**Other Abstract** | At Present，Digital Elevation Models(DEMs), especially the GRID DEMs, is the main
data sources for the research of soil erosion and terrain analysis. Numerous studies show that
the grid size of DEMs greatly influences the results of terrain analysis. However, up to now, it
is not clear and definite how large the grid size of the DEMs is required to demonstrate fully
and correctly the terrain information. This question has not answered clearly in the previously
studies. In another word, the relationship between the grid size of DEMs and the ability to
describe the terrain information of DEMs is still not clear. This problem to some extent leads
to 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 in
Yaojiawangou watershed, Suide, Shaanxi Province, and taking the gullies of loess plateau as
the studying objects, this paper uses the Fourier Analysis to solve the problem mentioned
above. Firstly, this paper discusses the relationship between the elevation error of DEMs and
grid size. According to this result, it analyzes the influence of grid size on the extraction of
erosion gullies network from DEM. Secondly, the paper analyzes the spatial frequency
characters of different grade gullies and summaries the spatial frequency range of each grade
gully with the analysis of spectrum on one dimension of 24 profiles lying on the gullies
network. At last, the paper extends the results of spectrum analysis of one dimension profiles
to two-dimensional spectral analysis, and then designs the two-dimensional frequency
low-pass filter to filter the information of each grade gully. According to the comparison
between the filtered DEMs and the original DEMs with different grid size, the study matches
spatial scale of erosion gullies with the grid size of DEMs. The conclusion of this paper solves
the 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: the
elevation on a grid is overestimated in down-slope direction, while underestimated in
up-slope direction. As a result, the value of elevation error on a grid is decided by the slope
gradient, while the complication of distribution of elevation error is decided by the size of
grid. The analysis of elevation error calculated from the DEMs series generated by sampling
directly shows that The RMSE and the maximum error of elevation DEMs increases with the
grid size in the way that is quadratic function which is very similar with a linear relationship.
The maximum error on a grid far exceeds the 2 times of RMSE.
2. The relationship between the characters of gullies network extracted from DEMs and
the grid size
By means of analyzing the characters of gullies network extracted from DEMs series
which have different grid sizes, the main influence of the grid size on the extraction of gullies
network is reflected in two aspects --- the flow direction and the threshold of flow
accumulation. The change of grid size causes the flow direction’s turning dramatically in the
complicated topography area. The relationship between the grid size and the threshold of flow
accumulation shows that the reasonable threshold decreases while the grid size increases by
the way of negative second order power function. Generally, the change of grid size doesn’t
influence the structure of the gullies network while the reasonable flow accumulation
threshold is determined. However, the position of gullies will move from the original position
and the shape of gullies will be changed. The larger the grid size is, the simpler the shape of
gullies.
3. One-dimensional spectrum of gullies based on profile data
1）The comparison between the spatial characters of profiles which are perpendicular
with the direction of the gullies and its One-dimensional spectrum shows that: the gully which
is in higher level, in general, has the greater depth and the wider interval, meanwhile, the
corresponding spatial frequency is lower, the period is longer and the amplitude is larger. On
the contrary, the lower the level of gully is, the smaller the fluctuation, the narrower the
interval, the higher the frequency, the shorter the period, the smaller the amplitude. The spectrum could reflect the spatial frequency with varying amplitude clearly, and could
correspond to the spatial characters of fluctuation of gullies.
2）According to the result of comparison between the profiles reconstructed by the
varying spatial frequency and the original profiles, the main spatial frequency range of
erosion gullies is concentrated in the low frequency part. The gullies with different grades
correspond to the different spatial frequency: The minor fluctuation including rills on the
slope has the frequency range higher than 0.1 cycle per meter; the frequency range of shallow
furrow is from 0.03 to 0.1 cycle per meter; the gully has the frequency range from 0.015 to
0.085 cycle per meter, while the gullies which are in higher grade correspond to the frequency
range lower than 0.015 cycle per meter.
3）In terms of the spectrum of profiles extracted from the DEMs data series which have
different grid size, the analysis of profile characters using Fourier transformation requires that
the 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 expression
ability 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, the
two-dimensional spectrum could reflect the scale of the relief. However, it is difficult to
obtain the corresponding relationship between the two-dimensional spectrum and spatial
characters. Therefore, according to the statistic of directions of gullies, the result of analysis
of one-dimensional spectrum is decomposed to the horizontal and vertical direction to achieve
the two-dimensional frequency range of gullies with different grades. This spatial range is the
basic parameter to design the two-dimensional frequency low-pass filter.
2）Using the two-dimensional frequency low-pass filters to remove the gullies
information step by step, three DEMs which do not contain the information of the rills ,
shallow furrows, gullies basically could be obtained respectively. The Comparison between
these three DEMs and the DEMs with varying grid sizes indicates the relationship between
the expression ability of the terrain information for DEMs and the grid size: the fluctuation
including rills nearly could not be described on the DEMs whose grid size is larger than about
2 meters. When the grid size of DEMs is larger than about 5 meters, the shallow furrows
could be represented clearly. If the grid size of DEMs is larger than about 10 meters, the
gullies information will nearly disappear. |

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