Towards a Robust Framework for Navigating Flood-Related Challenges: A Comprehensive Proposal for an Advanced Flood Risk Assessment Scale in the Slovak Republic

Abstract

This study presents a new multi-index hierarchical model for flood risk assessment which incorporates three indicator indexes—hazard, vulnerability, and exposure—to develop a five-level risk scale. The methodology is applied to historical data on flood events in The Slovak Republic between 2001 and 2010. The input values are characterized in more detail through the use of weighted values to provide a more balanced overall risk assessment. The original formula used to calculate the risk levels was found to produce results with overly high numerical values, and therefore the multiplication step of the formula was replaced by addition to insure greater simplicity and ease of use. This refined methodology introduces a novel quantitative approach to risk assessment, offering flexibility and variability in the indicator layer. The methodology can be adapted to assess risk at either the macro or micro scale and at more specific periods of time. The resulting risk values offer a nuanced understanding of risk levels across different indexes and underscores the method’s innovation.

1. Introduction

Each year, prolonged spells of heavy rain, winter thaws, and watercourse obstructions cause both major and minor flooding in Slovakia which results in millions of euros of damage. The precise impact of human activities in these flooding events is still the subject of much discussion, but considerable evidence suggests that river management decisions, changes in land use such as increased logging, and factors related to greenhouse gas emissions can contribute to a greater risk of floods [1]. The role of urbanization has also attracted growing interest in recent years, with several recent studies exploring issues such as low-impact development and urban storm water management [2].

Accurate and reliable assessments of flood risks are crucial in ensuring the sustainability of habitats. Methods should apply a multifunctional, adaptable framework to assess both natural and man-made risks, incorporating risk mitigation approaches and spatial planning to minimize potential damage; for example, a three-level paradigm for risk assessment which takes into consideration all potential relationships between risks and hazards [3].

Flood risk management typically involves the use of flood vulnerability assessments, the creation of flood risk maps, and financial analyses of flood management; these analytical hierarchical procedures apply frameworks for assessing flood risks which also evaluate specific factors such as flood characteristics, exposed elements and their value, and flood damage functions using up-to-date datasets and model approaches that link flood damage to other relevant variables [4,5]. As the consequences of climate change become more apparent, cities are expected to become increasingly vulnerable to a variety of environmental issues, among the foremost of which is flooding. Analyses of urban flood risks examining building characteristics and spatial concepts are directly incorporated into cities’ urban planning processes [6]. In a broader scope, however, national flood risk assessments can also assist in the development of state-level flood and management policies, the distribution of resources, and oversight of the execution of flood mitigation measures [7].

The Central European state of Slovakia has suffered several catastrophic flooding events in recent years, most notably in 2020, and a recent study by Leščešen et al. [8] suggests that the situation will worsen in the coming decades, especially in the case of destructive summer floods. Given this likelihood, there is a clear need for new methodologies of flood risk assessments in the Slovak context; the geographical diversity of the country also means that frameworks with the capacity to assess risk at the micro level of individualities would be particularly useful [9].

This study presents a new methodology for flood risk assessment in the specific context of The Slovak Republic which integrates data from historical data on flood events. This approach offers a flexible framework that considers dynamic climatic conditions and the increasing frequency of extreme weather events. Unlike existing probabilistic models, the proposed framework applies additive logic to maintain interpretability while allowing for regional comparisons. The study represents the first systematic application of such an integrated, multi-dimensional risk model within the Slovak national context. The combination of empirical data with an adaptable evaluation structure addresses a notable gap in Slovak flood risk research and contributes to the broader field of climate adaptation, spatial risk modeling, and disaster risk mitigation.

2. Materials and Methods

The study applies a multi-index hierarchical model stratification comprising three interconnected levels: the object layer, the index layer, and the indicator layer [10]. The initial layer, referred to as the object layer, located and defined the object or entity under scrutiny, in this case the specific region or area of The Slovak Republic.

The second layer, referred to as the index layer, is a critical component in risk quantification and establishes the key elements of the Hazard Index (H), the Vulnerability Index (V), and the Exposure Index (E). Hazard refers to the factors (indicators) that contribute to the probability of an adverse outcome occurring, while exposure gauges the degree of contact with potential risks [11]. The combination of these three variables provides a comprehensive risk profile that forms the basis for informed decision-making.

The results of the first two analyses are integrated into the third layer, the indicator layer, which incorporates 22 domain-specific flood risk indicators which have been systematically selected to reflect the spatial, physical, and socio-environmental determinants relevant to flood hazard modeling and risk quantification of the study area. While it is not entirely accurate to divide the index and indicator layers apart because of the overlap in their content, the indicators bring context to the indexes by encapsulating the nuances that provide the qualitative and quantitative data necessary for a nuanced risk assessment. For instance, within the hazard index, specific indicators might include environmental factors, technological considerations, or human elements that contribute to the overall risk landscape.

Through this holistic approach, organizations and individuals can navigate the intricate landscape of risk, making informed decisions and implementing targeted mitigation strategies.

