# Modeling and visualizing uncertainties of flood boundary delineation: algorithm for slope and DEM resolution dependencies of 1D hydraulic models

## Abstract

As flood inundation risk maps have become a central piece of information for both urban and risk management planning, also a need to assess the accuracies and uncertainties of these maps has emerged. Most maps show the inundation boundaries as crisp lines on visually appealing maps, whereby many planners and decision makers, among others, automatically believe the boundaries are both accurate and reliable. However, as this study shows, probably all such maps, even those that are based on high-resolution digital elevation models (DEMs), have immanent uncertainties which can be directly related to both DEM resolution and the steepness of terrain slopes perpendicular to the river flow direction. Based on a number of degenerated DEMs, covering areas along the Eskilstuna River, Sweden, these uncertainties have been quantified into an empirically-derived disparity distance equation, yielding values of distance between true and modeled inundation boundary location. Using the inundation polygon, the DEM, a value representing the DEM resolution, and the desired level of confidence as inputs in a new-developed algorithm that utilizes the disparity distance equation, the slope and DEM dependent uncertainties can be directly visualized on a map. The implications of this strategy should benefit planning and help reduce high costs of floods where infrastructure, etc., have been placed in flood-prone areas without enough consideration of map uncertainties.

### Keywords

1D hydraulic modeling River flood inundation Uncertainty Quantile regression Geographical information systems (GIS) Digital elevation model (DEM)## 1 Introduction

### 1.1 Background

Hydraulic modeling of river floods has received a significant boost during the last 10 years; not only thanks to improved computers and hydraulic modeling software, but also to the capabilities and user-friendliness of geographical information systems (GIS). During the same period, new legislation, such as EU’s flood directive, demands that flood risks are incorporated into risk and management plans, and together, this has led to production of numerous flood risk maps. Although these maps may have been produced by professionals who are aware of the different inaccuracies and uncertainties underlying the maps, they are often used by people who have little or no experience of neither hydraulic nor digital elevation modeling. Furthermore, as these maps tend to form the basis for many decisions in spatial and physical planning of the built environment, there is a need for tools that can communicate the intrinsic uncertainties always present in the maps.

There are different types of uncertainties involved in flood risk mapping (see e.g. Pappenberger et al. 2008 and Merwade et al. 2008, for general treatise on this subject). The most immediate is which model to be used (e.g. Wagener and Gupta 2005), but in practice, the most commonly treated uncertainty is which magnitude of flow to use for a certain flood return period. This can be handled by running the model with different water discharges and thereby get a range of flood inundation areas. Other ways of treating uncertainty is through Monte Carlo simulations, i.e., feeding the model with slightly varied input of all input parameters, and where the large number of output maps in turn can be related to flood prediction uncertainty on how accurate the modeled results are (e.g. Apel et al. 2008). Obviously, specific objects and parameters in the hydraulic model can also influence the accuracy of the produced results. For example Koivumäki et al. (2010) studied the effects of buildings in the model, Pappenberger et al. (2006) looked at the effects of boundary conditions and bridges, and Cook and Merwade (2009) and Castellarin et al. (2009) treated the effects of cross-section location and spacing. As the hydraulic modeling usually involves calibration against a previous flood event, the importance of roughness is also widely known. By varying river bed and floodplain roughness values in a range that theoretically can be expected, minimum and maximum extents of the flooded area can be modeled. Although modelers have acknowledged the implications of roughness for a long time, it is not until recent years that any efforts have been made to see how much this type of uncertainty affects the results (see e.g. Pappenberger et al. 2005; Werner et al. 2005; Casas et al. 2006; Schumann et al. 2007; Wilson and Atkinson 2007; Brandt 2009; Warmink et al. 2013; Wu in press). This is probably due to the type of uncertainty which earlier has been considered the main constraint for successful modeling, viz. the quality of the digital elevation model (DEM).

