Topographic Surface Interpolation with Geodesic Distance

Objectives: The Earth's surface is a non-Euclidean curved space with topographic relief. Due to the difficulty to measure surface distances, analysis and calculations on topographic surfaces rely extensively on approximate solutions, which hinders precise utilization of the topography models. In this paper, the exact geodesic distance mapping is introduced on the undulating medium and small-scale terrain, and the metric correction and accuracy evaluation of the surface interpolation model are investigated. Methods: In view of the non-positive definite autocorrelation matrix caused by non-Euclidean metric, this paper proposes a recalculation solution after the topographic projection through dimension reduction constrained by the geodesic distance, and rewrites the main interpolation algorithm. Then, on a relatively regular 10m-level horizontal resolution terrain grid, select elevation variables that are less affected by environmental interference to establish an interpolation benchmark test. Then, use the optimized model to perform interpolation tests on 0.3m topographic point clouds and 0.25 rad ERA surface air temperature, respectively, to explore the applicable scale of the geodesic distance. Resulsts: The results showed that the simple inverse distance weighting and natural neighbor interpolation accuracy under geodesic distance had a clear improvement compared to Euclidean distance; The accuracy of radial basis function and ordinary Kriging interpolation is generally better than that of simple interpolation under Euclidean distance. In terms of geodesic distance of projection, their RMSE is further improved by about 25% compared to themselves. Conclusions: This indicates that at small and medium scales with significant relieves, geodesic distance can often better utilize the accuracy information of terrain models. The related methods can be used to improve the accuracy of elevation interpolation and provide reference for surface modeling.