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Computation of the third-order partial derivatives from a digital elevation model

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Loci of extreme curvature of the topographic surface may be defined by the derivation function (T) depending on the first-, second-, and third-order partial derivatives of elevation. The loci may partially describe ridge and thalweg lines. The first- and second-order partial derivatives are commonly calculated from a digital elevation model (DEM) by fitting the second-order polynomial to a 3×3 window. This approach cannot be used to compute the third-order partial derivatives and T. We deduced formulae to estimate the first-, second-, and third-order partial derivatives from a DEM fitting the third-order polynomial to a 5×5 window. The polynomial is approximated to elevation values of the window. This leads to a local denoising that may enhance calculations. Under the same grid size of a DEM and root mean square error (RMSE) of elevation, calculation of the second-order partial derivatives by the method developed results in significantly lower RMSE of the derivatives than that using the second-order polynomial and the 3×3 window. An RMSE expression for the derivation function is deduced. The method proposed can be applied to derive any local topographic variable, such as slope gradient, aspect, curvatures, and T. Treatment of a DEM by the method developed demonstrated that T mapping may not substitute regional logistic algorithms to detect ridge/thalweg networks. However, the third-order partial derivatives of elevation can be used in digital terrain analysis, particularly, in landform classifications.

Keywords: Curvature; Digital terrain modelling; Drainage network; Surface

Document Type: Research Article


Affiliations: Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region 142290, Russia

Publication date: 2009-02-01

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