1 Introduction

In the context of earthquake loss modeling, downtime is generally defined as the time between the occurrence of an earthquake and the re-occupancy of a building. It includes both an irrational and a rational component (Cardone et al. 2019). Specifically, the irrational component is the time required to assess the state of the building, take decisions, and mobilize economic and human resources. The duration of this phase depends on the dimension of the area invested by the earthquake, the density of the built environment, and the resources available to perform the inspections. The rational component, instead, includes the time to repair or replace the building, if required. In the time interval between the main seismic shock and the inspection, emergency management is particularly challenging due to the uncertainty of the structural state. However, the decision-maker should promptly decide whether evacuate the building and suspend ordinary activities or not (Han et al. 2016; Thöns and Stewart 2020).

Due to emerging signal processing techniques and improvements in sensing systems that enable the acquisition of high-fidelity dynamic measurements at the occurrence of earthquakes, Seismic Structural Health Monitoring (S2HM) has grown considerably in the last few decades (Limongelli and Çelebi 2019). Specifically, S2HM algorithms generally identify and track structural damage-sensitive features [e.g., inter-story drifts (Dolce et al. 2017), resonant frequencies (Iacovino et al. 2018), or nonlinear normal mode shapes (Quqa et al. 2021)] calculated instantaneously in the time interval of the strong motion. These features can be directly employed to detect and possibly localize damage induced by the seismic event or used to update finite element models (Shiradhonkar and Shrikhande 2011; Bursi et al. 2018). S2HM may be particularly convenient compared to continuous Structural Health Monitoring (SHM) due to its robustness to temperature effects (Limongelli and Çelebi 2019). Although monitoring systems are becoming widespread for the integrity assessment of strategic and monumental structures, it is unclear if the investment, consisting of the cost of monitoring systems, is worthy. Therefore, decision-makers need tools to estimate the benefit of S2HM before its adoption.

The Value of Information (VoI) from Bayesian decision theory can be used for this purpose (Raiffa and Schlaifer 1961; Benjamin and Cornell 1970). It can be used to quantify the amount of money saved by the decision-maker when they use the information from monitoring systems to support their decisions. Therefore, it can be intended as the maximum amount of money that the decision-maker should invest for structural health information (Pozzi and Der Kiureghian 2011; Thöns and Faber 2013; Straub 2014; Thöns 2018).

The literature on the VoI in civil engineering mainly addresses bridges and wind turbine support structures (Zhang et al. 2021). The value of collecting information on the structural health of building to support seismic emergency management operations has been addressed in (Yeo 2005) and (Omenzetter et al. 2016). In particular, (Yeo 2005) applied the classical framework of VoI from Bayesian decision analysis to assess if the collection of information on structural conditions from a visual inspection is financially desirable in managing buildings affected by earthquakes. More recently, (Omenzetter et al. 2016) discussed the VoI from the joint utilization of SHM and visual inspection information.

In general terms, the VoI is obtained by comparing two situations, namely the situation in which the decision analysis is carried out using the available information on the state of the building at the time of the decision, the so-called “Prior” decision analysis, and the situation in which the decision analysis is performed considering the new information (before having it), referred to as “Pre-Posterior” decision analysis. Both decision analyses consist in the selection of the optimal action (e.g., between “Evacuate” or “Do not evacuate” the building), that is the one entailing the minimum expected costs. Usually, these costs depend on the probability that the damaged structure fails after selecting the action. The new information is modeled through likelihood functions, which express the probability of observing a certain monitoring outcome when the structure is in a given damage state.

In the case of seismic emergency management (Giordano and Limongelli 2020; Iannacone et al. 2021), the prior information on the structural conditions can be retrieved by so-called fragility curves, which provide the probability of reaching or exceeding a specified limit state as a function of the intensity of the damaging event. Considering aftershocks as a possible cause of the failure, the probability of failure can be obtained employing aftershock fragility functions, which express the probability of failure as a function of both the intensity of the damaging event (aftershock) and the state of the structure damaged by the mainshocks. Figure 1 shows the general flowchart of the VoI in this context, in which the decision-maker must make two decisions, namely, either installing or not the SHM system, and, if an earthquake occurs, evacuating or not the building considering aftershocks as a possible cause of the failure. If an S2HM system is installed, it can support decision-making in the aftermath of the earthquake; otherwise, the decision-maker selects the optimal emergency management action using their available knowledge. The main elements of a VoI analysis in this context are highlighted in the figure, i.e., the (mainshocks and aftershocks) fragility functions of the considered building, the seismic hazard model of the region, the direct and indirect costs of different decision alternatives (e.g., failure/loss of functionality), and the likelihood functions in the considered damage states.

Fig. 1
figure 1

VoI flowchart

Full size image

This paper intends to fill the above-mentioned research gap in the VoI literature by providing a comprehensive framework to quantify the VoI from S2HM in building management and investigate in which cases the monitoring information provides the maximum benefit in this context. The methodology is demonstrated using a reference case study consisting of a Reinforced Concrete (RC) building located in a seismic area. Three different heights of the building have been considered to investigate the sensitivity of the VoI to this parameter. As previously remarked, the estimation of the prior probabilities of the damage states requires the availability of aftershock fragility curves. Their evaluation has received limited attention in literature thereby the relevant literature on the topic is shortly overviewed.

The paper is organized as follows. Section 2 introduces mainshock and aftershock fragility functions, Sect. 3 presents S2HM parameters used to assess the performance of buildings in seismic prone areas. Section 4 is devoted to the description of the theoretical framework of the VoI. The reference case study is analyzed and discussed in Sect. 5, including different sensitivity analyses. The final remarks conclude the paper.

2 Seismic fragility

2.1 Mainshock fragility

The decisional framework adopted in this paper to compute the VoI from S2HM assumes that the prior knowledge on the structural conditions of buildings in the aftermath of a seismic event is derived from fragility functions (Kassem et al. 2020; Iacovino et al. 2021) which express the probability that an Engineering Demand Parameter (EDP)—e.g., peak inter-story drift—exceeds a threshold \(EDP_{l}\) related to the \(l\)th damage state of the structure \(DS_{l}\), given the intensity measure \(I_{m}\) of the mainshock. Fragility curves are generally expressed in the form:

$$P\left( {EDP \ge EDP_{l} \left| {I_{m} } \right.} \right) = \Phi \left[ {\frac{1}{{\beta_{tot} }}\ln \left( {\frac{{I_{m} }}{{I_{{DS_{l} }} }}} \right)} \right]$$
(1)

where \({\Phi }\) is the cumulative distribution function of the standard normal distribution, \(I_{{DS_{l} }}\) and \(\beta_{tot}\) are respectively the median value and the dispersion of the structural capacity associated with the considered limit state (represented by the value \(EDP_{l}\) of the \(EDP\)). In particular, \(\beta_{tot}\) is the total standard deviation which represents the uncertainties in the fragility function, calculated by combining the uncertainty of the demand (intensity measure) and capacity. This latter dispersion includes model, material, and geometric uncertainties.

