黄土高原小流域侵蚀沟道空间频谱分析
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摘要
目前，数字高程模型（DEM），特别是规则格网DEM，已经成为土壤侵蚀以及其他与地形相关的研究的主要数据源。众多的研究表明，DEM的格网尺寸大小对于地形分析结果有着重要的影响。但对于多大格网尺寸的DEM究竟能够表达怎样的地形信息，即DEM的格网尺寸与地形的表达能力之间的关系仍不明确。这一问题使得在应用DEM进行土壤侵蚀或地形分析中存在一定的盲目性。针对上述问题，本研究基于陕西省绥德县窑家湾沟的全数字摄影测量和野外实测的高精度DEM数据，以黄土高原侵蚀沟道为研究对象，应用傅里叶频谱分析方法，首先研究了DEM数据的高程精度与格网尺寸的关系，进而分析了利用DEM提取沟道网络时格网尺寸的影响。基于DEM提取的沟道数据，布设了24个剖面，并利用一维频谱分析方法，分析了不同等级沟道的空间频率特征，总结出了各侵蚀沟道的空间频率范围。最后将一维频谱分析结果推广至二维频谱分析，并以此为依据对实验数据进行二维频率低通滤波。通过对滤波后的DEM数据与各格网尺寸DEM进行对照，将侵蚀沟道的空间尺度与DEM格网尺度对应起来，以解决DEM的格网尺寸与地形的表达能力之间的对应关系。论文的主要研究结果如下：
1．DEM的高程精度与格网尺寸的关系
格网尺寸对规则格网DEM高程精度的影响表现在：上坡方向低估格网高程，下坡方向则高估。因此格网上的高程误差大小取决于地形坡度，其空间分布的复杂程度取决于格网尺寸大小。利用直接采样获得一系列DEM格网上高程误差的分析结果表明，DEM高程中误差和最大误差与格网尺寸存在近似线性的二次函数关系，随着格网尺寸增大，DEM高程误差增大，且格网上的最大高程误差远超出2倍中误差水平。
2．DEM格网尺寸与沟道网络提取及其基本特征的关系
利用不同格网尺寸DEM提取的沟道网络的特征分析的结果表明，DEM格网尺寸对沟道网络的提取主要体现在对水流方向的影响与水流累积阈值的影响两方面。在地形发生较为显著变化的部位，格网尺寸的变化会使水流方向有较为剧烈的变化；通过对水流累积阈值和格网尺寸关系的分析，认为合理的水流累积阈值与格网尺寸存在二次幂指数递减关系。总体来讲，当确定了适宜的水流累积阈值，格网尺寸的变化并不影响沟道网络的整体结构特征，但会使沟道的空间位置发生变化，具体形态特征也发生变化。格网尺寸越大，沟道网络形态越概括。
3．基于剖面的沟道一维频谱特征
1）通过对实验区域布设的基本垂直于沟道延伸方向的剖面的空间特征及其一维频谱特征的对照，级别较高的侵蚀沟道，其空间形态通常起伏较大，间隔较宽，对应的空间频率较低，周期较长，振幅较大；而级别较低的侵蚀沟道，其空间形态一般起伏较小，间隔较窄，其对应频率较高，周期越短，振幅较小。从频谱分析结果中可以清楚反映不同振幅的起伏变化的空间频率，并能与沟道的实际起伏情况对应。
2）通过对不同频率组分的空间信号重建结果与原始沟道剖面数据的比对，侵蚀沟道的主要空间频率集中在较为低频的部分。不同等级的沟道对应的空间频率范围不同。包括细沟在内微小的地形起伏，空间频率大于0.1周/米；在约0.03——0.1周/米之间，基本对应坡面上的各种浅沟；在约0.015——0.085周/米之间，基本对应各种规模切沟；小于约0.015周/米的，主要是各种冲沟及更大规模沟道。
3）根据对不同格网尺寸DEM提取的剖面数据的频谱特征分析，在利用傅里叶分析方法进行地形剖面特征分析时，需尽可能采用较小格网尺寸DEM，并尽可能布设较长剖面。
4．二维频谱特征分析及DEM格网尺寸与DEM沟道信息表达能力的关系
1）根据对实验区域DEM数据的二维频谱特征的分析，二维频谱能够反映地表的各种尺度与规模的地形起伏，但直接利用二维频谱特征与空间特征建立对应关系是较为困难的。因此，本研究结合沟道延伸方向的统计结果，将一维剖面频谱分析结果通过频率分解至水平与垂直方向，得出实验区域各等级沟道的二维空间频率范围，并以此为依据设计二维频域低通滤波器。
2）对实验区域DEM数据进行低通滤波，逐步滤掉包括细沟的微小起伏、浅沟及切沟信息。根据滤波结果与各格网尺寸DEM的对比，得出格网尺寸与DEM沟道信息表达能力的关系：大于等于2m的格网尺寸，无法表达地面上包括细沟在内的微小的地形起伏的；格网尺寸大于5m，则很难表现坡面上浅沟特征；格网尺寸超过10m，切沟的特征也基本消失。
At Present，Digital Elevation Models(DEMs), especially the GRID DEMs, is the maindata sources for the research of soil erosion and terrain analysis. Numerous studies show thatthe grid size of DEMs greatly influences the results of terrain analysis. However, up to now, itis not clear and definite how large the grid size of the DEMs is required to demonstrate fullyand correctly the terrain information. This question has not answered clearly in the previouslystudies. In another word, the relationship between the grid size of DEMs and the ability todescribe the terrain information of DEMs is still not clear. This problem to some extent leadsto the blindness in the practical application of DEMs in the analysis of soil erosion and terrain.Based on the DEM generated by the full digital photogrammetry and field measurements inYaojiawangou watershed, Suide, Shaanxi Province, and taking the gullies of loess plateau asthe studying objects, this paper uses the Fourier Analysis to solve the problem