Predicting approximate seismic responses in multistory buildings from real-time earthquake source information, for earthquake early warning applications

Regional earthquake early warning (EEW) alerts and related risk-mitigation actions are often triggered when the expected value of a ground-motion intensity measure (IM), computed from real-time magnitude and source location estimates, exceeds a predefined critical IM threshold. However, the shaking experienced in mid- to high-rise buildings may be significantly different from that on the ground, which could lead to sub-optimal decision-making (i.e., increased occurrences of false and missed EEW alarms) with the aforementioned strategy. This study facilitates an important advancement in EEW decision-support, by developing empirical models that directly relate earthquake source parameters to resulting approximate responses in multistory buildings. The proposed models can leverage real-time earthquake information provided by a regional EEW system, to provide rapid predictions of structure-specific engineering demand parameters that can be used to more accurately determine whether or not an alert is triggered. We use a simplified continuum building model consisting of a flexural/shear beam combination and vary its parameters to capture a wide range of deformation modes in different building types. We analyse the approximate responses for the building model variations, using Italian accelerometric data and corresponding source parameter information from 54 earthquakes. The resulting empirical prediction equations are incorporated in a real-time Bayesian framework that can be used for building-specific EEW applications, such as (1) early warning of floor-shaking sensed by occupants; and (2) elevator control. Finally, we demonstrate the improvement in EEW alert accuracy that can be achieved using the proposed models.


Introduction
Earthquake early warning (EEW) typically involves (1) detecting earthquakes in the early stages of fault rupture; (2) rapidly predicting the intensity of the impending shaking at selected target sites (or across regions); and (3) issuing alerts to end users that trigger important risk-mitigation actions before the arrival of the potentially damaging seismic motions (Allen and Melgar 2019). Depending on source-to-site distance (Wald 2020), EEW systems can provide up to a few tens of seconds of warning time. Examples of mitigating actions facilitated by EEW include the "drop, cover, and hold-on" manoeuvre (for avoiding injuries), the stopping of elevators at the nearest floor, the opening of firehouse doors, the slowing of high-speed trains, and the saving of vital computer information (e.g., Porter 2020).
We particularly focus on regional EEW systems in this study. These systems generally carry out step (2) of EEW by leveraging information provided in the early portion of near-fault seismic signals (e.g., 3-4 s of P-and/or S-waves) to determine source parameter estimates, which are used to predict resulting ground-motion intensity measures (IMs) at distant sites (e.g., Cremen and Galasso 2020). The uncertainty of this process may be captured by applying the well-known probabilistic seismic hazard analysis (PSHA ;Cornell 1968) framework in real-time (i.e., RTPSHA; e.g., Iervolino et al. 2006;Convertito et al. 2008), as follows: where f A (a|b) is the probability density function of a conditional on b, are physical characteristics of the early seismic signals, f M (m| ) and f R (r| ) are respectively the realtime probability distributions of magnitude and source-to-site distance, and f IM ( |m, r) is typically computed using a ground-motion model (GMM). Equation 1 produces realtime-dependent hazard curves, which are then used to determine whether an alert should be issued to relevant end users (in step 3). For instance, a simple decisional rule consists of issuing an EEW alarm at a target site if p IM ≥ IM cr is larger than a predefined threshold ( p cr ), where IM cr is a critical IM value for an asset of interest (Iervolino 2011).
The aforementioned strategy may lead to poor alert performance (i.e., frequent occurrences of costly false alarms and/or potentially deadly missed alarms) for structural applications, since the shaking (i.e., engineering demand) experienced in mid-to high-rise buildings is generally significantly different from that on the ground (e.g., Shome et al. 1998). To address this challenge,  recently proposed an engineering-oriented decision-making methodology for building-level EEW. This methodology leverages the performance-based earthquake early warning framework developed by Iervolino (2011) and explicitly uses risk-based predictions to determine whether an EEW alert should be triggered. However, the advanced earthquake-engineering assessment procedure adopted in the methodology requires a significant amount of structure-specific information, which makes the  approach unsuitable for application to large numbers of buildings in a region.
