Evaluation of the predictive Performance of Regional and Global Ground Motion Predictive Models against Greek Recorded Strong Motion Data.

8 The aim of the present work is to evaluate the Ground Motion Prediction Models (GMPMs), applicable in active shallow 9 crustal regions of Greece, for peak response parameters (PGA, PGV) and various ordinates of acceleration spectra (5% 10 damping), to be implemented in Probabilistic Seismic Hazard Analyses (PSHA). The evaluation is data-driven, taking 11 into account the most updated strong-motion dataset for Greece. According to the authors ’ knowledge and literature 12 review, such an effort has not been attempted before for region of interest, as the selection of GMPMs for PSHA was 13 made based on qualitative criteria. The steps which were followed to fulfill such a goal include: a) pre-selection of 14 regional, pan-European and global GMPMs, suitable for implementation in Greece; b) selection of appropriate data-driven 15 scoring methods; c) scoring of pre-selected GMPMs for both peak response intensity measures and spectral acceleration 16 values; d) final selection of GMPM suites and weighting. The final selection of GMPMs describes adequately the 17 epistemic uncertainty of strong-motion in Greece and provides a useful tool for addressing a major component of PSHA 18 studies, that is GMPM selection.


INTRODUCTION
Ground Motion Predictive Models (GMPMs) constitute one of the most important elements of modern seismic hazard assessment analysis.These models are utilized to estimate ground motion shaking intensity, expressed by peak response parameters, such as the peak ground acceleration (PGA) and velocity (PGV), as well as, spectral accelerations (Sa), from a number of independent variables related to earthquake magnitude, site-to-source distance and site effects.One of the main tasks in seismic hazard assessment is the selection of appropriate Ground Motion Predictive Models (GMPMs) to be used.Data-driven evaluation of GMPMs has become possible thanks to the increasing amount of available strongmotion data and can give valuable information regarding the ability of empirical models to predict ground motion in various regions (Drouet et al., 2007;Allen and Wald, 2009;Delavaud et al., 2012a and b;Bastias et al., 2015;Zafarani and Farhadi, 2017;Lanzano et al., 2020).
In the recommendation of the current seismic hazard map of Greece (EAK2000) Probabilistic Seismic Hazard Analysis (PSHA) results from multiple research teams were utilized.The GMPMs which were used for the estimation of PGA and PGV were the ones of Makropoulos (1978), Makropoulos and Burton (1984), Theodulidis and Papazachos (1990), Theodulidis and Papazachos (1992), Theodulidis and Papazachos (1994), Ambraseys et al. (1996), Margaris et al. (2002).The selection of GMPMs from each team relied on the fact that most of them were developed using the limited, by that time, recorded strong motion data in Greece.However, the GMPM selection was not supported by an objective measurement of the performance of the models with respect to recorded data, due to their inadequate number.Nowadays, considering that the strong motion network in Greece has expanded significantly and many strong earthquakes have been recorded, the number of available recorded data to test GMPMs has dramatically increased.Recently, Margaris et al. (2021), published the updated strong motion dataset for Greece.The data included within this dataset, along with the accompanying flat files, are utilized herein to evaluate the performance of multiple GMPMs for peak response (PGA, PGV) and spectral acceleration (Sa(T)) parameters.The current dataset has been utilized for the development of the most recent GMPM for Greece, published by Boore et al. (2021).Since the validation dataset is the same to the one used to develop this GMPM, this model has been excluded from the evaluation procedure and is only considered as a reference one.
The strategy followed in the selection of GMPMs is partially similar to the one implemented by Delavaud et al. (2012b) to construct the ground-motion logic tree for PSHA in Europe, which was the final product of the EU project SHARE (Woessner et al., 2015).The final outcome is presented as combination of two objective models testing procedures, in order to minimize the role of subjectivity in PSHA.In the subsequent paragraphs, the procedure to select and rank the GMPMs for Active Shallow Crustal Regions (ASCRs) is presented.