### 1.2 Previous research on delineation uncertainties related to DEMs

Before the advent of LiDAR, the results from hydraulic models, which could be based on detailed surveyed cross sections, were overlain on DEMs of poor resolution. In Sweden, e.g., up to only a couple of years ago, the only elevation database of national coverage has been Lantmäteriet’s (the Swedish mapping, cadastral and land registration authority) with 50 m cell resolution (other countries have had similar resolutions). Very rarely, there have been DEMs of higher quality available. Due to the poor quality of the elevation models, in Sweden all such maps were given a notification that they should not be used for detailed planning. Hence, there have been some studies with the specific objective to study how the quality of DEMs affects the accuracy of inundation boundary delineation from 1D hydraulic models, which end products are water levels at each modeled cross section. By comparing these modeled levels with measured levels, several studies have shown that the accuracy of predicting correct levels is surprisingly high, irrespectively of the quality of DEM (e.g. Casas et al. 2006; Yacoub and Sanner 2006; Brandt 2009). Only with poor DEMs (i.e., cell sizes bigger than 10–25 m) together with steep river slopes, or abrupt slope change, the water levels may deviate significantly between modeled and real conditions (Brandt 2009). However, when it comes to the spatial extent, which is important when the inundation extents are transferred to maps, high-resolution DEMs of high quality may also produce inaccurate results.

An early attempt to look at spatial deviations was done by Zhang and Montgomery (1994) on two areas in the USA. They gridded spot elevation data to DEMs of 2, 4, 10, 30, and 90 m resolution. They noticed that better resolution than 10 m lead to improved modeling results. However, the best two DEMs did not produce any significant improvements; most probably due to the catchments being characterized by moderately to steep terrain gradients. Later, Werner (2001) used laser altimetry data for a reach of the river Saar in Germany. The original cell resolution was 2.5 m, which then was aggregated by averaging neighboring cell values to cell sizes of 5, 10, and 25 m. He concluded that a cell resolution of 10 m indicated the break when flood extents started to deviate significantly.

^{2}, equivalent to 5.89 m cell size. By testing different threshold values of α, their recommendation is that α should be less than 4°, which in their case represented about 38 % of the original number of points, i.e., ca 0.0111 points/m

^{2}, equivalent to 9.50 m cell sizes.

Another study was undertaken by Casas et al. (2006). They looked at an area next to the Ter River, Spain, and tested different DEMs ranging from 1 to 4 m in cell size. The DEMs were derived from laser altimetry data, GPS surveyed data, 5 m contour data (scale 1:5000), as well as combinations between them, together with or without bathymetric data. They concluded that for a 500 m^{3}/s discharge, the 4 m resolution DEM yielded inundated area differences up to 7.3 %. However, if higher discharges were used (3000 m^{3}/s), the differences were reduced to 2.6 %. Therefore they argued that coarser resolution will have less consequence in floodplain areas.

Raber et al. (2007) looked at Reedy Fork Creek, North Carolina, and started with laser altimetry data with a mean point distance of 1.35 m, which later were filtered in several steps down to 9.64 m. By comparing statistics over the modeled inundated areas, they concluded that it is enough with 4 m mean point spacing. For better DEMs they did not see any significant differences between the model results.

Cook and Merwade (2009) studied the Brazos River, Texas, and Strouds Creek, North Carolina, for different resolutions (laser altimetry data of 3 m for Brazos River and 6 m for Strouds River, as well as 10 and 30 m USGS data for both rivers) combined with different qualities of cross-section resolutions. Although their research focus was on inundated area differences, they did notice that for the smaller Strouds River (with a width of 9.5 m during normal conditions) the average width of a modeled flood where 25 % wider when poor DEMs were used. Similarly, the larger Brazos River’s (with a width of 175 m during normal conditions) average width was 5 % wider. This effect was doubled when laser altimetry data were integrated in the cross-section profiles.

