1 Introduction and motivation

In recent decades, the importance of vertical ground motions in design of short-period critical structures (e.g., nuclear power plants) and lifeline systems has become a matter of debate (Bozorgnia and Campbell 2004; Gülerce and Abrahamson 2011). Several investigators examined the relationship between horizontal and vertical components of ground motions and their corresponding characteristics (e.g., Bozorgnia and Campbell 2004; Bindi et al. 2010; Edwards et al. 2011; Poggi et al. 2012). These researchers pointed out the need to enhance predictive equations for V/H spectral ratios.

The V/H spectral ratio can be estimated using independent ground-motion prediction models for vertical and horizontal components, developed from the same database and mathematical models (e.g., Soghrat and Ziyaeifar 2016a; Darzi et al. 2018; Zolfaghari and Darzi 2019). However, one of the main shortcomings identified in this approach is the difficulties in computation of V/H residuals and standard deviations. Another alternative approach is to develop empirical ground-motion models for V/H ratios directly (e.g., Bozorgnia and Campbell 2016; Akkar et al. 2014; Bommer et al. 2011) in which direct determination of residuals and standard deviations for the estimated V/H ratios can be achieved. Moreover, the consistency between horizontal and vertical components is better preserved in direct V/H models than those obtained from independent horizontal and vertical ground-motion models (Akkar et al. 2014).

Simplified period-dependent models for V/H ratios are embedded in some seismic design codes such as National Earthquake Hazards Reduction Program (NEHRP 2009), in order to estimate vertical components from corresponding horizontal spectrum. There are even seismic design codes proposing flat V/H ratios across all periods; for example, the Iran’s practical building code (standard no.2800) which recommends a constant value of two-thirds between vertical and horizontal design spectra over all range of periods. Several studies found that clear peak of V/H at short periods exceeds a value of two-thirds in near fault areas (Bozorgnia et al. 1995, 1996; Bozorgnia and Campbell 2004, 2016). Series of studies revealed strong dependency of V/H ratio to spectral period, site effect, and source-to-site distance, while there seems to be less dependency to magnitude and faulting mechanism (Bozorgnia and Campbell 2004). Such degree of dependencies of V/H ratios to site and path characteristics highlights the shortcomings of oversimplified V/H models used in certain design codes. For instance, one of the main deficiencies of such models is the lack of proper consideration of site effect, which in turn undermines the validity of such approaches for site-specific seismic hazard assessments (Bommer et al. 2011). The same applies to path effect (source-to-site distance) which has not been addressed in those simple models. These deficiencies highlight the need for a more rational approach for assessment of V/H ratios, where source, path, and site effects are all taken into account. A direct application of such models is in site-specific seismic hazard assessment as well as modification of seismic design guidelines and codes. Akkar et al. (2014) developed a model for Europe and the Middle East which can be used to estimate V/H ratios for all site conditions, including soil nonlinearity and as a continuous function of VS30. Development of empirical ground-motion prediction equations (GMPEs) for predicting V/H ratios is a preferred approach which allows direct assessment of V/H ratio as a function of path, site, and source characteristics (Akkar et al. 2014).

Recently, two models were developed by Soghrat and Ziyaeifar (2016b) and Zafarani et al. (2018), hereinafter called SZ16 and Zea18, respectively. The former model was tailored to the Northern Iran only. Both models predict spectral ordinates for periods shorter that 4 s. Both models also account for SoF effects; nevertheless, almost similar predictions were presented for strike–slip and reverse events. Neither Zea18 nor SZ16 include fault rupture in their predictive models.

Soghrat and Ziyaeifar (2016b) examined the applicability of some regional and global V/H ground-motion models (e.g., Gülerce and Abrahamson 2011; Akkar et al. 2014; and Bommer et al. 2011) for Northern Iran and reported weak correlations between recorded Iranian strong motion data and those predicted by the models. Therefore, the present study is an attempt to develop a new V/H ground-motion models based on local strong ground-motion database, recorded across all tectonic regions in Iran. The process in particular pay special attention to the behavior of V/H response spectra at short periods, taking into account that vertical natural periods for most structural components and systems are located at short periods.

Therefore, the proposed V/H ground-motion models predict PGA, PGV, and 5%-damped spectral ratios for a wide range of spectral ordinates from 0.01 to 10 s as functions of moment magnitude, source-to-site distance, style-of-faulting, and ground types. For this purpose, a large well-compiled Iranian ground-motion databank including 1871 three-component records from 420 earthquakes was compiled. The database is considered as an improved and enriched version of the Iranian strong ground-motion database, compared to those used in previous GMPEs for Iran. Extended efforts were invested to enhance the metadata with regard to available style-of-faulting and recording stations with measured VS30 which account for 98% (1860 records) and 75% (1399 records) respectively, while for Zea18, the measured VS30 values and SoF are available for only 40% and 34% of their dataset, respectively. Moreover, for about 60% of the events used by SZ16, the faulting mechanism is unknown and also only 45% of their selected records are classified with known VS30. For the remaining records in both studies, VS30 values were classified based on empirical methods. This study also uses an updated version of the Iranian strong motion database up to late 2014 which includes additional 38 earthquakes and 140 three-component records happened in 2014, among them 11 events (93 records) with Mw > 5.0.

We examined four different functional forms to explore the most appropriate ones for unbiased determination of V/H spectral ratios. All these models predict V/H ratios as functions of source-to-site distance, magnitude, soil conditions, and styles-of-faulting. The compiled database provides various source-to-site distance measures such as Repi (epicentral distance), Rhyp (hypocentral distance), RJB (Joyner-Boore distance), and Rrup (rupture distance) for each strong motion record. The availability of these distance metrics in our database enabled us to develop consistent V/H models for these distance measures, applicable for distance range of up to 200 km. The examination process includes comparison of median estimates of V/H ratios derived from RJB-, Repi-, Rrup-, and Rhyp-based models.