2.1. Study Area and Its Vulnerability to Flooding Events

The administrative delineation of Slovakia (Figure 1) into the regions of West Slovakia, Central Slovakia, and East Slovakia is rooted in a historical partition that dates back to the 13th century, but the contemporary regional structure, comprising eight upper territorial entities and seventy-nine districts, was established in 1996 and reflects the socioeconomic development and considerable variety in terms of population, population density, and geographical size [12,13]. The Slovak Republic is a landlocked nation with an area of 49,036 km². Due to its location in the western portion of the Eurasian continent, the climate has an oceanic character, with milder winters and cooler summers. The Pannonian Basin and the Carpathians mean that the country features considerable differences in relief, and the territory is traversed by the main European watershed which forms an important hydrological border. More than 96% of the land drains into the Black Sea via the River Danube and its tributaries, while the Dunajec and its tributaries empty the remaining area into the Baltic Sea, traversing a distance of 1950 km. The Slovak Republic is predominantly a land with a southern, southwestern, and southeastern slope, as is indicated by the surface area [14,15,16].

2.2. Conceptual Risk Framework

This study focuses on the development and application of a multi-index conceptual model based on the method described by Zhang et al. [10] to assess the flood risk assessment of the specific territory of The Slovak Republic (Figure 2).

An additive (summation-based) methodology is applied to the index aggregation in order to preserve interpretability and reduce the impact of outliers or extreme values on the final composite flood risk score. Multiplicative approaches can often exaggerate the overall risk due to the influence of a single high component (e.g., extreme hazard or vulnerability values), but the summation method enables a more balanced integration of the core components of hazard, exposure, and vulnerability.

Although numerous studies in Central Europe have developed flood risk assessment methods, most rely on probabilistic or multiplicative models. These approaches often generate disproportionately large numerical values and are less adaptable to regional socio-environmental contexts. In The Slovak Republic, flood risk assessments have so far been fragmented, focusing either on hydrological modeling or on selected socio-economic factors, without providing an integrated and transparent framework. To date, no methodology has systematically combined hazard, vulnerability, and exposure indicators into a multi-index structure that is both interpretable and adaptable for national and regional applications. This lack of an integrated, scalable, and easily applicable methodology represents a significant research gap. The present study addresses this gap by proposing and testing a novel additive, multi-index flood risk assessment scale specifically tailored to Slovak conditions, offering a transparent tool for both scientific analysis and policy decision-making.

Several other studies have highlighted the importance of manageable (i.e., included the replacement of the multiplication of index values with their sum), interpretable, and practically applicable composite indicators, particularly in risk assessments at the regional scale (e.g., [2,5,7,10,17,18,19,20,21]). For instance, a study by Kosztya et al. highlights the value of transparent and communicable risk indices [2], while Romali et al. demonstrated the advantages of additive approaches in regional flood risk modeling due to their simplicity in calibration and policy relevance [5]. Additive structures have also been successfully applied in other recent flood risk studies, such as that conducted in the Mai Hoa Commune in Vietnam where flood vulnerability was assessed using the summation of normalized sub-indices (exposure, susceptibility, and resilience) [19]. A similar assessment in Khyber Pakhtunkhwa, Pakistan applied additive methods due to their compensatory properties and reduced distortion [20], while a risk mapping process in urban areas of Spain used additive weighting in combination with open-source data to assist decision-making at a regional level [21].

A study by Lu et al. on flood risk assessments in the Qinghai–Tibet Plateau (QTP) have advanced our understanding of hazard drivers, socio-economic exposure, and vulnerability in a setting characterized by strong topographic gradients, glacier dynamics, permafrost, and rapid climate change. Most existing studies in the QTP focus on either hazard mapping using hydrological/geomorphic indicators or on vulnerability and exposure analyses using socio-economic proxies; however, fewer works integrate these components in a single, consistently interpretable framework and even fewer employ an additive, multi-index approach that yields a transparent, five-level risk scale suitable for decision support [22].

Additive approaches also enhance the adaptability of models across spatial units. As Hall et al. [7,18] and Zhang et al. [10] have shown, index-based additive models allow for more straightforward recalibrations which makes them particularly suitable for comparative assessments across diverse territorial contexts such as, for example, the eight administrative regions of Slovakia considered in this study.

A study by Rindsfüser et al. which reviews the methodologies of flood risk assessments found that a matrix incorporating a three-pronged approach of hazard, vulnerability, and exposure is the most effective means of assessing risk [23].

Our study contributes to this literature in several ways. First, we implement a three-level, multi-index framework (Hazard, Vulnerability, Exposure) with 22 clearly defined indicators, combined additively to produce a five-level flood risk scale. This contrasts with many QT-region studies that use either single indices or multiplicative aggregations, and it improves interpretability for regional and local decision-makers. Second, we explicitly discuss transferability to the QTP by outlining regionally relevant indicator categories (environmental, technical, social) and by describing a reproducible workflow: data sources, indicator definitions, normalization, weighting, and aggregation steps. Third, the approach supports macro- and micro-scale assessments and therefore can inform both regional planning and local risk mitigation in QTP-like settings, where data availability and spatial heterogeneity pose substantial challenges. Finally, while the present calibration uses historical floods from Slovakia (2001–2010), the methodological roadmap—indicator set, additive aggregation, and five-level framework—is readily adaptable to QTP with locally appropriate indicators and data.

2.3. Data Collection and Processing—Hazard, Vulnerability and Exposure Indexes

The methodology for processing the data and categorizing the layers is outlined in the following flowchart (Figure 3).