To facilitate more informed EEW decision-making for the latter case (i.e., building portfolios), this study employs simplified building models that can be used to approximate structural responses due to dynamic loadings (e.g., Hoenderkamp and Snijder 2000;Iwan 1997;Saiidi and Sozen 1981;Vukobratović and Fajfar 2016;Miranda and Taghavi 2005;Miranda and Akkar 2006). We specifically develop empirical equations that directly relate earthquake source and site information (M, R, and soil type) to estimates of Interstory Drift Ratio (IDR) and Peak Floor Acceleration (PFA) engineering demand parameters (EDPs) for a wide range of structures. Based on these equations, Eq. 1 can be modified to: where f EDP ( |m, r) is the joint PDF of the estimated EDP conditional on M and R. It is important to note that f EDP ( | ) could also be obtained using probabilistic seismic demand analysis (PSDA; e.g., Shome et al. 1998), in which RTPSHA would be coupled with a probabilistic seismic demand model (PSDM). PSDMs relate EDP(s) and selected ground-motion IM(s) based on nonlinear analysis of structural response using ground-motion records (e.g., Jalayer and Cornell 2009), from which f EDP ( | ) can be determined via statistical inference methods (e.g., linear regression). However, the introduction of f EDP ( | ) would require an IM integral in addition to those of Eq. 2, making PSDA a less efficient/suitable approach for the real-time (EEW) applications of interest in this study.
This paper is organized as follows. We first develop the empirical EDP prediction model, leveraging well-established methodologies for approximating seismic responses in multistory buildings. We next implement the EDP model in the real-time framework of Eq. 2, to demonstrate its application to EEW. We finally demonstrate the improved EEW alert accuracy obtained by triggering alarms based on EDPs instead of IMs.

Quantifying approximate seismic responses
We quantify the approximate elastic seismic responses of multistory buildings, using the simplified structural model presented in Miranda (1999), which has been widely applied in the literature (e.g., Galasso et al. 2013;Cheng et al. 2014;Neam and Taghikhany 2016;De Bortoli and Zareian 2018). This continuous model with uniform mass and lateral stiffness distribution comprises one flexural and one shear cantilever beam, which are connected by an infinite number of axially rigid link members that transmit horizontal forces. The base is assumed to be fixed, such that foundation flexibility is neglected and torsional deformations are ignored. While the assumption of constant mass along the height is reasonable for most structures, the assumption of consistent lateral stiffness is only valid for low-rise buildings (e.g., less than three stories). However, Miranda and Taghavi (2005) concluded that the dynamic characteristics of a uniform model sufficiently approximate those of non-uniform models in many cases.
The Miranda (1999) model requires three building-specific parameters -T 1 , i , and -to compute approximate demands for a given ground motion (i.e., mode shapes, modal participation factors, and the ratio of the period of vibration of higher modes to the fundamental period are fully defined by only three parameters). T 1 is the first mode period of vibration and i is the damping ratio of the ith mode. regulates the relative participation of shear and flexural deformations in the model, and is defined as follows: ( where GA is the shear rigidity of the shear beam, EI is the bending stiffness of the flexural beam, and H is the building height. Note that H is estimated in this study from T 1 , using the following relationship for generic structures provided in ASCE 7-2010 (2010): Figure 1 illustrates the influence of the parameter on first mode characteristics. We examine PFA, IDR, and maximum interstory drift ratio (MIDR) EDPs in this study. PFA is obtained using the methodology of Miranda and Taghavi (2005). IDR and MIDR are computed according to the procedure of Miranda and Akkar (2006). We exclusively account for the first six modes, given that their cumulative effective modal mass equates to more than 90% of the total for most real buildings (Galasso et al. 2013), and assume i = 5% for each. To capture the responses of a wide range of structures, we consider 15 values of T 1 from 0.1 s to 5 s (in increments of 0.1 up to 0.5 s, increments of 0.25 between 0.5 and 1 s, and increments of 0.5 for larger periods), three values of (0.1, 8, 30) that respectively represent shear wall buildings, dual systems, and moment-resisting frames (see Fig. 1), and 101 equally spaced values of relative height (i.e., nondimensional quantities that result from normalization by the total building height and vary from 0 to 1).

