Geometrical Analysis of the Inland Topography to Assess the Likely Response of Wave-Dominated Coastline to Sea Level: Application to Great Britain

Journal of Marine Science and Engineering

Article Geometrical Analysis of the Inland Topography to Assess the Likely Response of Wave-Dominated Coastline to Sea Level: Application to Great Britain

Andres Payo * , Chris Williams, Rowan Vernon, Andrew G. Hulbert, Kathryn A. Lee and Jonathan R. Lee British Geological Survey, Nicker Hill, Keyworth, Nottingham NG12 5GG, UK; [email protected] (C.W.); [email protected] (R.V.); [email protected] (A.G.H.); [email protected] (K.A.L.); [email protected] (J.R.L.) * Correspondence: [email protected]; Tel.: +44-0115-936-3103

Received: 24 September 2020; Accepted: 29 October 2020; Published: 31 October 2020

Abstract: The need for quantitative assessments at a large spatial scale (103 km) and over time horizons of the order 101 to 102 years have been reinforced by the 2019 Special Report on the Ocean and Cryosphere in a Changing Climate, which concluded that adaptation to a sea-level rise will be needed no matter what emission scenario is followed. Here, we used a simple geometrical analysis of the backshore topography to assess the likely response of any wave-dominated coastline to a sea-level rise, and we applied it along the entire Great Britain (GB) coastline, which is ca. 17,820 km long. We illustrated how the backshore geometry can be linked to the shoreline response (rate of change and net response: erosion or accretion) to a sea-level rise by using a generalized shoreline Exner equation, which includes the effect of the backshore slope and differences in sediment fractions within the nearshore. To apply this to the whole of GB, we developed an automated delineation approach to extract the main geometrical attributes. Our analysis suggests that 71% of the coast of GB is best described as gentle coast, including estuarine coastline or open coasts where back-barrier beaches can form. The remaining 39% is best described as cliff-type coastlines, for which the majority (57%) of the backshore slope values are negative, suggesting that a non-equilibrium trajectory will most likely be followed as a response to a rise in sea level. For the remaining 43% of the cliffed coast, we have provided regional statistics showing where the potential sinks and sources of sediment are likely to be.

Keywords: erosion; sea-level rise; nearshore

1. Introduction One of the most signiﬁcant challenges facing coastal geomorphology and engineering today is the need to improve our ability to make quantitative predictions of morphological change at a scale that is relevant to longer-term strategic coastal management [1]. Following [2], this scale is herein referred to as the mesoscale, and is characterised by time horizons of the order of 101 to 102 years. The need for these quantitative assessments at the mesoscale has been reinforced by the 2019 Special Report on the Ocean and Cryosphere in a Changing Climate [3], which concluded that adaptation to a sea-level rise (R) will be needed no matter what emission scenario is followed. As a response to rising sea levels, the location of the shoreline is anticipated to change, with the magnitude and direction of the change (transgression or progradation) being a continuum between two extreme behaviours: passive inundation and morphodynamic evolution. Passive inundation occurs when the land surface is static during transgression and therefore shoreline retreat follows the slope, S0, of the backshore topography. Morphodynamic evolution is usually accompanied by erosion and deposition, which drive morphologic changes that impact future retreat.

J. Mar. Sci. Eng. 2020, 8, 866; doi:10.3390/jmse8110866 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 866 2 of 26

Which of these two extreme behaviours is more likely to occur can be understood from knowledge of the backshore slope and the different instances of the Bruun rule [4]. The Bruun rule is the most common model used for practical predictions of shoreline retreat, particularly in the context of global warming. While simple to use, the Bruun rule makes simplifying assumptions, which are frequently misunderstood or ignored in practice [4]. The Bruun rule requires the following three assumptions: (1) the shoreface, beach, and substrate must have a homogeneous composition; (2) the shoreface and beach must maintain a fixed equilibrium profile; and (3) this profile must be closed with respect to external sources and sinks of sediment. Different instances (classical, back-barrier and cliff Bruun rules) can be obtained [4] by applying the conservation of mass, under the Bruun rule assumptions, over different control volumes to produce a relationship where the landward rate of movement of the shoreline, ε (distance/time), is a linear function of the sea-level rise itself (R, distance/time):

ε = R/S (1) where S represent the effective slope controlling the sea-level rise response. Which geometric boundaries accurately represent the effective slope, S, has been an issue of significant debate. Although some applications suggest that the effective slope should be the shoreface slope, Ss, or, for low-sloped coasts, some other modification accounting for barrier geometry [5], recent analytic research suggests that long-term equilibrium requires that the coastal recession rate follow the regional upland slope [6–8]. When the conservation of mass is applied to the shoreface, as shown in Figure1a, where S = SS under the Bruun essential assumptions provides the classic Bruun rule, the shoreface profile translates upwards following a sea-level rise and landwards following the backshore slope. However, on gentle coasts (S0 < Ss) the classic Bruun rule fails to account for back-barrier deposition. In this case, extending the Bruun profile to include the barrier and back-barrier (Figure1b) reduces the average profile slope S, giving a “barrier Bruun rule” where S < Ss. For a barrier of sufficient length, S = S0, and the barrier Bruun rule reduces to passive inundation, here accomplished by barrier rollover. Similarly, on steep coasts (S0 > Ss), extending the Bruun profile to include a cliff face (Figure1c), accounting for cliff erosion, increases the average profile slope S, giving a “cliff Bruun rule” where S > Ss, and for a tall-enough cliff, the cliff Bruun rule again reduces to passive inundation. Equation (1) shows how the backshore slope plays an important role in determining the coastal response to a sea-level rise, but caution is needed when applying the Bruun rule, as its assumptions are not always met in reality and have to be applied with care. Much of the controversy surrounding the Bruun rule assumptions concerns the validity of the closed equilibrium profile concept (Assumptions 2 and 3), but [7] argued that the least robust assumption is the assumption that negligible sediment flux across the shoreline is required to fully close the profile. The violation of the negligible sediment flux across the shoreline assumption leads to physically unreasonable long-term predictions for the Bruun shoreline implying cliff erosion or back-barrier deposition when applied to coasts steep or gentle relative to the shoreface slope. To overcome this limitation, a generalized Bruun rule that includes the landward sources and sinks of sediment was proposed by [7]. Here, we propose an innovative approach to assess the potential shoreline response to rising sea levels based only on the geometrical analysis of backshore topographical profiles, and we applied it to the whole of the Great Britain-and-isles coastline (hereinafter referred as GB). First, we show how Equation (1) was generalized by [7] to include the sources and sinks of sediment for steep and gentle backshore slopes to produce a generalized cliff and back-barrier Bruun rule, respectively. To be able to obtain an analytical solution of the generalized Bruun rule, [7] assumed that backshore relief can be characterized by a constant slope. To characterize the backshore topography of the length of the GB coastline (ca. 17,820 km), we first extracted the topographical profiles using an improved version of the automatic delineation approach proposed by [9] at 50 m intervals from the BlueSky 2 m Digital Terrain Model (DTM). We then extracted not only the backshore slope but also a set of geometrical attributes to assess how much the backshore relief deviated from the constant slope assumption. To facilitate the interpretation of the results, we clustered the profiles via a combination of statistical clustering and J. Mar. Sci. Eng. 2020, 8, 866 3 of 26 expert elucidation. This approach enabled (1) the classification of the types of coastline morphology that occur around the GB coastline; (2) an estimation of the main contributions to the generalized sediment conservation equation for each coastline type; and (3) the attribution of the entire Great BritainJ. Mar. Sci. coastline Eng. 2019 with, 7, x FOR these PEER coastline REVIEW types. 3 of 27

Figure 1. Applications of the essential Bruun rule to different control volumes produce different instantiations.Figure 1. Applications (a) Application of the to essential a shoreface Bruun and beach rule to gives different the classic control Bruun volumes rule. (b produce) Application different to ainstantiations. beach and shoreface (a) Application backed by to a barriera shoreface island and gives beach a barrier gives Bruunthe classic rule. Bruun (c) Application rule. (b) Application to a beach andto a shoreface beach and backed shoreface by a clibackedff gives by a a cli barrierff Bruun island rule. gives In each a barrier case, the Bruun implied rule. transgression (c) Application slope to a (blackbeach dashed and shoreface line) equals backed the averageby a cliff profile gives slopea cliff SBruun, but only rule. in In the each classic case Bruun, the implied rule is thistransgression equal to ̅ theslope nearshore (black dashed slope Ss line)(gold equals solid line).the average (Adapted profile by authorsslope 푆, from but only [7]). in the classic Bruun rule is this equal to the nearshore slope Ss (gold solid line). (Adapted by authors from [7].) The paper provides a detailed overview of the methodology and data employed (Section2) beforeHere outlining, we propose the main an results innovative of the approach study (Section to assess3). Thethe potential significance shoreline of the resultsresponse in relationto rising to sea producinglevels based a typology only on ofthe the geometrical Great Britain analysis coastline of backshore as well as topographical an assessment profiles of the likely, and response we applied to a it sea-levelto the whole rise is of discussed the Great in Britain Section-and4. -isles coastline (hereinafter referred as GB). First, we show how Equation (1) was generalized by [7] to include the sources and sinks of sediment for steep and gentle 2. Materials and Methods backshore slopes to produce a generalized cliff and back-barrier Bruun rule, respectively. To be able 2.1.to obtain Generalized an analytical Sediment solution Conservation of the Equation generalized for Steep Bruun and Gentlerule, [7] Backshore assumed Slopes that backshore relief can be characterized by a constant slope. To characterize the backshore topography of the length of the GB Fromcoastline the (ca. general 17,820 analysis km), we of first the extract conservationed the topographical of sediment over profiles a cross-shore using an improved profile z =versionη[x] at a fixed alongshore position y, using a shoreface control volume s x u (Figure2)[ 7] derived a of the automatic delineation approach proposed by [9] at 50 m intervals≤ ≤from the BlueSky 2 m Digital generalizedTerrain Model shoreline (DTM) transgression. We then extracted for any coastalnot only profile the backshore as: slope but also a set of geometrical attributes to assess how much the backshore relief deviated from the constant slope assumption. To ds facilitate the interpretationc0H of= the resultsqx,s , qwex,u clustered the ∂profilesyQy via a combinationcLR0 of statistical(2) dt − − − |{z} clustering and expert|{z elucidat} ion. This| approach{z } enabled (1) |{z}the classification of the types of coastline cross shore f lux divergence alongshore f lux divergence accomodation morphology thatshoreline occur migration around the− GB coastline; (2) an estimation of the main contributions to the generalizedEach variable sediment in Equation conservation (2) is equation described for in each Table coastline1, and each type; term and can (3) bethe understood attribution asof athe cross-sectionalentire Great Britain area of coastline sediment with produced these coastline (or consumed) types. per unit time. The left-hand side is sediment producedThe by paper coastal provides erosion a duedetailed to shoreline overview retreat of the (for methodologyds/dt < 0). The and right-hand data employed terms represent(Section 2) sedimentbefore outlining accumulation the main due results to net of cross- the study and along-shore (Section 3). The sediment significance flux divergences, of the result ands in sedimentrelation to consumptionproducing a requiredtypology to of keep the Great pace withBritain a sea-level coastline rise as well at an as eff anective assessment rate R’. Tableof the1 likelysummarizes response the to variablesa sea-level shown rise is in discussed Figure2. Notein Section that Equation4. (2) requires ηs zsea = const but is insensitive to the − particular datum used to define the shoreline s. The authors of [6] provided a full derivation of the shoreline2. Materials Equation and Methods (2) and discussed its significance generally.

2.1. Generalized Sediment Conservation Equation for Steep and Gentle Backshore Slopes From the general analysis of the conservation of sediment over a cross-shore profile z = η[x] at a fixed alongshore position y, using a shoreface control volume s ≤ x ≤ u (Figure 2) [7] derived a generalized shoreline transgression for any coastal profile as:

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Table 1. Shoreline sediment conservation equation parameters a.

