the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Scenariobased multirisk assessment from existing singlehazard vulnerability models. An application to consecutive earthquakes and tsunamis in Lima, Peru
Juan Camilo Gómez Zapata
Massimiliano Pittore
Nils Brinckmann
Juan LizarazoMarriaga
Sergio Medina
Nicola Tarque
Fabrice Cotton
Multihazard risk assessments for building portfolios exposed to earthquake shaking followed by a tsunami are usually based on empirical vulnerability models calibrated on postevent surveys of damaged buildings. The applicability of these models cannot easily be extrapolated to other regions of larger/smaller events. Moreover, the quantitative evaluation of the damages related to each of the hazard types (disaggregation) is impossible. To investigate cumulative damage on extended building portfolios, this study proposes an alternative and modular method to probabilistically integrate sets of singlehazard vulnerability models that are constantly being developed and calibrated by experts from various research fields to be used within a multirisk context. This method is based on the proposal of statedependent fragility functions for the triggered hazard to account for the preexisting damage and the harmonisation of building classes and damage states through their taxonomic characterisation, which is transversal to any hazarddependent vulnerability. This modular assemblage also allows us to separate the economic losses expected for each scenario on building portfolios subjected to cascading hazards. We demonstrate its application by assessing the economic losses expected for the residential building stock of Lima, Peru, a megacity commonly exposed to consecutive earthquake and tsunami scenarios. We show the importance of accounting for damage accumulation on extended building portfolios while observing a dependency between the earthquake magnitude and the direct economic losses derived for each hazard scenario. For the commonly exposed residential building stock of Lima exposed to both perils, we find that classical tsunami empirical fragility functions lead to underestimations of predicted losses for lower magnitudes (M_{w}) and large overestimations for larger M_{w} events in comparison to our statedependent models and cumulativedamage method.
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Cascading natural events, commonly defined as a primary hazard triggering a secondary one, have jointly induced large disasters (Gill and Malamud, 2016). In the case of earthquakes, between 25 % and 40 % of economic losses and deaths have been reported to result as a consequence of secondary effects, i.e. tsunamis, landslides, liquefaction, fire, and others (Daniell et al., 2017). Wellknown examples are the submarine earthquakes and the subsequent tsunamis that occurred in 2004 in the Indian Ocean, in 2011 in Japan, and in 2018 in Palu Bay in Indonesia (Goda et al., 2019). These events not only induced cumulative physical damage on the exposed infrastructure but also brought drastic socioeconomic cascading effects that are still perceptible today (de Ruiter et al., 2020; Suppasri et al., 2021). Despite the magnitude of such events, multihazard risk assessment remains a relatively new research field with still not unified terminologies and approaches (Pescaroli and Alexander, 2018; Tilloy et al., 2019). Nonetheless, a number of studies (e.g. Kappes et al., 2012; Komendantova et al., 2014; Gallina et al., 2016; Julià and Ferreira, 2021; De Angeli et al., 2022; Cremen et al., 2022) have unanimously agreed that more realistic multirisk evaluations can only be conducted if both (1) multihazard (e.g. Marzocchi et al., 2012; Liu et al., 2016) and (2) multivulnerability interactions (e.g. Zuccaro et al., 2008; Gehl et al., 2013) are considered altogether. While the former comprises the study of the conditional probabilities of the occurrences of these hazards and their combination, the study of the latter involves reviewing the many classes of vulnerabilities that are associated with an exposed territory.
Therefore, this study narrows down the scope of scenariobased multihazard risk by assuming that a second hazardous event is always triggered after the occurrence of the first one, thus eliminating the need to quantify the probability of this occurring. Thus, we will only focus on the dynamic physical vulnerability and related cumulative damage that a building stock exposed to a close succession of hazardous events might suffer. As a premise, this study contributes to the field by proposing a modular method to probabilistically integrate sets of singlehazard vulnerability models that are being constantly developed and calibrated by experts from various research fields to be used within a multirisk context. The rest of this Introduction discusses the stateoftheart exposure modelling for largescale building portfolios for multihazard risk assessment: focus is made on the underlying assumptions to propose generalised building typologies with associated fragility functions used to assess their physical vulnerabilities to earthquake and tsunamis. Having done that, the last part of the Introduction summarises the general scope and capabilities of the original method that will be described in detail afterward.
In exposure modelling for multihazard risk purposes, we can distinguish between two main approaches:

The first approach is using a single set of building classes, each employing as many fragility/vulnerability models as the natural hazards considered, for example, the HAZUSMH (FEMA, 2003, 2017; Dabbeek and Silva, 2020; Dabbeek et al., 2020). They typically have associated sets of fragility functions with equivalent damage states regardless of the hazard. Aligned with this philosophy, the vulnerability classes of the European Macroseismic Scale (EMS98) originally proposed by Grünthal (1998) were used by some authors not only to describe the likely damage due to seismic action, but also to classify likely ranges of vulnerabilities to other hazards based on the building's material types (Schwarz et al., 2019; Maiwald and Schwarz, 2019).

