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地球信息科学学报 >

2016 , Vol. 18 >Issue 4: 461 - 468

DOI: https://doi.org/10.3724/SP.J.1047.2016.00461

基岩露头采样密度对黄土古地形重建的影响

  • 段家朕 ,
  • 熊礼阳 , * ,
  • 汤国安
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  • 1. 南京师范大学 虚拟地理环境教育部重点实验室,南京 210023
  • 2. 江苏省地理环境演化国家重点实验室培育建设点,南京 210023
  • 3. 江苏省地理信息资源开发与利用协同创新中心,南京 210023
*通讯作者：熊礼阳（1989-）,男,江西南昌人,博士生,主要从事黄土继承性、DEM数字地形分析研究。E-mail: xiongliyang@163.com

作者简介：段家朕（1990-）,男,山东临沂人,硕士生,主要从事DEM数字地形分析研究。E-mail:

收稿日期: 2015-03-11

  要求修回日期: 2015-03-29

  网络出版日期: 2016-04-19

基金资助

国家自然科学基金项目（41171299、41271438、41471316）

江苏省高校自然科学研究重大项目（13KJA170001）

江苏高校优势学科建设工程资助项目

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Effect of Outcrop Sampling Density on the Underlying Terrain Reconstruction

  • DUAN Jiazhen ,
  • XIONG Liyang , * ,
  • TANG Guoan
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  • 1. Key Laboratory of Virtual Geographic Environment, Ministry of Education, Nanjing Normal University Nanjing, 210023, China
  • 2. State Key Laboratory Cultivation Base of Geographical Environment Evolution, Nanjing, 210023, China
  • 3. Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application, Nanjing, 210023, China
*Corresponding author: XIONG Liyang, E-mail:

Received date: 2015-03-11

  Request revised date: 2015-03-29

  Online published: 2016-04-19

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《地球信息科学学报》编辑部 所有
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摘要

下伏古地形对黄土地貌的形成、演化与发育具有重要的控制作用。基于下伏地层露头的采样数据,通过空间插值方法进行古地形数字模型(DEM)重建是研究黄土地貌的重要方法。其中,采样点密度是下伏地形DEM重建精度的主要影响因子。本文以1:20万绥德幅地质图所涉及区域为实验样区,研究采样点密度及样条函数方法对下伏古地形重建精度的影响。实验结果表明,在高密度样本条件下,采用规则样条插值方法进行古地形重建时,插值结果呈现显著的“龙格现象”,即多项式插值结果出现剧烈波动现象,且随着样本密度增加,样条插值结果的误差呈逐步趋缓的降低趋势,并逐步趋于稳定。同时,基于古地形DEM提取的特征点,其局部最高点和局部最低点的数目呈直线上升趋势,说明在一定程度上基于有限数据采用样条函数进行地下三维建模并不一定能获得平滑曲面。研究结果对如何选择合理的空间采样密度进行古地形DEM重建具有重要的指导意义。

关键词： DEM; 黄土地貌; 下伏古地形; 采样点密度; 样条空间插值

本文引用格式

段家朕 , 熊礼阳 , 汤国安 . 基岩露头采样密度对黄土古地形重建的影响[J]. 地球信息科学学报, 2016 , 18(4) : 461 -468 . DOI: 10.3724/SP.J.1047.2016.00461

Abstract

The Pre-Quaternary underlying terrain profoundly controls the evolution and formation of loess landform. Obvious relationships, i.e. the geomorphological inheritance, could be found between the underlying terrain and the modern terrain. As a consequence, the Pre-Quaternary underlying terrain in the Loess Plateau should be regarded as the key factor for the understanding of the loess landform evolution. Among numerous numerical calculation methods, spatial interpolation has been regarded as an important method to reconstruct the DEM of underlying terrain by using the sampled bedrock outcrop points selected from a geological map. However, the sampling density has a great impact on the accuracy of the reconstructed underlying terrain. In this paper, the Suide geological map area (1:200 000) was selected as the study area, and then the influence of sampling density on the accuracy of the reconstructed underlying terrain was investigated using spline method. By adopting cross-validation method to evaluate reconstructed underlying terrain, the result shows that, different interpolation methods cause uncertainties to different degrees during the reconstruction of underlying terrain, particularly the spline method. On a basis of high density outcrop points and spline function interpolation process, the morphology of underlying terrain exhibits a typical “Runge phenomenon”. This phenomenon was always resulted from a polynomial interpolation process. With an increased sampling density, the error in underlying terrain appears a slowly decrease tendency firstly, and then it keeps stable. Meanwhile, the number of the extracted features has a linear upward trend. The result also shows that the sampling density of 1.7-2.0 points per square kilometer could achieve a good balance between the accuracy and underlying terrain feature reservation. The aforementioned results adjust our previous understandings that spline function could smooth the interpolated surface to some extent. And the result also provides guidance for the selection of a reasonable spatial sampling density.