GROUND MOTION PREDICTIVE MODEL SELECTION FOR SHALLOW EARTHQUAKES
The evaluation of the predictive performance of various GMPMs against strong motion data for shallow events in Greece is undertaken.The proposed models were obtained from the list compiled by Douglas ((https ://www.gmpe.org.uk; the database was accessed in 2021).Although a recent GMPM has been recently published for Greece (Boore et al., 2021), considering of additional GMPMs would contribute to account for the epistemic and aleatory uncertainties inherent in strong motion prediction within the framework of PSHA.The evaluation procedure includes regional GMPMs, developed based on available strong motion data at the time in Greece, as well as, global and pan-European GMPMs, developed based on worldwide or European strong motion data.Since the majority of the regional GMPMs provides estimates of peak response parameters (e.g.PGA), the evaluation is initially limited to two intensity measures of ground motion, namely PGA and PGV.
Table 1 presents the GMPMs the predictive performance of which was investigated, along with some of their basic applicability features.The selection of this suite of models was based upon the criteria suggested by Cotton et al. (2006) and Bommer et al. (2010) regarding the tectonic regionalization.Although, some of the earlier regional models developed for Greece do not meet the aforementioned criteria, we believe that their inclusion in the evaluation procedure is important, since some of them are widely used in seismic hazard analyses, and also to highlight the development of GMPMs for Greece with time.Many of the regional models do not predict spectral accelerations, however, for the selection of the global or pan-European GMPMs, the capability of prediction spectral accelerations was set as a prerequisite.The evaluation of each model is made against strong motion data recorded in Greece by 471 earthquakes occurred between 1973 and 2015, published in Margaris et al. (2021).Regarding the model of Bindi et al. (2014), we chose the RJB distance attenuation model, however, the hypocentral distance metric was implemented as well, with similar results.Similarly, the GMPM of Akkar et al. (2014) was considered using the RJB distance metric, which exhibited the optimal performance against the recorded data.Moreover, the recent GMPM of Kotha et al. (2020), was implemented without using the regionally adjusted coefficients.
Figure 1 presents the distance of peak response and some spectral acceleration intensity measures for the candidate GMPMs, for earthquake magnitude M5.5, rock site conditions (VS30=800 m/s) and normal faulting.The choice of M5.5 is justified by the fact that the mean earthquake magnitude of the dataset presented in Margaris et al. (2021) is about 5.5.The basin depth, required for some GMPMs from the NGA-West2 database, is not available, hence, it was set as 'unknown'.As observed, the number of investigated GMPMs is larger for peak response intensities than spectral accelerations due to consideration of earlier local GMPMs which do not predict adopt spectral values (Theodulidis and Papazachos, 1992;Skarlatoudis et al., 2003;2007;Chousianitis et al., 2018).The rapid distance attenuation denoted by the GMPM of Boore et al. (2021) is a specific feature of ground motion in Greece, which may be attributed to either regional and/or soil-structure interaction effects.
Figure 2 presents the magnitude scaling of peak response and some spectral intensity measures for the candidate GMPMs, 88 for R JB equal to 20 km, rock site conditions and normal faulting.Apparently the latest GMPM for Greece, that is Boore 89 et al. ( 2021), presents the lowest magnitude scaling among the considered GMPMs.A curvature of the intensity measures 90 is observed for M>6), which is obvious for other GMPMs, as well.On the other hand, for peak response parameters, an 91 early GMPM for Greece, that is Theodulidis and Papazachos (1992), presents the highest magnitude scaling and exhibits 92 an almost linear increase of intensity measures with magnitude.This is also apparent for other early GMPMs for Greece, 93 which may have been limited by the amount of data used for their development.With respect to spectral accelerations, 94 the model of Danciu and Tselentis (2007) stands out, predicting higher accelerations for large magnitudes at periods of 95 0.4 and 1.0 seconds and for low magnitudes at 2.5 seconds.For larger distances the observations are similar, with the 96 differentiations posed above being amplified.97 98

EVALUATION DATASET 99
The dataset of recordings, which were used for the evaluation of the GMPMs previously described, is presented in Margaris et al. (2021).The database consists of 471 earthquakes, occurred between 1973 and 2015 and produced 2993 recordings from 333 sites.Figure 3 presents maps showing the distribution of shallow earthquakes and their distribution with respect to magnitude, focal depth and style of faulting.

