Applicability of Shallow Artificial Neural Networks on the Estimation of Frequency Content of Strong Ground Motion in Greece

Abstract

The frequency content of strong ground motion significantly affects the response of engineered systems under seismic excitation. Among some scalar parameters which exist in the literature, the mean period T_m has proved to be the most efficient. Ground Motion Predictive Equations (GMPEs) are usually developed for ground motion parameters through the calibration of coefficients of predefined functional forms, via linear or nonlinear regression, and based on recorded ground motion data. Such expressions of T_m are rare in the literature. Recently, the use of machine learning (ML) algorithms in earthquake engineering and engineering seismology has increased. The Artificial Neural Network (ANN) is an effective ML algorithm which has already been explored for the development of GMPEs for amplitude-based ground motion parameters. Within the work presented herein, multiple nonlinear regression (NLR)- and ANN-based GMPEs are developed for T_m using the latest strong motion database for shallow earthquakes in Greece. To the author’s knowledge, the implementation of ANN for producing GMPEs for T_m for shallow earthquake events has not been explored. Direct comparison between the NLR- and ANN-based GMPEs is performed, in terms of performance indexes, aleatory uncertainty, and working examples, as well as testing against earthquake events not included in the original dataset. The results reveal that the ANN-based GMPEs are useful in reducing aleatory uncertainty, although care should be taken in their implementation to avoid overfitting issues.

1. Introduction

A proper description of earthquake-induced strong ground motion requires the quantification of its amplitude, duration, and frequency content. More specifically, the effect of frequency content of ground motion has proven important for the seismic response of structures and geotechnical systems [1,2,3]. If the frequency content of ground motion, defined as the distribution of the signal’s energy with respect to frequency, is close to the fundamental frequencies of a structural or geotechnical system, then resonance occurs leading to increased seismic response.

The most complete description of the frequency content of ground motion is achieved through the Fourier amplitude spectrum (FAS), as well as the acceleration response spectrum. Nevertheless, in some cases, it is desirable to use scalar parameters which can effectively represent the frequency content of seismic excitation. There are a number of scalar frequency-content parameters of ground motion which have been proposed in the literature. The most common ones have been investigated by [1,4]. These include the mean period, T_m, the average spectral period, T_avg, the smoothed spectral predominant period, T_o, and the predominant period, T_p. Among these frequency-content parameters, T_m utilizes the FAS, whereas T_avg, T_o, and T_p are computed through the 5%-damped acceleration response spectrum of ground motion. T_m is preferred over the rest of the parameters due to its robustness and its ability to characterize the frequency content of acceleration time series directly [4]. Since then, numerous studies have been published examining the effect of T_m on the seismic response of engineered systems [3,5,6,7,8]. For example, T_m was considered as an important predictor variable for the estimation of seismic-induced slope displacements by Rathje et al. (2014) [9], whereas Jibson and Tanyaş (2020) [3] indicated that T_m is a good predictor variable for earthquake-induced landslide size distribution. Furthermore, Sotiriadis et al. (2019; 2020) [7,8] utilized T_m for characterizing the frequency content of ground motion and correlating it with kinematic soil-structure interaction (SSI) effects on buildings with embedded foundations. Moreover, the study of Song et al. (2014) [10] led to the conclusion that T_m affects the collapse capacity of post-mainshock buildings, while Sotiriadis et al. (2017) [6] used T_m to normalize the acceleration response spectra of ground motion and interpret the varying role of SSI on the seismic response of multi-storey buildings with respect to the fixed-base assumption.

In contrast to the amplitude and duration of strong ground motion, the development of Ground Motion Predictive Equations (GMPEs) for frequency-content parameters has received less attention. GMPEs are empirical models which provide estimates of ground motion parameters; they are calibrated through strong motion datasets and are essential for seismic hazard assessment, seismic design, and planning of emergency response. The usual predictor variables included in a GMPE are the earthquake magnitude (M), the seismic source-to-site distance (R), site effects proxies, and fault-type mechanisms. To the author’s knowledge, only a few GMPEs exist for T_m. Rathje et al. (2004) [4] updated the previously published model of Rathje et al. (1998) [1], using worldwide strong motion data to calibrate their model. Du (2017) [11] presented a robust GMPE for T_m using the expanded worldwide NGA-West2 ground motion database, including more site and faulting-type predictors. On a regional scale, Yaghmaei-Sabegh (2015) [12] and Lashgari and Jafarian (2022) [13] developed GMPEs for T_m based on Iranian data, whereas for Greece only Chousianitis et al. (2018) [14] included T_m in their suite of empirical models based on strong motion data in Greece. All of the aforementioned efforts included the calibration of coefficients of predefined functions forms, selected based on data trends or theoretical insights, through linear or nonlinear regression algorithms.

On the other hand, various machine learning (ML) algorithms have been gaining ground in the field of GMPE development over the last decade in an attempt to tackle the limitations of traditional methods, as reported in recently published extensive literature reviews [15,16,17,18]. The Artificial Neural Network (ANN) algorithm was incorporated to develop GMPEs for peak ground acceleration (PGA) and velocity (PGV) and spectral accelerations (Sa) using data recorded from European and California shallow crustal seismic events [19,20,21], estimating the computed errors which were lower than those derived from conventional regression analyses. In addition to this, Khosravikia et al. (2019) [22] proposed ANN-based GMPEs for the Oklahoma, US, area. Various strong motion parameters, including energy characteristics and strong motion duration, were adopted in deriving GMPEs applying five non-parametric machine learning models [23]. Moreover, for energetic characteristics, e.g., Cumulative Absolute Velocity (CAV), Kuran et al. (2023) [24] estimated GMPEs evaluating the machine learning approach and considered a Gradient Boosting Algorithm. Additionally, Deep Neural Networks (DNNs) using V_s profiles were applied to develop linear and nonlinear site amplification GMPEs [25]. For Mexico’s subducting earthquakes registered at rock sites, GMPE models were proposed by Ramos-Cruz and Ruiz-Garcia (2024) [26] utilizing Support Vector Machine (SVM) regression. For ground motion in Greece, Morfidis et al. (2024) [27] implemented the ANN algorithm to obtain GMPEs for PGA and PGV, which performed slightly better than the most updated regression-based model by utilizing a small number of neurons. Regarding the frequency content of ground motion, Sofiane et al. [28] developed a predictive model for T_m using feed-forward ANN and analyzed the effects of seismological parameters on it, with data collected from the Kik-Net database. This is the only existing work found on the implementation of ANNs for the prediction of frequency content of ground motion.