mentionedabove. Firstly, this paper discusses the relationship between the elevation error of DEMs andgrid size. According to this result, it analyzes the influence of grid size on the extraction oferosion gullies network from DEM. Secondly, the paper analyzes the spatial frequencycharacters of different grade gullies and summaries the spatial frequency range of each gradegully with the analysis of spectrum on one dimension of24profiles lying on the gulliesnetwork. At last, the paper extends the results of spectrum analysis of one dimension profilesto two-dimensional spectral analysis, and then designs the two-dimensional frequencylow-pass filter to filter the information of each grade gully. According to the comparisonbetween the filtered DEMs and the original DEMs with different grid size, the study matchesspatial scale of erosion gullies with the grid size of DEMs. The conclusion of this paper solvesthe corresponding relationship between the expression ability of the terrain information for DEMs and the grid size of DEMs.
The main conclusions of this paper are as follows:
1. The relationship between the elevation error of DEMs and the grid size
The influence of grid size on the elevation error of DEMs is shown as follows: theelevation on a grid is overestimated in down-slope direction, while underestimated inup-slope direction. As a result, the value of elevation error on a grid is decided by the slopegradient, while the complication of distribution of elevation error is decided by the size ofgrid. The analysis of elevation error calculated from the DEMs series generated by samplingdirectly shows that The RMSE and the maximum error of elevation DEMs increases with thegrid size in the way that is quadratic function which is very similar with a linear relationship.The maximum error on a grid far exceeds the2times of RMSE.
2. The relationship between the characters of gullies network extracted from DEMs andthe grid size
By means of analyzing the characters of gullies network extracted from DEMs serieswhich have different grid sizes, the main influence of the grid size on the extraction of gulliesnetwork is reflected in two aspects---the flow direction and the threshold of flowaccumulation. The change of grid size causes the flow direction’s turning dramatically in thecomplicated topography area. The relationship between the grid size and the threshold of flowaccumulation shows that the reasonable threshold decreases while the grid size increases bythe way of negative second order power function. Generally, the change of grid size doesn’tinfluence the structure of the gullies network while the reasonable flow accumulationthreshold is determined. However, the position of gullies will move from the original positionand the shape of gullies will be changed. The larger the grid size is, the simpler the shape ofgullies.