The intensity measure is quantified in the literature using different intensity parameters corresponding to peak (e.g., PGA, PGV, PGD) or spectral [e.g., \(Sa\left( T \right)\), \(Sv\left( T \right)\), \(Sd\left( T \right)\)] intensity measures. Fragility curves are constructed for selected damage states \(DS_{l}\), identifying different performance levels of the structure. The number of damage states (and consequently the number of limit states) depends on the damage scale used. For example, the HCR (Rossetto and Elnashai 2003) damage scale considers seven different \(DS_{l}\) (none, slight, light, moderate, extensive, partial failure, and failure) whereas five \(DS_{l}\) (no damage, slight damage, moderate damage, extensive damage and failure) are defined by HAZUS99 (FEMA-NIBS 2012).

2.2 Aftershock fragility

Buildings in seismically active regions could be subjected to more than one earthquake or mainshock-aftershock sequence in a short period. Structures that present minor damage after a mainshock can be severely damaged as a result of these sequences. Aftershock fragility curves consider that, at the occurrence of the aftershock, the structure might be already in a damaged state \(DS_{l}\), induced by the mainshock. With specific regard to the limit state of failure, the conditional probability that the structure (which is in a damaged state \(DS_{l}\) after the mainshock) fails due to an aftershock with an intensity measure \(I_{a}\) can be computed as:

$$P\left( {F\left| {I_{a} ,DS_{l} } \right.} \right) = \Phi \left[ {\frac{1}{{\beta_{tot} }}\ln \left( {\frac{{I_{a} }}{{I_{{F\left( {DS_{l} } \right)}} }}} \right)} \right]$$
(2)

where \(I_{{F\left( {DS_{l} } \right)}}\) and \(\beta_{tot}\) are, respectively, the median value and the dispersion of the structural capacity associated with failure when the structure is in a damaged state \(DS_{l}\) at the occurrence of the aftershock. The statistical properties of \(I_{a}\) depend on those of the mainshock.

There is ongoing research to assess the increased seismic vulnerability of damaged structures. To understand the structural behavior under earthquake sequences, some studies use nonlinear Multiple-Degree-of-Freedom (MDOF) models (Li et al. 2014), while many others employ Single-Degree-of-Freedom (SDOF) models (Hatzigeorgiou and Beskos 2009).

In recent research works, fragility relationships are derived by performing nonlinear time history analysis with a sequence of mainshock and aftershock ground motions (Luco et al. 2004; Uma et al. 2011), investigating also the effects of the design approaches (Abdelnaby 2018), the seismic region, the earthquake intensity measure (Hosseinpour and Abdelnaby 2017), the number of stories (Raghunandan et al. 2015).

Table 1 reports the main parameters characterizing the case studies (structural typology, number of stories, location) and the aftershock fragility curves (intensity measure of the aftershock \(I_{a}\), EDP), relating to different studies.

Table 1 Aftershock fragility curves parameters
Full size table

3 Seismic structural health monitoring of buildings: performance parameters

S2HM systems for civil buildings generally measure accelerations at a certain number of stories. Measurements are then processed to extract performance parameters able to provide information about damage states. Recent S2HM approaches extract instantaneous modal-based parameters from accelerations recorded during a seismic event (Aloisio et al. 2021; Ditommaso et al. 2012; Giordano et al. 2020; Limongelli and Çelebi 2019; Quqa et al. 2021), when the structures may experience large excursions in the nonlinear field. Ditommaso et al. (Ditommaso et al. 2021) used the Stockwell transform to monitor the modal curvature variation during seismic events. Quqa et al. used the modal assurance distribution to identify instantaneous nonlinear normal modes (Quqa et al. 2021). However, in general, these parameters can hardly be related to performance measures due to their strong dependency on the specific structure. Also, the same variation of modal parameters can correspond to several different damage types and damage states, with very different safety concerns. For example, a given variation of the first modal frequency may correspond to the settlement of a pillar, the formation of a plastic hinge at a critical section, or the cracking of non-structural elements.

As an alternative, the inter-story drift, obtained by double integration of story accelerations (Dolce et al. 2017), can be generally used as a seismic performance parameter. The advantage of inter-story drift is the possibility to define thresholds corresponding to specific limit states. For instance, American codes, such as FEMA-356 (FEMA 2000) and ASCE/SEI 41-13 (ASCE 2013) defined threshold values of the inter-story drifts corresponding to performance limit states, addressed to as “Immediate Occupancy”, “Life Safety”, “Collapse Prevention”, and “Collapse” (Roohi and Hernandez 2020). On the other hand, the main drawback of inter-story drifts is related to the difficulty of their practical estimation through double integration of acceleration measurements. These measurements are usually affected by noise due to monitoring devices. This drawback may be particularly important when low-cost sensors with a considerable noise floor level are employed. Indeed, acceleration noise (modeled as white noise), after integration, takes the form of a Brownian motion whose variance increases linearly with time (Kaya and Safak 2019). Nevertheless, pre-processing operations, such as bandpass filtering, can be applied to reduce this effect (Dolce et al. 2017). Also, the inter-story drift can be calculated by integrating the contribution of each mode separately (Kaya and Safak 2015), reducing the estimation error.

Assuming the use of suitable pre-processing operations, which are not the main focus of this study, the inter-story drift is here considered as the seismic performance parameter to manage the seismic emergency of buildings. It is assumed that at least two accelerometers are deployed at consecutive levels: for example, where a soft story behavior can be experienced, or at the top and at the bottom of the building to evaluate an average inter-story drift along with the building height.

4 Value of information analysis

The topic discussed in the previous sections, i.e., mainshock and aftershock fragility functions and S2HM outcomes, are incorporated in the framework of the VoI in the following.

The VoI is formulated in the context of the Bayesian decision theory (Raiffa and Schlaifer 1961) which deals with decision-making in uncertain environments and is grounded on the Bayesian interpretation of probability and the expected utility theory (Neumann and Morgenstern 1947). In the Bayesian framework, the probability \(P\left(S\right)\) assigned to a state \(S\) is intended as the degree of belief that such state is true. The expected utility theory by von Neuman and Morgenstern instead relates to the behavior of a decision-maker which, if they behave rationally, should select the action which maximizes their expected utility, i.e., minimum expected costs (Verzobio et al. 2021). The availability of additional information on the state \(S\) can change the degree of belief that such state is true and, as a consequence, the probability assigned to it. Ultimately, the new information can modify the selection of the optimal action.

