Performance-based earthquake early warning for tall buildings

Abstract

The ShakeAlert Earthquake Early Warning (EEW) system aims to issue an advance warning to residents on the West Coast of the United States seconds before the ground shaking arrives, if the expected ground shaking exceeds a certain threshold. However, residents in tall buildings may experience much greater motion due to the dynamic response of the buildings. Therefore, there is an ongoing effort to extend ShakeAlert to include the contribution of building response to provide a more accurate estimation of the expected shaking intensity for tall buildings. Currently, the supposedly ideal solution of analyzing detailed finite element models of buildings under predicted ground-motion time histories is not theoretically or practically feasible. The authors have recently investigated existing simple methods to estimate peak floor acceleration (PFA) and determined these simple formulas are not practically suitable. Instead, this article explores another approach by extending the Pacific Earthquake Engineering Research Center (PEER) performance-based earthquake engineering (PBEE) to EEW, considering that every component involved in building response prediction is uncertain in the EEW scenario. While this idea is not new and has been proposed by other researchers, it has two shortcomings: (1) the simple beam model used for response prediction is prone to modeling uncertainty, which has not been quantified, and (2) the ground motions used for probabilistic demand models are not suitable for EEW applications. In this article, we address these two issues by incorporating modeling errors into the parameters of the beam model and using a new set of ground motions, respectively. We demonstrate how this approach could practically work using data from a 52-story building in downtown Los Angeles. Using the criteria and thresholds employed by previous researchers, we show that if peak ground acceleration (PGA) is accurately estimated, this approach can predict the expected level of human comfort in tall buildings.

Keywords

Earthquake early warning tall buildings beam model modeling uncertainty peak floor acceleration

Introduction

After the initiation of an earthquake, ShakeAlert Earthquake Early Warning (EEW) system estimates the earthquake magnitude and location based on the signals recorded by seismic stations near the epicenter. It then predicts ground-motion intensities across the region using standard empirical ground-motion models. An early warning is issued if the predicted ground-motion intensity surpasses a predefined threshold for a given area, which can provide several seconds to tens of seconds of advance notice before the arrival of ground shaking, depending on the distance between the site and the earthquake epicenter (Given et al., 2014, 2018). More detailed information about the existing ShakeAlert workflow can be found in other published works, such as Böse et al. (2023), Kohler et al. (2020), and McGuire et al. (2021).

Currently, the earthquake intensity prediction is limited to ground-level motion, while residents in multi-story buildings may experience significantly different, and likely amplified, shaking (Çelebi et al., 2016a, 2016b; Çelebi et al., 2020, 2021). This amplification effect is particularly relevant because the EEW system has greater benefits for long-distance earthquakes (Wald, 2020), where the dominant long-period ground-motion components significantly impact the response of tall buildings. However, there is currently no ideal solution that involves precise ground-motion time history and accurate finite element (FE) models of buildings to carry out nonlinear time-history analysis for accurate building response prediction. Consequently, several researchers have attempted to explore alternative approaches to extend the EEW system to include building structures. In a recent study by Ghahari et al. (2022a), the empirical formulas available in FEMA P-58 (2018) and ASCE/SEI¹ 7-16 (2017) for estimating peak floor acceleration (PFA) were evaluated using data recorded from instrumented buildings through the California Strong Motion Instrumentation Program (CSMIP). It was demonstrated that these simple formulas are unable to reliably predict PFA, and their accuracy significantly diminishes when the peak ground acceleration (PGA) estimates are inaccurate. Another potential approach is the use of response spectrum (RS) methods; however, this requires an accurate estimation of the modal properties of buildings and the ground-motion RS. While such properties can be inferred from recorded data in instrumented buildings (see, e.g. Cruz and Miranda, 2021), this approach cannot be extended to other buildings that are not instrumented. In addition, the associated uncertainty in these estimations has not been quantified.

Alternatively, the extension of EEW to building response prediction can be considered within a framework similar to the performance-based earthquake engineering (PBEE) methodology of the Pacific Earthquake Engineering Research Center (PEER) (Porter, 2003), aiming to establish a probability-based decision-making framework (Grasso et al., 2007; Iervolino et al., 2007; Wu et al., 2013). Nonetheless, detailed numerical models of buildings on a regional scale are currently unavailable, and even if they were available, it would be computationally infeasible to individually run all of these models in the time-limited EEW setting. A potential solution to this problem is the use of simplified beam models, which can adequately represent the dynamic behavior of tall building structures (Ameri Fard Nasrand et al., 2023; Miranda, 1999; Rostami et al., 2021; Shirzad-Ghaleroudkhani et al., 2017; Taciroglu et al.,2016, 2017). For instance, Cheng et al. (2014) employed a coupled shear-flexural beam (Miranda and Taghavi, 2005) to obtain the probability density function (PDF) of engineering demand parameters (EDPs), such as PFA. Although this performance-based earthquake early warning (PBEEW) approach appears promising, it suffers from several shortcomings, some of which are mentioned in the work by Cheng et al. (2014) as follows:

The beam model is a simplified representation in which several factors, including torsional effects, structural nonlinearity, non-uniform mass/stiffness distribution, soil–structure interaction, variable modal damping ratios, and so on, are neglected. Its results are susceptible to modeling uncertainty, which has not been quantified.

The existing PDF of PFA conditioned on PGA, generated by Taghavi-Ardakan (2006), is developed by analyzing a deterministic beam model, that is, certain parameters, under a set of ground motions recorded at short source-to-site distances, which may not be appropriate for EEW applications.

There may be a more suitable intensity measure (IM) for PBEEW applications, considering that the seismic response of tall buildings is more sensitive to long-period components of the ground motion, which PGA may not adequately capture.

This article addresses the first two issues to enhance the applicability of PBEEW. The authors recently developed a method to quantify the modeling uncertainty associated with the beam model by incorporating errors into the parameters (Ghahari et al., 2022b). While the simple beam still suffers from the mentioned limitations, in this article, we employ this new technique to modify the PDF of PFA conditioned on PGA considering all these sources of uncertainties. Also, as this method is implemented within a Bayesian calibration, the parameter uncertainty is taken into account. It is worth mentioning that the proposed solution for considering modeling/parameter uncertainty is not limited to the beam model and can be applied to any numerical model of a building. Therefore, if detailed FE models of buildings and the necessary computational resources are available, these models can be used within this framework. However, as pointed out earlier, there is currently no such capability for a regional-scale application. To tackle the second issue, a new set of ground motions is collected, which will be used to generate a new set of PDFs. The third point, that is, the selection of an optimal IM, is beyond the scope of this article. The readers can find more details on the optimal IM selection in these references (Baker and Cornell, 2008; Dehghanpoor et al., 2021; Kazantzi and Vamvatsikos, 2015; Zhang et al., 2018).

In the following section, the overall PBEEW framework is described, starting with a concise overview of the PEER PBEE approach. Subsequently, the necessary components to construct this framework are presented. To illustrate the construction of the PDF of PFA conditioned on PGA, a 52-story building located in Downtown Los Angeles serves as the example. Actual strong-motion data recorded by this building during several significant events, particularly the M 7.1 Ridgecrest earthquake of 2019, are used for application, verification, and validation purposes. The article concludes by highlighting the main findings of this study and providing options for future improvements.

