Multi-Criteria Assessment of Flood Risk on Railroads Using a Machine Learning Approach: A Case Study of Railroads in Minas Gerais

Abstract

In a climate change scenario where extreme precipitation events occur more frequently and intensely, risk assessment plays a critical role in ensuring the safety and operational efficiency of facilities. This case study uses a combination of the multi-criteria analysis approach and hydrological studies that use machine learning algorithms to simulate new rainfall events in order to estimate the risk of flooding on railroads. Risk variables, including terrain, drainage capability, accumulated flow, and land use and land cover, will be weighed using the multicriteria approach. A methodical evaluation of the most vulnerable locations on the railroad network will be possible thanks to the analysis of these parameters based on the geographic information system (GIS) approach. In the meantime, historical precipitation, flow, and hydrological balance data will be used to calibrate and validate hydrological models. The database required for the machine learning model can be created with these hydrological data. The research regions are situated in the densely rail-networked state of Minas Gerais. The geographical and climatic diversity of Minas Gerais makes it the perfect place to test and validate the suggested approaches. The models evaluated included linear regression, random forest, decision tree, and support vector machines. Among the evaluated models, Linear Regression emerged as the best-performing model with an R² value of 0.999998, a mean squared error (MSE) of 0.018672, and a low tendency to overfitting (0.000011).

1. Introduction

In the current context of climate change, extreme precipitation events are increasing in frequency and intensity. It poses significant risks to the safety and operability of railroad infrastructure. These changes, driven by a complex interplay of atmospheric dynamics and human activities, amplify the severity and frequency of such events. Studies worldwide have shown unprecedented levels of rainfall and flooding, making it critical to understand and mitigate these impacts. Research in Serbia highlights how climate change exacerbates rainfall extremes, leading to significant pluvial flooding risks [1]. Further evidence from Vietnam illustrates the severe impacts of climate change on urban areas. In those areas, the increase in extreme precipitation has led to frequent and severe flooding, disrupting daily life and economic activities [2]. The evaluations that have been carried out to date suggest that the frequency and intensity of extreme rainfall events increase significantly due to global warming. For this reason, it is compulsory to implement better predictive models and mitigation strategies. Historically compiled statistical data on the behavior of climate trends in urban and suburban areas show the diverse impacts on the region, highlighting the need for adapted risk management approaches [3]. Studies conducted in a variety of geographic settings, such as plateau regions where grassland productivity is influenced by extreme rainfall and daytime temperature variations, highlight the broad applicability of these assessment methods [4].

Minas Gerais, a Brazilian state renowned for its extensive rail network and diverse geography and climate, provides an ideal setting for testing advanced risk assessment methods. Its varied topography and climatic conditions offer a unique opportunity to examine the impact of extreme weather events on infrastructure. Choosing Minas Gerais as a case study is particularly important because its diverse environments provide a representative example of how different climatic and geographical factors can influence the resilience of rail systems. To highlight the relevance of this case study, two examples of natural disasters triggered by heavy rainfall and potential drainage system failures will be presented. With this context in mind, the aim of this research is to strengthen the resilience of railroad infrastructure, drawing on insights from the state’s diverse geographical and climatic scenarios to better address the challenges posed by extreme weather.

The risk of flooding on railroads could have serious consequences such as service interruptions and physical damage to infrastructure. In turn, this severely affects the economies of the localities that require rail transport on a daily basis. Some research works emphasize the importance of applying multi-criteria analysis and advanced tools such as machine learning. This means improving the accuracy of prediction and management of this type of risk [5,6]. This is the case of the analytical hierarchy process (AHP) method which, when combined with machine learning, has proven to be effective in infrastructure risk management. It allows detailed and accurate assessments that integrate multiple risk factors [7,8].

This research integrates the AHP approach with hydrological studies supported by machine learning algorithms. This makes it possible to simulate rainfall events and carry out an assessment of the flood risk in the rail network of Minas Gerais. The integration of these approaches makes it possible to assess current risks and the projection of future scenarios.

This paper is divided into five parts. The first part, the Introduction, presents an overview of the Minas Gerais railroad context and outlines the general aspects of this paper. To frame the context of this research, a detailed analysis of previous work in Literature Review is presented. This section identifies trends, challenges, and potential opportunities for future exploration. The third part, Materials and Methods, details the methodologies and tools used throughout this research, starting with a description of the study area and the processes involved in data collection. It then delves into the application of geographic information systems (GISs) for spatial analysis and multi-criteria analysis (MCA) for decision-making. Additionally, it explains the design of the drainage system and the use of hydrological models, followed by an analysis of disasters triggered by intense rainfall on the railroads. This section also explores the implementation of machine learning algorithms to predict the flow rate from watershed contributions. The Results Section presents the findings from the multi-criteria analysis, hydrological analysis, and the performance metrics of the machine learning algorithms. Finally, the Discussion Section provides a critical examination of these results, highlighting their implications and potential applications for enhancing railroad resilience in Minas Gerais.

This study hypothesizes that the integration of the analytic hierarchy process with machine learning algorithms improves the accuracy and reliability of flood risk assessments in Brazil. For that reason, it is compulsory to address the following three questions shown in the next numbered list.

1.

What are the railroad’s main points of vulnerability in relation to extreme weather events, such as heavy rainfall?

2.

How might the estimated service life of the railroad be affected by climate change and an increase in the frequency of heavy rainfall?

3.

How can new approaches such as machine learning help evaluate the performance of existing drainage systems?

2. Literature Review

In recent years, there has been an increasing amount of literature on flood risk assessment, reflecting the critical need to address the growing impacts of climate change on extreme weather events. Techniques such as machine learning (ML) and multi-criteria analysis (MCA) have become central to infrastructure resilience and flood management strategies. Supervised learning models have been particularly effective, as demonstrated by [9,10], who used labeled datasets to predict flood risks in real-time. These models were further enriched by sensor networks, ensuring precise risk assessments. However, their reliance on labeled data poses challenges in dynamic urban settings.

