1 Introduction

Two key ingredients for seismic risk assessment are probabilistic seismic demand models (PSDMs) relating the response of a structural system, or infrastructure component, to a parameter representing the strength of earthquake events called the seismic intensity measure (IM), and the corresponding ground motion prediction equations (GMPEs) characterizing the probability model of the IM. The IMs selected to characterize the earthquake intensity and predict seismic behavior play a critical role in affecting the uncertainty propagated in the risk assessment and loss estimation framework (Mackie and Stojadinovic 2001, 2004; Shafieezadeh et al. 2012; Padgett et al. 2008; Luco and Cornell 2007). To predict infrastructure performance, the current practice of adopting hazard intensity measures based on integer order derivatives or integrals of the ground motion time history (e.g. peak ground acceleration or spectral acceleration) may not be efficient. Complex dynamic features of key infrastructure constituents under earthquake loads may not be optimally predicted using discrete states of acceleration, velocity, or displacement sensitivities which are the basis for the set of traditional earthquake IMs used in seismic risk assessment. Large levels of uncertainties in probabilistic models that predict both ground motion IMs and the associated physical response of structural systems induce significant uncertainties in the final outcome of risk analyses. This uncertainty can undermine confidence in the resulting risk estimates, and pose a significant challenge in implementing such methods for risk-informed decision making purposes.

A new perspective on characterizing earthquake intensity for modeling seismic risk to distributed infrastructure systems is required to enable robust analytical methods for reliable probabilistic characterization of earthquakes across a region, efficient modeling of the physical demand imparted on infrastructures, and increased confidence in resulting risk estimates. The introduction of fractional order calculus to uncover so-called “α-order” ground motion intensity can offer the basis of this transformation. Extending the integer order of differential operators (derivative and integral) to arbitrary real numbers in the context of fractional order calculus releases a major limitation of classical probabilistic and mechanistic modeling which has found applications in various fields of science and engineering such as visco-elastic material modeling (Müller et al. 2011; Mainardi 2010; Meral et al. 2010), control of dynamical systems (Li et al. 2010a, b; Monje et al. 2010; Tavazoei and Haeri 2008; Caponetto 2010; Odibat 2010) and structural engineering (Ruge and Trinks 2004). Several studies have introduced fractional order calculus within the domain of earthquake engineering, for example to model the response of structures with damping or isolation systems (Koh and Kelly 1990; Chang and Singh 2002; Singh et al. 2011), or to characterize the viscoelastic behavior of soils in the presence of seismic waves (Dikmen 2005; Lenti et al. 2012). Experimental studies by Oldham and Spanier (1974) revealed that the damping behavior of some materials in the frequency domain follows a power law model with non-integer orders. Gaul et al. (1991) in a study on simulation of wave propagation in soil domains showed that the frequency domain behavior of visco-elastic soil materials can be better described using fractional order models compared to conventional integer order models. Considering soil medium as a system responsible for propagation of waves from source to the surface, given the above noted fractional characteristics, it is expected that the outputs of the system i.e. ground motions at the surface and responses of structures to these excitations will have fractional order characteristics as well. Shafieezadeh et al. (2012) leveraged this feature and developed a series of fractional order intensity measures by primarily releasing the constraints of integer orders in conventional IMs. The performance of these IMs were evaluated against conventional intensity measures for a class of bridges that accounts for about 20% of the bridge inventory in the state of California. Results indicated that fractional order IMs can significantly reduce dispersions in probabilistic seismic demand models for engineering demand parameters of interest by as much as 36% compared to conventional IMs. In short, fractional order IMs can offer good predictive quality of structural response while reducing uncertainty introduced into the probabilistic analysis.

The study by Shafieezadeh et al. (2012) reveals the fact that there is a strong need for developing fractional order GMPEs because site-specific or regional seismic hazard analyses are required in order for fractional order IMs to be usable in practice. To do so, this paper proposes the first GMPE for estimating peak ground fractional order response (PGRα) IMs. In this respect, a ground-motion database is compiled from the PEER NGA-West2 project database (Ancheta et al. 2014) for developing the PGRα predictive model which is mainly appropriate for active shallow crustal earthquakes in California. Next, the PGRα earthquake intensities of the recordings in the ground-motion database are computed by utilizing a fractional order numerical approximation approach. The PGRα predictive model at peak ground acceleration (PGA; α = 0), peak ground velocity (PGV; α = −1) and various α values (between 0 and −1) are then derived by accounting for magnitude, distance, style-of-faulting, linear and nonlinear site, hanging wall, basin response, and anelastic distance attenuation effects in the ground-motion estimates. The proposed GMPE models the aleatory variability (i.e., standard deviation) by taking into account the magnitude and nonlinear site effects that provide more accurate realizations of the ground motion estimates.

2 Peak ground fractional order response IMs

In the literature, various definitions of fractional order operators have been proposed; among which the Grunwald–Letnikov, Riemann–Liouville, and Caputo’s definitions are widely used formulations for fractional order derivation and integration (Podlubny 1999). However, Caputo’s definition is the most preferred one in engineering applications because it gives bounded values for fractional derivatives of constants and includes physically interpretable initial conditions such as displacement and velocity. Based on Caputo’s definition, the α-derivative of a function f (t) defined on the interval [0, T] at time t ∈ (0, T] can be expressed in the form of a convolution integral as shown in Eq. (1). In this equation,\({}_{0}^{C} D_{t}^{\alpha }\) denotes the notion of the Caputo’s (i.e. C) differintegral fractional operator of a real valued order α with respect to the variable t, starting at time t = t 0  = 0 whereas p is an integer number that satisfies p−1 < α ≤ p and Γ(*) is Euler’s Gamma function.

$${}_{0}^{C} D_{t}^{\alpha } f\left( t \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{{\varGamma \left( {\alpha - p} \right)}}\mathop \int \limits_{0}^{t} \frac{{f\left( \tau \right)^{p} }}{{\left( {t - \tau } \right)^{{\alpha + 1 - p}} }}d\tau ;} \hfill & {\alpha \ge 0} \hfill \\ {\frac{1}{{\varGamma \left( { - \alpha } \right)}}\mathop \int \limits_{0}^{t} \frac{{f\left( \tau \right)}}{{\left( {t - \tau } \right)^{{\alpha + 1}} }}d\tau ;} \hfill & {\alpha < 0} \hfill \\ \end{array} } \right.$$
(1)

