# Simplified Uncertainty Bounding: An Approach for Estimating Flood Hazard Uncertainty

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Uncertainty in Hydraulic Model Results

#### 1.2. Approximation Methods

#### 1.3. Influence of Hydraulic Geometry

_{0}is the channel bed slope, S

_{f}is the friction slope, and g is the acceleration due to gravity. The friction slope is typically solved by an empirical roughness relation such as Manning’s equation:

_{0}; they contain two dependent variables: h and V. Area and friction slope are functions of flow depth. For irregular channels, empirically derived power functions can define the relationship between hydraulic geometry parameters (e.g., area and hydraulic radius) and flow depth, for example [36],

_{A}represents an error term. Thus, the solution to depth and velocity will be partially dependent on the relationship between depth and hydraulic geometry parameters, which can vary in space along a reach. Further, Equation (2) highlights the fact that the solution is also partially dependent on adjacent hydraulic controls, such as expansions or contractions that might be a result of bridges, a valley form, etc.

## 2. Materials and Methods

#### 2.1. Study Sites

#### 2.2. Hydraulic Model

_{e}is the energy loss between Cross Section 1 and Section 2. The specific energy at a cross section is calculated as

_{e}, is a function of the distance between the two cross sections, the expansion and contraction loss parameters, and the friction slope between the two cross-sections, S

_{f}:

#### 2.3. Monte Carlo Simulations

_{i}

_{,j}is the inundation status (1 = wet, 0 = dry) at a pixel for simulation j of n.

#### 2.4. The Simplified Point Approach

#### 2.5. Evaluation Metrics

#### 2.6. Impact of Hydraulic Structures

#### 2.7. Correlation Metrics

## 3. Results

#### 3.1. Inundated Area

#### 3.2. Relative Error in Depth and Top Width

#### 3.3. Hydraulic Structures

#### 3.4. Correlation Metrics

## 4. Discussion

## 5. Conclusions

- The accuracy of SUB to estimate uncertainty from MCS was variable among reaches, spatially within reaches, and across quantiles of the uncertainty distribution. However, accuracy generally increased with decreasing deviation from the mean, and accuracy decreased with increased variance in model inputs and parameters.
- The CSI and percentage of inundated area indicated that SUB was highly accurate, but the relative top width error indicated poorer performance, especially for some outliers.
- Hydraulic structures can significantly impact the accuracy of SUB but in a non-uniform manner. The direction of the error depends on the quantile and location. This suggests that structures present thresholds in the sensitivity of hydraulic response to uncertainty in model inputs and parameters.
- Results of the regression and trend analysis indicated varying influences of hydraulic and topographic metrics, highlighting the complexity of hydraulic processes. However, a positive relationship was consistently identified between the confinement ratio and the top width error, indicating that SUB will overestimate uncertainty in flood width for less confined floodplains. Consistent relationships with additional metrics were not identified among stream reaches.

## Supplementary Materials

## Author Contributions

## Funding

^{®}Initiative through the Cooperative Ecosystem Studies Unit Agreement W912HZ-20-2-0031. Travel for this work was funded by the Consortium of Universities for the Advancement of Hydrologic Science, Inc.’s 2016 Pathfinder Fellowship.

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Flow chart describing the process to create a 90% confidence interval (CI) of flood hazard estimates using MCS and the SUB method.

**Figure 3.**90% confidence intervals (CI) of flood inundation estimated by MCS and the SUB for (

**a**) the 1% AEP flood at Proctor Creek, (

**b**) the 2% AEP flood at Proctor Creek, (

**c**) the 1% AEP probability flood at Bronx Wash, and (

**d**) the 2% AEP flood at Bronx Wash. The 90% CI indicate areas that have a 5–95% chance of being inundated by the flood with a specific AEP.