Additional machine learning approaches such as support vector machines (SVMs) and logistic regression have also been widely adopted. The authors of ref. [11] employed logistic regression with remote sensing and topographical data to create flood susceptibility maps validated across various geographies. Similarly, linear regression models provided simpler yet valuable insights into hydrological relationships, such as precipitation–runoff correlations. For more complex datasets, deep learning techniques such as long short-term Memory (LSTM) networks have demonstrated high accuracy, as shown by [12], who used these models to predict water levels in flood-prone regions. These methods, though powerful, often demand significant computational resources.

Hybrid approaches integrating physical and data-driven models are gaining traction. For example, ref. [13] introduced a physics-guided ML model to simulate urban drainage systems, balancing computational efficiency with predictive accuracy. The authors of ref. [14] combined a GIS, remote sensing, and random forest models in Tiruchirappalli, India, optimizing wastewater drainage and treatment zones. Geospatial data integration has proven crucial, as [11,14] utilized a GIS for comprehensive flood-prone area assessments, prioritizing mitigation strategies by overlaying key risk factors such as slope and proximity to water bodies.

Hybrid and deep learning methods demonstrated exceptional capabilities in solving complex issues, enhancing disaster resilience. The authors of ref. [15] applied explainable ML techniques to predict hydrological responses in sustainable urban drainage systems (SuDS), achieving both accuracy and model interpretability.

Applications to infrastructure resilience further highlight the practical value of these methodologies. Physics-guided ML [13] and explainable ML [15] have been instrumental in real-time risk prediction and adaptive infrastructure management. These methods align closely with this study’s objectives: to assess and mitigate flood risks for railroads in Minas Gerais. Through the combination of MCA and ML, this research aims to enhance strategic planning and infrastructure resilience against climate-induced challenges.

Current tools such as the random forest model [16,17] and advanced deep learning techniques [18,19] have produced highly accurate flood risk maps. These maps serve as critical tools for infrastructure management and decision-making [20]. The integration of geospatial and Earth observation data with MCA facilitates real-time and comprehensive flood risk assessment [21,22]. Successfully applied globally, these methods enhance infrastructure resilience against extreme weather [23,24]. Implementing similar methodologies in Minas Gerais can inform mitigation strategies adaptable to other regions. This study seeks to improve railroad infrastructure resilience in Minas Gerais, ensuring safety and reliability amid climate-related challenges. Employing ML, MCA, and geospatial tools will optimize decision-making and strategic planning in the context of ongoing climate change.

3. Materials and Methods

3.1. Study Area

The state of Minas Gerais, where the area of interest of this study is located, is known for its leading role in the railroad sector in Brazil. The rail network in Minas Gerais is currently managed by four private concessionaires: Estrada de Ferro Vitória-Minas (EFVM), Ferrovia Centro Atlântica S.A (FCA), Minas Rio Logística S.A (MRS), and Rumo Malha Central (RMC). The total length is approximately 5000 km. According to the National Association of Brazilian Railroad Transporters (ANTF), rail transport accounts for 21.5% of the freight transport matrix in Brazil, while road transport predominates with 67.6%. The Brazilian rail network is only 30,810 km long, in contrast to the road network, which extends for approximately 1.7 million km. Minas Gerais stands out in this scenario due to its significant rail infrastructure, which is crucial for transporting commodities, consolidating the state as one of the country’s main rail transport hubs. The state of Minas Gerais is responsible for the largest volume of freight transported by rail in Brazil. In 2023 alone, 44% of the total useful tons were transported, which is equivalent to 235,000,972 tons of commodities [25,26,27,28].

Table 1. Commodities transported on railroads in Minas Gerais, Brazil.

Concessionaire	Products
RMC	Fertilizers, corn, wheat, soybeans, bran, vegetable oil, and sugar
EFVM	Iron ore, coal, and steel products—coil—bf
FCA	Soybeans, grains—corn, sugar, soybean meal, and iron ore
MRS	Iron ore, sugar, bulk cement, and steel products—others

shows the main products transported on these railroads.

The study area consists of 38 municipalities in the state of Minas Gerais, totaling 10. 367.36 km². The region is crossed by 447.54 km of operating railroads, 60.66 km of which is managed by Estrada de Ferro Vitória-Minas (EFVM), 276.21 km by Ferrovia Centro Atlântica S.A. (FCA), and 110.67 km by Minas Rio Logística S.A. (MRS) [29]. This area is representative of the state’s economic and geographical diversity, especially due to its intense mining and agricultural production, and its strategic role in the logistics of exporting mineral and agricultural commodities. Figure 1 depicts the area of interest in this study case.

The region is marked by a wide variety of geomorphological features. There are areas of plateau in the central zone, such as the Belo Horizonte Plateau, mountainous and rough terrain to the southwest, such as the geological zone known as Serra do Quadrilátero Ferrifero, and areas of depression to the north, close to the São Francisco River basin. The topography varies significantly, with altitudes between 628 and 1738 m above sea level. Hydrologically, the region is crossed by important rivers such as the Rio das Velhas and has several streams and dams, including the Pampulha, which are essential for water supply and flood control. The land cover in the urban areas is predominantly intense urbanization, while in the rural areas, agricultural areas and fragments of native vegetation, such as the Cerrado, predominate. Soil types include lateritic and clay soils, common in tropical and subtropical climate regions, with ferruginous soils due to the presence of iron minerals in the area [30,31,32,33].

3.2. Research Methodology

This study is organized into four main stages: data collection, multi-criteria analysis, hydrological modeling, and the application of machine learning algorithms for prediction. Figure 2 shows the flowchart of the steps described.

The first stage consists of the comprehensive collection of data relevant to the analysis. This includes meteorological, hydrological, and topographical data, as well as information on the existing infrastructure and geographical conditions of the study area. These data are obtained from a variety of sources, such as environmental sensors, historical records, and geospatial databases.

