Scale characters analysis for gully structure in the watersheds of loess landforms based on digital elevation models

Abstract

Scale is the basic attribute for expressing and describing spatial entity and phenomena. It offers theoretical significance in the study of gully structure information, variable characteristics of watershed morphology, and development evolution at different scales. This research selected five different areas in China’s Loess Plateau as the experimental region and used DEM data at different scales as the experimental data. First, the change rule of the characteristic parameters of the data at different scales was analyzed. The watershed structure information did not change along with a change in the data scale. This condition was proven by selecting indices of gully bifurcation ratio and fractal dimension as characteristic parameters of watershed structure information. Then, the change rule of the characteristic parameters of gully structure with different analysis scales was analyzed by setting the scale sequence of analysis at the extraction gully. The gully structure of the watershed changed with variations in the analysis scale, and the change rule was obvious when the gully level changed. Finally, the change rule of the characteristic parameters of the gully structure at different areas was analyzed. The gully fractal dimension showed a significant numerical difference in different areas, whereas the variation of the gully branch ratio was small. The change rule indicated that the development degree of the gully obviously varied in different regions, but the morphological structure was basically similar.

Keywords

watershed / scale features / gully structure / bifurcation ratio / fractal dimension / scale sequence

Introduction

A watershed is a natural geographic unit that is mainly characterized by surface runoff erosion, weathering, and block movement (^{Band, 1989}; ^{Lu et al., 1991}). The characteristics of a watershed, such as morphology, structure, and evolution, have always been a research hotspot, and related studies have achieved several notable results. Four laws on the geographical characteristics of watersheds were proposed by analyzing the quantity of morphology (^{Horton, 1945}). Based on the watershed structure characteristics, the division principle of stream ordering was proposed (^{Strahler, 1957}). Watershed research has generated results based on the theories of Horton and Strahler. Take for example the Loess Plateau watershed. The validity of Horton and Strahler’s theories was verified by establishing the watershed structure and quantitative indices (Cheng and Jiang, 1986). On the basis of stream ordering by Strahler’s method, the fractal characteristics of a watershed were analyzed (^{Altaf et al., 2014}). Drainage density was studied in broke flash flood areas (^{Oguchi, 2015}). River parameters were studied in tropical regions (^{Babu et al., 2016}). The fractal characteristics of river morphology and a proposed quantitative calculation of fractal dimensions were discussed by using the theory of fractal geometry (^{Strahler, 1958}; ^{Mandelbrot, 1967}; Mandelbrot and Wheeler, 1982; La Barbera and Rosso, 1989). Fractal dimensions can be used to quantify the watershed structure, which is thus explained according to the characteristic of watershed development morphology. In addition, the feature of stream networks is studied based on watershed evolution (^{Schuller et al., 2001}; ^{Shen et al., 2011}; ^{Tian et al., 2012}; ^{Pan et al., 2012}).

The Loess Plateau of China is the most typical loess landform in the world. It possesses a representative gully terrain area that attracts worldwide attention for its formation, geomorphological characteristics, evaluation, and significance to the local environment and human activities (^{Lu et al., 1991}; ^{Stevens et al., 2013}; ^{Xiong et al., 2016}; ^{Zhu et al., 2018}). A gully is the basic structure of catchment topography. The special analysis and application of watershed and gullies based on the morphological structure and hydrological properties of gullies have always been priorities in topography and watershed hydrology research. By using field measurements for loess hilly regions, gully types were divided (^{Chen, 1984}). The information about the gully head was studied based on DEM data (^{Zhu et al., 2014}). Gully evolution was investigated under paleotopographic controls (^{Xiong et al., 2017}). The development of gully systems was studied based on monsoonal climatic shift (^{Huang et al., 2012}). The experiment analyzed the structure of a gully that was influenced by artificial factors (^{Frankl et al., 2012}).

Scale is the limit of spatial entity when describing and analyzing all geographic objects, which include specific scope and frequency. Scale is also defined as space, time, and semantic dimension (^{Dungan et al., 2002; Li and Ying, 2005}). The current multiple definition of scale inevitably causes diversity and complexity in object scale characteristics. The optimal scale, scale effect, and scale dependency are necessary scientific propositions in the processes of description, analysis, and simulation. Optimal spatial scale ranges were determined by studying terrain slope and aspect (^{Gao, 1997}; ^{Tang et al., 2006}). A scale effect was proposed in the system of regional geography (^{Lu et al., 2000}; ^{Wang et al., 2014}). Scale dependency was studied by focusing on the role of vegetation in watershed formation, analysis of the spatial scale model, and scales of the imaging principle in the geological entity space (^{William, 2011}; Sen Roy and Rouault, 2013; ^{Bultreys et al., 2016}).

