Analysis of exponent K based on “SHARE” project data and its implications on importance factors of EN 1998–1

This contribution deals with seismic activity represented by a hazard curve through a single parameter–exponent k as given in EN 1998–1 and its implications on importance factors. We used the SHARE project dataset to calculate exponent k for the wider European area, and a limited number of separate national studies for comparison of the results since comparison to the SHARE results on the same dataset resulted in values of exponent k smaller by 1–1.5. The results indicate that the recommended value of exponent k of 3 is rather an exception than an expected value in seismically active regions and that, with the exclusion of the Vrancea zone, for the majority of Europe, exponent k is well below that assumed in EN 1998–1, which consequently indicates that the importance factors for these locations should be higher than those recommended in EN 1998–1.


Introduction
Exponent k is defined in the note of article 2.1(4) of EN 1998-1 (EN 2004 by means of an annual exceedance rate. As stated in this article, it is assumed that the annual rate of exceedance H(a gR ) may be taken to vary with a gR as: where a gR is the reference peak ground acceleration related to ground type A, and k is an exponent correlating the reference peak ground acceleration with the annual exceedance (1) H a gR ∼ k 0 a −k gR 1 3 rate. The article also states that exponent k is dependent on seismicity and that it is generally of the order of 3.
Analytical expression of seismic hazard results with an equation given in a closed form, preferably as simple as Eq. (1), has advantages  over the numerical approach of solving the hazard integral, as allows seismic hazard results to be further analytically treated in seismic demand or seismic risk problems SAC/ FEMA 2000a;Sewell et al. 1991;Yun et al. 2002).
Representation of seismic hazard in the form of Eq. (1) with reported values of exponent k for different sites dates back to at least 1994 (Kennedy and Short 1994). In this report, for numerous eastern and western sites in the United States, the exponent k is determined to be in the range from 1.7 to 3.3, calculated over the annual probability range of 10 −3 to 10 −5 . However, the idea itself of representing a hazard curve with a linear function in a log-log scale was not new at that time (Sewell et al. 1991).
It is also worth noting the FEMA-350 report from 2000 (SAC/FEMA 2000a), where the analytical approximation of the seismic hazard curve is treated identically to Eq. 1. This report also defines the default determination range on 2 and 10% probabilities of exceedance in 50 years. It suggests default values of exponent k in a tabulated form taking values of 1, 2, and 3. The value of 3 is assigned to seismically most active regions. When comparing exponents k given in different studies and reports, care should be taken since this parameter is sensitive to chosen spectral period and approximation range.
With the given simplified link between the hazard curve and exponent k defined in Eq. (1), the hazard curve is practically approximated by an exponent function, equivalent to a linear function on a log-log scale. The note does not provide further instructions on how to determine exponent k or in what range, but it is reasonable to assume that, in the particular case, the interest is on return periods, which are of concern for the design of structures and are predefined in EN 1998-1 by choice of importance factors, which will be discussed in more detail further in the text.
In addition, because of the characteristics of the probabilistic modeling of seismic hazard that is ultimately represented in the shape of an annual rate of exceedance curve, it is impossible to define a single k value that can represent the whole possible range of the curve. This implies that the k value is valid only within a limited determined range; the narrower the range, the more accurate the approximation is.
The importance factor γ I introduced in EN 1998-1 expresses the peak ground acceleration of the arbitrary return period T L related to the reference return period T LR as: and is also related to exponent k through different return periods. The idea behind this concept is to use a single seismic hazard map for design, corresponding to the reference return period (recommended as 475 years), while different return periods can be taken into account with importance factors calculated according to Eq. (2). Although this concept simplifies the design procedure, it contains drawbacks inherent in the approximation process.
In this study, we used the SHARE project dataset of seismic hazard data for the wider European area (Figs. 2,3,4 and 5) to calculate exponent k according to Eq. (1) and discuss the implications of the calculated values on the importance factors given in EN 1998-1.
