Comparison of the Entropy- and Copula-based Precipitation-Streamow Drought Indices for Assessing Agricultural Drought in Arid and Semi-arid Coastal Areas of Southern Iran

1 In arid and semi-arid regions, precipitation and seasonal streamflow are the two major sources of 2 water for vegetation. The scarcity of these water sources has a detrimental effect on vegetation 3 cover degradation. The purpose of this research is to study the effect of meteorological and 4 hydrological droughts, and also their combined effects, on vegetation changes in seven coastal 5 sub-basins in southern Iran (part of the Bandar-Sedij and Kol-Mehran catchment). To track 6 meteorological and hydrological droughts, the Standardized Precipitation Index (SPI) and the 7 Streamflow Drought Index (SDI) were used. The copula function and the entropy approach (which 8 is developed in this research) were used to blend individual meteorological and hydrological 9 drought indices, yielding hybrid indices called the Copula-based Drought Index and the Entropy- 10 based Drought Index (EnDI). The single (i.e., SPI and SDI) and hybrid drought indices (CoDI and 11 EnDI) were compared in terms of temporal behavior, drought severity and duration characteristics, 12 drought frequency, and a bivariate analysis of the drought severity-duration return period. The 13 results indicated that the rank correlation ( 𝑟 " ) between SPI and SDI ranged between 0.327 and 14 0.726 in the studied sub-basins. However, the two hybrid indices CoDI and EnDI had extremely 15 high correlations ( 𝑟 " ≥ 0.9 ). Despite the fact that meteorological droughts benefited both hybrid 16 drought indices more than hydrological droughts, the contribution of meteorological droughts to 17 EnDI was greater than that of CoDI. Over the study region, CoDI reported droughts that were both 18 longer and more severe than those recorded by EnDI. EnDI showed stronger associations with the 19 Normalized Vegetation Difference Index (NDVI) in nearly all the sub-basins, possibly because 20 precipitation has a greater effect on EnDI than it does on CoDI. EnDI was therefore recommended 21 as a superior index for estimating vegetation droughts throughout the research region.


Introduction 1
Drought is a complex, recurrent, and multifaceted environmental hazard that has a wide range of 2 effects on ecosystems and society (Van Loon, 2015). Drought is defined as a temporary and 3 continuous period of available water shortage and its spatial and temporal characteristics vary from 4 one region to another region (Tallaksen and Van Lanen, 2004). Drought is described as a lack of 5 water caused by a variety of factors that vary depending on the climate of the region, such as a 6 lack of precipitation, soil moisture, streamflow, and groundwater level (Mishra and Singh, 2010). 7 Drought periods are irregular in arid and semi-arid regions, and they can persist for a long time   Precipitation and seasonal streamflow are the two main sources of water supply for 7 vegetation in arid and semi-arid regions. In general, long-term scarcity of these water sources is 8 expected to put a considerable stress on vegetation cover (i.e., agricultural drought). Several 9 research on the link between meteorological, agricultural and hydrological droughts have been drought) with SPI is higher than SDI and about short-term droughts, 3-month time scale is 18 preferred to 1-and 2-month time scales.

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The aim of this study is to examine the individual impact of meteorological and hydrological 20 droughts and their combined effects on vegetation changes (agricultural drought) in seven coastal 21 sub-basins in southern part of Iran (part of the Great Kal, Mehran, and Bandar e Sadij). For this 22 purpose, SPI and SDI indices were employed to track meteorological and hydrological droughts, 23 respectively. NDVI was used to track droughts in vegetation cover (or agricultural droughts).

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Hybrid indices were developed utilizing two approaches of copula function and entropy weighting 2 which their capability to identify periods of drought is also examined and compared. Due to the 3 arid and semi-arid climate of the study area, it is expected that precipitation has a greater role in 4 vegetation changes than streamflow. Also, hybrid indices are expected to explain a higher 5 percentage of variability in vegetation due to the combined effect of SPI and SDI in their  The basin under consideration is part of Kal-Mehran-Bandar Sadij basins, which are located in 11 southern Iran (Fig.1). This basin has a total size of 71483.26 km 2 and is located between latitudes Kahourestan (KRB), and Sikhoran (SRB). The climate of region is warm and dry, with an annual 15 average temperature of 27°C and an annual average precipitation of 215.8 mm. The spatiotemporal 16 pattern of rainfall over the study area is irregular, and the majority of rivers are seasonal.

