# Study of Nonstationary Flood Frequency Analysis in Songhua River Basin

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## Abstract

**:**

## 1. Introduction

## 2. Study Areas and Data

#### 2.1. Study Areas

^{2}. Based on the terrain and river channel characteristics of the Songhua River, three sub-basins, the Hulan River basin, the Mayi River basin, and the Tangwang River basin, have been selected as typical basins from upstream to downstream, with their different latitudes and terrain differences. The Hulan River is the first tributary of the left bank of the midstream of the Songhua River; its basin area is about 36,800 km

^{2}. The proportion of mountainous areas and hilly and plain areas is 37% and 63%, respectively [20,21,22]. The Mayi River is a lower-latitude first tributary on the right bank of the Songhua River; its basin area is about 10,600 km

^{2}, and it is a mountainous river. The proportion of mountainous areas and hilly and plain areas is 76.9% and 23.1%, respectively [23]. The Tangwang River basin is the highest-latitude first tributary among these three typical river basins, with a total area of about 20,600 km

^{2}. It has many low mountains and hills, and the mountainous area accounts for 60.1%; it is a typical mountain stream forest basin [24]. However, there is no hydrological station at the export of the basin. Chenming Station is closest to the export; this station is used to determine the boundary of the basin. The locations of typical basins and hydrological stations within the watershed are shown in Figure 1.

#### 2.2. Data

_{1}), annual maximum three-day volume (W

_{3}), annual maximum seven-day volume (W

_{7}), annual maximum one-day accumulated precipitation (P

_{1}), and annual maximum three-day accumulated precipitation (P

_{3}) of stations in the Hulan River basin, Mayi River basin, and Tangwang River basin are excerpted from the Annual Hydrological Report P. R. China, Hydrological Data of Heilongjiang River Basin by annual maximum series methods (AMS) [25,26]. The historical flood data sources are excerpted from Historical Floods in Heilongjiang Province. The historical rainstorm data corresponding to the historical flood are excerpted from the dissertation [27]. In this dissertation, Haoqi Liu constructed a precipitation simulation model for stations in five basins of the Songhua River using DNN and DRF, which were used to forecast historical precipitation, and obtained a good result. The DEM used for mapping is taken from the Elevation of Chinese Provinces (https://www.resdc.cn/data.aspx?DATAID=284 (accessed on 3 April 2023)), whose data source is SRTM3 V4.1. The range of measured flood sequences and the year of historical flood events for each station are shown in Table 1.

## 3. Materials and Methods

#### 3.1. Mann–Kendall Mutation Test

_{k}is the cumulative number of values at the time i greater than the number of values at time j. Under the assumption of the event sequence independence, the statistic UF

_{k}is defined.

_{n}, x

_{n}

_{−1},…, x

_{1}, so that UB

_{k}= −UF

_{k}(k = n, n − 1,…, 1). Repeat the above operation according to the event sequence x in reverse order x

_{1}, x

_{2},..., x

_{n}, so that UB

_{k}= −UF

_{k}(k = n, n − 1,..., 1). Draw the UF

_{k}and UB

_{k}curves, and the moment corresponding to the intersection of the two is the moment of the mutation point.

#### 3.2. Pettitt Test

_{0}is located where U

_{t}

_{0}has its maximum.

#### 3.3. Generalized Additive Models for Location, Scale, and Shape Framework

_{t}at a time t (t = 1,2,…,n) obey the probability density function f (y

_{t}|θ

^{t}), where θ

^{t}= (θ

_{t1},θ

_{t2},…,θ

_{tm}) is the distribution parameter vector corresponding to time t, m is the number of distribution parameters, and n is the number of observations. In practical application, m is generally taken as 4 at most, θ

^{t}= (θ

_{t1}, θ

_{t2}, θ

_{t3}, θ

_{t4}) = (μ

_{t}, σ

_{t}, ν

_{t}, τ

_{t}), t = 1, 2, 3, 4. Let y = (y

_{1}, y

_{2},…, y

_{n})

^{T}be a time sequence composed of independent observations y

_{n}, and g

_{k}(·) can express the monotone link functional relationship with the corresponding explanatory variables and random effect terms, and its general expression is

_{k}is a vector of length n (k = 1, 2, 3, 4), β

_{k}= (β

_{1k}, β

_{2k}, …, β

_{Ikk})

^{T}is a regression parameter vector of length I

_{k}, J

_{k}is the number of random variables, X

_{k}is a covariate matrix of n × I

_{k}, Z

_{jk}is a fixed design matrix of n × q

_{jk}, γ

_{jk}is a random variable vector of q

_{jk}dimension obeying a normal distribution, and q

_{jk}represents the dimension of random influence factors in the j-th random effect. If we do not consider the influence of random effects on the distribution parameters, i.e., for k = 1, 2, 3, 4, and J

_{k}= 0, then GAMLSS can be transformed into a saturated model:

#### 3.4. Model Evaluation and Residual Analysis

_{i}are the ordered observations, and M

_{i}are the ordered statistic medians. $\overline{r}$ and $\overline{M}$ are the means of r

_{i}and M

_{i}. Based on the length of sequences, when r > 0.980, it passes the test with a significance level of 0.05. In the 95% confidence interval, the closer the R-value is to 1, the closer the residual sequence is to obeying the standard normal distribution, and the better the model simulation will be.

## 4. Results

#### 4.1. Mutation Test for Flood Extremum Sequences

_{7}sequences with the other three flood characteristic series at Shangzhi Station of the Mayi River are quite different. This may be caused by the different flood events extracted by different flood characteristics according to the AMS method. Overall, the mutation years of the flood characteristic sequences of the hydrological stations in these three typical basins are concentrated before the 1980s [36]. Especially between 1960 and 1980, due to the impression of the construction of water conservancy projects and the expansion of farms, the flood characteristic sequence was generally nonstationary. Therefore, analyzing the nonstationary flood frequency of these sequences is feasible.

