Topographic Surface Interpolation with Geodesic Distance
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摘要: 地球表面是个地形起伏的非欧氏空间，地图投影将大尺度的地球椭球面转换为容易分析计算的欧氏平面，但没有处理地形的不规则起伏。由于基本的表面距离不能量算，地形曲面上的空间分析广泛依靠近似计算，损害了地形模型的利用精度。本文在起伏明显的中小尺度地形上引入精确的测地距离，对密切依赖于空间距离的表面插值模型的度量改正方法及精度变化进行了研究。针对非欧氏度量引起的自相关矩阵非正定，提出测地距离限制的降维投影的解决办法，并以此改写主要插值算法。然后在一个10m级水平分辨率地形格网上，选择较少受环境干扰的高程插值建立基准实验，再以优选模型分别对0.3m级地形点云和0.25度级ERA地表气温进行插值检验，探讨测地距离的适用尺度。结果发现：测地距离下反距离权重与自然邻近插值精度较欧氏距离都有明确提升；径向基函数与普通克里金插值精度在欧氏距离下普遍较简单插值要好，在降维投影的测地距离上，其RMSE又分别较自身再下降约25%。这表明在起伏明显的中小尺度下，测地距离通常可以更好地利用地形模型的精度信息。相关方法可用于提升高程插值的精度，也可为地表建模提供参考。Abstract: 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.
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