Key words： DEM; Loess underlying terrain; sampling density; spline interpolation

1 引言

黄土高原下伏古地形的形态特征,在一定程度上控制着黄土地貌的形成过程与空间分布。刘东生认为现代黄土“塬、墚、峁”的地貌形态和下伏古地貌关系密切,存在显著的继承性特征[1],之后诸多学者也研究了黄土古地形特征,用于分析黄土地貌发育过程的继承性特征[2-5]。可见,黄土高原的古地形是科学认识黄土地貌发育与演化的关键因素。受限于测量数据和实验方法,上述对黄土古地形的研究仍处于定性描述和半定量分析的阶段。在黄土古地形重建过程中,古地形被黄土覆盖,仅有少数基岩在沟谷中露头,区域性下伏地层高程数据难以准确探测。随着数字地形分析方法的发展,采用多源信息采集下伏地层的高程信息,并运用空间插值方法获得区域性的古地形模拟表面成为可能。
空间插值作为一种根据已知采样点的高程值估计未知点高程值的数学方法[6],是在有限数据条件下进行曲面建模的最佳方案之一。插值方法的选择是构建DEM的关键环节。合理的插值方法能得出相对“最佳”的结果,从而提高DEM的插值精度。国内外众多学者对不同插值方法进行了大量的研究[7-14]。在黄土地形曲面建模研究中,付永恒和张锦明通过对比实验认为,在地形较为复杂的地区样条插值是较好的选择[15]。熊礼阳等通过对比不同插值方法,认为如果采用低阶多项式样条函数,样条插值可获得较小误差的插值结果,在低密度下对古地形的模拟重建结果相对较好[16]。王春等采用高程数值误差场和局地坡面形态误差场等分析技术,认为样条插值有很好的地面形态重构 精度[17]。
Fernando认为在插值过程中,样点密度对插值结果的影响大于插值方法[18]。通常情况下,随着样本密度的增加,插值精度会相应地提高[18]。但在高密度样本条件下,采用规则样条函数进行古地形模拟,结果往往出现明显的局部地形剧烈波动现象（图1）,即多项式插值结果出现“龙格现象”,结果导致地形模拟产生较为明显的区域失真问题[19],这既不符合插值函数对插值结果的预判,也与实际的地形特征存在差距。显然,这一现象与插值样点密度、插值函数类型密切相关。因此,探究插值样本密度、插值精度以及插值结果剧烈波动现象之间的关系,寻找样本点密度与插值结果精度之间的平衡点,从而选择合理的空间采样密度是保证黄土古地形重建精度的关键。
Fig. 1 Interpolation results of by Kriging function and regular spline function under high sampling density

图1 高密度样点条件下普通克里金函数与规则样条函数插值结果对比

2 数据与方法

2.1 实验样区与实验数据

实验样区为1:20万绥德幅J-49-(21)地质图所覆盖区域,面积达7300 km2。该地区属黄土丘陵沟壑区第一幅区[20],为典型的峁梁状黄土丘陵沟壑区。区域沟壑密度平均为5~6 km/km2,地面裂度为42%[21]。该千沟万壑的地形特征使得下伏地层在沟谷中露头众多,为下伏古地形重建提供了插值样本数据基础。实验数据包括：
(1) 地质图：采用1﹕20万绥德幅地质图,作为古地形基岩出露点位判读的数据源。
(2) DEM：采用与地质图区域相对应的25 m分辨率数据作为基本的高程信息源。
(3) 遥感影像：采用高清Google Earth影像图 （5 m分辨率）对地质图出露基岩点位进行位置校正。

2.2 实验方法

2.2.1 实验流程
实验流程设计如图2所示。
Fig. 2 Flow chart of this research

图2 实验流程图

2.2.2 基岩出露点数据采样
采集地质图中标注的不同地质年代及岩石类型的出露基岩点坐标位置及海拔高度[16]（图3）。采样步骤如下：（1）在地质图中找出基岩出露点;（2）通过遥感影像得到基岩点位,进行位置精校正;（3）基于DEM数据获取基岩露头点高程值。
Fig. 3 Geological map of the study area and the distribution of samplings

图3 实验样区地质图及样本点分布图

2.2.3 不同密度插值点与检验点分类
实验首先利用ArcGIS地统计分析模块中的创建子集方法提取检验样点（占原始样本点数据10%）和实验样点（占原始样本点数据90%）;其次,在实验样点中,以10%为增量提取不同密度下的样本点,得到原实验样本点数据量10%~100%的10个子样本点数据集（表1）;然后,以各样本点数据为基础,采用ArcGIS空间分析模块中的样条插值方法进行插值（搜索样点数为12,格网尺寸为25 m,权重取0.1）;最后,得到10个不同样点密度下的插值结果。
Tab. 1 Numbers of samplings and their equivalent density in each dataset

表1 各样本点集样本点数及等效密度表

样本集 n1 n2 n3 n4 n5 n6 n7 n8 n9 n10
点数 2531 5061 7592 10 122 12 653 15 183 17 714 20 244 22 775 25 305
等效密度/（个/km2） 0.346 0.693 1.040 1.386 1.733 2.080 2.427 2.773 3.120 3.466

