7.10. Average Annual Damage Probabilities

postprocessor.calculate_average_annual_damage_probability(fragility_array, hazard_array, return_period=1, max_return_period=5000)[source]

Calculate the Average Annual Damage State Probability (AADP) based on fragility and hazard curves.

This function estimates the average annual probability of damage states occurring over a given return period, using the fragility curve (which relates intensity measure levels to damage state probabilities) and the hazard curve (which relates intensity measure levels to annual rates of exceedance).

The calculation integrates the product of the fragility function and the hazard curve over the specified range of intensity measure levels, accounting for the return period and a maximum return period threshold.

Parameters:

  • fragility_array (2D array) – A 2D array where the first column contains intensity measure levels, and the second column contains the corresponding probabilities of exceedance for each intensity level.

  • hazard_array (2D array) – A 2D array where the first column contains intensity measure levels, and the second column contains the annual rates of exceedance (i.e., the probability of exceedance per year) for each intensity level.

  • return_period (float, optional, default=1) – The return period used to scale the hazard rate. This is the time span (in years) over which the average annual damage probability is calculated. A typical value is 1 year, but longer periods can be used for multi-year assessments.

  • max_return_period (float, optional, default=5000) – The maximum return period threshold used to filter out very low hazard rates. The hazard curve is truncated to include only intensity levels with exceedance rates above this threshold.

Returns:

average_annual_damage_probability – The average annual damage state probability, calculated by integrating the product of the fragility function and the hazard curve over the given intensity measure levels.

Return type:

float

Theoretical Background

The Average Annual Damage Probability (AADP) for a damage state \(ds\) is the expected frequency of exceeding that damage state per year, obtained by integrating the fragility curve over the seismic hazard (McGuire, 2004).

Classical integral

\[\text{AADP}_{ds} = \int_0^{\infty} P(\text{DS} \geq ds \mid \text{IM} = x)\; \left|\frac{d\lambda(x)}{dx}\right| dx\]

where:

  • \(P(\text{DS} \geq ds \mid \text{IM} = x)\) is the fragility curve giving the probability of exceeding damage state \(ds\) at intensity \(x\),

  • \(\lambda(x) = P(\text{IM} > x)\) is the mean annual rate of exceedance from the hazard curve, and

  • \(|d\lambda/dx|\) is the probability density of IM occurrences per year.

Discrete approximation

In practice the integral is evaluated numerically using midpoint quadrature over the IM bins of the hazard curve:

\[\text{AADP}_{ds} \approx \sum_{j} P(\text{DS} \geq ds \mid \text{IM} = \bar{x}_j) \cdot \Delta\lambda_j\]

where \(\bar{x}_j = (x_j + x_{j+1})/2\) is the midpoint of the \(j\)-th IM interval and \(\Delta\lambda_j = |\lambda(x_j) - \lambda(x_{j+1})|\) is the corresponding rate of occurrence.