高精度曲面模型解算改进的Gauss-Seidel法

New method for solving high accuracy surface modeling

为了降低HASM的时间复杂度,采用一种改进Gauss-Seidel(GS)算法(MGS)解算HASM方程组。首先,从理论上分析了MGS算法收敛速度快于GS算法,然后以高斯合成曲面作为研究对象,用四组模拟试验表明,相同的网格数、达到相同的计算精度,MGS算法计算时间小于GS算法,且两种算法时间差与模拟区域网格数呈二次线性相关;固定网格数,使用相同的内迭代或者外迭代次数,MGS算法精度高于GS算法,但增加内迭代或者外迭代次数,GS算法同样收敛;MGS算法计算时间与网格数呈线性相关。MGS算法能够有效解决HASM模拟大区域的计算时间瓶颈,提高HASM运算速度。以甘肃省董志塬某测区SRTM3作为研究对象,基于MGS的HASM用于模拟DEM表明,HASM精度要高于传统的插值方法。

High accuracy surface modelling（HASM） constructed based on the fundamental theorem of surface is more accurate than the classical methods,but the computational speed of HASM is proportional to the third power of the total number of grid cells in the computational domain.In order to decrease the computational cost and improve the accuracy of HASM,this paper employed a modified Gauss-Seidel（MGS） to solve HASM.The fact that MGS is more accurate and faster than GS is proved in terms of theorem.Gauss synthetic surface was employed to comparatively analyze the simulation errors and the computing time of MGS and GS.The numerical tests showed that under the same simulation accuracy,MGS is faster than GS,and the time difference between MGS and GS is approximately proportional to the second power of the total number of grid cells.Under the same outer or inner iterative cycles,MGS is more accurate than GS.The computing time of MGS is proportional to the first power of the total number of grid cells.Compared with the direct methods for solving HASM,MGS greatly shortens the computing time of HASM.SRTM3（36°—37°N,107°—108°E） of Dongzhi tableland located in Gansu province was employed as a real word example to validate the accuracy of HASM based on MGS.In the example,about 50% of SRTM3 was used as validation points,the others for DEM simulation.The results indicated that RMSE of HASM based on MGS is about 2.4,1.8,1.3,2.7 times less than those of KRIGING,IDW,TIN and NEAREST.