Estimating annualized earthquake loss for residential buildings in Tehran, Iran

A probabilistic seismic risk assessment is conducted in this study to quantify Annualized Earthquake Loss (AEL) of Tehran, the capital of Iran. To do so, a comprehensive review of three main components of risk assessment including the seismic hazard analysis, exposure model and vulnerability functions is performed. The classical seismic hazard analysis is carried out based on the most recent earthquake catalog by considering the available uncertainties. A high-quality building exposure model based on recent census data with geographic resolution of census blocks is also compiled. According to available information, buildings are classified into 19 groups by considering their construction material, quality and height. The results show that sum of AEL in Tehran is about 10,488 million USD, which is equivalent to 0.16% of exposed economic value. The spatial distribution of AEL indicated that value of AEL in the eastern districts is higher; however, the relative seismic risk in term of AELR in the southern districts of the city is much higher. In addition, a disaggregation analysis per building typology is performed to identify the contribution of different building taxonomies in AEL. The results show that the masonry and low-quality steel and concrete structures with poor designing specifications have the highest contribution in AEL. The finding from this study can be used by local authorities, government and insurance sector in order to develop effective risk mitigation plans and a robust insurance scheme for Tehran, Iran.


Introduction
Tehran is the capital of Iran, and located in the north central of country. According to the latest census conducted in 2016, Tehran has a population of 8,737,510 residing in 2,870,653 households (SCI 2016). Based on the information provided by the Statistical Center of Iran (SCI), Tehran contributes more than 30% of Iran's Gross Domestic Product (GDP). This issue demonstrates the main role of the city in the country's economy. In addition, the city serves as Iran's political, administrative, social, and cultural hub. Thus, its safety and security against natural and man-made hazards are of great concern to the government.
Tectonically, Tehran lies at the foothills of the Alborz Mountains, which are characterized by their shallow large earthquakes accompanied by surface faulting (Berberian and Yeats 1999). According to seismotectonic provinces delineated by Mirzaie et al. (1998), Tehran is also located at the northern boundary of the seismic zone of Central-East Iran (Fig. 1). This area is under compressive stress and is shortened by Arabia-Eurasia convergence (Jackson et al. 2000). This tectonic setting demonstrates the presence of several major faults around the city including Mosha, North-Tehran Fault, Rey, Kahrizak, Pishva, Garmsar, Eyvanaki, Eshtehard, and Taleghani faults (Ashtari et al. 2005). Figure 1 shows Tehran's boundaries in relation to the major active faults within a radius of 150 km that caused several devastating historical earthquakes (see Table 1). The most recent earthquake in this region was Lavasan earthquake with magnitude of 7.0-7.3 Mw on 27 March 1830. The absence of major earthquakes during the past decades and the results from past time-dependent seismic hazard analysis (e.g., by Jalalalhosseini et al. (2018)) suggest high probability of strong motion occurrence around the Tehran city in Fig. 1 Tehran's boundaries in relation to the major active faults within a radius of 150 km the near future. This issue also highlighted in a study of Berberian and Yeats (2017), which review the past earthquakes of Tehran. They stated that future earthquake in Iran is comparable with a time bomb.
Alongside the high seismicity of the region, the poor construction quality and inappropriate land management intensify the seismic risk of the Tehran metropolitan area. This issue is highlighted in several studies (JICA 2000;Mansouri et al. 2010;Firuzi et al. 2019). In the study done by the Japan International Cooperation Agency (JICA 2000), the seismic risk of Tehran is assessed in term of the probable number of causalities and economic losses in three different seismic scenarios including the rupture of Rey, Mosha, and North-Tehran faults. According to their results, the rupture of the Rey fault is the most vulnerable scenario, with a 55% building damage ratio and 6% fatality ratio. Mansouri et al. (2010) also provided an estimation of economic losses for Tehran. They compared the economic losses in Tehran using the hazard map for return period of 475 years and Rey seismic scenario. Their results show that relative area loss in case of rupture of Rey fault is about 65% which is five percent more than using the seismic hazard map for a return period of 475 years. They also used four distinct vulnerability curves to evaluate the sensitivity of their finding in relation to the vulnerability curves. Their results demonstrate that employing different vulnerability curves can result in a 5% variation in the outcomes. Firuzi et al. (2019) used the Monte Carlo simulation approach to estimate Annualized Earthquake Loss (AEL) for Tehran. They generate 250 synthetic earthquake catalogs with a length of 100,000-years and estimate the mean and standard deviation of AEL for residential buildings in Tehran. According to their analysis, the average AEL of building stock is roughly $0.331 billion, or 0.29 percent of the overall exposed economic value. Results of aforementioned studies show the high vulnerability of Tehran meteoroidal area in seismic events. Thus, it is important to provide a detailed seismic risk assessment to address the impact of earthquakes in the region. Such assessments can play a fundamental role in sustainable developing the urban area and providing local authorities valuable information for designing appropriate risk mitigation plans.
