Beyond correlation: quantifying compound hydro-climatic shocks with copulas

Abstract Compound temperature–precipitation shocks shape hydro-climatic impacts in semi-arid regions but are not fully represented by linear correlation. Using monthly observations from Tulcea, Dobrogea (1965–2019), we quantify contemporaneous dependence by isolating independent and identically distributed innovation shocks and modeling their joint distribution with copulas. Temperature is filtered with a periodic autoregressive moving average of orders 2 and 1 and a periodic Generalized Autoregressive Conditional Heteroskedasticity variance model, and fitted with a Student- t marginal. For precipitation, we use a periodic autoregressive of first-order model with a Generalized Extreme Value marginal. Probability-integral transforms (PIT) yield uniform innovations. Candidate copulas are estimated by maximum likelihood, selected via Akaike/Bayesian criteria (AIC/BIC), and checked with White’s information-matrix test. The Frank copula is selected for all four compound configurations (warm–dry, cold–wet, warm–wet, cold–dry), implying a zero asymptotic tail dependence but non-trivial finite quantile $$\left( {\tilde{t} \in \left( {0,1} \right)} \right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:mover> <mml:mi>t</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> co-occurrence. At $$\tilde{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>t</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> = 0.90 (the 90th percentile), the warm–dry and cold–wet co-occurrence ( $$\tilde{J}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>J</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> =0.0196) is nearly double the independence ( $$\tilde{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> =0.0100) expectation (a 97% increase with ( $$\tilde{R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>R</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> =1.969). By contrast, warm–wet and cold–dry are suppressed ( $$\tilde{J}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>J</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> =0.0015 and 0.0030) and co-occur well below independence (an 85 and 70% decrease with $$\tilde{R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>R</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> = 0.151 and $$\tilde{R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>R</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> = 0.303, respectively). The approach yields station-specific multipliers for compound event risk and a generalizable procedure that disentangles within-series dynamics from bivariate dependence.