Spatial-Temporal Change Analysis for Multivariate Drought Risk Based on Bayesian Copula: Application to the Balkhash Lake Basin

13 In this study, a spatial-temporal Bayesian copula (SBC) method is developed through integrating 14 spatial-temporal analysis and Bayesian copula into a general framework. SBC method can help 15 model dependence structures of variable pairs and handle the uncertainty caused by parameter in 16 copulas, and SBC can reveal the spatial and temporal changes of drought events. SBC is applied 17 to the Balkhash Lake Basin (in Central Asia) to analyze spatial-temporal characteristic and 18 drought risk in 1901-2020. Several findings can be summarized: (1) Balkhash Lake Basin 19 suffered 53 drought events in 1901-2020, and five typical severe drought events occurred in 2


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Drought is the most widely affected natural disaster in the world and has adverse impact on 37 agriculture, industrial production, urban water supply and ecological environment (Neisi et al.,38 2020). Over the past twenty years, climate change led to a 46% deterioration in drought 39 conditions worldwide, which caused economic loss of 124 billion dollars, affecting more than  Despite the widespread impact, drought identification and risk analysis are still challenging 43 because of its different definitions, multivariate characteristics and spatial-temporal variability 44 (Guo et al., 2018). Therefore, it is necessary to conduct monitoring and assessment in drought-45 prone areas to determine drought characteristic, spatial-temporal variation and multivariate 46 interaction.

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Over the past decades, many efforts have been devoted to drought monitoring and assessment, 49 and more than fifty drought indices have been developed which are applicable in different precipitation and water balance, which are widely applied to meteorological drought (Hamal et 55 al., 2020). HDI is developed by meteorological indicators and runoff, which represents a drought 56 that river runoff is below the normal level; and HDI is usually applied to hydrological drought 57 (Yang et al., 2020). On a large regional scale, areas with scarce precipitation and intense 58 evapotranspiration are usually characterized by meteorological drought. However, watersheds 59 that are significantly affected by seasonal changes in runoff are usually characterized by 60 hydrological drought (Hu et al., 2019;Dehghan et al., 2020). Thus, it is complicated to analyze 61 drought in a watershed with large seasonal variation in runoff which is located in arid area. Both 62 meteorological factors (e.g., precipitation and evaporation) and underlying surface factors (e.g., 63 soil moisture and runoff) needed to be considered. PDSI provides a water balance model that 64 includes precipitation, evapotranspiration, runoff and soil moisture to describe drought of the watershed in arid area comprehensively (Palmer, 1965  Generally, drought is a three-dimensional spatial-temporal phenomenon, and the variation of a 71 drought evolves both static and dynamic factors (Herrera et al., 2017;Diaz et al., 2020). 72 Specifically, duration, severity and peak are static factors of a drought, and centroid, the multivariate characteristic are simplified in dimensionality reduction, the spatial-temporal 86 correlation of drought is diluted. Therefore, more robust method is desired for accurately 87 describing a drought event from both static and dynamic perspective, as well as quantitatively 88 analyzing the interaction between multivariate factors. used Archimedean copula to fit severity-duration-frequency and severity-area-frequency 95 curves, and results revealed the multidimensional drought characteristics in northern Algeria.

