Table 2 Copula function family between the watershed lagged intensity and frequency.

Copula	Functional form	Relationship between $\theta$ and $\tau$
Gumbel copula	$F\left( {p,z} \right) = C\left( {u,v} \right) = \exp \left\{ { - \left[ {\left( { - \ln u} \right)^{\theta } + \left( { - \ln v} \right)^{\theta } } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}} } \right\}$	$\tau = 1 - \frac{1}{\theta },\theta \in [1,\infty )$
Clayton copula	$F\left( {p,z} \right) = C\left( {u,v} \right) = \left( {u^{ - \theta } + v^{ - \theta } - 1} \right)^{{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/ {\vphantom {{ - 1} \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}}$	$\tau = \frac{\theta }{2 + \theta },\theta \in (0,\infty )$
Frank copula	$F\left( {p,z} \right) = C\left( {u,v} \right) = - \frac{1}{\theta }\ln \left[ {1 + \frac{{\left( {\ell^{ - \theta u} - 1} \right)\left( {\ell^{ - \theta v} - 1} \right)}}{{\left( {\ell^{ - \theta } - 1} \right)}}} \right]$	$\tau = 1 + \frac{4}{\theta }\left[ {\frac{1}{\theta }\int_{0}^{\theta } {\frac{t}{{\ell^{t} - 1}}dt - 1} } \right],\theta \in R$

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Copula	Functional form	Relationship between \(\theta\) and \(\tau\)
Gumbel copula	\(F\left( {p,z} \right) = C\left( {u,v} \right) = \exp \left\{ { - \left[ {\left( { - \ln u} \right)^{\theta } + \left( { - \ln v} \right)^{\theta } } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}} } \right\}\)	\(\tau = 1 - \frac{1}{\theta },\theta \in [1,\infty )\)
Clayton copula	\(F\left( {p,z} \right) = C\left( {u,v} \right) = \left( {u^{ - \theta } + v^{ - \theta } - 1} \right)^{{{\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/ {\vphantom {{ - 1} \theta }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\theta $}}}}\)	\(\tau = \frac{\theta }{2 + \theta },\theta \in (0,\infty )\)
Frank copula	\(F\left( {p,z} \right) = C\left( {u,v} \right) = - \frac{1}{\theta }\ln \left[ {1 + \frac{{\left( {\ell^{ - \theta u} - 1} \right)\left( {\ell^{ - \theta v} - 1} \right)}}{{\left( {\ell^{ - \theta } - 1} \right)}}} \right]\)	\(\tau = 1 + \frac{4}{\theta }\left[ {\frac{1}{\theta }\int_{0}^{\theta } {\frac{t}{{\ell^{t} - 1}}dt - 1} } \right],\theta \in R\)

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