# A Non-Stationarity Analysis of Annual Maximum Floods: A Case Study of Campaspe River Basin, Australia

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

## 3. Methodology

- (1)
- The AMF trend analysis using the MK test.
- (2)
- The AMF change point analysis using the Pettitt test.
- (3)
- The stationary and non-stationary flood frequency analysis.

#### 3.1. AMF Trend Analysis Using MK Test

#### 3.2. AMF Change Point Analysis Using Pettitt Test

#### 3.3. Stationary and Non-Stationary Flood Frequency Analysis

#### 3.3.1. Stationary Flood Frequency Analysis

#### 3.3.2. Non-Stationary Flood Frequency Analysis

_{0}and μ

_{1}. If the scale parameter shows variation, the ln(scale) parameter (ϕ) is expressed according to ϕ

_{0}and ϕ

_{1}, as represented in the following equations:

_{0}(for location parameter) and ${\varphi}_{0}$ (for scale parameter), and ${\mu}_{1}$ and ${\varphi}_{1}$, which represent the covariate’s effect on the parameter. In a non-stationary model, it is assumed that the location and scale parameters show variation linearly with time and the physical covariate as adopted by many studies (e.g., [38,39,40]), since the linearity assumption gives the flexibility of easy model fitting and interpretable results. In all non-stationary models, the shape parameter is considered to be constant since high error is involved in estimating it to allow changing with a covariate [41]. A multivariate analysis of models with five covariates (SOI, DMI, TPI, SAM, and time) would result in a very large number of non-stationary models; therefore, the non-stationary models were developed through combinations including one covariate at a time in the location and/or scale parameter as shown in Table 2 for the sake of simplicity. The time covariate in the models corresponds to an index representing the water year, and the values are centred and scaled before being used in the non-stationary models with the time covariate. The R nonstat package [41] was used to develop stationary and non-stationary GEV models as well as MK trend and Pettitt change point tests.

#### 3.3.3. Model Selection Process

#### Statistical Methods

#### Graphical Methods

#### 3.3.4. Uncertainty

## 4. Results and Discussion

#### 4.1. AMF Trend Analysis

#### 4.2. AMF Change Point Analysis

#### 4.3. Stationary and Non-Stationary Flood Frequency Analysis

^{3}/s, 59.2 m

^{3}/s, 110.7 m

^{3}/s, and 245.5 m

^{3}/s, respectively. As the parameters of the SGEV model are constant, design flood estimates from the SGEV model do not show variation over time. This is specifically important for Stations 406235 and 406250, where NSGEV outperformed the SGEV counterpart. For example, the difference between 20-year design floods of SGEV and NSGEV models at Station 406235 is −74% in 2021. Table 7 indicates the change percentage of 2-, 10-, 20-, and 50-year design floods in 2021 between SGEV and the best fitting NSGEV for the stations where the NSGEV model showed better fitting than the SGEV (i.e., Stations 406235 and 406250).

## 5. Conclusions

- Statistically significant decreasing trends (at 0.01 and 0.05 significance levels) in AMFs were detected regarding almost all stations in Campaspe River Basin.
- The year 1996 was identified as the statistically significant change point at almost all stations.
- Non-stationary GEV models had a time covariate that outperformed the stationary counterparts for two stations (Stations 406235 and 406250).
- The difference between the design floods of SGEV and NSGEV is particularly important for the NSGEV15 model with time-varying location and scale parameters.
- There is not enough evidence to state that ENSO, SAM, IOD, or IPO had significant effects on AMF non-stationarity in the basin.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Design flood estimations from SGEV and NSGEV15 for 2-, 10-, 20-, and 50-year return periods at Station 406235.

Station Number | Sub-Basin Area (km^{2}) | Location | Available Data Period |
---|---|---|---|

406208 | 37.6 | 144.451° E, 37.388° S | 1970–2022 |

406213 | 633.8 | 144.539° E, 37.016° S | 1975–2022 |

406214 | 237 | 144.428° E, 36.774° S | 1972–2022 |

406226 | 171.6 | 144.650° E, 36.881° S | 1978–2022 |

406235 | 212.3 | 144.660° E, 36.948° S | 1980–2022 |

406250 | 77.5 | 144.370° E, 37.320° S | 1983–2022 |

Model Name | Parameters of Model | ||
---|---|---|---|

$\mathsf{\mu}$ | $\mathbf{l}\mathbf{n}\left(\mathsf{\sigma}\right)=\mathsf{\varphi}$ | $\mathsf{\xi}$ | |

NSGEV1 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{S}\mathrm{O}\mathrm{I}$ | Constant | Constant |

NSGEV2 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{T}\mathrm{P}\mathrm{I}$ | Constant | Constant |

NSGEV3 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{S}\mathrm{A}\mathrm{M}$ | Constant | Constant |

NSGEV4 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{D}\mathrm{M}\mathrm{I}$ | Constant | Constant |

NSGEV5 | $\mathsf{\mu}\left(\mathrm{t}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{t}$ | Constant | Constant |

