Evidence for conformal invariance of crop yields

The aim of this paper is to study the nature of spatial correlation of yields of agricultural crops. The focus is primarily on natural or non-anthropogenic spatial variation, patterns that cannot be explained by topography, by variety or treatment effects, or by agricultural practices. Conformal invariance implies stationarity and isotropy, and also determines the rate of decay of spatial correlations. The resulting Gaussian model is studied empirically to see whether it describes satisfactorily the pattern of spatial correlations observed in field trials of various crops. By embedding the law in a larger statistical model, a convolution of white noise and the Matérn class having a range parameter λ −1 and a smoothness parameter ν , and by gathering data of sufficient range and quantity, the model predictions were tested. Twenty-five examples of crop yields are studied, including cereals, root crops and other vegetables, nut, citrus and alfalfa yields. At the scale of typical field trials, we find that non-anthropogenic variation is reasonably close to isotropic. Furthermore, we find consistent evidence that the range parameter tends to be large and the smoothness parameter small. The large value of the range parameter confirms Fairfield Smith (Fairfield Smith 1938 J. Agric. Sci. 28 , 1–23), who found that spatial correlation in agricultural processes decreases with distance, but at a slower rate than exponential. The small value of the smoothness parameter means that, by Matérn standards, agricultural processes are rough. For each of the examples studied, the limiting model fits the data just as well as the full model, in reasonable agreement with the hypothesis of the conformal model that ( λ , ν )=(0, 0) for all crops in all seasons.