Existing methods for multi-hazard risk assessment can be classified into qualitative, semi-quantitative and quantitative (Kappes et al., 2012; Zharikova et al., 2020). For qualitative methods the hazard intensity and recurrence interval limits are predefined and are then used to classify each of the perils into several classes. The value boundaries are usually determined by the location or objectives of the assessment. However, it is difficult to compare information from different sources in these types of assessments since most assessments have different criteria (Marzocchi et al., 2009).

Semi-quantitative methods typically use index-based approaches to represent risk on a standardized numerical scale (Kappes et al., 2012). With indices, the risk is given a score usually on a continuum. For example, this score could be calculated by multiplying the frequency of the peril per annum by the number of people (or the area) affected by it. If the index is needed for multiple perils combined, they are usually summed together from the individual perils. Both qualitative and semi-quantitative methods treat multiple perils as being independent. It is therefore difficult to consider the relationship between the perils within these frameworks.

In the context of risk reduction, quantitative methods offer advantage with the ability to quantify impact or risks and evaluate risk reduction measures thereby supporting decision making. Tilloy et al. (2019) did an extensive review of quantitative methods dealing with interconnected hazard and classified quantitative approaches into three techniques: stochastic, empirical, and mechanistic. Stochastic methods consider the statistical dependency between occurrence and intensity of hazards. For stochastic methods, the purpose is either to model the distribution of each individual hazard (Hao and Singh, 2016; Liu et al., 2018), or to model the joint probability between hazards (Sadegh et al., 2018). The empirical method uses existing datasets to fit probability distributions. The accuracy of the fit can be quantified using dependence measures like Pearson or Spearman (Czado, 2019), or regressions (Petroliagkis et al., 2016). The limitation of the empirical method is its dependency on data, and data is often limited in number due to rarity of mega events occurrence and damage information recorded. Mechanistic methods have been widely used by risk scientists for computing the potential impact of natural hazard events. However, the intensive nature of the computation makes them difficult to use for probabilistic risk assessment and hence there can be bias toward a particular set of outcomes (Geist et al., 2009). Tocchi et al. (2023) propose a multi-risk framework by integrating probabilistic hazard data (earthquakes and floods) with exposure and physical/social vulnerability indicators at the municipal level. A weighted, normalized aggregation produces a composite risk index to support disaster risk reduction prioritization. Recent reviews have also highlighted the need for probabilistic frameworks capable of representing cascading hazard interactions and propagating uncertainty consistently across hazard, vulnerability, and loss models (Goda and De Risi, 2023; Behrens et al., 2021).

Existing probabilistic earthquake multi-hazard studies have progressively expanded the range of hazards considered; however, they generally involve a trade-off between the number of cascading hazards represented, the treatment of uncertainty, and the ability to quantify long-term probabilistic risk.

Probabilistic earthquake risk assessments generally focus on one earthquake peril, typically seismic stochastic models with many of the parameters constrained through empirical methods and, sometimes, physical modelling. For probabilistic seismic hazard assessments, it is common practice to model the expected level of ground shaking and its variability due to site effects and aleatory uncertainty using Ground Motion Prediction Equations (GMPEs) across a region. Examples of these types of models span local (e.g., Burbidge et al., 2019), national (e.g., Burbidge, 2012; Leonard et al., 2014; Gerstenberger et al., 2022) to global scale (e.g., Johnson et al., 2023).

It is much rarer for a probabilistic earthquake impact or risk assessment to consider a combination of earthquake induced perils. The most common method to incorporate the cascading effects of earthquakes is to focus on one or two earthquake scenarios and cascades (e.g., Power et al., 2018; Burbidge et al., 2019; Kianirad et al., 2019). These are often done for a particular cascade of secondary effects (e.g., Power et al., 2018) in a deterministic or expert judgment-based approach or are done probabilistically but without fully quantifying the uncertainty and variability in the final loss estimates (e.g., Kianirad et al., 2019). The paper by Iannacone et al. (2024) introduces a simulation-based methodology to generate multi-hazard event sets over a system's life cycle, accounting for Level 1 (occurrence-based) interactions. Utilizing competing Poisson processes and sequential Monte Carlo sampling, the approach models concurrent, triggering, and altering interactions among hazards, producing realistic event sequences.

