A framework to quantify the effectiveness of earthquake early warning in mitigating seismic risk

Earthquake early warning systems (EEWSs) aim to rapidly detect earthquakes and provide timely alerts, so that users can take protective actions prior to the onset of strong ground shaking. The promise and limitations of EEWSs have both been widely debated. On one hand, an operational EEWS could mitigate earthquake damage by triggering potentially cost- and life-saving actions. These range from automated system responses such as slowing down trains to the actions of individuals that receive the alerts and take protective measures. On the other hand, the effectiveness of an EEWS is conditional on the ability to issue warnings that are sufficiently accurate and timely to facilitate an appropriate action. The refinement of earthquake early warning (EEW) algorithms and the installation of denser and faster seismic networks have improved performance; however, the benefit in risk reduction that an EEWS could achieve remains unquantified. In this study, we leverage upon regional event-based probabilistic seismic risk assessment to devise a quantitative and fully customizable framework for evaluating the effectiveness of EEW in mitigating seismic risk. We demonstrate this framework using Switzerland as a testbed, for which we compute and contrast human loss exceedance curves with and without EEW.


Introduction
provided a detailed assessment of such handicaps. For instance, no-alert zones (where S-waves arrive before an alert is issued) are bound to exist around earthquake epicenters, where the separation between P-and S-waves is minimal. Kuyuk and Allen (2013) showed that increasing the density of the seismic network can reduce the size of noalert zones, but not fully eliminate them. This is troubling given that it is in these nearsource areas where the strongest ground shaking, and therefore damage, is expected to occur. In large earthquakes (M . 6.5), however, strong shaking and damage also occurs in broad areas around the fault rupture, which can extend over tens to hundreds of kilometers. These events clearly offer the best opportunities for EEW, because warning times to strongly affected areas could exceed more than 10 s (Bo¨se et al., 2018). Wald (2020) further stressed that for what concerns human responses (and to a lesser extent automated responses), the amount of warning time needed to impact the outcome is unclear. In fact, this time might be substantial, as one would need to become aware of the warning, recognize it, decide on a protective action, and then proceed with its execution. Porter and Jones (2018) assessed the time needed for the duck, cover, and hold on (DCHO) protective response by collecting survey responses; their analysis indicated a median DCHO reaction time of 8.8 s with a 0.4 logarithmic standard deviation. Adding to that, as the size of large earthquakes takes longer to be determined, Minson et al. (2018) concluded that EEWS could produce sufficient warning times only for users who set a low ground motion intensity threshold for receiving an alert (so that an alert is warranted before convergence to the final magnitude estimate). In a different study by Minson et al. (2019), a low alerting threshold was also deemed necessary to avoid excessive missed alerts as a result of ground motion variability. However, setting a conservative alerting threshold comes with the downside of more false alerts, something that might not be acceptable for certain users.
Under these challenges, assessing the feasibility of an EEWS, in terms of delivering its intended benefits, becomes crucial for both stakeholders seeking actionable information and scientists trying to design the system in the most effective way. To answer this need, a substantial number of feasibility studies have been undertaken in recent years to assess EEWSs in different parts of the world. However, as the EEWS performance is bound to depend on a multitude of uncertain factors, such efforts have, for the most part, understandably targeted individual system characteristics that are more testable, or adopted a narrower scope in their assessment. This usually translates to using proxy targets such as satisfactory alert accuracy (Pinsky, 2017), size of no-alert zones (Kuyuk and Allen, 2013;Picozzi et al., 2015), length of warning times (Bo¨se et al., 2022;Meier et al., 2020), or composite indices combining the above and/or other proxies (Cremen et al., 2022). Many of these studies have, in fact, been instrumental in enhancing particular EEWSs or improving the EEW science in general; yet, they do not provide directly actionable information to decision makers, that is, an assessment of whether EEW can be an effective driver of risk reduction. In this spirit, Auclair et al. (2015) state that ''an early warning should not be considered as an end in itself, and its utility has to be determined with regard to its effective use for risk mitigation.'' That said, several authors have taken a damage-or loss-oriented look at EEW, although usually from the viewpoint of real-time decision making (Cremen and Galasso, 2021;Iervolino, 2011;Iervolino et al., 2006Iervolino et al., , 2007Mitrani-Resier et al., 2016;Picozzi et al., 2013;Wu et al., 2013), rather than of assessing the long-term benefit. Exceptions to this are the studies of Veneziano and Papadimitriou (2003) and Le Guenan et al. (2016). The former proposed an EEWS optimization targeting the minimization of derailments of the highspeed Tohoku Shinkansen trains in Japan, as well as the reduction of unnecessary train delays due to false alarms. Le Guenan et al. (2016) developed a framework to quantify and optimize the long-term utility of an EEWS in the case of a toll bridge. While both studies investigated particular applications, their analyses aimed at assessing and/or optimizing the long-term EEW impact. One attempt to assess more wide-ranging effects of EEWS is the work of Bouta et al. (2020), who proposed a cost-benefit analysis for the implementation of an EEW system in Washington. Their approach involved several simplifications, for example, in the modeling of false/missed alerts, or by assessing a set of earthquake scenarios rather than incorporating a fully probabilistic modeling of earthquake occurrence. Regardless of these simplifications, the authors set up a promising framework for evaluating EEW benefits for utilities (electricity and water), essential facilities (hospitals and fire stations), or for the general public through personal actions.
