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.

The mathematical expression is:

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

As an alternative, the logistic S-function can be used.

The mathematical expression is:

where:

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:

In the figure: Y_m = 3.43, Y_n = 0.47, A = 0.112, B = -3.16, C = 1.42.

The third degree or cubic regression also offers a useful alternative.

The equation reads:

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.

• Maas–Hoffman model