The VoI is obtained by comparing the expected costs of the actions chosen through two types of decision analysis, the Prior and Pre-Posterior decision analyses. Both decision analyses are carried out before the new information on the state of the building is collected. However, the Prior decision analysis is performed considering only the (prior) knowledge of the decision-maker, that is it does not consider the monitoring information. The Pre-Posterior decision analysis is performed accounting for the monitoring information (“Posterior” to the future acquisition of the information) but before the information is collected (“Pre” with respect to the actual acquisition of the information). The Pre-Posterior decision analysis requires thus the modeling of the future monitoring information.

A framework to compute the VoI for seismic emergency management of bridges was presented in (Giordano and Limongelli 2020). In the following subsections, this framework is described and adapted to the case of buildings.

4.1 Decision analyses

It is assumed that, after a mainshock, the building can be in one of the \(L\) discrete damage states \(DS_{l}\), \(l = 1, \ldots ,L\), that cover the spectrum between no damage/minor damage up to severe damage. The decision-maker disposes of a set of decision alternatives constituted by several possible management actions \(A_{n}\), \(n = 1, \ldots ,N\). “Evacuate” or “Do not evacuate” are two of such decision alternatives. The decision analysis aims at selecting the action corresponding to the minimum expected management cost.

In the ideal case in which the damage state of the building is known with certainty, the decision-maker can compute the expected cost of each action \(A_{n}\), \(E\left[ {c\left( {A_{n} } \right)|DS_{l} } \right]\), which depends on \(DS_{l}\). This expected cost can be computed considering the probability of failure of the building in \(DS_{l}\) and the direct and indirect costs associated with each action \(A_{n}\). Assuming an aftershock as the possible cause of the failure, the expected cost of \(A_{n}\) is conditioned on the damage state \(DS_{l}\) of the building at the occurrence of the aftershock. Since the characteristics of the aftershock depend on those of the mainshock, the expected cost of \(A_{n}\) is also conditioned on the characteristics of the mainshock that will be represented through its magnitude \(M_{m}\) and epicentral distance \(R_{m}\) i.e., \(E\left[ {c\left( {A_{n} } \right)|M_{m} , R_{m} ,DS_{l} } \right]\). This expected cost can be computed as follows:

$$E\left[ {c\left( {A_{n} } \right)|M_{m} ,R_{m} ,DS_{l} } \right] = c_{F} \left( {A_{n} } \right)P\left( {F\left| {M_{m} ,R_{m} ,DS_{l} } \right.} \right) + c_{{\overline{F}}} \left( {A_{n} } \right)\left[ {1 - P\left( {F\left| {M_{m} ,R_{m} ,DS_{l} } \right.} \right)} \right]$$
(3)

where \(P\left( {F\left| {M_{m} ,R_{m} ,DS_{l} } \right.} \right)\) is the probability of failure due to aftershocks (see Sect. 4.4), \(c_{F} \left( {A_{n} } \right)\) and \(c_{{\overline{F}}} \left( {A_{n} } \right)\) are the cost of failure and survival, respectively, which depend on the action \(A_{n}\).

The actual damage state of the building after a mainshock is usually unknown, thereby the decision-maker must consider all the possible damage states \(DS_{l}\) and corresponding probabilities of occurrence \(P\left( {DS_{l} |I_{m} } \right)\), that are conditioned on \(I_{m}\). The prior probabilities can be obtained from mainshock fragility curves (see Sect. 2.1) as follows:

$$\left\{ {\begin{array}{*{20}l} {P\left( {DS_{l} |I_{m} } \right) = P\left( {EDP \ge EDP_{l} |I_{m} } \right) - P\left( {EDP \ge EDP_{l + 1} |I_{m} } \right)} \hfill & {for} \hfill & {l < L} \hfill \\ {P\left( {DS_{l} |I_{m} } \right) = P\left( {EDP \ge EDP_{l} |I_{m} } \right)} \hfill & {for} \hfill & {l = L} \hfill \\ \end{array} } \right.$$
(4)

where \(EDP_{1} = 0\). Considering all the possible damage states, the expected cost of the action can be computed using the total probability rule, as follows:

$$E[c\left( {A_{n} } \right)|{{\varvec{\Theta}}}_{{\mathbf{m}}} ] = \mathop \sum \limits_{l = 1}^{L} E[c\left( {A_{n} } \right)|M_{m} ,R_{m} ,DS_{l} ]P(DS_{l} |I_{m} )$$
(5)

where the vector \({{\varvec{\Theta}}}_{{\mathbf{m}}} = \left[ {M_{m} ,R_{m} ,I_{m} } \right]\) contains the parameters that describe the mainshock. The optimal action \(\hat{A}\) is chosen among all the possible decision alternatives \(A_{n}\) as the one corresponding to minimum expected costs \(c_{1} \left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right)\), as follows:

$$\hat{A} = \hat{A}\left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right) = arg\mathop {min}\limits_{n} E\left[ {c\left( {A_{n} } \right)\left| {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right.} \right]$$
(6)
$$c_{1} \left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right) = E\left[ {c\left( {\hat{A}} \right)|{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right] = \mathop \sum \limits_{l = 1}^{L} E\left[ {c\left( {\hat{A}} \right)|M_{m} ,R_{m} ,DS_{l} } \right]P\left( {DS_{l} |I_{m} } \right)$$
(7)

The suffix 1 in \(c_{1}\) indicates that this expected cost is computed through a Prior decision analysis that is without the support of the monitoring information.

When one of the possible outcomes of an SHM system \(O_{j}\), \(j = 1, \ldots ,J\) is available, the prior probabilities \(P\left( {DS_{l} |I_{m} } \right)\) of the damage states can be updated based on this new knowledge. It should be noted that, in this study, we assume the identified inter-story drift as an observation of the SHM system (see Sect. 3). The updating of the prior probabilities is done through the Bayes’ theorem, namely:

$$P\left( {DS_{l} \left| {O_{j} ,I_{m} } \right.} \right) = \frac{{P\left( {O_{j} \left| {DS_{l} } \right.} \right)P\left( {DS_{l} \left| {I_{m} } \right.} \right)}}{{P\left( {O_{j} \left| {I_{m} } \right.} \right)}}$$
(8)

where \(P\left( {DS_{l} \left| {O_{j} ,I_{m} } \right.} \right)\) is the probability of the damage state \(DS_{l}\) estimated accounting for the monitoring information. The term \(P\left( {O_{j} \left| {DS_{l} } \right.} \right)\) is the likelihood function, that is the probability of observing the outcome \(O_{j}\) when the building is in a damaged state \(DS_{l}\). The denominator \(P\left( {O_{j} \left| {I_{m} } \right.} \right)\) represents the total probability (e.g., computed across all the possible damage states) of observing the monitoring outcome \(O_{j}\) when an earthquake of intensity measure \(I_{m}\) occurs:

$$P\left( {O_{j} \left| {I_{m} } \right.} \right) = \mathop \sum \limits_{l = 1}^{L} P\left( {O_{j} \left| {DS_{l} } \right.} \right)P\left( {DS_{l} \left| {I_{m} } \right.} \right)$$
(9)

