The concept of social vulnerability (SV) is based on the selection of socio-economic variables that are related to each other. An increase in any of these variables should lead to higher vulnerability. The assigned variables should is categorized as either Sensitivity or Coping Capacity in the context of flood events. For the calculation of SV, we followed the procedure presented in von Szombathely et al. (2023), where SV is calculated considering the coping capacity and sensitivity, both represented with equal importance (0.5,0.5). These weights are based on an analytical hierarchy process (AHP) and local expert opinions. All weights in either the group of sensitivity or coping capacity have to add up to one. In the case study, sensitivity is defined by assigning a weight of 0.7 to children under 10 years old (C) and 0.3 to elderly singles over 65 years old (ES). For coping capacity, equal importance (0.5) are used between the number of residents who left school without a high school diploma within the previous three years (EDQ) as well as recipients of social welfare (WR).

Since we include several variables that contribute to a combined flood risk index, it is important to consider how these variables interact to create a comprehensive assessment of flood risk. While the alternatives can be easily ranked based on a single attribute, combining all attributes into a single index requires special techniques. One method in flood risk analysis is the Technique for Order Performance by Similarity to an Ideal Solution (TOPSIS) (Hwang and Yoon, 1981) which has been applied in several flood risk analysis' (Pathan et al., 2022; Ekmekcioğlu et al., 2021; Rafiei-Sardooi et al., 2021; Nguyen et al., 2020; Yang et al., 2018) and was applied in this work in addition to the methodology presented (von Szombathely et al., 2023). This technique quantifies for each alternative (A1–A8 in Fig. 3) the relative distance to the positive ideal (A+ in Fig. 3) and negative ideal (A- in Fig. 3) and ranks them accordingly.

The TOPSIS method can compare attributes with differing units and is able to incorporate specific weights for each attribute (derived from AHP, see above), which scales the respective axis (X, Y, Fig. 3) and adjusts the distance between the respective values and the maximum (A⁻) and minimum (A⁺). Therefore, the values assigned to each alternative can also be interpreted geometrically: for each attribute, the maximum (A⁻) and minimum (A⁺) values define two ideal points within a multidimensional space (in our example: two-dimensional, since SV consists of Coping Capacity and Sensitivity). The number of dimensions is equal to the number of different attributes. Then each alternative occupies a place in space. Lastly, we can calculate the relative Euclidean distance to the negative and positive ideal (A-, A+ see Fig. 3).

Heterogeneity may be a driving factor in the way attributes contribute to the final risk index. This is not considered in the TOPSIS approach but can be compensated by applying the Shannon Entropy method (Shannon, 1948). Other studies have already applied this concept in a flood risk context (Malekinezhad et al., 2021; Yang et al., 2018). In our analysis, both the TOPSIS and the Shannon Entropy methods are applied to estimate Sensitivity, Coping Capacity and the Social Vulnerability Index (SVI). In short, the weights determined using the AHP method (from von Szombatehly et al., 2023) are incorporated into the TOPSIS algorithm (as described above), and the TOPSIS results are then further weighted using the Shannon entropy approach.

The distribution of the Social Vulnerability Index depends on the distribution of the initial socio-economic data. Since we do not assume that flood risk scales linearly with the SVI, we include an additional transformation (flood sensitivity transformation). By considering hazard, exposure and social vulnerability as risk (IPCC, 2014) observations with “no” social vulnerability would correspond to zero risk. In the case of the proposed SVI depending on age, social welfare and the high school diploma, zero values of the SVI are possible on the building scale. To prevent that this leads to zero risk, a value of a quarter mean of the SV value is added to the SV value. Additionally, we place an additional emphasis on the most vulnerable groups by squaring the SVI. This is a subjective choice based on expert knowledge, which has been shown to improve statistical models in assessing flood related social vulnerability (Rufat et al., 2019). We acknowledge the possible variations of resulting risk based on this transformation and show the effect of varying transformation thresholds as well as squaring the SVI on the overall estimated risk within the sensitivity analysis (Sect. 4.3).