Data scale is crucial to determine the complexity of a watershed structure. Therefore, based on different scales, the most suitable cell size for hydrology was studied, and the extraction results of the watershed were analyzed in a study on water resources (^{Sharma et al., 2011; Lin et al., 2013}). Aside from the influence of data scale, the accumulation threshold value also changes the watershed structure. Therefore, the threshold value of flow accumulation is regarded as an analysis scale (^{Zhou et al., 2008}). To determine the optimal range for extracting gullies and the evolution of the watershed in different historical periods, many scholars studied the watershed structure indices under different analysis scales (^{Li et al., 2014}; ^{Xiong et al., 2016}; ^{Xiong et al., 2017}). In the analysis of current extraction results and watershed information, a watershed, as a natural geographical unit, must have its own characteristics and regularity in the expression of data scale and information analysis scale. Therefore, conducting a systematic and in-depth study on the scale characteristics of a watershed structure and studying the differences and similarities of the scale in the watershed structure information are necessary to further enrich and complete the theory on the system of digital watershed geomorphology. In the present study, multiple samples of watershed regions were obtained based on DEM data. Then a multi-representation of DEM data and the analysis scale sequence of watershed threshold values were separately built, and the watershed structure indicators of each sample of watershed region were calculated. The change rules of the quantitative indicators of watershed structure were analyzed in different data scales and analysis scales. The scale characteristics of watershed structure, along with the geographical features of watershed regions, were studied in depth.

Materials and methods

Study area

In the Loess Plateau of China, gully development is mature and representative. Thus, Yijun, Ganquan, Yanchuan, Suide, and Jiaxian, which are located in the Loess Plateau of Shaanxi Province, were selected as the study areas (Fig. 1(a)). Typical loess landforms and significantly developed gullies are distributed in these areas. Digital topographical maps in scales of 1:10,000 with a spatial resolution of 5 m and 1:50,000 with a spatial resolution of 25 m were used as experimental data. To ensure the integrity of the scientific research, an entire watershed of each test area was extracted using DEM data (Fig. 1(b)–Fig. 1(f)). The feature parameter of the watershed and the landform type of the test areas are shown in Table 1.

Experimental method

The flowchart of the experimental method is shown in Fig. 2.

In Fig. 2, the scale sequence includes multi-resolution data and the different watershed threshold values. The watershed structure information was extracted with the hydrological analysis method. The quantitative parameters for describing the watershed structure information were the bifurcation ratio (^{Horton, 1945}) and the fractal dimension (^{Mandelbrot and Wheeler, 1982}). The specific details of research method are described as follows.

Extraction and computation of watershed structure information

1) In this experiment, the watershed, measuring approximately 15–20 km², and its information were extracted by ArcGIS software. This step ensured the extraction yielded more and convenient analysis of abundant watershed information.

2) The watershed structure indices were computed. In the analysis of watershed structure, the gully levels were divided. Several criteria were considered in dividing the gully levels, and the improved method called Strahler’s sequence (^{Strahler, 1957}) was adopted. Among the information indexes for describing a watershed structure, the bifurcation ratio and the fractal dimension are important indices to describe the spatial structure and morphological information of the watershed. The bifurcation ratio is the ratio of a certain gully level number to a higher one. The calculation formula is expressed as follows (^{Cheng and Jiang, 1986}):

(1){{\displaystyle r}}_b={\displaystyle \frac{{{\displaystyle N}}_{\mu }}{{{\displaystyle N}}_{\mu +\mathrm{1}}}}.

In Eq. (1), N_m is the number of gullies, m is a certain level, N_m+1 is the higher-level number of gullies, and m+1is a higher level.

The fractal dimension can reflect the development stage and developmental morphology of gullies. The specific calculation method is shown by Eq. (2) (^{Hong and Hong, 1988}).

(2)D={\displaystyle \frac{{{\displaystyle \mathit{lg}r}}_b}{{{\displaystyle \mathit{lg}r}}_{\mathrm{1}}}}.

In Eq. (2), r_b is the mean bifurcation ratio of the gully and r₁ is the mean length ratio of the gully.

