Ground Motion Model for Spectral Displacement of Intermediate-Depth Earthquakes Generated by Vrancea Seismic Source

Abstract

In support of displacement-based design (DBD), an attenuation model for the prediction of the spectral displacement of intermediate-depth earthquakes generated by Vrancea source is proposed. DBD is an alternative to force-based design, the main benefits being a better and confident description of the structural response and the removal of some of the inconsistencies of force-based code design. The basic input for DBD is the displacement response spectrum (DRS). Vrancea intermediate-depth source is responsible for the seismic hazard for most of the Romanian territory. The source produces, on average, two or three earthquakes with M_W >7.0 per century, the prominent characteristics being the large displacement demand and large predominant periods (≈ 1.5 s) for sites located in the Romanian Plain. The model is applicable for sites positioned in front of the South-Eastern Carpathian Arc on type B and C soils. Equations predicting spectral displacement were developed by two-stage regression analysis, using a database containing national analog records of moderate-strong earthquakes and the available digital records, of smaller earthquakes. The model was extended for periods up to 8.0 s using national digital strong motion records and Japanese high-quality digital records of earthquakes triggered by a similar seismo-tectonic environment. The model successfully reproduced observed data, for both type B and C soils and the goodness of fit was tested using methods available in literature.

1. Introduction

Romania is a country exposed to seismic hazard throughout its territory, a fact that is supported by historical evidence spanning more than ten centuries. There are 14 known seismic sources affecting the territory of the country, nine of them located inside the country’s borders. All except one produce shallow earthquakes, most of them with maximum credible magnitudes less than 7.0, being of local interest. On the other hand, the Vrancea subcrustal seismic source, located where the East European plate and the Intra-Alpine and Moesian sub plates converge, is by far the most aggressive source and affects the whole country. Two or three major earthquakes are generated per century, with hypocenters at depths of 70–110 or 130–160 km and epicenters localized inside a small rectangular region with dimensions of about 80 × 40 km. Vrancea subcrustal earthquakes are felt at large distances over an extended area in South-East Europe.

Intermediate-depth earthquakes usually occur in subduction zones where two tectonic plates are in contact, one slipping and sinking underneath the other. For the Vrancea seismic zone there is evidence that subduction ceased 10 million years ago [1]. Since the 1970s, the researchers hypothesized that the intermediate-depth seismicity in the Vrancea area is related to the dipping of a portion of a tectonic plate into the mantle (asthenosphere). This would be the last stage of the subduction process. The nature of the plate (oceanic or continental) is, for the time being, a subject of debate among seismologists. The fact that there is a low seismic activity, located at depths of 40–70 km, has led to the idea that the plate fragment is already detached from the continental crust. Accordingly, the fragment, originally quasi-horizontal, has reached a nearly vertical position, and is colder and denser than the surrounding environment and descends under the action of gravity. The bottom of the descending fragment is located at a minimum depth of 350 km. Interaction between gravitational forces, buoyancy, viscous and friction forces produces shear forces large enough to trigger earthquakes in the descending body [2].

The focal mechanism for all M_W >7 earthquakes is reverse faulting with the fault plane oriented along the NE-SW direction [3]. At least one major event (4 March 1977, M_W = 7.4) was a multi-shock source, with a foreshock and a subsequent three shocks generated within a 20-s time interval. This feature is in agreement with a large release of energy over the surface of the rupture with local asperities or stress field non-uniformities.

As mentioned, the epicenters of the subcrustal Vrancea earthquakes are confined to a small rectangular region elongated along the NE-SW direction [4]. The alignment of the epicenters towards the NE-SW, as well as of the rupture plane for the major shocks, can partly explain the directional effects towards Bucharest (SW) or Moldova (NE). There is a tendency of increasing magnitude with increasing depth. This phenomenon was explained by the increase in resistance of the asperity cell along with the increase in the depth due to the lithostatic pressure.

Regression relations that correlate with the magnitude of the earthquake with the length of the rupture surface, and the surface area of the rupture [5] can provide the maximum magnitude of the source. The maximum values for the surface rupture length are 150–200 km and, for the surface of the rupture area, ≈ 8000 km². These values lead to a maximum credible earthquake magnitude of M_W ≈ 8.1 and a probable focal depth estimated between 140 and 170 km [4].

This paper focuses on the main input for DBD procedure, the relative displacement spectrum, for earthquakes of intermediate-depth and sites located in front of the Carpathian Arc, corresponding to the historical regions of Moldova, Muntenia and Dobrogea.

2. Ground Motion Model

Ground motion prediction equations (GMPE) quantitatively represent the way in which a parameter of the earthquake ground motion decreases with increasing source-site distance. The vast majority of prediction equations use the horizontal response spectral acceleration (SA) as a parameter that is representative of the seismic motion. In the 1990s, the concepts of performance-based design and displacement-based design were devised, to better control the structural behavior. These concepts have, as a starting point, the idea that is more meaningful to set and check specific performance levels (a certain earthquake intensity measure and a corresponding damage state), and the fact that the damage caused by earthquakes to building structures is better related to the peak relative displacements than to peak accelerations. Although GMPEs for peak ground displacement were available since 1974 [6,7], only in 1998 [8], then in 2004 [9], were significant efforts towards the development of GMPE for relative displacement response spectrum made. In recent years, the attenuation models for DRS began to receive researchers’ attention due to increasing DBD popularity, in the studies of Akkar and Bommer [10], Cauzzi et al. [11,12] and Faccioli et al. [11,12,13], followed more recently in [14,15]. Some of the aforementioned models provide the DRS with different values of damping, all of them are valid for shallow earthquakes (note that the GMPE in [15] uses a database which also includes Romanian strong motion records generated by shallow earthquakes).

