Influence of resolution on elevation and slope at watershed scale in Loess Plateau
- Wang Chunmei^{1},
- Yang Qinke^{1}Email author,
- Liu Hongyan^{2},
- Guo Weiling^{3},
- David LB Jupp^{4} and
- Li Rui^{3}
https://doi.org/10.1186/2193-1801-2-S1-S13
© Chunmei et al.; licensee Springer 2013
Published: 11 December 2013
Abstract
This article studied the effects of resolution on elevation and slope using Statistical and Geostatistical methods. Xian' Nan watershed in Loess Plateau was taking as the study area. The base data was a 1:10000 topographic map and the resolutions studied in this paper included 5 m, 25 m and 50 m. The results showed that: (1) for elevation and slope data, the mean value, STD value, histogram and semi-variogram changed with resolution reduction. The mean value, STD value became smaller in both elevation and slope cases. Histograms moved to the left which shows there was a decrease of elevation and slope with resolution. The sill for semi-variograms of elevation and slope decreased with resolution reduction; (2) the changes of Mean value; STD and histogram were greater in elevation data than in slope data. (3) By using the Independent Structure model, the semi-variogram could be modeled by 4 components for elevation data and the semi-variogram could be modeled by 3 components for slope data. There was more information in slope than in elevation in the components with short range (short wave-length) information. (4) The influence of resolution reduction was greater in the components with short range, so the degree of influence of resolution reduction was related to the amount of short wave-lengths information. The results of this paper had shown which information was lost with resolution reduction and the reason for the different changes on mean value, STD and histogram for elevation and slope. It could also be used to explain different scaling effects in different terrain areas in the future.
Keywords
Introduction
Digital Elevation Models (DEM) is widely used as a digital representation of terrain. Terrain models include important factors for soil erosion and hydrology, etc. [1–3]. DEM resolution is one of the most important factors that influence the ability of a DEM to represent terrain. That is because as resolution becomes coarser, many of the terrain indexes that derived from the DEM will change. Elevation and Slope are two of the most important terrain factors in many study fields [4, 5]. Many researchers have paid attention to the change of terrain factors with DEM resolution reduction such as Chang and Tsai (1991) [6], Gao (1997) [7], Zhang (1999) [8], Wolock (2000) [9] and Wu et al. (2008) [10]. Their results showed that slope tended to become smaller in most of areas with resolution reduction.
But there is not yet sufficient study on why there are differences between terrain factors derived from fine resolution DEM and from coarse resolution DEM and which part of the information has been lost with resolution reduction. Geostatistics has been widely applied to study fields such as vegetation investigation, soil characteristic analyses, etc. Some researchers have used geostatistical analyses to study spatial patterns in topography [11]. In this research the authors studied the structure of terrain and the change in terrain structure with resolution reduction by investigating changes in each component of the semi-variogram. The study uses Xian' Nan watershed as the study area. This watershed is located in Loess Hilly area in Loess Plateau in China. The authors studied the differences between elevation and slope derived from DEMs with resolutions of 5 m, 25 m and 50 m.
The aim of this research is to show which information "disappears" or "reduces" when resolution becomes coarser. This may help explain why terrain factors change with resolution and understand the terrain characteristics which change with resolution.
Material and methods
Study area
Base data and data processing
The base data is a topographic map at 1:10,000 scale issued by China's National Bureau of Surveying and Mapping in 1981 with a contour interval of 5 m which covers the whole study area. The data processing includes topographic map digitizing, projection transformation, check in elevation values, check in river directions and Lake Boundaries. The projection for the base data is Krasovsky_1940_Albers. Longitude of the Central Meridian is E105°. The first Standard Parallel is N25° and the second Standard Parallel is N47°. Latitude of projection Origin is 0°. False Easting and False Northing are both 0 m. The projection is an equal-area projection. The grid cells are "square" and in meters.
Parameters of ANUDEM
Maximum iterations | Profile curvature | Cell size (m) |
---|---|---|
40 | 0.7 | 5, 25, 50 |
Information statistics for elevation and slope
Slope calculation
Where "h" is the resolution step or grid cell size for the data.
Histogram intersection
Swain and Ballard [13] efficiently recognized objects by matching their color histograms using the histogram intersection (HI) method. In this research the author used Histogram Intersection (HI) to evaluate histogram similarity.
Where HI(X, Y) is Histogram Intersection of two histograms X and Y; ${x}_{i}$ and ${y}_{i}$ are frequency values of X and Y at slope value of i. The values for HI(X, Y) range from 0% to 100%. Histograms are more similar to each other if HI(X, Y) is larger. If two histograms are totally the same, HI(X, Y) equals to 100%.