In this work, we piece together concepts and methods of some of the aforementioned studies and the general framework used in earthquake risk modeling to develop a transparent, customizable, and probabilistic approach for quantifying the potential EEW-related risk reduction. We focus here on the reduction of casualty (injuries and fatalities) risk due to personal action, but our approach could also be applied to assess other EEW benefits. In the following, we: 1. Present and detail the overall proposed framework, 2. Offer a demonstration using Switzerland as a case study, and 3. Explore the sensitivity of the estimated EEW effectiveness to different uncertain and possibly dynamic in time parameters and assumptions.
A risk-based framework to assess the effectiveness of EEW Adopting a risk-based view when assessing EEW effectiveness allows an evaluation of the long-term EEW-related benefits that explicitly accounts for all the uncertainties that are inherent in the earthquake generation process, the ensuing strong ground motion, structural damage, and associated human losses. The following sections describe the main components of our methodology. In brief, it includes the modeling of earthquake risk, the computation of the event loss table (ELT), the determination of alert triggers for each event, and the estimation of expected event warning times to be used for the calculation of EEW-related casualty reduction ratios (CRRs). The latter can be used to update the ELT to reflect the effect of EEW, and derive traditional risk metrics, such as the average annual loss (AAL) or the so-called probable maximum loss (PML) curve (Figure 1), which associates a given level of loss to a return period of exceedance (see the ''Performance evaluation'' section). Note that in the context of this work, the term loss refers to human casualties, and more precisely to injuries and fatalities.

Earthquake risk modeling
Earthquake risk models typically rely on three main components: (1) a seismic hazard model, (2) an exposure model, and (3) a seismic vulnerability model. The seismic hazard model, roughly speaking, provides the frequency and spatial distribution of damaging earthquakes across the region of interest, as well as the joint-across-sites-of-interest probability of occurrence of some level of ground shaking intensity given an earthquake rupture of certain characteristics. The exposure model describes the quantity, value, spatial distribution, and structural characteristics of the various assets at risk (e.g. building stock, road networks, lifelines). Finally, the vulnerability model relates the level of ground shaking intensity to direct, indirect, or human losses for each type of asset in the exposure database. Due to the correlated nature of the loss experienced by different assets, a probabilistic event-based procedure is used to assess seismic risk when dealing with a spatially distributed exposure Crowley and Bommer, 2006;Mitchell-Wallace et al., 2017). This involves the following steps: 1. The generation of an n-year long stochastic earthquake catalog. The earthquake events are sampled using the rates and rupture characteristics defined in the hazard model. The length n of the catalog needs to be sufficient for ensuring statistical stability of the results (Silva, 2018). The above assumes a time-independent Poisson model for earthquake occurrence. Alternatively, m n-year long catalogs could be generated, where the length n is dictated by the variable of interest and the number m selected to ensure statistical stability. 2. Using a ground motion or intensity prediction equation (IPE), a random field of the chosen intensity measure (IM) across all exposure sites is simulated for each rupture generated in the previous step. The result is a possible realization of the spatial IM footprint for every rupture. 3. For each earthquake, the simulated IM values at the asset locations are fed into the relevant vulnerability functions to obtain the corresponding asset losses. If of interest, aggregated results (either for the entire exposure, or for a subset such as a particular city) can be computed from the sum of event losses obtained for the respective assets. The result is a so-called ELT (Mitchell-Wallace et al., 2017) containing the total event losses for the specified collection of assets (e.g. a given city) for each event in the catalog (Table 3). This sample of losses that comprises the ELT can then be processed to obtain any risk metric of interest such as the AAL or the PML curve.

Earthquake risk modeling under operational EEWS
The procedure described above can be employed, as is, for the assessment of seismic risk in the absence of an EEWS. An operational EEWS is expected to limit the losses induced for a given ground shaking intensity level. To this end, we adopt a practical adjustment that can be easily implemented on top of existing risk models, and be tuned in a transparent manner to reflect a range of different assumptions and expectations regarding the operation of the EEWS.
