J. Earth Syst. Sci. (2021) 130:3 Ó Indian Academy of Sciences

https://doi.org/10.1007/s12040-020-01490-1 (0123456789().,-volV)(0123456789().,-volV)

Prediction of water shortage loss in situations with small samples based on an improved Gumbel copula

1 2, 3 4 LONGXIA QIAN ,YONG ZHAO *, HONGRUI WANG and SUZHEN DANG 1School of Science, University of Posts and Telecommunications, Nanjing 210 023, . 2State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, 100 038, China. 3College of Water Sciences, Beijing Normal University, Key Laboratory for Water and Sediment Sciences, Ministry of Education, Beijing 100 875, China. 4Yellow River Institute of Hydraulic Research, Yellow River Conservancy Commission, Zhengzhou 45003, China. *Corresponding author. e-mail: [email protected]

MS received 25 May 2020; revised 4 August 2020; accepted 6 August 2020

Prediction of water shortage losses is of great importance for water resources management. A new mathematical expression of water shortage loss was proposed in order to describe the random uncertainty and economic attributes of water resources. Then, Gumbel copula with a new method of parameter estimation was introduced to model the joint probabilistic characteristics for water supply and water use in situations when sufBcient data is unavailable. The new parameter estimation method requires only the minimum and maximum values of two variables. The improved Gumbel copula was proved to be reliable based on the RMSEs (root mean square error) and AICs (Akaike information criterion), statis- tical tests and upper tail dependence tests. The potential water shortage losses for all the districts of were predicated. The water shortage loss in the Urban is highest (7.02 billion CNY), followed by the new district of and , while those in the and Ji County are very small. Keywords. Economic attributes; water shortage loss; insufBcient data; improved Gumbel copula; Tianjin.

1. Introduction predict the occurrence probability of water short- age risk or other risks. For example, the probabil- Water resources have multiple attributes, ities of water shortage were forecasted through a including natural and economic attributes. Many stochastic approach (Yu et al. 2015). The infor- countries in the world are facing water shortage, mation diffusion model was employed to predict which has caused considerable economic losses water shortage risk and compared with some tra- (Zhang et al. 2015;Liet al. 2016; Qian et al. 2016). ditional methods (Feng and Huang 2008). Yang Prediction of water shortage risk is of great et al. (2015) proposed an integrated method for importance for risk prevention and control. systematic quantitative risk analysis. Qian et al. Haimes (2009) pointed out that risk is a measure (2019) put forward a new model to predict water of the probability and severity of consequences. At shortage probability in situations with small sam- present, many researches concentrates on how to ple numbers. The probability of earthquake risk 3 Page 2 of 12 J. Earth Syst. Sci. (2021) 130:3 was forecasted through Markov chains (Fujinawa water shortage losses in all the districts of Tianjin 1991). are predicted by using these methods in section 3. Researches about how to deBne or predict water Some conclusions and future work are given in shortage loss are very scarce. Vulnerability was section 4. Brstly used to express the expected maximum severity of a sojourn into the set of unsatisfactory states (Hashimoto et al. 1982). But it is difBcult to 2. Methods estimate the probability of every unsatisfactory state. Qian et al. (2014)deBned potential economic In this section, some details about the expression of losses due to water shortage in terms of vulnera- water shortage loss and Gumbel copula maximum bility. However, the random uncertainty was not entropy estimation are presented. considered about water supply and water demand in this definition. Qian et al. (2016) constructed the joint distribution of water supply and water 2.1 Expression of water shortage loss demand by supposing that the relationship The amount of water shortage refers to the differ- between water supply and water demand is inde- ence between the amounts of water supply and pendent. However, there is a strong correlation water use in this paper. Water shortage loss is between water supply and water demand. Copulas referred to as the potential economic loss due to have been widely used to build the joint distribu- water shortage. Based on the definition of mathe- tion among variables. Gao et al. (2018) applied matical expectation, water shortage loss can be copulas to construct the joint probabilistic char- calculated quantitatively as follows: acteristics of water shortage. Zhang et al. (2016) Z Z used Student t-copula to build the joint distribu- b d tion of water supply and water demand. Ayantobo EL ¼ q Á fxðÞÁ; y ðÞy À x dxdy; ð1Þ et al. (2019) applied copulas to analyze the joint a c characteristics among three drought variables. where x and y are the amounts of water supply and Zhang et al. (2015) employed copulas to model the water use, and fxðÞ; y is the joint probability den- joint distribution of two major drought character- sity function of x and y; and q denotes the unit istics. Maximum likelihood estimation (MLE) is average economic beneBt of water resources (Qian often applied to estimate the parameters of copula et al. 2014); a and b are the possible range of water which requires a large amount of data (Qian et al. supply, and c and d are the possible range of water 2018). Unfortunately, data is often scarce when use. If the amount of water supply is greater than assessing water resources risk for many developing that of water use, i.e., x [ y, then EL =0. regions (Qian et al. 2019). Copulas with maximum likelihood estimation may be inapplicable in data- 2.2 Gumbel copula with maximum entropy scarce regions. Therefore, Gumbel copula with estimation maximum entropy estimation for small sample situations is introduced to model the joint proba- Gumbel copula, Clayton copula and Frank copula bilistic characteristics of water supply and water are often used to study the joint probabilistic use in situations with small samples. characteristics of random variables. Maximum There are some contributions in this paper. likelihood estimation or maximum pseudo-likeli- First, water shortage losses were deBned in terms of hood method are often applied to compute the mathematical expectation for the purpose of con- unknown parameters of copulas, while both meth- sideration of random uncertainty and economic ods require a great number of samples (Vergni attributes of water resources. Then, Gumbel cop- et al. 2015; Zhang et al. 2016). However, small ula with maximum entropy estimation was intro- samples are often available in water resources risk duced to model the joint distribution between (Qian et al. 2019). Qian et al. (2018) proposed a water supply and water use in situation with new method for estimating the parameter of insufBcient sample. The structure of this paper is Gumbel copula in situations when small samples as follows. Section 2 presents some methods pro- are available. The new method is called as maxi- posed and applied in this paper, including a new mum entropy estimation. The performance of expression about water shortage losses and Gumbel Gumbel copula with maximum entropy estimation copula with maximum entropy estimation. The is satisfying when small samples are used. J. Earth Syst. Sci. (2021) 130:3 Page 3 of 12 3

