# A New Ground-Motion Prediction Model for Shallow Crustal Earthquakes in Turkiye
## 3 Ground-Motion Prediction Models (GMPMs)  
The GMPM functional forms used in this study are composed of a set of GMPEs. The basic formulation of GMPEs is presented by defining the selected model components and their associated variables. The functional form includes the variables' moment magnitude \(M_{W}\), style-of-faulting (SoF), shear wave velocity (\(V_{S30}\)), distance (\(R_{JB}\)), and PSA value at the reference rock site condition (\(PSA_{T}\)). The function chosen for our GMPMs is given in Equation (1):  
\[ln(Y)=f_{M}(M_{w})+f_{D}\left(Z_{hypp}\right)+f_{SoF}(SoF)+f_{R}(M_{w},R)+f_{S }(V_{S30},PSA_{r})+\sigma \tag{1}\]  
In the ground-motion model, \(ln\) is the natural logarithm, \(Y\) is ground-motion intensity measure (PGA, PGV and 5%-damped PSA). \(Y\) has units of cm/s for PGV and cm/s\({}^{2}\) for PGA and PSA, \(f_{M}(Mw)\) represents the magnitude scaling of ground motion, \(f_{D}(Z_{HYP})\) is the depth function, \(f_{SoF}(SoF)\) is the style of faulting, \(f_{R}(Mw,R_{JB})\) includes both geometric spreading and anelastic attenuation and scales the ground motion with distance, \(f_{S}(V_{S30},PSA_{r})\) is site effect for the PSA at a reference rock site (\(V_{ref}\)), and, finally, \(\sigma\) is the total standard deviation of residual distribution. The general expression adopted for the magnitude scaling is adapted from [PERSON] et al. [37]. It involves trilinear magnitude scaling with a quadratic component. The regression coefficients (\(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\) and \(c_{5}\)) are calculated using mixed-effect regression. M is the moment magnitude (\(Mw\)) predictor variable, and \(M_{1}\) and \(M_{2}\) are the hinge magnitudes and are assumed to be 6.75 and 5.5 for \(M_{1}\) and \(M_{2}\), respectively. The magnitude function \(f_{M}(M)\) is defined in Equation (2):  
\[f_{M}(M)=\begin{cases}c_{1}+(c_{2})(M_{2}-M_{1})+(c_{3})(M-M_{2})+c_{4}(8.5-M _{2})^{2}&for\ M<M_{2}\\ c_{1}+(c_{2})(M-M_{1})+(c_{4})(8.5-M)^{2}&for\ M_{2}\leq M<Mis due to sparsity of data with \(Z_{TOR}\). The depth dependency function\(f_{D}(Z_{HYP})\) is given by Equation (3):  
\[f_{D}(Z_{HYP})= \begin{cases}0,&Z_{hyp}\leq 7\\ c_{6}\Big{(}Z_{hyp}-7\Big{)},&7<Z_{hyp}\leq 20\\ 13c_{6},&Z_{hyp}>20\end{cases}  \tag{3}\]  
where \(c_{6}\) is the regression coefficient. One of the key predictive variables used in our regression analysis is the style-of-faulting (SoF) term. Fault type refers to the categorization of events as \(SS\), \(NS\), or \(RS\). The SoF effect is addressed in the reference model using multiplicative coefficients on dummy variables. We did not assess the magnitude dependence for scaling with SoF. The fault type function is given in Equation (4), where \(c_{7}\) and \(c_{8}\) are coefficients that depend on the model and the \(f_{N}\) and \(f_{R}\) dummies for the \(NS\) and \(RS\) events, respectively. The style-of-faulting function is given by:  
\[f_{SoF}(SoF)=\left( \begin{array}{c}c_{7}\end{array} \right)f_{N}+\left(c_{8} \right)f_{R} \tag{4}\]  
Distance scaling consists of two components: geometric spreading and anelastic attenuation. The geometrical spreading effect increases with the earthquake magnitude. Moreover, for a large earthquake, the attenuation of ground motion with distance decreases more slowly than for a smaller earthquake. Therefore, the geometrical spreading term, as expressed in Equation (5a), is represented as a linear function dependent on the magnitude. The path function is given by:  
\[f_{R}\big{(}M_{w},R_{JB}\big{)}=f_{R,GS}\big{(}M_{w},R_{JB}\big{)}+f_{R,AA}(R) \tag{5}\]  
\[f_{R,GS}\big{(}M_{w},R_{JB}\big{)}=[d_{1}+d_{2}(M_{w}-6.75)]ln(R)\] (5a) in which the first term represents magnitude-dependent geometrical spreading (\(d_{2}\)) and is computed through regression, while the second term accounts for anelastic attenuation. \(d_{1}\) and \(d_{2}\) are the regression coefficients. We used the anelastic attenuation term and, since it is assumed to be more significant at longer distances, we decided to use this effect at long distances (\(R_{JB}>80\)).