To estimate the proportion of the population that would be able to successfully take protective action after receiving a warning, we adopt the approach proposed by Bouta et al. (2020), which relies on three key probability estimates: P s : the probability that an intended recipient receives and notices the warning message. P r : the probability that a recipient responds to the warning message, and P(CAjt w i ): the probability of casualty avoidance (CA) as a function of warning time t w i available for earthquake i at the area of interest Bouta et al. (2020) refer to these three parameters as sender-driven forces, receiverdriven forces, and value-of-warning-time. The percentage of the exposed population that receives the warning, acts upon it, and succeeds in protecting itself, herein termed as casualty reduction ratio (CRR), can therefore be computed as: This implies that the number of casualties C i alert for a given earthquake i, when an alert is issued, can be computed as: where C i 0 represents the number of casualties for earthquake i in the absence of EEW and can be obtained from the ELT. The above assumes that alerts are (1) always correctly issued and (2) that people are in a position to react to them. However, if an earthquake occurs while people are asleep, a timely response to an alert will be likely hard to attain. Therefore, we estimate the number C i EEWS of casualties for earthquake i in the presence of an operating EEWS as: where F i alert , F i day are flags defined as follows: Note that an alert is likely to have some limited effect even during the night; nevertheless, here we opt for a binary flag for simplicity. To do so, we assume that people sleep on average 7 h per day. Therefore (given the time-independence of earthquake occurrence), F i day can be simulated as 0 or 1 with 7/24 and 17/24 odds, respectively, for each earthquake.

Estimation of CRRs
The parameters involved in the computation of CRR, introduced in the previous section, are subject to many uncertainties. For instance, P s depends not only on the technical infrastructure in place to deliver warnings in time, but also on behavioral and social factors. P r, on the contrary, is heavily influenced by cognitive and social factors. Both P s and P r are also not static in time. P s can be increased through infrastructure upgrades that improve broadcasting reliability, or by introducing additional channels for delivering the alerts, so that more people can be immediately made aware. P r could increase as a result of communication campaigns, training, and experience, while it could decrease following frequent false alarms that undermine public trust in the system. The probability of CA given warning time P(CAjt w i ) could also depend significantly on a range of factors, while the empirical data to constrain it are scarce. Establishing a general base value requires data and may differ based on the local construction practice, time of the day when the earthquake occurs, population density, and various social factors. While some estimates have been proposed for these parameters (Bouta et al., 2020), they must be taken with caution, especially if they are to be applied to a country without a mature EEWS in place. Nonetheless, one can investigate the potential of EEWSs by adopting reasonable lower and upper bounds of these parameters. The sensitivity of the proposed framework on each of the three key parameters can also offer indications on the impact of policy actions, such as investing in the education of the public to recognize EEW alerts and to take the recommended actions in the event of an earthquake. Bouta et al. (2020) use the range of parameters shown in Table 1 for their analyses, as identified from literature relevant to EEW, and also to warnings for other perils such as tsunami or wildfires.
In accordance with Bouta et al. (2020), we use P s and P r values of 80% and 65% as base estimates for application to our testbed in Switzerland in a following section. However, these can be mostly seen as target values, rather than as a realistic current day assessment, given that the estimates in Table 1 have been proposed for Japan that operates a mature, well-known, and widely adopted EEWS. For what concerns the modeling of P(CAjt w i ), aside from the linear form used by Bouta et al. (2020), Fujinawa and Noda (2013) leverage upon past research in Japan and suggest fatality reductions of 25%, 80%, 90%, and 95% for lead times of 2, 5, 10, and 20 s, respectively. Given that that these estimates appear rather optimistic, here we also propose an analytical solution by combining literature data with some critical judgment and assumptions. More precisely, we assume that people, who have received the alert and are willing to take protective action (i.e. the proportion of the population after application of P s and P r in Equation 1), will either perform an attempt of evacuation (AE) or an attempt to seek cover according to the DCHO recommendations (ADCHO). This of course assumes that a proper education of the public about the DCHO protocol has taken place. Accordingly, P(CAjt w i ), effectively equivalent to the probability of CA given the available warning time t w i , can then, using the total probability theorem, be estimated as: Equation 7 can be further expanded as follows: where SDCHO and SE denote the successful outcomes of DCHO and evacuation, respectively. Although this may depend on the EEWS and the information delivered to alert recipients, here we assume that recipients are not provided with the lead time and hence take the probabilities P(ADCHOjt w i ) and P(AEjt w i ) equal to P(ADCHO) and P(AE), respectively. In the following paragraphs, we establish representative values for the  (2008); Fujinawa and Noda (2013) probability terms of Equation 7, by going through available literature data and using judgment when necessary. Base values that we use in our case study application for Switzerland are reported in Table 2, along with value ranges that we deem plausible and explore later on as part of a sensitivity analysis.
The initial reaction of people to earthquake warnings (Becker et al., 2020;Nakayachi et al., 2019) or ground shaking (Lindell et al., 2016;Prati et al., 2012Prati et al., , 2013 have been studied through surveys. The proportion of people that attempt to evacuate ranges from 2% to roughly 38% in the studies we identified, while about 30%-40% (see the study by Lindell et al., 2016) either ''freeze'' or take no action (which is generally consistent with the P r value range used by Bouta et al.). After adjusting the probability space to individuals that respond to the warning (i.e. do not freeze or choose to do nothing), we can estimate the probability of attempted evacuation P(AE)-given that a protective action is taken-to range anywhere in the 3%-63% range. For our case study, we choose a 30% value, while we assign the remaining 70% to attempted DCHO. This assumes that an effective outreach campaign has succeeded in altering public responses to largely comply with the general recommendations.
