# Estimation of the Peak over Threshold-Based Design Rainfall and Its Spatial Variability in the Upper Vistula River Basin, Poland

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

#### 2.2. Research Methods

#### 2.3. The POT-Based Estimation

#### 2.3.1. Selection of the POT Sample

#### 2.3.2. Testing the Independence, Randomness, and Homogeneity

#### 2.3.3. Testing the Goodness of Fit to the GPD Distribution

#### 2.3.4. Estimating the Design Rainfall Depth

#### 2.4. The AM-Based Estimation

#### 2.4.1. Testing the Homogeneity of AM

#### 2.4.2. Parameter Estimation and Verification of the Type of the Distribution

#### 2.4.3. Selection of the Best Distribution of the AM and Estimation of Design Rainfall Depths

#### 2.5. Selection of a Better Method from the POT-Based and AM-Based Methods

#### 2.6. Estimation of the Design Rainfalls for Long Return Periods and Their Confidence Intervals

#### 2.7. Associations between Distribution Characteristics and the Station’s Altitude

## 3. Results

#### 3.1. POT-Based Method

#### 3.1.1. Selection of the POT Sample and Estimation of the GPD Parameters

#### 3.1.2. Testing of Homogeneity, Independence, and Randomness

#### 3.1.3. The Anderson–Darling Test of Goodness of Fit

#### 3.1.4. The Design Rainfall Depth and Its Confidence Interval

#### 3.2. AM-Based Method

#### 3.2.1. Selection AM Sample and Testing the Homogeneity

#### 3.2.2. Verification of the Type of the Distribution

#### 3.2.3. Selection of the Best Distribution of the AM by Using the AIC Criterion

#### 3.3. Selection of a Better Method from the POT and AM Based on the Comparison between Rainfall Depths Derived from the POT and AM Series

#### 3.4. Associations between Distribution Characteristics and Station’s Altitude

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The location of the stations used in the study. 1. Raków, 2. Sandomierz, 3. Brynica, 4. Wolbrom, 5. Chrzanów, 6. Tarnów, 7. Pilzno, 8. Rzeszów-Jasionka, 9. Skoczów, 10. Stróża, 11. Nowy Sącz, 12. Ochotnica Górna, 13. Lesko, 14. Wisłok Wielki, 15. Kasprowy Wierch, 16. Cisna.

**Figure 2.**Flow diagram that visualizes the consecutive steps in the POT-based estimation (the left branch) and in the AM-based estimation (the right branch).

**Figure 7.**Design rainfall depth computed using the POT-based and AM-based methods. The circles represent the sample values, while the continuous lines are theoretical distributions: GPD (blue line, POT-based method) and the theoretical distribution that was selected by using the AIC (red line, AM-based method).

**Figure 8.**The scatterplot showing the relation between the station’s altitude and design rainfall depth for return periods of 100, 150, and 200 years. (

**a**) With the Kasprowy Wierch station (marked in red) (

**b**) Without the Kasprowy Wierch station. The blue or green area represents the confidence region around the regression line and its upper and lower bands are confidence limits of the predicted rainfall depths at the confidence level of $95\%$.

**Figure 9.**The scatterplot showing the relation between the station’s altitude and variance of the GPD. (

**a**) Including the Kasprowy Wierch station marked in red. (

**b**) Without the Kasprowy Wierch station. The blue or green area represents the confidence region around the regression line and its upper and lower bands are confidence limits of the predicted variance of rainfall depth at the confidence level of $95\%$.

No. | Station | Longitude | Latitude | Altitude [m. a.s.l.] | Mean dp ${}^{\left(1\right)}$ [mm] | Mean AM ${}^{\left(2\right)}$ [mm] |
---|---|---|---|---|---|---|

1 | Raków | 21${}^{\circ}$03′00″ | 50${}^{\circ}$41′00″ | 220 | 1.63 | 36.14 |

2 | Sandomierz | 21${}^{\circ}$42′57″ | 50${}^{\circ}$41′48″ | 217 | 1.55 | 38.44 |