For each outcome \(O_{j}\), the decision-maker selects the optimal action \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}_{{O_{j} }}\):

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}_{{O_{j} }} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} \left( {O_{j} ,{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right) = arg\mathop {min}\limits_{n} E\left[ {c\left( {A_{n} \left| {O_{j} ,{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right.} \right)} \right]$$
(10)

The corresponding minimum expected cost is:

$$E\left[ {c\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}_{{O_{j} }} } \right)|O_{j} ,{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right] = \mathop \sum \limits_{l = 1}^{L} E[c\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}_{{O_{j} }} } \right)|M_{m} ,R_{m} ,DS_{l} ]P(DS_{l} |O_{j} ,I_{m} )$$
(11)

According to Eq. 11, an optimal action corresponds to each outcome of the monitoring system. The Pre-Posterior decision analysis is carried out before the outcome is collected thereby the expected cost must be computed considering all the possible outcomes of the monitoring system, each weighted by its probability of occurrence for a given mainshock, \(P\left( {O_{j} |I_{m} } \right)\). The cost \(c_{0} \left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right)\) computed through the Pre-Posterior decision analysis (in this case, the suffix 0 is used) is given by:

$$c_{0} \left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right) = \mathop \sum \limits_{j = 1}^{J} E\left[ {c\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}_{{O_{j} }} } \right)|O_{j} ,{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right]P(O_{j} |I_{m} )$$
(12)

4.2 Value of information

The VoI for a mainshock defined by the vector \({{\varvec{\Theta}}}_{{\mathbf{m}}}\) is computed as the difference between the Prior decision analysis cost, \(c_{1} \left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right)\), and the Pre-Posterior cost \(c_{0} \left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right)\), namely:

$${\text{VoI}}\left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right) = c_{1} \left( {{{\varvec{\Theta}}}_{{\mathbf{m}}} } \right) - c_{0} \left( {\Theta _{{\mathbf{m}}} } \right)$$
(13)

Equation 13 gives the VoI provided by the monitoring system if a mainshock \({{\varvec{\Theta}}}_{{\mathbf{m}}}\) occurs. However, since the VoI is computed when the monitoring system is installed, that is before the mainshock happens, the probability distributions of the parameters \(M_{m}\), \(R_{m}\) and \(I_{m}\) (that characterize \({{{\varvec{\Theta}}}_{{\mathbf{m}}} }\)) must be accounted for. The VoI is obtained as follows:

$${\text{VoI}} = \iiint\limits_{{M_{m} ,R_{m} ,I_{m} }} {{\text{VoI}}\left( {M_{m} ,R_{m} ,I_{m} } \right)f\left( {I_{m} \left| {M_{m} ,R_{m} } \right.} \right)f\left( {M_{m} } \right)f\left( {R_{m} } \right)dI_{m} dM_{m} dR_{m} }$$
(14)

where \(f\left( {I_{m} \left| {M_{m} ,R_{m} } \right.} \right)\) is the Probability Density Function (PDF) of \(I_{m}\) conditioned on \(M_{m}\) and \(R_{m}\), \(f\left( {M_{m} } \right)\) and \(f\left( {R_{m} } \right)\) are the PDFs of the magnitude and the epicentral distance, respectively.

The VoI computed utilizing Eq. 14 relates to a single seismic event. However, the estimation of the VoI must be performed to decide how much can be invested in a monitoring system that will provide information for a period \(T_{LC}\). The VoI must thus be computed over the period \(T_{LC}\), for which monitoring is envisaged. To this aim, the Life Cycle VoILC can be used (Zonta et al. 2014):

$${\text{VoI}}_{{{\text{LC}}}} = \mathop \sum \limits_{i = 1}^{{T_{LC} }} \lambda_{m} \frac{{{\text{VoI}}}}{{(r + 1)^{i} }}$$
(15)

where \(\lambda_{m}\) is the annual rate of earthquake occurrence and \(r\) is the discount rate. The \({\text{VoI}}_{{{\text{LC}}}}\) can be interpreted as the expected reduction in emergency management costs in the period \(T_{LC}\), provided by the SHM system.

4.3 Seismic hazard

In the previous sections, the seismic hazard was integrated into the definition of the VoI through the probability distributions of the parameters of the mainshock and the aftershock. The relevant models adopted herein are provided in the following.

As for mainshocks, a stationary Poisson model is adopted for their occurrence (Anagnos and Kiremidjian 1988). The PDF of the mainshock magnitude, \(f\left( {M_{m} } \right)\), is modelled as truncated exponential function as follows:

$$f\left( {M_{m} } \right) = \frac{{\beta e^{{ - \beta M_{m} }} }}{{e^{{ - \beta M_{m,l} }} - e^{{ - \beta M_{m,u} }} }}$$
(16)

where \(M_{m,l}\) and \(M_{m,u}\) are the lower and upper bounds, respectively, of the mainshock magnitude, and \(\beta = b\ln 10\), where \(b\) is the Negative slope of the Gutenberg-Richter law. An area source model (Baker 2013) is used in which the earthquakes are generated with uniform probability inside the seismogenic area whose geometric properties determine the distribution of the source-to-site distances. The distribution of the intensity measure conditioned on the magnitude and the distance of the earthquake \(f\left( {I_{m} \left| {M_{m} ,R_{m} } \right.} \right)\) is determined using Ground Motion Prediction Equations (GMPEs) available in the literature (Douglas 2011). GMPEs allow accounting for the uncertainty in the physical parameters of fault rupture which have a strong influence on the uncertainties of structural conditions.

As for aftershocks, they are modeled as a nonhomogeneous Poisson process in which their occurrence rate decreases with increasing elapsed time after the mainshock. The magnitudes of the aftershocks are assumed to be bounded between a lower and an upper bound. The lower bound is taken equal to the lower bound of mainshock magnitudes whereas the upper bound is taken equal to the magnitude of the mainshock which caused the aftershocks. The mean number of aftershocks in the period \(\left[ {t;t + T} \right]\), following a mainshock of magnitude \(M_{m}\), reads:

$$N_{a} \left( {M_{m} ,t,T} \right) = \frac{{10^{{a + b\left( {M_{m} - M_{m,l} } \right)}} - 10^{a} }}{p - 1}\left[ {\left( {c + t} \right)^{1 - p} - \left( {c + t + T} \right)^{1 - p} } \right]$$
(17)

where \(a\), \(b\), \(c\), and \(p\) are parameters that depend on the seismic area and \(M_{m,l}\) is the lower bound of aftershock magnitudes.