The probability of successful DCHO (i.e. the probability that an individual has had enough time to drop and cover) as a function of warning time P(SDCHO|t w i , ADCHO) is Probability of successful DCHO given attempted DCHO and warning time modeled according to the suggestions of Porter and Jones (2018) assuming a lognormal distribution with median warning timet w DCHO needed to DCHO of 8.8 s and logarithmic standard deviation of 0.4. To compute the aforementioned probability, the cumulative distribution function (CDF) F SDCHO (t w i |ADCHO) of the lognormal distribution is used. Similarly, we use a lognormal distribution and its CDF F SE (t w i , AE) to model the probability of successful evacuation given warning time, t w i , as P(SE|t w i , AE). Given the lack of relevant data, we take here a median evacuation timet w evac of 25 s as base value, which appears realistic for application to a country like Switzerland with predominantly low-and midrise buildings. We also investigate values ranging from 15 to 45 s (Figures 2b and c and 7).  Table 2.
We set a logarithmic standard deviation of 0.8, higher compared to what was assigned for DCHO, to reflect the dependence of evacuation success on factors such as the size or the height of buildings.
Finally, the success rate of actions like DCHO and evacuation in preventing casualties needs to be set. Admittedly, the actual effectiveness of DCHO is unknown and can only be inferred by looking into the common causes of casualties claimed by earthquakes. Porter and Jones (2018) came up with an upper bound by assuming that all injuries pertaining to nonstructural objects, falls, and human behavior can be prevented. A similar approach is adopted here to establish a range of reasonable values with respect to the effectiveness of successful DCHO in preventing injuries and fatalities. Reviewing literature data (Johnston et al., 2014;Mahue-Giangreco et al., 2001;Peek-Asa et al., 1998;Petal, 2004;Shoaf et al., 1998;Spence and So, 2009;Wagner, 1996;Yeow et al., 2020), we decided on the value ranges reported in Table 2 as realistic estimates. The reader is referred to the Electronic Supplement for further details.
For what concerns the effectiveness of successful evacuation, we use our best judgment to assign probabilities of preventing fatalities and injuries. Evacuation could arguably be an effective strategy in parts of the world with predominantly low-rise construction. However, in most other cases it is not a recommended action, as it is likely to increase the risk of injury while one attempts to exit the building. If successfully conducted, that is, if an individual has had enough time to escape, we assume a 99.9% probability of preventing fatality. The remaining 0.1% is meant to represent risky behavior under panic (e.g. escaping from balconies) that might compromise the outcome. The effectiveness of evacuation in preventing injuries is harder to judge. On one hand, the rush to escape during strong ground shaking might increase the likelihood of injuries related to falls, even if the evacuation succeeds. On the other hand, the headstart granted by the alert (depending on the length of warning time which will vary by earthquake) might be enough to allow some people to move to safer places and potentially prevent injuries resulting from falling objects. Based on this, a wide range of probability estimates seem reasonable for injury avoidance given successful evacuation. However, for our base case that assumes 30%/70% proportion of evacuation/DCHO attempts and median evacuation time of 25 s, it appears that the probability of injury avoidance is dominated by the DCHO component (Figure 2c), and therefore the value assigned to P(CAjt w i , SE) has minimal impact. Factoring this in, we set a base value of 70% for P(CAjt w i , SE), while some sensitivity checks are also made using values ranging from 40% to 90%. Figure 2 illustrates the P(CAjt w i ) curves and how they are affected by the different probability estimates discussed above, as well as how they compare with the values used by Bouta et al. (2020) and Fujinawa and Noda (2013).

Estimation of event-site-specific warning times and alert triggers
Longer warning times provide recipients with more opportunity to take protective action and have a larger impact on casualty reduction (see previous section). For each rupture in the generated stochastic catalog, the expected warning time at each site of interest can be computed in line with the relevant algorithms employed by the EEWS. This estimate should reflect the time difference between the theoretically predicted P-wave arrival at a number of seismic sensors needed to trigger the EEW algorithm applied and the predicted S-wave arrival at the target exposure, while also accounting for data latencies and computational delays of the algorithm (Bo¨se et al., 2022). The computed warning times for each event-site combination are fed into Equation 7 to compute the relevant P(CAjt w i ).
Another critical component in the assessment of EEWS performance is accounting for the likelihood that an alert is missed. Herein, this is achieved through the flag F i alert introduced previously. Note that by missed alerts, we do not refer to cases in which the alert arrives late (no-alert zones). The latter are captured through the insufficient or negative warning times computed. Instead, we refer solely to the cases in which the EEWS falsely determines that an alert should not be issued because the expected ground motion intensity is too low.