### 1.3 Aim and objectives

Nowadays flood risk maps are usually based on DEMs with quite high quality. In Sweden, a new national elevation dataset of 2 m resolution is under production, and thanks to the detailed appearance of the maps, many users as well as hydraulic modelers tend to put high confidence in them and consider the results to be very accurate, i.e., with a flood-boundary position accuracy of just one or two raster cells. However, there are a few studies available that have shown that these maps may also suffer severely from DEM-derived uncertainties, but despite the recognition of the problem, it seems that practically no attempts have been made to actually visualize the uncertainties of these maps (cf. Lim et al. 2016). Considering the fact that there still are accuracy and uncertainty issues due to the quality of the DEMs, together with the absence of effective visualization techniques to represent these issues, the general aim of this paper is to provide insights into the importance of DEMs influence on 1D hydraulic modeling. The specific objectives are to produce: (1) a general equation capable of describing the uncertainties related to the DEM resolution and the floodplain characteristics, here represented by the slope perpendicular to the flow direction, and (2) an algorithm capable of illustrating the uncertainties of flood boundary mapping, related to the quality of the DEMs.

## 2 Prior studies of Eskilstuna and Testebo rivers

All previously mentioned studies focus on the DEMs global resolution, despite the relatively obvious influence of the local terrain; especially the slope characteristics perpendicular to the flow direction. Where cross sections have steep slopes, the inundation delineation is more certain than for cross sections with gentle slopes. Hence, river side areas with gentle perpendicular slopes call for elevation data of higher resolution to reduce the uncertainty of inundation extent delineation (Brandt 2005). This is also supported by Colby and Dobson (2010) who in their study on rivers in North Carolina concluded that the “extent and internal pattern of flooding in the low-relief coastal plains was found to be especially sensitive to the representation of terrain”. Therefore, a first attempt to look into this problem in detail was carried out for two areas of Eskilstuna River, Sweden (reported in Brandt 2009): one with relatively steep side slopes and one with relatively flat side slopes. These areas were modeled with the 1D hydraulic software HEC-RAS (Hydrologic Engineering Center 2008) for a steady state flow of 198 m^{3}/s and tested for inundation boundary delineation with several DEMs of varying cell resolution (besides this, that study also looked at the effect of systematic vertical errors of the surrounding terrain as well as the relative importance of errors between roughness and DEM resolution).

The northern investigated area in Eskilstuna River is 2241 m long, consists of two parts with a water power station in between, and has relatively steep side slopes. The original point cloud has a point spacing of 1.36 points/m^{2}. The southern area, on the other hand, is 1731 m long, has relatively gentle side slopes, and has a point spacing of 1.64 points/m^{2}. This is equivalent to raster DEMs of 0.86 and 0.78 m cell sizes, respectively. These datasets were then degraded step-by-step through both removal of points and by introducing random errors in order to fully simulate higher flight heights in accordance with the scanner equipment specifications (cf. Klang and Klang 2009, for full details).

## 3 Representing DEM-derived uncertainties of flood maps

This work has focused on two aspects of accuracies and uncertainties related to the elevation models that are used for inundation mapping. One is related to the cell resolution of the DEM and the other is related to the river channel and floodplain slope perpendicular to the flow direction.

### 3.1 DEMs used for creating an uncertainty equation

^{2}, corresponding to cell sizes of 0.78 and 0.86 m, respectively, had been degraded and downsampled to a number of different DEMs (Table 1) of gradually increased point spacing or cell size. The DEMs were the same as for the two areas shown in Fig. 3a (Brandt 2009).

Simulated flight height (m) | No. of points/m | Point spacing [Cell size] (m) | Introduced random errors of size 1 σ | |
---|---|---|---|---|