In Romania, the first studies on ground motion models were developed by Lungu 1994 [16] and Radu 1994 [17]. The models were azimuth-dependent and predicted the peak ground acceleration (PGA). Subsequent studies by Stamatovska and Petrovski [18] for PGA, Sokolov [19] for pseudo-spectral acceleration, peak ground velocity and Medvedev-Sponheuer-Karnik (MSK) scale seismic intensity, followed the approach of azimuth-dependent coefficients. In 2000, Lungu [20] developed another model based on the same functional form as the one in 1994, with coefficients which are not azimuth-dependent. An attenuation model for peak ground acceleration of Vrancea subcrustal earthquakes was elaborated [21] and used in the Romanian seismic design code, P100-1/2013, in order to estimate the horizontal forces needed for design. More recently, Vacareanu et al. [22], developed a new model for spectral acceleration in 2015 [23], introducing a zonation of Romanian territory in Fore-Arc and Back-Arc regions, accounting for the different attenuation and filtering, corresponding to the two regions located on the sides of the Carpathian Arc.

Theoretically, DRS needed for displacement-based design could be determined from the pseudo-acceleration spectra, provided that the spectral ordinates at intermediate-long periods are reliable. Response spectra derived from analog records are considered valid up to periods of 4 s. High-quality digital records provide valid spectral ordinates for periods exceeding 10 s.

The displacement spectra are more dependent on moment magnitude than the acceleration spectra, and the shape and ordinates of the DRS are much more sensitive to the way acceleration processing/correction was performed than the acceleration spectrum ordinates. A GMPE was developed for the relative displacement spectrum applicable in Romania, using a database of analog and digital records, by direct processing of DRS of the records.

2.1. Strong Motion Database. Processing of Records. Ground Types

In order to develop a GMPE, a database containing strong motion records was compiled. The databank used for this study contains 272 ground motion records (544 horizontal components) from 15 intermediate-depth earthquakes. A number of nine earthquakes were produced by Vrancea subcrustal source (235 records), while six were recorded in Japan. The database contains earthquakes with magnitudes 5.2≤ M_W≤ 7.4, with focal depths in the range 65–160km and partly covers the database considered by Vacareanu et al. in [23]. From the total number of records, 169 were from stations located on type C ground (62%), the remainder being recorded on ground type B. Digital records add to 57% of the total number and are generated by earthquakes having 5.2≤ M_W≤ 7.1. Therefore, most earthquakes in the database, with M_W >7.0, were analogical recorded in Romania. Figure 1 shows the database configuration; the size of the data point is correlated with its magnitude.

Because of the small number of ground motion recorded on firm ground (rock or rock-like formation including, at most, 5 m of weaker material at the surface and v_s30 >800 m/s, type A according to Eurocode 8) and the scarcity of strong ground motions behind the Carpathian Arc (Transylvania), these records were not selected in the database.

Figure 2a,b shows the main tectonic structures and the major recorded seismic events produced by the Vrancea intermediate-depth source.

The decision to include non-Romanian records was taken as a result of the lack of high-quality (digital) national records for earthquakes with M_W ≥6.0. The recommendation in the literature is to use records from other countries, when there are not enough local records available. Furthermore, it is desirable to extend the databases with "import" records to obtain GMPE, especially when local records do not cover the full range of magnitudes and distances for which the attenuation model is designed.

The Japanese earthquakes [24] have focal depth in the range of 65–125 km and magnitudes 6.0≤ M_W≤ 7.1. Unfortunately, the database of the two networks (KiK-net and K-NET) does not include records for seismic events with M_w >7.1 for depths between 60 and 200 km.

Table 1. Database structure.

Date (year/month/day)	Local Time	Latitude (°N)	Longitude (°E)	Focal Depth (km)	Mw	Number of Horizontal Components	Country of Origin
1977/03/04	19:21:54	45.77	26.76	94	7.4	4	RO
1986/08/30	21:28:37	45.52	26.49	131	7.1	70	RO
1990/05/30	10:40:06	45.83	26.89	91	6.9	92	RO
1990/05/31	00:17:48	45.85	26.91	87	6.4	66	RO
2004/10/27	20:34:36	45.84	26.63	105	6.0	92	RO
2005/05/14	01:53:21	45.64	26.53	149	5.5	14	RO
2005/06/18	15:16:42	45.72	26.66	154	5.2	14	RO
2009/04/25	17:18:48	45.68	26.62	110	5.4	10	RO
2013/10/06	01:37:21	45.67	26.58	135	5.2	108	RO
2001/12/02	22:02:00	39.40	141.26	122	6.4	6	JAP
2003/05/26	18:24:00	38.81	141.68	71	7.0	24	JAP
2005/07/23	16:35:00	36.58	140.14	73	6.0	6	JAP
2008/07/24	00:26:00	39.73	141.63	108	6.8	16	JAP
2011/04/07	23:32:00	38.20	141.92	66	7.1	18	JAP
2013/02/02	23:17:00	42.70	142.23	102	6.5	4	JAP

summarizes the main characteristics of the earthquakes selected in the database.

In [11,12], attention is drawn to the sensitivity of the displacement spectra to the quality of the recording (digital vs. analog) and the way the accelerograms are processed. Analog records obtained during seismic movements are affected by various types of errors (due to instrumentation and digitization, among others) that affect recording quality, especially at high (>20 Hz) and low (<0.5 Hz) frequencies. Low-frequency errors affect the history of velocities and displacements, while high frequency errors particularly affect the peak ground acceleration. To limit the effect of these errors, various corrections and filters are used. Filtering removes the errors, however, along with them, it eliminates the useful information present in the filtered frequency range [25].

The analog records were obtained in a form already processed; the waveforms were not further adjusted. The methodology used for filtering is described in [25]. The filtering procedure is not uniformly applied, the filter is of Ormsby type and the cutting thresholds are 0.15–0.25 Hz for low frequencies and 25–28 Hz for high frequencies. The digital records were processed by applying a fourth-order Butterworth filter with the lower threshold at 0.05 Hz while the higher is at 50 Hz.

