# Regional Frequency Analysis of Extreme Dry Spells during Rainy Season in the Wei River Basin, China

Authors: Dunxian She; Jun Xia; Yanjun Zhang; Lijie Shan
Journal: Advances in Meteorology
 (2016)

---

## Abstract

Our research analyzes the regional changes of extreme dry spell, represented by the annual maximum dry spell length (noted as AMDSL) during the rainy season in the Wei River Basin (WRB) of China for 1960–2014 using the L-moments method. The mean AMDSL values increase from the west to the east of the WRB, suggesting a high dry risk in the east compared to the west in the WRB. To investigate the regional frequency more reasonably, the WRB is clustered into four homogenous subregions via theK-means method and some subjective adjustments. The goodness-of-fit test shows that the GEV, PE3, and GLO distribution can be accepted as the “best-fit” model for subregions 1 and 4, subregion 2, and subregion 3, respectively. The quantiles of AMDSL under various return levels figure out a similar spatial distribution with mean AMDSL. We also find that the dry risk in subregion 2 and subregion 4 might be higher than that in subregion 1. The relationship between ENSO events and extreme dry spell events in the rainy season with cross wavelet analysis method proves that ENSO events play a critical role in triggering extreme dry events during rainy season in the WRB.


---

## Body

## 1. Introduction

The Fifth Intergovernmental Panel on Climate Change (IPCC) report confirms a clear tendency of the probable occurrence of more extreme events in many regions around the world [1]. Global warming can possibly intensify the hydrological cycle and thus can result in changes in the precipitation amount, frequency, magnitude, and intensity [2, 3]. Therefore, studying the changing patterns of precipitation is of great importance for the evaluation of hydrological and meteorological consequences, for example, drought and flooding events, associated with global warming. In particular, as one of the costliest natural disasters through the agrosocioeconomic aspects in many areas of the world [4–6], droughts are expected to occur more frequently, caused by the predicted changes in the hydrologic cycle [7, 8]. Meanwhile, they are also supposed to largely affect the socioeconomic activities, water resources management, agriculture policies, and so forth [3].Many studies have been performed for the research of drought, such as the definition and evaluation of droughts, including the spatiotemporal structure and statistical characteristic analysis, [3, 4, 9, 10] and the multivariable probability distributions jointly considering multiple drought variables (such as duration, severity, and magnitude) simultaneously [11–13]. Knowledge about the anticipated changes in drought frequency or magnitude by means of persistent daily precipitation deficiency (i.e., dry spell of precipitation) has attracted more and more interest [8, 14–17] and is considered to be necessary for the design and application of valid and reasonable drought resistance strategies. Here, dry spell, which may provide relevant information about drought regimes, can be defined as the uninterrupted period with daily precipitation amount less than a certain threshold, followed and preceded by at least one day with precipitation amount larger than the threshold [8, 15, 18]. It is worth noting that the drought frequency analysis using dry spell in many of the previous studies is on the basis of the single-site estimations. Nevertheless, Modarres [19] emphasized the importance of studying drought frequency on a regional scale, which groups similar stations together to be homogenous region and suggested that it should be specially considered in the regional drought evaluation. To date, the regional drought frequency analysis has drawn much interest [19–22]. However, such analysis, which makes use of the dry spell, particularly the extreme dry spell reflecting the variation of extreme drought condition, still needs to be further concerned and analyzed.The Wei River Basin (WRB), located in the middle reach of the Yellow River Basin in China, has been affected by many serious droughts with long duration and high severity since the end of the 1960s. The frequently occurring drought events will lead to the difficulty for the water resources to satisfy the requirement of the socioeconomic activities, human lives, and ecosystems of the WRB. Moreover, the Guanzhong Plain in the WRB, which is regarded as a very important economic development zone in this area, plays a critical role in the social and economic development of the surrounding areas. Nevertheless, in recent years, water demand in the Guanzhong region increases greatly due to the economy and population growth. At present, the local water of Guanzhong region is difficult to supply sufficient water resources for the regional economic and social development [23]. Therefore, considering the significance of water security in the WRB, increasing the knowledge and a further investigation about the spatial distribution and frequency changes of droughts are of significant importance to generate a scientific water resources management strategy. This work can also help us improve the adaptation and mitigation measures of drought in this basin.In the current research, specifically, the annual maximum dry spell length (noted as AMDSL) is utilized to represent the extreme dry spell events and analyze the regional frequency of extreme droughts in the WRB. The main objectives of this study are to(
1
) divide the whole WRB into several homogenous subregions, which are appropriate for the regional frequency analysis; (
2
) analyze the spatial variations of extreme dry spell and estimate the potential risk of the region which will be more prone to prolonged dry spells; (
3
) reveal the spatial variation of quantiles of AMDSL by the regional frequency method via L-moments method; (
4
) study the relationship between El Niño-Southern Oscillation (ENSO) events and extreme dry spells in the WRB using the cross wavelet analysis method.The contents of this study are given below. A brief description of the WRB and used data are presented in Section2. Section 3 defines the dry spell during the rainy season and introduces the L-moments method and also the cross wavelet analysis method. Section 4 gives the results and some discussions, including the relationship between ENSO events and extreme dry spells, followed by summary in Section 5.