The purpose of this study is estimating the AEL, which is an important index for insurance industry, in Tehran. There are two common probabilistic approaches to quantify the AEL: (1) the classical method, and (2) the event-based (Monte Carlo simulation) method. In the classical approach, the area under the loss exceedance curve, which is obtained from the convolution of the hazard curve and vulnerability curve, is representation of the AEL (Jaiswal et al. 2015). While in the event-based approach, the  Berberian and Yeats (1999) 1 3 AEL loss is determined by aggregating the impact of hundreds of synthetic events in the region (Salgado-Galvez 2017). Employing each of the aforementioned approaches depends on the application. The classical approach can be used if the main output of interest is AEL, or if a comparison analysis of risk at multiple sites is required. The event-based technique can be employed if the main result of analysis is the mean and standard deviation of AEL (Crowley 2014). The main goal of this study is to provide a comparative map of AEL for residential buildings within 22 municipal districts in Tehran. Thus, in the present study, the classical method is used for conducting the seismic risk. To the authors knowledge, there is not a peer-reviewed study in literature which provides a comparative map of AEL in Tehran using classical approach. To fill the above-mentioned technical gap, this study aims to estimate AEL of Tehran using classical probabilistic approach. Within this study, the most recent census data on buildings is employed, as well as an updated seismic hazard model. Throughout this paper, first, a description of seismic hazard model for Tehran is presented. Then, the procedure for compiling the exposure model and classification of the building is presented. Next, a description of the employed fragility and vulnerability curves are presented. Finally, an estimation of the possible losses is presented, along with a comprehensive discussion.

Seismic hazard model
The main output of interest from seismic hazard analysis is hazard curve. Hazard curve expresses the annual probability (or rate) of exceeding of a given level of ground shaking. In this study, the conventional Probabilistic Seismic Hazard Analysis (PSHA) proposed by Cornell (1968) is adopted to quantify the hazard curve. This method is composed from four major steps: (1) compiling a uniform earthquake catalog; (2) defining seismic sources and assigning their corresponding seismicity parameters; (3) selecting appropriate Ground Motion Prediction Equations (GMPEs), and (4) combining above steps using total probability theorem to obtain the hazard curve. Each step of PSHA analysis, as well as the procedure to capture uncertainty, will be discussed in the following sections.
Compiling earthquake catalog: Compiling a reliable and homogenous earthquake catalog is a crucial step for estimating the seismicity parameters. The homogenous earthquake catalogue provided by Mousavi-Bafrouei and Mahani (2020) is used herein, which is based on the compiled Iranian earthquake catalog from the local and international agencies. However, this catalog contains events with the moment magnitudes ranging from 2.6 to 8.1 until 31st of December 2018. In order to conclude the final catalogue used in this study, more recent earthquakes data were borrowed from Iranian Seismological Center (IRSC) and homogenized by the same relation presented in Mousavi-Bafrouei and Mahani (2020). As a result, the final instrumental earthquake catalog is composed from 3633 events with magnitude greater than 2.6 Mw, which its distribution is shown in Fig. 2 with the radius of 150 km around the center of Tehran city. In order to assess the completeness of catalog, the maximum curvature approach is adopted using ZMAP package (Wiemer (2001)). As shown in Table 2, the catalog's completeness prepared in this study shows comparable values compared to a recent study, which assess the completeness of earthquakes in Iran, by Khodaverdi et al. (2016). Furthermore, two approaches are adopted for considering the uncertainties regarding declustering the earthquake catalog. The time-space window approach proposed by Gardner and Knopoff (1974) is used as the first method for declustering, which identified 2592 events as independent events. The approach developed by Reasenberg (1985) is used as the second method of declustering that identified 3171 events as independent events. Catalogs from both of these methods are used for estimating the seismicity parameters.
