Assessment of Spatial-temporal Variation of Precipitation and Meteorological Drought in Shanxi Province, China

4 In this study, the spatial and temporal characteristics of rainfall and the risk of meteorological 5 drought based on precipitation data observed in 22 meteorological stations from 1961-2020 6 across Shanxi province in China were analyzed. Four precipitation indices and modified 7 Mann-Kendall test were used to analyze the patterns and trends of precipitation. Furthermore, 8 the risk analysis for drought duration and severity of meteorological drought were analyzed 9 with entropy copula. Results showed that the precipitation distribution is irregular and 10 precipitation in central northern parts of Shanxi is more concentrated than that in southern 11 Shanxi. The annual and seasonal precipitation concentration showed no significant change in 12 most stations, while the daily precipitation concentration decreased in a few areas. Results 13 also disclose that the number of droughts ranged from 61 to 80 during 1961-2020 and Shanxi 14 tends to dry. The most severe drought event lasts for 27 months and the largest severity is 15 29.7. The entropy copula is suitable for drought frequency analysis and return period 16 calculation and results indicate that the middle and northern parts of Shanxi are at high risk of 17 drought according to the entropy copula calculated joint return period. 18


Introduction
The temporal-spatial distribution of precipitation is of paramount importance with respect to water availability, drought and water resources management, and the risk of floods (Raziei 2018).The amount and regime of precipitation in a region are also important to the agricultural and economical practices.Under the background of global warming and climate changes, the rainfall regimes, such as rainfall, extreme rainfall, seasonal rainfall has been significantly altered (Feng et al. 2013;Huang et al. 2016;Deng et al. 2018).Furthermore, global climate change leads to greater uncertainties in spatial and temporal variability of precipitation, which plays a key role in the occurrence and propagation of droughts (Mishra and Singh 2010).Therefore, investigation of changes in precipitation as well as drought characteristics is important for water resource management and agricultural production.
A substantial body of work has focused on the variability of precipitation on various temporal scales.Martín-Vide (2004) proposed daily precipitation concentration index (DPCI) for evaluating the varying weight of daily precipitation, i.e., the contribution of rainy days with the largest amount of precipitation to the total amount of precipitation.It measures irregularities in rainfall distribution by determining the total amount of rainfall in each precipitation class.A higher DPCI value refers to a more heterogeneous precipitation distribution and is more prone to cause droughts and floods (Shi et al. 2013).Therefore, it is of great significance to investigate the precipitation concentration and structure based on daily precipitation data.Huang et al. (2016) computed DPCI values for the Wei River Basin and found the precipitation durations of 1-3-day events have the largest rate of contribution to the total annual precipitation amount.They concluded that the precipitation with a short duration is the major precipitation event in the basin.Deng et al. (2018) studied spatial and temporal variation of DPCI across Pearl River basin, China and found that the spatial patterns of the DPCI revealed an irregular rainfall distribution across the basin.
The concentration of rainfall in one year is another important aspect in this research area.
The precipitation concentration index (PCI) proposed by Oliver (1980) is a commonly used index to quantify the relative distribution of precipitation patterns, and to estimate the seasonality of the precipitation in a given study area (Coscarelli and Caloiero 2012;Ghaedi and Shojaian 2020).Using DPCI and PCI indices, Coscarelli & Caloiero (2012) analyzed the concentration of daily and monthly rainfall in Southern Italy and detected a very inhomogeneous temporal distribution of the daily rainfall in the eastern region.Huang et al. (2016) estimated the seasonal heterogeneity of precipitation amounts in Wei River Basin using PCI.A remarkable seasonality of precipitation in the western and northern of Wei River Basin is detected, and the monthly precipitation distribution in the basin tends to be more uniform because of the downward trend in PCI.The relative rainfall seasonality index (SI) proposed by Walsh & Lawler (1981) is also utilized for assessing the seasonal contrasts in rainfall amounts through the year (Bari et al. 2017;Ingle et al. 2018).
Moreover, a new rainfall seasonality index (RE) was developed based on a probabilistic interpretation of precipitation fractions and the concept of relative entropy borrowed from information theory (Feng et al. 2013).This index can describe the annual distribution of rainfall and the timing and duration of the wet season based on information theory indicators.Feng et al. (2013) used RE to identify regions across the tropics with highly seasonal rainfall regimes.Pascale et al. (2015) evaluated the RE by using precipitation gridded datasets and historical simulations from coupled atmosphere-ocean general circulation models.
Drought is a common natural hazard which results due to the deficit of rainfall compared to expected average value.It may occur in any part of the world, but duration and intensity of droughts vary greatly across different climatic zones.Moreover, drought may cause serious impacts on regional agriculture, water resources, and the environment (Zuo et al. 2018).
Droughts are classified into four categories according to various types of deficits i.e., meteorological drought, agricultural drought, hydrological drought and socio-economic drought (Nabaei et al. 2019).A number of drought indices have been developed by researchers for quantifying, monitoring, and analyzing drought (Waseem et al. 2015;Zhang et al. 2015;Tirivarombo 2018;Wang et al. 2020).The Standardized Precipitation Index (SPI) is a commonly used index to characterize meteorological droughts and it has been shown to be an effective tool for assessing meteorological drought (Mckee et al. 1993).Copula functions have been widely applied to evaluate the multivariate analysis of drought for its flexibility in choosing marginal distributions (Xu et al. 2015;Liu et al. 2011).For instance, Gu et al. (2020) incorporated copulas and a drought hazard propagation ratio to examine the drought propagation process that from meteorological to hydrological aspects.
The selection of marginal distributions is of crucial importance as it strongly impacts the performance of the copula in modeling multivariate variables.However, it is time-consuming to select the appropriate distribution from a large number of candidates and the results may be underestimated/overestimated if the distribution is misidentified (Guo et al. 2017).To overcome this problem, the maximum entropy principle was introduced in deriving probability distribution functions with a minimum of bias from limited information in a more objective way (Hao and Aghakouchak 2013).Consequently, the maximum entropy principle and copula were combined (hereafter, entropy copula) to take advantage of the strengths of both methods.Guo et al. (2017) analyzed the risks of flood and extreme precipitation events in two catchments of the Loess plateau, China with the use of entropy copula method.Yang et al. (2021) applied maximum entropy copula to assess bivariate drought risk of the Kaidu River Basin, China.
Although rainfall and drought have been studied at different spatial and temporal scales for many regions, very few studies have focused on assessing rainfall trends as well as the drought characteristics across Shanxi province.Therefore, the objectives of this study are to analyze (i) the spatial and temporal characteristics of daily, monthly, and seasonal rainfall, (ii) the bivariate drought risk of Shanxi province by using entropy copula function, the results are capable providing decision supports for drought management.The remainder of this paper is outlined as follows.The study area and data are presented in section 2. The precipitation indices and entropy copula are introduced in sections 3. Section 4 provides the results and discussion, and the main findings are summarized in Section 5.