Attempts are also made in this paper to examine the validity of the proposed models by studying the aleatory standard deviations and checking residuals, followed by their comparison with some previously published V/H models from local and regional databases.

2 Iranian ground-motion database

The provided Iranian ground-motion database comprised of 1871*3 waveforms from 420 earthquakes recorded at 668 Iranian strong motion stations. All unprocessed acceleration time histories were recorded by Iran Strong Motion Network (ISMN, http://ismn.bhrc.ac.ir/), operated, and maintained by the Building and House Research Center (BHRC 2005). Only three-component accelerograms were used in this study. We used direct Mw estimates from the Harvard global CMT catalog (Centroid Moment Tensor 2015), the NEIC global catalog (National Earthquake Information Center 2015), and where necessary (for 130 earthquakes, 289 records out of 1871) converted from Ms and mb using conversion formulas derived from ISC (International Seismological Centre 2015) catalog, proposed by Shahvar et al. (2013). This database includes earthquakes with Mw ≥ 4.5 and records with distance ≤ 200 km. Figure 1 shows the geographical distribution of these earthquakes (indicated by circles) and strong motion stations (shown by black triangles).

Fig. 1
figure 1

Geographic distribution of earthquakes and strong ground-motion recording stations across Iran. Stations with unknown VS30 (yellow squares) are excluded from the process for V/H model development. Earthquakes are colored by focal depth, as darker circles represent deeper events and their sizes are proportional to Mw. The gray lines indicate active faults (Zolfaghari and Darzi 2014)

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The focal mechanism of 251 earthquakes with 1421 records were identified through double-couple fault plane solution using strike, dip and rake angles of fault planes. The remaining 40% of earthquakes (24% of records) could not be associated to any faulting mechanism. Refined investigation was carried out to these data in order to determine the most probable faulting mechanisms through determination of corresponding causative faults. For this purpose, we studied and collected information about seismic characteristics of neighboring active faults (e.g., dip and strike angle) and their corresponding reliable mechanism. Additional information such as style-of-faulting of dependent shocks and their associated active faults, regional tectonic setting, topographic map of vicinity area, and information from previous earthquakes associated to the causative fault are used to assess fault characteristics. In the present study, vast majority of earthquakes (98%) were associated to focal mechanisms. This is a significant improvement compared to the earlier model of Zea18 for which the faulting mechanisms for about 66% of the studied earthquakes were unknown. As is seen in Fig. 2b, the database consists of 187 strike–slip events (811 records) and 217 reverse events (985 records). There were few observations with normal-faulting mechanism (11 out of 420 earthquakes). The Mw–RJB distribution of all records with respect to style-of-faulting is depicted in Fig. 2a. As inferred from this figure, records are well distributed except for 6.7 < Mw < 7.1 over all distance ranges and for Mw > 6.5 and RJB < 20 km.

Fig. 2
figure 2

a Magnitude–distance distribution of the compiled database in terms of style-of-faulting. b Histogram of number of earthquakes and records versus SoF shown in dark gray and light gray bars, respectively. Reverse faults (RV), SS, and NR stand for reverse, strike–slip, and normal faulting types, respectively

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The measured VS30 values were available for 433 recording stations (75% of total 1871 records), ranging from 155 to 2400 m/s which shows a notable enhancement compared to the recent model of Zea18, in which 306 stations (40% of their records) had measured VS30. These measurements were made for ISMN through seismic refraction technique by BHRC. In order to have a robust predictive model, records with unknown VS30 values (shown as yellow squares in Fig. 1) were removed from the database. Then, the remaining data was classified based on ground type definitions suggested by Iran’s building design code as follows: site class I (VS30 > 750 m/s), site class II (375 < VS30 < 750 m/s), site class III (175 < VS30 < 375 m/s), and site class IV (VS30 < 175 m/s). Only 4 stations with 5 recorded events are included in site class IV which were combined with station with type III classification. As shown in Fig. 3b, 35% of all accelerograms were recorded on site class I, 48% on site class II and 17% on site class III. The Mw–RJB distribution of the selected records grouped in different site classes is shown in Fig. 3a. The scatter plots of records associated to each ground type (I, II and III) show similar distribution for all magnitude–distance intervals.

Fig. 3
figure 3

a Magnitude–distance distribution of the compiled database in terms of local site classes. Data points are shown with different symbols for site classes (I, II, and III). b Histogram of number of stations and records versus site classes

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Focal depth measurements were derived from reliable local and global earthquake catalogs. The local catalog of IRSC (Iranian Seismological Centre, http://irsc.ut.ac.ir/) was given higher priority for determining earthquake focal depth. For 3 events with no reliable depth estimates, the Scherbaum et al. (2004) equations were applied to estimate focal depth using moment magnitude and style-of-faulting as inputs. In order to avoid unbiased results in terms of earthquake depth, only earthquakes with focal depth less than 40 km (92% of earthquakes) were allowed in the regression analyses. Figure 4 shows the distribution of selected earthquakes with respect to moment magnitude, focal depth, and number of recorded accelerograms per earthquake. As can be seen in this figure, a good portion of database is associated with events with multiple records for earthquakes with Mw > 6. About 13% of earthquakes are single-recorded events, among which 67% are with Mw > 5.

Fig. 4
figure 4

Magnitude–depth distribution for the selected database. Data points are classified based on number of accelerograms per earthquake

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In order to make the GMPEs applicable in probabilistic seismic hazard analyses, models based on both finite-fault (Rrup and RJB) and point source (Repi and Rhyp) distance metrics are presented in this paper. RJB is defined as the horizontal distance to the closest point on the surface projection of the fault rupture (Joyner and Boore 1981), and Rrup is the closest distance to fault rupture plane. The RJB, Repi, and Rhyp were first computed for all strong motion records in the database. Then, using the procedure described by Kaklamanos et al. (2011), rupture distances (Rrup) were estimated for all records in the database. These estimates are made as functions of RJB, dip angle, width of fault rupture plane, depth to top of rupture plane (ZTOR), and source-to-site azimuth.