Defining the seismic sources: Delineation of seismic sources is one of the most subjective steps in performing PSHA. One of the best practices to consider epistemic uncertainty is by using logic tree. In order to build the logic tree, four seismic source models are considered which are shown in Fig. 3. The first seismic source model is composed of 166 active faults (Fig. 3a). The second seismic source model consists of 15 area sources (Fig. 3b). These seismic source models  were borrowed from the Earthquake Model of the Middle East (EMME) , which was a collaborative project to evaluate seismic hazard and risk in the Middle Eastern region (Sesetyan et al. 2018). The third seismic source model was used from a model developed by Mahsuli et al. (2019) to provide seismic hazard in Iran using reliability method. This model is composed of three area sources and 32 active faults (Fig. 3c). The fourth seismic source model is composed of two area sources and 62 active faults in the region (Fig. 3d). This model is developed by the authors to consider distribution of past earthquakes and geological evidence.
A back ground seismicity model, consisting of a uniform grid cell of 0.1 × 0.1 degree, is defined in addition to the aforementioned seismic source models. It should be remarked that the seismicity parameters corresponding to seismic sources are estimated using the approach developed by Kijko and Sellevoll (1992). In the latest approach, the completeness of the earthquake catalog is considered.
GMPEs selection: GMPEs represent the distribution of ground motion value as a function of parameters such as earthquake magnitude, distance, fault mechanism, and site conditions. With respect to the extent of GMPEs available in literature, selecting appropriate models is of great concern. A list of published GMPEs in literature can be found in Douglas (2018). In the present study, the GPMEs proposed by Firuzi et al. (2020) is used, which assessed the suitability of 14 candidate GMPEs for Iran based on statistical tests. They used the likelihood (LH), log-likelihood (LLH) and Euclidean distancebased ranking (EDR) tests developed by Scherbaum et al. (2004Scherbaum et al. ( , 2009) and Kale and Akkar (2012) to evaluate the appropriateness of the candidate GMPEs. Their results indicated that relations of Kale et al. (2015), Kotha et al. (2016), Akkar and Bommer (2010), Zhao et al. (2006) and Idriss (2014) have the best performance. Firuzi et al. (2020) also proposed the corresponding weights of GMPEs to be employed in logic tree that used in this study and will be presented in the following section.
Logic Tree: There are two types of uncertainty: aleatory and epistemic variabilities. The aleatory uncertainty is related to the inherent nature of randomness of issue; while epistemic uncertainty is stemmed from inaccuracy of the model or incompleteness of data. The aleatory uncertainties are considered in analysis using integration over the probability distribution of parameters; however, the epistemic uncertainties are considered in analysis using logic tree (Bommer et al. 2005). As mentioned earlier, five seismic source models, two declustering earthquake catalog methods, three different seismic depths and five GMPEs is used in the logic tree. The final structure of logic tree is composed of 150 branches. Figure 4 shows the general structure of logic tree used in this study.
Computing seismic hazard: For evaluating the seismic hazard in the present study the open-source OpenQuake software with appropriate flexibility in defining seismic sources and employing the logic tree is used (Pagani et al. 2014). It should be noted that the local site condition is considered in this analysis based on the study from JICA (2004) that classifies the region into 41 soil types based on information of 450 boreholes. The amplification factor of each soil type is estimated by using one-dimensional response analysis. Figure 5 shows the seismic hazard map of Tehran for return periods of 475 and 2475 on engineering bed-rock and soil surface. As depicted, the acceleration on the bed-rock in the eastern and northern parts of Tehran is relatively higher than western and southern parts of Tehran ( Fig. 5a, b). This is due to the concentration of small magnitude and active faults of Mosha and Rey. By employing the local site condition, the acceleration in the southern part of the city is also increased (Fig. 5c, d). This

Developing exposure model
Developing an accurate and detailed exposure model is an essential phase of seismic risk assessment. A reliable exposure model contains information about the spatial distribution of both population and built assets. The main focus of this section is compiling a reliable database of residential buildings with their main structural characteristics in Tehran.