The second stage uses the hierarchical multi-criteria analysis approach. This technique allows multiple criteria to be assessed simultaneously, providing a holistic and integrated view of the factors that contribute to flood risk in any given area. The criteria considered include land use and cover, the drainage capacity of the region, and the slope of the land. This analysis aims to identify flood-prone areas.

The third stage involves hydrological modeling, where mathematical models are used to estimate the precipitation intensity and its behavior. These models are used to calculate the flow rate, which is the main variable of drainage design.

The fourth stage applies machine learning algorithms. The figure below illustrates the workflow for implementing a machine learning system applied to predicting flow rates and monitoring drainage systems. The process starts with data collection and proceeds through the steps outlined in Figure 3 below.

With the data collected and the information generated by the previous stages, these algorithms are trained to identify patterns and predict precipitation. By analyzing future rainfall volumes for the region, it will be possible to identify possible failures in the drainage assets. The proposed methodology consists of creating a database with future rainfall values, through which it will be possible to compare the flow values (calculated with current rainfall data) and thus assess whether the system remains safe, regardless of changes in the region’s rainfall pattern. This comparison will be performed in relation to the reference design flow, which is traditionally calculated in Brazil using the rational method.

3.3. Data Collection

Data collection is fundamental to this study. It provides the necessary inputs for both the MCA and machine learning models.

The topographic data were obtained from TOPODATA of the National Institute for Space Research (INPE) [34], a geophysical database that provides detailed digital elevation models (DEMs). These data are necessary for hydrological modeling (cumulative flow and flow direction) of the study area. They were used to assess the terrain’s drainage capability and susceptibility to flooding. Land use and land cover data were collected from ArcGIS Living Atlas of the World [35], a web map resource through which it is possible to download files in raster format containing classification and land use data. The map is derived from ESA Sentinel-2 images with a resolution of 10 m. It is a composite of land use and land cover (LULC) predictions for 10 classes throughout the year, in order to generate a representative image of 2023. In addition, precipitation data were obtained from the monthly CHIRPS (Climate Hazards Group InfraRed Precipitation with Station data) and MOD16 sets [36,37]. CHIRPS provides precipitation estimates with high spatial and temporal resolutions, combining satellite data with measurements from weather stations, while MOD16 offers information on evapotranspiration, which is useful for understanding the water balance in the region studied.

3.4. Geographic Information System (GIS)

The analysis techniques used in this study were developed using QGIS software, version 3.34.8 [38]. The main tools used were the spatial analyst commands contained in the hydrology menu. These commands allow the processing of shapefile-type vector files and raster-type geospatial files.

The GIS analysis process, which includes applying these tools to evaluate critical factors such as slope, flow direction, and water accumulation, is explained in detail below. The entire workflow is illustrated in the process flowchart in Figure 4.

The image presented is a detailed flowchart of the analysis process used in this study, illustrating the various stages and tools applied. The flowchart begins with three sets of input data: a digital elevation model (DEM) of the area of interest, the river flow direction, and the land use and land cover (LULC) in Minas Gerais, all indicated in light blue boxes (input). From the DEM, two analyses are performed: slope and flow direction, which result in slope and drainage angles, respectively. Both results are reclassified, as indicated by the purple boxes (reclassified output). The river flow direction is reclassified directly, while the LULC data are first clipped to the area of interest and then reclassified. All reclassified data (slope, flow direction, and LULC) are then combined using the weighted overlay technique, represented by an orange diamond (GIS process). The final result of this process is the identification of flood-prone areas, indicated by a green box (final output). The legend to the left of the flowchart clarifies the function of each shape and color: inputs (light blue boxes), GIS processes (yellow diamonds), outputs (red boxes), reclassified outputs (purple boxes), and the final result (green box).

The following steps describe the detailed GIS data processing used in the multi-criteria model to analyze and identify flood-prone areas around the railroads in the study region.

• Slope tool (spatial analyst): The slope is the maximum rate of change in the z-value (“elevation”) of each pixel in the raster image. For degrees, the values range from 0–90. This tool was used to determine the degrees of slopes in the digital elevation model. It is worth noting that the greater the degree of slopes, the greater the runoff, and the lower the degree of slopes of the surface, the greater the propensity for water to accumulate.
• Fill tool (spatial analyst): This tool was used to fill in all the depressions in the digital elevation model (DEM) in order to carry out the hydrological analysis. This ensures uniformity in the data to avoid areas of false depression. This enabled the appropriate flow direction to be generated within the basin, as well as determining the accumulation of water flow in the region studied.
• Flow direction tool (spatial analyst): The corrected DEM derived from the fill was used to produce a flow direction raster. The flow direction indicates the possible direction of water flow in the elevation model.
• Flow accumulation tool (spatial analyst): This tool was applied to assess flow accumulation from the flow direction raster as input. Output pixels with high flow accumulation are areas of concentrated flow and were used to create stream channels/networks.
• Reclassification tool (spatial analyst): this tool was used to reclassify all the factors that influence flooding, such as slope, flow direction, and land cover, on a common scale ranging from 1 to 4, where 1 represents very low risk and 4 high risk (see Table 2).
• Weighted overlay analysis: this technique was used to overlay the raster datasets of slope, flow direction, and land cover that were reclassified on a common measurement scale and weighted according to Equation (1) [39]:

$$\mathrm{Weights}={\sum }_{i=1}^{n}0.40\times {LULC}_i+0.35 \times {Flow Direction}_i+0.25\times {Slope}_i$$

(1)

• Buffer tool (vector/geoprocessing): This tool was used to create buffer zones around the railroad line. A buffer distance of 2 km was adopted to define the area of influence of the railroad.

3.5. Analytic Hierarchy Process (AHP) and Multi-Criteria Analysis (MCA)

Several studies consider precipitation as a risk criterion. In this article, for the purposes of simplification, one of the scenarios proposed by the methodology of [39] was used, which simulates two scenarios for assessing flood susceptibility in a given region: one with four criteria (LULC, slope, accumulated flow, and accumulated precipitation) and another that excludes precipitation data. The main advantage of this method is its ease of replication in strictly urban environments, as it considers land use and land cover as the primary factor. Although soil type influences water infiltration and consequently runoff, the method proposed by [39] simplifies the usual multi-criteria approach for flood analysis by incorporating soil influence into the weighting of the land use and land cover variable.