The final selected database for regression analyses consists of 1350 accelerographic records from 370 earthquakes, recorded during 1975 to 2014. Table 1 shows the proportion of compiled database which was removed to prevent rough records in the regression analyses, such as deep events and recording stations with unknown VS30 values.

Table 1 Distribution of data before and after applying filtering criteria for performing regression analyses
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All waveforms were uniformly corrected for baseline offset based on the approach proposed by Boore et al. (2002) and de-noised using the modified adaptive wavelet de-noising technique proposed by Ansari et al. (2007, 2010). Compared to conventional filtering approaches (e.g., Boore 2005; Boore and Bommer 2005), the applied approach eliminates nonstationary noises in the time domain and attenuates noises for all period range of engineering interest. This facilitates development of a ground-motion model applicable to the broad range of periods, while highly noisy waveforms can be corrected efficiently. Peak ground acceleration (PGA), peak ground velocity (PGV), and 5%-damped response spectral acceleration (SA), sampled at 60 equally log-spaced vibration periods from 0.01 to 10 s, are included for horizontal and vertical components. The average horizontal ground-motion intensity measures were computed by geometric mean calculation.

3 Functional forms of V/H models

To find the best functional form with optimum performance on the compiled ground-motion database and to avoid shortages caused by complex models, first the distribution of observed V/H ratios w.r.t magnitude and distance is explored. Then, the behavior of median V/H ratios as functions of independent variables and also the trends of V/H ratios for different scenarios of functional forms are investigated.

Figure 5 shows observed V/H spectral ratios against distance (top row) and magnitude (second row) for PGA and spectral periods of 0.1 and 2 s. Data points are plotted for reverse-faulting events with ground motion recorded on rock site. According to this figure (top row), as distance increases, the V/H ratios tend to decrease at PGA, remain constant at T = 0.1, and increase at T = 2 s. The distribution of V/H ratios, however, does not show strong dependency to moment magnitude as shown in the second row of this figure.

Fig. 5
figure 5

Distribution of actual V/H spectral ratios against distance (top row) and magnitude (second row) for PGA, T = 0.1 and 2 s. Reverse faulting events located on rock sites are plotted

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The median V/H ratios for PGA and response spectral ordinates of T = 0.1, 2.5, and 4 s are shown in Fig. 6. To eliminate the SoF effect, all recording data are initially scaled to reverse faulting through preliminary empirical scaling factors. To explore the effect of distance on the V/H ratios for a given period, data are grouped into three Mw intervals and by 10-km RJB bins as shown in first row of Fig. 6, illustrating median V/H ratio for each distance cluster plotted against corresponding standard deviation. Apparently, the median V/H ratios show decay for PGA with increasing distance, remain independent of RJB for T = 0.1 s and gradually increase with distance at longer periods. Similar results were also observed in Fig. 5.

Fig. 6
figure 6

Median V/H ratios of PGA, T = 0.1, 2.5, and 4 s as functions of RJB (first row) for three magnitude intervals and Mw (second row) for three distance intervals, for rock sites. Median V/H predictions (black solid lines) and their corresponding standard deviations (black dotted lines) obtained from the selected functional form are plotted against distance-magnitude cluster which will be discussed in Sect. 4

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In a similar manner, in order to investigate the magnitude dependency of V/H ratios, data are grouped into three RJB intervals and by magnitude bins of 0.1 Mw as illustrated in second row of Fig. 6 which shows cluster of the median V/H ratio for each distance against corresponding standard deviation. As discussed for Fig. 5, magnitude dependency of V/H ratios at different periods is almost negligible for all distance ranges. Similar results were observed by several studies such as Gülerce and Abrahamson (2011), Akkar et al. (2014), and Bommer et al. (2011).

In order to find the most suitable functional form for development of V/H ratio model, four different functional forms are examined based on the following generic form:

$$ {\log}_{10}\left(\mathrm{V}/\mathrm{H}\right)={b}_1+{b}_2{M}_{\mathrm{w}}+{b}_3{M}_{\mathrm{w}}^2+\left({f}_{R1}\mathrm{or}\ {f}_{R2}\right)+{b}_7{S}_{II}+{b}_8{S}_{II I}+{b}_9 RV+{b}_{10} SS+\varepsilon $$
(1)

where

$$ {f}_{R1}=\left({b}_4+{b}_5{M}_{\mathrm{w}}\right)\ {\log}_{10}\left({R}_1+{b}_6\right);{R}_1={R}_{\mathrm{rup}}\ \mathrm{or}\ {R}_{\mathrm{hyp}} $$
(2)
$$ {f}_{R2}=\left({b}_4+{b}_5{M}_{\mathrm{w}}\right)\ {\log}_{10}\left(\sqrt{{R_2}^2+{b_6}^2}\right);{R}_2={R}_{\mathrm{JB}}\ \mathrm{or}\ {R}_{\mathrm{epi}} $$
(3)

The proposed model includes a magnitude scaling term, a geometric decay term as a function of distance, a site amplification variable, and a style-of-faulting factor to estimate median vertical-to-horizontal ground-motion ratio (denoted as V/H in Eq. 1) which can be either PGV, PGA, or 5%-damped response SA at periods from 0.01 to 10 s. The \( {b}_3{M}_{\mathrm{w}}^2 \) term accounts for magnitude saturation effects. The geometric attenuation term of fR1 in Eq. (2) is considered for Rrup or Rhyp distance metrics, and the term fR2 in Eq. (3) is with respect to RJB or Repi distance metric. Contrary to our model, Akkar et al. (2014) used similar functional form for three RJB-, Repi-, and Rhyp-based GMPEs for Europe and the Middle East. The b5Mw term takes into account magnitude-dependent geometrical spreading. Note that in case of small events (Mw < 5), Rrup and RJB are almost equal to Rhyp and Repi, respectively.

