Horizontal PGA Estimates for Varying Deep Geological Conditions—A Case Study of Banja Luka

Abstract

In this study, the city of Banja Luka is used as a case study to evaluate horizontal PGA values in regions with a history of moderate to strong earthquakes and with different deep geological conditions. We present regional attenuation equations for PGA that can capture both the impacts of deep geology and local soil conditions. A PSHA study for a site in Banja Luka was carried out using the developed empirical scaling equations and compared to all previous seismic hazard estimations for the same region. The data indicate that variations in deep geological conditions may have a greater impact on PGA values than local soil effects. Given the scarcity of scaling equations that consider deep geology in addition to local soil conditions, we believe this case study is a step toward developing more accurate PGA estimates for comparable regions.

1. Introduction

Peak ground acceleration (PGA) is still a very important parameter in earthquake engineering [1,2,3], and it is imperative to estimate it as reliably as possible. Most empirical formulae for scaling PGA values only take into account the effects of local soil, ignoring the effects of deep geology [4]. There are, however, strong motion studies [5,6,7,8] that show that evaluating the severity of surface ground motion requires taking into account both deep geology and local soil conditions to minimize bias. The 2004 version of Eurocode 8 [9] also acknowledges the significance of deeper geological conditions, stating in Clause 3.1.2(1) that the National Annex may specify the classification scheme that will account for deep geology. The term “deep geology” refers to geological site surroundings on a scale of a few hundreds of meters or even kilometers [10], whereas “local soil” refers more to a geotechnical description on a scale of a few tens of meters, usually down to the first layer, with V_S ≥ 800 m/s.

Furthermore, most GMPEs (Ground Motion Prediction Equations) classify local soil conditions solely based on the top 30 m of the stratigraphic profile, regardless of research showing that the average shear-wave velocity of the top 30 m does not correlate with a site resonance [11,12], particularly for deep soil sites [13]. This method is still extensively employed, because data on deep geology is limited, and investigating a soil profile to depths of at least 100 m would be prohibitively costly for the design of common civil engineering structures.

In Bosnia and Herzegovina, seismic hazard maps for use in the application area of Eurocode 8 [14,15] are defined for ground type A, which is defined as a rock or “other rock-like geological formation” with up to 5 m of weaker surface material and V_S > 800 m/s in the top 30 m of the soil profile. The PGA value from the official hazard map is then multiplied by the soil factor, S, to obtain the PGA at the top of local soil.

Table 1. PGA for different ground types according to Eurocode 8 [9], where a_g represents the PGA value indicated on the accompanying seismic hazard map for the investigated site.

Eurocode 8 [9] Ground Types	Type 1 Spectrum: “Most Contributing” Earthquakes with MS > 5.5	Type 2 Spectrum: “Most Contributing” Earthquakes with MS ≤ 5.5
Ground type A, VS,30 > 800 m/s “Rock, at the surface up to 5 m of weaker material.”	ag	ag
Ground type B, VS,30 = 360–800 m/s “At least several tens of meters thick deposits. Very dense sand, gravel, or very stiff clay.”	ag × 1.2	ag × 1.35
Ground type C, VS,30 = 180–360 m/s “Deep deposits, several tens of meters up to hundreds of meters thick. Dense or medium dense sand, gravel, or stiff clay.”	ag × 1.15	ag × 1.5
Ground type D, VS,30 < 180 m/s “Deposits. Loose-to-medium cohesionless soil, or predominantly soft-to-firm cohesive soil.”	ag × 1.35	ag × 1.8
Ground type E “Alluvium layer at the surface, between 5 and 20 m thick, above stiffer material. VS,30 < 360 m/s.”	ag × 1.4	ag × 1.6

shows how S varies by ground type. S is additionally dependent on the type (Type 1 or Type 2) of spectrum, which is determined by the magnitude of the “most contributing earthquakes” [9]. It should be noted that Eurocode 8 defines the magnitudes of the most significant earthquakes using the M_S scale. However, as the majority of modern GMPEs and seismic source models utilize the M_W magnitude, the majority of seismic hazard maps in Europe were created using this magnitude. Furthermore, recent studies have demonstrated the M_W scale’s significant shortcomings and support the implementation of the M_wg scale in its place [16,17,18]. Since our hazard analyses utilize a European source zone model defined by other authors (see Section 4), converting magnitude scales will be outside the purview of this research. We do, however, intend to include magnitude conversion of the seismicity data into our upcoming studies.

The goal of this study was to estimate PGA values at different deep geological conditions for regions with moderate to strong earthquakes. The case-study region encompasses the city of Banja Luka, Bosnia and Herzegovina, for which our associated studies [19,20,21] examined vertical PGA values as well as uniform hazard spectral estimates.

