# A Systematic Analysis of the Interaction between Rain-on-Grid-Simulations and Spatial Resolution in 2D Hydrodynamic Modeling

^{*}

## Abstract

**:**

^{2}) in Central Germany. Suitable model settings and recommendations on model discretization and parametrization are derived therefrom. The sensitivity analysis focuses on the influence of the mesh resolution’s interaction with the spatial resolution of the underlying terrain model (‘subgrid’). Furthermore, the sensitivity of the parameters interplaying with spatial resolution, like the height of the laminar depth, surface roughness, model specific filter-settings and the precipitation input-data temporal distribution, is analyzed. The results are evaluated against a high-resolution benchmark run, and further criteria, such as 1. Nash–Sutcliffe efficiency, 2. water-surface elevation, 3. flooded area, 4. volume deficit, 5. volume balance and 6. computational time. The investigation showed that, based on the chosen criteria for this size and type of catchment, a mesh resolution between 3 m to 5 m, in combination with a DEM resolution from 0.25 m to 1 m, are recommendable. Furthermore, we show considerable scale effects on flooded areas for coarser meshing, due to low water levels in relation to topographic height.

## 1. Introduction

#### Motivation and Research Gap

- Zeiger & Hubbart (2021) [3] evaluated an integrated modeling approach, using a coupled-modeling routine. The river basin model SWAT (‘Soil and Water Assessment Tool’) was used to determine effective rainfall rates and HEC-RAS 2D was used as for the rain-on-grid simulations. The Hinkson Creek Watershed (232 km
^{2}) was discretized with 10 m mesh and 1 m DEM as underlying subgrid. - Krvavica & Rubinić (2020) [4] applied HEC-RAS with direct precipitation in a small ungauged catchment of 3.08 km
^{2}. The focus of the research project was the evaluation of the influence of different storm designs and rainfall durations on the catchment outflow. A mesh with an average grid size of 10 m and local refinements of 5 m was used with a 2 m subgrid DEM for topographic details. - Rangari et al. (2019) [16] applied HEC-RAS as HDRRM for different storm events in the highly urbanized area of Hyderabad. The model was set up for an area with 47.08 km
^{2}, a fixed mesh resolution and underlying DEM of each 10 m and 139,487 computational cells. - Caviedes-Voullième et al. (2020) [17] compared the impact of the application of zero-inertia (‘ZI’) and shallow-water (‘SW’) models. The results of six different benchmarking test cases were analyzed. One of the test cases included the application in an urban area in Glasgow with a catchment size of 0.4 km
^{2}. The study was conducted for four different (1. ‘very-coarse’—4 m, 2. ‘coarse’—3 m, 3. ‘medium’—2 m, 4.‘fine’—1 m) model-set-ups. - Tyrna et al. (2018) [8] applied the 2D hydraulic model FloodArea in an ungauged urban area with a study area of 144 km
^{2}to provide “large-scale high-resolution fluvial flood hazard mapping”. They presented a method that involves a precipitation model, a hydrological model, a digital elevation model and a hydraulic model component. The model set-up consisted of a 1 m raster-based model. - Pina et al. (2016) [18] used two case studies to compare a semi (‘SD’)—and a fully (‘FD’)—distributed model. The study consists of two model set-ups with the focus of the comparison of the SD and FD approaches and the integration of a sewer-system network. The first FD model has an average resolution of 61 m
^{2}with a catchment size of 8.5 km^{2}. The second FD model has a size of 1.5 km^{2}with an average cell size of 89 m^{2}. The focus of the modeling processes is the comparison of the two different model types. - Cea & Rodriguez (2016) [19] present the development of a 2D distributed hydrologic-hydraulic model (‘GUAD-2D’) with the objective to model the rainfall–runoff process within a catchment. The presented model was tested in a 12.97 km
^{2}large urban catchment, an ungauged area with a cell size of 4 m and a 500 year storm event. The model was evaluated against a model-set up without DRM. - Fraga et al. (2016) [20] conducted a global sensitivity and uncertainty analysis for a 2d-1d dual drainage model [21] using the GLUE (‘Generalized Likelihood Uncertainty Estimation’, compare Beven & Binley (1992) [22]) method which is developed for distributed models. For the surface model the sensitivity of the Manning’s n coefficient and the infiltration parameters were analyzed. The study was conducted in a motorway section with an area of 0.049 km
^{2}. The model geometry consists of an unstructured mesh with a fixed mesh resolution of circa 3 m. - Leandro et al. (2016) [23] introduced a methodology that stepwise increases the model complexity in different modeling levels to evaluate the impact of ‘spatial heterogeneity of urban key features’. The 2D overland flow model P-DWave [24] was set-up for a catchment area (‘Borbecker Mühlenbach’, 4.9 km
^{2}). The model was systematically extended in five stages with focus on the representation of buildings and land surfaces. The model geometry was created with a fixed grid with 2 m resolution. - Yu et al. (2015) [2] applied the hydro-inundation model FloodMap in an urbanized area in Kingston upon Hull (UK). The influence of improved urban and rural drainage and storage capacity was investigated in a stepwise manner. During the modeling process the model sensitivity to roughness and mesh resolution was determined.
- Néelz & Pender (2013) [25] analyzed various 2D hydrodynamic model in eight different benchmark test cases. The last test case includes an application of the 2D models by direct precipitation in a small urban area in Glasgow with a total size of 0.384 km
^{2}. The same test case was further evaluated for various models in [26] and for HEC-RAS in [27]. For most of the models, a fixed spatial resolution and roughness values were used with a 2 m grid. For HEC-RAS two different mesh resolution (2 m, 4 m) were evaluated, which showed minor sensitivity on model run times and sensitivity on water level time series [27]. - Chen et al. (2010) [28] applied the integrated 1D sewer and 2D overland flow model SIPSON/UIM in a small urban catchment area (‘Stockbridge’, ca. 0.18 km
^{2}) close to a riverside. The model is set-up with a fixed resolution of 2 m. The focus of the study is given on the impact of different design storms and floods for an area which is affected by combined pluvial and fluvial flooding.