## 2. Study Area and Data

The Wei River is the largest tributary of the Yellow River in China. It originates from the Niaoshu Mountain in Weiyuan County, Gansu province, and flows approximately 818 km through Gansu, Ningxia, and Shaanxi provinces in China and finally runs into the Yellow River at Tongguan (Figure1) [25]. The Wei River Basin locates between 103.5°E–110.5°E and 33.5°N–37.5°N, with a drainage area of about 1.34
×
10
5 km2. Topographically, the altitude in the western and northwestern mountainous areas is high and in the Guanzhong Plain and south portion of the WRB is low [23]. The Wei River has many branches (Figure 1), including the Jing River and Beiluo River. The Jing River is considered to be the largest tributary of the Wei River with a drainage area of approximately 45400 km2 (accounting for about 34% of the whole WRB). The Beiluo River is the second largest tributary of the Wei River and covers a drainage area of about 26900 km2 (contributing to about 20% of the whole WRB) [25].Figure 1
Location of the Wei River Basin. The black lines represent the boundary of the study area, and the blue dots denote the used meteorological stations.The WRB locates in the climatic transition area between semiarid and subhumid zones. The precipitation, temperature, evaporation, and runoff display an evident variation in space and time. The long term mean annual precipitation of the WRB is about 573 mm, gradually decreasing from southeast to northwest due to the monsoonal climate [26, 27]. The annual precipitation shows large seasonal variability that the precipitation amount during the rainy season (RS, May–October) accounts for 70–85% of the annual total precipitation (see Figure 2). The precipitation deficiency, particularly during the RS, will largely influence the water supply in this area and affect the socioeconomic activities, agriculture, hydrological policies, and so forth. Therefore, studying the regional variations of extreme dry spell during the RS in this area is of great importance and interest.Figure 2
The boxplots of long term mean monthly precipitation of the WRB during 1960–2014.Daily precipitation data from 1960 to 2014 in 28 meteorological stations, which was collected from China Meteorological Administration (available online athttp://www.cma.gov.cn/), is used in this study. The location and detailed information of the used stations are given in Figure 1 and Table 1. Additionally, the Nino 3.4 Index data during 1960–2014 were employed to analyze the relationship between ENSO events and the drought occurrence. These data were downloaded from the NOAA Earth System Research Laboratory (available online at http://www.esrl.noaa.gov/psd/data/climateindices/list/).Table 1
The detailed descriptions of meteorological stations in the WRB.
Station
Province
Latitude (N)
Longitude (E)
Elevation (m)
Precipitation during RS (mm)
Mean of AMDSL (days)
Lintao
Gansu
35.37
103.87
1886.6
459.9
15.42
Huajialin
Gansu
35.38
105.02
2450.6
417.8
15.49
Dingbian
Shaanxi
37.58
107.58
1360.3
288.2
23.62
Wuqi
Shaanxi
36.92
108.17
1272.6
411.9
18.93
Haiyuan
Ningxia
36.57
105.65
1854.2
332.3
20.11
Tongxin
Ningxia
36.97
105.90
1343.9
234.1
25.18
Guyuan
Ningxia
36.00
106.27
1779.9
398.7
18.40
Huanxian
Gansu
36.57
107.30
1543.6
378.2
20.35
Yan’an
Shaanxi
36.58
109.45
957.6
471.3
19.40
Xiji
Ningxia
35.97
105.72
1916.5
361.6
17.09
Pingliang
Gansu
35.55
106.67
1346.6
437.4
18.95
Xifengzhen
Gansu
35.73
107.63
1414.9
464.9
18.36
Changwu
Shaanxi
35.20
107.80
1206.3
479.2
17.80
Luochuan
Shaanxi
35.77
109.42
1158.3
510.3
17.16
Pucheng
Shaanxi
34.95
109.58
499.2
430.7
20.16
Hancheng
Shaanxi
35.47
110.45
457.9
456.7
22.24
Minxian
Gansu
34.43
104.02
2314.6
503.3
11.85
Longxian
Shaanxi
34.90
106.83
924.2
495.5
17.93
Tianshui
Gansu
34.57
105.87
1083.4
444.6
16.04
Fengxiang
Shaanxi
34.52
107.38
781.1
505.1
18.04
Taibai
Shaanxi
34.03
107.32
1543.1
637.2
14.33
Yongshou
Shaanxi
34.70
108.15
994.6
471.8
18.16
Wugong
Shaanxi
34.25
108.22
445.8
484.7
19.56
Qindu
Shaanxi
34.24
108.43
472.8
414.9
20.53
Huaxian
Shaanxi
34.52
109.73
341.5
469.3
20.89
Foping
Shaanxi
33.52
107.98
1087.7
781.7
13.22
Shangzhou
Shaanxi
33.87
109.97
742.2
553.7
18.35
Zhen’an
Shaanxi
33.43
109.15
693.7
648.6
15.53


## 3. Methods

### 3.1. Definition of Dry Spell during Rainy Season

In this research, the period from 1 May to 31 October is considered as the rainy season (RS), and the remaining period in one year is considered as the nonrainy season (non-RS). Dry spell refers to the consecutive period with daily precipitation amount that is no more than 1 mm/day. Meanwhile, the dry spell length (DSL) can be determined as the duration of the period. Similar to the descriptions of the identification of DSL in Giraldo Osorio and García Galiano [28], we provide six examples shown in Figure 3 to illustrate the identification of DSL during the RS. Once the dry spell occurred totally out of the RS, it will be rejected (event A in Figure 3). Otherwise, the dry spell can be accepted; however, the identification of the DSL differs slightly under various situations. For eventB, which occurs in the RS, the DSL will be identified without length correlation; that is, D
S
L
B
=
L
B. For event C orD, with the beginning or termination of the dry spell not in the RS, the DSLs ought to be revised to consider only the days in the RS; that is, D
S
L
C
=
L
C
–
L
C
2 or D
S
L
D
=
L
D-L
D
1. Finally, for the dry spell like event E orF, which occurs in days of non-RS or RS of several years, the DSLE (or DSLF) can be determined as L
E-L
E
1-L
E
2 (or L
F-L
F
1-L
F
2).Figure 3
Definition of dry spell length during rainy season (RS).L
RS and L
non-RS are the length of RS and non-RS period, respectively.In our research, the regional variability of the annual maximum dry spell length during the RS (AMDSL), that is, the extreme dry spell length, will be emphasized and investigated. The AMDSL of one year in a station can be defined as the maximum length of all dry spells at that year. Particularly, it should be clarified that the AMDSLs are obtained based on the identification of DSL during RS in our study.


### 3.2. Regional Frequency Analysis Method

#### 3.2.1. Brief Introduction of L-Moments Method

In this study, the L-moments method, proposed by Hosking [29] on the basis of the probability weighted method (PWM), is employed to investigate the regional frequency analysis in the WRB. This method has been widely applied in the regional flood analysis, for example, Kumar and Chatterjee [30]; Saf [31]; Noto and La Loggia [32]; Yang et al. [33]. The details of this method can be referred to Hosking and Wallis [34]. Generally speaking, there are several steps in regional frequency analysis to be undertaken by using the L-moments method, such as [34] (
1
) identifying the homogeneous subregions and evaluating the discordancy of the stations in all subregions; (
2
) defining and evaluating the heterogeneity of the subregions; (
3
) selecting an appropriate (or optimal) statistical probability distribution as the regional frequency model by means of the goodness-of-fit test; and (
4
) estimating the quantiles and analyzing the spatial variability of variables interested. The further introduction of foregoing steps is described in the upcoming sections.


#### 3.2.2. Identification of Homogeneous Regions

The delineation and determination of homogenous or similar region, which is to group the stations contributing to the homogeneity based on particular similarity measures and homogeneity criterion, are widely believed to be the most difficult task and need several subjective assumptions [35]. In this study, the cluster analysis method with theK-means method is used to divide all the stations into several separated groups. This procedure is performed directly in the software of MATLAB. Furthermore, for a preliminary homogenous region clustered by theK-means method in this study, there may be still some stations discordant in the study area; therefore, some subjective adjustments should always be undertaken to improve the regional physical coherence and to reduce the discordance of regions with the help of the discordance test [36]. These subjective adjustments, including (
1
) moving a station or a few stations from one subregion to another, (
2
) subdividing a relative large region into two or more small regions, (
3
) merging two or more small regions to a large region, (
4
) classifying the target region by reallocating its stations to the other regions, and (
5
) excluding a station or some stations from the considered dataset, will also be taken [34].Additionally, the controlling of the suitability of the division of the subregions is also necessary after generating the homogenous regions. A useful measure is the discordancy measure (D), computed by the sample L-moment ratios for different stations. The discordancy measure can be employed to determine whether the stations in a divided region in advance are conspicuously discordant with the whole group. The formal description of the discordancy measure can be referred to Hosking and Wallis [34]. Generally, if D
i is large enough, station i will be regarded as discordant. The description of “large” relies on the total number of used stations.