RV and SS are dummy variables taking the value of 1 for reverse and strike–slip faulting mechanism, respectively, and 0 for unknown SoF. SII and SIII are also dummy variables accounting for site categories defined by Iran’s building design code based on VS30 values, where SII = SIII = 0 represents site class I (rock site, VS30 > 750 m/s); SII = 1 and SIII = 0 are for site class II (375 < VS30 < 750 m/s); and SIII = 1 and SII = 0 are for site class III (VS30 < 375 m/s). In Eq. (1), ε represents the error term which is normally distributed in logarithmic units with standard deviation of sigma (\( {\sigma}_{\left({\log}_{10}\mathrm{V}/\mathrm{H}\right)} \)). The total sigma decomposed of between-event (τ , inter-event) and within-event (φ, intra-event) aleatory standard deviations resulting from the regression procedure, as given by the following expression:

$$ {\sigma}_{\left({\log}_{10}\mathrm{V}/\mathrm{H}\right)}=\sqrt{\tau_{\mathrm{V}/\mathrm{H}}^2+{\varphi}_{\mathrm{V}/\mathrm{H}}^2} $$
(4)

Based on the generic form shown in Eq. (1), four different scenarios of functional forms were investigated in this study. Functional form (1) excludes the magnitude-dependent geometric spreading coefficient (b5 = 0). Functional form (2) ignores the magnitude-dependent saturation coefficient (b3 = 0). Functional form (3) simplifies the base equation by removing both b5 and b3 coefficients (b5 = b3 = 0). Functional form (4) includes anelastic attenuation effect (mostly at long-distances R > 80 km) by adding (+b11R) term. We constrained b6 = 5 km while evaluating the functional forms and obtained other coefficients by conducting nonlinear least-square regression analyses to each functional form. Note that due to insignificant effect of SoF pointed out by SZ16 and Zea18, we did not consider such dependency.

Figure 7 compares the median V/H trends as a function of RJB for Mw 5, 6 and 7. Comparisons are made for PGA and spectral periods of 0.5 and 3 s along with V/H ratio observations for Mw 5 ± 0.2, 6 ± 0.2, and 7 ± 0.2 for site class II. Each row in Fig. 7 represents the median V/H estimates derived from one of the four functional forms described earlier. Figure 7a, b, c, and d are associated to functional forms (1), (2), (3), and (4), respectively.

Fig. 7
figure 7

Median V/H ratios trends against RJB for PGA and spectral periods 0.5 and 3 s as functions of Mw (5, 6, and 7) obtained from four alternative functional forms. a Functional form (1). b Functional form (2). c Functional form (3). d Functional form (4). V/H observations from site class II are plotted for Mw 5 ± 0.2, 6 ± 0.2, and 7 ± 0.2

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As shown in Figs. 5 and 6 and also pointed out by other researchers (e.g., Bommer et al. 2011; Gülerce and Abrahamson 2011), the V/H ratios tend to attenuate faster at very short periods and flatten and increase gradually towards longer periods. Detailed investigation on the reasons for such pattern is given by Gülerce and Abrahamson (2011), and the seismological explanations are offered by Beresnev et al. (2002) and Kawase and Aki (1990).

Functional form (1), including a quadratic dependence on Mw (Fig. 7a), shows similar trends as noted above. At very short periods, the V/H ratios tend to mildly attenuate with increasing distance; at T = 0.5 s, they are constant; and for T > 0.5 s, they rise smoothly with distance. Although for PGA and for events with Mw 5 and Mw 6, median curves are closer to each other; their overall behavior is quite reasonable and consistent with the actual data trends. At long periods, the median V/H ratios for events with Mw 7 overlap the corresponding V/H ratios of Mw 6 and Mw 5. Soghrat and Ziyaeifar (2016b) observed stronger magnitude dependency for data from Northern Iran than those pointed out by Bommer et al. (2011) for Europe and the Middle East; however, they could not find a meaningful trend and decided to discard b3 and b5 coefficients.

For functional form (2) (Fig. 7b) and for PGA and T = 0.5 s at RJB > 20 km, V/H curves corresponding to different magnitude levels tend to converge. Moreover, at long period (T > 0.5 s), the median V/H ratios for curves representing to Mw 5 overlap with those for Mw 6 and subsequently Mw 6 median curves overlap the Mw 7 median curve at RJB > 40 km. These trends are contrary to the actual data pattern. The results of functional form (3) (Fig. 7c), which is the simplest applied model, are similar to the first functional model, and they are both in good agreement with the V/H ratios of actual data.

As shown in Fig. 7d, the predicted median V/H ratios based on functional form (4) reveal an unreasonable behavior with decreasing trend at long distances. Thus, it is revealed that applied database in this study does not allow to capture magnitude-dependent geometric spreading term, neither additional anelastic attenuation term.

Comparing the stability and significance of the coefficients of the models derived from different functional forms, the coefficients associated to functional form (3) showed better stability and smoothness over the whole range of periods, while other models presented unpleasant and undesirable jaggedness for b1, b2, and b3, especially for T > 0.7 s. Besides, b4, as an important coefficient representing attenuation with distance, was completely unstable in functional form (2). Furthermore, we evaluated the residuals of different functional forms and found that functional forms (1) and (3) represent more unbiased models compared to functional forms (2) and (4). Altogether, we believe that functional forms (1) and (3) show better stability in the multi-step maximum-likelihood regressions procedure and, therefore, represent the most reasonable fits among four alternative forms. In conclusion, authors decided to examine performance of both functional forms through further investigations using standard deviations and residual analyses.