We calculated the seismic hazard for a location in Banja Luka using regional GMPEs for PGA and compared the results to all previous and current estimates of PGA and macroseismic intensity. In this study, local soil will be classified in compliance with Seed et al. [22,23]. The same local soil classification was used in former Yugoslavia to define site conditions for recorded strong motion data and to develop regional GMPEs [24,25,26]. According to this classification, “rock” soil sites are those with V_S > 800 m/s, stiff soil has a 15 to 75 m thick soil layer atop that with V_S > 800 m/s, while deep soil has a soil layer with a thickness of more than 100 m. For classifying the deep geology conditions, we will use the classification of Trifunac and Brady [10], who divided deep geology into three types: the geological rock, sediments, and complex or intermediate geological site surroundings.

2. Area Surrounding the Case-Study Site—Seismicity and Geology

2.1. Seismicity of the Area Surrounding the Analyzed Location in Banja Luka

The area surrounding Banja Luka is characterized by intraplate seismicity. The city of Banja Luka is located between the south-west part of the Pannonian basin, with a rare occurrence of stronger earthquakes [27,28] and the Dinaric Alps—see the top left plot in Figure 1. The Adriatic Platform and Dinarides collide, causing most regional earthquakes to occur along the Adriatic coast (see Figure 1). Moho depths in the western Balkans range from 25 km in the area of the Pannonian basin to 45 km below the Dinarides [29,30].

The metropolitan area of Banja Luka encompasses the Banja Luka valley and surrounding hills. The area has deep fault zones that are visible in the terrain [31,32,33]. The largest MCS intensity assessed in Banja Luka was VIII, caused by the M_W = 6.1 Banja Luka earthquake on 27 October 1969.

Figure 1. (Top left) Dinaric Alps and Pannonian basin, (top right) epicenters of the earthquakes with M_W ≥ 3 recorded between 1900 and 2025 [34], (bottom) the analyzed site (solid blue circle) and the epicenters of the largest historical earthquake in the vicinity of Banja Luka and two recent destructive earthquakes in neighboring Croatia.

Figure 1. (Top left) Dinaric Alps and Pannonian basin, (top right) epicenters of the earthquakes with M_W ≥ 3 recorded between 1900 and 2025 [34], (bottom) the analyzed site (solid blue circle) and the epicenters of the largest historical earthquake in the vicinity of Banja Luka and two recent destructive earthquakes in neighboring Croatia.

The 1969 earthquakes began with a series of small-magnitude earthquakes between late 1968 and early 1969. This activity resumed in September and culminated with the largest foreshock on 26 October 1969 (M_W = 6.1 and M_L = 6.0, epicentral intensity of VII–VIII °MCS) and the main shock the next day (27 October 1969, M_W = 6.1 and M_L = 6.4, epicentral intensity of VIII–IX °MCS). The earthquakes’ impact on Banja Luka (over 1117 people were injured and 15 were killed [32]) would have been even greater if each of the schools, firms, and private as well as public institutions had not been closed following the foreshock and if authorities had not prevented residents from returning to their homes [35]. The major earthquake was felt across a vast area. The VII °MCS isoseismal encompassed an area of 9000 km² [32], and 60% of all buildings were irreparably damaged, including the majority of schools. Over 76,000 people were rendered homeless, and financial losses were significant.

In neighboring Croatia, an M_W = 5.3 earthquake hit Croatia’s capital, the city of Zagreb, on 22 March 2020, with the epicenter only 7 km north [36] and a hypocentral depth of 10 km. This was the strongest event recorded near Zagreb in the past 140 years—an M_W = 6.3 earthquake was recorded in 1880. Later that very year, on 29 December 2020, yet another catastrophic earthquake hit Croatia, this time near the city of Petrinja, around 100 km northwest of Banja Luka and 40 km south of Zagreb, with M_W = 6.4 [37]. Although the maximum intensity of the 2020 Zagreb earthquake was not larger than VII, it still caused significant damage to the historical city center, with over 1900 buildings rendered uninhabitable as a result of the catastrophe. There were 27 people injured, and one of them died as a result of their injuries. Interestingly, even though the 1880 Zagreb earthquake was thought to be stronger (maximum felt intensity, VIII–IX) than the 2020 one, and the population within today’s city limits increased tenfold (from 80,000 to 800,000), the effects of the 1880 event were strikingly similar: approximately 1800 buildings were damaged, 29 people were seriously injured, and one person died as a result of injuries.

Today, the wider area of Banja Luka has a population of around 180,000 inhabitants. Although many buildings in the city center were built after the 1969 earthquake in compliance with earthquake-resistant design codes, in the wider area of Banja Luka, there are still many older and vulnerable buildings. Some of the recent regional seismic risk studies [38,39,40,41,42] reveal a significant proportion of buildings older than the 1960s in the entire region of former Yugoslavia (the first earthquake-resistant code was enacted in 1964 [43]). Similar structures are common in other countries of the Mediterranean and may sustain severe damage during moderate to large earthquakes [44,45,46,47,48].