PBEEW

PEER PBEE

Due to the massive economic losses after the 1994 Northridge earthquake, the seismic design philosophy evolved from force-based to performance-based seismic design (SEAOC, 1995). After deterministic approaches were used in seismic evaluation documents such as FEMA 356 (2000), PEER established a probabilistic framework for PBEE to account for uncertainties associated with various sources (Cornell and Krawinkler, 2000). The idea behind the PEER PBEE is to probabilistically estimate the structural performance by propagating all sources of uncertainties from seismic hazard to decision-making using the total probability theorem. The PEER PBEE process is shown in Figure 1 (Porter, 2003). The objective of this process is to estimate the frequencies at which a particular performance metric would exceed various levels for a given design of a structure. This process consists of four main steps. In the first step (hazard analysis), the seismic hazard at the structure site is evaluated considering the seismic environment (e.g. nearby faults), the structure location, and other information jointly denoted as D. The result of this probabilistic seismic hazard analysis (PSHA) is a hazard curve that describes the annual frequency at which an IM, for example, PGA, exceeds various levels. This term is mathematically represented by a conditional PDF $p (IM | D)$ , which reads as the probability density of IM conditioned on available information D. Assuming all aspects of possible seismic hazards are reflected in IM, in the second step (structural analysis), an EDP, for example, PFA, is estimated based on different levels of IMs. As a single IM cannot entirely represent a seismic hazard, the associated record-to-record variability is taken into account by defining $p (EDP | IM)$ and consequently obtaining $p (EDP | D)$ . This is usually obtained by running a numerical model of the structure under a set of ground motions conditioned on the design information D. Although Steps 2 and 3 are usually combined and treated as a single step (Baker, 2015), in the third step, EDP is used as input to a set of fragility functions that represent the probability of various levels of Damage Measures (DMs) conditioned on structural response, that is, $p (DM | EDP)$ . For example, the expected level of damage to an elevator when PFA exceeds various levels is a type of fragility function. Similar fragility functions can be obtained numerically or experimentally (Akkar et al., 2005; Jaiswal et al., 2011; Ramirez et al., 2012). In the last step, the probabilistic estimation of the performance of the structure (e.g. death, dollar, etc.), called decision variable (DV), conditioned on the DM, is added to the calculation to finally obtain $p (DV | D)$ . By probabilistically quantifying $p (DV | D)$ , estimates of the frequencies at which various levels of DVs are exceeded, and a variety of risk-management decisions can be made. Due to Markovian assumptions implicitly considered in these PDFs in that the conditional probabilities of each step are assumed to be dependent on only the previous step and no others, $p (DV | D)$ can be calculated as:

p (D V | D) = \int \int \int p (D V | D M) p (D M | E D P) p (E D P | I M) p (I M | D) d D M d E D P d I M .

(1)

Figure 1.

The general framework of PEER PBEE after Porter (2003).

PBEEW

As initially proposed by Grasso et al. (2007) and later extended by Cheng et al. (2014), the PEER PBEE can be modified for use in EEW, as shown in Figure 2. First, PBEEW begins once an earthquake event occurs, eliminating the need for PSHA since the seismic source and properties of the event, such as Magnitude (M), source-to-site distance (R), and site conditions, are known. Instead, a new module (EEW) is added to the PBEE framework, representing source estimation, that is, $p (M, R | D (t))$ . With this information, the ground-motion estimation component of the Hazard Analysis step, that is, $p (IM | D)$ , can be conducted through forward physics-based numerical simulations of earthquake ruptures (Graves and et al, 2011; Mccallen et al., 2020; Rodgers et al., 2018). However, because running such time-consuming and computationally expensive numerical analyses is not feasible for EEW applications, empirical ground-motion prediction equations (GMPEs) can be used instead (Boore et al., 2014). These models, which will be briefly described later, provide a log-normal distribution for IMs, such as PGA, using input variables, such as $M$ and $R$ ( $p (IM | M, R)$ ), as shown in Figure 2. The last two modules of the framework are disabled in this study because we are only interested in $p (EDP | D)$ , similar to the work by Cheng et al. (2014). The module that is extensively discussed and improved in this study is Structural Analysis, where $p (EDP | IM)$ is regenerated while considering the modeling uncertainty and EEW-suitable ground motions, as will be discussed in the following sections. It is worth mentioning that D is replaced with D(t) in the PBEEW because information about the seismic source is time dependent and continuously updated as more data are received. Earthquake magnitude grows as the event itself grows, so the estimate of magnitude for a large event will tend to increase with time. In modeling, the actual rupture dimensions require a finite-fault model which usually are not available when an early warning is issued. More importantly, the magnitude estimated from the initial portions of the recorded waveforms provides just an estimate of the actual magnitude of the event and it could be off by more than a unit or magnitude. In this study, we assume that accurate estimates of seismic source and distance are available, and $M$ and $R$ are certain. Therefore, we replace $p (IM | M, R) p (M, R | D (t))$ with $p (IM | D)$ and in which D encapsulates all information provided by the EEW module (e.g. earthquake magnitude, source-to-site distance, fault type, site conditions, etc.) needed to obtain $p (IM | D)$ , for example, through GMPEs as will be described later. With these simplifying assumptions, the PDF of an EDP of interest, such as PFA, can be expressed as:

p (EDP | D) = \int_{IM} p (EDP | IM) p (IM | D) d IM .

(2)

Figure 2.

The general framework of PBEEW after Cheng et al. (2014).

Theoretical background

Simple beam model

In this article, PFA is extracted from the absolute acceleration response of a floor and used as an EDP for EEW application. The absolute acceleration response of a building at a normalized height $x = \frac{h}{H} (0 \leq x \leq 1)$ , ${\overset{\cdot\cdot}{u}}^{t} (x, t)$ , where $h$ is the height of the floor from the ground level and $H$ is the total height of the building, can be calculated in the frequency domain as follows:

{\overset{\cdot\cdot}{U}}^{t} (x, ω) = \sum_{k = 1}^{m} [Γ_{k} φ_{k} (x) \frac{- ω^{2}}{ω_{k}^{2} - ω^{2} + 2 i ξ_{k} ω_{k} ω} + 1] {\overset{\cdot\cdot}{U}}_{g} (ω),

(3)

where $φ_{k} (x)$ is the amplitude of the $k$ th mode shape at height $x$ , and $Γ_{k}$ , $ω_{k}$ , and $ξ_{k}$ are the modal participation factor, natural frequency, and damping ratio of the $k$ th mode, respectively (Clough and Penzien, 2013). ${\overset{\cdot\cdot}{U}}_{g} (ω)$ represents the Fourier Transform of the base excitation, ${\overset{\cdot\cdot}{u}}_{g} (t)$ , and $m$ is the number of effective modes, which is set at 6 in this article. PGA, the IM in this study, is extracted from ${\overset{\cdot\cdot}{u}}_{g} (t)$ . To use this equation for response prediction, the modal properties of the building are required. As discussed earlier, the coupled beam model, which can be characterized by only a few parameters, can be employed to obtain the necessary modal properties. The concept of a coupled shear-flexural beam model to represent building structures was initially proposed in the work by Osawa (1965). Miranda (1999) revisited this idea to generalize the drift spectrum and also calculate PFA (Miranda and Taghavi, 2005; Taghavi and Miranda, 2005). The fundamental representation of this model can be fully characterized by a dimensionless parameter $α$ that indicates the relative contributions of flexural and shear behaviors, the first natural period $T_{1} = 2 π / ω_{1}$ , and the required modal damping ratios $ξ_{k}$ (Miranda, 1999). Once $ω_{1}$ is determined, higher natural frequencies are calculated as:

ω_{k}^{2} = \frac{γ_{k}^{2} {β_{k}}^{2}}{γ_{1}^{2} {β_{1}}^{2}} ω_{1}^{2},

(4)

where coefficients $β_{k}$ and $γ_{k}$ are calculated by solving the corresponding characteristic equation presented below:

2 + (2 + \frac{α^{4}}{γ_{k}^{2} {β_{k}}^{2}}) \cos (γ_{k}) \cosh (β_{k}) + \frac{α^{2}}{γ_{k} β_{k}} \sin (γ_{k}) \sinh (β_{k}) = 0 .