3 | Brynica | 19${}^{\circ}$00′00″ | 50${}^{\circ}$28′00″ | 285 | 2.03 | 41.53 |

4 | Wolbrom | 19${}^{\circ}$45′00″ | 50${}^{\circ}$23′00″ | 370 | 2.06 | 43.03 |

5 | Chrzanów | 19${}^{\circ}$23′00″ | 50${}^{\circ}$09′00″ | 295 | 2.10 | 42.60 |

6 | Tarnów | 20${}^{\circ}$59′04″ | 50${}^{\circ}$01′48″ | 209 | 1.97 | 49.71 |

7 | Pilzno | 21${}^{\circ}$18′00″ | 49${}^{\circ}$59′00″ | 210 | 1.98 | 46.64 |

8 | Rzeszów-Jasionka | 22${}^{\circ}$02′32″ | 50${}^{\circ}$06′39″ | 200 | 1.76 | 37.79 |

9 | Skoczów | 18${}^{\circ}$47′42″ | 49${}^{\circ}$47′56″ | 295 | 2.59 | 53.84 |

10 | Stróża | 19${}^{\circ}$56′00″ | 49${}^{\circ}$48′00″ | 307 | 2.54 | 51.37 |

11 | Nowy Sącz | 20${}^{\circ}$41′21″ | 49${}^{\circ}$37′38″ | 292 | 2.01 | 45.14 |

12 | Ochotnica Górna | 20${}^{\circ}$14′00″ | 49${}^{\circ}$32′00″ | 620 | 2.32 | 50.48 |

13 | Lesko | 22${}^{\circ}$20′30″ | 49${}^{\circ}$27′59″ | 420 | 2.24 | 42.96 |

14 | Wisłok Wielki | 21${}^{\circ}$59′57″ | 49${}^{\circ}$22′44″ | 550 | 2.64 | 49.95 |

15 | Kasprowy Wierch | 19${}^{\circ}$58′55″ | 49${}^{\circ}$13′57″ | 1991 | 4.89 | 83.01 |

16 | Cisna | 22${}^{\circ}$20′00″ | 49${}^{\circ}$13′00″ | 540 | 2.98 | 52.47 |

^{(1)}Daily precipitation

^{(2)}Annual maximum.

**Table 2.**The values of the ${x}_{t}$, $\widehat{\gamma}$, and $\widehat{\sigma}$ parameters for each station.

Station | ${\mathit{x}}_{\mathit{t}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\sigma}}$ | Distribution |
---|---|---|---|---|

1. Raków | 27.5 | 0.27 | 7.39 | GPD |

2. Sandomierz | 23.0 | 0.30 | 6.83 | GPD |

3. Brynica | 25.9 | 0.29 | 7.53 | GPD |

4. Wolbrom | 31.3 | 0.27 | 8.31 | GPD |

5. Chrzanów | 27.5 | 0.00 | 10.96 | EXP |

6. Tarnów | 28.0 | 0.00 | 14.85 | EXP |

7. Pilzno | 28.8 | 0.00 | 13.30 | EXP |

8. Rzeszów-Jasionka | 27.2 | 0.00 | 9.82 | EXP |

9. Skoczów | 38.5 | 0.28 | 10.96 | GPD |

10. Stróża | 32.0 | 0.31 | 9.92 | GPD |

11. Nowy Sącz | 32.0 | 0.00 | 10.94 | EXP |

12. Ochotnica Górna | 27.5 | 0.30 | 8.24 | GPD |

13. Lesko | 38.1 | 0.18 | 6.92 | GPD |

14. Wisłok Wielki | 34.7 | 0.25 | 8.83 | GPD |

15. Kasprowy Wierch | 52.3 | 0.30 | 16.62 | GPD |

16. Cisna | 42.1 | 0.20 | 8.60 | GPD |

**Table 3.**The p-values of the test of the Kendall’s $\tau $ (column 2), the MK test (column 3), and the ADU test statistics and the critical values ADU${}_{\mathrm{crit}}$ (column 4).

