# Unravelling the Importance of Uncertainties in Global-Scale Coastal Flood Risk Assessments under Sea Level Rise

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## Abstract

**:**

## 1. Introduction

**input variable**(i.e., scenario, model parameter, data, etc.) at a time [3,10,11]. Though such approaches can easily and efficiently provide a first order estimate of uncertainty, they only provide narrow insights into the contributions of each uncertain variable, because only a limited number of combinations of input variables are generally accounted for [12].

## 2. Materials and Methods

#### 2.1. DIVA Modelling Framework

#### 2.2. Global Sensitivity Analysis

_{i}. Formally, VBSA relies on the first-order Sobol’ indices (ranging between 0 and 1), which can be defined as:

_{i}corresponds to the main effect of ${X}_{i}$, i.e., the proportion of the variance reduction of Y (i.e., representing the uncertainty in Y) that is solely induced by varying ${X}_{i}$. The higher the influence of ${X}_{i}$, the lower the variance when fixing ${X}_{i}$ (corresponding to the term $\mathrm{Var}\left(Y|{X}_{i}\right)$ in Equation (1)), hence the closer ${S}_{i}$ to one. In general, the sum of all ${S}_{i}$ is ≤1 (i = 1, 2, …, n); the difference with one is a quantification of the higher order interactions (see [13]).

_{i}remains valid in any situation: ${S}_{i}$ can be used to identify the input variable that should be fixed (as pointed out by [28]). Whether dependencies are present or not, the index provides a measure of importance (i.e., a sensitivity measure) that is useful for the ranking of the different uncertainties described in Table 1. We adopt this viewpoint in the present study.

#### 2.3. Uncertain Variables and Values Considered

#### 2.4. Likelihoods of Variables Values

- The r-largest annual value rGEV = 5 is chosen as of higher likelihood based on the conclusions of [20];
- A preference is given to the original parametrization of the Logistic depth-damage curves by using a half-damage depth of 1.0 m;
- In light of recent observations by [39], subsidence in delta regions is considered a more plausible scenario;
- Likelihood weights are assigned to combinations of RCP/SSPs; this is further detailed below;
- No preference is given, neither for global population distribution, nor for the Assets-to-GDP ratio, in the absence of evidence favoring one scenario over another.

#### 2.5. Dependence between Uncertain Variables

_{{2.6,4.5,8.5}}|SSP

_{1–5}). We estimate P(RCP

_{{2.6,4.5,8.5}}|SSP

_{1–5}) based on the location of the RCP-SSP combination in the diagram “carbon intensity improve rate versus energy intensity improve rate” provided by [19]: the lower both rates, the easier and cheaper it is to transform the economy and hence the higher the conditional probability P(RCP

_{{2.6,4.5,8.5}}|SSP

_{1–5}) is. More formally, the conditional probability is assumed to be inversely proportional to the product of the carbon intensity and energy intensity improvement rates. Using a normalization so that the probabilities sum to one, this yields Table 3.

^{2}scenarios needs to be distinguished from the notion of infeasibility in the real world.” ([19], Page 165). And specifically for the combination of SSP3 and RCP2.6, they note that “infeasibility, in the case of SSP3, is thus rather an indication of increased risk that the required transformative changes may not be attainable due to technical or economic concerns.” ([19], Page 165). In any case, assuming a probability of 0 for the SSP3/RCP2.6 or SSP5/RCP2.6 combination would not make a big difference as those probabilities are anyway small according to our approach.

## 3. Results

#### 3.1. Equally Plausible Values

^{2}reaches values from 96.3 to 99.7% (with mtry = 8) for EAD over time, and >99.9% whatever the time instance for AC: this is strong evidence for the high predictive capability of the trained RF models, hence comforting our confidence in the replacement of the DIVA model by the RF models. To compute the sensitivity indices, we randomly generate 50,000 samples of the random variables (assigned to the uncertainties of Table 1) and compute the values of the output variables of interest using the trained RF. This dataset is then used to derive the sensitivity indices as described in Section 2.2. Preliminary convergence tests showed us that 50,000 random samples were sufficient to reach stable sensitivity indices’ values (this is also shown by the narrow width of the confidence intervals of the GSA results; see below in Figure 2).