There are a few studies in the literature discussed about compiling the exposure model in Tehran or the entire country. As a part of the EMME-GEM (Earthquake Model of the Middle East-Global Earthquake Model) projects, Mansouri and Hosseini (2014) provided residential building stock and population databases for Iran. They used the 2006 national census data at the city level as a primary source of information for compiling the exposure model. Then, the number of buildings located at the centroid of the city was re-distributed within the grid cell with resolution of 5 km, according to the population dataset LandScan TM developed by Oak Ridge national laboratory (Dobson, 2000), which provides population count estimates for a 30 arc-second grid. Motamed et al. (2019) also provided an exposure model for Iran to quantify the seismic risk. They used the 2011 national census as the main reference for compiling the exposure model. They classified buildings into 23 categories based on the construction materials, the height of the structure, the load lateral resistance system, and the year of construction. These databases have low geographic resolution of building inventory for seismic risk in a city. Review of above studies show that available exposure models have poor geographic resolution and limited structural characteristics for seismic risk assessment to be employed in the present study. Therefore, here, a reliable exposure model of residential buildings in Tehran using the latest census data is compiled. The essential factor in compiling the exposure model of residential buildings in Tehran is paying attention to extremely heterogeneous building inventories. The building inventory in Tehran covers a wide range of construction types, from non-engineered structures like old masonry structures to high-rise buildings built based on the most recent codes and standards. Ideally, information on each building should be separately collected and analyzed. Such precise information, however, is not available. Thus, buildings should be classified in groups which share common characteristics such as construction material, age, number of stories, load resistance system, ductility level, etc. Although some attributes are essential to define the vulnerability of a given building, the information to define them is often not available (e.g., type of load resistance system). Therefore, considering such attributes in building classification depends on the availability of information.
In the present study, information provided by SCI is adopted for building classification. SCI is Iran's official center for collecting information regarding the population and building inventories. This center represents information about the construction material, age of structure, built-in area, and population. From 1956 to 2006, this center conducted national census once a decade. After 2006, the census is performed national census every five years. Here, the latest information provided by SCI in 2016 is used. This is the most reliable and available information. It is noted, SCI provides the information in census block. This is the smallest geographic unit used by the SCI for gathering data. Figure 6 depicts the distribution of census blocks throughout Tehran. Bal et al. (2010) assess the impact of geographic resolution of urban exposure on economic losses for three different geographic resolution. Their results show that the total damage over an Fig. 6 Distribution of census blocks used by SCI throughout Tehran (SCI, 2016) urban area, expressed as a mean damage ratio (MDR), is rather insensitive to the spatial resolution of the exposure. Thus, the aleatory uncertainty regarding the spatial variation ground motion in census block do not impose significant uncertainty on the results. Therefore, in the present study, the center of census block used for estimation of ground motion value in analysis.
Here, building inventories are classified in terms of construction material, construction quality, and height. Although classifying the building inventories in such generic groups may impose on certainty to the results, this is in accordance with available information provided by SCI. From the construction material point of view, buildings are categorized into: reinforced-concrete, steel, and masonry. It is noted that all adobe, masonry, rubble stone, and other types of structures are considered as masonry class. Figure 7 shows the number of buildings constructed using different material. As depicted, most of structures are constructed using steel. The high number of masonry buildings in Tehran is of great concern. Three different level of construction qualities including low-code, mid-code, and high-code are also considered in building classifications. Quality of construction is related to the seismic regulation used for designing a building. The Iranian Code of Practice for Seismic Resistant Design of Building (Standard 2800) is the primary reference for seismic design of structures. At present, four versions of this standard are published. The first edition was presented in 1987; buildings constructed by this version are considered as low-code. The second edition was published in 1996; buildings designed by this version are considered as mid-code. The third and fourth versions were presented in 2005 and 2015; buildings designed by these versions are considered as high-code. Figure 7 also shows the number of buildings based on the year of construction. As depicted, most of buildings are constructed by the second edition of Standard 2800, which are categorized in mid-code.