METHODS OF ASSESSMENT OF PREDICTIVE PERFORMANCE
Assessment of the performance of GMPMs is traditionally based on residual analysis, where residuals are computed as the logarithmic difference between observations and predictions, normalized by the total standard deviation which are assumed to be normally distributed.Since the various scoring methods which have been utilized for selection of GMPMs may exhibit specific limitations (References), we decided to adopt two evaluation methods, which would capture different aspects and help towards minimizing the epistemic uncertainty associated to GMPMs.These are described in the following sections.Skarlatoudis et al. , 2003;2007;Bea21: Boore et al., 2021;Bssa14: Boore et al., 2014;Bindi14: Bindi et al., 2014;Akkar14: Akkar et al., 2014;CY14: Chiou and Youngs, 2014;ASK14: Abrahamson et al., 2014;CH18: Chousianitis et al., 2018;DaTs07: Danciu and Tselentis, 2007;Cauzzi15: Cauzzi et al., 2015;TP92: Theodulidis and Papazachos, 1992

Normalized residuals
The comparison of GMPMs against a set of ground motions can serve several purposes, the main of which is to realize the range to which the GMPM of interest represents the local properties of source, path and site scaling of the ground motion.When modelling both the expected ground motion and the aleatory variability, each GMPM is considered in the form of a probability lognormal distribution, as indicated by equation (1).
log   = (  ,   ,   ) +  ,, (1) In equation ( 1), yij represents the ground motion recorded at location j due to an event i.The term μ (mi, rij, pij) represents the expected ground motion from an earthquake of magnitude mi, recorded at distance rij and the term pij corresponds to other model parameters (e.g.site amplification, faulting type etc.).The total uncertainty, denoted by ZT,ij, is modelled as a normal distribution with a mean zero and a standard deviation equal to σT.Therefore, ZTij is the total normalized residual of the jth recording from the ith earthquake event: When calculating the normalized residuals, the term yij is the recorded ground motion, μ (mi, rij, pij) is the mean estimate of the GMPM and σT is the total standard deviation of the GMPM.From the above, it follows that a GMPM is considered as a good fit to the recorded data if its normalized residuals follow closely a standard normal distribution, with a mean zero and standard deviation equal to 1.0.Differences in the mean of the residuals may indicate a tendency for a GMPM to over-or under predict the records, whereas differences in the standard deviation may suggest an over-or underestimation of ground motion variability.

Log-Likelihood Approach
The analysis of GMPM residuals, which was presented earlier, is critical in understanding the possible biases that may exist when implementing a specific model to a region.However, in PSHA, it is necessary to decide, whether or not, a GMPM is suitable for application and if so, how the various GMPMs are to be weighted within the epistemic uncertainty analysis.The simple likelihood analysis (Scherbaum et al., 2004) may not be sufficient since they may be biased by the sample size and/or subjective perspective regarding the criteria used to assess the fit of model.One possible method to measure quantitatively and subjectively the goodness fit of GMPMs to a data set is by using of information theory and the log-likelihood (LLH) approach, proposed by Scherbaum et al. (2009).
The LLH approach is a data-driven evaluation method, which implements an information theoretic approach for the selection and ranking of GMPMs.The method derived a ranking criterion from the Kullback-Leibler (KL) divergence, which denotes information loss when a model g, defined as a distribution (given by a GMPM), is used to approximate a reference model, f (data-generation process, i.e. nature).According to Scherbaum et al. (2009), the LLH is finally represented by the negative average sample log-likelihood, according to equation (3), and is a measure of the distance between a model and the data-generation process.A small LLH indicates that the candidate model is close to the process which has generated the data while a large LLH corresponds to a model that is less likely to have generated the data.
In equation (3), N is the number of observations.Scherbaum et al. (2009) provided a potential means of deriving weights for each model within a suite of models, according to equation (4).In equation ( 4), K is the number of GMPMs.2020) present a good fit to the recorded data for both PGA and PGV, as well, with the former having a lower mean residual and a slightly larger standard deviation than the latter.Also, the GMPM of Campbell and Bozorgnia (2014) presents a comparable fit to the recorded data with the aforementioned models.The GMPM of Derras et al. (2014) shows a small mean residual, with increased standard deviation.The models of Akkar et al. (2014) and Cauzzi et al. (2015) shows a somewhat less good fit, with respect to NGA-2 GMPMS, with a mean residual value exceeding unity for PGA, however, its match to PGV data is similar to that of Abrahamson et al. (2014).Regarding past regional GMPM models (Theodulidis and Papazachos, 1992;Skarlatoudis et al., 2003;2007;Danciu and Tselentis, 2007;Chousianitis et al., 2018), they present large values of mean residuals and standard deviation for PGA, deviating significantly from the standard normal distribution.On the other hand, for PGV, their fit is improved and the GMPMs of Skarlatoudis et al. (2003) and Chousianitis et al. (2018) present an adequate accuracy, taking into account their simplicity.
It should be mentioned though that most of the GMPMs considered exhibit improved comparison to the recorded data for PGV in comparison to PGA.
It is apparent that the ranking depends on the intensity measure considered.However, the GMPMs of Chiou and Youngs (2014), Kotha et al. (2020) and Derras et al. (2014) are within the top 5 models for each intensity measure.In PSHA, different GMPMs can be used for different intensity measures, however, this would make analysis preparation more complex.Considering of equal importance for PGA and PGV as intensity measures, the corresponding residual values can be combined, so that a unique residual-based ranking can obtained, as shown in Table 4.  2020) is within the top 5.
Considering of equal importance PGA and PGV as intensity measures, the corresponding LLH values can be combined, so that a unique LLH-based ranking and weighting scheme can obtained, as shown in Table 7.