(5)

Finally, $k$ th mode shape can be reconstructed as follows:

φ_{k} (x) = A_{1} \sin (γ_{k} x) + A_{2} \cos (γ_{k} x) + A_{3} \sinh (β_{k} x) + A_{4} \cosh (β_{k} x)

(6)

where coefficients $A_{1}$ to $A_{4}$ are obtained by solving the following system of linear equations:

[\begin{matrix} 0 & 1 & 0 & 1 \\ γ_{k} & 0 & β_{k} & 0 \\ - γ_{k}^{2} \sin (γ_{k}) & - γ_{k}^{2} \cos (γ_{k}) & β_{k}^{2} \sinh (β_{k}) & β_{k}^{2} \cosh (β_{k}) \\ - γ_{k} {β_{k}}^{2} \cos (γ_{k}) & γ_{k} {β_{k}}^{2} \sin (γ_{k}) & γ_{k}^{2} β_{k} \cosh (β_{k}) & γ_{k}^{2} β_{k} \sinh (β_{k}) \end{matrix}] [\begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \end{matrix}] = 0 .

(7)

Having mode shapes calculated, modal contribution factors are then obtained as:

Γ_{k} = \frac{\int_{0}^{1} φ_{k} (x) dx}{\int_{0}^{1} φ_{k} {(x)}^{2} dx} .

(8)

GMPE

GMPEs, formerly known as attenuation relationships, have played an essential role in seismic hazard analysis (SHA) since 1960 (Esteva and Rosenblueth, 1964). Douglas (2021) summarized these equations, covering the period from 1964 to 2021. GMPEs are statistical models that provide the median (or mean of the logarithm) estimate of the expected ground-motion IM, such as PGA, and its standard deviation for a given earthquake scenario because observed ground-motion intensities exhibit a log-normal distribution. In other words, it assumes that the IM of interest follows a log-normal distribution expressed as:

p (\ln IM | θ_{e}) = \frac{1}{σ_{\ln IM (θ_{e})} \sqrt{2 π}} e^{- \frac{1}{2} {(\frac{\ln IM - μ_{\ln IM} (θ_{e})}{σ_{\ln IM} (θ_{e})})}^{2}},

(9)

where $μ_{\ln IM}$ and $σ_{\ln IM}$ are mean and standard deviation provided by GMPE for a given source/path/site parameter vector of $θ_{e}$ . The vector $θ_{e}$ usually contains parameters, such as earthquake magnitude, $M$ , source-to-site distance, $R$ , time-average shear wave velocity at the upper 30 m of the site, $V_{S 30}$ , and source mechanism. In this study, we use the GMPE of Boore et al. (2014) to estimate $p (\ln PGA | θ_{e})$ .

Bayesian model calibration with embedded model error

Based on the idea originally developed by Sargsyan et al. (2015, 2019), the authors proposed embedding the modeling error/uncertainty into the parameters of the beam model to quantify the uncertainty associated with any prediction generated by this simplified model (Ghahari et al., 2022b). In this approach, model parameters $θ = {[T_{1} \begin{matrix} α & ξ \end{matrix}]}^{T}$ (the superscript $T$ denotes the matrix/vector transpose operator) are augmented by an additive adjustment $δ$ to produce a new set of variables $\tilde{θ} :$

\tilde{θ} = θ + δ (ζ, ϑ),

(10)

where the embedded error term $δ (ζ, ϑ)$ is a stochastic term (indicated by the stochastic variable $ζ$ ) following a PDF $p_{δ} (., ϑ)$ , which is parameterized by a set of parameters $ϑ$ ². Therefore, to fully represent the beam model, a new parameter vector $\tilde{ϑ} = (θ, ϑ)$ must now be estimated. Similar to the work by Ghahari et al. (2022b), we choose a multivariate normal (MVN) distribution for embedded model error as a correction to nominal parameter $θ$ . This correction is in line with the original works of Sargsyan et al. (2015, 2019). Hence, we cast the embedding in the form (equation 11) as a triangular first-order Gauss-Hermite polynomial chaos expansion (PCE) which is equivalent to MVN and enables PCE machinery for uncertainty propagation. This means that,

{\tilde{θ}}_{j} = \sum_{q = 0}^{j} ϑ_{qj} ζ_{q}, for j = 1, \dots, n_{θ},

(11)

where $n_{θ}$ is the number of physical parameters, and $ζ = {[\begin{matrix} ζ_{1} & \dots & ζ_{n_{θ}} \end{matrix}]}^{T}$ is an independently and identically distributed random vector with a standard normal distribution. Consequently, the calibration problem is to find an unknown parameter vector ${\tilde{ϑ}}_{n_{θ} \times (3 + n_{θ}) / 2}$ to satisfy the following measurement equation:

y_{i} = h (x_{i}; \tilde{ϑ}) + e_{i} .

(12)

where, in this study, $y_{i}$ represents the measured acceleration amplification factor (AAF) at the normalized height $x_{i}$ defined as:

y_{i} = PFA (x_{i}) / PGA .

(13)

The function $h (x_{i}; \tilde{ϑ})$ denotes the AAF predicted by the beam model at the same normalized height and $e_{i}$ is the measurement noise, assumed to be an independently and identically distributed random variable vector with a zero-mean Gaussian distribution and known variance $σ_{n}^{2}$ estimated from the sensor’s specification or measured pre-event data. Assuming a non-informative prior distribution for unknown parameters, the posterior probability distribution of the parameters given data, $p (\tilde{ϑ} | D)$ , is proportional to the likelihood function according to the Bayes’ theorem (Bernardo and Smith, 2008). Therefore, it can be estimated through Markov chain Monte Carlo (MCMC) by evaluating the likelihood function. The likelihood function can be approximated through independent-normal approximation (Sargsyan et al., 2019) as follows:

p (y | \tilde{ϑ}) = {(2 π)}^{- \frac{N}{2}} Π_{i = 1}^{N} \frac{1}{σ_{i} (\tilde{ϑ})} \exp \frac{- {(y_{i} - {\bar{y}}_{i} (\tilde{ϑ}))}^{2}}{2 σ_{i} {(\tilde{ϑ})}^{2}},

(14)

where $y = {[\begin{matrix} y_{1} & \dots & y_{N} \end{matrix}]}^{T}$ . The mean ${\bar{y}}_{i} (\tilde{ϑ})$ and the standard deviation $σ_{i} (\tilde{ϑ})$ of the prediction can be calculated using PCE as follows, which is a much more efficient way than the nominally time-consuming sampling techniques:

{\bar{y}}_{i} (\tilde{ϑ}) = E_{ζ} {h (x_{i}; \tilde{θ})} ≅ h_{0} (x_{i}; \tilde{ϑ}),

(15)

σ_{i} {(\tilde{ϑ})}^{2} = E_{ζ} {{[h (x_{i}; \tilde{θ}) - {\bar{y}}_{i} (\tilde{ϑ})]}^{2}} ≅ \sum_{p = 1}^{K - 1} h_{p}^{2} (x_{i}; \tilde{ϑ}) {Ψ_{p}}^{2} + σ_{n}^{2} .

(16)

where $h_{p} (x_{i}; \tilde{ϑ})$ represents the coefficients of the PCE expansion of the model prediction, and $Ψ_{p} (.)$ refers to the Hermite polynomials (Wiener, 1938). In addition, $K = (n_{θ} + P)! / (n_{θ}! P!)$ where $P$ denotes the order of the output expansion, which is set to 2 in this article. Once the posterior PDF of parameters, that is, $p (\tilde{ϑ} | D)$ , is estimated, it can be propagated through the model to calculate any response function, along with its associated uncertainty.