Moreover, it is still reliable when the true joint parameter p can be obtained by solving the distribution of two variables is not Gumbel copula following equation (Qian et al. 2018). The principle of the maximum oEpðÞ entropy estimation for Gumbel copula is as follows. ¼ 0; ð3Þ Let X and Y be two random variables with op marginal distributions u ¼ G1ðÞx and v ¼ G2ðÞy , where an optimization model can be built based on the Z Z  maximum entropy principle (Qian et al. 2018): oEpðÞ a2 b2 ocuðÞ; v; p ¼ Á g ðÞÁx g ðÞy lnðcuðÞ; v; p Z Z op op 1 2 a2 b2 a1 b1  min À EpðÞ¼ cuðÞÁ; v; p g1ðÞÁx g2ðÞy ocuðÞ; v; p Ág ðÞÁx g ðÞÞþy dxdy; a1 b1 1 2 op  lnðÞcuðÞÁ; v; p g1ðÞÁx g2ðÞy dxdy;  ð2Þ oðÞcuðÞ; v; p C 0 C ¼ E þ E0 where p is the parameter of Gumbel copula to be op D D estimated, and C 0D À CD0 C ¼ þ E0; D2 D

8 hihi9 > 1 1 p 1 1 > < p p p p = C ðÞÀ log u þÀðÞlog v log ðÞÀ log u þÀðÞlog v Á C 0 ¼ hi: p2 > 1 1 > : ðÞÀ log u p logðÞþÀÀ log u ðÞlog v p logðÞÀ log v À logðÞ log u log v ; 8 hihi9 > 1 1 p 1 1 > < p p p p = À1 ðÞÀ log u þÀðÞlog v log ðÞÀ log u þÀðÞlog v Á D0 ¼ hi p2 > 1 1 > : ðÞÀ log u p logðÞþÀÀ log v ðÞlog v p logðÞÀ log v þ 1 ;