Careful examination of the existing literature on GMPEs derived through ML algorithms reveals that limited attempts have been made in developing such a model for scalar frequency-content parameters of ground motion. The present work aims to fill this gap by investigating the applicability of shallow Artificial Neural Networks on the production of a GMPE for Tm for strong motion in Greece, taking advantage of the most updated database which was recently published. At the same time, nonlinear regression-based models are developed using the same dataset as well in order to highlight the advantages and weaknesses of each algorithm in the prediction of the mean period of ground motion.

2. Strong Motion Data

The strong ground motion dataset adopted herein is the one published by Margaris et al. (2021) [29]. It is the most comprehensive and updated strong motion dataset for shallow earthquakes in Greece so far and includes 471 seismic events recorded between 1973 and 2015, which produced a total of 2993 recordings from 333 different sites. The dataset includes key source parameters, such as hypocenter locations, moment magnitudes (M_w), fault-plane solutions, finite-fault information, and reported values of the average shear wave velocity of the top 30 m, V_S30, for all 333 sites. Within the framework of this paper, it was decided to exclude data from earthquake events with a magnitude lower than M4.5, as this magnitude is the smallest one of usual engineering interest, as well as data from events with a focal depth (H) larger than 40 km in order to retain the characterization of shallow events in our work. Upon implementation of these criteria, a total of 2551 recordings were used—coming from 341 earthquakes recorded at 323 sites.

Figure 1 presents the distribution of the data used herein in terms of earthquake moment magnitude (M_w), Joyner–Boore source-to-site distance (R_JB), V_S30 at the recording stations, and focal depth (H), along with their marginal distributions. The earthquake magnitude spans from M4.5 to M7.0, with the majority of data belonging to events with magnitudes between M5.0 and M6.0. More specifically, 68% of the data correspond to earthquake magnitudes between M5.0 and M6.0, 14% between M4.5 and M5.0, and 18% to data from events with magnitudes between M6.0 and M7.0. The source-to-site distance, R_JB, ranges from 1 km to 300 km, with near-source data being limited. More specifically, 62% of the data correspond to an R_JB larger than 100 km, whereas the rest of the data are equally distributed to R_JB values lower than 50 km (19%) and between 50 and 100 km (19%). Furthermore, V_S30 ranges from 91 m/s to 1183 m/s, with the vast majority of the recording sites exhibiting V_S30 values between 200 and 600 m/s. For a more detailed description of the dataset, also including information on the processing of strong motion recordings, the readers could refer to [29]. It should be noted that no weighting or enhancing method was implemented to alleviate the inherent sparsity of the dataset.

The processed and filtered acceleration time series compiled by [29] were used to calculate the mean period, T_m, of each recording. More specifically, the two horizontal components of each record were used to calculate the rotD50 [30,31] component of the FAS according to the following [1]:

$$T_m={\displaystyle \frac{{\sum }_i{C}_i^2\left(\frac{1}{f_i}\right)}{{\sum }_i{C}_i^2}},with\mathsf{\Delta }f\le 0.05 \mathrm{H}\mathrm{z}$$

(1)

In Equation (1), C_i² represents the sum of the squared real and imaginary parts of the Fourier amplitude ordinates, f_i represents the discrete fast Fourier transform frequencies ranging from 0.25 to 20 Hz, and Δf denotes the frequency interval. Figure 2 demonstrates the FAS of two records of the current dataset, from earthquakes with similar magnitudes and sites with similar V_S30 values but at quite different source-to-site distances. The red part of the FAS highlights the required frequency range over which T_m is computed.

Figure 3 presents the variation in T_m, as calculated for the whole dataset, with respect to RJB, for three earthquake magnitude bins. It is evident that T_m increases in a quadratic manner as R_JB increases, whereas larger earthquake magnitudes lead to larger T_m as well.

The effect of R_JB on Tm is strong; thus, it may hinder the impact of site effects as represented by V_S30. Therefore, to highlight the variation in T_m with V_S30, the recorded data were divided into different earthquake magnitude and distance bins, as shown in Figure 4. Figure 4 shows that for R_JB < 50 km, the correlation between T_m and V_S30 is more evident compared to larger distances. This is supported by the fact that the R² coefficient of simple polynomial equations between T_m and V_S30 is significantly larger for small distances rather than larger ones. Moreover, the data suggest that for M_w > 5.0 and R_JB < 50 km, T_m does not vary significantly with increasing V_S30 when V_S30 exceeds approximately 800 m/s. Furthermore, it is evident that data recorded at distances R_JB > 50 km provide larger T_m values than their counterparts for R_JB < 50 km, highlighting the strong effect of R_JB.

The strong motion data used in this work are provided in (Supplementary Materials) electronic supplement (ES1_Strong_Motion_data.xlsx).

3. Development of Ground Motion Predictive Equations (GMPEs)

3.1. Nonlinear Regression (NLR) for Predefined Functional Forms

The main predictor variables usually included in GMPEs are the earthquake magnitude, a source-to-site distance metric, a parameter denoting the local site conditions of the recording stations, and optionally, a parameter denoting the faulting mechanism type of earthquake rupture. Herein, we adopt the moment magnitude (M_w) as representative of the total energy released during earthquake rupture, the Joyner–Boore distance (R_JB), and V_S30 as a common variable to represent the local site conditions at the recording station. Within this work, GMPEs for the mean period, T_m, were first developed through one-step nonlinear regression to compare with the Artificial Neural Network (ANN) models, which will be presented in the next section. The rotd50 component of 2551 recordings of the strong motion dataset was utilized for the calibration of the regression coefficients. For this, it was necessary to select appropriate functional forms. The first functional form was based on the work of Du (2017) [11] who presented a robust model for T_m, based on the NGA-West2 strong motion database. The functional form of [11] was adjusted to the dataset used herein and the final functional form is shown in Equations (2)–(5).

$$\ln \left(T_m\right)=f_{mag}+f_{dis}+f_{V_{S30}}$$

(2)

$${\mathrm{f}}_{mag}=\left\{\begin{array}{c}{c}_1+c_2·M_w; M_w<5\\ c_1+c_2·M_w+c_3·\left(M_w-5\right); 5\le M_w<7.3\end{array}\right.$$