4 Regression analyses

In this section, two-stage maximum-likelihood regression technique (Joyner and Boore 1993, 1994) was performed to compute the period-dependent coefficients and aleatory variability for selected functional forms. This technique was applied to functional forms (1) and (3) for four scenarios associated to different distance metrics as discussed earlier. In the first step, the regression analyses were undertaken on all data and site amplification coefficients (b9 and b10) and b4. Through this process, the within-event aleatory (φ) variability’s were determined from the first stage of regression, while magnitude scaling coefficients (b1, b2, and b3) and between-event variability (τ) were derived from the second stage. For detailed explanation on regression procedure, please refer to Darzi et al. (2018). Note that site class I is considered as reference rock ground type. In final step, by fixing b1and b2 (and b3 for functional form (1)) to the values derived from the previous step, the regression analyses were repeated to estimate the coefficients for focal mechanism predictors. In this way, predicted ground-motion intensities were scaled with regard to magnitudes to become equal for different faulting mechanisms (following Cauzzi and Faccioli 2008). In this step, regression analyses were performed to earthquakes with more than 3 ground-motion records with 5 < Mw < 6, which constitutes the majority of accelerograms with almost the same number of reverse and strike–slip events. This leads to reduction of between-event uncertainty component of the predictive model by average 20% over all vibration periods and up to 40% at some long periods. It should be noted that due to the small number of normal and unknown faulting earthquakes (29 records), they have been removed from this regression step. Another point to note is that all coefficients are period-dependent, except b6 (saturation term of fictitious depth, in Eqs. (2) and (3)), which is kept constant at 5 km.

Some recently developed V/H models (Akkar et al. 2014; Gülerce and Abrahamson 2011) considered nonlinear site effect, whereas Bommer et al. (2011) and Zea18 did not account for nonlinear site term. The nonlinear site effects, which mostly influenced the horizontal components of ground motions, have not been implemented in our models. This is due to the deficiency of our dataset with regard to factors controlling nonlinearity such as strong intensity measures at very short distances and at stations located on soft soil. Zea18 and Sedaghati and Pezeshk (2017) too neglected nonlinear site amplification effect in their recent GMPEs developed for horizontal component in Iran due to the same reasons as well as weak influence of soil nonlinearity in their dataset.

It should be highlighted that the V/H prediction models accounting for nonlinear site response can only be applied to the horizontal-GMPEs including nonlinear effects. Therefore, as mentioned earlier, due to lack of such horizontal ground-motion models for Iran, development of V/H model based on nonlinear site-response would not be applicable at the present time. Moreover, it is a common practice to explore nonlinear site response analysis as part of site specific PSHA studies. Nevertheless, the significance of soil nonlinearity in Iranian ground-motion database deserved to be studied in more detail, taking into account the seismicity of the country as well as relatively dense strong motion network in Iran.

The coefficients derived from regression analyses for the proposed V/H models and their corresponding uncertainty components for PGV, PGA, and 5%-damped response spectral acceleration from 0.01 to 10 s are all available as Electronic Supplement A to this paper. Table 2 lists the period-dependent coefficients, the intra- and inter-event uncertainly components, and total uncertainty for RJB-based model derived from functional form (1) for some selected vibration periods.

Table 2 Regression coefficients for the functional form (1) based on RJB
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Similar processes were taken for the functional form (3). For the sake of brevity and not to crowd the figures, hereinafter, all plots and results are shown for functional form (3), while only the results derived from functional form (1) are reported. Moreover, the V/H ground-motion models for Rrup, Rhyp, RJB, and Repi are developed following similar steps, except utilizing different geometric attenuation terms described in Sect. 3.

In Fig. 8, median V/H ratios and their corresponding standard deviations are compared against observed V/H ratios as a function of distance classified by Mw 5 ± 0.2, 6 ± 0.2, and 7 ± 0.2 in the top row, as well as w.r.t moment magnitude classified by RJB 30 ± 10, 60 ± 20, and 110 ± 30 km. Comparisons are made for PGA and spectral acceleration at T = 0.1, 1, and 3.5 s. As it can be seen, the proposed model agrees fairly well with the original V/H data. Similar results were also found for other site classes.

Fig. 8
figure 8

Comparison between observed V/H and predicted median V/H (solid black lines) and their standard deviations (black dashed lines) for rock sites and events of Mw 5, 6, and 7 (first row) and RJB 30 ± 10, 60 ± 20, and 110 ± 30 km (second row)

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Moreover, in Fig. 6, the predicted median V/H (black solid lines) for rock sites and their corresponding standard deviations (black dotted lines) obtained from functional form (3) are depicted for Mw 5, 6, and 7 in the top row and for RJB 30, 60, and 110 km, in the bottom row. Apparent from the figure, there is not much difference between the predicted and observed median V/H at different periods.

In Fig. 9, we compared period-dependent variations of within-event (φ), between-event (τ), and total (σ) standard deviations of the proposed model in this study based on functional form (3) with four local and regional models. These comparative V/H models are SZ16 for Northern Iran; Zea18 for all Iranian territory; Akkar et al. (2014), hereinafter ASA14; and Bommer et al. (2011), hereinafter BAK11 for Europe and the Middle East.

Fig. 9
figure 9

Comparison of the a within-event (φ), b between-event (τ), and c the total (σ) standard deviation for GMPEs proposed in this study based on functional form (3) (thick black lines) with SZ16 (Soghrat and Ziyaeifar 2016b), ASA14 (Akkar et al. 2014), BAK11 (Bommer et al. 2011), and Zea18 (Zafarani et al. 2018)

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In our model, τ ranges from 0.038 to 0.1 and has small contribution in total sigma, while φ ranges from 0.124 to 0.207 and is a major contributor to the total sigma values of our model. Also, total sigma tends to increase as shifting towards longer periods from 0.137 to maximum value of 0.216 at T = 8 s. The ranges of total sigma dependency to spectral periods for various V/H models are as follows: from 0.16 to 0.2 for ASA14, from 0.163 to 0.21 for BAK11, from 0.21 to 0.27 for SZ16, and from 0.18 to 0.23 for Zea18. Immediately apparent from this comparison is that the aleatory variability of our presented model yields the lowest values over the whole period ranges.

