## 2 Hazard Calculations for Areal Sources  
For traditional ergodic GMMs, the seismic hazard for an areal source is given by Equation (1) [28]. \[Haz(IM>z)=N(M_{min})\int_{M}\int_{R}f_{m}(M)f_{R}(R,M)P(IM>z|M,R,S)dRdM \tag{1}\]  
in which \(IM\) is the ground-motion intensity measure of interest, \(M\) is the magnitude, \(R\) is the distance metric, \(S\) is the site parameter, \(N(M_{min})\) is the rate of earthquakes in the source zone with \(M>M_{min}\), \(f_{m}(M)\) is the magnitude probability density function (pdf), \(f_{R}(R,M)\) is the pdf of the distances for a given magnitude from all possible locations of the earthquake in the source zone, and \(P(IM>z|M,R,S)\) is the conditional probability of the \(IM\) exceeding \(z\) given the scenario. For closest distance metrics, such as the rupture distance, \(R_{RUP}\), or the Joyner-Boore distance, \(R_{JB}\), the \(f_{R}(R,M)\) is magnitude dependent due to the magnitude dependence of the rupture dimension. For an ergodic GMM, the conditional probability of exceeding the ground motion for a given scenario is given by Equation (2) [29]  
\[P(IM>z|M,R,S)=1-\Phi\bigg{(}\frac{ln(z)-\mu_{erg}(M,R,S)}{\sigma_{erg}(M,R)} \bigg{)} \tag{2}\]  
in which \(\mu_{erg}(M,R,S)\) is the median \(IM\) from the GMM in natural log units, \(\sigma_{erg}(M,R,S)\) is the aleatory standard deviation of the GMM and \(\Phi()\) is the cumulative distribution function of a standard normal distribution. Computing the \(P(IM>z|M,R,S)\) term involves evaluating the GMM for the median and standard deviation of the ground motion and computing the \(\Phi\) function using a series approximation. For areal sources, most of the run time in a PSHA calculation is spent computing \(P(IM>z|M,R,S)\). The hazard integral accounts for all possible locations of the earthquakes in the zone. Because multiple locations in the zone are at the same distance from a site, a direct integration over the locations of the earthquakes involves repeated calculations of the \(P(IM>z|M,R,S)\) term for the same M and R. In the form of the hazard integral given in Equation (1), the \(f_{R}(R,M)\) term combines rates of the earthquakes in different parts of the areal source zone that are at the same distance. This leads to much faster calculations of the hazard by avoiding repeated calculations of the hazard for the same M and R combination. With a non-ergodic GMM, the median ground motion will vary by earthquake location, so the approach of simply summing the rates of earthquakes that are at the same distance to the site cannot be used to speed up the hazard calculation. The hazard integral can be written to directly integrate over all possible source locations in the areal source zone, replacing the \(f_{R}(R,M)\) with the pdfs for the location of the source in terms of the longitude and latitude and the depth [29]. \[Haz(IM>z)=N(M_{min})\int_{M}\int_{Lat_{e}}\int_{Lone}\int_{H}f_{m}(M)f_{loc}( Lat_{e},Lone_{e})f_{H}(H) \tag{3}\]  
\[P(IM>z|M,R(M,Lat_{e},Lone_{e},H),S)\ dH\ dLn_{e}\ dLat_{e}\ dM\]  
in which \(H\) is the depth of the earthquake, and \(Lat_{e}\) and \(Lone_{e}\) are the coordinates of the rupture. In this form, the distance to the site, \(R\), is computed given the source lo cation. The distance depends on magnitude to account for the rupture dimension for large-magnitude events. For a non-ergodic GMM, the median GMM can be written in terms of an adjustment, \(\Delta\mu\), to a reference ergodic GMM [29]. The non-ergodic adjustment term depends on both the site and source locations.