#### 3.2.3. Test of Homogeneous Regions

One of the important things for the assessment of the reliability of the regional frequency analysis is the definition of the degree of homogeneity [37]. Hosking and Wallis [34] recommended a test statistic as heterogeneity measure on the basis of the theory of L-moments, which compares the regional dispersion of L-moment ratios with the mean dispersion of the L-moment ratios calculated from N
sim simulations, to evaluate whether all the stations in a region may be regarded as belonging to a homogeneous region. The heterogeneity measure (H) is calculated as(1)
H
=
V
observe
-
μ
v
,
simulate
σ
v
,
simulate
,where μ
v
,
simulate and σ
v
,
simulate represent the mean and standard deviation of the N
sim simulated values of dispersions; V
observe represents the regional dispersions computed from the observations. Further introduction of the computation of heterogeneity measure can also be found in Hosking and Wallis [34].Generally, a region can be treated to be “acceptably homogeneous” whenH
<
1, “possibly heterogeneous” when 1
≤
H
<
2, and “definitely heterogeneous” when H
≥
2.


#### 3.2.4. Selection of the Best-Fit Regional Frequency Distribution

Selection of a suitable distribution is also considered to be an important step in the process of regional frequency analysis. Hosking and Wallis [34] constructed the Z
DIST measure to guide the selection of the optimal regional distribution.The fit of a candidate statistical probability distribution can be regarded as the true underlying frequency distribution ifZ
DIST is small and sufficiently close to 0, and Hosking and Wallis [34] suggested that a distribution is reasonably accepted when Z
DIST
≤
1.64. Additionally, if more than one statistical probability distribution satisfies the measure Z
DIST
≤
1.64 in the determination of best-fit regional distribution, the one with the smallest Z
DIST can be chosen as the optimal distribution. In our research, we also use the Z
DIST value to decide the appropriate regional distribution.


#### 3.2.5. The Estimation of Quantiles

In this research, five candidate statistical probability distributions (i.e., GLO: generalized logistic distribution, GPD: generalized Pareto distribution, GEV: generalized extreme value distribution, PE3: Pearson type III distribution, and GNO: generalized normal distribution) are used. For the cumulative probability density functions, the estimation of the parameters using the L-moments method is given in Hosking and Wallis [34]. Six various return levels, that is, 2, 5, 10, 25, 50, and 100 years, are utilized here. The quantiles under different return levels can be estimated on the basis of the optimal regional distribution to reveal the variations of extreme dry spells. For detailed estimation processes, the interested reader is also referred to Hosking and Wallis [34].


### 3.3. Cross Wavelet Analysis Method

Wavelet transforms have become a useful tool for investigating local variation in time series [24] and have been explored for use in hydrological and climatic time series analysis [38, 39]. In this study, for the purposes to understand how the changes of ENSO can affect the drought changes in the WRB, we investigate the relationships between the ENSO series and AMDSL series by using the cross wavelet analysis method [38, 40]. The details of the cross wavelet method are given below.For two time seriesA andB, we can define the cross wavelet spectrum as [24](2)
W
n
A
B
S
=
W
n
A
S
W
n
B
∗
S
,where W
n
A
S and W
n
B
(
S
) denote the wavelet transforms and W
n
B
∗
(
S
) is the complex conjugate of W
n
B
(
S
). Further, we can define the cross wavelet power as W
n
A
B
S.The theoretical probability distribution of cross wavelet power of background spectra of theP
k
A and P
k
B time series can be given as [24, 39](3)
D
W
n
A
B
S
σ
A
σ
B
<
p
=
Z
v
p
v
P
k
A
P
k
B
,where Z
v
(
p
) is the confidence level related to the probability p for a probability density function defined by the square root of the product of two χ
2 distributions.


## 3.1. Definition of Dry Spell during Rainy Season

In this research, the period from 1 May to 31 October is considered as the rainy season (RS), and the remaining period in one year is considered as the nonrainy season (non-RS). Dry spell refers to the consecutive period with daily precipitation amount that is no more than 1 mm/day. Meanwhile, the dry spell length (DSL) can be determined as the duration of the period. Similar to the descriptions of the identification of DSL in Giraldo Osorio and García Galiano [28], we provide six examples shown in Figure 3 to illustrate the identification of DSL during the RS. Once the dry spell occurred totally out of the RS, it will be rejected (event A in Figure 3). Otherwise, the dry spell can be accepted; however, the identification of the DSL differs slightly under various situations. For eventB, which occurs in the RS, the DSL will be identified without length correlation; that is, D
S
L
B
=
L
B. For event C orD, with the beginning or termination of the dry spell not in the RS, the DSLs ought to be revised to consider only the days in the RS; that is, D
S
L
C
=
L
C
–
L
C
2 or D
S
L
D
=
L
D-L
D
1. Finally, for the dry spell like event E orF, which occurs in days of non-RS or RS of several years, the DSLE (or DSLF) can be determined as L
E-L
E
1-L
E
2 (or L
F-L
F
1-L
F
2).Figure 3
Definition of dry spell length during rainy season (RS).L
RS and L
non-RS are the length of RS and non-RS period, respectively.In our research, the regional variability of the annual maximum dry spell length during the RS (AMDSL), that is, the extreme dry spell length, will be emphasized and investigated. The AMDSL of one year in a station can be defined as the maximum length of all dry spells at that year. Particularly, it should be clarified that the AMDSLs are obtained based on the identification of DSL during RS in our study.


## 3.2. Regional Frequency Analysis Method

### 3.2.1. Brief Introduction of L-Moments Method

In this study, the L-moments method, proposed by Hosking [29] on the basis of the probability weighted method (PWM), is employed to investigate the regional frequency analysis in the WRB. This method has been widely applied in the regional flood analysis, for example, Kumar and Chatterjee [30]; Saf [31]; Noto and La Loggia [32]; Yang et al. [33]. The details of this method can be referred to Hosking and Wallis [34]. Generally speaking, there are several steps in regional frequency analysis to be undertaken by using the L-moments method, such as [34] (
1
) identifying the homogeneous subregions and evaluating the discordancy of the stations in all subregions; (
2
) defining and evaluating the heterogeneity of the subregions; (
3
) selecting an appropriate (or optimal) statistical probability distribution as the regional frequency model by means of the goodness-of-fit test; and (
4
) estimating the quantiles and analyzing the spatial variability of variables interested. The further introduction of foregoing steps is described in the upcoming sections.


### 3.2.2. Identification of Homogeneous Regions

The delineation and determination of homogenous or similar region, which is to group the stations contributing to the homogeneity based on particular similarity measures and homogeneity criterion, are widely believed to be the most difficult task and need several subjective assumptions [35]. In this study, the cluster analysis method with theK-means method is used to divide all the stations into several separated groups. This procedure is performed directly in the software of MATLAB. Furthermore, for a preliminary homogenous region clustered by theK-means method in this study, there may be still some stations discordant in the study area; therefore, some subjective adjustments should always be undertaken to improve the regional physical coherence and to reduce the discordance of regions with the help of the discordance test [36]. These subjective adjustments, including (
1
) moving a station or a few stations from one subregion to another, (
2
) subdividing a relative large region into two or more small regions, (
3
) merging two or more small regions to a large region, (
4
) classifying the target region by reallocating its stations to the other regions, and (
5
) excluding a station or some stations from the considered dataset, will also be taken [34].Additionally, the controlling of the suitability of the division of the subregions is also necessary after generating the homogenous regions. A useful measure is the discordancy measure (D), computed by the sample L-moment ratios for different stations. The discordancy measure can be employed to determine whether the stations in a divided region in advance are conspicuously discordant with the whole group. The formal description of the discordancy measure can be referred to Hosking and Wallis [34]. Generally, if D
i is large enough, station i will be regarded as discordant. The description of “large” relies on the total number of used stations.


### 3.2.3. Test of Homogeneous Regions

One of the important things for the assessment of the reliability of the regional frequency analysis is the definition of the degree of homogeneity [37]. Hosking and Wallis [34] recommended a test statistic as heterogeneity measure on the basis of the theory of L-moments, which compares the regional dispersion of L-moment ratios with the mean dispersion of the L-moment ratios calculated from N
sim simulations, to evaluate whether all the stations in a region may be regarded as belonging to a homogeneous region. The heterogeneity measure (H) is calculated as(1)
H
=
V
observe
-
μ
v
,
simulate
σ
v
,
simulate
,where μ
v
,
simulate and σ
v
,
simulate represent the mean and standard deviation of the N
sim simulated values of dispersions; V
observe represents the regional dispersions computed from the observations. Further introduction of the computation of heterogeneity measure can also be found in Hosking and Wallis [34].Generally, a region can be treated to be “acceptably homogeneous” whenH
<
1, “possibly heterogeneous” when 1
≤
H
<
2, and “definitely heterogeneous” when H
≥
2.


### 3.2.4. Selection of the Best-Fit Regional Frequency Distribution

Selection of a suitable distribution is also considered to be an important step in the process of regional frequency analysis. Hosking and Wallis [34] constructed the Z
DIST measure to guide the selection of the optimal regional distribution.The fit of a candidate statistical probability distribution can be regarded as the true underlying frequency distribution ifZ
DIST is small and sufficiently close to 0, and Hosking and Wallis [34] suggested that a distribution is reasonably accepted when Z
DIST
≤
1.64. Additionally, if more than one statistical probability distribution satisfies the measure Z
DIST
≤
1.64 in the determination of best-fit regional distribution, the one with the smallest Z
DIST can be chosen as the optimal distribution. In our research, we also use the Z
DIST value to decide the appropriate regional distribution.


### 3.2.5. The Estimation of Quantiles

In this research, five candidate statistical probability distributions (i.e., GLO: generalized logistic distribution, GPD: generalized Pareto distribution, GEV: generalized extreme value distribution, PE3: Pearson type III distribution, and GNO: generalized normal distribution) are used. For the cumulative probability density functions, the estimation of the parameters using the L-moments method is given in Hosking and Wallis [34]. Six various return levels, that is, 2, 5, 10, 25, 50, and 100 years, are utilized here. The quantiles under different return levels can be estimated on the basis of the optimal regional distribution to reveal the variations of extreme dry spells. For detailed estimation processes, the interested reader is also referred to Hosking and Wallis [34].


## 3.2.1. Brief Introduction of L-Moments Method

In this study, the L-moments method, proposed by Hosking [29] on the basis of the probability weighted method (PWM), is employed to investigate the regional frequency analysis in the WRB. This method has been widely applied in the regional flood analysis, for example, Kumar and Chatterjee [30]; Saf [31]; Noto and La Loggia [32]; Yang et al. [33]. The details of this method can be referred to Hosking and Wallis [34]. Generally speaking, there are several steps in regional frequency analysis to be undertaken by using the L-moments method, such as [34] (
1
) identifying the homogeneous subregions and evaluating the discordancy of the stations in all subregions; (
2
) defining and evaluating the heterogeneity of the subregions; (
3
) selecting an appropriate (or optimal) statistical probability distribution as the regional frequency model by means of the goodness-of-fit test; and (
4
) estimating the quantiles and analyzing the spatial variability of variables interested. The further introduction of foregoing steps is described in the upcoming sections.


## 3.2.2. Identification of Homogeneous Regions

The delineation and determination of homogenous or similar region, which is to group the stations contributing to the homogeneity based on particular similarity measures and homogeneity criterion, are widely believed to be the most difficult task and need several subjective assumptions [35]. In this study, the cluster analysis method with theK-means method is used to divide all the stations into several separated groups. This procedure is performed directly in the software of MATLAB. Furthermore, for a preliminary homogenous region clustered by theK-means method in this study, there may be still some stations discordant in the study area; therefore, some subjective adjustments should always be undertaken to improve the regional physical coherence and to reduce the discordance of regions with the help of the discordance test [36]. These subjective adjustments, including (
1
) moving a station or a few stations from one subregion to another, (
2
) subdividing a relative large region into two or more small regions, (
3
) merging two or more small regions to a large region, (
4
) classifying the target region by reallocating its stations to the other regions, and (
5
) excluding a station or some stations from the considered dataset, will also be taken [34].Additionally, the controlling of the suitability of the division of the subregions is also necessary after generating the homogenous regions. A useful measure is the discordancy measure (D), computed by the sample L-moment ratios for different stations. The discordancy measure can be employed to determine whether the stations in a divided region in advance are conspicuously discordant with the whole group. The formal description of the discordancy measure can be referred to Hosking and Wallis [34]. Generally, if D
i is large enough, station i will be regarded as discordant. The description of “large” relies on the total number of used stations.


## 3.2.3. Test of Homogeneous Regions

One of the important things for the assessment of the reliability of the regional frequency analysis is the definition of the degree of homogeneity [37]. Hosking and Wallis [34] recommended a test statistic as heterogeneity measure on the basis of the theory of L-moments, which compares the regional dispersion of L-moment ratios with the mean dispersion of the L-moment ratios calculated from N
sim simulations, to evaluate whether all the stations in a region may be regarded as belonging to a homogeneous region. The heterogeneity measure (H) is calculated as(1)
H
=
V
observe
-
μ
v
,
simulate
σ
v
,
simulate
,where μ
v
,
simulate and σ
v
,
simulate represent the mean and standard deviation of the N
sim simulated values of dispersions; V
observe represents the regional dispersions computed from the observations. Further introduction of the computation of heterogeneity measure can also be found in Hosking and Wallis [34].Generally, a region can be treated to be “acceptably homogeneous” whenH
<
1, “possibly heterogeneous” when 1
≤
H
<
2, and “definitely heterogeneous” when H
≥
2.


## 3.2.4. Selection of the Best-Fit Regional Frequency Distribution

Selection of a suitable distribution is also considered to be an important step in the process of regional frequency analysis. Hosking and Wallis [34] constructed the Z
DIST measure to guide the selection of the optimal regional distribution.The fit of a candidate statistical probability distribution can be regarded as the true underlying frequency distribution ifZ
DIST is small and sufficiently close to 0, and Hosking and Wallis [34] suggested that a distribution is reasonably accepted when Z
DIST
≤
1.64. Additionally, if more than one statistical probability distribution satisfies the measure Z
DIST
≤
1.64 in the determination of best-fit regional distribution, the one with the smallest Z
DIST can be chosen as the optimal distribution. In our research, we also use the Z
DIST value to decide the appropriate regional distribution.


## 3.2.5. The Estimation of Quantiles

In this research, five candidate statistical probability distributions (i.e., GLO: generalized logistic distribution, GPD: generalized Pareto distribution, GEV: generalized extreme value distribution, PE3: Pearson type III distribution, and GNO: generalized normal distribution) are used. For the cumulative probability density functions, the estimation of the parameters using the L-moments method is given in Hosking and Wallis [34]. Six various return levels, that is, 2, 5, 10, 25, 50, and 100 years, are utilized here. The quantiles under different return levels can be estimated on the basis of the optimal regional distribution to reveal the variations of extreme dry spells. For detailed estimation processes, the interested reader is also referred to Hosking and Wallis [34].


## 3.3. Cross Wavelet Analysis Method

Wavelet transforms have become a useful tool for investigating local variation in time series [24] and have been explored for use in hydrological and climatic time series analysis [38, 39]. In this study, for the purposes to understand how the changes of ENSO can affect the drought changes in the WRB, we investigate the relationships between the ENSO series and AMDSL series by using the cross wavelet analysis method [38, 40]. The details of the cross wavelet method are given below.For two time seriesA andB, we can define the cross wavelet spectrum as [24](2)
W
n
A
B
S
=
W
n
A
S
W
n
B
∗
S
,where W
n
A
S and W
n
B
(
S
) denote the wavelet transforms and W
n
B
∗
(
S
) is the complex conjugate of W
n
B
(
S
). Further, we can define the cross wavelet power as W
n
A
B
S.The theoretical probability distribution of cross wavelet power of background spectra of theP
k
A and P
k
B time series can be given as [24, 39](3)
D
W
n
A
B
S
σ
A
σ
B
<
p
=
Z
v
p
v
P
k
A
P
k
B
,where Z
v
(
p
) is the confidence level related to the probability p for a probability density function defined by the square root of the product of two χ
2 distributions.


## 4. Results and Discussions

### 4.1. The Importance of Studying the Extreme Dry Spell during RS

In order to explain the importance and significance of studying the dry spell during RS, we firstly assess the proportion of the longest dry spell (LDS) in every year that occurred in RS (noted asp
1
×
100%), which can be computed as the total number of years exhibiting the LDS in RS divided by the total number of years in that station. For example, if a station contains a time period of 50 years in which the LDS occurs in RS during 10 of those years, the proportion of the LDS in RS can be determined as p
1 = 10/50 or 20%. In addition, the proportion of the LDS in every year which occurs in the non-RS can be accordingly determined as (1
-
p
1) × 100%. The results of the proportions for the LDS in RS and non-RS of all stations are concluded in Figure 4. It can be seen that, in the majority of stations, the LDS will occur during the non-RS period. The maximum proportion of the LDS in RS is only 33% which is observed in Tongxin station. However, because the total precipitation amount during the non-RS (accounting for about 15% of the annual total precipitation amount) is quite smaller compared to that during RS (accounting for about 85% of the annual total precipitation amount) in our study area (see Figure 2), the occurrence of prolonged dry period with daily precipitation lower than a certain threshold during the non-RS will have smaller impact when the dry spell with similar severity occurred during the RS. Particularly, as most of the total agricultural area in the region is rain-fed, the precipitation variability during the RS is especially important for the agricultural sector. On the other hand, since the RS is also the crops growth season in China, the persistence precipitation deficiency during this period will induce adversely impact agricultural yields, water resources, ecosystems, and human systems. Therefore, we should pay more attention to the LDS during the RS. The definition of the characteristics of the LDS, that is, AMDSL, can be referred to Section 3.1.Figure 4
The proportion of the longest dry spell occurred during RS and non-RS during 1960–2014 in the WRB.


### 4.2. Identification of Homogeneous Regions of the WRB

Six types of attributes for the stations, including latitude, longitude, elevation, total precipitation during RS period, and the mean AMDSL (see Table1), are chosen to determine the possible homogenous subregions via theK-means method in hierarchical clustering method. Figure 5 maps the division of the stations into several groups together with their boundaries. As introduced in Section 3.2, several subjective adjustments are performed to ensure the clusters are more appropriate. Finally, the whole study can be divided into four subregions (see Figure 5). Subregion 1 mainly belongs to the headwater of the WRB with a high elevation, subregion 2 is almost located in the northern part of the WRB with a small total annual precipitation amount, which is also the headwater of the Jing River Basin and the Beiluo River Basin (see Figure 1), subregion 3 is mostly distributed in the middle reach of the WRB, and subregion 4 is located in the lower reach of the WRB with a low elevation and also a high total annual precipitation amount. It is worth mentioning that different from the way to divide the WRB into four subbasins (see Figure 1) according to the topographic feature and hydrological and climatic conditions by the Ministry of Water Resources, China, we identify the subregion mainly on the basis of the attributes related to the attributes of dry spell; therefore, the division of the WRB identified above may be more appropriate for the performance of regional frequency study of AMDSL. Furthermore, the test of the homogeneity of the four subregions will be presented in Section 4.4.Figure 5
The division of four subregions of the WRB using theK-means method in the hierarchical cluster analysis.


### 4.3. Spatial Variation of the AMDSL

The AMDSL of every year in each station is computed and the mean AMDSL (MAMDSL) in all years during 1960–2014 can be obtained. Then, the MAMDSL in all stations is interpolated into the whole region using the software of ArcGIS (Figure6). The spatial variation of MAMDSL shows that the MAMDSL increases from the west to the east in the WRB, which is similar to the spatial structure of long term mean annual precipitation in this area. Specifically, the stations with maximum MAMDSL are mainly located in the north of subregion 2, while the stations with minimum MAMDSL values are principally found in the west of subregion 1. Such particular spatial distribution indicates that the eastern part may be more probable to undergo a prolonged dry period than the western part; thus the dry risk in the west is higher than in the east of the WRB.Figure 6
The spatial variation of MAMDSL in the WRB.Furthermore, we also present the main statistical characteristics of the AMDSL series in each subregion in the boxplot of Figure7 and in Table 2 in order to illustrate the differences of AMDSL among all subregions in detail. It can be observed that the subregion 1 presents smaller mean AMDSL (15.65 days) and also less variability (the standard deviation is 5.25), indicating a small probability of the long dry spell occurrence in this area (see Figure 7 and Table 2). The mean and standard deviation of AMDSL for the subregion 2 are widely larger than those in the other three subregions, suggesting a high occurrence probability of prolonged dry spells in this region. The maximum AMDSL in subregion 2 reaches 65 days (Table 2), which means that the period with precipitation amount lower than 1 mm/day sustains more than two months in that station. These statistical characteristics of AMDSL, reflected by the boxplot in Figure 7 and Table 2, are consistent with the results by the visual observation in Figure 6.Table 2
The statistical characteristics of AMDSL in each subregion of WRB.
Regions
Mean AMDSL (days)
Maximum AMDSL (days)
Minimum AMDSL (days)
Standard deviation
Subregion 1
15.65
41
6
5.25
Subregion 2
20.04
65
10
6.56
Subregion 3
17.75
44
8
5.35
Subregion 4
18.92
47
7
5.63Figure 7
Boxplots of AMSDL series in each subregion of the WRB.


