The choice of window size in approximating topographic surfaces from Digital Elevation Models

Abstract

Quantitative surface analysis through quadratic modelling of Digital Elevation Models (DEMs) is a promising tool for automatically describing the physical environment in ecological studies of terrestrial landscapes. Fundamental topographic variables such as slope, aspect, plan and profile curvature can be simply calculated from the parameters of a conic equation fitted to a DEM window through the least-squares method. The scale of the analysis, defined by the size of the DEM window used to fit the conic equation, affects both the estimated value of the topographic variables and the propagation of elevation errors to derived topographic variables. The least-squares method is amenable to the analytical treatment of the propagation of elevation errors to the derived topographical variables. A general analytical model of error propagation is presented that accounts for the effects of window size and of spatial autocorrelation in elevation errors. The method is based on the Taylor approximation of the least-square fitting equation and allows for the presence of stationary autocorrelation in the elevation errors. In numerical simulations with DEMs from British Columbia, Canada, it is shown that increasing the size of evaluation windows effectively reduces the propagation of elevation errors to the derived topographic variables. However, this was obtained at the expense of topographic detail. A methodology is proposed to evaluate quantitatively the loss of topographic detail through a χ ²-test of the corrected residuals in the immediate neighbourhood of the evaluation point. This methodology, in combination with the analytical model of error propagation, can be used to select the scale or range of scales at which to calculate topographic variables from a DEM.

Acknowledgements

The TRIM digital elevation models used were provided by the Fort St John Forest District, British Columbia Ministry of Forests, Canada, in the context of a research project funded by Forest Renewal BC. M. A. gratefully acknowledges support from the R. Howard Webster Foundation Fellowship for members of Green College UBC and from the Canfor Corporation Fellowship in Forest Ecosystem Management. The authors thank the three anonymous reviewers for comments, which improved the quality and clarity of the paper.

Notes

^*Present address: Department of Organismic & Evolutionary Biology, Harvard University, HUH 22 Divinity Avenue, Cambridge, MA 02138, USA; e-mail: [email protected]

Additional information

Notes on contributors

M. Albani*

*Present address: Department of Organismic & Evolutionary Biology, Harvard University, HUH 22 Divinity Avenue, Cambridge, MA 02138, USA; e-mail: [email protected]