In this study, the fundamental period of the structures is assumed to be known and certain to simplify the analyses. This is a reasonable assumption, as the fundamental period can be well predicted or measured. In the case of non-instrumented buildings, it is still recommended to exclude the fundamental period from the parameters into which the modeling error is embedded. This is because it is a physical parameter that can be observed, unlike modal damping ratios, which are inherently uncertain, and α, which is not a directly observable parameter.

AAF

Validation of the beam model

To validate the applicability of the beam model calibrated through the embedded model calibration method, we use data from a 52-story building located in Downtown Los Angeles, which is instrumented through CSMIP (CSMIP Station No. 24602) and has been well studied (Çelebi et al., 2016c; Ventura and Ding, 2000). This calibrated model will subsequently demonstrate the effectiveness of the proposed PBEEW work in real life.

The 52-story building, depicted in Figure 3 (left), was designed in 1990. It comprises 52 stories above the ground and includes 5 levels below the ground. The lateral force-resisting system consists of concentrically braced steel frames at the core, supplemented by outrigger moment-resisting frames. The building has a square-shaped plan, measuring $48 m \times 48 m$ up to the 45th floor, and gradually reducing in size toward the roof. The building is supported by reinforced concrete spread footings at level E. Instrumentation for the building was installed during its construction, and the system consists of 20 accelerometers as illustrated in Figure 3 (right). Table 1 provides a list of earthquake events recorded by this building since its construction. For this study, we focus on events with a distance of 100 km or greater, as these are suitable for EEW analysis.

Figure 3.

Left: The 52-story building in Downtown Los Angeles (Google view), and right: its seismic instrumentation layout (downloaded from the Center for Engineering Strong-Motion Data (CESMD)).

Table 1.

Earthquakes recorded by the 52-story building

No.	Earthquake	Date/time	Magnitude	Epicentral distance (km)
1	Sierra Madre	6/28/1991 14:43	5.8 Ml	33
2	Landers	6/28/1992 11:57	7.3 Ml	169
3	Big Bear	6/28/1992 15:05	6.5 Ml	133
4	Northridge	1/17/1994 12:30	6.4 Ml	31
5	Chino hills	7/29/2008 18:42	5.4 Mw	47
6	Encino	3/17/2014 13:25	4.4 Mw	23
7	La Habra	3/29/2014 4:09	5.1 Mw	34
8	Ridgecrest	7/4/2019 17:33	6.4 MW	196
9	Ridgecrest	7/6/2019 3:19	7.1 MW	200
10	South El Monte	9/19/2020 6:38	4.5 MW	17
11	Lennox	4/5/2021 11:44	4.0 MW	14

AAFs recorded in the east-west direction of the building during the 2019 M 7.1 Ridgecrest earthquake are used for the model calibration. Level “A” is considered the base of the building (zero normalized height) where PGA is taken from the signal recorded by Channel 5 and used to obtain AAF at Levels 14, 22, 35, 49, and the roof with normalized heights of 0.26, 0.41, 0.64, 0.89, and 1, respectively. Considering $T_{1} = 5.98 \sec$ ., which was estimated from 2019 M 7.1 Ridgecrest earthquake data through the Bayesian modal identification technique (Ghahari et al., 2022a), Figure 4(a) shows the estimated joint posterior PDF of parameters $α$ and $ξ$ and their marginal PDFs. It can be observed that the parameter $α$ receives most of the modeling uncertainty, while a damping ratio of around 1% with small uncertainty is estimated. Figure 4(b) illustrates the mean prediction over all MCMC samples and the area covered by one standard deviation around the prediction (dark blue). Due to the limited number of data points along the height of the building, the posterior uncertainty (light blue), which is a fraction of the total uncertainty and represents the estimation error, constitutes approximately 50% of the total uncertainty (modeling and estimation uncertainty). This figure clearly demonstrates the importance of uncertainty quantification and the capability of the proposed method to quantify this uncertainty. Notably, all data points fall approximately within one standard deviation around the prediction mean. The readers can find details about posterior uncertainty and total uncertainty in the work by Ghahari et al. (2022b).

Figure 4.

(a) The posterior PDF of the physical parameters, and (b) predicted AAF and the associated uncertainties obtained from the 2019 M 7.1 Ridgecrest earthquake.

Figure 5 displays the marginal PDF of AAF along the height of the building in which the PDF increases from blue to red. This figure highlights the areas along the building height where the beam model can predict AAF with high confidence, that is, larger PDF values, while AAF estimation at the roof level is susceptible to significant uncertainty.

Figure 5.

The marginal predictive PDF of AAF along the height of the building obtained from the 2019 M 7.1 Ridgecrest earthquake.

To validate the predictive performance of the beam model and assess the estimated uncertainty obtained from the M 7.1 Ridgecrest earthquake data, AAF is predicted under three additional long-distance earthquake events. Figure 6 presents the mean prediction (red) along with one standard deviation (blue). It can be observed that the actual recorded AAFs (markers) are largely encapsulated within the predicted uncertainty, which demonstrates the superior performance of this stochastic beam model for future predictions. It should also be emphasized that while this study is limited to instrumented buildings whose data from previous events can be used for the beam model calibration, the proposed PBEEW framework can be used for non-instrumented buildings skipping the calibration step. To do so, suitable PDFs of the beam model parameters, similar to Figure 4(a) can be estimated for different building classes using extensive numerical studies. In that case, the fundamental period of the building, $T_{1}$ , must be added to the list of parameters as well.

Figure 6.

AAF blind prediction for three different long-distance earthquakes (a: 1992 Landers, 1992 Big Bear, and M 6.4 Ridgecrest earthquakes) using the beam model calibrated using the data recorded in M 7.1 Ridgecrest earthquake, 2019.

Selected ground motions

Taghavi-Ardakan (2006) used 80 ground motions to develop the probability distribution of AAF using deterministic beam models. These PDFs were subsequently employed by Cheng et al. (2014) to propose the PBEEW framework. The ground motions were initially compiled by Medina (2003) and categorized into four bins based on earthquake magnitude and epicentral distance: “Small magnitude, small distance,”“small magnitude, large distance,”“large magnitude, small distance,” and “large magnitude, large distance.” Here, large magnitude refers to the moment magnitudes ranging from 6.6 to 6.9, and large distance denotes distances between 30 and 60 km. However, as previously mentioned, these ground motions are not suitable for EEW applications since they do not originate from large and long-distance events.

In this study, alongside the existing set of 80 ground motions collected by Taghavi-Ardakan (2006), which will be referred to as the “old set,” a new set of ground motions from California events is collected from NGA-West2 database (Ancheta et al., 2014) according to the following criteria:

Earthquake magnitude greater than M 6.0;

Source to site distance between 50 and 250 km;

$V_{S 30} in the range of 360 to 760 m / s, corresponding to site class C as per NEHRP (2003) .$

Using these criteria, 150 ground motions are selected from the following earthquake events: 1971 M 6.6 San Fernando Valley, 1989 M 6.9 Loma Prieta, 1992 M 7.3 Landers, 1992 M 6.5 Big Bear, 1994 M 6.7 Northridge, and 1999 M 7.1 Hector Mine earthquakes. The distribution of PGA versus earthquake magnitude and source-to-site distance of these records are shown in Figure 7. In this figure, the median PGA predicted by the GMPE used in this study by Boore et al. (2014) is also shown for all six earthquake magnitudes. In addition to the distribution of the records, this figure emphasizes the fact that the GMPE PGA is mostly underestimated at long distances which will be the case for the 52-story building under the 2019 M 7.1 Ridgecrest earthquake as will be shown later. Note that data from the two 2019 Ridgecrest earthquakes are excluded from this set and reserved for later use as test data. This newly collected set of ground motions is hereafter referred to as the “new set.” In addition, as these ground motions will be subsequently used to analyze the 52-story building with a fundamental period of around 6 s, a subset of 75 records with the lowest usable frequency of 0.1 Hz is chosen. Figure 8 illustrates the 5% damped spectral acceleration of both sets of records, with all records scaled to have PGA = 1. The figure also presents their mean spectrum. Notably, the new set exhibits greater richness in the mid- and long-period ranges compared to the old set. As will be shown later, this new set of ground motions results in a better PFA prediction. So, further studies on the ground-motion selection can improve the performance of the proposed PBEEW. For example, conditional spectrum (see, e.g. Baker and Lee, 2018) or scenario-based simulations (Aagaard et al., 2010) can be employed to collect more suitable ground motions.