- David & Schmalz (2020) [5] evaluated the application of the modeling software HEC-RAS in a low mountain-range study area with a catchment size of 38 km
^{2}. Various model settings were tested for the specific application of DRM and a model calibration was carried out based on the Manning’s n roughness. It has been shown that internal model parameters are sensitive for the application of hydrodynamic rainfall–runoff modeling and must be adapted to a different value range. Due to the size of the area and the resulting extensive computing times, no detailed sensitivity analysis was carried out. - Jia et al. (2019) [7] developed the model system for surface and channel runoff CCHE2D. It was applied to a subcatchment of the Mississippi River, with a catchment size of 18 km
^{2}. In the study a local sensitivity analysis for the Manning’s n roughness was carried out. The evaluated values were in a range between 0.03 and 0.3 s × m^{−1/3}. The model was set up with a fixed mesh resolution between 3.8 and 5 m. - Broich et al. (2019) [6] developed an approach for the implementation of the DRM in the 2D hydrodynamic model TELEMAC-2D. They applied the extended model to the catchment areas of Simbach (45.9 km
^{2}) and Triftern (90.1 km^{2}). Furthermore, alternative approaches for the roughness calculation for sheet water flow were implemented as new calculation routines. Additionally the impact of model intern (‘hidden’) parameters on the modeling results were evaluated. The model geometry was based on a 5 m DEM with 1 s timestep. - Hall (2015) [29] conducted a DRM application in the Birrega catchment with an area of ca. 185 km
^{2}. For the model application, the 2D model MIKE from the Danish Hydraulic Institute (‘DHI’) was used. The model geometry consists of a grid with a constant resolution of 20 m. In the modeling process, design floods with different return periods were evaluated. In a simplified local sensitivity analysis of the impact of 1. rainfall, 2. Manning’s roughness, 3. infiltration and 4. groundwater inundation. Each model set-up was tested for a large- (176 km^{2}) and small- (7 km^{2}) scale catchment area. - Cea & Bladé (2015) [30] developed a discretization scheme (‘Decoupled hydrological discretization’, DHD) to solve the 2D SWE for hydrodynamic rainfall–runoff applications. They applied the model to five test cases, where two test cases involved application in small rural basins. The first catchment has a size of 4 km
^{2}with an average cell size of 15.5 m. The second catchment has a size of 5 km^{2}with an average cell size of 20 m. The model was calibrated by the infiltration rate and the Manning’s n values. In the study, the alternative discretization scheme is evaluated against three other methods. - Clark et al. (2008) [31] compares the two 2D models TUFlow and SOBEK with a traditional lumped hydrological rainfall runoff model. In a 11.85 km
^{2}large catchment area a local sensitivity analysis is performed considering various parameters. The spatial resolution is evaluated for mesh resolution between 10 m and 100 m (TUFlow) and 5 m and 100 m (SOBEK). It has shown that both models are sensitive towards the mesh resolution.

^{2}to 232 km

^{2,}with spatial resolutions varying between 1 m and 30 m. The number of cells varies between 5444 and 144,000,000 cells. The parameter’s sensitivity and model output uncertainty is regarded sparely in the modeling process when applying the DRM in a new catchment. Some studies, such as Clark et al. (2008) [31], Hall et al. (2015) [29], Yu et al. (2015) [2], Fraga et al. (2016) [20], Jia et al. (2019) [7] and David & Schmalz (2020) [5] have evaluated the Manning’s n sensitivity. The studies of Clark et al. (2008) [31], Yu et al. (2015) [2] and Caviedes-Voullième et al. (2020) [17] evaluated different spatial resolutions during the modeling process and one study conducts a detailed global sensitivity and uncertainty analysis based on the GLUE [22] method. None of the studies has applied a sensitivity or uncertainty analysis together with the free software HEC-RAS from the U.S. Army Corps of Engineers [32].

## 2. Objectives

- To introduce a stepwise methodology which allows a systematic analysis of model behavior and parameter sensitivity when applying HEC-RAS and the DRM in a small rural catchment.
- To reduce the number of model runs in order to manually execute the methodology.
- To evaluate the parameter sensitivity of: 1. mesh resolution, 2. subgrid topographical data, 3. laminar depth, 4. Manning’s n values, 5. model-specific filter settings and 6. precipitation data.
- To give recommendations on suitable spatial resolution and identify sensitive model settings when applying the 2D model HEC-RAS for storm hazard analysis in small catchments of low mountain range areas.

## 3. Materials and Methods

#### 3.1. Model Behavior, Sensitivity Analysis and Model Uncertainty

#### 3.2. Systematic Analysis of Model Behavior

#### 3.3. Hardware

#### 3.4. Evaluation of Results

^{2′}, compare Equation (5)). A further three indices are evaluated based on absolute values: 4. volume deficit (‘VD’, compare Equation (6)), 5. volume balance (‘VB’, compare Equation (7)) and 6. computational time (‘CT’, compare Equation (8)). In Step 2 local sensitivities are determined based on absolute and relative sensitivity indices (compare Equations (9) and (10)).