3. One-dimensional spectrum of gullies based on profile data
1）The comparison between the spatial characters of profiles which are perpendicularwith the direction of the gullies and its One-dimensional spectrum shows that: the gully whichis in higher level, in general, has the greater depth and the wider interval, meanwhile, thecorresponding spatial frequency is lower, the period is longer and the amplitude is larger. Onthe contrary, the lower the level of gully is, the smaller the fluctuation, the narrower theinterval, the higher the frequency, the shorter the period, the smaller the amplitude. The spectrum could reflect the spatial frequency with varying amplitude clearly, and couldcorrespond to the spatial characters of fluctuation of gullies.
2）According to the result of comparison between the profiles reconstructed by thevarying spatial frequency and the original profiles, the main spatial frequency range oferosion gullies is concentrated in the low frequency part. The gullies with different gradescorrespond to the different spatial frequency: The minor fluctuation including rills on theslope has the frequency range higher than0.1cycle per meter; the frequency range of shallowfurrow is from0.03to0.1cycle per meter; the gully has the frequency range from0.015to0.085cycle per meter, while the gullies which are in higher grade correspond to the frequencyrange lower than0.015cycle per meter.
3）In terms of the spectrum of profiles extracted from the DEMs data series which havedifferent grid size, the analysis of profile characters using Fourier transformation requires thatthe DEMs should have smaller grid size, and that profile line should be as longer as possible.
4. Two-dimensional spectrum Analysis and the relationship between the expressionability of the terrain information for DEMs and the grid size
1）Based on the analysis of two-dimensional spectrum of DEMs of the studied area, thetwo-dimensional spectrum could reflect the scale of the relief. However, it is difficult toobtain the corresponding relationship between the two-dimensional spectrum and spatialcharacters. Therefore, according to the statistic of directions of gullies, the result of analysisof one-dimensional spectrum is decomposed to the horizontal and vertical direction to achievethe two-dimensional frequency range of gullies with different grades. This spatial range is thebasic parameter to design the two-dimensional frequency low-pass filter.
2） Using the two-dimensional frequency low-pass filters to remove the gulliesinformation step by step, three DEMs which do not contain the information of the rills,shallow furrows, gullies basically could be obtained respectively. The Comparison betweenthese three DEMs and the DEMs with varying grid sizes indicates the relationship betweenthe expression ability of the terrain information for DEMs and the grid size: the fluctuationincluding rills nearly could not be described on the DEMs whose grid size is larger than about2meters. When the grid size of DEMs is larger than about5meters, the shallow furrowscould be represented clearly. If the grid size of DEMs is larger than about10meters, thegullies information will nearly disappear.
1．DEM的高程精度与格网尺寸的关系