Determination of scale

Different spatial resolutions result in different descriptions of the spatial structure of watersheds. Therefore, the DEM data of 1:10,000 and 1:50,000 scales were used to extract the gully and explore the relationship between spatial data and watershed structure in the same area. Different accumulation threshold values were set to obtain the different gully structure-level information of the watershed and analyze the spatial patterns of the watershed structure information. The analysis scale subdivided the accumulation threshold value of the watershed. The spatial information on small watersheds extracted from the DEM data was more abundant in the1:10,000 scale than in the 1:50,000 scale. The analysis scale sequence is the sequence of the accumulation threshold values of watersheds. The watershed information with different accumulation threshold values was extracted by using a method similar to that used in watershed and gully extraction.

Mathematical methods for analyzing the relationship between watershed structure information and scale

 In this research, mathematical analysis included the correlation coefficient and variation coefficient calculation methods. Correlation coefficient is the statistical index used to reflect the correlation among variables. The specific calculation method is formulated as

(3)r(X,Y)={\displaystyle \frac{Cov(X,Y)}{\sqrt{Var[X]}Var[Y]}}.

In Eq. (3), Cov(X,Y) is the covariance between X and Y, Var[X] is the variance of X, and Var[Y] is the variance of Y.

The discrete degree of data can be determined by calculating the variation coefficient. The formula for calculating the variation coefficient is as follows:

(4)C={\displaystyle \frac{STD(X)}{\bar{X}}}

In Eq. (4), STD(X) is the standard deviation, and \bar{X} is the mean value of this group.

Experimental results and analyses

Result of extraction of watershed structure in different data scales

To analyze the spatial structure of the watershed, we studied the changes in the watershed structure at different data scales (spatial resolution). The data resolutions were established by the DEM data resampling at the 1:10,000 and 1:50,000 scales. Four data resolutions were used to extract the gully and explore the relationship between data scale and watershed structure in the same area. Given that a single watershed has limited information and cannot express the entire data features at a small scale, the entire DEM data were adopted to extract watershed information. As the spatial resolutions of the 1:10,000 and 1:50,000 scales maps were different, the pixel size of the1:50,000 scale map data was bigger than that of the 1:10,000 scale map data. Given the data scales of 1:10,000 (5 m) and 1:50,000 (25 m) as examples, the maximum flow accumulation of the 1:50,000 scale DEM data was equal to 1/25 of the 1:10,000 scale DEM data for determining the same water catchment area. In the extraction and analysis of the characteristics of the gully structure, the flow accumulation threshold of the two different DEM data scales was set to 1.0% of the maximum flow accumulation of each dataset. This parameter was obtained through many experiments of extraction of natural stream networks based on DEMs. The watershed structure information based on different data scales (spatial resolution) is shown in Table 2.

By analyzing the data in Table 2, we found that the parameters value describing the watershed structure information, which were extracted from the same map scale but different spatial resolutions data, were almost the same. Thus, the data scales of 1:10,000 (5 m) and 1:50,000 (25 m) were selected to analyze the watershed information. The results in the different test areas are shown in Figs. 3 and 4.

By analyzing Figs. 3 and 4, we found that the bifurcation ratio did not exhibit an obvious change in the five test areas, thereby indicating that the watershed information was not affected by the spatial resolution of data in different scales. The statistics of the fractal dimension values for Yanchuan, Jiaxian, and Yijun changed in the range of 0.16–0.42; Yanchuan’s values changed drastically. The statistics of the fractal dimension values for Suide and Ganquan were stable. The fractal dimension value of the 1:50,000 scale was greater than that of the 1:10,000 scale in those areas. Given that the DEM data of different scales had different spatial resolutions, the mean length ratio of the gullies extracted from the 1:50,000 scale DEM data was greater than that extracted from the 1:10,000 scale DEM data.

Result of extraction watershed structure in different analysis scales

Establishment of analysis scale sequence

Different accumulation threshold values were set to obtain the different structure level information of gullies of the watershed and analyze the spatial patterns of watershed structure information. Thus, the analysis scale subdivided the accumulation threshold value of the watershed. The spatial information on small watersheds extracted from the DEM data was more abundant for the1:10,000 scale than for the 1:50,000 scale. Therefore, we used the DEM data at the 1:10,000 scale to establish the analysis scale.