Model format
We develop empirical equations that directly relate the approximate building responses of Sect. 2.1 to earthquake source parameters that can be rapidly estimated with a regional EEW algorithm (i.e., typically magnitude and epicentral location), and site-specific soil characteristics. We use the functional form of the ground motion model (GMM) developed by Akkar and Bommer (2010), but substitute the Joyner-Boore distance metric with epicentral distance ( R epi ), since current investigations on the practical implementation of European EEW are focused on point-source algorithms . We also exclude the style-of-faulting indicators, given that the relevant data are not usually produced from an EEW algorithm (Minson et al. 2014). The final format of the equations to be developed is: where, for a prescribed value of and T 1 (and conditional on i = 5% for all modes), Y is the EDP of interest (i.e., PFA at a given floor height, IDR for a given story height, or MIDR), taken as the geometric mean of the values computed for the two horizontal components of a ground motion. M is moment magnitude. S s and S a are 0 except in the case of soft-soil ( S s = 1 ) or stiff-soil ( S a = 1 ) sites. N( , ) is a normal distribution with mean and standard deviation , is the inter-event standard deviation, is the intra-event standard deviation, and = √ 2 + 2 is the total standard deviation.

Model calibration
We calibrate the model using a subset of the ITACA v1.0 Italian strong-motion database , which contains 3955 three-component processed accelerograms from approximately 1800 M w ≤ 6.9 seismic events that occurred in Italy during the period 1972-2009. We select this database, given that its complete set of ground-motion records is rapidly accessible via the REXEL 3.5 software tool (Iervolino et al. 2010), whereas newer and/or alternative databases require tedious record-by-record downloading or impose strict time-based limits on data accessibility. We specifically consider records with R epi ≤ 200 km from M w ≥ 5 events for which the faulting style is known. This results in a final dataset of 580 two-component records from 54 seismic events (see Fig. 2). 74% of the considered records are from normal faulting earthquakes, 20% are from reverse faulting earthquakes, A summary of the records used to develop the empirical models, with respect to magnitude and epicentral distance 1 3 and 6% are from strike-slip events. 46% of the examined accelerograms were recorded on rock, 29% were recorded on stiff soil, and 25% were recorded on soft soil ( V s30 < 360 m/s).
We calculate the coefficients of Eq. 5 for each set of {T 1 , , h, i = 5%} as follows. We first determine a preliminary set of {b k } , using non-linear regression. We then obtain the resulting residuals, log 10 (Z p,q ) , according to: log 10 (edp obs,p,q ) is the logarithm of the geometric mean observed EDP of interest for the horizontal components of the qth recording from the pth event, and M p and R epi p,q are the corresponding observed magnitude and epicentral distance. All other variables are as defined in Eq. 5. We finally perform an iterative mixed-effects regression of log 10 (Z p,q ) according to the algorithm of Abrahamson and Youngs (1992), to determine maximum likelihood estimates of , , and {b k } . The parameters for each developed equation are provided in an electronic supplement.

Model evaluation
We first evaluate the distance-scaling of the developed equations. We specifically examine the variation of intra-event residuals ( p,q ) versus epicentral distance (Scasserra et al. 2009), given that consistency in this relationship would suggest a bias in the distance attenuation. Figure 3 provides the resulting plots for a hypothetical mid-rise building ( T 1 = 0.75 s) and all considered values of . It may be observed that there is no notable distance-dependence of the residuals, indicating that the distance-scaling of the developed equations is adequate for the presented cases. We quantitatively determine the presence of a trend in intra-event residuals versus distance for each developed equation, by using linear regression to fit: and testing the null hypothesis that the slope c R equals zero. (Note that p,q corresponds to the EDP of interest, a R is a constant, R p,q are residuals, and all other variables have been defined previously.) We find large p-values (i.e., ≥ 0.05) across all of the calibrated equations, implying that the aforementioned null hypothesis cannot be rejected and therefore that there is not a statistically significant relationship between the intra-event residuals and distance in any case.
