# Non-Stationary Bayesian Modeling of Annual Maximum Floods in a Changing Environment and Implications for Flood Management in the Kabul River Basin, Pakistan

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}) at the Saidu Sharif in Swat valley. For annual precipitation time series, statistically insignificant trends were revealed for the whole Swat River basin. However, significant positive increasing trends of precipitation (2.18 mm year

^{−1}) were observed in the Lower Swat basin. Saidu Sharif, Mardan, and Charsada stations showed significant positive trends (increased precipitation over time) at the 5% significance level in the annual precipitation time series [20]. The results of these studies revealed the presence of trends in precipitation, and their conclusions suggest an important link between the changes exhibited in hydro-climatic variables [21].

## 2. Study Area and Data Description

#### 2.1. Study Area

^{2}. The Kabul River starts at the base of Unai pass from the Hindu Kush Mountains in Afghanistan and flows eastward, covering a distance of 700 km to drain into the Indus River, Pakistan [43]. The entire basin covers an area of 87,499 km

^{2}. The elevation in the basin varies substantially from 249 m.a.s.l to 7603 m.a.s.l. High elevation mountains are mainly located in the north. The average temperature and average precipitation vary significantly across the River basin. The average temperature is about 13 °C. Most of the precipitation occurs in the northern mountain and highlands, reported up to 1600 mm. [44].

#### 2.2. Flood Data

#### 2.3. Flood Generating Mechanism in KRB

## 3. Methods

#### 3.1. Preliminary Analysis

#### 3.1.1. Trend Analysis

#### 3.1.2. Selection of Extreme Value Distribution

#### 3.1.3. Goodness of Fit Statistics to GEV Distribution

_{0}: The data follow the specified distribution;

_{a}: The data do not follow the specified distribution.

^{2}or D, was greater than the critical value of 2.5018 and 0.18482, respectively.

#### 3.2. Model Design

- (1)
- Stationary Case: all the model parameters were considered constant.
- (2)
- Non-stationary Case: the location parameter (µ) was considered a function of time, as shown in Equation (4), while scale and shape parameters were kept constant:$$\mu \left(t\right)={\mu}_{1}t+{\mu}_{0},$$
_{1}, µ_{0}) are the regression parameters [50,57,58,59,60]. The location parameter was calculated for each study site in the stationary case and non-stationary case.

#### 3.2.1. Bayes Theorem for GEV Distribution

_{1}, y

_{2}, …, y

_{n}} be the set of n observations. According to the Bayes theorem, the probability of θ given Y (posterior) is proportional to the product of the probability of θ (prior) and the probability of Y given θ (likelihood function). Assuming the independence between the observations, Y:

_{1}, µ

_{0}, σ, ξ).

#### 3.2.2. Prior Distribution

#### 3.2.3. Parameters Estimation and Convergence Criterion

#### 3.2.4. Model Evaluation

## 4. Results and Discussion

#### 4.1. Temporal and Spatial Trends in Flood Regime

#### 4.2. Evaluation of Goodness of Fit for Annual Extreme Data of Flood

^{3}s

^{−1}) as an outlier, as per evaluation criterion. Similar to site 6, site 9 also represented two outliers in its flood series. The outlier 1918 m

^{3}s

^{−1}represents the flood of 1992, and the corresponding value of 1602 m

^{3}s

^{−1}represents the flood of 1987 for site 9.

^{3}s

^{−1}value corresponds to the flood of 2016 in the Swat River basin, the sub-basin of the KRB. The flood discharge of 37 m

^{3}s

^{−1}at Lund Khawar West (Site 22) in 1997 was also recorded as an outlier. Despite the existence of outliers, the data series belongs to the GEV distribution as per AD and K-S test statistics.

#### 4.3. Regionalization of Shape Parameter for Flash Floods Across the KRB

#### 4.4. Comparison between Stationary and Non-Stationary Bayesian Models

^{3}s

^{−1}(+34.9%) for the Adezai River at Adezai Bridge (site 5). The value of the Bayes factor was 0.0058, which was less than 1, so the non-stationary model was favored. The maximum peak flood observed at the Adezai River was 2285 m

^{3}s

^{−1}, during the historic flood of 2010 in the KRB. The non-stationary model reasonably described the historical peak flood (Figure 3). Similar behavior was observed for the other study sites, 11, 18, 21, 22, and 24 (Table 6 and Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8).

