Dimension and entropy in the soil‐covered landscape

Abstract

AbstractBoth the Hausdorff dimension and the K‐entropy supply a measure of the irregularity of the landspace surface. The relationship between the two measures is investigated over a variety of terrains in Britain and a method of calculating the entropy is checked against an independent estimate of the dimension with reasonable agreement. The calculation of the K‐entropy requires that the landscape surface be represented by an homogenous ergodic random field. This condition is satisfied by the tendency of soil‐covered terrains to progressively approximate to a form well represented by a Gaussian field.Gaussian random fields can either be very smooth, possessing derivatives of all orders at every point or they are highly irregular and non‐differentiable everywhere. Within the regular conceptualization the Rice‐Kac theory is used to predict the numbers of crossing points and the extent of excursion sets. These predictions are tested against an example terrain from the High Weald of East Sussex with very good agreement, apart from predictions of local maxima. A worked example of the calculation of the K‐entropy is given as an appendix.The potential role of information theory in geomorphology extends beyond the use made of entropy in this investigation. In particular ergodic theory has important practical and theoretical implications.