Symbol Variable Units x cross-shore coordinate m y alongshore coordinate m z vertical coordinate m t time s η(x) surface elevation m cross-shore shoreline s m position cross-shore profile toe u m elevation ηs shoreline elevation m ηu profile toe elevation m 2 qx cross-shore sediment flux m /s alongshore sediment Q m3/s J. Mar. Sci. Eng. 2019, 7, x FORy PEER REVIEW discharge 4 of 27 ∆q net flux difference m2/s L 푑푠 profile length m 푐 퐻 = 푞 − 푞 − 휕 푄 − 푐⏟̅퐿푅′ ⏟0 푑푡 ⏟푥 , 푠 푥 ,푢 ⏟푦 푦 H 푐푟표푠푠−net푠ℎ표푟푒 profile푎푙표푛푔푠 reliefℎ표푟푒 푎푐푐표푚표푑푎푡𝑖표푛 m (2) 푠ℎ표푟푒푙𝑖푛푒 푓푙푢푥 푑𝑖푣푒푟푔푒푛푐푒 S 푚𝑖푔푟푎푡𝑖표푛 average profile푓푙푢푥 푑𝑖푣푒푟푔푒푛푐푒 slope m/m 2 2 Each variable in αEquation (2) is describedprofile in shape Table 1 factor, and each term can mbe understood/m as a cross- sectional area of sedimentR producedrelative (or consumed) sea-level per rise unit rate time. The left- mhand/s side is sediment produced by coastal R’erosion due toe shorelineffective sea-levelretreat (for rise ds/dt rate < 0). The right m-hand/s terms represent 3 3 sediment accumulationc due to net averagecross- and sand along concentration-shore sediment flux divergences,m /m and sediment 3 3 consumption requiredc0 to keep paceshoreline with a sea sand-level concentration rise at an effective ratem R’/.m Table 1 summarizes the variables shown in Figure 2. Note thata as Equation used in ( Figure2) requires2. ηs – zsea = const but is insensitive to the particular datum used to define the shoreline s. The authors of [6] provided a full derivation of the shoreline Equation (2) and discussed its significance generally.

Figure 2. TheFigure shoreline 2. The shoreline response response to ato sea-level a sea-level riserise for for both both passive passive inundation inundation and morphodynamic and morphodynamic evolutionevolution is best understood is best understood from afrom general a general analysis analysis of of the the conservation conservation of of sediments sediments over over a cross a cross-shore- shore profile z = η[x] at a fixed alongshore position y, using a shoreface control volume s ≤ x ≤ u. The profile z = η[x] at a fixed alongshore position y, using a shoreface control volume s x u. The sediment sediment conservation Equation (1) assumes the control volume contains a relatively homogeneous≤ ≤ conservationdeposit Equation with an average (1) assumes sediment the concentration control volume 푐̅, but allowscontains the possibilitya relatively of ahomogeneous compositional deposit with an averagechange at sediment the shoreline concentration s, where the sedimentc, but allows concentration the possibility c0 may differ of from a compositional c. In general, c may change at the represent bulk sediment or sediment of a particular size range. Symbols explained in Table 1. (Figure shoreline s, where the sediment concentration c0 may differ from c. In general, c may represent bulk sedimentadapted or sediment by authors of afrom particular [7].) size range. Symbols explained in Table1. (Figure adapted by authors fromThe [effective7]). sea-level rise rate R’ includes accommodation due to both a relative sea-level rise and changes in profile geometry.

푑퐿 푑푆̅ 푑훼 ⏟푅′ = ⏟푅 + (2훼 − 1)푆̅ + (훼 − 1)퐿 + 퐻 ⏟ 푑푡 ⏟ 푑푡 ⏟푑푡 푒푓푓푒푐푡𝑖푣푒 푟푒푙푎푡𝑖푣푒 (3) 푙푒푛푔푡ℎ 푠푙표푝푒 푠ℎ푎푝푒 푠푒푎 푙푒푣푒푙 푠푒푎 푙푒푣푒푙 where the bulk profile geometry is described completely by the length L, relief H, average slope 푆̅, and average shape a (Figure 2). These variables are related by 퐻 퐴 푆̅ = , 훼 = (4) 퐿 퐿퐻 Note that the order-one shape factor α depends only on the fractional sediment fill of the control volume bounding box. For smooth monotonic profiles, α also describes the profile curvature: α = 1/2 for a linear profile, while α < 1/2 for a concave profile, and α > 1/2 for a convex profile.

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The eﬀective sea-level rise rate R’ includes accommodation due to both a relative sea-level rise and changes in proﬁle geometry.

dL dS dα R0 = R + (2α 1)S + (α 1)L + H (3) |{z} |{z} − dt − dt dt e f f ective sea level relative sea level | {z } | {z } |{z} length slope shape where the bulk proﬁle geometry is described completely by the length L, relief H, average slope S, and average shape a (Figure2). These variables are related by

H A S = , α = (4) L LH Note that the order-one shape factor α depends only on the fractional sediment fill of the control volume bounding box. For smooth monotonic profiles, α also describes the profile curvature: α = 1/2 for a linear profile, while α < 1/2 for a concave profile, and α > 1/2 for a convex profile. The mathematical expression for the cliff shoreline evolution model shown in Figure3 is obtained by applying the shoreline Exner Equation (2) to the vertical cliff (i.e., zero length) and the nearshore region and then combining them into a net cliff-nearshore mass balance equation resulting in ! ! ds c L R c R = s s = s , (5) dt − c0 Hs + Hc − c0 S where S is the average slope of the combined cliff and nearshore profile (Figure3). This simplified ds model assumes cliff erosion ( dt < 0) produces a seaward flux of sediment (qs > 0) in concert with shoreline retreat and produces sediment that is instantly available to feed nearshore aggradation. For a rocky coast, cliff retreat occurs only after shoreline retreat undercuts the cliff, and produces rock fragments, which must be weathered before sand becomes available to the nearshore. In the special case where cliff and nearshore deposits are homogeneous, c0 = cs, Equation (5) reduces to the cliff Bruun rule of Equation (1) and Figure1c. However, the slope S will vary as the cliff relief Hc[t] changes through time.J. Mar. Sci. Eng. 2019, 7, x FOR PEER REVIEW 6 of 27

Figure 3. FigureIdealized 3. Idealized geometry geometry and and mass mass balancebalance on on a steep a steep coast. coast. Nearshore Nearshore geometry is geometry fixed but cliff is fixed but cliff heightheightHc canHc can vary vary inin time. Shoreline Shoreline s locateds located at cliff toe. at Beach cliff toe. top elevation Beach η tops rises elevation with sea levelηs rises with zsea, maintaining a fixed beach relief. Cliff erosion produces sediment flux qs, which fuels nearshore sea level zsea, maintaining a fixed beach relief. Cliff erosion produces sediment flux qs, which fuels aggradation. Sand fraction c0 of cliff deposits may differ from nearshore sand fraction cs. (Figure nearshoreadapted aggradation. by authors Sand from fraction[7].) c0 of cliff deposits may differ from nearshore sand fraction cs. (Figure adapted by authors from [7]). At any time t, the cliff relief Hc[t] is the difference in elevation between the backshore topography η0[s] and the shoreline elevation ηs[s]. If we define the origin of our (x,z) coordinate system to be the shoreline position (s, ηs) at time t = 0, then from the geometry of Figure 3, the cliff relief at time t is

퐻푐[푡] = 퐻푐,0 − 푆0푠 − 푅푡 , (6)

where 퐻푐,0 is the cliff relief at time t = 0. Differentiating Equation (6) with respect to time gives 푑푠 d퐻 푆 = − 푐 − 푅, (7) 0 푑푡 dt which can be combined with Equation (5) to eliminate ds/dt, giving an ordinary differential equation (ODE) for the time evolution of the cliff relief Hc[t]

푑퐻 퐻 −퐻 [푡] 푐 = R ( 푐,∞ 푐 ) , (8a) 푑푡 퐻푠−퐻푐[푡]

푐 퐻 = ( s) 푆 퐿 − 퐻 , 푐,∞ 푐 0 s s 0 (8b) 퐻푐[0] = 퐻푐,0 ,

where 퐻푐,∞ is the equilibrium cliff relief reached as t becomes large. Equations (8a)–(8b) constitute [7]’s model for shoreline retreat on steep coasts. On steep coasts, shoreline retreat may obey the classic Bruun model in the short term, but in the long run, shoreline retreat will always converge to the passive inundation model. A first-order estimate for the equilibrium transition is the time for the sea level to traverse the effective zone of erosion [8], which is determined by the tidal range in conjunction with wave variability. For example, with a 1 m and 10 m elevation of the zone of erosion, under a sea-level rise rate of 10 mm/year, it will take 100 to 1000 years to fully traverse this zone and reach equilibrium. To determine shoreline behaviour over intermediate timescales, we must solve Equation

(8a) for a time-varying cliff relief 퐻푐[푡], from which we can compute the shoreline trajectory s[t] using Equation (6). The solution to Equations (8a)–(8b) was given by [7] as

퐻푐,0−퐻푐,∞ 퐻푐,0−퐻푐,∞ 푅푡 퐻푐[푡] = 퐻푐,∞ + (퐻푠 + 퐻푐,∞) × W [(( ) 푒푥푝 [ ]) 푒푥푝 [− ]] , (9) 퐻푠+퐻푐,∞ 퐻푠+퐻푐,∞ 퐻푠+퐻푐,∞

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At any time t, the cliff relief Hc[t] is the difference in elevation between the backshore topography η0[s] and the shoreline elevation ηs[s]. If we define the origin of our (x,z) coordinate system to be the shoreline position (s, ηs) at time t = 0, then from the geometry of Figure3, the cli ff relief at time t is

Hc[t] = Hc S s Rt, (6) ,0 − 0 − where Hc,0 is the cliﬀ relief at time t = 0. Diﬀerentiating Equation (6) with respect to time gives

ds dH S = c R, (7) 0 dt − dt − which can be combined with Equation (5) to eliminate ds/dt, giving an ordinary differential equation (ODE) for the time evolution of the cliff relief Hc[t] ! dHc Hc, Hc[t] = R ∞ − , (8a) dt Hs Hc[t] − cs Hc, = S0Ls Hs, ∞ c0 − (8b) Hc[0] = Hc,0, where Hc, is the equilibrium cliff relief reached as t becomes large. Equations (8a)–(8b) constitute [7]’s ∞ model for shoreline retreat on steep coasts. On steep coasts, shoreline retreat may obey the classic Bruun model in the short term, but in the long run, shoreline retreat will always converge to the passive inundation model. A first-order estimate for the equilibrium transition is the time for the sea level to traverse the effective zone of erosion [8], which is determined by the tidal range in conjunction with wave variability. For example, with a 1 m and 10 m elevation of the zone of erosion, under a sea-level rise rate of 10 mm/year, it will take 100 to 1000 years to fully traverse this zone and reach equilibrium. To determine shoreline behaviour over intermediate timescales, we must solve Equation (8a) for a time-varying cliff relief Hc[t], from which we can compute the shoreline trajectory s[t] using Equation (6). The solution to Equation (8a,b) was given by [7] as " ! " #! " ## Hc,0 Hc, Hc,0 Hc, Rt Hc[t] = Hc, + (Hs + Hc, ) W − ∞ exp − ∞ exp , (9) ∞ ∞ × Hs + Hc, Hs + Hc, −Hs + Hc, ∞ ∞ ∞ where W[] is Lambert’s W function [10]. The mathematical expression for the back-barrier shoreline evolution model shown in Figure4 is obtained by applying the shoreline Exner Equation (2) to the shoreface, overwash zone and back-barrier regions and then combining them into a net backshore-barrier-island and nearshore mass balance equation resulting in " # " # ds Ls + Lw + Lb Lb = R + (1 2αb) , (10) dt − (c /cs)Hs H − (c /cs)Hs H 0 − b 0 − b J. Mar. Sci. Eng. 2019, 7, x FOR PEER REVIEW 7 of 27 where W[] is Lambert’s W function [10]. The mathematical expression for the back-barrier shoreline evolution model shown in Figure 4 is obtained by applying the shoreline Exner Equation (2) to the shoreface, overwash zone and back- barrier regions and then combining them into a net backshore-barrier-island and nearshore mass balance equation resulting in

푑푠 퐿푠+퐿푤+퐿푏 퐿푏 = − [ ] 푅 + (1 − 2훼푏) [ ] , (10) 푑푡 (푐0⁄푐s)퐻푠−퐻푏 (푐0⁄푐s)퐻푠−퐻푏

At any time t, the back-barrier relief Hb[t] is the difference in elevation between the backshore topography η0[b-Lb] and the shoreline elevation ηs[s]=zsea[t]. If we define the origin of our (x, z) coordinate system to be the shoreline position (s, ηs) at time t = 0, then from the geometry of Figure 4, the back-barrier relief at time t is

푆0푠+푅푡 퐻푏[푡] = 퐻푏,0 − , (11) 1+푆0⁄푆b where 퐻푏,0 is the initial back-barrier relief at time t = 0. Differentiating Equation (10) with respect to time gives

푑푠 d퐻 푆 = (1 + 푆 ⁄푆 ) 푏 − 푅, (12) 0 푑푡 0 b dt which can be combined with Equation (10) to eliminate ds/dt, giving an ordinary differential equation (ODE) for the time evolution of the back-barrier relief Hb[t]

푑퐻 퐻 −퐻 [푡] 푏 = R̃ ( 푏,∞ 푏 ) , (13a) 푑푡 퐻̃푠−퐻푏[푡]

(푐 ⁄푐 )퐻 −푆 (퐿 +퐿 ) 퐻 = 0 s s 0 s w , 푏,∞ 1+푆 ⁄푆 0 b (13b) 퐻푏[0] = 퐻푏,0, where 퐻푏,∞ is equilibrium back-barrier relief, and the effective shoreface relief 퐻̃푠 and sea-level rise rate R̃ are defined by −1 1−2훼푏 퐻̃푠 = 훽(푐0⁄푐s)퐻푠, R̃ = 훽푅, 훽 = (1 + ) , (14) 1+푆0⁄푆b 훽 J.where Mar. Sci. Eng.is a2020 shape, 8, 866 factor accounting for back-barrier curvature. In typical cases, where the curvature7 of 26 is mild (훼푏=1/2) and the back-barrier is relatively steep (푆b>>S0), this shape factor is near one.

