Deep uncertainties in shoreline change projections: an extra-probabilistic approach applied to sandy beaches

. Global mean sea-level rise and its acceleration are projected to aggravate coastal erosion over the 21 st 10 century, which constitutes a major challenge for coastal adaptation. Projections of shoreline retreat are highly uncertain, however, namely due to deeply uncertain mean sea-level projections and the absence of consensus on a coastal impact model. An improved understanding and a better quantification of these sources of deep uncertainty are hence required to improve coastal risk management and inform adaptation decisions. In this work we present and apply a new extra-probabilistic framework to develop shoreline change projections of sandy coasts that allows 15 considering intrinsic (or aleatory) and knowledge-based (or epistemic) uncertainties exhaustively and transparently. This framework builds upon an empirical shoreline change model to which we ascribe possibility functions to represent deeply uncertain variables. The model is applied to two local sites in Aquitaine (France) and Castellón (Spain). First, we validate the framework against historical shoreline observations and then develop shoreline change projections that account for possible (although unlikely) low-end and high-end mean sea-level 20 scenarios. Our high-end projections show for instance that shoreline retreats of up to 200m in Aquitaine and 130m in Castellón are plausible by 2100, while low-end projections revealed that 58m and 37m modest shoreline retreats, respectively, are also plausible. Such extended intervals of possible future shoreline changes reflect an ambiguity in the probabilistic description of shoreline change projections, which could be substantially reduced by better constraining SLR projections and improving coastal impact models. We found for instance that if mean sea-level 25 by 2100 does not exceed 1m, the ambiguity can be reduced by more than 50 %. This could be achieved through an ambitious climate mitigation policy and improved knowledge on ice-sheets. used herein to perform future shoreline change projections within the extra-probabilistic theory. real estate boom that occurred in the second half of the 20 th century exacerbated such imbalance, giving rise to constructions on beaches that were already in decline. Subsequently, to try to solve this problem, more actions were taken, including the construction of seawalls and jetties and replenishments. In their natural state, these are beaches of fine to medium sand with D 50 between 0.2-0.35 mm. Shoreline evolution in the Castellón-Sagunto stretch was retrieved using the CoastSat toolkit (Vos et al., 2019b) 255 based on monthly or bimonthly observations from Landsat 5, Landsat 8 and Sentinel 2. CoastSat has been shown to have a particularly high accuracy in microtidal environments (Vos et al., 2019a). For the Castellón-Sagunto stretch, the dataset retrieved by CoastSat has been validated against discrete profile surveys at some specific sites. The shoreline evolution over the period 1989-2019 for the profile studied in Chilches is shown on Fig. 4c. Over the 31-year period, 859 shoreline positions (orange timeseries) were retrieved for this profile, with an average of 260 25 (70) observational records per year before (after) 2017. The profile shows an average coastline retreat of 0.6 m/year over the period 1989-2019. 2012;Slangen et al., 2014;Gregory et al., 2019). Regional projections of the sterodynamic component, which corresponds to 285 changes in ocean density and circulation corrected from the inverse barometer effect, are derived from the outputs of the global climate model simulations performed within the 5 th phase of the Coupled Model Intercomparison Project (CMIP5) that are in the IPCC AR5 and SROCC reports. Note however among the 21 CMIP5 models available, MIROC-ESM and MIROC-ESM-CHEM models are discarded as they simulate unrealistic Here, we discuss the advantage of the use of possibilities in comparison to e.g. a modelling framework that would be fully probabilistic. To illustrate this, we re-calculate shoreline change projections with Eq. (1) in Aquitaine (site 1) in 2100 but assuming that ∆𝑅𝑆𝐿𝐶 and tan𝛼 follow normal distributions. We consider the RCP8.5 scenario with

and can be addressed quantitatively by extra-probabilistic methods (Dubois and Guyonnet, 2011). Extra-probabilistic theories of uncertainty recognize that several probabilistic laws may exist given the piece of information available. Instead of providing a single uncertainty (probabilistic) model, they deliver sets of plausible 130 probabilistic models. In the present study, we use the possibility theory to represent uncertainties of deeply uncertain variables (Dubois and Prade, 1988). The basic ingredient is the interval used for representing experts' knowledge. In most cases, however, experts may provide more information by expressing preferences within this interval. Such "nuanced" information can be conveyed using the possibility distributions, denoted π (Dubois and Prade, 1988), which describes the more or less plausible values of some uncertain quantity. The intervals defined 135 , e e are called α-cuts. They contain all the values that have a degree of possibility of at least α (lying between 0 and 1). The example of α-cut on a trapezoid possibility distribution is shown on Fig. 1a. The interval for α=0 and α =1 is called the support and the core, respectively. The α-cuts formally correspond to the confidence intervals 1-α as traditionally defined in the probability theory, i.e probabilistic and possibilistic theories was exploited by Le Cozannet et al. (2017) to derive a possibility 145 distribution to represent uncertainties on GSLR by 2100 conditional on RCP8.5 scenario.