Planar (m) | Vertical (m) | |||

1122 (Reference for southern area) | 1.64 | 0.78 | ||

1232 (Reference for northern area) | 1.36 | 0.86 | ||

1122 (Southern area) | 1.64 | 0.78 | 0.17 | 0.11 |

1232 (Northern area) | 1.36 | 0.86 | 0.18 | 0.12 |

1500 | 0.92 | 1.04 | 0.20 | 0.12 |

2000 | 0.52 | 1.39 | 0.25 | 0.13 |

2500 | 0.33 | 1.74 | 0.31 | 0.14 |

3000 | 0.23 | 2.09 | 0.36 | 0.16 |

3500 | 0.17 | 2.43 | 0.41 | 0.17 |

4000 | 0.13 | 2.77 | 0.47 | 0.19 |

4500 | 0.102 | 3.13 | 0.52 | 0.20 |

5000 | 0.083 | 3.47 | 0.57 | 0.22 |

5500 | 0.068 | 3.83 | 0.63 | 0.24 |

14,391 | 0.010 | 10.0 | 0.63 | 0.24 |

35,979 | 0.0016 | 25.0 | 0.63 | 0.24 |

71,960 | 0.0004 | 50.0 | 0.63 | 0.24 |

### 3.2 Estimating the uncertainties due to DEM resolution and floodplain slopes

_{d}) according to:

^{2}) (Fig. 5).

Disparity distances for the 1.04, 3.83, and 50 m resolution DEMs

Slope class center (m) | log (Slope class center) | Class boundaries | No. of obs. | Disparity distance (m) for percentile | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | ||||

[δ = 1.04 m] | ||||||||||||

0.0006 | −3.25 | (−3.5) to (−3.0) | 6 | 19.53 | 25.86 | 28.53 | 31.48 | 32.50 | 33.55 | 59.86 | 106.79 | 107.24 |

0.0018 | −2.75 | (−3.0) to (−2.5) | 9 | 6.93 | 7.67 | 8.40 | 8.85 | 8.89 | 9.02 | 12.60 | 15.97 | 20.93 |

0.0056 | −2.25 | (−2.5) to (−2.0) | 39 | 2.07 | 2.47 | 3.55 | 4.79 | 5.32 | 6.17 | 6.72 | 9.48 | 11.65 |

0.0178 | −1.75 | (−2.0) to (−1.5) | 113 | 0.78 | 1.21 | 1.58 | 1.99 | 2.40 | 2.80 | 3.59 | 5.02 | 6.92 |

0.0562 | −1.25 | (−1.5) to (−1.0) | 157 | 0.29 | 0.42 | 0.56 | 0.74 | 0.99 | 1.20 | 1.54 | 2.15 | 2.90 |

0.1778 | −0.75 | (−1.0) to (−0.5) | 261 | 0.09 | 0.15 | 0.22 | 0.29 | 0.36 | 0.46 | 0.58 | 0.80 | 1.14 |

0.5623 | −0.25 | (−0.5) to (0) | 116 | 0.05 | 0.07 | 0.12 | 0.15 | 0.21 | 0.26 | 0.31 | 0.38 | 0.54 |

[δ = 3.83 m] | ||||||||||||

0.0006 | −3.25 | (−3.5) to (−3.0) | 1 | 12.67 | 12.67 | 12.67 | 12.67 | 12.67 | 12.67 | 12.67 | 12.67 | 12.67 |

0.0018 | −2.75 | (−3.0) to (−2.5) | 15 | 5.48 | 7.73 | 9.75 | 17.74 | 22.77 | 23.30 | 24.20 | 31.65 | 35.53 |

0.0056 | −2.25 | (−2.5) to (−2.0) | 50 | 2.10 | 2.86 | 4.33 | 5.40 | 8.88 | 11.93 | 14.38 | 19.74 | 24.99 |

0.0178 | −1.75 | (−2.0) to (−1.5) | 136 | 1.09 | 2.02 | 3.09 | 4.17 | 4.69 | 5.47 | 7.05 | 8.38 | 11.39 |

0.0562 | −1.25 | (−1.5) to (−1.0) | 205 | 0.61 | 0.97 | 1.36 | 1.69 | 2.20 | 2.76 | 3.31 | 4.58 | 5.99 |

0.1778 | −0.75 | (−1.0) to (−0.5) | 234 | 0.22 | 0.40 | 0.64 | 0.88 | 1.11 | 1.35 | 1.68 | 2.12 | 2.77 |