3 实验结果与分析

3.1 古地形重建结果

在各样点密度下采用规则样条插值方法得到的古地形重建结果如图4所示。由图4可看出,随着样本点数量的增加,插值结果出现越来越剧烈的波动现象,“山峰”和“山谷”数量明显增加。
Fig. 4 Reconstructed results of the underlying terrain under different sampling density

图4 各样点密度古地形重建结果

3.2 结果分析

3.2.1 XY散点图分析
为揭示检验样点实测值与估计值的相关性以及插值结果的准确性,对检验样点的实测值与估计值做XY散点图,如图5所示。其中,X代表检验点样条插值结果高程,Y代表检验点基岩露头高程,按Y=aX+b计算得到线性关系和相关系数R²;参数a反映了实测值与估计值的相似性,当X、Y完全一致时,a=1;相关系数R2反映了实测值与估计值之间的整体精度。由数据可看出,随着样本点数据量的增加,相关性呈现增长趋缓的态势。当样本点数增加到总数的50%~60%左右(1.7~2.0个/km2)时,相关性指数增长趋势逐步趋于稳定（图6）。
Fig. 5 XY scatter diagram for the measured value and estimated value under different sampling density

图5 不同样本集下检验样点的实测值与估计值XY散点图

Fig. 6 Correlation coefficient for the measured value and the estimated value of the test samples

图6 检验样点的实测值与估计值相关系数

3.2.2 误差分析
本文采用交叉验证方法验证样条插值的效果,即通过计算检验样点的实测值与计算值的误差来评判插值结果的优劣[22-23]。实验采用平均绝对误差（Mean Absolute Error,MAE）、平均相对误差（Mean Relative Error, MRE）、均方根误差（Root Mean Square Error, RMSE）作为衡量样条插值精度评价指标。其中,MAE反映估计值的误差范围,MRE反映计算值对于实测值的准确度,RMSE反映计算值的灵敏度和极值情况[22]。其表达式如式(1)-(3)所示。
MAE = ∑ i = 1 n ABS ( Z a , i - Z e , i ) n （1）
MRE = 1 n ∑ i = 1 n | ABS ( Z a , i - Z e , i ) Z a , i | （2）
RMSE = 1 n - 1 ∑ i = 1 n ( Z a , i - Z e , i ) 2 （3）
式中： Z a , i 、 Z e , i 分别为第i个样点的实际测量值和插值的预测值;n为验证样点的点数。
利用检验样点分别计算各样点密度下插值结果的MAE、MRE、RMSE、局部最高点和局部最低点个数等统计值。结果如表2、图3和图7所示。
Tab. 2 Error statistics under different sampling density

表2 各样点密度下插值结果精度误差统计表

样本集 n1 n2 n3 n4 n5 n6 n7 n8 n9 n10
MAE 16.132 13.325 11.295 10.912 9.032 9.032 8.004 7.343 6.718 6.587
RMSE 31.647 26.701 21.030 25.904 17.708 18.607 16.358 13.654 12.612 13.060
MRE 0.018 0.015 0.013 0.012 0.010 0.010 0.009 0.008 0.007 0.007
Fig. 7 Error statistic under different sampling density

图7 各样点密度下精度误差

结果表明,随着样本点数据量的增加,各误差度量指标均呈现趋缓的下降趋势,即随着样本点密度增加,插值结果逐步逼近实际地形,但在样点密度达到一定程度后,插值结果误差趋于稳定,所构建的地形质量无明显上升。就本文而言,当样本点密度增加到2个/km2时,古地形插值质量已达到 稳定。
3.2.3 地形特征分析
为量化局部地形起伏现象,对不同样点密度下样条插值重建的下伏古地形提取局部最高点（Local Highest Point,LHP）和局部最低点（Local Lowest Point,LLP）数目[24],将其作为衡量插值结果局部起伏状态的指标。具体操作流程如图8所示。
Fig. 8 Flow chart of LHP extraction method based on reverse DEM

图8 反地形DEM局部最高点提取流程图

由图8可见,局部最低点和最高点直接采用原始DEM提取。同时,为降低边缘点不确定性造成的结果偏差,在提取局部最低点和最高点时,对边缘点进行删除,不计入最终结果。不同样点密度下样条插值结果的局部最低点和最高点个数统计如表3和图9所示。
Tab. 3 Number of LHPs and LLPs number with different sampling density

表3 各样点密度下局部最高点和最低点统计表

样本集 n1 n2 n3 n4 n5 n6 n7 n8 n9 n10
LHP 300 589 895 1234 1515 1853 2158 2457 2762 3097
LLP 370 761 1105 1493 1895 2308 2650 2968 3279 3641
Fig. 9 Number of LHPs and LLPs with different sampling density

