Multiscale sources of spatial variation in soil. I. The application of fractal concepts to nested levels of soil variation

Summary Soil variation has often been considered to be composed of‘functional’ or ‘systematic’ variation that can be explained, and random variation (‘noise’) that is unresolved. The distinction between systematic variation and noise is entirely scale dependent because increasing the scale of observation almost always reveals structure in the noise. The white noise concept of a normally distributed random function must be replaced to take into account the nested, autocorrelated and scale‐dependent nature of unresolved variations. Fractals are a means of studying these phenomena. The Hausdorff‐Besicovitch dimension D is introduced as a measure of the relative balance between long‐ and short‐range sources of variation; D can be estimated from the slope of a double logarithmic plot of the semivariogram. The family of Brownian linear fractals is introduced as the model of ideal stochastic fractals. Data from published and unpublished soil studies are examined and compared with other environmental data and simulated fractional Brownian series. The soil data are fractals because increasing the scale of observation continues to reveal more and more detail. But soil does not vary exactly as a Brownian fractal because its variation is controlled by many independent processes that can cause abrupt transitions or local second order stationarity. Estimates of D values show that soil data usually have a much higher proportion of short‐range variation than landform or ground water surfaces. The practical implication is that interpolation of soil property values based on observations from single 10 cm auger observations will be unsatisfactory and that some method of bulking or block kriging should be used whenever longrange variations need to be mapped.