Reduction rate of mean value and STD
Where ${\mathsf{\text{M}}}_{v}$ is Mean value Reduction Rate; ${\mathsf{\text{M}}}_{{r}^{\prime}}$ is mean value with coarser resolution of ${r}^{\prime}$; ${\mathsf{\text{M}}}_{r}$ is mean value with finer resolution of $\phantom{\rule{0.1em}{0ex}}r$; ${S}_{v}$ is STD Reduction Rate; ${\mathsf{\text{S}}}_{{r}^{\prime}}$ is STD value with coarser resolution of ${r}^{\prime}$; ${\mathsf{\text{S}}}_{r}$ is STD value with finer resolution of $\phantom{\rule{0.1em}{0ex}}r$.
Geostatistical analysis
Covariance and semi-variogram
$C\left({h}_{x}{h}_{y}\right)$ is the covariance function and $\gamma \left({h}_{x},{h}_{y}\right)$ is the semi-variance function; (x, y) stands for the spatial coordinators of the tested point of the slope; Z(x, y) stands for the data of the tested point; ${h}_{x}$ and ${h}_{y}$ stands for the interval in x and y direction between two tested points (in 1D cases $h={\left({{h}_{x}}^{2}+{{h}_{y}}^{2}\right)}^{1/2}$), m is the mean value over the image. We will assume that the covariance is stationary and the mean (m) is a constant over the image.
${\sigma}^{2}$ is the variance of the image data which needs to be spatially stationary [14].
Modeling covariance and semi-variogram
In this way, the first component is the one with greatest "roughness" or high spatial frequency content effect and the last is the one with greatest low frequency (regional) effect. This representation models the data with different "scale" components with ${Y}_{1}\phantom{\rule{0.3em}{0ex}}{Y}_{2}\phantom{\rule{0.3em}{0ex}}{Y}_{3}\dots \dots {Y}_{N}$ representing scale components with decreasing map scale.
The radial distance is $h=\sqrt{{h}_{x}^{2}+{h}_{y}^{2}}$; ${\sigma}_{j}^{2}$ is the variance of the component j; The quantity ${R}_{j}$ is taken as the range of component j and the component has the form of a correlation function.
Effect of filtering on the covariance and semi-variogram
Where ${C}_{FZ}\left(h\right)$ the covariance of the filtered functions and ${\gamma}_{FZ}\left(h\right)$ is the semivariogram of the filtered function.
FWHM (Full Width Half Maximum) is the width of the filter at half height. H is the height of the filter.
The covariance of the filtered function is the same type as before however it has reduced variance and increased range.
Model efficiency analysjs
In order to evaluate the result of the Independent Structures model of the semi-variogram, the model efficiency coefficient (ME) which was proposed by Nash and Sutcliffe [15] was used in this research.
Where ME is the model efficiency, ${Y}_{obs}$ is the observed value, ${Y}_{pred}$ is the predicted value, ${Y}_{mean}$ is the mean observed value. Values for ME range from -∞ to 1. The closer ME is to 1, the better the model will predict individual values.
Results and analysis
Mean elevation and slope change with resolution
Mean and Std of elevation and slope
Mean value | STD | |||
---|---|---|---|---|
Resolution (m) | Elevation (m) | Slope (°) | Elevation (m) | Slope (°) |
5 | 1220.1 | 28.5 | 74.0 | 11.1 |
25 | 1219.5 | 22.7 | 73.4 | 8.8 |
50 | 1218.9 | 18.5 | 72.9 | 7.4 |
Mean value and STD reduction rate
Mean value | STD | |||
---|---|---|---|---|
Elevation (m) | Slope (°) | Elevation (m) | Slope (°) | |
5 m-25 m | -0.03 | -0.29 | -0.03 | -0.12 |
25 m-50 m | -0.03 | -0.17 | -0.03 | -0.07 |
Histograms of elevation and slope changes with resolution
HI comparing with 5 m data (%)
5 m | 25 m | 50 m | |
---|---|---|---|
DEM | 100 | 96.258 | 94.0111 |
Slope | 100 | 73.774 | 55.7426 |
There seems to be a great difference between slope histograms with 25 m, 50 m resolution and the slope histogram with 5 m resolution (Figure 2b). As resolution reduces, the histograms of slope move to the left side which is lower. HI values between the slope histograms with 25 m and 50 m resolution and the slope histogram with 5 m resolution are 73.8% and 55.7% which shows the great influence of resolution on the histogram of slope.
