# Nonstationary Multivariate Hydrological Frequency Analysis in the Upper Zhanghe River Basin, China

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1. Study Area

^{2}, stretching between longitudes 112.44–114.06° E and latitudes 35.86–37.60° N, where mostly mountainous terrain with intermontane Shangdang Basin exists (Figure 1). The river system is arranged in a fan, thick branching pattern, flowing across the Taihang Mountains, with steep mountains, overlapping peaks and sparse vegetation on both sides. Only few river terrace and flood land is left as subsistence farmland resources for local residents along the river (around 133 to 200 m

^{2}/capita). At the beginning of this centenary, this region has 4.188 million population (mostly farmer); many reservoirs and irrigated districts; and a rapid industrial growth, with insufficient irrigation water (about 400 m

^{3}/capita/year) and a high industrial and agricultural output value ratio (4.29 in 2000), therefore water resources are in great demand; however, deficient and unevenly distributed in time and space. The regional annual mean precipitation is about 570.4 mm, approximately 73% of which concentrates in one or two heavy rains at late July or early August. Hence, the natural stream flow distribution disaccords with agricultural water demand, while the irrigation season is April-to-June and November, when are low water months. There are two main tributaries in the Upper Zhanghe River Basin, Zhuozhang River and Qingzhang River, each mainly gauged by one hydrological station at downstream section, the Houbi station (Shanxi) controls 11,206 km

^{2}area of Zhuozhang River Basin and the Kuangmenkou station (Hebei) 4159 km

^{2}area of Qingzhang River Basin. The Zhanghe River Upstream Authority manages the whole region through the Houbi-Kuangmenkou-Guantai Reach (HKGR in the paper), to be more exact, from the Houbi and Kuangmenkou sections on tributaries to Guantai section of mainstream. The HKGR is the major water resources consumption of the Upper Zhanghe River Basin, as the water source for 49 villages in three provinces, containing four big irrigation canals, the Red Flag Canal, Yuejin Canal, Dayuefeng Canal, and Xiaoyuefeng Canal. Complicated natural environmental conditions, severe water scarcity, upstream-downstream shared rivers and ever-growing large population drive the region into one of the most prominent areas with frequent water-related disputes and conflicts nationwide.

#### 2.2. Data and Restoring Calculation

^{2}, with longer complete observation records since 1956. There are three canals between two stations along the Zhuozhang River, whose diversion volumes were added up to the observed series at Tianqiaoduan section, to extend and restore the natural runoff at Houbi section. While for the Qingzhang River, the Xida power station, established in 1991, diverts streamflow upstream the Kuangmenkou section to generate electricity and drainages downstream in considered time horizon. The measured water amount of the Xida diversion canal was recovered to the dataset from the Kuangmenkou station.

## 3. Methods

#### 3.1. Statistic Detection Tests

#### 3.1.1. Abrupt Change Detection Tests

_{1}is the size of smaller population and n

_{2}the other, according to the sliding pointer τ, and W is the sum of ranks for smaller size sample. If the maximum absolute value of U (|U|

_{max}) > U

_{α}

_{/2}, then the ‘change’ hypothesis will be accepted, whose corresponding time is the change-point.

_{t}

_{,n},

_{0}, let |U

_{t}

_{,n}|

_{max}be k

_{0}, if ${P}_{0}=2exp\left\{-6{k}_{0}^{2}/\left({n}^{3}+{n}^{2}\right)\right\}$ ≤ 0.5, then the stationarity hypothesis will be rejected, and the corresponding time of t

_{0}is the change-point [17].

#### 3.1.2. Monotonic Trend Detection Tests

_{SRC}| < t

_{v}

_{,α/2.}

_{SRC}(Spearman rank correlation coefficient) is calculated by ${r}_{SRC}=1-\frac{6{\displaystyle \sum _{i=1}^{n}{{d}_{i}}^{2}}}{{n}^{3}-n}$, d

_{i}= R

_{i}− i, i represents the chronological order, R

_{i}the rank.

_{i}, x

_{j}are data and sgn ( ) is a sign function $\mathrm{sgn}({x}_{i}-{x}_{j})=\{\begin{array}{cc}-1& for({x}_{i}-{x}_{j})0\\ 0& for({x}_{i}-{x}_{j})=0\\ 1& for({x}_{i}-{x}_{j})0\end{array}$. For n ≥ 10, S is approximate normal distribution, and its standardized form Z is

_{α}

_{/2}.