0.5623 | −0.25 | (−0.5) to (0) | 123 | 0.11 | 0.25 | 0.42 | 0.55 | 0.69 | 0.86 | 1.02 | 1.21 | 1.64 |

[δ = 50 m] | ||||||||||||

0.0006 | −3.25 | (−3.5) to (−3.0) | 1 | 253.04 | 253.04 | 253.04 | 253.04 | 253.04 | 253.04 | 253.04 | 253.04 | 253.04 |

0.0018 | −2.75 | (−3.0) to (−2.5) | 2 | 9.58 | 10.31 | 11.10 | 11.94 | 12.85 | 13.83 | 14.88 | 16.01 | 17.23 |

0.0056 | −2.25 | (−2.5) to (−2.0) | 18 | 14.95 | 18.42 | 21.22 | 22.44 | 27.93 | 31.28 | 33.25 | 36.74 | 53.32 |

0.0178 | −1.75 | (−2.0) to (−1.5) | 96 | 5.54 | 7.03 | 8.56 | 10.63 | 15.14 | 18.59 | 25.37 | 33.62 | 40.17 |

0.0562 | −1.25 | (−1.5) to (−1.0) | 265 | 4.00 | 6.67 | 9.81 | 12.58 | 14.82 | 18.83 | 22.51 | 27.10 | 32.49 |

0.1778 | −0.75 | (−1.0) to (−0.5) | 230 | 2.10 | 4.80 | 6.40 | 7.71 | 9.62 | 12.08 | 14.34 | 17.36 | 26.42 |

0.5623 | −0.25 | (−0.5) to (0) | 30 | 0.91 | 2.16 | 2.57 | 2.92 | 3.17 | 4.90 | 5.60 | 6.85 | 7.89 |

## 4 Constructing the algorithm

### 4.1 Algorithm for visualizing flood boundary delineation uncertainties

_{d}equation is exceeded. As the areal extent of inundation is derived using a regular grid of raster cells, with individual elevation values, there is a need to translate the disparity distance locations of the cross sections to elevation values. At the coordinates of disparity distance exceedance, elevation values are sampled or calculated based on the D

_{d}equation, one lower elevation defining the inner boundary of uncertainty and one higher elevation defining the outer boundary of uncertainty (by using the D

_{d}equation, the algorithm can consider abrupt ends due to e.g. buildings giving a so called “wall effect”). (3) Next the nodes in all cross sections are populated with the uncertainty elevation values and two TINs are created; one with values representing the inner uncertainty boundary and the other the outer uncertainty boundary. These TINs are then rasterized (this point-to-TIN-to-raster conversion will ensure that water levels will always fall in the downstream direction whereas direct raster interpolation from points may produce elevation artifacts). (4) Finally the DEM can be compared with the uncertainty rasters yielding three distinctive areas: not flooded areas (at least according to the percentile used), areas uncertain to be flooded, and flooded areas (at least according to the percentile used).

### 4.2 Resulting pseudo code

The following variables are used:

Boolean: *flag*

Integers: *m* _{ max } (number of cross sections); *m* (cross section number), *n* (cell or point number in a cross section)

Float: *DI* _{ flag }, *DO* _{ flag }, (previous valid distances of inner and outer uncertain flood area, respectively); *ws* (water surface elevation); *c*, *z* (coefficient and exponent values taken from the *D* _{ d } equation); *δ* (cell resolution)

*x*,

*y,*

*h*(coordinates and elevations of cross section nodes/cells);

*DI*,

*DO*,

*SI*,

*SO*(inner and outer uncertainty distances and slopes, respectively);

*RXSI*,

*RXSO*,

*LXSI*,

*LXSO*(inner and outer uncertainty elevations for right and left cross sections, respectively

- 1.
Count the number of cross sections (

*m*_{ max }). Determine the cell resolution (*δ*) of the DEM and use the disparity equation (*D*_{ d }) for the desired probability. - 2.
Split all cross sections at river channel center. Define cross sections as