#### 4.2. FFA by Time-Covariate GAMLSS and Spatial Distribution of Optimal Theoretical Distribution

_{1}, W

_{3}, and W

_{7}of each station are selected with the minimum value of AIC as the evaluation criterion. The parameter estimation results of the optimal distribution obtained are shown in Table 4 (taking the stations in the Mayi River basin as an example; the stations in the Hulan River basin and Tangwang River basin are shown in Appendix A.1 and Appendix A.2). In these tables, cs and pb denote that the μ/σ parameter was modeled as a cubic spline or P-splines of time or precipitation [16,38]. The value 1 or 2 in parentheses denotes the degree of freedom. The generalized gamma distribution is a three-parameter distribution, the shape parameter ν (shown in Section 3.2) is a constant, and ν in the distribution is omitted in this paper. Then, according to the optimal distribution of each station, their respective residual evaluation indicators were calculated. Taking the flood sequence of the Mayi River—Lianhua station as an example, the QQ-normal diagram was drawn, and the Filliben coefficient was calculated. As shown in Figure 2.

#### 4.3. FFA by Precipitation-Covariate GAMLSS and Spatial Distribution of Optimal Theoretical Distribution

_{1}and P

_{3}in the model. The model selection still uses the AIC criterion as the evaluation criterion to calculate the theoretical distribution of the flood sequence of each station. It is worth noting that when plotting the centile curve cluster of the optimal distribution in each station, the centile curves of the flood characteristics at all stations changed with the precipitation covariates, which showed a resemblance to the change in time covariates, which was greatly affected by the maximum cumulative precipitation. As there are different correspondences between different degrees of cumulative rainfall and flood characteristics, P

_{1}corresponds better to Q and W

_{1}than P

_{3}. However, when the dependent variables are W

_{3}and W

_{7}, the precipitation parameter P

_{1}is less able to explain these two flood characteristics than P

_{3}. The AMS method was also used to select the rainfall sequences in this paper, so the fields selected for P

_{1}and P

_{3}events are not necessarily the same for each year, and P

_{3}is preferred for W

_{3}and W

_{7}.

_{1}, and not in W

_{3}and W

_{7}. It is seen that the covariate of the models Q and W

_{1}is P

_{1}, and that of W

_{3}and W

_{7}is P

_{3}, and we analyze the generation of abnormal states from the covariate point. The black circle in Figure 5a,b demonstrates that the abnormal point value of the P

_{1}sequence at Lianhua Station is larger (122.215 mm), but the corresponding flood feature sequence value is smaller (790 m

^{3}/s, 55.64 × 10

^{6}m

^{3}), and the red circle in Figure 5a,b shows that the second-largest point of the P

_{1}sequence value corresponds to a smaller flood characteristic as well. The year of occurrence of the P

_{3}maximum point in the W

_{3}and W

_{7}models is the same (1966) as the year of occurrence of the P

_{1}maximum point in the Q and W

_{1}models. However, the red circles in Figure 5c,d show that the flood characteristics corresponding to the second-largest value point of the P

_{3}sequence are much larger than those corresponding to the maximum value point, which is different from the models of Q and W

_{1}in Figure 5a,b, where the maximum point in P

_{1}leads to an abnormal downward trend in the second half of the centile curve of the precipitation-covariate GAMLSS and finally converges to this point. The abnormal point occurred in 1966, which is in the measured period of the sequences, and our data were extracted from the yearbook, which has high data reliability. Therefore, we did not process the data for the measured period before calculation, which led to the abnormal situation. This is also a reminder that variable–covariate correlations should be analyzed before model calculation and that outliers should be excluded in advance based on the historical situation to avoid the effect of abnormal data on the model calculation result. Based on the centile curve calculation results of all stations, it can be found that when the 90% quantile is selected, the deviation between the theoretical quantile frequency of the Tangwang River and the measured point frequency is the smallest, ranging from −5.0% to 3.4%. It is followed by the Mayi River basin, for which the deviation range is −7.4~2.5%. The largest deviation occurred in the Hulan River basin, which ranged from −9.6% to 5.7%. Compared with the time-covariate GAMLSS, the deviation range of the precipitation-covariate GAMLSS is slightly larger. The main reason may be the introduction of precipitation uncertainty information when the model is calculated with precipitation as a covariate. In the case of the time-covariate model, most of the data are measured. The occurred time, intensity, and flood characteristics of historical flood data have been verified by flood investigations, and the uncertainty is small. However, the situation for precipitation covariates is very different. The strength of the correspondence between precipitation data and floods, whether a single precipitation event can perfectly match each corresponding flood event, and the uncertainty of historical rainstorm data are greater. It is easy for abnormal points to affect the overall deviation. However, considering the above situation, the deviation of the simulation results is within a reasonable range. The spatial distribution of the optimal theoretical distribution of the GAMLSS preferences for different extreme flood characteristics at each station with precipitation as a covariate is shown in Figure 6.

_{3}residual point distribution and y = x straight line (obviously jump phenomenon) in Hulan River—Lianhe Station and Tangwang River—Wuying station, the residual sequences of the flood characteristics of other stations can be considered as an approximate standard normal distribution, and the fit result of GAMLSS is good. Figure 6 shows that the optimal distribution of the mainstream of the Hulan River is different from the optimal distribution of its tributaries; except for W

_{7}, the optimal distribution of all flood characteristics is the gamma distribution. Looking at the terrain and precipitation-covariate GAMLSS optimal distribution types of the three basins together, we can find that the mountainous terrain optimal distribution types are mostly of a lognormal distribution, including most of the mountainous basins or the upstream mountainous parts of the plain basins.

#### 4.4. FFA under Stationarity Assumption

_{1}, W

_{3}, and W

_{7}sequences fit by stationary GAMLSS are slightly worse than the flood characteristics fit by the P-Ⅲ distribution. It can be found from Table 5 that under the assumption of stationarity, from the results of AIC, the AIC values corresponding to the theoretical distribution of P-Ⅲ with an optimal value of 55.6% are smaller, and their fitting accuracy is similar to that of GAMLSS.