The limitation of integer order derivatives of ground displacement can be released by taking the α-order integral of ground acceleration (\(\ddot{x}_{g}\)) and presenting peak ground fractional order response (PGRα) as:

$$PGR_{\alpha } = \begin{array}{*{20}c} T \\ {max} \\ 0 \\ \end{array} \left( {\left| {{}_{0}^{C} D_{t}^{\alpha } \ddot{x}_{g} \left( t \right)} \right|} \right)$$
(2)

Among the various techniques that have been proposed for numerical realization of fractional order derivatives and integrals, the modified approach for Oustaloup recursive approximation to Caputo derivatives in the frequency domain (Oustaloup et al. 2000; Xue et al. 2006) is adopted given its accuracy at a reasonable computation time. The reader is referred to Shafieezadeh et al. (2012) for the details of this approximation, which is integrated in the SIMULINK environment of MATLAB (MathWorks 2015) and used for computing PGRα values herein. Illustrations of the application of the modified Oustaloup approximation technique to couple of accelerograms (Mw 6.69 Northridge Earthquake of 01/17/1994, Beverly Hills-14145 Mulhol Station, MUL009 horizontal component and Mw 4.27 Yorba Linda Earthquake of 09/03/2002, LA—Obregon Park Station, 24400360 horizontal component) are shown in Fig. 1. Results for PGRα are compared with conventional peak ground motion IMs for integer values of α. The results show the agreement of the approximated α-order IMs using the modified Oustaloup method with the traditional IMs computed from the accelerograms, and reinforce the definitions of α-order IMs at integer orders (e.g. PGA, PGV and PGD). They also illustrate the trend regarding variation of PGR with α for the selected motions.

Fig. 1
figure 1

Comparison of approximated PGRα values with conventional peak ground motion IMs

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Figure 2 shows another illustration for the first recording (Northridge Earthquake) to give a slight intuition about the physical interpretation of these IMs. The time series of the ground motion recording for different alpha levels [i.e. alpha = 0 (PGA), −0.25, −0.50, −1 (PGV)] is shown in this figure. The observation made from this figure is that the time-step at which the peak ground fractional responses occur do not coincide with the time step of conventional peak ground responses (i.e. PGA and PGV). This indicates that the peak ground fractional responses are related to the different frequency content of the ground motion data.

Fig. 2
figure 2

Comparison of approximated PGRα values with conventional peak ground motion IMs

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3 Ground motion database

The ground-motion database utilized in this study is a subset of the PEER NGA-West2 database (Ancheta et al. 2014) which was compiled from the worldwide recordings of shallow active crustal regions (e.g. California, Italy, Turkey, Japan, China, etc.). The subset of expanded PEER NGA-West2 database is extracted by considering inclusion criteria comparable to those in Campbell and Bozorgnia (2014). The selected events are mainshock earthquakes from the Western United States (US) with moment magnitude Mw ≥ 4.0 and extended-source distance metrics Joyner-Boore (RJB) ≤ 300 km and rupture (RRUP) ≤ 300 km. Another selection criterion is the presence of known style-of-faulting information based on rake angles and measured or inferred values of VS30-based site classes of the strong motion stations. Only good quality recordings (e.g. spectral shape, well-triggered, and free field) are included in the database. In addition, a filter is applied to the events in order to select only well-recorded earthquakes with respect to minimum number of recordings (N) criteria: N < 5 for Mw < 5.5, N < 3 for 5.5 ≤ Mw < 6.5. A number of additional accelerograms that potentially satisfy the above set of criteria were not included in the database because of systematic problems downloading them from the PEER ground motion database.

The application of the aforementioned criteria results in compilation of a ground-motion database with the total number of 4491 recordings from 82 earthquakes and 1366 strong motion stations. In the final database, 2366 accelerograms recorded from 73 earthquakes can be classified as near-source ground motion following the criterion of RJB and RRUP < 80 km. This threshold can be considered as a limit for earthquakes to have significance in probabilistic seismic hazard analysis and structural engineering applications. ESM_1.1 lists the details of ground motion database in terms of selected earthquakes, recording sites and distance classes.

The distribution of RRUP versus Mw along with their individual distributions for the ground motions in the database are shown in Fig. 3. There are sufficient data to model the magnitude effects in the predictive model except for the very largest magnitude bin (Mw > 7.5) and partially one of the moderate magnitude bins (6.0 < Mw < 6.5). The extent of data for the close distance range (i.e. RRUP < 20 km) is less than that for the rest of distance ranges in the database but the number of recordings is sufficient to satisfactorily model the distance effects in the GMPE.

Fig. 3
figure 3

The distribution of ground motion data: a RRUP versus Mw scatter plots, b Mw histograms, and c RRUP histograms

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Figure 4 displays the RRUP versus Mw distribution of ground motion data with respect to style-of-faulting (strike-slip-SS; normal-NM; reverse-RV) and site class. The National Earthquake Hazards Reduction Program (NEHRP; Building Seismic Safety Council [BSSC], 2009) site classification is used in the comparative plots. The SS and RV ground motion data are well represented in the database, whereas the database lacks a sufficient number of accelerograms from NM faulting earthquakes; however, NM faulting earthquakes are not characteristic of the regional seismicity. The database includes measured and inferred VS30 values for NEHRP site classifications and for modeling the linear and nonlinear site effects of the predictive model. A significant amount of data in the database is coming from strong-motion stations located on the NEHRP-C and -D site categories (Fig. 5). Beyond this, an adequate number of rock site accelerograms are included in the database.

Fig. 4
figure 4

Distribution of the recordings: a style-of-faulting and b NEHRP site classification

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Fig. 5
figure 5

VS30 histograms of the ground-motion database (Note One record has VS30 value beyond the upper range of the plot corresponding to NEHRP-A site class, VS30 = 2016 m/s)

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4 Selection of the main functional form of the GMPE

The regression analysis to develop the PGRα predictive model is conducted in multiple steps. To do so, the near-source database defined by RJB and RRUP < 80 km and the entire ground motion database (i.e. RJB and RRUP < 300 km) are utilized. As mentioned in the previous section, a distance of 80 km has been suggested as the approximate limit of interest for considering the significant effects of ground excitations on structural components (Campbell and Bozorgnia 2014). By using the near-source database limited at this distance, the fictitious depth term is computed and the terms of the model for the style-of-faulting are derived. Next, the rest of the regression coefficients such as magnitude scaling, hanging wall, site effects, etc. are obtained except for the constant term, magnitude-independent distance scaling and anelastic attenuation terms. To prevent jagged variations of spectral estimates, a limited amount of smoothing is applied to the regression coefficients. The rest of coefficients are computed by utilizing the entire ground motion database. Components of the aleatory variability model are obtained in a separate step using the entire ground motion database. The final set of coefficients and standard deviation terms are derived using the random effects regression algorithm proposed by Abrahamson and Youngs (1992).