Setting-up shoreline change projections within the extra-probabilistic framework
In principle, the extra-probabilistic framework can be used with any shoreline change model. In this study, we adopt the perspective of coastal adaptation practitioners that generally rely on empirical models that extrapolate observed shoreline changes to anticipate better their future evolution (Peter et al., 2003;Le Cozannet et al., 150 2019a;Vousdoukas et al., 2020;Cowell et al., 2003). In the absence of estuaries or other major sediment sources or sinks, our empirical model expresses shoreline change ΔS following Eq. (1): where − expresses the change in shoreline position in the cross-shore direction from reference time t0 to time t, ∆ quantifies the contribution of sea level rise to shoreline changes, which takes the form of the Bruun 155 rule (Bruun, 1962); • gives an estimate of the multi-decadal shoreline changes; characterizes the interannual-to-decadal variability of shoreline change. These terms, which are described further in the following, include intrinsic and knowledge uncertainties that need to be adequately represented as input and then propagated.  As a first step (Fig. 2a,b), uncertainty distribution of inputs are constructed. For instance, in Eq. (1), • and are both derived from past shoreline change observations. Note that and terms do not describe a 165 single physical process but rather a combination of processes that operate at different timescales including waves climates, sediment budgets, effects of longshore gradients in sediment transport or anthropogenic actions. These processes are recognized complex and difficult to model with reduced complexity models (Montaño et al., https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License. 2020; Vitousek et al., 2017). By using the empirical model, our objective is to reproduce the observed trends and modes of variability without trying to model the physical processes explicitly, while keeping a low computation 170 time (see Helgeson et al. (2020) for a broader discussion of this approach). The term • is the product of the number of years n since the reference year and the multi-decadal linear trend derived from observations after substracting the effect of sea-level rise. The residuals of the linear regression to compute are then used to derive . We sample residuals that are distant by a gap of N years (with 1 < N < 10 as we focus on interannual-todecadal timescales) and compute their standard deviation. This procedure is repeated for all possible combinations 175 of residuals separated by N years. Finally is determined as the maximum standard deviation value obtained among all samples. Note that is found to maximize for N ≥ 5 years. Since and are derived assuming that errors of the linear regression are normally distributed, they are both prescribed as probability distributions.
In contrast, terms accounting for future sea level (ΔRSLR) and its impact on shoreline change (1/tan α) are both sources of deep uncertainty and are therefore too imprecise given the current knowledge to be constrained by 180 probability distributions. For instance, to reflect the full range of current uncertainty, ΔRSLR should consider projections that are either below or beyond the likely-range provided by the IPCC, but for which probability are not well established. Regarding the coastal impact model, under the Bruun rule (Bruun, 1962), tan α corresponds to the slope of the active profile from the depth of closure to the top of the upper shoreface. The Bruun rule underlying assumptions include considering that sediment transport only occurs perpendicularly to the shoreline, 185 thus neglecting any tri-dimensional variability, and assuming that the coastal profile is an equilibrium profile that has uniform sediment size. An alternative to the Bruun rule was proposed through the Probabilistic Coastline Recession (PCR) model (Ranasinghe et al., 2012). The PCR model quantifies sediment losses at the dune toe during storms, as well as the nourishment of the dune by aeolian sediment transport processes between storms.