Spectral ordinates
The evaluation procedure shown for PGA and PGV has also been performed for spectral acceleration ordinates at periods of 0.15 sec, 0.4 sec, 1.0 sec and 2.5 sec.From this assessment, the GMPMs of Skarlatoudis et al. (2003;2007), Theodulidis and Papazachos (1992) and Chousianitis et al. (2018), have been excluded since they do not give estimates for spectral accelerations.
Figures 6, 7, 8 and 9 present the distribution of normalized residuals for the spectral acceleration values mentioned.As expected, the GMPM of Boore et al. (2021) exhibits the lowest mean residuals and standard deviations.The GMPMs of Kotha et al (2020), Abrahamson et al. (2014) and Chiou and Youngs (2014) present a consistent behaviour with respect to the evaluation dataset being within the top 5 of models for all the spectral acceleration ordinates considered.Furthermore, the model of Cauzzi et al. (2015) is presented within the top 5 with the exception for the spectral acceleration at period equal to 0.15 sec.The GMPM of Boore et al. (2014) performs very well in low periods (0.15 and 0.4 sec) being very high in the corresponding ranking, however, its efficiency is reduced for larger periods.2014) are ranked at the lowest places for all the spectral acceleration ordinates considered.Figure 10 presents the mean residual Z* with respect to structural period, for every GMPM considered.Table 8 presents the ranking of GMPMS (excluding the model of Boore et al., 2021) based on the mean residuals Z* from all the spectral ordinates.2015) demonstrate a consistent performance with respect to the evaluation dataset, being within the top 5 models for all the periods, with the latter model being more accurate for longer periods.Also, the model of Chiou and Youngs (2014) presents an almost constant LLH value and lies within the top 5 models for all the periods except for the 2.5 sec.