A simple statistical study is performed to determine the most suitable functional form for the PGRα predictive model. An important point in selecting the functional form to develop a new GMPE is the ability of the model to take into account some crucial effects such as magnitude scaling, magnitude-dependent geometrical spreading, etc. (Bommer et al. 2010). Existing well-structured functional forms already include these key effects in their mathematical expressions. A number of candidate functional forms are considered in this study for their potential to predict PGRα including those utilized by Abrahamson et al. (2014), ASK14; Boore et al. (2014), BSSA14; and Campbell and Bozorgnia (2014), CB14. These functional forms are evaluated only for modeling the magnitude scaling (f mag ), distance scaling (f dis ), style-of-faulting (f flt ), anelastic attenuation (f atn ) and linear site (f site ) effects in the GMPEs, in order to prevent complexity in preliminary assessments. In addition, a homoscedastic aleatory variability model is considered for the preliminary GMPEs.

Equations 3, 9 and 12 present the simplified functional forms of ASK14, BSSA14 and CB14, respectively which are referred to as basic form GMPEs. Magnitude scaling, distance scaling and style-of-faulting effect components of ASK14 and BSSA14 functions are given in Eqs. (4)–(8), 10, 11, respectively. The corresponding components for CB14 form are explained in detail in the next section (Median predictive model). The same linear site model given in Eq. (13) is considered for all functional forms.

$$\ln Y_{ASK14} = f_{mag} + f_{dis} + f_{flt} + f_{site}$$
(3)
$$ f_{mag} = \left\{ \begin{array}{ll} a_{1} + a_{5} \left( {M_{w} -6.75} \right) + a_{8} \left( {8.5 - M_{w} } \right)^{2} ; & \quad M_{w} > 6.75 \\ a_{1} + a_{4} \left( {M_{w} - 6.75} \right) + a_{8} \left( {8.5 - M_{w} } \right)^{2} ; & \quad 5.0 < M_{w} < 6.75 \\ a_{1} + a_{4} \left( {5.0 - 6.75} \right) + a_{8} \left( {8.5 - 5.0} \right)^{2} + a_{6} \left( {M_{w} - 5.0} \right) + a_{4} \left( {M_{w} - 6.75} \right) + a_{7} \left( {M_{w} - 5.0} \right)^{2} ; & \quad M_{w} < 5.0 \\ \end{array} \right.$$
(4)
$$f_{dis} = \left\{ {\begin{array}{*{20}l} {\left[ {a_{2} + a_{3} (M_{w} - 6.75)} \right] \ln \left( {\sqrt {R_{RUP}^{2} + h^{2} } } \right) + a_{17} R_{RUP} ;} \hfill & {M_{w} > 5.0} \hfill \\ {\left[ {a_{2} + a_{3} \left( {5.0 - 6.75} \right)} \right] \ln \left( {\sqrt {R_{RUP}^{2} + h^{2} } } \right) + a_{17} R_{RUP} ;} \hfill & {M_{w} \le 5.0} \hfill \\ \end{array} } \right.$$
(5)
$$f_{flt} = f_{flt,F}\;f_{flt,M}$$
(6)
$$f_{flt,F} = a_{8} F_{RV} + a_{9} F_{NM}$$
(7)
$$f_{flt,M} = \left\{ {\begin{array}{*{20}l} {0;} \hfill & {M_{w} \le 4.5} \hfill \\ {M_{w} - 4.5;} \hfill & {4.5 < M_{w} \le 5.5} \hfill \\ {1;} \hfill & {M_{w} > 5.5} \hfill \\ \end{array} } \right.$$
(8)

where FNM and FRV are dummy variables equal to unity for normal (NM) and reverse faults (RV), respectively. For strike-slip (SS) events, FNM and FRV are zero.

$$\ln Y_{BSSA14} = f_{mag\& flt} + f_{dis} + f_{site}$$
(9)
$$f_{mag\& flt} = \left\{ {\begin{array}{*{20}l} {a_{1} SS + a_{2} NS + a_{3} RV + a_{4} \left( {M_{w} - 5.5} \right) + a_{5} \left( {M_{w} - 5.5} \right)^{2} ;} \hfill & {M_{w} \le 5.5} \hfill \\ {a_{1} SS + a_{2} NS + a_{3} RV + a_{6} \left( {M_{w} - 5.5} \right);} \hfill & {M_{w} \le 5.5} \hfill \\ \end{array} } \right.$$
(10)
$$f_{dis} = \left[ {a_{7} + a_{8} \left( {M_{w} - 4.5} \right)} \right] \ln \left( {\sqrt {R_{JB}^{2} + a_{7}^{2} } } \right) + a_{9} \left( {\sqrt {R_{JB}^{2} + a_{7}^{2} } - 1} \right)$$
(11)
$$\ln Y_{CB14} = f_{mag} + f_{dis} + f_{flt} + f_{atn} + f_{site}$$
(12)
$$f_{site} = s_{1} \hbox{min} \left\{ {\ln \left( {{\raise0.7ex\hbox{${V_{S30} }$} \!\mathord{\left/ {\vphantom {{V_{S30} } {1130}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${1130}$}}} \right),0} \right\}$$
(13)

Although the use of simplified functional forms may be considered as a cursory approach, the aim of this evaluation is to decrease the level of subjectivity by invoking statistical tools to select the functional form among the host of alternatives. The preliminary GMPEs are developed for peak ground fractional order IMs for the horizontal component of ground motions at α = 0 (PGA), −0.25, −0.50, −0.75 and −1 (PGV). These IMs are computed using the average horizontal component definition referred to as RotD50 (Boore 2010). Figure 6 shows the magnitude scaling comparison between the candidate functional form GMPEs of this study and the NGA-West2 predictive models of ASK14, BSSA14, CB14 and CY14 (Chiou and Youngs 2014) at α = 0 (PGA) and −1 (PGV) for an earthquake scenario with strike-slip faulting, RRUP = 10 km and rock site condition (VS30 = 760 m/s). Results indicate that ground motion estimates from the basic form GMPEs are comparable with the spectral ordinates of the NGA-West2 predictive models. This agreement enhances confidence in the statistical tests conducted to select the main functional form of the GMPE.