The problem of fitting hazard curves is addressed at a general level in more detail in Vamvatsikos (2014), where the hazard curves are approximated by functions of the first (2) I ∼ T LR T L − 1 k and the second order and inherent errors of the process are evaluated, where the first-order function is a straight line in the log-log space, identical to the EN 1998-1 definition and Eq. (1) of this manuscript.
Unfortunately, only a few studies have addressed the problem of fitting the hazard curve data and exponent k to a given dataset, particularly for a wider area. Apart from the results of the SHARE study (Deliverable 2.7; Weatherill et al. 2013) that used the dataset on the wider European area (Figs. 2, 3, 4 and 5), the same as in this manuscript, to our best knowledge, only Lubkowski (2010) addressed this topic to a wider scale, namely, for 57 worldwide regions that include all continents covering low, moderate and high seismic hazard regions. Other studies that are worthy of being mentioned in the context of the results of this study addressed this problem either at a national level (Pavel et al. 2016;Solakov 2009;Larsson and Magnusson 2017;Schmitt 2020) or for particular sites of interest (Bursi et al. 2016).

Importance factors and exponent K
EN 1998-1 categorizes buildings into four different importance classes-from I to IV. Importance class I corresponds to buildings of minor importance to public safety, and class IV refers to buildings of vital importance for civil protection. Importance factors are ascribed to each importance class (EN 1998-1/article 4.2.5) with recommended values of 0.8, 1.0, 1.2, and 1.4 for importance classes I to IV, subjected to changes in national annexes. Importance class II is related to ordinary buildings for which the importance factor related to the reference peak ground acceleration of 475 years is 1.0.
Given the predefined value of exponent k and the recommended importance factors, it is possible to determine the return periods T L corresponding to each importance factor by using Eq. (2), as presented in the first column of Table 1. The importance classes for the I to IV return periods are 243, 475, 821, and 1303 years, respectively. The return periods calculated in such a way can be used to calculate the importance factors for other values of exponent k, as presented in Table 1.
The calculated results indicate that, in less seismically active regions, which correspond to smaller values of exponent k, larger values of importance factors are required for importance classes III and IV in order to preserve the same level of safety (expressed in return periods) as in areas with assumed seismicity represented by exponent factor of 3. On the other hand, in areas of higher seismicity, lower values of importance factors can be used for the same level of safety of importance classes III and IV.
The calculation is also presented in Fig. 1, where, for various values of exponent k, the relationship between the return period and the importance factor is presented. The appearance of the rotating effect of the curves around the focal point of the importance factor equal to 1 suggests that less seismically active areas require more time to express their full earthquake potential compared to areas of higher seismic activity. For instance, in low seismicity areas represented by exponent k of 1.5, the importance factor of γ I = 1.4 would correspond to the peak ground acceleration for a return period of 787 years instead of 1303 years. Therefore, keeping the recommended importance factor in less seismically active areas represented by an exponent factor lower than 3 would lead to underestimation of the design forces for structures of importance classes III and IV, i.e., it would be as if we have used the acceleration values from the seismic hazard map for a return period of 787 years, instead of 1303 years. The inverse effect is presented for importance factor of 1 3 γ I = 0.8; however, the difference is not so prominent, and this importance factor is related to structures of minor importance for the public safety.