(3)

$${\mathrm{f}}_{dis}=\left\{\begin{array}{c}{c}_4·R_{JB}; R_{JB}\le 100 \mathrm{km}\\ c_4·R_{JB}+c_5·\left(R_{JB}-100\right); 100 \mathrm{km}<R_{JB}\le 200 \mathrm{km}\\ c_4·R_{JB}+c_5·\left(R_{JB}-100\right)+c_6·\left(R_{JB}-200\right); R_{JB}>200 \mathrm{km}\end{array}\right.$$

(4)

$${\mathrm{f}}_{V_{S30}}=\left\{\begin{array}{c}{c}_7·\mathrm{l}\mathrm{n}\left(V_{S30}\right); V_{S30}\le 800 m/s\\ c_7·\ln \left(800\right); V_{S30}>800 m/s\end{array}\right.$$

(5)

The model which is developed through this functional form is now denoted as Du17, for brevity. With respect to [11], the following adjustments have been made:

• f_mag does not contain the third branch which is present in [11] for earthquake magnitude larger than M7.3, as no event with M > M7.0 exists in the database used herein.
• R_JB is used instead of R_rup, which is the closest distance to the rupture. This change was made as not all data had available R_rup values in the current dataset.
• f_VS30 replaces the term f_VS30,Z1 which was present in [11] and included an additional term for the depth-to-shear-wave velocity equal to 1 km/s. Such information is not available in the dataset used herein; thus, it was omitted.
• The GMPE of [11] also included a depth-to-top-of-rupture term as well as a directivity term, which were not definable in the current dataset. Therefore, they were not included in the selected functional form.

A second functional form was also investigated, which was based on the functional form of Boore et al. (2021) [32]. Boore et al. (2021) [32] presented the most recent GMPE for PGA, PGV, and PSA at a 5% damping ratio, for shallow earthquakes in Greece, using the strong motion dataset of [29], which is used in this work. The functional form adopted is shown in Equation (6).

$$\begin{gathered} \ln \left(T_m\right)=b1\ast U+b2\ast SS+b3\ast NS+b4\ast RS+b5·\left(M_w-M_h\right)+b6·{\left(M_w-M_h\right)}^2+b7·\ln \left({\displaystyle \frac{\sqrt{R_{JB}^2+h^2}}{R_{ref}}}\right) \\ +b8·\left(\sqrt{R_{JB}^2+h^2}-R_{ref}\right)+b9\ast \ln \left({\displaystyle \frac{V_{S30}}{V_{ref}}}\right) \end{gathered}$$

(6)

Factors b1–b6 represent the contribution of the earthquake event, factors b7–b8 represent the contribution of the seismic wave propagation path, and factor b9 is for the contribution of site effects to the ground motion. As suggested by [32], parameter h is a finite-fault factor which was chosen to be equal to the mean hypocenter depth of the dataset, that is, 13.2 km. The dummy variables U, SS, NS, and RS are the fault-type predictor variables and have values of 1.0 for unspecified, strike-slip, normal-slip, and reverse-slip rupture mechanisms, respectively, and 0.0 otherwise. The inclusion of the fault-type mechanism is one key difference of this functional form with that of Du17. The parameters M_h, M_ref, and R_ref were adopted from [32] and take values equal to 6.2, 4.5, and 1.0, respectively. Moreover, the reference shear wave velocity, V_ref, was adopted from [32] as equal to 760 m/s. It should be noted that the path term associated with coefficient b7 initially included a term involving the earthquake magnitude. The p-value test for the regression coefficient associated with that term was much larger than 0.05; thus, it was finally omitted. The model developed through this functional form is now denoted as Bea21, for brevity.

Table 1. Regression coefficients and performance indexes of the nonlinear regression-based GMPE for T_m, based on the functional form Du17.

Coefficient	Best Estimate	p-Value
c1	−3.414	4.44 × 10−18
c2	0.8284	1.38 × 10−27
c3	−0.5580	1.29 × 10−11
c4	0.0062	2.11 × 10−81
c5	−0.0016	1.31 × 10−3
c6	−0.0035	1.05 × 10−9
c7	−0.3474	2.77 × 10−40
R2	0.60	-
RMSE	0.3482	-
AIC	1805.1	-
BIC	1845.8	-

and

Table 2. Regression coefficients and performance indexes of the nonlinear regression-based GMPE for T_m, based on the functional form Bea21.

Coefficient	Best Estimate	p-Value
b1	−2.4664	1.11 × 10−11
b2	−2.4924	8.60 × 10−156
b3	−2.4613	1.23 × 10−152
b4	−0.8443	4.47 × 10−155
b5	0.7044	4.68 × 10−37
b6	−0.1667	1.80 × 10−12
b7	0.1994	8.96 × 10−16
b8	0.0025	7.80 × 10−24
b9	−0.3310	1.89 × 10−37
Mh	6.2	-
Mref	4.5	-
Rref	1.0	-
Vref	760.0	-
R2	0.59	-
RMSE	0.3534	-
AIC	1880.2	-
BIC	1932.5	-

present the best estimates of the nonlinear regression coefficients and their p-values of the functional forms Du17 and Bea21, respectively. Nonlinear regression was performed using MATLAB version 9.12.0 (R2022a), Natick, Massachusetts:, The Mathworks Inc., 2022 [33]. Moreover, Table 1 and Table 2 include the R², the root-mean-squared error (RMSE), the Akaike’s Information Criterion (AIC), and the Bayesian Information Criterion (BIC), which work as performance indexes of the regressed models. Higher R² and lower RMSE, AIC, and BIC values denote a more accurate model. Comparison between Table 1 and Table 2 reveals that model Du17 performs slightly better than Bea21, exhibiting lower AIC, BIC, and RMSE values but practically similar R² coefficients.

3.2. Artificial Neural Networks (ANNs)

The input variables for the development of ANN-based predictive models for T_m included M_w, ln (R_JB), and ln (V_S30). The output or target value was defined as the natural logarithm of the rotD50 mean period, ln (T_m). The inclusion of the fault-type mechanism as an input variable did not provide any improvement in prediction; hence, it was decided to omit it. This observation is in line with [11] who did not consider the fault-type mechanism as an input variable. AΝΝ algorithms are sensitive to unscaled numerical variables which may lead to slow training. Therefore, each input parameter was normalized to the range of [0, 1]. It was decided to develop ANNs with only one hidden layer to avoid overfitting the networks to the training data and to generate models that could be written as easily-handled closed-form equations. As an additional measure to reduce the risk of overfitting during the development of ANN models, the holdout validation method was used in which the dataset was randomly divided into three subsets. In total, 70% of the data was used for training the ANN, 15% was used for validation, and the remaining 15% was used for testing. Alternatively, the k-fold cross-validation method could be used. The training process of ANNs is affected by a number of input parameters called hyperparameters. The hyperparameters are chosen by the user and should be kept constant during training. The process of selecting the optimal hyperparameters is called hyperparameter optimization or tuning. The hyperparameters investigated herein are the training algorithm, the activation or transfer function of the hidden layer, and the number of neurons of the hidden layer. The activation (transfer) functions and training algorithms investigated are shown in

Table 3. Activation (transfer) functions which were investigated during the hyperparameter tuning of ANN training for T_m GMPE.