Station | Kendall’s $\mathit{\tau}$ p-Value (lag = 1) | MK p-Value | ADU / ADU${}_{\mathbf{crit}}$ |
---|---|---|---|

1. Raków | 0.282 | 0.960 | 0.50/1.31 |

2. Sandomierz | 0.438 | 0.359 | 0.25/1.27 |

3. Brynica | 0.240 | 0.694 | 0.61/1.30 |

4. Wolbrom | 0.520 | 0.247 | 0.26/1.33 |

5. Chrzanów | 1.000 | 0.816 | 0.15/1.30 |

6. Tarnów | 0.425 | 0.887 | 0.10/1.30 |

7. Pilzno | 0.430 | 0.839 | 0.29/1.32 |

8. Rzeszów-Jasionka | 0.495 | 0.332 | 0.31/1.26 |

9. Skoczów | 0.054 | 0.578 | 0.27/1.30 |

10. Stróża | 0.505 | 0.218 | 0.11/1.34 |

11. Nowy Sącz | 0.179 | 0.374 | 0.14/1.31 |

12. Ochotnica Górna | 0.969 | 0.904 | 0.23/1.29 |

13. Lesko | 0.710 | 0.383 | 0.14/1.25 |

14. Wisłok Wielki | 0.332 | 0.054 | 0.19/1.35 |

15. Kasprowy Wierch | 0.715 | 0.185 | 0.27/1.31 |

16. Cisna | 0.744 | 0.121 | 0.20/1.32 |

**Table 4.**The design rainfall depths of return period T = 100 years and T = 200 years, and its $95\%$ confidence intervals.

Station | ${\mathit{P}}_{\mathit{T}=100}$ | ${\mathit{CI}}_{{\mathit{P}}_{\mathit{T}=100}}$ | ${\mathit{P}}_{\mathit{T}=200}$ | ${\mathit{CI}}_{{\mathit{P}}_{\mathit{T}=200}}$ |
---|---|---|---|---|

1. Raków | 106.36 | (82.06; 142.38) | 128.14 | (95.39; 178.56) |

2. Sandomierz | 120.95 | (94.31; 158.27) | 148.57 | (112.33; 201.01) |

3. Brynica | 132.41 | (104.02; 172.61) | 161.98 | (123.52; 218.19) |

4. Wolbrom | 124.62 | (97.63; 163.78) | 149.82 | (113.61; 204.19) |

5. Chrzanów | 88.54 | (79.27; 99.05) | 96.14 | (85.72; 107.09) |

6. Tarnów | 111.14 | (99.05; 124.66) | 121.44 | (107.85; 136.63) |

7. Pilzno | 101.50 | (89.94; 113.98) | 110.72 | (97.70; 124.78) |

8. Rzeszów-Jasionka | 79.85 | (71.22; 89.45) | 86.65 | (76.91; 97.49) |

9. Skoczów | 171.88 | (131.77; 228.23) | 209.37 | (154.99; 288.61) |

10. Stróża | 179.30 | (138.73; 238.47) | 222.26 | (166.57; 306.31) |

11. Nowy Sącz | 91.27 | (81.81; 101.73) | 98.86 | (88.19; 110.65) |

12. Ochotnica Górna | 159.67 | (127.07; 203.82) | 196.51 | (152.23; 258.19) |

13. Lesko | 91.34 | (75.56; 113.02) | 103.60 | (83.38; 132.17) |

14. Wisłok Wielki | 140.18 | (112.20; 176.44 | 167.22 | (130.12; 216.68) |

15. Kasprowy Wierch | 271.52 | (211.75; 358.95) | 333.97 | (252.44; 457.28) |

16. Cisna | 119.74 | (98.44; 147.69) | 137.96 | (110.45; 175.07) |

**Table 5.**The AIC values for the GEV, GA2, GA3, LOG, WE2, WE3, and GUM distributions. The lowest AIC values (in boxes) show the distribution functions that provide the best fit.