Study area
Shanxi province (Fig. 1) is located in the loess plateau the west of North China with a total area of 156700km 2 .It belongs to the temperate continental monsoon climate zone with annual mean temperature ranging from 8°C to 14°C and long-term annual average precipitation ranging from 400 to 650 mm.The seasonal distribution of precipitation in Shanxi is uneven, which is mainly concentrated in June to August (Dong and Yu 2022).Topographically, the altitude decreases from northeast mountainous areas to the southern edge of the province.
Therefore, the spatial distribution of precipitation in the province is greatly affected by the topography.Drought is a major meteorological disaster in this region and it occurs frequently with damaging effects on the economy and ecological system.

Study data
Daily precipitation data observed at 22 meteorological stations (Fig. 1) from 1961-2020 was acquired from National Meteorological Information Center (NMIC) of the China Meteorological Administration, and was available at http://data.cma.cn/en.The data quality was strictly checked by NMIC before the dataset was released.

Daily precipitation concentration index (DPCI)
To assess the contribution of the various daily precipitation classes to the total precipitation, it is necessary to analyze the accumulated percentages (Y) of precipitation corresponding to the cumulative percentage of rainy days (X) (Martín-Vide 2004;Wang et al. 2013).The relationship between X and Y can be expressed as an exponential curve (Martín-Vide 2004): where a and b are the regression constants computed by the method of least squares.The DPCI can be calculated by: where S is the area enclosed by the concentration curve and the bisector of the quadrant and it can be calculated as: A higher DPCI value reflects that the precipitation is more concentrated in a few rainy days during the year for a meteorological station, and vice versa.

Precipitation concentration index (PCI)
PCI is used to investigate monthly heterogeneity of rainfall amounts and is computed by (Coscarelli and Caloiero 2012): where Pi is the monthly precipitation amount of the ith month.As described by Oliver (1980), PCI values that are less than 10 suggest uniform precipitation distribution throughout the year, whereas values from 11 to 20 denote a seasonal distribution.Values above 20 represent marked seasonal differences (De et al. 2011).