#### 3.2. Hydrological Frequency Analysis Methods

#### 3.2.1. p-III Distribution

#### 3.2.2. Copulas

_{X}

_{,Y}(x,y) is a bivariate distribution function with marginal cumulative distribution functions F

_{X}(x) and F

_{Y}(y), a copula C can be denoted as

^{−1}is its pseudo-inverse function such that ϕ

^{−1}(∞) = 0, ϕ

^{−1}(0) = 1 [25]. Most common Archimedean copulas are Gumbel Copula, Frank Copula, Clayton Copula, etc., based on their generators (Table 1). All these three copula types have been considered in this research.

## 4. Results and Discussion

#### 4.1. Stationarity Test

_{max}(4.90) was larger than 1.96 in the MRS test and ${P}_{0}$ (3.38 × 10

^{−5}) was less than 0.5 in the Pettitt test. While even though the valid results of two tests were different at Kuangmenkou section, both test statistics appeared similar values in 1977 and 1978, the difference is less than 0.22% (0.01 out of 4.56 (|U|

_{max})) in MRS test, and 0.80% (5 out of 627 (|U

_{t}

_{,n}|

_{max})) in Pettitt test, whose ${P}_{0}$ were 1.18 × 10

^{−4}. Therefore, the population may consider varies since 1978, the research time horizon was partitioned into two sub-periods, pre-change (1956–1977) and post-change (1978–2017) periods. The test statistics of the two methods are shown in Figure 2.

_{20,0.05/2}and t

_{38,0.05/2}represent the critical value for the pre- and post-period separately, in Table 2.

#### 4.2. Runoff Decline Reason Analysis

^{3}(from 984 to 437), while for Qingzhang River the value is 286 million m

^{3}(from 550 to 264). Runoff is extremely sensitive to precipitation, secondly to that is rainfall-runoff relationship, thus the reason of runoff decreasing was discussed from these aspects. Stated another way, climate change and human activities are commonly considered as two major factors influence runoff trends, precipitation as the most sensitive climate factor to runoff, its variation implies the climate contribution; and its relationship with streamflow reflects the underlying surface conditions, which are the most striking effects from human activities.

#### 4.2.1. Precipitation Variation

#### 4.2.2. Rainfall-Runoff Relationship

^{3}, controlling over 95 percent of the valley area, supplying 36 large-sized irrigation districts. A rapidly growing population and increasing industrial and agricultural water consumption boost water withdrawal. To make matters worse, among the supplies, most of the water (about 70% to 80%) goes to agriculture with high loss ratio, more evaporation and less drainage. In addition, groundwater overdraft and coal mining increases leakage loss of surface water, especially rivers, which result in a sharp surface runoff reduction. Zeng et al. [31] separated the effects of climate change and human activities on runoff in the Zhanghe River Basin, and indicated that the climate change contributes more than human activities on annual runoff changes, while the major factor was changeable at seasonal and monthly scales.

#### 4.3. Probability Distribution Analysis

#### 4.3.1. Marginal Probability Distribution

#### 4.3.2. Joint Probability Distribution

_{C}, F

_{F}, and F

_{G}) were further analyzed, by the goodness-of-fit evaluation method, comparing with the empirical cumulative distribution F

_{emp}(x,y). As we proposed design runoff to guide future allocation, the period since 1978 was selected for its better representation. The correlation coefficient squares were all about 0.98, witnessed an overall good performance of the selected copulas, and Frank copula showed a slightly better behavior in Figure 5. Figure 6 illustrates the joint distribution in the case of Frank copula.

^{8}m

^{3}means the AR should be 90%, while copula offered even larger value at the frequency of 95% on 3.39 × 10

^{8}m

^{3}, so at this magnitude of runoff it must take the pre-arranged planning for extremely low flow at 95% rather than the 90%’s, otherwise water deficits would appear due to under-designing. Third, the joint distribution was discussed above, and how it was connected to the marginal distributions was revealed in Table 7. For instance, runoff of both tributaries at 75% DF represents an event that two tributaries upstream witness two low flows encountering at the confluence, and their joint probability was calculated to be 64.40% by Frank copula, the slip told that the situation would happen more frequently than thought traditionally. In general, each type of dry situations is more likely to happen in HKGR than sub-basins. Then take one asynchronous encounter as example, DF at Houbi section is 50% and Kuangmenkou section is 95%, their joint probability is 49.58%. This is a normal-extreme dry encounter event, in which the major question would be whether it is a relief for the worse side. The results in this study only told us that the flow after merging slightly changed by the extreme dry tributary, thus the HKGR and Zhuozhang River still have normal runoff volume to supply, while about relieving the situation of Qingzhang River, engineering allocation should be taken into account.