#### 4.5. An Attempt to Apply NS-FFA in the Work of River Management Scope Demarcation

_{1}, W

_{3}, W

_{7}) calculation frequency p = 10% as an example, the calculation results and accuracy analysis are shown in Table 6. The values in the columns “stationarity assumption” and “nonstationarity assumption” of Table 6 are the simulated values of the model at corresponding frequencies, and the numbers in brackets are relative errors (used to express the calculation accuracy). Table 6 shows that, for a p = 10% flood, the calculated value of GAMLSS considering time covariates under the nonstationarity assumption is closer to the measured extremum sequence, absolute value of relative error is 0.12~15.78%, and the simulation accuracy is the highest among the four models. This indicates that using this model to consider the influence of time covariates is beneficial to improving the accuracy of the flood extremum frequency calculation results. In a frequency analysis, the model does not need to divide historical floods and measured floods but only needs to directly input the time corresponding to the flood extremum into the model. Compared with the traditional method with a P-III distribution, the hydrological frequency calculation process is simple and can be used as a basis for checking the rationality of design flood calculation results. The accuracy of GAMLSS under the stationarity assumption is equivalent to that of P-III, and the calculated values are larger than the measured values. The calculated results of this model are slightly better but inferior to those of GAMLSS with time covariates. It can be seen from the simulation results of GAMLSS with precipitation covariates that the influence of precipitation on the calculation results of this model is unstable. This might be because there is an error in adding some historical precipitation interpolation, or the precipitation events might not exactly match the flood events that correspond to the extreme value of the flood. However, the accuracy of the occurrence time of historical floods is relatively higher, which leads to the calculation accuracy of the precipitation-covariate model being relatively lower than that of GAMLSS with time covariates. In addition to the p = 10% frequency case, some flood characteristics at p = 20% frequency were simulated in these models. The conclusions drawn from the time-covariate model simulation results for all basins are consistent with those drawn in the case of a frequency of p = 10%, with the exception of the Hulan River basin, where the time-covariate model simulation performs poorly. This suggests that using time-covariate NS-FFA to demarcate the scope of river management in typical basins is more reliable. Even if there are unique frequencies, as in this work, NS-FFA can also be used by selecting other suitable models with a better simulation effect first.

## 5. Conclusions and Discussion

#### 5.1. Conclusions

_{1}, W

_{3}, W

_{7}) in three typical basins of the Songhua River, and the mutation points were concentrated from 1960 to 1980 by the results of the Mann–Kendall mutation test and referring to the Pettitt test. The accuracy of the theoretical distribution in the nonstationary GAMLSS considering time and precipitation covariates is improved compared with both case models under the stationarity assumption. The optimal theoretical distribution considering time covariates fits all empirical frequencies better than the P-Ⅲ distribution, and fitting accuracy for optimal theoretical distribution considering precipitation-covariates is higher by 60% than the P-Ⅲ distribution. In the case of time-covariate models, the optimal distribution of the flood extremum in each typical basin is mainly LN (with 63.75%), followed by the WEI distribution (with 18.75%), and the optimal distributions of a few stations are the GG and GA distributions. In the case of precipitation covariates, the optimal distribution of the flood extremum in each typical basin is also mainly LN (with 57.5%). For all typical basins where covariates influence the theoretical distribution type of the flood extremum variables, the optimal theoretical distribution type considering time covariates can improve the accuracy of the flood frequency calculation results.

#### 5.2. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Optimal Distribution of Flood Extreme Value Considering Time Covariates at Stations in Hulan River Basin

Characteristic | Station | Fitted Distribution | Location Parameter β | Scale Parameter σ | AIC | SBC |

Q | Beiguan | GG | 33.199 − 0.014 × t | −0.383 | 761.0 | 769.5 |

Chenjiadian | LN | −1.199 + 0.002 × cs(t) | −0.435 | 444.4 | 455.5 | |

Lanxi | LN | 32.901 − 0.013 × cs(t,2) | −88.292 + 0.044 × cs(t,2) | 1052.7 | 1070.5 | |

Lianhe | WEI | 49.277 − 0.022 × cs(t,1) | 168.934 − 0.085 × cs(t,1) | 525.5 | 536.1 | |

Nihe | WEI | 49.610 − 0.023 × cs(t,2) | −7.317 + 0.004 × cs(t,2) | 610.8 | 627.8 | |

Ougenhe | LN | 52.896 − 0.024 × cs(t,2) | −93.045 + 0.047 × cs(t,2) | 613.2 | 628.3 | |

Qinjia | LN | 43.224 − 0.019 × pb(t,2) | −22.244 + 0.011 × pb(t,2) | 990.1 | 1004.9 | |

Qinggang | LN | 30.922 − 0.013 × cs(t,2) | −94.675 + 0.047 × cs(t,2) | 580.7 | 595.4 | |

Qing’anzhen | WEI | 63.869 − 0.030 × cs(t,1) | 45.780 − 0.023 × cs(t,1) | 472.1 | 483.2 | |

Tieli | WEI | 25.759 − 0.010 × cs(t,2) | 58.709 − 0.029 × cs(t,2) | 876.7 | 894.5 | |

W_{1} | Beiguan | GG | 29.009−0.014 × t | −0.459 | 428.4 | 436.9 |

Chenjiadian | LN | −3.504 + 0.002 × cs(t) | −0.587 | 187.7 | 198.8 | |

Lanxi | LN | 30.416 − 0.013 × cs(t,2) | −88.636 + 0.045 × cs(t,2) | 717.8 | 735.6 | |

Lianhe | WEI | 47.539 − 0.023 × cs(t,1) | 164.278 − 0.082 × cs(t,1) | 310.5 | 321.1 | |

Nihe | WEI | 46.855 − 0.023 × cs(t,2) | −7.646 + 0.004 × cs(t,2) | 288.0 | 305.1 | |

Ougenhe | LN | 48.903 − 0.023 × cs(t,2) | −93.097 + 0.047 × cs(t,2) | 366.1 | 381.2 | |

Qinjia | LN | 41.257 − 0.019 × pb(t) | −22.466 + 0.011 × pb(t) | 661.8 | 676.5 | |

Qinggang | LN | 28.487 − 0.013 × cs(t,2) | −94.912 + 0.048 × cs(t,2) | 352.9 | 367.6 | |

Qing’anzhen | WEI | 58.358 − 0.029 × cs(t,1) | 53.450 − 0.027 × cs(t,1) | 227.3 | 238.4 | |

Tieli | WEI | 20.827 − 0.009 × cs(t,2) | 59.334 − 0.030 × cs(t,2) | 519.8 | 537.5 | |

W_{3} | Beiguan | GG | −0.662 | 518.3 | 526.8 | |

Chenjiadian | LN | 1.570 + 0.0002 × cs(t) | −0.643 | 268.4 | 279.5 | |

Lanxi | LN | 30.966 − 0.013 × cs(t,2) | −88.954 + 0.045 × cs(t,2) | 860.3 | 878.1 | |