Figure 4. Idealized geometry and mass balance on a gentle coast. Barrier island consists of subaqueous shoreface,Figure 4. Idealized subaerial geometry island (overwash and mass zone)balance and on subaqueous a gentle coast. back-barrier. Barrier island Island consists bounded of subaqueous seaward byshoreface, shoreline subaerials and landward island (overwash by backshore zone)b ,and both subaqueous at sea level backzsea.-barrier. Shoreface Island and bounded island have seaward fixed geometry.by shoreline Back-barrier s and landward slope Sbyb andbackshore shape αbb, areboth fixed, at sea but level back-barrier zsea. Shoreface length andLb islandand relief haveH bfixedcan varygeometry. in time. Back Island-barrier maintained slope Sb byand overwash shape αb flux areq fixed,s across but the back shoreline.-barrier Back-barrierlength Lb and growth relief fuelledHb can by overwash flux q across the backshore. Sand fraction c of barrier island deposits may differ from b s sand fraction c0 of backshore deposits. (Figure adapted by authors from [7]).

At any time t, the back-barrier relief Hb[t] is the difference in elevation between the backshore topography η0[b-Lb] and the shoreline elevation ηs[s] = zsea[t]. If we define the origin of our (x, z) coordinate system to be the shoreline position (s, ηs) at time t = 0, then from the geometry of Figure4, the back-barrier relief at time t is S0s + Rt Hb[t] = Hb,0 , (11) − 1 + S0/Sb where Hb,0 is the initial back-barrier relief at time t = 0. Differentiating Equation (10) with respect to time gives ds dH S = (1 + S /S ) b R, (12) 0 dt 0 b dt − which can be combined with Equation (10) to eliminate ds/dt, giving an ordinary differential equation (ODE) for the time evolution of the back-barrier relief Hb[t]   dHb Hb, Hb[t]  = eR  ∞ −   , (13a) dt Hfs H [t] − b

(c0/cs)Hs S0(Ls+Lw) Hb, = +S− S , ∞ 1 0/ b (13b) Hb[0] = Hb,0, where Hb, is equilibrium back-barrier relief, and the eﬀective shoreface relief Hfs and sea-level rise ∞ rate eR are deﬁned by

! 1 1 2αb − Hfs = β(c0/cs)Hs, eR = βR, β = 1 + − , (14) 1 + S0/Sb where β is a shape factor accounting for back-barrier curvature. In typical cases, where the curvature is mild (αb = 1/2) and the back-barrier is relatively steep (Sb >> S0), this shape factor is near one. Equations (13) and (14) constitute our model for shoreline retreat on gentle coasts. From Equation (13), we can see immediately that as in the steep coast case, in the long-term limit J. Mar. Sci. Eng. 2020, 8, 866 8 of 26

of an equilibrium back-barrier (Hb = Hb, ), shoreline transgression converges to passive inundation. ∞ Similarly,to determine shoreline behavior over intermediate timescales, we must solve Equation (13a) for time-varying back-barrier relief Hb[t] and then compute the shoreline trajectory s[t] using Equation (11). As shown by [7], the solution to Equation (13a–c) is given by       Hb,0 Hb,  Hb,0 Hb,   eRt  Hb[t] = Hb, Hfs Hb, W − ∞ exp − ∞ exp  , (15) ∞ ∞       − − × Hfs Hb, Hfs Hb, −Hfs + Hb, − ∞ − ∞ ∞ where W[] is Lambert’s W function [10]. In our gentle coast model, the fundamental requirement for barrier island formation is net onshore sediment transport (qs < 0). As shown by [7], for an equilibrium barrier, we can use Equations (2) applied to the shoreface shown in Figure4 and (12) to express this as ! ! c0 cs cs qs,eq = RHs , Ss > S0, (16) − S0 − Ss c0

Hence, even if Ss > S0, barrier formation may be impossible on a sand-poor substrate (c0 < cs). On the other hand, onshore sediment transport does not guarantee barrier island formation. An equilibrium back-barrier is only possible if Hb, > 0, which requires ∞

qs,eq > csLwR, (17)

The onshore ﬂux, qs,eq, must be suﬃcient to aggrade the island while leaving some sediment left over for back-barrier aggradation [7]; otherwise, a barrier beach will form rather than a true barrier island. More generally, we can interpret Lw as a maximum overwash distance. Physical limitations on overwash processes will always impose a maximum length, Lw,max, as a function of both storm and wave climate (e.g., surge heights) and sediment supply limitations, such that n o Lw,max = min Lw,max, qs,eq/(csR) , (18) −

When Equation (17) is satisﬁed, we get a barrier island with Lw = Lw,max and a subaqueous back-barrier, but when Equation (17) is violated, we get a barrier beach with Lw < Lw,max and no back-barrier. Combining Equations (16) and (17) shows that a true barrier island will form only if

! cs Lw Ss > 1 + , (19) c0 Ls

Equations (16) and (19) delineate a continuum of coastal landforms that can occur on gentle coasts experiencing a sea-level rise. If Equation (16) is violated, we get a cliﬀ. Otherwise we get a barrier beach with a length given by Equation (18), or if Equation (19) is also satisﬁed, we get a barrier island. The role of backshore topography in shoreline change due to a sea-level rise can be inferred from Equations (9) and (15). We start by noticing that the two real branches Wk=0,1 of the Lambert function W have a shape as shown in Figure5. Any initial state value that falls within the domain of the principal branch, Wk=0, will evolve from the initial Hc,0 and Hb,0 towards the equilibrium Hc, and ∞ Hb, , respectively. Any initial state value that falls within the domain of the branch Wk=1 will evolve ∞ towards and will never reach equilibrium. The slope of the W function at t = 0 indicates the rate of −∞ change of Hc[t] and Hb[t] over time. From Figure5, it can be seen that the slope of W(x) is maximal and near vertical at the x = 1/e value (i.e., where the two branches coincide); the slope is approximately − unity near x = 0 and smaller than unity for x >> 0. The initial value of x on W(x) can be obtained from z the identity W(x = ze ) = z of the Lambert Wk=0,1 branches as cs H H Hc,0 S0Ls + SsLs H + S L + c,0 c, − c0 c,0 s s S Ss zc = − ∞ = = 1 = 1, S = Hc,0/Ls, (20) + cs cs cs Hs Hc, S Ls S Ls − S − ∞ c0 0 c0 0 c0 0 J. Mar. Sci. Eng. 2019, 7, x FOR PEER REVIEW 9 of 27

푐s 퐻푐,0−( )푆0퐿s+푆s퐿s ̅ 퐻푐,0−퐻푐,∞ 푐0 퐻푐,0+푆s퐿s 푆+푆s ̅ z푐 = = 푐s = 푐s − 1 = 푐s − 1 , 푆 = 퐻푐,0⁄퐿s, (20) 퐻푠+퐻푐,∞ ( )푆0퐿s ( )푆0퐿s ( )푆0 푐0 푐0 푐0 J. Mar. Sci. Eng. 2020, 8, 866 9 of 26 퐻푏,0−퐻푏,∞ 퐻푏,0−퐻푏,∞ z푏 = = . (21) 퐻̃푠−퐻푏,∞ 훽(푐0⁄푐s)퐻푠−퐻푏,∞ Hb,0 Hb, Hb,0 Hb, where z and z are the initial zvalue= s of z− in x=∞ze=z for the cliff− and∞ back.-barrier models of Equations (21) 푐 푏 b (c c )H H Hfs Hb, β 0/ s s b, (9) and (15), respectively, and are function− s ∞of both the geometry− ∞ of the coast and the sediment z fractionwhere zcomposition.c and zb are Equations the initial (2 values0) and (21) of z providein x = aze simplefor thegeometrical cliff and relationship back-barrier between models the of topographyEquations(9) and and the (15), sediment respectively, fraction and ratio are for functions z푐 and of z푏 both to be the equal geometry to −1, of 0 the and coast > 0, andwhich the corresponsedimentds fraction with the composition. x values x = Equations -1/e, x = 0 (20)and andx >> (21)0, with provide the maximum, a simple geometrical median and relationship minimum characteristicbetween the rates topography of change and of theW with sediment x. From fraction the expected ratio for rateszc and of changezb to be of equal 퐻푐[푡] to and1, 퐻 0푏 and[푡] and> 0, 푑푠 − 푑푠 Equationswhich corresponds (7) and (12) with, we thecan xalso values inferx if= the1 shoreline/e, x = 0 and positionx >> will0, with retreat the ( maximum,< 0) or advance median ( and> − 푑푡 푑푡 H [td]퐻푐 0minimum) as the sea characteristic-level rises. We rates first of notice change that of the W with condition x. From for thethe expectedshoreline rates to advance of change requires of c and< H [t] ds dt b andd퐻푏 Equations (7) and (12), we can also infer if the shoreline position will retreat ( dt < 0) or 0 and >ds 0 for our cliff and back-barrier models, respectively. Hereinafter, we will only consider advancedt ( dt > 0) as the sea-level rises. We first notice that the condition for the shoreline to advance dH therequires main branchdHc < 0 Wandk=0 solutionb > 0 for, which our cli willff and continuously back-barrier evolve models, towards respectively. equilibrium Hereinafter,, and therefore we will, the conditiondt for shorelinedt advance under a sea-level rise can be expressed as only consider the main branch Wk=0 solution, which will continuously evolve towards equilibrium, and therefore, the condition for shoreline 퐻푐,0 advance> 퐻푐,∞ ; under퐻푏,0 < a퐻 sea-level푏,∞, rise can be expressed as (22) Equipped with Equations (20)–(22), we can now infer if the shoreline is likely to respond to a Hc,0 > Hc, ; Hb,0 < Hb, , (22) sea-level rise by eroding or advancing and at∞ what rate of∞ change (i.e., maximum, medium or minimum).

