&mu_cr=e^-1.109Fs (1) where &mu_cr is the membership grade of the accumulative suitability scores Fs for crop cr. The accumulative scores are obtained from the summation of all the suitability scores on land characteristics such as, topsoil and subsoil characteristics, drainage, flooding, texture, stoniness, depth of water penetration, furrow slice CEC, topsoil and subsoil salinity, topsoil and subsoil sodicity (ESP), etc.). The suitability scores are classified as follows: S1 low limitation accumulative score of >27, S2 slight limitation with accumulative score of 15-27, moderate limitation 5-14 and severe limitation with an accumulative score of <5. However, we use the fuzzy membership grades are meant to accommodate the indeterminate boundaries between these classes. Now to determine the actual land versatility, which we define as the potential suitability of a particular soil or tract of land to a variety of utilizations, we need to derive an aggregated membership grade for all of the utilization types. This needs a powerful aggregation operator. Choquet integrals are powerful aggregation operators used in multi-attribute decision-making models. It extends the usual weighted mean or averaging that permits the user to express interactions among criteria being aggregated. This could be achieved by means of fuzzy measures or other measures such us ordinal values. But the emphasis in this explanation is based on fuzzy measures. Let us suppose that X = {x1, …, xn} is a finite subset, and P(X) be power set of X, then a fuzzy measure on subset X is a function &mu: (X) → [0, 1] (2)

such that the following conditions are met:

i) &mu(f) ii) A &le B implies &mu(A)&le &mu(B) iii) &mu(A) = 1 (boundary conditions)

Now suppose that we want to aggregate certain number of criteria on the subset X such as fuzzified soil quality indicators on individual soil profiles and associated fuzzy measure , then the Choquet integral of the function f : X → with respect to R (the fuzzy measure) is defined by:

CI_&mu (f)=&sum[f(Xsub>S(i) - f(Xsub>S(i-1))]&mu(A_S(i)) (3)

Equation (3) can also be denoted as:

&int_Afd&mu (4)

Equation (4), the Choquet integral, is actually an expression of linear combination of the values, with particular sets of weights &mu. It reduces down to weighted mean when fuzzy set is additive (as in classical probability) such that &mu(AÙB) = &mu(A) + &mu(B)and if &mu(A&PiB) = 0; it also reduces to ordered weighted averaging (OWA) operator if &mu(A) = &mu(B)

The aggregated scores were mapped into a continuous digital GIS layer for further operations. Now going back to the core issue of pedodiversity, the functional pedodiversity is calculated as within-block variance of the aggregated versatility membership grade. In comparative terms the higher the variance, the larger the functional diversity of the soil. We further compare the functional pedodiversity with the widely used taxonomic pedodiversity as measured by abundance and richness of the soil classes grouped at the family level of the Australian Soil Classification of Isbell (1996). The computed functional pedodiversity provide a new way of looking at the soil in terms of its agro-ecosystem function. This has the potential to be expanded to include additional functional properties of the soil. Reference: Isbell, R.F. (1996) The Australian Soil Classification. CSIRO Publishing, Melbourne, Australia.