When an earthquake occurs and is detected by the EEWS, an alert is typically only issued if certain criteria are met. This is meant to prevent over-alerting, that is, issuing warnings for small earthquakes that are unlikely to cause damage and casualties. Several alternative alerting criteria have been adopted in the literature (Iervolino et al., 2006;Minson et al., 2019;Pinsky, 2017;Zollo et al., 2010), the most common of which rely on either ground motion/macroseismic intensity I thresholds, or magnitude-distance windows (Cremen and Galasso, 2020). Here, for each earthquake in the stochastic catalog, we can determine whether an alert is issued or not, according to chosen alerting criteria. In the case of intensity thresholds, the rupture of earthquake i and its distance to the site are used along with a ground motion or macroseismic intensity prediction equation to obtain either the expected value E(I) or the probability of exceedance of a certain intensity level (Saunders et al., 2020). If the obtained value is above the assumed trigger threshold, then an alert is issued (F i alert = 1), otherwise it is not (F i alert = 0). In the case of magnitudedistance windows, alerts at a given site are issued (F i alert = 1) when an earthquake that falls within a predefined magnitude-distance window has been detected. Note that the above assumes that the earthquake source is accurately determined by the EEWS, therefore missed alerts result only from a mismatch between expected and simulated intensity. Uncertainty in the detection or characterization of the earthquake source at the time of the event could also be accounted for (it is well known that the source characterization of a large earthquake with long rupture duration can take significant time). Such an extension is not explored here as we focus on the application of our framework to Switzerland that comprises a moderate seismicity region; therein we expect the accuracy of alerts to be dictated by the variability in ground shaking intensity (Iervolino et al., 2009). Table 3 presents an example of an ELT with the EEWS-related variables described in the previous sections. The values shown are just indicative. According to the proposed workflow, the F i alert =F i day flags, warning times t w i , and associated CRR i (t w i ) are assessed for each event i in the original ELT and used to adjust its casualty estimates (Table 3, Figure 1).

Performance evaluation
The performance of an EEWS can then be evaluated by contrasting risk metrics in its presence and absence. We look at two risk products, namely the AAL and the PML curve, with loss here referring to the number of casualties. As the name suggests, AAL is the expected (mean) number of earthquake-induced annual casualties, and can be computed by summing up all the loss values in the ELT and dividing the sum with the length n (in years) of the stochastic catalog used to generate the ELT. The PML curve associates a level of loss to a return period of exceedance. If the ELT event losses are ordered from high to low, the return period of the exceedance of a loss value equal to one of the xth event can be computed as n/x (Mitchell-Wallace et al., 2017). The envisaged comparison entails computing the AAL and PML from the original ELT (that assumes no EEW effect) and the new ELT, whose loss values have been adjusted according to Equation 3.

EEWS and earthquake risk model for Switzerland
As a demonstration of the methodology, we apply our approach to Switzerland, a country exposed to moderate seismic hazard (Wiemer et al., 2016). The Swiss Seismological Service (SED) at ETH Zurich operates a non-public EEW demonstration system for Switzerland (Massin et al., 2021) that is based on a low-latency seismic network of 300 permanent stations and uses the Virtual Seismologist (Cua et al., 2009) and FinDer (Bo¨se et al., 2018) EEW algorithms. Depending on their respective configuration, these algorithms require station triggers (i.e. P-wave arrivals) at four to six stations. Efforts are underway to include a single-sensor based EEW algorithm in the near future. In this work, warning times are computed at the centroid of each municipality of interest as the time difference between P-wave arrival at the required number of stations (herein, we investigate all one-, four-, and six-station cases) and the S-wave arrival at each site of interest. To this end, we adopt P-and S-wave velocities of 5.8 km/s and 3.3 km/s, values that are deemed representative for Switzerland (Diehl et al., 2021). We add 2 s to this estimate to reflect expected latencies of the operated seismic stations (Massin et al., 2021). As mentioned earlier, our analysis assumes that each algorithm performs perfectly well and determines earthquake source parameters without any errors. According to the theoretical study of Trugman et al. (2019), it takes less than 3 s to characterize earthquakes with M \ 6.5, and 4 s for 6.5 \ M < 7.0. Since our analyses are mostly conducted for EEW algorithms requiring four to six stations, we consider it a valid assumption that the nearest station has recorded sufficiently long data when the P-wave reaches the fourth or sixth closest station. Furthermore, we mainly adopt a low alert trigger threshold (intensity 4.0), which is likely to be exceeded even if the magnitude is initially underestimated. For more details, the reader is referred to the study by Bo¨se et al. (2022), in which the available warning times for 11 large Swiss cities are systematically analyzed for different loss classes and different EEWS configurations using the same stochastic earthquake catalog as used here. For the triggering of alerts given an earthquake rupture, for most calculations, we use an intensity-based threshold, assuming an alert will only be issued if the predicted mean intensity E(I EMS-98 ) is above 4.0 at the site of interest, where I EMS-98 is the EMS-98 macroseismic intensity (Gru¨nthal, 1998). We also explore a few alternative options, based on either I EMS-98 thresholds or magnitude M-epicentral distance R epi windows (Table 4), and study their impact on risk reduction. Of course, more or less conservative alerting criteria can be adopted depending on the user preferences.