图9 各样点密度下局部最高点与局部最低点

局部最高点和局部最低点数目呈现出1﹕1的比例关系,且随着样本点数量的增加（图9）,局部最高点和局部最低点呈明显的线性增长关系。即随着样本点密度的增加,插值结果反映的地形出现更多的局部“山地和谷地”。
为对比插值精度与地形起伏变化之间的关系,对MAE、MRE、RMSE等指数分别与局部最高点和局部最低点数据做相关分析,如图10所示。由于局部最高点和局部最低点随样点密度的增加呈1﹕1比例关系,此处对二者取平均值作各精度指标与局部极值点相关图中的Y值,X值为各误差指数值。
如图10所示,随着样点密度的增加,各项误差指数与局部极值点平均数成指数增长关系,即随着样点密度的增加,样条插值精度提高速率远远低于局部地形起伏态势的增加趋势。而结合各样点密度下最大值、最小值、平均值与标准差等统计值,局部地形起伏态势为在数量上的增加,而在垂直方向（即高差）上没有明显的变化。
为更直观地对上述指标相关性进行对比分析,对各项指标进行归一化处理,结果如图11所示。
Fig. 10 Relation between local extreme value and each precision index

图10 各精度指标与局部极值点相关图

Fig. 11 Each precision result normalization with different sampling density

图11 不同样点密度下各精度结果归一化图

由图11可见,就整体实验数据而言,随着样本点数据量的增加,样条插值结果精度提高,且效率逐步趋缓,但局部范围内地形起伏现象却呈增加趋势,且与样本点数目成线性关系,即随着样本点密度的增加,样条插值结果虽然在整体趋势上较好地反映了古地形,但是在局部地形中存在剧烈波动现象呈线性增加态势。结合前人的结论[16],黄土高原下伏古地形作为地形基础,对黄土地貌发育起到重要控制作用,对比现今绥德地区地貌,这种局部地形剧烈波动现象是插值结果表现出的不稳定现象。进一步分析可知,当样本点数目控制在50%~60%（1.7~2.0个/km2）时,可得到精度较高的插值精度结果,同时可较好地控制局部地形起伏现象。

4 结论与展望

基于基岩露头信息与空间插值方法的黄土高原下伏古地形重建,是揭示宏观尺度黄土下伏古地形特征的重要方法。本文基于不同密度的下伏地层露头样点,采用样条插值方法重建了黄土下伏古地形,主要结论如下：
（1）不同插值方法在黄土古地形重建中具有不确定因素。尤其在高密度样本条件下,使用样条插值方法,其插值结果呈现显著的剧烈波动现象,即“龙格现象”。 该现象说明在一定程度上,基于有限数据采用样条函数进行地下三维建模并不一定能获得平滑曲面。
（2）对于本研究实验样区的古地形重建,当样本点密度增加,样条插值的多项式“龙格现象”效应更加突出,在一定程度上影响插值结果的稳定性。随着样本点密度的增加,误差呈现逐步降低的特征,降低速率表现为先快速,后缓慢,最终趋于平缓;而地形特征的局部最高点和局部最低点的数目呈现直线上升趋势;且误差与地形特征点数量呈一定的指数关系。
（3）当布设样本点密度为1.7~2.0个/km2时,样条插值结果精度和局部起伏现象达到平衡,既能够相对最优地减少插值误差,也能够最大化地保留原始地形特征。该结果为相关研究中如何合理选择空间采样密度提供一定参考。

The authors have declared that no competing interests exist.

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夏正楷. 黄土高原第四纪期间水土流失的地质记录和基本规律[J].水土保持研究,1999,6(4):49-53.黄土高原保存有许多第四纪期间水土流失的地质记录，这些记录表明，５０万年以来，黄土高原至少有５个水土流失十分强烈的埋藏，它们分别出现在０．７０～０．５０ＭａＢ，Ｐ、０．２～０．２５ＭａＢ．Ｐ、０．１４～０．８０ＭａＢ．Ｐ、０．０１～０．０１５ＭａＢ．Ｐ和０．００６～０．００２ＭａＢ。Ｐ等抉温暖湿润的阶段，说明影响黄土高原水土流失的主要原因是气候条件。

DOI

[ Xia Z K.The records of quaternary soil erosion in the Loess Plateau[J]. Research of Soil and Water Conservation, 1999,6(4):49-53. ]

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桑广书,陈雄,陈小宁,等.黄土丘陵地貌形成模式与地貌演变[J].干旱区地理,2007,30(3):375-380.通过对陕西北部典型黄土丘陵区地貌调查,提出了黄土丘陵地貌形成 模式,并对地貌演变进行了探讨.黄土高原原生黄土丘陵分布广泛,主要在下伏古丘陵基础上由黄土加积而成;次生黄土丘陵是黄土塬、黄土台塬等经沟谷侵蚀、改 造而成.黄土丘陵区河流阶地和沟谷层状地貌反映了河流、沟谷的形成与演变.黄河一级支流形成于早更新世末;早中更新世末、晚中更新世初黄河一级支流和较大 的二级支流形成了较完整的水系;中更新世末如韭园沟等较大的沟谷形成;晚更新世末河网、沟谷格局与现在已基本一致;尚未切入基岩的冲沟多形成于全新世;长 度百米至数十米的切沟多是历史时期或近几十年来形成的.