#### 4.4. Synchronous-Asynchronous Encounter Risk Analysis

^{8}m

^{3}(p

_{x}= 62.5%), 4.64 × 10

^{8}m

^{3}(p

_{x}= 37.5%) at Houbi section (X) and 1.63 × 10

^{8}m

^{3}(p

_{y}= 62.5%), 2.55 × 10

^{8}m

^{3}(p

_{y}= 37.5%) at Kuangmenkou section (Y). The synchronous-asynchronous encounter situations of rich-poor annual runoff of tributaries could be classified as (with unit 10

^{8}m

^{3}): Rich-rich encounter probability: p

_{rr}= P (X ≥ 4.64, Y ≥ 2.55); Rich-normal encounter probability: p

_{rn}= P (X ≥ 4.64, 1.63 ≤ Y <2.55); Rich-poor encounter probability: p

_{rp}= P (X ≥ 4.64, Y < 1.63); Normal-rich encounter probability: p

_{nr}= P (3.44 ≤ X < 4.64, Y ≥ 2.55); Normal-normal encounter probability: p

_{nn}= P (3.44 ≤ X < 4.64, 1.63 ≤ Y <2.55); Normal-poor encounter probability: p

_{np}= P (3.44 ≤ X < 4.64, Y < 1.63); Poor-rich encounter probability: p

_{pr}= P (X < 3.44, Y ≥ 2.55); Poor-normal encounter probability: p

_{pn}= P (X < 3.44, 1.63 ≤ Y < 2.55); Poor-poor encounter probability: p

_{pp}= P (X < 3.44, Y <1.63). The Clayton copula offered an extreme option in bounding case, thus was also selected, to compare with Frank copula and research the sensitivity of the choices of copulas. The contour lines of the joint cumulative distribution for annual runoff of two tributaries and their encounter situations are displayed by Frank and Clayton copulas in Figure 8, and the results are in Table 8.

_{xy}for two tributaries, p

_{xy_f}for results by Frank and p

_{xy_c}by Clayton, in Figure 8. Frank copula agreed well with Clayton in wet year (p

_{xy}≤ 37.5%), and the dot of Clayton started to leave Frank’s solid line ever since, especially for normal to extremely dry year (p

_{xy}≥ 50%), where the runoff values of p

_{xy_c}were obviously smaller than that of p

_{xy_f}. Table 8 well illustrates the sensitivity of choices among copulas. We knew that Clayton offered smaller runoff estimations before this results, but did not have clear picture about the consequences. In this table, the two copulas obtained totally opposite conclusion about synchronous and asynchronous proportion. Through Frank copula, there is about 60% chance that the tributaries would have normal year or suffer from floods and droughts simultaneously, and over a quarter for each disaster situation; still 40% that one tributary would complement the other in need, which was ignored in the traditional HFA by assuming that tributaries share same DF. While this proportion reached 60% by Clayton copula, and the most desirable poor-rich encounter situation accounted for 11%, overwhelming the 3.7% by Frank, which would result in unprepared water shortage problems. Additionally, the worst synchronous poor-poor encounter risk given by Clayton was less than 17%, much smaller than it should be, comparing to 25.6% by Frank, and the authority may not pay enough attention to the situation on this basis. Therefore, the under-designing the probability of extreme runoff situation by Clayton copula could lead to severe water crisis in the study area, where is under high water stress and suffers from physical water scarcity.