### 4.4. Regional Frequency Analysis of the AMDSL Series

For this study, the discordancy of all stations in the four subregions in the WRB is firstly checked according to the discordancy measure introduced earlier. Table3 presents the D
i values for subregion 1 (5 stations), subregion 2 (5 stations), subregion 3 (6 stations), and subregion 4 (6 stations). According to Hosking and Wallis [41] and Hosking and Wallis [34], the critical values of discordancy measures should be determined as 1.333 (number of stations = 5) and 1.648 (number of stations = 6) for the subregions of our study. Table 3 shows that the values of discordancy measures of all used stations in each subregion are all below the corresponding critical D value, indicating the reasonable acceptance of all of these stations to be discordant, and the division of the subregions in our study is appropriate for the further regional frequency analysis.Table 3
Values of the discordancy measure of the four subregions in the WRB.
Regions
Station
Discordancy measurement
Subregion 1
Lintao
1.2750
Huajialin
1.3221
Guyuan
0.7334
Xiji
0.5472
Minxian
1.1223
Subregion 2
Dingbian
1.1166
Wuqi
1.3232
Huanxian
0.1947
Pingliang
1.3312
Xifengzhen
1.0344
Subregion 3
Yan’an
0.9705
Changwu
1.6161
Luochuan
0.1911
Longxian
1.2628
Tianshui
1.0010
Yongshou
0.9185
Subregion 4
Pucheng
1.0781
Fengxiang
0.3013
Taibai
1.5532
Wugong
1.0969
Qindu
0.6791
Huaxian
1.2914
Note: the critical value ofD is 1.333 and 1.648, when the number of stations in the region is 5 and 6, respectively. That is to say, the critical value of D is 1.333 for subregion 1 and subregion 2 and 1.648 for subregion 3 and subregion 4.Moreover, the homogeneities of the four subregions of the WRB are tested with the homogeneity measures, that is,H, and the results are given in Table 4. The H values of all subregions are less than one; thus these areas can be accepted as homogenous.Table 4
The homogeneity measures of the four subregions in the WRB.
Subregion 1
Subregion 2
Subregion 3
Subregion 4
H
0.19
0.24
-
0.33
-
1.49After the identification of the homogeneity of the obtained subregions, we can choose the optimal regional distributions withZ
DIST values following the procedure described in Section 3.2.4. The Z
DIST values of the four subregions are given in Table 5. We can find that two or more candidate distributions (Z
DIST
<
1.64) may fit the AMDSL series very well but the optimal distribution, which will be finally utilized, is the one with Z
DIST value closest to zero. For subregion 1, Z
DIST values of most distributions with Z
DIST
<
1.64 suggest that these distributions can be treated as appropriate regional distribution except GPD. However, GEV is identified as the optimal one since it has the minimum Z
DIST. Therefore, such result indicates that it is of great importance to undertake thorough distributions selection processes rather than only subjectively fit one single distribution for all AMDSL series so as to reduce the uncertainty originated from the choice of the “best-fit” distribution for the considered AMDSL series. In addition, following the similar procedure of the determination of the “best-fit” distribution in subregion 1, PE3, GLO, and GEV can be accepted as the optimal regional model for subregion 2, subregion 3, and subregion 4, respectively.Table 5
TheZ
D
I
S
T of the five candidate distributions in each subregion in the WRB. The bold values mean that the distribution is the optimal model for the corresponding subregion.
Distributions
Subregion 1
Subregion 2
Subregion 3
Subregion 4
GLO
0.96
2.43
0.41
1.06
GEV
−0.31
0.83
-
1.23
−0.80
GNO
-
0.62
0.46
-
1.47
-
0.96
GPD
-
3.24
-
2.84
-
4.86
-
4.81
PE3
-
1.27
−0.33
-
2.07
-
1.51Meanwhile, the L-moment ratio diagram, which maps the sample and theoretical L-moments ratios, that is, L-skewness and L-kurtosis, together in one figure [42], is also employed to check the performance of the optimal distribution chosen earlier (Figure 8). It has been proved to be an efficient tool for distinguishing candidate statistical distributions for the description of regional data [29, 34] and has been frequently applied in the selection process of the optimal statistical probability distribution functions [21, 22, 42]. In Figure 8, the circle point and solid black point represent the L-moments ratios of each station and the average regional L-skewness and L-kurtosis, respectively. Figure 8(a) shows that GEV and GNO fit AMDSL series better compared to other distributions for most stations in subregion 1, yet the average regional value implies that GEV is the optimal regional distribution for subregion 1. Such result is consistent with Table 5. In addition, Figures 8(b)–8(d) also show similar consistency with the conclusions given in Table 5 for the optimal distribution selection in subregion 2 to subregion 4.Figure 8
L-moments ratio diagram for subregion 1 (a), subregion 2 (b), subregion 3 (c), and subregion 4 (d) in the WRB. The circle point and solid black point represent the L-moments ratios of each station and the average regional L-skewness and L-kurtosis, respectively.
(a)
(b)
(c)
(d)After the determination of the “best-fit” regional statistical probability distributions for every subregion, we compute the quantiles of AMDSL under six return periods (2, 5, 10, 25, 50, and 100 years). The L-moments method is employed to estimate the parameters of the regional distributions. The spatial variations of the quantiles in the WRB under various return periods are presented in Figure9. A similar spatial distribution can be observed between different return periods; that is, the quantiles of AMDSL increase from west to east of the region. Such changes of AMDSL across space in Figure 9 are also similar to that of MAMDSL revealed in Figure 6. It can also indicate that, among the four subregions, the dry risk in subregion 2 and subregion 4 might be higher than that in subregion 1 provided that the quantiles in these two subregions are larger.Figure 9
The spatial variations of quantiles of AMDSL with the return periods of (a) 2, (b) 5, (c) 10, (d) 25, (e) 50, and (f) 100 years.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.5. Relationship between ENSO Events and Dry Spell Events

ENSO event is a very important factor influencing globe climate and weather [43–45] and shows strong correlation with dry events in many regions of the world [46]. Therefore, analyzing the relationship between ENSO events and extreme dry spells is significant for drought resistance in a specific region. In our research, the cross wavelet analysis is utilized to study the relationship between ENSO events and AMDSL variability in the stations of Huajianlin, Huanxian, Changwu, and Huaxian, which are located in subregion 1, subregion 2, subregion 3, and subregion 4, respectively (Figure 10). ENSO events show a statistically positive relationship with AMDSL variability in subregion 1 with a 2-3-year signal in 1969–1973, a 4–6-year signal in 1975–1985 and 1990–2000, and a 12–15-year signal in 1980–2005 at the 5% significance level (Figure 10(a)). Such statistically evident positive correlations directly prove that ENSO events play a critical role in triggering AMDSL during RS period in the WRB. Figures 10(b)–10(d) show the similar results with Figure 10(a) generally indicating that ENSO events have strong influence on causing extreme dry spells during RS period in the WRB.Figure 10
The cross wavelet transforms between the ENSO events and AMDSL series in stations Huajianlin (a), Huanxian (b), Changwu (c), and Huaxian (d). The meanings of the symbol can be referred to Torrence and Compo [24].
(a)
(b)
(c)
(d)