Fig. 6
figure 6

Magnitude scaling comparison between the basic models and NGA-West2 GMPEs. The GMPE is evaluated for strike-slip faulting, RRUP = 10 km, VS30 = 760 m/s, and dip = 90°. Default values of depth to shear wave velocity of 1.0 km/s (Z1.0) and 2.5 km/s (Z2.5) at the site, depth to top of rupture, hypocentral depth and rupture width are calculated from Kaklamanos et al. (2011). The RJB values corresponding to RRUP = 10 km are calculated from the virtual fault plane

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A number of ground motion data-driven testing tools are considered for the evaluation of GMPEs; the tools include Nash and Sutcliffe Efficiency Coefficient (NSE; Nash and Sutcliffe 1970), Likelihood (LH; Scherbaum et al. 2004), Log-likelihood (LLH; Scherbaum et al. 2009) and Euclidean Distance Based Ranking (EDR; Kale and Akkar 2013; Akkar and Kale 2014). These tools are commonly used in testing the appropriateness of candidate predictive models for site specific or regional seismic hazard applications; however, they also provide ranking of GMPEs under a ground motion database. In addition, Akaike’s and Bayesian Information Criterion (AIC and BIC, respectively) are also considered to rank the candidate functional forms. In this study, the NSE, LH, LLH and EDR ranking indices, and AIC and BIC statistical values are computed for the basic form GMPEs under the compiled ground motion database for α = 0 (PGA), −0.25, −0.50, −0.75 and −1 (PGV). Table 1 lists the average scores of ranking indices that represent the overall performances of the basic form GMPEs for the considered α range. Among these indices, higher values of NSE, lower values of LLH, EDR, AIC and BIC, and values of LH around 0.5 indicate the best performance of GMPEs (or functional forms) under the considered ground motion database. The best performance of a predictive model could be interpreted as the effectiveness of the selected functional form to describe the characteristic features of the subject ground motion database. The underlined italic values in Table 1 indicate the best performing model (or models) for the implemented statistical methods. The basic functional form of CB14 shows the best performance among all data-driven testing tools and gives comparable criterion values among the statistical tools and thus this functional form is selected to develop the PGRα predictive model.

Table 1 Average data-driven testing scores of the GMPEs with selected functional forms
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5 Median predictive model

The final functional form of the PGRα predictive model accounts for magnitude, distance, style-of-faulting, linear and nonlinear site, hanging wall, basin response, and anelastic distance attenuation effects. Equation 14 shows the complete functional form for estimating the average horizontal component (RotD50; Boore 2010) median peak ground fractional responses (cm/s2+α) at 21 α values ranging from 0 (PGA) to −1 (PGV) in natural logarithm unit. Fractional orders less than −1 are not considered because the low frequency content of accelerograms starts to become important in the range of PGD, which is highly sensitive to the low-cut filtering process of ground motion recordings. Each component of final functional form in Eq. (14) is briefly explained in the following paragraphs.

$$\ln Y = f_{mag} + f_{dis} + f_{flt} + f_{hng} + f_{sed} + f_{atn} + f_{site}$$
(14)

The magnitude scaling model includes three breaks at moment magnitudes Mw 4.5, 5.5 and 6.5 to characterize the ground motions as follows:

$$f_{mag} = \left\{ {\begin{array}{*{20}l} {a_{0} + a_{1} M_{w} ;} \hfill & {M_{w} \le 4.5} \hfill \\ {a_{0} + a_{1} M_{w} + a_{2} \left( {M_{w} - 4.5} \right);} \hfill & {4.5 < M_{w} \le 5.5} \hfill \\ {a_{0} + a_{1} M_{w} + a_{2} \left( {M_{w} - 4.5} \right) + a_{3} \left( {M_{w} - 5.5} \right);} \hfill & {5.5 < M_{w} \le 6.5} \hfill \\ {a_{0} + a_{1} M_{w} + a_{2} \left( {M_{w} - 4.5} \right) + a_{3} \left( {M_{w} - 5.5} \right) + a_{4} \left( {M_{w} - 6.5} \right);} \hfill & {M_{w} > 6.5} \hfill \\ \end{array} } \right.$$
(15)

The distance scaling (geometrical spreading) model is presented in Eq. (16). This model implements the RRUP distance metric as the main distance measure.

$$f_{dis} = \left( {a_{5} + a_{6} M_{w} } \right) \ln \left( {\sqrt {R_{RUP}^{2} + a_{7}^{2} } } \right)$$
(16)

where a7 coefficient is the fictitious depth term. The style-of-faulting model given in Eqs. (17)–(19) is magnitude dependent. FNM and FRV are dummy variables equal to unity for normal (NM) and reverse faults (RV), respectively. For strike-slip (SS) events, FNM and FRV are zero. No correction is suggested for reverse events as no difference is observed in the spectral trends between SS and RV ground motion data in the database.

$$f_{flt} = f_{flt,F}\;f_{flt,M}$$
(17)
$$f_{flt,F} = a_{8} F_{RV} + a_{9} F_{NM}$$
(18)
$$f_{flt,M} = \left\{ {\begin{array}{*{20}l} {0;} \hfill & {M_{w} \le 4.5} \hfill \\ {M_{w} - 4.5;} \hfill & {4.5 < M_{w} \le 5.5} \hfill \\ {1;} \hfill & {M_{w} > 5.5} \hfill \\ \end{array} } \right.$$
(19)

The hanging wall model of Donahue and Abrahamson (2014) with some modifications by Abrahamson et al. (2014) is employed in the predictive model as shown in Eq. (20). Some of the model coefficients of Donahue and Abrahamson (2014) are period dependent. The modifications performed by Abrahamson et al. (2014) on the original model of Donahue and Abrahamson (2014) provide period-independent hanging wall model coefficients (except for a10 coefficient), rendering the implementation of this model for the PGRα GMPE feasible. The hanging wall model considers the effects of dip angle (Eq. 21), Mw (Eq. 22 where magnitude scaling hanging wall term, aHW = 0.2), RX (closest distance to the surface projection of the top edge of fault rupture, Eqs. (23)–(25) where h1 = 0.25, h2 = 1.5 and h3 = −0.75, and W is the fault rupture width), ZTOR (depth to top of fault rupture, Eq. 26), and RJB (Joyner-Boore distance, Eq. 27) for sites located on hanging wall side of the fault rupture (i.e. FHW = 1). For other site locations (i.e. FHW = 0), hanging wall effects are not considered in the ground motion estimates.