Table 2 outlines the flood susceptibility based on three factors, land use and land cover, slope, and accumulated flow, each classified into levels of flood vulnerability ranging from very low to high. Areas with vegetation (Class 1) are the least vulnerable due to their natural ability to absorb and retain water, while areas with regenerating forests and planted trees (Class 2) have slightly higher vulnerability but still provide good protection. Agricultural lands and pastures (Class 3) are moderately vulnerable due to vegetation cover and compacted soil while exposed soil and urban areas (Class 4) are highly vulnerable because of the lack of vegetation, impermeable surfaces, and potential drainage issues. For slope, sloping areas greater than 51 degrees (Class 1) have very low susceptibility as water quickly runs off, slopes ranging from 23 to 51 degrees (Class 2) have low susceptibility, slopes from 9 to 23 degrees (Class 3) have moderate susceptibility as water may accumulate more easily, while slopes between 0 to 9 degrees (Class 4) are highly susceptible since water has more time to infiltrate and accumulate. In terms of accumulated flow, areas with flow from 0 to 1600 units (Class 1) have very low susceptibility due to minimal water accumulation, areas with flow between 1601 and 5500 units (Class 2) have low susceptibility, areas where the flow ranges from 5501 to 18,000 units (Class 3) have moderate susceptibility, while areas with flow greater than 18,000 units (Class 4) are highly susceptible due to significant water accumulation.

Table 2. Multicriteria classes for flood susceptibility assessment.

Land Use and Land Cover	Class	Susceptibility
Native vegetation	1	Very Low
Areas in regeneration and planted forests	2	Low
Agricultural areas and pastures	3	Moderate
Exposed soil and urban area	4	High
Slope (Degrees)	Class	Susceptibility
>51	1	Very Low
23 to 51	2	Low
9 to 23	3	Moderate
0 to 9	4	High
Accumulated Flow	Class	Susceptibility
0 to 1600	1	Very Low
1601 to 5500	2	Low
5501 to 18,000	3	Moderate
>18,000	4	High

Table 2. Multicriteria classes for flood susceptibility assessment.

Land Use and Land Cover	Class	Susceptibility
Native vegetation	1	Very Low
Areas in regeneration and planted forests	2	Low
Agricultural areas and pastures	3	Moderate
Exposed soil and urban area	4	High
Slope (Degrees)	Class	Susceptibility
>51	1	Very Low
23 to 51	2	Low
9 to 23	3	Moderate
0 to 9	4	High
Accumulated Flow	Class	Susceptibility
0 to 1600	1	Very Low
1601 to 5500	2	Low
5501 to 18,000	3	Moderate
>18,000	4	High

The multi-criteria analysis (MCA) framework used in this study is based on the analytic hierarchy process (AHP). MCA was chosen for its ability to handle multiple criteria and provide a clear rationale for weighting these criteria. Various criteria were selected based on their relevance to flood risk assessment. It includes terrain characteristics (slope and elevation), drainage capability (soil permeability and land cover), the historical frequency of flooding, and the economic impact of service interruptions. In the use of MCA, each criterion was compared pairwise to determine their relative importance. This process involved expert judgment and was facilitated by creating a comparison matrix.

3.6. Drainage System Design and Hydrological Models

The drainage system is developed using the data obtained from hydrological studies (precipitation and basin contribution modeling), including the design and placement of the assets that comprise the drainage system. This phase is where the drainage assets are calculated and detailed. Generally, the purpose of drainage assets is to capture and direct all the water that reaches the railroad track, in any form, to the appropriate location, whether it is surface water (precipitation) or deep water (groundwater). In Brazil, the drainage system in linear infrastructure works is subdivided into two categories: surface drainage and deep drainage. The aim of a surface drainage system is to design assets with the capacity to collect and convey the water that precipitates on the railroad and its surroundings to a suitable drainage point, ensuring the integrity of the infrastructure and the safe continuous flow of vehicles. Deep drainage, on the other hand, aims to intercept, collect, conduct, and remove water from the groundwater and from surface infiltration in the structural layers of the pavement, through properly dimensioned assets, in order to protect and guarantee the estimated useful life of the rail track bed [40].

Due to the complexity of the analysis, some simplifications were adopted for the development of the models in this study. Firstly, the deep drainage aspects were not evaluated. The main focus here is the evaluation of design flows used to dimension surface drainage elements. It was decided to assess only the cut and embankment gutters, as well as the embankment ditches, primarily due to the complexity of evaluating the drainage system as a whole and the lack of information on the design parameters of the studied railroads. Furthermore, ditches and gutters are some of the most important elements of the surface drainage system and play a crucial role in combating pavement erosion. The evaluation will be carried out by calculating the flow rate of contributing basins, using data collected in the previous stages of this study. The database created has been used in a prediction model capable of estimating flow values from the input of rainfall data. The model aims to be a tool for analyzing drainage assets, especially in areas that are more susceptible to flooding.

According to Brazilian regulations, surface drainage assets are designed by determining the design flow rate, calculated using the Manning formula (Equation (2)) associated with the continuity equation (Equation (4)). The Manning equation is shown below:

$$\mathrm{V}={\displaystyle \frac{R^{2/3}\times I^{1/2} }{\eta }}$$

(2)

where:

• V = flow velocity, in m/s;
• R = hydraulic radius, in m;
• I = longitudinal slope of the asset, in m/m
• η = Manning’s roughness coefficient, dimensionless, depending on the type of lining used.