Given a certain amount of sea-level rise, the response of the PCR model is less erosive than the Bruun rule by one 190 order of magnitude. While the use of the PCR model is rather expensive computationally, Le Cozannet et al. (2019a) demonstrated that, in a first approximation, the equilibrium response of the PCR model can be emulated in Eq. (1) by replacing the nearshore slopes (or Bruun slopes) by the foreshore slopes. Bruun and PCR models are however both difficult to validate because of the scarcity of coastal data and the complexity of the hydrosedimentary processes involved. This constitutes one of the source of deep uncertainty. Hence, to reflect the 195 absence of consensus on coastal erosion induced by sea-level rise, neither surrogate PCR model nor Bruun rule should be discarded in our uncertainty propagation. To account for the limited knowledge of future sea level and its impact on shoreline change, we construct ΔRSLR and 1/tan α terms as trapezoidal possibility distribution (see also sections 4a and b).
As a second step (Fig. 2b), to propagate the heterogeneous uncertainty nature of the terms in Eq. (1), we used the 200 HYRISK R package (Rohmer et al., 2017). HYRISK software is designed to jointly propagate probability and  whereas aleatory uncertainty is represented by the overall tilt of the p-box. The gap between the upper and lower CDF can be considered as "what is unknown" and represents the imperfect state of knowledge (Rohmer et al., 2019). To quantify this deep uncertainty, we use an indicator termed as "ambiguity" and defined as the width (in meter) between medians of the upper and lower CDF. In addition, we define the low-end threshold (i.e. minimum 215 adaptation needs, shown in green) as the shoreline change value for which there is a chance smaller than a to be reached under the less impacting (i.e. upper) CDF. In other words, the low-end value corresponds to a threshold, which is very likely to be exceeded. Finally, we define the high-end threshold (i.e. high risk-adverse applications, shown in red) as the value below which there is still more than b chance for the projections to hold under the most impacting (i.e. lower) CDF. In this case, the high-end threshold corresponds to a value which can be possibly but 220 unlikely exceeded. As an example, we define a and b as 0.4 and 0.6, respectively, although these thresholds are meant to be tailored to user needs depending on their risk aversion.

Case studies and data
In this work, the extra-probabilistic approach to perform shoreline change projections is applied in two coastal sites where the coastline is largely dominated by sandy beaches (Fig. 3) but (i) for which we have different sources 225 of shoreline change observations and sampling, and (ii) that have highly distinct geomorphologic and hydrodynamical characteristics.

Case studies and shoreline change observational records
The first site studied (Site 1) is located in the municipality of Naujac-sur-Mer, which belongs to the Aquitaine coast (Fig. 3). The Aquitaine is a 230 km long sandy coast located in south-western France constituted by high- As a result, the coast shifted from the state of dynamic equilibrium with intense longshore transport and continuous sediment intake to imbalance, with the same longshore transport intensity but without any sediment contribution updrift. This resulted in the chronic recession of the beaches sheltered by the structures, and the accretion of the 250 beaches located downdrift. Besides, the real estate boom that occurred in the second half of the 20 th century exacerbated such imbalance, giving rise to constructions on beaches that were already in decline. Subsequently, to try to solve this problem, more actions were taken, including the construction of seawalls and jetties and replenishments. In their natural state, these are beaches of fine to medium sand with D50 between 0.2-0.35 mm.
Shoreline evolution in the Castellón-Sagunto stretch was retrieved using the CoastSat toolkit (Vos et al., 2019b) 255 based on monthly or bimonthly observations from Landsat 5, Landsat 8 and Sentinel 2. CoastSat has been shown to have a particularly high accuracy in microtidal environments (Vos et al., 2019a). For the Castellón-Sagunto stretch, the dataset retrieved by CoastSat has been validated against discrete profile surveys at some specific sites.
The shoreline evolution over the period 1989-2019 for the profile studied in Chilches is shown on Fig. 4c. Over the 31-year period, 859 shoreline positions (orange timeseries) were retrieved for this profile, with an average of 260 In Aquitaine (Site 1), topographic and bathymetric surveys recorded nearshore slopes comprised in the range 1.2%-1.5% that can occasionally be as mild as 1% (Bernon et al., 2016), and slopes of the upper shoreface that can be as steep as ~10% (Bulteau et al., 2014). In Castellón, beach slopes have been determined by combining two 270 datasets: a topography dataset from the Spanish Geographic Institute (Instituto Geográfico Nacional, IGN); and a bathymetry dataset from the Spanish Ministry for the Ecological Transition and Demographic Challenge (Ministerio para la Transición Ecológica y el Reto Demográfico, MITERD). Specifically in Site 2, a nearshore slope of 3.1% was retrieved.