Activation Function	Function
Symmetric sigmoid transfer function	‘tansig’
Logarithmic sigmoid transfer function	‘logsig’
Radial basis normalized transfer function	‘radbasn’
Soft max transfer function	‘softmax’
Inverse transfer function	‘netinv’
Symmetric saturating linear transfer function	‘satlins’
Linear transfer function	‘purelin’

and

Table 4. Training algorithms which were investigated during the hyperparameter tuning of ANN training for T_m GMPE.

Training Algorithm	Function
Levenberg–Marquardt	‘trainlm’
Bayesian Regularization	‘trainbr’
Scaled Conjugate Gradient	‘trainscg’
BFGS Quasi-Newton	‘trainbfg’

, respectively. The training of ANNs was performed in MATLAB [33]. For each training session, 100 runs were made with the same hyperparameters and the one among them exhibiting the lowest RMSE was kept.

Figure 5 presents the investigation made regarding the optimal activation (or transfer) function of the hidden layer. As shown in the figure, multiple trial ANNs were trained with varying numbers of neurons in the hidden layer. The investigation revealed that the tansig transfer function is optimal, exhibiting the lowest RMSE values for most of the trained ANNs. The radbasn transfer function follows closely; however, it was decided to fix the activation function as tansig. Figure 6 shows the investigation of the optimal training algorithm. The best training algorithm, which led to the minimum RMSE, was Bayesian Regularization (trainbr). However, training with this algorithm was too time-consuming. Thus, the second-best training algorithm was preferred, that is, Levenberg–Marquardt (trainlm) which performed fairly. Training of ANNs stops when the maximum number of epochs is reached (1000 here) or the validation performance has increased more than 6 times since the last time it decreased. These conditions refer to early stopping criteria. Furthermore, both Figure 5 and Figure 6 show that increasing the layer size is meaningful for a reduction in RMSE for up to 10–15 neurons. Therefore, it was decided to proceed to the training of ANNs with fixed hyperparameters and a layer size ranging between 1 and 15 neurons. The structure of the investigated ANNs is shown in Figure 7.

Table 5. Performance indexes of trained ANNs with varying hidden layer size.

Number of Neurons	R2	RMSE	AIC	BIC
1	0.57	0.3623	2005.7	2040.6
2	0.60	0.3461	1790.1	1854.0
3	0.62	0.3406	1720.8	1813.8
4	0.63	0.3366	1671.9	1793.9
5	0.64	0.3325	1622.0	1773.1
10	0.65	0.3278	1601.7	1898.1
15	0.65	0.3248	1607.3	2049.0

presents the performance indexes of the trained ANNs with varying numbers of neurons of the hidden layer. The R² and RMSE improve with the increase in hidden layer size, although the rate of improvement decreases as the number of neurons increases. A similar trend is observed for AIC and BIC, up to the 5-neuron ANN. Up to this ANN, increasing the number of neurons leads to a relative decrease in AIC and BIC. However, for the 10-neuron ANN, while the AIC decreases with respect to the 5-neuron ANN, the BIC is increased. Moreover, for the 15-neuron ANN, both AIC and BIC are larger than the corresponding values of the 10-neuron ANN. Since both AIC and BIC are likelihood-based measures of model fit which include a penalty for model complexity, it may be deduced that ANNs with more than five neurons are not that effective in reducing the prediction errors, while maintaining a rational number of parameters. The comparison between Table 1, Table 2, and Table 5 provides a first comparison between the nonlinear regression-based (NLR) GMPEs and the ANN-based ones. In terms of all of the performance indexes, both NLR models are quite close to the 2-neuron ANN, with the latter exhibiting slightly better metrics.

3.3. Mixed-Effects Residual Analysis and Uncertainty Modeling

The scope of the mixed-effects residual analysis is to produce an aleatory variability model, which describes the between-event and within-event variability of an empirical model, through the residuals (Res) between the observed ground motion (lnT_mobs) and the mean estimates of the predicted values (μlnT_mpr), according to Equation (7). The indexes i and j in Equation (7) indicate a seismic event and a recording station, respectively. Mixed-effects analysis, as suggested by Abrahamson and Youngs (1992) [34], is implemented so that the residuals are distinguished in between- and within-event residuals, according to Equation (8). In Equation (8), B is the overall bias of the residuals, while η_i and ε_ij are the between-event and within-event residuals. The latter (η_i and ε_ij) are normally distributed with zero mean value and standard deviation equal to τ and φ. Then, the total standard deviation (σ) is calculated, according to Equation (9).

$${Res}_{ij}=\ln T_{mobs}-\mu \ln T_{mpr}$$

(7)

$${Res}_{ij}=B+{\eta }_i+{\epsilon }_{ij}$$

(8)

$$\sigma =\sqrt{{\tau }^2+{\phi }^2}$$

(9)

Figure 8 presents the results of the mixed-effects residual analysis, that is, the between-event (η) and within-event (ε) residuals for both the ANN-NLR models, for T_m. The between-event residuals are plotted against the earthquake magnitude, whereas within-event residuals are plotted against R_JB and V_S30. All of the developed models exhibit acceptably low within- and between-event residuals with almost no bias or trend with respect to the predictor variables, especially in ranges with sufficient data.

Based on the mixed-effects residual analysis, the aleatory variability or uncertainty models of the GMPEs developed herein are proposed in Figure 9. More specifically, Figure 9 presents the variation in the standard deviation of between- and within-event residuals, namely τ and φ, as well as the total standard deviation for each of the produced models. It is noted that between-event variability is uniform along the different earthquake magnitudes, whereas the within-event variability is magnitude-dependent. The values of τ and φ are also given in tabulated form in

Table 6. Parameters of aleatory variability of the proposed GMPEs and comparison with existing models.

