in sediment deposition.This steep slope can often fail, lowering the profile and thereafter building up again and prograding through these successions of avalanching (Pratson et al., 2007).With increasing sediment held in suspension the deposition reflects a more gradual depositional profile with topset accumulation inhibited by hydrodynamic factors and limited bottomset accumulation reflecting decrease in sediment supply, generating a sigmoidal clinoform (Patruno & Helland-Hansen, 2018;Pellegrini et al., 2018;Pirmez et al., 1998;Swenson et al., 2005).Meanwhile studies have also shown that delta morphology and rollover point configurations can change rapidly, and deltas are dynamic systems with varying physiography related to multiple influencing factors (Trincardi et al., 2020).
Sigmoidal profiles are suggested to represent depositional equilibrium of prograding shelf margins on stable substrate, leading to high stable slope angles (O'Grady et al., 2000).The present study also finds Gaussian profiles associated with short and steep slopes, unlike a separate study (Adams & Kenter, 2014) which observed that many Gaussian profiles developed on low angle slopes of 1-3°.
Gaussian (sigmoidal) and inverse quadratic slopes, with long gentle rollovers, are herein observed related to relatively deep inflection points, while exponential and quadratic more often develop in shallower water (Table 1).Another key difference lies in the length of the apron segments, which is typically short or very short for the slopes with longer rollovers.Along continental margins, there is a strong correlation between slopes scarred by multiple seafloor failures and significant accumulations of turbidite deposits along the continental rise (Ericson et al., 1961;Normark et al., 1993;Walker, 1978).This could be indicative of erosional processes creating more abrupt shelf-edges.The generally shallower inflection points and lower shelf angles for exponential and quadratic slopes would also mean sea-level variations would be more likely to expose the shelf-edge and the propensity for canyon formation.Canyons are an important morphological feature on slopes, both those connected directly onshore across the shelf and those forming at the shelf-break, along with those developing on the mid to lower slope through seafloor failure.There is some debate over their formation.One school of thought suggesting they form during sealevel low-stands when sediment is delivered at or near the shelf-break generating erosional turbidity currents (Kolla & Macurda, 1988;Rasmussen, 1994), the other that canyons are initiated by various processes leading to slope failure (Pratson et al., 2007).Canyons can also be fed by longshore currents cascading over the shelf edge (Puig et al., 2008).
Inverse quadratic slopes display an increasing gradient and largely lack slope apron deposits.Given the current train of discussion this would imply factors creating shelf and upper slope deposition with limited sediment reaching the lower slope or bypassing to the floor.
4.2 | Sediment flux, slope angle and the formation of linear slopes Linear profiles have previously been interpreted to result from excess sediment creating an angle of repose system (Adams & Schlager, 2000).One study found linear slopes, or more accurately linear segments of slopes, to be steeper than their counterparts and associated with progradation following periods of aggradation (Adams & Kenter, 2014).The present study suggests linear slopes are the lowest angle slopes, similarly to observations by O' Grady et al. (2000).Their study found linear slopes to be associated with smooth and gentle profiles, previously referred to as abyssal aprons (Emery & Uchupi, 1965), and common offshore large river systems with high modern sediment input such as the Amazon, Congo and Niger.High sediment input in weak substrate lowers the gradient through margin collapse and fills in any antecedent canyons (O'Grady et al., 2000).Mathematical modelling supports higher sediment rate leading to lower slope angles (Pirmez et al., 1998).Strongly influential on slope angle is lithology and degree of cohesion, whereby muddy slopes tend to be both gentler and less curved (Adams & Kenter, 2014;Swenson et al., 2005).Meanwhile there are many factors which are thought to influence the angle of submarine slopes.Carbonate systems are generally assumed to develop steeper slopes than their siliciclastic counterparts as a direct result of varying petrophysical properties (such as greater shear-strength), early lithification and cementation, and in-situ carbonate production (Adams & Kenter, 2014;Playton et al., 2010;Schlager & Camber, 1986).The pre-existing physiography and antecedent geology also clearly influence the slope angle (Brothers et al., 2013) as well as over-steepening or slope failure (Pratson et al., 2007).Sediment behaviour on slopes is strongly controlled by the degree of cohesion, where individual grains rolls in non-cohesive sediments, while cohesive (>50% clay) are dominated by sliding.In slopes, thin layers of clay can cause sliding in otherwise non-cohesive sediment.Linear segments on slopes have been reported to correlate fairly well with angle of repose of sediment type, 30-35° for non-cohesive, 5-8° for cohesive, with intermediate values for mixtures (Adams & Kenter, 2014, and refs therein).Mud, the predominant lithology, when saturated has an angle of repose is around 12.5-17.5°(Allen, 1985), which is much steeper than most slopes observed on continental margins.Slopes can fail at much shallower angles due to earthquake shaking, sediment loading induced increase in pore-pressure, flowing pore-water adding to the pull of gravity and deformation of underlying salt or mud, with failure of slopes as low as 2-4° (Iverson & Major, 1986;O'Grady et al., 2000;Pratson et al., 2007;Prior & Suhayda, 1981).There is also on many continental margins a close correlation between the slope and the propagation angle of the internal tide suggesting that the internal tidal wave inhibits deposition as the slope steepens towards the propagation angle, and thus increasing bottom shear stress (Cacchione et al., 2002).In deltas, steep foresets are found associated with low-energy environments, where sediment transport is thought to reflect gravity-driven and mass-flow processes (Pirmez et al., 1998;Syvitski et al., 1988).Modelling on the other hand, suggests advection dominated deposition leads to steeper slopes, while diffusion creates gentler foresets (Driscoll & Karner, 1999).The same study also indicates that without increasing sediment rate, clinoforms become shorter and steeper during sea-level rise to match the rate of aggradation.
By and large sediment accumulation on a continental margin will govern the extent of the shelf and therefore also the greater the depth into which the slope extends (Jervey, 1988).Control over shelf width is also exerted by thermal subsidence, with younger rapidly cooling and subsiding lithosphere, often exceeding sediment supply rate leading to a narrowing or aggradational development of the shelf as sedimentation shifts landward (Reynolds et al., 1991).Conversely, on older margins, sediment supply can outpace thermal subsidence and the shelf will widen and slope lengthen during progradation.Isostatic subsidence also controls accommodation and affects the slope more locally dependent on the mass of the load and the rigidity of the lithosphere (Watts & Ryan, 1976).Sediment deposited on lithosphere with high rigidity will extend the load across a wider area facilitating the development of a broader margin (Pratson et al., 2007).Isostatic subsidence and compaction are greatest where most sediment accumulation occurs, near the shelf break, which creates a positive feedback loop in creating more space for further sediment accumulation and thus stabilising the position of the shelf break.This influence decreases with increasing sediment supply which leads to progradation (Pratson et al., 2007).Pilot studies have, however, suggested that the location of the rollover/inflection in shelf clinoforms, is largely the effect of accommodation space, and sedimentation controls the slope angle and sediment distribution (Anell & Midtkandal, 2017;Anell & Wallace, 2020).This would be in agreement with the generally narrow shelves | 11 of 15 EAGE ANELL associated with the high sediment, long low angle linear profiles.Overall narrow margins are more commonly associated with steep slopes and limited sediment supply (O'Grady et al., 2000), although there is clearly much disparity.Although lithology and sediment supply strongly influence the gradient of many submarine slopes, given the number of other influencing factors, it is difficult to use only the angle as a predictive tool.

