  Mathematical problems of digital terrain analysis include interpolation of
digital elevation models (DEMs), DEM generalization and denoising, and
computation of morphometric variables by calculation of partial derivatives of
elevation. Traditionally, these procedures are based on numerical treatments of
two-variable discrete functions of elevation. We developed a spectral
analytical method and algorithm based on high-order orthogonal expansions using
the Chebyshev polynomials of the first kind with the subsequent Fejer
summation. The method and algorithm are intended for DEM analytical treatment,
such as, DEM global approximation, denoising, and generalization as well as
computation of morphometric variables by analytical calculation of partial
derivatives. To test the method and algorithm, we used a DEM of the Northern
Andes including 230,880 points (the elevation matrix 480 $\times$ 481). DEMs
were reconstructed with 480, 240, 120, 60, and 30 expansion coefficients. The
first and second partial derivatives of elevation were analytically calculated
from the reconstructed DEMs. Models of horizontal curvature ($k_h$) were then
computed with the derivatives. A set of elevation and $k_h$ maps related to
different number of expansion coefficients well illustrates data generalization
effects, denoising, and removal of artifacts contained in the original DEM. The
test results demonstrated a good performance of the developed method and
algorithm. They can be utilized as a universal tool for analytical treatment in
digital terrain modeling.