Figure 5. Graph of the two real branches of Lambert Wk=0,1 function and geometrical interpretation based on our simple cliff-shoreface model assuming (c0/cs) = 1 or that cliff and shoreface sediment Figure 5. Graph of the two real branches of Lambert 푊 function and geometrical interpretation composition are the same. The black arrows indicate푘= the0,1 direction towards which an initial state based on our simple cliff-shoreface model assuming (푐0⁄푐s) = 1 or that cliff and shoreface sediment will evolve over time. Points I, II and III are all on the principal branch Wk=0, and all tend toward composition are the same. The black arrows indicate the direction towards which an initial state will equilibrium (x = 0), while Point IV is the only one over Wk=1 and represents an unstable point that will evolve over time. Points I, II and III are all on the principal branch 푊 , and all tend toward evolve towards - over time. 푘=0 equilibrium (x = 0)∞, while Point IV is the only one over 푊푘=1 and represents an unstable point that willEquipped evolve towards with Equations -∞ over time. (20)–(22), we can now infer if the shoreline is likely to respond to a sea-level rise by eroding or advancing and at what rate of change (i.e., maximum, medium or minimum). 2.2. Characterization of Great Britain-and-Isles Cross-Shore Profiles: Overview 2.2. Characterization of Great Britain-and-Isles Cross-Shore Profiles: Overview We followed a five-step methodology to extract and classify the elevation transects for the GB coastlineWe followedas illustrated a five-step in Figure methodology 6. Steps 1 to extract3 extract anded classifythe elevation the elevation profiles transectsin a traceable for the and GB repeatablecoastline asway illustrated, while Steps in Figure 4 and6 .5 cluster Steps 1ed to the 3 extracted extracted theprofiles elevation into types profiles of elevation in a traceable transects and thatrepeatable could way,be linked while Stepswith the4 and anticipated 5 clustered geomorphological the extracted profiles change into types due of to elevationa sea-level transects rise. T thathe numbercould be of linked elevation with transect the anticipateds will depe geomorphologicalnd on the resolution change of due the toDigital a sea-level Elevation rise. TheModel number (DEM of) elevation transects will depend on the resolution of the Digital Elevation Model (DEM) and the coastline length and location, and the actual elevation transect geometry will depend on how the orthogonal transects are defined. The method chosen will need to be able to effectively handle a large number J. Mar. Sci. Eng. 2020, 8, 866 10 of 26

J. Mar. Sci. Eng. 2019, 7, x FOR PEER REVIEW 10 of 27 (in the order of millions of transects) to cover the extent of the whole coastline of GB, including the and the coastline length and location, and the actual elevation transect geometry will depend on how islands, which is 31,368 km, according to Ordnance Survey (OS). To delineate the orthogonal transects, the orthogonal transects are defined. The method chosen will need to be able to effectively handle a we chose the same approach used in the CliffMetric algorithm proposed by [9], which was specifically large number (in the order of millions of transects) to cover the extent of the whole coastline of GB, designedincluding to resolve the islands, coastlines which with is 31,368km, very irregular according shapes to Ordnance such as the Survey GB coastline. (OS). To delineate As our goal the was to assessorthogonal the likely transects response, we chose of thethe same different approach elevation used in transects the CliffMetric to a sea-level algorithm rise, proposed based by uniquely [9], on backshorewhich was topographical specifically designed data, we to selected resolve coastlines the OS High with Water very irregular Line as shapesthe preferred such as coastline the GB for generatingcoastline. the As orthogonal our goal was transects. to assess Thethe likely CliffMetric response code of the proposed different by elevation [9] automatically transects to a delineates sea- the coastlinelevel rise, for based a given uniquely still on water backshore level but topographical does not allow data, we the selected user to the define OS High a coastline Water Line as an as input. Our firstthe preferred step was coastline then to for modify generat theing Cli theff orthogonalMetric code transects. to add The the CliffMetric option of acode user-defined proposed by coastline. [9] Thisautomatically improved version delineates of Clitheff coastlineMetric wasfor a given then s usedtill water to extract level but the does elevation not allow profiles the user to from define a DTM a coastline as an input. Our first step was then to modify the CliffMetric code to add the option of a provided by BlueSky International Limited (5 m resolution) and the most up-to-date available DTM user-defined coastline. This improved version of CliffMetric was then used to extract the elevation for GB (BlueSky is a commercial product subject to license). The dimensionality of the transects profiles from a DTM provided by BlueSky International Limited (5 m resolution) and the most up- extractedto-date was available then reduced DTM using for GB Principal (BlueSky Component is a commercial Analysis product (PCA) [ subject11]. The to reduced license). dimension The set ofdimensionality the elevation of transects the transects was then extracted clustered was then using reduced the K-means-partitional using Principal Component clustering Analysis approach in MATLAB(PCA) [11] [12. The]. We reduced chose dimension a partitional set of clustering the elevation approach transect overs was a then hierarchical clustered approachusing the K due- to the largemeans number-partitional of transectsclustering onapproach the order in MATLAB of 106 that [12] we. We needed chose a to partitional cluster: theclustering number approach of distance calculationsover a hierarchical for hierarchical approach approaches due to the large increases number geometrically of transects on with the order the numberof 106 that of we observations. needed The partitionalto cluster: the approach number is of an distance iterative calculations process in for which hierarchical experts approaches have to assess increases thenumber geometrically of clusters and thewith algorithm the number iteratively of observations. guesses The the partitional centroids orapproach central is point an iterative in each process cluster, in and which assign experts points to have to assess the number of clusters and the algorithm iteratively guesses the centroids or central the cluster of their nearest centroid. The sensitivity of the number of clusters and location was tested point in each cluster, and assign points to the cluster of their nearest centroid. The sensitivity of the by using different number of clusters from 5 to 10. We found that ten clusters were the minimum number of clusters and location was tested by using different number of clusters from 5 to 10. We requiredfound to that separate ten clusters all types were of environments; the minimum inrequired particular, to separate we needed all types to increase of environments from 9 to; 10in to be able toparticular separate, we the needed low-lying to increase cliff of from East 9 Englandto 10 to be (Cellable to 3) separate from the the abundant low-lying cluster cliff of East 8. The England sensitivity to the(Cell seed 3)centroid from the wasabundant tested cluster by running 8. The sensitivity the clustering to the ten seed times centroid for Nc was= tested10, mapping by running the the clusters overclustering aerial photography ten times for as Nc shown = 10, mapping in Figure the12 clust anders confirming over aerial photography that the regional as shown statistics in Figure remained 12 unchangedand confirming (i.e., that that the the top regional five dominant statistics clusters remained per unchanged region remained (i.e., that dominant). the top five dominant clusters per region remained dominant).

FigureFigure 6. Methodological 6. Methodological approach approach followed followed to classify the the elevation elevation profiles profiles along along the Great the Great Britain Britain coastline.coastline The. scriptsThe scripts used for used the for different the different steps are steps indicated are indicated and included and included as supplementary as supplementary information. information. 2.3. Datasets: BlueSky DTM and HWM Coastline To deﬁne the morphology of the GB coastline, elevation data were extracted using a DTM provided by BlueSky International Limited (referred to from this point as the BlueSky DTM). This is a 5 m-resolution DTM comprised of both aerial photogrammetry and airborne LiDAR data that covers J. Mar. Sci. Eng. 2020, 8, 866 11 of 26

GB and the islands. The dataset has a multi-temporal resolution from 2006, with the majority of the data being available post-2015 to 2017. The vertical uncertainty of the dataset is estimated as being 0.5 m. ± The transects used in this study to constrain the backshore topography orthogonal to the coastline were based on the vectorised high water mark (HWM) as extracted from the OS OpenMap Local dataset, version January 2020. In England and Wales, this is the mean level of all the high tides; in Scotland, this is the mean level of the spring high tides. In places where there is no foreshore (e.g., vertical cliffs), the tidal boundary is classified as the high water mark. This dataset has a recommended viewing scale of 1:3000 to 1:20,000. As with the elevation data, these were available as 100 km2 tiles of vector polygon data. To extract the coastline from this dataset, the 100 km2 tiles were dissolved into a single vector dataset. The merged polygon data were then converted into a vector line format, following which the outer extents of all features were removed, leaving a single polyline representation of the coastline. This polyline representation of the data was required, as for this study, the improved CliffMetrics program, which was used to facilitate the data extraction, was extended to take in this externally defined coastline in vector polyline format (see Section 2.4). The manipulation of the OS OpenMap Local data to extract the HWM polyline as described means that we represented the coastline as being the border of polygons available within the OS OpenMap Local dataset. Given the finest scale recommended for viewing this dataset of 1:3000, we estimated that this related to a potential uncertainty of 1.5 m associated with the position of the HWM line, considering that the Minimum ± Discernable Mark (MDM) of a 1:3000 scale dataset would be about 1.5 m [13]. The purpose of the coastline transects developed and used in this study was to provide a general characterization of the coast. The vertical ( 0.5 m) and temporal uncertainty concerning the temporal resolution of the ± elevation data regarding their acquisition period (2006–2017), combined with the uncertainty relating to the positioning of the HWM ( 1.5 m), are not believed to be of concern and are consequently not ± considered further in this manuscript.

2.4. Improved CliffMetric Algorithm To extract the elevation profiles, we used an improved version of the CliffMetric algorithm for automatic cliff top and toe delineation [9]. The CliffMetric algorithm builds upon existing methods but is specifically designed to resolve very irregular planform coastlines with many bays and capes, such as parts of the coastline of Great Britain. The original algorithm automatically and sequentially delineates and smooths shoreline vectors, generates orthogonal transects and elevation profiles with a minimum spacing equal to the DEM resolution, and extracts the position and elevation of the cliff top and toe. The outputs include the non-smoothed raster and smoothed vector coastlines, transects orthogonal (hereinafter referred to as transects) to the coastline (as vector shape files), xyz profiles (as comma-separated-value, CSV,files), and the cliff top and toe (as point shape files). The algorithm also automatically assesses the validity of the profile and omits profiles that are too short (i.e., the extraction of the cliff top and toe is not possible when the profile only contains two points). The length of the profiles is initially defined by the user, and the algorithm will attempt to generate the elevation along the full length at every point along the coast. For very irregular coastlines, with many capes and bays, the transects often cross over the coastline before reaching the full (pre-defined) length. For those profiles that cross the shoreline, their length will be reduced to the location where the coastline is crossed. The number of points along the transect will depend on the DEM resolution and orientation of the transect relative to the grid. We used profile lengths of 500 m (i.e., as adopted in the Eurosion project [14]) and a DEM of a 5 m resolution, which means that we had circa 100 points per full-length transect. The elevation of the points along the transects was obtained as the elevation value of the DEM cell whose centroid was closest along the transect. If the algorithm only found two points along a non-full-length normal, the normal was marked as non-valid and no elevation profile was extracted. We improved the CliffMetric algorithm by allowing the user to define a vector coastline along which the user would like the transect to be generated instead of using the automatically delineated coastline vector. The user needs to enter two additional parameters (Figure S7), which are the shape J. Mar. Sci. Eng. 2020, 8, 866 12 of 26

file with the user-defined coastline that will be used to generate the orthogonal transects and the sea handiness, indicating on which side of the coastline the sea is: left or right when traversing the points along the coastline. The user-defined coastline needs to be provided in shape file format and point geometry. The algorithm will traverse the coastline, sequentially, from the second feature (i.e., point) in the geometry layer until the second-to-last feature (i.e., ascending ordinal). The orthogonal transect is obtained as the line defined by the centroid of the raster cell closest to the coastline point (i.e., chainage = 0) and the landward point of the transect (i.e., chainage user defined length + one diagonal cell). ≤ The location of the landward point is calculated as the end of a line orthogonal to the local coastline orientation: the coastal orientation is obtained as the straight line linking the coastline points before and after “this” coastline point. The second new input parameter, the sea handiness, is then used to decide on which side of the coastal orientation the landward point of the profile is located. Notice that, opposite to when the coastline is automatically delineated, the improved CliffMetric version uses the user-defined coastline as given without smoothing. We ran the improved CliffMetric algorithm on the BGS-High Performance Computer, which can process the DEM tiles and the outputs. The extracted CSV files with the elevation profiles contain the following columns: “Dist”, “X”, “Y”, ”Z” and “detrendZ”, which correspond with the chainage in DEM units (e.g., metres); the coordinates X,Y in the DEM coordinate system (i.e., the British National Grid, EPSG: 27700) of the cell centroids from which the elevation, Z, was extracted; and the de-trended elevation, also in DEM units, respectively.