Historical sea level and projections 275
For both sites, the absolute sea-level time evolution in the past is constructed from tide gauge records which are To obtain RSLC regional projections, we sum up the future regional contributions of sterodynamic effects, melting of mountain glaciers and ice sheets, land water and glacial isostatic adjustment (Slangen et al., 2012;Slangen et al., 2014;Gregory et al., 2019). Regional projections of the sterodynamic component, which corresponds to 285 changes in ocean density and circulation corrected from the inverse barometer effect, are derived from the outputs of the global climate model simulations performed within the 5 th phase of the Coupled Model Intercomparison Project (CMIP5) that are used in the IPCC AR5 and SROCC reports. Note however that among the 21 CMIP5 models available, MIROC-ESM and MIROC-ESM-CHEM models are discarded as they simulate unrealistic https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License. sterodynamic contributions (Thiéblemont et al., 2019). Furthermore, in the semi-enclosed basins (e.g. 290 Mediterranean Sea), the rather coarse resolution of AOGCMs prevents an accurate representation of small-scale processes (e.g. water exchange at Gibraltar), which in turn affects regional sea-level estimates (Marcos and Tsimplis, 2008;Slangen et al., 2017). The Mediterranean sterodynamic sea-level projections are therefore estimated by relying on those of the Atlantic area near Gibraltar, which is the Mediterranean Sea entry point. For other mass contributions to sea level (i.e. glaciers, ice sheets, land water), regional changes are obtained by 295 downscaling global estimates using barystatic-GRD fingerprints . Past shoreline changes are investigated first to ensure the validity of the modelling framework. Table 1 summarizes how uncertainties of each variables of Eq. 1 are defined (using either probability distributions or possibility distributions as introduced in Sect. 2) to model past shoreline changes in Sites 1 and 2, respectively. Over the historical period, mean sea level uncertainty for these two sites is assumed to be well represented by a normal probability distribution. For vertical ground motion (VGM), the sites that are investigated in our study have no 305 statistically significant trends identified. Therefore, uncertainties due to VGM were prescribed as a centered normal distribution with a standard deviation of 2 mm/yr, as retrieved by the analysis of trends computed from the coastal permanent GNSS stations in the SONEL database (Wöppelmann and Marcos, 2016). and are also prescribed as normal probability distribution since they were derived assuming that errors of the linear regression are normally distributed (see section 2.b). Finally, as described in section 2.b, there is no consensus on the model 310 to be used to project shoreline change in response to SLR. The design of the possibilistic distribution of the beach slope should therefore reflect this unknown by considering both the Bruun and the PCR model. The upper shoreface slopes are generally steeper than the nearshore slopes (e.g. 5-13% versus 1-2% in Aquitaine), applying the surrogate of the PCR model leads to reduced shoreline retreat estimates in comparison with the Bruun rule estimates (see section 2.b). Therefore, we defined the beach slope as an imprecise parameter which follows a 315 possibilistic trapezoid distribution that span values ranging from the mildest records of the nearshore slope to steeper upper shoreface slopes. For Site 1, this leads to a core of the trapezoid in the range 1.2%-1.5% and a mildest slope of 1% (defining the origin of the support; see Table 1). For Site 2, the core of the trapezoid is in the range 2%-3.5% and the mildest slope is 1.5%. Finally, in absence of precise estimate of upper shoreface slopes sites 1 and 2, we use a uniform 10% slope as upper point of the trapezoid (Table 1)   The results are derived from the uncertainty propagation scheme using 5000 random draw based on the uncertainty 325 definition of each term of coastal impact model described in Table 1. For ease of comparison between the two sites, probability boxes are shown for a period of 10 years (gold) and 29-30 years (red) with respect to observational record references that are 2014 in Aquitaine and 2019 in Castellón.
