1 Introduction

In last two decades, some great earthquakes (e.g., Kobe 1995, Chi-chi 1999, and Bam 2003) have shown the importance of vertical component of earthquake in inflicting damage on a variety of structural systems (e.g., Palaskas et al. 1996). Saadeghvaziri and Foutch (1991) have shown that axial load variation in columns due to vertical excitation can even cause instability in certain type of structural systems. Yu et al. (1997) have reported about 21 % increase in axial load of columns and about 7 % variation in their moments. In another work, however, Shakib and Fuladgar (2003) have studied a base isolated structure subjected to vertical and horizontal component of earthquake in which axial load in columns of the studied structures was up to three times of what it was expected (without considering the vertical component of the earthquakes). Generally, it is believed that the effects of vertical component of earthquake in structural responses are more pronounced in the near fault regions (e.g., Button et al. 2002; Kunnath et al. 2008; Bommer et al. 2011; Gülerce and Abrahamson 2011).

According to above results, it seems that, in the structural design procedure, both horizontal and vertical design spectra are required in the process to reduce the vulnerability of the structural systems to seismic hazards. However, some of the existing design codes/guidelines are providing the designers only with the horizontal spectrum (such as Standard 2800 2014, Iranian code of practice for seismic restraint design of buildings). In some others, the vertical spectrum is defined using a ratio of 2/3 with respect to the horizontal one (e.g., FEMA 356 2000; ASCE 41 2013). The new trend is toward introducing vertical spectrum using vertical-to-horizontal spectral ratios (such as Eurocode 8 2004; FEMA P-750 2009). Vertical spectrum in Eurocode 8 (2004) is partially based on the model proposed by Elnashai and Papazoglou (1997) and the one introduced in FEMA P-750 (2009) is on the basis of the methodology developed by Bozorgnia and Campbell (2004). Elnashai and Papazoglou (1997) introduced the vertical design spectrum with a flat plateau at short periods (0.05–0.15 s). Bozorgnia and Campbell (2004) proposed a simplified vertical to horizontal spectral ratio (V/H) for scaling the horizontal spectrum to the vertical one.

Typically, there are two different approaches in obtaining the vertical component of response spectrum using ground motion prediction equations (GMPEs). The differences between these approaches are based on the method of using GMPEs in development of vertical spectrum. The first approach is a direct application of GMPEs for vertical component of earthquake, and in the second one, the attenuation model is in the form of vertical to horizontal spectral ratio function. The attenuation model in this case will be used to scale the horizontal spectrum to the vertical one (Ambraseys and Douglas 2003; Bommer et al. 2011; Gülerce and Abrahamson 2011; Soghrat and Ziyaeifar 2016). While V/H ratio usually scales down the horizontal spectrum, it may scale up the spectrum in near distances particularly for the short period range of the response spectrum.

GMPEs have a key role in seismic hazard evaluation for site-specific spectra. To propose a GMPE for any specific region, the magnitude, source-to-site distance, and peak ground characteristics of the earthquakes in that region are required. In addition, some other parameters such as site class, faulting mechanism, and so on might be considered necessary in development of GMPEs for a particular region. GMPEs can be developed using physical or mathematical approaches. For the regions with limited records of ground motion, the application of physical models will be essential for successful prediction of ground motions. In this approach, limited records are employed for the physical model calibration. These models usually have been developed in the context of the random vibration theory and the stochastic modeling approach (see, e.g., Halldorsson and Papageorgiou 2005; Soghrat et al. 2012). One of the most remarkable advantages of physical models relative to mathematical models is that source, path, and site effects may be derived from available limited data through calibration of physical models (in this approach, using ground motion records in back-calculation studies results in finding the ground motion unknown parameters and leads to development of the GMPEs).

Mathematical method, based on regression analyses, is used when abundant records of ground motion are available for the studied region. For regions where there are sufficient data, these mathematical methods are suitable and have been successfully developed. Data sufficiency, type of regression technique, and the classification of data are effective in validity and accuracy of these methods (Soghrat et al. 2012).