Although the Romanian seismic code in force evaluates soil conditions following the approach of Lungu, which is based on control periods [26], for important structures, the code recommends studies to characterize field conditions: the shear and compression wave velocity profile, v_s and v_p, down to the base rock or minimum for the first 30 m and the terrain stratification (thickness, density, type). Then, the weighted average value v_s on the considered stratification is calculated and the soil is classified according to Eurocode 8.

In this study, Eurocode 8 [27] terminology was used. v_s,30 has the advantage that it is a proven method (used in countries like US, Japan), is recommended by national seismic design code and can be applied relatively easily. Within the BIGSEES project (a Romanian multidisciplinary study aimed at improving earthquake risk mitigation in a Eurocode 8 framework), a database containing stratifications and compression wave velocities was created. Most borehole measurements were conducted in the 1970s, and information on shear wave velocities is no longer available [28]. There is a small number of boreholes with depths between 13 and 150 m, located in Bucharest, for which there is a complete set of data. In the study of Allen and Wald [29], a methodology is proposed to obtain information on shear wave velocity using topographic slope data. Using a correlation between the topographic slope and the data recorded by v_s,30 in several locations in the United States, Taiwan, Italy, Puerto Rico, New Zealand and Japan, the v_s,30 data from the slope topography survey can be used to describe the soil conditions at a regional level. Following this methodology, a map was created [28], which allows the assessment of soil conditions for Romanian territory.

Studies by Neagu and Aldea, [28], using data from 19 bore holes in Bucharest showed a good correlation between values given by the Allen and Wald study [29] and field measurements. The differences between the two datasets are, on average, 12% (the slope method slightly underestimating the shear wave velocity), with a maximum difference of 28%. In spite of all these differences, ground type classification is the same for both methods for the surveyed sites.

In this study, the values of the shear velocities for the first 30 m, for locations where the waveforms were recorded, are according to [29] and available on the United States Geological Survey website [30]. Japanese sites, are usually assigned v_s,30 values [24]. However, some sites do not have boreholes extending to the depth of 30m, so the values for v_s,30 were determined using the methodologies described in [31] and using the database files available in [32].

2.2. Ground Motion Prediction Equation

The coefficients of the attenuation model for relative displacement response spectrum ordinates were determined using two-stage regression analysis, following the methodology given in Joyner and Boore [33,34]. Two-stage regression is used in order to uncouple magnitude scaling and distance scaling. The method is used extensively in determining the coefficients of attenuation laws, and is based on maximizing the likelihood of the set of observations.

The first step of the two-stage regression algorithm consists of determining the coefficients which give the distance dependence, and an array of deviations (for each record). In the second step, coefficients expressing magnitude dependence are determined by maximizing the likelihood of the set of observations.

The functional form of the GMPE, given in [33], is based on the random-effects model of Brillinger and Preisler [35]

$$\lg (SD)=a+b(M_W-6)-\lg \sqrt{D_{epi}{}^2+h^2}+c\sqrt{D_{epi}{}^2+h^2}+{\epsilon }_r+{\epsilon }_e$$

(1)

where SD (cm) is the spectral ordinate of relative displacement (as a geometrical mean of two perpendicular horizontal components) for 5% damping, M_W is the moment magnitude of the earthquake, D_epi (km) is the epicentral distance, a, b, c and h are coefficients which are determined through regression, ε_r is an independent random variable normal distributed, which takes values for every record, ε_e is an independent random variable normal distributed with values for every earthquake and lg denotes base 10 logarithm. Random variable ε_r has the mean equal to 0 and variance σ_r², represents the variability between seismic stations (intra-event), while random variable ε_e has 0 mean and variance σ_e², representing the variability between seismic events (inter-event). Total variance is

$${\sigma }^2={\sigma }_r^2+{\sigma }_e^2$$

(2)

Original functional form [33] uses the Joyner-Boore distance (the shortest distance from the seismic station to the vertical projection of the ruptured surface) as a metric instead of D_epi, which was used in this study. Because Joyner-Boore distance is not available for intermediate-depth earthquakes generated by the Vrancea source, epicentral distance was chosen as a predictor variable.

The attenuation model used for peak ground acceleration in P100-1/2013 zonation uses as a distance metric the hypocentral (focal) distance, R_hypo. For this study, epicentral distance was proved to be similarly correlated with computed spectral displacements as the hypocentral distance. Moreover, formally, the significance of the epicentral distance is closer to the Joyner-Boore than the focal distance. Figure 3 and Figure 4 presents, side by side, the correlation factors of spectral displacements (for earthquakes with M ≥6 recorded in Romania and Japan) with D_epi and R_hypo for T = 1.0 s.

The records were arranged in three bins, according to their magnitude and soil type. One can notice, for soil type B and C, the epicentral distance correlates with the logarithm of spectral displacement as the focal distance does, with the exception of strong earthquakes and soft soil (type C).

The first two terms of the GMPE take into account the quasilinear variation of the logarithm of amplitude with magnitude, with the moment magnitude scale being selected to express the size of the earthquakes used in the database. The third term corresponds to the geometric attenuation of the seismic waves, which decreases proportionally with the inverse of the distance. The fourth term corresponds to the anelastic attenuation, due to the media traversed by the seismic waves.

As expected, there is a strong correlation between the moment magnitude and logarithm of spectral displacements, as shown in Figure 5. For the moderate-large national records set, the records were sorted in three bins, according to their epicentral distance. A slightly better correlation is observed for a quadratic expression of the lg(SD) variation, especially for the case of sites located at epicentral distances between 100 and 150 km. Considering a quadratic dependence of lg(SD) with magnitude would require an additional term in the functional form.

In this study, the relative displacement spectrum ordinates are expressed as the geometric mean of two perpendicular horizontal components. It is preferred to perform the regression analysis based on this quantity for it is regarded as statistically representative for any random direction. Most attenuation models use the geometric mean instead of the maximum value as the expected parameter.

