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Using ambient vibration measurements for risk assessment at an urban scale: from numerical proof of concept to Beirut case study (Lebanon)
 Christelle Salameh^{1}Email author,
 PierreYves Bard^{1},
 Bertrand Guillier^{1},
 Jacques Harb^{2},
 Cécile Cornou^{1},
 Jocelyne Gérard^{3} and
 Michelle Almakari^{4}
 Received: 1 January 2017
 Accepted: 12 April 2017
 Published: 26 April 2017
Abstract
Keywords
 Spectral coincidence
 Nonlinearity
 Seismic damage
 Microtremor
 Beirut
 Neural network
Background
It has been often observed that buildings suffer larger damage from earthquake shaking if their fundamental frequency is close to that of soil (Takewaki 1998; Guéguen et al. 1998; Hellel et al. 2010). However, these past lessons of soil and building spectral coincidence leading to greater damage are generally not taken directly into account, neither in presentday seismic regulations at small scale, nor in largescale seismic risk analysis. Concerning the former, the characteristics of seismic loading are generally derived from the code, i.e., based on very crude site classification, without specific attention to site frequency and associated amplification. Recent findings by De Biasio et al. (2015) or Perrault and Guéguen (2015) have already concluded that the ground motion intensity offering the best correlation with damage in a nonlinear and realistic building model is the spectral ordinate around the fundamental building frequency. As to the largescale risk estimates, there often exists a lack of consistency between seismic hazard studies which do take into consideration the actual soil frequencies and the largescale vulnerability estimation and seismic risk maps which pay only poor attention to building frequencies. The latter generally follow a statistical approach because of the lack of detailed information about existing buildings. They use more traditional specifications such as structural typology, age of the building and maintenance level, together with empirical formulas relating damage to simple seismic loading parameters: A large part of the more detailed outcomes of local hazard analysis is therefore most often poorly accounted for. These empirical methods for vulnerability estimation are essentially based on the feedback from past earthquakes and a crude appraisal of structural characteristics from rapid visual inspections. The scope of this paper is therefore to contribute to the improvement of largescale risk assessment methodology, with an attempt to account simultaneously for the dynamic characteristics of the buildings and of the soil (i.e., their natural frequencies) and to investigate whether it helps in better predicting the structural damage.
The main challenge is to propose and test damage assessment approaches that could combine the spectral contents of the ground motion and the dynamic behavior of the buildings, while being simple and robust enough to be applied at a large scale (typically urban scale). The challenge comes from the fact that the building and soil characteristics exhibit a very large spatial variability due to building heterogeneities (height, material, age, etc) and geological conditions, while the goal is to obtain reliable estimates for a whole urban area.
In that aim, we performed a comprehensive parameter study to investigate the seismic demand for a set of 141 SDOF elastoplastic oscillators with mechanical characteristics (frequency, elastic limit) representative of most common building typologies and height (Lagomarsino and Giovinazzi 2006), lying at the surface of a large set (887) of realistic soil profiles with linear behavior compiled from European, Japanese and Californian sources, for a series of 60 reference rock motions covering a wide range of magnitude, distance and PGA values. For each case, the simulated damage is quantified on a continuous scale between 0 (no damage) and 4 (collapse) according to the EMS98 intensity scale. Then, a “damage increment” is evaluated for the same oscillator resting on the soil profile compared to spatially uniform rock conditions. The extensive simulated database (over 7 million models in total) has been analyzed using an artificial neural network approach to correlate this rocktosoil damage increment to several simple and macroscopic explanatory variables linked to soil behavior (PGA, fundamental frequency, velocity contrast) as well as building characteristics (frequency, ductility), without any a priori information on the functional form of such relationships (Dreyfus 2005). All this “conceptual framework” is shortly described in the first section.
