A Comparative Analysis of Overland Flow Models for Urban Flood Risk Assessment in Norway: A Fuzzy Approach

Data

We use a digital terrain model (DTM) obtained from the Norwegian national topographic data server (Kartverket, n.d.-b.). This high-resolution DTM, with a one-meter resolution (DTM1), provides detailed information about the terrain and elevation of the study area.

In addition to the DTM, we incorporate surface permeability data. Two vector layers were acquired at Kartverket (n.d.-a). These layers contain information on the distribution of permeable surfaces (e.g., soil) and impermeable surfaces (e.g., buildings and roads) within the study area. We use the permeability data to determine Manning’s coefficient for further analyses.

Flusstools

Flusstools (v1.1.8) is a Python package developed by the University of Stuttgart for fuzzy comparison of continuous valued raster datasets (Negreiros et al., 2022). By incorporating fuzzy spatial relationships, Flusstools enables more accurate comparisons of raster data, even when slight spatial misalignments exist. This approach accounts for the inherent imprecision in spatial data and improves the robustness of analyses (Negreiros et al., 2022; Hagen-Zanker, 2006). Fuzzy tolerance also accommodates algorithms with different input handling and parameter settings.

The concept of fuzziness of location in Flusstools means that a pixel in one simulated map can be considered similar not only to its exact counterpart in another map but also to neighboring pixels (Negreiros et al., 2022). To achieve this, Flusstools introduces the concept of fuzzy pixels, where the neighborhood of each fuzzy pixel is defined by a finite distance, called the neighborhood radius (rad). Rad represents a window of N pixels, including the central pixel (Negreiros et al., 2022).

The degree to which neighboring pixels contribute to the fuzzy representation of a central pixel is determined by a distance decay membership function (Negreiros et al., 2022; Hagen-Zanker, 2006). This function involves the use of the halving distance (dhalv), which is the distance in pixels at which the membership function decays to half its maximum value at the central pixel (Negreiros et al., 2022; Hagen-Zanker, 2006) (Figure 2).

Figure 2.

Create a Hybrid DTM

To account for the influence of urban patterns, such as roads and buildings, on overland flow modeling, we utilize the v.extrude module in GRASS GIS. This module transforms the 2D layer representing house roofs into 3D features by assigning specific elevation values to each vertex of the extruded features using the elevation parameter (GRASS Development Team, 2025). The result is a hybrid DTM that incorporates buildings as 3D features in the overland flow modeling.

Delineation of the Watershed Using Hydrological Analysis

We employ GRASS GIS to delineate the study area watershed using a DTM1. We utilize the r.watershed module to calculate flow accumulation, flow direction, and stream lines. Subsequently, We use the r.water.outlet module in conjunction with the x and y coordinates of the RURS point to determine the extent of the watershed. Finally, We use the r.stream.extract to create the waterways vector and the r.mask module to create watershed boundary mask for subsequent analyses (GRASS Development Team, 2025)

Calculation of a Single Urban Flood Event Using Itzï, Rsim and Fastflood Algorithms

To model a single urban flow event using the three algorithms described in the software section, I employ the following inputs and parameters:

The modeling process involved two steps. The Itzï and Rsim modules were executed within GRASS GIS, while the Fastflood algorithm was run on a web-based application optimized for its specific functionality (Van den Bout, B. et al., 2023b). In all three cases, infiltration is not considered due to lack of information. Manning coefficients are 0.5 for impermeable surfaces and 0.25 for permeable surfaces.

Evaluation of the Depth and Velocity Raster Layers Using a Numerical Fuzzy Map Comparison Method

To evaluate the performance of the flood modeling algorithms, we employ Flusstools to compare the generated water depth and velocity raster maps. For this study, we selected a radius (rad) of eight meters, which is twice the width of the local roads in the study area, and a half-distance (dhalv) of four meters, representing the approximate road width (Figure 3). While the fuzzy map comparison evaluates the entire raster, the roads, buildings, and their neighboring areas are of particular interest in urban flood modeling. This is because the spatial heterogeneity of roads and buildings, also known as the urban pattern, significantly influences overland flow patterns.

Calculation and Evaluation of the Depth–Velocity Number (DV Number) for Each Model

The DV number is the direct multiplication of the water’s depth and its velocity (D * V). It provides a basic measure of the momentum and force of flowing water. It is primarily used for assessing flood hazard severity and helps determine the potential for floodwaters to cause damage or pose a risk to safety. We calculate the DV number using the D and V simulated by each of the models and evaluate them visually to determine differences in the dimension and spatial distribution of the DV number.

Fuzzy Numerical Map Comparison of the Water Depths

The results of the depth and velocity raster comparison using the fuzzy numerical approach shows that Itzï and Rsim have the highest probability of similarity (0.59). The lowest probability of similarity occurs between Rsim and Fastflood (Table 1).

A visual comparison of water depth rasters in Figure 5 shows that Fastflood and Itzï produce more similar water flow patterns than Itzï and Rsim. However, further analysis of results of the depth raster comparison using histograms (Figure 6) confirm the quantitative findings in Table 1, providing additional insights into the probability distribution of water depth modeling across Areas A, B, and C.

The probability distribution for the Itzï–Rsim comparison, shown in Figure 6, indicates that the two models produce similar water depth results. The left-skewed distribution suggests that most raster cells in Areas A and B—except for the outlet (Area C)—have similarity values clustered toward the higher end of the scale, closer to 1.0, reflecting strong agreement between the models.

Figure 5.