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Abrupt change detection for annual runoff series during 1956–2017 of two tributaries at Houbi (Hb) and Kuangmenkou (Kmk) sections, with maximum values of |U| in MRS test and |Ut,n| in Pettitt test both appearing in 1977 at Hb section, as for Kmk section, 1977 and 1978 showed similar extreme values in both tests, though with different results. (

**a**) MRS test statistic |U|, based on formula (1); (

**b**) Pettitt test statistic |Ut,n|, based on formula (2).

**Figure 3.**Variation analysis of annual area and station precipitation. (

**a**) National meteorological stations selected and their distribution, in green and red. All have contributed to annual area rainfall calculation for two sub-basins (

**b**,

**c**), and three in red were also tested to analyze trends in the Zhuozhang River basin (

**d**,

**e**); (

**b**) MRS test (|U|) for area precipitation (1956–2017) of Houbi(|U|_AP_Hb) and Kuangmenkou(|U|_AP_Kmk); (

**c**) Pettitt test (|Ut,n|) for area precipitation of two sub-basins during the same period; (

**d**) MRS test for station precipitation of Changzhi (|U|_SP_Cz 1973–2017) at south source, Yushe (|U|_SP_Ys 1957–2017) at north source, and Xiangyuan (|U|_SP_Xy 1957–2017) at the merge of south and west source of the Zhuozhang River; (

**e**) Pettitt test for station precipitation of three stations in different source regions in the Zhuozhang River basin.

**Figure 4.**The univariate p-III distribution of annual runoff subseries of two tributaries during the pre-change and post-change periods. (

**a**) Annual runoff distribution at Houbi (Hb) section; (

**b**) Annual runoff distribution at Kuangmenkou (Kmk) section.

**Figure 5.**Empirical cumulative distribution F

_{emp}(x,y) versus theoretical cumulative distribution F(x,y) for the three types of copulas, Clayton, Frank, and Gumbel.

**Figure 6.**Joint distribution in the case of Frank copula. (

**a**) Probability density distribution; (

**b**) Cumulative probability distribution.

**Figure 7.**Cumulative exceedance probability curves in the cases of Independence, Clayton, Frank and Gumbel copulas.

**Figure 8.**Joint cumulative exceedance probability curves in the cases of Clayton and Frank copulas for annual runoff of tributaries, and synchronous-asynchronous encounter probability of rich-poor annual runoff at Houbi and Kuangmenkou sections (9 districts represent different encounter situations separately, depend on two annual runoff values of each stream at frequencies of 37.5% and 62.5%). Each district amounts to the projected area of cumulative distributions.

Bivariate Copula | Generator ϕ(t) | Parameter $\mathit{\alpha}$ | CDF ^{1} ${\mathit{C}}_{\mathit{\phi}}\left(\mathit{u},\mathit{v}\right)$ |
---|---|---|---|

Clayton [26] | t^{−α} − 1 | α > 0 | ${\left({u}^{-\alpha}+{v}^{-\alpha}-1\right)}^{-1/\alpha}$ |

Frank | $-\mathrm{ln}\left(\frac{{e}^{-\alpha t}-1}{{e}^{-\alpha}-1}\right)$ | α ≠ 0 | $-\frac{1}{\alpha}\mathrm{ln}\left(1+\frac{\left({e}^{-\alpha u}-1\right)\left({e}^{-\alpha v}-1\right)}{{e}^{-\alpha}-1}\right)$ |

Gumbel | (−ln t)^{α} | α ≥ 1 | $\mathrm{exp}\{-{({\left(-\mathrm{ln}u\right)}^{\alpha}+{\left(-\mathrm{ln}v\right)}^{\alpha})}^{1/\alpha}\}$ |

^{1}CDF is the abbreviation for Cumulative Distribution Function.

**Table 2.**Monotonic trend detection for annual runoff subseries during pre-change and post-change periods.

Station | SRC Test | MK Test | Result | |||||
---|---|---|---|---|---|---|---|---|

|t_{SRC}|_pre ^{1} | t_{20,0.05/2} ^{2} | |t_{SRC}|_post | t_{38,0.05/2} | |Z|_pre | |Z|_post | Z_{0.05/2} ^{3} | ||

Houbi | 1.045 | 2.086 | 0.233 | 2.024 | 1.015 | 0.739 | 1.96 | no trend |

Kuangmenkou | 0.013 | 0.665 | 0 | 0.676 | no trend |

^{1}_pre and _post represent pre-change and post-change subseries;

^{2}critical value t

_{v}

_{,}

_{α}

_{/2};

^{3}critical value Z

_{α}

_{/2}.