## 4.1. The Importance of Studying the Extreme Dry Spell during RS

In order to explain the importance and significance of studying the dry spell during RS, we firstly assess the proportion of the longest dry spell (LDS) in every year that occurred in RS (noted asp
1
×
100%), which can be computed as the total number of years exhibiting the LDS in RS divided by the total number of years in that station. For example, if a station contains a time period of 50 years in which the LDS occurs in RS during 10 of those years, the proportion of the LDS in RS can be determined as p
1 = 10/50 or 20%. In addition, the proportion of the LDS in every year which occurs in the non-RS can be accordingly determined as (1
-
p
1) × 100%. The results of the proportions for the LDS in RS and non-RS of all stations are concluded in Figure 4. It can be seen that, in the majority of stations, the LDS will occur during the non-RS period. The maximum proportion of the LDS in RS is only 33% which is observed in Tongxin station. However, because the total precipitation amount during the non-RS (accounting for about 15% of the annual total precipitation amount) is quite smaller compared to that during RS (accounting for about 85% of the annual total precipitation amount) in our study area (see Figure 2), the occurrence of prolonged dry period with daily precipitation lower than a certain threshold during the non-RS will have smaller impact when the dry spell with similar severity occurred during the RS. Particularly, as most of the total agricultural area in the region is rain-fed, the precipitation variability during the RS is especially important for the agricultural sector. On the other hand, since the RS is also the crops growth season in China, the persistence precipitation deficiency during this period will induce adversely impact agricultural yields, water resources, ecosystems, and human systems. Therefore, we should pay more attention to the LDS during the RS. The definition of the characteristics of the LDS, that is, AMDSL, can be referred to Section 3.1.Figure 4
The proportion of the longest dry spell occurred during RS and non-RS during 1960–2014 in the WRB.


## 4.2. Identification of Homogeneous Regions of the WRB

Six types of attributes for the stations, including latitude, longitude, elevation, total precipitation during RS period, and the mean AMDSL (see Table1), are chosen to determine the possible homogenous subregions via theK-means method in hierarchical clustering method. Figure 5 maps the division of the stations into several groups together with their boundaries. As introduced in Section 3.2, several subjective adjustments are performed to ensure the clusters are more appropriate. Finally, the whole study can be divided into four subregions (see Figure 5). Subregion 1 mainly belongs to the headwater of the WRB with a high elevation, subregion 2 is almost located in the northern part of the WRB with a small total annual precipitation amount, which is also the headwater of the Jing River Basin and the Beiluo River Basin (see Figure 1), subregion 3 is mostly distributed in the middle reach of the WRB, and subregion 4 is located in the lower reach of the WRB with a low elevation and also a high total annual precipitation amount. It is worth mentioning that different from the way to divide the WRB into four subbasins (see Figure 1) according to the topographic feature and hydrological and climatic conditions by the Ministry of Water Resources, China, we identify the subregion mainly on the basis of the attributes related to the attributes of dry spell; therefore, the division of the WRB identified above may be more appropriate for the performance of regional frequency study of AMDSL. Furthermore, the test of the homogeneity of the four subregions will be presented in Section 4.4.Figure 5
The division of four subregions of the WRB using theK-means method in the hierarchical cluster analysis.


## 4.3. Spatial Variation of the AMDSL

The AMDSL of every year in each station is computed and the mean AMDSL (MAMDSL) in all years during 1960–2014 can be obtained. Then, the MAMDSL in all stations is interpolated into the whole region using the software of ArcGIS (Figure6). The spatial variation of MAMDSL shows that the MAMDSL increases from the west to the east in the WRB, which is similar to the spatial structure of long term mean annual precipitation in this area. Specifically, the stations with maximum MAMDSL are mainly located in the north of subregion 2, while the stations with minimum MAMDSL values are principally found in the west of subregion 1. Such particular spatial distribution indicates that the eastern part may be more probable to undergo a prolonged dry period than the western part; thus the dry risk in the west is higher than in the east of the WRB.Figure 6
The spatial variation of MAMDSL in the WRB.Furthermore, we also present the main statistical characteristics of the AMDSL series in each subregion in the boxplot of Figure7 and in Table 2 in order to illustrate the differences of AMDSL among all subregions in detail. It can be observed that the subregion 1 presents smaller mean AMDSL (15.65 days) and also less variability (the standard deviation is 5.25), indicating a small probability of the long dry spell occurrence in this area (see Figure 7 and Table 2). The mean and standard deviation of AMDSL for the subregion 2 are widely larger than those in the other three subregions, suggesting a high occurrence probability of prolonged dry spells in this region. The maximum AMDSL in subregion 2 reaches 65 days (Table 2), which means that the period with precipitation amount lower than 1 mm/day sustains more than two months in that station. These statistical characteristics of AMDSL, reflected by the boxplot in Figure 7 and Table 2, are consistent with the results by the visual observation in Figure 6.Table 2
The statistical characteristics of AMDSL in each subregion of WRB.
Regions
Mean AMDSL (days)
Maximum AMDSL (days)
Minimum AMDSL (days)
Standard deviation
Subregion 1
15.65
41
6
5.25
Subregion 2
20.04
65
10
6.56
Subregion 3
17.75
44
8
5.35
Subregion 4
18.92
47
7
5.63Figure 7
Boxplots of AMSDL series in each subregion of the WRB.