Calculation of exponent factor K from efehr data
Data for the calculation have been downloaded from the EFEHR website (www. efehr. org) using the provided MATLAB script. The requested data correspond to rock conditions (v s, 30 > 800 m/s), arithmetic mean aggregation level, and peak ground acceleration consistent with the EN 1998-1 requirements for hazard maps. It should be noted that the hazard values at the EFEHR website are calculated for the probability of exceedance in 50 years-not for the annual exceedance rate. This can be verified by using an online request for a hazard curve for an arbitrary calculation point, calculating the PGA value for a return period of 475 years, and comparing the result to the value provided on the website. Therefore, to determine exponent k according to the EN 1998-1 requirements, it is first necessary to transform the data from the probability of exceedance in 50 years into data calculated for the annual exceedance rate. Another question arises in the approximation process -the choice of the approximation range. As stated before, due to the specific shape of the hazard curve, it is not possible to accurately approximate the hazard curve using Eq. (1) over the entire range of calculated values. This leads to the conclusion that the fewer points are included in the approximation, the more accurate the approximation is. The approximation range and the narrowing of the number of approximation points must be justified by design application of the approximated range, which can be roughly estimated from the calculated return periods presented in Table 1 (if the recommended reference return period of 475 years is used). Also, the SHARE Deliverable (Deliverable 2.7) recommendation was considered as to which periods between 70 and 5000 years were to be used. Similarly to that proposition, in this study, the start year (data of the first year used in the approximation process) was considered to be not less than 70 years and the end year (data from the last year used in the approximation process) to be not greater than 5000 years, as the years vary between the calculation points. As it can be seen from the histogram and the distribution map in Fig. 2, for more than 80% of the calculation points, the start year corresponds to the range between 70 and 115 years (right-hand side of the y-axis), although the start year can be as high as 1000 years. Such is the case with a small number of points concentrated in areas of very low seismic hazard where even the lowest values of acceleration used in the calculation require high return periods. These points are concentrated on the southeastern border of Finland and Russia, the eastern part of Ukraine, and the western Ireland. Nonetheless, for the vast majority of calculation points, the start year is well below 243 years, which corresponds to an importance factor of 0.8, or likewise, well below 475 years, i.e., the start year can be considered relevant for the approximation depending on the purpose of exponent k.
The end year for the approximation range was defined as data corresponding to the first year from the calculation dataset that is lower or equal to 5000 years. As the calculation points tend to be coarser with increasing the acceleration values for high hazard areas, the end year tends to be smaller in areas with lower seismic hazard. The geographical distribution map and the histogram are presented in Fig. 3, from which it can be noticed that the distribution of the end years is more even. Considering the validity of the used minimum end year regarding the maximum year covered by the importance factor of 1.4, it can be concluded that all points are greater than 1303 years, corresponding to the recurrence period bonded to the importance factor of γ I = 1.4 (importance class IV), i.e., the selection is relevant for approximation.
One more relevant issue that should be mentioned in the approximation process is the number of points included in the approximation for each calculated hazard point presented in Fig. 4. The number of points used in the approximation is to some degree related to the inherent error in the approximation process; however, the overall accuracy also depends on the range between the start year and the end year. As can be seen in the geographic distribution map, the more seismically active area is-the less points are used in the approximation. This is inevitably the result of the chosen points of acceleration in the calculation of the seismic hazard, which are given as results at the EFEHR website and cannot be changed, and the chosen boundaries for the start and the end year of approximation. The total number of points varies between 3 and 14. Seismically active areas include less points, and vice versa, the less seismically active areas include more points.
The final step is a linear approximation of the hazard curve on a log-log scale in a given range, which is equivalent to the exponential function of approximation given in Eq. (1).

Results
The calculated results for exponent k for the arithmetic mean values of PGA related to ground type A are presented in Fig. 5. Unlike the statement made in EN 1998-1/article 2.1(4) that for most sites, exponent k is generally of the order of 3, the results indicate that the value of exponent factor k of 3 is rather an exception than an expected value. The value of exponent factor of the order near 3, but still lower than 3, is expected only in the wider Vrancea region, originating from the Vrancea intermediate seismic zone. For the seismically active parts of the Mediterranean and Iceland, exponent k is of the order of 2.5. At the same time, for the north and northwestern Balkan Peninsula, it is of the order of 2. For the Iberian Peninsula, its values vary between 1.2 and 2.2. There is a general trend of decreasing values of exponent k in the continental part of Europe with the increase of latitude. Western Scandinavia has values of exponent k exceeding 1.5, reaching even the value of 2 at some points. In contrast, in eastern Scandinavia, the values fall below 1. The lowest k values of the order of 0.5 have been calculated for the southeastern border zone between Russia and Finland.