Although, various GMPEs have been developed for horizontal component of earthquake in the region (using both physical and mathematical approaches as reported by Sinaiean 2006; Ghodrati-Amiri et al. 2009; Ghasemi et al. 2009; Soghrat et al. 2012; Zafarani and Soghrat 2012), there are no reliable GMPEs for vertical component of earthquake in northern Iran or the whole Iranian plateau based on the recorded data in this large area.

In the current work, using a mathematical approach based on the recorded data and events in the studied region, a database consisting of selected records and events is developed. Among the available functional form for the GMPEs, the one that shows a better fitness to the local data is chosen. Later, the regression coefficients for both horizontal and vertical GMPEs were determined independently. After evaluating the integrity of results, a comparison with some other studies has been carried out to show the accuracy of the models proposed in this work.

2 Database and record processing

Iranian plateau is considered as a large earthquake prone zone, and it is divided into five different tectonic regions (Azerbaijan-Alborz, Kopeh Dagh, Zagros, Makran and central–east Iran classified by Mirzaei et al. 1998). These regions are all shown in Fig. 1. In this work, northern Iran including Azerbaijan-Alborz and Kope-Dagh regions are considered for further studies.

Fig. 1
figure 1

Five tectonic regions of Iran (Mirzaei et al. 1998) and distribution of the studied earthquakes

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The strong motion data for these regions has been recorded by the Iranian Strong Motion Network (ISMN) that can be accessed through the Building and Housing Research Center (BHRC). This network consists of more than 1100 stations and has more than 10,000 three-component accelerograms (analog and digital recordings) in various active seismic regions of the country.

For reducing uncertainties, only those records with available average S-wave velocity in the depth of 30 m (Vs30) are chosen from the database. Four site groups in terms of Vs30 have been adopted in this study (I, Vs30 > 750; II, 375 < Vs30 < 750; III, 175 < Vs30 < 375; and IV, Vs30 < 175). This classification is based on the Standard 2800 (2014) (Iranian Code of Practice for Seismic Resistant design of Building) which is similar to the one proposed by Eurocode 8 (2004) (A, Vs30 > 800; B, 360 < Vs30 < 800; C, 180 < Vs30 < 360; and D, Vs30 < 180). Altogether, 325 acceleration time histories (listed in Appendix Table 7) from 55 earthquakes (tabulated in Table 1) are used in the database. According to this dataset, site classifications for both standards (2800 and Eurocode) are the same for 98 % of the records (318 of 325 records).

Table 1 The list of earthquakes used in analyses procedure
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The range of moment magnitude (M w ) of these events is about 4.1–7.3. Figure 1 shows the locations of the epicenters of these earthquakes. It should be noted that, the moment magnitude of 10 earthquakes (the event number of 5, 7, 9, 10, 19, 35, 36, 37, 38 and 54 in Table 1) with 26 records (about 20 % of events and 8 % of records) are estimated from the magnitude conversion relations reported by Shahvar et al. (2013).

The magnitude-distance distribution for these events is illustrated in Fig. 2. According to this figure, the database is well suited for the records with the magnitudes of 6.5 and lower (and distances less than 200 km). Since the epicentral distance for all the records are available in the dataset, if the Joyner-Boore distance (Rjb) is not available, the epicentral distance is used instead. It should be mentioned that, for the earthquakes with M w  ≥ 6, the Joyner-Boore distance is available for all cases.

Fig. 2
figure 2

Distribution of moment magnitude-distance versus site classification (Standard 2800 2014)

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Histograms for the number of records versus distance and magnitude are shown in Fig. 3. According to these histograms, the dataset is dominated by the records with distances of 50–100 km (about 35 % of records) as well as by those with the moment magnitude of 6–6.5 (about 38 % records).

Fig. 3
figure 3

Histograms of records versus distance and magnitude

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In about 52 % of the events, there is no information (NA) about their faulting mechanisms in the studied region (attributed to the events with M w  ≤ 5.2). The events with reverse and strike-slip faulting are representing about 27 and 21 % of the incidents, respectively. From the viewpoint of record frequency, about 50 % of the records are considered reverse faulting, 20 % are strike-slip faulting and the rest are assumed as unknown.