cuðÞ; v; p More details about the maximum entropy nohi p estimation can be found in Qian et al. (2018). 1 1 1À1 exp ÀÀðÞlog u pþÀðÞlog v p ðÞlog u Á log v p RMSE (root mean square error) and AIC criteria ¼ hi 1 1 2Àp (Akaike information criterion) are employed to uv ðÞÀ log u pþÀðÞlog v p evaluate the goodness of Bt. They are expressed as hi follows: 1 1 p 1 ÂÀðÞlog u pþÀðÞlog v p þ À 1 ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Xn 1 2 RMSE ¼ ðÞEi À Ti ; ð4Þ n i 1 ln cuðÞ; v; p ¼ hi 1 1 p 1 AIC ¼ n lnðÞþMSE 2m; ð5Þ ¼ÀðÞ Àlog u pþÀðÞlog v p þ À 1 logðÞ log u Á log v p hi where Ei and Ti are the joint empirical and theo- 1 1 p 1 þ log ðÞÀ log u pþÀðÞlog v p þ À 1 À logðÞuv retical frequency of the ith sample respectively, n is hip samples size and m is the number of parameters, 1 1 P À ðÞ2 À p log ðÞÀ log u pþÀðÞlog v p : 1 n 2 and MSE ¼ n i¼1 ðÞEi À Ti .

Moreover, a1 and a2 denote the minimum and 2.3 Genetic algorithm maximum of X, and b1 and b2 are those of Y , and g1ðÞx and g2ðÞy are probability density functions Genetic algorithm is often applied to solute an (PDFs) of X and Y , respectively. The optimal optimization model. Based on the idea of natural 3 Page 4 of 12 J. Earth Syst. Sci. (2021) 130:3 genetics and biological evolution, genetic algorithm , , new district of works with a number of solution sets over the Binhai, Wuqing district, Baodi district, Ninghe search domain rather than a single one and can district, and Ji County (Bgure 1). avoid local optima (Goldberg and Holland 1988). It belongs to a warm temperate semi-humid con- Genetic algorithm includes the following steps. tinental monsoon climate. The average annual First, initial population is generated based on some precipitation is 574.9 mm, increasing from south to rules. Second, the quality of the individual is north. The summer precipitation accounts for evaluated according to its Btness computed by the about 80% of the annual precipitation. The annual value of the objective function. Individuals with average surface water resource of the city is about higher Btness will have more opportunities to 1.07 billion m3, and the annual average under- reproduce new generations. Then, new generations ground water resources is about 590 million m3,of will be produced by selection operation, crossover which the exploitable resources are 450 million m3. operation and mutation operation. More details The annual average water resources in Tianjin is about genetic algorithm can be found in the liter- about 1.6 billion m3, and the annual average net ature (Goldberg and Holland 1988). This paper inCow is 2.4 billion m3. The amount of annual used genetic algorithm to search global optimal water entering into the sea is about 1.5 million m3. parameter of the improved Gumbel copula. In 2018, the amount of water resources per capita was only 113 m3, and water shortage is very serious in Tianjin. External water sources are the main 3. Case study water supply, such as Luanhe River and South-to- North Transferred Water. In addition, other The water shortage losses in 2020 in all the districts unconventional sources, e.g., reclaimed water and of Tianjin were estimated based on the new expres- seawater desalination are also employed for coping sion of water shortage loss and in this section. with water scarcity. The sequences of water supply (without consid- 3.1 Study area and data eration of transferred and unconventional water) and water use during the period of 2008–2016 in all Tianjin is located in the northeast of the North the districts of Tianjin were used and provided by China plain. It consists of the following districts: Tianjin Water Bureau. Urban district, , ,

3.2 Model construction

3.2.1 Construction of marginal distributions

Three distributions such as Inverse Gaussian (IG), Log-logistic (LL) and Gamma (GM) are applied to model the marginal distributions of sequences of water supply and water use in all the districts of Tianjin. The Btting of three probability density functions for water supply are compared, as shown in Bgures 2 and 3. Comparisons of the Btting of three probability density functions for water use are given in Bgures 4 and 5. The best distribution is selected based on the results of the K–S tests, as shown in tables 1 and 2. According to Bgures 2–5 and table 1, the best marginal distribution functions for water supply and water use in all the districts of Tianjin can be obtained, as shown in table 2. Taking Dongli dis- trict for example, the p-values of Inverse Gaussian (IG), Log-logistic (LL) and Gamma (GM) for water supply are 0.8, 0 and 0, respectively. Thus Figure 1. The administrative map of Tianjin. the best distribution for the sequence of water J. Earth Syst. Sci. (2021) 130:3 Page 5 of 12 3