Figure 7.

The magnitude-distance-PGA distribution of the selected ground motions, and corresponding median PGA, predicted by GMPE (Boore et al., 2014) with the site condition of the 52-story building.

Figure 8.

Scaled 5% damped acceleration response spectra of all ground motions in (a) Taghavi’s (old) and (b) this study’s (new) sets.

To observe the impact of record-to-record variability on the response of the beam model, deterministic beam models with various $α$ and $T_{1}$ are analyzed under both sets of ground records. The models maintain a constant damping ratio of 2%. Figures 9 and 10 present the results for the old and new sets of ground motions, respectively. It can be observed that both sets of records yield a similar level of mean amplification in the short-period range. However, the new set of ground motions exhibits larger mean amplification in the long-period range due to their long-period content. Furthermore, both sets of results demonstrate a comparable level of variability, as depicted in this figure.

Figure 9.

Mean and coefficient of variation (COV) of AAF at the roof (a and b) and mid-height, $x = 0.5$ , (c and d) levels under the old set of ground motions.

Figure 10.

Mean and COV of AAF at the roof (a and b) and mid-height, $x = 0.5$ , (c and d) levels under the new set of ground motions.

Peak roof acceleration conditioned on EEW data

PDF of AAF

Assuming the beam model as a deterministic model, Taghavi-Ardakan (2006) developed PDFs of AAF for various $T_{1}$ , $α$ , and $ξ$ values. However, as mentioned previously, these PDFs solely reflect the uncertainty present in the ground motions, neglecting the additional uncertainty stemming from modeling errors. By employing a stochastic beam model that incorporates modeling errors, we can regenerate these PDFs, accounting for both modeling and record-to-record variabilities. Figure 11 illustrates these new PDFs for two levels of the 52-story building. The original PDFs developed by Taghavi-Ardakan (2006), that is, computed using a deterministic beam with $T_{1} = 5.98 \sec .$ , $α = 12$ , and $ξ = 1 %$ correpsoding to a maximum a posteriori (MAP) estimations of parameters (see Figure 4(a)), are also shown in black. As anticipated, the PDFs generated in this study under both sets of ground motions exhibit greater width due to the inclusion of modeling uncertainty. Moreover, as expected, the PDFs generated under the new set of ground motions show MAP probability at larger AAF values. As observed in Figure 11, all PDFs can be reasonably approximated by log-normal distributions. Although such an approximation is not essential for the PBEEW approach, it facilitates the integration of equation 2. This will be further discussed in the subsequent sections.

Figure 11.

The estimated PDF of AAF at two levels of the 52-story building obtained using old and new sets of ground motions. The parameters of the new log-normal distributions are shown on each subplot.

Before showcasing the final outcome of the proposed PBEEW framework in the next section, Figure 12 is presented to demonstrate the variation of the AAF PDFs along the height of the building. This figure illustrates the median AAF and logarithmic standard deviation of AAF for both sets of ground motions. It can be observed that while both sets of motions exhibit the same total variability, encompassing modeling and record-to-record variability, the median AAF is higher under the new set of ground motions.

Figure 12.

The estimated median and standard deviation of PDF of AAF along the height of the 52-story building under old and new sets of ground motions.

PDF of PFA

Having the log-normal distribution of $p (\ln \frac{PFA}{PGA} | \tilde{ϑ})$ , $p (\ln PFA | \ln PGA, \tilde{ϑ})$ can be expressed as:

p (\ln PFA | \ln PGA, \tilde{ϑ}) = \frac{1}{σ_{\ln AAF} \sqrt{2 π}} e^{- \frac{1}{2} {(\frac{\ln PFA - \ln PGA - μ_{\ln AAF}}{σ_{\ln AAF}})}^{2}},

(17)

where $e^{μ_{\ln AAF}}$ and $σ_{\ln AAF}$ are already shown in Figure 12. By substituting equations 17 and 9 into equation 2, we obtain:

p (\ln PFA | D) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{1}{2} {(\frac{\ln PFA - μ}{σ})}^{2}},

(18)

where $μ = μ_{\ln PGA} + μ_{\ln AAF}$ and $σ = \sqrt{σ_{\ln PGA}^{2} + σ_{\ln AAF}^{2}}$ . Equation 18 is based on the fact that PGA and AAF are independent, which is not always true due to the probable nonlinear behavior of the structure. However, in the EEW application, structures are subjected to long-distance events under which the level of structural nonlinearity would be limited. Figure 13 shows equation 18 for the roof and 22nd floor PFA (not $\ln PFA$ ) of the 52-story building in the 2019 M 7.1 Ridgecrest earthquake. In this figure, PDFs obtained using old and new sets of ground motions are shown in subplots a/c and b/d, respectively. This figure is presented to show how the uncertainties in each of the three components involved in response prediction, that is, ground-motion representation, building model, and response prediction method, affect the final PFA prediction. The black curve represents the PFA PDF obtained by running the stochastic beam model under the recorded ground motion. In other words, this curve is the PDF shown in Figure 5 at the roof and 22nd levels, adjusted to the exact PGA which is 0.02 g. So, the only source of uncertainty in this curve is the modeling uncertainty coming from the beam model. When we use equation 18, even by assuming the PGA is exactly predicted ( $PGA = 0.02 g)$ and it is certain ( $σ_{\ln PGA}^{2} = 0$ ), the PDF changes to the blue curve. That is, both the mean and variance change due to the record-to-record variability introduced through the response prediction method. If the GMPE-estimated PGA is used instead of the actual PGA, the PDF further changes to the yellow curve, indicating a much wider probability distribution (more uncertainty) due to both error and uncertainty in the ground-motion representation. The green curve represents the case when the GMPE PGA is used while assuming its uncertainty is zero, that is, $σ_{\ln PGA}^{2} = 0$ . To obtain the GMPE estimated PGA (0.016 g) and its logarithmic standard deviation (0.653), the equation developed by Boore et al. (2014) with the parameters listed in Table 2 is used to create Figure 13. In addition, for comparison, the actual recorded PFAs at the roof and 22nd levels are shown in this figure by the vertical dashed lines. As seen in subplots (b) and (d), the usage of the new set of ground motions, as observed in Figure 12, slightly shifts the PDFs to the right and closer to the actual PFAs. The median ( $μ$ ) and standard deviation ( $σ_{\ln AAF}$ ) of all cases presented in Figure 13 are presented in Table 3 for the reader’s convenience.

Figure 13.

PDF of PFA conditioned on 2019 M 7.1 Ridgecrest earthquake scenario: (a) roof level/old set, (b) roof level/new set, (c) 22nd floor/old set, and (d) 22nd floor/new set.

Table 2.

Parameters used to estimate PGA at the 52-story building site in the 2019 M 7.1 Ridgecrest earthquake

No.	Magnitude(Mw)	Region	Fault type	$R_{JB}$ (km)	$V_{S 30} (\frac{m}{s})$	$z_{1}$ (km)
1	7.1	1	1	185	356	0.305

Table 3.