The second approach is jointly applying a number of different building classifications per individual hazard to the same exposed buildings (e.g. GomezZapata et al., 2021e; Arrighi et al., 2022). Their associated fragility functions may have different sets of damage states (differing in number and description). Notably, these models are constantly developed and individually validated by experts of each research field.
Although the first type might be useful in the assessment of risk arising from independent hazards, their related sets of fragility models lack multihazard calibration and validation and, therefore, do not offer sufficient inputs for assessing the increasing damage from cascading events (Ward et al., 2020).
Moreover, the definition of the damage scale depends on the building type (Hill and Rossetto, 2008) and the likely failure mechanisms that it can experience under the action of specific hazard intensity measures (IMs) (Vamvatsikos et al., 2010; Selva, 2013). Therefore, the observable damage features on individual structural or nonstructural components that jointly describe a certain damage state can have contrasting descriptions across various hazarddependent vulnerability types (Gehl and D'Ayala, 2018; Figueiredo et al., 2021), and there is often not a 1:1 relation between them, i.e. for the case of earthquakes and tsunamis (Bonacho and Oliveira, 2018; Lahcene et al., 2021). The reasons behind such a mismatching between the definitions of damage states may arise from the absence of standard formats for damage data collection across regions and across the several vulnerability types of interest (Mas et al., 2020; Frucht et al., 2021). Notably, the study of Negulescu et al. (2020) found this aspect to be particularly significant for the multihazard risk context, stating that the damage states of earthquake and tsunami fragility models can have variable levels of compatibility. This assumption led to contrasting loss estimates with respect the US HAZUS approach, which is based on the complete equivalence between damage grades. This background portrays the need to standardise the description of the physical damage through harmonising scales across several hazarddependent vulnerabilities, which are inputs for unified methods in multihazard risk (Ward et al., 2022).
The earthquake engineering community has investigated the cumulative damage expected during seismic sequences (e.g. Papadopoulos and Bazzurro, 2021; Karapetrou et al., 2016; Trevlopoulos et al., 2020), but this concept is rarely considered in other research disciplines. For instance, the physical vulnerability of building portfolios to tsunamis has been typically evaluated through empirical fragility functions derived from postnearfield tsunami surveys. A drawback of these functions is that they have been presented solely as tsunami fragility functions in terms of the inundation depth, when in reality these surveys encompassed assets that experienced cumulative damage due to the joint effect of the tsunamigenerating earthquake and the tsunami itself (Charvet et al., 2017). Due to this limitation, analytical fragility functions were recently proposed for individual structures (e.g. Attary et al., 2017; Petrone et al., 2017) and for largescale building stocks with generalised typologies (Belliazzi et al., 2021). However, as remarked by Attary et al. (2021), using these functions for loss estimation should only be valid for farfield tsunamis, and for nearfield events the damage induced by shaking before the tsunami strikes must still be addressed.
To the best of the authors' knowledge, only a few studies have investigated the performance of heterogeneous and largescale building portfolios for risk estimates subjected to consecutive ground shaking and tsunamis. Hereby, we summarise some of them. In Goda and De Risi (2018), a rationale was proposed for adopting the larger value of the damage ratios from independent earthquake and tsunami risk computations. In Park et al. (2019), a probabilistic multirisk approach was presented for a building stock in the US subjected to spatially uncorrelated seismic ground motions and subsequent tsunamis. This study showed the disaggregation of losses per hazard and per materialbased building type across several return periods while assuming statistical independence between their respective damage states. As a common denominator of the aforementioned studies, the cumulative damage and losses from a building portfolio were not assessed. Since these metrics cannot be obtained as the sum of the effects from each individual hazardous event (Bernal et al., 2017; Terzi et al., 2019), it is rather necessary to address the nonlinear damage accumulation on the same exposed assets during the multihazard sequences (Merz et al., 2020).
This study proposes a modular method to probabilistically integrate existing sets of singlehazard vulnerability models (or “reference schemes”). For this aim, this method comprises four main modules. The first two refer to sets of compatibilities between the vulnerability models selected for each singlehazard vulnerability (e.g. between existing seismic and tsunami building classification schemes). The first probabilistic compatibility set is obtained between (1) building classes (as presented in Sect. 2.1), while the second is obtained between (2) damage states (Sect. 2.2). These two conversions are done through the use of taxonomic attributes that are independent of the definition of the reference schemes. This is done with the purpose of representing the damage distribution resulting after the first hazard (i.e. earthquake) through a damageupdated exposure model whose damage scale is dependent on the classification scheme required for assessing the vulnerability to a triggered event (i.e. tsunami). The third module results from the need to perform risk assessment for the triggered hazard using the damageupdated exposure model that is now represented in terms of the second vulnerability scheme (e.g. building classes and damage states for tsunami fragility). Hence, this module comprises the proposal of (3) sets of statedependent fragility functions for the second hazard (e.g. tsunami), as presented in Sect. 2.3. These three modules are valuable inputs for ultimately assessing the expected cumulative damage. They are later complemented by a last fourth module: (4) a consequence model to assess the incremental direct economic losses (Sect. 2.4) that are expected from consecutive hazard scenarios.
In the application section of this paper (Sect. 3), we demonstrate the application of this method by investigating the likely cumulative damage on the residential buildings of Lima (Peru) by considering this city's exposure to six megathrust earthquake scenarios (main shock) and subsequent tsunamis. This is done using existing vulnerability models per hazard and addressing the probabilistic compatibilities between building classes and damage states. Complementarily, a set of tsunami statedependent fragility functions that are obtained through the use of simple ad hoc scaling factors are proposed. Nonetheless, as it will be discussed, these functions can and should be replaced by other sets of statedependent tsunami fragility functions derived from more sophisticated methods when they become available. Every damage distribution is translated into direct economic losses to gain a comparative risk metric and disaggregate the contribution of each hazard scenario.
To assess the cumulative damage that is expected to be experienced by a building portfolio during hazardous event sequences, we rely on the principle that its related exposure model is represented by jointly applying existing building classification schemes, one per each individual hazardous scenario of the cascading sequence. For example, one building that is expected to be affected by a first hazard intensity measure IM^{A} (here A refers to an IM used to model ground shaking, for example, peak ground acceleration (PGA) in g) and a second one IM^{B} (B refers to an IM used to model inland tsunami inundation, for example, inundation depth in m) is actually classified under two exposure classification schemes (${T}_{k}^{\mathrm{A}}$ and ${T}_{j}^{\mathrm{B}}$), respectively, which have attached their related vulnerability models. Each scheme contains a set of mutually exclusive, collectively exhaustive building classes, $k=\left\{{k}_{\mathrm{1}},\mathrm{\dots},{k}_{n}\right\}$ and $j=\left\{{j}_{\mathrm{1}},\mathrm{\dots},{j}_{n}\right\}$, correspondingly.
To assess the expected damage state after the first hazardous event (e.g. ground shaking), we apply their fragility function $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{\text{IM}}^{\mathrm{A}}\right)$, which gives us the probability that a building k, typically assumed to be in an undamaged state (${D}_{k\mathrm{0}}^{\mathrm{A}}$), changes to a progressive state z due to a hazard intensity IM^{A} (green part in Fig. 1b). For risk assessment, this is completed by the consequence model, $p\left(L\mathrm{}{D}_{kz}^{\mathrm{A}}\right)$, which assigns a loss ratio L of the total replacement cost of building class k given the occurrence of a damage state. Thus, the expected loss given a hazard intensity IM^{A} is calculated considering the contributions from all possible damage states and their probabilities, as per Eq. (1).
If this damaged building portfolio is subjected to the action of a second scenario with a hazard intensity IM^{B}, it would experience cumulative damage moving from a damage state $z\left({D}_{kz}^{\mathrm{A}}\right)$ to a damage state w (but in the domain of the second vulnerability scheme: ${D}_{jw}^{\mathrm{B}}$). Due to this differential scheme classification, their respective set of damage states may not have trivial equivalences because they can also have different observable damage features. Therefore, we propose integrating a set of modular components:

The first module is interscheme compatibilities between each hazarddependent exposure classification scheme $p\left({T}_{k}^{\mathrm{A}}\mathrm{}{T}_{j}^{\mathrm{B}}\right)$ (i.e. purple part/doublelined box in Fig. 1b). Its derivation follows the method originally proposed in Gómez Zapata et al. (2022b), and it is summarised herein in Sect. 2.1. By reusing this approach, a building stock that was initially classified for a first hazard vulnerability (i.e. earthquakeoriented (EQ) typologies) can now be probabilistically represented by other predefined classes (e.g. tsunamioriented (TS) typologies).