Figure 2. Comparisons of three probability density functions for water supply in all the districts of Tianji: (1) Urban district; (2) Dongli district; (3) Xiqing district; (4) Jinnan district; (5) Beichen district; (6) New district of Binhai. supply in Xiqing district is Inverse Gaussian. By concluded. The parameters of the best marginal analogy, the best distributions for all the other distributions in table 2 are computed by maximum sequences of water supply and water use can be likelihood estimation. 3 Page 6 of 12 J. Earth Syst. Sci. (2021) 130:3

Figure 3. Comparisons of three probability density functions for water supply in all the districts of Tianji: (7) Wuqing district; (8) Baodi district; (9) ; (10) Jinghai district; (11) Ji County.

3.2.2 Construction of joint distributions the districts of Tianjin. Because the number of samples during the period of 2008–2016 is only 9, Gumbel copula is used to build the joint distribu- maximum entropy estimation was applied to tion functions of water supply and water use in all compute the parameters of the Gumbel copulas in J. Earth Syst. Sci. (2021) 130:3 Page 7 of 12 3

Figure 4. Comparisons of three probability density functions for water use in all the districts of Tianji: (1) Urban district; (2) Dongli district; (3) Xiqing district; (4) Jinnan district; (5) Beichen district; (6) new district of Binhai. all the districts of Tianjin. The process of esti- determine whether ÀEpðÞ can reach minimum mating the parameter by maximum entropy esti- when p belongs toðŠ 0; 1 (Bgure 6). Figure 6 shows mation is as follows, taking Ji County for example. that equation (3) has a unique solution. Substi- First, the trend of ÀEpðÞ was observed to tuting the possible ranges of water supply and 3 Page 8 of 12 J. Earth Syst. Sci. (2021) 130:3

Figure 5. Comparisons of three probability density functions for water use in all the districts of Tianji: (7) Wuqing district; (8) Baodi district; (9) Ninghe district; (10) Jinghai district; (11) Ji County. water use during the period of 2008–2016 (equa- Figure 7 indicates that the optimal parameter is tion 3), the optimal parameter can be searched obtained in the 61st generation, and the average through the genetic algorithm, as shown in distance between individuals converges to 0. Sim- Bgure 7. ilarly, the optimal parameters of Gumbel copulas J. Earth Syst. Sci. (2021) 130:3 Page 9 of 12 3

Table 1. The results of K–S tests for three distributions for water supply (WS) and water use (WU) in all the districts of Tianjin. p-value Urban Dongli Xiqing Jinnan Beichen Binhai Wuqing Baodi Ninghe Jinghai Ji IG WS 0.1 0.8 0.5 0.8 0.4 0.9 0.5 0 0.2 0.2 0.7 WU 0.5 0.9 0.7 0.03 0.3 0.6 0.4 0.4 0.1 0.1 0.5 LL WS 0 0 0 0 0 0 0.6 0 0.3 0 0.7 WU 0.8 0 0.3 0 0 0.3 0.7 0.7 0.7 0 0.5 GM WS 0 0 0 0 0 0 0.1 0.3 0 0 0 WU 0 0 0 0 0.2 0 0 0 0 0 0

Table 2. The best marginal distribution function of water supply and water use in all the districts of Tianjin.

District Urban Dongli Xiqing Jinnan Beichen Binhai Wuqing Baodi Ninghe Jinghai Ji Water supply IG IG IG IG IG IG LL GM LL IG LL Water use LL IG LL IG IG IG IG LL LL IG LL