The data used in the study is taken from Reports of the Progress and Consequences of Floods in the Territory of The Slovak Republic (Informácie o priebehu a následkoch povodní na území SR) from the period 2001 to 2010 [24]. These annual reports combine local records from weather stations and analyze satellite data to catalog and quantify flood damage across each of Slovakia’s eight self-governing regions incorporating raster and land cover (Corine) methodologies to determine land use classification [25]. The indicators for the indexes of hazard, vulnerability, and exposure were determined according to a methodology by Zelenákova et al. [26]. The weighting assigned to each of these three index components is supported by findings from the relevant literature [2,17] and also integrated consultations with experts in the field drawing upon the core principles of multicriteria decision analysis (MCDA). Although a fully formalized MCDA protocol was not applied, the relative importance of each component was assessed qualitatively, with experts’ judgment ensuring that the final configuration aligns with practical relevance and contextual understanding of flood dynamics. While this approach inherently incurs a degree of subjectivity, it also reflects widely accepted practices used in regional-scale risk assessments such as our Slovak dataset in which empirical calibrations are limited by data availability. The selected weighting scheme was therefore intended to reflect the relative contribution of each component to overall flood risk under the specific socio-environmental context of Slovakia.

The primary objective at this stage was to establish a functional, adaptable framework for flood risk comparison across regions, and therefore the establishment of a more comprehensive sensitivity analysis of alternative weighting scenarios is beyond the scope of the current research. However, the modular nature of the composite index allows for the straightforward adjustment of weights and recalculations of the risk score, offering a flexibility that makes the model well-suited for future testing and stakeholder-driven customization.

Three main hazard indicators were identified: the extent of the flooded area, monthly precipitation, and the number of times a Level Three severity of flood activity was declared. The “extent of the flooded area” indicator contains three additional geographical sub-categories: mixed urban, forest area, and urban area.

Table 1. Schematic representation of the hazard index.

HI1.1	vIH1.1 × xIH1.1	yIH1.1
HI1.2	vIH1.2 × xIH1.2	yIH1.2
HI1.3	vIH1.3 × xIH1.3	yIH1.3
HI2	vIH2 × xIH2	yIH2
HI3	vIH3 × xIH3	yIH3
ΣRIH		yIHn

Notes: v_IHn—weight of the hazard; x_IHn—value of the indicator; y_IHn—value of the hazard index.

depicts the hazard indicators as H_In, where H denotes the index of the hazard, I stands for “index”, and n denotes the number of indicators in the hazard index. In designing the methodology, weights were assigned to the indexes based on a simple mathematical division of the input value of 100 points (or 100%) across the three indexes. The hazard index has a total of thirty points; these are further subjectively distributed amongst the indicators according to their weighting, i.e., their potential contribution to the occurrence of a flood event. The weight and value of the hazard indicator are denoted as v_IHn and x_IHn, where v denotes the weight and x denotes the value of the indicator. The hazard index in Table 1 is expressed as the product of the weight v_IHn and the indicator value x_IHn. The resulting value is shown in the third column, denoted as y_IHn, where y indicates the value of the hazard index. After mathematically expressing all the values of the hazard index, the values of y_IHn are summed to give ΣR_IH. The sum ΣR_IH indicates the resulting hazard index value for each year R. The same procedure for calculating the resulting index value applies to the vulnerability and exposure indexes.

The vulnerability determination defines elements that are affected by a hazard or the consequence of a hazard [27,28] and relates to the ability to adapt to, cope with, or recover from the hazard [29].

Table 2. Schematic representation of the vulnerability index.

VI1.1	vIV1.1 × xIV1.1	yIV1.1
VI1.2	vIV1.2 × xIV1.2	yIV1.2
VI1.3	vIV1.3 × xIV1.3	yIV1.3
VI1.4	vIV1.4 × xIV1.4	yIV1.4
VI2	vIV2 × xIV2	yIV2
VI3	vIV3 × xIV3	yIV3
ΣRIV		yIVn

Notes: v_IVn—weight of vulnerability; x_IVn—value of the indicator; y_IVn—value of the vulnerability index.

describes a similar procedure for calculating ΣR_IV. In the case of the vulnerability index, the indicators are denoted as V_IN. The vulnerability index has a total value of forty based on the potential contribution to the selected vulnerability elements; it is equally divided among its three indicators of affected population, protected landscapes, and the condition of flood protection facilities. The weight and value of the indicator are denoted in Table 2 as v_IvN and x_IvN respectively, and the resulting value is denoted as y_IvN.

The exposure index describes the direct exposure to the adverse effects of a risk; for example, in the case of human health assessment, it could relate to the health risk of exposure to drinking water contamination [29], the impact of contaminants in the home [30] or the impact of mineral oils in food [31]. In flood risk assessments in coastal cities, the exposed element is mainly deemed to be densely populated coastlines [32], while other exposed elements may be the landscape and its use [33,34,35] or the potential risk of fatality during a flood event [17].

In our methodology, the E_I exposure index has a value of thirty across five indicators, the highest number in the framework: flooded residential buildings, flooded non-residential buildings, affected engineering networks, affected animals, and other—a final category of “damages” which consists of twenty-six sub-indicators. However, to ensure clarity,

Table 3. Schematic representation of the exposure index.

EI1	vIE1 × xIE1	yIE1
EI2	vIE2 × xIE2	yIE2
EI3	vIE3 × xIE3	yIE3
EI4	vIE4 × xIE4	yIE4
EI5	vIE5 × xIE5	yIE5
ΣRIE		yIEn

Notes: v_IEn—weight of exposure; x_IEn—value of the indicator; y_IEn—value of the exposure index.

expresses the notation of the ΣR_IE calculation for the five main indicators. As in the previous two cases, the indicator weight is denoted as v_IEn, the indicator value as x_Ien, and the resulting indicator value as y_IEn.