For our risk assessment, we leverage upon preliminary data sets currently being developed for the Earthquake Risk Model Switzerland ERM-CH (Roth et al., 2018) that is expected to be released in 2023, and assemble a simple risk model to run with the OpenQuake engine (Pagani et al., 2014). A brief description of the model used here is given below, while some additional material is provided in the Electronic Supplement. Note, however, that at the time of writing, the data sets underpinning the model remain preliminary, and the focus here is placed primarily on demonstrating the proposed methodology. Therefore, while we do expect that certain trends and relative differences reported here would also apply using the final model, caution should be taken.
The earthquake rupture forecast is adopted from the Swiss national seismic hazard model SUIhaz2015 (Wiemer et al., 2016). For the sake of simplicity, we collapse the various branches of the logic tree into a single average one and we further adopt a minimum magnitude of 5. Ground shaking is estimated in terms of EMS-98 macroseismic intensity (Gru¨nthal, 1998). To this end, we test two alternative models: (1) an IPE developed for Switzerland (Fa¨h et al., 2011), hereinafter referred to as ECOS09, and (b) the combination of the Switzerland-specific ground motion prediction equations (GMPEs) of Edwards and Fa¨h (2013) for Alpine and Foreland regions and assuming a 60 bar stress drop and the ground motion intensity conversion equation (GMICE) proposed by Faenza and Michelini (2010), hereinafter referred to as FM. The second model (FM) allows a differentiation between the alpine and foreland regions which are thought to differ in terms of ground motion attenuation with distance. More precisely, a shallower decay of ground motion amplitudes is predicted in the Swiss foreland as opposed to the alpine regions where the decay is predicted to be steeper. More details on the intrinsic differences and impact of the two models on the risk estimates can be found in the study by Bo¨se et al. (2022). A two-sigma truncation is applied on the modeled normal distribution to avoid sampling unrealistic intensity estimates. The mean intensity given by the models is amplified to reflect local site conditions according to the amplification model currently in use for the ShakeMap system in Switzerland (Cauzzi et al., 2014). No spatial correlation of ground motion residuals is considered.
We employ an exposure model for the residential and commercial Swiss building stock (industrial buildings are excluded from this analysis) underpinned by an extensive geolocalized building database compiled by the Swiss Federal Office for the Environment Macroseismic intensity E(I EMS-98 ) ø 3.5, 4.0, 5.0 Magnitude-distance windows M ø 6.0, R epi < 250 km or M ø 5.5, R epi < 200 km or M ø 5.0, R epi < 150 km (BAFU). The number of occupants in each building is set to a time-averaged fixed number, that is, it is assumed time-invariant for this work. The original data set that numbers over 2 million buildings is subsequently aggregated on a 1 km 3 1 km grid to ease the computational cost of the calculations. The lateral load-resisting system of each building, not included in the aforementioned database, is randomly assigned using statistics obtained from ground surveys (Diana et al., 2019;Lestuzzi et al., 2016). Vulnerability curves in terms of fatality and injury loss ratio to macroseismic intensity were derived by integrating fragility functions for each building typology with a human loss consequence model. The fragility functions were developed largely based on the framework proposed by Lagomarsino and Giovinazzi (2006), adjusted to the Swiss building stock, while the consequence model was obtained by combining the HAZUS (Federal Emergency Management Agency (FEMA), 2010) and Spence (2007) models. The adopted taxonomy, as well as plots of the employed vulnerability functions, is given in the Electronic Supplement.