Exponential and quadratic functions
Beyond the rollover, be it long and gentle or abrupt, most of the slopes are characterised by a decreasing angle towards the abyssal plain.This reflects the exponential decrease of sediment, and decay of gravity transport along the slope (Adams & Kenter, 2014).Quadratic curvature has previously not been considered for curvefitting of slope margins, and yet it is the most common type of curvature observed.The key difference between exponential and quadratic functions is the slope (first derivative) of the curves, which is linear for quadratic slopes, and exponential for exponential slopes.For continental margins, this means that slopes fitting a quadratic curve show a constant rate of change of the slope angle, while the exponential curve changes at a proportional rate.
There is a misconception that continental slopes are largely sites of erosion and bypass, as opposed to deposition (Pratson, 2001).Transport and deposition on continental slopes is gravity-dominated, and both erosive and depositional processes operate along the slopes.Slides, debris flows and turbidity currents commonly operate, along with along-slope transport and bottom-currents (Gauchery et al., 2021;Weaver et al., 2000).Turbidity currents, the dominant mechanism on continental margins, can be initiated by a number of triggering mechanisms and move downslope under the influence of gravity.Whether a turbidity current ignites and gains momentum depends on erosion exceeding deposition thus triggering a positive feedback loop (Parker et al., 1986).If it loses mass deposition will fall off exponentially with distance downslope.Theoretical calculations and experimental modelling have shown that on steep slopes ignition of repeated turbidity currents will eventually lower the gradient, while repeated waning seaward thinning turbidites on low slope gradients will steepen the slope.Both processes suggest final angles around 4° (Gerber et al., 2004;Kostic et al., 2002).
An agent in shaping the deep-sea morphology of continental slopes is contour currents, which are efficient agents of sediment transport and deposition (Faugères & Mulder, 2011).Contour currents redistribute and resuspend fine-grained sediment and can transport them thousands of kilometres.Their velocity, and thus ability of erode and transport sediment is linked to the steepness of the slope, with lower velocities associated with gentler slopes (Faugères & Mulder, 2011).There are many examples of mixed process margins where shorter lived gravity-driven slumps, slides, debris flows and turbidity currents interact with long-lived sustained contour currents (Rodrigues et al., 2022).The contour currents rework submarine fans and lobes and create various depositional drift features (Faugères & Mulder, 2011;Faugères & Stow, 2008).
Although contourites and mixed deposit systems are still not completely understood (Faugères & Mulder, 2011;Faugères & Stow, 2008;Gauchery et al., 2021;Rodrigues et al., 2022;Shanmugam, 2017), it is possible to speculate on the potential effect on slope angle and curvature.Contourite deposits might for example increase the length and lower the angle of the continental rise, create small-scale topography and surface heterogeneity and thus diminish the overall R 2 fit to a mathematical function, or increase base of slope deposition on an otherwise sediment-starved margin thus indicating a non-existent upper-lower slope interaction.In addition, if vigorous enough erosive ocean currents can generate sediment instability and potentially mass movement (Gauchery et al., 2021;Maselli & Kneller, 2018;Mulder et al., 2003;Rodrigues et al., 2022).Contourites themselves are also prone to failure resulting from low-permeability, high plasticity and high pore-water content (Bryn et al., 2005;Laberg & Camerlenghi, 2008).
It has been proposed that many slopes develop their shape in response to slope readjustment, whereby oversteepening leads to bypass of the slope and aggradation at the base of slope and minor progradation atop the shelf, which builds up to merge and form a new equilibrium profile (Galloway, 1998; Figure 6).This would be similar in morphology to the modelled shoreline progradation of deltas, with (fluvial) aggradation and steep foresets driven by gravity deposits and mass flow (Swenson et al., 2005).It is possible that quadratic slopes represent a profile forming dominantly through deposition, with systematically decreasing amount of sediment towards the sea-floor, while exponential profiles represent a state of disequilibrium, with dominantly bypass of the steeper upper slope and accumulation within the slope apron (Figure 6).
The final slope conundrum lies with the inverse quadratic, whose shape, while steep, reflects very limited slope bypass or deposition towards the abyssal plain.As previously discussed, the very long gentle rollover appears to be a depositional feature, rather than | 13 of 15 EAGE ANELL reworking.Covault et al. (2011) studied canyon-channel systems along continental margins, and found six corresponding to convex profiles.They proposed the form was due to active tectonic uplift and deformation, such as steep fronted accretionary wedges, and that turbidites were trapped in pockets of accommodation alongslope (Beaubouef & Friedmann, 2000;Booth et al., 2000;Covault et al., 2011).The influence of antecedent topography seems plausible to explain the dominant form of the profile (Brothers et al., 2013), and deposition with some form of sediment trapping controlling the formation of a long gentle rollover may explain the formation of the inverse quadratic slope with their abrupt sea-floor termination.

