# A Regional Application of Bayesian Modeling for Coastal Erosion and Sand Nourishment Management

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## Abstract

**:**

## 1. Introduction

## 2. Bayesian Modeling for Coastal Erosion Management

_{i}, given a particular set of observations, O

_{j}(Pearl, 1988). The first term on the right-hand side is the likelihood of observations O

_{j}given that the forecast F

_{i}is true. The second term on the right is the prior probability distribution of F

_{i}(e.g., the probability of a given forecast based on the entire training dataset, in the absence of any additional observations). The denominator on the right side is the prior probability distribution of O

_{j}.

## 3. Study Area

^{3}/year (Figure 1). In general, the natural supply of sediment to the coast is very limited and, consequently, the coastline is retreating [4,27].

^{3}to 6 million m

^{3}of sand in 1990, and then to 12 million m

^{3}in 2001 [29]. Even higher volumes might be necessary in the future to cope with the more severe predicted sea level rise scenarios. The total nourishment volumes implemented along the Holland Coast between 1965–1990, 1991–2000, and 2001–2016, and divided for different nourishment types (beach nourishments, shoreface nourishments, dune nourishments, and others), are shown in Figure 2. Beach and dune nourishments are generally implemented directly on the beach or dunes. Shoreface nourishments are implemented in proximity of the breaker bars (≈3 to 5 m water depth). As shown in Figure 2, the total volume of shoreface nourishments has been increasing since 1990, due to their lower cost (relative to beach nourishments) and lower interference with the coastal environment.

## 4. Material and Methods

#### 4.1. Data Availability

^{3}of sand nourishment per linear meter of coastal length where the nourishment was built.

- -
- Changes in MCL (momentary coastline) position, defining the position of the coastline as a function of the volumes of sand in the near shore zone, approximately between the dune foot (+3 m NAP, where NAP ≈ mean sea level) and −5 m NAP (Figure 4) [32]. Positions are given with respect to predefined reference points at each transect (i.e., the RSP points “RijkStrandPalen” = “Beach Poles”).
- -
- Changes in DF (dune foot) position, defining the position of the dune foot, and estimated as the most seaward intersection of the +3 m NAP line and the cross-shore profile.

#### 4.2. Bayesian Network

- ▪
- Time interval and spatial characterization of the study area (in yellow);
- ▪
- nourishment type and volume (in purple);
- ▪
- effects on the morphological indicators (in green).

#### 4.2.1. Time Interval and Spatial Characterization of the Study Area

- ▪
**Time interval**: 1965–1990; 1991–2000; 2001–2016. Time intervals have been chosen in order to discriminate different periods in which the nourishment policy has been adapted (i.e., in 1990 and 2000).- ▪
**Area**: North Holland, Rijland, and Delfland. These are the three coastal sections in which the Holland coast is divided.

#### 4.2.2. Nourishment Type and Volume

- ▪
**Nourishment volume**(in m^{3}/m/year): Yearly nourishment volume divided by the length of the nourishment.- ▪
**Nourishment type**: Beach or dune nourishment; shoreface nourishment; no nourishment; more than one type of nourishment at the same transect. It describes the nourishment type.- ▪
**Nourishment**: Yes; no. To discriminate transects which have been nourished at least once during the entire period (1965–2016) from the ones which have never been nourished.

#### 4.2.3. Effects on the Morphological Indicators

- ▪
**MCL change**(m/year): To quantify changes in coastline (MCL) position.- ▪
**Dune foot (DF) change**(m/year): To quantify changes in dune foot position.- ▪
**Percentage of transects in which the momentary coastline (MCL) moves**: Landward; seaward. To quantify the percentage of transects in which MCL has a positive (seaward), negative (landward) shift as a result of the effects of natural morphological changes and nourishments.- ▪
**Percentage of transects in which the dune foot (DF) moves**: Landward; seaward. To quantify the percentage of transects in which the dune foot has a positive (seaward), negative (landward) shift as a result of the effects of natural morphological changes and nourishments.

