## 2 Materials and Methods
### Evaluation of the Predictive Perfomance of GMPEs--GMPE Ranking  
GMPEs capture the ground motion variability concerning distance, earthquake magnitude, source type, and site properties. However, they are formulated using limited knowledge concerning the complex earthquake source and wave propagation characteristics. As a result, they provide an approximation of the \"true\" ground motion values. For this reason, the appropriate selection of GMPEs and the determination of their predictive performance against real strong motion data is a challenging, albeit necessary, task that tends to become common practice in PSHA studies (e.g., [83; 84]). A comprehensive effort of GMPE ranking for Greece was not performed until very recently, when [PERSON] and [PERSON] [85] used the most up-to-date Greek strong motion dataset and evaluated a number of GMPEs using three different approaches. The most widely used goodness-of-fit measures to evaluate the performance of GMPEs (also employed by [PERSON] and [PERSON] [85]) are the Log-Likelihood (\(LLH\)) method, proposed by [PERSON] et al. [86], and the Euclidean Distance-Based Ranking (EDR) scheme, suggested by [PERSON] and [PERSON] [87]. Every approach has its strengths and inherent deficiencies; hence, the results should be used as an advisory tool for experts, rather than a black-box routine. In the present study, the \(LLH\) method was used to evaluate the predictive performance of the selected GMPEs and assign the proper weights in a logic tree approach, which is described below (Section 2.5). The \(LLH\) method is a metric, based on information theory, that uses the log-likelihood fit to estimate the misfit between two probability functions. In this case, the divergence between the log-normal distribution of the predicted ground motion values from the selected GMPEs, \(g(x)\), and the recorded data, \(f(x)\), is expressed by the average \(LLH\) values derived by Equation (1):\[LLH(g,x)=-\frac{1}{N}{\sum_{i=1}^{N}log_{2}(g(x_{i}))} \tag{1}\]  
where \(N\) is the total number of observations \(x_{i}\). Lower \(LLH\) values indicate that a model is closer to the real values, whereas larger \(LLH\) values correspond to one that is more unlikely to reproduce the observed data. Next, Equation (2), provided by [PERSON] et al. [88], was used in order to assign the appropriate weights for each GMPE based on the computed \(LLH\) values:  
\[w_{j}=\frac{2^{-log_{2}LLH_{j}}}{\sum_{j=1}^{k}2^{-log_{2}LLH_{j}}} \tag{2}\]  
The first prerequisite to evaluate the suitability of GMPEs in our study area is the compilation of a reliable database of records. This data set should incorporate information on earthquake parameters, station characteristics, and record information. For this purpose, strong-motion records and all relevant information were acquired from the Engineering Strong-Motion (ESM) database [89]. In theory, the LLH method can be implemented regardless of the size of the data; however, [PERSON] et al. [90] indicated through synthetic tests that \(\sim\)40 records are required to achieve a rather stable assessment. Given the relatively limited extent of the study area and the scarcity of strong-motion data, we also included records from neighboring regions that have been proven to considerably affect the seismic hazard in the area (e.g., the 1981 eastern Gulf of Corinth earthquake sequence). The final dataset was composed of 158 three-component accelerograms from 68 earthquakes, covering the time period from 1981 to 2022, in which 53 records were available from 14 instruments installed in the Attica prefecture. It should be mentioned that the database was selected to be compatible with the validity range of candidate GMPEs (in terms of earthquake magnitude and distance ranges). The functional form of the adopted GMPEs, along with the results of their ranking, expressed by their corresponding weights, are shown in Table 1. The results indicate that the GMPE of SAKK2016 performed best in capturing the ground motion variability in the study area compared to the other candidate GMPEs.