Seasonality index (SI)
The rainfall seasonality index (SI) is calculated as follow (Walsh and Lawler 1981): (5 where xi is the monthly precipitation for month i; and R is the annual precipitation.The higher the index, the greater is the departure of rainfall distribution in the given year (Table 1) (Walsh and Lawler 1981).

Relative entropy seasonality index (RE)
The relative entropy seasonality index (RE) is a measure to quantify the difference between the true sequence of monthly rainfall fractions pk,m and the uniform monthly precipitation sequence qm=1/12 (m=1,2,…,12) (Feng et al. 2013).It can be calculated as: where pk,m is the monthly probability distribution,  , =  ,   ⁄ ( , is the monthly precipitation in the mth month for hydrological year k; and is the total annual precipitation of hydrological year k).The RE attains its maximum value (log212) when the total annual precipitation is concentrated in a single month and equals to zero for a uniform precipitation sequence.As discussed in Pascale et al. (Pascale et al. 2016), the RE can be used to define the number of months of the wet season as  ′ = 12 * 2  .Thus, the number of dry months nk for each hydrological year is Changes in RE imply changes in the number of wet and dry months.In particular, areas with increasing RE has a reduction of the number of wet months and an increase in the number of dry months.

The Modified Mann-Kendall (MMK) test
The Modified Mann-Kendall (MMK) trend test was proposed to eliminate the effects of the series autocorrelation in Mann-Kendall (MK) trend test (Hamed and Rao 1998)

Standardized precipitation index (SPI)
SPI is calculated only by the use of monthly precipitation data and it can be calculated for different time scales to express drought, such as monthly (SPI1), seasonal (SPI3), and annual (SPI12) (Yerdelen et al. 2021).The detailed formulation of SPI calculation can refer to Naeini et al.(2021).Drought classification based on SPI values is shown in Table 2.

Drought Characteristics and distributions
Drought duration (D) and severity (S) can be extracted from the SPI based on the Run theory proposed by Yevjevich (1967).D means consecutive months in which the SPI value is less than threshold level.S is the absolute value of cumulative values of SPI in D when SPI values are consistently less than threshold level.The D and S were assumed to follow exponential and gamma distributions, respectively (Liu et al. 2011).In this study, the parameters are estimated for D and S using maximum likelihood estimation (MLE) method.

Entropy copula
The maximum entropy copula has been developed based on the maximum entropy theory and copula theory (Li and Zheng 2016).The entropy of a bivariate copula can be expressed as: where u and v denoting realizations of U and V of the marginal probabilities; c(u, v) is the probability density function of copula.
The constraints can be expressed as (Aghakouchak 2014;Li and Zheng 2016): ∫ ∫   (, ) ∫ ∫ (, ) where r=1,2 ,..., m, and m is the maximum order of the moment, which is considered 2 in this study; h(u, v) is a function of the marginal u and v; which can be related to a certain dependence structure; ρ is the Spearman rank correlation coefficient when h(u, v)=uv.
The density function of entropy copula can be obtained by maximizing the entropy in Eq.
(8) subject to the constraints in Eqs.( 9)-( 12), and the result can be expressed as: where λ0, …, λ2m+1 are the Lagrange parameters, and λ0 can be expressed as: The parameters can be estimated by minimizing a convex function in Eq. ( 15) by a numerical method like Newton-Raphson iteration method.For more details about the estimation of parameters please refer to Hao and Singh (2013).

Drought frequency analysis
Drought frequency refers to the frequency of drought occurs in a certain period of time (Yang et al. 2022).It is calculated by the following equation: where Pf is drought frequency; n is the number of years with drought; N is the total study years.The higher the value of Pf indicates a higher drought frequency.

Bivariate return periods
In this study, the joint return period of drought duration and severity are calculated in two cases: either drought duration or drought severity exceeds a certain value at the same time (D≥d∪S≥s) which is denoted as "OR" case, and drought duration and drought severity both exceed a certain value at the same time (D≥d∩S≥s) which is denoted as "AND" case.The bivariate joint probability for two variables (D and S) can be described as follow (Shiau and Shen 2001): The "AND" and "OR" return period are calculated respectively by the following equations: where T AND and T OR represent the "AND" and "OR" return period, respectively; E is the    ).All stations have average SI values between 0.80 (53588) and 0.97 (53594 and 53673), implying a marked seasonal precipitation regime with a long dry season (Table 1).As was the case for PCI (Fig. 3(a)), the SI value increases northward and seven stations with larger SI values (SI>0.90) are located in north central Shanxi.

Time variability and trend analysis of the indices
The time series of the monthly seasonality indices (PCI, SI, and RE) of six stations (53478, 53487, 53490, 53578, 53588 and 53593) are illustrated in Fig. 4. As is evident, the variation of the indices re-scaled between 0 and 1 has significant concordance.Thus, they can be interchangeably used for identifying precipitation characteristics and seasonality at a given station.The MMK statistics of DPCI are listed in the last column in Table 4 as well.The time variability of DPCI index of the six stations is illustrated in Fig. 5, with the Zs values shown in the graphs.Consistent with the results found in PCI, SI and RE, the PDCI of station 53478 exhibits a significant decreasing tendency at 0.05 significance level.Nonetheless, the significant decreasing trend is not found in station 53578, indicating that the coincidence of the daily and monthly concentration indices is not always expected.
Additionally, according to Table 4, the DPCI series of six stations (53478, 53490, 53663, 53673, 53853 and 53959) exhibit significant decreasing trend at 0.05 significance level, and the DPCI time series of stations 53775 and 53877 show significant increasing trend.The results of MMK test of SPI3 concerning every month at 22 meteorological stations are exhibited in Fig. 7.The stations in Fig. 7 are arranged according to their latitude, from south to north order.Fig. 7 can reflect the spatial-temporal changing characteristics of dry and wet conditions in Shanxi province.It can be observed that the number of months having a wet trend in northern part is more than that of southern region.In southern area, the months in autumn and winter have a trend towards dry.Thus, the relevant authorities should pay attention to the potential drought risk.The distribution of the dry periods according to the classification in Table 2 is given in Fig. 8.It is determined that the most common drought class at all station is mild drought, and this event occurred at a maximum of 60% at station 53673 locating in north Shanxi and minimum of 49% at station 53853 locating in south Shanxi.The highest moderate drought occurrence rate is 30% at a northern station 53487, while the highest severe drought is 16% at station 53853 locating in south Shanxi.The highest rate of occurrence of extreme drought is 8% at station 53787 locating in central Shanxi.

Drought characteristics
The SPI3 is used for extracting D and S and the threshold level is considered as 0. The number and average of severity and duration of drought have been presented A strong relationship is determined between drought duration and severity throughout the province according to Kendall's τ, Spearman's ρ and Pearson.For all stations, the values of τ, ρ and Pearson range from -0.79~-0.93,-0.82~-0.85 and -0.67~-0.69,respectively.
Therefore, entropy copula can then be used to describe the dependence between drought duration and severity.

Bivariate return periods
With the help of entropy copula, the T AND and T OR return periods of D and S are calculated using Eqs.10 and 19, in which D and S were assumed to follow exponential and gamma distributions, respectively.The entropy copula with parameter m=2 is fitted for the combination of D and S of each station.
The bivariate return periods T AND and T OR are determined using the D and S values calculated from univariate 5-, 10-20-, 50-and 100-year return periods.Fig. 9 and Fig. 10 presents the univariate and the corresponding bivariate return periods T AND and T OR , respectively.Generally, the bivariate return periods increase when the univariate return periods increase.T AND is longer than the corresponding univariate return period, while the T OR is less.Accordingly, it can be reached that there is significant difference between univariate return periods and bivariate return periods in the province.Furthermore,stations 53478,53487,53593,53594,53578,53663 and 53588,located in the middle and northern parts of Shanxi, are at high risk of drought according to T AND .
In practical application, T AND can be considered as the upper bound of the target year of the return period and T OR can be considered as the lower bound.In Shanxi province, drought duration and severity are always at high levels.The results of drought risk analysis will probably be inaccurate if only the absolute value of a single drought variable is considered and the interactions between different variables are ignored.The actual drought risk would be underestimated if only the 'AND' case joint return period is considered, whereas drought risk would be overestimated if only the 'OR' case is considered.which may cause the risk of failure of drought control measures and drought mitigation techniques increase to some extent, which may cause the failure of drought control measures and drought mitigation techniques to some extent.