NSGEV6 | Constant | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{S}\mathrm{O}\mathrm{I}$ | Constant |

NSGEV7 | Constant | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{T}\mathrm{P}\mathrm{I}$ | Constant |

NSGEV8 | Constant | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{S}\mathrm{A}\mathrm{M}$ | Constant |

NSGEV9 | Constant | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{D}\mathrm{M}\mathrm{I}$ | Constant |

NSGEV10 | Constant | $\mathsf{\varphi}\left(\mathrm{t}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{t}$ | Constant |

NSGEV11 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{S}\mathrm{O}\mathrm{I}$ | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{S}\mathrm{O}\mathrm{I}$ | Constant |

NSGEV12 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{T}\mathrm{P}\mathrm{I}$ | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{T}\mathrm{P}\mathrm{I}$ | Constant |

NSGEV13 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{S}\mathrm{A}\mathrm{M}$ | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{S}\mathrm{A}\mathrm{M}$ | Constant |

NSGEV14 | $\mathsf{\mu}\left(\mathrm{x}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{D}\mathrm{M}\mathrm{I}$ | $\mathsf{\varphi}\left(\mathrm{x}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{D}\mathrm{M}\mathrm{I}$ | Constant |

NSGEV15 | $\mathsf{\mu}\left(\mathrm{t}\right)={\mathsf{\mu}}_{0}+{\mathsf{\mu}}_{1}\times \mathrm{t}$ | $\mathsf{\varphi}\left(\mathrm{t}\right)={\mathsf{\varphi}}_{0}+{\mathsf{\varphi}}_{1}\times \mathrm{t}$ | Constant |

Station Number | z-Score | 2-Sided p-Value | Outcome |
---|---|---|---|

406208 | −3.66157 | 0.000250671 | S (0.01) |

406213 | −2.36704 | 0.017931094 | S (0.05) |

406214 | −3.66343 | 0.00024886 | S (0.01) |

406226 | −2.60603 | 0.009159779 | S (0.01) |

406235 | −2.59458 | 0.00947072 | S (0.01) |

406250 | −1.45834 | 0.144746587 | NS |

Station Number | p-Value | Outcome | Year of Change |
---|---|---|---|

406208 | 0.000295 | S (0.01) | 1996 |

406213 | 0.010492 | S (0.05) | 1996 |

406214 | 0.002299 | S (0.01) | 1996 |

406226 | 0.015921 | S (0.05) | 1996 |

406235 | 0.016234 | S (0.05) | 1996 |

406250 | 0.088723 | NS | 2000 |

Gauge Number | Model | $\mathbf{Location}{\mathit{\mu}}_{0}$ | $\mathbf{Location}{\mathit{\mu}}_{1}$ | $\mathbf{Ln}\left(\mathbf{Scale}\right){\mathit{\varphi}}_{0}$ | $\mathbf{Ln}\left(\mathbf{Scale}\right){\mathit{\varphi}}_{1}$ | Shape | AIC | BIC |
---|---|---|---|---|---|---|---|---|

406208 | SGEV | 0.9068 | 0.4415 | 1.4555 | 226 | 231 | ||

406208 | NSGEV5 | 0.8803 | 0.0226 | 0.4322 | 1.4738 | 228 | 235 | |

406208 | NSGEV15 | 1.4105 | −0.6666 | 0.7438 | −0.6579 | 1.0779 | 225 | 233 |

406213 | SGEV | 12.9959 | 2.8779 | 1.0512 | 406 | 411 | ||

406213 | NSGEV5 | 12.6147 | 0.5300 | 2.8679 | 1.0701 | 408 | 415 | |

406213 | NSGEV15 | 15.3868 | −4.1344 | 3.0092 | −0.4580 | 0.8058 | 406 | 414 |

406214 | SGEV | 1.4513 | 1.1221 | 2.026 | 300 | 305 | ||

406214 | NSGEV5 | 1.2899 | 0.0499 | 1.0587 | 2.1561 | 301 | 308 | |

406214 | NSGEV15 | 2.3495 | −0.9221 | 1.4599 | −0.6661 | 1.3696 | 301 | 309 |

406226 | SGEV | 2.1333 | 1.2295 | 1.2554 | 282 | 287 | ||

406226 | NSGEV5 | 2.1347 | −0.0004 | 1.2300 | 1.2552 | 284 | 290 | |

406226 | NSGEV15 | 2.9686 | −1.2884 | 1.3622 | −0.5455 | 0.8445 | 280 | 288 |

406235 | SGEV | 6.1003 | 2.0935 | 0.8171 | 334 | 339 | ||

406235 | NSGEV5 | 6.5611 | −0.4283 | 2.1262 | 0.7407 | 335 | 342 | |

406235 | NSGEV15 | 7.7278 | −3.4482 | 2.1293 | −0.4784 | 0.5273 | 330 | 338 |

406250 | SGEV | 5.8260 | 1.5449 | 0.1154 | 249 | 253 | ||

406250 | NSGEV5 | 5.8056 | −1.9601 | 1.4617 | 0.1721 | 247 | 253 | |

406250 | NSGEV15 | 5.7582 | −2.1543 | 1.4527 | 0.1318 | 0.1695 | 248 | 256 |

Year | NSGEV15 (2-Year) | NSGEV15 (10-Year) | NSGEV15 (20-Year) | NSGEV15 (50-Year) |
---|---|---|---|---|