Lianhe | WEI | 49.792 − 0.023 × cs(t,1) | 161.483 − 0.081 × cs(t,1) | 398.8 | 409.3 | |

Nihe | WEI | 44.319 − 0.021 × cs(t,2) | −7.795 + 0.004 × cs(t,2) | 394.0 | 411.0 | |

Ougenhe | LN | 46.176 − 0.021 × cs(t,2) | −94.088 + 0.047 × cs(t,2) | 455.9 | 471.0 | |

Qinjia | LN | 42.423 − 0.019 × pb(t) | −23.773 + 0.012 × pb(t) | 796.4 | 811.0 | |

Qinggang | LN | 29.497 − 0.013 × cs(t,2) | −95.310 + 0.048 × cs(t,2) | 449.5 | 464.1 | |

Qing’anzhen | WEI | 56.775 − 0.027 × cs(t,1) | 55.404 − 0.028 × cs(t,1) | 310.6 | 321.7 | |

Tieli | WEI | 20.538 − 0.008 × cs(t,2) | 58.328 − 0.029 × cs(t,2) | 636.7 | 654.5 | |

W_{7} | Beiguan | GG | 14.859 − 0.0003 × bfp(t,2) | −0.803 | 572.8 | 581.2 |

Chenjiadian | GA | −0.082 + 0.001 × cs(t) | −0.666 | 317.8 | 328.9 | |

Lanxi | LN | 31.515 − 0.013 × cs(t,2) | −90.416 + 0.045 × cs(t,2) | 958.9 | 976.6 | |

Lianhe | LN | 56.087 − 0.026 × cs(t,1) | −124.62 + 0.062 × cs(t,1) | 458.0 | 468.5 | |

Nihe | WEI | 41.249 − 0.019 × cs(t,2) | −4.250 + 0.002 × cs(t,2) | 449.9 | 466.9 | |

Ougenhe | LN | 42.091 − 0.019 × cs(t,2) | −94.626 + 0.047 × cs(t,2) | 510.7 | 525.8 | |

Qinjia | LN | 43.256 − 0.019 × pb(t) | −25.114 + 0.013 × pb(t) | 883.3 | 897.7 | |

Qinggang | LN | 30.062 − 0.013 × cs(t,2) | −95.860 + 0.048 × cs(t,2) | 516.1 | 530.8 | |

Qing’anzhen | WEI | 54.494 − 0.026 × cs(t,1) | 59.978 − 0.030 × cs(t,1) | 353.7 | 364.8 | |

Tieli | WEI | 19.522 − 0.008 × cs(t,2) | 58.085 − 0.029 × cs(t,2) | 702.7 | 720.5 |

#### Appendix A.2. Optimal Distribution of Flood Extreme Value Considering Time Covariates at Stations in Tangwang River Basin

Characteristic | Station | Fitted Distribution | Location Parameter β | Scale Parameter σ | AIC | SBC |

Q | Chenming | LN | 27.311 − 0.010 × pb(t) | −0.445 | 1083.7 | 1092.9 |

Dailing | GG | 24.092 − 0.010 × cs(t) | −0.565 | 668.2 | 682.6 | |

Nancha | LN | 5.661 | −12.681 + 0.006 × cs(t) | 818.6 | 831.3 | |

Wuying | GG | 17.659 − 0.006 × t | −0.783 | 779.8 | 788.2 | |

Yichun | LN | 30.590 − 0.012 × pb(t) | −2.680 + 0.001 × pb(t) | 847.7 | 859.0 | |

W_{1} | Chenming | GA | 26.682 − 0.011 × pb(t) | −0.518 | 751.0 | 760.1 |

Dailing | GG | 14.653 − 0.006 × cs(t) | −0.551 | 341.1 | 355.6 | |

Nancha | LN | 9.929 − 0.004 × cs(t) | −0.455 | 489.9 | 502.6 | |

Wuying | GG | 14.925 − 0.006 × pb(t) | −0.805 | 475.7 | 484.1 | |

Yichun | LN | 21.341 − 0.009 × pb(t) | −3.957 + 0.002 × pb(t) | 514.7 | 525.7 | |

W_{3} | Chenming | GA | 26.819 − 0.011 × pb(t) | −0.539 | 879.4 | 888.5 |

Dailing | LN | 11.687 − 0.004 × cs(t) | −0.543 | 432.3 | 444.7 | |

Nancha | LN | 8.864 − 0.003 × cs(t) | −0.499 | 592.7 | 605.3 | |

Wuying | LN | 16.036 − 0.006 × pb(t) | 14.707 − 0.008 × pb(t) | 592.0 | 602.7 | |

Yichun | LN | 3.952 | −5.410 + 0.002 × cs(t,1) | 606.6 | 615.0 | |

W_{7} | Chenming | GA | 26.156 − 0.010 × pb(t) | −0.565 | 960.7 | 969.8 |

Dailing | LN | 11.020 − 0.004 × cs(t) | −0.576 | 493.8 | 506.1 | |

Nancha | LN | 9.494 − 0.003 × cs(t) | −0.530 | 661.4 | 674.0 | |

Wuying | LN | 4.778 | 11.547 − 0.006 × pb(t) | 662.9 | 669.8 | |

Yichun | LN | 4.479 | −7.616 + 0.004vcs(t,1) | 663.6 | 672.1 |

#### Appendix A.3. Theoretical Optimal Time-Covariate Distribution Quantile Curves in Hulan, Tangwang, and Mayi River Basins

#### Appendix A.4. Optimal Probability Distribution Results of Flood Extremum in Hulan River Basin under Stationary Condition

Characteristic | Station | Fitting Results ofP-Ⅲ Distribution | Fitting Results of Stationary GAMLSS | |||||

