The Loess Plateau of China has drawn the attention of many researchers worldwide for its unique geographical condition, geomorphologic research values, loess landform feature, and its special natural and cultural landscapes [1
]. Many previous studies about the Loess Plateau have been carried out to solve scientific problems regarding its formation process [3
], landform shape [5
], gully development [8
], soil erosion [11
], environmental evolution [13
] etc., and many outstanding research results have been achieved. In recent years, the development of Digital Elevation Model (DEM) applications and Geographic Information System (GIS) spatial analysis techniques has greatly promoted the study of the land surface analysis of the Loess Plateau. Digital land surface analysis has been used to extract the basic terrain derivatives (e.g., slope and aspect) [15
] and to study complicated applications (e.g., paleo-topographic reconstruction and landform classification) [17
] of the Loess Plateau. The terrain points [22
], lines [25
], and surfaces [28
] of the Loess Plateau have been well studied in terms of extraction methods based on DEMs. However, these studies mainly focused on research with the view of developing and evaluating technologies and methods, and there is a lack of research and discussion on the role of terrain data in the study of the topography and geomorphology of the Loess Plateau. Tang et al. [1
] established the theory of the slope topo-sequence, which they successfully applied to study the Loess landform. The corresponding slope topo-sequence, which showed an obvious spatial distribution, of any of the different Loess landforms could be found [31
]. The topo-sequence of hypsometric integrals (HI) was applied to the loess watershed landform in an area of the Loess Plateau affected by severe soil erosion, based on a digital land surface analysis and geo-informatics method named Tupu [32
]. Furthermore, an existing method has been put forward to study the spatial distribution based on the landform planation index (LPI), which is the terrain derivative extracted from DEMs [33
]. In spite of the fact that these methods realized a quantitative analysis of the macroscopic differentiation rules of the Loess Plateau, these methods are not sensitive to the local topographic changes in the Loess Plateau, which cover up the local variation of the terrain.
The Loess shoulder-line is one of the most significant topographic structures in the Loess Plateau, which separates the complicated physiognomy of the plateau distinctly into positive (inter-gully area) and negative terrains (inner-gully area) (P-N terrains) in terms of the formation and topographic feature of the landforms. This shoulder-line plays important roles in the research of soil and water conservation. As the most vital boundaries, extraction methods [34
] and efficient calculations [35
] have mainly been applied to the Loess shoulder lines in many studies. Few researchers have applied the quantitative index of the shoulder lines to study the spatial distribution of the Loess Plateau land at present. The shoulder lines formed and developed at the slope scale, meandering in the whole watershed, and carved out the spatial pattern of Loess gullies and Loess tableland, Loess ridges, and Loess hills. The regional difference in the spatial morphological characteristics of Loess shoulder lines in different landform types has become an important basis for geomorphological regionalization.
Fractal theory has been used to address the limitation of traditional quantitative methods on geomorphic features, and has opened up a new approach to generally describe the geomorphic features of a drainage basin and its geomorphic evolution [37
]. More specifically, fractal theory has provided a way to quantitatively describe self-similar or self-affine landforms [39
]. The classic application of fractals to geomorphology focused on line elements such as coastlines, rivers, and faults [18
]. Based on grid DEMs, the fractal dimension was shown to characterize more than mere terrain “irregularity” or “roughness” [40
]; instead, it can also be employed to describe the changes in the variability of the topography in relation to the distance [41
]. The relationship between the fractal parameters and the physical processes needs further research. For a loess landform, the fine fractal structure of a single watershed varies for different landform types and developmental processes. Thus, an analysis of the geometric singularity distribution of loess shoulder lines in different loess landforms to obtain the fractal parameters of these shoulder lines could provide theoretical and methodological support for further research on scientific and reasonable classification of loess geomorphic regions.
The research described in this paper is based on elevation data, which was obtained from the actual direction of the curve at a certain distance along shoulder lines extracted from DEMs. The elevation data, as with most time series data, also consist of nonlinear and non-stationary data capable of reflecting the topographic spatial variation and relief of the area surrounding the shoulder lines. The main difference between the time series data and elevation data is that the two kinds of data are based on the time and sampling distance as independent variables, respectively. Due to a high degree of similarity within time series, it is feasible to use the elevation sampling data of shoulder lines to apply to land surface analysis.
In this study, we use the multi-fractal detrended fluctuation analysis (MF-DFA) [42
] developed in recent years to study topographic data series extracted from shoulder lines. MF-DFA has successfully been applied to fields as diverse as finance and stock markets [43
], seismicity [46
], mineral grade detection [49
], climate change [50
], traffic flow [55
], speech signal characteristics [57
], plant species identification [58
], air pollution [59
], and heart rate dynamics [61
]. It is very valuable to introduce this method into the analysis of shoulder-lines. In this study, MF-DFA can distinguish different landform types of local or global changes of the land surface features.
This research mainly focuses on the introduction of the multi-fractal spectrum exponent and Hurst exponent in order to describe the topographic variation law of different landform types that can reflect the overall macro- and local micro-pattern of the land surface of the Loess Plateau. The research results provide quantitative information in support of the loess landform classification.