2.2. Geological Surroundings of the Analyzed Location in Banja Luka

Banja Luka is situated on the south-west boundary of the Pannonian basin and the north boundary of the Dinaric Alps (see Figure 1). The basin encompasses parts of nine different countries, including northeast Croatia. The deep geological sediments of the Pannonian basin date back to the Pannonian Sea and have a thickness of up to several kilometers [49]. When the Pannonian Sea emptied (after reaching its maximum geographical extent during the Pliocene), the basin’s lowlands formed the remaining plain. The basin represents a back-arc basin filled with sediments, and it spread during the Miocene [50].

In Banja Luka, besides geological sediments, there are also scattered outcrops of geological rock and areas of complex deep geological strata. In the vicinity of the analyzed location, the geological sediments have thicknesses of up to several hundred meters and consist of marls, clays, and sands, with a river terrace atop it, while a spilite formation represents geological rock [51]. There is also a diabase-cherty formation (or metamorphic ophiolitic melange), which can be classified as intermediate deep geology [10]. For a more detailed geological description of the Banja Luka region, please refer to [21,51,52,53].

2.3. Official PGA Estimates for Banja Luka

The ex-SFRY issued its first seismic-resistant design code in 1964 [43]. There was a seismic zoning map developed in 1950 based on the maximum (MCS—Mercalli–Cancani–Sieberg Scale) intensities observed at the time, which was utilized in conjunction with the 1964 code. In 1981, a new earthquake-resistant design code was approved [54], and a year later, a seismic zoning map was developed to be used temporarily with the new code [55]. The 1982 map was based on the maximum measured MCS intensities once more. In 1990, the 1981 code was modified with six new seismic zoning maps (for 50, 100, 200, 500, 1000, and 10,000 year return periods) [56]. Although the seismic hazard was again expressed in MCS intensity and for average soil conditions, these were the first seismic hazard maps in ex-Yugoslavia created utilizing the PSHA (Probabilistic Seismic Hazard Assessment) approach. Please take note that we do not have return periods for the 1950 and 1982 maps, because they were developed using the largest observed intensities rather than a probabilistic hazard analysis.

Table 2. Macroseismic (MCS) intensity degrees given in the official seismic zoning maps of former Yugoslavia [43,54,55,56] for the selected location in Banja Luka, together with the associated median − σ and median + σ empirical PGA estimates computed using Equation (1). The PGA values of the 2018 official maps, which were used in conjunction with Eurocode 8 [14], are also shown.

Map	I [°MCS]	PGA
1950	VII	0.081–0.102 g
1982	VIII	0.159–0.199 g
1990—50 years	VII	0.081–0.102 g
1990—100 years	VIII	0.159–0.199 g
1990—200 years	IX	0.309–0.388 g
1990—500 years	IX	0.309–0.388 g
1990—1000 years	IX	0.309–0.388 g
1990—10,000 years	IX	0.309–0.388 g
2018—95 years	VI–VII	0.080 g
2018—475 years	VIII	0.170 g

shows MCS intensities for Banja Luka, as given in 1950, 1982, and 1990 official hazard maps. Table 2 also shows the corresponding empirical PGA estimates, calculated using the following equation [24]:

$$\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{P}\mathrm{G}\mathrm{A}\right)=-0.079+0.290\cdot I\pm P\cdot \sigma ,\sigma =0.049,$$

(1)

where PGA is the horizontal peak ground acceleration [cm/s²], I is the macroseismic intensity [°MCS], σ represents the standard deviation, and P = 0 for median estimates. For each intensity degree, Table 2 shows the PGA values calculated by Equation (1) for P = −1 and P = 1.

In 2018, two new hazard maps were included in Bosnia and Herzegovina’s National Annex to Eurocode 8 [14]. The maps were created with the PSHA approach. The first map is for a 95-year return period, or a 10% chance of exceeding in 10 years, while the second is for a 475-year return period, or a 10% chance of exceeding within 50 years. The 95-year map correlates to Eurocode 8’s “damage limitation” requirement, whilst the 475-year map corresponds to the “no-collapse” requirement. The 2018 maps were generated for ground type A (see Table 1), and on these two maps, it is the horizontal PGA that represents seismic hazard numerically. Furthermore, both maps disregard how deep geology affects surface ground motion. On these maps, the horizontal PGA values for the analyzed location in Banja Luka are 0.08 g for the 95-year and 0.17 g for the 475-year return period. We added these values in Table 2, together with the corresponding intensities estimated using Equation (1). Table 2 shows that official hazard estimates from 1950 and 1982 are similar to those from 2018. However, the empirical PGA estimates based on the intensities provided in the ex-Yugoslav 1990 hazard map for the 500-year return period are substantially larger than the 475-year PGA value from the 2018 hazard map, even after multiplying it by the greatest soil factors from Table 1. We presume that following the disastrous 1969 Banja Luka earthquake, the hazard level for Banja Luka was purposefully increased several-fold on the 100-, 200-, and 500-year official maps from 1990 [56] to assure better seismic safety in the future.

