Probabilistic seismic hazard analysis

Definition

Fix a site and a ground motion measure $Y$, such as peak ground acceleration. Let there be finitely many seismic sources $k = 1, \dots, K$. Source $k$ produces earthquakes with magnitude at least $m_0$ at the annual rate $\nu_k$. Their magnitudes have density $f_{M,k}$, often a truncated Gutenberg–Richter law, and their distances to the site have density $f_{R,k}$. A ground motion model gives $P(Y > y \mid m, r)$ for an earthquake of magnitude $m$ at distance $r$. Probabilistic seismic hazard analysis computes the hazard curve $$\lambda(y) = \sum_{k=1}^{K} \nu_k \iint P(Y > y \mid m, r)\, f_{M,k}(m)\, f_{R,k}(r)\, dm\, dr,$$ the annual rate of earthquakes that produce $Y > y$ at the site. If these exceedances form a Poisson process in time, the probability of at least one exceedance in $t$ years is $1 - e^{-\lambda(y) t}$.