# An Approach Using a 1D Hydraulic Model, Landsat Imaging and Generalized Likelihood Uncertainty Estimation for an Approximation of Flood Discharge

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## Abstract

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## 1. Introduction

## 2. Study Area and Data Set

**Figure 1.**Study reaches. (

**a**) The Wabash River at Montezuma, IN, USA (The Montezuma reach); (

**b**) The Missouri River at Nebraska City, NE, USA (The Nebraska reach).

Study Reach | River length (km) | Mean bed slope (m/km) | Number of cross-section | Mean width of cross-section (km) | Mean spacing cross-section (km) |
---|---|---|---|---|---|

Montezuma | 9 | 0.25 | 18 | 3.61 | 0.53 |

Nebraska | 19 | 0.21 | 11 | 12.84 | 1.90 |

2001 NLCD Classification | Manning’s n | Source | ||
---|---|---|---|---|

Minimum | Normal | Maximum | ||

Open Water | 0.025 | 0.030 | 0.033 | [35] |

Developed, Open Space | 0.010 | 0.013 | 0.160 | [36] |

Developed, Low Intensity | 0.038 | 0.050 | 0.063 | [36] |

Developed, Medium Intensity | 0.056 | 0.075 | 0.094 | [36] |

Developed, High Intensity | 0.075 | 0.100 | 0.125 | [36] |

Barren Land | 0.025 | 0.030 | 0.035 | [35] |

Deciduous Forest | 0.100 | 0.120 | 0.160 | [35] |

Evergreen Forest | 0.100 | 0.120 | 0.160 | [35] |

Mixed Forest | 0.100 | 0.120 | 0.160 | [35] |

Scrub/Shrub | 0.035 | 0.050 | 0.070 | [35] |

Grassland/Herbaceous | 0.025 | 0.030 | 0.035 | [35] |

Pasture/Hay | 0.030 | 0.040 | 0.050 | [35] |

Cultivated Crops | 0.025 | 0.035 | 0.045 | [35] |

Woody Wetlands | 0.080 | 0.100 | 0.120 | [35] |

Emergent Herbaceous Wetland | 0.075 | 0.100 | 0.150 | [35] |

**Table 3.**The observed discharge at USGS gauge stations for flood discharge on Landsat and peak flow in a flood event.

Study Reach | USGS gauge station | For flood discharge on Landsat imagery | For a peak flow of the flood event | ||
---|---|---|---|---|---|

Date of Landsat image | Discharge at gauge station (m^{3}/s) | Date of peak flow | Discharge at gauge station (m^{3}/s) | ||

Montezuma | 03340500 | 11 June 2008 | 1450 | 8 June 2008 | 2197 |

Nebraska | 06807000 | 9 July 2011 | 6031 | 7 July 2011 | 6258 |

**Figure 2.**Geometric shape of cross-sections for study reaches. (

**a**) Montezuma reach; (

**b**) Nebraska reach.

## 3. Methodology

#### 3.1. Extraction of the Observed Data from Landsat 5 TM Satellite Imagery

#### 3.2. Approximation of Flood Discharge Using HEC-RAS and the GLUE Methodology

#### 3.2.1. Monte Carlo Simulation Using HEC-RAS

_{1}, Y

_{2}are the depth of water; V

_{1}, V

_{2}are the average velocities; Z

_{1}, Z

_{2}are the elevation of the main channel; α

_{1}, α

_{2}are the velocity weighting coefficients; g is the gravitational acceleration; and h

_{e}is the energy head loss. In addition, 1 and 2 refer to the upstream and downstream cross section, respectively.

_{f}is the friction slope; and R is the hydraulic radius.

^{3}/s to 5000 m

^{3}/s for the Montezuma reach, and from 1000 m

^{3}/s to 10,000 m

^{3}/s for the Nebraska reach. Here, the ranges of discharge for both study reaches are determined to sufficiently cover the discharge on the target flood events, by considering the flood extend width and the water depth obtained from satellite imagery and DEM; (4) Manning’s n value is generally considered as a primary parameter in calibrating HEC-RAS [50,51]. However, in this study, the roughness coefficients extracted from NLCD 2001 (Table 2) are used for calculation of the water surface elevation without calibration of the model, due to the assumption that there is no gauged information regarding the discharge relationship. Downstream boundary condition is represented by normal depth.

#### 3.2.2. Approximation of Flood Discharge Using the GLUE Methodology

_{L}is posterior likelihood values; and X is the value of x simulated by model; L[M(Θ,I)] is a likelihood measure by model prediction (M) for given parameter (Θ) and set of input data (I); P

_{i}is a penalty function; i is iteration; and r is a cut-off threshold.