^{3}s

^{−1}(−1.19%) as compared to the non-stationary model (Figure 9, the Chitral River at Chitral). The value of the Bayes factor was 0.068, which was less than 1, so the non-stationary model was favored. This behavior was obvious for study sites 6 and 19 (Table 6, Figure 9 and Figure 10).

#### 4.5. Performance of Bayesian Models to Predict the Extreme Floods

## 5. Conclusions

- Trend analysis showed a mixture of increasing and decreasing trends at different gauges in the KRB at α = 0.05. The Chitral River, Kalpani River, Main Swat River, and Bara River basins showed significant increasing trends, and the Panjkora River basin displayed a moderate increasing trend in its annual maximum flood regime. However, the Lower Swat and Kabul sub-basins showed decreasing trends, except for the Adezai River in the Kabul sub-basin, which showed a significant increasing trend.
- The overall basin was under critical change and signals of clear non-stationarity in the flood regime were evident at various spatial scales throughout the basin.
- The presence of a significant trend and significant difference in flood estimates for 100-year flood between stationary and non-stationary FFCs were found that represent the clear violation from the so-called stationary assumption.
- The non-stationary Bayesian model was found to be reliable for the study sites that had a significant trend at α = 0.05, while the stationary model overestimated or underestimated the flood risk for these sites. On the other hand, the stationary Bayesian model performed better for the study sites for trends at α = 0.1, while the non-stationary Bayesian model overestimated or underestimated the flood risk for such sites.
- The use of informed priors on the shape parameter based on regional information improved the estimation of flood quantiles and reduced the uncertainty.
- Proper consideration should be given to identify the outliers while using Bayesian models.
- The presence of non-stationarity in the flood regime of the KRB has substantial implications for flood management and water resources development. A design with stationary assumption will cause two major concerns: under estimation or overestimation of design for structural and non-structural measures in the KRB. An event-based design may also overestimate or underestimate the risk in hydraulic design that was intended. Some previous studies in other parts of world also provided similar results [1,13,31,84,85,86].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Description of Location, SRTM-DEM (Shuttle Radar Topography Mission–digital elevation model, meters) and flow gauge stations for the Kabul River basin (KRB), Pakistan.

**Figure 3.**Comparison between non-stationary and stationary flood frequency curves (FFCs) at site 5, along with non-stationary uncertainty bounds for the year 2016.

**Figure 4.**Comparison between non-stationary and stationary FFCs at site 11, along with non-stationary uncertainty bounds for the year 2016.

**Figure 5.**Comparison between non-stationary and stationary FFCs at site 18, along with non-stationary uncertainty bounds for the year 2016.

**Figure 6.**Comparison between non-stationary and stationary FFCs at site 21, along with non-stationary uncertainty bounds for the year 2016.

**Figure 7.**Comparison between non-stationary and stationary FFCs at site 22, along with non-stationary uncertainty bounds for the year 2016.

**Figure 8.**Comparison between non-stationary and stationary FFCs at site 24, along with non-stationary uncertainty bounds for the year 2016.

**Figure 9.**Comparison between non-stationary and stationary FFCs at site 6, along with non-stationary uncertainty bounds for the year 2013.

**Figure 10.**Comparison between non-stationary and stationary FFCs at site 19, along with non-stationary uncertainty bounds for the year 2016.

**Figure 11.**Comparison between non-stationary and stationary FFCs at site 9, along with non-stationary uncertainty bounds for the year 2009.

**Figure 12.**Comparison between non-stationary and stationary FFCs at site 16, along with non-stationary uncertainty bounds for the year 2016.

**Figure 14.**Comparison between non-stationary and stationary FFCs at site 6, along with non-stationary uncertainty bounds for the year 2004.

**Figure 16.**Comparison between non-stationary and stationary FFCs at site 11, along with non-stationary uncertainty bounds for the year 2015.

**Figure 18.**Comparison between non-stationary and stationary FFCs at site 18, along with non-stationary uncertainty bounds for the year 2009.

**Figure 19.**Comparison between non-stationary and stationary FFCs at site 19, along with non-stationary uncertainty bounds for the year 2009.

**Figure 20.**Comparison between non-stationary and stationary FFCs at site 21, along with non-stationary uncertainty bounds for the year 2009.