2.5. Cluster Analysis of Elevation Profiles Due to the large dimensionality, (ca. 4 million points x ca. 100 elevation values per transect), some pre-processing was needed to be able to perform a cluster analysis of the extracted transects. The automatic extraction procedure produced a set of transects where most of the transects were of the same length (i.e., 500 m for this work) or shorter (i.e., if the normal was shortened by the algorithm) but did not necessarily have the same number of cross-shore points (i.e., it depended on how many raster cells were underneath each coastline transect). To ensure that all the profiles had the same number of cross-shore points, we normalized the length of all the profiles to the maximum length and resampled at nx = 100 cross-shore locations, obtaining a matrix of nx np, where np is the number of × transects used for this analysis. By normalizing the profile length, we are implicitly assuming that the variance of the length is non-informative, which seems reasonable given the transect length is a user pre-defined variable. We used the de-trended elevation at each of the nx cross-shore locations to ensure that the elevation at the seaward limit of the transect (x/xmax = 0) and landward limit of the transect (x/xmax = 1) was the same: this ensured that the data were periodic and easier to fit to a set of orthogonal Eigen functions. The matrix dimension was reduced to a matrix of nc np, where nc nx is the number of main × ≤ Principal Components (PCs) obtained from applying the Principal Component Analysis [15] that capture a given percentage of the total database variance. Principal Component Analysis (PCA) is a technique for reducing the dimensions of large datasets, increasing interpretability but at the same time minimizing information loss. The empirically generated PCs have a series of properties that make them suitable for reducing the dimensionality of a dataset: (1) they are a reduced set of variables that are optimal predictors for the original variables in the sense of the square minimums, and (2) the new variables are obtained as an original combination of the former ones and are not correlated. By reducing the dimensions of the database using the PCs, we avoided including some of the non-informative small profile differences in the cluster analysis. The matrix algebra required is standard on commercially available routines (e.g., the pca function in MATLAB); thus, it is not discussed in more detail here. For the interested reader, the MATLAB script used is provided as supplementary material. A detailed description of cluster analysis is beyond the scope of this paper, and the reader is referred to Hennig, et al. [16] for a general description and to [17,18] for updated references to coastal applications. In brief, it entails the calculation of distances (interpreted as the similarity) between all objects in a data matrix, on the basis that those closer together are more alike than those are J. Mar. Sci. Eng. 2020, 8, 866 13 of 26 further apart. The clustering techniques can be generally classified as hierarchical or partitional approaches [12]. Hierarchical, agglomerative clustering is a bottom-up approach where objects and then clusters of objects are progressively combined on the basis of a linkage algorithm (or dendogram) that uses the distance measures to determine the proximity of objects and then clusters to each other. The number of distance calculations in hierarchical approaches increases geometrically with the number of observations, making this method unsuitable for very large datasets. Partitional methods have advantages in applications involving large datasets for which the construction of a dendogram is computationally prohibitive. A problem accompanying the use of a partitional algorithm is the choice of the number of desired output clusters. The partitional technique usually produces clusters by optimizing a criterion function defined either locally or globally. A combinatorial search of the set of possible labelling for an optimum value of a criterion is clearly computationally prohibitive. In practice, therefore, the algorithm is typically run multiple times with different starting states, and the best configuration obtained from all of the runs is issued as the output clustering. One such partitional technique is the K-means approach [12]. K-means is one of the simplest unsupervised learning algorithms that solves the well-known clustering problem. The procedure is a simple and easy way to classify a given dataset through a certain number of clusters (assume k clusters) fixed a priori. The main idea is to define k centroids, one for each cluster. The next step is to take each point belonging to a given dataset and associate it to the nearest centroid. When no point is pending, the first step is completed and an early group is performed. At this point, it is necessary to re-calculate k new centroids as centres of the clusters resulting from the previous step. After these k new centroids, a new binding has to be performed between the same data points and the nearest new centroid. A loop is generated. As a result of this loop, it may be noticed that the k centroids change their location step by step until no more changes are made. In other words, centroids do not move any more. We used the “Manhattan distance metric” as the distance metric because this metric provides more meaningful results from both the theoretical and empirical perspectives for large datasets [19].

3. Results

3.1. CliffMetrics and Principal Component Analysis Outputs Out of the 4,405,501 points from which the user-defined coastline for GB is made, we obtained 4,289,708 profiles. There are 115,793 profiles fewer (2.6%) than the number of coastal points and profiles. The majority (97%) of the profiles or 4,153,505 were considered valid, and only 136,203 profiles (3%) were non-valid because they were too short to be included in the analysis: a profile was considered too short if the length was no longer than one diagonal cell or 7.07 m for the 5 5 m cell size used in × this study. As expected, the median length of the valid profiles was 500 m, with a minimum length of 10 m and, unexpectedly, a maximum length of 2144 m: we found that 52 profiles were longer than 508 m. The minimum expected elevation for the BlueSky DTM was 5 m, with no data being defined − as 3.402e+38. For the profile cluster analysis, we ensured that we used only profiles that passed the − quality check and did not include flat profiles or elevation profiles with data gaps. The total number of valid transects was reduced to 3,912,935 after we eliminated those profiles whose minimum elevation and elevation at the landward end were larger than 0.1 and 5 m, respectively, keeping the transects of − length smaller than or equal to the user-defined length (508 m for our study). The majority (99%) of the variability observed in the de-trended elevation profiles is explained by a reduced set of ten PCs. Figure7a shows the cumulative percentage explained by the first 20 PCs obtained by applying the PCA to all 3,912,935 valid profiles. The cumulative percentage of variability explained increases rapidly with the first few PCs, with the first two PCAs alone explaining 86% of the total observed variability. The first 10 PCs explain 99% of the total variability. The mean differences between the original elevation profile and the re-constructed elevation profile using the first 10 PCs are less than 15cm, as shown for the profile in Figure7b. Using the first 10 PCs allows us to reduce the ± J. Mar. Sci. Eng. 2019, 7, x FOR PEER REVIEW 14 of 27

explained increases rapidly with the first few PCs, with the first two PCAs alone explaining 86% of J. Mar.the Sci. total Eng. observed2020, 8, 866 variability. The first 10 PCs explain 99% of the total variability. The14 meanof 26 differences between the original elevation profile and the re-constructed elevation profile using the first 10 PCs are less than ±15cm, as shown for the profile in Figure 7b. Using the first 10 PCs allows dimensionsus to reduce of thethe observationdimensions matrixof the byobservation a factor of matrix 10, from by 100a factor observations of 10, from for each100 observations transect to just for 10each PCs, transect while explaining to just 10 PCs, 99% while of the explaining total variability. 99% of the total variability.

Figure 7. The Principal Component Analysis (PCA) suggests that we can capture the majority of Figure 7. The Principal Component Analysis (PCA) suggests that we can capture the majority of the the variability observed on the de-trended elevation profiles with a reduced number of Principal variability observed on the de-trended elevation profiles with a reduced number of Principal Components (PCs): (a) percentage of variability explained by the first 20 PCs. The percent of Components (PCs): (a) percentage of variability explained by the first 20 PCs. The percent of cumulative variance explained is on the y-axis, and the Principal Component (PC) number is on the cumulative variance explained is on the y-axis, and the Principal Component (PC) number is on the x-axis; (b) comparison of one of the original elevation profiles, x, versus the reconstructed, ex, using the x-axis; (b) comparison of one of the original elevation profiles, x, versus the reconstructed, 푥̃, using first 10 PCs. Top panel shows the elevation profile, and bottom panel, the absolute difference between the first 10 PCs. Top panel shows the elevation profile, and bottom panel, the absolute difference the original and reconstructed. between the original and reconstructed. Figure8 shows the cross-shore eigenvector values for the first three PCs. Note that the first eigenvectorFigure of 8 the shows three the modes cross accounts-shore eigenvector for the 59% of values the total for variability. the first three This PC results. Note is not that surprising the first sinceeigenvector we used of the the non-normalized three modes accounts de-trended for elevation the 59% profiles, of the total and, variability. consequently, This the result centroid, is not definedsurprising as the since mean we of used the variables, the non- cannormalized explain mostde-trended of the data. elevation For that profile reason,s, and, the firstconsequently, eigenvectors the arecentroid, often highly defined correlated as the mean with theof the mean variables, vectors. can Thus, explain the combination most of the ofdata. the For first that eigenvectors reason, the gives first aeigenvectors representation are of often the mean highly de-trended correlated elevation with the profile. mean vectors. The convex Thus, parabolic the combination shape of theof the mean first de-trendedeigenvectors profile gives suggests a repre thatsentation a large of percentage the mean ofde profiles-trended do elevation not increase profile. in elevation The convex linearly parabolic from theshape seaside of the landwards mean de but-trended increase profile at di ffsuggestserent rates. that Thea large location percentage of the maximumof profilesof do the not de-trended increase in profileelevation indicates linearly the fromcliff top the location. seaside The landwards second butand increase the third at PCs different explain rates. 27% and The 5% location of the totalof the variabilitymaximum each. of the These de-trended two PCs profile can be understoodindicates the as cliff the deviationtop location. from The the second mean de-trended and the third profile PCs representedexplain 27% by and the 5% first of PC.the Thetotal firstvariability three PCs each. explain These 91%two PCs of the can total be understood variability. as The the closer deviation the scorefrom is the to zeromean for de each-trended one ofprofile the PCs, represente the closerd by to the a flat first profile PC. The the first de-trended three PCs elevation explain profile91% of is,the whichtotal meansvariability. that theTheelevation closer the increases score is to linearly zero for from each the one seaside of the landwards. PCs, the closer to a flat profile the de-trended elevation profile is, which means that the elevation increases linearly from the seaside landwards.

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FigureFigure 8. 8.Cross-shore Cross-shore eigenvectors eigenvectors for forthe ﬁrst the threefirst three PCs. The PCs. percentage The percentage shown in shown the legend in the indicates legend Figure 8. Cross-shore eigenvectors for the first three PCs. The percentage shown in the legend theindicates percentage the percentage of variability of variability explained explained by each PC. by each PC. indicates the percentage of variability explained by each PC.

FigureFigure9 9shows shows the the centroid centroid and and variability variability of each of each one ofone the of ten the di tenfferent diffe clustersrent clusters obtained obtained when Figure 9 shows the centroid and variability of each one of the ten different clusters obtained plottedwhen plotted against againstthe first theand first second and PCs. second Cluster PCs #8. Cluster is the most #8 is abundant the most type abundant of profile, type representing of profile, when plotted against the first and second PCs. Cluster #8 is the most abundant type of profile, 48%representing of all the 48 profiles,% of all whilethe profiles, the remaining while the clusters remaining combined clusters represent combined the represent remaining the 42% remaining of the representing 48% of all the profiles, while the remaining clusters combined represent the remaining profiles.42% of the The profiles. spreading The of spreading Cluster #8 of is Cluster also significantly #8 is also smallersignificantly that the smaller spreading that the of the spreading other clusters, of the 42% of the profiles. The spreading of Cluster #8 is also significantly smaller that the spreading of the asother shown clusters by the, as horizontal shown by and the vertical horizontal lines and that vertical represent lines twice that the represent standard twice deviation the standard for each other clusters, as shown by the horizontal and vertical lines that represent twice the standard cluster.deviation The for low each scores cluster. for The Cluster low #8,scores for bothfor Cluster PCs, suggest #8, for thatboth profilesPCs, suggest belonging that profiles to this cluster belonging are deviation for each cluster. The low scores for Cluster #8, for both PCs, suggest that profiles belonging closeto this to clusterelevation are profiles close to with elevation constant profiles slopes with and low constant elevation. slope Thes and large low spreading elevation. of The both large PC to this cluster are close to elevation profiles with constant slopes and low elevation. The large scoresspreading for the of otherboth clustersPC scores suggests for the that other there clusters is a significant suggests variabilitythat there inis botha significant the cliff topvariability elevation in spreading of both PC scores for the other clusters suggests that there is a significant variability in andboth horizontal the cliff top location. elevation and horizontal location. both the cliff top elevation and horizontal location.