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Copula has the main advantage of reveal drought risk by quantifying the correlation among 97 factors which affect drought event, and it is convenient when modeling marginal distributions 98 and multivariate dependence structures . However, copula also suffers several 99 drawbacks such as verification of the optimal marginal distribution, enormous uncertainty of Bayesian copula into a general framework, which provided an efficient and accurate method for 108 fitting optimal marginal distribution. Overall, using Bayesian inference to improve copula can 109 minimize the uncertainty in parameter estimation. This study aims to develop a spatial-temporal Bayesian copula (SBC) method for analyzing 112 drought risk, through integrating spatial-temporal analysis and Bayesian copula into a general 113 framework. The main novelty and contribution of this study can be listed as: (1) this is the first 114 attempt to develop an integrated SBC method for analyzing multivariate (duration, severity and 115 affected area) drought risk; (2) SBC is capable of modeling dependence structures of variable 116 pairs and dealing with the uncertainty caused by parameter in copulas; (3) SBC can reveal the 117 spatial and temporal changes of drought events; (4) SBC is applied to the Balkhash Lake Basin 118 (in Central Asia) for drought risk analysis from 1901 to 2020; (5) the findings will be helpful to 119 disclose drought risk of Balkhash Lake Basin in the past century. The SBC method integrates spatial-temporal analysis and Bayesian copula into a general 123 framework ( Figure 1). In detail, drought variables (e.g., duration, severity, affected area) are 124 identified by using scPDSI and runs-theory. The correlation between drought variables are tested 125 based on Pearson, Kendall and Spearman coefficients. Marginal distribution of drought variable 126 is fitted by gamma, generalized extreme value, inverse Gaussian, log logistic, lognormal and 127 Weibull. Four Archimedean copulas (i.e., Clayton, Frank, Gumbel, Joe) are employed to model 128 dependence structures of variable pairs. The optimal marginal distribution and copula can be 129 selected based on goodness-of-fit tests. Bayesian inference is used for dealing with uncertain 130 parameters in copulas. Drought centroid and displacement direction are used for revealing the 131 spatial-temporal changes of drought. Multivariate drought risk of the Balkhash Lake Basin is 132 analyzed based on joint return periods and joint probabilities at different guarantee levels.
The longitude and latitude of Ps and Pe can be expressed as:  Copula is applied to model dependence structures among correlated variable pairs. Based on 166 Sklar theory, for a n-dimensional distribution function F, with univariate marginal F1, …, Fn, a 167 multivariate copula function C exists: The probability density of a copula can be expressed as: where E(L) represents the mean interval time of two consecutive drought events. Therefore, the 205 bivariate risk indictor R is defined as: Balkhash Lake is a closed terminal lake located at 73°20'E-79°12'E, 45°00'N-46°44'N in Central

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Asia. The lake stretches from east to west over 600 km, and width of the lake is 9-19 km in the   and drought area ratio. The characteristic information identified in Table 1 would be helpful to 280 understand the historical period of the drought in the Balkhash Lake Basin, and is also the basis 281 for the next step of multivariate analysis. Place Table 1 here Five typical severe drought events (SDE) are highlighted in Table 1, which indicates the in the north of Balkhash Lake were affected. Generally, extreme droughts occurred within the 308 periods of the severe droughts. By comparing the mean self-calibrating PDSI of each decade, the 309 periods of 1911-1920, 1921-1930, 1931-1940, 1961-1970, 1971-1980 and 1991-2000 were in 310 drought state, because the average annual self-calibrating PDSI of each period was less than -1.0.   Table 2 presents the correlation test results, which indicates that the 374 correlation coefficient of variable pairs of duration-severity is the highest, followed by severity-375 area and duration-area. All these three pairs pass the significant test at 5% level. Consequently, it 376 is necessary to consider the influence of the interaction among variables when analyzing drought 377 risk, otherwise the results are likely to be biased.

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Place Table 2 here show good agreement between the theoretical and empirical distributions. Thus, AICc test was 388 applied to select the optimal distribution for Bayesian copula. Table 3 shows that GEV is the  Clayton is the optimal copula to model dependence structure of duration-severity due to the 412 minimum AIC (-239.60) and BIC (-237.63). Similarly, Clayton is also the optimal copula to 413 model dependence structures of duration-area and severity-area.

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Place Figure 8 and Table 4 here   Place Figure 9 and Table 5 here Therefore, joint probability of duration, severity and area can be used to analyze the drought risk.

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The actual drought risk would be underestimated if only Tand

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In this study, a spatial-temporal Bayesian copula (SBC) method has been developed through

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The authors declare that they have no known competing financial interests or personal 510 relationships that could have appeared to influence the work reported in this paper. Step 1: Model sample data Step 2: Add latitude and longitude information to sample_1.txt Step 3: Rasterize the sample_1.txt by ASCII code in ArcGIS, and output it as sample_1.tif 548 549 Step 4: Load a raster file with projection information, and define the projection on the 550 example_1.tif 551 552 Step 5:

Duration-Area
Blue: Empircal Color lines: Fitted