3. GMPEs for Horizontal PGA and Varying Deep Geology

The northwestern Balkans are one of the few areas in the entire world with available information on deep geological conditions from numerous recording stations. Several researchers used those regional strong motion data to develop GMPEs that take into consideration both local soil (up to 100 m in depth) and deep geology [7,25,26]. The GMPEs were employed in a series of recent microzonation studies and shown to be in great agreement with recorded macroseismic intensities [6,52,57,58,59,60,61,62,63,64,65].

We are now going to present regional empirical equations for estimating PGA values that account for both local soil conditions and deep geological site environment. The attenuation equations will be defined in this mathematical form:

$$\begin{gathered} \log [\mathrm{P}\mathrm{G}\mathrm{A}]=c_1+c_2\cdot M+c_3\cdot \log (\sqrt{R^2+{R_0\left(T\right)}^2})+c_4\cdot S_{L1}+c_5\cdot S_{L2}+c_6 \\ \cdot S_{G1}+c_7\cdot S_{G2}+{\sigma }_{\log }\cdot P. \end{gathered}$$

(2)

In Equation (2), PGA denotes horizontal peak ground accelerations (in g), M is the earthquake magnitude (refer to [66] for more information on magnitude type), and R is the epicentral distance. Equations for the hypocentral distance were also developed in our previous studies [57,58,59,60]. S_L1 and S_L2 are categorical variables for local soil, while S_G1 and S_G2 are for deep geological conditions.

Table 3. Categorical variables used in this study for different types of local soil and deep geological site settings.

Local Soil Parameters	Local Soil Categorical Variables	Deep Geology Parameters	Deep Geology Categorical Variables
“Rock” soil sites: sL = 0	SL1 = SL2 = 0	Geological rock: s = 2	SG1 = SG2 = 0
Stiff soil sites: sL = 1	SL1 = 1 and SL2 = 0	Intermediate (or complex) sites: s = 1	SG1 = 1 and SG2 = 0
Deep soil sites: sL = 2	SL1 = 0 and SL2 = 1	(Deep geological) sediments: s = 0	SG1 = 0 and SG2 = 1

summarizes these categorical variables. In Table 3, we use the same values for the categorical (dummy) variables (S_L1, S_L2, S_G1, and S_G2) and site parameters (s_L and s) as defined in [8]. We assumed that the data had a log-normal distribution. Consequently, σ_log is the standard deviation of the common logarithm of PGA, and ε is equal to 0 for the median estimate and ±1 for the median ±1 σ_log estimates.

The database for prediction equations includes 436 horizontal components of strong-motion accelerograms from 112 earthquakes with a magnitude of 3 < M ≤ 6.8. Our analysis is predominantly based on the EQINFOS database [67], which contains records from earthquakes in the northwest Balkan region. The authors of the database used the M_L magnitude as defined by Lee et al. [66] for all events with a magnitude smaller than around 6.0, while only for the strongest earthquakes, they used the M_S magnitude. They purposefully chose the local magnitude, M_L, for the majority of events, since it was thought to be most suited for the so-called “strong motion”—the frequency content between 0.1 and 30 Hz that civil engineers are most interested in [68]. According to a subsequent study that converted the M_L magnitudes of the earthquakes in Croatia and the adjacent areas to moment magnitudes (M_W) [69], the difference between M_L and M_W was less than 2% for magnitudes between 3.5 and 6.5. Furthermore, Scordilis [70] found that the gap between M_S and M_W magnitudes is roughly 0.2% for magnitudes between 6.2 and 7.0. Therefore, we believe that while accounting for other limitations of the provided database, we can use our GMPEs in seismic hazard calculations with the seismic source zones identified by using the M_W-based seismicity data. Every one of the accelerograms in our database was recorded in the northwestern Balkans. The majority (418) were recorded between 1976 and 1987 [67,71,72], with the remaining (18) in 2010. Maps that show the locations of recording stations and the epicenters of contributing earthquakes can be found in our previous papers [7,58,59].

MATLAB^® (version 8.5) scripts were used for calculating the GMPE scaling coefficients c₁, c₂, c₃, c₄, c₅, c₆, c₇, and σ_log by multiple linear regression analyses. For maximizing the R² statistics, R₀ was iteratively adjusted. Because around two thirds of the data came from distances shorter than around 30 km, we performed a supplementary analysis that only included data collected at epicentral distances of less than 30 km. This was conducted with the intention of determining whether more reliable GMPEs might be obtained for shorter source-to-site distances. Finally, because the regression was also performed for pseudo-spectral accelerations across different vibration periods, the scaling coefficients were smoothed using the MATLAB^® function “smooth” and the weighted linear least squares and a second-degree polynomial model.