It should be noted that SCI only provides the number of residential units and not the number of stories. Thus, there is uncertainty about the number of building's stories in the database. In the present study, the number of stories is estimated based on the additional available real estate database. This database provides the number of stories in 22 municipal districts Tehran. Figure 8 shows distribution of the final building types throughout Tehran. As depicted, buildings located in the southern part of the city are masonry and low-quality steel and concrete structures with poor designing specification. While quality of construction at the central and northern parts of the city is significantly higher. The compiled building inventories used for quantifying seismic risk in Tehran.

Vulnerability/fragility curves
Seismic risk assessment requires a set of fragility or vulnerability curves. Fragility curves describe the probability of exceeding different damage states given a level of ground shaking. While vulnerability curves provide the probability of losses given a level of ground shaking. Generally, the former is used for estimating damages; whereas the latter is used for estimating economic losses. It should be remarked that vulnerability curves can be derived from fragility curves (Yepes-Estrada et al. 2017). In this procedure, the loss ratio for a given intensity measure is determined by the sum of the product of probability of different damage states and their corresponding mean damage factor. This explanation formulated in the following equation.
In the above equation, LR is the loss ratio, ds is different damage states, P[ds = DS] is probability of various damage states and MDF ds is mean damage factor. Based on aforementioned explanations, for economic loss estimation, appropriate vulnerability (or converted fragility curves) should be adopted. The key point in selecting vulnerability curves is the compatibility of the vulnerability models with pre-defined building typologies. There are several vulnerability curves in literature which developed for various building structures in Iran. Most of these fragility curves are empirically developed based on data of past earthquakes.
Preliminary efforts for developing fragility curves of building stock in Iran was carried out by Tavakoli and Favakoli (1993). They developed fragility curves for residential buildings in Iran regardless of the year of construction, quality of construction, and construction material using data of past earthquakes. In a study done by JICA (2000)  buildings based on their construction material and age. As a part of EMME-GEM project, Mansouri and Amini-Hosseini (2013) developed fragility curves for 10 building typologies in Iran based on EMS-98 procedure. They considered construction material and quality of construction in definition of building classes. Omidvar et al. (2012) also proposed empirical fragility curves for engineered steel, and RC structures in Iran. Sadeghi et al. (2015) derived a set of vulnerability curves for the 42 Iranian building types by combining the existing fragility/vulnerability in literature. They considered parameters such as seismic load resistance, construction material, height, and quality of construction for defining building taxonomies.  1) is used. Figure 9 shows the final vulnerability curves. As depicted, masonry represents the largest loss ratios, while reinforced-concrete structures are the least vulnerable class.

AEL estimation
In the present study, the AEL is estimated based on the classical probability procedure. In this approach, the Loss Exceedance Curve (LEC), which describes the annual probability of economic losses of structures for a given ground motion level, is derived from the convolution of the hazard curve and vulnerability curve. The area under the LEC provides the Fig. 9 The converted fragility curves of Fallah Tafti et al. (2020) to vulnerability curve a reinforced concrete and masonry structure; and steel structure (abbreviations are as follow: ST: steel, RC: concrete, MA: masonry, LR: low-rise, MR: mid-rise, HR: high-rise, LQ: low-quality, MQ: mid-quality, and HQ: highquality) annual repair to replacement cost ratio. The general framework of aforementioned procedure is illustrated in Fig. 10.