4.1. Influence Factors of Loess Shoulder-Line
The analytic result shows the MF-DFA methods achieve a much deeper understanding of the data structure than other data analysis methods. Since the multi-fractal analysis method can be successfully applied to many complex systems, it has great potential to be applied to the land surface structure of loess shoulder lines in the Loess Plateau. The influence factors and the physical meaning of the multi-fractal characteristics of the loess shoulder-line land elevation sampled must be considered to be important.
a shows the multi-fractal spectrum obtained by MF-DFA analysis of loess shoulder-line land elevation sampling at the Chunhua, Ganquan, and Suide study sites. This figure shows that the three curves of the multi-fractal spectrum are very similar to the curve with a long tail to the right. Determining the factors causing these results is a problem worthy of further discussion with respect to the method of land surface analysis in the Loess gully areas. First, the extraction accuracy of the Loess shoulder line is one of the important factors to influence the study. Many researchers [25
] have performed an in-depth study on the extraction methods and the accuracy of the Loess shoulder line. This study is based on previous researchers’ high-accuracy extraction of the Loess shoulder line in the study area and to ensure that Loess shoulder-line data can accurately reflect the topography change of the study site. The Loess shoulder-line land elevation sample data in our study is a one-dimensional elevation series, as in other research fields, such as the time series of climate change, hydrological time series, and seismic data series. There must be noise in the series data, which may be another significant factor to cause the deviation (error) in the result of the topographic analysis of the Loess shoulder line. Thus, noise filtering of the data series is very important when using the MF-DFA method to analyze the topographic changes of the Loess shoulder-line. After the noise removal process described in Section 2.3.1
, the Loess shoulder-line land elevation sample series data is analyzed by the MF-DFA method. Figure 11
shows a comparison between the multi-fractal spectrum obtained before and after noise filtering, which differs completely. The results obtained with raw data without any noise removal process, displayed in Figure 11
a, show the similarity of the multi-fractal characteristics of the three study sites. Figure 11
b shows that the changes in the land surface of the Loess shoulder line are different for the three study sites, and that the topographic change reflected in the multi-fractal spectrum of the Loess shoulder-line is consistent with the actual terrain. A detailed description of the loess shoulder-line topographic variation at the three sites is presented in the Results Section. This is reflected in the land surface analysis of the loess shoulder line, where the noise in the data is a non-negligible impact factor.
4.2. Correlation Analysis with Other Geographical Agents
The results of Suide show that the width of the multifractal spectrum of the Suide site is 0.6479, which indicates that the internal variation of the terrain of this site is relatively uniform. This may be related to the land use type of the Suide site, which is mainly artificial terraced fields. The Suide site is characterized by a large number of artificial terraces of various types, which have a great effect on reducing the soil erosion [68
]. In addition, the site also has a large number of check-dams, which also has a great effect on weakening soil erosion [69
]. The check-dam intercepts a large amount of sediment from an upstream channel, and the slope in combination with the siltation of the check-dam, gradually forms a new equilibrium profile upstream, and elevates the erosion datum of its control site. The erosion energy of the upper reaches of the dam and the slope on both sides of the dam gradually decrease; therefore, the erosion effect is weakened, forcing the evolution of the small watershed to accelerate the transition into “old age.”
The results of the Ganquan site show that the multifractal spectral width is 0.8945, which indicates that the variation in terrain changes along the Ganquan site is very different and the terrain change is more intense. This may also be related to the land use type, rainfall intensity, and so on. The main land use type in the Ganquan site is grassland, but the coverage is very low, and many places are bare loess, with a weak effect on soil erosion. The average annual precipitation at the Ganquan site is ~500 mm, which is mainly concentrated in the form of heavy rain in summer, and soil erosion is very serious. Unlike the Suide site, Ganquan has few such soil and water conservation facilities. Heavy rain in summer could cause serious soil erosion, so the gullies develop very actively, resulting in the intensive terrain variation of the Ganquan site.
The results of the Chunhua site show that the multifractal spectral width is 0.6852, which indicates that the difference in terrain variation along the shoulder line is similar. The landform type of the Chunhua site is dominated by loess tableland, and the ratio of hilly to gully sites is about 8:2. Although the annual precipitation is 600.6 mm, the land use type is mainly garden and woodland, and the vegetation cover is more pronounced. In addition, the loess soil layer at the Chunhua site is thinner and the bedrock is exposed, leading to a high erosion base as well as weak soil erosion.
A multifractal, also known as a multiple fractal measure, reveals a class of complexities and singularity. A whole site with a complex fractal characteristic can be divided into many small sites with different singularities, so that the internal fine structure of the fractal can be understood more thoroughly.
Loess shoulder-line land surface variations series data from the Suide, Ganquan, and Chunhua sites were selected and the MF-DFA method was used to study the variations in these three land sites. The following conclusions were drawn. The loess shoulder-line topographic variations have strong multi-fractal characteristics at the three sites. The strongest multi-fractal characteristics were found at the Ganquan site, followed by the Chunhua site, with the weakest multi-fractal characteristics at the Suide site. The noise and analysis scale of sampling data have a strong influence on the analysis of the loess shoulder line, and the best scale values of the three areas are different, namely 7, 17, and 11 for the Suide, Ganquan, and Chunhua sites, respectively. The topographic spatial variations of the loess shoulder-line are mainly related to other geographic agents, such as lithology, precipitation, and land use. At the Suide site, artificial landforms (such as artificial terraces and check-dams) are the main factor; at the Ganquan site, precipitation; and at the Chunhua site, lithology. The results showed that the combination of EEMD, MF-DFA, and DCCA achieved success in revealing the terrain variations on a watershed. The results could provide theoretical and methodological support for further research on scientific and reasonable classification of loess geomorphic regions. We are optimistic about the potential use of our methods.