**Figure 22.**Comparison between non-stationary and stationary FFCs at site 24, along with non-stationary uncertainty bounds for the year 2009.

Site# | Sub Basin and Flow Gauge Stations | Basin Area (km^{2}) | Coefficient of Variation (Cv) | Number of Years of Record |
---|---|---|---|---|

Kabul River Basin | 87,499 | |||

1 | Kabul River at Warsak | 0.292 | 52 (1965–2016) | |

2 | Kabul River at Nowshera | 0.433 | 55 (1962–2016) | |

3 | Shahalam River | 0.724 | 30 (1987–2016) | |

4 | Naguman River | 0.829 | 30 (1987–2016) | |

5 | Adezai River | 0.739 | 30 (1987–2016) | |

Chitral River Basin | 11,396 | |||

6 | Chitral River | 0.2 | 50 (1964–2013) | |

Panjkora River Basin | 5917 | |||

7 | Panjkora River | 0.859 | 33 (1984–2016) | |

Main Swat River Basin | 6066 | |||

8 | Swat River at Kalam | 0.2 | 59 (1961–2009) | |

9 | Swat River at Chakdara | 0.336 | 49 (1961–2009) | |

10 | Swat River at Khawazakela | 0.84 | 34 (1983–2016) | |

11 | Swat River at Ningolai | 1.425 | 31(1986–2016) | |

Lower Swat River Basin | 2685 | |||

12 | Swat River at Munda Head Works | 0.744 | 55 (1962–2016) | |

13 | Khiyali River at Charsada Road | 0.815 | 48 (1969–2016) | |

14 | Jundi Nullah at Tangi | 3.06 | 37 (1974–2011) | |

Jindi River Basin | 13 | |||

15 | Jindi River | 0.684 | 48 (1969–2016) | |

Kalpani River Basin | 2830 | |||

16 | Naranji Nullah | 0.975 | 49 (1968–2016) | |

17 | Badri Nullah | 0.893 | 45 (1966–2010) | |

18 | Kalpani River at Mardan | 1.476 | 33 (1984–2016) | |

19 | Kalpani River at Risalpur | 0.752 | 33 (1984–2016) | |

20 | Dagi Nullah | 1.01 | 33 (1984–2016) | |

21 | Bagiari Nullah | 0.917 | 30 (1987–2016) | |

22 | Lund Khawar West | 1.13 | 30 (1987–2016) | |

Bara River Basin | 3388 | |||

23 | Budni Nullah | 1.28 | 43 (1974–2016) | |

24 | Bara River at Kohat Bridge | 1.69 | 34 (1983–2016) | |

25 | Khuderzai Nullah | 1.65 | 32 (1980–2011) | |

26 | Chillah Nullah at Pabi | 1.15 | 32 (1980–2011) | |

27 | Hakim Garhi Nullah | 0.6 | 31 (1980–2010) | |

28 | Wazir Garhi Nullah | 1.69 | 30 (1981–2010) | |

29 | Muqam Nullah | 0.781 | 30 (1981–2010) |

Site # | Mann–Kendall (Test-Z) | Site # | Mann–Kendall (Test-Z) | Site # | Mann–Kendall (Test-Z) |
---|---|---|---|---|---|

1 | −1.54 | 11 | 4.78 *** | 21 | 3.28 ** |

2 | −0.35 | 12 | −0.89 | 22 | 2.83 ** |

3 | 0.41 | 13 | 1.18 | 23 | −1.28 |

4 | −2.02 * | 14 | 0.86 | 24 | 2.28 * |

5 | 2.61 ** | 15 | −0.37 | 25 | −1.19 |

6 | 2.80 ** | 16 | 1.79 ^{+} | 26 | −0.67 |

7 | 0.93 | 17 | −3.07 ** | 27 | 0.34 |

8 | −1.36 | 18 | 3.24 ** | 28 | −0.54 |

9 | 1.73 ^{+} | 19 | 2.13 * | 29 | −0.83 |

10 | −2.36 * | 20 | 0.16 |

^{+}Trend significant at α = 0.1.

**Table 3.**The goodness of fit statistics of annual maximum daily peak flow to generalized extreme value (GEV) distribution.