Two earthquake-induced perils that have been combined to calculate the combined probabilistic losses are those from strong ground shaking and tsunami (e.g., Goda 2020). In Goda (2020) the earthquake occurrence was represented by a set of multiple renewal models, implemented using a logic-tree approach, whereas earthquake rupture characterization is based on stochastic source models with variable fault geometry and heterogeneous slip distribution. By integrating these hazard components with seismic and tsunami fragility functions, the author calculated the time dependent multi-hazard loss potential from the combined impact of ground shaking and tsunami for some coastal communities in Japan. However, the framework was limited to the direct impacts of ground shaking and tsunami, and did not account for earthquake-induced ground failure processes such as liquefaction, lateral spreading, and landslides. Consequently, it did not capture the full cascade of earthquake-induced hazards that can substantially influence losses, particularly in regions such as New Zealand where these secondary hazards are common. More recently, Goda and De Risi (2023) reviewed the evolution of earthquake–tsunami catastrophe modelling and identified the explicit representation of cascading and compounding hazards as one of the key directions for future probabilistic risk assessment.

One probabilistic approach that captures the uncertainties in the ground motion modelling and its variability across the sites, the triggered cascading hazards, and the uncertainties in the asset's performance for a specific earthquake is presented by Moratalla and Uma (2023). In that study they first created an event tree of possible cascades from a Mw 8.4 earthquake offshore Napier and then used Monte Carlo Simulation to determine a range of possible series of disruption outcomes to the road network for that specific earthquake. The study included the combined effect of shaking, liquefaction, lateral spreading, landslides and building collapse on the road network, but did not include tsunamis or consider a range of possible earthquakes. It thus could not calculate impact metrics such as annualized probabilistic loss or risk exceedance curves from the combined effects of these earthquakes induced perils which require assessment of the risk across a range of earthquakes, rather than just one scenario.

Dunant et al. (2021) proposed a novel method for probabilistic cascading multi-hazard risk assessments using graph theory. The framework was tested in Franz Josef, New Zealand, and included impacts from earthquakes, landslides, and floods to housing and road infrastructure. One major advance provided by that study was the inclusion of non-dependent perils such as geological and meteorological hazards in a multi-hazard approach capable of combining them. However, the emphasis of that framework was on representing interactions between different hazard types rather than preserving the realization-by-realization conditional dependence between earthquake source characteristics, cascading hazard intensities, and asset damage.

This study develops a probabilistic framework for quantifying the combined physical damage and loss ratios arising from earthquake-induced cascading hazards, including ground shaking, tsunami inundation, liquefaction, lateral spreading, and landslides. The framework propagates uncertainties consistently through the hazard and damage modelling chain using a fully correlated Monte Carlo simulation, enabling the estimation of physically consistent multi-hazard risk metrics from a stochastic earthquake catalogue.

The framework is demonstrated through a case study of residential buildings in Napier City, New Zealand, exposed to earthquakes generated by the Hikurangi Subduction Zone (Sect. 3). The resulting risk metrics support applications including land-use planning, emergency management, resilience assessment, mitigation prioritization, and insurance decision-making.

In this study, we propose a Monte Carlo Simulation (MCS)-based framework to estimate the impact of earthquakes and their cascading hazards on the built environment. The framework is designed to model the effects of ground shaking, liquefaction, tsunami inundation, lateral spreading, and earthquake-induced landslides, using a synthetic catalogue of earthquakes. The results generated include Damage States (DS), which represent discrete levels of structural damage ranging from slight to complete. These are harmonized across all perils using a unified five-level classification developed in this study, based on existing fragility models such as those in FEMA (2013) and Suppasri et al. (2012). In addition, the framework estimates Mean Damage Ratios (MDRs) – representing the proportion of repair cost relative to full replacement – assigned to each harmonized DS using values derived from the HAZUS methodology.

The key advantage, and novelty, of the proposed framework lies in the generation of hazard intensities that are internally consistent across all perils within each simulated earthquake event. Here, internal consistency refers to the conditional dependence of all cascading hazards on a common set of event-specific ground motion parameters and spatial footprint, ensuring coherence across perils and locations for a given realisation. In contrast to approaches that sample hazards independently, this framework derives secondary perils (e.g., liquefaction, landslides, lateral spreading) directly from the primary hazard fields (ground shaking and tsunami), which serve as inputs to both hazard-specific models and subsequent fragility assessments.