Another comparison can be made with the map of exponent k given in Fig. 3. 13a in Deliverable (Deliverable 2.7) of the SHARE project. The obtained results show around 1-1.5 higher values of exponent k than those in this study. The difference is due to the incorrect determination of exponent k in the SHARE study on a hazard curve given for the probability of exceedance in 50 years instead of an annual rate of exceedance hazard curve. This is clearly noticeable in Fig. 3. 12 of the mentioned deliverable with the presented hazard curves for the chosen cities. Namely, the y axis is labeled as "annual rate of exceedance," however, by obtaining the results from the EFEHR website, it is easily noticeable that the given results represent probability values of exceedance in 50 years. Hence, exponent k in the SHARE deliverable was determined on the unrepresentative part of the hazard curve, leading to a false conclusion that the recommended value of exponent k in the EN 1998-1 is valid for most of the seismically active areas of Europe.
The values of exponent k can be perceived in more detail in Table 2 for the chosen sites across Europe. Sites have been chosen to correspond to the one defined in SHARE Deliverable Deliverable 2.7. Again, most of the values of exponent k in the SHARE study are stacked around the value of 3, while the results of this study indicate much lower values.
The hazard curves for the chosen sites are presented in Fig. 6, with the determination range for exponent k used in this study and the SHARE project. Instead of the stated range (Deliverable 2.7) between 70 and 5000 years, the SHARE study determined exponent k in the range between 3475 and 249,975 years where the peak ground acceleration takes values of around 1 g. 1 3

Values of exponent K in national studies
To some extent, the values of exponent k are available from various national studies of seismic hazard that are a good source for cross-validation of the results. Considering the most active seismic zone in Europe (based on the SHARE study results (Deliverable 2.7)-the Vrancea intermediate depth seismic zone, national studies of exponent k known to authors are available for Romania and Bulgaria.
The seismic hazard analysis for Romania (Pavel et al. 2016) deals with this topic in some detail. It provides separate figures for exponent k for the national and SHARE studies. For the national study, the determination range of exponent k is between 30 and 2475 years, which is somewhat different from the SHARE study, but not in a way that could compromise the comparison of the results. The article demonstrated that, in the SHARE study, the total territorial coverage with maximum exponent k due to the Vrancea source zone is much smaller than in the national study.
The seismic hazard study of Bulgaria (Solakov 2009) shows values of exponent k ranging from 1.6 to 4.8, the latter being controlled by the Vrancea intermediate seismic hazard zone. The maximum values of exponent k obtained in the national study are much higher than those in the SHARE study (Deliverable 2.7), particularly in northern Bulgaria, where the values of exponent k are approximately 2.2. The high values of exponent k in the national study also cover a significant part of the northern border zone of Romania.
For Italy, the seismically active country that should be representative of the recommended value of exponent k given in the EN 1998-1, interactive results on seismic hazard are available online via the INGV service (http:// esse1-gis. mi. ingv. it/), which can be used for the determination of exponent k. For the 50% percentile and available range of data that correspond to return periods between 30 and 2500 years, the values of exponent k in Fig. 6 Hazard curves for the chosen sites across Europe with shaded determination ranges for exponent k in the SHARE study, and this study the areas of the highest hazard are of the order of 2.5, which is similar to the results in this study. Considering Table 1, the difference between the importance factors for categories III and IV is 3 and 7%. Also, according to Bursi et al. (2016), for the seismic zone in Sicily, the estimated value of exponent k equals 2.
In his work, Schmitt (2020) partially addresses the topic of exponent k and notes that, for Germany, the evaluations of exponent k indicate that, for many seismic regions, the values are lower than 3. Further, the author presents the hazard curves for the cities of Heilbronn and Karlsruhe with the remark that exponent k of 2 instead of 3 would be suitable for the city of Karlsruhe.
