Table 3 Base models and tested dynamic equations derived from the generalized algebraic difference approach (GADA) (M1–M5)
Base equation | Site-related parameters | Solution for X with initial conditions (\({H}_{0}\), \({T}_{0}\)) | Dynamic form | |
|---|---|---|---|---|
Hossfeld IV \(H=\frac{b\cdot {T}^{c}}{{T}^{c}+a}\) | \(a={\beta }_{1}{{R}_{0}}^{-1}\) \(b = {\beta }_{2}+{R}_{0}\) | \({X}_{0}={H}_{0}-{\beta }_{1}+\sqrt{{({H}_{0}-{\beta }_{1})}^{2}+\frac{2{H}_{0}{e}^{{\beta }_{2}}}{{T}_{0}^{c}}}\) | \(H={H}_{0}\cdot \frac{{T}^{c}\left({T}_{0}^{c} {X}_{0}+{e}^{\beta_2}\right)}{{T}_{0}^{c}\left({T}^{c}{X}_{0}+{e}^{\beta_2}\right)}\) | M1 |
Chapman-Richards: \(H={a\cdot (1-{e}^{-c\cdot T})}^{b}\) | \({\text{a}}={{\text{e}}}^{{X}_{0}}\) \(b={\beta }_{1}+\frac{{\beta }_{2}}{{X}_{0}}\) | \({X}_{o}=\frac{\left({\text{ln}}{H}_{0}-{\beta }_{1}{L}_{0}\right)+\sqrt{{\left({\text{ln}}{H}_{0}-{\beta }_{1}{L}_{0}\right)}^{2}-4{\beta }_{2}{L}_{0}}}{2}\) \({L}_{0}={\text{ln}}(1-{e}^{-c{T}_{0}})\) | \(H={H}_{0}\cdot {\left(\frac{1-{e}^{-cT}}{1-{e}^{-c{T}_{0}}}\right)}^{{\beta }_{1}+\frac{{\beta }_{2}}{{X}_{0}}}\) | M2 |
\({\text{a}}={{\text{e}}}^{{X}_{0}}\) \(b={\beta }_{1}+\frac{{\beta }_{2}}{{X}_{0}}\) | \({X}_{o}=\frac{\left({\text{ln}}{H}_{0}-{\beta }_{1}{\text{ln}}(1-{e}^{-c{T}_{0}})\right)}{1+{\beta }_{2}{\text{ln}}(1-{e}^{-c{T}_{0}})}\) | \(H={H}_{0}\cdot {\left(\frac{1-{e}^{-cT}}{1-{e}^{-c{T}_{0}}}\right)}^{{\beta }_{1}+{\beta }_{2}\cdot {X}_{0}}\) | M3 | |
Lundqvist-Korf: \(H=a \cdot {e}^{{-b\cdot T}^{-c}}\) | \({\text{a}}={{\text{e}}}^{{X}_{0}}\) \(b={\beta }_{1}+\frac{{\beta }_{2}}{{X}_{0}}\) | \({X}_{o}=\frac{{\beta }_{1}\bullet {T}^{-c}+{\text{ln}}{H}_{0}+{L}_{0}}{2}\) \({L}_{0}=\sqrt{{\left({\beta }_{1}{\cdot {T}_{0}}^{-c}+{\text{ln}}{H}_{0}\right)}^{2}+4{\beta }_{2}\cdot {{T}_{0}}^{-c}}\) | \(H={e}^{{X}_{0}}\cdot {e}^{-({\beta }_{1}+\frac{{\beta }_{2}}{{X}_{0}})\cdot {T}^{-c}}\) | M4 |
Weibull: \(H =a\cdot (1-{e}^{-b\cdot {T}^{c}})\) | \(a={X}_{0}\) \(b={\beta }_{1}+{\beta }_{2}{X}_{0}\) | \({X}_{0}=\frac{{\text{ln}}{H}_{0}-{\beta }_{1}\bullet {\text{ln}}(1-{e}^{-{T}^{c}})}{1+{\beta }_{2}\bullet {\text{ln}}(1-{e}^{-{T}^{c}})}\) | \(H={e}^{{X}_{0}+\left({\beta }_{1}+{\beta }_{2}{X}_{0}\right)\cdot {\text{ln}}\left(1-{e}^{-{T}^{c}}\right)}\) | M5 |