As a result, all sampled hazard intensities and damage states follow a correlated cascade, where dependencies are preserved throughout the modelling chain. For example, damage arising from landslides and lateral spreading is jointly conditioned on the same underlying ground motion intensity (e.g., PGA), rather than being treated as independent realisations. This approach reduces artificial variability introduced by independent sampling and produces more physically consistent event scenarios. The process is repeated across all stochastic event sets (SES), enabling the construction of long-term, statistically robust hazard and damage distributions that capture both frequency and interdependence among hazards.

The implementation of the framework requires the 6 following components:

Earthquake Catalogue. A stochastic catalogue of earthquakes affecting the study area, including rupture geometries, magnitudes, and recurrence information. For example, in the case study presented in this study we use a 100 000-year synthetic catalogue of Hikurangi Subduction Zone (HSZ) earthquakes, developed using Event-Based Probabilistic Seismic Hazard Assessment (PSHA), the magnitude-frequency distribution (MFD) based on the 2022 New Zealand National Seismic Hazard Model (Gerstenberger et al., 2022) and implemented in OpenQuake (GEM, 2019).

Each earthquake in the catalogue is analysed using an MCS-based approach, which simulates damage from each individual peril and the combined effects of cascading hazards. Across all modules, hazard intensities are derived consistently from the same simulated earthquake event. Ground shaking intensities (e.g., PGA) modelled act as inputs for ground-failure-related hazards such as liquefaction, lateral spreading, and landslides, while tsunami inundation is modelled directly from the earthquake source parameters. A detailed description of the hazard-specific models and their parameterisation is provided in the Supplement. The overall procedure, detailed in steps 1–9 and illustrated in Fig. 2, is as follows:

Event Selection. A single earthquake event is selected from the stochastic catalogue (see Sect. 3.1).

For each event in the catalogue, multiple MCSs are performed to sample hazard intensities from their respective uncertainty distributions. Damage states are then sampled from the corresponding damage state probabilities, resulting in a range of possible damage scenarios for each event. This probabilistic sampling approach enables the representation of uncertainty in both hazard and impact, generating multiple realisations of physically consistent damage outcomes. A detailed description of the hazard-specific models and their parameterization is provided in the Supplement.

The number of MCSs required to achieve stable mean damage estimates for each event in the catalogue can be determined via convergence analysis. Due to the inclusion of multiple hazards and uncertainty sources, the minimum number of scenarios (N) needed for convergence is typically higher than in single-peril models.

Napier City was selected as a pilot study area to test the cascading hazards framework developed in this work. Situated on the eastern coast of New Zealand's North Island, the city lies directly above the Hikurangi Subduction Zone interface (Williams et al., 2013) and less than 200 km from the subduction trench, making it highly exposed to seismic and tsunami hazards, as well as to cascading effects arising from intense ground shaking (as shown by the historical examples in Sect. S1 in the Supplement, Callaghan, 1933; Dowrick, 1998).

For the purpose of the case study, the HSZ was modelled as the sole seismic source in order to isolate and quantify the contribution of subduction interface earthquakes to seismic and tsunami risk in Napier. The adopted parameters and recurrence characteristics follow the latest New Zealand National Seismic Hazard Model framework (Gerstenberger et al., 2022), in which the Hikurangi subduction interface is identified as a major contributor to seismic hazard in the Hawke's Bay region.

Liquefaction represents a major threat due to the prevalence of loose sedimentary deposits and a high groundwater table (Fairless and Berrill, 1984; Dellow et al., 1999), while the proximity of water channels to residential areas enhances the potential for lateral spreading. The city also exhibits potential for landslides triggered by seismic shaking, particularly on surrounding slopes. Furthermore, its coastal position facing the Hikurangi Subduction Zone increases vulnerability to earthquake-induced tsunamis, as well as to possible submarine landslide-generated tsunamis. The coexistence and interdependence of these hazards underscore Napier's pronounced susceptibility to cascading impacts, justifying its selection for the application and validation of the proposed framework. Sect. 3, along with the corresponding Supplement (Sects. S1 to S8), include the details of the hazards, exposure and fragility modelling for this case study.

A convergence analysis was conducted to assess the statistical stability of MDRs derived from the stochastic seismic simulations. Convergence refers to the point at which the cumulative mean MDR stabilizes with minimal variation as the number of simulations increases – a principle rooted in Monte Carlo–based uncertainty quantification (Burt and Garman, 1971; Ata, 2007). In the multi-hazard context, this analysis is particularly important due to the concatenation of uncertainties across cascading perils, which generally require a larger number of simulations compared to single-hazard models. Two convergence tests were performed.