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Monthly precipitation data from 25 rain gauge stations and average monthly streamflow 18 volume data from 13 hydrometric stations (placed inside the basin) were utilized in this   in Colorado. SPI requires only monthly precipitation data as input. The capability to calculate at 19 multiple time scales is a key aspect of SPI. Short-term SPI time scales (e.g., three months, the time 20 period utilized in this study) influence vegetation, whereas longer SPI time scales affect surface 21 and groundwater resources. The SPI is calculated by calculating precipitation probability in each 22 time frame. Therefore, the first step in calculating this index is to fit an appropriate statistical 23 distribution to precipitation data. According to Edwards and McKee (1997) studies, the gamma 1 distribution fitted properly to the US monthly precipitation data and the theory underlying the SPI 2 was initially presented on the basis of this distribution. The gamma probability density function is 3 defined as follows: where, is the shape parameter, is the scale parameter, is the precipitation value, and ( ) is 6 the gamma function. In the next step, the cumulative probability distribution of precipitation is 7 calculated as below: Due to the fact that the gamma function is not specified for = 0 (zero precipitation) and the 10 distribution of precipitation may include zero values, the total cumulative probability function, 11 which also contains zero precipitation, will be as follows: where is the probability of zero precipitation. After calculating the total cumulative function 14 ( ) ( )), the equal probability transformation from ) ( ) to the standard normal distribution Z 15 (or SPI) with a mean of zero and a variance of one is carried out according to the following formula: where MN is the cumulative precipitation in the th month and th year and 6@ (. ) is the inverse 18 standard normal function. on the fit of the two-parameter log-normal distribution to monthly streamflow data. The SDI 4 equation is as follows: where MN is the natural logarithm of the cumulative streamflow volume in month and year ; Y N 7 is the mean, and N is the standard deviation of MN . In this study, SDI was calculated as SPI in a 3-8 month time window. According to Table 1, the SDI is classified the same as the SPI.  Table 2. The empirical copula is used in this study to construct a hybrid  Table 1.  The hybrid drought index proposed in this study is based on the SPI and SDI entropy weights.

Entropy-based Precipitation-Streamflow Drought Index
2 First, the weight of each variable in each month of the year is obtained based on the entropy method 3 from the following equation (Shannon, 1948;Singh, 2015): where, N€ is the weight corresponds to the th variable (SPI or SDI) in the th month and satisfy 6 for the condition ∑ N€ = 1 k €v@ for month . N€ is entropy which is calculated from the following 7 equation: where MN€ is the normalized value of the th variable (SPI or SDI) in the th month and th year, 10 as calculated from the following equation: Entropy-based precipitation-streamflow Drought Index (EnD) is then calculated as follows: in which N@ and Nk are the weights of SPI and SDI indices in the th month, respectively. Table   16 1 can be used for classifying EnDI. In this study, Spearman correlation coefficient was used to analyze the correlation between 2 different drought indices, between NDVI and LST, between the single and hybrid drought indices, 3 and between NDVI and each of four drought indices used in this study. In addition, Spearman 4 correlation is directly related to the copula function; hence, the larger the value of this measure, 5 the stronger the correlation between the two copula marginal functions. The value of Spearman 6 correlation varies between -1 to +1. Positive correlation indicates a direct relationship between two 7 random variables and negative correlation shows an inverse relationship.

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A drought phenomenon happens when a specific drought index is less than zero. Accepting this 10 threshold allows for the determination of the two most important features of any drought 11 phenomenon, namely the duration (D) and severity (S) of the drought (Fig. 2). The duration of a 12 drought, according to this definition, is equal to the number of months in which the drought index 13 is less than zero. Drought severity is also defined as the area below the zero threshold of the 14 drought index. The severity and duration of a drought are two dependent variables, so as the 15 duration of the drought increases, the severity is expected to increase. Therefore, this dependency 16 provides the possibility to use copula function for modeling the relationship between the drought 17 duration and severity and to calculate the drought return period. In this study, six probability    SPI and SDI were calculated using the best distributions fitted to the 3-month precipitation 21 and streamflow data for the studied sub-basins, respectively. Fig. 3 depicts the results of calculating the two indices. The SPI fluctuates more regularly than the SDI in all sub-basins. 1 Instead, SDI revealed prolonged drought/wet periods in some sub-basins that were not observed 2 in SPI time series. For example, the HRB sub-basin experienced a 4-year streamflow drought 3 (2013-2016), whereas the SPI was often higher than average over the same time period. Similarly, 4 prior to 2000, the SDI-based BRB sub-basin consistently exhibited normal or higher-than-normal 5 conditions, but the SPI-based sub-basin alternated between drought and wet years. However, 6 adaptability to dry and wet periods increased in this sub-basin after 2000. There appears to be a 7 strong negative decreasing trend in the SDI series, which has resulted in normal and higher-than- year, however, is not significantly different from other wet years in the whole record period.

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The Spearman rank correlation coefficient between SDI and SPI was computed for each sub-   correlation. However, while the lowest correlation between SPI and SDI was observed in MRB 4 sub-basin, the lowest correlation between the hybrid indices was related to HRB sub-basin. 5 Therefore, it can be concluded that with higher correlation between individual indices, it is not 6 expected that the hybrid indices developed in this study also have a high correlation.