| CONCLUSIONS
Curvature analysis of over 150 continental margin profiles, and measurements of various parameters of 295 slopes, reveals that unlike previously thought, there are actually four mathematical functions which can define slope curvature.Of these, the fourth slope, the quadratic, is in fact the most common.Several factors influence which mathematical function fits best, one important aspect being the length of the rollover.Long rollovers typically create Gaussian (Sigmoidal) slopes, or inverse quadratic ones.Gaussian slopes are often short and steep, while abrupt rollovers are associated with increased base of slope accumulations, suggesting their formation is associated with erosion and slope failure.
There is a fundamental difference between exponential and quadratic slopes in the rate at which the slope angle decreases.The even decrease of quadratic slopes, with the 1st derivative forming a line, suggests sediment is partitioned systematically.The current study proposes that this is linked to exponential decay of carrying capacity downslope, which has previously been inferred for exponential slopes.Exponential slopes, on the other hand, typically display a more marked steep section and a gentle slope apron, and are therefore suggested to relate to over-steepening, erosion and bypass of the upper slope and increased lower slope accumulation during slope readjustment.
Wholly planar slopes, defined by linear equations, are relatively rare.They generally form very long, very low angle slopes.These slopes are suggested to derive from high sedimentation.Of note are the often narrow shelves associated with these slopes, which suggests that the position of the continental margin break is probably quite stable and in many cases controlled by accommodation rather than sedimentation.