It is interesting to compare studies for areas with lower seismic hazard to the results of this study. For countries where no separate published studies on exponent k were found, we used the values of the importance factors in the National Annexes for comparison. For instance, in the Norwegian National Annex for EN 1998-1 (EN 1998-1:2004+A1:2013+NA:2014, Norwegian National Annex for EN 1998, the values of importance factors of 0.7, 1.0, 1.4, and 2.0 correspond to the value of exponent k of approximately 1.5, which is in good correlation with the results of this study based on the SHARE data (Table 2). Sweden has not yet issued a national annex for EN 1998-1. However, in the study (Larsson and Magnusson 2017), exponent k has been calculated for the city of Lund, for which a value of 1.4 has been obtained, similar to the results of this study (Fig. 5). On the other hand, Germany (DIN EN 1998-1/NA, German National Annex for EN 1998-1)-adopted the recommended values of importance factors in their National Annex, although the seismic activity should lead to much higher importance factors. For example, the values of importance factors for the city of Cologne (Table 2) are thus underestimated by 16% and 32% for importance classes III and IV. Likewise, the National Annex for France (NF EN 1998-1/NA, French National Annex for EN 1998EN -1, 2007 leaves to the municipal authorities to determine the values of the importance factors, not mentioning that the values of exponent k dictate the values of the importance factors-i.e., that it is not a purely political decision. From (La nouvelle 2011), it seems that a single categorization has been used for the entire country of France. United Kingdom (BS NA EN 1998-1, National Annex for United Kingdom) has used the base map for 2500 years and the importance factor of 1.0. However, in cases where T NCR is determined on a project-specific basis, the importance factor should be determined, as well.

Discussion
The final goal of calculating exponent k based on a seismic hazard study is to determine the importance factors (Eq. 2). To preserve the same level of safety as the recommended value of exponent k given in EN 1998-1, the importance factors should be proportionally scaled (Eq. 2) to match the difference between the local seismic activity and the recommended one. Based on the results presented in this study and studies (Deliverable 2.7; Pavel et al. 2016;Solakov 2009;NS-EN 2014;Lubkowski 2010;Schmitt 2020;Bursi et al. 2016;Weatherill et al. 2013), a conclusion is drawn that, considering the territory of a country, a single value of exponent k can rarely be ascribed to the whole national territory with fair accuracy. At best, it is more likely that several values must be ascribed to different parts of the national territory, which can be accomplished by applying different values of exponent k (importance factors) to different seismic zones. However, to our best knowledge, this has not yet been applied in any of the EN 1998-1 National Annexes. The most accurate solution would be to ascribe different k values to each seismic hazard calculation point, but strictly speaking, this is not possible according to EN 1998-1. Further research is required to address the adequacy of the first order fit of the hazard curve as required by EN 1998-1 (Eq. 1), as according to Vamvatsikos (2014), errors of the first order fit can be less than 20%, while the second order fit manages excellent predictions with less than 5% error regardless the curvature. The errors are obviously associated with the approximation range that also needs to be clearly defined in EN 1998-1 to conclude this topic.
To minimize the error related to large areas that include various levels of seismic hazard, one solution for approximation with Eq. (1) could be to determine different values of exponent k for grouped datasets based on a discrete level of seismic hazard as in Lubkowski (2010) where the results of the approximation are given for different values of PGA, for a return period of 475, ranging from 0.05 to 0.40 g.
The exponent k determined in this study is much lower than the SHARE study results (Deliverable 2.7) on the same dataset. In the SHARE study, exponent k is mistakenly determined on a non-representative part of the hazard curve. For most locations, the difference is of the order of 1-1.5. Further, a comparison of the results of this study with values of exponent k given in other studies (Pavel et al. 2016;Solakov 2009;Larsson and Magnusson 2017;Weatherill et al. 2013;Lubkowski 2010;Schmitt 2020;Bursi et al. 2016) that used national seismic hazard results, i.e., different datasets, indicate that there is a good agreement between the values of exponent k. Therefore, for the majority of locations in Europe, exponent k is well below the recommended value (EN 2004) of exponent k of 3.0, which further indicates that this value is probably too high to be recommended for most of Europe's active seismic zones except for the Vrancea intermediate depth seismic zone.
Considering the results of this study, it is more likely that exponent k of the order of 2 is more representative for the seismic activity of the European continent. The consequence of the previously stated is that assuming the recommended value of exponent k, i.e., accepting the recommended values of importance factors, would lead to underestimating the seismic hazard for the importance classes III and IV for as much as 40% for importance class IV.