The PDF of the aftershock magnitude depends on the magnitude of the mainshock, and it is expressed as follows:

$$f\left( {M_{a} \left| {M_{m} } \right.} \right) = \frac{{\beta e^{{ - \beta \left( {M_{a} - M_{m,l} } \right)}} }}{{1 - e^{{ - \beta \left( {M_{m} - M_{m,l} } \right)}} }}$$
(18)

where \(\beta = b\ln 10\).

It is assumed that aftershocks occur with uniform probability in a circular region centered at the mainshock location. The area of this region is expressed as a function of \(M_{m}\) (Utsu 1970), as follows:

$$S_{a} = 10^{{M_{m} - 4.1}}$$
(19)

Hence, the distribution of source-to-site distances for aftershock \(f\left( {R_{a} \left| {M_{m} ,R_{m} } \right.} \right)\) is conditioned on both \(M_{m}\) and \(R_{m}\).

The GMPE used for mainshocks is also used for aftershocks, providing \(f\left( {I_{a} \left| {M_{a} ,R_{a} } \right.} \right)\). The PDF of \(I_{a}\) can be conditioned on \(M_{m}\) and \(R_{m}\) as follows:

$$f\left( {I_{a} \left| {M_{m} ,R_{m} } \right.} \right) = \iint\limits_{{M_{a} ,R_{a} }} {f\left( {I_{a} \left| {M_{a} ,R_{a} } \right.} \right)f\left( {M_{a} \left| {M_{m} } \right.} \right)f\left( {R_{a} \left| {M_{m} ,R_{m} } \right.} \right)dM_{a} dR_{a} }$$
(20)

4.4 Probability of failure

The probability of failure due to aftershocks depends on the conditions of the building after the mainshock and the characteristics of the aftershocks, which in turn depend on those of the mainshock, i.e., \(P\left( {F\left| {M_{m} ,R_{m} ,DS_{l} } \right.} \right)\). First, the probability of failure due to a single aftershock, \(P^{*} \left( {F\left| {M_{m} ,R_{m} ,DS_{l} } \right.} \right)\), must be evaluated, which reads:

$$P^{*} \left( {F\left| {M_{m} ,R_{m} ,DS_{l} } \right.} \right) = \int\limits_{{I_{a} }} {P\left( {F\left| {I_{a} ,DS_{l} } \right.} \right)f\left( {I_{a} \left| {M_{m} ,R_{m} } \right.} \right)dI_{a} }$$
(21)

where \(P\left( {F\left| {I_{a} ,DS_{l} } \right.} \right)\) is the aftershock fragility function (see Sect. 2.2) and the term \(f\left( {I_{a} \left| {M_{m} ,R_{m} } \right.} \right)\) is obtained using Eq. 20. The probability of failure due to a single aftershock is then employed to obtain the mean number of aftershocks \(N_{F} \left( {M_{m} ,R_{m} ,DS_{l} ,t,T} \right)\) that induce the failure of the structure in the time interval \(\left[ {t;t + T} \right]\), as follows:

$$N_{F} \left( {M_{m} ,R_{m} ,DS_{l} ,t,T} \right) = P^{*} \left( {F\left| {M_{m} ,R_{m} ,DS_{l} } \right.} \right)N_{a} \left( {M_{m} ,t,T} \right)$$
(22)

where \(N_{a} \left( {M_{m} ,t,T} \right)\) is the mean number of aftershocks in the period \(\left[ {t;t + T} \right]\) defined by Eq. 17 Finally, the probability of failure in the period \(\left[ {t;t + T} \right]\) is obtained as:

$$P\left( {F\left| {M_{m} } \right.,R_{m} ,DS_{l} } \right) = 1 - e^{{ - N_{F} \left( {M_{m} ,R_{m} DS_{l} ,t,T} \right)}}$$
(23)

5 Application

The theoretical framework of the VoI described in the previous sections is applied to an exemplary case study introduced to investigate the effect of different parameters on the VoI. The following decision problem is considered. The manager of a building in a seismic-prone area is considering the adoption of an S2HM system to support the management of the emergency phase following an earthquake. After the mainshock, the building could be damaged, and the decision-maker must select the optimal action between “Evacuate” and “Do not evacuate” the building. The decision-maker must consider the costs of the two decision alternatives. The evacuation of the building after the mainshock involves indirect losses, due for instance to business interruption and reallocation costs. On the other hand, if the building is not evacuated, it might fail due to aftershocks and cause fatalities. When the S2HM information is not available, decision-making is performed based on prior knowledge (i.e., fragility curves and GMPE). The S2HM system provides information on the structural state, thus reducing the uncertainty that affects the decision problem. A sensitivity analysis with respect to several parameters characterizing the case study is performed to demonstrate how they affect the VoI.

5.1 Description of the case study

5.1.1 Fragility model

Three building models are considered in this study. They were originally developed in (Raghunandan 2012; Raghunandan et al. 2015). The buildings consist of different RC moment frame structures of varying height and number of floors designed according to the modern US seismic design regulations. The main properties of these buildings are displayed in Table 2. The rationale behind the selection of these case studies relates to the availability of the parameters of mainshock and aftershock fragility functions for different levels of damage, defined in terms of maximum Inter-Story Drift (ISD) ratio.

Table 2 Information on the buildings
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It is assumed that, after the mainshock, the structures can be in one of the three discrete damage states \(DS_{l}\), defined as follows: \(DS_{1}\) for \(ISD < 0.020\), \(DS_{2}\) for \(0.020 \le ISD < 0.040\), \(DS_{3}\) for \(ISD \ge 0.040.\) The parameters that describe the mainshock and aftershock fragility functions, i.e., median capacities and lognormal standard deviations (see Sects. 2.1., 2.2.) are displayed in Table 3. The inelastic spectral displacement \(S_{di}\) is the intensity measure adopted to describe the ground motion intensity, which is the peak displacement of a bilinear SDOF oscillator when subjected to the ground motion of interest (Tothong and Luco 2007).

Table 3 Parameters of mainshock and aftershock fragility curves (from (Raghunandan 2012))
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The probabilities of failure due to aftershocks are computed as functions of \(M_{m}\) and \(R_{m}\) for each threshold, defined in terms of ISD. In turn, the probability of failure for the damage states—each delimited by two thresholds—is computed as the mean value of the probabilities of failure relevant to the two thresholds.