Figure 9. Centroids and spreading of scores for all the six clusters obtained. The coloured circle Figure 9. Centroids and spreading of scores for all the six clusters obtained. The coloured circle Figurerepresents 9. Centroids the centroid and of spreading the cluster, of and scores the solid for all vertical the six and clusters horizontal obtained. lines The represent coloured twice circle the represents the centroid of the cluster, and the solid vertical and horizontal lines represent twice the representsstandard deviation the centroid of eachof the PC. cluster The, percentageand the solid of vertical profiles and belonging horizontal to eachlines clusterrepresent is showntwice the in standard deviation of each PC. The percentage of profiles belonging to each cluster is shown in parenthesesstandard deviation in the legend of each together PC. The with percentage the cluster of number. profiles belonging to each cluster is shown in parentheses in the legend together with the cluster number. parentheses in the legend together with the cluster number. Figure 10a shows the distribution of the maximum profile elevation per cluster as a violin plot. Figure 10a shows the distribution of the maximum profile elevation per cluster as a violin plot. As expectedFigure 1 from0a shows the PCA the results,distribution the maximum of the maximum elevation profile of the elevation Cluster #8 per profiles cluster is theas a lowest violin ofplot. all As expected from the PCA results, the maximum elevation of the Cluster #8 profiles is the lowest of Asthe expected clusters, withfrom athe modal PCA maximum-profile-elevationresults, the maximum elevation value of of the 1 m.Cluster The frequency#8 profiles distributionis the lowest per of all the clusters, with a modal maximum-profile-elevation value of 1 m. The frequency distribution all the clusters, with a modal maximum-profile-elevation value of 1 m. The frequency distribution per cluster shown in the violin plots suggests that the most frequent (mode) and the average (mean per cluster shown in the violin plots suggests that the most frequent (mode) and the average (mean

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J. Mar. Sci. Eng. 2019, 7, x FOR PEER REVIEW 16 of 27 cluster shown in the violin plots suggests that the most frequent (mode) and the average (mean and median)and median) profiles profiles are not theare same.not the The same. most The frequent most profilefrequent elevation profile iselevation consistently is consistently smaller than smaller the medianthan the and median mean values and mean for all values clusters. for Theall clusters. differences The betweendifferences the between most frequent the most and frequent averaged and elevationaveraged profiles elevation is shown profiles in Figure is shown 10b,c, in which Figure illustrate 10b,c, which the closest-to-centroid illustrate the closest and most-frequent-to-centroid and elevationmost-frequent profiles forelevation each cluster. profiles The for elevation each cluster. profile The for elevation Cluster #8 profile is the flattestfor Cluster of all # the8 is de-trendedthe flattest of elevationall the de profiles,-trended irrespectively elevation profiles of if showing, irrespectively the closest-to-centroid of if showing the (Figure closest 10-tob)-centroid or most-frequent (Figure 10b) (Figureor most 10c)-frequent profile. (Figure The elevation 10c) profile. for the The closest-to-centroid elevation for the closest profiles-to for-centroid all clusters, profiles but for Cluster all clusters #8, , presentsbut Cluster a maximum #8, present elevations a maximum between the elevation coast point between (relative the chainagecoast point= 0%) (relative and the ch mostainage landward = 0%) and pointthe ofmost the landward profile (relative point of chainage the profile= 100%) (relative (Figure chainage 10b). = The 100 elevation%) (Figure profile 10b). forThe Clusters elevation # profile 1, 2, 3, 4,for 5, Clusters 6, 7, 9 and # 1, 10 2, di 3,ff 4,ers 5, by6, 7, both 9 and the 10 location differs ofby the both maximum the location and of the the elevation maximum of and the maximum.the elevation In chainageof the maximum. relative units,In chainage the location relative of units, the maximum the location of theof the de-trended maximum elevation of the de profiles-trended for elevation each clusterprofiles varies for fromeach 5%cluster for Clustervaries from #1 to 5% 85% for for Cluster Cluster #1 #7. to 85% for Cluster #7.

Figure 10. Statistical comparison between clusters: (a) violin plot of maximum elevation per cluster; Figure 10. Statistical comparison between clusters: (a) violin plot of maximum elevation per cluster; (b,c) elevation profiles for each cluster showing the profile closest to the centroid (b) and most frequent (b,c) elevation profiles for each cluster showing the profile closest to the centroid (b) and most profile (c). The percentage of profiles belonging to each cluster is shown in parentheses in the legend frequent profile (c). The percentage of profiles belonging to each cluster is shown in parentheses in together with the cluster number. the legend together with the cluster number. Table1 and Table S2 show the elevation and de-trended elevation profiles for each cluster for a representativeTable 1 profileand Table per S cluster.2 show Thethe elevation representative and de profile-trended was elevation chosenas profiles either for the each one closestcluster tofor a therepresentative cluster centroid profile (Figure per S2) cluster. or the The one representative closest to the most profile frequent was chosen value (Figureas either S3). the Despite one closest this to variability,the cluster when centroid the clusters (Figure are S2 mapped) or the one over closest an aerial to the view most of the frequent backshore value topography (Figure S3). (Figure Despite 11 ),this it seemsvariability, clear that when Clusters the cluster 1, 5 ands are 8 correspond mapped over with an either aerial gentle view profiles of the backshore or back-barrier-type topography profiles. (Figure Examples11), it seems of back-barrier clear that C beachlusters along 1, 5 and the 8 GB correspond coastline canwith be either found gentle at Chesil profiles beach or back and- Blakeney,barrier-type profiles. Examples of back-barrier beach along the GB coastline can be found at Chesil beach and Blakeney, which was classified mostly as Clusters 8, 1 and, to a lesser degree, 5. The variations of the cluster type observed at the Blakeney area seem to be related with the relief at the end of the shoreline

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J. Mar.which Sci. wasEng. classified2019, 7, x FOR mostly PEER asREVIEW Clusters 8, 1 and, to a lesser degree, 5. The variations of the cluster17 of type 27 observed at the Blakeney area seem to be related with the relief at the end of the shoreline normal. normal.The Wash The is Wash a very is low-lying a very low area-lying that wasarea alsothat dominantly was also dominantly classified as classified Clusters 8as and Cluster 1. Thes 8 high and cli 1ff. s Thearound high St.cliffs Bees around and Flamborough St. Bees and headFlamborough were mostly head classified were mostly as Clusters classified 2, 3 and, as C mostly,lusters 6.2, No3 and clear, mostlypattern, 6. in No the cle spatialar pattern distribution in the spatial was observed distribution for Clusters was observed 4, 7, 9 and for 10.Clusters The spatial 4, 7, 9 distributionand 10. The of spatialthe topographical distribution clusters of the istopographical best understood clusters by plotting is best the understood frequency distribution by plotting per the region. frequency distribution per region.

FigureFigure 11. 11. ClusterCluster types types mapped mapped to to the the coast coast point point along along the high waterwater markmark line line used used to to delineate delineate the thetransects. transects. Names Name ofs theof the locations locations around around the GB the coastline GB coastline indicated indicated as text. as Source text. Source of aerial of imagery: aerial imagery:Esri, Maxar, Esri, GeoEye,Maxar, GeoEye, Earthstar Earthstar Geographics, Geographics, CNES/Airbus CNES/Airbus DS, USDA, DS, USGS, USDA, AeroGRID, USGS, AeroGRID, IGN, and the IGN,GIS and User the Community, GIS User Community, v10.6. v10.6.

WeWe used used the the eleven eleven sediment sediment cells cells defined defined by by the the Shoreline Shoreline Management Management Plans Plans (second (second revision) revision) forfor England England and and Wales Wales and and the the eleven eleven administrative administrative units units of ofScottish Scottish marine marine regions regions (see (see Figure Figure 12 12) ) toto extract extract regional regional statistics statistics.. The The Scottish Scottish Marine Marine Regions Regions are are 11 11 areas areas established established for for the the purpose purpose of of regionalregional marine marine planning, planning, defined defined by by The The Scottish Scottish Marine Marine Regions Regions (SMR) (SMR) Order Order 2015. 2015. These These SMR SMR regionsregions are are sub sub-areas-areas of of both both the the "Scottish "Scottish marine marine area" area" defined defined in in the the Marine Marine Act Act 2010 2010 and and "Scottish "Scottish inshoreinshore region" region" defined defined in thethe MarineMarine and and Coastal Coastal Access Access Act Act 2009. 2009 From. From the regionalthe regional counts counts on cluster on clustertypes types (Supplementary (Supplementary Materials), Materials it is), clear it is that clear Cluster that Cluster 8 is the 8 most is the abundant most abundant cluster typecluster for type all 22 forregions all 22 regi (Figureons (Figure12). Clusters 12). Clusters 3, 6 and 3, 7 6 are and not 7 presentare not present on all regions. on all regions.

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Figure 12. Example of the regional statistics that we extracted for the 22 regions into which we divided FigureFigure 12. Example 12. Example of the of regional the regional statistics statistics that that we we extracted extracted forfor the 2222 regions regions into into which which we we divided divided the GB coastline: (a) shows the region coverage; (b–g) show the counts of topographical cluster types the GBthe coastline GB coastline: (a) shows: (a) shows the regionthe region coverage coverage; (;b (–bg–)g )show show thethe counts of of topographical topographical cluster cluster types types for some regions (all other regions are shown in Supplementary Materials). Source of aerial imagery: for somefor regionsome regions (all sother (all other regions regions are are shown shown in i nSupplementary Supplementary Materials).) .Source Source of ofaerial aerial imagery: imagery: Esri, Maxar, GeoEye, Earthstar Geographics, CNES/Airbus DS, USDA, USGS, AeroGRID, IGN, and the Esri, Maxar,Esri, Maxar, GeoEye, GeoEye, Earthstar Earthstar Geographics, Geographics, CNES/Airbus CNES/Airbus DS,DS, USDA, USGS, USGS, AeroGRID, AeroGRID, IGN, IGN, and and GIS User Community, v10.6. the GISthe User GIS CoUsermmunity, Community, v10.6. v10.6.

3.2. Main3.2. Main Topographical Topographical Characteristics Characteristics 3.2. Main Topographical Characteristics We extractedWe extracted the geometrical the geometrical attributes attributes of of the the backshorebackshore topographical topographical profile profile as as shown shown in in We extracted the geometrical attributes of the backshore topographical profile as shown in FigureFigure 13 for 1 all3 for transects all transects and and calculated calculated the the regional regional statistics.statistics. The The backshore backshore shape shape parameter parameter,, 훼0, α0, 훼 Figuredefineddefined 13 for for the allfor domain transectsthe domain and −<∞ calculated α< α01≤represents1 representthe regional hows how statistics. much much thethe The area backshore under under the the backshoreshape backshore parameter elevation elevation, 0, −∞ 0 ≤ definedprofileprofile from for the thefrom domain coast the coast to the− ∞to end the< α end of0 ≤ the of1 transecttherepresent transect disff differershow froms much from a rectanglea the rectangle area thatunder that has has the thethe backshore samesame length elevation and and an profilean from elevation the coast equal to to the the end maximum of the transectprofile elevation, differs from 푧푚푎푥 a. Noterectangle that we that used has 푧 the푚푎푥 sameinstead length of the and elevation equal to the maximum profile elevation, zmax. Note that we used zmax instead of the elevation 푧 푧 ≤ an elevationelevation equal at the to end the of maximum the profile, profile푒푛푑, because elevation, as we saw푧푚푎 푥fro. Notem the that cluster we type used 7 (Figure 푧푚푎푥 instead10), 푒푛푑 of the at the end of the profile, zend, because as we saw from the cluster type 7 (Figure 10), zend zmax, as the 푧푚푎푥, as the elevation does not grow continuously backshore. This shape parameter will be≤ close to 1 elevation doesat the not end grow of the continuously profile, 푧푒푛푑 backshore., because as This we shapesaw fro parameterm the cluster will betype close 7 (Figure to 1 if the10), shape 푧푒푛푑 ≤ is if the shape is similar to a rectangle and smaller if it is not. Negative values of 훼0 can occur for 푧푚푎푥, as the elevation does not grow continuously backshore. This shape parameter will be close to 1 similarprofiles to a rectangle where the and elevation smaller is ifnot it always is not. larger Negative than values or equal of toα 0thecan elevation occur for at profilesthe initial where point the if the shape is similar to a rectangle and smaller if it is not. Negative values of 훼0 can occur for elevation(x=0) is. We not also always extracted larger the than backshore or equal slope, to the 푆0 elevation; cliff face slope, at the initial푆푐; and pointinitial (cliffx = 0).relief, We 퐻 also푐,0. Figure extracted profiles where the elevation is not always larger than or equal to the elevation at the initial point the backshore14 shows slope,the regionalS0; cli statisticsff face slope,for the Sclustersc; and interpreted initial cliff asrelief, cliff-typeHc,0 .(i.e. Figure, Clusters 14 shows 2 to 4, 6, the 7, 9 regional and (x=0)statistics. We10).for alsoWe the observed extracted clusters that the interpreted there backshore was a as small slope, cliff -typepercentage 푆0; (i.e.,cliff ,face Clustersof 4% slope,, of 2the to푆 푐data 4,; and 6, tha 7, initialt 9 produced and cliff 10). relief, negative We observed 퐻푐 ,initial0. Figure that 1there4 shows wascliff the relief a small regional, suggesting percentage, statistics that ofthey for 4%, the did of clusters not the represent data interpreted that a producedcliff coast as cliff either negative-type, and (i.e. initialthey, Clusters were cliff omittedrelief, 2 to 4, suggesting for6, 7, our 9 and 10).that We theyassessment observed did not of that represent the therecliff coast awa cli sresponse ffa coastsmall either,topercentage a rising and sea they, of level 4% were around, of the omitted thedata GB tha for coastline.t our produced assessment negative of the initial cliff clcoastiff relief response, suggesting to a rising that sea they level did around not represent the GB coastline.a cliff coast either, and they were omitted for our assessment of the cliff coast response to a rising sea level around the GB coastline.