To reduce the uncertainty in results, the records with the following features are omitted from the dataset: data from instruments that triggered during the S-wave train; stations with no information on the site conditions; data with poor quality; records with only a single horizontal component; and the events that recorded by only one station.

The uncorrected (unprocessed) acceleration time series were corrected by a new multiresolution wavelet analysis proposed by Ansari et al. (2010) to remove undesirable noise from the recorded signals.

3 Functional forms

The form of a ground motion model is usually a function of magnitude, distance, site class, and other available parameters (e.g., style of faulting). Several functional forms have been examined to find a reliable one with the least possible error to fit our dataset. Among different classes of functional forms for modeling the ground motion, the following one is selected for further studies. This functional form is similar to the one proposed by Akkar and Bommer (2007, 2010).

$$ Lo{g}_{10}(Y)={b}_1+{b}_2Mw+{b}_3M{w}^2+\left({b}_4+{b}_5Mw\right){ \log}_{10}\sqrt{R^2+{b_6}^2}+{S}_i+{f}_s{F}_S+{f}_{rv}{F}_{RV}+{f}_{uu}{F}_U $$
(1)

where Y is the response variable (PGA and PSA in cm/s2); Mw, the moment magnitude; and R is the distance. b 1 to b 6 are the regression coefficients. S i (i = 1, 2, 3, and 4) is the coefficient for site classes I, II, III, and IV. Unlike some other works (e.g., Ghasemi et al. 2009; Soghrat et al. 2012; Ghodrati-Amiri et al. 2014), in this study, the style-of-faulting is also included in the functional form. The faulting mechanism coefficients are labeled f s , f rv , and f uu for strike-slip (F S  = 1, F RV  = F U  = 0), reverse (F RV  = 1, F S  = F U  = 0) and unknown faulting (F U  = 1, F S  = F RV  = 0), respectively.

To generalize the functional form for other definitions of site classification, another model based on Vs30 is also proposed in this work. In this model (represented by Eq. 2), the site effect is studied in terms of Vs30 instead of S i in model 1 (Eq. 1). In this functional form (model 2), γ is the regression coefficient for the site effect and V ref is assumed to be equivalent to 760 m/s.

$$ \begin{array}{l}Lo{g}_{10}(Y)={b}_1+{b}_2Mw+{b}_3M{w}^2+\left({b}_4+{b}_5Mw\right){ \log}_{10}\sqrt{R^2+{b_6}^2}+\gamma { \log}_{10}\left({V}_{s30}/{V}_{ref}\right)+\\ {}{f}_s{F}_S+{f}_{rv}{F}_{RV}+{f}_{uu}{F}_U\end{array} $$
(2)

4 Regression analyses and results

For development of GMPEs, two different approaches are typically applicable in the regression analyses, the two-step method and the random effects approach (Abrahamson and Youngs 1992). In this work, the regression analyses have been performed using random effects approach in which the residuals are divided into inter-event (between-event) and intra-event (within-event) terms. The regression model is given as follows:

$$ \log {y}_{ij}= \log {\mu}_{ij}+{\eta}_i+{\varepsilon}_{ij}, $$
(3)

in which y ij represents the response variable from the observed data (the geometric mean spectrum is used for the horizontal component). The μ ij indicates the model prediction for data point j from event i. It is assumed that the intra-event (ε ij ) and inter-event (η i ) residuals are independent and normally distributed with variances σ 2 and τ 2, respectively.