The next two sections present two additional developments aiming at improving the field applicability of the approach and testing its robustness. The first one deals with the testing of a simple and easytoobtain parameter as a relevant proxy for site amplification in order to replace the velocity contrast considered in the first section. Therefore, we used the same simulation results and the same neural network approach, to investigate the performance of other site amplification proxies (such as H/V amplitude, V _{S30}, V _{S10}), which would be easily available and economically affordable. A good candidate for such a proxy is the “A _{0HV}” amplitude, i.e., the amplitude of the H/V ratio derived from microtremor recordings. The other development deals with the changes brought by the consideration of nonlinearity in site response, such as a shift to lower frequency and decrease in amplitude, as results of the first section correspond only to linear site response. New neural networks are built for the modified rocktosoil damage increment resulting from the nonlinear site response, and the results are compared to the linear site response cases.
Finally, the ultimate step is to sensecheck this approach on an example case: The expected damage in Beirut, Lebanon, in terms either of soiltorock damage increment or of absolute damage, is mapped for various reference rock PGA values on the basis of the available information, i.e., building inventory, microtremor measurements and H/V processing, and instrumentally derived relationships providing building frequency as a function of story number, typology and foundation soil.
Conceptual model, numerical simulations and typical results
Model parameters
 1.
The seismic excitation consists of a series of 60 synthetic realistic accelerograms for different scenarios (magnitude between 3 and 7, distance between 5 and 100 km, PGA from 0.02 to 8.6 m/s^{2}). They are obtained using the GMPElike approach originally proposed by Sabetta and Pugliese (1996), which allows to obtain realistic time domain waveforms presenting both frequency and nonstationary characteristics representative of real accelerograms.
 2.
A set of 887 multilayered soil profiles corresponding to real sites was considered in this work. It consists of 614 Japanese KiKnet sites, 251 sites from the USA made available by D. Boore (http://quake.usgs.gov/~boore) and 22 European sites measured during the NERIES project (Di Giulio et al. 2012). The KiKnet velocity profiles were directly collected at http://www.kyoshin.bosai.go.jp and consist of surfacetodownhole measurements of S and P wave velocities. Some of these profiles were modified as explained in Salameh (2016) in order to have a minimum velocity of 800 m/s in the underlying halfspace, considered as seismic bedrock. The main characteristics of the resulting profiles are presented in Salameh (2016) and Almakari et al. (2016) (refer to “Appendix”): The profile database mainly consists of usual stiff soils, with shallow to intermediate thickness (smaller than 200 m in most cases, with only few sites—<6%—with fundamental frequency below 1 Hz), and “normally hard” to very hard underlying bedrock. V _{S30} is lower than 200 m/s for only 40 sites, while it exceeds 800 m/s for about 150 sites. Each soil profile is defined by its number of layers and their thickness, P and S wave velocity (V _{P}, V _{S}) and mass density. In the absence of any further information about the quality factors Q _{S} and Q _{P}, we assumed Q _{S} = V _{S}/10, with V _{s} in m/s, and Q _{P} = 2Qs (Aki and Richards 1980; Fukushima et al. 1995).
 3.One hundred and fortyone SDOF oscillators with an elastoplastic behavior and realistic properties (Fig. 1) were considered according to the outcomes of the RiskUE European project. Lagomarsino and Giovinazzi (2006) compiled the main characteristics of the various EuroMediterranean building types such as fundamental periods, elastic yield displacement d _{y} and ductility ratios d _{u}/d _{y}, where d _{u} is the ultimate displacement. All these structures are classified into five main building classes aggregating different typologies: masonry (class 1), nondesigned reinforced concrete (RC) (class 2), designed RC with low (class 3), medium (class 4) and high (class 5) ductility.
Numerical simulation and damage estimates
This equation is solved with a computation code written in Fortran language that proceeds using the stepbystep Newmark method, taking into account the yield limit that determines the elastoplastic behavior of the perfect elastoplastic oscillator.