The framework is applied to historical data on flood events in Slovakia drawn from official statistics [24]. The data available from this source provides information from the period 2001 to 2022, including that for the exceptional year 2020 which saw extensive heavy flooding in many regions of the country.

Our study uses data from a more restricted time period, more specifically that between 2001 and 2010;

Table 4. The resultant matrix of hazard, vulnerability, and exposure index values.

HIn	Σ01IH	Σ02IH	Σ03IH	Σ04IH	Σ05IH	Σ06IH	Σ07IH	Σ08IH	Σ09IH	Σ10IH
VIn	Σ01IV	Σ02IV	Σ03IV	Σ04IV	Σ05IV	Σ06IV	Σ07IV	Σ08IV	Σ09IV	Σ10IV
EIn	Σ01IE	Σ02IE	Σ03IE	Σ04IE	Σ05IE	Σ06IE	Σ07IE	Σ08IE	Σ09IE	Σ10IE

describes the matrix of the resulting hazard, vulnerability, and exposure index values for these years. The years are denoted by the last two digits of the millennium in order to simplify the notation. After populating this matrix with the actual values, the final risk measure will be classified using selected mathematical and statistical procedures into five levels of risk: none, negligible, low, medium, and high.

After populating this matrix with the actual values for each year, the resulting risk values are calculated using the following formula:

R = H × V × E

(1)

To obtain the final value, a procedure is proposed to sum up the values of the hazard, vulnerability, and exposure indexes for each single year, as is shown in

Table 5. Calculation of hazard, vulnerability, and exposure index values for the period considered.

HIn	Σ01IH	Σ02IH	Σ03IH	Σ04IH	Σ05IH	Σ06IH	Σ07IH	Σ08IH	Σ09IH	Σ10IH
VIn	Σ01IV	Σ02IV	Σ03IV	Σ04IV	Σ05IV	Σ06IV	Σ07IV	Σ08IV	Σ09IV	Σ10IV
EIn	Σ01IE	Σ02IE	Σ03IE	Σ04IE	Σ05IE	Σ06IE	Σ07IE	Σ08IE	Σ09IE	Σ10IE
Σ	${\sum }_{i=1}^{nHY}{\sum }_{j=1}^{nVZ}{\sum }_{k=1}^{nEY}$

.

This calculation then allows the final values for each index for each year to be listed, as is shown in

Table 6. Index values for the time frame under consideration.

H	H01	H02	H03	H04	H05	H06	H07	H08	H09	H10
V	V01	V02	V03	V04	V05	V06	V07	V08	V09	V10
E	E01	E02	E03	E04	E05	E06	E07	E08	E09	E10

.

The obtained values are then integrated into a risk scale with five levels of risk: no risk, negligible risk, low risk, medium risk, and high risk. This scale was implemented for each index separately, and the final risk measure was obtained by the product of these values according to Formula (1).

The final risk scale is given as follows:

• Lowest (i) as the lowest value for the i-th index from Table 6 (for each index separately).
• Highest (i) as the highest value for the i-th index from Table 6 (for each index separately).
• Median (a,b) as the median of values a and b (for each index separately).

The risk measures can then have the following mathematical expressions:

• No Risk (i) = Lowest (i)
• High Risk (i) = Highest (i)
• Low Risk (i) = Median (Lowest (i), Highest (i))
• Negligible Risk (i) = Median (Lowest (i), Median (Lowest (i), Highest (i)))
• Medium Risk (i) = Median (Median (Lowest (i), Highest (i)), Highest (i))

where

• No Risk—the no risk measure is determined by selecting the lowest value.
• High Risk—the high-risk measure is determined by selecting the highest value.
• Low Risk—the low-risk measure is determined for each index by taking the median of the lowest and highest value.
• Negligible Risk—the negligible risk measure at the second level is determined by taking the median of the lowest value and the median of the third level.
• Medium Risk—the medium risk measured at the fourth level is determined by taking the median of the third level and the fifth highest value.

This method is shown in

Table 7. Expression of risk scale boundaries.

	I. No Risk	II. Negligible Risk	III. Low Risk	IV. Medium Risk	V. High Risk
Hazard	Hlow	Hmed(low,med(low-high)	Hmed(low,high)	Hmed(med(low,high),high)	Hhigh
Vulnerability	Vlow	Vmed(low,med(low-high)	Vmed(low,high)	Vmed(med(low,high),high)	Ehigh
Exposure	Elow	Emed(low,med(low-high)	Emed(low,high)	Emed(med(low,high),high)	Ehigh

.

Based on Formula (1), the resulting risk will be the product of the hazard, vulnerability, and exposure indexes, and the resulting risk measure is therefore the product of the values for each risk level individually (

Table 8. Resulting risk.

