Multi-hazard risk assessments for building portfolios
exposed to earthquake shaking followed by a tsunami are usually based on
empirical vulnerability models calibrated on post-event 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 single-hazard vulnerability models
that are constantly being developed and calibrated by experts from various
research fields to be used within a multi-risk context. This method is based
on the proposal of state-dependent fragility functions for the triggered
hazard to account for the pre-existing damage and the harmonisation of
building classes and damage states through their taxonomic characterisation, which is transversal to any hazard-dependent 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 (

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). Well-known 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, multi-hazard 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 multi-risk evaluations can only be conducted if both (1) multi-hazard (e.g. Marzocchi et al., 2012; Liu et al., 2016) and (2) multi-vulnerability 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 scenario-based multi-hazard 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 single-hazard vulnerability models that are being constantly developed and calibrated by experts from various research fields to be used within a multi-risk context. The rest of this Introduction discusses the state-of-the-art exposure modelling for large-scale building portfolios for multi-hazard 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 multi-hazard 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 HAZUS-MH (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 (EMS-98) 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. Gomez-Zapata 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.

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 non-structural components that jointly describe a certain damage state can have contrasting descriptions across various hazard-dependent vulnerability types (Gehl
and D'Ayala, 2018; Figueiredo et al., 2021), and there is often not a

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 post-near-field 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 tsunami-generating 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 large-scale 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 far-field tsunamis, and for near-field 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 large-scale 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 multi-risk 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 material-based 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 non-linear damage accumulation on the same exposed assets during the multi-hazard sequences (Merz et al., 2020).

This study proposes a modular method to probabilistically integrate existing sets of single-hazard 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 single-hazard 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 damage-updated 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 damage-updated 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 state-dependent 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 mega-thrust 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 state-dependent 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 state-dependent 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

To assess the expected damage state after the first hazardous event (e.g.
ground shaking), we apply their fragility function

If this damaged building portfolio is subjected to the action of a second
scenario with a hazard intensity

The first module is inter-scheme compatibilities between each hazard-dependent exposure classification scheme

The second module is the related compatibility levels between inter-scheme damage states

This second risk calculation is performed by using a third module that refers to generic state-dependent tsunami fragility functions (i.e. with non-zero initial damage states made of new curves that represent the permissible damage progression). Since the resultant earthquake-induced damages are formerly expressed in the tsunami vulnerability domain (step 2), the non-zero damage limit states of this set of state-dependent tsunami fragility functions will implicitly account for such pre-existing damage. The joint ensemble of these three components can be ultimately used to calculate the cumulative expected damages after the triggered event with

For multi-risk assessment a fourth module that represents the incremental loss obtained from the economic consequence model attached to the classification scheme

These four modules are described hereafter.

The classified building stock under the first hazard-dependent
classification scheme

As shown in Pittore et al. (2018), every building class

We consider how the fragility functions associated with

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

Scale to assess the damage level on buildings as proposed by the AeDES form. Reprinted from Baggio et al. (2007).

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

We assume that the representations of damage states within the two
considered schemes are conditionally independent (

We obtain a probabilistic compatibility degree between damage states
(

The terms

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

The former expression defines a probabilistic state-dependent fragility
function composed of transition probabilities between increasing damage
states (for instance, for scheme B, this description follows:

A visual example of such transition probabilities within fragility functions
for several hazard-dependent models (also including

Example of a set of damage-state-dependent fragility functions for several single-hazard fragility functions comprising progressive transition probabilities. Figure modified from Gómez Zapata et al. (2020).

Only for the overall scope of this paper do we propose that state-dependent
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,

In this example, these six state-dependent transition values are included
within the

The reader should note that in this approach, the

We propose a simple economical consequence model that assigns the replacement cost ratios to every damage state of the building classes

Combining the two inter-scheme compatibility matrices, (

Equation (11) represents the disaggregated loss caused by the triggered event upon the buildings with a pre-existing damage (induced by

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 one-third 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 run-ups of over 24 m (Kulikov et al., 2005). The 1746
earthquake, with an estimated magnitude of

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 non-linear 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.

We use the dataset compiled by Gomez-Zapata et al. (2021e), which is composed of six earthquakes with moment magnitudes ranging from 8.5 to 9.0

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

Expected tsunami inundation heights (TIHs) in metres (m)
for three out of the six considered scenarios per moment magnitude
(

We make use of the existing building exposure models that represent the residential building stock of metropolitan Lima for ground-shaking vulnerability that were developed by Gomez-Zapata et al. (2021e) and are available from Gomez-Zapata 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 (Yepes-Estrada et al., 2017) through a mapping scheme proposed from expert elicitation (GEM, 2014). Since that information has been provided for dwellings, the so-called “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.

SARA building classes proposed for the residential building stock of metropolitan Lima and Callao, with their respective replacement costs per building unit (repl. cost USD per bdg.) as reported in Yepes-Estrada et al. (2017) in the frame of the SARA model released by the Global Earthquake Model (GEM) in 2015, which was based on official census data reported by INEI (2007). The intensity measures (IMs) of the associated seismic fragility functions to each building class, as reported in Villar-Vega et al. (2017), are also provided.

