The fractal properties of topography: A comparison of methods

Klinkenberg, Brian; Goodchild, Michael F.

doi:10.1002/esp.3290170303

Cited by 176 publications

(137 citation statements)

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“…To account for this problem it is assumed that topography is fractal. Following Klinkenberg and Goodchild (1992) and Zhang et al (1999), slope can be expressed as a function of the spatial scale by applying the variogram equation. The variogram equation is used to approximate the fractal dimension of topography and is expressed as follows:…”

Section: Scaling Slope According To the Fractal Methodsmentioning

confidence: 99%

Improving the global applicability of the RUSLE model – adjustment of the topographical and rainfall erosivity factors

et al. 2015

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Abstract. Large uncertainties exist in estimated rates and the extent of soil erosion by surface runoff on a global scale. This limits our understanding of the global impact that soil erosion might have on agriculture and climate. The Revised Universal Soil Loss Equation (RUSLE) model is, due to its simple structure and empirical basis, a frequently used tool in estimating average annual soil erosion rates at regional to global scales. However, large spatial-scale applications often rely on coarse data input, which is not compatible with the local scale on which the model is parameterized. Our study aims at providing the first steps in improving the global applicability of the RUSLE model in order to derive more accurate global soil erosion rates.We adjusted the topographical and rainfall erosivity factors of the RUSLE model and compared the resulting erosion rates to extensive empirical databases from the USA and Europe. By scaling the slope according to the fractal method to adjust the topographical factor, we managed to improve the topographical detail in a coarse resolution global digital elevation model.Applying the linear multiple regression method to adjust rainfall erosivity for various climate zones resulted in values that compared well to high resolution erosivity data for different regions. However, this method needs to be extended to tropical climates, for which erosivity is biased due to the lack of high resolution erosivity data.After applying the adjusted and the unadjusted versions of the RUSLE model on a global scale we find that the adjusted version shows a global higher mean erosion rate and more variability in the erosion rates. Comparison to empirical data sets of the USA and Europe shows that the adjusted RUSLE model is able to decrease the very high erosion rates in hilly regions that are observed in the unadjusted RUSLE model results. Although there are still some regional differences with the empirical databases, the results indicate that the methods used here seem to be a promising tool in improving the applicability of the RUSLE model at coarse resolution on a global scale.

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Section: Scaling Slope According To the Fractal Methodsmentioning

confidence: 99%

Improving the global applicability of the RUSLE model – adjustment of the topographical and rainfall erosivity factors

et al. 2015

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“…The surface roughness strongly affects the redistribution of snow by wind and gravitational processes (Jost et al, 2007), and can be seen as the capability of the surface to trap snow. As suggested by Lehning et al (2011), we also tested the fractal dimension D and the ordinal intercept γ of the semi-variogram, which have been identified as good measures for the surface roughness (Goodchild and Mark, 1987;Power and Tullis, 1991;Klinkenberg, 1992;Klinkenberg and Goodchild, 1992;Xu et al, 1993;Sun et al, 2006;Taud and Parrot, 2006).…”

Section: Statistical Modelsmentioning

confidence: 99%

Statistical modelling of the snow depth distribution in open alpine terrain

Grünewald¹,

Stötter

Pomeroy

et al. 2013

Hydrol. Earth Syst. Sci.

120

142

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Abstract. The spatial distribution of alpine snow covers is characterised by large variability. Taking this variability into account is important for many tasks including hydrology, glaciology, ecology or natural hazards. Statistical modelling is frequently applied to assess the spatial variability of the snow cover. For this study, we assembled seven data sets of high-resolution snow-depth measurements from different mountain regions around the world. All data were obtained from airborne laser scanning near the time of maximum seasonal snow accumulation. Topographic parameters were used to model the snow depth distribution on the catchment-scale by applying multiple linear regressions. We found that by averaging out the substantial spatial heterogeneity at the metre scales, i.e. individual drifts and aggregating snow accumulation at the landscape or hydrological response unit scale (cell size 400 m), that 30 to 91 % of the snow depth variability can be explained by models that are calibrated to local conditions at the single study areas. As all sites were sparsely vegetated, only a few topographic variables were included as explanatory variables, including elevation, slope, the deviation of the aspect from north (northing), and a wind sheltering parameter. In most cases, elevation, slope and northing are very good predictors of snow distribution. A comparison of the models showed that importance of parameters and their coefficients differed among the catchments. A "global" model, combining all the data from all areas investigated, could only explain 23 % of the variability. It appears that local statistical models cannot be transferred to different regions. However, models developed on one peak snow season are good predictors for other peak snow seasons.

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“…It is probable that the inclusion of these phenomena will lead to stationary states characterized by more a complex invariant measure than those above, thus producing a more complete stochastic theory of complex SOC systems, with complex stationary states. Indeed studies of DEMs show that topography exhibits fractal; see Klinkenberg and Goodchild [40], and multi-fractal; see Lavallée, Lovejoy and Schertzer [43], structure.…”

Section: More General Landsurfacesmentioning

confidence: 99%

The Stochastic Theory of Fluvial Landsurfaces

2006

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A stochastic theory of fluvial landsurfaces is developed for transport-limited erosion, using well-established models for the water and sediment fluxes. The mathematical models and analysis is developed showing that some aspects of landsurface evolution can be described by Markovian stochastic processes. The landsurfaces are described by nondeterministic stochastic processes, characterized by a statistical quantity the variogram, that exhibits characteristic scalings. Thus the landsurfaces are shown to be SOC (Selforganized-critical) systems, possessing both an initial transient state and a stationary state, characterized by respectively temporal and spatial scalings. The mathematical theory of SOC systems is developed and used to identify three stochastic processes that shape the * and the University of Iceland, 107 Reykjavík 1 surface. The SOC theory of landsurfaces reproduces established numerical results and measurements from DEMs (digital elevation models).

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The fractal properties of topography: A comparison of methods

Cited by 176 publications

References 50 publications

Improving the global applicability of the RUSLE model – adjustment of the topographical and rainfall erosivity factors

Improving the global applicability of the RUSLE model – adjustment of the topographical and rainfall erosivity factors

Statistical modelling of the snow depth distribution in open alpine terrain

The Stochastic Theory of Fluvial Landsurfaces

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