Drought is a multi-scale hydrometeorological hazard driven by the compound interplay of water deficits in space and time. While univariate indices capture individual drought characteristics, they cannot resolve the joint behaviour of severity and duration that determines cumulative water demand and exposure time. This study uses a deliberately bivariate copula formulation because the operational risk functional is the joint exceedance probability <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p_{sd}=\mathbb {P}\{S\ge s, D\ge d\}\)</EquationSource> </InlineEquation>, where S is event severity and D is event duration. This probability is the quantity required for severity–duration–frequency design, and any higher-dimensional model would have to be marginalized back to this same S, D law before the return period could be computed. Compound drought risk is quantified across 20 meteorological stations in Punjab, Pakistan, over 1981–2020. Drought events are extracted from the SPI-3 series via run theory, and the dependence between severity and duration is modeled using five parametric copulas and an entropy-based copula, with performance evaluated against empirical AND-exceedance return periods. The entropy copula yields lower mean absolute error at sixteen of the twenty stations. Joint return period contours, tile-plot heatmaps, severity–duration–frequency curves, and finite-threshold Kato tail-concentration analysis reveal that compound extreme events with severity and duration jointly above the 80th percentile occur at four to five times the rate of compound mild events at all stations. Spatially, northern and eastern stations show compressed return periods. At the same time, the southern Cotton–Wheat Zone exhibits expanded return periods but larger joint severity–duration magnitudes and strengthening upper-tail concentration, indicating a concentration of risk into fewer but more intense events. The entropy copula yields substantial gains at stations with nonlinear dependence, whereas parametric copulas suffice where dependence is regular, supporting a staged operational modelling strategy. Sensitivity tests across three SPI thresholds confirm the robustness of the severity–duration coupling. The results demonstrate that compound drought risk in Punjab is non-stationary in time, strongly structured in space, and requires zone-differentiated water management that accounts explicitly for the joint behaviour of drought severity and duration.

Assessing drought dynamics across climatic zones using parametric and entropy copula models

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发表日期：

2026年7月28日

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Muhammad Owais Khan,
Luca Di Persio,
Rizwan Niaz,
Hefa Cheng,
Laila A. Al-Essa,
Hanen Louati,
Mhassen. E. E. Dalam

摘要

Drought is a multi-scale hydrometeorological hazard driven by the compound interplay of water deficits in space and time. While univariate indices capture individual drought characteristics, they cannot resolve the joint behaviour of severity and duration that determines cumulative water demand and exposure time. This study uses a deliberately bivariate copula formulation because the operational risk functional is the joint exceedance probability \(p_{sd}=\mathbb {P}\{S\ge s, D\ge d\}\) , where S is event severity and D is event duration. This probability is the quantity required for severity–duration–frequency design, and any higher-dimensional model would have to be marginalized back to this same S, D law before the return period could be computed. Compound drought risk is quantified across 20 meteorological stations in Punjab, Pakistan, over 1981–2020. Drought events are extracted from the SPI-3 series via run theory, and the dependence between severity and duration is modeled using five parametric copulas and an entropy-based copula, with performance evaluated against empirical AND-exceedance return periods. The entropy copula yields lower mean absolute error at sixteen of the twenty stations. Joint return period contours, tile-plot heatmaps, severity–duration–frequency curves, and finite-threshold Kato tail-concentration analysis reveal that compound extreme events with severity and duration jointly above the 80th percentile occur at four to five times the rate of compound mild events at all stations. Spatially, northern and eastern stations show compressed return periods. At the same time, the southern Cotton–Wheat Zone exhibits expanded return periods but larger joint severity–duration magnitudes and strengthening upper-tail concentration, indicating a concentration of risk into fewer but more intense events. The entropy copula yields substantial gains at stations with nonlinear dependence, whereas parametric copulas suffice where dependence is regular, supporting a staged operational modelling strategy. Sensitivity tests across three SPI thresholds confirm the robustness of the severity–duration coupling. The results demonstrate that compound drought risk in Punjab is non-stationary in time, strongly structured in space, and requires zone-differentiated water management that accounts explicitly for the joint behaviour of drought severity and duration.