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To get an overview, Fig. 5 shows the 95% confidence level of the mean correlation 8 coefficients between different drought indices across the studied sub-basins. According to this 9 figure, the mean correlation coefficient between EnDI and SPI is higher than that between EnDI 10 and SDI. Similar connections also exist between CoDI with each of the single indices. Therefore, 11 it is possible to conclude that both CoDI and EnDI give more contribution to SPI than SDI. SPI, 12 however, contributes more to the construction of EnDI than CoDI. The contribution of SDI, on the 13 other hand, is more evident in construction of CoDI than EnDI.

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The two hybrid indices CoDI and EnDI were also compared in terms of drought severity and 16 duration characteristics (Fig. 6). As seen in this figure, CoDI has documented longer and more 17 severe droughts than EnDI, particularly in the HRB, KRB and DZRB sub-basins. For example, 18 CoDI recorded a 30-month drought period (06/2014-11/2016) in HRB sub-basin, the longest 19 drought duration experienced across the study area. When using EnDI, the longest drought was 20 split into three smaller drought episodes. Another example is the CoDI-identified 21-month 21 drought period (07/2007-03/2009), which EnDI has split it into three shorter drought episodes.

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However, the scatter diagram of drought severity vs. drought duration for both hybrid indices are almost identical in the BRB, DERB and MRB sub-basins. In the SRB sub-basin, the two hybrid 1 indices are not significantly different in terms of drought duration, but CoDI has reported more 2 severe droughts than EnDI.

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Two hybrid drought indices used in this study were examined in terms of univariate and bivariate 16 return period analysis of drought severity and duration. To this end, first, the marginal probability 17 distribution of each drought characteristic (i.e., severity and duration) was determined. Then, five 18 well-known bivariate copulas (Table 2) were fitted to the dependence structure of drought severity 19 and duration, and one with the lowest value of the Cramer-von Mises test (Genest and Favre, 2007) 20 was selected as the best copula among the others. The chosen copulas for the studied sub-basins 21 are presented in Table 3. The table indicates that the best copula function was not the same in all 22 sub-basins. The Normal copula (in MRB sub-basin) and the Joe copula (in SRB sub-basin) were introduced as the best models fitting the drought severity and duration derived from CoDI and 1 EnDI. In other sub-basins, the two hybrid indices differed in terms of the type of selected copula 2 function. All five copula functions used in this study were optimal in at least one case.

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In Table 4, for both CoDI and EnDI, the drought severity ( ) and the drought duration ( ) 4 for six return period levels of 2, 5, 10, 20, 50, and 100 years (column 2 in Table 3) were estimated 5 from the best marginal distributions fitted to the drought characteristics and . In bivariate case, 6 joint return period of and was estimated for the condition ≥ ≥ using Equation 7 15. For example, based on CoDI calculated in HRB sub-basin, the return period of ≥ 3.4 is 2 8 years, and the return period of ≥ 6.0 also is 2 years. For concurrent occurrence of two events 9 ≥ 3.4 and ≥ 6.0, the bivariate return period (column 5 in  allowing us to assess vegetation response to drought stress across the study area. The correlation 2 coefficients range from -0.3 (HRB sub-basin) to -0.6 (DZRB), and they are all statistically 3 significant at the 95% confidence level.

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The results of the correlation analysis between NDVI and ground-based drought indices 5 (SPI, SDI, CoDI, and EnDI) in the studied sub-basins are presented in Table 5. As seen in the

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In this work, the efficiency of two single drought indices, SPI and SDI, and two hybrid indices,

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CoDI and EnDI, for assessing drought impacts on vegetation cover in seven coastal sub-basins in 20 southern Iran was investigated. Preliminary data analysis revealed that the Gamma (Log-Normal) 21 distributions fit the cumulative 3-month precipitation (streamflow) data the best in nearly all of the 22 examined sub-basins. Because of the seasonality of river flows in the sub-basins of interest, moderate to reasonably high correlations between SPI and SDI were detected. In this work, a 1 hybrid drought index called Entropy-based Drought Index (EnDI) was established, which is based 2 on a combination of SPI and SDI and uses the entropy-weighting approach. Furthermore, a 3 Copula-based Drought Index (CoDI) was constructed by combining SPI and SDI with the best 4 copula model fitted to these indices. In this study, the drought indices utilized were compared in Hormozgan University) at the University of Tehran, which is hereby thanked and appreciated. Readers can contact authors for availability of data and materials.      (1 + k / ) 6 Ÿr@ k > 2, ∈ [0,1]   with the return period of bivariate drought severity-duration (T(S,D)) based on CoDI and EnDI in 2 the sub-basins. Table 5. Spearman correlation coefficient between NDVI and drought indices with zero-month 1 lag (Lag = 0) and one-month lag (Lag = 1) in the studied sub-basins.    indices across the studied sub-basins. correlations are significant at the 95% confidence level.