D0 (no damage): For d _{max} < 0.7d _{y}, DI = d _{max}/(0.7d_{y}); for this no damage state, DI ranges from 0 to 1.

D1 (slight damage): For 0.7d _{y} < d _{max} < 1.5d _{y}, DI = 1 + (d _{max} − 0.7d _{y})/(0.8d _{y}); for the slight damage state, DI ranges from 1 to 2.

D2 (moderate damage): For d _{max} between 1.5d _{y} and 0.5(d _{y} + d _{u}), DI = 2 + (d _{max} − 1.5d _{y})/[0.5(d _{y} + d _{u}) − 1.5d _{y}]; for the moderate damage state, DI ranges from 2 to 3.

D3 (extensive damage): For d _{max} between 0.5(d _{y} + d _{u}) and d _{u}; DI = 3 + [d _{max} − 0.5(d _{y} + d _{u})] [d _{u} − 0.5(d _{u} − d _{y})], for the extensive damage state, DI ranges from 3 to 4.

D4 (collapse): For d _{max} > d _{u}, DI = 4 for complete collapse damage state.
The final step is to compute a “damage increment index” Δ(DI) corresponding to the damage index variation for an identical oscillator loaded by the site motion a _{ S,i }(t) and the corresponding rock motion a _{ R,i }(t): Δ(DI) = DI(soil) − DI(rock). This damage increment is thus estimated for a total of 7,504,020 combinations (887 soil profiles, 60 input signals, 141 oscillators), except for the cases where either DI (rock) or DI (soil) is equal to 4 (i.e., the maximum displacement of the considered oscillator exceeds d _{u}), which were discarded in order not to introduce any “saturation” effect with the DI upper limit of 4. On the other extreme, the “no damage” state D0 was assigned a nonzero value from 0 to 1 to keep track of the consequence of site effects even within the undamaged elastic domain of building behavior. This elimination left a total number of 6,501,701 combinations, i.e., a reduction of 13.4%. This proportion is relatively stable for all building classes, except for class 1—masonry—for which it reaches 29%, as it is the most vulnerable class.
Analysis of results: neural network approach
The outcomes of such an approach may be analyzed in different ways.
Performance indicators (RMSE, R ^{2}) and synaptic weights (lines) for the three ANN models (columns) for building class 3
Model 1 Velocity contrast, linear site response  Model 2 H/V amplitude, linear site response  Model 3 Velocity contrast, nonlinear site response  

Site amplification proxy  C = V _{smax}/V _{smin}  A _{0HV}  C = V _{smax}/V _{smin} 
Performance indicators  
Δ(DI) standard deviation (initial value: 0.1724)  0.112  0.099  0.103 
Standard deviation reduction (%)  35%  43%  40% 
Coefficient of determination R ^{2}  0.81  0.86  0.82 
Synaptic weights  
f _{struct}/f _{soil}  0.51  0.51  0.51 
Site amplification proxy  0.19  0.20  0.16 
PGA  0.30  0.29  0.33 
These results—which are typical of what is obtained also for other building classes—emphasize the major impact of the spectral coincidence between soil and building natural frequencies. The synaptic weight around 50% for the frequency ratio f _{struct}/f _{soil} is associated with a marked peak of the damage increment around f _{struct}/f _{soil} = 1, with values exceeding 2 for moderate to strong motion and large velocity contrast, i.e., large site amplification. Such a conceptually simple model thus allows to quantify what had been repeatedly observed in postseismic investigations: Having a fundamental frequency close to that of soil leads to increased damage. Conversely, if the building frequency is lower than the soil frequency, the building does not “feel” the effect of the site resonance, and the damage level is comparable on soil and rock: This was observed for instance in Mexico City in 1985 for very highrise buildings with frequencies much below 0.5 Hz. This result might, however, be modified for multidegreeoffreedom oscillators if some overtone is coinciding with site frequency. A second observation is related to the impact of velocity contrast: The stronger the site amplification, the larger the damage increment Δ(DI). It is finally worth noticing the damage increment is found to be rather low for low PGA levels, then to increase significantly for moderate PGA levels (~0.2–0.4 g) and then to saturate or even decrease for very large PGA levels. This is related to: (1) the nonlinear behavior in the structure activated only beyond some threshold pga value, (2) the definition of the damage index, with very low values as long as the structure remains elastic and (3) the significant damage on rock for very strong motion.
