7.11. Average Annual Loss

postprocessor.calculate_average_annual_loss(vulnerability_array, hazard_array, return_period=1, max_return_period=5000)[source]

Calculate the Average Annual Loss Ratio (AALR) based on vulnerability and hazard curves.

This function estimates the average loss ratio occurring over a given return period (typically annual where return_period = 1), using the vulnerability curve (which relates intensity measure levels to an expected loss ratio) and the hazard curve (which relates intensity measure levels to annual rates of exceedance).

The calculation integrates the product of the vulnerability 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:

  • vulnerability_array (2D array) – A 2D array where the first column contains intensity measure levels, and the second column contains the corresponding expected loss ratios 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_loss_ratio – The average annual loss ratio, calculated by integrating the product of the vulnerability function and the hazard curve over the given intensity measure levels.

Return type:

float

Theoretical Background

The Average Annual Loss Ratio (AALR) is the expected loss per year normalised by the replacement cost, obtained by integrating the vulnerability function over the seismic hazard (Cornell & Krawinkler, 2000).

Classical integral

\[\text{AALR} = \int_0^{\infty} E[L \mid \text{IM} = x]\; \left|\frac{d\lambda(x)}{dx}\right| dx\]

where:

  • \(E[L \mid \text{IM} = x]\) is the expected loss ratio at intensity \(x\) (the vulnerability function),

  • \(\lambda(x) = P(\text{IM} > x)\) 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:

\[\text{AALR} \approx \sum_{j} E[L \mid \text{IM} = \bar{x}_j] \cdot \Delta\lambda_j\]

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.

Intensity levels with exceedance rates below \(1/T_{\max}\) (where \(T_{\max}\) is the maximum return period) are excluded to avoid numerical instability from very rare events.