In line with the IPCC framework, we define exposure as the presence of people in areas that may be adversely affected by pluvial flooding. In this study, exposure is assessed in relation to the residential population and their places of residence. Since different flood hazards have varying impacts, we differentiate between hazards to mobility restrictions and accessibility and hazards to well-being, leading to two distinct exposure concepts: 1.

Exposure related to mobility and accessibility (EMA). Includes all residents of a given building, as flooding can affect their ability to enter or exit the premises.

In this study, we consider urban pluvial flooding as the hazard. We define water level thresholds based on existing studies on pluvial flooding (Lazzarin et al., 2022; Bhola et al., 2020; Calianno et al., 2013;) and building regulations (Bignami et al., 2019). Following the two distinct exposure concepts we define two pluvial flood hazards: 1.

Hazard related to well-being (HWB). Water levels between 30 cm and 1 m with 10 cm increments directly adjacent to buildings, to assess the potential for damage and danger to well-being.

Building on the previously established data indices, flood risk at the building level is assessed using the IPCC Risk Assessment framework (IPCC, 2014), which integrates hazard, exposure, and vulnerability to quantify risks associated with pluvial flooding. This approach enables a detailed evaluation of flood-related impacts on mobility and accessibility and well-being in Hamburg. Based on this conceptual decision to define two different approaches to hazard and exposure, we ultimately calculate two different risk indices: 1.

Pluvial flood risk to well-being (PFRWB)

The SV toolset calculates Sensitivity, Coping Capacity, and the SV index using TOPSIS, and its transformed version using the flood sensitivity transformation tool. Each vulnerability tool allows adjustment of the weights for Sensitivity and Coping Capacity. Within each group, the weights must sum to one and can be modified in the toolbox (Eqs. 1–3). 1SVi=FTOPSISCCi,Si,0.5,0.5,2CCi=FTOPSISCi,ESi,0.3,0.7,3Si=FTOPSISWRi,EDQi,0.5,0.5, where the TOPSIS procedure is represented as FTOPSIS for the coping capacity (CC_i), and Sensitivity (Si), for each alternative (building) i.

In more detail, to apply TOPSIS, all attributes are normalized (Eq. 4), and then multiplied with the specific weight. The normalization of each alternative can be expressed as: 4zij=xij∑i=1mxij,j∈1,…,n,i∈1,…,m where zij is the normalized attribute and xij reflects the unweighted value for the ith alternative (buildings) and jth attribute (relative share of C, ES, WR, EDQ).

The maximum and minimum values for each attribute define two ideal points in a multidimensional space (Eq. 5). We can calculate the relative Euclidean distance to this negative and positive ideal (A-, A+ see Fig. 3). Generally speaking for alternative Ai with n different attributes, let zij be the normalized attribute then: 5FTOPSISzi1,…,zinw1,…,wn=∑j=1nwj2min⁡i(zij)-zj2∑j=1nwj2min⁡i(zij)-zj2+∑j=1nwj2max⁡i(zij)-zj2 with n attributes, where zij is the normalized attribute and j reflects the unweighted attribute for the i alternative.

Additionally, we apply the Shannon Entropy method. The idea of the Shannon Entropy method is to multiply the final risk index with an entropy-index between 1 and 2, where 1 indicates complete homogeneity and 2 indicates total inhomogeneity across different attributes. More specifically, for m alternatives with n attributes, where zij is the normalized attribute and j reflects the unweighted attribute for the ith alternative, the entropy Ui is calculated following Eq. (6): 6Ui=2+1ln⁡(n)∑j=1n(zij∑j=1nzijln⁡zij∑j=1nzij),i∈1,…,m The effect of heterogeneity depends on the specific context and its application should be evaluated individually for each hazard scenario. Hence, our toolbox offers the user to choose if Shannon Entropy shall be applied (please see additional screenshots of the tool user interfaces provided in Fig. S3a to c).

