The Van Genuchten–Gupta model is an inverted S-curve applicable to crop yield and soil salinity relations. It is named after Martinus Theodore van Genuchten and Satyandra K. Gupta's work from the 1990s.
Equation The mathematical expression is:
Y = Y m 1 + ( C / C 50 ) P {\displaystyle Y={\frac {Y_{\rm {m}}}{1+(C/C_{50})^{P}}}}
where Y is the yield, Ym is the maximum yield of the model, C is salt concentration of the soil, C50 is the C value at 50% yield, and P is an exponent to be found by optimization and maximizing the model's goodness of fit to the data. In the figure: Ym = 3.1, C50 = 12.4, P = 3.75
Alternative one
As an alternative, the logistic S-function can be used. The mathematical expression is:
Y ∧ = 1 1 + exp ( A X C + B ) {\displaystyle Y^{\wedge }={\frac {1}{1+\exp(AX^{C}+B)}}}
where:
Y ∧ = Y − Y n Y m − Y n {\displaystyle Y^{\wedge }={\frac {Y-Y_{\rm {n}}}{Y_{\rm {m}}-Y_{\rm {n}}}}}
with Y being the yield, Yn the minimum Y, Ym the maximum Y, X the salt concentration of the soil, while A, B and C are constants to be determined by optimization and maximizing the model's goodness of fit to the data. If the minimum Yn=0 then the expression can be simplified to:
Y = Y m 1 + exp ( A X C + B ) {\displaystyle Y={\frac {Y_{\rm {m}}}{1+\exp(AX^{C}+B)}}}
In the figure: Ym = 3.43, Yn = 0.47, A = 0.112, B = -3.16, C = 1.42.
Alternative two
The third degree or cubic regression also offers a useful alternative. The equation reads:
Y = A X 3 + B X 2 + C X + D {\displaystyle Y=AX^{3}+BX^{2}+CX+D}
with Y the yield, X the salt concentration of the soil, while A, B, C and D are constants to be determined by the regression. In the figure: A = 0.0017, B = 0.0604, C=0.3874, D = 2.3788. These values were calculated with Microsoft Excel The curvature is more pronounced than in the other models.
See also Maas–Hoffman model
References