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Influence of DEM orientation on the error of slope calculation

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Abstract

Slope information is usually derived from digital elevation model (DEM) data. However, the orientation of DEM grids can affect the accuracy of slope estimation, which will in turn have an influence on the accuracy of other derived information based on slope. This research evaluates six commonly used algorithms for slope estimation: the second-order finite difference, the third-order finite difference, the third-order finite difference weighted by reciprocal of squared distance, the third-order finite difference weighted by reciprocal of distance and the frame finite difference and the simple difference. With synthetic data and systematic experiments by rotating DEM grids, the accuracy of the algorithms was analysed, empirically and theoretically. The main findings are as follows. When the DEM resolution is constant and one of first five algorithms is used, the error of slope calculation caused by orientation can be described by a sine function with a minimal positive period of π/2. For the last algorithm, i.e., the frame finite difference, the error of slope calculation could not be described by any periodic function. These theoretical error quantification functions were tested and verified using synthetic data sets based on six different Gaussian surfaces. The root mean square error of slope calculation by the first five algorithms can also be described by a sine function with a minimal positive period of π/4. The study also shows that the accuracy of the theoretical conclusions can be guaranteed when the expressions are expanded to the third-order by using the Taylor formula.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (40930531, 41001254, 41171320), the Resources and Environment Information System State Key Laboratory Open Fund Projects (2010KF0002SA). And the authors are grateful to the reviewers’ suggestions.

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Authors and Affiliations

  1. Spatial Information Research Center, Fuzhou University, Fuzhou, China

    Nan Chen & Chongcheng Chen

  2. Key Lab of Virtual Geographical Environment, Ministry of Education, Nanjing Normal University, Nanjing, China

    Guo-an Tang

  3. Department of Geography, University of South Carolina, Columbia, SC, USA

    Diansheng Guo

Authors
  1. Nan Chen
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  2. Guo-an Tang
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  3. Diansheng Guo
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  4. Chongcheng Chen
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Corresponding author

Correspondence to Nan Chen.

Additional information

Communicated by: H. A. Babaie

Appendix

Appendix

\( {\widehat{f}}_x \) of 2FD is:

$$ \begin{array}{l}{\widehat{f}}_y=\frac{1}{2g}\left({z}_8^{\prime }-{z}_2^{\prime}\right)=-{f}_x \sin t+{f}_y \cos t-\frac{1}{6}{g}^2{f}_{xxx}{\left( \sin t\right)}^3+\frac{1}{2}{g}^2{f}_{xxy}{\left( \sin t\right)}^2 \cos t\hfill \\ {}-\frac{1}{2}{g}^2{f}_{xyy}{\left( \cos t\right)}^2 \sin t+\frac{1}{6}{g}^2{f}_{yyy}{\left( \cos t\right)}^3\hfill \end{array} $$

\( {\widehat{f}}_x \) of 3FD:

$$ \begin{array}{l}-\frac{1}{24}{g}^2{f}_{xxx} \sin 3t-\frac{5}{24}{f}_{xxx}{g}^2 \sin t-\frac{5}{24}{f}_{xyy}{g}^2 \sin t+\frac{1}{8}{g}^2{f}_{xyy} \sin 3t-{f}_x \sin t+{f}_y \cos t+\hfill \\ {}\frac{5}{24}{f}_{xxy}{g}^2 \cos t+\frac{1}{8}{g}^2{f}_{xxy} \cos 3t-\frac{1}{24}{g}^2{f}_{yyy} \cos 3t+\frac{5}{24}{f}_{yyy}{g}^2 \cos t\hfill \end{array} $$

\( {\widehat{f}}_x \) of 3FDWRSD:

$$ \begin{array}{l}-\frac{1}{6}{f}_{xxx}{g}^2{\left( \sin t\right)}^3+\frac{1}{6}{f}_{yyy}{g}^2{\left( \cos t\right)}^3-\frac{1}{4}{f}_{xyy}{g}^2{\left( \sin t\right)}^3+\frac{1}{4}{f}_{xxy}{g}^2{\left( \cos t\right)}^3+{f}_y \cos t\hfill \\ {}-{f}_x \sin t+\frac{1}{4}{f}_{yyy}{g}^2{\left( \sin t\right)}^2 \cos t-\frac{1}{4}{f}_{xxx}{g}^2{\left( \cos t\right)}^2 \sin t\hfill \end{array} $$

\( {\widehat{f}}_x \) of 3FDWRD:

$$ \begin{array}{l}0.094670{g}^2{f}_{xyy} \sin 3t-0.198223{g}^2{f}_{xyy} \sin t+0.198223{g}^2{f}_{xxy} \cos t+\hfill \\ {}0.094670{g}^2{f}_{xxy} \cos 3t-0.031557{g}^2{f}_{yyy} \cos 3t+0.198223{g}^2{f}_{yyy} \cos t-\hfill \\ {}0.031557{g}^2{f}_{xxx} \sin 3t-0.198223{g}^2{f}_{xxx} \sin t+1.000000{f}_y \cos t-1.000000{f}_x \sin t\hfill \end{array} $$

\( {\widehat{f}}_x \) of FFD:

$$ \begin{array}{l}\frac{1}{4}{g}^2{f}_{xxy} \cos 3t-\frac{1}{4}{f}_{xyy}{g}^2 \sin t+\frac{1}{4}{g}^2{f}_{xyy} \sin 3t-{f}_x \sin t+\frac{1}{4}{f}_{xxy}{g}^2 \cos t+{f}_y \cos t\hfill \\ {}-\frac{1}{12}{g}^2{f}_{yyy} \cos 3t+\frac{1}{4}{f}_{yyy}{g}^2 \cos t-\frac{1}{12}{g}^2{f}_{xxx} \sin 3t-\frac{1}{4}{f}_{xxx}{g}^2 \sin t\hfill \end{array} $$