$$f_{hng} = a_{10} F_{HW}\;f_{hng,dip}\;f_{{hng,M_{w} }}\;f_{{hng,R_{X} }}\;f_{{hng,Z_{TOR} }}\;f_{{hng,R_{JB} }}$$
(20)
$$f_{{hng,dip}} = \left\{ {\begin{array}{*{20}l} {{\raise0.7ex\hbox{${\left( {90 - dip} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {90 - dip} \right)} {45}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${45}$}};} \hfill & {dip > 30} \hfill \\ {{\raise0.7ex\hbox{${60}$} \!\mathord{\left/ {\vphantom {{60} {45}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${45}$}};} \hfill & {dip \le 30} \hfill \\ \end{array} } \right.$$
(21)
$$f_{{hng,M_{w} }} = \left\{ {\begin{array}{*{20}l} {1 + a_{HW} \left( {M_{w} - 6.5} \right); } \hfill & {M_{w} \ge 6.5} \hfill \\ {1 + a_{HW} \left( {M_{w} - 6.5} \right) - \left( {1 - a_{HW} } \right)\left( {M_{w} - 6.5} \right)^{2} ;} \hfill & {5.5 < M_{w} < 6.5} \hfill \\ {0;} \hfill & {M_{w} \le 5.5} \hfill \\ \end{array} } \right.$$
(22)
$$f_{{hng,R_{X} }} = \left\{ {\begin{array}{*{20}l} {h_{1} + h_{2} \left( {{\raise0.7ex\hbox{${R_{X} }$} \!\mathord{\left/ {\vphantom {{R_{X} } {R_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{1} }$}}} \right) + h_{3} \left( {{\raise0.7ex\hbox{${R_{X} }$} \!\mathord{\left/ {\vphantom {{R_{X} } {R_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{1} }$}}} \right)^{2} ; } \hfill & { R_{X} < R_{1} } \hfill \\ {1 - \left( {\frac{{R_{X} - R_{1} }}{{R_{2} - R_{1} }}} \right); } \hfill & {R_{1} \le R_{X} \le R_{2} } \hfill \\ {0;} \hfill & {R_{X} > R_{2} } \hfill \\ \end{array} } \right.$$
(23)
$$R_{1} = W \cos \left( {dip} \right)$$
(24)
$$R_{2} = 3R_{1}$$
(25)
$$f_{{hng,Z_{TOR} }} = \left\{ {\begin{array}{*{20}l} {1 - \frac{{Z_{TOR}^{2} }}{100};} \hfill & {Z_{TOR} \le 10} \hfill \\ {0;} \hfill & {Z_{TOR} > 10} \hfill \\ \end{array} } \right.$$
(26)
$$f_{{hng,R_{JB} }} = \left\{ {\begin{array}{*{20}l} {1;} \hfill & {R_{JB} = 0} \hfill \\ {1 - {\raise0.7ex\hbox{${R_{JB} }$} \!\mathord{\left/ {\vphantom {{R_{JB} } {30}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${30}$}};} \hfill & {R_{JB} < 30} \hfill \\ {0;} \hfill & {R_{JB} \ge 30} \hfill \\ \end{array} } \right.$$
(27)

The basin response model is given in Eq. (28) which is based on the depth to the 2.5 km/s shear wave velocity horizon beneath the site (Z2.5). The k coefficient in this equation is equal to 1.88. In case Z2.5 parameter is not available, its value can be inferred from the simple empirical relationship proposed by Campbell and Bozorgnia (2014) for Californian sites as given in Eq. (29).

$$f_{sed} = \left\{ {\begin{array}{*{20}l} {a_{11} \left( {Z_{2.5} - 1} \right); } \hfill & {Z_{2.5} \le 1} \hfill \\ {0;} \hfill & {1 < Z_{2.5} \le 3} \hfill \\ {a_{12}\, k\, e^{ - 0.75} \left( {1 - e^{{ - 0.25\left( {Z_{2.5} - 1} \right)}} } \right);} \hfill & {Z_{2.5} > 3} \hfill \\ \end{array} } \right.$$
(28)
$$Z_{2.5} = \exp \left( {7.089 - 1.144 \ln \left( {V_{S30} } \right)} \right)$$
(29)

The anelastic attenuation model is given in Eq. (30) which reflects attenuation effects in the ground motion estimates for sites with RRUP > 80 km.

$$f_{atn} = \left\{ {\begin{array}{*{20}l} {a_{13} \left( {R_{RUP} - 80} \right); } \hfill & {R_{RUP} > 80} \hfill \\ {0;} \hfill & {R_{RUP} \le 80} \hfill \\ \end{array} } \right.$$
(30)

The site response model is given in Eq. (31) which is adapted from Chiou and Youngs (2014) primarily due to its simplicity. In this equation, flin and fnl represent the linear and nonlinear components of site amplification, respectively. The linear site effects are taken into account using Eq. (32) where s1 describes the VS30 scaling whereas VS30 = 1130 m/s is the reference site condition beyond which the site amplification is not considered. The nonlinear site function is given in Eq. (33) where s2 represents the degree of nonlinearity as a function of VS30 and can be obtained from Eq. (34). In Eq. (33), PGR1130 is the median peak ground fractional response computed from Eq. (14) for the reference rock site condition (i.e. VS30 = 1130 m/s).

$$f_{site} = f_{lin} + f_{nl}$$
(31)
$$f_{lin} = s_{1} \hbox{min} \left\{ {\ln \left( {{\raise0.7ex\hbox{${V_{S30} }$} \!\mathord{\left/ {\vphantom {{V_{S30} } {1130}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${1130}$}}} \right),0} \right\}$$
(32)
$$f_{nl} = s_{2} \ln \left( {\frac{{PGR_{1130} + s_{3} }}{{s_{3} }}} \right)$$
(33)
$$s_{2} = s_{4} \left[ {e^{{s_{5} \left( {\hbox{min} \left( {V_{S30} ,1130} \right) - 360} \right)}} - e^{{s_{5} \left( {1130 - 360} \right)}} } \right]$$
(34)