We evaluate the magnitude-scaling of the developed equations, by examining the variation of inter-event residuals ( p ) versus magnitude (Scasserra et al. 2009); see Fig. 4, which uses the same hypothetical structure as Fig. 3. There is no discernible magnitude-dependent trend in the plotted residuals, suggesting that the magnitude-scaling of the developed equations is appropriate for the highlighted cases. We quantitatively define this trend for all calibrated equations, by performing a linear regression according to: and testing the null hypothesis that c M is zero. ( p corresponds to the EDP of interest, a M is a constant, M p are residuals, and all other variables are as defined previously.) We find (6) log 10 (Z p,q ) = log 10 (edp obs,p, Based on the above evaluations, we ultimately conclude that the equations are a satisfactory fit to the underlying EDP and source data. Figures 6, 7, 8 demonstrate the close alignment between equation predictions and corresponding observed data, using . Note that appears to have a minimal impact on predicted amplitudes for the examined T 1 . For example, MIDR on a rock site at R epi = 10 km is only 7% larger for = 30 than for = 0.1 . However, plays a more significant role at other building

Real-time applications of the models
We now apply the developed empirical models to EEW. We use the real-time analysis framework presented in Eq. 2 and case-study earthquakes at a location in the Campania region of south-west Italy (see Iervolino et al. 2009, and Fig. 9), where an EEW system is already undergoing real-time (non-public) testing (e.g., Satriano et al. 2011;Velazquez et al. 2020).
Note that we neglect distance uncertainty in this case, given its insignificance compared to that of magnitude and ground shaking (Iervolino et al. 2009), which simplifies the Eq. 2 framework to: where R epi t is the true epicentral distance to the target site of interest. We adopt the Bayesian formulation for f (m| ) provided in Iervolino et al. (2009), i.e.,: where k is the maximum predominant period measured within four seconds of the P-wave arrival at the kth seismic station, n is the total number of seismic stations for which k has been recorded at a given time from the event, M min is the minimum considered magnitude, M max is the maximum magnitude, = b∕log 10 (e) , and b is the slope  Allen and Kanamori (2003). Our analyses assume M min = 4 , M max = 7 , and = 1.69 , in line with Iervolino et al. (2009). We leverage the Irpinia Seismic Network (ISNet) of 27 stations (Fig. 9), which is the same array used to test EEW in the region.
We first examine a target site in Naples (where R epi t = 124 km) and focus on the evolution of EDP estimates as an increasing number of ISNet stations record i from an incoming M w 6.9 event. i values are assumed to be independent and identically distributed lognormal random variables (in line with Iervolino et al. 2006), and are therefore simulated according to: where M t is the true event magnitude (= 6.9) and all other variables are as defined previously. These simulated values are then used to calculate f EDP ( | ) , according to Eq. 9. We assume the structure of interest has T 1 = 3 s, to reflect the high-rise buildings located in the newly developed business area of the city, and consider all values for which the models were developed. These kinds of calculations can be leveraged for building-specific EEW applications, such as early warning of floor-shaking experienced by occupants and elevator control (Cheng et al. 2014). Figure 10 provide estimates of PFA (top floor), IDR (top story), and MIDR, averaged over 100 simulated sets of i for different values of n, as well as the distributions obtained if M t is directly input to the relevant empirical model. Also shown is the time required to process information from each set of n triggered stations ( t p n ), which is calculated as follows: where R epi n is the maximum epicentral distance associated with all n triggered stations, d is the hypocentral depth of the earthquake (assumed = 12 km), t is the processing time (= 5 s, from Iervolino et al. 2009), and V p is P-wave velocity. V p is taken to be 5.88 km/s, based on the S-wave velocity ( V s ) of 3.5 km/s provided for the same region in Iervolino et al. (2006) and the V p V s ratio of 1.68 used for the same earthquake location in Iervolino et al. (2009). It is observed that the predictions of Eq. 9 improve as more triggered stations contribute data to the calculations, since this increases the accuracy of f M (m| ) (Iervolino et al. 2006;see Fig. 14a). It is also seen that reasonably accurate EDP estimates are obtained for n = 18 (i.e., within 12 seconds) in all cases. This result is consistent with the findings of Iervolino et al. (2009) for PGA estimates at the same target site, who used the same EEW algorithm and an M w 6 event at the examined epicentral location.