## 5. Results

#### 5.1. Prior Probability Distributions

#### 5.2. Assessment of the Effectiveness of Sand Nourishments Against Erosion

#### 5.3. Assessment of the Effectiveness of Different Nourishment Designs

#### 5.4. Application of BERM-N as Predictive Tool to Achieve a Predefined Coastal Management Objective

^{3}/m/year would be required, distributed over 13.7% of the transects. Considering a total length of the coastline equal to 117 km, this would correspond to a yearly volume of about 2.5 million m

^{3}of sand/year (i.e., 161 m

^{3}/m/year × 117,000 m × 0.137). This is less than the current nourishment volume applied along the Holland coast, which currently results in an average accreting coastline as a result of the large nourishment volumes applied yearly. These nourishments are applied not just to preserve the current position of the coastline, but to maintain the entire coastal foundation in future sea level rise scenarios, as well as to provide the boundary conditions for the developments of additional functions, such as wide beach and dunes for nature and recreation (Section 3).

## 6. Discussions

#### 6.1. General Assumptions Related to the Construction of the BERM-N

#### 6.2. Alongshore Effects of Sand Nourishments

#### 6.3. BERM-N Tool for Coastal Erosion Management of Past and Future Conditions

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Map of Holland, the Netherlands, including the three subregions considered in this study: Delfland, Rijnland, and North Holland. Net yearly alongshore sediment transport rates are also shown [25].

**Figure 2.**Nourishment volumes (millions m

^{3}/year) at the Holland coast for the three periods: 1965–1990, 1991–2000, and 2001–2016. Note that the North Holland coast is ≈55 km long, Rijnland ≈41 km, and Delfland ≈21 km.

**Figure 3.**Example of the morphological development of a single cross-shore JarKus profile (transect 11,301) located at Delfland, between 1965 and 2015.

**Figure 4.**Computation of the momentary coastline (MCL) volume for a given JarKus transect. A is the area used to compute the Momentary Coastline position. A is delimited by an upper boundary, corresponding to the dune foot position, and a lower boundary, at a distance equal to 2 x H from the dune foot position. H is defined as the distance between the dune foot position and the mean low water line. RSP is the reference point from which distances are computed (“rijksstrandpalen”). Therefore, the MCL position can be estimated as MCL = (A/2H) + x, with x being the distance between the RSP line and the dune foot position.

**Figure 5.**Visualization of BERM-N. Nodes have been grouped in three categories according to the color. Yellow is used for the nodes describing the spatial characterization of the study area and the time interval; purple for the nodes describing the nourishment types and volumes; and green for the nodes describing the effects on the morphological indicators. At each node, the first column indicates the chosen discretization intervals, while the second column (adjacent to the histogram) is the percentage of the prior distribution in each bin. The last line indicates the mean of the prior distribution ± one standard deviation.

**Figure 6.**BERM-N application to assess the effectiveness of sand nourishments. Panel above (

**a**): BERM-N constrained in order to consider only transects which have not been nourished (see red box). Panel below (

**b**): BERM-N constrained in order to consider only transects which have been nourished (see red box).

**Figure 7.**Effects of no nourishments, shoreface, and beach nourishments on MCL (

**upper panel**) and dune foot (DF) (

**lower panel**) indicators. The red color represents the probability of a landward displacement of the indicator, whereas a green color represents the probability of a seaward displacement. Values indicate the mean values of the distributions.

**Figure 8.**Effects of no nourishments, shoreface, and beach nourishments on MCL (

**upper panel**) and DF (

**lower panel**) indicators. The effects are shown in three different columns, to indicate the effects one year, five years and ten years after implementation of a nourishment.

**Figure 9.**BERM-N application as predictive tool to assess the required nourishment volume in order to reach a predefined coastal erosion management objective: Reduce to zero the erosion over the entire coastline, only allowing for accretion. The red boxes show the two nodes constrained to a condition of “seaward” movement at the two coastline indicators.

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**MDPI and ACS Style**

Giardino, A.; Diamantidou, E.; Pearson, S.; Santinelli, G.; Den Heijer, K.
A Regional Application of Bayesian Modeling for Coastal Erosion and Sand Nourishment Management. *Water* **2019**, *11*, 61.
https://doi.org/10.3390/w11010061

**AMA Style**

Giardino A, Diamantidou E, Pearson S, Santinelli G, Den Heijer K.
A Regional Application of Bayesian Modeling for Coastal Erosion and Sand Nourishment Management. *Water*. 2019; 11(1):61.
https://doi.org/10.3390/w11010061

**Chicago/Turabian Style**

Giardino, Alessio, Eleni Diamantidou, Stuart Pearson, Giorgio Santinelli, and Kees Den Heijer.
2019. "A Regional Application of Bayesian Modeling for Coastal Erosion and Sand Nourishment Management" *Water* 11, no. 1: 61.
https://doi.org/10.3390/w11010061