Results
In the following, we quantify the reduction of the estimated risk from operation of an EEWS. Except for when stated otherwise, any results presented in this and the following section are computed using the ECOS09 IPE, the four-station algorithm for event detection, a I trigger EMSÀ98 = 4:0 alerting threshold, and the base values of the parameters given in Table 2. Figure 3 contrasts the injury exceedance curves for the municipality of Zurich with and without an operating EEWS in place. Conditional to the assigned parameter values, this analysis suggests that operating an EEWS in Zurich could reduce the injury risk estimates by about 20%-25% for short return periods and by 15%-20% over longer return periods. A second observation is that any warning time improvement from using fewer stations for the event detection confers some but not significant gains in risk reduction. The estimated impact of the number of stations remains consistent across different cities in Switzerland that we tested, but not shown here for brevity. Note that any potential increase in missed alerts due to inaccurate source characterization when using a single-station algorithm is not reflected in these results, where we assume the source is immediately and correctly characterized by each algorithm. Figure 4 provides an overview of the estimated EEWS-related reduction in injury and fatality exceedance by return period for each of 11 large Swiss municipalities (see map in Electronic supplement), analyzed using both the ECOS09 and FM ground shaking intensity models and assuming a four-station EEWS configuration. We only present casualty reduction estimates for return periods for which the original estimate amounted to at least one casualty. Moreover, we should note that the relatively small fluctuations at long return periods are likely to be the result of insufficient convergence, that is, smoother but not too Results are provided using two different models to predict intensity (ECOS09 and FM). Reduction estimates are reported only for return periods associated with exceedance of at least one casualty. different estimates would be expected had we simulated more stochastic catalogs. A few noteworthy observations are summarized below: Fatality reduction is estimated at around 8%-10% for most cities and return periods, while injury reduction is generally in the order of 15%-20%. Both fatality and injury reduction rates have a decreasing trend with increasing return period. This is expected, as the events associated with higher casualties are most likely earthquakes occurring in close proximity to each city; therefore, warning times are short or even zero. The aforementioned trend is even more evident for cities within regions with higher seismic activity (see Basel and Sion). These cities are more likely to be affected by earthquakes nearby rather than by earthquakes further away. Overall, the ECOS09 and FM models give similar results, albeit some differences are evident. Most importantly, the casualty reduction is lower (or even zero for long return periods) with the FM model in cities affected more by alpine seismic sources (e.g. Sion). For alpine sources, the intensity attenuation with distance of the FM model is much faster (and as a result, the most damaging earthquakes are likely to occur near the city and allow for only minor if any warning time).
Similarly, Figure 5 presents the estimated average annual injuries and fatalities for each city, with and without EEW. As also reflected in the casualty exceedance curves in Figure 4, we observe a marked disparity among the various cities. In Zurich for instance, when using the ECOS09 IPE, average annual injuries decrease by 15.8% with EEW, while average annual fatalities decrease by about 7.5%. On the contrary, in Basel, the reductions are in the order of 6.6% for injuries and of 1.9% for fatalities. The average annual casualty estimates are primarily influenced by frequent (low return period) events, for which the casualty estimates in both the EEWS and no-EEWS cases are zero. This results in a generally lower estimated reduction compared to what was found for casualties at moderate and higher return periods.

Parameter sensitivity
Next, we investigate the effect of some of the modeling choices described earlier. In Figure 6, we compare casualty reduction estimates in each of Sion and Zurich computed after assigning different values to P(CA|t w i , SDCHO), as well as when the approaches of Bouta et al. (2020) and Fujinawa and Noda (2013) are adopted for the computation of P(CAjt w i ). These plots indicate that if the effectiveness of successful DCHO in preventing injuries and fatalities is indeed within the 0.5-0.8 and 0.1-0.4 ranges that we deemed realistic, a maximum swing of about five percentage units is obtained when comparing the two extreme bounds. In addition, as expected, our predictions with Equation 7 are more conservative compared with those obtained using the recommendations of Fujinawa and Noda (2013) or the upper bound case of Bouta et al. (2020).
In Figure 7, we explore the impact of the median time to DCHOt w DCHO and evacuatioñ t w evac , of the probabilities P s and P r , as well as of the ratio of DCHO and evacuation attempts. In each case, all the other parameters are kept equal to the base values given in Table 2. It appears that the median evacuation time has a generally indiscernible impact on the estimated casualty reduction (unless extreme values are chosen), partly because of the choice to assign P(AE) = 30% and partly because of the generally longer times needed to perform this action. Furthermore, the impact of evacuation time modification appears to saturate if very long evacuation times (e.g. . 30 s in Figure 7) are needed, presumably as warning times are unlikely to be sufficient regardless of the exact chosen value, and any residual casualty reduction comes mainly from DCHO (along with some contributions from the tail end of the evacuation time distribution). This is also in line with Figure 2b, where the increments between the curves become smaller, as we increase the median evacuation time. A sensitivity check where we varyt w evac together with P(AE) and P(CA|SE) is given in the Electronic supplement. The median time to DCHO seems to have a non-negligible effect on injury reduction, but less so on fatality reduction. The latter is a result of our assumption of low effectiveness of DCHO in preventing fatalities, that is, P(CA|SDCHO) = 25%.