Our second case study involves a region-wide real-time application of the proposed empirical models. We specifically examine the spatial distribution of median EDP estimates for M t = 6 and a hypothetical mid-rise building ( T 1 = 0.75 s, = 8 ), using one set of simulated i values (from Eq. 11) for various n. Figures 11, 12, 13 respectively provide these estimates for PFA, IDR, and MIDR, as well as the corresponding median values calculated for the true magnitude. It can be seen that early real-time calculations significantly underestimate median EDP values (particularly at close-source distances). This may be explained by the fact that for low values of n, f (m| ) is dominated by its Gutenberg-Richter prior component (per Eq. 10), which provides an expected magnitude value that is significantly smaller than M t in this case (Iervolino et al. 2006; see Fig. 14b). Sufficiently accurate estimates are obtained in most of the region for approximately 18 triggered stations, across all EDPs. Evolutionary regional mapping of projected EDP estimates may be helpful for better informing the early stages of emergency response strategies that significantly depend on the functionality of target assets within the urban system (i.e., critical infrastructure).

Demonstrating the benefits of EDP-based EEW trigger thresholds for buildings
We now provide a simple demonstration of the improvement in alert accuracy offered by making decisions to trigger EEW alarms based on EDP instead of IM. We use a hypothetical mid-rise building ( T 1 = 1 s, = 30 ), and assume that an EEW alert should be issued if the PFA caused by the impending earthquake at the top, middle or bottom floor will  ) and c, f, i MIDR for a hypothetical high-rise building ( T 1 = 3 s) at the chosen target site, compared with those obtained using the true (i.e., final) value of magnitude ( M w = 6.9 ), for a-c = 0.1 , d-f = 8 , and g-i = 30 exceed 0.05 g, which is the threshold at which acceleration becomes "very annoying" from a human comfort perspective (Cheng et al. 2014, see Table 1). A reasonable IM-driven decisional rule in this case would be to trigger an alarm if the median prediction of PGA at the target site exceeds 0.05 g. We compare alert performance for this rule with that achieved for a comparable EDP-based decisional rule that triggers an alarm if the median PFA prediction at the top, middle or ground floor of the building exceeds 0.05 g. We quantify alert accuracy using the methodology of Minson et al. (2019), and randomly sample 10,000 sets of top-, middle-, and bottom-floor PFA values for the building according to the corresponding empirical prediction equations developed, for a range of magnitudes and epicentral distances (note that we neglect magnitude and epicentral distance uncertainty for this demonstration). A missed alarm occurs if the median prediction of PGA or PFA does not exceed 0.05 g and at least one simulated value does, a false alarm occurs in the opposite case, and a correct decision occurs if both the median and simulated values fall above or below 0.05 g. Note that we use the epicentral distance model of the Bindi et al. (2009) GMM introduced in Sect. 2.3, for the IM-driven decisional rule. We acknowledge that more modern versions of this GMM are available (Bindi et al. 2010(Bindi et al. , 2011, but both of these incorporate style-of-faulting information and/or rely on Joyner-Boore distance measurements in certain cases, which (as explained in Sect. 2.2) are typically not appropriate considerations for real-time EEW applications. The results of these analyses are presented in Fig. 15 (for M w = 6.5 , across a range of R epi values) and Fig. 16 (for R epi = 60 km, across a range of M w values) For both decisional rules, it is seen that false alerts can occur for low values of R epi and relatively high values of M w (i.e., conditions under which governing median predictions are higher than 0.05 g), whereas missed alerts can occur for high values of R epi and relatively low values of M w (i.e., conditions under which governing predictions are less than 0.05g). Proportions of false and missed alerts are greatest for cases in which the corresponding governing predictions are respectively just greater than or just less than 0.05 g. In all other cases (as intuitively expected), governing predictions and simulated EDPs of interest are in better agreement with respect to this decision threshold. It can be observed that the EDP-based decisional rule provides significantly better alert accuracy than that based on IM; it reduces missed alerts by 42% for the M w = 6.5 case and by 66% for the R epi = 60 km case. This example underlines the notable benefit of integrating EDP predictions in building-focused applications of EEW. It is important to note that the EDP rule transforms some IM-related missed alerts into false alerts in both cases. However, these system malfunctions are generally accepted by end users (at least in regions with public EEW) as opportunities to perform unscheduled earthquake preparation exercises (Velazquez et al. 2020).