_{i}) is calculated by the ith likelihood measure (L

_{i}), the minimum likelihood measure (L

_{MIN}), and the maximum likelihood measure (L

_{MAX}).

_{m,i}and E

_{o}represent the ith iteration of the modeled water surface elevation and the observed water surface elevation, respectively, for the jth among a total N of cross-sections.

_{o}indicates the observed inundation area; A

_{p}refers to the predicted flood inundation area; and A

_{op}represents an overlap of both the observed and the predicted inundation areas.

_{p}) of Equation (6) is based on results obtained from HEC-RAS, which provides water surface elevations only along cross sections. To produce the flood inundation area from these discrete model outputs, the DEM is subtracted from the water surface area, which is derived by the inverse distance weight (IDW) interpolation of the water surface elevations at each cross section. Equation (7) (EF likelihood measure) is a likelihood measure proposed in this study, which considers both the vertical and spatial differences due to the availability of the water surface elevation and the flood extents as observations.

## 4. Results and Discussion

#### 4.1. Extraction of the Observed Data from Landsat 5 TM Satellite Imagery

^{2}for the Montezuma reach and 105.0 km

^{2}for the Nebraska reach, and is used as the observed flood inundation maps to estimate the F likelihood measures in the GLUE methodology.

**Figure 3.**Three-dimensional representations of water-body overlaid on DEM extraction using ISODATA (Montezuma reach, 11 June 2008). Water-body (blue solid) extracted from Landsat image on DEM.

**Figure 4.**Intersection points for reading water surface elevation from DEM (Montezuma reach, 11 June 2008).

**Figure 5.**The water surface elevation (WSE) obtained from Landsat image and water surface elevation from Monte Carlo (MC) simulations. (

**a**) For the Montezuma reach; (

**b**) For the Nebraska reach.

#### 4.2. Approximation of Flood Discharge Using HEC-RAS and the GLUE Methodology

^{2}and 16.8 km

^{2}for the Montezuma reach, and 63.9 km

^{2}and 144.1 km

^{2}for the Nebraska reach, respectively. The Nebraska reach has a twice-longer river than the Montezuma reach. However, although minimum random discharge (1000 m

^{3}/s) in the Nebraska reach is much less than maximum (5000 m

^{3}/s) in the Montezuma reach, the minimum inundated area (63.9 km

^{2}) of the Nebraska reach is about four times larger than the maximum (16.8 km

^{2}) of the Montezuma reach. From these results, it can be expected that the flood inundation depth in the Nebraska reach is shallower than that in the Montezuma reach.

**Figure 6.**PDFs of discharge based on the rescaled likelihood measure. F likelihood measure: (

**a1**) Montezuma Reach; (

**b1**) Nebraska Reach; E likelihood measure: (

**a2**) Montezuma Reach; (

**b2**) Nebraska Reach; The combination of E and F likelihood measure (EF likelihood measure): (

**a3**) Montezuma Reach; (

**b3**) Nebraska Reach.

^{3}/s for the F likelihood measure, 4471 to 6344 m

^{3}/s for the E likelihood, and 4576 to 6412 m

^{3}/s for the EF likelihood measure. The 50% of the CDF giving the deterministic model output, ranges from 5346 to 7911 m

^{3}/s. Considering the gauged discharge of 6030 m

^{3}/s, the relative errors of approximated discharge are in the range of 10% (617 m

^{3}/s) to 31% (−2830 m

^{3}/s). In the case of the Montezuma reach, the flood discharge over all likelihood measures is approximated in the range of 687 m

^{3}/s to 2769 m

^{3}/s. Considering the gauged discharge of 1450 m

^{3}/s, the relative errors range from 14% (205 m

^{3}/s) to 55% (−797 m

^{3}/s). From these results, E likelihood based on the elevational difference, commonly produced a much better approximation of discharge than the F likelihood measure, based on spatial differences for both study reaches. The use of the EF likelihood measure improves the approximation of discharge, by reducing the relative error of 1% for the Nebraska reach and 4% for the Montezuma reach. A relative error of 10% in the Nebraska reach indicates a better approximation of discharge than the 14% in the Montezuma reach. The gauged discharge used in this study is estimated from a stage-discharge rating curve involving uncertainty. For comparing results from the suggested method with uncertainty in rating curves, the regression equation for the stage-discharge rating curve based on peak flows provided by USGS is developed assuming a Student’s t-distribution (Figure 8). Using a 95% confidence interval, uncertainty for discharge of 1450 m

^{3}/s at USGS gauge station in Montezuma ranges from 938 to 2241 m

^{3}/s and uncertainty for discharge of 6030 m

^{3}/s at USGS gauge station in Nebraska city rages from 2263 to 16,048 m

^{3}/s. These results show that almost all discharge estimated by the suggested method are in the bound of uncertainty in discharge estimated from rating equations (Table 4). Figure 9 shows the simulated flood inundation maps for the approximated discharge, which are deterministic representation values (50% of CDF) based on the EF likelihood measure. Among the three different likelihood measures used in this study, EF likelihood measures commonly produced the best approximation of discharge for both study reaches.