Changes in semi-variogram of elevation and slope with resolution
Independent structures of semi-variogram
Sill and range of elevation semi-variogram
Resolution (m) | Range | Sill | |
---|---|---|---|
5 | 140.24 | 328.23 | |
Y1 | 25 | 148.64 | 292.19 |
50 | 164.54 | 238.44 | |
5 | 404.38 | 1629.60 | |
Y2 | 25 | 407.37 | 1605.78 |
50 | 413.44 | 1558.99 | |
5 | 1866.01 | 977.49 | |
Y3 | 25 | 1866.66 | 976.81 |
50 | 1867.99 | 975.42 | |
5 | 50315.37 | 35390.90 | |
Y4 | 25 | 5212.52 | 1133.83 |
50 | 5058.91 | 977.95 | |
5 | 2410.63 | 2935.33 | |
Model-Sum | 25 | 2422.67 | 2874.78 |
50 | 2445.98 | 2772.84 |
Sill and range of slope semi-variogram
Resolution (m) | Range | Sill | |
---|---|---|---|
5 | 32.88 | 61.71 | |
Y1 | 25 | 39.58 | 23.90 |
50 | 49.68 | 11.18 | |
5 | 150.19 | 43.51 | |
Y2 | 25 | 207.23 | 40.44 |
50 | 284.18 | 35.76 | |
5 | 997.53 | 17.97 | |
Y3 | 25 | 2503.71 | 17.94 |
50 | 4006.32 | 17.88 | |
5 | 1180.60 | 123.18 | |
Model-Sum | 25 | 2750.52 | 82.27 |
50 | 4340.18 | 64.83 |
The effect of resolution on the independent structures of semi-variogram
ME of independent structure model
Resolution | Elevation | Slope |
---|---|---|
5 | 0.9997 | 0.9920 |
25 | 0.9994 | 0.8231 |
50 | 0.9987 | 0.6743 |
The influence of resolution reduction is smallest on long wave-length components and is largest for Y1 in both elevation and slope cases. Table 5 and Figure 3a show that in the elevation case, the sill value of Y1 reduced to 72.6% of the 5 m case when resolution is 50 m. Y2 and Y3 seem to have be stable and the reduction of sill when the resolution is 50 m is less than 5% of the 5 m case. Table VI and Figure 3b show that in slope case, the sill value of Y1 reduced to 18.1% of the 5 m case when resolution is 50 m. The sill value of Y2 reduced to 82.2% of the 5 m case when the resolution is 50 m and the sill value of Y3 reduced to99.5% of the 5 m case when the resolution is 50 m.
Comparing the elevation and slope cases, the influence of resolution reduction on slope is much larger than on elevation especially in the Y1 components. The Sill (Model sum) of elevation reduced only to 94.4% of the 5 m case but the sill (Model sum) of slope reduced to 52.6% of the 5 m case. The reason is that the process of slope calculation acts as a high pass filter. More short wave-length information is left in the image after the slope is calculated. Therefore there is much more short wave-length information in slope data than in elevation data. The low pass filter and the resolution reduction effect greatly influence the short wave-length information.
Discussion
ME of slope is lower than ME of elevation. It may because that the Gaussian Model is not the most appropriate one for slope. In this paper the main purpose is to show the scale effect which has been done. Work on appropriate model for slope is investigated by the authors.
In this paper the authors modeled the effects of resolution reduction on the semi-variogram using a Gaussian filter. Although the way ANUDEM changes with scale do not behave exactly the same way as a Gaussian filter, the model can work quite well. In the future, more effort is being put into the study of the differences between the two.
The semi-variograms in this paper has been modeled using Independent structure model. The model allows us to clarify the way resolution influence DEM data. In further study this would be applied to different terrain types.
In further study, the semi-variogram of each component will be mapped out using kriging to show the nature of different information in each component.
Conclusions
In this paper the independent structure model was used to model the semi-variograms of elevation and slope data with resolution of 5 m, 25 m and 50 m. The results showed how the short wave-length information disappeared or weakened as resolution reduced. These results can explain which component of the information has changed with resolution reduction. By calculating the mean value, STD and histogram of both elevation and slope at different resolutions, it is clear that the influence of resolution on elevation is less than on slope. The reason is that the short wave-length information accounts for more variance in slope data than in elevation data. Resolution reduction or low pass filtering influence the short wave-length information the most.
Declarations
Acknowledgements
This research is supported by Research on Spatial Frequency of Erosional Terrain in Loess Hilly Region(NSFC Project, 41301284) and Study on theoretical distribution model of Slope (NSFC Project, 41371274) . Part of the research was finished during Chunmei Wang’s visit to CSIRO.
Declarations
The publication costs for this article were funded by Scientific & Technical Development Inc.
This article has been published as part of SpringerPlus Volume 2 Supplement 1, 2013: Proceedings of the 2010 International Conference on Combating Land Degradation in Agricultural Areas (ICCLD'10). The full contents of the supplement are available online at http://www.springerplus.com/supplements/2/S1.
Authors’ Affiliations
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