\( {\widehat{f}}_x \) of SD:

$$ \begin{array}{l}-{f}_x \sin t+{f}_y \cos t-\frac{1}{2}{f}_{xx}g{\left( \sin t\right)}^2+g{f}_{xy} \sin t \cos t-\frac{1}{2}{f}_{yy}g{\left( \cos t\right)}^2-\frac{1}{6}{g}^2{f}_{xx x}{\left( \sin t\right)}^3\hfill \\ {}+\frac{1}{2}{g}^2{f}_{xx y}{\left( \sin t\right)}^2 \cos t-\frac{1}{2}{g}^2{f}_{xy y}{\left( \cos t\right)}^2 \sin t+\frac{1}{6}{g}^2{f}_{yy y}{\left( \cos t\right)}^3\hfill \end{array} $$

\( {\widehat{f}}_y \) of 2FD:

$$ \begin{array}{l}{f}_x \cos t+{f}_y \sin t+\frac{1}{6}{g}^2{f}_{xxx}{\left( \cos t\right)}^3+\frac{1}{2}{g}^2{f}_{xxy}{\left( \cos t\right)}^2 \sin t+\frac{1}{2}{g}^2{f}_{xyy}{\left( \sin t\right)}^2 \cos t\hfill \\ {}+\frac{1}{6}{g}^2{f}_{yyy}{\left( \sin t\right)}^3\hfill \end{array} $$

\( {\widehat{f}}_y \) of 3FD:

$$ \begin{array}{l}{f}_x \cos t+{f}_y \sin t+\frac{1}{6}{g}^2{f}_{xxx}{\left( \cos t\right)}^3+\frac{1}{2}{g}^2{f}_{xxy}{\left( \cos t\right)}^2 \sin t+\frac{1}{2}{g}^2{f}_{xyy}{\left( \sin t\right)}^2 \cos t\hfill \\ {}+\frac{1}{6}{g}^2{f}_{yyy}{\left( \sin t\right)}^3\hfill \end{array} $$

\( {\widehat{f}}_y \) of 3FDWRSD:

$$ \begin{array}{l}\frac{1}{4}{f}_{yyy}{g}^2{\left( \cos t\right)}^2 \sin t+\frac{1}{4}{f}_{xxx}{g}^2{\left( \sin t\right)}^2 \cos t+\frac{1}{4}{f}_{xyy}{g}^2{\left( \cos t\right)}^3+\frac{1}{4}{f}_{xxy}{g}^2{\left( \sin t\right)}^3\\ {}+\frac{1}{6}{g}^2{f}_{yyy}{\left( \sin t\right)}^3+\frac{1}{6}{g}^2{f}_{xxx}{\left( \cos t\right)}^3+{f}_y \sin t+{f}_x \cos t\end{array} $$

\( {\widehat{f}}_y \) of 3FDWRD:

$$ \begin{array}{l}-0.094670{g}^2{f}_{xxy} \sin 3t+0.198223{g}^2{f}_{xxy} \sin t+0.198223{g}^2{f}_{xyy} \cos t+\hfill \\ {}0.094670{g}^2{f}_{xyy} \cos 3t+0.031557{g}^2{f}_{yyy} \sin 3t+0.198223{g}^2{f}_{yyy} \sin t-\hfill \\ {}0.031557{g}^2{f}_{xxx} \cos 3t+0.198223{g}^2{f}_{xxx} \cos t+1.000000{f}_y \sin t+1.000000{f}_x \cos t\hfill \end{array} $$

\( {\widehat{f}}_y \) of FFD:

$$ \begin{array}{l}\frac{1}{12}{g}^2{f}_{yyy} \sin 3t+\frac{1}{4}{f}_{yyy}{g}^2 \sin t+\frac{1}{4}{f}_{xyy}{g}^2 \cos t+\frac{1}{4}{g}^2{f}_{xyy} \cos 3t+\frac{1}{4}{f}_{xxx}{g}^2 \cos t\hfill \\ {}+{f}_y \sin t+{f}_x \cos t+\frac{1}{4}{f}_{xxy}{g}^2 \sin t-\frac{1}{4}{g}^2{f}_{xxy} \sin 3t-\frac{1}{12}{g}^2{f}_{xxx} \cos 3t\hfill \end{array} $$

\( {\widehat{f}}_y \) of SD:

$$ \begin{array}{l}{f}_x \cos t+{f}_y \sin t-\frac{1}{2}{f}_{xx}g{\left( \cos t\right)}^2-g{f}_{xy} \sin t \cos t-\frac{1}{2}{f}_{yy}g{\left( \sin t\right)}^2+\hfill \\ {}\frac{1}{6}{g}^2{f}_{xx x}{\left( \cos t\right)}^3+\frac{1}{2}{g}^2{f}_{xx y}{\left( \cos t\right)}^2 \sin t+\frac{1}{2}{g}^2{f}_{xy y}{\left( \sin t\right)}^2 \cos t+\frac{1}{6}{g}^2{\left( \sin t\right)}^3{f}_{yy y}\hfill \end{array} $$

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Chen, N., Tang, Ga., Guo, D. et al. Influence of DEM orientation on the error of slope calculation. Earth Sci Inform 7, 277–285 (2014). https://doi.org/10.1007/s12145-014-0154-6

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