5.1.2 Hazard model

It is assumed that the buildings are in the center of the circular seismogenic area shown in Fig. 2 which produces earthquakes randomly and with equal probability in any location. The parameters defining the mainshocks and aftershocks are reported in Table 4. In particular, the aftershock parameters are selected for a generic aftershock sequence in California (Reasenberg and Jones, 1989).

Fig. 2
figure 2

Location of the building and seismogenic area

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Table 4 Parameters defining mainshocks and aftershocks
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The inelastic spectral displacement \(S_{di}\), used as intensity measure in the fragilities previously described, can be expressed as a function of the ductility demand \(\mu\) and the elastic displacement \(u_{y}\) specifically, \(S_{di} = u_{y} \mu\). In turn, the ductility demand \(\mu\) due to earthquakes is obtained using the GMPE proposed in (Rupakhety and Sigbjörnsson 2009) in the form:

$$\log_{10} \left( \mu \right) = b_{1} + b_{2} M_{w} + b_{3} \log_{10} \sqrt {d^{2} + b_{4}^{2} } + b_{5} S + \varepsilon$$
(24)

where \(b_{1}\), \(b_{2}\), \(b_{3}\), \(b_{4}\), and \(b_{5}\) are regression coefficients [tabulated in (Rupakhety and Sigbjörnsson 2009)] depending on the undamped natural period \(T\), the critical damping ratio \(\xi\), and mass normalized yield strength \(C_{y} = u_{y} \left( {2\pi /T} \right)^{2}\), \(S\) is a parameter depending on the soil type (assumed equal to 1), and \(\varepsilon\) is a zero mean normal random variable with standard deviation \(\sigma_{\varepsilon }\).

5.1.3 Information model

The S2HM outcome employed in this study is the maximum ISD ratio measured during the strong motion. For simplicity of notation, in the following, the maximum ISD ratio is indicated simply as ISD. It must be considered that: (1) by definition of the damage states \(DS_{l}\), a range of ISD values corresponds to each of them, and (2) the observed values of ISD are affected by uncertainty.

To simplify the information modeling, it is herein assumed that \(K\) values of \(ISD_{k}\) \(\left( {k = 1,2, \ldots K} \right)\) correspond to each damage state \(DS_{l}\). The probability that the S2HM provides an outcome \(O_{j}\), when the structure is in damage state \(DS_{l}\) can be expressed as follows:

$$P\left( {O_{j} |DS_{l} } \right) = \mathop \sum \limits_{k = 1}^{K} P\left( {O_{j} |ISD_{k} } \right)P\left( {ISD_{k} |DS_{l} } \right)$$
(25)

where \(P\left( {O_{j} |ISD_{k} } \right)\) is the probability to observe \(O_{j}\) when the real value of the ISD is \(ISD_{k}\) and \(P\left( {ISD_{k} |DS_{l} } \right)\) is the probability that \(ISD_{k}\) occurs when the structure is in the damage state \(DS_{l}\).

It is mentioned that the inter-story drift and its observations are continuous variables and therefore the integral version of Eq. 25 should be considered. Herein it is assumed that the values \(ISD\) are uniformly distributed within each damage state, see Fig. 3. For instance, for \(DS_{2}\), each of the values of \(ISD\) in the range \(0.02 - 0.04\) has the same value of PDF, equal to \(1/\left( {0.04 - 0.02} \right) = 50\).

Fig. 3
figure 3

Distribution of the ISD ratio in the three damage states

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The distribution of the observation of a given value of the inter-story drift\(aa\) is modeled through a Normal distribution with mean equal to the “real” value of the inter-story drift \(ISD\) and standard deviation \(\sigma = 0.05 \cdot ISD\).

The probability distributions of the S2HM outcomes in the three damage states are reported in Fig. 4.

Fig. 4
figure 4

Distribution of the S2HM outcome in the three damage states

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The proportionality assumed for σ and ISD implies that the uncertainty in the value of the S2HM observation increases with damage. The rationale behind this assumption is that the error in the identified parameters generally grows with the structural nonlinearities (Shan et al. 2015), which are typically amplified by damage (Kerschen et al. 2006). Most identification methods are indeed based on the assumption of structural linear behavior. Thus, they accumulate inaccuracies as the structure undergoes damage. While in the case of linear structures, the numerical integration method has shown to accumulate within 1–4% error at peak displacement prediction (Skolnik and Wallace 2010), structures undergoing inelastic deformations, generally accumulate considerable residual displacement (Shan et al. 2015).

5.1.4 Costs

As for the costs involved in the decision analysis, it is supposed that the number of people per floor, \(N_{pf}\) [person/floor], is 20. The number of floors is indicated as \(N_{f}\). The duration of the emergency is indicated as \(T\) [day]. The reference cost of each fatality, \(F\)[$/person], is approximated to 2 million dollars (Wong et al. 2005). This cost should be intended as the statistical value of generic human life, used for instance by an insurance company for recompensing purposes. Initially, it is assumed that:

  • If the building is not evacuated and fails because of an aftershock, all the people in the building die.

  • If the building is not evacuated and it does not fail, it is assumed that there are no costs to pay.

  • If the building is evacuated, there are indirect costs that do not depend on the state of the building. These costs are defined by the daily indirect cost per floor \(C\) [$/floor/day].

The costs of failure, \(c_{F} \left( {A_{n} } \right)\), and survival, \(c_{{\overline{F}}} \left( {A_{n} } \right)\), as a function of the selected action (see Eq. 3) are displayed in Table 5. The cost of the action “Evacuate” is indicated as \(IC = TN_{f} C\) (Indirect Cost) whereas \(DC = FN_{f} N_{pf}\) (Direct Cost) indicates the cost associated with failure when the structure is not evacuated.

Table 5 Cost of failure and survival
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5.2 Results

The VoI analysis is carried out to investigate the impact of the seismicity of the region, the magnitude of the indirect costs, the emergency duration, the prior knowledge, and the indirect costs related to the failure. The results are presented and discussed in the following sections.

5.2.1 Variation of VoI with the seismicity of the region

First, three cases corresponding to different hazard levels are described considering different values of the indirect costs. They are all relevant to a 12-story building and a reference period \(T = 60 days\).