I. No Risk	II. Negligible Risk	III. Low Risk	IV. Medium Risk	V. High Risk
Hlow	Hmed(low,med(low-high)	Hmed(low,high)	Hmed(med(low,high),high)	Hhigh
Vlow	Vmed(low,med(low-high)	Vmed(low,high)	Vmed(med(low,high),high)	Vhigh
Elow	Emed(low,med(low-high)	Emed(low,high)	Emed(med(low,high),high)	Ehigh
Hlow × Vlow × Elow	Hmed(low,med(low-high) × Vmed(low,med(low-high) × Emed(low,med(low-high)	Hmed(low,high) × Vmed(low,high) × Emed(low,high)	Hmed(med(low,high),high) × Vmed(med(low,high),high) × Emed(med(low,high),high)	Hhigh × Vhigh × Ehigh

).

In the following section, we will outline the results of our application of the empirical data to the proposed assessment framework.

3. Results

This section presents a comprehensive overview of the findings, delving into the quantification of flood damage, the specificities of self-governing regions, and the methodical development of hazard, vulnerability, and exposure indexes. The results of the current study provide valuable insights into the intricacies of flood risk assessment in The Slovak Republic. Through the meticulous utilization [26], a robust foundation for analysis has been established. This section presents a comprehensive overview of the findings, delving into the quantification of flood damage, the specificities of self-governing regions, and the methodical development of hazard, vulnerability, and exposure indexes. The ensuing exploration of the results sheds a light on the effectiveness of the developed methodology and contributes to a deeper understanding of flood risk in The Slovak Republic.

Rating Scale Creation

To create the rating scale, the input data were collected from historical data on floods that occurred in The Slovak Republic from 2001 to 2010. The data are matched with indicators in the individual hazard, vulnerability, and the exposure indexes.

Table 9. Values for the hazard index.

Indexes	Flood Risk Classes
	I.	II.	III.	IV.	V.
HI1.1	0.5	561.83	1123.16	1684.49	2245.82
HI1.2	0.4	8726.3	17,452.2	26,178.1	34,904
HI1.3	0.1	354.33	708.55	1062.78	1417
HI2	164	299.88	435.75	571.63	707.5
HI3	0.25	106.31	212.38	318.44	424.5

Notes: H_I1—flooded area [km²]; H_I1.1—mixed urban land; H_I1.2—agricultural land; H_I1.3—forest.

shows the calculated values for the hazard index for all proposed risk levels, from the lowest to the highest value based on the technique described previously by Zelenákova et al. [26]. Table 9 shows only the input values for the hazard index; the same way the values for the vulnerability and exposure index are collected and ranked, they are shown below.

The values are the results of the selection of input values based on the statistical analysis described in Table 7, and the analysis benefited from the large quantity of data available for each of the indicators for the studied period. All values are ranked by magnitude following the procedure described above.

An identical approach was applied to the vulnerability index, with the results being listed in

Table 10. Value for the vulnerability index.

Indexes	Flood Risk Classes
	I.	II.	III.	IV.	V.
VI1.1	4	24	44	64	84
VI1.2	1	9.63	18.25	596.63	1175
VI1.3	1	3.25	5.5	7.75	10
VI1.4	1	1.13	1.25	3.13	5
VI2	18,832.5	30,729	42,625.5	88,109.5	133,593.5
VI3	8	45.5	83	264	445

Notes: V_I1—affected residents; V_I1.1—people who have lost homes; V_I1.2—evacuated people; V_I1.3—injured people; V_I1.4—dead or missing people; V_I2—affected land with nature protection/CHKO; V_I1.3—condition of flood defenses.

. In selecting the vulnerability elements to be applied into the proposed methodology, indicators that exhibit susceptibility to the effects of a flood event, including factors such as the impact on affected populations, were deliberately chosen for the purpose. Furthermore, consideration was given to the indicators reflecting the capacity for recovery from hazards, as is exemplified by the inclusion of protected landscapes. The methodology also incorporates indicators representing the ability to mitigate the impact of adverse situations, more specifically those addressing the condition of flood protection facilities.

Table 11. Value for the exposure index.

Index	Flood Risk Classes
	I.	II.	III.	IV.	V.
EI1.1	1	1085.5	2170	3254.5	4339
E1.2	1	1497.5	2994	4490.5	5987
EI1.3	1	116	231	346	461
EI2.1	1	110.75	220.5	330.25	440
EI2.2	1	126	251	376	501
EI2.3	-	-	-	-	-
EI2.4	1	21.75	42.5	63.25	84
EI2.5	1	307.25	613.5	919.75	1226
EI3.1	1	111.75	222.5	333.25	444
EI3.2	0	0.5	1	1.5	2
EI3.3	0.05	55,750.04	11,500.03	167,250.01	223,000
EI3.4	0.07	78.99	157.92	2595.79	5033.65
EI3.5	0.33	2350.25	4700.17	7050.08	9400
EI3.6	0.33	1333.57	2666.82	4000.06	5333.3
EI3.7	1	2125.75	4250.5	6375.25	8500
EI3.8	1	2125.75	4250.5	6375.25	8500
EI3.9	1	6214	12,428	18,642	24,856
EI3.10	1	1555.25	3109.5	4663.75	6218
EI3.11	-	-	-	-	-
EI3.12	1	375.75	751.5	1127.25	1503
EI4.1	2	438.75	875.5	1312.25	1749
EI4.2	3	2635.75	5268.5	7901.25	10,534
EI5.1	0.5	87.13	173.75	260.38	347
EI5.2	1	388.75	777.5	1166.25	1555
EI5.3	1	71.75	143.5	215.25	287
EI5.4	2	94.24	186.5	278.75	371