#### 3.4.1. Nash–Sutcliffe Efficiency (NSE)

_{M}) will be set in relation to the results of the high-resolution benchmark run (Q

_{B}). The NSE index is criticized to overestimate the peak value [49] as cited in [46]. Since the peak valued plays an important role in storm hazard analysis, this index was chosen to analyze the results. Furthermore this index allows to compare the model output over the entire simulation time with the benchmark run. The NSE for the stream discharge will be computed at two different locations, the NSE

_{outlet}(Equation (1)) is computed at the outlet of the catchment (compare Figure 2), the NSE

_{village}(Equation (2)) is computed at the control point in the village in the upper part of the catchment.

#### 3.4.2. Difference in Maximum Water-Surface Elevation (ΔWSE)

_{max,stream}, Equation (3)) in the stream, close to the village. The second location (ΔWSE

_{max,road}, Equation (4)) is on the main road in the village. To determine the change of water-surface elevation the maximum value during the simulation time of the benchmark run (WSE

_{max,benchmark}) is subtracted from the maximum water-surface elevation of the model run (WSE

_{max,model run}). The index was chosen as further criteria to compare the differences of water depth of the evaluated model configurations.

#### 3.4.3. Flooded Area (F^{2})

^{2}based on Bates & de Roo (2000) [50], as cited in Aronica et al. (2002) [51], ref. [46] is added to evaluate the spatial distribution of the model results. This index (Equation (5)) is seen to be important since the spatial distribution of flooded area in the entire catchment is considered. Whereas the first two indices of NSE and ΔWSE only consider the model output at single locations. F

^{2}is determined for the maximum water depth larger than 0.05 m.

_{i}

^{B1M1}—inundated pixel present in the model M1 and present in the benchmark run B1.

_{i}

^{B0M1}—inundated pixel present in the model M1 and absent benchmark run B0.

#### 3.4.4. Volume Deficit (VD)

_{in}) in comparison to the accumulated output volume (V

_{out}) at the end of the simulation. The index is seen important to identify the water volume, which is kept in the catchment or lost during the simulation. It is calculated using the following equation (Equation (6)).

#### 3.4.5. Volume Balance (VB)

_{out}) plus the water volume which is kept in the model area (V

_{area}) at the end of the simulation by the total input volume (V

_{in}). It is calculated using the following equation (Equation (7)).

#### 3.4.6. Computational Time (CT)

_{modeled}) in relation to the real time (CT

_{realtime}). All simulations were calculated for a real time of 24 h.

#### 3.4.7. Local Model Sensitivity (e)

**e2**(Equation (10)) is determined based on the absolute change of model output.

^{2}) in comparison to the corresponding model run with the same spatial resolution of Step 01.

#### 3.5. 2D Hydrodynamic Model: HEC-RAS

## 4. Case Study, Data and Model Set-up

#### 4.1. Messbach Catchment

^{2}(Figure 2) and is part of the larger river system of the Gersprenz river. It is located in the south of Hesse in the low mountain range of the Odin forest (germ. ‘Odenwald’). The Messbach is a small, ungauged creek of ca. 0.7 to 1.5 m channel width and around 1860 m channel length. It forms an inflow to the tributary of the Fischbach tributary of the Gersprenz river which was subject to former study in [5,39,40,41].

^{3}was taken in operation on the main stream of the Fischbach River. There are no retention basins in the Messbach catchment itself.

^{®}) provided by HVBG was used [57]. A summary of the different land-use categories within the catchment is presented in Table 4 and Figure 4.

#### 4.2. Model Set-up

^{®}landuse data and then added to the original surface DEM. This procedure was made for reasons of comparability so that there is for each created DEM only one consistent raster resolution. In Figure 7 there is shown the DEM with included buildings with a resolution of 0.25 m in comparison to the DEM with 2 m.

_{a}= 0.05. Soil and landuse data (compare Table 4) is preprocessed using the ArcGIS plugin from HECGeoHMS [69]. The CN-value is aggregated to one single value of CN = 69 for the catchment. All simulations were run with the same sum of excess rainfall for the event.

## 5. Results and Discussion

#### 5.1. Pre-Study: Comparison of HEC-RAS 5.0.7 and 6.0

#### 5.2. Step 1—DEM vs. Mesh Resolution

#### 5.2.1. Nash–Sutcliffe Efficiency (NSE)

- a.
- NSE
_{outlet}

_{outlet}≥ 0.98) for all seven terrain models at the outlet of the catchment (Figure 9). There was no sensitivity with regard to the various terrain models. The runoff dynamics and the peak values (Q

_{max}= 4.24 m

^{3}/s) of the seven model runs are almost identical with the benchmark run. The group of simulations, with a computational grid of 3 m, show very good agreement with the benchmark (NSE ≥ 0.92) as well. For this group of simulation it was found that the simulations with a DEM between 1 m and 4 m reproduce well the runoff dynamics of the catchment. It is better reproduced than for the very finely resolved DEM of 0.25 m and 0.5 m. The maximum runoff decreases and is in the order of 3.65 m

^{3}/s (4 m DEM) and 3.93 m

^{3}/s (2 m DEM). For the group of simulations with a computational grid of 4 m and 5 m, the values for NSE decrease in an order of magnitude of approx. 0.05. All simulations are above the value of NSE > 0.8, which can be regarded as good agreement for hydrological calculations. Furthermore, the simulations with a coarser DEM have slightly better values for NSE (0.87 for the 0.25 m DEM and 0.90 for the 5 m DEM with a 4 m calculation grid). For the simulations with a calculation grid coarser than 10 m, there is a clear decrease in the correspondence with the benchmark. The reason for this jump is also because that there is a larger jump from the 5 m to the 10 m calculation grid. Here, the hydrographs only have a correspondence of 0.62 for the 0.25 m terrain model and 0.66 for the 5 m terrain model. The same tendency can be seen for the calculation runs with a mesh resolution of 20 m and 30 m. For the simulations between 2 m and 5 m the NSE value decreases by approx. 0.05 for each model group when the calculation grid is increased by 1 m. For the simulations between 10 m and 30 m the NSE value decreases by approx. 0.25 for each model group when the calculation grid is increased by 10 m.

