Risk Metrics

Introduction

Risk metrics quantify the expected losses and consequences from seismic hazards over time. This section provides interactive tools to compute and visualize key risk indicators for cities across 12 global regions. In performance-based earthquake engineering and regional risk applications, the Average Annual Loss Ratio (AALR) quantifies the expected loss ratio due to earthquake-induced damage over one year:

\[AALR = \int_0^{\infty} LR(IM) \cdot \frac{d\lambda(IM)}{dIM} \, dIM\]

where:

  • \(LR(IM)\) is the loss ratio from the vulnerability function

  • \(\lambda(IM)\) is the annual frequency of exceedance from the hazard curve

In practice, AALR is approximated using discrete intensity levels:

\[AALR \approx \sum_{i=1}^{N} LR(IM_i) \cdot [\lambda(IM_i) - \lambda(IM_{i+1})]\]

This section reports the following risk metrics for the supported vulnerability classes:

  • Structural AALR: Expected annual structural damage as a fraction of replacement cost

  • Non-Structural AALR: Expected annual non-structural damage

  • Contents AALR: Expected annual contents damage

  • Building AALR: Combined structural and non-structural losses

  • Total AALR: Combined structural, non-structural, and contents losses

  • Fatalities AALR: Expected annual fatality rate per building

  • Average Annual Collapse Probability (AACP): Annual probability of structural collapse

All AALR metrics use the efficient intensity measure for each building class — the IMT that minimises average dispersion across damage states, reducing uncertainty in the estimates.

Methodology

Risk Calculation

The Average Annual Loss (AAL) is computed by integrating the product of the vulnerability function and the hazard curve:

\[AAL = \int_0^{\infty} V(IM) \cdot \left| \frac{d\lambda(IM)}{dIM} \right| \, dIM\]

where:

  • \(V(IM)\) is the mean loss ratio from the vulnerability function at intensity \(IM\)

  • \(\lambda(IM)\) is the annual rate of exceedance from the hazard curve

  • The derivative represents the annual rate of occurrence at each intensity level

Numerical Integration

The integral is approximated using the trapezoidal rule:

\[AAL \approx \sum_{i=1}^{n} \frac{V(IM_i) + V(IM_{i-1})}{2} \cdot |\lambda(IM_{i-1}) - \lambda(IM_i)|\]

For collapse probability, the calculation integrates the collapse fragility function with the hazard curve:

\[AACP = \int_0^{\infty} P(C|IM) \cdot \left| \frac{d\lambda(IM)}{dIM} \right| \, dIM\]

Interactive Viewer

Use the interactive viewer below to explore precompiled risk metrics for a specific building class within a selected region. The bar charts show values across cities.