For a given set of regression parameters (represented by vector θ), the residuals are written as follows.

$$ {e}_{ij}\left(\theta \right)= \log\;\frac{y_{ij}}{\mu_{ij}\left(\theta \right)} $$
(4)

A positive sign for the residuals implies under-prediction results with respect to the observed data. The total standard deviation for the residuals in each spectral period is given as

$$ {\sigma}_T={\left({\sigma}^2+{\tau}^2\right)}^{0.5} $$
(5)

Each iteration of random effects procedure requires minimization of the residual e ij (θ) with respect to θ. For minimization of the residuals, the following norm proposed by Abrahamson and Youngs (1992) is used.

$$ \varGamma ={\left({\displaystyle \sum_{i=1}^{N_e}{\displaystyle \sum_{j=1}^{N_{is}.{N}_T}{\left[{e}_{ij}\left(\theta \right)\right]}^2}}\right)}^{0.5} $$
(6)

where N e indicates the number of earthquake events in the analysis, N is is the number of observations, and N T is assumed as the number of discrete spectral periods. In this investigation, N T is chosen equivalent to 31 for the periods between 0 and 4 s.

For model 1, based on Eq. 1, the regression coefficients for the median ground motion prediction and their corresponding standard deviations are given in Tables 2 and 3 for horizontal and vertical components, respectively. To stabilize the results, after some trial and error, the value of b 6 is assumed as a period-independent parameter (b 6  = 7.5).

Table 2 Regression coefficients for horizontal GMPE in model 1
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Table 3 Regression coefficients for vertical GMPE in model 1
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In case of using model 2, the regression coefficients for the median ground motion prediction are derived separately and shown in Tables 4 and 5. Again, similar to the model 1, the b 6 coefficient is assumed equal to 7.5 as a period-independent parameter.

Table 4 Regression coefficients for horizontal GMPE in model 2 using continuous Vs30
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Table 5 Regression coefficients for vertical GMPE in model 2 using continuous Vs30
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The total standard deviations for both horizontal and vertical components are shown in Fig. 4. The average total standard deviations in the range of period 0 to 4 s using model 1 are estimated about 0.312 and 0.326 for horizontal and vertical GMPEs, respectively. In model 2, these values are about 0.312 and 0.329 for horizontal and vertical components without significant differences comparing with model 1. These values are smaller than those resulting from some other models in the studied region. The average total standard deviation for horizontal GMPEs in this region are reported as 0.42, 0.334, and 0.328 by Motazedian (2006), Ghasemi et al. (2009), and Saffari et al. (2012), respectively.

Fig. 4
figure 4

Total standard deviation for models 1 and 2 in terms of period for horizontal and vertical components

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The inter-event and intra-event residuals of the proposed models for both horizontal and vertical components of earthquakes are shown in Figs. 5, 6, 7, and 8. As shown in figures, there is not a noticeable trend between inter-event residuals and moment magnitude of earthquakes for both horizontal and vertical components. Also, according to these figures, the intra-event residuals are unbiased with respect to the moment magnitude and distance parameters. To show the residual bias in terms of distance and magnitude, average of residuals in different magnitude and distance ranges are also shown in the same figures (using cross symbol). In this case, the magnitude bin is chosen 0.5 (∆Mw) and for the distance, ∆R is selected as 25 km. A line fitting is also performed on inter-event and intra-event residuals to find the bias of the proposed model with respect to distance and magnitude parameters.

Fig. 5
figure 5

The error terms for horizontal GMPEs in model 1: a intra-event residuals as a function of distance and magnitude and b inter-event residuals as a function of magnitude

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Fig. 6
figure 6

The error terms for vertical GMPEs in model 1: a intra-event residuals as a function of distance and magnitude and b inter-event residuals as a function of magnitude

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Fig. 7
figure 7

The error terms for horizontal GMPEs in model 2: a Intra-event residuals as a function of distance and magnitude and b inter-event residuals as a function of magnitude

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Fig. 8
figure 8

The error terms for vertical GMPEs in model 2: a intra-event residuals as a function of distance and magnitude and b inter-event residuals as a function of magnitude

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These observations show that the data and the functional forms used in this study are consistent and the models fit the data appropriately. It should be noted that the GMPEs for horizontal and vertical components of ground motion at site class IV may not be considered accurate due to the lack of sufficient data (less than 2 % of records are related to site class IV).