DOI

[ Sang G S, Chen X, Chen X N, et al.Formation model and geomorphic evolution of loess hilly landforms[J]. Arid Land Geography, 2007,30(3):375-380. ]

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程彦培,石建省,杨振京,等.古地形对黄土区岩土侵蚀趋势的控制作用[J].干旱区地理,2010,33(3):334-339.利用高精度的GPS（卫星全球定位系统）结合RS（遥感）与GIS（地理信息系统）重建了陕北府谷黄土区一个小流域的侏罗纪（J）、第三纪（N）时期古地形数字模型（DTM）,与该区现代小流域地形和土壤侵蚀的现状进行了分析对比,发现府谷黄土区经历了古准平面-古湖泊洼地-高原沟谷三大地貌发育过程,而且该小流域现代地形与侏罗纪、第三纪古侵蚀面之间具有很高的继承性,可见古地形在陕北黄土地区的岩土侵蚀过程中有着重要的控制作用,研究结果可以更好地为水土保持提供科学依据。

[ Cheng Y P, Shi J S, Yang Z J, et al.Control of ancient landform on rock-soil erosion in loess area[J]. Arid Land Geography, 2010,33(3):334-339. ]

[5]
袁宝印,郭正堂,郝青振,等.天水-秦安一带中新世黄土堆积区沉积-地貌演化[J].第四纪研究,2007,27(2):161-171.<p>前人的构造地质学研究,将天水-秦安一带的中新世黄土分布区划归两个不同的构造单元.文章基于野外调查和已有年代地层学工作,结合前人成果,对该区新生代沉积-地貌演化历史进行研究,并划分为以下主要阶段:1)古近纪初南部秦岭山地的剥蚀,使本区在原有基岩准平原地形的基础上,形成以冲洪积平原为主的地形.古近纪末-新近纪初的构造活动使冲洪积平原解体,在秦安地区形成基岩台地与沉陷盆地相间、天水-西和地区形成拉分盆地与隆起山地交错的地貌景观,这些高地为中新世黄土堆积提供了地形基础.2)中新世从22Ma到Ma,基岩台地和相对平缓的高地上堆积典型黄土-古土壤序列,盆地内则主要发育次生黄土等洼地沉积,表明研究区类似于今天的黄土高原.3)中新世晚期约Ma起发生的一次侵蚀事件,使研究区的一些小盆地内发育河流相和间歇性浅湖相沉积,秦安一带的黄土堆积也遭到侵蚀,形成的洼地内发育黄土状土或洼地静水沉积,其中包含较多哺乳动物化石,而大范围的相对平坦高地上一直继续发育黄土-古土壤序列.这次侵蚀对本区内甘肃群的沉积多样性有重要贡献,但一直没有深水湖泊发育的条件.4)发生于3.5Ma以后的另一次重大侵蚀,奠定了该区今天狭窄长墚地形的基础,是第四纪黄土堆积在本区保存较差的主要原因.</p>

[ Yuan B Y, Guo Z T, Hao Q Z, et al.Cenozoic evolution of geomorphic and sedimentary environments in the tianshui-qin’an regions[J]. Quaternary Sciences, 2007,27(2):161-171. ]

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李新,程国栋,卢玲.空间内插方法比较[J].地球科学进展,2000,15(3):260-265.<p>空间内插可以分为几何方法、统计方法、空间统计方法、函数方法、随机模拟方法、物理模型模拟方法和综合方法。介绍了每一种方法的适用范围、算法和优缺点。指出没有绝对最优的空间内插方法,必须对数据进行空间探索分析,根据数据的特点,选择最优方法;同时,应对内插结果做严格的检验。开发通用空间内插软件、智能化内插以及加强相关基础研究将是空间内插研究的重点。</p>

[ Li X, Cheng G D, Lu L.Comparison of spatial interpolation methods[J]. Advance in Earthences, 2000,15(3):260-265. ]

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李志林,朱庆.数字高程模型[M].武汉:武汉大学出版社,2000:125-139.

[ Li Z L, Zhu Q.Digital elevation model[M]. Wuhan: Wuhan University Press, 2000:125-139. ]

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Lam N S.Spatial interpolation methods: A review[J]. The American Cartographer, 1983,10(2):129-149.Two forms of spatial interpolation, the interpolation of point and areal data, are distinguished. Traditionally, point interpolation is applied to isarithmic, that is, contour mapping and areal interpolation to isopleth mapping. Recently, areal interpolation techniques have been used to obtain data for a set of administrative or political districts from another set of districts whose boundaries do not coincide. For point interpolation, the numerous methods may further be classified into exact and approximate. Exact methods include most distance-weighting methods, Kriging, spline interpolation, interpolating polynomials, and finite-difference methods. Approximate methods include power-series trend models, Fourier models, distance-weighted least-squares, and least-squares fitting with splines. Areal interpolation methods, on the other hand, are classified according to whether they preserve volume. Traditional areal interpolation methods which utilize point interpolation procedures are not volume-preserving, whereas the map overlay and pycnophylactic methods are. It is shown that methods possessing the volume-preserving property generally outperform those that do not.

