Van Genuchten–Gupta model explained

The Van Genuchten–Gupta model is an inverted S-curve applicable to crop yield and soil salinity relations.^[1] It is named after Martinus Theodore van Genuchten and Satyandra K. Gupta's work from the 1990s.

Equation

The mathematical expression is:

Y = (Y _{\rm})/(1+ (C / C _{50} ) ^{P})

where Y is the yield, Y_m 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: Y_m = 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 ^{\wedge} = (1)/(1+ \exp(A X ^{C} + B)) where:Y ^{\wedge} = (Y - Y _{\rm})/(Y _{\rm} - Y _{\rm})

with Y being the yield, Y_n the minimum Y, Y_m 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 Y_n=0 then the expression can be simplified to:

Y = (Y _{\rm})/(1+ \exp(A X ^{C} + B))

In the figure: Y_m = 3.43, Y_n = 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 = 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

Notes and References

1. M. Th. van Genuchten and S.K. Gupta, 1993. USDA-ARS, U.S. Salinity Laboratory 4500 Glenwood Drive, Riverside, California, USA, 92501. A reassessment of the Crop Tolerance Response Function. Journal of the Indian Society of Soil Science, Vol. 41, No. 4, pp 730–737.