Stations | Runoff Depth/mm | Mean Annual Runoff Coefficient | ||||
---|---|---|---|---|---|---|

1956–1977 | 1978–2017 | 1956–1977 | 1978–2017 | Decrease | Rate/% | |

Houbi | 88.6 | 39.3 | 0.148 | 0.075 | 0.073 | 49.0 |

Kuangmenkou | 110.5 | 53.0 | 0.188 | 0.103 | 0.085 | 45.4 |

Station | Period | Parameters | Design Annual Runoff at DF ^{1}/10^{8} m^{3} | |||||
---|---|---|---|---|---|---|---|---|

μ | σ | γ | p = 50% | p = 75% | p = 90% | p = 95% | ||

Houbi | 1956–1977 | 9.84 | 0.62 | 1.70 | 8.21 | 5.41 | 3.92 | 3.41 |

1978–2017 | 4.37 | 0.46 | 1.12 | 4.00 | 2.89 | 2.15 | 1.81 | |

Kuangmenkou | 1956–1977 | 5.50 | 0.66 | 1.98 | 4.39 | 2.91 | 2.23 | 2.03 |

1978–2017 | 2.64 | 0.71 | 2.18 | 2.03 | 1.33 | 1.05 | 0.98 |

^{1}DF is the abbreviation for Design Frequency.

Period | Type | Clayton | Frank | Gumbel | Empirical |
---|---|---|---|---|---|

1956–1977 | Kendall | 0.4712 | 0.6248 | 0.5812 | 0.6190 |

Spearman | 0.6496 | 0.8244 | 0.7695 | 0.7888 | |

1978–2017 | Kendall | 0.3474 | 0.4580 | 0.3927 | 0.4503 |

Spearman | 0.4969 | 0.6450 | 0.5535 | 0.6546 |

Exceedance Probability | Clayton | Frank | Gumbel | Independence |
---|---|---|---|---|

95% | 3.14 | 3.39 | 3.49 | 2.79 |

90% | 3.74 | 3.87 | 4.03 | 3.20 |

75% | 5.22 | 5.02 | 5.27 | 4.22 |

50% | 7.62 | 7.13 | 7.28 | 6.03 |

**Table 7.**The joint probabilities of annual runoff of two tributaries at given design frequencies by Frank copula (%).

DF of Houbi Station | DF of Kuangmenkou Station | |||
---|---|---|---|---|

50% | 75% | 90% | 95% | |

50% | 37.75 | 46.56 | 49.05 | 49.58 |

75% | 46.56 | 64.40 | 71.66 | 73.47 |

90% | 49.05 | 71.66 | 83.40 | 86.84 |

95% | 49.58 | 73.47 | 86.84 | 91.01 |

**Table 8.**The frequency analysis of synchronous-asynchronous encounter of rich-poor annual runoff of tributaries in case of Frank and Clayton copulas (%).

Copula | Synchronous Frequency | Asynchronous Frequency ^{1} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Poor-Poor | Normal-Normal | Rich-Rich | Total | Poor-Normal | Poor-Rich | Normal-Poor | Normal-Rich | Rich-Poor | Rich-Normal | Total | |

Frank | 25.61 | 8.69 | 25.61 | 59.91 | 8.15 | 3.74 | 8.15 | 8.15 | 3.74 | 8.15 | 40.09 |

Clayton | 16.98 | 6.38 | 16.98 | 40.34 | 9.31 | 11.21 | 9.31 | 9.31 | 11.21 | 9.31 | 59.66 |

^{1}Houbi v.s. Kuangmenkou.

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## Share and Cite

**MDPI and ACS Style**

Gu, H.; Yu, Z.; Li, G.; Ju, Q.
Nonstationary Multivariate Hydrological Frequency Analysis in the Upper Zhanghe River Basin, China. *Water* **2018**, *10*, 772.
https://doi.org/10.3390/w10060772

**AMA Style**

Gu H, Yu Z, Li G, Ju Q.
Nonstationary Multivariate Hydrological Frequency Analysis in the Upper Zhanghe River Basin, China. *Water*. 2018; 10(6):772.
https://doi.org/10.3390/w10060772

**Chicago/Turabian Style**

Gu, Henan, Zhongbo Yu, Guofang Li, and Qin Ju.
2018. "Nonstationary Multivariate Hydrological Frequency Analysis in the Upper Zhanghe River Basin, China" *Water* 10, no. 6: 772.
https://doi.org/10.3390/w10060772