# Effects of Probability-Distributed Losses on Flood Estimates Using Event-Based Rainfall-Runoff Models

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## Abstract

**:**

## 1. Introduction

## 2. Bootstrap-Based Goodness-of-Fit Procedure

#### 2.1. Probability Distributions

#### 2.2. Model Selection

- Kolmogorov Smirnov (K-S) test statistic:$$D=max\left(\underset{1\le i\le n}{\mathrm{max}}\left(\frac{i}{n}-{\widehat{Z}}_{i}\right),\underset{1\le i\le n}{\mathrm{max}}\left({\widehat{Z}}_{i}-\frac{i-1}{n}\right)\right)$$
- Cramér–von Mises (C–vM) test statistic:$${W}^{2}=\frac{1}{12n}+{{\displaystyle \sum}}_{i=1}^{n}{\left({\widehat{Z}}_{i}-\frac{2i-1}{2n}\right)}^{2}$$
- Anderson-Darling (A-D) test statistic:$${A}^{2}=-n-\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\left(2i-1\right)\left(log\left({\widehat{Z}}_{i}\right)+log\left(1-{\widehat{Z}}_{\left(n-i+1\right)}\right)\right)$$

- Estimate the model parameters ($\widehat{\vartheta}$) from $\left\{{x}_{1},\cdots ,{x}_{n}\right\}$ and construct the cumulative density function (CDF) ${\widehat{M}}_{\widehat{\vartheta}}$
- Evaluate $\overline{D}$, ${\overline{W}}^{2}$ and ${\overline{A}}^{2}$ (Equations (4)–(6)), where ${\widehat{Z}}_{i}={\widehat{M}}_{\widehat{\vartheta}}\left({x}_{i}\right)$
- Generate $B$ bootstrap samples of size $n$ from ${\widehat{M}}_{\widehat{\vartheta}}$, denoted as $\left\{{x}_{1,b}^{*},\cdots ,{x}_{n,b}^{*}\right\}$ given $b=\left\{1,\cdots ,B\right\}$
- Estimate ${\widehat{\vartheta}}_{b}^{*}$ from $\{{x}_{1,b}^{*},\cdots ,{x}_{n,b}^{*}\}$ and construct the CDF ${\widehat{M}}_{{\widehat{\vartheta}}_{b}^{*}}$
- Evaluate ${\overline{D}}_{b}^{*}$, ${({\overline{W}}^{2})}_{b}^{*}$ and ${({\overline{A}}^{2})}_{b}^{*}$ (Equations (1)–(3)), where ${\widehat{Z}}_{i}={\widehat{M}}_{{\widehat{\vartheta}}_{b}^{*}}\left({x}_{i,b}^{*}\right)$

## 3. Rainfall-Runoff Modeling

#### 3.1. Rainfall-Runoff Model

#### 3.2. Model Calibration

#### 3.3. Stochastic Design Storms

#### 3.4. Design Flood Estimation

## 4. Study Area and Data

#### 4.1. Historic Data Sets

#### 4.2. Study Catchments

## 5. Results

#### 5.1. Probability-Distributed Initial Losses

#### 5.2. Effect of Functional Form of Initial Losses on Flood Estimates

_{i}the ith candidate initial loss distribution given {i = 1, ⋯, m}.

#### 5.3. Effect of Statistical Uncertainties in Initial Losses on Flood Estimates

#### 5.4. Effect of Regionalization of Initial Loss on Flood Estimates

- Regionalization of the probability distribution (dist): each candidate distribution is assumed to be the true underlying distribution at the site of interest, the first two statistical moments are derived from observed data and then the distribution parameters are estimated using the MoM.
- Regionalization of distribution parameters (param): candidate distributions are assumed to be the true underlying distribution and parameters are estimated as the average within a hydrologically similar region (where individual catchment parameters were estimated using the MLE).
- Regionalization of statistical moments (stat): the candidate distribution is assumed to be representative of the true underlying distribution, the first two statistical moments are regionalized within a hydrologically similar region and then the distribution parameters are estimated using the MoM.

- Regionalization of the probability distribution (dist): a standardized KDE distribution is assumed to be representative of the underlying distribution and the median value is derived from the observed data; the standardized KDE distribution is derived from individual KDE distributions within a hydrologically similar region, each distribution is normalized by the median value and the average distribution is determined.
- Regionalization of the probability distribution and median value (dist + median): a standardized KDE distribution is assumed to be the true underlying distribution and the median value is estimated from within a hydrologically similar region.

