# Preliminary Seismic Hazard Analyses for the Ugandan Region
## 3 Materials and Methods
### Identifying and Characterizing Seismic Sources  
The first steps in PSHA require an identification of all potential seismic sources. This typically involves compiling and earthquake catalog in the region of interest. In this study, earthquakes were taken from the International Seismological Center (ISC) Reviewed Bulletin, the United States Geological Survey (USGS) published by the National Earthquake Information Center (NEIC) and Incorporated Research Institutions in Seismology (IRIS) online databases. These organizations offer online access to earthquake data. During compilation, it was discovered that a majority of the earthquakes that were recorded in the NEIC and IRIS databases were also in the ISC Reviewed Bulletin [17; 18; 19]. The region of interest is defined as 1.9\({}^{\circ}\) S to 4.6\({}^{\circ}\) N and longitudes 28.8\({}^{\circ}\) E to 36.2\({}^{\circ}\) E and limited to crustal earthquakes. The epicenters in the earthquake catalog will help define seismic sources, while the magnitudes will help define an earthquake recurrence law. One of the most widely used earthquake recurrence laws is typically expressed as [20]:  
\[\log\lambda_{m}=a-bm \tag{2}\]  
where \(\lambda_{m}\) = annual rate of exceedance for a minimum magnitude \(m\) or the average number of earthquakes per year with a magnitude greater than \(m\), \(a\), and \(b\) = are regressed constants per source or region of interest. This relationship requires the magnitudes used for regression be of the same magnitude type. This requires several magnitude homogenization techniques. The most common magnitude types include moment magnitude, \(M_{W}\), surface wave magnitude, \(M_{S}\), body wave magnitude, \(m_{b}\), local magnitude, \(M_{L}\)[20; 21; 22]. Additionally, there is a duration magnitude, \(M_{D}\), and a regional or more local magnitude type, \(M_{N}\) or \(m_{BLG}\)[23]. Modern GMPEs typically take \(M_{W}\) as input and it is the magnitude type that does not saturate, therefore magnitude homogenization is directed towards \(M_{W}\). The magnitude homogenization relationships considered in this study include [24]:  
\[M_{W} =0.590M_{S}+0.246 \tag{3}\] \[M_{W} =1.104m_{b}-0.194\] (4) \[M_{W} =0.764M_{D}-1.379 \tag{5}\]However, the proposed magnitude relationship between \(M_{W}\) and \(M_{L}\) is taken in steps, where \(M_{L}\) is first converted to an equivalent \(M_{N}\), and then from \(M_{N}\) to \(M_{L}\)[25]:  
\[M_{N}=0.900M_{L}+0.510 \tag{6}\]  
\[M_{W}=0.670M_{N}+1.730 \tag{7}\]  
Earthquakes may have more than one recorded magnitude type. When this is the case, priority will be given to \(M_{S}\), and then in descending order, \(m_{b}\), \(M_{N}\), \(M_{L}\), and \(M_{D}\). Once a catalog of earthquakes relevant to the Ugandan region is homogenized to one magnitude type, MW, then the catalog needs to be filtered to remove foreshocks and aftershocks. This is because the hazard integral shown in Equation (1) is based on independent events and therefore is more suitable for main shock events. A method that is considered conservative for filtering foreshocks and aftershocks in an earthquake catalog will be used [26]. This relationship essentially considers any earthquake event within a specified space and time window centered on the main shock is a foreshock or aftershock. These windows are magnitude dependent and are presented in Table 2.  
The hazard integral asks for minimum and maximum magnitudes for each seismic source. The minimum magnitude considered for this study is \(M_{W}\) = 4.0, as smaller earthquakes are unlikely to cause significant damage to infrastructure even when they are poorly built [27]. Therefore, \(M_{W,min}\) = 4.0 for all seismic sources in this study. However, the maximum magnitude will be estimated through [28]:  
\[M_{max}=M_{max,observed}+\Delta M \tag{8}\]  
where \(M_{max,observed}\) = maximum observed magnitude and \(\Delta M\) is considered 0.5 in this study.