Variability between seismic stations (intra-event), expressed through variance σ_r² can be computed, adapted after [36,37], as

$${\sigma }_r^2={\sigma }_1^2+{\sigma }_c^2$$

(3)

$${\sigma }_c^2=\frac{1}{no.rec}{\displaystyle \sum _{j=1}^{no.rec}\frac{{\left(\lg Y_{1j}-\lg Y_{2j}\right)}^2}{4}}$$

(4)

where σ₁² is the variance computed in the first stage of regression, and indexes 1 and 2 are the horizontal perpendicular components of the record j. The original expression has natural logarithms instead of decimal logarithms at the right side of the equation, and the GMPE was also expressed in terms of natural logarithms. σ_c² is a correction applied through variance in order to take into account that a randomly oriented horizontal component could be larger than the computed geometric mean.

The coefficients of the attenuation model were determined separately for ground type B and ground type C, due to much smaller ordinates and different spectral shapes of displacement spectra computed on ground type B.

Aiming at a better prediction of the response for large earthquakes (M_W >7.1), the opportunity of adding a quadratic term to the basic attenuation model was explored, leading to the following equation

$$\lg (SD)=a+b(M_W-6)+d{(M_W-6)}^2-\lg \sqrt{D_{epi}{}^2+h^2}+c\sqrt{D_{epi}{}^2+h^2}+{\epsilon }_r+{\epsilon }_e$$

(5)

which has indeed led to the improvement in the predictions of the DRS for the earthquake recorded on 4th of March 1977 and has, to a certain degree, reduced the residual values.

Due to the fact that all the moderate and major earthquakes in Romania are analog-recorded, the calculated values of the displacement spectra can be considered valid up to periods of maximum 4 s. Efforts have been made to predict spectral values up to 8 s. National records after 2004 are digital and of high quality, but were produced by earthquakes with M_W≤ 6.0. This was one of the reasons why Japanese intermediate-depth earthquakes, with magnitudes close to the major events in Romania, were added to the database. Recordings from Japan are high-quality digital records that can be considered reliable for periods exceeding 10 s.

3. Results and Discussion

To investigate the database dependence of the attenuation model, the regression was performed on three sets of data, one of which contains national records of moderate-large earthquakes of 1977, 1986 and 1990; the second set of data was made up of digital records only (national, with 5.2 ≤ M_W ≤ 6.0 and Japanese, from Kik-Net and K-NET networks, with 6.0 ≤ M_W ≤ 7.1) with coefficients calculated up to periods of 8 s and, finally, a set which contains the entire database and coefficients calculated for periods in the range 0.1–4.0s.

The GMPE is valid in the area in front of the Carpathian Arc: Moldova, Muntenia and Dobrogea on ground types B and C.

3.1. Moderate-Strong Set of Records

The first dataset includes records from 4th of March 1977 earthquake (M_W =7.4), 31st of August 1986 (M_W = 7.1), 30th and 31th of May 1990 (M_W = 6.9 and M_W = 6.4). The main features are the large displacement demands imposed on high rise structures (T >1.0 s) located on ground type C, this characteristic being common amongst historical earthquakes, such as the massive 1940 earthquake, which inflicted heavy damage on tall buildings in Bucharest, and the devastating earthquake of 1802, that caused the collapse of Coltei Tower and the majority of bell towers in Bucharest.

Due to the fact that the set of accelerograms from these earthquakes had already been processed [25], no further adjustments were made. The records were then sorted by the ground type and then the computation of elastic displacement spectra for a 5% damping was performed. The software used for spectra calculation was Seismosignal [38] for a range of periods between 0.025 and 4.000 s, with 0.025 s increment. The geometrical mean of spectral displacement for every period and record in the set was then computed. Two examples are given in Figure 6 to illustrate the difference in DRS shape and a period of peak value in terms of soil type category and epicentral distance.

One can observe the large difference between the maximum displacement values (48.5 cm at INCERC station (National Institute of Research and Development in Constructions), compared to 3.3 cm in Chisinau) and between the spectral forms, although the maximum values of the displacements are located within the same period interval: 1.5–2.0 s.

As expected, the spectral maximum value scales with the magnitude, as can be seen in

Table 2. Magnitude scaling of displacement spectra, INCERC site

Event (year/month/day)	MW	PGA, cm/s2		SDmax, cm
		EW	NS	EW	NS
1977.03.04	7.4	188	207	32.4	48.4
1986.08.31	7.1	109	96	8.8	12.4
1990.05.30	6.9	99	66	9.3	3.4

for INCERC recordings for the earthquakes in this dataset (table includes also the peak ground acceleration (PGA) values).

Note that, for an increase in magnitude from 7.1 to 7.4, the PGA increases two-fold, while the maximum spectral displacement increases four times and, for a magnitude increase from 6.9 to 7.1, PGA increases by 25% and displacements double (the geometric mean of the two components). The difference in magnitude affects the long period components of the ground motions while PGA is related to short period components.

After calculating geometric mean and intra-event variance, the two-step regression was performed following the procedure described in [33,34], with the coefficients of the attenuation model determined by maximizing likelihood. Coefficients were determined for periods ranging from 0.10 to 4.00 s with an increment of 0.1 s and are presented in Appendix A.

Figure 7 presents the outcome of predictions of recorded seismic ground motions found in the database against computed DRS, as a geometrical mean of the two horizontal components of the actual records. The top of the figure shows spectra reproduced on type B soils: at the left are Onesti station and 1990 earthquakes, while at the right are the results for Cahul and Valenii de Munte stations. The lower side of the figure presents DRS reproduced on type C soil for the 1986 and 1990 earthquakes for Ramnicu Sarat (left), respectively, Otopeni and Peris stations (right). The stations were selected so that the epicentral distance can be the same.

The thin lines represent the geometric mean of the two horizontal components for each record. The thick red line designates the median value of the predicted spectrum, whilst the grey zone encompasses a zone ± 1σ from the median values.

Using the attenuation relationship, the change in displacement spectra with changing magnitude, ground type and epicentral distance was analyzed. There is a narrowing of the area of large amplifications of the displacement spectrum with increasing magnitude of the earthquake, especially for soil type C. Smaller earthquakes tend to have quasi-flat areas over an extended range of periods, as confirmed by the displacement spectra calculated for the 1986 and 1990 earthquakes. There is a large difference between displacement demands of earthquakes separated by one degree of moment magnitude, particularly for type C soil, as shown in Figure 8.

