Table 6.

Estimations of annual actual evapotranspiration and elasticity coefficient based on the Budyko hypothesis.

Source(Year)	Equation	Elasticity coefficient
Budyko (1974)	$\frac{E{T}_a}{P}=\sqrt{\frac{E{T}_p}{P}\tanh \left(\frac{P}{E{T}_p}\right)\left[1-\exp \left(-\frac{E{T}_p}{P}\right)\right]}$	${\mathrm{\epsilon }}_P=1+\frac{0.5\varphi {\left[\varphi \tanh \left(\frac{1}{\varphi }\right)\left(1-e^{-\varphi }\right)\right]}^{-0.5}\left[\left(\tanh \left(\frac{1}{\varphi }\right)-\frac{1}{\varphi }{\mathrm{sech}}^2(\frac{1}{\varphi })\right)\left(1-e^{-\varphi }\right)+\varphi \tanh \left(\frac{1}{\varphi }\right)e^{-\varphi }\right]}{1-{\left[\varphi \tanh \left(\frac{1}{\varphi }\right)\left(1-e^{-\varphi }\right)\right]}^{0.5}}$
		${\mathrm{\epsilon }}_{E{T}_p}=\frac{0.5\varphi {\left[\varphi \tanh \left(\frac{1}{\varphi }\right)\left(1-e^{-\varphi }\right)\right]}^{-0.5}\left[\left(\tanh \left(\frac{1}{\varphi }\right)-\frac{1}{\varphi }{\mathrm{sech}}^2\left(\frac{1}{\varphi }\right)\right)\left(1-e^{-\varphi }\right)+\varphi \tanh \left(\frac{1}{\varphi }\right)e^{-\varphi }\right]}{{\left[\varphi \tanh \left(\frac{1}{\varphi }\right)\left(1-e^{-\varphi }\right)\right]}^{0.5}-1}$
Fu et al. (1981)	$\frac{E{T}_a}{P}=1+\frac{E{T}_p}{P}-{\left[1+{\left(\frac{E{T}_p}{P}\right)}^m\right]}^{\frac{1}{m}}$	${\mathrm{\epsilon }}_P=\frac{{(1+{\varphi }^m)}^{(\frac{1}{m}-1)}}{{(1+{\varphi }^m)}^{(\frac{1}{m})}-\varphi }{\mathrm{\epsilon }}_{E{T}_p}=\frac{{\varphi }^m(1+{\varphi }^m)\left(\frac{1}{m}-1\right)-\varphi }{{(1+{\varphi }^m)}^{(\frac{1}{m})}-\varphi }$
		${\mathrm{\epsilon }}_m=-{(1+{\varphi }^m)}^{\left(\frac{1}{m}\right)}\left(\frac{\ln (1+{\varphi }^m)}{m^2}-\frac{{\varphi }^m\ln (\varphi )}{m(1+{\varphi }^m)}\right)$
Choudhury (1999)	$\frac{E{T}_a}{P}=\frac{E{T}_p}{{\left(P^n+E{T_p}^n\right)}^{\frac{1}{n}}}$	${\mathrm{\epsilon }}_P=\frac{{(1+{\varphi }^{n_0})}^{(1/n_0+1)}-{\varphi }^{n_0+1}}{(1+{\varphi }^{n_0})\left[{\left(1+{\varphi }^{n_0}\right)}^{(1/n_0)}-\varphi \right]}{\mathrm{\epsilon }}_{E{T}_p}=\frac{1}{(1+{\varphi }^{n_0})\left[1-{(1+{\varphi }^{-n_0})}^{\left(1/n_0\right)}\right]}$
		${\mathrm{\epsilon }}_{n_0}=\frac{\ln (1+{\varphi }^{n_0})+{\varphi }^{n_0}\ln (1+{\varphi }^{-n_0})}{n_0(1+{\varphi }^{n_0})[1-{(1+{\varphi }^{-n_0})}^{(1/n_0)}]}$