DOI

[9]
Robeson S M.Spherical methods for spatial interpolation: Review and evaluation[J]. Cartography and Geographic Information Systems, 1997,24(1):3-20.ABSTRACT Global change research has placed new demands on methods of spatial analysis. In particular, spherical methods for spatial interpolation are required when spatial analyses are performed over large areas of the Earth's surface. In this article, spherical spatial interpolation procedures are reviewed, compared, and evaluated. Three classes of spherical interpolants are evaluated in detail: distance weighting, functional minimization, and tesselation. The strengths and weaknesses of a method from each of these classes&mdash;inverse-distance weighting, thin-plate splines, and surfaces fit to triangulated patches&mdash;are evaluated using a hypothetical mathematical surface and a global scale representation of topography. For smooth functions, such as the hypothetical mathematical surface, thin-plate splines produce a visually pleasing surface and have low interpolation error. For non-smooth surfaces, such as global topography, inverse-distance weighting, interpolating thin-plate splines, and triangulated C0 patches appear to handle rapid surface changes well. When choosing a spherical interpolant, the properties of the data being analyzed (e.g., smoothness, spatial coherence, etc.) must be taken into account. In addition, multivariate interpolation should be considered when related, ancillary data are available at higher spatial resolution than the original data.

DOI

[10]
Burrough P A, McDonnell R A. Principles of geographical information systems[M]. New York: Oxford University Press, 1998:333.

[11]
Franke R.Scattered data interpolation: Tests of some methods[J]. Mathematics of Computation, 1982,38(157):181-200.Let G be the group of isometrics of the 2-sphere, the Euclidean plane or the hyperbolic plane, the group of similarities of the Euclidean plane or the group of Mobius transformations of the 2-sphere. In each instance we determine which conjugacy classes in G are amalgamated when we allow conjugation of the elements of G by homeomorphisms of the space on which G acts. Our results are related to recent work on the homeomeric classification of two-dimensional patterns.

DOI

[12]
Weber D, Englund E.Evaluation and comparison of spatial interpolators II[J]. Mathematical Geology, 1994,26(5):589-603.The performance of several variations on ordinary kriging and inverse distance estimators is evaluated. Mean squared errors (MSE) were calculated for estimates made on multiple resamplings from five exhaustive data bases representing two distinctly different types of estimation problem. Ordinary kriging, when performed with variograms estimated from the sample data, was more robust than inverse-distance methods to the type of estimation problem, and to the choice of estimation parameters such as number of neighbors .

DOI

[13]
Declercq F A N. Interpolation methods for scattered sample data: Accuracy, spatial patterns, processing time[J]. Cartography and Geographic Information Systems, 1996,23(3):128-144.Interpolation routines based on polynomials, splines, linear triangulation, proximation, distance weighting, and kriging are tested on their efficacy to visualize spatial patterns. Implementations in commonly available software packages are used in order to yield practical recommendations on the application of current information technology. Two data sets of physical variables containing irregularly distributed sample point values are used as input data. Accuracy of predicted values at unvisited points, preservation of distinct spatial patterns (established from map use tasks), and processing time, are used as criteria to determine the merits of the various interpolation methods. It was found that highly accurate interpolations do not always produce realistic spatial patterns. Effectiveness of distance weighting and kriging methods was found to be largely dependent on the number of neighbors used. For both gradually and abruptly changing data, geographic reality was visualized most satisfactorily with the squared inverse distance weighting (w=d-2) method using respectively few (four to eight) and many (16 to 24) neighbors.

DOI

[14]
Yang X, Hodler T.Visual and statistical comparisons of surface modeling techniques for point-based environmental data[J]. Cartography and Geographic Information Science, 2000,27(2):165-175.Existing studies on spatial interpolation tend to overplay statistical perspective, paying little attention the locality and the visual performance of generated surface models. In an attempt to bridge these gaps in literatures, the authors compared the performance of five surface modelling methods, using a set of integrative criteria including absolute and relative statistical accuracy, visual pleasantness and faithfulness of generated surface models, sensitivity to changing sample size and search conditions, and computational intensity. The modeling methods used were: inverse distance, kriging, linear triangulation, minimum curvature, and radial basis functions. Because terrain relief is one of few environmental attributes whose continuous surfaces can be directly observed through appropriate procedures, we used as input data two sets of elevation points sampled irregularly from a USGS 1:24,000 topographical map covering a hilly area. We found that surface modeling methods, even if statistically accurate...