The finalized GMPE is as follows:

$$\begin{gathered} \log [\mathrm{P}\mathrm{G}\mathrm{A}]=-1.2957+0.3946\cdot M-1.3818\cdot \log (\sqrt{R^2+{19.5}^2})+0.1772\cdot S_{L1} \\ -0.0953\cdot S_{L2}-0.1469\cdot S_{G1}-0.1059\cdot S_{G2}+0.2691\cdot P, \end{gathered}$$

(3)

The GMPE based on data from epicentral distances shorter than 30 km is as follows:

$$\begin{gathered} \log [\mathrm{P}\mathrm{G}\mathrm{A}]=3.1868+0.3919\cdot M-3.8728\cdot \log (\sqrt{R^2+{40.0}^2})+0.1276\cdot S_{L1}-0.0953 \\ \cdot S_{L2}-0.2229\cdot S_{G1}-0.0899\cdot S_{G2}+0.2675\cdot P. \end{gathered}$$

(4)

By comparing Equations (3) and (4), we can observe that using only data recorded at distances less than 30 km results in a slightly smaller standard deviation.

Figure 2 displays the attenuation of PGA as a function of distance, as calculated with Equations (3) and (4). The coefficients pertaining to S_L and S_G can be used to compute the variations in values calculated by the same equation under different site conditions (refer to Table 3). For example, the ratio between PGA at the geological rock (s = 2) and intermediate deep geology sites (s = 1) is calculated as 1/10^c6, while the ratio between the geological rock (s = 2) and deep geological sediments (s = 0) is calculated as 1/10^c7. If we use Equation (3), the PGA estimates at the geological rock will be 1/10^−0.1469 = 1.40 times larger than at intermediate sites and 1/10^−0.1059 = 1.28 times bigger than at geological sediments. The fact that short-period waves travel faster through harder and more compact geological formations (like basalts and granites) than through sediments may explain the amplification at geological rock sites.

If the deep geology conditions remain constant, we can calculate ratios between the PGA values at stiff soil (s_L = 1) and “rock” soil (s_L = 0) by calculating 10^c4, while the PGA ratios between deep soil (s_L = 2) and “rock” soil (s_L = 0) is calculated as 10^c5. If Equation (3) is used, the PGA value for stiff soil sites is 10^0.1772 = 1.50 times larger than for “rock” soil sites. Horizontal PGA for deep soil sites will be 10^−0.0953 = 0.80 times the PGA for “rock” soil, according to the same equation. Put another way, horizontal short-period seismic waves at the surface will be less amplified (or even de-amplified) at deep soil sites than at “rock” soil sites. This implies that the energy dissipation of short-period waves in deep soils may outweigh a local soil amplification generated by the impedance difference between the deep soil and the harder rocks beneath.

What should be noted here, regardless of the type of deep geological formation, is that the effects of deep geology are comparable in scale to those of local soil and must not be overlooked.

Figure 2 illustrates PGA ranges (median ± one standard deviation) calculated with Equation (1) and VIII °MCS, which was the estimated predominant intensity in Banja Luka during the 1969 earthquake with the epicenter around 10 km from the city center. We can observe that only the combined effects of stiff soil sites and deep geological rock can result in median empirical PGA values comparable to the ones associated with VIII °MCS.

Figure 3 compares the horizontal PGA values recorded in Banja Luka at four separate accelerograph stations [67] with the median and median ± σ_log empirical PGA estimates that were computed using Equation (3). As seen in Figure 3, the recorded horizontal PGA values and the empirical predictions coincide very well.

The presented scaling equations do not account for potential PGA variations as a function of rupture direction relative to the investigated site. According to a recent study, the directivity phenomenon (the presence of high-velocity pulses) can have a significant impact on masonry structures, such as those in Banja Luka [73]. Additionally, near-field and far-field earthquake effects on low-rise buildings, such as those that are common in Banja Luka, are found to be dependent on local soil characteristics [74]. However, due to a lack of sufficient directivity and near-field data in regional strong motion recordings, this will fall outside the scope of our current research.

In this study, we used empirical GMPEs to directly predict site response. Though we did not compare our PGA estimates to the soil response modeled by pattern recognition methods [75] or linear and nonlinear soil response analyses [13,76,77,78,79,80,81], we intend to do so in the future using geotechnical data. Because Banja Luka lacks a network of soil profiles, we may need to interpolate a few available soil profile data [82] to cover the city’s territory.