Regarding the soil conditions, there are very high values of the relative amplification between the expected spectral values on soil C with respect to soil B. However, for moderate magnitudes (M_W ≈ 7.0), these are reduced to values found in the literature [11]. These large values of amplification are supported by the computed DRS for seismic stations located at the same epicentral distance, for 1986 and 1990 earthquakes. For smaller magnitudes, M_W ≤6.5, there is only a small increase (15–30%) in displacement for sites located on ground type C relative to sites on ground type B.

The epicentral distance variation is analyzed in Figure 9. For soil type C, the pronounced peak is near 2.30 s and has a very limited tendency to migrate to longer periods with increasing epicentral distance. Soil type B is characterized by much lower spectral displacements, with a pronounced peak at around 1.70 s followed by a relatively flat area.

For every 50 km increase in epicentral distance, the spectral displacement values drop by 1/3. The increase in epicentral distance flattens the peaks and smoothens the spectra.

In order to bring the GMPE outcomes closer to the data collected during strong earthquakes (M_W ≥ 7.1), it was attempted to introduce a quadratic term in the attenuation model. Figure 10 shows the reproduced and computed DRS spectra for Vrancea largest recorded earthquakes of 1977 and 1986.

It is noted from Figure 10 that the median value ± 1σ envelopes the two components of each record for these two large earthquakes. Figure 11 presents the predicted spectra using the GMPE with quadratic term for soil type C, a subcrustal seismic event with M_W =7.5 and epicentral distances of 100 and 150 km.

The displacement demand reaches very large values, for both median and median + 1σ, for spectral periods larger than 2 s. The two predictions highlight peaks at 2.3–2.4 s for median and 2.6 s for median + 1σ, and a “sombrero” shape for the DRS. A recent study [39] on the design displacement spectra, using stochastic finite-fault predictions, pointed to mean displacements in excess of 100 cm (at 2.0–2.5 s) for sites located in the Bucharest area, for an event with M_W = 7.5. Using the GMPE with a quadratic term, similar values for the median spectra were found.

Due to the quadratic term, the attenuation model reaches an extremum point. Its numerical value, M_W^sat, is obtained by deriving the expression of the GMPE in relation to the magnitude, zeroing and solving the equation [23].

For ground type B, an upper limit magnitude must be set for the whole range of periods. Unfortunately, M_W^sat is in the magnitude range of the dataset analyzed. Therefore, the quadratic term attenuation law for soil type B is

$$\begin{array}{l}\lg (SD)=a+b(M_W-6)+d{(M_W-6)}^2-\lg \sqrt{D_{epi}{}^2+h^2}+c\sqrt{D_{epi}{}^2+h^2}+{\epsilon }_r+{\epsilon }_e\\ use M_w=7.00 \mathrm{for} M_w>7.00, 0.0\le T\le 4.0s\end{array}$$

(6)

For ground type C, for T ≤0.2 s, M_W^sat is larger than 7.60. For T >0.2 s, M_W^sat is lower than 6.40. Therefore, the GMPE with quadratic term, for soil type C is

$$\begin{array}{l}\lg (SD)=a+b(M_W-6)+d{(M_W-6)}^2-\lg \sqrt{D_{epi}{}^2+h^2}+c\sqrt{D_{epi}{}^2+h^2}+{\epsilon }_r+{\epsilon }_e\\ use M_w=7.60 \mathrm{for} M_w>7.60, T\le 0.20s\\ use M_w=6.40 \mathrm{for} M_w<6.40, T>0.20s\end{array}$$

(7)

In fact, with the exception of spectral periods between 0.2 and 0.4 s, M_W^sat is larger than 7.9, so the attenuation model with quadratic term could be extended to this magnitude.

3.2. The Set of High-Quality Digital Records

This set includes only digital records from earthquakes with magnitudes in the range 5.2 ≤ M_W ≤ 7.1, which occurred at depths between 66 and 135 km in Romania and Japan. After filtering the records according to the procedure described above, the DRS were calculated for periods up to 8.0 s using Seismosignal [38] and ViewWave [40] software. The purpose for which the investigation was pushed to such large period values was to map the area of the spectrum beyond 4 s, where reliable information from the large and moderately analog recorded earthquakes is not available. There is information suggesting that spectral peaks have higher ordinates than those present in the relative displacement spectrum at 2.0 s for large earthquakes and locations in the Romanian Plain. Both seismological (Brune Model) and geotechnical considerations lead to the conclusion that such peaks would be around 5–6 s. Unfortunately, this study could only highlight this to a small extent.

Up to 4.00 s, the increment was 0.10 s, then it was set to 0.20 s for periods between 4.0 and 6.0 s and finally, for periods up to 8.00 s, the regression coefficients were determined at a 0.40 s increment. Figure 12 presents the prediction of the DRS for two sites located on ground type B and C, for a M_W = 6.0 earthquake. The prediction is compared to the computed displacement spectra from the records.

After 4.00 s, which is the last period for which the analog records are considered reliable, there are no relevant peaks, just flat zones; probably, earthquakes in this database were not strong enough to excite the layers of sediment that have fundamental periods between 4.0 and 8.0 s. Some records have small peaks around 6–7 s. Spectral values are lower than those corresponding to the first peak, which is around 1.20 s for ground type C (instead of about 2.00–2.30 s for the first set of records).

However, the model manages to reasonably predict the spectrum of a record that was not included in this regression dataset, EREN 1986, presented in Figure 13. The reproduced spectrum, according to the GMPE coefficients derived using this dataset, is steeper for this record and yields plateau values larger than those predicted using the GMPE derived using a moderate-strong national set, which was computed directly from the record. The steeper slope is distinctive for records of small magnitude earthquakes, for which this dataset is very rich, whilst the large values in the constant displacement region are due to Japanese records, with a higher displacement demand for the same earthquake magnitude.