Field applicability: testing H / V amplitude as a site amplification proxy
On the other hand, the comparison of amplitude extracted from the transfer functions A _{0t} and the A _{0HV} extracted from H/V curve derived from noise synthetics (Fig. 7b) exhibits a much larger scatter around the line A _{0t} = A _{0HV}; however, the two values are definitely correlated in a statistical sense with R ^{2} = 0.7. As noticed in BonnefoyClaudet et al. (2008), A _{0HV} are larger than A _{0HV} which is due to the choice of pure random orientation of excitation forces at the surface.
A few other usual site proxies were also tested in Salameh (2016): V _{S30} and V _{S10} which exhibit a comparable performance, though slightly lower than A _{0HV}. We therefore conclude from this comprehensive numerical investigation that the simplicity, low cost and short duration of H/V measurements contribute in making A _{0HV} a satisfactory site amplification proxy with a remarkable quality/cost ratio.
Robustness study: impact of soil nonlinearity
The concept of spectral coincidence implicitly assumes some kind of quasilinear behavior, with limited changes in natural frequencies of soil and/or structure. The results of the two previous sections, involving elastoplastic oscillators with variable ductility ratio, indicate that this concept is robust enough to withstand strong nonlinearity in the structure. The scope of this section is to investigate whether the nonlinearity in the site response can significantly weaken the dominant importance of spectral coincidence effects, as it may significantly change the site frequency and the associated amplification. The soil nonlinear behavior is most often characterized by a degradation of its mechanical properties, involving a shear modulus reduction and a damping increase with increasing shear strain; it basically results in deamplifying the ground motion and shifting the frequency response toward lower values (Bonilla et al. 2005; Régnier et al. 2013, 2016).
The realistic soil profiles adopted previously are thus modeled using nonlinear site response analysis, and the response analysis of the comprehensive set of SDOF oscillators is repeated here for the modified loading including the effects of soil nonlinearity. We used the code “NOAH” (NOnlinear Anelastic Hysteretic method, Bonilla 2001): This finite difference code propagates vertically incident plane SH ground motion in a soil deposit, by integrating the dynamic equation of motion at each time step. This method is more appropriate than the equivalent linear approach, especially for severe shaking with large strains.
Nonlinear material properties are actually unknown in the original site database, which contains only the “elastic” parameters. It is therefore needed to assign to each layer of each profile some realistic values for the nonlinear parameters (the shear degradation modulus and the damping). Depending on the code, they may be specified either through shear strength profile, or through degradation curves, describing the dependency of G/G _{max} and damping as a function of shear strain for each layer. For instance, the latter are required for most linear equivalent software like SHAKE, while the former is required by the NOAH code used in this study, through cohesion and friction angle profiles. We thus had to associate a friction angle and a cohesion to each layer for the NOAH input files: This was done using specific “rules” applied to the shear wave velocity profiles. We simply followed the approach used by PEER (“Pacific Earthquake Engineering Research Center”) for the incorporation of a NL term in the recent revision of Ground Motion Prediction Equation for Western United States (NGAW2 project, Kamai et al. 2014): They used two sets of generic, depthdependent NL degradation curves depending on the soil cohesion: The “EPRI” curves are assigned to cohesionless soils, while “IV” (Imperial Valley) curves are assigned to cohesive soils. Following the PEER approach, all soil profiles having a V _{S30} value lower than 190 m/s were considered as cohesive (“IV”), while sites with V _{S30} higher than 190 m/s were considered as cohesionless, in both cases from the surface down to the bedrock. This assignment procedure is very arbitrary, and we think, however, it provides a first set of results on the impact of soil nonlinearity, that is worth being analyzed in a statistical sense. More details on these NL site response computations and their main results can be found in Almakari et al. (2016).