The initial scale of analysis was determined according to the maximum flow accumulations of 1.0%, 5.0%, 10%, 15%, and 20%. Taking the small watershed of Ganquan county as an example, when the scale of analysis was set to the maximum flow accumulation of 5.0%, limited watershed information was required to acquire the watershed characteristics (Fig. 5(a)), and the gully was composed of only two levels. The two-level gully cannot reflect the spatial distribution of the entire watershed morphology. Figure 5(b) shows the gully extraction result when the scale of analysis was set to the maximum flow accumulation of 1.0% and the number of gully levels was four. However, the four-level gully did not cover the entire area of the DEM, and the analysis scale must be further subdivided. When the scale of analysis was set to the maximum flow accumulation of 0.1%, the gully structure could be extended to the entire region of DEM data (Fig. 5(c)).

The information of the gully structure when the analysis scale was less than the maximum flow accumulation of 0.1% was extracted. The result is shown as Fig. 6. A large number of redundant phenomena, which were inconsistent with the real terrain and extended to the slope through visual interpretation, appeared in the gully structure. The extraction result contradicted the actual terrain structure in the study area to a certain extent. This result would influence the quantitative analysis of the gully structure. Therefore, the initial scale of the scale sequence was determined to be the maximum flow accumulation of 0.1% to extract information of the watershed gully. It accurately determined the terminate scale of analysis sequence by changing the accumulation threshold value of the watershed. The specific experimental result is shown in Table 3. The index of the gully structure requested the extraction result to be at least a three-level Eq. (2). However, the gully levels of Yijun, Yanchuan, and Jiaxian converted from three levels to two levels ranged from 2.00% to 2.20%, that of Ganquan ranged from 3.40% to 3.60%, and that of Suide ranged from 4.60% to 4.80%. In summary, the terminate scale of the scale sequence was determined to be 2.00%, and the gully levels of the five research areas were set to three to calculate the indices of the gully structure. The rate of the analysis scale sequence was 0.2%, because this parameter was obtained through many experiments of extraction of natural stream networks based on DEMs. Therefore, the sequence of the accumulation threshold value of the watershed was set with a maximum flow accumulation ranging from 0.1% to 2%. The accumulation threshold interval value was 0.2%.

Analysis of watershed structure in different analysis scales

According to the analysis scale sequences of flow accumulation from 0.1% to 2% with the interval value of 0.2%, the fractal dimension index values of the different watersheds in the test areas were extracted via hydrological analysis in ArcGIS on the basis of the 1:10,000 DEM data scale. The results of the gully fractal dimensions are listed in Table 4.

The line graphs shown in Fig. 7 were drawn to determine the trend of the fractal dimension of the five test areas.

The findings of the analyses shown in Table 4 and Fig. 7 are summarized as follows: (i) The fractal dimensions of the different watershed areas changed in the scale range of 0.1% to 0.5% and shifted from the highest gully level to the lowest gully level, that is, from five levels to four levels. The second transformation of the gully level from four levels to three levels occurred in the scale range of 0.7% to 1.3%. The fractal dimension of the watershed also changed. (ii) As shown in Fig. 7, the fractal dimension value changed minimally at the same gully level in the analysis scale range of 0.2% to 2.0%, except for that in 1.3%. The gully level change from three levels to four levels started at the analysis scale of 1.3%. The gully structure was unstable, and so, the number of gully levels was three in the scale range of 1.5% to 2.0%. (iii) The watershed dimension values exhibited an increasing trend with the analysis scale decreasing in the five experimental areas. This result indicated that the watershed structure would gradually undergo refinement, and the gully would be increasingly complex.

The extraction results for the bifurcation ratio are shown in Table 5, and the trend of the bifurcation ratio is shown in Fig. 8.

The findings from the analyses in Table 5 and Fig. 8 are summarized as follows. When the analysis scale was set to the flow accumulation of 0.90%, the bifurcation ratio values of Suide, Jiaxian, Yanchuan, and Yijun reached the maximum. The gully bifurcation ratio decreased with the decline of the analysis scale, and the gully structure gradually stabilized. The gully bifurcation ratio value of Ganquan did not reach the maximum value at the analysis scale of 0.90% but did reached the maximum at the scale of 1.30%. An analysis of the change in the gully level at different analysis scale intervals showed that the level of Ganquan’s gully change ranged from three to four at a scale interval of 1.10% to 1.30%; the changes of the interval scale in the other experiment areas were in the range of 0.70% to 0.90%. These findings indicate that the gully level change in Ganquan was preferable to that in the other areas with the decrease of analysis scales.

