Exploring scaling laws in surface topography

Surface topography affects many soil properties and processes, particularly surface water storage and runoff. Application of fractal analysis helps understand the scaling laws inherent in surface topography at a wide range of spatial scales and climatic regimes. In this research, a high resolution digital elevation model with a 3 mm resolution on one side of the spectrum and large scale DEMs, with a 500 m spatial resolution on the other side were used to explore scaling laws in surface topography. With appropriate exploratory spatial data analysis of both types of data sets, two conventional computational procedures – variogram and Box Counting Methods (BCM) – address scaling laws in surface topography. The results respect scaling laws in surface topography to some extent as neither the plot treatment nor the direction treatment has a significant impact on fractal dimension variability. While in the variogram method, the change in slope in Richardson’s plots appears to be the norm rather than the exception; Richardson’s plots resulting from box counting implementation lack such mathematical behavior. These breaks in slope might have useful implications for delineating homogeneous hydrologic units and detecting change in trend in hydrologic time series. Furthermore, it is shown that fractal dimension cannot be used to capture anisotropic variabilities both within and among micro-plots. In addition, its numerical value remains insignificant at the 5% level in moving from one direction to another and also from one spatial scale to another while the ordinate intercept could discriminate the surface roughness variability from one spatial scale to another.