FINAL SELECTION OF GMPMS AND WEIGHTING
The methods of evaluation, which were described in section 2 and implemented in section 3, constitute an objective way of ranking the pre-selected GMPMs against the recorded strong motion data in Greece.As observed, different evaluation approaches lead to different rankings.Therefore, the final selection of a suite of GMPMs for PSHA and the corresponding weighting, will result from a combination of the rankings presented in Tables 4 and 7 for PGA and PGV and in Tables 8  and 9 for spectral accelerations.
The GMPM of Boore et al. (2021) is the most updated model for Greece, regarding shallow earthquakes and has been based on the most recent strong motion data set (Margaris et al., 2021).Therefore, its selection is out of question, as well as, the fact that it will be granted with the largest weighting factor.Thus, the authors deemed that a weighting factor of 0.5 was appropriate to be assigned to the GMPM of Boore et al. (2021), whereas the remaining weight is distributed to the best performing models, as described below.It should be noted that the selection of 0.5 as the weighting factor for Boore et al. (2021) model was a clearly subjective decision.
For peak response parameters (PGA, PGV), looking back at Tables 4 and 7, it is noteworthy that the GMPMs of Chiou and Youngs (2014), Kotha et al. (2020) and Derras et al. (2014) make it to the top 5 according to both evaluation approaches.Thus, their selection is justified in a more confident way.
The weighting factors of the final GMPMs will be made separately for each evaluation method.Equation (4) applies for the LLH approach, whereas for the residual-based approach the weighting factors are computed in a similar way, according to equation ( 6).
The weighting factors for the selected suite of GMPMs and peak response parameters is presented in Table 10.For spectral acceleration ordinates, looking back at Tables 8 and 9, it is important to say that the GMPMs of Chiou and Youngs (2014), Kotha et al. (2020), Cauzzi et al. (2015), Abrahamson et al. (2014) make it to the top 5 according to both evaluation approaches.Thus, their selection is justified in a more confident way.
The weighting factors of the final GMPMs will be separately attained for each evaluation method.Equation ( 4) applies for the LLH approach, whereas for the residual-based approach the weighting factors are computed in a similar way, according to equation ( 6).
The weighting factors for the selected suite of GMPMs and spectral accelerations is presented in Table 11.Thus, the 5 models presented in Table 11, along with their weighting factors, are proposed for seismic hazard assessment.The selected GMPMs, exhibiting the best forecasting ability, are calibrated on different datasets, since the model of Boore et al. ( 2021) is calibrated against regional data, the one of Kotha et al. ( 2020) is calibrated against a European dataset whereas Chiou and Youngs (2014), Abrahamson et al. (2014) and Cauzzi et al. (2015) are global GMPMs.Figure 12 shows the 5 selected GMPMs (red curves) with respect to the unselected models (grey curves).For M4.0 events and short distances the median prediction of selected models captures adequately the variability of all GMPMs.As the distance increases, and propagation path effects have a significant contribution, deviations are presented between the selected models and the worst performing GMPMs.For M5.5 events the variability of all GMPMs is described satisfactorily by the selected GMPMs, except for the long distances where, the least performing GMPMs (Danciu and Tselentis, 2007;Bindi et al., 2014) predict significantly higher spectral accelerations.Red curves correspond to the selected GMPMs whereas the grey curves correspond to the rest of the GMPMs For M7.0 events the discrepancy between the selected models and the worst performing GMPMs is more pronounced at short distances and is smoothed for larger distances.

Investigation of faulting mechanism on final GMPM selection and weights
The approach which was followed for the selection of GMPMs and the derivation of appropriate weights for PSHA analysis, included the evaluation of models against the whole strong-motion dataset for Greece.The same procedure was implemented again separately for normal-slip, reverse-slip and strike-slip faulting mechanisms.Within the evaluation dataset, 35%, 17% and 48% of data come from normal-slip, reverse-slip and strike-slip faulting mechanism events correspondingly.
The results for PGA and PGV are shown in Figure 13.For normal-slip faulting mechanism, the final selection of GMPMs is differentiated with respect to the other faulting mechanisms and the all-types-of-faults case.Instead of the GMPM of Derras et al. (2014) the model of Cauzzi et al. (2015) is promoted, giving better evaluation results.The GMPM of Derras et al. (2014) seems to provide more accurate predictions for reverse-slip faults, followed by its predictions for strike-slip faults.However, it seems that its performance against normal-slip events significantly affects the final weight for the alltypes-of-faults case.The GMPM of Kotha et al. (2020) exhibits an almost stable weight value irrespectively of all types of faults.This stands also for the Chiou and Youngs (2014) GMPM, except for reverse-slip events, where its performance is poorer compared to other faulting mechanisms.(2014).This is due to the fact that the evaluation results for normal-slip events did not justified the selection of five GMPMs.The reduction in the number of selected GMPMS for normal-slip events, leads to the apparent increase of the weights of the rest of the selected models.In general, Figure 14 shows a quite stable value of weights for the selected GMPMs.