格网尺寸对规则格网DEM高程精度的影响表现在：上坡方向低估格网高程，下坡方向则高估。因此格网上的高程误差大小取决于地形坡度，其空间分布的复杂程度取决于格网尺寸大小。利用直接采样获得一系列DEM格网上高程误差的分析结果表明，DEM高程中误差和最大误差与格网尺寸存在近似线性的二次函数关系，随着格网尺寸增大，DEM高程误差增大，且格网上的最大高程误差远超出2倍中误差水平。
2．DEM格网尺寸与沟道网络提取及其基本特征的关系
利用不同格网尺寸DEM提取的沟道网络的特征分析的结果表明，DEM格网尺寸对沟道网络的提取主要体现在对水流方向的影响与水流累积阈值的影响两方面。在地形发生较为显著变化的部位，格网尺寸的变化会使水流方向有较为剧烈的变化；通过对水流累积阈值和格网尺寸关系的分析，认为合理的水流累积阈值与格网尺寸存在二次幂指数递减关系。总体来讲，当确定了适宜的水流累积阈值，格网尺寸的变化并不影响沟道网络的整体结构特征，但会使沟道的空间位置发生变化，具体形态特征也发生变化。格网尺寸越大，沟道网络形态越概括。
3．基于剖面的沟道一维频谱特征
1）通过对实验区域布设的基本垂直于沟道延伸方向的剖面的空间特征及其一维频谱特征的对照，级别较高的侵蚀沟道，其空间形态通常起伏较大，间隔较宽，对应的空间频率较低，周期较长，振幅较大；而级别较低的侵蚀沟道，其空间形态一般起伏较小，间隔较窄，其对应频率较高，周期越短，振幅较小。从频谱分析结果中可以清楚反映不同振幅的起伏变化的空间频率，并能与沟道的实际起伏情况对应。
2）通过对不同频率组分的空间信号重建结果与原始沟道剖面数据的比对，侵蚀沟道的主要空间频率集中在较为低频的部分。不同等级的沟道对应的空间频率范围不同。包括细沟在内微小的地形起伏，空间频率大于0.1周/米；在约0.03——0.1周/米之间，基本对应坡面上的各种浅沟；在约0.015——0.085周/米之间，基本对应各种规模切沟；小于约0.015周/米的，主要是各种冲沟及更大规模沟道。
3）根据对不同格网尺寸DEM提取的剖面数据的频谱特征分析，在利用傅里叶分析方法进行地形剖面特征分析时，需尽可能采用较小格网尺寸DEM，并尽可能布设较长剖面。
4．二维频谱特征分析及DEM格网尺寸与DEM沟道信息表达能力的关系
1）根据对实验区域DEM数据的二维频谱特征的分析，二维频谱能够反映地表的各种尺度与规模的地形起伏，但直接利用二维频谱特征与空间特征建立对应关系是较为困难的。因此，本研究结合沟道延伸方向的统计结果，将一维剖面频谱分析结果通过频率分解至水平与垂直方向，得出实验区域各等级沟道的二维空间频率范围，并以此为依据设计二维频域低通滤波器。
2）对实验区域DEM数据进行低通滤波，逐步滤掉包括细沟的微小起伏、浅沟及切沟信息。根据滤波结果与各格网尺寸DEM的对比，得出格网尺寸与DEM沟道信息表达能力的关系：大于等于2m的格网尺寸，无法表达地面上包括细沟在内的微小的地形起伏的；格网尺寸大于5m，则很难表现坡面上浅沟特征；格网尺寸超过10m，切沟的特征也基本消失。
At Present，Digital Elevation Models(DEMs), especially the GRID DEMs, is the maindata sources for the research of soil erosion and terrain analysis. Numerous studies show thatthe grid size of DEMs greatly influences the results of terrain analysis. However, up to now, itis not clear and definite how large the grid size of the DEMs is required to demonstrate fullyand correctly the terrain information. This question has not answered clearly in the previouslystudies. In another word, the relationship between the grid size of DEMs and the ability todescribe the terrain information of DEMs is still not clear. This problem to some extent leadsto the blindness in the practical application of DEMs in the analysis of soil erosion and terrain.Based on the DEM generated by the full digital photogrammetry and field measurements inYaojiawangou watershed, Suide, Shaanxi Province, and taking the gullies of loess plateau asthe studying objects, this paper uses the Fourier Analysis to solve the problem mentionedabove. Firstly, this paper discusses the relationship between the elevation error of DEMs andgrid size. According to this result, it analyzes the influence of grid size on the extraction oferosion gullies network from DEM. Secondly, the paper analyzes the spatial frequencycharacters of different grade gullies and summaries the spatial frequency range of each gradegully with the analysis of spectrum on one dimension of24profiles lying on the gulliesnetwork. At last, the paper extends the results of spectrum analysis of one dimension profilesto two-dimensional spectral analysis, and then designs the two-dimensional frequencylow-pass filter to filter the information of each grade gully. According to the comparisonbetween the filtered DEMs and the original DEMs with different grid size, the study matchesspatial scale of erosion gullies with the grid size of DEMs. The conclusion of this paper solvesthe corresponding relationship between the expression ability of the terrain information for DEMs and the grid size of DEMs.
