# The Dune Engineering Demand Parameter and Applications to Forecasting Dune Impacts

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

- The parameters account for either the dune resilience parameters or the storm intensity, but not both;
- Provide a categorical response only (e.g., the scale is not a continuous number which limits the usefulness in statistical fitting);
- Incomplete parameterization of the resilience terms (e.g., consider dune volume but not position of the dune);
- Incomplete consideration for storm intensity parameters (e.g., account for peak storm intensity but not duration effects).

#### 1.2. Quantifying the Combined Effects of the Beach Width and Dune Volume

#### 1.3. Resilence or Vulnerbility Assessors

_{c}) in reference to mean high water.

#### 1.4. Storm Intensity Measures

#### 1.5. Storm Erosion Index (SEI)

_{d}):

_{b}is the depth-limited breaking wave height (H

_{b}= 0.8 h

_{b}), W* is the width of the active surfzone (approximated as the distance to the breakpoint), B is the berm height, S is the water level height above the mean sea level, and t

_{i}is a time index.

## 2. Study Area and Storm Climate

## 3. Methodology

- Development of the EDP;
- Application of the EDP to develop probabilistic fragility curves of dune impacts in storm events.

_{f}) parameters (e.g., beach characteristics including width, volume, etc.) and the Impact (i.e., observed dune volume loss for a given storm). The data are recast into the EDP by combining the IM and R

_{f}from previously published datasets. The EDP is then used as a physically based parameter to fit fragility curves to predict the observed Impact classes. The EDP is the argument, and the probability of class is the return.

#### 3.1. Development of the Engineering Demand Parameter

_{f}’). The general form of the EDP is presented in Equation (3) with arbitrary variables. In theory, any combination of any R

_{f}or IM parameters may be included in an EDP with corresponding power variables (e.g., ‘i’, ‘j’, ‘k’) selected to ensure non-dimensionality.

_{f}and Impact parameters were considered. Reduction of variables was selected based on two guiding principles. The first was to ensure the most physically meaningful parameters were included both from a qualitative and quantitative means. The second was to the extent possible, include only parameters that are easily measured or estimated without specialized tools or equipment. This is a necessary compromise to make the curves more useful in real-time forecasting with limited time and resources to collect data.

#### 3.2. Physical Meaning of Various EDP Cases

^{2}) and moment (L

^{3}). Cases 7 and 8 model the dune as a mass-moment of inertia (L

^{4}) with the distinction being Case 7 that uses a simplified measure of the moment arm (neglects vertical component of the moment arm), whereas Case 8 estimates the centroid of the dune by modeling the dune as a triangular shape. The vertical location of the centroid is then estimated as one third the difference between the crest and toe elevations. The resultant centroid is estimated using the Pythagoras theorem; adding the berm width and the crest width together. The equation shown in Case 8 in Table 5 shows the redundant square of the square-root for clarity. This is a slightly simplified approach of the Erosive Resistance (ER) parameter introduced by Judge, Overton and Fisher [21].

#### 3.3. The Fragility Model

#### 3.4. Fragility Function and Fitting Techniques

- The confidence of the fit distribution is proportional to the number of samples at failure. With the consensus that more than 20 samples are needed to have reasonable performance. Each sample must be independent.
- The samples must be tested to failure, with the demand (i.e., value of ‘x’) at failure known.

#### 3.5. Data Screening, Fitting Techniques and Sensitivity Analysis

#### 3.6. Comparision of the EDPs and Evaluation of Curves

#### 3.6.1. Log-Normal and MSA Curves

#### 3.6.2. Weibull Distribution

## 4. Results

#### 4.1. Preliminary Analysis

#### 4.2. EDP Curves

#### 4.3. Comparison of EDPs and Fragility Functions

## 5. Discussion

#### 5.1. General Assessment of the EDP

#### 5.2. Fragility Model Performance

#### 5.3. Limitations of the Curves and Potential Improvements

#### 5.4. Comparing EDP Cases and the Physical Meaning

## 6. Conclusions

#### 6.1. Thematic

- The EDP must represent the ratio of storm intensity over the resilience terms;
- Terms in numerator and denominator must be combined in a way that produces a non-dimensional parameter.