Notes: E_I1—flooded buildings [no.], E_I1.1—apartment buildings; E_1.2—family houses; E_I1.3—other residential buildings; E_I2—flooded non-residential buildings [no.]; E_I2.1—industrial buildings and warehouses, tanks, and silos; E_I2.2—agricultural buildings and warehouses, stables, and barns; E_I2.3—cultural monuments; E_I2.4—hospitals and health or social facilities; E_I2.5—other non-residential buildings; E_I3—damaged engineering networks; E_I3.1—the railway, cable cars, and other tracks [km]; E_I3.2—highways and main roads [km]; E_I3.3—minor roads and pavements [km]; E_I3.4—local, special-purpose, and forestry roads [km]; E_I3.5—drainage canal, sewerage, culverts, wastewater treatment plants [no.]; E_I3.6—water sources, water treatment [no.]; E_I3.7—long-distance oil and gas pipelines [km]; E_I3.8—local gas pipelines [km]; E_I3.9—long-distance and local water and steam pipes [km]; E_I3.10—long-distance and local electricity lines [km]; E_I3.11—timber yards [no.]; E_I3.12—other engineering networks [no.]; E_I4—number of injured animals [no.]; E_I4.1—evacuated livestock, poultry, and small animals; E_I4.2—dead livestock, poultry, and small animals; E_I5—other flood damage; E_I5.1—weight of evacuated material [tons]; E_I5.2—flooded transport facilities [no.]; E_I5.3—driftwood [m³]; E_I5.4—undermined retaining walls [km].

shows the values for the exposure index. This analytical approach facilitated the systematic identification and selection of exposed elements within the flood risk landscape that were amenable to quantification, thereby ensuring the inclusion of tangible factors that contribute to a more nuanced understanding of the potential impact of flood events.

The exposure index values presented in Table 11 form the culmination of this rigorous methodological process, reflecting the quantification of a broad range of elements identified during the analysis. These values serve as integral components within the broader framework of the flood risk assessment, contributing to a comprehensive evaluation of the exposure dimension.

Table 12. The values of the hazard indicators with associated weights.

Indexes	Weight	Flood Risk Classes
		I.	II.	III.	IV.	V.
HI1.1	5	2.5	2809.15	5615.8	8422.45	11,229.1
HI1.2	3	1.2	26,178.9	52,356.6	78,534.3	104,712
HI1.3	2	0.2	708.66	1417.1	2125.56	2834
HI2	10	1640	2998.8	4357.5	5716.3	7075
HI3	10	2.5	1063.1	2123.8	3184.4	4245
Σ	30	1646.4	33,758.61	65,870.8	97,983.01	130,095.1

and

Table 13. The values of the vulnerability indicators with associated weights.

Indexes	Weight	Flood Risk Classes
		I.	II.	III.	IV.	V.
VI1.1	5	20	120	220	320	420
VI1.2	5	5	48.15	91.25	2983.15	5875
VI1.3	5	5	16.25	27.5	38.75	50
VI1.4	5	5	5.65	6.25	15.65	25
VI2	10	188,325	307,290	426,255	881,095	1335,935
VI3	10	80	455	830	2640	4450
Σ	40	188,440	307,935.1	427,430	887,092.6	1,346,755

. show the data for hazard and vulnerability indexes after mathematical procedures wherein the index indicators were multiplied by their assigned weights (Table 1, Table 2 and Table 3). This process of value expression was executed for each of the studied years, with the cumulative values over the entire period constituting the data matrix being subsequently refined according to Table 5. This refinement adhered to the statistical methodologies outlined in the preceding chapter. It should be noted that the data collection process does not constitute the central focus of this paper, serving instead as an integral part of a broader dissertation that encapsulates the entire methodology for developing flood risk assessments.

The weighted values that are essential to the assessment are outlined by presenting the exposure indicators along with their corresponding weights (

Table 14. The values of the exposure indicators with associated weights.

Index	Weight	Flood Risk Classes
		I.	II.	III.	IV.	V.
EI1.1	2	2	2171	4340	6509	8678
E1.2	2	2	2995	5988	8981	11,974
EI1.3	2	2	232	462	692	922
EI2.1	1.2	1.2	132.9	264.6	396.3	528
EI2.2	1.2	1.2	151.2	301.2	451.2	601.2
EI2.3	1.2	-	-	-	-	-
EI2.4	1.2	1.2	26.1	51	75.9	100.8
EI2.5	1.2	1.2	368.7	736.2	1103.7	1471.2
EI3.1	0.5	0.5	55.875	111.25	166.625	222
EI3.2	0.5	0	0.25	0.5	0.75	1
EI3.3	0.5	0.025	27,875.02	55,750.02	83,625.01	111,500
EI3.4	0.5	0.035	39.495	78.96	1297.895	2516.825
EI3.5	0.5	0.165	1175.125	2350.085	3525.04	4700
EI3.6	0.5	0.165	666.785	1333.41	2000.03	2666.05
EI3.7	0.5	0.5	1062.875	2125.25	3187.625	4250
EI3.8	0.5	0.5	1062.875	2125.25	3187.625	4250
EI3.9	0.5	0.5	3107	6214	9321	12,428
EI3.10	0.5	0.5	777.625	1554.75	2331.875	3109
EI3.11	0.5	-	-	-	-	-
EI3.12	0.5	0.5	187.875	375.75	563.625	751.5
EI4.1	3	6	1316.25	2626.5	3936.75	5247
EI4.2	3	9	7907.25	15,805.5	23,703.75	31,602
EI5.1	1.5	0.75	130.695	260.625	390.57	520.5
EI5.2	1.5	1.5	583.125	1166.25	1749.375	2332.5
EI5.3	1.5	1.5	107.625	215.25	322.875	430.5
EI5.4	1.5	3	141.375	279.75	418.125	556.5
Σ	30	35.94	52,274.02	104,516.1	157,937.6	211,359.2

).