DOI

[15]
张锦明,游雄,万刚.DEM插值参数优选的试验研究[J].测绘学报,2014,43(2):178-185.根据插值算法权重确定方法的差异，分别选取反距离加权插值算法、径向基函数插值算法和普通克 里格插值算法中相关插值参数，进行“优选”试验研究。首先根据插值参数对插值精度的不同影响，选择相关插值参数作为试验研究对象，选择6种不同地貌类型地 区的稀疏分布的离散采样点作为试验数据源。然后运用交叉验证法、相关分析、趋势面分析和方差分析等一系列试验方法，系统研究并给出试验插值参数的“最优” 取值区间，消除插值参数选择的随意性，更好地指导DEM建模的运用。

DOI

[ Zhang J M, You X, Wan G.Experimental research on optimization of DEM interpolation parameters[J]. Acta Geodaetica et Cartographica Sinica, 2014,43(2):178-185. ]

[16]
熊礼阳,汤国安,袁宝印,等.基于DEM的黄土高原(重点流失区)地貌演化的继承性研究[J].中国科学:地球科学,2014,44(2):313-321.黄土堆积过程如何继承下伏古地 形,即黄土地貌演化的继承性研究,是反映黄土高原200余万年来发育、演化的关键环节,也是黄土沉积地貌的研究热点.本文以地质图、DEM、遥感影像及实 测数据为依据,运用GIS空间分析方法,得出黄土高原(重点流失区)第四纪黄土堆积以前原始地形面的数字高程模型.通过地形剖面图、面积高程积分、地形剖 面比降及地形剖面凸凹度等方法与指标,分析现今地形与古地形的空间关系.实验结果显示:两种地形的高程、比降、凸凹度呈现明显的线性正相关关系,特别在黄 土沉积较为完善的区域表现为相当完好的继承关系.尽管现今黄土高原部分地区受流水侵蚀,地表形态较为破碎,但仍然可以反映出,黄土堆积过程的总体趋势是缓 和了原始地貌的地形起伏程度.该结果深化了对黄土地貌发育演化继承性的认识.

[ Xiong L Y, Tang G A, Yuan B Y, et al.Geomorphological inheritance for loess landform evolution in a severe soil erosion region of Loess Plateau of China based on digital elevation models[J]. Science China: Earth Sciences, 2014,44(2):313-321. ]

[17]
王春,江岭,徐静,等.DEM地面形态重构方法的精度差异特征研究[J].地理与地理信息科学,2014,30(4):18-21,26.依据DEM地形建模过程,阐述了DEM地面形态重构概念,并以黄土丘陵1∶5万DEM数据为 例,采用高程数值误差场和局地坡面形态误差场相结合的分析技术,实验分析了双线性、三次卷积、局部二次多项式、规则样条函数等常用插值方法在基于加密格网 的DEM地面形态重构过程中的精度差异,以及DEM地面形态可重构的基本条件等问题.研究表明：在高程数值误差的极值大小、离散程度、空间分布等方面,规 则样条函数插值法具有最好的地面形态重构精度,其次为局部二次多项式、三次卷积和双线性插值法;对于已确定综合尺度的地形,存在满足高保真地面形态描述要 求的最佳DEM格网分辨率阈值,只有当实际DEM格网分辨率等于或高于该阈值时,才有可能无歧义、高精度地重构出DEM所描述的综合地形的地面形态.

DOI

[ Wang C, Jiang L, Jing X U, et al.Accuracy differences among approaches of DEM surface morphology reconstruction[J]. Geography and Geo-Information Science, 2014,30(4):18-21,26. ]

[18]
Aguilar F J, Agüera F, Aguilar M A, et al.Effects of terrain morphology, sampling density, and interpolation methods on grid DEM accuracy[J]. Photogrammetric Engineering & Remote Sensing, 2005,71(7):805-816.ABSTRACT This paper explores the effects of terrain morphology, sampling density, and interpolation methods for scattered sample data on the accuracy of interpolated heights in grid Digital Elevation Models (DEM). Sampled data were collected with a 2 by 2 meters sampling interval from seven different morphologies, applying digital photogrammetric methods to large scale aerial stereo imagery (1:5000). The experimental design was outlined using a factorial scheme, and an analysis of variance was carried out. This analysis yielded the following main conclusions: DEM accuracy (RMSE) is affected significantly by the variables studied in this paper according to &ldquo;morphology 17 sampling density 17 interpolation&rdquo; method. Multiquadric Radial Basis Function (RBF) was rated as the best interpolation method, although Multilog RBF performed similarly for most morphologies. The rest of RBF interpolants tested (Natural Cubic Splines, Inverse Multiquadric, and Thin Plate Splines) showed numerical instability working with low smoothing factors. Inverse Distance Weighted interpolant performed worse than RBF Multiquadric or RBF Multilog. In addition, it is found that the relationship between the RMSE and the sampling density N is adjusted to a decreasing potential function that may be expressed as RMSE/Sdz 17 0.1906(N/M)170.5684 (R2 17 0.8578), being Sdz the standard deviation of the heights of the M check points used for accuracy estimation, and N the number of sampling points used for creating the DEM. The results obtained in this study allow us to observe the possibility of establishing empirical relationships between the RMSE expected in the interpolation of a Grid DEM and such variables as terrain ruggedness, sampling density, and the interpolation method, among others that could be added. Therefore, it would be possible to establish a priori the optimum grid size required to generate or storage a DEM of a particular accuracy, with the economy in computing time and file size that this would signify for the digital flow of the mapping information.