The parameters ( $μ / σ) of predicted p (\ln P F A | D)$ using the 2019 M 7.1 Ridgecrest earthquake

Level	GM set	Exact GM	Exact PGA	GMPE	GMPEWO uncertainty
Roof	Old	−2.84/0.213	−3.23/0.357	−3.45/0.745	−3.45/0.357
Roof	New	−2.84/0.213	−2.96/0.336	−3.17/0.735	−3.17/0.336
22nd floor	Old	−3.45/0.203	−3.99/0.379	−4.20/0.755	−4.20/0.379
22nd floor	New	−3.45/0.203	−3.65/0.374	−3.86/0.753	−3.86/0.374

While this study is limited to demand estimation, and a full PBEEW framework is beyond the scope of this work, herein we present a few ways in which these demand PDFs can be directly used for real EEW. As discussed by Cheng et al. (2014), the estimated floor PFA can be used as follows:

Alert residents if floor acceleration exceeds the human comfort level (Boggs and Petersen, 1997), or if non-structural damage is expected and, thus the residents should drop, cover, and hold on;

Stop the elevators if the acceleration exceeds the serviceability limit (Kubo et al., 2011);

Halt acceleration-sensitive equipment. such as magnetic resonance imaging units in hospitals.

Here, as an illustration, we use the estimated PDFs to determine if a human comfort warning would have been issued for the 52-story building in Downtown Los Angeles if the PBEEW was in effect during the 2019 M 7.1 earthquake. According to the results obtained from wind tests (Griffis, 1993), Cheng et al. (2014) adopted the following human comfort classes: “Not perceptible,”“Threshold of perceptible,”“Annoying,” and “Very annoying” for PFA values less than 0.5%g, between 0.5%g and 1.5%g, between 1.5%g and 5%g, and above 5%g, respectively. We employ both the “mean-value” and “most-probable” methods to obtain the expected PFA from the estimated PDFs and compare them with the specified thresholds. In the “mean-value” method, the PFA at the 50% percentile of the cumulative distribution function (CDF) of $p (\ln PFA | D)$ is selected. In the “most-probable” method, the total area under $p (\ln PFA | D)$ is divided at PFA values of 0%g, 0.5%g, 1.5%g, 5%g, and ∞%g, and the segment with the largest area determines the comfort level. Figure 14 displays the CDF corresponding to the PDFs shown in Figure 13. Based on this figure, the PFA at the 50% percentile is identified and reported in Table 4. As seen in this table, the median PFA prediction accuracy degrades when using less accurate ground-motion estimation (moving from “Exact GM” to “GMPE” estimation); however, the results obtained from the new set of motions are closer to the recorded values. Also, the level of error is higher at the roof level because of the limitation associated with the beam model (e.g. neglecting the higher mode’s contribution, assuming uniform mass/stiffness distribution, etc.). However, we should emphasize that the purpose of this study is to propose a practical EEW solution for a regional scale, otherwise, a detailed FE model would make a better prediction. The corresponding human comfort classes, based on the aforementioned thresholds, are provided in Table 5. As observed, when the exact ground motion is used, the expected comfort level aligns with what actually occurred based on the recorded data. The PGA is still able to represent ground motion for this purpose, provided that the new set of ground motions is used for calculations. However, the GMPE-estimated PGA fails to provide a reliable prediction of the comfort level at the roof level. It is important to note that both the GMPE and GMPE without uncertainty yield the same median, so a single value is reported.

Figure 14.

Cumulative distribution functions (CDFs) of PFA conditioned on 2019 M 7.1 Ridgecrest earthquake scenario: (a) roof level/old set, (b) roof level/new set, (c) 22nd floor/old set, and (d) 22nd floor/new set.

Table 4.

The predicted PFA (%g) at the 50% percentile of PFA CDF conditioned on the 2019 M 7.1 Ridgecrest earthquake

Level	GM set	Exact GM	Exact PGA	GMPE	Recorded
Roof	Old	5.8	3.9	3.2	7.1
Roof	New	5.8	5.2	4.2	7.1
22nd floor	Old	3.2	1.9	1.5	3.2
22nd floor	New	3.2	2.6	2.1	3.2

Table 5.

The predicted comfort level in two levels of the 52-story building during the 2019 M 7.1 Ridgecrest earthquake estimated using the mean value method

Level	GM set	Exact GM	Exact PGA	GMPE	Recorded
Roof	Old	Very annoying	Annoying	Annoying	Very annoying
Roof	New	Very annoying	Very annoying	Annoying	Very annoying
22nd Floor	Old	Annoying	Annoying	Threshold of perceptible/annoying	Annoying
22nd Floor	New	Annoying	Annoying	Annoying	Annoying

The alternative approach to determine the comfort level is to use the “most-probable” method, which calculates the area of the PDF between those thresholds. Figure 15 displays these areas for the roof level in various scenarios. While there is no direct ground-truth decision that can be considered for this method, assuming “Very annoying” as the correct comfort level at the roof level using a single measurement (see the last column of Table 5), the stochastic beam model, under exact ground motion or exact PGA (provided that the demand model is generated using the new set of motions), is capable of providing such a prediction. Although using the new set of ground motions enhances the prediction using GMPE estimated PGA, it still predicts an “Annoying” level of motion at the roof. A similar figure for the 22nd floor is shown in Figure 16. Accepting the “Annoying” level as the actual occurrence at this level during the 2019 M 7.1 earthquake event, all methods yield such prediction, but the method based on the GMPE-estimated PGA performs better when the new set of ground motions is used in comparison to the old set.

Figure 15.

The most-probable comfort level at the roof of the 52-story building during the 2019 M 7.1 Ridgecrest earthquake estimated by various methods using (a) old and (b) new sets.

Figure 16.

The most-probable comfort level on the 22nd floor of the 52-story building during the 2019 M 7.1 Ridgecrest earthquake estimated by various methods using (a) old and (b) new sets.

We conclude this section by presenting Figure 17. This figure illustrates the predicted probability of exceedance (1-CDF) of PFA along the height of the building in the M 7.1 Ridgecrest earthquake, which was derived from the estimated PGA using the GMPE and the new set of records. Contrary to Figures 12 and 13 which are limited to two specific levels, this plot represents the likelihood of encountering PFA exceeding specific levels at different heights that provide valuable insights for future studies and establishing thresholds. For instance, the red dashed line represents the probability of observing a PFA equal to or greater than 2%g along the height of the building. The intensity of the color at the position of this line indicates the varying probability of observing PFA exceeding 2%g at different heights, ranging from 10% at lower levels to approximately 50% in the middle, and nearly 100% at the roof.

Figure 17.

The probability of exceedance of PFA along the height of the 52-story building obtained using GMPE-estimated PGA and the new set of ground motions. As an example, the probability of observing PFA greater than 2%g along the height of the building varies from 10% at the lower level to around 50% at the middle to almost 100% at the roof.