It is worth noting that, thanks to the abundance of reliable instrumental data collected from the earthquakes in this dataset, some anomalies were identified. There are significant differences between the spectral ordinates of the computed spectra for sites with a similar epicentral distance and the soil conditions located in Dobrogea versus sites located in Moldova or Romanian Plain. There were also important amplifications (by a factor of three to six) of the ground motion at large distances from the epicenter (Singureni station, 2013 event and Fulga de Sus, Petresti stations, 2004 earthquake). These features were also observed for the 1986 earthquake, for Otopeni station. Possible causes may be local tectonics and the existence of unknown faults or adverse site effects. Locally, these could channel the seismic energy different than estimated through the attenuation model. Nevertheless, the anomalies occur for a rather small number of stations.

3.3. Analysis of the Complete Data Set

The set includes all records in the database: 272 pairs of perpendicular horizontal components. With the increase in the number of records (especially those of small magnitude earthquakes, M_W = 5.0–6.0), the variability increases. The recorded ground motions of the 4 March 1977 earthquake, which imposes the highest displacement demands, loses its share from the first set (containing only 116 pairs of components), which is reflected in the shape and spectral values generated by this model.

Figure 14 shows a prediction of the median displacement spectra corresponding to a seismic event of M_W = 7.5, located at an epicentral distance of 150 km; all three sets of regression coefficients for both C and B soils are analyzed. For ground type C, the similarity between the shapes and values of sets one and three (the complete set) to 2.00 s is observed, after which they evolve separately, both having peaks at approximately 2.30 s. The second set, containing only small and moderate earthquakes, has a different spectral shape, with peaks between 1.20 and 1.60 s.

As pointed out before, the sets two and three, containing large amounts of records from smaller earthquakes, have steeper slopes. Dependence on database results is more evident in type B soil for sets one and three, where, although spectral shapes are similar and reach their maximum between 1.30 and 1.50 s, the plateaus which occur after the peak are at completely different levels.

After examining the above figures, the following conclusions can be drawn:

• For Type B soils, the coefficients of the attenuation model corresponding to the complete set seem to be an appropriate trade-off with respect to the values for the other two groups of records;
• For prediction of DRS generated by very strong earthquakes, M_W ≥ 7.40, the GMPE with the coefficients resulting from the regression of the first set is closer to the observed data for type C soils. The GMPE with quadratic term matches both the spectral shapes and the maximum displacement reasonably well and leads to greater displacement demand than the one shown in Figure 14. A design spectrum should envelope all relevant shapes, considering the appropriate ordinates of the peaks and the variability, through an appropriate number of standard deviations.

3.4. Model Testing

Once the regression coefficients have been calculated, it is important to check that the data provided by the attenuation law are reliable and whether the attenuation model can generate useful information from a dataset other than that used for regression. An important role in model testing is played by residuals, with quantities resulting from the differences between the recorded values and the values predicted by the attenuation model. Positive values of residual indicate underestimation of the seismic motion amplitudes, with negative ones indicating overestimation. Normalized residuals (NRES) are traditionally defined as

$$\epsilon =\frac{Y_{es}-{\mu }_{es}}{\sigma }$$

(8)

with ε being the normalized residual, Y_es is the logarithm of the amplitude of ground motion recorded during the earthquake e at station s, μ_es is the logarithm of the median value provided by the GMPE, and σ is the standard deviation of the attenuation model. Figure 15 shows the distribution of normalized residuals for four periods.

Figure 16 presents the distribution of normalized residuals compared to those produced by a standard normal distribution using Q-Q (quantile-quantile) plots. A Q-Q plot is a meaningful way to assess graphically two probability distributions, helping to check that the two populations have the same statistical distribution. The more normalized the residuals are when approaching the line that passes through the origin, the better the normal distribution describes the residual distribution. We see a distribution close to the normal one, both in histograms and in the alignment of residuals with the line.

The plots for 1.0, 1.5 and 2.0 s have most of the residuals between −1 and + 1, and they are close to the theoretical distribution. A small part deviates from the expected repartition, usually the ones that are larger than + 1.5. Therefore, there is a small tendency of the GMPE to underestimate the DRS ordinates for those periods, as shown in Figure 17. Keeping in mind that the numerical values of the residuals are reasonably small (smaller than + 2), the deviation could be accepted. Both positive and negative outliers are usually from earthquakes with a low magnitude (5.2, 5.5).

Two quantities can be introduced: the inter-event and the intra-event residuals. Inter-event residuals are calculated [41] using the following equation

$$\delta B_e=\frac{1}{N_s}{\displaystyle \sum _{s=1}^{N_S}(Y_{es}-}{\mu }_{es})$$

(9)

where N_s is the number of stations, and Y_es and μ_es are previously defined. Intra-event residuals are given by

$$\delta W_{es}=Y_{es}-\left({\mu }_{es}+\delta B_e\right)$$

(10)

The evaluation of these parameters allows verification of the distribution of inter-event residuals with the magnitude and distribution of intra-event residuals with distance. Figure 18 presents the distribution for two periods of 0.50 and 1.00 s, for the third set of data and soil type C.

The correlation between the moment magnitude and the value of the inter-event residuals for both periods is small and can be neglected. This is due to the relatively small number of earthquakes in the set. The digital set has also a negligible correlation with regression lines with minor slopes. Considering the data presented in the literature, the results shown above can be considered satisfactory.

From Figure 19, one can see that there is no correlation between residuals and distance. The above figures are representative of the entire range of periods covered by the GMPE.

With the values of the residuals, it is possible to calculate some statistical parameters by which the quality of the attenuation relation can be assessed. Those were proposed in [42], and they are median of the normalized residuals (MEDNR), mean of the normalized residuals (MEANNR) and standard deviation of the normalized residuals (STDNR). Depending on these indicators and limit values, the attenuation models are grouped into four categories of confidence, rated from A (best) to D (those not recommended to apply). The three statistical indicators are calculated for each period, for each set of data, and point towards an A rating for all sets and a large majority of spectral periods.