4. PSHA Analysis for the Case Study Location in Banja Luka

Using Equation (3) as the GMPE and the SHARE Project’s pan-European seismic source zone model [83,84,85], we performed a PSHA analysis for the site with coordinates 44°46.5′ N, 17°15′ E, denoted by the solid blue circle in Figure 1 and Figure 4. The PSHA analysis will be based on Cornell’s [86] and McGuire’s [87] approaches and will adhere to Chioccarelli et al.’s [88] procedure. For all hazard calculations, we used the REASSESS V2.1 software [88]. The total hazard is the sum of the contributions from each source zone, i (from the full set of zones, I). Figure 4 depicts the borders of the selected seismic zones used for hazard calculations in this research. Although not the subject of this investigation, seismic source zones can be established using publicly accessible regional seismological data [89,90,91]. In this particular study, we use the source zone model that was based on the new homogeneous earthquake catalogue (“SHARE European Earthquake Catalogue”—SHEEC [92]), created for the scope of the SHARE Project. The SHEEC catalogue’s completeness, homogeneity of magnitudes with respect to M_W, and other significant aspects are explained in depth in [84,93,94].

Figure 4 displays the source zones included in the hazard calculations for Banja Luka. The two larger circles, with radii of 111.75 and 81.75 km, represent the source-to-site distances that must be included in the PSHA computations to ensure 1% accuracy of computed PGA estimates with a probability of 10% in 10 and 50 years (the 95- and 475-year return periods), respectively. The two smaller circles, with radii of 57.75 and 40.00 km, indicate the maximum distances that will ensure a PSHA accuracy of 10% for the same two probabilities. Figure 4 also depicts the epicenters of several of the major historical earthquakes as well as a few recent disastrous earthquakes in the northwestern Balkan region [34]. As illustrated in Figure 4, local seismicity dominates in the PGA probabilistic estimates, and only the M_W = 6.1 1969 Banja Luka earthquake and M_W = 6.4 2020 Petrinja earthquake are within the depicted radii, while all other shown events occur in areas that do not significantly contribute to the seismic hazard. It is also worth mentioning that some recent regional studies have found that PGA amplitudes are primarily influenced by local seismicity and are not particularly affected by very strong and distant events, such as those in Romania’s Vrancea source zone [61,63,64,65,95,96]. This is because high-frequency seismic waves attenuate rapidly. As a result, while traveling from great distances, their final contribution is less than that of the waves created by local events. Seismic microzonation studies for the cities in Serbia and North Macedonia that are all closer to the Vrancea zone than Banja Luka demonstrate that the contribution of Vrancea earthquakes to PGA values is negligible when compared to local seismicity [61,63,64,65]. As a result, in this study, we will also overlook the Vrancea source zone.

For the calculation of the total hazard, we will employ the law (or formula) of total probability [97], which defines the probability of an event, A, in case that a series of n mutually exclusive events, H_i, occurs, and H_i comprises a full system of hypotheses about the event, A, as follows:

$$P(A)={\sum }_{i=1}^{n}P\left(A\right|H_i)\cdot P(H_i).$$

(5)

Based on Equation (5), we can apply the following equation to calculate the average yearly rate of occurrence of seismic events that may cause PGA to exceed the expected pga:

$$\begin{gathered} N\left(pga\right)={\sum }_{i\in I}{\nu }_i{\int }_{M_{min}}^{M_{max}}{\int }_{R_{min}}^{R_{max}}{f}_{gmpe}\left(\mathrm{P}\mathrm{G}\mathrm{A}>pga|M,R\right)\cdot f_{m_i}\left(M\right)\cdot \\ {f}_{r_i|m_i}\left(R|M\right)dMdR, \end{gathered}$$

(6)

where i denotes the seismic source zone serial number (from the set I), and ν is the annual rate of events exceeding M_min, which is defined as follows:

$$\nu =e^{a\mathit{ln}10-\left(b\mathit{ln}10\right)M_{min}}.$$

(7)

In Equation (7), a and b are the coefficients that characterize the total degree of seismicity in the analyzed area and the ratio between events of smaller and larger sizes, respectively, and are defined as follows:

$$\log N_{GR}\left(M\right)=a-b\cdot M,$$

(8)

where N_GR represents the rate of occurrence of earthquakes of different magnitudes in each zone, i; M_max is the maximum magnitude for each zone, i; M_min is the minimum magnitude (although the SHARE seismic-hazard map for all of Europe was calculated with M_min = 4.5 [84], we chose to use M_min = 4.0 for all zones in this study to obtain more conservative PGA estimates for shorter return periods); and R_min and R_max are the minimum and maximum epicentral distances equal to 0 and 300 km, respectively.

Furthermore, f_gmpe is the conditional cumulative distribution function, which is based on the GMPE and determines the likelihood that PGA will surpass pga if an event of magnitude, M, occurs at a source-to-site distance, R; f_r|m is the probability density function for distance; and f_m is the probability density function for magnitudes, which is defined as follows:

$$f_m\left(M\right)=\beta {\displaystyle \frac{e^{-\beta \left(M-M_{min}\right)}}{1-e^{-\beta \left(M_{max-}{M}_{min}\right)}}},M_{min}\le M\le M_{max},$$

(9)

where β = (ln10)b.