Compared to both local models of SZ16 and Zea18, the within-event standard deviation of our model is significantly lower, while the between-event variability is similar to others. This is due to application of a more comprehensive and reliable metadata and quite uniform distribution of data in terms of distance, especially RJB and Rrup. We should highlight that specifically larger number of records with known ground types is provided in this study and more importantly only records with measured VS30 values are used in the regression analyses. This had a significant role in reduction of within-event variability of our model. Unlike present study, both SZ16 and Zea18 models used empirical methods to assign site classes for a large portion of records with unknown site classes. Compared to regional and global models, the observed discrepancies may also stem from the database differences, different distance metrics, metadata completion, and insufficient small magnitude events.

It should be noted that there are minor discrepancies between the standard deviations calculated from functional forms (1) and (3). Moreover, due to imperceptible discrepancies among the total sigma of four considered distance measures, their corresponding curves overlap each other completely. Thus, we decided not to plot them herein. Derived values are tabulated as Electronic Supplement A to this paper.

5 Attenuation of V/H ratios w.r.t various distance measures

In Fig. 10, we explore the differences between the V/H spectral ratios predicted from four individual V/H ground-motion models developed based on distance metrics of RJB, Repi, Rhyp, and Rrup using similar ground-motion database.

Fig. 10
figure 10

Comparison of median V/H estimated from models developed based on RJB, Repi, Rhyp, and Rrup; as a function of period for R 20 km (b), and as a function of distance, at T = 0.05 s (a) and T = 3 s (c). V/H ratios are estimated for site class (SC) II, reverse faults (RV) and for Mw 6. All models derived from the same database

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Figure 10 b compares the variation of median V/H ratios against periods for various source-to-site measures at 20 km. The predictions are for site class II and reverse earthquakes with Mw 6. As shown here, the largest discrepancies between the V/H ratios corresponding to different distance metrics are immediately apparent at short periods (T < 0.1 s) and 2.5 < T < 6 s. Otherwise, there is negligible differences between different models. Akkar et al. (2014) found similar patterns towards large magnitudes with less than 5% discrepancies between median V/H values estimated from RJB, Repi, and Rhyp-based models for Europe and the Middle East. In Fig. 10, as magnitude increases, the differences between the plotted models increase as well. This is probably due to effect of large rupture dimensions on seismic energy propagation, especially for large earthquakes.

Figure 10a and c show the trend of median V/H ratios against distance for periods at 0.05 and 3 s, respectively. As illustrated in these figures, the curves converge as distances increase, so that for distances greater than 60 km, the V/H predictions are not sensitive to distance metrics which is due to reduction in the effect of focal depth and rupture dimensions. Comparing Fig. 10a–c, it is apparent that at short periods the Rhyp-based model predicts the highest V/H ratios and the RJB-based model estimates the lowest values, whereas, at T = 3 s, the results reversed.

6 Validation of V/H models

In this section, conventional residual analysis was carried out to evaluate the validity of the proposed V/H models. The V/H predictions made in this section are all calculated from the RJB-based model. Residual of V/H spectral ratio, denoted as res (T)i in the following equation, is calculated from the observed ground-motion amplitudes of V/H spectral ratios (denoted as obs (T)i) with respect to its corresponding predictions derived from proposed model (denoted as pre (T)i), as follows:

$$ \mathrm{res}{(T)}_i={\log}_{10}\left(\mathrm{obs}\ {(T)}_i\right)-{\log}_{10}\left(\mathrm{pre}\ {(T)}_i\right) $$
(5)

Residuals are decomposed to between- and within-event residuals which are assumed to have a normal distribution with zero mean and standard deviation of τ and φ, respectively, as presented in Eq. (4).

Figure 11 shows the between-event residuals against Mw for PGA, PGV, and spectral periods of T = 0.1, 0.5, 1, and 5 s. Note that positive and negative residuals represent underestimation and overestimation of our proposed model respectively. The two black solid lines in each panel represent the ±1σ bounds around the medians. The black squares indicate average values of residuals computed at different magnitude bins. Clustering of black squares close to zero reveals unbiased prediction of the V/H values in terms of their corresponding estimator parameters. According to the results indicated in Fig. 11, there is not any model misfit in terms of Mw. Only slight dispersive behavior of residuals is apparent for large magnitudes at all periods which is most likely due to poor distribution of data. To better visualize the possible trends in terms of depth, residual points are classified by earthquake depth as indicated in legend. This figure does not show any apparent trends for deeper events.

Fig. 11
figure 11

Between-event residuals as a function of Mw, determined using the proposed RJB-based V/H model, shown for PGA, four spectral periods (T = 0.1, 0.5, 1, and 5 s), and PGV. Residual points are classified according to the focal depth. Two horizontal lines in each panel represent ±1σ bounds of the predictions. Black squares represent mean residuals at magnitude bins

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Figure 12 shows the within-event residuals plotted as a function of RJB for PGA, PGV, and spectral ordinates of 0.1, 0.5, 1, and 5 s. The residuals do not show any apparent distance dependency, especially for RJB < 150 km, as the average residual values attain a zero value. Despite poor distribution of data at short distances (RJB < 10 km), their corresponding mean values of residuals are zero indicating unbiased trend. Moreover, residuals are classified in three site classes (i.e., I, II, and III) to explore any observable trend. The uniform distribution of residuals within ±1σ bounds confirms an acceptable level of reliability in the V/H estimated from the site terms of the proposed model. In general, the overall behavior of all residuals over the whole period is satisfactory.