^{3}/s) and reproduced with a time delay of more than one hour in comparison to the benchmark. The model thus shows a strong sensitivity in the reproduction of the hydrograph depending on the individual cell resolution. It can be generalized that cell resolutions larger than 10 m result in a significantly delayed and flattened hydrograph. This effect is caused by the changed model geometry and the resulting different detection of the terrain geometry. It cannot be traced back to different physical characteristics in the catchment.

- b.
- NSE
_{village}

^{3}/s (Figure 12). At the same time, the time of concentration increases up to a maximum of approx. 1.8 h for the 30 m mesh resolution. This leads to a decrease of the NSE from 0.98–0.99 (DEM 0.25 m to 5 m, 2 m mesh resolution) to 0.80–0.85 (DEM 0.25 m to 5 m, 5 m mesh resolution). For the coarser resolutions of 10 m to 30 m, the lowest values of NSE are determined in a range of NSE from 0.67 (10 m mesh, 3 m DEM) to the lowest value of 0.3 (30 m mesh, 0.25 m DEM). In summary, it can be said that the runoff dynamics up to a cell resolution of 5 m is in a comparable range with the benchmark (NSE > 0.8).

#### 5.2.2. Difference in Maximum Water-Surface Elevation (ΔWSE)

- a.
- ΔWSE
_{stream}

- b.
- ΔWSEroad

#### 5.2.3. Flooded Area (F^{2})

^{2}compares the flooded area with the benchmark run. Generally, the correspondence of the flooded area is relatively poor for all simulation runs (Figure 15). In addition, also the model runs with fine meshing have low correspondence in a range between 0.3 and 0.5 (“1” stands for the identical flood plains as the benchmark). Furthermore, for a grid resolution between 2 m and 5 m there is a clear decrease in the correspondence of the floodplain areas. The best match with the benchmark run is F

^{2}= 0.55 for the model with a cell resolution of 2 m and a terrain model of 0.25 m.

^{2}also includes differences of very shallow water depths in individual cells. Most of the cells have water depths lower than 5 cm, which is a great requirement of accuracy for the surface runoff model. In areas of larger sinks or the course of the river itself, there is a fundamental correspondence with the benchmark run-up to a cell resolution of 5 m. The main reason for the large deviation F

^{2}is due to the large number of flooded individual cells in the benchmark itself. In addition, there is an increase in the deviation of the floodplain areas around the buildings. Since the buildings were inserted into the terrain model as block elements with a height of 5 m, interpolation is made in the model between the wet cells on the building itself and the closest cells close to the ground. This leads to artificial moats around the building, the size of which depends on the cell size. Therefore, the informative value of F

^{2}is only possible if the cause of the deviation and the comparison of the actual flood areas are taken into account.

#### 5.2.4. Volume Deficit (VD)

^{3}water volume (Figure 18). This is held back in the catchment area because the volume control for all simulations is maintained (compare results for the volume balance VB). The benchmark run with the very fine resolution of the terrain model and the coarsest mesh resolution of 30 m calculation grid (approx. 6.4%, 1380 m

^{3}) have the highest retention volume in the catchment area. Furthermore, with the same mesh resolution, the finer terrain models have a larger proportion of water that remains in the catchment area than the coarser terrain models. The evaluation of the results show that a consideration of the characteristic values alone is not meaningful, as they do not provide any information about how and where the water is held back in the area.

^{3}is trapped in a cell of 30 m resolution. Thereby, a coarse computational grid in combination with a fine resolution of the terrain model creates artificial depressions in the terrain.

#### 5.2.5. Volume Balance (VB)

#### 5.2.6. Computational Time in Comparison to Real Time (CT)

^{−3.308}(R

^{2}= 0.99) if the mean values of the individual cell resolutions are used as a basis.

#### 5.3. Criteria for the Selection of Suitable Model Configurations

- NSE
_{village/outlet}= 0.8, - ΔWSE
_{road}= 0.03 cm, - Δ WSE
_{stream}= 0.05 cm, - VD = 5%, VB = 95%,
- CT = 1

^{2}, the criterion had to be reduced due to the poor agreement with the benchmark. Since the qualitative evaluation showed that there is agreement of the inundation areas up to a resolution of 5 m and 1 m DEM, F

^{2}= 0.28 was set. This corresponds to the top third of the simulations with the best agreement with the benchmark. Based on the criteria introduced, the following model configurations were selected for further investigation:

- 3 m mesh with 0.25 m, 0.5 m, 1 m terrain model,
- 4 m mesh with 0.25 m, 0.5 m, 1 m terrain model,
- 5 m mesh with 0.25 m, 0.5 m, 1 m terrain model.