We will now explain the alternative metrics for seismic hazard, shown in

Table 4. Four different probability metrics: P(pga) is the yearly probability of at least one exceedance of the expectation pga; p(pga) is the likelihood of at least one exceedance of the expectation pga during t = 10 or 50 years, and Tr(pga) is the “return period”.

P(pga)	p(pga) [%] in t = 10 years	p(pga) [%] in t = 50 years	$\mathit{T}\mathit{r}(\mathit{p}\mathit{g}\mathit{a})=\frac{1}{\mathit{N}\left(\mathit{p}\mathit{g}\mathit{a}\right)}\cong \frac{1}{\mathit{P}\left(\mathit{p}\mathit{g}\mathit{a}\right)}=\frac{1}{1-\sqrt[\mathit{t}]{1-\mathit{p}\left(\mathit{p}\mathit{g}\mathit{a}\right)}}$ [years]
0.020000	18.29	63.58	50.00
0.010481	10.00	40.95	95.41
0.010000	9.56	39.50	100
0.005000	4.89	22.17	200.00
0.002105	2.09	10.00	475.06
0.002000	1.98	9.52	500
0.001025	1.02	5.00	975.29
0.001000	1.00	4.88	1000.00
0.000404	0.40	2.00	2475.42
0.000100	0.10	0.50	10,000.00

. The so-called “return period” is calculated as follows:

$$Tr=N(pga{)}^{-1}.$$

(10)

Although Tr is commonly used in the engineering literature as an instinctive measure of hazard levels (a larger Tr indicates stronger events that occur less often and vice versa), it lacks a distinct physical meaning in the sense that it does not (generally) correspond to a single earthquake of engineering concern. This is because N(pga) for a single location is calculated by combining and summing data from all analyzed seismic source zones across all magnitudes and distances.

When homogeneous Poisson distributions are assumed, the annual probability, or the likelihood of at least one yearly exceedance of the expectation pga, may be expressed as follows [97]:

$$P\left(pga\right)=1-e^{-N\left(pga\right)}.$$

(11)

Finally, if a binomial distribution is assumed, the probability that the expectation pga will be exceeded at least once during t years can be computed as follows [97]:

$$p\left(pga\right)=1-[1-P(pga){]}^t.$$

(12)

A procedure that is called “seismic hazard disaggregation” can be used to identify the earthquakes that contribute the most to the computed hazard [98,99]. It is effectively the inverse of the process described in Equation (6), and it allows us to distinguish independent contributions from distinct magnitude and distance combinations.

The four plots of Figure 5 show the PSHA disaggregation for 44°46.5′ N, 17°15′ E (refer to the solid blue circles in Figure 1 and Figure 4) and Tr = 95, 475, 975, and 2475 years. These plots show that the most contributing magnitudes increase with the return period, Tr. The cumulative disaggregation for M and four different return periods is depicted in the top left plot of Figure 6. For the return periods of 95, 475, 975, and 2475, the magnitudes that contribute to 50% of the anticipated likelihood of exceedance are equal to 4.9, 5.2, 5.35, and 5.5, respectively. The three other plots in Figure 6 show the magnitude recurrence relations for the three most contributing source zones (in Figure 4, these three zones are designated with serial numbers). As can be seen from these plots, the real recurrence (return) periods of the earthquakes that contribute the most vary with the “return period”, Tr, which represents just a reciprocal value of N(a) (see Equations (6) and (10)). What is even more important, the real return periods are significantly shorter than the corresponding return periods, Tr. This is very important to understand, because Eurocode 8 [9] defines Types 1 and 2 spectra not in terms of maximum credible magnitudes but rather in terms of those magnitudes that “contribute most to the seismic hazard defined for the site for the purpose of probabilistic hazard assessment” (please refer to Eurocode 8 [9]: 3.2.2.1, (4), Note 2 and 3.2.2.2, (2)P, Note 1). The selection of the spectral type based on the most contributing earthquakes can be extremely confusing for average civil engineers. The type of spectrum is not specified in Bosnia and Herzegovina’s National Annex to Eurocode 8 [14], and users are free to select it without being given access to the hazard disaggregation data. Given Banja Luka’s history of experiencing earthquakes with magnitudes in excess of 6.0, it stands to reason that the majority of the contributing magnitudes would be greater than 5.5. However, Figure 5 and Figure 6 show that the magnitudes of the most contributing earthquakes are smaller than 5.5, with the exception of the most contributing magnitude for Tr = 2475 years. Even for Tr = 2475 years, this magnitude is equal to 5.5 or little less than 5.5.

The seismic hazard curves for 44°46.5′ N, 17°15′ E (see Figure 1 and Figure 4) were compared to the PGA values calculated based on regulations in former Yugoslavia [56] and the values from current official seismic hazard maps for Bosnia and Herzegovina [14] to validate the findings of our PSHA study, as shown in Figure 7.