Table A1. Coefficients of the GMPE for set 1, Equation (1), ground type B

T (s)	a	b	c	h	σr2	σe2	σ2	Meannr	Mednr	Stdnr
0.20	0.9743	0.4724	5.39 × 10−4	85.2	5.78 × 10−2	1.37 × 10−2	7.15 × 10−2	0.0304	0.0943	0.9110
0.40	2.3198	0.5288	−2.20 × 10−3	198.2	6.71 × 10−2	6.64 × 10−3	7.38 × 10−2	0.0414	0.0750	0.9411
0.60	1.9598	0.5393	−8.33 × 10−4	105.8	1.12 × 10−1	2.48 × 10−3	1.14 × 10−1	0.0123	0.0605	0.9680
0.80	1.9577	0.5734	−4.98 × 10−4	102.1	1.06 × 10−1	1.99 × 10−3	1.08 × 10−1	0.0089	0.0544	0.9637
1.00	1.8047	0.5086	5.82 × 10−4	59.6	1.04 × 10−1	1.13 × 10−2	1.15 × 10−1	0.0404	0.0896	0.9163
1.50	1.9139	0.5507	3.91 × 10−4	52.8	8.72 × 10−2	6.62 × 10−3	9.38 × 10−2	0.0264	0.0738	0.9369
2.00	2.0080	0.4867	3.40 × 10−4	54.8	7.55 × 10−2	1.70 × 10−2	9.26 × 10−2	0.0709	0.2011	0.8895
2.50	2.1979	0.3974	−1.02 × 10−4	64.1	6.25 × 10−2	2.33 × 10−2	8.59 × 10−2	0.0994	0.1528	0.8700
3.00	2.2554	0.3990	−3.15 × 10−4	70.2	5.48 × 10−2	2.61 × 10−2	8.10 × 10−2	0.1131	0.1188	0.8616
4.00	2.4494	0.3673	−6.72 × 10−4	103.0	7.30 × 10−2	1.46 × 10−2	8.76 × 10−2	0.0691	0.0614	0.9025

,

Table A2. Coefficients of the GMPE for set 1, Equation (1), ground type C.

T (s)	a	b	c	h	σr2	σe2	σ2	Meannr	Mednr	Stdnr
0.20	1.0593	0.5428	−3.93 × 10−4	102.9	4.08 × 10−2	1.60 × 10−2	5.67 × 10−2	0.0534	0.0326	0.9321
0.40	1.6011	0.7340	−7.56 × 10−4	112.7	4.29 × 10−2	3.19 × 10−3	4.61 × 10−2	0.0211	0.0292	0.9278
0.60	2.2368	0.8472	−2.93 × 10−3	137.3	4.15 × 10−2	1.91 × 10−2	6.07 × 10−2	0.0502	0.1198	0.8650
0.80	2.1411	1.0527	−2.79 × 10−3	126.3	4.20 × 10−2	1.09 × 10−2	5.29 × 10−2	0.0039	0.0290	0.8338
1.00	1.7332	1.1249	−1.18 × 10−3	62.5	3.42 × 10−2	3.75 × 10−2	7.17 × 10−2	0.0722	0.1226	0.8098
1.50	2.5936	1.1916	−3.63 × 10−3	155.6	4.96 × 10−2	1.27 × 10−2	6.23 × 10−2	0.0606	0.0235	0.7777
2.00	2.2520	1.2897	−2.83 × 10−3	103.4	4.90 × 10−2	3.84 × 10−2	8.74 × 10−2	0.1454	0.1444	0.7392
2.50	2.1502	1.2317	−2.00 × 10−3	81.7	4.52 × 10−2	6.80 × 10−2	1.13 × 10−1	0.2015	0.1721	0.7079
3.00	2.2834	1.0479	−1.70 × 10−3	88.9	5.57 × 10−2	6.86 × 10−2	1.24 × 10−1	0.1907	0.1798	0.7256
4.00	2.3475	0.8141	−1.30 × 10−3	90.7	6.84 × 10−2	4.72 × 10−2	1.16 × 10−1	0.1346	0.2371	0.7931

,

Table A3. Coefficients of the GMPE for set 3, Equation (1), ground type B.

T (s)	a	b	c	h	σr2	σe2	σ2	Meannr	Mednr	Stdnr
0.20	1.0841	0.5578	−2.23 × 10−4	88.3	8.15 × 10−2	3.76 × 10−2	1.19 × 10−1	0.0336	0.0661	0.8095
0.40	1.5353	0.6640	−4.13 × 10−4	90.0	1.03 × 10−1	3.47 × 10−2	1.38 × 10−1	0.0484	0.0169	0.8504
0.60	1.7131	0.6964	−5.89 × 10−4	80.9	1.19 × 10−1	2.52 × 10−2	1.44 × 10−1	0.0323	0.0618	0.8826
0.80	1.7831	0.7259	−3.70 × 10−4	78.4	1.12 × 10−1	4.62 × 10−2	1.58 × 10−1	0.0221	0.0008	0.8135
1.00	1.6559	0.7388	3.97 × 10−4	49.1	1.08 × 10−1	4.19 × 10−2	1.50 × 10−1	0.0205	0.0369	0.8197
1.50	1.7453	0.8348	−5.91 × 10−6	51.2	8.64 × 10−2	3.68 × 10−3	9.00 × 10−2	0.0107	0.1041	0.9654
2.00	1.7491	0.8445	−5.95 × 10−5	52.0	8.04 × 10−2	3.63 × 10−2	1.17 × 10−1	0.0733	0.2136	0.8254
2.50	1.8158	0.8383	−3.87 × 10−4	62.6	6.95 × 10−2	5.32 × 10−2	1.23 × 10−1	0.1175	0.1905	0.7903
3.00	1.8308	0.8688	−5.16 × 10−4	68.4	6.50 × 10−2	5.79 × 10−2	1.23 × 10−1	0.1359	0.2026	0.7784
4.00	1.8552	0.8807	−5.74 × 10−4	80.9	7.61 × 10−2	6.92 × 10−2	1.45 × 10−1	0.1532	0.2343	0.7880