One considered strategy for determining site response model coefficients would be to directly adopt Chiou and Youngs (2014) nonlinear model coefficients (s3, s4 and s5) while obtaining the linear site model coefficient (s1) from the regression analysis to reflect the site effects of the chosen ground motion database in the functional form. However this process is not feasible since nonlinear site model coefficients are not available for fractional orders except for α = 0 (i.e. PGA) and α = −1 (i.e. PGV). As an alternative, a separate regression analysis is performed to derive the site coefficients for intermediate fractional orders (i.e. α = −0.05 to −0.95) by implementing the basic functional form GMPE as in Sect. 4 (i.e. Selection of the main functional form of the GMPE). The regression analysis is constrained to derive the same nonlinear coefficients (i.e. s3, s4 and s5) as those given in Chiou and Youngs (2014) for PGA and PGV, therefore compatible coefficients for the intermediate fractional orders are derived. Then these nonlinear site coefficients are directly used in the site response model of the final predictive model. The values of s1 for the linear site model coefficient are derived again in the final regressions to consider the effects of the complete functional form.

The full set of α-independent and -dependent regression coefficients for the median PGRα predictive model is provided in Tables 2, 3 and 4.

Table 2 α-independent constant regression coefficients
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Table 3 α-dependent magnitude and distance scaling model regression coefficients
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Table 4 α-dependent hanging wall, basin response, anelastic attenuation and site response model regression coefficients
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6 Aleatory variability model

The total aleatory variability (σ) can be partitioned into the between-event (τ, Mw-dependent) and within-event (ϕ, Mw- and VS30-dependent) standard deviation terms as given in Eq. (35).

$$\sigma \left( {M_{w} ,V_{S30} } \right) = \sqrt {\tau^{2} \left( {M_{w} } \right) + \phi^{2} \left( {M_{w} ,V_{S30} } \right)}$$
(35)

The Mw-dependent between-event standard deviation term is modeled by:

$$\tau \left( {M_{w} } \right) = \left\{ {\begin{array}{*{20}l} {\tau_{1} ;} \hfill & {M_{w} \le 4.5} \hfill \\ {\tau_{1} + \left( {\tau_{2} - \tau_{1} } \right)\left( {M_{w} - 4.5} \right);} \hfill & {4.5 < M_{w} < 5.5} \hfill \\ {\tau_{2} ;} \hfill & {M_{w} \ge 5.5} \hfill \\ \end{array} } \right.$$
(36)

The Mw- and VS30-dependent within-event standard deviation term is modeled by:

$$\phi \left( {M_{w} ,V_{S30} } \right) = \left\{ {\begin{array}{*{20}l} {\phi \left( {M_{w} } \right); } \hfill & {V_{S30} \ge V_{2} } \hfill \\ {\phi \left( {M_{w} } \right) - \Delta \phi_{V} \left( {\frac{{\ln \left( {{{V_{2} } \mathord{\left/ {\vphantom {{V_{2} } {V_{S30} }}} \right. \kern-0pt} {V_{S30} }}} \right)}}{{\ln \left( {{{V_{2} } \mathord{\left/ {\vphantom {{V_{2} } {V_{1} }}} \right. \kern-0pt} {V_{1} }}} \right)}}} \right); } \hfill & {V_{1} < V_{S30} < V_{2} } \hfill \\ {\phi \left( {M_{w} } \right) - \Delta \phi_{V} ;} \hfill & {V_{S30} \le V_{1} } \hfill \\ \end{array} } \right.$$
(37)

where V1 = 225 m/s and V2 = 300 m/s are selected from the residual trends and ϕ(Mw) is computed by:

$$\phi \left( {M_{w} } \right) = \left\{ {\begin{array}{*{20}l} {\phi_{1} ; } \hfill & {M_{w} \le 4.5} \hfill \\ {\phi_{1} + \left( {\phi_{2} - \phi_{1} } \right)\left( {M_{w} - 4.5} \right);} \hfill & {4.5 < M_{w} < 5.5} \hfill \\ {\phi_{2} ;} \hfill & {M_{w} \ge 5.5} \hfill \\ \end{array} } \right.$$
(38)

The aleatory variability model is developed by evaluating the Mw bins of between-event residuals (ηi) to obtain between-event standard deviations τ, and the Mw, RRUP and VS30 bins of within-event residuals (εij) to obtain within-event standard deviations ϕ. The τ and ϕ values are computed corresponding to the generated bins from the random effects regression algorithm (Abrahamson and Youngs 1992). Figures 7 and 8 display the aleatory variability terms calculated for these bins as well as their 95% confidence intervals and final aleatory variability model fits (solid linear or trilinear functions) for the representative fractional orders. Figure 7 shows the magnitude-dependent main τ and ϕ models. Two breaks are selected at Mw = 4.5 and 5.5 from the general trends of the binned aleatory variability terms. In Eqs. (36) and (38), τ1−ϕ1 and τ2−ϕ2 pairs represent the aleatory variability terms for Mw ≤ 4.5 and Mw ≥ 5.5, respectively. Path and site effects on the within-event aleatory variability term are investigated from Fig. 8. The binned ϕ trends in the upper row of Fig. 8 show no dependence of ϕ upon RRUP for path effects whereas the binned ϕ trends in the bottom row of this figure indicate that the final within-event standard deviation term should account for site effects given observed dependence of ϕ upon VS30.