Before we compare different hazard estimates, it is important to remember that the PGA values on the hazard maps provided in the National Annex to Eurocode 8 are for “rock or comparable geological formations” [9]. Under former Yugoslavian standards, official hazard maps were created for average local soil conditions, and the hazard was calculated using the MCS intensity scale degrees [43,54,55,56]. The impact of deep geologic conditions was not explored in the creation of any of these maps.

As shown in Figure 6, our PSHA estimates of horizontal PGAs are in fair agreement with those from 1990 macroseismic intensities [56] only for the return periods of 50 and 10,000 years. For all other return periods (100, 200, 500, and 1000 years), our estimates are much smaller. However, our estimates in the case of (deep) geological rock and stiff soil sites are in good agreement with the PGA values for Banja Luka from the 2018 official maps of Bosnia and Herzegovina [14].

5. Discussion and Conclusions

In our previous studies, we either exclusively examined the geological rock sites that exist in Kopaonik, Serbia [6], or the deep soil sites that are found in Osijek, Croatia [57,58,59,60]. This is the first time we have examined a region that has varying deep geological and local soil conditions. Our study’s primary weakness is the size of our database, which only includes 436 horizontal components of regional strong motion records. Unfortunately, no other sets of data in the world include deep geology data in parallel with local soil data, with the exception of California, which has different seismo-tectonic properties than the Balkans [8]. We are, therefore, constrained to rely on our current database.

In this study, we examined horizontal PGAs for Banja Luka, which served as a case study area for regions with varying deep geological conditions and with a history of catastrophic earthquakes. This city is situated between the Pannonian basin and the Dinaric Alps. In addition to deep geological sediments, Banja Luka has scattered outcrops of geological rock and also areas with complex deep geology. This city was also devastated by the 27 October 1969 Banja Luka earthquake with M_W = 6.1, and the epicenter around 10 km from the city center. The estimated predominant intensity in Banja Luka during the 1969 earthquake was VIII °MCS.

Regional empirical GMPEs for horizontal PGAs are first presented. These GMPEs consider the effects of local soil and deep geology simultaneously. Despite the modest number of the strong-motion accelerograms from former Yugoslavia, which were used to derive the GMPEs, the empirical PGA estimates are shown to be in excellent agreement with the PGA values that were recorded in the past in Banja Luka.

Our conclusions can be summarized as follows:

• Regardless of the type of deep geological formation, the effects of deep geology are comparable in scale to those of local soil and must not be overlooked.
• Only the combined effects of stiff soil sites and (deep) geological rock can result in median empirical PGA estimates comparable to the ones associated with the intensity of the 1969 Banja Luka earthquake (VIII °MCS).
• Real re-occurrence (return) periods of the earthquakes that are most contributing to seismic hazard vary with the so-called “return period”, Tr, which represents just a reciprocal value of N(a) (see Equations (6) and (10)); moreover, real return periods are significantly shorter than the corresponding return periods, Tr.
• PGA probabilistic hazard estimates are dominated by local seismicity and are less influenced by earthquakes at distances larger than around 100 km.
• The latest official (2018) hazard maps [14] provide PGA estimates that are comparable to our estimates of horizontal PGAs for Banja Luka.

Table 5. Empirical ratios between the PGA for different site conditions, calculated using scaling coefficients of Equations (3) and (4) for local soil (c₄/c₅) and deep geology (c₆/c₇).

	Stiff Soil sL = 1/ “Rock” Soil sL = 0	“Rock” Soil sL = 0/ Stiff Soil sL = 1	Deep Soil sL = 2/ “Rock” Soil sL = 0	“Rock” Soil sL = 0/ Deep Soil sL = 2	Intermed. Sites: s = 1/ Geological Rock: s = 2	Geological Rock: s = 2/ Intermed. Sites, s = 1	Deep Geol. Sediments: s = 0/ Geological Rock: s = 2	Geological Rock: s = 2/ Deep Geol. Sediments: s = 0
GMPE	10c4	1/10c4	10c5	1/10c5	10c6	1/10c6	10c7	1/10c7
Equation (3)	1.50	0.66	0.80	1.25	0.71	1.40	0.78	1.28
Equation (4), R < 30 km	1.34	0.75	0.80	1.25	0.60	1.67	0.81	1.23

shows that depending on the GMPE used (Equation (3) or Equation (4)), the effects of deep geology may even outweigh the effects of local soil. For example, if Equation (4) is used, the largest site amplification of 1.67 is obtained for deep geology when comparing the PGA at geological rock to that at intermediate sites. Compared to the local soil amplification of 1.34, which is obtained at stiff soil sites (compared to the “rock” soil), the deep geology amplification of 1.67 is 25% higher.

We believe that our PSHA estimates, which were computed using regional GMPEs, can be perceived as a first step toward defining more realistic PGA probabilistic estimates in Banja Luka. When the number of recorded acceleration time histories increases in this region, the obtained scaling equations will be further assessed and, if necessary, calibrated or modified, as well as the PSHA analyses.