Fig. 12
Region-wide estimates of IDR (top story) for a hypothetical mid-rise building ( T 1 = 0.75 s, = 8 ) during an M w 6 event when a n = 2 , b n = 9 , c n = 18 , d n = 27 , and (e) the true (i.e., final) value of magnitude is obtained Fig. 13 Region-wide estimates of MIDR for a hypothetical mid-rise building ( T 1 = 0.75 s, = 8 ) during an M w 6 event when a n = 2 , b n = 9 , c n = 18 , d n = 27 , and e the true (i.e., final) value of magnitude is obtained Fig. 14 f (m| ) for (a) an M w 6.9 event (depicted in Fig. 10) and (b) an M w 6 event (depicted in Figures 11,  12, 13) when n = 2, 9, 18 , and 27. Also shown are the true magnitudes

Conclusions
This study has developed empirical EDP prediction models for use in real-time buildinglevel applications of EEW. These models directly relate approximate seismic responses of multi-story buildings (in terms of PFA, IDR, and MIDR) to site-specific characteristics and earthquake source parameters that can be rapidly estimated by regional EEW algorithms. We used Italian ground-motion data and well-known simplified response quantification methods (Miranda and Taghavi 2005;Miranda and Akkar 2006) to calibrate the models for three structural systems and a wide array of corresponding first mode periods. We integrated the developed models into a real-time Bayesian framework that incorporates the evolutionary uncertainties inherent in EEW measurements. Local and regional applications of the framework for case study earthquakes in Southern Italy revealed that reasonably accurate EDP estimates could be obtained if the data from approximately 18 stations were used to estimate the magnitude of the incoming event (although these findings significantly depend on the underlying EEW algorithm employed). The EDP-based Bayesian framework may be used to design bespoke EEW alerts and determine preliminary earthquake impact estimates for targeted structures within a region. Finally, we quantified the improvement in EEW alert accuracy obtained if the developed models were used to determine alert thresholds instead of the IM-based prediction methods typically adopted. Using a sample structure and a hypothetical alert trigger criterion, we found that an EDP-based decisional rule could reduce the proportion of missed alerts associated with a PGA-driven threshold by more than 45% on average, across a series of scenario earthquakes. In summary, the results of this study facilitate a key advancement in EEW decision support, by providing an important link between the typical seismological parameters computed by EEW algorithms and their ultimate effect on the built environment.
There are some limitations associated with this study, however. Firstly, the structural model used assumes linear elastic behaviour and classical damping. This means that the empirical prediction equations are only appropriate for performance levels in which a building is expected to remain elastic or exhibit minimal nonlinear behaviour. However, nonlinear extensions of the employed structural model are available (e.g., Xu et al. 2014;Xiong et al. 2016) and could be used to derive similar prediction equations for buildingspecific EEW applications involving high-intensity ground motions. Secondly, the empirical models were determined using data from Italy only and may not be directly applicable outside this context (although the proposed methodology for model development is general and can be applied to data from any region). Thirdly, the models do not account for correlations of EDPs, either within the same structure or between closely located buildings (Sun et al. 2018). To overcome this shortcoming, (1) median model predictions could be used to quantify intra-building correlations according to the EDP correlation methodology described in Cremen and Baker (2018) and (2) inter-building EDP correlations could be quantified with model residuals, in line with procedures used for assessing spatial correlations of ground-motion intensities (e.g., Jayaram and Baker 2009;Loth and Baker 2013).
On a final note, future work will consider expanding the Eq. 2 framework to account for losses, which contain the highest level of information for decision making about alarm issuance (e.g., Iervolino et al. 2007). This effort would represent a significant step towards achieving a portfolio-level analogue of the engineering-driven EEW decision-making methodology proposed by .