Finally, we perform two additional transformations, addressing the specific data distribution of SV (flood sensitivity transformation): (a) to prevent that zero SV values lead to zero risk, a value of a quarter mean of the SV value is added to the SV value, and (b) to mitigate the skewed social vulnerability data, we square social vulnerability values to emphasize on the most vulnerable (see Eq. 7). This threshold should be evaluated individually for different hazards or cities. Hence, our toolbox offers the user to apply different thresholds if needed (no, half or one mean value) 7SVPF=(Ui⋅SV+MeanUi⋅SV4)2, with the Entropy Ui, and the previously calculated social vulnerability value (SV).

All four tools are coded in Python and were incorporated within the ArcGIS framework. An exemplary outline of the calculated values is depicted in Fig. S4. We show the sensitivity to changes of this threshold within the sensitivity analysis.

The calculation of Exposure contains one tool (see Fig. 4) including several sub-tools which are linked to one-another. The full workflow exported from the toolbox can be viewed in Fig. S5. First, both input data, the available data based on the statistical unit and the building information are combined (see Fig. S6a to c). Then, the available living area per building is calculated based on the following equation (Fig. S6d and e) (Eq. 8). 8AB=Fl-B-1⋅AP where AB is the area of each building (in m²), Fl being the number of floors of the corresponding building and B the building type, (B = 1 for residential, B = 2 for mixed use, containing no residents on the base floor level but in the upper levels), and AP for the area (in m²) of each building.

Following the two previously mentioned exposure concepts of EMA, including all residents of a given building and EWB, considering only individuals residing on the ground floor level, the subsequent Eqs. (9) and (10) are used to obtain EMA and EWB. 9EMA=ABALSU×RSU with AB being the area in m² of each house, ALSU the total living area in m² within the statistical unit and RSU, the overall number of residents living in the corresponding statistical unit (SU). 10EWB=-(B-2)⋅EMAFl With building type (B = 1 for residential, B = 2 for mixed use) and the number of floors in the corresponding house (Fl).

The corresponding tools are shown in Fig. S6e to i. An excerpt of the resulting attribute table after implementing the exposure calculation is depicted in Fig. S7.

The calculation of the hazard is two-fold, one toolset is developed for the calculation of the hazard to mobility and accessibility and one for the hazard to well-being.

The assessment of the hazard to well-being is carried out in two stages (see Fig. S8). The first tool computes the areas of intersection between all flood-depth layers between 30 and 100 cm (with 10 cm increments) and the affected buildings (see Fig. S8a and c). To account for potential water intrusion near buildings, we applied a 2 m buffer around all buildings, which was determined based on the native 1 m resolution of the flood simulation, expert knowledge, and stakeholder workshops in Hamburg, as well as following the approach of von Szombathely et al. (2025).

The second tool (as shown in Fig. S8b) calculates the hazard to well-being (HWB) following a cumulative distribution function (CDF; Φ) of the log-normal distribution with μ = 0 and σ = 0.25. 11FX(x)=Φln⁡(x)-μσ We derive the hazard index with 12HWB=∑i=30100FX4⋅A2m,i,i∈30,40,50,60,70,80,90,100 where A2m,i as the fraction of the flooded area within a 2 m buffer at the water level i (in centimeter).

The toolset of the well-being hazard considers all potential flood-layers included in the corresponding input folder. In the sensitivity analysis, we therefore discuss the sensitivity of the applied method using a lower flooding threshold (20 cm). An example of the resulting output table is shown in Fig. S9.

The assessment of the hazard to mobility and accessibility includes two tools. Using the first tool (see Fig. S10a and c), we determine the areas of intersection, expressed in square meters (labelled with A) and percentages (labelled with P), between the flood hazard layer and (a) a 5 m buffer around each building, representing the feasibility of accessing the building, and (b) 15 m and (c) 15-to-30 m buffers around affected buildings, capturing potential intersections with the road network.

The second tool (Fig. S10b) includes the calculation of the actual hazard as described in von Szombathely et al. (2025), where the hazard index related to mobility and accessibility (HMA) is calculated using the maximum value derived from the Eq. (13). 13HMA=Max[FX4⋅A5m,30cm,FX4⋅R15m,30cm,FX4⋅R30m,30cm] There, A5m,30cm refers to the fraction of area within a 5 m buffer, which is flooded above a 30 cm water level. R15m,30cm and R30m,30cm describe the fraction of road area within a 15 m buffer and 15-to-30 m buffer, respectively, which are flooded above a 30 cm water level.

