A mathematical framework for quantifying physical damage over time from concurrent and consecutive hazards

Borre, Alessandro; Ottonelli, Daria; Trasforini, Eva; Ghizzoni, Tatiana; Zoppi, Giacomo; Boni, Giorgio; De Angeli, Silvia

doi:https://doi.org/10.5194/egusphere-2025-2379

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https://doi.org/10.5194/egusphere-2025-2379

Preprints

03 Jun 2025

| 03 Jun 2025

Status: this preprint is open for discussion and under review for Natural Hazards and Earth System Sciences (NHESS).

A mathematical framework for quantifying physical damage over time from concurrent and consecutive hazards

Alessandro Borre, Daria Ottonelli, Eva Trasforini, Tatiana Ghizzoni, Giacomo Zoppi, Giorgio Boni, and Silvia De Angeli

Abstract. Space and time play a crucial role in multi-hazard impact assessment. When two or more natural hazards occur at the same location simultaneously or within a short time frame, the physical integrity of assets and infrastructures can be compromised and the resulting damage can be greater than the one generated by individual hazards occurring in isolation. The current literature highlights the lack of quantitative standardised frameworks for multi-hazard impact assessment. This research presents a generalised mathematical framework for quantitatively assessing multi-hazard physical damage on exposed assets, such as buildings or critical infrastructures, over time. The proposed framework covers both concurrent and consecutive hazards, by modelling: (i) the increased damage resulting from the combined impact of two or more concurrent hazards that overlap in space and time, and (ii) the effects of cumulative damage on asset vulnerability and the recovery dynamics in case of consecutive hazards that overlap in space. The framework is applied to a real-world case study in Puerto Rico, including the concurrent wind and flood impacts generated by the passage of Hurricane Maria, as well as the consecutive impacts caused by the subsequent seismic sequence of 2019–2020. Based on simulations performed on a building portfolio, we found that neglecting residual damages caused by the hurricane when assessing the impacts of the subsequent earthquake would lead to a significant underestimation of the overall damage experienced by the assets. By providing a generalised formalisation to perform quantitative multi-hazard impact assessment, able to account for amplification phenomena and recovery dynamics, the framework can offer scientists and decision-makers a comprehensive and deeper understanding of the impacts caused by compound and consecutive events.

Received: 20 May 2025 – Discussion started: 03 Jun 2025

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Alessandro Borre, Daria Ottonelli, Eva Trasforini, Tatiana Ghizzoni, Giacomo Zoppi, Giorgio Boni, and Silvia De Angeli

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RC1: 'Comment on egusphere-2025-2379', Anonymous Referee #1, 18 Aug 2025 reply

Summary
This manuscript presents a mathematical framework for assessing multi-hazard risks that explicitly incorporates recovery dynamics and the temporal evolution of asset vulnerability. The authors argue that their primary contribution is a generalised mathematical formalization of multi-hazard risk quantification, with particular emphasis on modeling how concurrent and consecutive hazard events affect vulnerability to subsequent events. They demonstrate their approach using a case study of hurricanes and earthquakes in Puerto Rico, showing how hurricane damage modifies earthquake vulnerability over time.

Overall, this addresses an important gap in disaster risk assessment, which typically treats hazards independently without considering temporal interactions. However, while the systematic approach to multi-hazard impact assessment has merit, the current framing oversells the contribution while underselling significant limitations. The authors should address fundamental questions about operational utility and reposition their work to reflect its actual capabilities and constraints more accurately. Therefore, I recommend major revision of this manuscript.

Major Concerns

Core Contribution and Scientific Value
My primary concern relates to the fundamental claims about the manuscript's contribution and its practical utility. The authors position their work as providing a "generalised mathematical framework" for multi-hazard risk assessment, but this claim requires careful examination.

Mathematical Formalization: The mathematical framework presented is essentially a straightforward extension of existing single-hazard approaches, where total damage becomes a function of multiple hazards and recovery states. While the authors present this as novel, the mathematical formulation represents a straightforward extension. The real challenges in multi-hazard assessment lie not in the mathematical abstraction but in the empirical quantification of recovery functions, hazard dependencies, and vulnerability transitions over time - as the authors rightly state in sections 4.1 and 4.2.

Operational Utility Limitations: More critically, the framework suffers from a fundamental limitation that severely constrains its practical applicability. The Puerto Rico case study illustrates this problem clearly. The authors demonstrate how Hurricane Maria affected earthquake vulnerability by using post-hurricane damage data to calibrate vulnerability adjustments. However, this approach is inherently backward-looking and requires empirical damage data that would not be available for forward-looking risk assessments.

Consider a practical scenario: if a hurricane were to strike Puerto Rico tomorrow, practitioners using this framework would need to wait for post-hurricane damage assessments before they could adjust earthquake vulnerability functions. This severely limits the framework's utility for operational risk management, emergency planning, or prospective risk assessment. The authors do not adequately address how practitioners would estimate vulnerability changes in real-time or predictive applications without extensive post-event calibration data.

