A universal spectral analytical method for digital terrain modeling

ABSTRACT

There are three major mathematical problems in digital terrain analysis: (1) interpolation of digital elevation models (DEMs); (2) DEM generalization and denoising; and (3) computation of morphometric variables through calculating partial derivatives of elevation. Traditionally, these three problems are solved separately by means of procedures implemented in different methods and algorithms. In this article, we present a universal spectral analytical method based on high-order orthogonal expansions using the Chebyshev polynomials of the first kind with the subsequent Fejér summation. The method is intended for the processing of regularly spaced DEMs within a single framework including DEM global approximation, denoising, generalization, as well as calculating the partial derivatives of elevation and local morphometric variables.

The method is exemplified by a portion of the Great Rift Valley and central Kenyan highlands. A DEM of this territory (the matrix 480 × 481 with a grid spacing of 30″) was extracted from the global DEM SRTM30_PLUS. We evaluated various sets of expansion coefficients (up to 7000) to approximate and reconstruct DEMs with and without the Fejér summation. Digital models of horizontal and vertical curvatures were computed using the first and second partial derivatives of elevation derived from the reconstructed DEMs. To evaluate the approximation accuracy, digital models of residuals (differences between the reconstructed DEMs and the initial one) were calculated. The test results demonstrated that the method is characterized by a good performance (i.e., a distinct monotonic convergence of the approximation) and a high speed of data processing. The method can become an effective alternative to common techniques of DEM processing.

KEYWORDS:

• Chebyshev polynomials
• Fejér summation
• generalization
• denoising
• partial derivatives

Acknowledgement

The study was supported by RFBR grant 15-07-02484. The authors are grateful to Dr. I. Evans for the detailed review of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Russian Foundation for Basic Research (RFBR) [15-07-02484].