As expected, P s ÁP r has a sizable and linear effect on casualty reduction as it increases the pool of alert recipients that are likely to benefit from the warning. Therefore, successful dissemination of the alerts on one hand and education and/or training of the public on the other could arguably yield a significant increase of the system's effectiveness by pushing up the P s and P r values (on top of also reducing DCHO and evacuation times). Finally, the ratio of DCHO and evacuation attempts has an opposite trend on injuries and fatalities. It should be stressed though that this is a product of our modeling choice to assume P(CA|SDCHO) = 25%, which is not based on actual data. These trends are also seen for other cities (see Electronic supplement). Moreover, given that we do not account for delays in source characterization, a sensitivity plot where we decrease the available warning times is also provided in the Electronic Supplement. There, we can see that longer warning times naturally reduce any EEW-related benefit, highlighting the (1) importance of accurate determination of warning times in more thorough studies than the example provided herein and (2) the potential value of dense seismic networks and efficient algorithms.  Table 4. Similar plots for other Swiss cities can be found in the Electronic supplement. As already detailed, the effect of the choice of alerting criteria and the associated frequency of missed alerts is expressed through the flag F i alert , which essentially sets the casualty reduction to zero for earthquakes that do not trigger an alert. Setting more conservative alerting thresholds minimizes the number of these events, which then translates to higher casualty reduction metrics, and therefore better apparent EEWS performance. Based on Figure 8, it appears that if the triggering threshold is set reasonably low, the effectiveness of EEWS, as quantified here, is not significantly compromised. This is because casualties are mostly expected to occur at intensities that are much higher than commonly set triggering thresholds.
However, lower alerting thresholds also lead to more frequent false alerts, which can be problematic. On one hand, they can be disruptive if they occur too often. On the other hand, they could also either dissuade a range of users from adopting EEW protocols or at the very least reduce the public's confidence and hence responsiveness to alerts. Finally, they could even result in some casualties due to panic reactions, something that was deemed rare enough to not be modeled in this work. That said, surveys in countries with operational EEW systems like Japan and Mexico, indicated a strong support of the public even when carried out after earthquake sequences characterized by false, delayed, or missed alerts (Allen and Melgar, 2019). This bodes well for a sustained trust of the public in EEW systems. In any case, it is important to complement the risk-reduction performance assessed here with an assessment of the EEWS accuracy. To find the proper balance between missed/false alerts, analyses such as those described by Meier (2017) and Minson et al. (2019) can be conducted to support the choice of an alerting threshold.

Limitations
The primary objective of the case study was to serve as a proof-of-concept application and illustrate the potential of adopting a quantitative approach in assessing EEW systems. Therefore, as stated in previous sections, several simplifications were made, the most important of which are also summarized here for better clarity: (1) several parameter values describing human reactions and the effectiveness of specific protective actions were set based on limited information and judgment; (2) the source parameters are assumed to be immediately and correctly characterized once the necessary amount of stations have  Table 4 on injury estimates by return period for the municipality of Zurich. detected the earthquake; (3) the underlying risk model used here is rather simple, relies on macroseismic intensity as IM, and is not validated; (4) the number of occupants in each of the buildings comprising the exposure model is assumed fixed in time; (5) the study focuses only on the public alerting side of EEW, that is, we neglect any possible benefits from customized automated responses. Addressing these limitations for a more robust evaluation of EEW in Switzerland comprises the topic of future work.

Summary and conclusion
This article introduces a quantitative framework for assessing the effectiveness of EEW in mitigating earthquake risk. We rely on concepts widely used for regional event-based probabilistic earthquake risk assessment to compute standard risk metrics, and subsequently employ a logical procedure to adjust them under the hypothesis of an operational EEWS. A comparison between risk estimates with and without EEW can guide decisionmakers in making a case-and application-specific assessment on the effectiveness of a potential EEWS.
We demonstrate the proposed methodology using Switzerland as a case study. The question of EEWS effectiveness is particularly relevant in countries exposed to moderate crustal seismicity like Switzerland, for which the feasibility of EEW has often been questioned (Auclair et al., 2015). While the scope of this work is primarily conceptual and the results conditional on multiple assumptions, our findings suggest that the operation of an EEWS in Switzerland could confer a moderate but significant reduction in injury and fatality risk (in the order of 10% for fatalities 20% for injuries across most return periods of interest). These gains could be expected to increase in other areas that are affected by offshore seismicity that allows for longer warning times. It should also be noted that this study has only focused on public alerting and the personal action side of EEW. User-customized applications and automated responses offer further avenues for risk mitigation.
While any analysis of this kind is subject to numerous uncertainties, our quantitative modeling of the benefits incurred by EEW provides the opportunity to explore the influence of different factors and system design choices. For instance, we demonstrate that public awareness of EEW appears to be crucial to untap its full potential. By including parameters that track the response of individuals when they receive an alert, we can highlight the importance of public outreach efforts and compare their effectiveness in an EEWS alongside factors more commonly recognized as critical, such as scientific and technical improvements. In the context of feasibility studies, such parameters could potentially be seen as targets rather than as uncertain variables, with resources being allocated, for example, to training campaigns. Similarly, what-if analyses can be undertaken in a straightforward manner to provide support on decisions such as the choice of alerttriggering criteria, or the installation of new seismic stations (Bo¨se et al., 2022).
Overall, this study makes the case for a more risk-oriented view when assessing the performance and feasibility of EEWSs. A practical and quantitative framework has been formalized and illustrated for Switzerland. This framework is highly customizable, can be extended to various EEW applications, while it can also serve as a stepping stone for a downstream cost-benefit analysis.

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.