For both sites, the gap between the lower and the upper bounds (i.e. the ambiguity) increases when moving increasingly backward in time (away from the reference year). This is expected and simply reflects the fact that 330 uncertainty increases when exploring them further away from the reference date. In Aquitaine, the observed anomalous shoreline position for 1984 and 2004 are -14 m and -5 m, respectively (Fig. 4a). According to the associated p-boxes (Fig. 5a), the probability of exceedance of these two observed values are in the ranges 86%-92% and 65%-72%, hence well embedded within possibilistic bounds but also consistent with the fact that these observations appear to be well above (especially in 1984, upper ranges) the regression estimate (Fig. 4a). In 335 Castellón, observed shoreline positions in 1990 (2009) are -13 (-8) m, which correspond to probability of exceedance in the ranges 73%-88% (29%-40%). Expanding our analysis to the entire profile of site 1, we found that 55% of the observations fall within the 25%-75% probability bounds and 100% within the 5%-95% confidence limit. For site 2, we found that 78% of the observations fall within the 25-75% probability bounds and 96% within the 5%-95% confidence limit. This hindcast analysis hence suggests that our modelling framework is valid against 340 observational historical records. https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License.

Future projections of shoreline change
In contrast with the historical period for which observations of the mean sea-level are available and its uncertainty well quantified, projections of mean sea-level are deeply uncertain (see also introductory paragraph). This deep uncertainty source needs to be prescribed as input and, therefore, can no longer be considered as following a normal probabilistic distribution (as shown in Table 1). The relative sea-level change (RSLC) is defined as an 350 imprecise input variable, which follows a trapezoidal possibility distribution, while all others inputs are taken identical to       in Aquitaine (Fig. 6a). This result is consistent with the fact that SLR projections start to increasingly diverge after 2050 between the three future scenarios (Garner et al., 2018;Hinkel et al., 2019). In Castellón, small inter-scenario https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License. changes are also found (Fig. 6c) but lower median bounds are under those of the Aquitaine site; i.e. ~18 m for the Castellón site against ~25 m for the Aquitaine site. The upper median bound is also substantially more expanded 380 for the Aquitaine site (60 m) than the Castellón site (40 m) when considering the RCP8.5 scenario. Therefore, while scenario choice remains a modest source of uncertainty of shoreline projections by 2050, potential differences in nearshore slope and coastal impact models are already prominent. In 2100, ambiguity difference between RCP scenarios is strongly enhanced (see also Table 3). The upper uncertainty bound of the RCP8.5 scenario more than double those of the RCP2.6 scenario in both sites. 385 Table 3 provides the shoreline retreat thresholds of high-end and low-end scenarios associated with the probability boxes (and thresholds a and b) displayed on Fig. 6. Although defined arbitrarily, these two thresholds represent possible -but unlikely -"optimistic" and "pessimistic" future projections than can be considered as references to design minimum adaptation and maximum protection needs, respectively. In site 1 (site 2) in 2050, whatever the scenario, it appears that the shoreline could be retreating between ~24m (~16m) for a low-impact scenario and 390 more than 50 m (40 m) for a high-impact scenario. High-end values strongly increase in 2100, and could reach up to almost 200 m in site 1 and more than 130 m in site 2 under the RCP8.5 scenario. Under low-end scenarios, in 2100, 58 m and 37 m could still be lost in site 1 and site 2, respectively.