Cv | Cs | AIC | SBC | Fitted Distribution | AIC | SBC | ||

Q | Beiguan | 1.293 | 2.627 | 772.66 | 766.33 | GG | 766.1 | 772.4 |

Chenjiadian | 0.947 | 1.969 | 445.12 | 450.67 | LN | 445.3 | 449.0 | |

Lanxi | 1.041 | 2.111 | 1097.71 | 1104.36 | LN | 1091.2 | 1095.6 | |

Lianhe | 1.411 | 2.842 | 537.30 | 542.58 | GG | 543.1 | 548.4 | |

Nihe | 1.570 | 3.099 | 621.46 | 627.84 | LN | 623.0 | 627.2 | |

Ougenhe | 1.734 | 3.476 | 636.53 | 642.20 | GG | 641.3 | 647.0 | |

Qinjia | 0.981 | 1.999 | 1012.50 | 1019.07 | LN | 1007.2 | 1011.6 | |

Qinggang | 1.623 | 3.253 | 590.84 | 596.33 | GG | 597.1 | 602.6 | |

Qing’anzhen | 1.680 | 3.347 | 489.65 | 495.20 | GG | 490.5 | 496.1 | |

Tieli | 1.006 | 2.081 | 889.24 | 895.90 | LN | 893.6 | 898.0 | |

W_{1} | Beiguan | 1.100 | 2.119 | 439.87 | 433.54 | LN | 435.8 | 440.1 |

Chenjiadian | 1.162 | 1.682 | 189.76 | 195.31 | LN | 189.0 | 192.7 | |

Lanxi | 0.998 | 2.006 | 764.43 | 771.09 | LN | 756.3 | 760.8 | |

Lianhe | 1.853 | 3.533 | 328.04 | 333.33 | GG | 329.5 | 334.8 | |

Nihe | 1.469 | 2.502 | 298.42 | 304.80 | LN | 302.6 | 306.8 | |

Ougenhe | 1.834 | 3.512 | 400.66 | 406.34 | GG | 394.6 | 400.3 | |

Qinjia | 1.000 | 2.005 | 684.32 | 690.89 | LN | 679.0 | 683.4 | |

Qinggang | 1.568 | 3.054 | 363.42 | 368.90 | GG | 369.4 | 374.8 | |

Qing’anzhen | 2.020 | 3.543 | 252.19 | 257.74 | GG | 246.7 | 250.4 | |

Tieli | 1.046 | 2.051 | 532.83 | 539.49 | LN | 536.6 | 541.1 | |

W_{3} | Beiguan | 1.017 | 2.071 | 530.96 | 524.63 | LN | 528.0 | 532.2 |

Chenjiadian | 0.863 | 1.655 | 269.37 | 274.92 | LN | 269.5 | 273.2 | |

Lanxi | 0.995 | 2.013 | 905.35 | 912.01 | LN | 898.7 | 903.1 | |

Lianhe | 1.743 | 3.448 | 416.76 | 422.04 | GG | 419.4 | 424.7 | |

Nihe | 1.364 | 2.536 | 401.64 | 408.02 | LN | 410.3 | 414.5 | |

Ougenhe | 1.704 | 3.363 | 493.62 | 499.29 | LN | 485.1 | 488.9 | |

Qinjia | 0.952 | 1.925 | 820.32 | 826.89 | LN | 814.5 | 818.8 | |

Qinggang | 1.502 | 2.988 | 460.21 | 465.70 | GG | 466.1 | 471.6 | |

Qing’anzhen | 1.641 | 3.156 | 327.82 | 333.37 | LN | 328.5 | 332.2 | |

Tieli | 1.032 | 2.107 | 643.24 | 649.90 | LN | 653.4 | 657.8 | |

W_{7} | Beiguan | 0.946 | 2.003 | 576.47 | 582.80 | GG | 579.7 | 586.0 |

Chenjiadian | 0.891 | 1.799 | 317.78 | 323.33 | LN | 318.9 | 322.6 | |

Lanxi | 0.993 | 2.016 | 1003.49 | 1010.15 | LN | 997.9 | 1002.3 | |

Lianhe | 1.526 | 3.052 | 471.83 | 477.11 | GG | 478.6 | 483.9 | |

Nihe | 1.251 | 2.383 | 470.19 | 476.57 | LN | 467.7 | 472.0 | |

Ougenhe | 1.571 | 3.139 | 541.53 | 547.20 | LN | 539.0 | 542.8 | |

Qinjia | 0.909 | 1.843 | 907.10 | 913.67 | LN | 902.2 | 906.6 | |

Qinggang | 1.950 | 3.886 | 530.84 | 536.33 | GG | 533.8 | 539.3 | |

Qing’anzhen | 2.077 | 3.971 | 373.88 | 379.43 | LN | 369.6 | 373.3 | |

Tieli | 0.969 | 2.005 | 717.55 | 724.21 | LN | 720.9 | 725.3 |

#### Appendix A.5. Optimal Probability Distribution Results of Flood Extremum in Tangwang River Basin under Stationary Condition

Characteristic | Station | Fitting Results ofP-Ⅲ Distribution | Fitting Results of Stationary GAMLSS | |||||

Cv | Cs | AIC | SBC | Fitted Distribution | AIC | SBC | ||

Q | Chenming | 0.968 | 2.055 | 1085.47 | 1092.04 | LN | 1091.0 | 1095.4 |

Dailing | 1.363 | 2.775 | 683.76 | 689.94 | LN | 672.6 | 676.8 | |

Nancha | 1.100 | 2.310 | 813.27 | 819.60 | LN | 818.3 | 822.5 | |

Wuying | 1.850 | 3.832 | 827.67 | 833.95 | GG | 780.7 | 787.0 | |

Yichun | 0.823 | 1.717 | 852.71 | 859.04 | LN | 852.5 | 856.8 | |

W_{1} | Chenming | 0.972 | 2.052 | 752.32 | 758.89 | LN | 758.8 | 763.2 |

Dailing | 0.890 | 1.649 | 345.14 | 351.32 | LN | 343.4 | 347.5 | |

Nancha | 0.841 | 1.719 | 488.68 | 495.01 | LN | 488.8 | 493.0 | |

Wuying | 1.849 | 3.592 | 521.34 | 527.63 | GG | 476.6 | 482.9 | |

Yichun | 0.824 | 1.677 | 515.70 | 522.03 | LN | 516.2 | 520.4 | |

W_{3} | Chenming | 0.832 | 1.775 | 887.82 | 894.39 | LN | 887.1 | 891.4 |

Dailing | 0.825 | 1.681 | 434.87 | 441.05 | LN | 435.9 | 440.0 | |

Nancha | 0.791 | 1.680 | 590.60 | 596.93 | LN | 592.5 | 596.7 | |

Wuying | 1.822 | 3.708 | 646.57 | 652.85 | GG | 593.6 | 599.9 | |

Yichun | 0.758 | 1.596 | 608.17 | 614.51 | LN | 608.4 | 612.6 | |

W_{7} | Chenming | 0.810 | 1.742 | 968.66 | 975.23 | LN | 967.8 | 972.2 |

Dailing | 0.852 | 1.794 | 495.69 | 501.87 | LN | 498.1 | 502.2 | |

Nancha | 0.551 | 1.010 | 663.39 | 669.72 | GA | 659.8 | 664.0 | |

Wuying | 1.590 | 3.317 | 711.85 | 718.13 | LN | 662.8 | 667.0 | |

Yichun | 1.230 | 0.702 | 677.03 | 683.36 | LN | 665.2 | 669.4 |

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**Figure 1.**Locations and terrain of the river basins and hydrological stations. “