Conclusion

The existing ShakeAlert EEW system provides a warning to residents on the West Coast of the United States if the expected ground shaking intensity (Modified Mercalli intensity) exceeds a certain threshold. However, individuals inside buildings are likely to experience different, usually higher, levels of shaking (acceleration, velocity, displacement) due to the dynamic response of the buildings. Amplification levels are typically much greater for residents of tall buildings during long-distance earthquake events, for which the EEW system works better. The recent 2019 Ridgecrest earthquake events clearly highlighted this limitation of the existing warning strategy. The ideal solution would involve running detailed numerical models of the affected buildings, considering the expected ground-motion time history, to predict building responses and issue appropriate alerts. However, currently, such an analysis is theoretically and practically unfeasible. The expected ground-motion time histories cannot be accurately predicted, numerical models of the buildings might be unavailable, and time-history analysis, especially on a regional scale, is laborious, time-consuming, and undermines the purpose of EEW. A recent study by the same team revealed that the available simple formulas are unsuitable for estimating PFA unless modal properties of the building, and accurate ground-motion response spectra, are available. This article takes a different approach and attempts to extend the PEER PBEE to EEW, considering the fact that every component involved in building prediction is uncertain. This idea is not new and was originally proposed by other researchers, but it suffered from two shortcomings: (1) the simple beam model used for response prediction is susceptible to modeling uncertainty (due to, e.g. assuming linear elastic behavior, neglecting 3D behavior, neglecting soil–structure interaction effects, etc.) which had not been quantified, and (2) the ground motions used for the probabilistic demand model (the probabilistic relationship between IM and EDP) were not suitable for EEW applications.

In this article, we addressed these two issues by incorporating modeling errors into the parameters of the beam model and using a new set of ground motions, respectively. While validating this extended PBEEW requires studying numerous instrumented tall buildings, we used data from a 52-story building in Downtown Los Angeles to demonstrate the practicality of this approach. The criteria and thresholds for comparing the estimated PFA are beyond the scope of this study, but using those employed by previous researchers shows that if PGA, the IM used in this and previous studies, can be accurately estimated, this approach can predict the correct level of human comfort in tall buildings. It is noteworthy to mention that the current EEW system provides the PGA at regions expecting shaking. Additionally, the mean and standard deviation of AAF PDF at every building could be pre-calculated. So, the only operation that needs to be done to obtain the mean and total uncertainty of PFA is a simple addition as shown in equation 18. Thus, the proposed extension does not contribute to additional processing time.

Several remaining issues need to be addressed for the PBEEW approach to be implemented in real-life situations. To apply this approach on a regional scale, stochastic beam models of various types of buildings are necessary. This can be accomplished using data from available instrumented tall buildings and employing data analysis techniques to determine hyperparameters that control the distributions of the beam model’s parameters. PGA is not a sufficient and efficient IM, especially when used for tall buildings under potentially long-distance events because it is a high-frequency IM. So, a different IM may improve the results and reduce the record-to-record variability. Finally, regardless of the chosen IM, the results would depend on the accuracy of its value and its associated uncertainty. One possible solution is to leverage physics-based simulations to generate scenario-based ground motions, which can be used to develop more accurate GMPEs specifically designed for EEW applications, considering that the probable seismic sources and target areas are mostly known. In addition, such specific ground motions can be used to develop more accurate demand models in conjunction with stochastic beam models.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported in part by the United States Geological Survey (USGS) and the California Geological Survey (CGS) which are gratefully acknowledged. Any use of trade, firm, or product names is for the descriptive purposes only and does not imply endorsement by the US Government. Any opinions, findings, and conclusions expressed in this material are solely those of the authors and do not reflect the views of the California Department of Conservation. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.

ORCID iDs

Farid Ghahari

Grace A Parker

Ertugrul Taciroglu

Notes

References

Aagaard

Graves

Rodgers

Brocher

Simpson

Dreger

Petersson

Larsen

Jachens

(2010) Ground-motion modeling of Hayward fault scenario earthquakes, Part II: Simulation of long-period and broadband ground motions. Bulletin of the Seismological Society of America 100(6): 2945–2977.

Akkar

Sucuoğlu

Yakut

(2005) Displacement-based fragility functions for low-and mid-rise ordinary concrete buildings. Earthquake Spectra 21(4): 901–927.

Ameri Fard Nasrand

Mahsuli

Ghahari

Taciroglu

(2023) Bayesian model selection considering model complexity using stochastic filtering. Earthquake Engineering & Structural Dynamics 52: 3120–3148.

Ancheta

Darragh

Stewart

Seyhan

Silva

Chiou

BSJ

Wooddell

Graves

Kottke

Boore

Kishida

(2014) NGA-West2 database. Earthquake Spectra 30(3): 989–1005.

ASCE (2017) Minimum Design Loads and Associated Criteria for Buildings and Other Structures. Reston, VA: ASCE.

Baker

(2015) Efficient analytical fragility function fitting using dynamic structural analysis. Earthquake Spectra 31(1): 579–599.

Baker

Cornell

(2008) Vector-valued intensity measures incorporating spectral shape for prediction of structural response. Journal of Earthquake Engineering 12(4): 534–554.

Baker

Lee

(2018) An improved algorithm for selecting ground motions to match a conditional spectrum. Journal of Earthquake Engineering 22(4): 708–723.

Bernardo

Smith

AFM

(2008) Bayesian Theory. Hoboken, NJ: Wiley.

10.

Boggs

Petersen

(1997) Acceleration indexes for human comfort in tall buildings–Peak or RMS. CTBUH Monograph 13: 1–21.

11.

Boore

Stewart

Seyhan

Atkinson

(2014) NGA-West2 equations for predicting PGA, PGV, and 5% damped PSA for shallow crustal earthquakes. Earthquake Spectra 30: 1057–1085.

12.

Böse

Andrews

O’Rourke

Kilb

Lux

Bunn

McGuire

(2023) Testing the ShakeAlert earthquake early warning system using synthesized earthquake sequences. Seismological Society of America 94(1): 243–259.

13.

Çelebi

Ghahari

Taciroğlu

(2016b) Significance of beating observed in earthquake responses of buildings. In: 16th U.S.-Japan-N.Z. Workshop on the Improvement of Structural Engineering and ResiliencyAt: Todaiji Temple Cultural Center, Nara, Japan, 27–29 June.

14.

Çelebi

Ghahari

Haddadi

Taciroglu

(2020) Response study of the tallest California building inferred from the Mw7.1 Ridgecrest, California earthquake of 5 July 2019 and ambient motions. Earthquake Spectra 36: 1096–1118.

15.

Çelebi

Kashima

Ghahari

Abazarsa

Taciroglu

(2016a) Responses of a tall building with US code-type instrumentation in Tokyo, Japan, to events before, during, and after the Tohoku earthquake of 11 March 2011. Earthquake Spectra 32(1): 497–522.

16.

Çelebi

Swensen

Haddadi

(2021) Response study of a 51-story-tall Los Angeles, California building inferred from motions of the Mw7.1 July 5, 2019 Ridgecrest, California Earthquake. Bulletin of Earthquake Engineering 19(4): 1797–1814.

17.

Çelebi

Ulusoy

Nakata

(2016c) Responses of a tall building in Los Angeles, California, as inferred from local and distant earthquakes. Earthquake Spectra 32(3): 1821–1843.

18.

Cheng

Heaton

Beck

(2014) Earthquake early warning application to buildings. Engineering Structures 60: 155–164.

19.

Clough

Penzien

(2013) Dynamics of Structures. New York: McGraw-Hill.

20.

Cornell

Krawinkler

(2000) Progress and challenges in seismic performance assessment. PEER Center News. Available at: https://apps.peer.berkeley.edu/news/2000spring/performance.html (accessed 22 February 2024).

21.

Cruz

Miranda

(2021) Damping ratios of the first mode for the seismic analysis of buildings. Journal of Structural Engineering 147(1): 04020300.

22.

Dehghanpoor

Thambiratnam

Zhang

Chan

Taciroglu

(2021) An extended probabilistic demand model with optimal intensity measures for seismic performance characterization of isolated bridges under coupled horizontal and vertical motions. Bulletin of Earthquake Engineering 19: 2291–2323.

23.

Douglas

(2021) Ground motion prediction equations 1964–2021. Department of Civil and Environmental Engineering University of Strathclyde, Glasgow.

24.