Fig. 12
figure 12

Within-event residuals as a function of RJB, determined using the proposed RJB-based V/H model, shown for PGA, four spectral periods (T = 0.1, 0.5, 1, and 5 s) and PGV. Residuals are classified according to the site classes. Two horizontal solid lines represent ±1σ bounds of the predictions. Black squares represent mean residuals at distance bins

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Figure 12 was plotted with respect to three other models of Rrup, Rhyp, and Repi, and similar results were observed. For the sake of brevity, they are not reproduced here.

The within-event residuals as a function of VS30 are presented for PGA and T = 0.2 and 3 s in Fig. 13. Residuals are classified by site classes as indicated in the legend. As illustrated, despite clustering of residual points at long periods, the overall observations do not show any systematic trends, suggesting a good performance in the predicted median V/H computed from the developed model. Furthermore, residuals tend to scatter slightly for VS30 > 900 m/s especially at long periods which may be the results of sparse data. However, the variation of residuals is weak and can be neglected. As apparent from residuals distribution depicted in Fig. 13, the variation of V/H data shows no biased trend which could imply independency of the model to nonlinear soil behavior as also shown by Zea18.

Fig. 13
figure 13

Within-event residuals of V/H value as a function of VS30 and site classes (I, II, and III), shown for PGA, T = 0.2 and 6 s. Two horizontal solid lines represent ±1σ bounds of the predictions. Black squares represent mean residuals at VS30 bins

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As expected, the residuals suggest that V/H models presented in this study are well calibrated so that the assumed V/H models are in good agreement with the observed data in Iran. It should be noted that all residuals are plotted for V/H predictions of functional form (1), and similar results have been captured for functional form (3) as well. Due to negligible differences of two equations, results are not plotted in this paper.

7 Evaluation of predicted V/H spectral ratios

In this section, the effect of explanatory model predictors (i.e., site classes, Mw, RJB, and SoF) is explored under different scenarios. Furthermore, detailed comparisons are made between the predicted V/H ratios made by this study against a recently published model of Zea18. Similar definition of average horizontal component is used for both models.

Figure 14 illustrates the median V/H over the entire periods for the specific combinations of Mw and RJB. The V/H ratios are evaluated for rock site condition (class I or VS30 = 800 m/s, if required) and soft soils (class III or VS30 = 400 m/s, if required) which are shown in top and bottom rows, respectively. To investigate the impact of RJB and Mw, the V/H spectral ratios are plotted for RJB [5, 50] km and for Mw 5 in the left panels and for Mw (Bindi et al. 2010; Boore 2005) and RJB 10 km in the right panels along with their comparison with Zea18 indicated as gray curves. All models are for strike–slip faulting.

Fig. 14
figure 14

The median V/H ratios obtained from proposed RJB-based model (black lines) compared against those from Zea18; for site class I (top rows) and site class III (second rows), and for Mw 5 and RJB [5, 50] km (left panels), and for Mw (Bindi et al. 2010; Boore 2005) and RJB 10 km (right panels). SS stands for strike–slip faulting

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The V/H ratios show the highest value at short periods (0.04 to 0.08 s) and the lowest value at periods of 0.2 to 0.35 s (Fig. 14). As inferred from Fig. 14a and c, the peaks increase slightly as distance increases (~ 0.05 to 0.08 s). Similar to finding by Zea18, V/H estimations show higher dependency to RJB than to Mw. However, V/H dependency to Mw for our model is not as weak as SZ16 model for Northern Iran. As indicated, at 0.3 < T < 2.5 s, V/H estimates increase smoothly with increasing period similar to BAK11 model for Europe and the Middle East and also the global model proposed by Bozorgnia and Campbell (2004), while for longer periods, they decrease gradually, similar to the Bozorgnia and Campbell (2016) global model. At long distances, larger V/H estimates are apparent which is most probably caused by small magnitude events. Zea18 (gray curves) estimates approximately similar V/H ratios, except for T > 0.15 s, on rock site whose predictions are lower than ours.

Peak amplitude of V/H spectral curve at short period exceeds 1.0 at near-source fields (black solid curve in Fig. 14a) and large magnitude events (black dashed curve in Fig. 14b), as similarly observed by several other researches (e.g., Bommer et al. 2011; Bozorgnia and Campbell 2004, 2016). This is thoroughly contrary to Iran’s practical building design code which considers constant ratio of two-thirds for all periods. This value is lower on soil sites (Fig. 14d) than rock sites (Fig. 14b).

Figure 15a compares the median V/H spectral predictions for different SoF (Fig. 15a) and ground types (I, II, and III) (Fig. 15b) for Mw 5. The modeled site is located at RJB of 10 km. According to Fig. 15a, the predicted median V/H ratios resulting from strike–slip earthquakes are dominant at short-period ranges (T < 0.2 s), while estimations obtained from reverse-thrust events increase smoothly when moving towards longer periods, consistent with SoF pattern appeared in predictive V/H models of Gülerce and Abrahamson (2011) and BAK11. Compared to our model, Zea18 predicts almost equal V/H ratios for T < 0.7 s with no differences for different SoF and for longer periods predicts lower amplitudes.