Then, the artificial neural network approach was used again with the same procedure as in the previous sections, to correlate the new damage increment values taking into account the nonlinear site response with the same three input parameters: ratio of structural and soil (linear) frequencies f _{struct}/f _{soil}, reference rock PGA and velocity contrast C = V _{max}/V _{min}. The whole set of new results is discussed in more detail in Salameh (2016); we will focus here only on one example comparison, for RC buildings with low ductility (class 3).
The last column of Table 1 (model 3) lists the performance indicators and the synaptic weights for this last model: It is worth noticing that the standard deviation reduction (40%) is comparable and even slightly larger than in the linear site response case, while the dominant synaptic weight still remains associated with the frequency ratio f _{struct}/f _{soil}: Even for nonlinear site and building response, the values of linear elastic frequencies are a key parameter for the damage assessment. Another remarkable result is the slight increase in the synaptic weight of PGA, associated with a slight decrease of the velocity contrast synaptic weight, which are both consistent and related to the nonlinear site response, which lowers the amplification at high PGA, especially for soft soils with low velocities (and high velocity contrast).
The nonlinear site response thus impacts the rocktosoil damage increment: The displacement at the top of an elastoplastic oscillator is lower when considering a nonlinear behavior for the soil compared to the linear case, especially when the oscillator frequency is larger than the site frequency. The higher the level of nonlinearity in the site response (i.e., the larger the strains in the soil), the larger is the reduction of the oscillator displacement. Spectral coincidence is found to still have a dominant role, but in a more complicated way. Soil–structure resonance can take place when the new shifted site frequency coincides with the structural frequency: The “spectral coincidence” effect is thus shifted toward f _{struct}/f _{soil} values lower than 1.
Application to Beirut
Lebanon is a small Middle East country crossed by the 1200kmlong Levant Fault System (LFS). In Lebanon, the LFS splits into three main branches, on land and offshore: the left lateral strikeslip Yammouneh Fault (last known rupture: 1202, M7.5), the Beirut–Tripoli Thrust (551, M7+) and the Rachaya–Serghaya Fault (1759, M6.5). The 551 event was accompanied by a tsunami hitting the whole eastern Mediterranean coast, destroying several cities including Beirut, Tripoli, Saida and Tyre. The most recent large earthquake is the double shock of March 16, 1956 (Ms = 4.8–5.1) which killed 136 people, destroyed 6000 houses and damaged 17,000 houses (Brazee and Cloud 1984; Khair et al. 2000). Seismological trenches have shown that the return period of devastating earthquakes is about 1100 years along the Yammouneh Fault (Daëron et al. 2007), 1500–1750 years along the Mount Lebanon Thrust (Elias et al. 2007) and ~1300 years along Serghaya Faults (Gomez et al. 2003).
The objective is to map the rocktosoil damage increment for the Beirut City area, for several groundshaking scenarios (from weak to very strong) that could help to establish priorities within the framework of an improved seismic prevention policy. The basic tool is the neural network approach developed in the previous sections, especially the second one based on the structuretosoil frequency ratio, PGA level and H/V amplitude. In that aim, the required information is an inventory of buildings with their typology and frequencies, together with the measurement of the soil frequencies and of the corresponding A _{0HV} amplitudes.