Model	τ	φ	σ
NLR-Bea21	0.082	M4.5: 0.233 ≥M6.5: 0.369	M4.5: 0.247 ≥M6.5: 0.378
NLR-Du17	0.089	M4.5: 0.229 ≥M6.5: 0.360	M4.5: 0.246 ≥M6.5: 0.371
ANN—2 neurons	0.088	M4.5: 0.210 ≥M6.5: 0.379	M4.5: 0.228 ≥M6.5: 0.389
ANN—3 neurons	0.086	M4.5: 0.235 ≥M6.5: 0.346	M4.5: 0.250 ≥M6.5: 0.357
ANN—4 neurons	0.075	M4.5: 0.238 ≥M6.5: 0.346	M4.5: 0.249 ≥M6.5: 0.354
ANN—5 neurons	0.073	M4.5: 0.237 ≥M6.5: 0.341	M4.5: 0.248 ≥M6.5: 0.349
ANN—10 neurons	0.069	M4.5: 0.232 ≥M6.5: 0.331	M4.5: 0.242 ≥M6.5: 0.338
ANN—15 neurons	0.07	M4.5: 0.238 ≥M6.5: 0.321	M4.5: 0.248 ≥M6.5: 0.329
Rathje et al. (2004) [4]	0.17	0.31–0.42	0.354–0.453
Du (2017) [11]	0.217	≤M5.0: 0.312 ≥M6.0: 0.403	M5.0: 0.380 ≥M6.0: 0.458
Chousianitis et al. (2018) [14]	-	-	0.371

. Moreover, Table 6 includes the aleatory uncertainty parameters of three existing GMPEs for T_m, namely Rathje et al. (2004) [4], Du (2017) [11], and Chousianitis et al. (2018) [14]. The between-event standard deviation, τ, of the NLR models is similar to those of the 2- and 3-neuron ANN models. When more neurons are included in the ANN-based GMPEs, τ decreases noticeably.

More specifically, the reduction in τ of the 4–15-neuron ANN-based models ranges between 3 and 22%, with respect to the NLR models. Additionally, the values of τ reported for the models developed herein is significantly improved compared to other existing models, as suggested in Table 6. Furthermore, the within-event variability for small earthquake magnitudes does not vary significantly among the produced models, though all of them present lower values than the existing GMPEs. With respect to large earthquake magnitudes, the within-event variability is similar for the NLR-based models and the 2- and 3-neuron ANN-based GMPEs, while it seems to constantly decrease for increasing numbers of neurons. The decrease in φ for large-magnitude events varies between 3.9 and 11%. Again, φ for large magnitudes for the proposed models is improved with respect to the existing ones. Similar observations are made when comparing the total standard deviation, as well.

4. Comparison Between NLR- ANN-Based and Existing GMPEs with Recorded Data

The performance indexes and the results of the mixed-effects residual analysis adequately depict the predictive capabilities of the developed GMPEs within the perspective of statistical manipulation of residuals between recorded data and predictions. However, it is also essential to see how these models work in practice, what their prediction trends are, and how they compare to data and to predictions of existing GMPEs.

Two existing GMPEs are used for comparison, namely those of Du (2017) [11] and Chousianitis et al. (2018) [14]. The proper implementation of these GMPEs to compare with the proposed ones requires some clarifications and assumptions. The model in [11] uses the Euclidean Norm (or Squared Root of the Sum of Squares—SRSS) of the horizontal components, while that in [14] uses the geometric mean (Geomean) of the horizontal components. It is emphasized that the rotD50 component was utilized for the proposed GMPEs. Additional investigation of the current dataset, which included the computation of the SRSS and the geometrical mean component of T_m, revealed that no significant difference occurs between rotD50 T_m and SRSS T_m or Geomean T_m. Figures which prove this are provided in Appendix A. Furthermore, the model in [11] uses the closest-to-rupture distance (R_rup) and that in [14] uses the epicentral distance (R_epi) as a source-to-site distance metric. In the current study, the Joyner–Boore distance was used (R_JB). For the comparison to be possible, scatter plots between R_JB and R_rup, as well as R_JB and R_epi, were made and linear regression expressions were created, with high R² values. The corresponding graphs and linear regression expressions are included in Appendix A.

4.1. Distance Scaling of Ground Motion

In this section, the developed GMPEs are evaluated in terms of distance scaling, that is, how they predict the variation in T_m with source-to-site distance. The evaluation is performed separately for the NLR- and ANN-based GMPEs so that the relevant observations are clear. Figure 10 presents the distance scaling of T_m for two magnitude ranges, M5–M6 and M6–M7, as posed by the recorded data and predicted by the NLR-based proposed GMPEs and the existing GMPEs mentioned above. The mean V_S30 value denoted by the dataset for each magnitude range was considered, whereas the faulting mechanism, wherever it was necessary as input, was set to normal, which is the most common faulting type in the current dataset. The trends denoted by the data are adequately reproduced by the proposed GMPEs. The Bea21 functional form provides higher estimates than Du17, for M < 6.0. For distances larger than 200 km, the Du17 functional form exhibits a break point which leads to more accurate capture of T_m trends and lowest values. For M > 6.0 and distances between 100 and 230 km, Du17 provides higher estimates than Bea21. Regarding the existing GMPEs, the predictions of Chousianitis et al. (2018) [14] for the smaller magnitude range are consistent with the proposed ones, though relatively lower. On the contrary, the GMPE of Du (2017) [11] provides larger estimates for the small-distance range and similar for the mid- and large-distance range, compared to the proposed GMPEs. At the larger magnitude range, the GMPE of Chousianitis et al. (2018) [14] exhibits similar predictions to the proposed GMPEs; however, at distances larger than 200 km it provides significantly overestimated values for T_m. On the other hand, the GMPE of Du (2017) [11] shows similar values to the proposed GMPEs’ predictions for small distances (<50 km) and lower estimates for the rest of the distance range considered. Additionally, the estimates of two existing T_m models for ground motion in Iran (Yaghmaei-Sabegh, 2015 [12]; Lashgari and Jafarian, 2022 [13]) are depicted. Differences are spotted between the proposed GMPEs for Greece and those for Iran, especially for large earthquake magnitudes, which highlight the regional frequency-content characteristics of ground motion in Greece.

Figure 11 presents the same information as Figure 10, with the NLR-based models and the existing GMPEs replaced by the developed ANN-based GMPEs for T_m.