1982 | 19.7 | 90.2 | 143.4 | 258.8 |

1983 | 19.1 | 86.5 | 138.6 | 249.0 |

1984 | 18.6 | 83.5 | 133.0 | 241.0 |

1985 | 18.1 | 80.1 | 128.9 | 231.1 |

1986 | 17.6 | 77.2 | 124.5 | 220.8 |

1987 | 17.1 | 74.6 | 120.1 | 212.6 |

1988 | 16.5 | 72.0 | 115.7 | 204.2 |

1989 | 16.0 | 69.3 | 110.6 | 195.7 |

1990 | 15.5 | 66.5 | 106.5 | 188.2 |

1991 | 15.0 | 64.3 | 102.5 | 180.4 |

1992 | 14.5 | 61.8 | 98.5 | 172.7 |

1993 | 14.1 | 59.4 | 94.2 | 165.0 |

1994 | 13.6 | 57.3 | 90.6 | 157.6 |

1995 | 13.2 | 55.2 | 87.2 | 151.8 |

1996 | 12.8 | 53.2 | 83.4 | 146.2 |

1997 | 12.4 | 51.0 | 80.6 | 139.9 |

1998 | 12.0 | 48.9 | 77.5 | 134.8 |

1999 | 11.6 | 46.9 | 74.6 | 129.7 |

2000 | 11.2 | 45.1 | 72.0 | 125.3 |

2001 | 10.8 | 43.3 | 68.6 | 120.5 |

2002 | 10.4 | 41.7 | 66.0 | 114.8 |

2003 | 10.0 | 39.9 | 63.2 | 110.0 |

2004 | 9.7 | 38.5 | 60.5 | 105.6 |

2005 | 9.3 | 36.8 | 58.1 | 101.4 |

2006 | 8.9 | 35.4 | 55.4 | 97.4 |

2007 | 8.5 | 33.9 | 53.1 | 93.0 |

2008 | 8.1 | 32.4 | 51.0 | 89.5 |

2009 | 7.7 | 30.9 | 48.7 | 85.7 |

2010 | 7.4 | 29.6 | 46.8 | 81.8 |

2011 | 7.0 | 28.4 | 44.8 | 78.9 |

2012 | 6.6 | 27.3 | 43.0 | 75.3 |

2013 | 6.3 | 26.1 | 41.2 | 71.7 |

2014 | 5.9 | 24.9 | 39.6 | 69.0 |

2015 | 5.5 | 23.8 | 37.8 | 66.3 |

2016 | 5.2 | 22.8 | 36.1 | 63.8 |

2017 | 4.8 | 21.7 | 34.6 | 61.4 |

2018 | 4.4 | 20.5 | 33.3 | 58.7 |

2019 | 4.1 | 19.6 | 31.9 | 56.3 |

2020 | 3.8 | 18.6 | 30.3 | 54.2 |

2021 | 3.4 | 17.8 | 28.7 | 51.8 |

Station 406235 | Station 406250 | |||||
---|---|---|---|---|---|---|

Stationary | Non-Stationary | Change (%) | Stationary | Non-Stationary | Change (%) | |

2-year Design Flood (m ^{3}/s) | 9.5 | 3.4 | −64 | 7.6 | 4.1 | −46 |

10-year Design Flood (m ^{3}/s) | 59.2 | 17.8 | −70 | 17.6 | 13.9 | −21 |

20-year Design Flood (m ^{3}/s) | 110.7 | 28.7 | −74 | 21.9 | 18.6 | −15 |

50-year Design Flood (m ^{3}/s) | 245.5 | 51.8 | −79 | 28.1 | 25.8 | −8 |

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**MDPI and ACS Style**

Yilmaz, A.G.; Imteaz, M.A.; Shanableh, A.; Al-Ruzouq, R.; Atabay, S.; Haddad, K.
A Non-Stationarity Analysis of Annual Maximum Floods: A Case Study of Campaspe River Basin, Australia. *Water* **2023**, *15*, 3683.
https://doi.org/10.3390/w15203683

**AMA Style**

Yilmaz AG, Imteaz MA, Shanableh A, Al-Ruzouq R, Atabay S, Haddad K.
A Non-Stationarity Analysis of Annual Maximum Floods: A Case Study of Campaspe River Basin, Australia. *Water*. 2023; 15(20):3683.
https://doi.org/10.3390/w15203683

**Chicago/Turabian Style**

Yilmaz, Abdullah Gokhan, Monzur Alam Imteaz, Abdallah Shanableh, Rami Al-Ruzouq, Serter Atabay, and Khaled Haddad.
2023. "A Non-Stationarity Analysis of Annual Maximum Floods: A Case Study of Campaspe River Basin, Australia" *Water* 15, no. 20: 3683.
https://doi.org/10.3390/w15203683