Esteva

Rosenblueth

(1964) Espectros de temblores a distancias moderadas y grandes. Boletin Sociedad Mexicana De Ingenieria Sesmica 2(1): 1–18.

25.

FEMA 356FE (2000) Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Washington, DC: Federal Emergency Management Agency.

26.

FEMA P-58 (2018) Seismic performance assessment of buildings volume 1—methodology. Technical Report, FEMA P-58-1. Redwood City, CA: Applied Technology Council.

27.

Ghahari

Baltay

Çelebi

Parker

Mcguire

Taciroglu

(2022a) Earthquake early warning for estimating floor shaking levels of tall buildings. Bulletin of the Seismological Society of America 112: 820–849.

28.

Ghahari

Sargsyan

Çelebi

Taciroglu

(2022b) Quantifying modeling uncertainty in simplified beam models for building response prediction. Structural Control Health Monitoring 29(11): e3078.

29.

Given

Allen

Baltay

Bodin

Cochran

Creager

de Groot

Gee

Hauksson

Heaton

Hellweg

Murray

Thomas

Toomey

Yelin

(2018) Revised technical implementation plan for the ShakeAlert system—An earthquake early warning system for the West Coast of the United States. Reston, VA: U.S. Geological Survey.

30.

Given

Cochran

Heaton

Hauksson

Allen

Hellweg

Vidale

Bodin

(2014) Technical implementation plan for the ShakeAlert production system: An earthquake early warning system for the west coast of the United States. U.S. Department of the Interior, U.S. Geological Survey, Reston, VA.

31.

Grasso

Beck

Manfredi

(2007) Automated decision procedure for earthquake early warning. Engineering Structures 29(12): 3455–3463.

32.

Graves

Jordan

Callaghan

Deelman

Field

Juve

Kesselman

Maechling

Mehta

Milner

Okaya

Small

Vahi

(2011) CyberShake: A physics-based seismic hazard model for Southern California. Pure and Applied Geophysics 168(3): 367–381.

33.

Griffis

(1993) Serviceability limit states under wind load. Engineering Journal 30(1): 1–16.

34.

Iervolino

Giorgio

Manfredi

(2007) Expected loss-based alarm threshold set for earthquake early warning systems. Earthquake Engineering & Structural Dynamics 36(9): 1151–1168.

35.

Jaiswal

Wald

D’Ayala

(2011) Developing empirical collapse fragility functions for global building types. Earthquake Spectra 27(3): 775–795.

36.

Kazantzi

Vamvatsikos

(2015) Intensity measure selection for vulnerability studies of building classes. Earthquake Engineering & Structural Dynamics 44(15): 2677–2694.

37.

Kohler

Smith

Andrews

Chung

Hartog

Henson

Given

de Groot

Guiwits

(2020) Earthquake early warning Shakealert 2.0: Public rollout. Seismological Research Letters 91: 1763–1775.

38.

Kubo

Hisada

Murakami

Kosuge

Hamano

(2011) Application of an earthquake early warning system and a real-time strong motion monitoring system in emergency response in a high-rise building. Soil Dynamics and Earthquake Engineering 31(2): 231–239.

39.

Mccallen

Petersson

Rodgers

Miah

Pitarka

Petrone

Tang

(2020) The Earthquake Simulation (EQSIM) Framework for Physics-Based Fault-to-Structure Simulations. Livermore, CA: Lawrence Livermore National Laboratory.

40.

McGuire

Smith

Frankel

Wirth

McBride

de Groot

(2021) Expected Warning Times from the ShakeAlert Earthquake Early Warning System for Earthquakes in the Pacific Northwest. Reston, VA: U.S. Geological Survey Open-File.

41.

Medina

(2003) Seismic Demands for Nondeteriorating frame Structures and their Dependence on Ground Motions. Stanford, CA: Stanford University.

42.

Miranda

(1999) Approximate seismic lateral deformation demands in multistory buildings. Journal of Structural Engineering 125(4): 417–425.

43.

Miranda

Taghavi

(2005) Approximate floor acceleration demands in multistory buildings. I: Formulation. Journal of Structural Engineering 131(2): 203–211.

44.

NEHRP-National Earthquake Hazards Reduction Program (2003) Recommended provisions for seismic regulations for new buildings and other structures part 1: Provisions. Available at: https://www.fema.gov/sites/default/files/2020-10/fema_2020-nehrp-provisions_part-1-and-part-2.pdf (accessed 22 February 2024).

45.

Osawa

(1965) Response analysis of tall buildings to strong earthquake motions. Part 1: Linear response of core-wall buildings. Bulletin of the Earthquake Research Institute 43: 803–817.

46.

Porter

(2003) An overview of PEER’s performance-based earthquake engineering methodology. International Conference on Applications of Statistics and Probability in Civil Engineering273(1995): 973–980.

47.

Ramirez

Lignos

Miranda

Kolios

(2012) Fragility functions for pre-Northridge welded steel moment-resisting beam-to-column connections. Engineering Structures 45: 574–584.

48.

Rodgers

Pitarka

Petersson

Sjögreen

McCallen

(2018) Broadband (0–4 Hz) ground motions for a magnitude 7.0 Hayward fault earthquake with three-dimensional structure and topography. Geophysical Research Letters 45(2): 739–747.

49.

Rostami

Mahsuli

Ghahari

Taciroglu

(2021) Bayesian joint state-parameter-input estimation of flexible-base buildings from sparse measurements using Timoshenko Beam Models. Journal of Structural Engineering 147(10): 04021151.

50.

Sargsyan

Huan

Najm

(2019) Embedded model error representation for Bayesian model calibration. International Journal for Uncertainty Quantification 9(4): 365–394.

51.

Sargsyan

Najm

Ghanem

(2015) On the statistical calibration of physical models. International Journal of Chemical Kinetics 47(4): 246–276.

52.

SEAOC Vision (1995) A Framework for Performance Based Earthquake Engineering. Sacramento, CA: Structural Engineers Association of California.

53.

Shirzad-Ghaleroudkhani

Mahsuli

Ghahari

Taciroglu

(2017) Bayesian identification of soil-foundation stiffness of building structures. Structural Control Health Monitoring 25: e2090.

54.

Taciroglu

Çelebi

Ghahari

Abazarsa

(2016) An investigation of soil-structure interaction effects observed at the MIT green building. Earthquake Spectra 32(4): 2425–2448.

55.

Taciroglu

Ghahari

Abazarsa

(2017) Efficient model updating of a multi-story frame and its foundation stiffness from earthquake records using a Timoshenko beam model. Soil Dynamics and Earthquake Engineering 92: 25–35.

56.

Taghavi

Miranda

(2005) Approximate floor acceleration demands in multistory buildings. II: Applications. Journal of Structural Engineering 131(2): 212–220.

57.

Taghavi-Ardakan

(2006) Probabilistic Seismic Assessment of Floor Acceleration Demands in Multi-Story Buildings. Stanford, CA: Stanford University.

58.

Ventura

Ding

(2000) Linear and nonlinear seismic response of a 52-storey steel frame building. Journal of the Structural Design of Tall Buildings 9: 25–45.

59.

Wald

(2020) Practical limitations of earthquake early warning. Earthquake Spectra 36: 1412–1447.

60.

Wiener

(1938) The homogeneous chaos. American Journal of Mathematics 60(4): 897–936.

61.

Beck

Heaton

(2013) ePAD: Earthquake probability-based automated decision-making framework for earthquake early warning. Computer-aided Civil and Infrastructure Engineering 28(10): 737–752.

62.

Zhang

Yang

(2018) A spectral-velocity-based combination-type earthquake intensity measure for super high-rise buildings. Bulletin of Earthquake Engineering 16: 643–677.