Figure 13. Main geometrical attributes extracted from the elevation profiles along the GB coastline.

FigureFigure 13. 13. MainMain geometrical geometrical attributes attributes extracted extracted from from the the elevation elevation profiles proﬁles along the GB coastline.

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FigureFigure 14. 14. Example of of the the regional regional statistics statistics that that we weextracted extracted for the for 22 the regions 22 regions into which into whichwe divided we dividedthe GB thecoastline. GB coastline. As an example As an example,, we show we here show the stats here for the region stats for “Cell region 3”, and “Cell all 3”, other and regions all other are regionsin Supplementary are in Supplementary Materials; Materials; panels show panels theshow counts the per counts region per of region the cliff of thetop clielevationﬀ top elevation (a), slope (a ),of slopecliff offace cli (ﬀbface), shape (b), of shape the inland of the inlandtopography topography (c), inland (c), inland slope ( sloped) and (d a)rea and per area unit per length unit length (e). These (e). Thesegeometrical geometrical attribute attributess were were calculated calculated as shown as shown in Figure in Figure 14. Region14. Region area area is shown is shown in Figure in Figure 13. 13.

Figures 15 and 16 show the regional statistics for the initial cliff relief Hc,0 and backshore slope, S0, Figures 15 and 16 show the regional statistics for the initial cliff relief 퐻푐,0 and backshore slope, for all the interpreted cliff profiles. To obtain these stats, we did not include clusters that corresponded 푆0 , for all the interpreted cliff profiles. To obtain these stats, we did not include clusters that withcorresponded non-cliff coastlines with non- (Clusterscliff coastlines 1, 5 and (Clusters 8). We 1, observe 5 and 8). that We theobserve majority that (57%)the majority of the (57%) backshore of the slopebackshore values slope are negative values are and negative 43% have and positive 43% have slopes. positive The slopes. modal, The median modal, and median standard and deviation standard values for the positive and negative slopes are of the order of (3.3 10 2,, 9.6 10 4 and 4.1 103) deviation values for the positive and negative slopes are of the order× of −(3.3x10×-2,, 9.6−x10-4 and 4.1x10× 3) and ( 4.8 10 4, 2.9 10 2 and 3.4 103), respectively. As we show in Figure5 and Equation (9), and− (-4.8x10× -4−, -2.9x10− ×-2 and− -3.4x10− 3), respectively.× As we show in Figure 5 and Equation (9), profiles profileswith negative with negative backshore backshore slopes slopes will not will evolve not evolve towards towards an equilibrium an equilibrium cliff cli reliefff relief but but will will tend tend to to decrease towards at the fastest rate of change. decrease towards −−∞∞ at the fastest rate of change. Figure 17 shows the zc for those transects interpreted as cliff types and whose backshore slopes are positive and therefore will evolve towards an equilibrium relief. We calculated the zc values 3 using Equation (20) and assuming the values of the shoreface length, Ls = 10 m, and shoreface relief, 1 Hs = 10 m, as representative values for the GB coastline. All the zc values obtained are positive, with a mode, median and standard deviation equal to 13.9, 1.1 and 2.5 106, respectively. The modal value × being of the order of 101 suggests that the most likely state of the profiles is to be not in equilibrium yet. The large values also indicate that the rate of change is smaller than the one that will be realised as the cliff relief gets closer to the equilibrium relief. The median value being close to 1 indicates that half of the locations are close to equilibrium and in the region where the rate of change is mostly determined by the backshore slope.

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Figure 15. Regional statistics of initial cliff relief 퐻푐,0 along GB coastline. The central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, Figurerespectively. 15. Regional The whiskers statistics ofextend initial to cli theﬀ relief mostH extremec,0 along data GB coastline. points not The considered central mark outliers, indicates and the the median,outliers and are the plotted bottom individually and top edges using of the box“+” indicatesymbol. theSee 25th Table and S1 75th for percentiles,correspondence respectively. between Figure 15. Regional statistics of initial cliff relief 퐻푐,0 along GB coastline. The central mark indicates Theregion whiskers number extend and admin to the unit most used. extreme data points not considered outliers, and the outliers are plottedthe median, individually and the using bottom the “ + and” symbol. top edges SeeTable of the S1 boxfor correspondence indicate the 25th between and 75th region percentiles, number andrespectively. admin unit The used. whiskers extend to the most extreme data points not considered outliers, and the outliers are plotted individually using the “+” symbol. See Table S1 for correspondence between region number and admin unit used.

Figure 16. Regional statistics of initial backshore slope S0 along GB coastline. The central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. TheFigure whiskers 16. Regional extend to statistics the most of extreme initial backshore data points slope not considered푆0 along GB outliers, coastline. and The the centraloutliers mark are plottedindicates individually the median, using and the bottom “+” symbol. and top To edges obtain of thesethe box stats, indicate we did the not25th include and 75th clusters percentiles, that

correspondedrespectively. withThe whiskers non-cliﬀ coastlinesextend to (Clustersthe most 1,extreme 5 and 8)data and points 4% of not other considered clusters that outliers, had H andc,0 < the0. Seeoutliers Table S1are for plotted correspondence individually between using region the “+” number symbol. and To admin obtain unit these used. stats, we did not include Figure 16. Regional statistics of initial backshore slope 푆0 along GB coastline. The central mark clusters that corresponded with non-cliff coastlines (Clusters 1, 5 and 8) and 4% of other clusters that indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, had 퐻 < 0. See Table S1 for correspondence between region number and admin unit used. respectively.푐,0 The whiskers extend to the most extreme data points not considered outliers, and the outliers are plotted individually using the “+” symbol. To obtain these stats, we did not include clusters that corresponded with non-cliff coastlines (Clusters 1, 5 and 8) and 4% of other clusters that

had 퐻푐,0 < 0. See Table S1 for correspondence between region number and admin unit used.

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Figure 17 shows the 푧푐 for those transects interpreted as cliff types and whose backshore slopes are positive and therefore will evolve towards an equilibrium relief. We calculated the 푧푐 values 3 using Equation (20) and assuming the values of the shoreface length, 퐿푠 = 10 metres, and shoreface 1 relief, 퐻푠 = 10 metres, as representative values for the GB coastline. All the 푧푐 values obtained are positive, with a mode, median and standard deviation equal to 13.9, 1.1 and 2.5 × 106, respectively. The modal value being of the order of 101 suggests that the most likely state of the profiles is to be not in equilibrium yet. The large values also indicate that the rate of change is smaller than the one that will be realised as the cliff relief gets closer to the equilibrium relief. The median value being close to 1 indicates that half of the locations are close to equilibrium and in the region where the rate J. Mar. Sci. Eng. 2020 8 of change is mostly, , 866 determined by the backshore slope. 21 of 26

Figure 17. Regional statistics of zc along GB coastline. The central mark indicates the median, and the bottomFigure and17. Regional top edges statistics of the box of 푧 indicate푐 along theGB 25thcoastline. and 75th The percentiles,central mark respectively. indicates the The median, whiskers and extendthe bottom to the and most top extreme edges of data the points box indicate not considered the 25th outliers, and 75th and percentiles, the outliers respectively. are plotted The individually whiskers usingextend the to “ + the2” symbol. most extreme To obtain data these points stats, not we considered did not include outliers, clusters and thatthe outlierscorresponded are plotted with individually using the “+2” symbol. To obtain these stats, we did not include clusters that non-cliff coastlines (Clusters 1, 5 and 8) and 4% of other clusters that had Hc,0 < 0 and S0 0. We used corresponded with non-cliff coastlines (Clusters 1, 5 and 8) and 4% of other clusters that≥ had 퐻 <30 Equation (20) and assumed a shoreface slope representative of GB’s open coast to be equal to푐 10,0 − . Seeand Table 푆0 ≥ S10. forWe correspondence used Equation (20) between and assumed region number a shoreface and admin slope representative unit used. of GB’s open coast to be equal to 10-3. See Table S1 for correspondence between region number and admin unit used. 4. Discussion 4. Discussion 4.1. Interpretation of the Different Clusters 4.1. IEachnterpretation cluster canof the be D describedifferent Clusters using a number of parameters including the shorezone width, inclinationEach ofcluster the backshore can be described slope, minimum using a and number maximum of parameters elevations including of the profiles, the shorezone profile curvature width, andinclination distance fromof the the backshore high-water slope, mark to minimum cliff top. These and maximum are parameters elevations that we extractedof the profiles, for all profiles profile andcurvature analysed and based distance on the from mean the andhigh range-water of mark each to parameter cliff top. These for each are cluster. parameters Below, that we we summarise extracted thefor characteristicsall profiles and of analysed each cluster, based the on broad the mean spatial and extent range and of frequencyeach parameter of occurrence for each ofcluster. each cluster, Below, andwe summarise the dominant the geologycharacteristics or coastal of each environments cluster, the associatedbroad spatial with extent each and cluster. frequency Figures of occurrence S1 and S2 showof each the cluster, closest-to-centre-point and the dominant and geology the most-frequent or coastal environments profiles for each associated cluster and with are each helpful cluster. in visualisingFigures S1 theand geometry S2 show the of theclosest clusters.-to-centre-point and the most-frequent profiles for each cluster and are helpful in visualising the geometry of the clusters. Cluster 1—Low-Coastline Back-Barrier Type (Shallow Angle) Cluster 1—Low-Coastline Back-Barrier Type (Shallow Angle) Generally characterized by a narrow shore, a gently inclined slope that reaches a very low elevation a shortGenerally distance from characterized the HWM, by a very a narrow low profile shore, curvature a gently and inclined low maximum slope that elevations. reaches This a very cluster low elevation a short distance from the HWM, a very low profile curvature and low maximum elevations. accounts for 15% of the UK coastline and is typically well distributed around the coastline. It is more prevalent in areas such as eastern England and the north coast of Wales. It is commonly found in estuarine regions, where soft superficial deposits dominate the geology.

Cluster 2—High Cliﬀ Coastline

Generally characterized by a minimal shore, steep and high cliffs, a moderate distance from the HWM to the cliff top, a moderate profile curvature and high maximum elevations. This cluster accounts for 4% of the UK coastline and is more common on the west coasts of England, Wales and Scotland. These parts of the coastline are typically dominated by hard, strong bedrock. J. Mar. Sci. Eng. 2020, 8, 866 22 of 26