#### 5.4. Step 2—Further Parameter Sensitivity

#### 5.4.1. Laminar Depth

^{2}, Figure 26) with the respective initial run with corresponding spatial resolution. It has been shown that the height of the laminar flow depth can be viewed as weakly sensitive for the parameter maximum outflow. The elasticity ratio for the total mean values is in the order of e = 0.32. By reducing the laminar flow depth by 2 cm, the flow peak is reduced by approx. 0.4 m

^{3}/s and increasing the laminar flow depth by 2 cm it is increased by approx. 0.4 m

^{3}/s. This corresponds to an average of 10–15%. A change in the laminar flow depth has hardly any effect on the volume deficit. The elasticity ratio is only on the order of e = 0.08. For the pixel-by-pixel comparison of the floodplain areas, the elasticity ratio is in the order of e = 0.13 to e = 0.21. The comparison between the model resolutions shows that the computational grids and terrain models have model sensitivities of a similar order of magnitude. Only a low sensitivity with regard to the spatial resolution is found here. The distributions can be described well with an approximation function (R

^{2}S-Qmax = 0.97, R

^{2}S-VD = 0.99) in order to describe the basic characteristics of the model behavior with regard to the change in the laminar flow depth in combination with the different spatial resolutions. In general, it can be summarized that the determination of the laminar flow depth for the model results must be given significantly more consideration than is currently the case. On the basis of a series of test runs in a runoff simulation flume, [14] showed that the simulated thin-layer runoff resulted in laminar runoff. This coincides with the Reynolds numbers determined over a large area from [5] for simulated and calibrated storm events. The sensitivity analysis carried out shows that the determination of the laminar flow depth in a value range of 4–10 cm has an influence on the model results. This agrees with the information from [14] and recommendations from [13]. The simulated thin-layer runoff needs to be adapted and checked in relation to the implemented roughness approaches and basic flow characteristics [13].

#### 5.4.2. Manning’s n Roughness Values

^{2}= 0.98) is determined for the mean values of all simulations. With regard to the volume deficit S-VD (Figure 29), it has been shown that there is a significantly lower sensitivity (e = 0.12). A reduction in the roughness values results in a reduction in the volume deficit, as the water can flow away faster. Conversely, an increase in surface roughness leads to slower drainage and thus greater amounts of water remaining in the catchment area after the simulation has been completed. The evaluation also showed that the influence of the change in roughness values on the floodplain is rather small and is only in the order of magnitude of e = 0.15–0.2 (Figure 30). The evaluation showed that the influence of the roughness values on the model results of Qmax (Figure 28) is greatest. The actual roughness values should therefore be taken into account in the calibration process and in the subsequent uncertainty analysis. The influence for the model geometries with a 3 m computational grid is somewhat greater when the Manning’s n values are reduced. Overall, however, there is no clear tendency for the three quality criteria with regard to differences in terms of the various spatial resolutions.

#### 5.4.3. Filter Settings

^{3}/s and is classified as very low (Figure 32). In [5], a significant influence of the filter settings on the model results with regard to the volume deficit and corresponding reduction of the flow peak was found. There, however, the spatial resolution of the computational grid was about 30 m in the catchment area and 3 m along the stream, so that there may be a cell size dependency here. Therefore, the statement that these model settings are not significantly sensitive only applies to the model simulations of 3 m, 4 m, and 5 m computational grids performed in this study. With regard to the volume difference, this tendency can be identified as well when the values of the filter settings are reduced. For the standard settings, the volume deficit (S-VD, Figure 33) increases by an order of magnitude of 70–170%, whereas for the settings with ft = 0.003 m it only increases by an order of magnitude of 30–40%. The absolute value ranges for the volume deficit are then 2.9–3.0% for the standard settings and approx. 1.8% for ct = 0.003 m. This effect is significantly higher for the standard settings and the 5 m calculation grid with values between 140–170% increase in volume deficit compared to the group of simulations with 3 m calculation grid. From these results it can be seen that there is a dependency of the mesh resolution and the terrain model, which should be recorded in the process of calibration, especially with computational grids larger than 5 m resolution. With respect to the floodplains, there are only minor deviations from the respective comparison run in the order of magnitude of 1–1.5%, which is the smallest change compared to the other sensitivity studies conducted (Figure 34).

#### 5.4.4. Precipitation Input

^{3}/s) occur with the end distributed precipitation event. The outflow increases in an order of magnitude of up to 23%, especially for the cells with a lower resolution of 3 m. At the same time, the time of peak is delayed by approx. 25 min (Figure 35). With the coarser cell resolutions, the increase in the flood peak is only in an order of magnitude of 9–12%. In this sense, it can be stated that the precipitation distribution has a greater influence on the finer grid. For the block precipitation event there is a slight delay of the flood peak by approx. 15 min. At the same time, the flood peak increases by 2–9% for the various spatial resolutions, resulting in increased absolute peak flow of 3.37 to 4.0 m

^{3}/s (Figure 36). For the initial stressed precipitation event, there is the least change in the resulting runoff hydrograph. The resulting peak flow changes by 1–5%. The comparability is given since the initial stressed precipitation event most closely corresponds to the Euler-II model rain. For the volume deficit, there are only minor deviations from the initial run and no systematic change with regard to the model sensitivity can be identified (Figure 37). Three simulations have slightly larger deviations. However, since these are not systematic, they can be more related to the numerical calculation method rather than a systematic model response with respect to the precipitation input data. For F

^{2}similar effects can be recognized as for the change of maximum discharge (Figure 38). For the end distributed rainfall event, the highest deviation to the initial run is recognized with a value between 5.5–7%. For the block rainfall event there are deviations from 4–7% recognized in comparison to the Euler II floodplains. For the initial distributed rainfall event, the deviations in F

^{2}are in the order of magnitude between 2.5–3.8%. For all three rainfall events there is a slight dependency on the spatial resolution. It can be said that the finer the mesh and the DEM the higher the sensitivity is for the three evaluated rainfall distributions. In summary, it can be stated that the temporal distribution of precipitation has a relatively high influence on the model results with regard to the general runoff dynamics. The results of this study are in accordance with the BLUE hyetograph theory by Villani & Veneziano (1999) for linear basin response and low correlated space-time precipitation [70]. In [70] the hyetograph is “the mirror image of the instantaneous unit hydrograph”. This agrees also to the results founded in [4]. However, a closer comparison with the results from [4] shows the duration dependency of the resulting hydrograph. For the hyetograph with 1 h duration, the Huff Curve, an end based precipitation event has the largest flow rates. Whereas for the 6 h rainfall event the Euler II hyetograph has largest flow rates [4].