The second module is the related compatibility levels between interscheme damage states $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{D}_{jy}^{\mathrm{B}}\right)$ (i.e. red part/dashed box in Fig. 1b, explained in Sect. 2.2) that is needed when the fragility models attached to such schemes have different numbers of damage states and descriptions.
The two aforementioned conversions are represented through two sets of “compatibility matrices” that are probabilistically generated. The advantage of using these matrices is that through these conversions, the damageupdated exposure model resulting from the action of IM^{A} can be represented in the domain of the reference scheme attached to the second vulnerability to be analysed. Once this change of reference scheme is obtained, the damageupdated exposure model can be directly used for a second risk computation (e.g. for tsunami risk, addressing cumulative damage when the buildings experienced previous damage due to seismic ground shaking):
 3.
This second risk calculation is performed by using a third module that refers to generic statedependent tsunami fragility functions (i.e. with nonzero initial damage states made of new curves that represent the permissible damage progression). Since the resultant earthquakeinduced damages are formerly expressed in the tsunami vulnerability domain (step 2), the nonzero damage limit states of this set of statedependent tsunami fragility functions will implicitly account for such preexisting damage. The joint ensemble of these three components can be ultimately used to calculate the cumulative expected damages after the triggered event with IM^{B}, while accounting for the preceding induced by IM^{A} (i.e. green part/dotted box in Fig. 1b, developed in Sect. 2.3).
 4.
For multirisk assessment a fourth module that represents the incremental loss obtained from the economic consequence model attached to the classification scheme ${T}_{j}^{\mathrm{B}}$ (i.e. replacement costs and related loss ratios per damage state of the second scheme, ${D}_{jy}^{\mathrm{B}}$) is integrated. This is represented by the last box made of continuous (blue) lines in Fig. 1b, explained in Sect. 2.4.
These four modules are described hereafter.
2.1 Exposure modelling: taxonomic description, interscheme conversion, and spatial aggregation of building classes
The classified building stock under the first hazarddependent classification scheme ${T}_{k}^{\mathrm{A}}$ is spatially aggregated into a set of geocells that capture the local spatial variations of the hazards' IM of interest across the study area. For such a purpose, we recommend using variableresolution exposure models in the form of central Voronoi tessellations (CVTs). Besides spatially representing the building portfolio, they also provide a representative IM per geocell for reliable and computationally efficient vulnerability estimations (Pittore et al., 2020; GomezZapata et al., 2021e). They also implicitly serve as common minimum reference unit (MRU) aggregation entities between exposure and hazard (Zuccaro et al., 2018). This is because for their derivation, one can consider the combination of local variations of the hazard intensity measures (IMs) and certain exposure proxies (e.g. population density) across the same area. CVTbased models may be useful in a multihazard risk context where the spatial correlation of various IMs can differ (e.g. ground shaking and tsunami inundation).
As shown in Pittore et al. (2018), every building class k that belongs to one scheme A (in this case, earthquake) and every building class j that belongs to one scheme B (in this case, tsunami) can be described in terms of basic observable features F_{m} within a faceted taxonomy, that is, a building classification schema in which building classes result from the characterisation of individual attributes or facets (Brzev et al., 2013; Silva et al., 2018, 2022). This disaggregation is the common underlying vocabulary to obtain the probability that a building class within the source scheme (${T}_{k}^{\mathrm{A}})$ corresponds to another class within the target scheme $\left({T}_{j}^{\mathrm{B}}\right)$. As proposed in Gómez Zapata et al. (2022b), the degree of compatibility between the building classes belonging to both schemes can be represented by a compatibility matrix $p\left({T}_{k}^{\mathrm{A}}\mathrm{}{T}_{j}^{\mathrm{B}}\right)$ to account for the uncertainties when there is not a trivial (onetoone) mapping. Knowing in advance certain exposure metrics of the source scheme ${\left\{R\right\}}_{{T}_{k}^{\mathrm{A}}}$ (i.e. building counts), the respective values of the target scheme ${\left\{R\right\}}_{{T}_{j}^{\mathrm{B}}}$ can be obtained by applying the dot product (Eq. 2).
2.2 The probabilistic description and compatibility of interscheme damage states
We consider how the fragility functions associated with ${T}_{k}^{\mathrm{A}}$ (earthquake) and ${T}_{j}^{\mathrm{B}}$ (tsunami) may have diverse numbers and descriptions of damage states per considered hazarddependent vulnerability scheme (${D}_{kz}^{\mathrm{A}}z={z}_{\mathrm{1}},\mathrm{\dots},{z}_{{N}_{\mathrm{A}}}$ and ${D}_{jy}^{\mathrm{B}}y={y}_{\mathrm{1}},\mathrm{\dots},{y}_{{N}_{\mathrm{B}}}$). To harmonise their equivalence, we propose obtaining their probabilistic interscheme compatibility as a set of matrices $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{D}_{jy}^{\mathrm{B}}\right)$. This is achieved after having evaluated how the likely observable characteristics linked to each damage state within ${D}_{kz}^{\mathrm{A}}$ and ${D}_{jy}^{\mathrm{B}}$ can be expressed in terms of another one. For this aim, we first propose the use of the Agibilità E Danno in Emergenza Sismica form (AeDES; usability and damage in seismic emergency) of the Italian Civil Protection (Baggio et al., 2007) as a standard scoring system to create a synthetic dataset based on the likely observable damage on individual building components. Although the AeDES form was originally proposed for postearthquake damage data collection, we propose to transversally use it to describe every damage state z and y of ${D}_{kz}^{\mathrm{A}}$ (due to seismic ground shaking) and ${D}_{jy}^{\mathrm{B}}$ (due to tsunami inundation), respectively.
Expert elicitation is used on the AeDES form to create heuristics evaluating the expected damage extension per building type and each of the damage limit states defined within their respective fragility functions. For this aim, we make use of its implicit scale within a range of 0= L to 9= A over the building components n (lowlevel taxonomic attributes) as shown in Fig. 2. We decided to only include four out of these six components that can be found in any building type as listed in Eq. (3), as stairs and preexisting damage are not always present in all buildings. The importance of such building components for assessing their physical vulnerability has been documented in previous studies on ground shaking (e.g. Lagomarsino et al., 2021) and tsunamis (e.g. Del Zoppo et al., 2021).
A heuristic is generated by scoring the four components in Eq. (3) per damage state, per fragility function, and per building class of both exposure classification schemes. This is done through expert elicitation and establishes a training dataset of the possible observable damage extent {OD}_{n} in a harmonised manner. For instance, one set of {OD}_{n} (for a given damage state and building type) is made up of a set of four numbers from 0 to 9, e.g. $n=\left\{\mathrm{1},\mathrm{2},\mathrm{1},\mathrm{3}\right\}$, meaning level I for VS and RF, level H for FL, and level G for IP (Eq. 3). Thereafter, using the total probability theorem, the probability that the damage state z of building class j in scheme A corresponds to the damage state y of building class j in scheme B can be calculated by Eq. (4).
We assume that the representations of damage states within the two considered schemes are conditionally independent ($\u27c2\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\u27c2$). Thereby, given the information of the scored observable damage on the individual components {OD}_{n}, we can describe the source damage scheme ${D}_{kz}^{\mathrm{A}}$ to be modelled in terms of {OD}_{n} that jointly compose the target scheme ${D}_{jy}^{\mathrm{B}}:$ ${D}_{jz}^{\mathrm{A}}\u27c2\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\u27c2{D}_{jy}^{\mathrm{B}}\mathrm{}{\left\{\text{OD}\right\}}_{n}$. Thus, Eq. (4) can be expressed as a product, given by Eq. (5).
We obtain a probabilistic compatibility degree between damage states (${D}_{kz}^{\mathrm{A}}z={z}_{\mathrm{1}},\mathrm{\dots},{z}_{{N}_{\mathrm{A}}}$ and ${D}_{jy}^{\mathrm{B}}y={y}_{\mathrm{1}},\mathrm{\dots},{y}_{{N}_{\mathrm{B}}}$) for every pair of combination of building classes ${T}_{k}^{\mathrm{A}}$ and ${T}_{j}^{\mathrm{B}}$ through a Bayesian formulation as presented in Eq. (6).
The terms $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{\left\{\text{OD}\right\}}_{n}\right)$ and $p\left({D}_{jy}^{\mathrm{B}}\mathrm{}{\left\{\text{OD}\right\}}_{n}\right)$ in Eq. (6) can be solved through supervised machine learning techniques for classification (e.g. logistic regression, naive Bayes, decision trees) to predict the probabilities between the training sets and a synthetic testing dataset. The selection of the machine learning technique, naturally, carries epistemic uncertainties (Mangalathu et al., 2020) whose investigation is beyond the scope of this study. The testing dataset is obtained after generating random numbers of all the possible combinations of the AeDESbased scores. With this dataset we express the conditional probabilities of having damage states ${D}_{kz}^{\mathrm{A}}z$ and ${D}_{jy}^{\mathrm{B}}y$ (for each building class within schemes A and B) given {OD}_{n}. The term p({OD}_{n}) is a marginal probability that can be assumed to represent the proportion of one observation out of exhaustive combinations of {OD}_{n}. Lastly, $p\left({D}_{jy}^{\mathrm{B}}\right)$ describes the proportion of each damage state y within each building class k in the training dataset for scheme B. Once Eq. (6) is solved, the expression $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{D}_{jy}^{\mathrm{B}}\right)$ is obtained, which stems from the probabilistic interscheme damage compatibility matrix for each possible pair of buildings within schemes A and B. After having established the compatibility between building classes and damage states, a special set of fragility functions is needed to follow the damage progression inflicted by the second hazard. They are explained hereafter.
2.3 Statedependent fragility functions
The next steps of the method are carried out within the reference vulnerability scheme of the second hazard (i.e. tsunami). Let us suppose that the fragility functions ${F}_{{T}_{r}^{i},{q}_{{N}_{i}}}$ (for a set of building types T_{r} and composed by a set of ${q}_{{N}_{i}}$ damage limit states) are assumed to be modelled by cumulative lognormal distributions. They are defined by their respective logarithmic means ${\mathit{\mu}}_{{q}_{\mathrm{0}}}\left({T}_{r}^{i}\right)$ and their logarithmic standard deviations, for which we assume that their initial damage states ${q}_{{N}_{i}}$ are all represented by a zero q_{0} (for a pristine, intact structure). For a set of damage states ${q}_{{N}_{i}}$ in pristine structures, there is a corresponding set of values ${\mathit{\lambda}}_{{q}_{\mathrm{0}}}=\left[{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{1}}}},{\mathit{\lambda}}_{{q}_{y}},{\mathit{\lambda}}_{{q}_{w}},\mathrm{\dots},{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{i}}}\right]$. With this, let us assume that the damage state w belongs to ${D}_{jy}^{\mathrm{B}},y={y}_{\mathrm{1}},\mathrm{\dots},w\mathrm{\dots},{y}_{{N}_{\mathrm{B}}}$. Equation (7) represents the conditional probability that the building j (of scheme B) can move to a progressive state w given the action of IM^{B} when it already presented a damage state y due to the action of IM^{A}. For such a process, note that it was already classified in terms of scheme B, thanks to the compatibilities between damage states.