Misalignment of Claimed vs. Actual Contributions:
The manuscript lists several important developments as "limitations" (section 4.1) or "future research directions" (section 4.2) that would actually constitute the real scientific advances needed in multi-hazard assessment. These include empirically-derived vulnerability transition functions, standardized recovery curves based on extensive post-disaster data, and predictive models for hazard-induced vulnerability changes. Only after solving these challenges would a general mathematical framework provide meaningful operational value.
Specifically, I was excited to see the authors propose an approach to adjust vulnerability curves to account for pre-existing damage conditions (lines 394-396). Such an approach represents a potentially valuable contribution to the field. However, the implementation still relies entirely on post-event calibration data, making it neither operational for forward-looking assessments nor generalizable across different contexts without extensive empirical datasets.

In conclusion, current framework appears most suited for retrospective analysis and systematic post-disaster impact assessment rather than the forward-looking risk management applications that the authors suggest. This represents a significant gap between the claimed contribution and the demonstrated capabilities.

Language and Presentation
The manuscript suffers from clarity and communication issues that hinder comprehension of the technical content. The writing style relies on unnecessarily complex sentence structures that obscure rather than illuminate key concepts. Many sentences contain multiple subordinate clauses that could be simplified without losing technical precision.

Additionally, the manuscript exhibits significant redundancy, with core concepts repeated across sections without advancing the argument or providing new information. For example, the distinction between "concurrent and consecutive hazards" is mentioned repeatedly in the abstract, introduction, and methodology sections without substantive development of how the framework addresses each case differently.

The technical exposition would benefit from more precise language and clearer logical flow. Terms like "generalised framework" are used extensively without clear definition of what makes the approach "general" compared to existing methods.

Recommendations for Improvement
The manuscript addresses an important problem in disaster risk science, but it requires substantial revision to align claims with demonstrated capabilities. The authors should consider reframing their contribution more modestly and honestly. Rather than claiming a key contribution in operational multi-hazard assessment, they could position their work as a systematic methodology for post-disaster impact assessment or as a research template for understanding multi-hazard interactions in well-documented cases.

The framework has genuine value as a foundation for systematic vulnerability state tracking and post-disaster learning, but the authors should acknowledge its current limitations for predictive applications. Future work should focus on developing the empirical foundations needed to make such a framework operationally useful, including physics-based vulnerability transition models and standardized recovery parameters that can be estimated without extensive post-event data.

The language and presentation issues require comprehensive editing to improve clarity and eliminate redundancy.

Minor comments
L30: "hazard" in brackets?
L 48: "underlined" seems to be a word choice error. Consider "highlighted", "identified", "outlined", "emphasized".
Line 105: "occurs integrally at the beginning" is unclear - suggest "occurs entirely at the beginning" or "occurs instantaneously at the beginning" to clarify that all damage happens at once rather than gradually.
L 110: To my knowledge, figures should be referenced in sequential order (Fig. 1, then Fig. 2, then Fig. 3, etc.) as they appear in the text. Fig. 3 is introduce before Fig. 1 and 2.
L 118, 128: Avoid describing figure elements by color ("blue line", "green line"). Use descriptive labels instead for accessibility and clarity (e.g., "the horizontal line representing the response phase"). Related and for figures in general: Ensure all figure elements are clearly labeled in the legend. Figure 3 legend: "The second event in temporal order" is redundant - "second event" already implies temporal sequence. Suggest removing "in temporal order" throughout.

L 179: Remove "fortunate" - scientific writing should avoid value judgments.
L 279: Typo in "build" back better

L 407, Table 2: The values in Table 2 seem to be in USD, not "thousand USD". Please double check. Also check the formatting of the values in Table 2. Table 2/Figure 7: Both show the same results. Consider moving one of the two elements to the supplementary material. L 408: Line X: The damage-to-loss conversion methodology needs brief explanation rather than just a citation. Readers should understand the key assumptions without consulting external sources.

L 409: "Predefined loss ratios" needs clarification - predefined by whom, based on what data, and are they appropriate for Puerto Rico conditions? Specify the source and empirical basis.
L 415: The claim of "clear non-linear trend" needs quantitative support. How was non-linearity assessed? Provide statistical analysis (R², trend coefficients) rather than visual inspection and isolated examples.
L 421/423: "1,700 thousand" "1,742 thousand" is confusing and not scientific. Replace with scientific notation or standard units.

Throughout: Use standard terminology "slow-onset hazards" rather than "long-onset hazards" to align with established disaster risk literature (e.g., UNDRR, IPCC terminology).

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Citation: https://doi.org/10.5194/egusphere-2025-2379-RC1

Alessandro Borre, Daria Ottonelli, Eva Trasforini, Tatiana Ghizzoni, Giacomo Zoppi, Giorgio Boni, and Silvia De Angeli

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Short summary

This study introduces a generalized mathematical framework to quantify physical damage to assets from concurrent and consecutive hazards over time. Applied to a Puerto Rico case study (Hurricane Maria and the 2019–2020 earthquakes), it shows that ignoring residual damage significantly underestimates total impacts caused by consecutive events. By incorporating amplification effects and recovery dynamics, the framework improves multi-hazard damage assessments for researchers and decision-makers.


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