It should be mentioned the aforementioned procedure provides the repair to replacement cost ratio per year. To convert the annual repair to replacement cost ratio to economic value, the replacement cost of different building typologies is required. Due to the high inflation rate in Iran, there is uncertainty about the value of reconstructing the building. In this study, the value provided by two different references is considered for quantifying the AEL. The value provided by the Construction Engineering Organization (CEO) is used as Fig. 10 The general framework for quantifying the AEL in classical probabilistic seismic loss model the primary reference. CEO annually presents the construction expenses per square meter in Iran. In Table 3 the recommended value of this center for 2021 is presented (Iran Construction Engineering Organization (IRCEO), 2022). It should be mentioned the value provided by CEO is related to the entire country, while the expenses in Tehran metropolitan are much higher. A statistical study done by a group of experts shows that the construction expenses in Tehran has been rapidly increased in recent years (Jadval zarb 2022). The recommended reconstructing prices by this group is also presented in Table 3. In this study, both references are used to estimate AEL in Tehran. AEL based on the above explanation is estimated for census blocks, the aggregated value of AEL in 22 municipal districts of Tehran regarding two different replacement costs is shown in Fig. 11. The results show that the higher value of AEL is mostly dominated in the east, south-east, and north-east parts of Tehran including districts number 1, 2, 3, 4, 5, Fig. 11 The spatial distribution of AEL in 22 municipal districts of Tehran by the replacement cost provided by a CEO; and b expert opinions and 15. The seismic hazard in these districts is higher due to the exitance of active faults and soil condition (Fig. 5). On the contrary, the western part of Tehran has the lower value of AEL (e.g., district numbers 9, 21, and 22); the majority of building in these regions are recently constructed based on high-quality engineering regulation. The total sum of AEL for the whole of Tehran based on the replacement cost of provided by the CEO and expert opinions are 4659 and 10,488 million USD, respectively. The estimated AEL in the present study is much higher than the value provided by Firuzi et al. (2019) and Motamed et al. (2019). Firuzi et al. (2019) provide an estimation of AEL for residential buildings in Tehran based on the event-based approach. Their analysis shows that AEL in Tehran is about 331 million USD. Motamed et al. (2019) also provide an estimation of AEL for different provinces of Iran. Their analysis shows that the AEL for Tehran province is about 161-467 million USD. Differences in AEL are mainly related to the rapid increasement of construction expenses in Iran. According to STATISTA data, Iran's inflation rate in 2019 and 2020 were about 34.62% and 36.44%, respectively (STATISTA, 2022)Thus, the replacement cost of building stocks has grown, and AEL has increased as well.
It should be remarked that AEL does not necessarily correspond to seismic risk. The AEL may identify regions with high concentration of exposed economic value. The normalized of AEL to exposed economic value is an indicator of risk. AEL Ratio is a dimensionless parameter that represents the building vulnerability or relative risk. This is an appropriate index to provide a comparison of seismic risk in different regions. Figure 12 shows the spatial distribution of AELR in 22 municipal districts of Tehran. As depicted, the southern parts of the city are at higher seismic risk. There is an interesting contradiction between maps of AEL and AELR (Figs. 11 and 12). On one hand, AEL in the northern and eastern parts of Tehran is higher; on the other hand, the AELR in the southern parts of the city is higher. As a case in point, the number of buildings and AEL in districts 2 and 5 is higher than districts 9, 16, and 17. However, the AELR in districts 9, 16, and 17 is higher. This is related to the existence of many weak masonry structures in the southern part of the city. This information can be used by local authorities to provide higher priority measures in retrofitting and risk reduction projects. The AELR for the whole of Tehran Fig. 12 The spatial distribution of AELR in 22 municipal districts of Tehran is 0.161%. This value is lower than the values provided by Firuzi et al. (2019) and Motamed et al. (2019). They estimated the AELR about 0.29%. The lower value of AELR in the present study compared to the aforementioned studies may related to their differences in adopted methodologies and the region of study. It should be noted that Motamed et al. (2019) assessed the seismic risk for Tehran province; however, the study area in this study is restricted to the Tehran city. There are a high number of low-quality buildings in Tehran's suburbs. These structures increased the relative risk of Tehran province.
To identify the contribution of different building classes in AEL, a disaggregation per building typology is performed. The result is shown in Fig. 13. As depicted, the masonry and low-quality steel and reinforced-concrete structures show the highest contribution to the seismic AEL. This is evident due to the poor performance of these structures under the seismic loads which show the high loss ratio even in low hazard intensities. It should be remarked that similar result is presented by Firuzi et al. (2019) and Motamed et al. (2019) which assessed the seismic risk for Tehran and the whole of Iran, respectively. This issue highlights the importance of adopting appropriate plan by local government to retrofit such structures.