#### 6.2. Recommended EDP and Fragility Function

#### 6.3. Limitations, Compromises and Potential Improvements

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Study area location. Alternating red and yellow lines depict NJ Shoreline Segments utilized by Lemke and Miller [38] to produce the storm erosion potential climatology. Segments are numbered from north to south with #1 at Sandy Hook and #13 at Cape May. NJ coastal are counties labeled accordingly. Black + markers depict NJBPN survey locations. Map coordinate system is NAD83/UTM Zone 18N in kilometers. Courtesy of [20].

**Figure 2.**Spatial variance for select Resilience parameters by segment. (

**a**) Dune crest elevation; (

**b**) Dune volume; (

**c**) Berm width; (

**d**) Beach slope. Presented by segment, south (13) to north (1). Red horizontal bar—median value, blue box—25th and 75th percentiles. Outliers defined by 1.5 × IQR away from 25th/75th percentiles. Courtesy of [48].

**Figure 3.**Aggregated summary of quantity of profiles available and corresponding dune volume loss percentage by storm for the New Jersey coast. Courtesy of [20].

**Figure 4.**Conceptual flowchart to highlight sources of previously published data used to generate the EDP and subsequent fragility curves.

**Figure 5.**Observed dune loss percentage as a function of storm intensity (PEI) alone; each data point represents observations from a single profile. X-axis is storm intensity (PEI). Storm categories are denoted by colored vertical dashed lines. The Y-axis is dune loss percentage. Red circles indicate Major loss (>40%), blue circles indicate Moderate (>5% but <40%). Negative dune loss indicates accretion.

**Figure 6.**Observed dune loss percentage as a function of EDP Case 7; each data point represents observations from a single profile. X-axis is the EDP. Storm categories are denoted by colored vertical dashed lines. Storm category PEI values converted to an EDP using median resilience parameters. The Y-axis is dune loss percentage. Red circles indicate Major loss (>40%), blue circles indicate Moderate (>5% but <40%). Negative dune loss indicates accretion.

**Figure 7.**Weibull distribution quantile plots for all observations corresponding to Moderate classification. (

**a**) EDP 1—Berm Width; (

**b**) EDP 2—Berm and Dune width; (

**c**) EDP 3—Dune Volume; (

**d**) EDP 4—Foredune Volume; (

**e**) EDP 5—Shear; (

**f**) EDP 6—Moment; (

**g**) EDP 7—Simplified Mass-moment of Inertia; (

**h**) EDP 8—Mass-moment of Inertia.

**Figure 8.**Weibull distribution quantile plots for all observations corresponding to Major classification. (

**a**) EDP 1—Berm Width; (

**b**) EDP 2—Berm and Dune width; (

**c**) EDP 3—Dune Volume; (

**d**) EDP 4—Foredune Volume; (

**e**) EDP 5—Shear; (

**f**) EDP 6—Moment; (

**g**) EDP 7—Simplified Mass-moment of Inertia; (

**h**) EDP 8—Mass-moment of Inertia.

**Figure 9.**Weibull distribution quantile plots removing maximum observation corresponding to Major classification with a single datapoint removed. Only EDP cases 5 to 8 are shown for clarity. (

**a**) EDP 5—Shear; (

**b**) EDP 6—Moment; (

**c**) EDP 7—Simplified Mass-moment of Inertia; (

**d**) EDP 8—Mass-moment of Inertia.

**Figure 10.**Example log-normal distribution for EDP 7 for calibration/training (triangular shapes) and test datasets (square/diamond). Dashed curves represent MLE log-normal fit curves with red denoting Major and blue denoting Moderate classification.

**Figure 11.**Comparation of Weibull (solid line) vs. log-normal distribution (dashed) for EDP 7. Observed fractions of classification are denoted by shapes and retained from previous figure to illustrate performance. Red denotes Major, Blue denotes Moderate classification.

**Figure 12.**Log-normal predicted and observed fractions of damage classification by storm category (vertical columns). Each subplot corresponds to a damage classification: (

**a**) Minimal; (

**b**) Moderate; (

**c**) Major. Vertical bars correspond to EDP case (1 to 8 in each box), whiskers are standard deviation in predicted probabilities for storms within the Category. Dashed red horizontal bar is the fraction of observed classifications corresponding to the appropriate storm category.

**Figure 13.**Comparison of Weibull (green/right) and log-normal (blue/left) models by storm category (vertical columns). Each subplot corresponds to a damage classification: (

**a**) Minimal; (

**b**) Moderate; (

**c**) Major. Each vertical bar corresponds to an EDP case and model. Note only EDPs 5–8 are shown. Whiskers are standard deviation in predicted probabilities for the corresponding category of storms. Dashed red horizontal bar is the actual observed fraction corresponding to the appropriate storm category.