Figure 4 provides a visual representation of the flood risk assessment results, succinctly summarizing the values of the computed indexes after the multiplication process with their respective weights. This bar chart serves as an illustrative tool, offering a clear and accessible depiction of the relative magnitudes of hazards, vulnerability, and exposure across the analyzed period.

In accordance with Formula (1), the resultant risk is articulated as the product of the indexes pertaining to hazard, vulnerability, and exposure. However, the intricacy associated with acquiring these values becomes apparent upon the scrutiny of Figure 5.

It is clearly apparent here that the magnitudes of some of the calculated values exceed six decimal places, and this computational intricacy is further compounded by Formula (1) which multiplies the obtained values by their associating weights. However, for the sake of computational efficiency and methodological clarity, it was decided to refine Formula (1) with the aim of optimizing the mathematical representation and streamlining the evaluation process (Formula (2)):

R = H + V + E

(2)

In which the values are summed rather than multiplied; all other procedures for obtaining values to determine the boundaries of the risk scale remain unchanged. This change is represented in Figure 6.

Figure 5 elucidates the resultant risk measure in accordance with the risk assessment methodology delineated, which is predicated upon a comprehensive integration of hazard, vulnerability, and exposure indexes. The amalgamation of these indexes, each quantifying distinct facets of risk components, constitutes an analytical framework.

Table 15. The values of the exposure indicators with associated weights.

Level Scale of Risk Classification	Regional Division
	Bratislavský	Trnavský	Trenčiansky	Nitriansky	Žilinský	Bansko Bystrický	Prešovský	Košický
	Scale Values for the Period Between 2001 and 2010
I. no risk	3713.28	6139.46	1867.66	1775.42	1, 501.98	3480.57	2474.11	2052.19
II. negligible risk	5773.56	18,372.44	9014.99	3912.86	14,241.22	18,883.28	9890.42	6609.40
III. low risk	7076.19	30,230.94	15,041.30	5525.07	27,084.38	34,292.31	16,338.86	11,133.33
IV. medium risk	7296.77	41,297.64	22,621.16	8326.32	38,653.39	48,827.83	23,837.83	13,688.64
V. high risk	11,043.64	52,790.36	25,237.79	10,172.07	53,482.09	66,437.52	29,401.68	17,359.29

offers an overview of the results in numerical and graphical form. The process of scaling the risk classifications was based on the identification of boundary values: the initial boundary was defined by the lowest and the final by the highest value of the set. The middle boundary was determined as the median of these two extremes, while the other two intermediate values were determined by calculating the medians between the lower boundary and the middle, and between the upper boundary and the middle levels.

The delineations formed a five-level evaluation scale, which ensured an even and methodologically consistent distribution of data into individual risk levels. The classification process was applied for each self-governing region and for all of the evaluated indices, thereby ensuring the consistency of the calculations. The methodology was applied to data for the period 2011–2020 in order to test the validity of the methodology and demonstrate its practical applicability in the long-term monitoring of trends in the development of flood risk. The validation results are presented in graphical form, confirming that the proposed procedure provides a reliable framework for the assessment and interpretation of the spatial distribution of risk.

In 2011, a low flood risk level was identified in the Bratislava, Trnava, and Nitra self-governing regions, while a second-level risk was found in the assessment of the Trenčín, Žilina, Banská Bystrica, and Prešov self-governing regions and a third level of risk for the Košice region (Figure 7a). The results from 2012 (Figure 7b) show a first-level risk for the Bratislava, Trnava, and Nitra self-governing regions, while all other regions in the country register a second-level risk. A third-level risk was identified for the Banská Bystrica region in 2013, while the level for the Nitra region rose from the first to the second level (Figure 7c). In 2014 (Figure 7d) a fourth-level risk was reported for the Košice region, while the Bratislava region faced a fifth-level risk, the highest risk level; second levels of flood risk were identified for the self-governing regions of Trenčín, Nitra, Žilina, Banská Bystrica, and Prešov. The following year, 2015 (Figure 7e), saw a first level of flood risk for Bratislava, Trnava, and Nitra (Figure 7e), and a second level of risk was identified for the Nitra region in 2016 (Figure 7f) which rose to the fourth level in 2017 (Figure 7g). The results from 2018 (Figure 7h) showed an identical result to that found for 2012 (Figure 7b) and in 2015 (Figure 7e), where a first level of risk was identified for the Bratislava, Trnava and Nitra regions. In 2019 (Figure 7i) three different levels of risk were recorded: first levels for the Bratislava and Trnava regions, second levels for the Trenčín, Nitra, Žilina, Banská Bystrica, and Prešov regions, and a third level for the Košice self-governing region; the level for the Košice region rose to the fourth classification in the final assessed year of 2020 (Figure 7j), together with second levels for the Trenčín, Žilina, Banská Bystrica, and Prešov regions, with the rest of the country experiencing first levels of risk.