Cluster 3—High, Sheer Cliff Coastline Generally characterized by a minimal shore, very steep and high cliffs, a short distance from the HWM to the cliff top, a very high profile curvature and high maximum elevations. This cluster accounts for 2% of the UK coastline and is well distributed around the coast but is more common in the southwest and around northern Scotland. Typically, these parts of the coastline are dominated by hard, strong bedrock. Cluster 4—Narrow Beach and Cliff Coastline Generally characterized by a narrow shore, steep and high cliffs, a moderate distance from the HWM to the cliff top, a moderate profile curvature and high maximum elevations. This cluster accounts for 4% of the UK coastline and is found more around southwest England and the west coast of Scotland. These parts of the coastline are dominated by hard, strong bedrock. Cluster 5—Low Coastline Back-Barrier Type (Moderate Angle) Generally characterized by a narrow shore, a moderately steep and low cliff, a short distance from the cliff top to the HWM, a low profile curvature and low maximum elevations. This cluster accounts for 8% of the UK coastline and is well distributed around the coastline, although it is more common along the coast of east and northeast England. These coastlines are commonly straight and dominated by weak, poorly consolidated superficial deposits. Cluster 6—Plunging Coastline Generally characterized by a minimal shore, steep and moderately high cliffs, a very short distance from the HWM to the cliff top, a very high profile curvature and moderate maximum elevations. This cluster accounts for 3% of the UK coastline and is most common in the Pembrokeshire region of west Wales and along the NW coast of Cornwall. These parts of the coastline are typically formed of hard, strong, consolidated bedrock that is highly resistant to erosion. Cluster 7—Wide Shore Coastline Generally characterized by a wide shore, a steep and moderately high cliff, a long distance from the HWM to the cliff top, a low profile curvature and moderate maximum elevations. This cluster accounts for 6% of the UK coastline and is mostly found along the west coast of Scotland and around the Scottish isles. The coastline in these areas is typically formed of hard, strong (commonly metamorphic) bedrock, which often forms wide wave-cut platforms and complex shaped inlets. Cluster 8—Flat Coastline Generally characterized by a narrow shore, a flat or very gently rising slope, a short distance from the HWM to the top of the slope, a very low profile curvature and very low maximum elevations. This cluster accounts for 48% of the UK coastline, being by far the most prevalent cluster, and is predominantly found around estuaries, such as the Medway and Humber. Cluster 9—Retrograde Slope Coastline Generally characterized by a narrow shore, a steep and moderately high cliff, a long distance from the HWM to the cliff top, a low profile curvature, high maximum elevations and a retrograde slope after maximum elevation. This cluster accounts for 4% of the UK coastline and is most common along the west coast of Scotland and along the south coast of Cornwall. These regions of the coastline are commonly dominated by hard, strong bedrock. Cluster 10—High Rising Coastline Generally characterized by a wide shore, a steep and moderately high slope, a long distance from the HWM to the cliff top, a low profile curvature and moderate maximum elevations. This cluster accounts for 5% of the UK coastline and is most common along the west coast of Scotland and along the south coast of Cornwall. These regions of the coast are commonly dominated by hard, strong bedrock. J. Mar. Sci. Eng. 2020, 8, 866 23 of 26

4.2. Likely Response to Sea-Level Rise of GB Coastline Our cluster analysis (Figure 10) and interpretation (Figure 11 and Section 4.1) suggest that 71% of the coast of GB is best described as gentle coast, including estuarine coastline or open coasts where back-barrier beaches can form. The remaining 39% is best described as cliff-type coastlines. We have shown how using the geometrical relationships obtained from applying sediment conservation over a control volume that includes the backshore topography and sand sediment fraction (i.e., Equations (9) and (12)), we can infer the likely response of any gentle and steep profile to a sea-level rise. In particular, we have shown how we can qualitatively assess the rate of change (Figure5) and the net behaviour of the shoreline (i.e., erosion or accretion) by assessing the disequilibrium state of the backshore profile (Equation (22)). We observed that the majority (57%) of the backshore slope values (as defined in Figure 13) are negative and 43% positive. This observation suggests that a large proportion of GB’s coastline will respond by very rapidly retreating, with insufficient local source material to counteract the rapid regression, and will therefore require non-local sources of sand and other beach materials to maintain the actual shoreline position as the sea level rises. The number of potential sources of sand can be inferred by assessing the condition of the shoreline accretion of cliffed coast from Equation (22) as shown in Figure 18. If the initial cliff relief, Hc,0, is larger than the equilibrium cliff relief, Hc, , the necessary but not sufficient condition for shoreline advance from ∞ Equation (7) is met: the section will act as a sediment source, and if the net gained area is larger than the area lost due to a sea-level rise, then the shoreline will advance. On the contrary, if Hc,0 < Hc, , ∞ the only possible shoreline response is to retreat and the coast section will act as a sediment sink. The percentages of sections acting as potential sources of sand are the highest (80% and 73%) in Cell 3 and Cell 11, respectively, and the percentages of sections acting as potential sand sinks are the highest (67% and 66%) in the Clyde and West Highlands. The amount of sediment released per unit of cross-shore length also varies across regions (Figure S4), with median and standard deviation values equal to 11 and 23.3 m2/m, respectively. A first-order estimate of the sand required by a sea-level rise R, assuming that the cliff and shoreface have similar sand fractions, c0 = cs, can be obtained from 3 Equation (2), as RxLs, where Ls is the shoreface length. Assuming Ls = 10 metres and R = 0.5 metres, it will demand 5 102 m2/m, which will be equivalent to ca. 500 m/11 m2/m 50 m retreat. How this × material is released will depend, among other things, on the shape of the backshore profile, which we found (Figure S5) to vary across regions, with a median and standard deviation of 0.69 and 0.16, with a percentageJ. Mar. Sci. of lessEng. 2019 than, 7, 1%x FOR being PEER REVIEW negative. 24 of 27

Figure 18. Cont.

Figure 18. Regional statistics of 퐻푐,0 − 퐻푐,∞ along GB coastline. To obtain these stats, we did not include clusters that corresponded with non-cliff coastlines (Clusters 1, 5 and 8) and 4% of other

clusters that had 퐻푐,0 < 0 and 푆0 ≥ 0 . We used Equation (8b) and assumed a shoreface slope -2 representative of GB’s open coast to be equal to 10 . Grey bins represent section counts where 퐻푐,0 −

퐻푐,∞ > 0 (i.e., potential sediment sources), and red bins represent section counts for which 퐻푐,0 −

퐻푐,∞ < 0 (i.e., potential sediment sinks).

4.3. Limitations of This Study The results presented in this work need to be interpreted with caution and an awareness of the limitations of our approach. The number of elevation transects is very sensitive to the DEM resolution and the vector coastline used to delineate the orthogonal transects. The coastline of GB, including the islands, is 31,368 km, according to Ordnance Survey (OS), with the mainland making up 17,819 km. Other institutions have the figure lower—the CIA Fact book says 12,429 km, and the World Resources Institute says 19,717 km. We used a total of 3,912,935 valid transects, which represented ca. 19,565 km and were within the expected length based on other sources. The differences in the coastline length estimations are due to the coastline paradox [20]. The coastline paradox states that a coastline does not have a well-defined length. The measurements of the length of a coastline behave like a fractal, being different at different scale intervals (the distances between points on the coastline at which

J. Mar. Sci. Eng. 2019, 7, x FOR PEER REVIEW 24 of 27

J. Mar. Sci. Eng. 2020, 8, 866 24 of 26

Figure 18. Regional statistics of Hc,0 Hc, along GB coastline. To obtain these stats, we did not Figure 18. Regional statistics of 퐻− − 퐻∞ along GB coastline. To obtain these stats, we did not include clusters that corresponded with푐,0 non-cli푐,∞ﬀ coastlines (Clusters 1, 5 and 8) and 4% of other clusters include clusters that corresponded with non-cliff coastlines (Clusters 1, 5 and 8) and 4% of other that had H < 0 and S 0. We used Equation (8b) and assumed a shoreface slope representative c,0 퐻0 < 0 푆 ≥ 0 clusters that had 푐≥,0 and 0 2 . We used Equation (8b) and assumed a shoreface slope of GB’s open coast to be equal to 10− . Grey bins-2 represent section counts where Hc,0 Hc, > 0 representative of GB’s open coast to be equal to 10 . Grey bins represent section counts where− 퐻푐,0∞− (i.e., potential sediment sources), and red bins represent section counts for which Hc,0 Hc, < 0 퐻푐,∞ > 0 (i.e., potential sediment sources), and red bins represent section counts for which− 퐻푐,0∞− (i.e., potential퐻푐,∞ < 0 sediment(i.e., potential sinks). sediment sinks).

4.3. Limitations4.3. Limitations of This of T Studyhis Study The resultsThe results presented presented in thisin this work work need need to to be be interpreted interpreted with with caution caution and and an anaware awarenessness of the of the limitationslimitation of ours of approach.our approach. The The number number of of elevation elevation transectstransects is is very very sensitive sensitive to tothe the DEM DEM resolution resolution and theand vector the vector coastline coastline used used to delineateto delineate the the orthogonal orthogonal transects.transects. The The coastline coastline of ofGB, GB, including including the the islands,islands, is 31,368 is 31,368 km, km, according according to Ordnanceto Ordnance Survey Survey (OS),(OS), with the the ma mainlandinland making making up up17,819 17,819 km. km. OtherOther institutions institutions have have the the figure figure lower—the lower—the CIA CIA Fact Fact bookbook says 12,429 12,429 km km,, and and the the World World Resources Resources InstituteInstitute says 19,717says 19,717 km. km. We We used used a total a total of of 3,912,935 3,912,935 valid valid transects, which which represented represented ca. ca.19,565 19,565 km km and were within the expected length based on other sources. The differences in the coastline length and were within the expected length based on other sources. The differences in the coastline length estimations are due to the coastline paradox [20]. The coastline paradox states that a coastline does estimations are due to the coastline paradox [20]. The coastline paradox states that a coastline does not have a well-defined length. The measurements of the length of a coastline behave like a fractal, not havebeing a well-defineddifferent at different length. scale The intervals measurements (the distance of thes between length ofpoints a coastline on the coas behavetline likeat which a fractal, being different at different scale intervals (the distances between points on the coastline at which measurements are taken). The smaller the scale interval (meaning the more detailed the measurement), the longer the coastline. This “magnifying” effect is greater for convoluted coastlines such as the GB coastline than for relatively smooth ones. Our simplified framework provides valuable insights into coastal evolution, and has the particular advantage of providing simple geometrical reasoning to assess long-term shoreline retreat. However, this approach necessarily neglects many issues that could limit its applicability. Our models generalize the Bruun rule but still assume alongshore homogeneity, a fixed shoreface profile and closure depth, and a sandy nearshore. Further discussion of these assumptions is beyond the scope of this paper, but as noted by [7], we also note that the framework of the shoreline Exner Equation (2) is general enough to accommodate alongshore variability, dynamic profiles, shoreface effluxes and patchy sediment cover. Our geometric analysis ignored the likelihood that on steep coasts, cliff retreat may become weathering-limited in the presence of hard crystalline rock, which will reduce the numbers of both potential sinks and sources, as shown in Figure 18. We also assumed c0 = cs, which in reality will vary with the lithology that has been affected by coastal erosion. From the geometrical relationships, the reader can infer in which direction our assumptions will change as the c0/cs ratio increases and decreases. J. Mar. Sci. Eng. 2020, 8, 866 25 of 26

5. Conclusions Reliable approaches for assessing the likely shoreline change in response to a sea-level rise over time scales from decades to centuries at the national scale are needed worldwide for long-term coastal management and ensuring that the natural and built environment are adaptable to today’s and tomorrow’s climate. Here, we used a simple geometrical analysis of the backshore topography to assess the likely response of any wave-dominated coastline to a sea-level rise, and we applied it along the entire Great Britain (GB) coastline, which is ca. 17,820 km long. We illustrated how the backshore geometry can be linked to the shoreline response (rate of change and net response: erosion or accretion) to a sea-level rise by using a generalized shoreline Exner equation, which includes the effect of the backshore slope and differences in sediment fractions within the nearshore. Our analysis suggests that 71% of the coast of GB is best described as gentle coast, including estuarine coastline or open coasts where back-barrier beaches can form. The remaining 39% is best described as cliff-type coastlines, for which the majority (57%) of the backshore slope values are negative, suggesting that a non-equilibrium trajectory will most likely be followed as a response to a rise in sea level. For the remaining 43% of the cliffed coast, we have provided regional statistics showing where the potential sinks and sources of sediment are likely to be. Our findings have implications for both coastal engineers and stakeholders. For coastal engineers, we have illustrated how the relative importance of both the backshore topography and sediment composition differences between the nearshore and backshore can be assessed using widely available topographical information, sediment conservation principles and first order of magnitude analysis. Coastal stakeholders who manage the risk of coastal erosion and adaptation to rising sea levels can now better quantify at a regional level the percentage and location of potential sections that are likely to behave as sediment sinks and sources to better manage the limited amount of sediment available to compensate the area lost due to a sea-level rise.

Supplementary Materials: The following are available online at http://www.mdpi.com/2077-1312/8/11/866/s1. CliffMetric.Zip: improved CliffMetric algorithm used for this study. Author Contributions: Conceptualization, A.P., C.W. and K.A.L.; methodology, A.P., C.W. and A.G.H.; software, A.P.; validation, R.V., A.G.H. and J.R.L.; formal analysis, A.L.L.; investigation, A.L.L.; resources, K.A.L.; data curation, A.P., C.W. and A.G.H.; writing—original draft preparation, A.P.; writing—review and editing, J.R.L., R.V. and C.W.; visualization, A.P.; project administration, K.A.L. and R.V.; funding acquisition, K.A.L. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: This manuscript was prepared, verified and approved for publication by the British Geological Survey. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

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