**Table 1.**Summary of available measures of Storm Intensity and key characteristics. Note variables are identified in ‘Parameters Considered’ column.

Parameter | Storm Type | Intensity Measure | Equation | Output | Units | Parameters Considered | Source |
---|---|---|---|---|---|---|---|

Saffir–Simpson Hurricane Wind Scale | Tropical | Peak | N/A | Category (1–5) | - | Windspeed | Schott et al. (2012) [26] |

Hurricane Intensity Index (HII) & Hurricane Hazard Index (HHI) | Tropical | Peak | $HII={\left(\frac{{V}_{max}}{{V}_{max0}}\right)}^{2}$ $HHI={\left(\frac{R}{{R}_{0}}\right)}^{2}{\left(\frac{{V}_{max}}{{V}_{max0}}\right)}^{3}\left(\frac{{S}_{0}}{S}\right)$ | - | - | Windspeed (V), Radius (R) and translation speed (S), 0 designates reference values | Kantha (2006) [27] |

Integrated Kinetic Energy (IKE) | Tropical | Peak | $IKE={{\displaystyle \int}}_{v}^{}\frac{1}{2}\rho {U}^{2}dV$ | - | - | Wind speed (U), Storm Volume (V) | Powell and Reinhold (2007) [28] |

Hurricane Severity Index | Tropical | Peak | Size + Intensity | 50-point scale | - | Wind speed, Radius | Hebert et al. (2010) [29] |

Surge Scale (SS) | Tropical | Peak | $SS=\left(2.43E-4\right)\u2206p{L}_{30m}{\psi}_{x}\left(\frac{{R}_{33}}{{L}_{30m}}\right)$ | surge | m | $\mathrm{Pressure}(\Delta p),\mathrm{Wind}\mathrm{Radius}\left({R}_{33}\right),\mathrm{Bathymetry}\left(L{30}_{m}\right),\mathrm{Storm}\mathrm{size}(\psi $) | Irish and Resio (2010) [30] |

Cyclone Damage Potential Index (CDP) | Tropical | Peak | $CDP=4\frac{\left[{\left(\frac{{v}_{m}}{65}\right)}^{3}+5\left(\frac{{R}_{h}}{50}\right)\right]}{{v}_{t}}$ | - | - | Wind speed (v_{m}), Radius (R_{h}), forward speed (v_{t}) | Done et al. (2015) [31] |

Dolan and Davis Scale | Extra-tropical | Peak | $P={\left({H}_{s}\right)}^{2}{t}_{d}$ | - | m^{2}-s | Wave Height (H_{s}) duration (t_{d}) | Dolan and Davis (1992) [32] Mendoza et al. (2011) [33] |

Shoreline Risk Index | Extra-tropical | Peak | $I=SH{\left({t}_{D}\right)}^{0.3}$ | - | m^{2}-s^{0.3} | Wave height (H), Surge (S), number of tidal cycles (t_{D}) | Kriebel et al. (1996) [22] |

Storm Erosion Potential Index (SEIP) | Extra-tropical | Cumulative | $SEPI={\displaystyle {\displaystyle \sum}_{t=0}^{{t}_{D}}}{S}_{2SD}\left(t\right){H}_{MHHW}\left(t\right)\u2206t$ | - | m^{3} | Storm surge (S and H), tide duration (t) | Zhang et al. (2001) [34] |

Balsillie Regression Analysis | Both | Peak | ${Q}_{avg}=\frac{1}{1622}{\left({g}^{1/2}{t}_{r}{S}^{2}\right)}^{4/5}$ | Volume (Q) | m^{3} | Surge (S), Tide Rise (t_{r}) | Balsillie (1986) [35] |

Maximum Wave Run-up (R_{max}) | Both | Peak | ${R}_{max}={H}_{o}2.32{\mathsf{\xi}}_{o}$ ${\mathrm{where}\mathsf{\xi}}_{o}=\mathrm{tan}\frac{\beta}{\sqrt{\raisebox{1ex}{${H}_{o}$}\!\left/ \!\raisebox{-1ex}{${L}_{o}$}\right.}}$ | Wave Run-up (R) | m | $\mathrm{Wave}\mathrm{height}\left({H}_{o}\right),\mathrm{period}\left({T}_{p}\right),\mathrm{water}\mathrm{level},\mathrm{shoreline}\mathrm{slope}(\beta $) | Kraus and Wise (1993) [36] |