In contrast, the Itzï–Fastflood and Fastflood–Rsim comparisons reveal that approximately 10,800 cells exhibit non-similarity, accounting for about 2.8% of the total surface area (i.e., the total number of cells for the entire raster are 385714). These low-similarity regions are primarily located near houses, roads, and raster borders. The discrepancies stem from the Fastflood model, which uses a multi-directional sweeping approach to simulate water accumulation over large areas. While efficient, this method struggles with abrupt slope changes, a common feature in urban environments, leading to reduced accuracy in those zones.

Figure 6.

In both comparisons, the histograms display a right-skewed, asymmetrical shape, with a significant number of cells showing low similarity values (close to 0). Nonetheless, the peak of the distribution is centered around a similarity value of approximately 0.49, indicating that most cells still exhibit moderate agreement between the models.

Fuzzy Numerical Map Comparison of the Water Velocities

The comparison of velocities between Rsim and Fastflood shows the highest mean probability of similarity (0.53), followed by Itzï and Fastflood with average probability of similarity of 0.52. The lowest score shows between Itzï and Rsim with a probability of similarity of 0.44 (Table 2).

The depth–velocity relationship at the outlet (Area C) reveals a consistent trend between Rsim and Fastflood. Both models show an increase in water velocity with increasing depth. Rsim demonstrates a near-linear correlation, while Fastflood, despite exhibiting some degree of random variation, maintains a similar overall trend. Itzï’s results in Area C diverge significantly from Rsim and Fastflood, indicating an inverse relationship where water velocity decreases as depth increases. This observation is further supported by the velocity–depth panel charts in Figure 7.

Figure 7.

Figure 8 presents a visual comparison of water velocity rasters generated by three models: Rsim, Fastflood, and Itzï. Rsim predicts notably higher water velocities in ponding zones across Areas A and B, suggesting a strong sensitivity to localized water accumulation. Fastflood also indicates elevated velocities in these regions, though to a lesser degree. Its velocity increases are primarily concentrated near buildings and at the outlet (Area C). In contrast, Rsim highlights additional velocity increases in smaller impoundments along roads, intersections, and within Area C.

Itzï, however, exhibits a distinct pattern. It identifies ponding primarily at the outlet (Area C) and near buildings in Area B, while uniquely predicting a decrease in water velocity with increasing depth. This inverse relationship, not observed in the other models, suggests that Itzï may offer a more realistic representation of overland flow dynamics, particularly in areas with significant water accumulation.

Further spatial insight is gained from the raster maps comparison for water velocities. The cell count distribution of similarity probabilities (Figure 9) complements the numerical results in Table 2. While a substantial number of cells indicate dissimilarity, the Itzï–Fastflood comparison shows higher overall similarity compared to Itzï–Rsim. Furthermore, Rsim and Fastflood exhibit the greatest similarity to each other, reinforcing their closer alignment in water velocity prediction.

Figure 8.

Figure 9.

DV Number Evaluation in Itzï, Rsim, and Fastflood

The depth–velocity (DV) number demonstrates significant variability across different models, primarily driven by variations in velocity and depth inputs. This variability directly influences both the assigned susceptibility levels and the spatial delineation of susceptible areas, ultimately affecting the perceived safety of those areas. Furthermore, the DV number is sensitive to modeled flood patterns. Notably, models with similar flow patterns exhibit comparable DV numbers, even when water velocity and depth inputs differ substantially (Figure 10). While Itzï and Fastflood show consistent potential flow hazard categories, Rsim tends to either underestimate or overestimate these hazards. Despite these differences, all three models display spatial variability in high and medium DV numbers.

Figure 10.

Water Depth

Itzï and Rsim demonstrated the highest average fuzzy similarity in water depth predictions, suggesting a close alignment in their outputs. However, visual inspection of the depth rasters revealed that Fastflood and Itzï shared more similar spatial patterns. This discrepancy underscores the importance of integrating both quantitative and qualitative assessments in model evaluation. Histogram analyses of the fuzzy comparison rasters further supported these findings, showing that Areas A and B had higher similarity between Itzï and Rsim, while Area C (the outlet) exhibited greater variability.

Water Velocity

Rsim and Fastflood showed the highest average fuzzy similarity in velocity predictions, followed by Itzï and Fastflood. This suggests that Fastflood’s velocity outputs align more closely with both models. However, the depth–velocity relationship at the outlet (Area C) revealed a significant divergence in Itzï’s results. While Rsim and Fastflood showed increasing velocity with depth, Itzï exhibited an inverse relationship, indicating a potentially more realistic representation of overland flow dynamics in areas with significant water accumulation. Visual comparisons of velocity rasters highlighted further differences:

Depth–Velocity (DV) Number

The DV number, a key indicator for flood hazard assessment, varied significantly across the models. These differences were primarily driven by variations in depth and velocity inputs, which influenced the spatial distribution of susceptibility levels. Despite these differences, models with similar flow patterns tended to produce comparable DV numbers. Notably, Itzï and Fastflood showed consistent hazard classifications, while Rsim tended to underestimate or overestimate flood hazards.

Methodological Insights

The fuzzy map comparison approach proved effective in accounting for spatial uncertainty and quantitative inaccuracies. The use of a radius (rad) of eight meters and half-distance (dhalv) of four meters, based on local road widths, enabled a focused evaluation of model performance in urban flood-prone areas such as roads. Dividing the study area into three zones (A, B, and C) allowed for a more nuanced understanding of model behavior across varying terrain types.

Implications and Future Work

These findings have important implications for the implementation of the Norwegian Building Regulations. The comparative evaluation provides valuable insights into the strengths and limitations of each model, supporting informed decision-making in stormwater management planning and design. The study also highlights the value of combining quantitative metrics with visual assessments, and the need to account for spatial uncertainty in model evaluation.

Future research should explore: