Simulating Precision Farming: An Application of Mathematica

Michael Weiss

Precision farming is an important new "hi-tech" approach to farm management with the potential to revolutionize traditional agricultural practices. This emerging technology allows farmers to create finely detailed maps, expressible mathematically as surfaces, that describe important characteristics of a farm field, such as fertilizer requirements, by specific location. Computer-guided machinery keyed to such maps can assist a farmer in managing a farm field while responding to the field's spatially variable characteristics. Before the advent of precision technology, large farm fields had to be managed in a spatially uniform manner. The new approach may lead to more efficient application of chemical inputs and decreased runoff of excess fertilizer into the environment. Mathematica has proved to be an excellent tool for bridging the gap between a formal mathematical model of precision farming and the empirical data needed to test theoretical conjectures against real-world farm fields.

This talk describes a complex Mathematica simulation model developed to study the economic and environmental implications of precision farming. The model's crucial reliance on Mathematica's symbolic, numerical, and graphical capabilities will be illustrated. As an example, to capture the way in which the surface that gives the fertilizer requirement at each location depends, in turn, on the surface that represents the amount of fertilizer already in the soil, this model defines a mathematical operator that, when given one surface as an input, generates another as an output. This operator is purely symbolic, as are the subsequent formulas in which it appears. Thus, the operator and the formulas containing it can be evaluated at any desired soil fertilizer surface. Elsewhere in the model, Mathematica's random number generator is used for the Monte Carlo numerical integration of empirically generated surfaces in order to arrive at total quantities. Thus, for instance, the Monte Carlo integral of the crop-yield surface is calculated in order to obtain the total quantity of crop production. Using another numerical technique, the model also generates, and calculates with, not only random numbers but also random surfaces. Mathematica's graphics capabilities allow the rendering of various unusual images that arise in this subject. For example, an illustration of the traditionally recommended Illinois fertilizer application rule requires a surface cut by two planes. In this and other cases, Mathematica's ability to combine multiple surfaces with correct masking permits difficult concepts to be visualized in three dimensions. This presentation is intended for Mathematica experts and novices alike.