For this purpose, 7362 buildings located in downtown Beirut were surveyed by Saint Joseph University (USJ) staff within the framework of the LIBRIS ANR project, gathering information on the age of construction, number of floors and position of each building. In addition, the same project allowed performing ambient vibration measurements in a set of 330 buildings, to establish reliable relationships between building frequency and number of floors (Salameh et al. 2016), highlighting a slight difference between rock and soil. A frequency could then be estimated for each of the sosurveyed 7692 buildings, together with their typology class defined from their age as follows: (1) Buildings constructed before 1950 with a number of floors <4 are considered as masonry (i.e., class 1), (2) buildings constructed between 1950 and 2005 (introduction of the Lebanese seismic code) are considered as nondesigned RC (i.e., class 2), and (3) those constructed after 2005 are considered as designed RC with low ductility class (i.e., class 3).
Regarding site information, a total of 827 sites were measured for the Beirut City area (Brax 2013; Salloum et al. 2014; Salameh et al. 2016) to obtain the soil resonance frequency map of Beirut using HVSR (horizontal to vertical spectral ratio) approach (Nakamura 1989). The large density of measurements allowed to derive interpolated maps of both soil frequency and A _{0HV} amplitude, which could in turn be used to estimate the f _{struct}/f _{soil} and A _{0HV} values for each of the 7692 buildings.
 (a)
The presence of different geological settings in Beirut with a higher seismic risk is witnessed by the existence of red areas with damage increment between 1 and 2 damage grades. This is the case especially for Badaro, Bourj Hammoud and the NahrBeirut river areas in the eastern and southern parts: These areas correspond to younger and thicker quaternary sediments with low resonance frequencies and higher H/V amplitude (Brax 2013; Cornou et al. 2014). These three areas correspond to deposits of the river of Beirut, with moderate to large amplification at frequencies between 1 and 4 Hz often corresponding to the frequencies of the medium to highrise buildings located in those areas.
 (b)
The blue zones with negligible damage increment correspond to two rocky geological formation, rather than areas where building frequencies would be systematically lower than soil frequencies.

0 < DI < 1: no damage (blue)

1 < DI < 2: slight (green)

2 < DI < 3: moderate (yellow)

3 < DI < 4: extensive (orange)
Note also that, as mentioned in the first section, we do not consider buildings collapsing in the distribution of damage.
Conclusions
The work presented here was an attempt to test with conceptually simple models the damaging impact of spectral coincidence between structure and site, and to use comprehensive numerical simulation to propose easytouse tools to estimate damage increment due to site conditions at an urban scale, including an example application (here for the city of Beirut, Lebanon).
The impact of spectral coincidence, which seems obvious in case of tuned resonance of linear systems (soil and building), could be, however, considerably modified and attenuated in case of strong nonlinearities either in the soil or in the building. One must nevertheless mention recent trends in buildingspecific vulnerability estimations that take into account the spectral content of the input motion around the building fundamental frequency (De Biasio et al. 2015; Perrault and Guéguen 2015). The present work goes one step further along the same direction in considering also nonlinearities in the site response, but with a different approach as (a) it focuses on the damage differences between identical buildings located on a given site and on the reference rock, instead of considering only the “absolute” damage level, and (b) it considers multiple explanatory variables, mixing the rock loading level, site conditions and building characteristics through the peak ground acceleration on standard rock, a site amplification proxy and the ratio of elastic frequencies of building and soil, f _{struct}/f _{soil}. It is found that even in the case of soil or building nonlinearities, the elastic frequency ratio f _{struct}/f _{soil} has the predominant role in controlling the rocktosoil damage increment, ahead of the loading level summarized by the rock PGA and the site amplification proxy.
