ON THE VARIATION OF YIELD VARIANCE WITH PLOT SIZE

The problem examined is that of evaluating the spatial covariance function of yield density, from a knowledge of the way yield variance varies with plot size and shape. Results are obtained in ? 3 for several kinds of plot. Results are also obtained (?4) on the dependence of the yield variance on plot geometry for very small and very large plots. Special attention is paid to the case for which the covariance follows a power law at large distances. It is well known that, in order to explain the observed variation of yield variance with size and shape of plot, it is necessary to allow the possibility of correlation between yield densities at any two points in the plot. (We shall restrict ourselves to the stationary case, for which the expected yield density is constant over the area.) Moreover, it appears that this spatial correlation must often fall off relatively slowly with increasing distance between the two points; as a power function of the distance rather than as an exponential. The same behaviour is shown by observations on yarn diameter, flood height (Feller, 1951), and response from population samples. The calculations of this article will apply to these cases, too, but for concreteness we shall continue to speak of plots and yields (although we shall sonmetimes use the word 'region' instead of 'plot ', indicating that we do not confine ourselves to two dimensions). The type of calculation most usually made is to evaluate the yield variance for a plot of definite size and shape, and for a given spatial covariance function p(s). However, the inverse calculation would probably be more useful in general: to determine the covariance function from a knowledge of yield variance as a function of plot geometry. Such a procedure would enable one to make use of experimental results to obtain at least a partial estimate of p(s). We consider this question in ? 3. A solution of one form of problein is most easily reached by using a Mellin transform, so that the covariance which falls off as a power of the distance,