Site # | Gauge Stations | GEV Parameters | Anderson–Darling Test | Kolmogorov–Smirnov Test | |
---|---|---|---|---|---|

A-D Statistics | K-S Statistics | p-Value | |||

5 | Adezai River | ξ = 0.07899 σ = 454.66 µ = 521.18 | 0.6903 | 0.15394 | 0.43251 |

6 | Chitral River | ξ = 0.00307 σ = 143.37 µ = 1026.5 | 0.22503 | 0.06435 | 0.97732 |

9 | Swat River at Chakdara | ξ = 0.13247 σ = 152.1 µ = 646.8 | 0.66053 | 0.10305 | 0.59055 |

11 | Swat River at Ningolai | ξ = 0.52162 σ = 103.82 µ = 83.499 | 0.60066 | 0.1453 | 0.48501 |

16 | Naranji Nullah | ξ = 0.25789 σ = 77.168 µ = 81.939 | 0.19219 | 0.06263 | 0.98424 |

18 | Kalpani River at Mardan | ξ = 0.55205 σ = 106.9 µ = 77.204 | 1.2218 | 0.17818 | 0.21796 |

19 | Kalpani River at Risalpur | ξ = 0.20781 σ = 441.0 µ = 604.88 | 0.42201 | 0.10944 | 0.7987 |

21 | Bagiari Nullah | ξ = 0.06073 σ = 112.05 µ = 94.608 | 1.838 | 0.22761 | 0.08399 |

22 | Lund Khawar West | ξ = 0.37899 σ = 3.7993 µ = 3.164 | 0.48511 | 0.12903 | 0.7523 |

24 | Bara River at Kohat Bridge | ξ = 0.57308 σ = 16.871 µ = 9.4788 | 1.2595 | 0.15782 | 0.33006 |

Site # | Station Name | Historical Extreme (Outliers) | Observed Value | Critical Value |
---|---|---|---|---|

5 | Adezai River | 2285 | 2.449 | 2.394 |

6 | Chitral River | 1633/1603 | 2.941/2.76 | 2.576 |

9 | Swat River at Chakdara | 1918/1602 | 4.6/3.35 | 2.576 |

11 | Swat River at Ningolai | 1475 | 3.447 | 2.406 |

16 | Naranji Nullah | 850 | 4.748 | 2.576 |

18 | Kalpani River at Mardan | 1499 | 3.182 | 2.429 |

19 | Kalpani River at Risalpur | 3358 | 3.316 | 2.418 |

21 | Bagiari Nullah | 473 | 2.102 | 2.394 |

22 | Lund Khawar West | 37 | 3.235 | 2.394 |

24 | Bara River at Kohat Bridge | 331 | 4.234 | 2.44 |

Site # | 5 | 6 | 9 | 11 | 16 | 18 | 19 | 21 | 22 | 24 |
---|---|---|---|---|---|---|---|---|---|---|

5 | 1 | 0.24 | −0.04 | 0.63 | 0.35 | 0.61 | 0.14 | 0.25 | 0.39 | 0.48 |

6 | 0.24 | 1 | 0.29 | 0.11 | 0.42 | 0.33 | 0.42 | 0.37 | 0.38 | 0.41 |

9 | −0.04 | 0.29 | 1 | 0.11 | 0.11 | −0.05 | −0.22 | −0.02 | −0.17 | 0.04 |

11 | 0.63 | 0.11 | 0.11 | 1 | 0.2 | 0.59 | 0.04 | 0.49 | 0.6 | 0.21 |

16 | 0.35 | 0.42 | 0.12 | 0.2 | 1 | 0.41 | 0.47 | 0.29 | 0.32 | 0.63 |

18 | 0.61 | 0.33 | −0.05 | 0.59 | 0.41 | 1 | 0.63 | 0.53 | 0.52 | 0.54 |

19 | 0.14 | 0.42 | −0.22 | 0.04 | 0.47 | 0.63 | 1 | 0.65 | 0.64 | 0.42 |

21 | 0.25 | 0.37 | −0.02 | 0.49 | 0.29 | 0.53 | 0.65 | 1 | 0.41 | 0.2 |

22 | 0.39 | 0.38 | −0.17 | 0.6 | 0.32 | 0.52 | 0.64 | 0.41 | 1 | 0.47 |

24 | 0.48 | 0.41 | 0.04 | 0.21 | 0.63 | 0.54 | 0.42 | 0.2 | 0.47 | 1 |

**Table 6.**Comparison between 100-year flood estimates using the stationary and non-stationary Bayesian models for “at site modeling” for the KRB, Pakistan.