To investigate the suitability of the dataset used in this study, it is decided to exclude ten events without a reported moment magnitude (8 % of the records discussed earlier in this work) from the database and repeat the procedure again. Results showed no practical changes in the model prediction.

Figures 9 and 10 are representing the predicted values for peak ground acceleration and PSA at periods of 0.5, 1, and 3 s versus distance. These figures are based on the moment magnitude of 6 and site class II (Vs30 is equal to 500 m/s) for both horizontal and vertical components. Model 1 and model 2 are also compared with each other in the same figures. In addition to the median values, the median plus/minus a standard deviation is also shown for the comparison. The figures show a close similarity in predicted values between the models. According to the figure, saturation of spectral amplitudes in near distances is quite similar to those reported in common literature (e.g., Campbell and Bozorgnia 2014; Boore et al. 2013; Abrahamson et al. 2014; Chiou and Youngs 2014; Idriss 2014).

Fig. 9
figure 9

Attenuation prediction of ground motion for the horizontal component assuming M w  = 6.0, and SC = II (or Vs30 = 500 m/s) a reverse faulting and b strike-slip faulting

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Fig. 10
figure 10

Attenuation prediction of ground motion for the vertical component assuming M w  = 6.0, and SC = II (or Vs30 = 500 m/s) a reverse faulting and b strike-slip faulting

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Figures 9a and 10a show the observed values in case of reverse faulting for two events with the moment magnitude close to 6 in the dataset (M w  = 5.7 and 6.3). For the strike-slip faulting (shown in Figs. 9b and 10b), there is only one event with M w  = 6.0 in the dataset and therefore, the observed data and predicted values are in a better agreement. Generally speaking, it is believed that the observed values for PGA and PSAs are consistent with the predicted values.

A comparison between horizontal and vertical components of response spectra at magnitudes of 5, 6, and 7 and distances of 15, 60, and 100 km for the site class II is shown in Fig. 11. In addition to the horizontal and vertical components, the vertical to horizontal spectral ratio (V/H) is also plotted in the same figure.

Fig. 11
figure 11

Predicted horizontal and vertical PSA for M w  = 5, 6, 7, and distances of 15, 60, and 100 km at site class II

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The figure shows that the spectral amplitudes decrease as the distance increases in both horizontal and vertical components. According to the figure, in the range of short periods, vertical component of spectral amplitude in near distance is larger than the horizontal one. This is in agreement with the observed data and the studies reported by Bozorgnia and Campbell (2004) as well as Kunnath et al. (2008).

As shown in the figure, the peak value in PSA is usually shifted to the shorter range of periods for the vertical component of ground motion as reported by Kunnath et al. (2008). This shift in the peak value is much more noticeable in near distances and larger magnitudes. This feature of vertical PSA is quite important in designing of buildings in the near fault regions and shows that the ratio of 2/3 recommended for scaling the horizontal spectrum to the vertical one is not credible for all ranges of spectral periods. In fact, at periods less than 0.1 s, the V/H ratio is usually greater than 2/3 (particularly for short distances). Such results are discussed by some other researchers as well (e.g., Bommer et al. 2011; Gülerce and Abrahamson 2011; Akkar et al. 2014). According to these results, vertical component of earthquake is considered important in determination of structural responses, especially in short spectral periods where the ratio of vertical-to-horizontal spectrum is high.

5 Comparisons with local and global predictive models

A comparison between the median predicted horizontal and vertical components of PSA using the proposed ground-motion models with those by other available GMPEs is necessary for validation of our models. Table 6 shows the characteristics of these GMPEs.

Table 6 Characteristics of selected GMPEs for comparison
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All the models prediction (including the model 1 proposed in this work) for PGA and PSA (at periods of 0.5, 1, and 3 s) in site class II with M w  = 6 are presented in Figs. 12 and 13 (for both horizontal and vertical components of ground motion). The figures are based on Vs30 = 500 m/s and reverse faulting, if needed.

Fig. 12
figure 12

Comparing all the models prediction considering M w  = 6, SC = II (Vs30 = 500) in case of reverse faulting for the horizontal component

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Fig. 13
figure 13

Comparing all the models prediction considering M w  = 6, SC = II (Vs30 = 500) in case of reverse faulting for the vertical component

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The horizontal GMPEs developed in this study for distances less than 10 km is similar to the one reported by Soghrat et al. (2012), particularly at PGA and at T = 1 s. It should be noted that the period range of this model (Soghrat et al. 2012) for the comparison is up to 2 s, but the present study extends this range to 4 s.

For distances more than 10 km, the present model is similar to the models proposed by Soghrat et al. (2012) and Akkar and Bommer (2010) at PGA and at T = 0.5 s (low and middle periods). Also, the model is similar to the model proposed by Akkar and Bommer (2010) at T = 1 and 3 s (long periods). Although the methodologies in development of the current model and the one proposed by Soghrat et al. (2012) are quite different, the similarity between them is due to the fact that both of these models are developed using regional dataset (for northern Iran). According to Fig. 12, the local and regional GMPEs (Soghrat et al. 2012; Ghasemi et al. 2009; Akkar and Bommer 2010) are more compatible with the proposed models if they compared with the models developed using worldwide data (Boore et al. 2013; Campbell and Bozorgnia 2014).

Since there are no spectral GMPEs for vertical component of earthquake in this region (only a vertical GMPE for PGA was proposed by Nowroozi 2005 in this area), having a comparison with the local GMPEs is not possible. As shown in Fig. 13 for distances less than 10 km, there is a similarity between the proposed model for PGA with the one introduced by Nowroozi (2005) and Bindi et al. (2011). At periods of 0.5 and 1 s, the proposed model is almost similar to the one developed by Bindi et al. 2010.

For distances more than 10 km, the model in current study is similar to all the models for PGA except the one by Bindi et al. (2010). At T = 0.5 and 1 s, the proposed model is similar to those of Bindi et al. (2010, 2011). In these spectral periods, the model proposed by Campbell and Bozorgnia (2003) has much higher amplitude comparing with other available models in this study due to using a dataset with a distance range less than 60 km.

Some of the differences in predictions of the models are related to the definition used for soil categories and distance parameters. The rest of the differences are presumed to be related to data density and the types of functional form and faulting mechanism.

6 Discussion and conclusions

In this study, 325 records from 55 events with moment magnitude greater than 4 and distances less than 300 km have been used for development of vertical and horizontal GMPEs for northern Iran. The style-of-faulting is also included in regression analyses for both horizontal and vertical GMPEs.

Site classification in this work is based on the Iranian Code of Practice for Seismic Resistant design of Buildings (Standard 2800 2014). This site classification is similar to the one used in Eurocode 8 (2004). In addition, instead of using site groups (site classes I, II, III, and IV), another model for GMPEs based on using Vs30 for site effects consideration is also introduced in this work. The study shows that the predicted ground motions are in a good agreement using either approach. However, the results of regression analyses for the site classification IV in this study may be not considered reliable due to lack of records in the studied region.

The vertical GMPEs in the form of spectral amplitude proposed in this work is considered among the first attempts in development of such models in this region. The total standard deviations of proposed models for both horizontal and vertical components of ground motion represent their higher accuracy if they compare with other available local models in the studied region. The intra- and inter-event errors of these models are stable without any dependency on distance and magnitude parameters in terms of spectral periods.

Although the trends of the main features of the proposed models are similar to those reported in common literature, the spectral amplitudes of these models are not similar to other available models (due to using regional dataset).

The proposed model for the horizontal GMPE is similar to the models suggested by Soghrat et al. (2012) and Akkar and Bommer (2010). The model introduced for vertical component has a close similarity with the models proposed by Bindi et al. (2011, 2011) and Nowroozi (2005) wherever a comparison was possible.

To be accurate, the proposed models may not be used for distances more than 200 km and magnitude more than 7.3 (due to insufficiency of data in this region).