#### 5.5. Conclusions

- The functional form of the initial loss data in Australia can be approximated using the Beta and Gamma models. These results are in line with differing outcomes in previous studies, for instance Rahman et al. [12] recommend the use of the Beta distribution to represent initial losses, while Caballero and Rahman [28] determine the Gamma distribution to be most appropriate for initial loss data.
- By setting differing thresholds for the minimum sample size of initial loss data used to derive probability distributions, mixed results were found. This highlights the importance of sample size in deriving probability distributions of initial losses.
- Design flood estimates from successive candidate distributions showed relative errors typically within ±3%, similarly by shifting the distribution mean relative errors within approximately ±20% and a change in the distribution variance also resulted in relative errors within ±3%. These results lead us to conclude that knowledge of the central tendency of distributed initial losses is more important than knowledge of the true functional form of the initial loss data.
- Larger parameter sets generally present issues in regionalization; however, the comparison of the 4-parameter Beta and 2-parameter Gamma distributions showed that this was not the case in this study. Under the assumption that the catchments were ungauged, the relative errors of the design floods using the two distributions were comparable, with the Beta model producing slightly less uncertainty.
- Regionalization of the initial loss distributions led to design floods being estimated to within ±15% accuracy, with the largest uncertainty coming from errors in regionalization of the central tendency of initial loss data. This leads us to conclude that initial loss distributions can be regionalized relatively easily; however, difficulties lie in accurate estimation of the central tendency of the distribution.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Number of catchments in which the null hypothesis is rejected by each goodness-of-fit test, for each study dataset.

**Figure 3.**Percentage of catchments (with sample size greater than 10) in which each candidate distribution is selected by each goodness-of-fit test, for each study dataset.

**Figure 4.**Percentage of catchments (with sample size greater than 30) in which each candidate distribution is selected by each goodness-of-fit test, for each study dataset.

**Figure 5.**Distribution of relative errors in design flood estimates for each quantile across the eight study catchments using five candidate distributions to represent variability in initial losses. Here KDE (kernel density estimate) represents nonparametric method and AEP stands for annual exceedance probability.

**Figure 6.**Distribution of relative errors in design flood estimates for each flood quantile across the eight study catchments using distributions with the means shifted by specified percentages (Beta distribution).

**Figure 7.**Distribution of relative errors in design flood estimates for each flood quantile across the eight study catchments using various methods to regionalize probability-distributed initial losses (parametric models). (Here, Beta and Gamma distributions are used to regionalize initial loss distribution).

**Figure 8.**Distribution of relative errors in design flood estimates for each flood quantile across the eight study catchments using various methods to regionalize probability-distributed initial losses (nonparametric model).

**Table 1.**Geographic and meteorological catchment characteristics, including data availability for the selected study catchments.

Catchment | Drainage Area (km^{2}) | Mean Annual Rainfall (mm) | Mean Annual Runoff (mm) | Catchment Relief (mAHD) | Concurrent Rainfall-Runoff Record Length (Year) | Number of Events |
---|---|---|---|---|---|---|

Bielsdown | 82 | 1835 | 1424 | 362 | 29 (1971–1999) | 57 |

Leycester | 179 | 1541 | 525 | 943 | 45 (1967–2011) | 48 |

Macquarie | 35 | 1700 | 738 | 753 | 44 (1962–2005) | 26 |

Nowendoc | 218 | 972 | 298 | 595 | 41 (1971–2011) | 37 |

Orara | 135 | 1890 | 946 | 801 | 40 (1970–2009) | 37 |

Ourimbah | 83 | 1356 | 227 | 334 | 35 (1976–2010) | 38 |

Oxley | 218 | 1531 | 721 | 1179 | 47 (1965–2011) | 53 |

Pokolbin | 25 | 802 | 52 | 442 | 49 (1963–2011) | 36 |

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**MDPI and ACS Style**

Loveridge, M.; Rahman, A.
Effects of Probability-Distributed Losses on Flood Estimates Using Event-Based Rainfall-Runoff Models. *Water* **2021**, *13*, 2049.
https://doi.org/10.3390/w13152049

**AMA Style**

Loveridge M, Rahman A.
Effects of Probability-Distributed Losses on Flood Estimates Using Event-Based Rainfall-Runoff Models. *Water*. 2021; 13(15):2049.
https://doi.org/10.3390/w13152049

**Chicago/Turabian Style**

Loveridge, Melanie, and Ataur Rahman.
2021. "Effects of Probability-Distributed Losses on Flood Estimates Using Event-Based Rainfall-Runoff Models" *Water* 13, no. 15: 2049.
https://doi.org/10.3390/w13152049