The hydraulic radius R is defined as the ratio of the flow area to the wetted perimeter of the conduit in which the fluid is flowing. This relationship is an important measure in hydraulic analysis, particularly in studying the behavior of flows in channels and pipes. It is represented by the formula below:

$$R={\displaystyle \frac{A}{P_w}}$$

(3)

where:

• A = flow area, in m²;
• P_w = wetted perimeter, in m

The continuity equation is presented as follows:

$$Q_{adm}=A_c\times V$$

(4)

where:

• V = flow velocity, in m/s;
• A_c = contributing area, in m²;
• Q_adm = admissible flow or drainage capacity of the asset, in m³/s

The contributing area A_c refers to the portion of land or surface that directs runoff or water flow towards a specific point, structure, or drainage system. In railroad engineering, it typically includes the finished half-platform and any additional device width, accounting for critical lengths that depend on the grade conditions. To simplify the calculation of the contributing area, the criterion recommended by [41] was adopted, which specifies that the “Contribution area, in m², is determined through topographic, aerial photogrammetric, or expedited surveys”. Using this approach and estimating the area of interest through a digital elevation model (DEM) file, the contributing area was determined to be 2 ha. The calculated values for the afferent flow rate Q are provided in Appendix A.

It is important to note that the equation used to calculate the flow rate basin considers the rainfall intensity for the study site. In Brazil, the method developed in 1957 by engineer Otto Pfafstetter is applied to process rainfall data and determine precipitation intensity for the purposes of surface drainage design. The formula is the result of the compilation of rainfall records from 98 stations of the Meteorological Service of the Ministry of Agriculture [42]. The equation of the Otto Pfafstetter method is characterized by Equation (5).

$$i= {\displaystyle \frac{K\times T^a}{{(t+b)}^c}}$$

(5)

where:

• i is the maximum average intensity (mm/min);
• t is the rainfall duration (minutes);
• T is the return period (year);
• K, a, b, and c are specific constants for each rainfall station.

The parameters K, a, b, and c are adjusted considering the rainfall intensity of each region. In this study, the parameters were considered for each of the 38 municipalities comprising the area of interest. Generally, in Brazil, the parameters available in the database of the free software Plúvio 2.1 [43].

For the hydraulic design of the ditches and gutters, the contribution discharge is calculated using the rational method, the formula of which is shown below in Equation (6)

$$Q={\displaystyle \frac{C\times i\times A_c}{3.6\times {10}^6}}$$

(6)

where:

• Q = contributing flow in m /s;
• C = runoff coefficient/deflection coefficient, dimensionless, set according to the soil–vegetation cover complex and slope of the land, depending on the type of table to be used;
• i = rainfall intensity, in mm/h for the design rainfall, fixed in the hydrological study;
• A_c = area of contribution, in ha

Finally, in hydraulic design, the admissible flow (Q_adm) calculated in Equation (4), which measures the hydraulic capacity of the protective ditch or gutter, must be greater than the flow in Equation (6), which is the contribution flow [41]. Therefore:

$$Q_{adm}>Q$$

(7)

The Brazilian standard specifies mandatory cases for the inclusion of ditches and gutters in highway and railroad projects [43]. Below are the situations that require the discharge of intercepted and conducted water volumes by the ditch and gutter:

• Extensive cuts with a small slope, where the critical length of the ditch is reached, and increasing flow capacity requires larger sections;
• The presence of a well-defined secondary valley, causing water to concentrate in a single location;
• A sinuous longitudinal profile of the ditch with several low points, requiring great depths to ensure continuous flow. In these cases, a water outlet device, known as a downspout, is used to protect the rail track;
• The gutters should be located at all cuts, being constructed along the shoulder, terminating at convenient discharge points (cut-to-embankment transition points or catch basins).

Although the guidelines for including ditches and gutters in railroad projects and design calculation parameters are described, there are limitations that must be considered in this study. The lack of real design data makes it impossible to accurately assess the capacity of existing drainage assets in the study area. However, the importance of a detailed analysis of flood risks and the drainage capacity of infrastructures cannot be underestimated, as failures in the drainage system can result in serious consequences, especially during heavy rainfall events. Cases of natural disasters related to extreme rainfall and drainage system failures will be discussed in detail in the following section, highlighting the need for risk assessment studies focused on transportation infrastructure.

To proceed with this study, the reference values established by the standard [41] will be adopted, which specifies critical hydraulic parameters for the design and evaluation of drainage systems such as ditches and gutters. According to the standard, the maximum slope (I) is 0.006 m/m, the maximum flow velocity V_max is 1.42 m/s, and the maximum allowable flow rate (Q) is 0.98 m³/s. These values will be used as a basis for simulations and evaluations of the drainage system’s capacity in the study area, ensuring that the calculations are conducted according to the parameters recommended by the standard.

3.7. Disasters Triggered by Intense Rainfall on Minas Gerais Railroads

A recent study conducted by the Brazilian government identified landslides, erosions, and heat waves as the main risk factors for rail infrastructure in Brazil. The report highlights Minas Gerais railroads as among the most vulnerable to landslides: the Estrada de Ferro do Litoral (connecting the Iron Quadrangle to the Ports of Santos, operated by MRS Logística), the Estrada de Ferro Minas-Rio (also operated by MRS Logística), and the Estrada de Ferro Vitória a Minas (connecting the interior of Minas Gerais to the port of Espírito Santo) sections are at high risk of landslides. In the case of erosion, the Vitória-Minas Railroad and the Estrada de Ferro Minas-Rio Railroad are indicated as having the highest climate risk indices. As for the effects of heat waves, the most susceptible are the Minas-Rio Railroad and the Vitória-Minas Railroad. However, flooding events also represent a significant risk. Between 2011 and 2021, 2522 occurrences were recorded due to rain and storms on railroads throughout Brazil, with 254 of these being flooding events [44,45].

To illustrate the importance of the study topic, two examples of natural disasters resulting from the combination of heavy rainfall and potential failures in the drainage system will be presented. These events occurred in the years 2011 and 2022 in Minas Gerais, highlighting the vulnerability of local infrastructure to these extreme climatic factors.