The former expression defines a probabilistic statedependent fragility function composed of transition probabilities between increasing damage states (for instance, for scheme B, this description follows: ${y}_{{N}_{\mathrm{B}}}{y}_{{N}_{\mathrm{B}}\left({N}_{\mathrm{B}}\mathrm{1}\right)},{y}_{{N}_{\mathrm{B}}}{y}_{{N}_{\mathrm{B}}\left({N}_{\mathrm{B}}\mathrm{2}\right)}\mathrm{\dots})$. For a fragility model ${D}_{{T}_{r}^{i}}$ designed for a set of building types T_{r}, and composed of ${q}_{{N}_{i}}$ damage states (for any hazard of interest i), the required set of transition probabilities for a given range of hazard intensities are completely defined by a triangular number G_{f} as expressed in Eq. (8).
A visual example of such transition probabilities within fragility functions for several hazarddependent models (also including ${T}_{k}^{\mathrm{A}}$ and ${T}_{j}^{\mathrm{B}}$ as well as their respective sets of damage states ${z}_{{N}_{\mathrm{A}}}$ and ${y}_{{N}_{\mathrm{B}}}$) is presented in Fig. 3.
Only for the overall scope of this paper do we propose that statedependent fragility functions can be simplified by using ad hoc calibration parameters to modify these logarithmic mean values. For such a modification, we propose applying to them the exponential operator to obtain the physically accountable mean IM (hazard intensity measures). That is, ${\mathit{\lambda}}_{{q}_{\mathrm{0}}}\left({T}_{r}^{i}\right)$ defines each damage state as ${\mathit{\lambda}}_{{q}_{\mathrm{0}}}\left({T}_{r}^{i}\right)={e}^{{\mathit{\mu}}_{{q}_{\mathrm{0}}}\left({T}_{r}^{i}\right)}$. Subsequently we propose to obtain their respective differences $\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{\mathrm{0}}}$. For example, if a fragility function is composed of ${q}_{{N}_{i}}=\mathrm{4}$ damage states (excluding damage state 0, equivalent to no damage), there will be a set of damage states ${\mathit{\lambda}}_{{q}_{\mathrm{0}}}=\left[{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{1}}}},{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{2}}}},{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{3}}}},\mathrm{\dots},{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{i}}}\right]$ for which we should obtain the differences between all the possible top and bottom damage states and for which we must obtain six values: $\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{\mathrm{0}}}=\left[\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{1},\mathrm{2}}}},\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{1},\mathrm{3}}}},\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{1},\mathrm{4}}}},\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{2},\mathrm{3}}}},\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{2},\mathrm{4}}}},\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{{\mathrm{0}}_{\mathrm{3},\mathrm{4}}}}\right]$.
In this example, these six statedependent transition values are included within the G_{f}=10 triangular number (i.e. 4 from 0, 3 from 1, 2 from 2, 1 from 3) given by Eq. (8). Thereby, for each ${T}_{r}^{i}$, it is still necessary to determine the probabilistic representation (log mean and log standard deviation) of every damage state transition $\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{\mathrm{0}}}$. To do so, the ${\mathit{\lambda}}_{{q}_{\mathrm{0}}}\left({T}_{r}^{i}\right)$ values are proposed to be multiplied by the $\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{\mathrm{0}}}$ factors, and, by assuming that this product is a lognormally distributed random variable (ln operator), the values δ_{wy} will have a normal distribution. This is expressed as given by Eq. (9):
The reader should note that in this approach, the $\mathrm{\Delta}{\mathit{\lambda}}_{{q}_{\mathrm{0}}}$ values are a set of ad hoc calibration parameters or scaling factors that are applied directly to the ${\mathit{\lambda}}_{{q}_{\mathrm{0}}}$ for which each damage limit state was originally derived. The δ_{wy} values form the lognormal mean of the statedependent fragility functions. A similar approach was followed by Rao et al. (2017). The fragility functions used to constrain the statedependent fragility functions should have been derived only for the actual second acting hazard (i.e. farfield tsunamis). Thus, the use of those derived analytically is advised over empirical ones (which implicitly had the damage induced by ground shaking in their derivation). Further details about this approach and model assumptions to find the ad hoc calibration parameters are provided for the example case in the data repository in Gómez Zapata et al. (2022a).
2.4 Loss assessment for sequences of cascading hazard scenarios
We propose a simple economical consequence model that assigns the replacement cost ratios to every damage state of the building classes ${T}_{j}^{\mathrm{B}}$. The incremental economic loss, defined as the difference in the expected loss resultant from the initial damage state and final damage state, is calculated in terms of the reference scheme B as
Combining the two interscheme compatibility matrices, ($p\left({T}_{k}^{\mathrm{A}}\mathrm{}{T}_{j}^{\mathrm{B}}\right)$ and $\left({D}_{kz}^{\mathrm{A}}\mathrm{}{D}_{jy}^{\mathrm{B}}\right)$, along with Eqs. (7) and (10), we obtain the formulation in Eq. (11), which is identical to the one in Fig. 1b. This allows us to calculate the probability of observing an incremental loss due to the cumulative damage during the sequence of hazard scenarios.
Equation (11) represents the disaggregated loss caused by the triggered event upon the buildings with a preexisting damage (induced by IM^{A}). Finally, the likely loss for the entire sequence can be obtained by summing up Eqs. (1) and (11).
3.1 Context of the study area: metropolitan Lima, Peru
In 2022, Peru had a population of around 33 million people, with nearly 58 % of the population living in coastal communities (INEI, 2022). In Løvholt et al. (2014) it was stated that this country has the largest population exposed to tsunamis on the American continent. Lima, its capital, with nearly 10 million inhabitants (around onethird of the country's population), is home to the most important political, industrial, and economic activities of the country. Lima is ranked as the capital city exposed to the largest seismic hazard in South America (Petersen et al., 2018) and as the second city in the world in terms of the value of working days lost relative to the national economy due to earthquakes (Schelske et al., 2014). This city has suffered devastating disasters in the past. For instance, in 1586 and 1724 earthquakes triggered tsunami runups of over 24 m (Kulikov et al., 2005). The 1746 earthquake, with an estimated magnitude of M_{w} 8.8 (Jimenez et al., 2013), produced a tsunami with a local height of 15 to 20 m (Dorbath et al., 1990) and destroyed the city. In 1974, a M_{w} 8.1 event produced widespread damage and caused losses of ∼ USD 7.5 billion. Since then, the city has been experiencing continuous urbanisation with generally poor structural design (Tarque et al., 2019).
A tsunami event with similar characteristics as the one triggered by the 1746 earthquake was used for context in Adriano et al. (2014) to estimate the damage probabilities of the residential building stock of Callao (part of the metropolitan area of Lima), using the empirical tsunami fragility functions of Suppasri et al. (2013) for four building types. More recently, Ordaz et al. (2019) developed probabilistic earthquake and tsunami risk forecasts for Callao. However, that study neither described the vulnerability models used nor discussed the method employed to address the nonlinear damage accumulation. To the best of the authors' knowledge, neither cumulative damages due to earthquake and tsunami scenarios nor the use of analytical tsunami fragility functions for Lima have been reported in the scientific literature.
3.2 Scenarios of earthquakes and tsunamis for Lima
We use the dataset compiled by GomezZapata et al. (2021e), which is composed of six earthquakes with moment magnitudes ranging from 8.5 to 9.0 M_{w}, which were made available in GomezZapata et al. (2021c). In that dataset, each event is represented by an associated 1000 realisations of crosscorrelated ground motion fields (GMFs) for peak ground acceleration (PGA) and spectral accelerations (S.A.) at 0.3 and 1.0 s. The selection of these spectral periods depends on the fragility function's IM associated with the building classes of the exposure model (Sect. 3.3). The simulated GMFs were obtained making use of the ground motion prediction equation (GMPE) proposed in Montalva et al. (2017) and the spatially crosscorrelation model of Markhvida et al. (2018) employing the OpenQuake Engine (Pagani et al., 2014). For the site term of the GMPE, the dataset reported in Ceferino et al. (2018), which combined the slopebased ${V}_{{\mathrm{s}}_{\mathrm{30}}}$ values of Allen and Wald (2007) and a seismic microzonation (Aguilar et al., 2013), was used.
On the tsunami modelling side, we reuse the data repository of Harig and Rakowsky (2021) that compiles tsunami inundations for each of the mentioned six earthquakes using the finite element model TsunAWI. Similarly as performed by Harig et al. (2020), the inundation values were interpolated to a raster file with grid cell dimensions of 10×10 m. Figure 4 shows three of the tsunami inundation scenarios for the study area.
3.3 Exposure modelling: taxonomic description, interscheme conversion, and spatial aggregation of building classes for Lima
We make use of the existing building exposure models that represent the residential building stock of metropolitan Lima for groundshaking vulnerability that were developed by GomezZapata et al. (2021e) and are available from GomezZapata et al. (2021b). Such a building classification was defined by relating some covariates included within the last official Peruvian census from 2017 (INEI, 2017) at the block level with respect to 21 classes proposed by the South American Risk Assessment (SARA) project (YepesEstrada et al., 2017) through a mapping scheme proposed from expert elicitation (GEM, 2014). Since that information has been provided for dwellings, the socalled “dwelling ratios” proposed by SARA has also been implemented to obtain the building counts per class. A description of these building classes is presented in Table 1.
It is worth noting that although these typologies are similar to those of the SARA exposure model, there are differences between the building counts reported by that project and the adopted model. This might be due to the vintage of the input census datasets (2007 vs. 2017, respectively), the thematic detail induced by the spatial aggregation entities (districts/blocks/CVTs), having merged some building classes in terms of similar heights, and having reduced the number of unknown (UNK) types (∼91 % with respect the SARA model). The resultant exposure model is made up of ∼1 657 635 residential buildings, a 25 % increase with respect to the SAA model. However, as observed in Gómez Zapata et al. (2022b), this scheme does not properly capture the presence of highrise buildings, underestimating their presence, while overestimating the wooden types.
These SARA buildings are spatially aggregated onto central Voronoi tessellations (CVTs) to form seismicoriented exposure models. It is worth noting that the construction of such heterogeneous aggregation units was based on the selection of an underlying distribution that spatially combined and normalised two weighted map layers, namely (1) a tsunami inundation depth from a M_{w} 9.0 scenario (70 % weight) and (2) the population density at the block level (30 %). The resulting model provides higherresolution cells where both conditions are maximised, while coarser geocells occur when one can expect their absence. Further details about these models are available in GomezZapata et al. (2021a, b). Figure 5 shows the percentage of building typologies grouped by their main structural materials expected within each geocell.
The analytically derived set of seismic fragility functions by VillarVega et al. (2017) are assigned to every SARA class. They will be used to obtain the damage distribution for the crosscorrelated ground motions per earthquake scenario (Sect. 3.2). For this vulnerability assessment, we use the replacement cost as given by YepesEstrada et al. (2017) presented in Table 1. For their damage states, we assumed loss ratios of 2 %, 10 %, 50 %, and 100 %.
