基于DEM的对称13点局部窗口坡度算法分析
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摘要
局部曲面拟合法是基于DEM的计算坡度的常用方法之一,采用不同的拟合曲面和局部移动窗口将得到不同精度的计算模型,探索不同的局部窗口类型可以丰富局部曲面拟合法在数字地形分析中的应用。本文考虑不同距离临近点的高程信息,基于3×3局部窗口提出了一种由两种不同倍率、与主轴(x轴、y轴)呈45°夹角的格网点构成的新窗口——对称13点局部窗口,推导了三阶曲面拟合该窗口的坡度计算公式,并利用典型数学曲面对该模型的计算精度进行了分析。研究分析认为,相对于现有的坡度计算模型,三阶曲面拟合对称13点窗口模型能显著地提高坡度计算精度,但对误差的平滑能力较差,适用于高精度DEM。本文的研究扩展了曲面拟合法在数字地形分析中的应用。
Local surface fitting is one of the common methods to calculate the slope based on grid DEM. Two major factors of this method affect slope estimation accuracy including polynomial and local moving windows. In order to have a better slope calculation formula and to analyze the influence of the factors for calculation accuracy, a new local moving window, called 13 nodes local moving window, is established based on 3×3 local moving window in this paper. It is a symmetric window contained 13 grid nodes with two different grid sizes and intersecting with the principal axis on angles of 45 degree. A new slope computation formula is obtained by fitting the developed window with third order polynomial. To simplify the terrain surface and practical DEM data, a discretization mathematical surface simulates two kind of DEM data. one is high precision DEM data without error, while another with error by adding random noise to the mathematical surface. Based on the simulated DEM data, we analyze the calculation accuracy of the new formula, third order difference, third order difference with inverse square distance weighting and third-order polynomial fitting with 5×5 local window model through comparisons of the true value and calculated value of slope. The results show that third-order polynomial fitting with the symmetric 13 nodes local window model significantly improves the accuracy of the slope computation with the high precision DEM data. However, it has a poor capability for smoothness of error with the DEM data with error compared with other model. This study extends surface fitting method to apply in digital terrain analysis.
Local surface fitting is one of the common methods to calculate the slope based on grid DEM. Two major factors of this method affect slope estimation accuracy including polynomial and local moving windows. In order to have a better slope calculation formula and to analyze the influence of the factors for calculation accuracy, a new local moving window, called 13 nodes local moving window, is established based on 3×3 local moving window in this paper. It is a symmetric window contained 13 grid nodes with two different grid sizes and intersecting with the principal axis on angles of 45 degree. A new slope computation formula is obtained by fitting the developed window with third order polynomial. To simplify the terrain surface and practical DEM data, a discretization mathematical surface simulates two kind of DEM data. one is high precision DEM data without error, while another with error by adding random noise to the mathematical surface. Based on the simulated DEM data, we analyze the calculation accuracy of the new formula, third order difference, third order difference with inverse square distance weighting and third-order polynomial fitting with 5×5 local window model through comparisons of the true value and calculated value of slope. The results show that third-order polynomial fitting with the symmetric 13 nodes local window model significantly improves the accuracy of the slope computation with the high precision DEM data. However, it has a poor capability for smoothness of error with the DEM data with error compared with other model. This study extends surface fitting method to apply in digital terrain analysis.
引文
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[18]EVANS I S.An integrated system of terrain analysis and slope mapping.Final report on grant DA-ERO-591-73-G0040[M].Durham:University of Durham,1979.
[19]TANG G A.Research on the Accuracy of Digital Elevation Models[M].Beijing:Science Press Beijing New York,2000.