Figure 11a,b include the comparison between recorded data and the ANN-based GMPEs for earthquake magnitudes between M5 and M6. Figure 11b is actually a zoomed version of Figure 11a, which depicts only data for R_JB up to 100 km. The mean trends shown by all of the ANN-based models are satisfactory for distances larger than 6 km, capturing the nonlinear relationship between T_m and R_JB. The differences between the various ANNs seem to increase as R_JB increases. For distances smaller than or equal to 6 km (as shown more clearly in Figure 11b), the 2-, 3-, and 4-neuron ANNs provide stable estimates. On the contrary, unrealistic trends are observed for the rest of the ANN-based models. Both the 5-neuron and 10-neuron ANNs present increased T_m for smaller distances, whereas the 15-neuron ANN presents a sudden drop. Apparently, the lack of data at these small distances affects ANN training unfavorably. Hence, this is (possibly) an overfitting problem. It could be argued that the applicability of these models for 5 < M < 6 is constrained for the lowest R_JB = 6 km.

Figure 11c,d include the comparison between recorded data and the ANN-based GMPEs for earthquake magnitudes between M6 and M7, with the latter being a zoomed version of the former, depicting only data for R_JB up to 100 km. For distances larger than 60 km, all of the ANN-based models provide reasonable trends of increasing T_m with distance and similar estimates. For R_JB larger than 200 km, the 15-neuron ANN deviates profoundly from the rest of the ANNs—predicting lower T_m than the others. The 2-, 3-, and 4-neuron ANNs provide stable estimates of T_m for the whole distance range considered. On the other hand, the 5-neuron ANN presents a smooth response for a minimum R_JB equal to 20 km. For smaller distances, it predicts larger T_m for decreasing distance which is not theoretically coherent. For distances lower than 60 km, unrealistic trends are observed for the 10- and 15-neuron ANNs. Both of them present increased T_m for smaller distances, whereas at the distance of 6 km the 15-neuron ANN presents a sudden drop and the 10-neuron ANN presents a sudden increase. This abnormal response could be attributed to the lack of data at small distances which leads to an overfitting problem for these two ANN models.

4.2. Magnitude Scaling of Ground Motion

In this section, the developed GMPEs are evaluated in terms of earthquake magnitude scaling, that is, how they predict the variation in T_m with earthquake magnitude. In contrast to the previous section, the following figures include only the predictions of the proposed and existing GMPEs for a clearer comparison. Figure 12 presents the variation in T_m with earthquake magnitude for three values of R_JB, as predicted by the two proposed NLR-based GMPEs (Bea21, Du17) and two existing GMPEs (Du, 2017 [11]; Chousianitis et al., 2018 [14]). For R_JB = 10 km, the mean estimates of the proposed GMPEs are close. The highest estimates along the small- and high-magnitude range (M4.5–M5.5 and M6.0–M7.0) come from the Du17 functional form, which provides an almost linear variation in T_m with magnitude. Bea21 provides a nonlinear relationship between T_m and M due to the hinge magnitude, M_h, and the squared term in the functional form which includes M_h. The hinge magnitude, M_h, for Bea21 is at M6.2, whereas Du17 suggests one at M5 and one at M7.3. The latter does not apply here due to max M = 7.0 in the current dataset. Similar observations are made for R_JB = 50 and 150 km, with the differences between the functional forms increasing for longer distances. Regarding the existing GMPEs, Du (2017) [11] exhibits significantly overestimated mean periods for R_JB = 10 km, especially at small earthquake magnitudes. Its differences with respect to the proposed NLR-based GMPEs decrease as RJB increases and eventually it provides lower T_m estimates for long source-to-site distance and earthquake magnitude greater than M6.0. The GMPE proposed by Chousianitis et al. (2018) [14] presents similar trends to Du17, but with increased values of T_m for the whole magnitude range considered and up to R_JB = 50 km. For longer, it provides lower estimates than Du17.

Figure 13a presents the magnitude scaling of T_m, as predicted by the 2-, 3-, and 4-neuron ANN-based GMPEs. The mean trends of the models are quite theoretically consistent, as well as in accordance to the predictions of the NLR-based GMPEs, shown in Figure 11. The 2- and 3-neuron ANNs present smooth magnitude scaling curves, which capture the nonlinear relationship between T_m and M_w. Nevertheless, the 4-neuron ANN depicts a jagged-shaped curve, which also captures the associated nonlinearity. The magnitude scaling of the 4-neuron ANN is theoretically consistent for R_JB = 10 and 50 km; however, for R_JB = 150 km, above M6.0 it presents a descending branch which is counter-intuitive. This abnormal trend is possibly due to the lack of data for large-magnitude events, which leads to signs of overfitting. The estimates of T_m with respect to the NLR-based GMPEs are quite consistent. Figure 13b presents the magnitude scaling of T_m, as predicted by the 5-, 10-, and 15-neuron ANN-based GMPEs. The curves obtained through the implementation of these models do not demonstrate specific trends and they are clearly affected by the attempt of the ANNs to follow closely the data. Hence, overfitting is an issue in these models especially above M5.8-M6.1, depending on the value of R_JB.

4.3. V_S30 Scaling of Ground Motion

In this section, the developed GMPEs are evaluated in terms of V_S30 scaling, that is, how they predict the variation in T_m with varying V_S30. Figure 14a presents the V_S30 scaling of the proposed NLR-based GMPEs, along with the existing GMPE of Du (2017) [11] for earthquake magnitude M5.5. The existing GMPE of [14] has been excluded from this comparison as it includes site effects through dummy variables and not through a continuous V_S30 function. The proposed GMPEs present similar V_S30 scaling for the two distances considered. They predict decreasing T_m with increasing V_S30, which is consistent to the recorded observations (Figure 4). For V_S30 equal to or larger than 800 m/s, the decrease in T_m is negligible.

The GMPE of Du (2017) [11] presents similar trends; however, it exhibits higher amplification of T_m at low distances than the proposed GMPEs, whereas the opposite stands for long distances. Figure 14b is similar to Figure 13a with the 2-, 3-, and 4-neuron ANNs’ estimates plotted instead of the NLR-based GMPEs. The 2-neuron ANN presents similar trends to the NLR-based GMPEs. The 3- and 4-neuron ANNs exhibit different V_S30 scaling than NLR-based GMPEs and the 2-neuron ANN, which is, nevertheless, theoretically consistent. Up to V_S30 = 350–400 m/s, the decrease in T_m due to increasing V_S30 is mild; then an abrupt decrease in T_m occurs up to V_S30 = 450–500 m/s; and then an almost constant value of T_m follows up to the maximum V_S30 considered. The observations made for Figure 13a and Figure 13b also apply to Figure 14c and Figure 14d, respectively, which refer to earthquake magnitude M6.5.

