Table 6 Estimations of annual actual evapotranspiration and elasticity coefficient based on the Budyko hypothesis.

Budyko (1974)

\(\frac{E{T}_{a}}{P}=\sqrt{\frac{E{T}_{p}}{P}\,\tanh (\frac{P}{E{T}_{p}})[1-\exp (-\frac{E{T}_{p}}{P})]}\)

\({{\rm{\varepsilon }}}_{P}=1+\frac{0.5\phi {[\phi \tanh (\frac{1}{\phi })(1-{e}^{-\phi })]}^{-0.5}[(\tanh (\frac{1}{\phi })-\frac{1}{\phi }{{\rm{sech}}}^{2}(\frac{1}{\phi }))(1-{e}^{-\phi })+\phi \,\tanh (\frac{1}{\phi }){e}^{-\phi }]}{1-{[\phi \tanh (\frac{1}{\phi })(1-{e}^{-\phi })]}^{0.5}}\)

\({{\rm{\varepsilon }}}_{E{T}_{p}}=\frac{0.5\phi {[\phi \tanh (\frac{1}{\phi })(1-{e}^{-\phi })]}^{-0.5}[(\tanh (\frac{1}{\phi })-\frac{1}{\phi }{{\rm{sech}}}^{2}(\frac{1}{\phi }))(1-{e}^{-\phi })+\phi \,\tanh (\frac{1}{\phi }){e}^{-\phi }]}{{[\phi \tanh (\frac{1}{\phi })(1-{e}^{-\phi })]}^{0.5}-1}\)

Fu et al. (1981)

\(\frac{E{T}_{a}}{P}=1+\frac{E{T}_{p}}{P}-{[1+{(\frac{E{T}_{p}}{P})}^{m}]}^{\frac{1}{m}}\)

\({{\rm{\varepsilon }}}_{P}=\frac{{(1+{\phi }^{m})}^{(\frac{1}{m}-1)}}{{(1+{\phi }^{m})}^{(\frac{1}{m})}-\phi }{{\rm{\varepsilon }}}_{E{T}_{p}}=\frac{{\phi }^{m}(1+{\phi }^{m})(\frac{1}{m}-1)-\phi }{{(1+{\phi }^{m})}^{(\frac{1}{m})}-\phi }\)

\({{\rm{\varepsilon }}}_{m}=-{(1+{\phi }^{m})}^{(\frac{1}{m})}(\frac{\mathrm{ln}(1+{\phi }^{m})}{{m}^{2}}-\frac{{\phi }^{m}\,\mathrm{ln}(\phi )}{m(1+{\phi }^{m})})\)

Choudhury (1999)

\(\frac{E{T}_{a}}{P}=\frac{E{T}_{p}}{{({P}^{n}+E{{T}_{p}}^{n})}^{\frac{1}{n}}}\)

\({{\rm{\varepsilon }}}_{P}=\frac{{(1+{\phi }^{{n}_{0}})}^{(1/{n}_{0}+1)}-{\phi }^{{n}_{0}+1}}{(1+{\phi }^{{n}_{0}})[{(1+{\phi }^{{n}_{0}})}^{(1/{n}_{0})}-\phi ]}{{\rm{\varepsilon }}}_{E{T}_{p}}=\frac{1}{(1+{\phi }^{{n}_{0}})[1-{(1+{\phi }^{-{n}_{0}})}^{(1/{n}_{0})}]}\)

\({{\rm{\varepsilon }}}_{{n}_{0}}=\frac{\mathrm{ln}(1+{\phi }^{{n}_{0}})+{\phi }^{{n}_{0}}\,\mathrm{ln}(1+{\phi }^{-{n}_{0}})}{{n}_{0}(1+{\phi }^{{n}_{0}})[1-{(1+{\phi }^{-{n}_{0}})}^{(1/{n}_{0})}]}\)