On the tsunami vulnerability side, we represent the commonly exposed residential building stock to earthquakes and tsunamis in terms of two classification schemes, namely the Suppasri et al. (2013) and Medina (2019) schemes, which provide sets of empirical and analytical fragility curves, respectively. The former one was made available for Lima in GomezZapata et al. (2021b) and is comprised of six typologies. Notably, its corresponding set of empirical tsunami fragility functions (with six damage states) was derived by implicitly addressing the damage induced by the ground shaking after the M_{w} 9.1 2011 Japan earthquake and tsunami. Due to this reason, the steps outlined in Sect. 2.2 and 2.3 are not developed for the Suppasri et al. (2013) scheme. Their related direct scenariobased loss estimates were reported in GomezZapata et al. (2021e) from the variations obtained from seven geographical entities used to spatially aggregate the residential building portfolio of Lima and were presented in Sect. 4 for comparative purposes in contrast with the offered method applied to the Medina (2019) scheme. This second type of classification is to the authors' knowledge the only available model that provides analytical farfield tsunami fragility functions for the South American Pacific Coast. It includes six typical buildings located in Tumaco (Colombia) initially defined in Medina (2019), which are generalised in this study. They are MPN (wooden), MMP (masonry), MPCP1T1 (framed reinforced concrete (RC), one storey with a similar length $/$ width ratio), MPCP1T2 (framed RC, one storey with a higher length $/$ width ratio), MPCP2 (framed RC, two storeys), and MPCP3 (framed RC, three or more storeys). Their associated set of fragility functions was developed following the method proposed in Medina et al. (2019) to define the structural fragility due to tsunami forces. A summary that regards the structural characteristics of these building types and the method adopted in deriving these models are provided in the data repository (Gómez Zapata et al., 2022c).
As explained in Sect. 2.1, every building class within the three schemes of interest is disaggregated into attributes within the GEM v.2.0 faceted taxonomy. As done in Gómez Zapata et al. (2022b), fuzzy compatibility levels between the attribute values and building classes are assigned through expert elicitation. Thereby, synthetic surveys based on the possible combinations of attributes that every building class may describe are employed to solve the compatibility scores and obtain the probabilistic interscheme compatibility matrices expressed by $p\left({T}_{k}^{\mathrm{A}}\mathrm{}{T}_{j}^{\mathrm{B}}\right)$ that is shown in Fig. 6b. By applying a dot product (Eq. 2) between this matrix and the building counts of the source scheme SARA, ${\left\{R\right\}}_{{T}_{k}^{\mathrm{A}}}$, (Fig. 6a), we can obtain the building counts under the tsunami classification scheme of Medina (2019) ${\left\{R\right\}}_{{T}_{j}^{\mathrm{B}}}$ as shown in Fig. 6c.
The interscheme conversion between SARA and the Suppasri et al. (2013) classes for Lima was reported in GomezZapata et al. (2021e). The replacement cost values of the building classes within the Medina (2019) scheme are assumed to be the same as the SARA class for which the largest compatibility value was obtained from the interscheme compatibility matrix (Fig. 6b). We have adopted identical loss ratios per damage limit state as the ones assumed for earthquake vulnerability. This decision is aligned with previous related studies; i.e. similar loss ratios were also adopted in Antoncecchi et al. (2020) to assess the vulnerability of buildings to tsunamis using empirical fragility functions. It is worth noting that only the commonly exposed buildings to each pair of hazard scenarios (i.e. intersection between the IM of Figs. 4 and 5) are considered for the assessment of cumulative damage after the cascading sequence.
3.4 The probabilistic description and compatibility of interscheme damage states for Lima
We obtain the interscheme damage compatibility matrices, $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{D}_{jy}^{\mathrm{B}}\right)$, following the method presented in Sect. 2.2 to probabilistically harmonise the damage states that define the fragility functions of A (SARA) and B (Medina). It is worth noting that although A and B comprise four damage states, they do not have a trivial equivalence. A defines a single damage criterion for the entire set of building classes closely following the proposal by Lagomarsino and Giovinazzi (2006) as a function of the yielding and ultimate spectral displacements. Conversely, B uses a buildingclassdependent parameterisation based on the HAZUS interstorey drift ratios to define the structural damage levels on precode structures.
First, we use the AeDES scale to score the admissible observable damage extension on individual building components (n in Eq. 3) through expert elicitation, which can jointly describe each buildingspecific damage state of every scheme's fragility functions (${D}_{kz}^{\mathrm{A}}$, ${D}_{jy}^{\mathrm{B}}$). Some examples of this procedure are displayed in Fig. 7. These heuristics configure the training datasets. Subsequently, we have configured the testing datasets. They are composed of a synthetic dataset of 10 000 exhaustive possible combinations of the observable AeDES scorebased damage extension {OD}_{n}. Thereby, the likelihood terms and $p\left({D}_{jy}^{\mathrm{B}}\mathrm{}{\left\{\text{OD}\right\}}_{n}\right)$ in Eq. (6) represent the probability of classifying each damage state ${D}_{kz}^{\mathrm{A}}$ and ${D}_{jy}^{\mathrm{B}}$ given the set of scored building components {OD}_{n}.
To obtain the likelihood terms of in Eq. (6), we have decided to use the Gaussian naïve Bayes supervised machine learning classification algorithm. It is available in the free software library scikitlearn for the Python programming language (Buitinck et al., 2013). This selection is suitable for our classification problem because the observable damage heuristics can be assumed as normally distributed continuous data. This can be intuitively observed from the heuristic shown in Fig. 7 where the central damage states (i.e. moderate and extensive) show wider ranges of combinations of observable damage with respect to the lowest (slight) and largest (collapse) states. For illustrative purposes, in Fig. 8 we show one of the possible sets for the likelihood probabilities predicted for each damage state described in terms of observable damage extension with respect to the AeDES scale upon two building components (VS, IP) for two materialbased typologies in the commonly exposed area to both perils, i.e. masonry and wooden structures (see Fig. 6a, c).
The marginal probability in Eq. (6), p({OD}_{n}), is assumed to be the proportion between one observation and the exhaustive combinations ($\mathrm{1}/\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{000}$). Thereafter, we have obtained the probabilistic interscheme damage matrix $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{D}_{jy}^{\mathrm{B}}\right)$ for each combination of building types from the two schemas (i.e. 21 SARA classes to 6 Medina classes =126 conversion matrices). Examples of the interscheme damage matrices are shown in Fig. 9 for three pairs of building types that had the highest interscheme compatibility values in Fig. 6b. Each of the 126 matrices that relates the damage states for each possible combination of building classes from the two schemas is subsequently weighted by the corresponding value of $p\left({T}_{k}^{\mathrm{A}}\mathrm{}{T}_{j}^{\mathrm{B}}\right)$, that is, by the probability of the building classes of the two schemas actually being descriptive of the same actual building (i.e. Fig. 6b). When considered in Eq. (11), the damagerelated matrices are maximised by the most compatible pairs of interschema building matrices (Fig. 6b). The scripts, heuristics, the final set of likelihoods, and the compatibility matrices are provided in Gómez Zapata et al. (2022c).
3.5 Tsunami statedependent fragility functions for Lima
We have followed the method presented in Sect. 2.3 to configure the statedependent fragility functions based on Scheme B (Medina) with associated analytical farfield tsunami fragility functions. The parameters that define the lognormal cumulative distributions for the four original damage states (assuming an initial undamaged state) as well as for the set of G_{f}=10 transition probabilities (from Eq. 8) are provided in the data repository (Gómez Zapata et al., 2022a). Figure 10 shows the analytical tsunami fragility functions (continuous lines) and statedependent fragility curves with their respective damage transitions (noncontinuous lines) for the six building classes.
From Fig. 10 it is possible to observe some features of the tsunami damagedstate fragility functions based on ad hoc calibration parameters (Sect. 2.3). For example, the masonry building class is the one most fragile to tsunami forces when in an undamaged state. Consequently, their associated statedependent fragilities are shifted towards the left side of the plot in quite an extreme fashion (Fig. 10a). This means that for that building type there is a higher probability for it to follow a longer damage progression after having been strongly affected by the seismic ground shaking (dotted and dashed lines). Conversely, for the wooden buildings (Fig. 10b), these are more likely to follow a damage progression than other classes if they were slightly affected by the shaking (see dashed lines). For the two onestorey RC building types assessed (MPCP1T1 and MPCP1T2), there are negligible differences between the transition probabilities D_{2}−D_{3} and D_{3}−D_{4}, as well as between D_{1}−D_{3} and D_{2}−D_{4}. Notably, the interdistances between these pairs of sets (of damage states) are of a similar order as the ones comprised by one and two damage state(s), respectively. This feature is neither present for the other RC buildings with increasing heights nor present for the wooden types. It is worth noting that these observations are fully dependent on the specific fragility models that we have implemented and the assumptions adopted to derive them (Eq. 10), and no further generalisations should be done until they can be further compared with other approaches and validated through other means.
3.6 Cumulative damage from consecutive ground shaking and tsunami scenarios in Lima
The spatially crosscorrelated ground motion fields (Sect. 3.2, Fig. 11a, b), along with the exposure model for seismic vulnerability and its corresponding sets of fragility functions (Sect. 3.3, Fig. 11d, e), are the first sets of inputs required by the engine DEUS (Brinckmann et al., 2021) to estimate the damage distribution and direct economic losses for the residential building stock of Lima after each of the six earthquake scenarios considered. DEUS is a software designed to compute scenariobased risk from any type of natural hazard over spatially aggregated building portfolios. This version of DEUS is an opensource Python programme whose number of executions is proportional to the consecutive risk scenarios.
As shown in Fig. 11f, g, the resulting damaged exposure model (after ground shaking) is used as input for a second execution to account for the cumulative damage induced by the corresponding tsunami scenarios. DEUS makes use of the two sets of interscheme compatibility matrices for buildings (Sect. 3.3) and damage states (Sect. 3.4) to change from the source earthquake reference scheme to the target tsunami reference scheme (see Fig. 11g). These are inputs together with the tsunami inundation heights (Sect. 3.2, Fig. 11c) and statedependent tsunami fragility functions (Sect. 3.5, Fig. 11h) for the second run of DEUS. This time, the damage states are updated in the building exposure model, delivering only the disaggregated damage and losses expected from the tsunami. Finally, the cumulative distribution of losses is obtained by adding the latter disaggregated tsunami losses with the initial results derived from the earthquake ground shaking (Fig. 11i).
The generated results are presented in the form of loss exceedance curves in Fig. 12. This figure reports the probability of exceeding the selected loss metric (replacement cost in USD) for the six earthquake and tsunami scenarios that might impact the portion of the residential building stock of Lima that is commonly exposed to each pair of hazard scenarios. This figure shows five sets of curves, hereby described:

Earthquake groundshakinginduced loss (blue curves). They represent the direct losses due only to seismic ground shaking using the SARA scheme (VillarVega et al., 2017). They are obtained through 1000 realisations of crosscorrelated seismic ground motion fields using the models described in Sect. 3.2.

Losses obtained from the sole use of empirical fragility functions as simulating a nearfield tsunami (red curves). These curves represent the losses from the cumulative effects of the shaking and the tsunami (without any possibility to separate both effects). Such loss prediction may be biased, since the empirical fragility functions of Suppasri et al. (2013) assuming an initial undamaged state (${D}_{k\mathrm{0}}^{\mathrm{A}}$) have not been validated for smaller or larger events. Similarly to what was concluded in GomezZapata et al. (2021e), we have also observed that as the earthquake magnitude increases, the differences between the two largest loss values in the curve (from the two finestresolution entities) are reduced.

Losses obtained from the sole use of analytical fragility functions as simulating a farfield tsunami (purple curves). They represent the direct losses obtained solely through the implementation of the analytical tsunami fragility tsunami Medina (2019), while assuming an initial undamaged state (${D}_{k\mathrm{0}}^{\mathrm{A}}$), thus neglecting seismic ground shaking. Similarly, as done for the former case (empirical functions), the reduced variability of these results was accounted for through computations using seven exposure models with variable spatial resolutions obtained from a recent study (GomezZapata et al., 2021e). This is a result of the lack of variability in the seismogenic parameters to vary only M_{w} and not having assumed distributions for slip rates but single values.

The losses related to the tsunami event obtained after using our method (statedependent fragility functions, orange curves). They represent the direct losses, which were only derived from the updated exposure model (i.e. with nonzero damage states). This means that these curves only represent tsunamiinduced losses for buildings that have already experienced earthquakerelated damage. These loss exceedance curves are constructed using Eq. (11). Thus, this procedure implied the interscheme building conversion $p\left({T}_{k}^{\mathrm{A}}\mathrm{}{T}_{j}^{\mathrm{B}}\right)$ derived in Sect. 3.3 and the interscheme damage state conversion $p\left({D}_{kz}^{\mathrm{A}}\mathrm{}{D}_{jy}^{\mathrm{B}}\right)$ obtained in Sect. 3.4, as well as the statedependent tsunami fragility functions constrained in Sect. 3.5.

Cumulative losses (our method) induced by the ground shaking and tsunami sequence (green curves). They represent the losses obtained by adding the shakinginduced losses (blue curves) with the aforementioned disaggregated tsunamiinduced losses (orange curves), that is, the outcome of the method proposed in this paper. These green curves represent, according to our approach, the likely losses that would be expected from each magnitudedependent scenariobased cascading sequence over the considered building stock.
Hereafter we describe some observations that arise from Fig. 12:

The resultant losses obtained after having used the two sets (empirical or analytical) of tsunami fragility functions (while assuming initial undamaged states) are profoundly different. As expected, the use of the empirical tsunami fragility model (red curves), for all the magnitudes, leads to larger values in comparison with the values obtained from the analytically derived fragility functions (purple). These differences increase with magnitude. This feature might arise from the fact that empirical fragility functions consider both earthquake and tsunami actions, while the purple curves consider only the effects of the tsunami, as well as because empirical fragility functions only account for flow depth as the IM. Conversely, the analytical fragility functions implemented were derived using the theoretical forces associated with the flow velocity tsunami waves as input in the generating numerical model. Similar observations regarding the reduction in the loss estimations when flow velocity is included have been drawn by other studies (e.g. Attary et al., 2021; Park et al., 2017).

We observe that the ground shaking dominates the losses at lower magnitudes (M_{w} 8.5, 8.6), while the tsunami, either from analytical (emulating farfield tsunamis) or empirical fragility functions (nearfield tsunamis), controls the losses for the rest of the scenarios with larger magnitudes. The former is aligned with the observations of Goda and De Risi (2018) and GomezZapata et al. (2021e) for the case of empirical tsunami models. Moreover, a similar trend is observed for the disaggregated tsunamiinduced losses (assuming initial nonzero damage) whose respective loss values (orange curves) are larger than the shakinginduced losses for M_{w} 8.8, 8.9, and 9.0. Hence, these features highlight that as the magnitude increases, there is an increasing comparative importance of the tsunami risk within the considered sequence of hazards.

Expected loss values from cumulative damages based on singlehazard vulnerability models (our method, green curves) are clearly different from the one produced by classical empirical tsunami models. Classical empirical tsunami fragility functions lead to considerably lower loss estimations for the lowmagnitude earthquakes and substantial larger estimations for the larger ones.

The differences between the loss exceedance curves derived from both sets of analytical fragility models (either from undamaged or with preexisting damage) are larger for the lower magnitudes (M_{w} 8.5, 8.6) and decrease with increasing magnitude. As the magnitude increases, there is an increasing tendency of convergence between these two loss curves (M_{w} 8.9, 9.0).