The main conclusions of this paper are as follows:
1. The relationship between the elevation error of DEMs and the grid size
The influence of grid size on the elevation error of DEMs is shown as follows: theelevation on a grid is overestimated in down-slope direction, while underestimated inup-slope direction. As a result, the value of elevation error on a grid is decided by the slopegradient, while the complication of distribution of elevation error is decided by the size ofgrid. The analysis of elevation error calculated from the DEMs series generated by samplingdirectly shows that The RMSE and the maximum error of elevation DEMs increases with thegrid size in the way that is quadratic function which is very similar with a linear relationship.The maximum error on a grid far exceeds the2times of RMSE.
2. The relationship between the characters of gullies network extracted from DEMs andthe grid size
By means of analyzing the characters of gullies network extracted from DEMs serieswhich have different grid sizes, the main influence of the grid size on the extraction of gulliesnetwork is reflected in two aspects---the flow direction and the threshold of flowaccumulation. The change of grid size causes the flow direction’s turning dramatically in thecomplicated topography area. The relationship between the grid size and the threshold of flowaccumulation shows that the reasonable threshold decreases while the grid size increases bythe way of negative second order power function. Generally, the change of grid size doesn’tinfluence the structure of the gullies network while the reasonable flow accumulationthreshold is determined. However, the position of gullies will move from the original positionand the shape of gullies will be changed. The larger the grid size is, the simpler the shape ofgullies.
3. One-dimensional spectrum of gullies based on profile data
1）The comparison between the spatial characters of profiles which are perpendicularwith the direction of the gullies and its One-dimensional spectrum shows that: the gully whichis in higher level, in general, has the greater depth and the wider interval, meanwhile, thecorresponding spatial frequency is lower, the period is longer and the amplitude is larger. Onthe contrary, the lower the level of gully is, the smaller the fluctuation, the narrower theinterval, the higher the frequency, the shorter the period, the smaller the amplitude. The spectrum could reflect the spatial frequency with varying amplitude clearly, and couldcorrespond to the spatial characters of fluctuation of gullies.
2）According to the result of comparison between the profiles reconstructed by thevarying spatial frequency and the original profiles, the main spatial frequency range oferosion gullies is concentrated in the low frequency part. The gullies with different gradescorrespond to the different spatial frequency: The minor fluctuation including rills on theslope has the frequency range higher than0.1cycle per meter; the frequency range of shallowfurrow is from0.03to0.1cycle per meter; the gully has the frequency range from0.015to0.085cycle per meter, while the gullies which are in higher grade correspond to the frequencyrange lower than0.015cycle per meter.
3）In terms of the spectrum of profiles extracted from the DEMs data series which havedifferent grid size, the analysis of profile characters using Fourier transformation requires thatthe DEMs should have smaller grid size, and that profile line should be as longer as possible.
4. Two-dimensional spectrum Analysis and the relationship between the expressionability of the terrain information for DEMs and the grid size
1）Based on the analysis of two-dimensional spectrum of DEMs of the studied area, thetwo-dimensional spectrum could reflect the scale of the relief. However, it is difficult toobtain the corresponding relationship between the two-dimensional spectrum and spatialcharacters. Therefore, according to the statistic of directions of gullies, the result of analysisof one-dimensional spectrum is decomposed to the horizontal and vertical direction to achievethe two-dimensional frequency range of gullies with different grades. This spatial range is thebasic parameter to design the two-dimensional frequency low-pass filter.
2） Using the two-dimensional frequency low-pass filters to remove the gulliesinformation step by step, three DEMs which do not contain the information of the rills,shallow furrows, gullies basically could be obtained respectively. The Comparison betweenthese three DEMs and the DEMs with varying grid sizes indicates the relationship betweenthe expression ability of the terrain information for DEMs and the grid size: the fluctuationincluding rills nearly could not be described on the DEMs whose grid size is larger than about2meters. When the grid size of DEMs is larger than about5meters, the shallow furrowscould be represented clearly. If the grid size of DEMs is larger than about10meters, thegullies information will nearly disappear.
引文
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