**Figure 4.**(

**a**) Critical success index comparing SUB against MCS for each reach and flood frequency and (

**b**) the ratio of inundated area estimated by SUB relative to MCS. McMullen Creek from Stephens and Bledsoe [3].

**Figure 5.**Relative depth errors in SUB relative to MCS for (

**a**) the 1% AEP flood at Proctor Creek, (

**b**) the 2% AEP flood at Proctor Creek, (

**c**) the 1% AEP probability flood at Bronx Wash, and (

**d**) the 2% AEP flood at Bronx Wash. Vertical gray lines indicate hydraulic structure locations.

**Figure 6.**Inundated top width relative errors in SUB relative to MCS for (

**a**) the 1% AEP flood at Proctor Creek, (

**b**) the 2% AEP flood at Proctor Creek, (

**c**) the 1% AEP probability flood at Bronx Wash, and (

**d**) the 2% AEP flood at Bronx Wash. Vertical gray lines indicate hydraulic structure locations.

**Figure 7.**Cumulative distribution of the absolute depth (

**left**) and inundated top width (

**right**) errors in SUB relative to MCS.

**Figure 8.**Difference in the absolute value of relative error between simulations of SUB with structures and without structures for the (

**a**) 1% AEP depth, (

**b**) 2% AEP depth, (

**c**) 1% AEP top width, and (

**d**) 2% AEP top width.

Drainage Area (km ^{2}) | Mean Basin Slope (%) | % Imperviousness | % Developed | |
---|---|---|---|---|

Bronx Wash | 3.2 | 1.89 | 46 | 100 |

Proctor Creek | 42 | 9.76 | 35 | 84 |

Bronx Wash | Distribution | µ | σ |
---|---|---|---|

Flood quantile 0.99 (cms) | Lognormal | 16.9 | 4.6 |

Flood quantile 0.98 (cms) | Lognormal | 13.2 | 3.7 |

Manning’s n-value (channel) | Lognormal | 0.017–0.035 | 0.09–0.12 |

Manning’s n-value (floodplain) | Lognormal | 0.025–0.055 | 0.11–0.15 |

Channel change (m) | Normal | 0 | 0.61 |

Proctor Creek | |||

Flood quantile 0.99 (cms) | Lognormal | 221.8 | 33.3 |

Flood quantile 0.98 (cms) | Lognormal | 189.4 | 23.9 |

Manning’s n-value (channel) | Lognormal | 0.04–0.065 | 0.13–0.15 |

Manning’s n-value (floodplain) | Lognormal | 0.07–0.11 | 0.16–0.18 |

Channel change (m) | Normal | 0 | 0.09 |

**Table 3.**General trends between top width errors in SUB and physical and hydraulic metrics. (+) indicates a positive trend, (−) indicates a negative trend, and (0) indicates no apparent trend.

Predictor | Proctor Creek | Bronx Wash | ||
---|---|---|---|---|

1% AEP | 2% AEP | 1% AEP | 2% AEP | |

Confinement Ratio | + | + | + | + |

Froude Number | − | − | 0 | 0 |

Friction Slope | − | 0 | 0 | 0 |

**Table 4.**General trends between depth errors in SUB and physical and hydraulic metrics. (+) indicates a positive trend, (−) indicates a negative trend, and (0) indicates no apparent trend.

Predictor | Proctor Creek Q | Bronx Wash | ||
---|---|---|---|---|

1% AEP | 2% AEP | 1% AEP | 2% AEP | |

Confinement Ratio | 0 | 0 | 0 | 0 |

Froude Number | 0 | 0 | 0 | 0 |

Friction Slope | − | − | + | + |

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**MDPI and ACS Style**

Stephens, T.; Bledsoe, B.
Simplified Uncertainty Bounding: An Approach for Estimating Flood Hazard Uncertainty. *Water* **2022**, *14*, 1618.
https://doi.org/10.3390/w14101618

**AMA Style**

Stephens T, Bledsoe B.
Simplified Uncertainty Bounding: An Approach for Estimating Flood Hazard Uncertainty. *Water*. 2022; 14(10):1618.
https://doi.org/10.3390/w14101618

**Chicago/Turabian Style**

Stephens, Tim, and Brian Bledsoe.
2022. "Simplified Uncertainty Bounding: An Approach for Estimating Flood Hazard Uncertainty" *Water* 14, no. 10: 1618.
https://doi.org/10.3390/w14101618