#### 5.5. Discussion of the Results in the Context of Rain-on-Grid Simulations

#### 5.6. Comparison with Determined Unit—Hydrograph

## 6. Conclusions

^{2}as well as to point-by-point evaluation at different control points in the catchment via the NSE. The main results of this study are presented below:

- For the coarser grids from 10 m mesh, the runoff is significantly delayed at the catchment outlet. The results showed that this effect is caused by a very low water layer that is computationally kept in the cells. The comparison with the benchmark run showed that comparable results could be achieved up to a mesh resolution of 5 m and a terrain model up to 1 m (NSE ≥ 0.8).
- For the water levels in the channel, there is very good agreement (Δ WSE ≤ 0.03 m) with the benchmark run up to a spatial resolution of 5 m mesh.
- The area-based index F
^{2}shows large deviations from the benchmark for all simulations. The study has made evident that deviations are sourcing mainly from a large number of microsinks in the 0.25 m DEM. The latter are mapped well using a 1 m mesh. The detailed display of microsinks reduces with increasing DEM and mesh resolutions. - The study emphasized that the volume deficit as indicator was only partially meaningful. A qualitative analysis was additionally necessary to interpret the results adequately. It is shown that artificial depressions are detected for the mesh resolution coarser than 10 m. These are caused due to the very low water levels (<10 cm) in comparison to changes in topography.
- With regard to the computing time, model resolutions in the order of 3 m or higher are considered acceptable for a catchment area of this size (A = 2.13 km
^{2}, CT < 1 and thus smaller than 24 h of computing time). - The laminar flow depth is viewed as weakly sensitive with regard to the maximum discharge height. The roughness values are considered sensitive with respect to the discharge height. The filter settings show only a very low sensitivity for the calculated resolutions with respect to the runoff height and the floodplains. For the precipitation distributions, it is shown that the initially stressed precipitation event has only a very small effect on the runoff height. For the 1 h-event, the end distributed rainfall event results in the highest peak flow.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Topography of the Messbach catchment and location of Messbach village [56].

**Figure 7.**Buildings in the modeling domain using different DEM resolution: on the left: 0.25 m resolution, on the right 2 m resolution.

**Figure 8.**Temporal rainfall distributions (

**a**): Euler Type II, (

**b**): Initial, (

**c**): Block, (

**d**): End distribution.

**Figure 9.**Overview of results of NSE (flow hydrograph) at the outlet of the catchment for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 10.**Overview of hydrographs at the outlet of the catchment for DEM 0.25 m and different mesh resolutions (mesh: colored legend).

**Figure 11.**Overview of results of NSE (flow hydrograph) at the control point in the village for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 12.**Overview of hydrographs at the outlet of the catchment for DEM 0.25 m and different mesh resolutions (mesh: colored legend).

**Figure 13.**Overview of results of ΔWSE at the control point in the stream for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 14.**Overview of results of ΔWSE at the control point on the road for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 15.**Overview of results of flooded area F

^{2}for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 16.**Qualitative comparison of flooded area (max. flood depth during overall simulation time) for four different mesh resolution (

**a**): 1 m, (

**b**): 2 m, (

**c**): 5 m, (

**d**): 10 m and 0.25 m DEM.

**Figure 17.**Qualitative comparison of flooded area (max. flood depth during overall simulation time) for four different mesh resolution (

**a**): 2 m, (

**b**): 5 m, (

**c**): 10 m, (

**d**): 30 m and 1 m DEM.

**Figure 18.**Overview of results of VD for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 19.**Remaining water in microrelief (

**a**): 1 m, (

**b**): 5 m mesh of the 0.25 m DEM and shallow artificial sinks (

**c**): 10 m, (

**d**): 30 m mesh at the end of the simulation time.

**Figure 20.**Identified sinks for DEM with (

**a**): 0.25 m, (

**b**): 0.5 m, (

**c**): 2 m and (

**d**): 5 m spatial resolution.

**Figure 21.**Overview of results of VB for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 22.**Overview of results of computational time CT for the different mesh and DEM resolutions (DEM: colored legend).

**Figure 23.**Qualitative comparison of the hydrographs and model sensitivity for laminar depth at the catchment outlet for 3 m mesh and 0.5 m DEM.

**Figure 24.**Comparison of the model sensitivity (S-Qmax) for laminar depth and the evaluated mesh and DEM resolutions with the averaged elasticity ratio e.

**Figure 25.**Comparison of the model sensitivity (S-VD) for laminar depth and the evaluated mesh and DEM resolutions with the averaged elasticity ratio e.

**Figure 26.**Comparison of the model sensitivity (S-F

^{2}) for laminar depth and the evaluated mesh and DEM resolutions.

**Figure 27.**Qualitative comparison of the hydrographs and model sensitivity for Manning’s n values at the catchment outlet for 3 m mesh and 0.5 m DEM.

**Figure 28.**Comparison of the model sensitivity (S-Qmax) for Manning’s n values and the evaluated mesh and DEM resolutions with the averaged elasticity ratio e.

**Figure 29.**Comparison of the model sensitivity (S-VD) for Manning’s n values and the evaluated mesh and DEM resolutions with the averaged elasticity ratio e.

**Figure 30.**Comparison of the model sensitivity (S-F

^{2}) for Manning’s n values and the evaluated mesh and DEM resolutions.

**Figure 31.**Qualitative comparison of the hydrographs and model sensitivity for the filter settings at the catchment outlet for 3 m mesh and 0.5 m DEM.

**Figure 32.**Comparison of the model sensitivity (S-Qmax) for the filter settings and the evaluated mesh and DEM resolutions.