Figure 15 presents the V_S30 scaling of ANN-based models with 5, 10, and 15 neurons in the hidden layer, for earthquake magnitudes M5.5 (Figure 15a) and M6.5 (Figure 15b) and two values of R_JB. The 5-neuron ANN presents a consistent V_S30 scaling and is similar to the 3- and 4-neuron ANNs, both in values and trends, for both earthquake magnitudes. On the other hand, the 10- and 15-neuron models exhibit theoretically non-consistent trends, which is possibly due to overfitting of the ANNs to the dataset.

5. Testing of Developed GMPEs Through Residual Analysis on Unseen Data

This section aims to further test the developed NLR- and ANN-based GMPEs for T_m against unseen data, that is, ground motion data which have not been used to calibrate the regression coefficients of the former nor to train the latter. For this, strong motion data from three earthquakes which occurred in Greece afterward are utilized. The first earthquake considered is that of 30 October 2020, the M7.0 Samos Island (Aegean Sea) earthquake, which affected both Greece and Turkey. The strong motion data (77 records) related to this event were retrieved from Askan et al. (2022) [35], where one can refer to obtain more information. The other two earthquakes come from the Thessaly earthquake sequence which occurred in March of 2021 and included two main shocks, namely one M6.3 on March 3 and one M6.0 on March 4. The strong motion data related to these events (42 records each) were retrieved from the technical report of Margaris et al. (2022) [36]. The horizontal components of the ground motion records from these three earthquake events were used to compute the rotD50 Fourier spectrum and then calculate T_m according to Equation (1).

The evaluation procedure followed includes the statistical manipulation of the residuals between the recorded data of the additional earthquake events and the predictions of the developed GMPEs. It is especially interesting to see how the models characterized by overfitting in the previous section, namely the 10- and 15-neuron ANNs, respond to unseen data. Additionally, the unseen data come from earthquake events with magnitudes that lie at the edge or near the edge of the developed GMPEs’ applicability range.

Figure 16 presents the normalized residuals between the observed T_m values and the developed GMPEs’ predictions for the 2020 M7.0 Samos Island earthquake, with respect to R_JB. The normalized residuals are defined as the residuals computed through Equation (7), normalized to the standard deviation of each GMPE. Bea21, Du17, and the 2- and 3-neuron ANNs present a minor bias of normalized residuals with distance, whereas the absolute value of the mean trend is close to zero. On the other hand, the 4-, 5-, and 10-neuron ANNs exhibit increased residuals for source-to-site distances smaller than 100 km. The normalized residuals of the 15-neuron ANN do not depict a significant trend with R_JB; however, a large overall offset from zero is observed.

Figure 17 presents the normalized residuals between the observed T_m values and the developed GMPEs’ predictions for the 2021 M6.3 Thessaly earthquake, with respect to R_JB. Bea21, Du17, and the 2-, 3-, 4-, and 5-neuron ANNs present some bias of normalized residuals with distance, whereas the absolute value of the mean trend is reasonably close to zero. The observed trend is due to the increased residuals for some close-source recorded ground motion. The 10-neuron ANN exhibits an improved response compared to the former GMPEs, with a minor trend and low absolute values of normalized residuals with respect to zero. Finally, the 15-neuron ANN presents no trend of normalized residuals with respect to R_JB and a relatively low mean offset from zero, providing good predictions of T_m.

Figure 18 presents the normalized residuals between the observed T_m values and the developed GMPEs’ predictions for the 2021 M6.0 Thessaly earthquake, with respect to R_JB. Bea21, Du17, and the 2-, 3-, 4-, and 5-neuron ANN present some bias of normalized residuals with distance, whereas the absolute value of the mean trend is reasonably close to zero. Nevertheless, the 4- and 5-neuron ANNs seem to respond better than the 2- and 3- neuron ANNs for small distances. The 10-neuron ANNs exhibit a significant trend with respect to R_JB and quite an offset from zero at small distances. Finally, the 15-neuron ANN presents a minor trend of normalized residuals with respect to R_JB and an adequate response in T_m predictions.

To evaluate the proposed GMPEs for the whole testing dataset used in this section, the Multivariate Loglikelihood (MLLH) approach, proposed by Mak et al. (2017) [37], is implemented herein. The MLLH score accounts for the hierarchical nature of modern GMPEs, as well as the ground motion correlation. The lower the MLLH score, the closer the model’s predictions to the observations. The presentation of the equations of the MLLH approach falls beyond the scope of this work. Therefore, the reader may refer to [37] for further details. Figure 19 presents the MLLH scores for the proposed GMPEs. It is observed that the NLR-based and 2–5-neuron ANN models exhibit similar values. On the other hand, the 10- and, more intensely, the 15-neuron ANNs depict noticeably higher MLLH values, proving that their predictive capability is inferior to the rest of the GMPEs on the unseen data.

6. Discussion

The current work focused on developing new GMPEs for the mean period, T_m, of strong motion in Greece. T_m has been proven to be the most effective scalar parameter for characterizing the frequency content of strong ground motion. The significance in developing updated GMPEs for T_m is highlighted by its use in multiple applications, such as the prediction of seismically-induced slope displacements, evaluation of kinematic soil-structure interaction effects, interpretation of seismic damage on structures, etc. The mean period may not be considered as a major ground motion parameter in seismic hazard assessment. However, it could act as an additional ground motion characteristic to the common amplitude-based parameters (e.g., PGA, S_a). A vector of ground motion parameters which includes T_m and any amplitude-based intensity measure should be suitable in some of the applications noted above. Anyway, the advantages of a vector-valued seismic hazard assessment have been highlighted by several studies (e.g., [38,39]). T_m could also be used as a criterion for the selection of ground motion records in nonlinear response history analyses of structures.

The new GMPEs were developed by calibrating the regression coefficients of predefined functional forms through nonlinear regression (NLR), as well as, through training of shallow Artificial Neural Networks (ANNs) utilizing the most updated strong motion database for Greece. The newly developed models are also provided in electronic supplement ES2. To the author’s knowledge, this is the first attempt to use ANNs for the prediction of T_m. A thorough investigation for the selection of ANN hyperparameters was performed and it was decided to train the ANN with up to 15 neurons for the hidden layer. The dataset was randomly split into three parts. In total, 70% of the data was used for ANN training, 15% was used for validation, and the remaining 15% was used for testing in order to avoid overfitting the ANN-based models.