Coastal Storm Impulse Parameter (COSI) | Both | Cumulative | ${I}_{s}=\int \left[f{p}_{\left(t\right)}+{M}_{\left(t\right)}\right]{d}_{t}$ | Impulse (I_{s}) | kg-s per m | Total Horizontal momentum, surge (fp_{(t)}) and Wave (M_{(t)}) (wave height, period, surge, duration) | Basco and Mhmoudpour (2012) [25] |

Storm Erosion Potential Index | Both | Both | $\mathrm{SEI}={{\displaystyle \int}}_{t}^{}\mathrm{IEI}\left({t}_{i}\right)={W}_{*}\left({t}_{i}\right)\left[\frac{0.068{H}_{b}\left({t}_{i}\right)+S\left({t}_{i}\right)}{B+1.28{H}_{b}\left({t}_{i}\right)}\right]$ | cross-shore distance | m | Water level (S), wave height (H_{b}) and duration (t), Berm elevation (B) | Miller and Livermont (2008) [37] Lemke and Miller (2020) [38] |

Event | SEI | SEI t_{r} [yr] | PEI | PEI t_{r} [yr] |
---|---|---|---|---|

October 1991 | 1534 | 5.2 | 67.3 | 5.2 |

January 1992 | 1017 | 2.2 | 64.4 | 4.1 |

December 1992 | 3326 | 24 | 90.8 | 20 |

December 1994 | 869 | 1.7 | 57.5 | 2.3 |

January 1996 | 719 | 1.3 | 51.1 | 1.4 |

February 1998 | 1434 | 4.5 | 67.0 | 5.0 |

September 2003 | 1040 | 2.3 | 53.8 | 1.8 |

12 October 2005 | 1779 | 6.8 | 54.3 | 1.8 |

25 October 2005 | 866 | 1.7 | 60.0 | 1.4 |

September 2006 | 964 | 2.0 | 50.3 | 1.4 |

November 2007 | 670 | 1.2 | 60.4 | 2.9 |

May 2008 | 998 | 2.1 | 53.4 | 1.7 |

September 2008 | 1714 | 6.4 | 54.3 | 1.8 |

November 2009 (Vets) | 2986 | 18 | 74.8 | 9.1 |

March 2010 | 1220 | 3.2 | 58.7 | 2.5 |

September 2010 | 581 | 1.0 | 62.9 | 3.6 |

August 2011 (Irene) | 788 | 1.4 | 73.3 | 8.2 |

October 2012 (Sandy) | 3056 | 19 | 119 | 49 |

SEI | PEI | |||||
---|---|---|---|---|---|---|

Category | Value | Tr | Annual Prob. | Value | Tr | Annual Prob. |

2 | 730 | 1 | 77.1% | 23 | <1.0 | >99.9% |

3 | 1510 | 5 | 19.8% | 58 | 2 | 41.5% |

4 | 2300 | 10 | 9.5% | 93 | 22 | 4.7% |

5 | 3080 | 19 | 5.2% | 128 | 62 | 1.6% |

**Table 4.**Reduced parameters of interest. Note FB denotes freeboard or vertical distance between parameter and mean sea level, WL—water level, vol—volume, Hb—breaking wave height, cumE—cumulative energy (H

^{2}), Z denotes elevation, Max denotes the maximum hourly value over a single storm duration and Bslope/Islope denote beach and intertidal slope.

Type | Parameter Name | Symbol/Abbreviation | Units | Highly Correlated |
---|---|---|---|---|

Intensity Measure | Peak Erosion Intensity | PEI | Length (m) | MaxWL, MaxHb, cumE, DL, DLvol |

Resilience | Berm Width | Bwidth | Length (m) | Berm Vol |

Resilience | Crest Width (cross-shore distance from dune toe to crest) | Fwidth | Length (m) | Fvol, Dvol, Fslope^{−1} |

Resilience | Dune volume (volume from toe to heel) | Dvol | Volume per length (m^{3}/m) | CrestZ, ToeZ, CrestFB |

Resilience | Foredune volume (volume from toe to crest) | Fvol | Volume per length (m^{3}/m) | Fwidth, CrestZ, Dvol, CrestFB, Fslope^{−1} |

Resilience | Dune Crest Elevation | CrestZ | Length (m) | Dvol, Fvol, ToeZ, CrestFB, ToeFB, Bslope, |