Fig. 7
figure 7

Binned aleatory variability terms and their 95% confidence intervals for fractional order, α = −0.25 (left column) and −0.75 (right column), and between-event standard deviation, τ, (upper row) and within-event standard deviation, ϕ, (bottom row) against Mw

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Fig. 8
figure 8

Binned aleatory variability terms and their 95% confidence intervals for fractional order, α = −0.25 (left column) and −0.75 (right column), and within-event standard deviation, ϕ, against RRUP (upper row) and within-event standard deviation, ϕ, against VS30 (bottom row)

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The binned ϕ trends (bottom row of Fig. 8) indicate that ϕ is constant for small values of VS30, increases afterwards with VS30 up to a certain value and then follows an almost stable trend. Two breaks are selected at VS30 = 225 m/s and 300 m/s to fit a suitable model to within-event standard deviations by evaluating the VS30 versus εij residual trends in conjunction with the VS30 versus ϕ behaviors. These selections also comply with the observations of Boore et al. (2014). Thus a reduction to ϕ values calculated from Eq. (38) is proposed for sites with VS30 < V2 = 300 m/s. The application of this reduction is proposed by the introduction of a Δϕv parameter in Eq. (37) which has a similar form as the equation in Boore et al. (2014) predictive model. To obtain a generic Δϕv parameter, the within-event standard deviation terms are computed for Mw ≤ 4.5 (i.e. ϕ1) and Mw ≥ 5.5 (i.e. ϕ2) separately corresponding to εij for no VS30 limitation (i.e. complete residuals), εij for VS30 < 225 m/s and εij for VS30 > 300 m/s. The upper row of Fig. 9 shows the variations of the corresponding ϕ1 and ϕ2 values with respect to the fractional order. The calculated ϕ1 and ϕ2 values of the complete residuals and the residuals for VS30 > 300 m/s almost overlap each other whereas ϕ1 and ϕ2 values of the residuals for VS30 < 225 m/s differ from the complete standard deviations. The differences in ϕ values for the latter case (i.e. VS30 < 225 m/s) are calculated and the variations for Mw ≤ 4.5 (i.e. Δϕ1) and Mw ≥ 5.5 (i.e. Δϕ2) are shown in the bottom row of Fig. 9. The variations in Δϕ1 and Δϕ2 suggest a considerable reduction in within-event standard deviation of small magnitude ground motion data whereas the reduction is modest for the ones of moderate-to-large magnitudes. To simplify the aleatory variability model, a magnitude-independent Δϕv reduction parameter is proposed based on the weighted average of Δϕ1 and Δϕ2 based upon the number of ground motion data included in the bins of Mw ≤ 4.5 and Mw ≥ 5.5. The variation of the proposed reduction parameter Δϕv with respect to fractional order is shown in Fig. 9. Fractional order dependent between-event and within-event standard deviations together with the reduction parameter are listed in Table 5.

Fig. 9
figure 9

Effects of VS30 on within-event standard deviation terms (upper row) and the proposed reduction parameter (Δϕv) for ground motion data with VS30 ≤ 300 m/s (bottom row)

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Table 5 α-dependent standard deviations and reduction parameter
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7 Evaluation of the proposed GMPE

First, the residuals (i.e. the difference between median prediction and empirical data) of the predictive model computed from the random effect regression analysis are evaluated to scrutinize the deviation between ground motion estimates of the proposed GMPE and the empirical observations from the model development database. Figures 10, 11 and 12 display between-event (or inter-event) residuals (ηi) versus Mw, and within-event (or intra-event) residuals (εij) versus RRUP and VS30 for fractional orders, α = 0 (PGA), −0.25, −0.50 and −1 (PGV). The between-event residuals for Mw and within-event residuals for RRUP and VS30 do not indicate any systematic variations that point out biased estimates for the ground motion database. The corresponding mean binned residuals and their confidence limits also fluctuate about zero which implies unbiased ground motion estimates of the proposed GMPE with respect to the main estimator parameters (i.e. Mw, RRUP and VS30).

Fig. 10
figure 10

Between-event residual (ηi) distribution against Mw of the predictive model for fractional orders, α = 0, −0.25, −0.50 and −1. The error bars indicate 95% confidence intervals of the mean binned residuals

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Fig. 11
figure 11

Within-event residual (εij) distribution against RRUP of the predictive model for fractional orders, α = 0, −0.25, −0.50 and −1. The error bars indicate 95% confidence intervals of the mean binned residuals

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Fig. 12
figure 12

Within-event residual (εij) distribution against VS30 of the predictive model for fractional orders, α = 0, −0.25, −0.50 and −1. The error bars indicate 95% confidence intervals of the mean binned residuals

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A number of illustrative examples are provided to present scaling of the proposed GMPE with main estimator parameters (i.e. Mw, RRUP and VS30) and α values as well as comparisons with the ground motion estimates and aleatory variability components of the NGA-West2 predictive models. An earthquake scenario with vertical strike-slip fault mechanism is designated for all cases. To calculate the secondary estimator parameters (e.g. Z1.0, Z2.5, ZTOR, ZHYP, W, etc.), the relationships proposed in Kaklamanos et al. (2011) are utilized. The RJB extended-source distances corresponding to the selected RRUP metrics are calculated from the virtual fault planes considering the magnitude-dependent ZTOR values.

Figures 13 and 14 compare the magnitude and distance scaling of the proposed predictive model with the NGA-West2 GMPEs. The comparisons are given at RRUP = 10 km and 40 km for distance scaling, and Mw = 5.5 and 7.5 for magnitude scaling for a site condition with VS30 = 760 m/s. The comparisons are made for PGA and PGV which are the common IMs between the proposed model and the NGA-West2 GMPEs. In Fig. 14, the distance scaling curves start at the smallest possible RRUP value, given its relation to the magnitude-dependent ZTOR value which is deep for small magnitude earthquakes but shallow (even zero) for large magnitude events. The magnitude and distance scaling of the proposed GMPE show good agreement with the ground motion estimates of the NGA-West2 models for the selected earthquake scenarios. For small magnitude ranges (Mw < 5) in Fig. 13, ground motion estimates from the proposed model are slightly larger than estimates from the other models. The difference in the lower magnitude bounds of the proposed GMPE (Mw = 4) and the others (Mw ~ 3) may be one of the reasons for this observation. Bommer et al. (2007) highlighted that minimum magnitude (Mmin) limits of databases would control the spectral estimates at small magnitudes; the predictive models that use a ground motion database with lower Mmin may yield smaller ground motion estimates than those developed from a ground motion database with higher Mmin. Overall, given that the PGRα GMPE was developed to also consider the effect of α, the proposed predictive model offers a comparable estimate of integer order PGRs across Mw and RRUP to those proposed by NGA-West2 models while also enabling assessment of fractional order behavior.