The concerning point regarding Tehran is that the majority of buildings in the southern part of Tehran are from masonry and low-quality steel and reinforced-concrete structures, where accommodating low-income residents and exposed to high seismic hazard. This issue highlights the necessity of providing incentive measures such as low-profit loans or tax exemption for those who retrofit their vulnerable buildings in those regions.

Limitation and drawbacks
Classical probabilistic seismic loss approach used in the present study is associated with limitations which imposed uncertainties on results. In this study, the local site condition is considered in the analysis as a deterministic parameter provided by study of JICA (2000). However, there is uncertainty regarding this value. In the study done by Ordaz  (2022) three approaches are proposed for considering such variability in analysis. The last approach, which is the most sophisticated and comprehensive one, used the correlation model like those from Baker and Cornell (2006) or Jaimes and Candia (2019) for providing different realizations of Uniform Hazard Spectrum (UHS) on the rock. Then, by employing an amplification function, which is function of acceleration on the rock, the probability distribution of UHS on the soil surface is derived. It should be mentioned, this approach is developed based on UHS. However, in the present study, we require to have probability distribution of hazard curve on the rock to employ the local site condition. The variability of hazard curve on the rock can be obtained from the logic tree. Then, by employing the amplification factor on different branches of log tree, distribution of acceleration of the soil surface can be determined.
Proper considering near source effects such as velocity pulses or directivity is the other challenge of seismic risk assessment using classical method. The most relevant method for taking the near field impact into account in the analysis is using complicated GMPEs like NGA-west2 models. In the present study, GMPEs are taken from the statistical tests, which assess the suitability of different GMPE models based on the observed ground motion values in the past earthquakes of Iran, performed by Firuzi et al. (2019). In that study, the NGA-west2 GMPEs are among the candidate models. The results show that the relation of Kale et al. (2015), Kotha et al. (2016), Akkar and Bommer (2010), Zhao et al. (2006) and Idriss (2014) show the best performance. Thus, in the present study, one of the NGA-west2 models are considered in the analysis. In addition, the near source effects can be considered in analysis using the approaches like stochastic-finite-fault model with dynamic corner frequency. This is a useful tool for addressing source, path and near source effects. Motazedian and Atkinson (2005) by employing the method proposed by Mavroeidis and Papageorgiou (2003) provide formulation for generating the impulse long-period velocity pulses. However, due to lack of instrumental strong earthquakes records on Tehran's faults, providing an accurate estimation of required parameters such as stress drop is associated with uncertainties.
In addition, the seismic hazard in this study estimated by performing uncorrelated PSHA at several locations (i.e., for spatially distributed exposure). However, the spatial correlation may have significant impact on seismic risk assessment for a spatially distributed exposure. This issue is evaluated in several studies. Weatherill et al. (2015) shows that disregarding the spatial correlation in seismic risk assessment resulted in underestimated losses, especially in long return periods. Although, considering the spatial correlation in classical PSHA is a challenge, Wesson and Perkins (2001) using the concept of joint probability hazard curve proposed an approach for considering the spatial correlation in conventional PSHA. In spite of that this method provides more realistic results, this is beyond the scope of the paper.
There are also limitations about the input parameters which their improvement may increase the accuracy of results. As a case in point, the building taxonomy in the present study is defined based on parameters such as construction material, construction quality and height. It is believed that the greatest part of uncertainty in the studies of vulnerability and of damage are of epistemic origin due to the fact of classifying very different buildings in generic models of structures. The best practice for considering such variability in the analysis is using logic tree with different fragility or vulnerability curves. However, restriction of available information about the buildings limits the authors to use fragility curves developed by Fallah Tafti et al. (2020). Clearly, limitation of building parameters and available fragility curves in the analysis will impose uncertainty on the results.
Alongside the above limitation, employing methods like dynamic exposure model (Pittore et al. 2015), machine learning methods (Riedel et al. 2015), cost-benefit models (Riedel and Gueguen 2018), and PSHA testing methods (Stirling and Gerstenberger 2010) will improve the results of this study. In the light of aforementioned uncertainties and restrictions, the AEL provided in this study can be used by local authorities and insurance sector to provide appropriate risk mitigation plans and insurance scheme.

Conclusion
In this study, classical probabilistic seismic loss approach is used to quantify the AEL in Tehran metropolitan. This is an important parameter for the insurer industry to set premiums. The normalized of AEL to the exposed economic value is also an important parameter for local authorities to identify the high seismic risk regions and provide appropriate risk mitigation plans. The analysis is done by a comprehensive review of the main three components of seismic loss assessment (seismic hazard, exposure model, and vulnerability curves).
The convectional PSHA proposed by Cornell (1968) is used to provide the hazard curve in the centroid of census block. Through the PSHA, the epistemic uncertainty regarding defining seismic source models, declustering the earthquake catalog, selecting appropriate GMPEs, and assigning appropriate seismicity parameters are considered in logic tree. The final logic tree composed of 150 branches. The median of hazard curves in census block is used for loss estimation. The exposure model is compiled based on the latest census data. SCI provides information about the material and year of construction. The height of buildings is also considered in defining building classes using additional database. The final building classes composed of 19 taxonomies. It should be mentioned the geographic resolution of the exposure model is census block. This is the smallest available information available in Tehran. A set of vulnerability curves provided by Fallah Tafti et al. (2020) is used for loss estimation. The vulnerability curves provided by Fallah Tafti et al. (2020) are compatible with pre-defined building taxonomies.
Based on aforementioned factors, the aggregated value of AEL in 22 municipal districts of Tehran is derived. The higher value of AEL is related to the eastern and northern districts of the city including districts number 1, 2, 3, 4, 5 and 15. This is due to high seismic hazard and concentration of exposed economic in these districts. Although, the AEL in eastern districts is higher, the relative seismic risk in term of AELR in the southern districts of the city is much higher. This is an intersecting result, on one hand, the AEL in eastern and northern districts of Tehran is higher; on the other hand, AELR in southern districts of the city is higher. This issue indicates that the number of buildings or exposed economic in northern and eastern parts of the city is higher; the quality of structures in these regions also is higher in relative to the southern parts of the city. For example, the number of buildings in districts 2 and 5 is higher than districts 9, 16, and 17. However, the relative risk in districts 9, 16, and 17 is higher. This is due to dominating of many weak masonry buildings in south of Tehran. This issue shows the importance of adopting appropriate plans by local authorities and decision-makers to provide appropriate plans for retrofitting the buildings of these districts.
The sum of AEL for the whole of Tehran based on two different references of replacement cost are about 4659 and 10,488 million USD, respectively. This is equivalent to 0.161% of exposed economic value. The estimated AEL in the present study is higher than studies of Firuzi et al. (2019) and Motamed et al. (2019); however, the AELR is lower. The higher prediction of AEL in this study is related to the increasement in replacement cost in recent years. The lower value of AELR in the present study compared to the aforementioned studies may attributed to differences in adopted methodologies and the region of study.
In addition, a disaggregation analysis of AEL per building class is performed to identify the contribution of different building classes in AEL. The results show that the masonry and low-quality steel and concrete structures have the highest contribution in AEL. Those building typologies form the majority of building classes in the southern part of the city that are mainly the settlement of low-income people. Providing incentive measures such as low-profit loans with long-term repayment can provide motivation for habitant of those regions.
It should be mentioned that the estimated AEL in the present study is associated with limitation and drawbacks. Disregarding factors like available variabilities in V S30 , correlation of ground motion values, epistemic uncertainties of fragility curves, and near source effects impose uncertainties to the results. In addition to these types of uncertainties, there are some limitations and inaccuracies regarding the exposure model. Thus, any improvements in those parameters will enhance the results. Despite aforementioned uncertainties and restrictions, the AEL provided in this study can be applied to develop an effective insurance scheme and risk reduction programs for Tehran.