*LEFT*or*RIGHT*. - 3.
Let

*m*= 1. - 4.Repeat until number of cross sections has been exceeded, i.e., m >
*m*_{ max }.- 4.1.If RIGHT cross section (cross section should be seen looking in the downstream direction, where cells are ordered from left (negative cell positions in the river channel) to right (positive cell positions on the ground).
- 4.1.1.
Determine the coordinates, i.e.,

*x*(0) and*y*(0), and elevation,*h*(0), of the DEM cell, where the modeled flood boundary intersects the cross section, i.e., the raster cell (or point) Cell (0). If no intersection exists, exclude that cross section from the analysis. - 4.1.2.
Let

*n*= 0,*flag*= false, DI(0) = 0. - 4.1.3.Repeat until the distance of the inner flood area,
*DI*, for Cell (n − 1), is longer than the*D*_{ d }for the same slope of that between the Cell (n − 1) and Cell (0), or that all cells in the cross section has been treated:- 4.1.3.1.
Get the coordinates and elevation of Cell (n)’s neighboring cell, i.e.

*x*(*n*− 1),*y*(*n*− 1), and*h*(*n*− 1) at Cell (n − 1), and calculate the distance,*DI*(*n*− 1), and slope,*SI*(*n*− 1), from Cell (0) to Cell (n − 1). - 4.1.3.2.
If

*SI*(*n*− 1) > 0 (i.e. the terrain is higher at Cell (n − 1), creating an island) then*n*=*n*− 1, else*flag*= true,*DI*_{ flag }=*DI*(*n*), and compare*DI*(*n*− 1) with the*D*_{ d }for desired probability for the absolute value of slope of*SI*(*n*− 1). If*D*_{ d }is exceeded then stop, else*n*=*n*− 1.

- 4.1.3.1.
- 4.1.4.
If

*flag*= false then*DI*_{ flag }=*δ*/2 (This accounts for cross sections where no cells in the inner part of the cross section have negative slopes, i.e. no certain flooding seems to occur). - 4.1.5.
If

*ws*(*m*) – exp[(ln*DI*_{ flag }– ln*c*)/*z*]**DI*_{ flag }>*h*(*n*− 1) then inner uncertainty boundary elevation*RXSI*(*m*) =*ws*(*m*) – exp[(ln*DI*_{ flag }– ln*c*)/*z*]**DI*_{ flag }else*RXSI*(*m*) =*h*(*n*− 1) (This accounts for the wall effect; see Sect. 4.3). - 4.1.6.
Let

*n*= 0,*flag*= false, DO(0) = 0. - 4.1.7.Repeat until the distance of the outer flood area,
*DO*(*n*+ 1), for Cell (n + 1), is longer than the*D*_{ d }for the same slope of that between the Cell (n + 1) and Cell (0), or that all cells in the cross section has been treated:- 4.1.7.1.
Get the coordinates and elevation of Cell (n)’s neighboring cell, i.e.

*x*(*n*+ 1),*y*(*n*+ 1), and*h*(*n*+ 1) at Cell (n + 1), and calculate the distance,*DO*(*n*+ 1), and slope,*SO*(*n*+ 1), from Cell (0) to Cell (n + 1). - 4.1.7.2.
If

*SO*(*n*+ 1) < 0 (i.e. the terrain is lower at Cell (n − 1), creating a water pond) then*n*=*n*+ 1, else*flag*= true,*DO*_{ flag }=*DO*(*n*), and compare*DO*(*n*+ 1) with the*D*_{ d }for desired probability for the absolute value of slope of*SO*(*n*+ 1). If*D*_{ d }is exceeded then stop, else*n*=*n*+ 1.