The first evaluated site–event level convergence, where a single building was repeatedly exposed to a given earthquake event to determine the number of realizations required for MDR stability. The mean MDR stabilized to within ±2 % after approximately 120–180 simulations, indicating convergence (Fig. 5 shows an example for a Mw 7.5 earthquake). The ±2 % value was used only as a representative indicator of convergence and not as a stopping criterion. Accordingly, 200 simulations per event were performed to provide an additional margin beyond the observed convergence point (Fig. 5 shows an example for a Mw 7.5 earthquake).

The second test assessed catalogue-level convergence, examining how many unique events were required for the cumulative MDR to stabilize across the stochastic earthquake catalogue. The results show that stability was achieved within ±2 % after 20 000–30 000 events, with full convergence reached at 45 000 events (Fig. 6). These findings confirm that the 100 000-year event catalogue provides a statistically sufficient basis for robust risk estimation.

The framework proposed in this work can provide damage state (DS) level results and DRs at each asset location. Results can then be processed to present area-wide outputs such as regional damage state vs. return period, MDRs from combined and disaggregated perils, or annual average MDRs. Results for the case study proposed – impacts from the Hikurangi Subduction Zone earthquakes and cascading hazards to Napier City's residential building portfolio- are presented as follows:

Regional damage states. Return period DSs can be used to characterize urban resilience by quantifying the probabilities of structural damage over time. Results can be presented in the form of exceedance probability of DSs and the proportion of buildings under a certain damage state at different return periods. The DS exceedance probability per year (P) is calculated as follows:1PDS≥x=Nbuildings(DS≥x)NbuildingsThen, the return periods are obtained as:2RPi=Nyears+1iWhere i is the ith largest exceedance probability for damage state DS ≥x.

Figure 7 illustrates the distribution of exclusive damage states (DS0–DS4) in Napier City as a function of return period for earthquakes on the Hikurangi Subduction Zone. The estimated number of buildings in each damage state for each return period is provided in Table S2 of the Supplement. The results show a progressive shift towards higher damage states with increasing return period. Minor damage (DS1) becomes noticeable for return periods greater than approximately 100 years, whereas extensive damage (DS3) and complete damage (DS4) remain uncommon until return periods exceed approximately 500 years. Beyond approximately 2500 years, the proportion of undamaged buildings (DS0) falls below 20 %, while the proportion of buildings in DS4 increases rapidly. At return periods between 5000 and 10 000 years, DS3 and DS4 become the dominant damage states, indicating that severe structural damage and collapse are expected for the largest earthquake scenarios. At the longest return periods considered (10 000–100 000 years), the proportion of buildings in DS3 and DS4 approaches or exceeds 50 %.

To account for uncertainty, the complete stochastic event catalogue was simulated 200 times, with uncertainty sampled and propagated through the ground-motion, cascading-hazard, and damage-modelling chain during each realization. The results presented correspond to the median (50th percentile) damage-state distribution across the 200 stochastic catalogue realizations. The relative variability (coefficient of variation) generally decreases with increasing return period, reflecting the more spatially consistent and severe damage associated with larger earthquake events.

The proposed framework has limitations that should be acknowledged. Path-dependent damage is not represented, such that structural damage caused by earthquake shaking does not modify the vulnerability of buildings to subsequent hazards (e.g., tsunami), owing to the limited availability of fragility models describing sequential hazard interactions. The tsunami modelling does not account for debris impact, tidal stage, storm surge, or long-term sea-level rise. The earthquake-induced landslide model considers only co-seismic triggering and therefore does not account for antecedent rainfall, seasonal groundwater conditions, or other hydrological controls on slope stability. The liquefaction and lateral spreading assessments rely on empirical relationships, while the harmonization of damage states across hazards represents a necessary simplification to enable the integration of different fragility models within a unified probabilistic framework. The adopted harmonization procedure also introduces uncertainty into the estimated losses, and a systematic sensitivity analysis of alternative mapping approaches is recommended as future work. Validation of the integrated framework is inherently challenging owing to the scarcity of observational datasets documenting cascading earthquake hazards and their impacts, particularly in New Zealand. Owing to the modular structure of the framework, these components can be updated as improved hazard and fragility models become available.