9 C1ðÞu; v or C2ðÞu; v is uniformly distributed in 8 ½Š0; 1 . More details can be found in Qian et al. 7 (2018). The results of K–S test are shown in table 5. 6 Table 5 indicates that all the probability values 5 4 are greater than the significance level (0.05). It is -E(p) 3 concluded that the joint distributions of water 2 supply and water use for all the districts of Tianjin 1 can be simulated by the improved Gumbel copula. 0 The empirical upper tail dependence coefBcients -1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 for water supply and water use for all the districts p were estimated in terms of the following equation Figure 6. The trend of ÀEpðÞin Ji County. (Frahm et al. 2005). 8 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 39 < XN log 1 log 1 = 1 Ui Vi in other districts can be calculated, and the k^CFG 2 2 exp log4 5 : U ¼ À : 1 ; parameters in all the districts are shown in table 3. N log 2 i¼1 maxðÞUi;Vi The RMSEs and AICs between the joint empir- 7 ical and theoretical frequencies of Gumbel copula ð Þ in all the districts of Tianjin are computed based on The theoretical upper tail dependence equations (4 and 5), as shown in table 5. The joint coefBcients of the improved copulas for all the empirical frequency is computed as follows (Qian districts were computed according to the following et al. 2018). equation. m À 0:44 k ¼ 2 À 2p: ð8Þ H ¼ pXðÞ¼ x ; Y  y i : ð6Þ U i i i N 0:12 þ Comparisons of empirical and theoretical upper Table 4 shows that the RMSEs in most of the tail dependence coefBcients for all the districts districts of Tianjin are close to 0.3. The performance of Tianjin are shown in table 6. Table 6 shows of the improved Gumbel copula is satisfying, that most of empirical upper tail dependence considering that only four data are used for coefBcients are very close to those theoretical parameter estimation. Moreover, we applied a values, although there are some differences for goodness-of-Bt test to determine whether the CDF some districts. Therefore, improved Gumbel copula of water supply and water use have been can be used to model the joint probabilistic appropriately modelled by the improved Gumbel characteristics for water supply and water use for copula. The goodness-of-Bt test is proposed by Ma all the districts of Tianjin according to RMSEs, et al. (2012) and belongs to a univariate distribution AICs and statistical test and upper tail dependence test. K–S test was used to determine whether tests. 3 Page 10 of 12 J. Earth Syst. Sci. (2021) 130:3

4 0.25 Best fitness Mean fitness 0.2 3

0.15 2 0.1 Fitness value 1

Current best individual 0.05

0 0 0 102030405060708090100 1 Generation Number of variables (1) 0.25 7

6 0.2 5

0.15 4

0.1 3 Expectation 2 Average Distance 0.05 1

0 0 10 20 30 40 50 60 70 80 90 100 -1.21414 -1.21412 -1.2141 -1.21408 -1.21406 -1.21404 -1.21402 -1.214 -1.21398 Generation Raw scores 10-3

Figure 7. The result of the optimal parameter in Ji County.

Table 3. The values of parameters of Gumbel copulas in all the districts of Tianjin.

District Urban Dongli Xiqing Jinnan Beichen Binhai Wuqing Baodi Ninghe Jinghai Ji Value 0.78 0.88 0.89 0.87 0.51 0.91 0.5 0.62 0.61 0.28 0.21

Table 4. RMSEs and AICs between the joint empirical and theoretical frequencies of Gumbel copula in all the districts of Tianjin.

District Urban Dongli Xiqing Jinnan Beichen Binhai Wuqing Baodi Ninghe Jinghai Ji RMSE 0.25 0.31 0.29 0.24 0.31 0.26 0.30 0.33 0.34 0.42 0.28 AIC –22.7 –18.9 –20.0 –23.1 –18.8 –21.8 –19.2 –17.5 –17.2 –13.6 –20.4

Table 5. K–S tests for the improved Gumbel copula in all the districts of Tianjin. p-value Urban Dongli Xiqing Jinnan Beichen Binhai Wuqing Baodi Ninghe Jinghai Ji c1(u,v) 0.2 0.6 0.6 0.7 0.1 0.9 0.5 0.1 0.9 0.3 0.1 c2(u,v) 0.1 0.5 0.5 0.9 0.7 0.6 0.3 0.1 0.8 0.4 0.1

Table 6. Comparisons of empirical and theoretical upper tail dependence coefBcients for all the districts of Tianjin.

Districts Urban Dongli Xiqing Jinnan Beichen Binhai Wuqing Baodi Ninghe Jinghai Ji Empirical 0.20 0.25 0.23 0.17 0.30 0.32 0.51 0.46 0.67 0.86 0.81 Theoretical 0.28 0.15 0.15 0.15 0.58 0.13 0.58 0.48 0.47 0.79 0.84

Table 7. The values of water shortage loss in all the districts of Tianjin (billion CNY).