Fig. 15
figure 15

Median V/H ratios obtained from proposed RJB-based model. a Reverse and strike–slip faulting mechanism compared with the results of Zea18. b For site class I (solid lines), II (dashed lines), and III (dot-dashed lines); for Mw 5 and RJB 10 km

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The immediate observation from the comparative plots of Fig. 15b is that our predictions are the largest for rock ground types for almost all periods, except for 0.06 < T < 0.2 s. Note that if soil nonlinearity were considered in our model, we would have expected larger V/H predictions at short spectral periods for large magnitudes at soil sites and short source-to-site distances. Notwithstanding minor discrepancies of the median V/H predictions at short periods, site term influence becomes apparent at intermediate-to-long periods for Iranian database, similar to what was observed by Zea18 and BAK11. Unlike our model, Bozorgnia and Campbell (2016) found significant differences for different VS30 values (see Fig. 9) in NGA-West2 dataset, especially at short periods. At longer periods, as sites are representing softer layers, they are expected to have a lower V/H which. This trend is consistent with SZ16 and BAK11 models which are developed for Northern Iran and Europe and the Middle East, respectively.

8 Comparison with other studies

Figure 16 presents the comparison between the predicted median V/H spectral ratios made by this study with two recently published local models of SZ16 and Zea18 for Iran as well as two regional V/H models of BAK11 and ASA14 for Europe and the Middle East. Comparisons are made reverse-faulting events of Mw 5 (first column) and Mw 7 (second column) and for RJB 10 km (first row) and RJB 50 km (second row). Note that vertical axis has similar scale for all panels.

Fig. 16
figure 16

Comparison of predicted median V/H by this study (solid black lines) with those predicted by selected V/H models of SZ16 (Soghrat and Ziyaeifar 2016b), BAK11 (Bommer et al. 2011), ASA14 (Akkar et al. 2014), and Zea18 (Zafarani et al. 2018) for rock site (class I, VS30 = 800 m/s, if needed). The plots are for reverse events with Mw 5 (left panels) and 7 (right panels), at RJB 10 km (first row) and 50 km (second row)

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In terms of site term definition, all comparative models are consistent as they all have site class terms, except for ASA14 and Zea18 functions which use continuous VS30 measures. In order to plot ASA14 and Zea18 for rock-like site, VS30 value of 800 is suggested. BAK11 uses the NEHRP site classification with fairly similar definition to the site classification scheme defined by Iran’s practical building code used here and by SZ16. Since the distance metric used by all comparative models is RJB, the proposed RJB-based model proposed by this study is used.

As shown in Fig. 16, all comparative models have similar shapes and trends over the whole period ranges with V/H peaks locating at very short periods (~ 0.05 s), except at Mw 7 for ASA14 and BAK11 which show small differences at longer periods (~ 0.075 and 0.1 s, respectively). In general, our model predicts the highest V/H spectral ratios at intermediate to long periods compare to other models, except for ASA14 which estimates higher V/H ratios at far distances by a maximum of 18% at T = 1 s. According to Fig. 16, the short-period predictions of our proposed model are approximately consistent with Zea18, especially at short distances. Apparently, for 0.7 < T < 2 s, SZ16 model provides relatively consistent results with our model with minor differences at Mw 5 which shows smooth decreasing trend from 17% at T = 0.7 to 6% at T = 2 s. SZ16 predictions are not smooth over all period ranges. At T > 0.4 s, BAK11 and Zea18 indicates the biggest differences of 40% and 32% respectively with respect to our predictions.

The median V/H estimates from our model follow each other closely for soil and rock site conditions. At short periods (T < 0.2 s), our predictions are approximately consistent with other models, while at longer periods, our proposed model predicts notably larger V/H ratios than other comparative models as the differences increases for small-magnitude events. All comparative models exhibit almost flat median V/H predictions for T < 0.2 s and for Mw 5 at far distances, while the V/H predictions increase sharply towards longer periods. Generally, V/H ratios are expected to reach their greatest amplitudes at short periods with large magnitude and short distance (Bozorgnia and Campbell 2004). As it is expected, such effect is well pronounced in Fig. 16b as maximum V/H ratios at short period reach as high as 1.2 for RJB 10 km and Mw 7. This value increases for shorter distance or/and larger magnitude.

9 Conclusion

In present study, we developed a prediction model for the ratios of vertical-to-horizontal for 5%-damped response spectral accelerations in the vibration period range of 0.01 to 10 s, PGA and PGV, using Iranian ground-motion database recorded during shallow crustal earthquakes. Proposed models are applicable for the magnitude range of 4.5 to 7.4 and distances of up to 200 km. This broad range of magnitude and distance functionality allows us to prevent extrapolation usually performed for engineering practical purposes to estimate V/H ratios beyond the valid range of parameters. Four versions of the base functional form are fitted to actual data to explore the most appropriate ones. Among them, two functional forms are selected which both exclude the magnitude-dependent geometric spreading coefficient (b5 in Eq. 1) and one ignores the magnitude-dependent saturation coefficient as well (b3 in Eq. 1). Unlike most recently developed models, magnitude saturation effects have been constrained in our models through second-order magnitude term. The selected equations yield unbiased predictions of V/H ratios as a function of explanatory variables of distance, magnitude, site class, and style of faulting. Two main faulting mechanisms of strike–slip and reverse types are employed in the presented functional forms.

We aimed to improve the applicability of the new GMPEs in estimating vertical response spectrums and believe that the new functions could eliminate unnecessary computational effort for assessment of virtual components in practical seismic hazard studies. For this reason, one of the main advances of this research is the development of consistent models based on four distance metrics of point- (Repi and Rhyp) and finite-fault (RJB and Rrup) models. Compared to four recently developed local and regional V/H models, our proposed models yield the lowermost standard deviations over the whole period range. We verified our models through inspection of between- and within-event residuals analyses and found stable and unbiased trends at all periods, magnitudes, distances, and VS30 ranges. The residuals, thus, indicate that the V/H models are consistent with the empirical data.

The V/H estimates appeared to be notably dependent on spectral periods, distance, and site amplification terms. We demonstrate the significant effect of ground type for intermediate-to-long periods. According to our models, higher V/H values are expected at short periods, short distances (at short periods), and rock site condition (T > 0.2 s). We conclude that for short spectral periods, the predictions of V/H ratios are more sensitive to distance rather than magnitude.