The comprehensive set of numerical simulations analyzed with a neural network approach led to relatively easytouse mathematical relationships (even though without any “physical” functional form) that were applied to the case of Beirut city (Lebanon). The soil and building frequencies are derived from ambient vibration measurements, which are numerous enough to map f _{0soil} over the whole downtown area of Beirut and to derive f _{0struct} simply from the number of floors and the nature of the foundation site (rock of soil). In addition, the use of the amplitude of the H/V ratio as a site amplification proxy is shown—on a numerical basis—to provide as reliable damage estimates as a series of more “physical” proxies as the velocity contrast, which is in practice very expensive and difficult to obtain at a whole city scale. It was therefore possible to map the expected rocktosoil damage increment for different shaking scenarios characterized by a rock PGA from 0.05 to 0.5 g, and also to map the estimated absolute damage level with a similar neural network approach and for the same shaking scenario. The proposed method proves to be easy to use for damage assessment at urban scale: It requires the usual type of information gathered for risk assessment, i.e., a seismic microzonation (here based only on dense microtremor measurements and H/V processing), a building inventory (here based primarily on construction date and height) and reliable estimates of building elastic frequencies (here based on specific relationships for the local building stock derived from a large number—several hundred—of ambient vibration measurements on building top floor).
We thus consider this attempt was successful in demonstrating the relevancy and the feasibility of the approach, which allows in some way to benefit from the advantages of the “mechanical” approach typically used for the design of specific buildings, at an urban scale where usually only empirical, macroseismic methods based on semiqualitative/semiquantitative information are used. However, this is only a first attempt, which would deserve many further checks and developments. While it is likely that refinements in the site or building response models would not basically change the key features of the results from a qualitative viewpoint, it might significantly change the quantitative relationships describing damage increment or absolute damage as a function of explanatory variables, which would in turn impact the applications to practical cases. Among the possible or needed improvements, some deal with the site response, in particular with more realistic NL site parameters (as shown in Almakari et al. (2016), it is likely that the very arbitrary model used here overestimates NL behavior for deep deposits and underestimates it for shallow deposits), and may be also with more regionspecific input waveforms (a sensitivity study showed, however, that the overall results are not changed statistically speaking with replacing the set of synthetic time histories by a set of real accelerograms). Some other deal with the building response, especially with the consideration of multidegreeoffreedom oscillators, especially for highrise, intermediate to longperiod buildings, and the incorporation of more elaborated nonlinear behavior, which would better reproduce the loading and unloading phases for reinforced concrete or masonry buildings than a simple elastoplastic oscillator. It is worth mentioning that a sensitivity study with multidegreeoffreedom (MDOF) elastoplastic oscillators (Neziri 2016) indicated some significant changes corresponding to coincidence between soil fundamental frequency and building overtones, mainly at low PGA levels (linear response), and smaller changes at higher PGA because of the lesser importance of higher modes in case of larger drifts and larger damping.
Finally, it would be worth also exploring the performance of other sets of explanatory variables (for instance, the actual site amplification A_{0T}, or some other ground motion intensity measures like PGV instead of the too crude PGA parameter) in order to further reduce the standard deviation of residuals and identify the best input parameters: Our bias here was to deliberately consider only a very small number of simple, easytoobtain parameters.
In any case, the best test for such an approach would be to check its results for cities with significant site amplification having already been hit by damaging earthquakes, and where effects of such spectral coincidence could be checked on actually observed damage (e.g., Mexico City 1985; Kathmandu, Nepal 2015; Puerto Viejo and other coastal cities, Ecuador 2016).
Declarations
Authors’ contributions
Most of the scientific and technical work has been carried out by the first author (CS) during her Ph.D. work under the scientific supervision of the four next coauthors. JG contributed to the compilation of Beirut buildings database. MA contributed to the nonlinear site response during her Master internships. The redaction has been shared among the first five coauthors. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the LIBRIS ANR project (2010–2014) funded by the French national Research agency in collaboration between ISTerre Laboratory (Grenoble, France) and Notre Dame UniversityLouaize (NDU, Lebanon). It was also partially funded by IRD (Institut de recherche pour le développement) through a Ph.D. fellowship and Labex OSUG@2020 (Investissements d’avenir—ANR10 LABX56). The authors also thank Boumédiène Derras for his help and advice about the artificial neural network approach and particularly the reviewers for their fruitful comments to improve this work.
Competing interests
The authors declare that they have no competing interests.
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