District Urban Dongli Xiqing Jinnan Beichen Binhai Wuqing Baodi Ninghe Jinghai Ji Losses 7.02 0.81 1.17 0.35 0.46 3.5 2.05 1e-3 0.39 0.42 1e-6 J. Earth Syst. Sci. (2021) 130:3 Page 11 of 12 3

3.3 Prediction of water shortage loss water supply may lead to high risk of some districts. For example, the amount of water demand in 2020 The joint probability density functions (fxðÞ; y in will be as much as 0.97 billion m3 in the Urban equation 1) of water supply and water use in all district and 1.07 billion m3 in the new district of districts of Tianjin can be obtained based on the Binhai, while those of surface water supply and results of the best marginal distributions (table 3) groundwater supply in 2020 are only 0.07 billion m3 and Gumbel copulas (table 4). Taking Ji County and 0.11 billion m3. Industrial water demand and for example, the joint probability density function domestic water demand in the new district of Bin- for water supply and water use (fxðÞ; y )isas hai increases rapidly, with an average annual follows: growth of 9.9% and 12.6%, respectively. Therefore, the water shortage loss will continue to increase in fxðÞ¼; y cuðÞÁ; v; p g1ðÞÁx g2ðÞy ; ð9Þ new district of Binhai. Moreover, the current where average economic beneBt of water resources will be much greater than that used in this paper, so all the cuðÞ¼; v; p districts of Tianjin would suffer greater loss. nohi p 1 1 1À1 exp ÀÀðÞlog u pþÀðÞlog v p ðÞlog u Á log v p hi 1 1 2Àp 4. Conclusions uv ðÞÀ log u pþÀðÞlog v p hi 1 1 p 1 A new mathematical expression of water shortage ÂÀðÞlog u pþÀðÞlog v p þ À 1 ; p loss was proposed in order to describe the random uncertainty and economic attributes of water and resources. Then, the Gumbel copula with maximum  entropy estimation was introduced to construct the logðÞÀx l1 1 1 e r1 joint distribution function for water supply and g1ðÞ¼x !x [ 0; water use in the case of small samples. The r x 2 1 logðÞÀx l1 improved Gumbel copula requires only the mini- 1 þ e r1 mum and maximum values of two variables for parameter estimation. The improved Gumbel cop- and  ula was proved to be reliable based on the RMSEs, AICs and statistical test and upper tail dependence logðÞÀy l2 1 1 e r2 tests. The potential water shortage losses for all the  g2ðÞ¼y !2 y [ 0; districts of Tianjin were predicated. The water r2 x logðÞÀy l2 r shortage loss for Urban district is as much as 7.02 1 þ e 2 billion CNY and highest, followed by new district of R R x y Binhai and Wuqing district, while those in the and u ¼ 0 g1ðÞx dx, and v ¼ 0 g2ðÞy dy, and Baodi district and Ji County are very small. l1 ¼ 0:34, and r1 ¼ 0:01, and l2 ¼ 0:51, and Although the improved Gumbel copula requires r2 ¼ 0:01, and p ¼ 0:21. only the possible ranges of two variables, the error The average economic beneBt of water resources would increase when samples are too sparse. Fur- 3 of Tianjin is about 18.97 CNY/m (Gan et al. thermore, the estimation of economic beneBtof 2008). Substituting the average economic beneBtof water resources is also very important for the water resources of Tianjin, fxðÞ; y and the possible prediction of water shortage loss. This paper ranges of water supply and water use in 2020 into employed the result of economic beneBt in the equation (1), the potential water shortage loss for existing literature. The method about how to all the districts can be calculated by numerical estimate the economic beneBt of water resources integration, as shown in table 7. will be studied in future work. Table 7 shows that the water shortage loss for Urban district is as much as 7.02 billion CNY and highest, followed by new district of Binhai (3.5 Acknowledgements billion CNY) and Wuqing district (2.05 billion CNY), while those in Baodi district and Ji County The study was supported by the Open Research are very small. High water demand and insufBcient Fund of State Key Laboratory of Simulation and 3 Page 12 of 12 J. Earth Syst. Sci. (2021) 130:3

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Corresponding editor: SUBIMAL GHOSH