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
(

These SARA buildings are spatially aggregated onto central Voronoi
tessellations (CVTs) to form seismic-oriented 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

Spatial distribution of the percentage of the main structural material of the residential buildings in metropolitan Lima in each central Voronoi tessellation (CVT) geocell using the dataset of Gomez-Zapata et al. (2021b). The
colour scale represents the material type:

The analytically derived set of seismic fragility functions by Villar-Vega et al. (2017) are assigned to every SARA class. They will be used to obtain the damage distribution for the cross-correlated ground motions per earthquake scenario (Sect. 3.2). For this vulnerability assessment, we use the replacement cost as given by Yepes-Estrada 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 Gomez-Zapata 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

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 inter-scheme compatibility matrices expressed by

Classification of the buildings in the maximum-exposed
area to both perils (

The inter-scheme conversion between SARA and the Suppasri et al. (2013) classes for Lima was reported in Gomez-Zapata 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 inter-scheme 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.

We obtain the inter-scheme damage compatibility matrices,

First, we use the AeDES scale to score the admissible observable damage
extension on individual building components (

Examples of the AeDES-based heuristics (see original AeDES form of Baggio et al., 2007, on Fig. 2) that describe the expected observable damage on the four selected building components listed in Eq. (3) (vertical structure, VS; floor, FL; roof, RF; infills and partitions, IPs) using the scale from I–A (i.e. I

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 scikit-learn 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 material-based typologies in the commonly exposed area to both perils, i.e. masonry and wooden structures (see Fig. 6a, c).

Predicted likelihood probabilities of classifying each
damage state of two building types that belong to the earthquake-oriented
(EQ) vulnerability scheme SARA (

The marginal probability in Eq. (6),

Probabilistic inter-scheme damage compatibility matrices
for three pairs of building classes:

We have followed the method presented in Sect. 2.3 to configure the state-dependent fragility functions based on Scheme B
(Medina) with associated analytical far-field 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

Analytical tsunami fragility functions with initial undamaged states as proposed by Medina (2019) (continuous lines) and derived state-dependent fragility curves (non-continuous lines) in terms of flow depth (m) as IM for six building classes:

From Fig. 10 it is possible to observe some features of the tsunami damaged-state 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 state-dependent 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 one-storey RC building types assessed (M-PCP1-T1 and M-PCP1-T2), there are negligible differences between the transition probabilities

The spatially cross-correlated 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 scenario-based risk from any type of natural hazard over spatially aggregated building portfolios. This version of DEUS is an open-source Python programme whose number of executions is proportional to the consecutive risk scenarios.

Proposed workflow for multi-risk assessment in Lima from
each pair of consecutive earthquake and tsunami scenarios. A

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 inter-scheme 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 state-dependent 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:

Five loss exceedance curves for the residential building portfolio of Lima are presented in six subplots per earthquake magnitude scenario (

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 (

Expected loss values from cumulative damages based on single-hazard 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 low-magnitude 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 pre-existing damage) are larger for the lower magnitudes (

Consequently, since tsunami-induced losses either from analytical fragilities (initial undamaged states) or from state-dependent and inter-scheme models converge for the larger magnitudes (

Conversely, considering observation three, (i.e. as the magnitude decreases, the differences between purple curves and orange curves increase), their respective summations with the shaking-induced 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 lower-magnitude earthquakes we have considered (i.e.

When we consider analytical fragility functions with

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 inter-scheme 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 high-quality empirical data collection and analytical models become available, they could be used to constrain the relationships between the failure mechanisms and attribute relevance to hazard-related susceptibilities. This might contribute to enhancing the construction of heuristics that characterise the likely observable damage extent (per damage limit state, building type, and hazard-dependent 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 state-dependent 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 multi-hazard 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 large-scale building exposure models.

The derivation of the hazard intensities could also benefit from future enhancements. For instance, the GMPE-based seismic accelerations derived from a simplified

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 cross-correlated 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 multi-risk 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 (

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 Gomez-Zapata 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

We have proposed a modular method that allows us to consistently re-use existing single-hazard fragility models that are being developed by experts in various research fields and integrate them for multi-hazard risk assessment for extended building portfolios. This integration aims for the probabilistic harmonisation of diverse hazard-dependent 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. HAZUS-MH; 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 multi-hazard 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 hazard-dependent vulnerability schemes are selected to model the building portfolio under each hazard-dependent 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 state-dependent 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 hazard-dependent vulnerability schemes through the probabilistic inter-scheme compatibility matrix proposed in Gómez Zapata et al. (2022b).

The exposure models are spatially aggregated into optimal geographic entities (i.e. CVT-based models) that account for the spatial variability of low-correlated hazard IM in their derivation (Gomez-Zapata et al., 2021e). This selection was taken due to performance purposes only, but a more refined block-based 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 inter-scheme 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 hazard-dependent 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 multi-hazard sequence while allowing us to exploit existing hazard-specific risk-oriented 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 near-field 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
larger-magnitude scenarios in comparison with the state-dependent 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 lower-magnitude
scenarios than we have considered (

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 multi-hazard scenario-based 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 multi-vulnerability 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 scenario-based ground motions and tsunami inundation maps are available in Gomez-Zapata et al. (2021c,

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 “Multi-risk 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 Páez-Ramírez, Juan Palomino, Matthias Rüster, and Sandra Santa-Cruz 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. 03G0905A-H and 03G0876A-J). These projects are part of the funding programme CLIENT II – International Partnerships for Sustainable Innovations.The article processing charges for this open-access 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.