Highlights•
A fourth mathematical function (quadratic) is found to be the most common fit of continental slopes.• Quadratic slopes decrease evenly compared to exponentially decreasing slopes.• Two different decreasing functions may show systematic versus erosional/depositional profiles.• The quintessential s-shaped sigmoidal profile may be the fundamental depositional form.• Linear slopes display long low angle profiles and develop in response to high sedimentation.F I G U R E 1 The full-scale conveyor belt of sediment forming clinoforms on ever-increasing scales towards the continental margin.Modern settings rarely comprise the shelf-scale clinoform, these are more common in ancient non continental-scale margin settings.The lower right figure shows the formation of compound and hybrid clinoforms (here at delta and shelf scale, but can apply to other forms) as systems merge or detach during large-scale relative sea-level changes.Redrawn from Anell et al. (2023), Patruno and Helland-Hansen (2018).

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I G U R E 3 The upper figure shows the parameters measured for each continental margin with the mathematically defined upper and lower inflection, and upper rollover point.Note the 'lower rollover' is based on interpretation.Below this are the five types of mathematical functions applied for curve fitting of the margins, with the basic equation that defines each slope (O'Grady et al., 2000).

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I G U R E 4 The figures show examples of several selected profiles which match a mathematical function (stippled).The vertical and horizontal scale are the same for all the examples.The abbreviated name is study specific and matches a profile on the map in Figure 5.The number beneath the name shows the R 2 fit for each slope for their respective mathematical function.

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The upper left figure shows two normalised (length and height set to 100) examples of each type of curvature.The following figures show a normalised example of each type of curvature, with the mathematical function stippled and the R 2 value displayed.For the Exponential, Quadratic and Gaussian functions, the near-fitting but clearly different mathematical function is also displayed for comparison.Within the smaller square are several other examples shown in actual height and length.Below each is the actual profile with the slope portion (>1% declivity) outlined in a box.The map at the base shows the location of all of the examples displayed in Figures 4 and 5.

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I G U R E 6 Cartoon sketch of some proposed development scenarios based on the observations from this study.(a) Shows the proposed disequilibrium profile creating exponential slopes, with erosion or bypass of the upper slope and accumulation at the toe.(b) Shows the eventual development of an equilibrium profile and continued sedimentation creating an even decrease in sediment along the slope and formation of a quadratic or sigmoidal profile (a and b are redrawn based on Brothers et al., 2013).(c) Depicts the potential formation of abrupt compared to smooth rollovers, with slope failure and erosion creating an abrupt edge and increased base of slope accumulation.(d) Shows the proposed generation of linear slopes, with a relatively stationary shelf break and high sedimentation driving increased slope accumulation and formation of long low angle slopes.The cartoons below each scenario depicts sediment accumulation rate along the profile and the resulting shape of the slope.The one below (b) is redrawn based on Walsh et al. (2004).