Site # | Station Name | Historical Extreme m^{3} s^{−1} | Stationary m^{3} s^{−1} | Non-Stationary m^{3} s^{−1} | Difference b/w Stationary & Non-Stationary m^{3} s^{−1} | Percent Difference (%) | Bayes Factor | % Difference between Preferred Model and Historical Extreme |
---|---|---|---|---|---|---|---|---|

5 | Adezai River | 2285 | 4276 | 2782 | 1494 | 34.9 | 0.0058 | 17.86 |

6 | Chitral River | 1633 | 1895 | 1918 | −23 | −1.19 | 0.068 | 14.85 |

9 | Swat River at Chakdara | 1918 | 1991 | 2686 | −695 | −25.8 | 7.06 | 3.8 |

11 | Swat River at Ningolai | 1475 | 2891 | 2528 | 363 | 12.5 | 0.0065 | 41.65 |

16 | Naranji Nullah | 850 | 1127 | 1222 | −95 | −7.7 | 9.55 | 24.6 |

18 | Kalpani River at Mardan | 1499 | 3881 | 2887 | 1054 | 27.15 | −Infinity | 48.14 |

19 | Kalpani River at Risalpur | 3358 | 4918 | 5140 | −222 | −4.31 | 0.4348 | 34.66 |

21 | Bagiari Nullah | 473 | 1666 | 819 | 847 | 50.8 | 0.0321 | 42.24 |

22 | Lund Khawar West | 37 | 76 | 51 | 25 | 32.89 | 0.11 | 27.45 |

24 | Bara River at Kohat Bridge | 331 | 686.7 | 357.5 | 330.9 | 48.18 | −Infinity | 7.2 |

Site # | Time Series Length | Extreme Event (Year) | Mann–Kendall (Test-Z) | Stationary m^{3} s^{−1} | Non-Stationary m^{3} s^{−1} | Difference between Stationary and Non-Stationary m^{3} s^{−1} | Percent Difference (%) | Bayes Factor |
---|---|---|---|---|---|---|---|---|

5 | 1987–2009 | 2010 | −0.05 | 3300 | N/A | N/A | N/A | N/A |

6 | 1964–2004 | 2005/2010 | 2.88 ** | 1701 | 1978 | 277 | 14 | 0.0054 |

9 | 1961–1991 | 1992 | 1.42 | 1746 | N/A | N/A | N/A | N/A |

11 | 1986–2015 | 2016 | 4.49 *** | 2211 | 1295 | 916 | 41.43 | 15.167 |

16 | 1968–2009 | 2010 | 1.31 | 850.8 | N/A | N/A | N/A | N/A |

18 | 1984–2009 | 2010 | 2.29 * | 1472 | 1085 | 387 | 26.3 | 0.1208 |

19 | 1984–2009 | 2010 | 2.76 ** | 4580 | 3595 | 990 | 21.6 | 0.008 |

21 | 1987–2009 | 2010 | 2.2 * | 1469 | 704 | 765 | 52 | 0.016 |

22 | 1987–1996 | 1997 | −0.28 | 30.38 | N/A | N/A | N/A | N/A |

24 | 1983–2009 | 2010 | 1.15 | 339.6 | 355 | 15.4 | 4.33 | +Infinity |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mehmood, A.; Jia, S.; Mahmood, R.; Yan, J.; Ahsan, M.
Non-Stationary Bayesian Modeling of Annual Maximum Floods in a Changing Environment and Implications for Flood Management in the Kabul River Basin, Pakistan. *Water* **2019**, *11*, 1246.
https://doi.org/10.3390/w11061246

**AMA Style**

Mehmood A, Jia S, Mahmood R, Yan J, Ahsan M.
Non-Stationary Bayesian Modeling of Annual Maximum Floods in a Changing Environment and Implications for Flood Management in the Kabul River Basin, Pakistan. *Water*. 2019; 11(6):1246.
https://doi.org/10.3390/w11061246

**Chicago/Turabian Style**

Mehmood, Asif, Shaofeng Jia, Rashid Mahmood, Jiabao Yan, and Moien Ahsan.
2019. "Non-Stationary Bayesian Modeling of Annual Maximum Floods in a Changing Environment and Implications for Flood Management in the Kabul River Basin, Pakistan" *Water* 11, no. 6: 1246.
https://doi.org/10.3390/w11061246