CONCLUSIONS
Within the framework of PSHA, particular attention is devoted in the selection of GMPMs.In the work presented herein, the reliability of the prediction accuracy of a pre-selected suite of GMPMs, against observed strong motion data of shallow Greek earthquakes, is evaluated.Special attention is given in the prediction of peak response parameters (PGA, PGV), as well as spectral acceleration ordinates.Although, a recent GMPM (Boore et al., 2021), predicting peak response and spectral acceleration intensity measures, has been proposed for Greece, in order to reduce the epistemic uncertainties in PSHA studies, it is customary to use a set of empirical models rather than a single model.Most of the models included in the pre-selected suite of GMPMs satisfy tectonic regionalization criteria.However, some earlier regional models are included even though they do not fully comply with the criteria posed by Cotton et al. (2006) and Bommer et al. (2010), highlighting the development of GMPMs in Greece with time.
Two evaluation methods are implemented to test the pre-selected GMPMs, namely the normalized residuals Z* and the LLH method.The use of two instead of one evaluation method aims at minimizing the inherent deficiencies of each method.The empirical dataset which was used to assess the predictive performance of GMPMs is the most updated for Greece and has been presented in Margaris et al. (2021).Separate evaluation procedures are conducted for peak ground motion parameters (PGA, PGV) and response spectral accelerations.Therefore, two sets of GMPMs, comprised by four to five best performing models are selected and weighted according to the scoring results of the two applied methods.The recent GMPM of Boore et al. (2021), developed for Greece, is assigned the largest weighting factor and is excluded from the evaluation procedure, as it has been developed based on the evaluation dataset.The rest of the selected models are the ones of Kotha et al. (2020), Chiou and Youngs (2014) and Derras et al. (2014) for peak ground motion parameters and the GMPMs of Kotha et al. (2020), Chiou and Youngs (2014), Abrahamson et al. (2014) and Cauzzi et al. (2015) for response spectral accelerations.Investigation on the effect of the faulting mechanism on the GMPM selection and weighting is also performed.The evaluation results for each faulting mechanism reveal that the selection made for the all-types-of-faults case is consistent to all separate faulting mechanisms evaluations, with some minor exception for normal-slip events.

Figure 3 :
Figure 2: Magnitude scaling of the peak response and spectral acceleration predictions of the candidate GMPMs, assuming rock site conditions, RJB distance equal to 20 km and normal style of fault (Skea03:Skarlatoudis et al. , 2003;   2007; Bea21: Boore et al., 2021; Bssa14: Boore et al., 2014; Bindi14: Bindi et al., 2014; Akkar14: Akkar et al., 2014;   CY14: Chiou and Youngs, 2014; ASK14: Abrahamson et al., 2014; CH18: Chousianitis et al., 2018; DaTs07: Danciu and   Tselentis, 2007; Cauzzi15: Cauzzi et al., 2015; TP92: Theodulidis and Papazachos, 1992; CB14: Campbell and   Bozorgnia, 2014; Kot20: Kotha et al., 2020; Derras14: Derras et al., 2014) Figures 4 and 5 present the PGA and PGV normalized residual distribution for the record set and the GMPMs considered.As expected, in both figures, the model ofBoore et al. (2021) exhibits a distribution of residuals which is very close to a standard normal distribution, a fact that indicates a good fit to the data.The models ofChiou and Youngs (2014),Abrahamson et al. (2014) andKotha et al. (2020) present a good fit to the recorded data for both PGA and PGV, as well, with the former having a lower mean residual and a slightly larger standard deviation than the latter.Also, the GMPM ofCampbell and Bozorgnia (2014) presents a comparable fit to the recorded data with the aforementioned models.The GMPM ofDerras et al. (2014) shows a small mean residual, with increased standard deviation.The models ofAkkar et al. (2014) andCauzzi et al. (2015) shows a somewhat less good fit, with respect to NGA-2 GMPMS, with a mean residual value exceeding unity for PGA, however, its match to PGV data is similar to that ofAbrahamson et al. (2014).Regarding past regional GMPM models(Theodulidis and Papazachos, 1992;Skarlatoudis et al., 2003; 2007;Danciu and Tselentis, 2007;Chousianitis et al., 2018), they present large values of mean residuals and standard deviation for PGA, deviating significantly from the standard normal distribution.On the other hand, for PGV, their fit is improved and the GMPMs ofSkarlatoudis et al. (2003) andChousianitis et al. (2018) present an adequate accuracy, taking into account their simplicity.It should be mentioned though that most of the GMPMs considered exhibit improved comparison to the recorded data for PGV in comparison to PGA.