Figures 5 and 6 report the results relevant to Case 1: high seismicity region \(\left( {M_{m,u} = 9} \right)\) and low indirect costs \(\left( {C = 5000 \$ } \right)\). Figure 5a illustrates the outcome of the Prior decision analysis (that is performed without the support of the S2HM information) in terms of the optimal action chosen as a function of \(M_{m}\) and \(R_{m}\). As expected, the action “Do not evacuate” is preferable for mainshocks of low magnitude with an epicenter far from the building (low right end corner of the figure) whereas the optimal action is “Evacuate” when high magnitude earthquakes occur close to the building (upper left corner of the figure). The boundary between the two optimal actions corresponds to earthquakes (e.g., combinations of \(M_{m}\) and \(R_{m}\)) for which the expected costs of the two actions are very close to each other. This boundary moves toward higher values of \(M_{m}\) at the increase of the epicentral distance \(R_{m}\). The VoI, reported in Fig. 5b reaches the maximum values in a narrow region around this boundary. Indeed, where the two management actions have similar expected costs, the information from the S2HM system provides the highest benefit. This aspect is highlighted in Fig. 6, which report the VoI (Fig. 6a) and the expected costs of the two emergency management actions (Fig. 6b) as a function of the event magnitude \(M_{m}\). The grey region in Fig. 6b indicates, for each value of the magnitude \(M_{m}\), the expected cost relevant to different values of the epicentral distance \(R_{m}\) in the range (0–50 km). The upper and lower boundaries of the grey region correspond respectively to \(R_{m} = 0 km\) (maximum probability of failure and therefore maximum expected cost for a given magnitude) and \(R_{m} = 50 km\) (minimum probability of failure and therefore minimum expected cost for a given magnitude). The dash-dotted line represents the—constant in this case—expected cost of the action “Evacuate”. The maximum VoI is located in the magnitude range 6.5–7.5, roughly, corresponding to the intersection between the dash-dotted line and the grey region: here the two decision alternatives lead to equal prior expected costs.

Fig. 5
figure 5

Results for \(M_{m,u} = 9\) and C = 5,000 $: a Optimal action according to the Prior decision analysis; b \({\text{VoI}}\left( {R_{m} ,M_{m} } \right)\)

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Fig. 6
figure 6

Results for \(M_{m,u} = 9\) and C = 5000 $: a \({\text{VoI}}\left( {M_{m} } \right)\) and b prior expected costs

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Figure 7 shows the results obtained for Case 2: high seismic hazard and high indirect costs: \(C = 50,000\)$. In this case, the action “Evacuate” becomes optimal in a range of higher magnitudes with respect to Case 1 (Figs. 6a, 7a). This is due to the higher values of C with respect to Case 1, which leads to higher indirect costs related to the action “Evacuate”. On the other hand, the expected cost of the action “Do not evacuate” is the same as Case 1 since it mainly depends on the probability of failure and thereby on the seismicity of the region. Graphically, the dashed-dot horizontal line moves up and crosses the curves corresponding to the direct costs (grey region) in the range of higher magnitudes, around 8–9. In this same region, the VoI reaches its maximum values Fig. 7a and presents higher values with respect to Case 1. This is due to the higher expected costs of the two management actions that lead to a higher value of their difference representing the benefit achieved when the decision-maker changes her decision thanks to the S2HM information.

Fig. 7
figure 7

Results for \(M_{m,u} = 9\) and C = 50,000 $: a Optimal action according to the Prior decision analysis, and b \({\text{VoI}}\left( {R_{m} ,M_{m} } \right)\)

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Figure 8 shows the VoI as a function of \(M_{m}\) (Fig. 8a), computed by marginalizing out the term \(R_{m}\) from \({\text{VoI}}\left( {R_{m} ,M_{m} } \right)\), and the expected costs of the two actions (Fig. 8b). The VoI is again maximum where the expected costs of the two actions are similar which, in this case, happens in the magnitude range 7.5–8.5. It is underlined that Figs. 6b and 8b differ in the position of the horizontal line represented by the (fixed) expected cost of the action “Evacuate” whereas the grey region corresponding to the expected cost of the action “Do not evacuate” is the same (the same direct costs are considered).

Fig. 8
figure 8

Results for \(M_{m,u} = 9\) and C = 50,000 $: a \({\text{VoI}}\left( {M_{m} } \right)\) and b prior expected costs

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Case 3 is relevant to a region at medium seismic hazard \(\left( {M_{m,u} = 8} \right)\) and high indirect costs \(\left( {C = 50,000 \$ } \right)\). The meaning of the figures is the same as for the previous cases. Figure 9 shows that the optimal action identified based on Prior decision analysis is “Evacuate” only for a small area on the upper left corner of the plot, due to the lower direct costs with respect to Case 2. Consequently, significative values of the VoI are reached only for high values of magnitude.

Fig. 9
figure 9

Results for \(M_{m,u} = 8\), C = 50,000 $, and \(N_{f} = 12\): a Optimal action according to the Prior decision analysis, and b \({\text{VoI}}\left( {R_{m} ,M_{m} } \right)\)

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Figure 10 shows the VoI as a function of \(M_{m}\) (Fig. 10a) and the prior expected costs of the two actions (Fig. 10b). The prior expected costs of the action “Do not evacuate” (direct costs) are generally lower with respect to Case 2 due to lower seismic demand, leading to a lower probability of failure. Figure 10a shows that the VoI monotonically increases over the range of magnitudes since there is no range of magnitude in which the expected cost of the action “Do not evacuate” is always higher than the expected cost of the action “Evacuate”.

Fig. 10
figure 10

Results for \(M_{m,u} = 8\), C = 50,000 $, and \(N_{f} = 12\): a \({\text{VoI}}\left( {M_{m} } \right)\) and b prior expected costs

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5.2.2 Variation of VoI with the indirect costs

To investigate the impact of the daily indirect costs \(C\) on the VoI, the analysis is repeated for several different values of this parameter and the results are displayed in Fig. 11. The analysis is carried out considering buildings of a different number of floors in different seismogenic areas. Specifically, three different upper bound for magnitudes \(M_{m,u}\) have been considered, from 7 to 9 (see Sect. 4.3 and Table 4). The duration of the emergency duration is fixed to \(T = 60 days\). Different values of \(C\) represent different intended uses of the buildings. For instance, low values of \(C\) can be associated with residential use when indirect losses are related to the cost of alternative accommodations such as hotels. Instead, high values of \(C\) can be associated with cases where a business interruption may have a higher cost.