Resilience | Dune Toe Elevation | ToeZ | Length (m) | CrestFB, ToeFB, d50, ISlope, BSlope, CrestZ |

Impact | Dune Loss Percent | DL | % (-) | DLvol |

Case | Parameters | EDP | Physical Proxy |
---|---|---|---|

1 | PEI, Berm Width | $\frac{\mathrm{PEI}}{\left(Bwidth\right)}$ | Setback |

2 | PEI, Berm Width, Dune Crest Width | $\frac{\mathrm{PEI}}{\left(Bwidth+Fwidth\right)}$ | Setback |

3 | PEI, Dune Volume | $\frac{{\mathrm{PEI}}^{2}}{\left(Dvol\right)}$ | Volume |

4 | PEI, Foredune Volume | $\frac{{\mathrm{PEI}}^{2}}{\left(Fvol\right)}$ | Volume | ‘540-rule’ |

5 | PEI, Berm Width and Dune Volume | $\frac{{\mathrm{PEI}}^{2}}{\left(Bwidt{h}^{2}+Dvol\right)}$ | Shear |

6 | PEI, Berm Width and Dune Volume | $\frac{{\mathrm{PEI}}^{3}}{\left(Bwidth\times Dvol\right)}$ | Moment |

7 | PEI, Berm Width and Dune Volume | $\frac{{\mathrm{PEI}}^{4}}{\left(Bwidt{h}^{2}\times Dvol\right)}$ | Simplified Mass-moment of Inertia |

8 | PEI, Berm Width and Dune Volume | $\frac{{\mathrm{PEI}}^{4}}{\left[{\left(\sqrt{{\left(Bwidth+Fwidth\right)}^{2}+{\left(\frac{1}{3}\left(CrestZ-ToeZ\right)\right)}^{2}}\right)}^{2}\times Dvol\right]}$ | Mass-moment of Inertia |

Damage Class | Definition |
---|---|

Major | Dune volume loss > 40% |

Moderate | Dune volume loss 5–40% |

Minor | Dune volume loss < 5% |

EDP | Moderate | Major | ||
---|---|---|---|---|

θ (Median) | β (Dispersion) | θ (Median) | β (dispersion) | |

5 | 1220 | 1.80 | 2058 | 1.10 |

6 | 702 | 1.51 | 1694 | 1.15 |

7 | 1839 | 1.99 | 45629 | 2.50 |

8 | 1260 | 2.34 | 7915 | 2.05 |

EDP | Moderate | Major | ||
---|---|---|---|---|

λ (Scale) | κ (Shape) | λ (Scale) | κ (Shape) | |

5 | 366 | 1.30 | 635 | 3.04 |

6 | 282 | 1.78 | 1012 | 2.16 |

7 | 570 | 1.14 | 4428 | 1.32 |

8 | 243 | 1.13 | 1697 | 1.35 |

MAE | Bias | RMSE | ||||
---|---|---|---|---|---|---|

Moderate | Major | Moderate | Major | Moderate | Major | |

1 | 26% | 16% | −21% | −16% | 35% | 26% |

2 | 23% | 13% | −19% | −13% | 30% | 20% |

3 | 26% | 17% | −19% | −16% | 32% | 24% |

4 | 26% | 15% | −19% | −14% | 33% | 21% |

5 | 25% | 16% | −20% | −16% | 33% | 25% |

6 | 15% | 9% | −11% | −8% | 19% | 12% |

7 | 15% | 18% | −12% | −18% | 19% | 28% |

8 | 19% | 15% | −16% | −15% | 25% | 22% |

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## Share and Cite

**MDPI and ACS Style**

Janssen, M.S.; Miller, J.K.
The Dune Engineering Demand Parameter and Applications to Forecasting Dune Impacts. *J. Mar. Sci. Eng.* **2022**, *10*, 234.
https://doi.org/10.3390/jmse10020234

**AMA Style**

Janssen MS, Miller JK.
The Dune Engineering Demand Parameter and Applications to Forecasting Dune Impacts. *Journal of Marine Science and Engineering*. 2022; 10(2):234.
https://doi.org/10.3390/jmse10020234

**Chicago/Turabian Style**

Janssen, Matthew S., and Jon K. Miller.
2022. "The Dune Engineering Demand Parameter and Applications to Forecasting Dune Impacts" *Journal of Marine Science and Engineering* 10, no. 2: 234.
https://doi.org/10.3390/jmse10020234