Consequently, since tsunamiinduced losses either from analytical fragilities (initial undamaged states) or from statedependent and interscheme models converge for the larger magnitudes (M_{w} 8.9, 9.0), their respective summations with the shakinginduced losses would be approximately similar at the largest probabilities of exceedance. Nevertheless, this observation needs to be better investigated through more exhaustive simulations of tsunami inundation per considered scenario.

Conversely, considering observation three, (i.e. as the magnitude decreases, the differences between purple curves and orange curves increase), their respective summations with the shakinginduced losses will lead to very different results. Hence, this observation suggests that, although earthquake and tsunami structural responses can be separately approximated for very large magnitudes, it is still required to address cumulative damages from the vulnerability interactions that are expected on the lowermagnitude earthquakes we have considered (i.e. M_{w} 8.5, 8.6).
When we consider analytical fragility functions with ${D}_{k\mathrm{0}}^{\mathrm{A}}$ that only emulate the damaging actions of farfield tsunamis (without any ground shaking), we observe that as the magnitude increases, their respective loss exceedance curves converge with the ones that assumed state dependency (${D}_{kz}^{\mathrm{A}})$. This is because, for the largermagnitude events, the damaging actions due to seismic ground shaking will correspondingly increase. Hence, the available probabilistic damage transitions from the damage states within the earthquake (source) to tsunami (target) schemes will consequently be reduced. Therefore, we observe that if farfield analytical tsunami fragility functions are used, their corresponding results will be very much alike, regardless of whether they are considered undamaged (${D}_{k\mathrm{0}}^{\mathrm{A}}$) or having preexisting damage $\left({D}_{kz}^{\mathrm{A}}\right)$. Therefore, for these largermagnitude events, regardless of which curve is summed up with the shakinginduced losses, the resulting loss distributions for the hazard sequence would lead to quite similar results. Thus, the implementation of state dependency on tsunami fragility may not be fully necessary to be addressed for very large earthquake magnitudes (M_{w} 8.9, 9.0). This observation is aligned with studies (i.e. Petrone et al., 2020; Rossetto et al., 2018) that suggest that earthquake and tsunami structural responses can be separately approximated. However, the former statement would not apply to the lowmagnitude earthquakes investigated in Lima for which the preexisting damage due to earthquakes must be addressed. No generalisations should therefore be made in this regard, with sensitivity analyses needing to be carried out in the future.
This study has proposed a modular method to disaggregate the direct losses expected for building portfolios exposed to consecutive hazardous scenarios of different natures in which their individual components could be individually improved. Therefore, future sensitivity analyses on some of the modules related to damage states would benefit from the understanding of how their embedded uncertainties would impact their corresponding results. We can mention the following:

The disaggregation of building classes into taxonomic attributes as presented in Sect. 2.1 is an important input to obtain the probabilistic interscheme compatibility matrices based on Gómez Zapata et al. (2022b). However, it is worth noting the shortcoming described by Charvet et al. (2017), referring to the generally poor taxonomic building characterisations of the currently available tsunami fragility models. They are, most of the time, only based on their main construction material, although sometimes they include the number of storeys, and rarely do they include other attributes such as the date of construction (e.g. Suppasri et al., 2015). When more enriched descriptions for tsunami vulnerability become available in the future, this approach will remain useful for similar purposes.

When/if local highquality empirical data collection and analytical models become available, they could be used to constrain the relationships between the failure mechanisms and attribute relevance to hazardrelated susceptibilities. This might contribute to enhancing the construction of heuristics that characterise the likely observable damage extent (per damage limit state, building type, and hazarddependent fragility model) that could be obtained through more refined approaches such as unsupervised machine learning. Its use applied on real datasets that document observations on building components (even different from the ones presented in Eq. 3) could contribute to refining statedependent tsunami fragility functions and restricting the heuristics on the likely observable damage (Sect. 2.2) and thus minimising subjective expert judgment. In this sense, it is worth noting that the set of predicted likelihood probabilities in the probabilistic compatibility degree between damage states from different hazard fragility functions that we derived from the synthetic datasets created through the heuristics and the AeDES scoring system is not unique, as they depend on the choice of machine learning technique and on the heuristics derived through expert elicitation. In this sense, we have documented a preliminary sensitivity analysis on such parameterisation in Gómez Zapata et al. (2022c). However, further investigation of the impact of such parameterisation is still advised.

As described by Hill and Rossetto (2008), we have observed that, when characterising damage states due to the impacts of natural hazards on buildings, there is still the need for standardisation in describing observable physical damage after any kind of hazardous event through the harmonisation of damage scales for data collection, not only regarding entire building units but also regarding the particular damage (and extent) experienced by certain individual components. In this regard, although we have used the AeDES scale, other damage scales could be more suitable to describe the observable damage to some building classes than for others (Hill and Rossetto, 2008; Turchi et al., 2022). Nonetheless, the choice of a standard scale to transversally describe any observable set of damage on buildings will benefit the research in multihazard vulnerabilities.

The integration of economic consequence models for physical vulnerability based on the replacement costs as a function of the buildings' area, as for instance presented in Triantafyllou et al. (2019), for tsunami vulnerability is worth testing. This also depends on the available data, and it is out of the scope of this paper, but it would be worth exploring their contribution once more refined estimations about replacement cost are available for Lima. Nevertheless, one should be aware of the uncertainties involved in largescale building exposure models.

The derivation of the hazard intensities could also benefit from future enhancements. For instance, the GMPEbased seismic accelerations derived from a simplified ${V}_{{\mathrm{s}}_{\mathrm{30}}}$ site grid of ∼1 km might be too coarse to capture local site effects in the expected ground motions. However, the performance of siteresponse analyses that account for the local geotechnical soil properties of sitespecific soil profiles, as for instance reported by Aguilar et al. (2019), is a computationally demanding task that is out of the scope of this study, but when integrated it could benefit the overall quality of seismic risk calculations for the study area. Complementarily, we strongly advise to generate exhaustive sets of crosscorrelated ground motion fields (at the required spectral periods by the buildings classes) to address their aleatory uncertainty. The selection of this model, among the available ones, carries epistemic uncertainties.
It is worth noting that the variability of the loss exceedance curves obtained for the cumulative damage (due to tsunamis) was derived from the damaged exposure models subjected to each realisation of crosscorrelated ground motion fields (i.e. orange curves in Fig. 12). Therefore, investigating the impact of other tsunami vulnerability and hazard data products (Behrens et al., 2021), which was beyond the scope of this paper, is nonetheless worth exploring. When such parameterisation in the tsunami data products becomes available for Lima, future studies could provide dimensionality of the contribution of the tsunami hazard upon the outlined method for scenario multirisk estimates.
For the commonly exposed residential building stock of Lima exposed to both perils, we have observed that assuming initial undamaged states in the selected tsunami empirical fragility functions leads to large underestimations for lower magnitudes (M_{w}) and large overestimations for larger M_{w} events in comparison to when statedependent models were used. Hence, the initial undamaged state assumption used to assess the tsunami vulnerability in former studies (e.g. Adriano et al., 2014; GomezZapata et al., 2021e) may not be completely accurate to represent the losses expected after this type of cascading sequence. This is because such an assumption misses the calculation of earthquakerelated damage, which is an important input needed to assess cumulative damage and losses through statedependent analytical fragility models. On the other hand, adopting the larger value between independent earthquake and tsunami risk computations proposed by Goda and De Risi (2018) may lead to better correspondence with our model (mostly for the lower M_{w} events) than the sole use of the selected nonstatedependent analytical fragility functions.
To give a perspective on the importance of addressing cumulative damage and losses for building stocks, let us recall some of the findings of the available studies of GomezZapata et al. (2021e) and Markhvida et al. (2017). They investigated the likely economic losses of the entire residential building portfolio of Lima and Callao solely after seismic ground motion from a M_{w} 8.8 scenario, addressing the variability induced by the same crosscorrelation model we have implemented herein. In the first study, ∼1 657 635 residential buildings were considered, and both studies considered the SARA building classes and fragility functions, similar to what we have done. Both studies reported mean loss values of around USD 7 billion and a maximum of around USD 35 billion (among a stochastic sample of events). It is then interesting to compare such a range of values with the mean loss values reported for a similar M_{w} (Fig. 11d). Notably, the forecasted losses per event (shaking and tsunami) and inferred from cumulative damage were derived from the much smaller commonly exposed building stock to each pair of hazard scenarios (see Fig. 8c), which constitute ∼21 209 buildings. This means that the building count for the entire residential stock of Lima (Fig. 6a) is around 78 times larger than that commonly exposed to both perils (Fig. 6b). Hence, we can note the important role of tsunamiinduced losses in the study area. The mean losses expected from the cascading sequence of that M_{w} 8.8 (i.e. value for the 50th percentile on the green curve in Fig. 11b) is ∼ USD 0.75 billion and a maximum of around USD 0.94 billion. Therefore, given the difference between the size of both building portfolios, finding out that the losses for the entire city are expected to be only 9 times larger than the ones forecasted after the action of both earthquakes and tsunamis tells us the crucial importance of carefully addressing the cumulative damage due to tsunamis in the study area. Moreover, this tell us that, besides all of the secondary effects of the tsunami, these types of future scenarios in Lima will constitute not only a huge source of direct economic losses for building portfolios but also uncertainties due to the lack of data to calibrate or validate these types of risk assessment after the action of cascading hazards.
We have proposed a modular method that allows us to consistently reuse existing singlehazard fragility models that are being developed by experts in various research fields and integrate them for multihazard risk assessment for extended building portfolios. This integration aims for the probabilistic harmonisation of diverse hazarddependent building classes and damage states, which are included in their associated fragility functions. Through this integration, we aim to provide an alternative approach to conventional ones (e.g. HAZUSMH; FEMA, 2003, 2017) that considers a single building class with sets of fragility functions for a variety of hazards. In this sense, the method we have developed can be particularly useful to assess the cumulative damage in hazard sequences of different natures and forces that might induce various failure mechanisms upon the exposed buildings. Thereby, the presented integrative method contributes to reducing the existing gaps due to the typical lack of collective calibration and validation of multihazard risk methods. This is due to, for instance, when triggered events act on damaged assets right after the first hazard or even simultaneously, thus experiencing compound hazards with no time for damage reconnaissance or disaggregation of the damage features induced by the individual hazards.
We have proposed a modular method composed of the following components:

Existing hazarddependent vulnerability schemes are selected to model the building portfolio under each hazarddependent vulnerability scheme of interest. They contain sets of building classes and associated fragility functions. To model the physical vulnerability of the building portfolio towards the triggering event (in this case, earthquake), there is no preference on whether empirical or analytical fragility functions should be used.

On the other hand, to model the physical vulnerability of the building stock towards the triggered event, sets of statedependent fragility functions must be derived for each building type within the selected scheme. For this purpose, it is important to use models that do not involve the damaging effects of the triggered event as the starting point (i.e. avoiding empirical models and using analytical ones). This proposal overcomes the assumption of initial undamaged states for the structures exposed to the triggered event and allows us to account for the differential cumulative damage between hazards.

The building classes are characterised through their disaggregation into building taxonomic attributes. This description allows for the harmonisation between the building classes belonging to different hazarddependent vulnerability schemes through the probabilistic interscheme compatibility matrix proposed in Gómez Zapata et al. (2022b).

The exposure models are spatially aggregated into optimal geographic entities (i.e. CVTbased models) that account for the spatial variability of lowcorrelated hazard IM in their derivation (GomezZapata et al., 2021e). This selection was taken due to performance purposes only, but a more refined blockbased model could also have been used.

A generalised description of the damage states is given based on a set of observable damage types on individual building components. This is done through a scoring system based on an underlying common scale (employing, for example, the AeDES form) that ultimately allows us to get the damage state interscheme conversion. We use the total probability theorem, a Bayesian formulation, and machine learning techniques.

A vulnerability assessment is conducted for sequences of cascading hazard scenarios through the proposal of consistent economical consequence models across hazarddependent vulnerability schemes. They must define replacement cost ratios per damage state and per fragility function associated with each vulnerability scheme.
The joint combination of these components creates a method to update the damage states throughout the multihazard sequence while allowing us to exploit existing hazardspecific riskoriented taxonomies (i.e. building classifications with corresponding fragility functions and defined damage states) available in the literature for a wide range of natural hazards. This is a modular method in which each one of their individual components can be separately customised when seeking future improvements.
When applying this method on the residential building stock of Lima (Peru), we have observed, on the one hand, that considering the risk metrics from tsunami vulnerability only from the selected set of empirical fragility functions (derived from nearfield tsunamis) as representative of the shaking and tsunami sequences leads to underestimations for the lower magnitudes. On the other hand, we have observed overestimations for the largermagnitude scenarios in comparison with the statedependent method that accounts for the accumulated damage due to the former earthquake solicitations. We have observed that the use of the proposed method to assess the cumulative damage is more relevant for the lowermagnitude scenarios than we have considered (M_{w} 8.5 and 8.6). This might be due to the greater damage extension on the exposed buildings that is expected from the seismic demands in comparison with those imposed by their corresponding tsunamis, and thus there is a greater chance to obtain cumulative damage. On the contrary, for larger magnitudes, the use of statedependent fragilities and analytical functions assuming no preexisting damage is converging, and thus the importance of assessing statedependency is reduced.
Considering the limitations and simplifications assumed in this study, we are not claiming that the resulting economic losses we have calculated for the residential building stock of Lima from multihazard scenariobased risk computations are totally exhaustive. Thus, caution should be taken with the interpretation and extrapolation of these conclusions to other study areas and combinations of models. Nevertheless, awareness of these uncertainties for the reliable quantification of risk towards these cascading hazards is increasingly important to enhance mitigation strategies for disaster risk reduction (Imamura et al., 2019). Furthermore, it is worth recalling that the method herein proposed has been exclusively designed for spatially extended residential buildings as proof of a concept for integrating existing fragility models. We do not provide a complete validation of multivulnerability approaches, but rather we offer a holistic and novel harmonising method to track such dynamics in a consistent manner. Hence, our method is not meant to replace more detailed analytical analyses required to determine the structural response of individual buildings subjected to seismic and tsunami loading (e.g. Petrone et al., 2017; Rossetto et al., 2019).
The data used in the elaboration of this study are available in open repositories. The scenariobased ground motions and tsunami inundation maps are available in GomezZapata et al. (2021c, https://doi.org/10.5880/riesgos.2021.008) and Harig and Rakowsky (2021, https://doi.org/10.5880/riesgos.2021.010), respectively. The first set was calculated making use of the Shakyground script (Weatherill et al., 2021, https://doi.org/10.5880/riesgos.2021.004), which relies on the OpenQuake Engine (Pagani et al., 2014, https://doi.org/10.1785/0220130087), while the second set was calculated using the TsunAWI software. The exposure and fragility models for both hazardvulnerability schemes (earthquake and tsunami) are available in GomezZapata et al. (2021a, https://doi.org/10.5880/riesgos.2021.006; b, https://doi.org/10.5880/riesgos.2021.007) and were adapted to fulfil the data formats required by the scripts provided by AssetMaster and ModelProp (Pittore et al., 2021, https://doi.org/10.5880/riesgos.2021.005). They were used as inputs for the scenariobased seismic risk assessment (Sect. 3.6) using the DEUS software (Brinckmann et al., 2021, https://doi.org/10.5880/riesgos.2021.011). The scenariobased risk estimates for earthquakes and tsunamis using analytical and empirical fragility functions, respectively, are provided in GomezZapata et al. (2021d, https://doi.org/10.5880/riesgos.2021.009).
JCGZ and MP conceived the study. JCGZ wrote the manuscript. JCGZ conducted the data processing and formal analysis. NB contributed to software development. JLM and SM developed analytical tsunami fragility functions. NT contributed to the synthetic scorings of observable damage per building typology. MP and FC supervised the study and reviewed the manuscript. All authors agreed to the published version of the manuscript.
The contact author has declared that none of the authors has any competing interests.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the special issue “Multirisk assessment in the Andes region”. It is not associated with a conference.
We want to express our gratitude to Andrey Babeyko, Michael Haas, Michael Langbein, Jörn Lauterjung, Giuseppe Nicodemo, Juan PáezRamírez, Juan Palomino, Matthias Rüster, and Sandra SantaCruz for their support during the elaboration of this study. Thanks to Sven Harig and Natalja Rakowsky for having provided us with the tsunami inundation models for Lima. We also thank Henning Lilienkamp and Graeme Weatherill for their support with the simulation of spatially correlated ground motion fields and machine learning techniques. We thank Kevin Fleming for the careful proofreading in a previous draft of the manuscript.
This research has been supported by the German Federal Ministry of Education and Research (BMBF, Bundesministerium für Bildung und Forschung) through the RIESGOS and RIESGOS 2.0 projects (grant nos. 03G0905AH and 03G0876AJ). These projects are part of the funding programme CLIENT II – International
Partnerships for Sustainable Innovations.
The article processing charges for this openaccess publication were covered by the Helmholtz Centre Potsdam – GFZ German Research Centre for Geosciences.
This paper was edited by Elisabeth Schoepfer and reviewed by two anonymous referees.
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