Conclusions
In this study, daily and monthly total precipitation of 22 stations with relatively regular distribution over Shanxi was used to investigate the spatial and temporal variability of rainfall and the meteorological drought across the Shanxi province in China.The main conclusions are as follows: (1) In Shanxi province, PCI values ranged from 17.1 to 21.1, indicating an irregular precipitation distribution.The high values of SI and RE revealed a seasonal precipitation pattern with a long dry season in Shanxi, and the number of dry months increases northward.
The amount of annual precipitation increases from north to south, and the precipitation in central northern parts of Shanxi is more concentrated than that in southern Shanxi.
(2) Both increasing and decreasing trends were found in annual precipitation amount based on MMK trend test, while only six stations exhibit a significant decreasing trend.The annual and seasonal precipitation concentration in most stations did not change significantly according to the variation of precipitation indices PCI, SI and RE.Annual and seasonal daily precipitation concentration decreased significantly in a few areas.
(3) The variation trend of SPI1, SPI3 and SPI12 series shows that Shanxi tends to dry, but the trend is not significant.Based on SPI3, the number of months having a wet trend in northern part is more than that of southern region.The months in autumn and winter have a trend towards dry in southern area.
(4) Shanxi experienced frequent drought events in 1961-2020 with the number of droughts ranged from 61 to 80 in 22 stations.The mean of the maximum drought duration is 15.7 months, and the most severe drought event lasts for 27 months.As a result of spatial analysis of the drought duration and severity in Shanxi province with entropy copula, it is seen that the central and northern parts of the province have a high risk in terms of bivariate return periods.

Fig. 1
Fig.1 Location of the Shanxi province and its related meteorological stations expectation of the drought interval; u=FD(d) and v=FS(s) are the marginal cumulative distribution function of D and S, respectively; C(u,v) is the cumulative distribution function of entropy copula.the precipitation characteristics The spatial distributions of annual mean precipitation and mean number of rainy days relative to 1961-2020 are depicted in Fig.2.The annual mean precipitation ranges from 382 mm (53487) to 753 mm (53588) and the mean number of rainy days varied between 73 (53664) and 121 (53588) days.It should be noted that station 53588 locating in the eastern foothills of Wutai mountain in central northern Shanxi has the largest annual precipitation amount and the longest rainy days.Except station 53588, the spatial variety of annual precipitation presented a latitudinal gradient with the lower precipitation values being observed in northern Shanxi while the higher precipitation values are distributed in southern region.

Fig. 2 .
Fig.2.The mean annual precipitation and the number of rainy days during 1961-2020 at each station Results of MMK test for annual precipitation over the period 1961-2020 are presented in

Fig. 3
Fig.3shows the spatial pattern of mean PCI, SI, DPCI, RE and the number of dry months

Fig. 3
Fig.3 Spatial pattern of (a) PCI, (b) SI, (c) DPCI, (d) RE and (e) the number of dry months across Shanxi Spatial distribution of DPCI values is depicted in Fig.3(c).A value of 0.61 is a threshold

Fig. 4
Fig.4 Time series of the standardized monthly seasonality indices of the selected stations for 1960-2020 period

Fig. 5
Fig.5 The DPCI time series of the selected stations for 1960-2020 period.The Zs values over the graphs correspond to MMK test 4.4 Drought analysis 4.4.1 Variation of SPI

Fig. 7
Fig.7 The drought trend of 12 months at every meteorological station

Fig. 8
Fig.8 Percentage of occurrence for different drought classifications

Fig. 9
Fig.9 Spatial distribution of the bivariate return period T AND corresponding to various univariate return periods

Table 1
(Walsh and Lawler 1981)regimes indicated by the seasonality index (SI)(Walsh and Lawler 1981) Silva et al. 2015)es the lag-i autocorrelation of the hydro-meteorological series into account and it is robust for the trends in hydro-meteorological series (DaSilva et al. 2015).A positive value of the test statistic Z indicates an upward trend in the time series, and a negative value Wang et al.(2019)downward trend.For a given significance level α, the original hypothesis (notrend) is rejected if || >   2⁄ .The significance level was set at a 5% and the null hypothesis is rejected if |Z|≥1.96.The calculation procedure of the MMK method is described inWang et al.(2019).

Table 2
Drought classification based on SPI values(McKee et al.1993)

Table 3
Results of the MMK test for annual precipitation over period 1961-2020 *Statistically significant trends at the 5% significance level.

Table 4
presents the statistics of the trend analysis applied to the 22 stations.PCI, SI and RE showed no significant trend in most stations based on MMK trend test, but they all tended to decrease significantly at stations 53478, 53578 and 53594 locating in the northern Shanxi.Contrarily, these three indices show positive trend in stations 53588 and 53775.

Table 4
MMK statistics Zs of the monthly seasonality indices relative to the selected stations

Table 5
Number and characteristics of moderate or worse droughts in stations of Shanxi province 360Table5