Figure 5a illustrates a case of a flood that resulted in the erosion of part of the Centro Atlântica Railroad (FCA) track. Both events were caused by an intense volume of precipitation that hit the region of Lavras, in Minas Gerais, in January 2011. The force of the water caused the embankment to collapse, damaging approximately 30 m of the railroad line and interrupting traffic. This incident occurred on the urban perimeter of the railroad, causing significant infrastructure losses [46].

The second event, shown in Figure 5b, happened in January 2022 in the municipality of Araguari, in Minas Gerais. According to information provided by the concession company responsible for managing the railroad, the erosion had already been identified and was being monitored. Nevertheless, the heavy precipitation that has hit the region has significantly aggravated the situation. At the time, daily losses were estimated at around USD 5000 per day [47,48].

The examples mentioned above emphasize the importance of both appropriate drainage systems and maintenance of the assets, as well as continuous monitoring of the railroad infrastructure. The implementation of effective drainage systems can prevent erosion and flooding, while regular maintenance and constant monitoring ensure early identification of problems, allowing rapid interventions and minimizing economic losses and safety risks.

3.8. Machine Learning Algorithms

Machine learning (ML) is a discipline of computer science that allows computers to have the ability to learn without having to be programmed manually. ML mathematical models use statistical theories to forecast labeled data, then pre-processing is usually required to structure the raw data. There are three categories for ML models: supervised (regression and classification), unsupervised, and reinforced learning. All models applied in this analysis are supervised. A supervised learning model is any algorithm that gives insights from data with explicit supervision, i.e., the data are previously trained. A supervised regression problem aims to predict a continuous value, while the classification type is used to predict qualitative responses [49,50,51].

For all models, the data were divided into training, testing, and validation sets in a ratio of 7:2:1. This subdivision is a common practice in machine learning model development, ensuring that the model can generalize effectively to new data. The data are separated into distinct subsets, allowing the model to learn, be adjusted, and finally be tested. The first one, known as the training set, is used to teach the model by allowing it to learn the patterns and relationships within the data. The second subset, referred to as the validation set, is used to fine-tune the model’s hyperparameters and adjusts these during the training process, helping to assess the model’s performance metrics. Finally, the test set is used to evaluate the model’s final performance on unseen data, providing an estimate of how it will behave in real-world scenarios.

The main concepts of the methods employed in this study are briefly presented as follows:

• Linear regression (LR): It is described as a first-degree equation. A linear regression model finds a linear relationship between an independent variable (X) and a dependent variable (Y). It predicts the target variable (Y) through the weighted sum of the characteristic variables (X), also known as features. The values calculated by the model are approximating a straight line. The linearity between its variables makes it an accessible model to interpret [52].
• Support vector machine (SVM): This algorithm was developed by [53]. This algorithm produces a series of hyperplanes to transform non-linear features into linear ones. Its goal is to identify a hyperplane with a small norm at the same time as minimizing the sum of the distances between the data points and the hyperplane. In regression problems, the aim is to find a hyperplane that is “near“ as many data points as possible [54,55].
• Decision tree (DT): It is one of the supervised machine learning methods known as tree-based methods. These are based on segmentation and stratification of the predictor space into simple regions, to categorize or predict the target answer. The method detects the potential structure to describe the model adopting recursive segmentation to continuously split the data space into diverse subsets. Thus, it can determine important relationships between the variables and patterns of behavior from them [56].
• Random forest (RF): According to [57], it is an algorithm that creates a combination of predictors called trees, such that each depends on the value of an independent vector sampled randomly. These vectors follow the same distribution for all trees in the forest. The generalization error obtained from a forest of tree classifiers depends on the correlation between them. The algorithm is commonly used because it is simple to understand and presents good results in different types of machine learning analysis.

To evaluate the performance of all the models, the following metrics were used: R² score, MAE, MSE, and RMSE. The R² score is a measure of fit that evaluates the linear relationship between the independent variables (X) and the target variable (Y). The parameter takes on values ranging from 0 to 1, which allows the quality of the model to be assessed, with 1 being the most appropriate fit value. Therefore, it allows how well the test data samples are likely to be predicted by the model to be measured through the proportion of variation in the training data. The mean absolute value (MAE) measures the error between the real variable and the predicted variable. MAE values approaching zero suggest increased correctness in the predicted values [52,58]. The mean square error (MSE), as the name implies, can be defined as the mean squared error of the model. The root mean square error (RMSE) is by definition the square root of the MSE. The MSE is measured in units that are the square of the target variable, resulting in larger values for larger errors, which somewhat penalizes less accurate models. On the other hand, the RMSE is measured in the same units as the target variable.

The machine learning model employed in this study was developed to predict the flow rate from the watershed contribution. This parameter is essential for designing drainage systems, as it is the admissible reference value (see Equation (7)). The dataset includes hydrological information such as evapotranspiration, air humidity, average temperature, and precipitation from the study area. Other variables are determined using Equations (2)–(6) presented previously. It is important to note that, despite the limitations, the purpose of this model is to support the analysis of the critical points of the infrastructure by considering design factors. This aims to provide an overview of areas that may require attention or improvement, helping to optimize resources and decision-making.

4. Results

4.1. Multi-Criteria Analysis

As shown in Figure 6, the area of interest is characterized by a significant variation in altitude (Figure 6a) and dense urbanization (Figure 6d), mainly in the central part of the study area near the city of Belo Horizonte. In the central zone, plateau areas also stand out. Mountainous terrain is found towards the south. To the north, depressed areas predominate. Despite the variations in altitude, the topography of the land in the study area is predominantly flat with a low slope (<9%), which makes the region of interest highly vulnerable to flooding according to the slope criterion. Elevation inversely affects flooding, meaning that as elevation decreases, the frequency of flooding increases, making lower elevations more susceptible to flooding.