Figure 4 :Figure 5 :
Figure 4: PGA residual distribution for the record set and the GMPMs considered.The red lines indicate the probability density function fit to the data, whilst the black lines correspond to the standard normal density function

Figure 6 :Figure 7 :Figure 8 :
Figure 6: Residual distribution for the record set and the GMPMs considered for Sa (T=0.15 sec).The red lines indicate the probability density function fit to the data, whilst the black lines correspond to the standard normal density function

Figure 9 :Figure 10 :
Figure 9: Residual distribution for the record set and the GMPMs considered for Sa (T=2.5 sec).The red lines indicate the probability density function fit to the data, whilst the black lines correspond to the standard normal density function

Figure 11
Figure 11 presents the computed LLH indices for every GMPM considered, with respect to period of vibration.The GMPM of Boore et al. (2021) exhibits an almost constant value of LLH for all the periods considered.The model of Derras et al. (2014) presents very low LLH values for every structural period, being ranked in the 1 st or 2 nd place.This is somewhat different to what was observed based on the normalized residuals.Moreover, the GMPMs of Kothat et al. (2020) and Cauzzi et al. (2015) demonstrate a consistent performance with respect to the evaluation dataset, being within the top 5 models for all the periods, with the latter model being more accurate for longer periods.Also, the model ofChiou and Youngs (2014) presents an almost constant LLH value and lies within the top 5 models for all the periods except for the 2.5 sec.

Figure 11 :
Figure 11: LLH with respect to structural period for every GMPM considered Table 9 presents the ranking of GMPMs (excluding Boore et al., 2021) based on the LLH value, reckoning with all the spectral acceleration ordinates.

Figure 12 :
Figure 12: Trellis plot of the median predictions of the pre-selected GMPMs, for rock site conditions and normal faulting.Red curves correspond to the selected GMPMs whereas the grey curves correspond to the rest of the GMPMs

Figure 13 :
Figure 13: Effect of faulting mechanism on final selection of GMPMs and weights for PGA and PGV

Figure 14 :
Figure 14: Effect of faulting mechanism on final selection of GMPMs and weights for spectral accelerations

Table 2 :
Ranking of GMPMs based on PGA mean residuals and corresponding standard deviation

Table 3 :
Ranking of GMPMs based on PGV mean residuals and corresponding standard deviation

Table 4 :
Ranking of selected GMPMs based on combined PGA and PGV residuals .(2014)takefirst and second places for both intensity measures, whereas the model ofKotha et al. ( Cauzzi et al. (2015)the ranking of the selected GMPMs based on LLH value for PGA and PGV respectively.The GMPM ofBoore et al. (2021)is excluded from the ranking as it has been developed based on the data set ofMargaris et  al. (2021)which is used as evaluation data set herein.It is observed that the GMPMs ofCauzzi et al. (2015)and Derras et al

Table 5 :
Ranking of selected GMPMs based on LLH for PGA.

Table 6 :
Ranking of selected GMPMs based on LLH for PGV

Table 7 :
Ranking of selected GMPMs based on combined LLH for PGA and PGV

Table 8 :
Ranking of selected GMPMs based on combined residuals for all spectral acceleration ordinates considered

Table 9 :
Ranking of selected GMPMs based on combined LLH values for all spectral acceleration ordinates considered

Table 10 :
Final weights for PGA and PGV of selected GMPMs based on both evaluation approaches

Table 11 :
Final weights for spectral accelerations of selected GMPMs based on both evaluation approaches