The performance indexes of both NLR- and ANN-based GMPEs were adequate. A noticeable difference between the performance indexes of the NLR- and ANN-based GMPEs was observed when using more than two neurons in the hidden layer. Moreover, the mixed-effects residual analysis revealed low between-event standard deviation and reasonable values of within-event standard deviation for all of the developed models. All of the components of standard deviation, namely between- and within-event and total, of the proposed GMPEs were lower than those of the existing GMPEs for T_m.

Section 4 presented how the proposed GMPEs work in terms of distance, magnitude, and V_S30 scaling, through comparison with the recorded data as well as with existing GMPEs. All of the proposed GMPEs satisfactorily followed the trends of recorded data and were consistent to existing GMPEs, as well. The NLR-based and the 2-, 3-, and 4-neuron ANN-based GMPEs produced stable predictions, capturing the associated nonlinearities between the predictor variables and T_m. On the other hand, the 5-, 10-, and 15-neuron ANN-based GMPEs presented some abnormal predictions, which, in some cases, did not agree with basic theory. These abnormalities were possibly due to overfitting of these models and occurred in cases where the lack of data is evident (e.g., large earthquake magnitudes recorded at small distances). An attempt was made to check if these abnormal predictions are indeed due to overfitting or due to physical heterogeneity, which may be captured by including additional path and rupture parameters. Therefore, it was decided to explore alternative versions of the ANN-based GMPEs, which included the closest-to-rupture distance, R_rup; the horizontal distance from the top edge of the rupture, measured perpendicular to the fault strike, R_x; and the horizontal distance off the end of the rupture measured parallel to strike, R_y0. The performance indexes of these alternative ANN-based GMPEs were similar to the original ones and the abnormal predictions did not improve. Therefore, it was concluded that, indeed, the abnormal predictions of the original 5-, 10-, and 15-neuron ANNs were not due to poor selection of predictor variables. Hence, overfitting was not totally avoided, although holdout validation was implemented during the training of ANNs. By taking into consideration only the performance indexes and the residual analysis results of the developed models, one could conclude that ANNs with increasing numbers of neurons provide lower error metrics and, hence, better predictive capabilities. However, this section showed that the ANN-based models for T_m should be carefully evaluated in the way they work and how they compare to physical-based models. Among the proposed GMPEs developed herein, the NLR-based models, as well as the 2–4-neuron ANN-based models, may be used along the complete range of seismotectonic and site features (M, R_JB, V_S30) which were considered within the dataset. Moreover, the 5-neuron ANN is recommended to be used for R_JB values larger than or equal to 20 km. On the other hand, it is recommended that 10- and 15-neuron ANN-based GMPEs should be avoided, as their predictions were unstable, especially at large earthquake magnitudes.

Although an overfitted model provides reduced prediction errors within the training dataset, it fails to generalize and to give accurate estimates for unseen data. Therefore, it was decided in Section 5 to compare the predictions of all of the proposed GMPEs with recorded data from recent earthquakes in Greece, which were not included in the training-calibration dataset. The comparison revealed adequate predictive capability of the NLR- and 2-, 3-, and 4-neuron ANN-based GMPEs. Additionally, even the 5-neuron ANN-based model provided estimates which did not deviate significantly from the GMPEs mentioned above. On the other hand, the residuals of the 10- and 15-neuron ANN-based GMPEs exhibited either significant trends with source-to-site distance, or significant offsets from zero, especially for the M7.0 Samos Island earthquake. Nevertheless, the response of the 10- and 15-neuron models against the recorded data of the M6.0 and M6.3 earthquakes of Thessaly was adequate. The difference in evaluation of the latter ANN-based models against the M7.0 and the M6.3 and M6.0 earthquake data may be attributed to the fact that the training dataset contained more data in the M6.0–M6.5 magnitude range than in the M6.5–M7.0. It should be noted that the holdout validation method was utilized herein to tackle any overfitting issues for the ANN-based models. However, the alternative method of k-fold cross-validation method may provide improved generalization for them and is worth investigating in future works.

The implementation of ANNs in developing GMPEs for T_m is useful as it brings enhanced predictive capabilities, reduces prediction uncertainty, and captures nonlinearities between the predictor and the response variables, provided that the training dataset is complete. Otherwise, care should be taken when implementing complex ANN-based models, especially when considering cases which lie on the edge of models’ applicability. The more traditional regression-based GMPEs are still important, as their functional form represents relationships inspired by fundamental theory physics-based processes, as well as earthquake data observations. A combination of these model development techniques may be optimal, especially in cases or data regions where the amount of data is limited. Future research may attempt to include directivity effects, as well as a residual analysis of station-to-station variability. The reasons for not addressing these issues herein is the lack of directivity effects proxies in the dataset and that the majority of the stations of the dataset had fewer than five recordings, so the robustness of station-to-station variability may be questionable, as also discussed in [32].

7. Conclusions

In conclusion, the present work’s contributions may be summarized in the following:

• New, updated GMPEs for estimating T_m were developed for strong ground motion in Greece. The significance of this contribution is proved by the multiple earthquake engineering applications of T_m, where the effect of frequency content is important for the seismic response of engineered systems. (geotechnical or structural).
• Training of the Artificial Neural Network (ANN), which is a well-known machine learning algorithm, was implemented for the first time to produce a Ground Motion Predictive Equation for T_m.
• A direct comparison between nonlinear regression- and ANN-based models is performed, not only in terms of their error statistics but also in the way they capture the relationship between the predictor variables, namely earthquake magnitude, source-to-site distance and site effects, and the response variable (T_m).
• The effect of overfitting, which was detected in some ANN-based models during the investigation of how they work, was examined by implementing these models on unseen ground motion data. It was found that the ANN-based models which exhibited overfitting issues responded poorly against the M7.0 earthquake in the unseen data. Earthquakes of that magnitude were not represented adequately within the training dataset. On the contrary, the same ANN-based models exhibited a similar response to the rest of the NLR- or ANN-based GMPEs against the M6.3 and M6.0 earthquakes in the unseen data. Earthquakes of that magnitude were better represented in the training dataset than the M7.0 earthquake.
• Hence, the applicability of shallow ANNs in the estimation of T_m is possible and comes with many advantages, regarding the capturing of the link between the involved parameters and the decrease in epistemic and aleatory uncertainty. However, care should be taken not to create overly complex models, which despite reducing the aleatory uncertainty may violate theoretical principles and suffer from overfitting, as evidenced herein for the used dataset.
• The conclusions made herein for the performance of the proposed NLR- and ANN-based GMPEs are related to the dataset used. As the dataset of strong ground motion in Greece (and worldwide) increases over time, it is believed that modeling of ground motion will continue to improve.