Sensitivity analysis
Shoreline change projections shown in Fig. 6 reveal that the uncertainty strongly amplifies with distant time horizons, in particular under high global warming scenarios. From a coastal planning perspective, such large uncertainties can be considered as unhelpful and not be used as such to support the decision making process 400 (Rohmer et al., 2019). In this case, it is particularly relevant to determine which uncertainty contributes the most to the total uncertainty in order to anticipate how foreseen improvements in the understanding of the physical system could reduce the uncertainty of projections. To this end, we performed a sensitivity analysis based on the pinching method (Tucker and Ferson, 2006). The pinching method consists of quantifying how the p-box changes if uncertain input parameters are pinched to a fixed value, i.e. assuming that the new knowledge context enables 405 to remove the corresponding epistemic uncertainty. The uncertain parameter leading to the maximum changes in the p-box is the one with the largest impact, i.e. the one that deserves further investigation in priority. Here, we pinch one parameter of Eq. (1) at a time and quantify the resulting effect on the ambiguity and high-end values. Fig. 7 shows the results of the sensitivity analysis applied to site 1 for the RCP8.5 scenario in 2100. Note that this analysis has been extended to all scenarios and site 2 and revealed close results, leading to similar conclusions. 410 The figure reads e.g. as follows: assuming that the sea level off the Aquitaine coast would rise by 0.37 m in 2100 https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License. only (a very low estimate), the ambiguity (Fig. 7a) and high-end estimate (Fig. 7b) of shoreline change projection would both reduce by more than 50%. These results show that ambiguity and high-end estimate are primarily sensitive to uncertainty in SLR and beach slope. Ambiguity and high-end estimate in shoreline change projections increase linearly with increasing SLR and decrease more abruptly (following an inverse function, consistent with 415 Eq. (1)) with increasing beach slope. In comparison, the Tx and Lvar uncertainties have practically no effect on the ambiguity of shoreline change projections but show some influence on high-end estimates. The high-end shoreline change sensitivity to Tx and Lvar is also more pronounced in 2050 (not shown). This sensitivity analysis therefore suggests that improving both SLR projections and the understanding of their impact on shoreline could lead to a substantial reduction of uncertainty of future shoreline change. It should be 425 emphasized that in the event that future SLR would not exceed the likely range (i.e. ~1 m), the ambiguity would be lowered by more than 50 %. Similarly, knowing exactly the nearshore slope contributes to drastically reduce the shoreline change uncertainty, in particular if this nearshore slope is steep (i.e. > 2%). Fixing the beach slope value in our simplified shoreline change equation implicitly suggests, though, that the coastal impact model is also well defined. The latter underlying assumption is however erroneous as reviewed previously (e.g. section 2.2). In 430 the discussion, we explore in more details how shoreline change uncertainty is sensitive to the coastal impact model. To address this question, we have changed the nearshore slope definition as input of our model. Results are shown on Fig. 8. To consider solely the Bruun model, beach slopes are defined as trapezoid considering the range 1.2%-1.5% for the core and 1%-1.6% for the support. Note that the 1.2% and 1.5% beach slopes correspond to the interval of foreshore slope from the dune toe to the depth of closure in Aquitaine (i.e. Bruun slopes). For the PCR model emulation, we adopted the approach of Le Cozannet et al. (2019a), where the slopes of the upper shoreface 445 are substituted to the Bruun slopes. In site 1, slopes of the upper shoreface are comprised between 5% and 13%.
Therefore, the PCR model was emulated by defining beach slopes as trapezoid considering the core 5. 1%-12.9% and the support 5%-13%. the reference model in its lower bounds and has an area four times smaller than for the Bruun model. Therefore, considering the PCR model leads to a strong reduction of the uncertainty of SLR-induced shoreline change but also to a sharp decrease of projected coastline retreat. This is due to the fact that the SLR-induced shoreline change is proportional to the inverse of the beach slope, which varies weakly on the range of beach slopes 5%-13%.
Conversely, the Bruun model exacerbates shoreline change ambiguity and shoreline change sensitivity to SLR 460 uncertainties.

Considering anthropization
Along the Castellón coastal stretch, most sectors have been affected by human intervention. This implies that great caution is needed when applying our simple shoreline change model for this area. For instance, in Almardà (South https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License.
of Chilches), beach nourishments have been carried out over the 1995-1998 and 2010-2013 periods, resulting in 465 an overall beach accretion of 1.5 m/year over the 1989-2019 period as shown on Fig. 9a. Outside beach nourishment periods though, shoreline retreat is observed as revealed by positive trend displayed in red. Although our shoreline change model does not explicitly include past anthropogenic influences, effects such as beach nourishment can be implicitly accounted for in the and terms. For instance, for Almadarà (Fig. 9a), the is negative (i.e. beach accretion) due to beach nourishment. Therefore, shoreline change projections made for 470 this site would assume that beach nourishment will be pursued in the future at the same rates and frequency. In such a case, our projections show that by 2100 and even under the RCP8.5, shoreline is expected to further progress toward the sea, with a very large uncertainty though as revealed by the black p-box Fig. 9b. Assuming that beach nourishment will continue is however strongly uncertain and should be avoided. In this regard, we derived the term by relying only on periods outside beach nourishment shown by the red segments. 480 This leads to a weighted mean of 0.83 m/year. The resulting projections in 2100 under the RCP8.5 scenario are shown by the red p-box, which in this case clearly indicates that in absence of future beach nourishment, the shoreline is projected to retreat in face of sea-level rise. The ambiguity remains very similar, indicating that accounting for beach nourishment simply translates the p-box. Nonetheless, we note that when the nourishment is not included, the p-box is more tilted, which is due to the higher standard error associated to the term. 485

Advantages of extra-probabilistic approaches
Here, we discuss the advantage of the use of possibilities in comparison to e.g. a modelling framework that would be fully probabilistic. To illustrate this, we re-calculate shoreline change projections with Eq. (1) in Aquitaine (site 1) in 2100 but assuming that ∆ and tan follow normal distributions. We consider the RCP8.5 scenario with https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License.
∆ defined as 0.69m ± 0.24m and the Bruun rule with tan defined as 1.35% ± 0.15%. The resulting shoreline 490 change projections are normally distributed with 5 th and 95 th percentiles of 71m and 152m, respectively. Within a probabilistic approach, these left and right tails can be reasonably associated to low and high-end projections. The comparison with extra-probabilistic low and high-end projections in Table 3 (i.e. 66m and 196m, respectively) shows substantial differences, and in particular that high-end values obtained within the probabilistic theory are much lower. More importantly, we found that the high-end projection obtained with the possibilistic framework 495 is not even achievable under the probabilistic model built here; hence indicating that the probability-based highend scenario is too optimistic in the sense that it fails to reflect deep uncertainty. One should thus design dedicated (and separated) high-end scenarios to explore such projections that may appear arbitrary.
Finally, the problem of model uncertainty related to the use of the Bruun or the surrogate PCR model provides a good illustration of how the quantified measure of ambiguity in the projection can de decomposed. The use of 500 possibilities allows making very transparent the ambiguity thanks to the p-boxes graphical representation. This has also the advantage of showing how future progress in the system knowledge may contribute reducing deep uncertainty. From a decision-making perspective, the extra-probabilistic approach thus allows a transparent and exhaustive consideration of uncertainties. One should nonetheless bear in mind that in case where knowledge uncertainty becomes very prominent and requires an extensive use of possibility distribution as input, the 505 ambiguity in the outcome may be considered by end-users as too large to be informative and useful.

Conclusion
The approach presented in this paper provides a framework for assessing deep uncertainties in shoreline change projections. This framework is versatile since the definition of input variables and their distribution can be adapted easily to the characteristics of a local site, its data coverage and the degree of knowledge of hydrosedimentary 510 processes acting locally. Furthermore, this extra-probabilistic approach that we here apply to an empirical shoreline evolution model can be actually replicated to any of the available models of shoreline evolution (Montaño et al., 2020).
In our approach, residual uncertainties that have not been integrated quantitatively still remain. For example, the Bruun rule and the PCR models are not the only plausible models for shoreline change reconstructions. Similarly, 515 our high-end sea-level rise estimates might be exceeded by 2050 according to recent expert elicitation of the future contribution of Greenland and Antarctica ice-sheets to sea-level rise (Bamber et al., 2019). The approach consisting in summing up the different modes of variability of shoreline change can also be challenged on the ground. For example, coastal defenses may limit the potential retreat of shorelines in other areas. Finally, future adaptation is unknown and could limit or favor coastal erosion and shoreline changes. 520 Despite these limitations, our approach is potentially useful to determine to which extent reducing our uncertainties on e.g. future sea-level rise or coastal impact models can help improving the precision of future shoreline change projections. For example, we have shown that if sea-level rise does not exceed 1m, shoreline change uncertainties will be reduced significantly. This could be achieved through an ambitious climate mitigation policy and improved knowledge on ice-sheets. While there remain the issue of the long term commitment to sea-level rise (Clark et al., 525 2016), reducing this source of deep uncertainties would grant more time for coastal adaptation. https://doi.org/10.5194/nhess-2020-412 Preprint. Discussion started: 13 January 2021 c Author(s) 2021. CC BY 4.0 License.