This validation is only a preliminary effort to verify the accuracy of the methodology; a future study will offer a fuller evaluation and verification of the novel approach.

4. Discussion

The formulation of an evaluation matrix requires an overall approach based on an in-depth exploration of different options. In this study, historical data on flood events in Slovakia were applied to a matrix using the mathematical operation to produce a final result. The data is input into the methodology through the initial acquisition of the index values which are then multiplied by the assigned weight to obtain the input values for the final expression of the risk matrix. The proposed methodology also allows for the variability of the indicators. However, the mathematical operation of multiplication can be used either to increase or decrease the indicator values as required. By multiplying the index values, a set of resulting values which defined the boundaries of the five risk levels was created. The initial proposal for the development of the rating scale was based on existing frameworks of risk analysis [36]. The resulting values can determine a risk value which is determined by the risk scale as was also applied in other studies [37,38].

After applying the proposed procedure, the numerical difficulties of the resulting values became apparent. After adding the three values, the resulting figures were ten or more digits long, a fact which greatly complicated the resulting risk matrix. One of the main aims of our methodology for assessing flood risks using the available data was that it should have a simple approach, and the proposed procedure for multiplying the values did not meet this requirement. Therefore, a new procedure was proposed to determine the risk matrix and the limits of the individual risk stages by adding the index values together rather than multiplying them. The new procedure reduced the number of digits in the resulting values and greatly enhanced the overall simplicity and usability of the framework.

The risk matrix enables risk assessments that can define individual levels of calculated risk, but without an input matrix and thresholds that delimit the individual risk levels; indicators that exceed these thresholds or which may be significantly undervalued may invalidate the assessment. It is therefore important to reconcile the elements entering the risk assessment with elements that determine the assessment rate [39,40]. The input indicators can be varied for assessment at macro and micro scales, and visualization tools can also be employed to identify critical and vulnerable areas [41]. The presented methodology sets the framework for risk assessment over a 10-year period for the whole territory of The Slovak Republic. Given the granular nature of the dataset used in the study, the risk measure can be shifted to a smaller scale, for example, a specific city or district, or to a particular time range. In such cases, the numerical complexity of the boundaries of the individual risk levels or the weighting applied to individual indexes may need to be amended accordingly [18,34,35,42]. The resulting values in the risk matrix can be further modified and adjusted by other procedures to ensure that the resulting value is presented in single or double digits.

The methodology can be used in short-term risk analyses of flood risks in The Slovak Republic at the macro scale for the period under consideration, but it can also be used for long-term assessments by being modified for micro periods and micro measures in order to achieve the simplest possible values of the resulting risk matrix. The proposed methodology can also serve as a theoretical basis for the development of an improved risk matrix. In addition to modifying the indicators, it is also possible to modify the thresholds and risk grades. In the framework presented here, five flood risk levels are considered: no risk, negligible risk, low risk, medium risk, and high risk [7,43], but future modifications could adjust the rating scale to fewer levels, or even use a mathematical expression of the level of risk rather than a verbal description.

Considering the complexity and wide range of the input data, the mathematical operation of summing the resulting values was adopted to simplify the approach and permit the possibility of further modifications. Although the model provides a structured approach to risk mapping, future research may focus on refining the indicators and expanding data availability to enhance accuracy.

No external validation of the model was performed using official flood hazard maps, primarily because within the Slovak context, such maps would need to be generated using the same input datasets and methodological structure. However, the model’s structure is designed to support such validation efforts in the future, especially through retrospective comparisons with updated flood records or newly developed hazard maps. Similarly, the initial map-based visualization of the aggregated risk scores incorporated into this study is only a preliminary effort to demonstrate the findings of the methodology; our forthcoming study aims to explore that aspect of the research in more comprehensive detail.

Moreover, although a full sensitivity analysis was not included in this stage, the model was intentionally designed with structural flexibility as the main focus. This would allow for variation in input parameters and weights, supporting the recalibration and testing of different scenarios. Future research may incorporate structured sensitivity testing into the methodology in order to evaluate the robustness of model outputs and a fuller understanding of the influence of individual indicators on the overall risk classification.

5. Conclusions

This study has presented the design and development of an assessment scale that defines boundaries for five flood risk levels using historical data on Slovak flood events. The building block of the methodology is the initial formulation of risk, as determined by the indexes of hazard and vulnerability as variables of probability and consequence. The results of these calculations allowed a rating scale methodology to be derived in which the input index values delineated the boundaries for each risk level. However, this approach ultimately proved to be inapplicable due to the large numerical values generated for the individual thresholds. As a result, the mathematical operation of the product was replaced by the sum of these values, an adjustment which helped to simplify the resulting values, one of the main aims of the development of the framework. The intention of the methodology is to develop a simple flood risk assessment approach, and therefore, a simple assessment tool is required. The methodology can be used in both theoretical and practical terms; the framework uses an initial theoretical basis which can be extended to macro or micro levels, while the practical solution can contribute to the evaluation of flood risk for The Slovak Republic for the studied period. Although new approaches are constantly emerging in flood risk assessment, each offers its own perspective on the phenomenon, and the assessment issues can be improved and refined. This study contributes to this dynamic landscape, recognizing that ongoing refinement and improvement are essential to our collective understanding of this complex phenomenon. By embracing these advances, the flood risk assessment methodology presented here serves as a testament to the ongoing pursuit of clarity and efficacy in understanding and managing flood risks.