## 4.4. Regional Frequency Analysis of the AMDSL Series

For this study, the discordancy of all stations in the four subregions in the WRB is firstly checked according to the discordancy measure introduced earlier. Table3 presents the D
i values for subregion 1 (5 stations), subregion 2 (5 stations), subregion 3 (6 stations), and subregion 4 (6 stations). According to Hosking and Wallis [41] and Hosking and Wallis [34], the critical values of discordancy measures should be determined as 1.333 (number of stations = 5) and 1.648 (number of stations = 6) for the subregions of our study. Table 3 shows that the values of discordancy measures of all used stations in each subregion are all below the corresponding critical D value, indicating the reasonable acceptance of all of these stations to be discordant, and the division of the subregions in our study is appropriate for the further regional frequency analysis.Table 3
Values of the discordancy measure of the four subregions in the WRB.
Regions
Station
Discordancy measurement
Subregion 1
Lintao
1.2750
Huajialin
1.3221
Guyuan
0.7334
Xiji
0.5472
Minxian
1.1223
Subregion 2
Dingbian
1.1166
Wuqi
1.3232
Huanxian
0.1947
Pingliang
1.3312
Xifengzhen
1.0344
Subregion 3
Yan’an
0.9705
Changwu
1.6161
Luochuan
0.1911
Longxian
1.2628
Tianshui
1.0010
Yongshou
0.9185
Subregion 4
Pucheng
1.0781
Fengxiang
0.3013
Taibai
1.5532
Wugong
1.0969
Qindu
0.6791
Huaxian
1.2914
Note: the critical value ofD is 1.333 and 1.648, when the number of stations in the region is 5 and 6, respectively. That is to say, the critical value of D is 1.333 for subregion 1 and subregion 2 and 1.648 for subregion 3 and subregion 4.Moreover, the homogeneities of the four subregions of the WRB are tested with the homogeneity measures, that is,H, and the results are given in Table 4. The H values of all subregions are less than one; thus these areas can be accepted as homogenous.Table 4
The homogeneity measures of the four subregions in the WRB.
Subregion 1
Subregion 2
Subregion 3
Subregion 4
H
0.19
0.24
-
0.33
-
1.49After the identification of the homogeneity of the obtained subregions, we can choose the optimal regional distributions withZ
DIST values following the procedure described in Section 3.2.4. The Z
DIST values of the four subregions are given in Table 5. We can find that two or more candidate distributions (Z
DIST
<
1.64) may fit the AMDSL series very well but the optimal distribution, which will be finally utilized, is the one with Z
DIST value closest to zero. For subregion 1, Z
DIST values of most distributions with Z
DIST
<
1.64 suggest that these distributions can be treated as appropriate regional distribution except GPD. However, GEV is identified as the optimal one since it has the minimum Z
DIST. Therefore, such result indicates that it is of great importance to undertake thorough distributions selection processes rather than only subjectively fit one single distribution for all AMDSL series so as to reduce the uncertainty originated from the choice of the “best-fit” distribution for the considered AMDSL series. In addition, following the similar procedure of the determination of the “best-fit” distribution in subregion 1, PE3, GLO, and GEV can be accepted as the optimal regional model for subregion 2, subregion 3, and subregion 4, respectively.Table 5
TheZ
D
I
S
T of the five candidate distributions in each subregion in the WRB. The bold values mean that the distribution is the optimal model for the corresponding subregion.
Distributions
Subregion 1
Subregion 2
Subregion 3
Subregion 4
GLO
0.96
2.43
0.41
1.06
GEV
−0.31
0.83
-
1.23
−0.80
GNO
-
0.62
0.46
-
1.47
-
0.96
GPD
-
3.24
-
2.84
-
4.86
-
4.81
PE3
-
1.27
−0.33
-
2.07
-
1.51Meanwhile, the L-moment ratio diagram, which maps the sample and theoretical L-moments ratios, that is, L-skewness and L-kurtosis, together in one figure [42], is also employed to check the performance of the optimal distribution chosen earlier (Figure 8). It has been proved to be an efficient tool for distinguishing candidate statistical distributions for the description of regional data [29, 34] and has been frequently applied in the selection process of the optimal statistical probability distribution functions [21, 22, 42]. In Figure 8, the circle point and solid black point represent the L-moments ratios of each station and the average regional L-skewness and L-kurtosis, respectively. Figure 8(a) shows that GEV and GNO fit AMDSL series better compared to other distributions for most stations in subregion 1, yet the average regional value implies that GEV is the optimal regional distribution for subregion 1. Such result is consistent with Table 5. In addition, Figures 8(b)–8(d) also show similar consistency with the conclusions given in Table 5 for the optimal distribution selection in subregion 2 to subregion 4.Figure 8
L-moments ratio diagram for subregion 1 (a), subregion 2 (b), subregion 3 (c), and subregion 4 (d) in the WRB. The circle point and solid black point represent the L-moments ratios of each station and the average regional L-skewness and L-kurtosis, respectively.
(a)
(b)
(c)
(d)After the determination of the “best-fit” regional statistical probability distributions for every subregion, we compute the quantiles of AMDSL under six return periods (2, 5, 10, 25, 50, and 100 years). The L-moments method is employed to estimate the parameters of the regional distributions. The spatial variations of the quantiles in the WRB under various return periods are presented in Figure9. A similar spatial distribution can be observed between different return periods; that is, the quantiles of AMDSL increase from west to east of the region. Such changes of AMDSL across space in Figure 9 are also similar to that of MAMDSL revealed in Figure 6. It can also indicate that, among the four subregions, the dry risk in subregion 2 and subregion 4 might be higher than that in subregion 1 provided that the quantiles in these two subregions are larger.Figure 9
The spatial variations of quantiles of AMDSL with the return periods of (a) 2, (b) 5, (c) 10, (d) 25, (e) 50, and (f) 100 years.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.5. Relationship between ENSO Events and Dry Spell Events

ENSO event is a very important factor influencing globe climate and weather [43–45] and shows strong correlation with dry events in many regions of the world [46]. Therefore, analyzing the relationship between ENSO events and extreme dry spells is significant for drought resistance in a specific region. In our research, the cross wavelet analysis is utilized to study the relationship between ENSO events and AMDSL variability in the stations of Huajianlin, Huanxian, Changwu, and Huaxian, which are located in subregion 1, subregion 2, subregion 3, and subregion 4, respectively (Figure 10). ENSO events show a statistically positive relationship with AMDSL variability in subregion 1 with a 2-3-year signal in 1969–1973, a 4–6-year signal in 1975–1985 and 1990–2000, and a 12–15-year signal in 1980–2005 at the 5% significance level (Figure 10(a)). Such statistically evident positive correlations directly prove that ENSO events play a critical role in triggering AMDSL during RS period in the WRB. Figures 10(b)–10(d) show the similar results with Figure 10(a) generally indicating that ENSO events have strong influence on causing extreme dry spells during RS period in the WRB.Figure 10
The cross wavelet transforms between the ENSO events and AMDSL series in stations Huajianlin (a), Huanxian (b), Changwu (c), and Huaxian (d). The meanings of the symbol can be referred to Torrence and Compo [24].
(a)
(b)
(c)
(d)


## 5. Summary and Conclusions

Understanding change pattern of drought from the viewpoint of persistent daily precipitation deficiency is of great significance for drought control and resisting, agricultural activities, water resources management, and so forth. In our study, drought during rainy season (RS) in the WRB is studied using the concept of dry spell, which refers to the continuous period with daily precipitation amount less than 1 mm/day. The regional variability of the extreme dry spells, represented by the AMDSL during the RS, is further analyzed on the basis of the L-moments method. The primary conclusions are listed below.(1)
With the consideration of the attributes of topography, precipitation, and spatial pattern of dry spells, the whole study area can be clustered into four subregions. The discordancy measure shows that no station has appeared to be discordant under such classification. Furthermore, the homogeneity measures indicate that all subregions are accepted as homogenous and thus are suitable for the regional frequency analysis.
(2)
The goodness-of-fit test shows that GEV is the “best-fit” distribution of subregions1 and 4, whereas PE3 and GLO distribution can be regarded as the optimal distribution of subregions 2 and 3, respectively. On the other hand, the results also demonstrate the importance of considering sufficient number of candidate distributions rather than one single distribution to reduce the uncertainty caused by the choice of the optimal distribution.
(3)
The spatial variation of MAMDSL shows that the MAMDSL increases from the west to the east in the WRB. In addition, a similar spatial distribution can be observed under various return periods; that is, the AMDSL values increase from west to east of the WRB. It can also indicate that, among the four subregions, the dry risk in subregion2 and subregion 4 might be higher than that in subregion 1 provided that the quantiles in these two subregions are larger.



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*Source: 6427568-2016-09-05.xml*