Fig. 11
figure 11

a VoI for \(M_{m,u} = 9\), b \(M_{m,u} = 8\), and c \(M_{m,u} = 7\)

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Results show that, for the same building, the VoI increases with the seismicity of the area, that is for increasing values of \(M_{m,u}\). Furthermore, the VoI increases with the number of floors. Specifically, increasing values of \(M_{m,u}\) imply the increase of the seismic exposure of the buildings meaning higher probabilities of failure (for a given couple \(M_{m}\) and \(R_{m}\)) and probabilities of occurrence of earthquakes with high magnitudes. The increasing number of floors, instead, involves the increase of both direct and indirect costs. It is highlighted that the VoI is negligible for very low or very high values of \(C\), which is when the indirect costs are very low or very high. In this situation, the prior expected costs of the two decision alternatives (“Evacuate” or “Do not evacuate”) are very different and, as found in other studies (Giordano et al. 2020; Giordano and Limongelli 2020), this leads to negligible values of the VoI. The reason is that in this situation, the information from the S2HM does not change the behavior of the decision-maker. For low values of \(C\) (low indirect costs) the action “Evacuate” appears very convenient; for high values of \(C\) (high indirect costs) the action “Evacuate” is very expensive and therefore “Do not Evacuate” becomes the optimal action, regardless of the state of the system. If the state of the system does not affect the decision, its knowledge and thereby the information from the S2HM, becomes irrelevant. To improve clarity, a second horizontal axis has been included in the three plots in Fig. 11 showing the values of the ratio \(IC/DC\) that correspond to each \(C\). For values of \(IC/DC > 1\) the VoI is null since the action “Do not evacuate” is always the optimal action independently of the state of the system. For \(IC/DC\) close to zero the indirect costs are much lower with respect to the direct costs and so the action evacuate is the optimal, irrespective of the building state.

5.2.3 Variation of VoI with the emergency duration

Figure 12 shows the results for different values of \(T\), i.e., the duration of the emergency period. The increase of \(T\) leads to the increase of the indirect costs that depend linearly on \(T\) (see Table 5), and of the direct costs, due to the increase of the probability of failure (see Sects. 4.3., 4.4.). Results indicate that the VoI increases with the increasing duration of the emergency period since both direct and indirect costs increase. Furthermore, the peak of the VoI moves towards lower values of \(C\) which indicates that the indirect costs increase more rapidly with \(C\) with respect to the direct costs. However, the maximum VoI does not change significantly at the increase of the reference emergency period from 30 to 90 days.

Fig. 12
figure 12

VoI as a function of \(C\) and duration of the emergency \(T\) for \(M_{m,u} = 8\) and \(N_{f} = 12\)

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5.2.4 Variation of VoI with the prior knowledge

In the previous sections, the VoI provided by the S2HM was computed assuming that the decision-maker disposes of (aftershock) fragility curves as a support to manage the emergency. The information provided by the S2HM enables the improvement of the knowledge on the system with respect to that provided by the fragility curves. However, the knowledge of the aftershock fragility curves is not always available, as shown by the literature survey reported in Sect. 2.2. To investigate the impact of the prior knowledge provided by the fragility curves, in this section, the VoI is computed attributing the same prior probability to each damage state \(P\left( {DS_{1} } \right) = P\left( {DS_{2} } \right) = P\left( {DS_{3} } \right) = 1/3\) (non-informative prior probabilities). Figure 13a shows the VoI as a function of the daily indirect costs \(C\) obtained considering prior probabilities from fragility functions and non-informative prior probabilities. The analysis parameters are \(T = 60 \;days\), \(M_{m,u} = 9\), and \(N_{f} = 12\). The VoI relating to the case when fragility functions are available (red dashed line) is the same shown in Fig. 11a and is reported for comparison purposes. The VoI obtained considering non-informative prior probabilities is represented by the blue solid line and, for a given \(C\), is generally higher than the VoI obtained considering prior probabilities from fragility curves. The use of the fragility functions is a way to include information in the decision-making process. Therefore, if this information is not included in the analysis, the knowledge acquired by means of the S2HM system provides a higher benefit.

Fig. 13
figure 13

\({\text{VoI}}\) as a function of \(C\): a effect of non-informative prior probabilities and b combined effect of non-informative prior probabilities and additional failure costs

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5.2.5 Variation of VoI with indirect costs associated with failure

In the previous sections, it has been assumed that the action “Do not evacuate” involves direct costs related to the fatalities but not indirect costs. However, in this situation, the collapse of the building entails not only the loss of the building itself but also of resources (human and material) needed for business continuity. To account for this aspect, in this section, an additional cost is introduced through a downtime period \(T^{*}\) intended as the time required to re-organize and re-allocate the activity carried out in the building. The relevant indirect costs are expressed as \(T^{*} N_{f} C\); in the analysis the value \(T^{*} = 365 days\) has been considered for demonstration.

Figure 13b displays the VoI computed considering non-informative prior probabilities and additional indirect costs associated with the structural failure, according to Table 6, considering the following parameters: \(T = 60 days\), \(M_{m,u} = 9\), and \(N_{f} = 12\).

Table 6 Modified cost of failure and survival
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The VoI continuously increases with the indirect costs (represented by the daily indirect costs \(C\)). In this case, the VoI is not null for high values of the daily indirect costs. The reason is that the daily indirect costs affect the expected costs of both actions and thereby “Do not Evacuate” is not necessarilty the optimal action for high values of \(C\) (see Sect. 5.2.2). This result further confirms that the VoI strongly depends on the magnitude of the costs involved in the decision analysis and the difference between the expected costs of the management actions.

6 Conclusion

In this paper, the benefit of installing a permanent S2HM on buildings to support emergency management is explored. To estimate this benefit, a framework based on the Value of Information (VoI) from Bayesian decision analysis is proposed and applied to reference case studies consisting of RC buildings with a different number of floors.

The computation of the VoI in this context has been carried out considering the availability: of (1) the seismic hazard model of the region where the building is located; (2) fragility curves that give insights into the structural conditions without SHM information; (3) likelihood functions to model the outcome of the S2HM system in different system states; (4) consequences of failure and survival for different management actions.

A sensitivity analysis is carried out concerning: (1) the height of the building; (2) the seismicity of the region; (3) the ratio between indirect and direct consequences; (4) the duration of the emergency, which is defined as the interval between the mainshock, and the inspection performed by an expert technician; (5) the prior knowledge; (6) the indirect losses related to the failure of the building. Results show that:

  • the VoI increases with the seismicity of the region and the number of building floors. Both factors are related to the expected costs of different actions. The higher the seismicity, the greater the probability of failure and the probability of being in a severe damage state. Thus, the expected cost of the action “Do not Evacuate” increases. A higher number of floors, instead, involves the increase of both direct and indirect costs and thus of the expected costs of both the action “Do not Evacuate” and the action “Evacuate”.

  • higher values of the VoI are obtained when different management actions have similar prior expected costs. In these cases, the optimal action is more uncertain and the information from the monitoring system strongly impacts the decision-making.

  • the VoI is negligible when the prior expected costs of different management actions are very different from each other since the information from the S2HM system does not modify the behavior of the decision-makers.

  • the ratio between indirect and direct costs is the parameter that mostly influences the value of the VoI.