Uncertainty interval estimates for computing slope and aspect from a gridded digital elevation model

Abstract

The first order derivatives of a Digital Elevation Model (DEM) defined over a regular grid are usually computed without an uncertainty estimate. The standard procedure involves a compact 3 × 3 window. Using a Taylor expansion, an uncertainty interval for each partial derivative as a function of the cell size was devised using two estimates, either of different resolution or of different order. The intervals for slope and aspect can be derived afterwards. We carried out an experiment comparing some different estimates of the slope and aspect over a synthetic surface representative of a real topography and amenable to offer an exact derivative. The partial derivatives were numerically estimated with four different procedures: the Simple procedure defined by Jones over a 2 × 2 window, the Evans–Young procedure using a centered difference over a 3 × 3 window, and using a 5 × 5 window both with an extrapolated Evans–Young procedure and the expression due to Florinsky. The results confirm that intervals for both slope and aspect always included the exact value even after drastically increasing the cell size. Finally, a real case with an integer-valued DEM was considered, illustrating the combined effect of Representation and Truncation error.

Keywords:

• Slope
• aspect
• uncertainty
• accuracy
• DEM
• DTM
• geomorphometry

Acknowledgements

This research was motivated by the ‘Digital Elevation Model Intercomparison eXperiment’ (DEMIX) initiative, promoted by the Terrain Mapping Sub-Group (TMSG) of the Working Group on Calibration and Validation within the Committee on Earth Observation Satellites (CEOS-WGCV). Their interest is gratefully acknowledged. Early suggestions and data for the El Hierro test site by Prof. Peter Guth must also be acknowledged.

Disclosure statement

There are no relevant financial or non-financial competing interests to report.

Data and codes availability statement

The code and data that support the findings of this study are openly available in https://doi.org/10.6084/m9.figshare.19314302.v1.

Additional information

Notes on contributors

Carlos López-Vázquez

Carlos López-Vázquez: Industrial Engineer and M.Sc. in Fluid Mechanics by the Universidad de la República (Uruguay). Ph.D. in Geoinformatics by the Royal Institute of Technology (Sweden). Interested in geographic data quality, accuracy improvement of geographic data, Spatial Data Infrastructures, geometric conflation and numerical analysis. President of the Cartography Commission 2013–2017, Pan American Institute of Geography and History.