Discussion

The correlation between the gully bifurcation ratio and the gully fractal dimension values was analyzed, and the parameters of the correlation and variation coefficients were calculated. The results are shown in Table 6.

By analyzing the results of the extraction of watershed structure information in multiple data scales and different analysis scales, we arrived at the following findings. (i) The data scales, which include map scale and spatial resolution, exerted certain effects on the extraction of gully structure information, but the entire watershed structure remained similar and stable. (ii) The fractal dimension value showed the increasing tendency of the gully structure in the test areas with decreasing analytical scale. The data presented in Table 6 revealed a significant negative correlation between the fractal dimension value and analysis scale. The fractal dimension values of the watershed network of the same levels were similar, but the fractal dimension values significantly changed as the gully level varied. (iii) According to the analysis of the statistics shown in Fig. 8, the changes in the bifurcation ratio values of Suide, Jiaxian, Yanchuan, and Yijun were consistent. Furthermore, the bifurcation ratios of the different watershed areas showed the same change tendency. Given that the different gully levels reflect watershed gully development, the gully structure changed with the decreasing analysis scale, and the entire gully structure changed according to the optimal analysis scale. According to the results analysis in Section 3.2.2, the maximum value of the bifurcation ratio appeared in the analysis scale of 0.9%, and the gully levels changed from three to four in the same analysis scale. Thus, an optimal analysis scale should be established to study gully structure.

The findings of the analyses of the geographic features of the watershed, the gully fractal dimension, and the gully bifurcation ratio are summarized below.

(i) The mean values of the fractal dimension of the watersheds of Yijun, Ganquan, Jiaxian, and Suide were close; Yanchuan had the lowest mean value. This result is explained as follows. The landform types of Yijun and Ganquan resemble a loess ridge and gully, and the watershed gullies of these two areas are in the same development stage. Therefore, slight difference was noted between the mean values of the study areas (Table 6). The landform types of Jiaxian and Suide were similar to a loess hill and gully. Thus, the watershed gullies of the two areas were consistent, and the mean values of the test areas were nearly equal. The test area of Yanchuan also belongs to loess hill and gully; this area had the lowest mean value, and the structural development of the gully was incomplete, as shown in Table 6. The development of loess hill and gully at the gully structure lags behind that of loess ridge and gully.

(ii) As shown in Table 6, the mean values of the bifurcation ratios of the five watershed areas were in the range of 4‒5, which indicated that the bifurcation ratio values of the watersheds in different landform areas were similar and showed minimal variation. In addition, changes in the bifurcation ratios were not obvious at different scales of analysis. Therefore, bifurcation ratio is not significantly correlated with the scale of analysis.

(iii) The variation and correlation coefficients of fractal dimensions are smaller in Ganquan than in the other areas. The change trend is shown in Fig.7(b). The fractal dimension of the gully of Ganquan was not significantly correlated with the analysis scale. The major reason for this observation is that the gully is structurally stable, and the topographic development is relatively mature in Ganquan.

Conclusions

On the basis of the DEM data of five different areas of China’s Loess Plateau at 1:10,000 and 1:50,000 scales, we set the bifurcation ratios and fractal dimensions of watershed structure information as the research objects. We also analyzed the characteristics and change rule of watershed structure information at different scales of data and analysis. A preliminary discussion on the regional changes in watershed structure information with different natural geographical features in the test area was presented. The main results and conclusions of this study are as follows.

(i) By analyzing the structure information indices of the DEM data at two types of scales, we found that the watershed structure information did not change along with changes in data scale. (ii) The structure changed significantly with the change in the scale of analysis. The fractal dimension of the watershed gully structure and analytical scale showed a significantly negative correlation, and the index value changed obviously when the gully level changed. (iii) The fractal dimension showed great diversity in the watershed development of the gully structure in the different test areas. The bifurcation ratio had a small variation, which indicated that the morphological features of the gully structure were basically similar.

Given the experimental sample region and the influence of different DEM data scales, the conclusions of this study must be further tested and verified. The characteristic indices of watershed structure must also be supplemented. Future studies should consider the landscape type, geological tectonic activities, and many other factors such as surface cover and artificial facilities. The development evolution and characteristics of a watershed require a thorough discussion on the information indices of watershed structure.

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