To ensure relevant insights into surrounding road conditions and to exclude greater hazard ratings due to small flooded road areas, only flooded road areas exceeding 4 m² are included. A schematic drawing (Fig. 5) depicts the applied buffers and intersections applied. Figure S11 shows an excerpt of the resulting HMA values.

In a final calculation, the calculated sub-indices for social vulnerability, exposure and hazard are linked to obtain two specific risks during pluvial flooding events, one related to well-being (PFR_WB) and the other one related to mobility and accessibility (PFR_MA).

The PFR_WB is calculated as: 14PFRWB=SVPFa⋅EWBb⋅HWBc where SV_PF reflects the social and economic characteristics that influence flood resilience, EWB considers the number of ground-floor residents, as those are most vulnerable to direct flood impacts, and HWB measures the likelihood and severity of floodwater intrusion at the ground floor level. Exponents a, b, and c reflects their corresponding (optional) weights.

The PFR_MA is calculated as: 15PFRMA=SVPFa⋅EMAb⋅HMAc With SV_PF reflecting the social and economic characteristics influencing flood resilience, EMA considering the number of residents, and HMA representing how flooding impacts movement and access in affected areas. Within the toolbox, we provide further options by selecting the exponent for each sub-index (hazard, exposure and social vulnerability), which allows for further specific weighting of the sub-indices in the final risk assessment (see Fig. S12). An excerpt of the resulting risk values and related sub-indices is shown in Fig. S13. Additionally, we provide a tool to estimate the visualization classes (no risk to very high risk) based on an iterative mean-based filtering process. In this context, the absolute values themselves are not decisive, rather, the relative gradations between them determine the risk structure (see Fig. S14). This approach ensures that the classification of risk classes reflects the data's internal structure.

To contextualize the results, we qualitatively compare the simulated rainfall scenario to two recent heavy rainfall events in Hamburg (years of 2018 and 2024). The event in Lohbrügge (10 May 2018) was a local convective downpour with intensities above the 100-year return period. The event caused severe flooding, infrastructure damage, major traffic disruptions and evacuations due to sinkholes (Deutscher Wetterdienst (DWD), 2021). The Barmbek event (2 June 2024) reached intensities of the 100-year return period used in this study and slightly above triggering over 800 operations of the local fire brigades. There, traffic disruptions, power outages, and flooded basements, stores and underground parking garages were reported (Gaertner et al., 2024; Scheiwe and Schwieger, 2024)

The development of the presented toolbox was developed in close cooperation with city stakeholders in Hamburg. Various agencies (see Table S3 for details) were actively involved in its development through three stakeholder workshops and are the main beneficiaries of the toolbox. The toolbox is therefore mainly designed for cities, where similar data is available. In Germany, high-resolution pluvial flood simulations are or will soon be available across the country (Wimmer and Hovenbitzer, 2025; Wimmer et al., 2023). In addition, high-resolution urban flood maps are available from the EU project REACHOUT for the cities of Athens, Milano, Logrono, Gdynia (Staccione and Pal, 2024). Beyond Europe, there are high-resolution simulations available for New York (Department of Environmental Protection NYC (DEP), 2024), Los Angeles (Sanders et al., 2022), Miami (Sanders et al., 2025), and Houston (Schubert et al., 2022).

Recent advances enabling faster flood simulations (Van Den Bout et al., 2023) and flood simulations driven by artificial intelligence (Li et al., 2025) may further increase data availability and therefore transferability of our framework globally. For the availability of population data and social data, we acknowledge limitations due to data protection and privacy, which vary by country or commune, while high-resolution data might be available at least internally within the local authorities. Recent developments of global building data, including the area and height of buildings on a meter resolution (Zhu et al., 2025) could support future adaptations of the toolbox.