**·**”: 1. Lanxi Station; 2. Qinggang Station; 3. Nihe Station; 4. Qinjia Station; 5. Qing’an Town Station; 6. Ougenhe Station; 7. Tieli Station; 8. Beiguan Station; 9. Lianhe Station; 10. Chenjiadian Station; 11. Chenming Station; 12. Dailing Station; 13. Nancha Station; 14. Yichun Station; 15. Wuying Station; 16. Lianhua Station; 17. Zhonghe Station; 18. Yanshou Station; 19. Yangshu Station; 20. Shangzhi Station.

**Figure 2.**QQ-normal diagram and Filliben coefficients for optimal models of 4 flood characteristics in (

**a**–

**d**) in Lianhua Station. (

**a**) Q, (

**b**) W

_{1}, (

**c**) W

_{3}, (

**d**) W

_{7}.

**Figure 3.**Centile curves of the optimal model under time covariates for the 4 flood characteristics. (

**a**) Q, (

**b**) W

_{1}, (

**c**) W

_{3}, (

**d**) W

_{7}.

**Figure 4.**Spatial distribution diagram of optimal theoretical distributions of different flood extremum considering time covariates. (

**a**) Optimal models of Q, (

**b**) optimal models of W

_{1}, (

**c**) optimal models of W

_{3}, (

**d**) optimal models of W

_{7}.

**Figure 5.**Centile curves of the optimal models under precipitation covariate for the 4 flood characteristics. (

**a**) Q, (

**b**) W

_{1}, (

**c**) W

_{3}, (

**d**) W

_{7}.

**Figure 6.**Spatial distribution diagram of optimal theoretical distributions of different flood extremum considering precipitation covariates. (

**a**) Optimal models of Q, (

**b**) optimal models of W

_{1}, (

**c**) optimal models of W

_{3}, (

**d**) optimal models of W

_{7}.

Station | Years of Dataset | Year(s) of Historical Flood Event(s) | Station | Years of Dataset | Year(s) of Historical Flood Event(s) |
---|---|---|---|---|---|

Beiguan | 1956~2016 | / | Chenming | 1954~2016 | 1911, 1945, 1951 |

Chenjiadian | 1970~2016 | / | Dailing | 1959~2016 | / |

Lanxi | 1953~2016 | 1851, 1911, 1932, 1945 | Nancha | 1956~2016 | / |

Lianhe | 1976~2016 | 1911, 1915 | Wuying | 1957~2016 | / |

Nihe | 1957~2016 | 1911, 1932 | Yichun | 1957~2016 | 1955 |

Ougenhe | 1971~2016 | 1911, 1932, 1945 | Lianhua | 1957~2016 | 1932 |

Qinjia | 1955~2016 | 1911, 1912, 1932, 1945 | Shangzhi | 1955~2016 | 1932 |

Qinggang | 1974~2016 | 1911, 1945, 1962 | Yanshou | 1958~2016 | 1932 |

Qing’anzhen | 1972~2016 | 1911, 1932 | Yangshu | 1957~2016 | 1932 |

Tieli | 1952~2016 | 1911, 1919, 1932 | Zhonghe | 1957~2016 | 1932 |

Name | Probability Density Functions (pdf) | Parameter Link Functions |
---|---|---|

GA | $f\left(y\left|\right)separators="|">\mu ,\sigma \right)$ $y>0,\mu >0,\sigma >0$ $E\left(Y\right)=\mu ,Var\left(Y\right)=\mu \sigma $ | ${g}_{1}\left(\mu \right)=\mathit{ln}\left(\mu \right)$ ${g}_{2}\left(\sigma \right)=\mathit{ln}\left(\sigma \right)$ |

LN | $f\left(y\left|\right)separators="|">\mu ,\sigma \right)$ $y>0,-\infty <\mu <\infty ,\sigma >0$ $E\left(Y\right)=\mathit{exp}\left(\mu +\frac{{\sigma}^{2}}{2}\right)$ $Var\left(Y\right)={E}^{2}\left(Y\right)\xb7\left[\mathit{exp}\left({\sigma}^{2}\right)-1\right]$ | ${g}_{1}\left(\mu \right)=\mathit{ln}\left(\mu \right)$ ${g}_{2}\left(\sigma \right)=\mathit{ln}\left(\sigma \right)$ |

WEI | $f\left(y\left|\right)separators="|">\mu ,\sigma \right)$ $y>0,\mu >0,\sigma >0$ $E\left(Y\right)=\mu \Gamma \left(\frac{1}{\sigma}+1\right)$ $Var\left(Y\right)={\mu}^{2}\left\{\Gamma \left(\frac{2}{\sigma}+1\right)-{\left[\Gamma \left(\frac{1}{\sigma}+1\right)\right]}^{2}\right\}$ | ${g}_{1}\left(\mu \right)=\mathit{ln}\left(\mu \right)$ ${g}_{2}\left(\sigma \right)=\mathit{ln}\left(\sigma \right)$ |

GG | $f\left(y\left|\right)separators="|">\mu ,\sigma ,\nu \right)$ $0<y<\infty ,0<\mu <\infty ,\sigma >0,-\infty <\nu <\infty ,\nu \ne 0$ $E\left(Y\right)=\mu ,Var\left(Y\right)={\sigma}^{2}$ | ${g}_{1}\left(\mu \right)=\mathit{ln}\left(\mu \right)$ ${g}_{2}\left(\sigma \right)=\mathit{ln}\left(\sigma \right)$ ${g}_{3}\left(\nu \right)=\nu $ |

