高精度曲面建模优化方案

HASM optimization based on the improved difference scheme

目的为了进一步提高高精度曲面建模(HASM)方法的模拟精度和计算速度,进而拓宽该模型的应用领域,提出了新的HASM模型算法。方法采用新的差分格式计算HASM高斯方程中的一阶偏导数,以HASM预处理共轭梯度算法为例分析改进的差分格式对HASM的优化效果。结果数值实验表明:在计算耗时及内存需求不变的情况下,采用新的差分格式的HASM算法可以显著提高单次迭代的模拟精度,同时能够降低关键采样点缺失对模拟结果精度的影响。进一步研究发现,当HASM采用新差分格式与原始差分格式(中心差分)交替迭代时,能够快速降低模拟结果的误差。结论本文算法当达到指定的精度条时能够显著减小计算耗时,同时还能降低关键采样点缺失对模拟结果的影响。

Objective In order to solve the error problem in the spatial surface simulation, the high accuracy surface model- ing （HASM） method, which is based on the fundamental theorem of the surface theory, has been developed during the past decades. Numerical tests have already shown that HASM is much more accurate than the classical methods such as Kring, inverse weighting （IDW） method, and splines. In addition, surface modeling of digital elevation model and spatial simula- tion of soil properties also indicate that the HASM indeed increases interpolation accuracy. However, the huge time con- sumption of the HASM limits its application at large scales. Numerical analysis have found that the main reason that lead to large time consumption is that the out-iteration process consume too much time, so an effective way to improve the simula- tion efficiency is speed up the out-iteration process. Therefore we propose a new method to construct the difference equa- tions of the HASM in this paper. Method The new difference equation employs five grids to compute the first difference, while the original method only uses three grids. First, we prove that the modified difference equation could really improve the simulation accuracy by formula derivation. Then we design several numerical tests to verify the simulation accuracy of the new method. Sampling rate and the spatial position of the sample points are considered in the numerical tests. Three kinds of sampling rate are designed （0. 5% , 1% , and 2% ） to analysis the new method＇s advantage for different sampling rates. One of the numerical tests removes key points of the study area （valley bottom of the simulation surface） to analysisthe influence of the missing of key sampling points to the origin and new methods. Result numerical test results show that HASM that adopted the new difference equation can improve the single iteration accuracy significantly. At the same time, the contrast experiments indicate that the modified HASM could decreases the influence of the key missing sample points, which is very important in the practical application because it is often impossible to identify all the key sample points. It should be pointed that the modified HASM does not require increasing memory to obtain these advantages. Furthermore the study reveals that the out-iteration process that employed the modified difference equations and the origin difference equa- tions in turn could speed up the increasing of the simulation accuracy. This trait is of great help to arrive at the specified simulation accuracy quickly in practical application. Conclusion Compared to the classical interpolation methods, HASM possess significant advantages on simulation accuracy. However the huge time consumption limits its wide application. Ai- ming to perfect HASM＇s simulation efficiency, we propose the modified HASM algorithm by building new difference equa- tions. Compared to the origin one, the modified HASM has three advantages： First, the modified HASM could improve the single iteration accuracy significantly. Second, the modified HASM could get better simulation results when some key sam- ple points are missing. Third we could accelerate the error decreasing in the out-iteration process when using the modified and origin HASM.