and

Table A4. Coefficients of the GMPE for set 3, Equation (1), ground type C

T (s)	a	b	c	h	σr2	σe2	σ2	Meannr	Mednr	Stdnr
0.20	3.0994	0.6661	−5.70 × 10−3	278.0	4.47 × 10−2	3.08 × 10−2	7.55 × 10−2	0.0610	0.2030	0.9760
0.40	6.9703	0.8816	−1.00 × 10−2	483.3	4.16 × 10−2	5.17 × 10−2	9.34 × 10−2	0.1374	0.3661	0.9482
0.60	9.1643	0.9443	−1.26 × 10−2	534.6	3.82 × 10−2	7.56 × 10−2	1.14 × 10−1	0.0743	0.1374	0.8699
0.80	3.5193	1.0032	−5.55 × 10−3	239.0	4.33 × 10−2	8.46 × 10−2	1.28 × 10−1	0.0086	0.0062	0.8242
1.00	2.8424	1.0528	−4.04 × 10−3	169.3	3.99 × 10−2	9.75 × 10−2	1.37 × 10−1	0.0835	0.0959	0.8115
1.50	3.3158	1.1715	−5.05 × 10−3	205.3	4.37 × 10−2	7.10 × 10−2	1.15 × 10−1	0.0748	0.1774	0.8594
2.00	3.0047	1.2101	−4.54 × 10−3	177.3	4.15 × 10−2	5.09 × 10−2	9.24 × 10−2	0.0080	0.1074	0.9204
2.50	2.8699	1.2264	−4.20 × 10−3	167.4	4.10 × 10−2	5.52 × 10−2	9.63 × 10−2	0.0166	0.1358	0.9195
3.00	2.7517	1.2277	−3.79 × 10−3	165.2	4.73 × 10−2	5.38 × 10−2	1.01 × 10−1	0.0133	0.0967	0.9319
4.00	2.9025	1.2132	−4.11 × 10−3	187.1	5.49 × 10−2	5.50 × 10−2	1.10 × 10−1	0.0208	0.0342	0.9336

present the coefficients of the GMPE along with the MEANNR, MEDNR and STDNR indexes for each spectral period.

Figure 20 presents the distribution of the normalized residuals as a function of magnitude. For all periods, the residuals are uniformly distributed for all magnitudes, except for a minor concentration for 5.2 earthquakes for small periods (0.3–0.5 s), as also shown in Figure 16a.

The distribution of NRES with respect to epicentral distance is shown in Figure 21. For all periods, the residuals are very uniformly distributed.

The normalized residual’s analysis, both in terms of magnitude and distance, showed that the GMPE provides non-biased estimates of the DRS of the records found in the database.

Figure 22 and Figure 23 show the attenuation of the model with the distance, along with a comparison with two other GMPE for spectral displacement. Unfortunately, the two models are calibrated against data from shallow earthquakes. The reference models are Cauzzi and Faccioli [11], presented with an orange dotted line, and Hassani et al. [14], drawn with a blue dashed line. Measured data are shown with blue dots, and are from a M = 6.9 earthquake in Figure 22 and two M = 7.1 earthquakes in Figure 23. The thick red line is the median of the proposed GMPE, the shaded band delimitates a region ± 1σ. Figure 22 shows the attenuation on type B soil, while Figure 23 displays the model on type C soil. For soil class B, the proposed model shows a similar attenuation with GMPE by Hassani et al., especially for periods of 1.0 and 1.5 s, while the Cauzzi and Faccioli model attenuates at a higher rate. For soil class C, the attenuation of the proposed model is higher than for type B soil for epicentral distances larger than 100 km, and almost inexistent for distances less than 100 km for all three studied periods. This is consistent with the attenuation provided by the model of Vacareanu et al. [23], which showed limited or no reduction in spectral acceleration for epicentral distances up to 100 km. It is significantly different than the one corresponding to the other two models. The Cauzzi and Faccioli model is increasingly attenuating with the distance, while Hassani et al. has a rather constant attenuation. The model fits well the measured data for type B soil. For soil type C, for 0.3 and 1.0 s, there is a group of records with virtually the same distance, located in Bucharest, for which the GMPE overestimates the displacement. Bearing in mind the fact that the outlying measured data are very localized and the whole dataset is from two earthquakes only, this situation might not be relevant.

The median values for the total standard deviation are in the range 0.30–0.35 (log10 units) for type B soil, and usually smaller for type C soil. These values are smaller than the ones reported in the literature [11,23], due to the limited number and region-specificity of the records.

4. Conclusions

Using a database of strong motion records of intermediate depth earthquakes from Romania and Japan, an attenuation model for spectral displacement is developed. The main variables are earthquake magnitude, epicentral distance and soil type. The coefficients of the GMPE are determined for ground types B and C (classified according to Eurocode 8), for periods between 0.10 and 4.00 s with an increment of 0.10 s, and it is applicable for sites located in front of the Carpathian Arc, at epicentral distances between 30 to 300 km. The coefficients were determined through two-stage regression. Using a set of digital strong motion records, a GMPE with coefficients determined up to 8.0 s was created. In order to predict the DRS of very strong earthquakes, a quadratic term was added to the original equation, which significantly improved the prediction of such strong seismic events.

As our investigation shows, magnitude and soil type have the largest impact, while the epicentral distance has a smaller influence. Increasing the magnitude or diminishing the epicentral distance leads to a shrinking of the zone with high amplifications. Especially for moderate–strong seismic events, large amplifications were found to occur on type C sites. The models were tested with good results by appraising inter-event and intra-event residuals, according to the relevant scientific literature.

Although the present work aims at providing input data for DBD, it is also useful for assessing the displacement demand for seismic design, for a specific earthquake scenario, described by simple parameters, such as ground type, moment magnitude and epicentral distance.