GU | $f\left(y\left|\right)separators="|">\mu ,\sigma \right)$ $y>0,-\infty <\mu <\infty ,\sigma >0$ $E\left(Y\right)=\mathit{exp}\left(\mu +\frac{{\sigma}^{2}}{2}\right),Var\left(Y\right)={E}^{2}\left(Y\right)\xb7\left[\mathit{exp}\left({\sigma}^{2}\right)-1\right]$ | ${g}_{1}\left(\mu \right)=\mu $ ${g}_{2}\left(\sigma \right)=\mathit{ln}\left(\sigma \right)$ |

Flood Characteristics | Hydrological Stations in Hulan River Basin | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Lanxi | Qinggang | Nihe | Qinjia | Qing’an Zhen | Ougenhe | Tieli | Beiguan | Lianhe | Chenjiadian | |

Q | 1967/1974 | 1981/1983 | 1998/1998 | 1969/1974 | 1998/1999 | 1988/1999 | 1969/1960 | 1981/1989 | 1982/1983 | 1973/2004 |

W_{1} | 1967/1967 | 1981/1983 | 1997/1998 | 1969/1974 | 1998/1999 | 1988/1999 | 1969/1974 | 1981/1990 | 1982/1983 | 1973/2004 |

W_{3} | 1967/1967 | 1979/1983 | 1998/1998 | 1969/1974 | 1998/1999 | 1988/1999 | 1966/1974 | 1981/1990 | 1982/1983 | 1973/2004 |

W_{7} | 1967/1974 | 1981/1983 | 1998/1998 | 1969/1974 | 1999/1999 | 1988/1999 | 1969/1974 | 1979/1990 | 1980/1983 | 1973/2008 |

Flood Characteristics | Hydrological Stations in Tangwang River Basin | Hydrological Stations in Mayi River Basin | ||||||||

Chenming | Dailing | Nancha | Yichun | Wuying | Lianhua | Zhonghe | Yanshou | Yangshu | Shangzhi | |

Q | 1974/1975 | 1971/1974 | 1973/1974 | 1988/1991 | 1966/1975 | 1995/1998 | 1968/1998 | 1966/1998 | 1971/1992 | 1967/1975 |

W_{1} | 1974/1975 | 1968/1974 | 1974/1974 | 1990/1991 | 1972/1975 | 1995/1998 | 1969/1967 | 1966/1998 | 1971/1992 | 1967/1975 |

W_{3} | 1974/1975 | 1964/1975 | 1974/1974 | 1990/1992 | 1972/1975 | 1994/1998 | 1968/1967 | 1966/1998 | 1971/1992 | 1966/1975 |

W_{7} | 1974/1975 | 1964/1975 | 1974/1974 | 1991/1992 | 1966/1975 | 1995/1998 | 1967/1998 | 1966/1998 | 1971/1992 | 1960/1967 |

**Table 4.**Optimal distribution of flood extremes at each station in the Mayi River basin considering the time-covariate GAMLSS and the analytic expression of the parameters.

Flood Characteristic | Station | Optimal Distribution | Location Parameter β | Scale Parameter σ | AIC | SBC |
---|---|---|---|---|---|---|

Q | Lianhua | LN | 29.955 − 0.012 × cs(t) | −0.298 | 921.7 | 934.4 |

Shangzhi | LN | 34.507 − 0.015 × pb(t) | −0.301 | 835.5 | 844.2 | |

Yanshou | LN | 33.975 − 0.014 × cs(t) | −0.277 | 860.2 | 872.8 | |

Yangshu | LN | 38.396 − 0.017 × pb(t) | −0.029 | 697.9 | 705.6 | |

zhonghe | LN | 29.962 − 0.012 × cs(t) | −0.283 | 872.8 | 885.5 | |

W_{1} | Lianhua | LN | 27.507 − 0.012 × cs(t) | −0.328 | 613.0 | 625.7 |

Shangzhi | LN | 33.475 − 0.015 × cs(t,2) | −12.730 + 0.006 × cs(t,2) | 517.7 | 534.9 | |

Yanshou | LN | 31.388 − 0.014 × cs(t) | −0.317 | 552.3 | 564.9 | |

Yangshu | LN | 34.550 − 0.017 × pb(t) | −0.061 | 381.8 | 389.6 | |

zhonghe | LN | 16.055 − 0.007 × pb(t) | −0.449 | 474.4 | 481.3 | |

W_{3} | Lianhua | LN | 28.917 − 0.012 × cs(t) | −0.361 | 733.0 | 745.6 |

Shangzhi | LN | 28.229 − 0.012 × pb(t) | −0.392 | 621.7 | 630.8 | |

Yanshou | LN | 31.122 − 0.013 × cs(t) | −0.378 | 660.2 | 672.7 | |

Yangshu | GG | 31.284 − 0.015 × cs(t,1) | −0.216 | 490.7 | 501.3 | |

zhonghe | LN | 16.899 − 0.007 × pb(t) | −0.506 | 581.6 | 588.8 | |

W_{7} | Lianhua | LN | 29.649 − 0.012 × cs(t) | −0.433 | 802.0 | 814.7 |

Shangzhi | LN | 24.923 − 0.010 × cs(t,2) | −9.153 + 0.004 × cs(t,2) | 677.7 | 694.9 | |

Yanshou | LN | 30.507 − 0.013 × cs(t) | −0.453 | 718.2 | 730.7 | |

Yangshu | GG | 32.119 − 0.015 × cs(t) | −0.391 | 541.4 | 556.2 | |

zhonghe | LN | 15.503 − 0.006 × pb(t) | −0.598 | 640.6 | 647.9 |

**Table 5.**Optimal probability distribution results of flood extremum in Mayi River basin under stationarity assumption.