Another important factor in assessing flood susceptibility is flow accumulation. In the methodology proposed in this study, flow accumulation was determined from the flow direction raster. In the flow accumulation raster, each pixel contains information on the amount of water flowing at a given point, acting as a discharge profile. Therefore, an increase in flow accumulation suggests an increase in flood susceptibility. However, considering that the study area has a very low risk in relation to flow accumulation and an elevation ranging from 628 m to 1738 m, we can conclude that there is little flow convergence in the region. This significant variation in altitude, together with areas of low to moderate slope (ranging from 9° to 23°), leads to rapid water runoff in some parts and slower runoff in others, resulting in a low amount of water accumulating at specific locations. Consequently, this reduces the probability of significant flooding in the area, since the topography and slope contribute to the dispersion of water flow.

Analysis of land use and its correlation with susceptibility to flooding reveals that areas with predominantly native vegetation have very low susceptibility, due to the vegetation cover that helps absorb rainwater. Agricultural areas and pastures, however, are moderately susceptible, as densely compacted soil can interfere with water infiltration. On the other hand, densely urbanized areas with exposed soil have the highest susceptibility to flooding due to soil impermeability resulting from the construction of infrastructure. This scenario is especially critical in the center of the study area, where the city of Belo Horizonte, with its high urbanization density, significantly increases the risk factors, increasing the frequency and severity of flooding. The most susceptible areas are between the cities of Belo Horizonte, Contagem, Ibirité, and Betim, where urban segments of the Ferrovia Centro Atlântica (FCA) are located. In contrast, the least vulnerable sections, in green, are to the east, in the cities of Caeté and Barão de Cocais, in the eastern region of the study area, where sections of the Vitória-Minas Railroad (EFVM) are situated.

4.2. Hydrological Analysis

The hydrological studies carried out in this study are based on the guidelines established by the National Department of Transportation Infrastructure (DNIT), as described in the DNIT Drainage Manual [59]. These regulations require a detailed analysis of the watersheds and calculations of precipitation and flow rates, which vary according to the area of the watershed. The watershed or drainage basin is defined as the area that collects precipitation and contributes to the flow of watercourses and their effluents. For the drainage design of road infrastructures, the regulations require the delimitation of the area of each contributing sub-basin, subdividing the watershed into regions that supply different drainage assets. The boundaries of a contributing basin are determined by the water dividers or spikes that separate it from adjacent basins. These areas must be calculated or measured planimetrically, with the results expressed in hectares [60]. The delineation of drainage sub-basins and the other geoprocessing steps involved were developed using QGIS 3.34.8 software [38].

The study area comprises four river basins: the Rio Doce Basin, the Pará-São Francisco Basin, the Rio Paraopeba, and the Rio das Velhas, as illustrated in the Figure 7 below.

The sequence of steps required to draw up a railroad drainage system indicates that the number and positioning of assets depend mainly on the area of contribution of the basins. To reduce the complexity of this case study, the largest contributing area (A = 20 ha. The maximum admissible for the rational method is 100 ha) was adopted as a constant for all analyzed sections of the railroad. The coefficient of runoff (C) adopted was 0.80, as established by the standard [54] for impermeable surfaces. Finally, the concentration time was set at 5 min. According to the rational method (see Equation (6)), the average rainfall intensity over the basin (i) is determined by Equation (5). For this determination, the time of concentration of the basin and the time of recurrence appropriate to the asset to be designed are considered. According to the Brazilian standard [60,61], a return time of 5 or 10 years is generally adopted. However, in this paper, in order to simulate different scenarios and understand the variation in the design flow rate pattern over the years, rainfall was calculated for the following return times: 5, 25, 50, and 75 years. The results are presented in Appendix A.

The return time is the variable that indicates the statistical probability of an intense rainfall event occurring within a specific period. For example, a 5-year return time means that an event has a 1 in 5 (or 20%) probability of occurring in any given year, while a 75-year return time indicates a 1 in 75 (or 1.33%) chance of occurrence each year. In drainage systems, the return time is essentially employed to design the assets of the drainage and water collection system, such as culverts, channels, and ditches, to ensure that they can handle the volume of water from precipitation events. In addition, carrying out risk analyses based on the assessment of return time has become a fundamental concept in flood risk studies, as it allows the potential impacts on existing infrastructure to be quantified.

A closer examination of the table reveals that the flow calculated by the rational method varied by an average percentage of 68.23% over the study period, which covers return times of 5 to 75 years. During the first 20 years (between the return times of 5 and 25 years), there was an average increase in flow of 30%. In contrast, in the final interval (between 50 and 75 years), the increases were in the order of 10%. The city of Itaúna registered the highest percentage variation, with a significant increase of 91.54% over the years. This indicates that the flow rate calculated for a return time of 5 years, which was initially 7.06 m³/s, almost doubled in 70 years, reaching 13.53 m³/s.

Although the rail infrastructure in the Minas Gerais region was designed assuming a probability of rainfall in the range of 270 to 300 mm, the city of Itaúna is already experiencing rainfall that reaches or exceeds this magnitude. In October 2023, the city recorded accumulated rainfall of 97 mm, a figure more than 10 times higher than the expected precipitation intensity shown by Equation (5) (8.67 mm/min for TR = 75 years) [62,63]. This event demonstrates that the existing infrastructure may not be sufficient to deal with the rainfall intensities that are already occurring, highlighting the need to review and consequently adapt the drainage and water management systems in the region.

Figure 8 shows the results of the MCA, with a special focus on the city of Itaúna.

These results are consistent with the multi-criteria analysis obtained earlier. As illustrated, several segments along the FCA railroad that passes through the city are identified as high-risk areas, surrounded by regions of moderate risk. This distribution highlights the vulnerability along the section of the railroad as indicated by the rainfall pattern analysis using the rational method. It seems possible that these results are due to massive urbanization along the railroad, as well as other factors such as land cover and use, topography, and soil composition, all of which contribute to the overall assessment of flood risk.

4.3. Machine Learning Metrics

The results of the regression models used in this study are summarized in

Table 3. Metrics of Machine Learning models.