DOI

[19]
王春,刘学军,汤国安,等.格网DEM地形模拟的形态保真度研究[J].武汉大学学报(信息科学版),2009,34(2):146-149.分析了现有格网DEM地形模拟的失真现象，研究DEM地形模拟失真的根源。提出了DEM地形形态保真度的概念，探讨了建设高保真DEM必须解决的问题。

[ Wang C, Liu X J, Tang G A, et al.Morphologic fidelity of grid digital elevation model[J]. Geomatics & Information Science of Wuhan University, 2009,34(2):146-149. ]

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黄秉维. 编制黄河中游流域土壤侵蚀分区图的经验教训[J].科学通报,1955,12:15-21,14.正 一根治黄河水害和開發黄河水利的綜合規劃是在苏联專家組帮助下拟訂的最徹底的战勝數千年來为害不絕的黄河,並且也是最充分的利用它來为我國人民服务的偉大計劃。这个規劃的原則和基本內容以及鄧子恢副總理關於該規劃的報告已在第一届全國人民代表大会第二次会議上通过了。全国人民代表大会的决議代表着我國六億人民的愿望、决心和力量。我們已开始向黄河進軍,我們将大規校地徹底地改造黄河。变害河为利河的无限美好的远景,只要幾十年的時間就可以完全实现了。

[ Huang B W.The Experience and lessons of the compilation of soil erosion zoning map in the middle reaches of the Yellow River[J]. Chinese Science Bulletin, 1955,12:15-21,14. ]

[21]
田剑,汤国安,周毅,等.黄土高原沟谷密度空间分异特征研究[J].地理科学,2013,33(5):622-628.以5 m分辨率DEM为信息源,借助样方分析思想,运用数字地形分析方法和克里格插值模型,获得黄土高原全区的沟谷密度分布图。在此基础上,探讨黄土高原沟谷的 空间分异特征及影响因素。实验结果表明,黄土高原沟谷密度空间分异明显,沟谷密度在陕北的绥德-米脂一带达到高峰,由北向南递减。以六盘山和吕梁山为界, 沟谷密度有三种变化特征,六盘山以西地区,沟谷密度较低且变化平稳,六盘山以东吕梁山以西地区的沟谷密度由北向南呈现梯度显著下降变化,吕梁山以东地区, 沟谷密度呈现起伏变化,沟谷密度值分布在1.7~6.4 km/km2范围内。在宏观上,由陇西盆地、鄂尔多斯地台和汾渭裂谷等地质构造控制沟谷空间分布态势;降雨强度因素对沟谷侵蚀作用显著,加剧了沟谷密度空 间分异特征;植被条件和地面组成物质呈现由西北向东南变化制约着沟谷发育。土壤侵蚀方面,沟谷密度与输沙模数空间耦合性较强,存在明显的正相关。

[ Tian J, Tang G A, Zhou Y, et al.Spatial variation of gully density in the Loess Plateau[J]. Scientia Geographica Sinica, 2013,33(5):622-628. ]

[22]
邬伦,吴小娟,肖晨超,等.五种常用降水量插值方法误差时空分布特征研究——以深圳市为例[J].地理与地理信息科学,2010,26(3):19-24.在洪水、滑坡等地质灾害预警预 报中,通常需要对多个时点、多个站点的降水量观测数据进行高精度插值,雨量插值精度对灾害预警预报的可靠度具有很大的影响,因此研究降水量插值方法误差的 时空分布特征具有重要的科研和实用价值。该文以深圳市2008年6月12日至14日百年一遇的强降水过程为例,采用交叉验证方法对反距离权重法、普通克里 金法、全局多项式法、局部多项式法和径向基函数法五种常用空间插值方法误差的时空分布特征进行分析,研究成果可为根据雨量时空分布特点选取适用雨量插值模 型提供相关依据,并为相关研究提供借鉴。

[ Wu L, Wu X J, Xiao C C, et al.On temporal and spatial error distributions of five precipitation interpolation models: a case of Shenzhen[J]. Geography and Geo-Information Science, 2010,26(3):19-24. ]

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DOI

[ Wang C, Gu L W, Tao Y, et al.A new calculation model of DEM terrain description error[J]. Journal of Geo-information Science, 2014,16(5):699-706. ]

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[ Zhong T, Tang G A, Zhou Y, et al.Method of extracting surface peaks based on reverse DEMs[J]. Bulletin of Surveying & Mapping,2009,4:35-37. ]

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