Station | GEV | GA2 | GA3 | LOG | WE2 | WE3 | GUM |
---|---|---|---|---|---|---|---|

1. Raków | 483.79 | 493.34 | 470.83 | 486.68 | 510.92 | 479.29 | 486.86 |

2. Sandomierz | 496.87 | 507.19 | 488.86 | 498.82 | 527.11 | 503.14 | 499.39 |

3. Brynica | 499.35 | 501.98 | 500.70 | 498.12 | 518.05 | 503.40 | 497.84 |

4. Wolbrom | 509.04 | 508.94 | 508.90 | 507.14 | 517.54 | 509.70 | 507.13 |

5. Chrzanów | 504.91 | 503.50 | 504.31 | 502.73 | 509.68 | 504.82 | 502.93 |

6. Tarnów | 533.53 | 531.91 | 533.07 | 531.44 | 538.21 | 533.43 | 531.55 |

7. Pilzno | 507.91 | 508.10 | 506.97 | 506.16 | 517.38 | 507.62 | 506.02 |

8. Rzeszów-Jasionka | 483.19 | 481.69 | 483.48 | 481.67 | 487.11 | 485.27 | 481.66 |

9. Skoczów | 541.56 | 544.23 | 542.15 | 540.52 | 555.86 | 544.30 | 540.66 |

10. Stróża | 535.30 | 545.45 | 537.62 | 538.10 | 561.31 | 540.89 | 538.87 |

11. Nowy Sącz | 494.12 | 494.73 | 491.73 | 492.74 | 504.25 | 490.52 | 492.26 |

12. Ochotnica Górna | 525.93 | 540.82 | 532.97 | 531.81 | 558.98 | 539.42 | 531.65 |

13. Lesko | 467.97 | 469.10 | 466.80 | 467.08 | 481.15 | 467.06 | 466.00 |

14. Wisłok Wielki | ne * | ne | ne | ne | ne | ne | ne |

15. Kasprowy Wierch | 602.26 | 605.08 | 576.67 | 601.08 | 616.41 | 601.40 | 602.18 |

16. Cisna | 501.26 | 504.43 | 499.48 | 501.58 | 517.20 | 499.54 | 499.92 |

**Table 6.**The scores ${d}_{1},{d}_{2}$ assigned to the stations based on the comparison between the $\mathit{MDE}$ and $\mathit{RMSE}$ from the POT and AM methods. The number 1 shows a better fit. The final score d is shown under the table.

Case 1 | Case 2 | |||||||
---|---|---|---|---|---|---|---|---|

Station | ${\mathrm{MDE}}_{\mathrm{POT}}$ | ${\mathrm{MDE}}_{\mathrm{AM}}$ | ${\mathrm{RMSE}}_{\mathrm{POT}}$ | ${\mathrm{RMSE}}_{\mathrm{AM}}$ | ${\mathrm{MDE}}_{\mathrm{POT}}$ | ${\mathrm{MDE}}_{\mathrm{AM}}$ | ${\mathrm{RMSE}}_{\mathrm{POT}}$ | ${\mathrm{RMSE}}_{\mathrm{AM}}$ |

1. Raków | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

2. Sandomierz | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

3. Brynica | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 |

4. Wolbrom | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |

5. Chrzanów | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

6. Tarnów | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

7. Pilzno | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

8. Rzeszów-J. | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |

9. Skoczów | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

10. Stróża | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

11. Nowy Sącz | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

12. Ochotnica G. | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

13. Lesko | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

14. Wisłok Wielki | - | - | - | - | - | - | 0 | - |

15. Kasprowy W. | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

16. Cisna | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |

d${}_{\mathrm{POT}}$ = 25 | d${}_{\mathrm{AM}}$ = 5 | d${}_{\mathrm{POT}}$ = 25 | d${}_{\mathrm{AM}}$ = 5 |

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## Share and Cite

**MDPI and ACS Style**

Kołodziejczyk, K.; Rutkowska, A. Estimation of the Peak over Threshold-Based Design Rainfall and Its Spatial Variability in the Upper Vistula River Basin, Poland. *Water* **2023**, *15*, 1316.
https://doi.org/10.3390/w15071316

**AMA Style**

Kołodziejczyk K, Rutkowska A. Estimation of the Peak over Threshold-Based Design Rainfall and Its Spatial Variability in the Upper Vistula River Basin, Poland. *Water*. 2023; 15(7):1316.
https://doi.org/10.3390/w15071316

**Chicago/Turabian Style**

Kołodziejczyk, Katarzyna, and Agnieszka Rutkowska. 2023. "Estimation of the Peak over Threshold-Based Design Rainfall and Its Spatial Variability in the Upper Vistula River Basin, Poland" *Water* 15, no. 7: 1316.
https://doi.org/10.3390/w15071316