[20]晋蓓,刘学军,张宏鸣,等.基于格网DEM的等高线提取误差传播模型[J].人民黄河,2017,39(02):21-24.[JIN Bei,LIUXuejun,ZHAO Hongming,et al.Error propagation models of contour interpolation based on gird DEM[J].Yellow river,2017,39(2):24-27]
[2]卢华兴,刘学军,王永君,等.插值条件下格网DEM坡度计算模型的噪声误差分析[J].测绘学报,2012,41(06):926-932.[LU Huaxing,LIU Xuejun,WANG Yongjun,et al.Noise error analysis of slope algorithms based on grid DEM derived from interpolation[J].Acta Geodaetica et Cartographica Sinica,2012,41(06):926-932]
[3]陈楠.基于信息损失量的坡度精度分析[J].地球信息科学学报,2014,16(6):852-858.[CHEN Nan.Study on slope accuracy based on information loss[J].Journal of Geo-information Science,2014,16(6):852-858]
[4]CHANG K,TSAI B.The Effect of DEM Resolution on Slope and Aspect Mapping[J].Cartography and Geo-graphic Information Systems,1991,18:69-77
[5]刘学军,龚健雅,周启鸣,等.基于DEM坡度坡向算法精度的分析研究[J].测绘学报,2004,33(3):258-263.[LIU Xuejun,GONG Jianya,ZHOU Qiming,et al.A study of accuracy and algorithms for calculating slope and aspect based on grid digital elevation model(DEM)[J].Acta Geodaetica et Cartographica Sinica,2004,33(3):258-263]
[6]FLORINSKYI V.Accuracy of local topographic variables derived from digital elevation models[J].International Journal of Geographical Information Science,1998,12(1):47-61
[7]刘学军,卞路,卢华兴,等.顾及DEM误差自相关的坡度计算模型精度分析[J].测绘学报,2008,37(2):200-206.[LIUXuejun,BIAN Lu,LU Huaxing,et al.The accuracy assessment on slope algorithms with DEM error spatial autocorrelation[J].Acta Geodaetica et Cartographica Sinica,2008,37(2):200-206]
[8]ISSACSON D L,RIPPLE W J.Comparison of 7.5 minute and 1degree digital elevation models[J].Photogrammetric Engineering and Remote Sensing,1991,56:1523-1527
[9]LEE J,SNYDER P K,FISHER P F.Modeling the effect of data errors on feature extraction from digital elevation models[J].Photogrammetric Engineering and Remote Sensing,1992,58(10):1461-1467
[10]张玉伦,王叶堂.低山丘陵区多源数字高程模型误差分析[J].遥感技术与应用,2018,33(6):1112-1121[ZHANG Yulun,WANG Yetang.Accuracy assessment of the ASTER GDEM and SRTM DEM in the low mountain and hilly region[J].Remote Sensing Technology and Application,2018,33(6):1112-1121]
[11]刘晓,赵荣,梁勇,等.顾及地貌与DEM分辨率的坡度算法适应性研究[J].测绘科学,2017,42(3):29-34.[LIU Xiao,ZHAORong,LIANG Yong,et al.Research of algorithms for deriving slope based on different geomorphic types and DEM resolution[J].Science of Surveying and Mapping,2017,42(3):29-34]
[12]周启鸣,刘学军.数字地形分析[M].北京:科学出版社,2006:87-93.[ZHOU Qiming,LIU Xuejun.Digital terrain analysis[M].Beijing:Science Press,2016:87-93]
[13]刘学军,张平,朱莹.DEM坡度计算的适宜窗口分析[J],测绘学报,2009,38(3):264-281.[LIU Xuejun,ZHANG Ping,ZHU Ying.Suitable window size of terrain parameters derived from grid-based DEM[J].Acta Geodaetica et Cartographica Sinica,2009,38(3):264-281]
[14]ALBANI M,KLINKENBERG B,ANDISON D W,et al.The Choice of Window Size in Approximating Topographic Surfaces from Digital Elevation Models[J].International Journal of Geographical Information Science,2004,18(6):577-593
[15]ZEVENBERGEN L W,THORNEV C R.Quantitative analysis of land surface topography[J].Earth Surface Processes and Landforms,1987,12(1):47-56
[16]FLORINSKY I V.Computation of the third-order partial derivatives from a digital elevation mode[J].International Journal of Geographical Information Science,2009,23(2):213-231
[17]赵明伟,汤国安,张磊,等.基于DEM的5节点二阶差分坡度算法模型与实验分析[J].地球信息科学学报,2012,14(3):286-291.[ZHAO Mingwei,TANG Guoan,ZHANG Lei,et al.The DEM based 5-node second-order finite difference model for slope computation[J].Journal of Geo-information Science,2012,14(3):286-291]
[18]EVANS I S.An integrated system of terrain analysis and slope mapping.Final report on grant DA-ERO-591-73-G0040[M].Durham:University of Durham,1979.
[19]TANG G A.Research on the Accuracy of Digital Elevation Models[M].Beijing:Science Press Beijing New York,2000.
[20]晋蓓,刘学军,张宏鸣,等.基于格网DEM的等高线提取误差传播模型[J].人民黄河,2017,39(02):21-24.[JIN Bei,LIUXuejun,ZHAO Hongming,et al.Error propagation models of contour interpolation based on gird DEM[J].Yellow river,2017,39(2):24-27]