**Figure 33.**Comparison of the model sensitivity (S-VD) for the filter settings and the evaluated mesh and DEM resolutions.

**Figure 34.**Comparison of the model sensitivity (S-F

^{2}) for the filter settings and the evaluated mesh and DEM resolutions.

**Figure 35.**Qualitative comparison of the hydrographs and model sensitivity for the precipitation distribution at the catchment outlet for 3 m mesh and 0.5 m DEM.

**Figure 36.**Comparison of the model sensitivity (S-Qmax) for the precipitation distribution and the evaluated mesh and DEM resolutions.

**Figure 37.**Comparison of the model sensitivity (S-VD) for the precipitation distribution and the evaluated mesh and DEM resolutions.

**Figure 38.**Comparison of the model sensitivity (S-F

^{2}) for the filter settings and the evaluated mesh and DEM resolutions.

**Table 1.**Overview of case studies 2D rainfall–runoff modeling (‘Direct Rainfall Method’)—Urban applications.

Reference | 2D Model (s) | Catchment: Size | Spatial Resolution (Number of Cells) | Rainfall Input | Rainfall Loss Approach | Roughness Values [s×m^{−1/3}] | Sensitivity Analysis | Calibration Data |
---|---|---|---|---|---|---|---|---|

Zeiger & Hubbart (2021) [3] | SWAT/ HEC-RAS | Hinkson Creek, MO: 232 km^{2} | DEM: 1 m Mesh: 10 m | 10 historical storm events | long term hydrological modeling (SWAT) SCS CN approach | Manning formula n from 0.003 (barren land) to 0.092 (Herbaceous); calibrated values | Computational interval | gauging stations |

Krvavica & Rubinić (2020) [4] | HEC-RAS | Novigrad: 3.08 km ^{2} | DEM: 2 m Mesh: 10 m, 5 m refinement (38,499 cells) | 2 historical events 6 statistical events | SCS CN approach | Manning formula n = 0.015 (roads) n = 0.2 (agriculture) | 6 different design storms 4 different rainfall durations | none |

Caviedes-Voullième et al. (2020) [17] | in-house development | Glasgow: 0.384 km ^{2} | 4 m (24,100) 3 m (42,693) 2 m (96,400) 1 m (384,237) | 1 rainfall event | none | Manning formula n = 0.02 (roads) n = 0.05 (other) | SWE and ZI (‘zero inertia’) solver mesh resolution | none |

Rangari et al. (2019) [16] | HEC-RAS | Hyderabad: 47.08 km ^{2} | DEM: 10 m Mesh: 10 m (139,487) | 3 historical events 3 statistical events | no information | Manning formula n = 0.025 | none | none |

Tyrna et al. (2018) [8] | FloodArea | Unna: 144 km ^{2} | 1 m (144,000,000) | 1 statistical event 1 fictional event | simplified physical approach based on Green and Ampt and Darcy | Manning formula from n = 0.013 (roads) to n = 0.250 (forest) | none | none |

Pina et al. (2016) [18] | Infoworks ICM v.5.5 | Cranbrook: 8.5 km ^{2}Zona central: ca. 1.5 km ^{2} | Cranbrook: avrg. 8.5 m (117,712) Zona central: avrg. 11.8 m (10,741) | Cranbrook: 3 historical events, 5 statistical events Zona central: 4 historical events, 6 statistical events | fixed runoff coefficient | no information | SD/FD approach different design storms | gauging station |

Cea & Rodriguez (2016) [19] | GUAD-2D | Alginet: 12.97 km ^{2} | 4 m (ca. 810,625) | 500-year event | SCS CN approach Green-Ampt Horton Philip | Manning formula | hydraulic-hydrological calculation | none |

Fraga et al. (2016) [20] | in-house development | Motorway section: 0.049 km ^{2} | avrg. 3 m (ca. 5444) | 7 historical events | initial- constant approach | Manning formula n = 0.02–0.1 (impervious surface) n = 0.02–0.5 (pervious surface) n = 0.008–0.025 (conduits) | Manning’s n Infiltration rates Discharge coefficients | discharge data |

Leandro et al. (2016) [23] | P-DWave | Borbecker Mühlenbach: 4.9 km ^{2} | 2 m (ca. 1,225,000) | 1 historical event | Green-Ampt | Manning formula | 5 modeling levels, increasing complexity of key urban features | none |

Yu et al. (2015), [2] | FloodMap | City of Kingston/Hull: - | 10 m 20 m | 1 historical event | Green-Ampt | Manning formula | Mesh resolution Manning’s n Hydraulic conductivity | inundation areas |

Néelz & Pender (2013) [25] Karl Broich et al. (2018) [26] Brunner (2018) [27] | Various | Glasgow: 0.384 km ^{2} | 2 m (ca. 97,000) HEC-RAS: DEM: 0.5 m Mesh: 2 m, 4 m | 1 event | none | Manning formula 0.02 (roads) 0.05 (area) | HEC-RAS: mesh resolution | none |

Chen et al. (2010) [28] | Sipson/UIM | Stockbridge: ca. 0.18 km ^{2} | 2 m (ca. 45,000) | statistical events | no information | no information | design storms flood types | none |

**Table 2.**Overview of case studies 2D rainfall–runoff modeling (‘Direct Rainfall Method’)—Rural applications.

Reference | 2D Model (s) | Catchment: Size | Spatial Resolution (Number of Cells) | Rainfall Input | Rainfall Loss Approach | Roughness Values [s×m−1/3] | Sensitivity Analysis | Calibration Data |
---|---|---|---|---|---|---|---|---|

