Guidelines for the application of the probabilistic framework

These guidelines guide the designer in the operational application of the probabilistic framework for the performance assessment of coastal and harbor defense structures under present and future climate conditions.

1. Case study set-up

• Select the site and the type of coastal defense structure to be analyzed
• Define the global simulation parameters:
  • P_f,Lmax — maximum admissible probability of failure at the end of the service life
  • L — service life of the structure [years]
  • CV — target coefficient of variation of the estimated probability of failure
• Compute the number of life cycles to be simulated using Monte Carlo analysis (n_lc=(1-P_f,Lmax)/(CV²* P_f,Lmax))​ where CV controls the statistical accuracy of the simulation (smaller CV values require a larger number of Monte Carlo realizations).

2. Collection of input data

• Structural and geometric data: structure geometry, structural materials, state of conservation and level of damage
• Bathymetric and meteocean data: coastal bathymetry and topography, wave data, storm surge data, sea level rise scenarios (SLR)

3. Definition of design conditions

• Selection of limit states: ULS – Ultimate Limit State, SLs – Serviceability Limit State
  • definition of the reliability function Z by rewriting the design formula associated with the failure mechanism in the form: Z=R-S or Z=R/S
  • probabilistic characterization of the variables included in R and S

• Definition of climate scenarios
  • Present climate scenario
  • Future climate scenario including SLR

4. Random generation of the storm-by-storm climate

• Create the storm dataset using a Peak Over Threshold (POT) analysis of the historical significant wave height time series.
• Perform extreme value analysis
• Estimate the mean annual storm frequency λ
• Estimate the mean annual storm frequency λ
• Calibrate the site-specific empirical relationships linking the following variables to the significant wave height, estimating both the coefficients and their standard deviations:
  • Storm surge (h_SS)
  • Mean and peak wave periods (T_m, T_p)
  • Storm duration (d_s)
• Generate n_SS = λ x n_lc storm events for each life cycle:
  • ​Offshore significant wave height (H_s0) sampled from the extreme value distribution
  • Storm surge (h_SS) as a function of H_s0
  • Wave periods (T_m, T_p) as a function of H_s0
  • Storm duration (d_s) as a function of H_s0
  • Water level at the structure toe (h) sampled from a normal distribution with mean equal to the mean sea level (with or without SLR) and standard deviation equal to half of the astronomical tidal ranges
• Propagate H_s0 to the structure toe using:
  • linear shoaling
  • wave breaking control

The random realizations of each variable are stored in a matrix with n_lc rows and n_SS columns.

5. Random generation of geometry and material properties

Generate n_lc random realizations of the parameters describing the geometry and material properties of the coastal or harbor defense structure.

6. Probabilistic analysis

• Compute the reliability function (Z) associated with the considered limit state for each generated sea state
• Compute the cumulative probability of failure P_f(t) as the ratio between the number of life cycles composed of t years containing at least one failure and the total number of simulated life cylces n_lc, for t=1, …., L

7. Performance assessment and design decision

On the basis of the results of the probabilistic analysis, indicators are obtained which allow the comparison between different design configurations and climate scenarios, supporting the designer in selecting the most effective solution in terms of performance, risk reduction and economic sustainability.

• Performance indexes based on the P_f(t) curves associated to ULS and SLS
  • Index r = P_f(L) / P_f,Lmax where P_f(L) is the cumulative probability of failure at the end of the service life of the structure and P_f,Lmax is the maximum admissible probability of failure (r<1 is the acceptability condition)
  • Index s = dP_f(t)/dt represents the rate of increase of the probability of failure and is estimated as the slope of the linear regression of the curve P_f(t) over time. This index describes the rate of risk growth during the service life of the structure

• Economic indexes
  • Repair costs
  • Costs due to the loss of functionality of the defense work