- 4.1.7.1.
- 4.1.8.
If

*flag*= false then*DO*_{ flag }=*δ*/2 (This accounts for cross sections where no cells in the outer part of the cross section have positive slopes, i.e. no certain dry ground seems to occur). - 4.1.9.
If

*ws*(*m*) + exp[(ln*DO*_{ flag }– ln*c*)/*z*]**DO*_{ flag }<*h*(*n*+ 1) then outer uncertainty boundary elevation*RXSO*(*m*) =*ws*(*m*) + exp[(ln*DO*_{ flag }– ln*c*)/*z*]**DO*_{ flag }else*RXSO*(*m*) =*h*(*n*+ 1) (This accounts for the wall effect).

- 4.1.1.
- 4.2.
*Else LEFT cross section*(*follow 4.1, but beware of sign changes,*<*/*>*changes, and change of RXSI/RXSO to LXSI/LXSO.* - 4.3.
Let m = m + 1.

- 4.1.
- 5.
Assign cross sections, i.e. both LEFT and RIGHT, with elevation values from XSI and XSO elevation values.

- 6.
Create two point themes where the point locations are taken from the nodes in the cross section lines. One theme with associated inner elevation values and one with outer elevation values.

- 7.
Create two TINs from the point themes. One based on inner elevation values and one based on outer elevation values. Rasterize the TINs to the same extent as the DEM to create uncertainty rasters.

- 8.
Compare the uncertainty rasters with the DEM. If cell elevations in uncertainty raster for outer elevations < DEM then cell is not flooded. If cell elevations in uncertainty raster for inner elevations > DEM then cell is flooded. Other alternatives results in uncertain with respect to flooded/not flooded.

### 4.3 Calculation of wall effect

_{d}for the same slope (i.e., between location of the modeled flood boundary location and the cell C(0)), the uncertainty elevation (from the D

_{d}equation) of cell C(n) is compared with the ground elevation of cell C(n + 1). The elevation can be calculated by adding the water surface elevation to the product of the distance and slope from Eq. 2:

### 4.4 Applying the algorithm

_{d}equation and the algorithm, two DEMs were chosen, 3.83 m and 50 m, respectively, with a 95 percentile confidence. The results can be seen in Fig. 10, where red areas represent the areal range that is uncertain to be flooded. It is 95 % certain that the flood boundary will be within this area. The remaining 5 % are divided into blue areas that are almost certain to be flooded and areas outside of the red areas that are almost certain not to be flooded.

## 5 Discussion and conclusion

Previous research has produced relatively consistent recommendations from 4 to 10 m DEM resolution. Better resolution than that has not produced any significant improvements in inundation boundary prediction. Yet there are numerous real case examples on disparities where the models clearly have failed and where the water boundary may be several hundreds of meters wrong, even when LiDAR data has been used as input (also cf. e.g. Croke et al. 2014 who stressed the importance of geomorphological understanding during flood risk management). Casas et al. (2006) argued that coarser resolution will have less consequence when discharge increases, i.e., floodplain areas will be more accurately mapped than near river areas. This may be true with respect to difference in total surface area of the true and modeled inundation. But, even if the flow rates are bigger, the flood boundary perimeter should be of roughly the same length, provided the river flows through a pronounced river or floodplain valley, and therefore the uncertainty problem will still persist at the modeled inundation boundaries. To overcome unpleasant surprises, extra caution should therefore be taken when either the DEMs are of poor resolution or that the terrain adjacent to the modeled inundation boundary is flat.

It is generally possible to construct an equation for the uncertainty of floodplain boundaries depending on DEM resolution and terrain slope. Except for the relation between z exponent and resolution, high correlation coefficients were yielded for all coefficients and exponents. This is probably due to the topographical character of the areas the modeling is based on; or in other words, the geographical areas of this study are not enough flat for sufficiently long distances. Hence, the main weakness in the data lies in the number of observations for the poorest DEM resolution together with the characteristics of the geographical area. Although there are very flat areas, these are not very big, making it impossible to get big disparities. This will also impact the equation. Most probable, the z exponent is too big, i.e., the slopes of the lines in Fig. 7 should be sloping steeper in the negative direction. Therefore, to have a more conservative and “safe” estimate on risks, the z exponent could be set to the value for the highest resolution (~–0.72), irrespectively of DEM resolution. Furthermore, as the same transects, with approximately 10 m spacing, are used for all DEM resolutions, the same DEM cells are used several times for calculation of the transect slopes of the coarse resolution DEMs. This is not the case for fine-resolution DEMs. Another problem may be the usage of retransforming logged values in the regression analysis (cf. Granger and Newbold 1976 or, for a hydrological example, Jansson 1985). No correction has been applied in this work and therefore some of the coefficients and exponents are probably underestimated.