Fig. 13
figure 13

Magnitude scaling comparison of the proposed GMPE with the NGA-West2 models for strike-slip events and VS30 = 760 m/s: a PGA, RRUP = 10 km; b PGA, RRUP = 40 km; c PGV, RRUP = 10 km; d PGV, RRUP = 40 km

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Fig. 14
figure 14

Distance scaling comparison of the proposed GMPE with the NGA-West2 models for strike-slip events and VS30 = 760 m/s: a PGA, Mw = 5.5; b PGA, Mw = 7.5; c PGV, Mw = 5.5; d PGV, Mw = 7.5

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Fractional order scaling of the proposed GMPE with magnitude for VS30 = 760 m/s and 255 m/s, and VS30 for Mw = 4.5 and 7.5 is represented in Figs. 15 and 16 at distance level of RRUP = 10 km for strike-slip earthquakes. The magnitude effects for small-to-large magnitude levels of Mw = 4.5, 5.5, 6.5 and 7.5 are explored whereas the importance of VS30 is examined for soft-to-rock site conditions with VS30 = 255, 520, 760 and 1130 m/s. The plots of fractional order scaling with magnitude (Fig. 15) depict the fact that the decrease in the PGRα values for small magnitudes (Mw = 4.5 and 5.5) is quadratic; however, this behavior changes gradually (turns into cubic) for both site classes as the magnitude increases. In addition, the large PGRα differences between small magnitudes become narrower through moderate-to-large magnitudes due to magnitude saturation effects up to around α = −0.5 (after that level, magnitude saturation effects are diminished; see Fig. 13c). While soil nonlinearity effects are not observed in fractional order scaling with VS30 plots for small magnitude events (Fig. 16a), those effects are detected in the evaluations of scaling for large magnitude (Fig. 16b) earthquakes for soft and stiff sites (i.e. VS30 = 255 and 520 m/s, respectively), and α values between 0 and −0.5.

Fig. 15
figure 15

Fractional order scaling with magnitude of the proposed predictive model for strike-slip earthquakes and RRUP = 10 km: a VS30 = 760 m/s, b VS30 = 255 m/s

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Fig. 16
figure 16

Fractional order scaling with VS30 of the proposed predictive model for strike-slip earthquakes and RRUP = 10 km: a Mw = 4.5, b Mw = 7.5

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Figure 17 presents the comparison of aleatory variability components (i.e. between-event, τ, and within-event, ϕ, standard deviations) of the proposed predictive model and selected NGA-West2 GMPEs for PGA and PGV levels as well as the τ and ϕ values of the proposed model at the intermediate α levels. The comparisons are made for two different magnitude and VS30 levels because of the Mw and VS30 dependency of the proposed aleatory variability model (see Eqs. 35–38). Figure 17a shows a case for Mw = 7.5 and VS30 = 760 m/s whereas Fig. 17b displays a comparison for Mw = 4.5 and VS30 = 255 m/s. In general, the values of aleatory variability components of the proposed GMPE are within the corresponding bounds of the NGA-West2 models. The slight dissimilarities at PGA and PGV levels can be attributed to the differences of the size of the model development databases (NGA-West2 models have larger ground motion databases than the GMPE in this study) and lower magnitude bounds (NGA-West2 models have lower threshold magnitude, Mw ~ 3.0, than the predictive model in this study) between the proposed GMPE and NGA-West2 models.

Fig. 17
figure 17

Comparisons of aleatory variability components of the proposed predictive model with the NGA-West2 GMPEs: a Mw = 7.5 and VS30 = 760 m/s, b Mw = 4.5 and VS30 = 255 m/s

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8 Summary and conclusions

To develop a new ground-motion predictive model for a set of novel ground motion intensity measures—peak ground fractional order responses—a subset of the PEER NGA-West2 database is extracted by considering various inclusion criteria (e.g. only mainshock earthquakes from Western US; moment magnitude Mw ≥ 4.0; extended-source distance metrics ≤ 300 km; known values of site class and style-of-faulting; well-recorded earthquakes with respect to minimum number of recordings criteria such as N < 5 for Mw < 5.5, N < 3 for 5.5 ≤ Mw < 6.5). The application of the above criteria results in the compilation of a ground-motion database with a total of 4491 recordings from 82 different earthquakes. Caputo’s definition with modified Outsalup’s recursive approximations in the frequency domain was used to compute the PGRα values of the ground motion recordings. The main functional form of the predictive model was selected by implementing rational data-driven testing tools from the literature, which compared alternative functional forms from the NGA-West2 predictive models. The final PGRα GMPE accounts for magnitude, distance, style-of-faulting, linear and nonlinear site, hanging wall, basin response, and anelastic distance attenuation effects through the selected functional form. The site response and hanging wall effects of this predictive model were imported from the well-known models developed in the NGA-West2 project. To obtain the site response and hanging wall model coefficients that correspond to intermediate α values between PGA (α = 0) and PGV (α = −1), regression analyses were conducted. The final predictive model produces PGA (α = 0), PGV (α = −1), and peak ground fractional order responses at 19 α’s ranging from −0.05 to −0.95.

Through the derivation of the newly proposed GMPE for PGRα, insights are gleamed regarding the influence of various predictors (e.g. magnitude, distance, site effects) on the median value and aleatory variability of fractional order responses and how scaling effects vary as a function of α. For example, the fractional order scaling of the proposed GMPE exhibits different behaviors depending on magnitude level, and the soil nonlinearity effects are more pronounced with increasing level of spectral intensities. The resulting GMPE reveals high predictive quality across a range of fractional orders, while also taking care to account for various sources of aleatory variability. Given that predictive models of PGRα are now available, and that existing work has suggested such fractional order IMs can provide more accurate predictions of the seismic response of structures, the outcomes of this study can enable utilization of a new class of ground motion intensity measures that take advantage of concepts of fractional order calculus for seismic risk assessment. On the other hand, there is still a strong need for work at the probabilistic seismic response level that compares optimal alpha level for a range of realistic structural systems and idealized/generalized cases. Future research should focus on investigation of fractional order IMs for a range of structural systems as well as exploration of the overall impact of integrating the proposed fractional order GMPE on resulting risk evaluations and loss estimates for distributed infrastructure systems. Beyond this, additional fractional order predictive models developed from different ground motion databases are needed to properly model the epistemic uncertainty in probabilistic seismic hazard analysis and to account for the regional variability while calculating the fractional order ground motion estimates.

9 Data and resources

This study uses a subset of the NGA-West 2 database (Ancheta et al. 2014). The flatfile and ground motion accelerograms can be accessed through the websites of NGA-West2 Project (http://peer.berkeley.edu/ngawest2/, last accessed June 2016) and PEER Ground Motion Database (http://ngawest2.berkeley.edu/, last accessed June 2016), respectively.