David & Schmalz (2020), [5] | HEC-RAS | Fischbach: 38 km ^{2} | DEM: 1 m Mesh: 100 m, 5 m refinement (246,100) 30 m, 3 m refinement (687,800) | 3 historical events | constant psi SCS CN approach with modification [33] | Manning formula final range from n = 0.07 for pastures to n = 0.11 for forest-covered areas | Manning’s n model specific parameters: computational and filter tolerances | gauging station |

Jia et al. (2019), [7] | CCHE2D | Howden Lake: 18 km ^{2} | from 3.76 to 4.98 m (ca. 942,600) | historical events | no loss (clayey soils) | Manning formula initial value from 0.03 to 0.3 final value n = 0.3 (catchment area), n = 0.16 (channel) | Manning’s n | gauging station |

Broich et al. (2019), [6] | TELEMAC 2D | Simbach a. Inn: 45.9 km^{2}Triftern: 90.1 km ^{2} | 5 m (ca. 1,836,000) 5 m (ca. 3,604,000) | 1 historical event | SCS CN approach | DWA, (2020), [34] Machiels et al. (2009), [35] Lawrence, (1996), [36] Lindner, (1982), [37] | model specific parameters: fricti.f (H0), steep slope correction (SSC) | gauging station |

Hall (2015), [29] | MIKE Flood | Birrega: 185 km ^{2} | 20 m (ca. 462,500) | 2 historical events 5 statistical events | constant infiltration rate | Manning formula calibrated n from n = 0.022 (roads) to n = 0.059 (urban/native vegetation) | rainfall depth Manning’s n Infiltration rate Groundwater inundation | gauging station |

Cea & Bladé (2015), [30] | in-house development | Solivella: 4 km ^{2}Maior River: 5 km ^{2} | Solivella: avrg. ca. 15.5 m (17,926) Maior River: avrg. ca. 20 m (24,676) | Solivella: 1 fictional storm event Maior River: 1 historical event | Solivella: no infiltration (fully saturated soil) Maior river: constant infiltration rate | Manning formula Solivella: n = 0.15 Maior river: from n = 0.3 to n = 0.5 | four different discretization schemes | none |

Clark et al. (2008), [31] | TUFLOW, SOBEK | Boembee Valley: 11.85 km^{2} | 5 m, 10 m, 20 m, 50 m, 100 m | design storm: 100-year event, 2 h | constant infiltration rate | constant n-values: 0.04, 0.06, 0.08 | timestep Manning’s n mesh resolution run length slope return period | none |

PC-1 | PC-2 | PC-3 | PC-4 | |
---|---|---|---|---|

System | ThinkStation P330 | ThinkStation P520c | ThinkStation P330 | ThinkStation P520c |

CPU | I9-9900, 3.10 GHz | Xeon W-2125, 4.00 GHz | i7-9700, 3.00 GHz | Xeon W-2125, 4.00 GHz |

GPU | Nvidia Quadro P2000 | Nvidia Quadro P2000 | Nvidia Quadro P620 | Nvidia Quadro P2000 |

RAM | 16 GB | 16 GB | 16 GB | 32 GB |

Used for | DEM vs. Mesh resolution Precipitation data | Filter parameters | Manning’s n | Laminar depth |

Landuse Classification [57] | Soil Classification [58] | ||||
---|---|---|---|---|---|

Description | Area [km^{2}] | Percentage [%]* | Description | Area [km^{2}] | Percentage [%]* |

Wooded area | 1.15 | 54.0 | sandy loamy silt | 1.25 | 58.7 |

Farmland: field crops | 0.40 | 18.8 | medium clayey silt | 0.69 | 32.4 |

Farmland: grassland | 0.39 | 18.3 | clayey silt | 0.13 | 6.1 |

Trails | 0.09 | 4.2 | Not classified | 0.06 | 2.8 |

Settlements Thereof buildings | 0.04 0.01 | 1.9 0.5 | |||

Roads | 0.03 | 1.4 | |||

Water bodies | 0.02 | 0.9 |

^{2}).

# | File Name* | Mesh Resolution | Number of Cells |
---|---|---|---|

1 | 0_25 m_1 m | 1 m | 2,128,354 |

2 | 0_25 m_2 m | 2 m | 531,489 |

3 | 0_25 m_3 m | 3 m | 236,035 |

4 | 0_25 m_4 m | 4 m | 132,652 |

5 | 0_25 m_5 m | 5 m | 84,810 |

6 | 0_25 m_10 m | 10 m | 21,102 |

7 | 0_25 m_20 m | 20 m | 5236 |

8 | 0_25 m_30 m | 30 m | 2299 |

*****for 0.25 m DEM.

Landuse Category | Manning’s n Value |
---|---|

wooded area | 0.198 |

farmland: arable land | 0.18 |

farmland: grassland | 0.15 |

trails | 0.03 |

settlements, buildings | 0.1, 0.013 |

roads | 0.013 |

river | 0.029 |

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**MDPI and ACS Style**

David, A.; Schmalz, B. A Systematic Analysis of the Interaction between Rain-on-Grid-Simulations and Spatial Resolution in 2D Hydrodynamic Modeling. *Water* **2021**, *13*, 2346.
https://doi.org/10.3390/w13172346

**AMA Style**

David A, Schmalz B. A Systematic Analysis of the Interaction between Rain-on-Grid-Simulations and Spatial Resolution in 2D Hydrodynamic Modeling. *Water*. 2021; 13(17):2346.
https://doi.org/10.3390/w13172346

**Chicago/Turabian Style**

David, Amrei, and Britta Schmalz. 2021. "A Systematic Analysis of the Interaction between Rain-on-Grid-Simulations and Spatial Resolution in 2D Hydrodynamic Modeling" *Water* 13, no. 17: 2346.
https://doi.org/10.3390/w13172346