- In the short term (before 2030–2040, leftmost part of Figure 3a), the uncertainty is mainly controlled by two sources of uncertainty, namely the value of the Assets-to-GDP ratio (rose-colored envelope), the rGEV parameter of the extreme value analysis (purple-colored envelope) with sensitivity indices of ~25%, and ~75% respectively;
- After this date, the time evolution of their influence differs: Figure 3a shows the rapidly decreasing rGEV influence to low value <1% by 2050, whereas the influence of A:GDPr decreases less abruptly, down to values of ~10% by 2050; This decrease is relative to the total uncertainty, which is increasing rapidly after 2040; hence, this decrease simply reflects the growing importance of other sources of uncertainties after 2050;
- The importance of SSP scenarios (blue-colored envelope) rapidly increases over time: the sensitivity index reaches large values above 50% after 2030, until driving the whole uncertainty by 2100 with a contribution >50%; again, the slow decrease of the relative contribution of SSP to EAD uncertainty after 2050 is due to RCP becoming a significant source of uncertainty. This large impact of SSP is expected because different SSPs have large impacts in the development of the global floodplains;
- The importance of RCP scenarios (in orange) is only slowly increasing over time reaching a low index value >5% by 2065–2070; this late emergence of RCP scenarios as a source of uncertainty in EAD is consistent with the projected times of divergence of sea-level projections: until about 2050, sea-level projections are almost the same whatever the RCP scenario due to their high dependence on past greenhouse gas emissions [40];
- The role of the choice in the population database (in yellow) and of RSLR (in green) remain moderate with a sensitivity index of 5–8%, and the role of subsidence in delta regions is minor (in red).

- In the short term (leftmost part of Figure 3b), the uncertainty related to RSLR (in green) dominates (with an index of >40%);
- The three other most important uncertainties are the population database (in yellow), the extreme modelling parameter (in purple), and the RCP scenario (in blue);
- Over time, the importance of RCP rapidly exceeds the one of RSLR, and reaches >60% by 2050, whereas the one of RSLR drops down to <25% after this date;
- The sensitivity index of POP and of rGEV rapidly decreases down to low value <5% by 2040.

#### 3.2. With Most Likely Values

## 4. Discussion and Conclusion

#### 4.1. Summary and Implications

#### 4.2. Residual Uncertainties

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Random Forest Regression

_{i}the result of the DIVA simulation given the i

^{th}vector of p input variables: ${X}_{i}=\left({X}_{i}^{\left(1\right)},\dots ,{X}_{i}^{\left(p\right)}\right)$ with i = 1, …, n. A cut of a given area A is defined by the pair (j, z) where j = (1, …, p) and z is the position along the j

^{th}coordinate given the limits of A. Let us define C

_{A}the set of all possible cuts for A. The partition aims at maximizing the following criterion over C

_{A}:

_{i}that belong to A (respectively A

_{L}and A

_{R}), and I

_{(B)}is the indicator function that reaches 1 if B is true and 0 otherwise.

_{size}is reached), a constant value of the response variable is predicted within each area (mean value for regression).

_{tree}times, i.e., the RF model is set up using an ensemble of n

_{tree}different tree models.

^{2}coefficient defined as follows:

^{2}to one, the more satisfactory the validation, the higher the predictive capability of the RF model. To further increase the predictive capability, we tune the value of mtry to reach the maximum Q

^{2}value. Figure A1 depicts the time evolution of Q

^{2}as a function of mtry. This shows that mtry = 8 leads to a very satisfactory predictive capability of the RF model for both EAD and AC: Q

^{2}ranges from 96.3 to 99.7% for EAD over time, and >99.9% whatever the time instance for AC. The influence of the other RF parameters (i.e., n

_{tree}and n

_{size}) on the predictive capability was also tested and appeared to be minor compared to the one of mtry: they were fixed to constant values n

_{tree}= 1000 and n

_{size}= 5.

**Figure A1.**Time evolution of the performance indicator Q

^{2}estimated using a 10-fold cross-validation procedure for EAD (

**a**) and for AC (

**b**) considering different mtry values.

## Appendix B. Filtering Algorithm for Estimating the Sensitivity Indices

- Random sampling of the uncertainty input space. No particular algorithm is here required, and dependencies can here be taken into account using appropriate random sampling scheme;
- Partition of the input parameter space into clusters. This can be done in different manners, for instance using an equi-probable partitioning [30]. Alternatives can focus on clustering techniques; for instance K-means algorithm with a fixed number of samples, or a simple systematic regular partition. In the present study, the partition corresponds, by construction, to the selection of a given modelling scenario;
- Computation of the local conditional variance, i.e., $\mathrm{Var}(Y|{X}_{i})$ for each cluster. This provides a quantification of the evolution of the local variance in the domain of variation of the input parameters;
- Estimation of the local Sobol’ indices, i.e., $1-\mathrm{Var}(Y|{X}_{i})/\mathrm{Var}$(Y). Depending on the value at which the parameter is fixed, the local variance reduction can be high or low;
- The Sobol’ indices are then obtained by computation of the expectation values of the local Sobol’ Indices as described by Equation (1).

## Appendix C. Projections of Extreme Sea Levels

_{m}, can be calculated as follows:

## Appendix D. Coastal Flood Risk in DIVA

_{p}) that gives the number of people living below a given elevation level x is constructed by superimposing a DEM with a spatial population dataset and interpolating piecewise linearly between the given data points. Only grid cells that are hydrologically connected to the coast are considered. From areas below 1 m of elevation, the areas covered by coastal wetlands are subtracted, because these are uninhabitable. For each segment, a cumulative asset exposure function (denoted e

_{a}) is obtained by applying subnational per capita GDP rates to the population data multiplied by an empirically estimated Assets-to-GDP ratio (A:GDPr) [5]. Future exposure is calculated by applying national population and GDP growth rates of the shared socioeconomic paythway scenarios (SSP) to the coastal segments.

_{dyke}is the dyke height, and x

_{max}is the maximum extreme sea level to be taken into account (here assumed to be the 10,000-year return level). The probability density function f(.) is the one of extreme sea levels and is derived from the analysis described in Appendix C.

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**Figure 1.**Time evolution of the expected annual damage cost EAD (Top) and of the adaptation cost AC (Bottom) expressed in % of the Gross Domestic Product of 2014 assuming equally plausible modelling scenarios:: (

**a**,

**c**) Example of five random realizations; (

**b**,

**d**) Mean value (calculated for 50,000 random realizations). The bounds of the uncertainty band are defined at −/+ one standard deviation (red envelope). Note that the vertical y-axis of the AC Figures does not start at zero.

**Figure 2.**Sensitivity indices for EAD (

**a**) and for AC (

**b**) by 2100 considering the two assumptions regarding scenarios’ likelihood (“equally plausible”-red and with different likelihood weights -blue). The error-bars indicate the 95% confidence interval derived from a bootstrap-based approach (using 250 bootstrap random replicates).

**Figure 3.**Time evolution of the sensitivity indices (denoted S) for EAD (

**a**) and for AC (

**b**) considering the situation of equally plausible modelling scenarios. A rescaling has been applied so that the sum of the indices is one.

**Figure 4.**Time evolution of the sensitivity indices (denoted S) for EAD (

**a**) and for AC (

**b**) considering the situation where most likely scenarios have been identified. A small rescaling has been made so that the sum of the indices is one.

**Table 1.**Uncertain variables and their possible values considered in the global sensitivity analysis.

Modelling Uncertainty | Values | Number of Values |
---|---|---|

Socio-economic development (SSP) | SSP1-5 | 5 |

Greenhouse gas concentrations (RCP) | RCP2.6, 4.5, 8.5 | 3 |

Global population distribution (POP) | GPW4 or Landscan | 2 |

Magnitude of the Regional Sea-Level Rise (RSLR) | Median value, 5th and 95th percentile given RCP scenario | 3 |

Logistic depth-damage curves (DF) | Half-damage depth 1 or 1.5 m | 2 |

r-largest annual value (rGEV) | Number of r largest values used to fit GEV: 1, 2 or 5 | 3 |

Subsidence in delta region (SUBS) | Included or not | 2 |

Assets-to-GDP ratio (A:GDPr) | Value of 2.8 or 3.8 | 2 |

Uncertain Variable | Most Likely Value | Likelihood Weight |
---|---|---|

Logistic depth-damage curves | Half-damage depth at 1.0 m | 0.75 |

Extreme value modelling | rGEV = 5 | 0.66 |

Subsidence in delta region | Included | 0.75 |

Socio-economic & Greenhouse Gas concentrations | See Table 3 | See Table 3 |

**Table 3.**Definition of the occurrence probability for the Representative Concentration Pathways (RCP) scenarios conditional on the shared socioeconomic pathways (SSP) scenarios, i.e., each probability value is interpreted as the occurrence of the considered RCP scenario given the considered SSP scenario. These values should not be interpreted as the probability of SSP given RCP.

RCP2.6 | RCP4.5 | RCP8.5 | |
---|---|---|---|

SSP1 | 0.18 | 0.34 | 0.48 |

SSP2 | 0.09 | 0.12 | 0.79 |

SSP3 | <0.01 | 0.11 | 0.89 |

SSP4 | 0.12 | 0.18 | 0.70 |

SSP5 | 0.05 | 0.085 | 0.865 |

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**MDPI and ACS Style**

Rohmer, J.; Lincke, D.; Hinkel, J.; Le Cozannet, G.; Lambert, E.; Vafeidis, A.T. Unravelling the Importance of Uncertainties in Global-Scale Coastal Flood Risk Assessments under Sea Level Rise. *Water* **2021**, *13*, 774.
https://doi.org/10.3390/w13060774

**AMA Style**

Rohmer J, Lincke D, Hinkel J, Le Cozannet G, Lambert E, Vafeidis AT. Unravelling the Importance of Uncertainties in Global-Scale Coastal Flood Risk Assessments under Sea Level Rise. *Water*. 2021; 13(6):774.
https://doi.org/10.3390/w13060774

**Chicago/Turabian Style**

Rohmer, Jeremy, Daniel Lincke, Jochen Hinkel, Gonéri Le Cozannet, Erwin Lambert, and Athanasios T. Vafeidis. 2021. "Unravelling the Importance of Uncertainties in Global-Scale Coastal Flood Risk Assessments under Sea Level Rise" *Water* 13, no. 6: 774.
https://doi.org/10.3390/w13060774