**Figure 7.**Cumulative density function (CDF) obtained by taking top 30% of likelihood measure and the 5%, 50%, and 95% boundaries of the approximated flood discharge. Based on F likelihood measure: (

**a1**) Montezuma Reach; (

**b1**) Nebraska Reach; Based on E likelihood measure: (

**a2**) Montezuma Reach; (

**b2**) Nebraska Reach; Based on the combination of E and F likelihood measure (EF likelihood measure): (

**a3**) Montezuma Reach: (

**b3**) Nebraska Reach.

Likelihood Measure | CDF | Discharge (m^{3}/s) | |
---|---|---|---|

Montezuma | Nebraska | ||

F | 0.05 | 1752 | 6938 |

0.50 | 2247 | 7911 | |

0.95 | 2769 | 8862 | |

E | 0.05 | 687 | 4471 |

0.50 | 1179 | 5346 | |

0.95 | 1788 | 6344 | |

EF | 0.05 | 801 | 4576 |

0.50 | 1245 | 5413 | |

0.95 | 1827 | 6412 | |

Observation | 1450 | 6030 |

**Figure 8.**Stage-discharge rating curve based on peak flows provided by USGS. (

**a**) Montezuma reach; (

**b**) Nebraska reach.

**Figure 9.**The simulated flood inundation maps for the approximated discharge (50% of CDF based on EF likelihood measure). (

**a**) Montezuma (Q = 1245 m

^{3}/s); (

**b**) Nebraska (Q = 5413 m

^{3}/s).

## 5. Summary and Conclusions

- This study demonstrates that Landsat imagery can be used as secondary source for discharge estimation in a data-poor environment. The water-body extracted from the Landsat imagery can be used in conjunction with a hydraulic model to estimate flood discharge. However, the use of Landsat imagery in a small-scale study can produce relatively more uncertainty in reading water surface elevation from a DEM (10 m × 10 m) than in a large-scale study, due to the coarse resolution (30 m × 30 m) of a Landsat image. Therefore, flood information obtained from Landsat imagery in planning flood risk management in a data-poor environment is more appropriate for larger rivers. The approximated flood discharge estimated for the Nebraska reach is 5413 m
^{3}/s, and 1245 m^{3}/s for the Montezuma reach. The relative errors between the gauged data and the approximations are 10% for the Nebraska reach and 14% for the Montezuma reach, respectively. - In the GLUE methodology, the different results between E Likelihood measure and F likelihood measure showed subjectivity on the selection of criteria meeting informal likelihood measure. However, when considering the physical conditions of the study reach, such as the shape of the valley, size of the reach, and the flood intensity, the informal likelihood measure in the GLUE methodology can enhance the ability of finding improved flood information in data-poor environment. In addition, each likelihood measure is differently responded corresponding to the random discharge, but produces common results in flood discharge estimation for both study reaches. For example, the approximated discharge for both reaches is overestimated for F likelihood measure on the spatial flood extent and underestimated for E likelihood measure on the observed water surface elevation. In addition, the combination (EF likelihood) of two likelihood measures estimates discharge closest to the observed discharge at gauge station.

## Acknowledgments

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Jung, Y.; Merwade, V.; Yeo, K.; Shin, Y.; Lee, S.O.
An Approach Using a 1D Hydraulic Model, Landsat Imaging and Generalized Likelihood Uncertainty Estimation for an Approximation of Flood Discharge. *Water* **2013**, *5*, 1598-1621.
https://doi.org/10.3390/w5041598

**AMA Style**

Jung Y, Merwade V, Yeo K, Shin Y, Lee SO.
An Approach Using a 1D Hydraulic Model, Landsat Imaging and Generalized Likelihood Uncertainty Estimation for an Approximation of Flood Discharge. *Water*. 2013; 5(4):1598-1621.
https://doi.org/10.3390/w5041598

**Chicago/Turabian Style**

Jung, Younghun, Venkatesh Merwade, Kyudong Yeo, Yongchul Shin, and Seung Oh Lee.
2013. "An Approach Using a 1D Hydraulic Model, Landsat Imaging and Generalized Likelihood Uncertainty Estimation for an Approximation of Flood Discharge" *Water* 5, no. 4: 1598-1621.
https://doi.org/10.3390/w5041598