Flood Characteristic | Station | Fitting Results of P-Ⅲ Distribution | Fitting Results of Stationary GAMLSS | |||||
---|---|---|---|---|---|---|---|---|

Cv | Cs | AIC | SBC | Best Fit Distribution | AIC | SBC | ||

Q | Lianhua | 1.163 | 2.379 | 926.96 | 933.29 | GG | 927.5 | 933.9 |

Shangzhi | 1.071 | 2.201 | 836.69 | 843.12 | LN | 843.3 | 847.6 | |

Yanshou | 1.111 | 2.276 | 862.79 | 869.07 | GG | 866.5 | 872.8 | |

Yangshu | 1.252 | 2.519 | 692.33 | 698.66 | LN | 702.5 | 706.7 | |

Zhonghe | 0.992 | 2.055 | 874.12 | 880.46 | LN | 875.7 | 879.9 | |

W_{1} | Lianhua | 1.229 | 2.484 | 610.81 | 617.14 | GG | 619.4 | 625.7 |

Shangzhi | 1.001 | 2.001 | 524.67 | 531.10 | LN | 525.7 | 530.0 | |

Yanshou | 1.116 | 2.246 | 554.25 | 560.53 | LN | 559.5 | 563.7 | |

Yangshu | 1.220 | 2.223 | 379.26 | 385.60 | GG | 386.0 | 392.3 | |

Zhonghe | 0.861 | 1.757 | 475.73 | 482.07 | LN | 474.7 | 478.9 | |

W_{3} | Lianhua | 1.279 | 2.617 | 721.90 | 728.24 | GG | 740.3 | 746.7 |

Shangzhi | 0.991 | 2.045 | 627.19 | 633.62 | LN | 629.5 | 633.8 | |

Yanshou | 1.088 | 2.238 | 660.12 | 666.40 | LN | 667.3 | 671.5 | |

Yangshu | 1.177 | 2.300 | 483.25 | 489.58 | GG | 494.4 | 500.8 | |

Zhonghe | 0.826 | 1.766 | 582.06 | 588.39 | LN | 582.2 | 586.4 | |

W_{7} | Lianhua | 1.195 | 2.481 | 790.90 | 797.23 | LN | 809.9 | 814.1 |

Shangzhi | 0.888 | 1.883 | 683.23 | 689.66 | LN | 684.7 | 689.0 | |

Yanshou | 0.976 | 2.045 | 723.05 | 729.34 | LN | 725.5 | 729.7 | |

Yangshu | 1.058 | 2.121 | 544.10 | 550.43 | GG | 545.6 | 552.0 | |

Zhonghe | 0.836 | 1.847 | 638.67 | 645.00 | LN | 640.9 | 645.1 |

**Table 6.**Comparison of Q, W

_{1}, W

_{3}, and W

_{7}flood extremum and simulated values in typical Songhua River basin. The measured extremum sequence consists of the measured point of flood frequency closest to the frequency quantile (10%), and the empirical frequency value corresponding to the measured extremum is in brackets. * indicates the case where the optimal distribution corresponds to the lowest relative error in four cases.

Flood Characteristic | Typical Basin | p = 10% Flood’s Extreme Value | Measured Sequence Extremum | |||
---|---|---|---|---|---|---|

Stationarity Assumption | Nonstationarity Assumption | |||||

P-Ⅲ | GAMLSS-Stationary | GAMLSS-Time | GAMLSS-Precipitation | |||

Q (m ^{3}/s) | Hulan | 2668.21 (+32.75%) | 2376.811 (+18.25%) | 1882.199 (−6.36%) * | 2641.498 (+31.42%) | 2010 (11.4%) |

Tangwang | 3694.11 (+27.38%) | 3143.928 (+8.41%) * | 2595.534 (−10.50%) | 4211.814 (+45.23%) | 2900 (10.4%) | |

Mayi | 2038.91 (+23.57%) | 1715.71 (+3.98%) | 1644.117 (−0.36%) * | 3165.404 (+91.84%) | 1650 (9.28%) | |

W_{1}(10 ^{6} m^{3}) | Hulan | 222.98 (+31.00%) | 202.676 (+19.08%) | 160.3656 (−5.78%) * | 209.260 (+22.94%) | 170.208 (11.4%) |

Tangwang | 301.4 (+37.88%) | 254.205 (+16.29%) | 195.572 (−10.53%) * | 332.363 (+52.05%) | 218.592 (10.4%) | |

Mayi | 178.74 (+36.10%) | 136.591 (+4.01%) | 131.482 (+0.12%) * | 247.202 (+88.23%) | 131.328 (9.28%) | |

W_{3}(10 ^{6} m^{3}) | Hulan | 630.26 (+27.08%) | 576.242 (+16.19%) | 453.966 (−8.46%) * | 445.035 (−10.26%) | 495.936 (11.4%) |

Tangwang | 749.19 (+19.60%) | 673.833 (+7.57%)* | 527.561 (−15.78%) | 679.182 (+8.43%) | 626.4 (10.4%) | |

Mayi | 486.91 (+44.50%) | 365.005 (+8.32%) | 348.004 (+3.28%) * | 369.042 (+9.52%) | 336.96 (9.28%) | |

W_{7}(10 ^{6} m^{3}) | Hulan | 1299.3 (+24.28%) | 1191.927 (+14.01%) | 936.618 (−10.41%) | 950.234 (−9.11%) * | 1045.44 (11.4%) |

Tangwang | 1389.76 (+26.65%) | 1249.309 (+13.85%) | 991.495 (−9.65%) * | 1258.202 (+14.66%) | 1097.366 (10.4%) | |

Mayi | 845.45 (+46.71%) | 643.986 (+11.75%) | 597.305 (+3.65%) * | 637.124 (+10.56%) | 576.288 (9.28%) |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Liu, M.; Xing, Z.; Liu, H.; Song, J.; Hou, Q.; Xu, Y.
Study of Nonstationary Flood Frequency Analysis in Songhua River Basin. *Water* **2023**, *15*, 3443.
https://doi.org/10.3390/w15193443

**AMA Style**

Wang Y, Liu M, Xing Z, Liu H, Song J, Hou Q, Xu Y.
Study of Nonstationary Flood Frequency Analysis in Songhua River Basin. *Water*. 2023; 15(19):3443.
https://doi.org/10.3390/w15193443

**Chicago/Turabian Style**

Wang, Yinan, Mingyang Liu, Zhenxiang Xing, Haoqi Liu, Jian Song, Quanying Hou, and Yuan Xu.
2023. "Study of Nonstationary Flood Frequency Analysis in Songhua River Basin" *Water* 15, no. 19: 3443.
https://doi.org/10.3390/w15193443