As the disparity distance equation has been developed from elevation and slope values taken from the reference DEM, this may give erroneous estimates when applying the algorithm on DEMs of lower qualities. Therefore, in future research an equation based on the slopes derived from the lower quality DEMs will be created to see if that may affect the results. Also important is not to use the equation for higher confidences than 95 %. For example it is possible to get computed disparity distances that are shorter than those that were actually observed, even when 100 % is used as input. As can be seen in Fig. 7, there are not exactly 10 % of observations above and below the 90 percentile lines for the three DEMs shown. This arises mainly due to the combination of the 14 DEMs. The magnitudes of these errors (which can be seen as the uncertainty of the disparities for a given probability) have not been calculated in this study, as the primary focus have been on modeling and visualizing the uncertainties of the flood boundary lines, not the uncertainties of the uncertainties. However, when more robust data have been gathered for a wider range of geographical settings, it will be of interest to also study this (cf. Roscoe et al. 2012 for such a procedure).

High confidence percentages have to be treated using extreme value statistics. If more data, especially coarse resolution data, had been available, another option would be to use envelope curves. In this study envelope curves were created through increasing the 95 % confidence level by one magnitude. When comparing these envelope curves with the visually determined curves in Brandt and Lim (2012), there are strong indications that the z exponent is too big (cf. Fig. 4b), again calling for a z value closer to −0.72, also for poorer resolutions. Furthermore, the importance of regularly, and not too sparsely, placed cross sections can be seen in Fig. 10. For the 50 m DEM case, the inundated areas do not follow the trunk river in the western (left) part. This can be attributed to cross sections not expanding long enough, making them disqualified in the algorithm which in the end therefore produced too low water levels for this area.

To test the applicability of the equation and resulting algorithm, future research will look at another river where river side slopes are much gentler, as already now it can be seen that the disparities are higher for the Testebo River than the Eskilstuna Rivers, for the same slope. However, the disparities at Testebo River are compared against an actual flood event that may not have been mapped good enough, whereas Eskilstuna River’s are compared against a reference DEM. This means that the D_{d} equation developed for Eskilstuna River provides a measure of uncertainties only related to the DEM, whereas the disparities for Testebo also contain other types of uncertainties. Also, higher precision than cm on elevations has to be used to avoid the linear patterns seen in Fig. 5. This will also reduce the risk of getting 0 m/m slopes that make the use of logarithm functions problematic. More studies on other rivers, following both the Eskilstuna case of totally focusing on the DEM, as well as other cases that are compared with actual flood events, should both verify the approach as well as making adjustment of the equation possible. Finally, as this kind of uncertainty only represents one type of uncertainty, i.e., random errors in the DEM, other types including friction parameter errors, systematic errors of DEMs, rain and reach input of water flow, model structure, operator errors, etc. should also be considered. Therefore, being humble describing the uncertainties of the presented flood risk maps is to be recommended.

## Notes

### Acknowledgments

Thanks are due to Nancy Joy Lim, both for commenting on the manuscript as well as discussing the entire project, and the anonymous reviewers for their constructive criticism. The study has been financially supported by the EU through Tillväxtverket (Project GLOBES 2, number 170430) and Lantmäteriet (Project: “Kvalitetsbeskrivning av geografisk information vid översvämningskartering”).

### Compliance with ethical standards

### Conflict of interest

The author declares that he has no conflict of interest.

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