Model	R2	MSE	RMSE	MAE	Overfitting
Linear Regression	0.999998	0.018672	0.136645	0.065880	0.000011
Ridge Regression	0.999998	0.018672	0.136647	0.065892	0.000011
Lasso Regression	0.999998	0.018571	0.136275	0.064732	0.000012
Random Forest	0.991793	63.041223	7.939850	4.988735	0.007527
Decision Tree	0.986610	102.851934	10.141594	6.778889	0.013390
Support Vector Machine	0.646779	2713.268202	52.089041	25.934691	0.013329

.

Firstly, it is important to notice that ridge and lasso regression are techniques employed in linear regression models to mitigate overfitting and improve the ability of the model to generalize the data.

Based on the metrics provided in Table 3, we can conclude that the linear models are the best prediction approaches. Firstly, linear, ridge, and lasso regression have the lowest mean squared errors (MSEs) compared to the other models, indicating a better ability to adjust to the training data and a lower likelihood of overfitting. In addition, their coefficients of determination (R²) are extremely close to 1, which suggests that the models explain practically all of the variability in the data accurately. This contrasts with the performance of the support vector machine model, which has a significantly lower R² of less than 0.70. It also presents the highest values for mean absolute error (MAE) and root mean squared error (RMSE), indicating worse accuracy in predicting precipitation.

Regarding the random forest and decision tree models, the results show that although these have relatively good R² scores, they have very high MSE and MAE values. In addition, these models tend to suffer from overfitting, which indicates that they overfit the training data and may perform poorly on new data.

Overfitting is a common term used to describe excessive learning in machine learning models. This occurs when the model fits the training data very accurately while failing to generalize to new data. In other words, the model becomes highly specific to the training dataset, even capturing noise and particularities that are not representative of other datasets. Generally, overfitting models have high R² values, close to 1, which can be misleading as it indicates an almost perfect fit to the training data. However, when compared to new data, the model is unable to maintain the same level of accuracy since instead of learning generalizable patterns, it has simply memorized the training data. This memorization causes the model to reproduce the results on test data, but fail when applied to real-world problems. As a result, overfitting leads to inaccurate or misleading predictions, which cannot be reliably applied in practical situations.

To detect overfitting, a common approach is to compare the model’s performance on training and validation datasets. It is assessed by the difference between the R² obtained during training and the R² of the test data.

Figure 9 shows the graphs of the models comparing the validation values (actual values from the 10% dataset) and the forecast values, visually illustrating the discrepancy between the forecasts and the actual data for each model evaluated.

In conclusion, the linear regression models (linear regression, ridge regression, and lasso regression) showed the best results in terms of predictive error (MSE, RMSE, and MAE) and goodness of fit (R²), with an insignificant tendency towards overfitting. Random forest and decision tree models also performed well when only considering their R² scores, as they produced greater errors and had a slight tendency towards overfitting. The support vector machine model performed the worst, with very high errors and a strong tendency toward overfitting, making it less suitable for this dataset.

5. Discussion

The results of this study demonstrate the potential of combining multi-criteria analysis (MCA), geographic information systems (GISs), and machine learning algorithms for assessing flood risks within the railroad network of Minas Gerais, Brazil. This comprehensive methodology offers a nuanced view of flood susceptibility by systematically evaluating diverse risk factors, such as land use, topography, and precipitation patterns, facilitating targeted and informed mitigation efforts. Similar studies have highlighted the utility of multi-criteria frameworks for managing flood risk in infrastructure planning, reinforcing the validity of this approach in high-risk regions [64,65]. The insights gained here not only align with but also extend current knowledge by integrating machine learning models, which are increasingly recognized as pivotal for improving predictive accuracy in hydrological and infrastructure resilience studies [66].

Machine learning, specifically linear regression in this study, proved exceptionally reliable, achieving a high R² value (0.999998) and low mean squared error (MSE) (0.018672), suggesting that simpler models may be sufficient when relationships between variables are largely linear. However, given the complexities of flood prediction and the increasing unpredictability introduced by climate change, it would be valuable to explore non-linear models such as random forests and deep neural networks, which have shown promise in capturing intricate patterns in flood-related data [67]. These models may address limitations in the linear model, especially in non-linear or highly variable climate scenarios.

While the findings indicate that the MCA and linear regression models provide a solid foundation for flood risk assessment, certain limitations warrant consideration. First, this study relied on historical precipitation and land cover data, which may not fully capture recent climate trends or short-term extreme events. Additionally, while the model accurately predicts flood risks based on existing data, the static nature of the MCA criteria means that dynamic changes, such as urban expansion or vegetation loss, are not fully accounted for. These findings suggest that incorporating real-time satellite and sensor data could significantly enhance the model’s adaptability and relevance in rapidly evolving landscapes.

In terms of future research directions, there is a clear opportunity to incorporate more sophisticated machine learning algorithms, such as convolutional neural networks (CNNs) or recurrent neural networks (RNNs), which could capture spatial and temporal dependencies within flood data more effectively than linear models. Additionally, leveraging real-time environmental monitoring data—such as rainfall intensity, soil moisture, and temperature—could provide a more dynamic and responsive model, addressing some of the static limitations observed in this study. Implementing these advances could lead to predictive models that not only anticipate flood events but also adapt to evolving risk factors, offering infrastructure planners more reliable tools for long-term resilience planning.

Moreover, the practical implications of these findings highlight the need for a balanced approach to model complexity. While simpler models such as linear regression offer high accuracy with minimal overfitting (observed at only 0.000011), their applicability may be limited in regions with non-linear flood dynamics. Thus, a hybrid approach that combines linear models for routine assessment with non-linear models for high-risk areas could offer an optimal solution.

Finally, this study contributes valuable insights into the use of MCA and machine learning for flood risk assessment in railroad infrastructure. While the linear models used here were highly effective, future studies could benefit from a comparative analysis of different machine learning approaches across various geographic and climatic settings. Additionally, incorporating adaptive algorithms and real-time data streams could refine flood risk assessment, enhancing the resilience of infrastructure in Minas Gerais and other regions facing similar environmental challenges. This expanded approach will ensure that infrastructure systems are better equipped to withstand the growing threats posed by climate-induced hazards, supporting sustainable and resilient development.