Regional features of topographic relief over the Loess Plateau, China: evidence from ensemble empirical mode decomposition

Abstract

Landforms with similar surface matter compositions, endogenic and exogenic forces, and development histories tend to exhibit significant degrees of self-similarity in morphology and spatial variation. In loess hill–gully areas, ridges and hills have similar topographic relief characteristics and present nearly periodic variations of similar repeating structures at certain spatial scales, which is termed the topographic relief period (TRP). This is a relatively new concept, which is different from the degree of relief, and describes the fluctuations of the terrain from both horizontal and vertical (cross-section) perspectives, which can be used for in-depth analysis of 2-D topographic relief features. This technique provides a new perspective for understanding the macro characteristics and differentiation patterns of loess landforms. We investigate TRP variation features of different landforms on the Loess Plateau, China, by extracting catchment boundary profiles (CBPs) from 5 m resolution digital elevation model (DEM) data. These profiles were subjected to temporal-frequency analysis using the ensemble empirical mode decomposition (EEMD) method. The results showed that loess landforms are characterized by significant regional topographic relief; the CBP of 14 sample areas exhibited an overall pattern of decreasing TRPs and increasing topographic relief spatial frequencies from south to north. According to the TRPs and topographic relief characteristics, the topographic relief of the Loess Plateau was divided into four types that have obvious regional differences. The findings of this study enrich the theories and methods for digital terrain data analysis of the Loess Plateau. Future study should undertake a more in-depth investigation regarding the complexity of the region and to address the limitations of the EEMD method.

Keywords

catchment boundary profile / topographic relief period / ensemble empirical mode decomposition / Loess Plateau

Introduction

Landform morphology is the most basic and important geographic parameter. Landforms such as mountains, hills, plains, and basins are the result of complex processes, during which the Earth’s surface undergoes changes directed by the effects of endogenic and exogenic forces (^{MacMillan and Shary, 2009}; ^{Wyrick et al., 2014}; ^{Tammelin and Kauppila, 2018}). The nonuniform spatial distribution of these forces is the root cause behind the formation of various complex landforms. The variation that exists in topographic relief features plays a decisive role in shaping patterns of matter flow and energy conversion on the Earth’s surface, and also makes a considerable contribution to constraining the scale and layout of human activities (Black and Perron, 2017; ^{Cowley et al., 2017}; ^{Cao et al., 2019}).

Landforms that formed under relatively stable conditions of matter, energy, and time tend to exhibit self-similarities (^{Zhou et al., 2010}; ^{Bollati et al., 2017}; ^{Yang and Zhou, 2017}; ^{Serrano et al., 2018}). In particular, rolling landforms (e.g., barchans and undulating folded mountains) tend to exhibit relatively regular spatial wave-like patterns. For such landforms, the mean distance between two adjacent peaks is referred to as the topographic relief period (TRP). In the loess hill-gully areas of the Loess Plateau, rainfall erosion is the major landform-shaping force, which is due to the uniform composition of the Earth’s surface in the region and the relative stability of the Ordos platform (^{Leger, 1990}; ^{Bi et al., 2012}; ^{Wang et al., 2010}; ^{Xiong et al., 2016}; Li et al., 2018; ^{Na et al., 2018}). Mature loess landforms also exhibit substantial degrees of self-similarity in topographic relief. Thus, it is of theoretical significance to use the TRP to define and quantitatively describe the regular patterns of rolling topographic landforms, especially the distinctive landforms of the Loess Plateau.

In-depth research of topographic relief has been conducted from different perspectives, including the descriptive representation of topographic relief features (^{Mackay and Band, 1998}; ^{Collier et al., 2003}; ^{Marston and Jenny, 2015}; ^{Orengo and Petrie, 2018}), typology of landforms (^{Gomez, 1984}; ^{Orr et al., 2008}; de Matos-Machado et al., 2019), the impact on socioeconomic development (^{Bumbak and Ilies, 2014}; ^{Allen, 2009}; ^{Telbisz et al., 2014}; ^{Zhang et al., 2019}), ecological environment evaluation (^{Wang et al., 2015}; ^{Persendt and Gomez, 2016}; ^{Savvides et al., 2016}), and the relationship with natural disasters (^{Latrubesse et al., 2009}; ^{Iverson et al., 2015}). Descriptive representations of topographic relief features have been explored in terms of the extraction method (^{Xiong et al., 2017}; ^{Orengo and Petrie, 2018}; ^{Gailleton et al., 2019}; ^{Yang et al., 2019}), the minimum computational unit (^{Khan et al., 2016}; ^{Meng et al., 2018}; ^{Gruber et al., 2019}), a suitable representation window (^{Li, 2015}), and prediction modeling (^{Nunalee et al., 2015}; Li et al., 2018). Moreover, combinations of measures—such as relative relief, altitude above sea-level, and topographic waviness—have been used to define the boundaries between China’s three terrains, the topology and boundaries of mountainous terrains, the topology of landforms (^{Clarke and Romero, 2017}), and the relationship between topographic waviness and administrative divisions (^{Robl et al., 2017}; ^{Zhang et al., 2017}). Furthermore, the impact of topographic relief on economic development, such as population distribution (^{Liu et al., 2015}; ^{Badgley et al., 2017}), human settlement distribution (^{Hui et al., 2015}; ^{Xi et al., 2018}), and the spatial distribution of poverty (Zhou et al., 2018), has been analyzed at various spatial scales using spatial auto-correlation and coupling methods. Human ecological environments—such as the spatiotemporal distributions of atmospheric pollution (^{Zhao et al., 2019}), the value of ecosystem service (^{Dymond et al., 2017}), soil erosion (Gu et al., 2016; Li et al., 2018), climate change (^{Karim et al., 2016}), land utilization change (^{Machowski et al., 2016}), and farmland productivity—have been evaluated using methods such as the modeling of positive-negative factors and spatial metrics, the topographic gradient effect (^{Martel, 2016}), geographic information system-enabled spatial analysis, topographic heterogeneity (^{Diefenderfer et al., 2018}), sunshine duration (^{Ambreen et al., 2015}), and the topographic control effect. The quantitative relationship between landslide and topographic waviness has also been obtained using rectangular neighborhood analysis window and mean change point analysis methods.

In summary, topographic relief has a significant impact on the human living environment and production activities, and also provides a major measure of landform typology. The most commonly used measure of topographic relief features in the literature is topographic waviness, which is defined as the difference between the altitudes above sea level of the highest and lowest points for a given area. It is a measure for describing regional topographic features at the macro scale, but considers only the vertical dimension of topographic relief, not the horizontal dimension, which can be measured by the TRP.

TRPs reflect the tectonic movements that underlie landform features as well as the overall peripheral structure of regional landforms. This is particularly so on the Loess Plateau, where severe water and soil losses are experienced as a result of runoff erosion; hence, landform features at catchment boundaries reflect pre-erosion topography to a certain degree (^{Li et al., 2017}). Catchment boundaries (i.e., divides) are the highest edge line of a given catchment, and have relatively stable topographic features because they are generally almost immune to runoff erosion (the so called ‘erosion-free zone’). Although a catchment boundary profile is not a topographic profile in the general sense, it can be considered to be a spectral line. In contrast to general topographic profiles, vertical catchment boundary profiles are more representative of a catchment’s basic topographic features because they i) have a spatially determinate position, ii) are not anisotropic, and iii) reflect the overall topography of a catchment to a certain degree. A preliminary study showed that catchment boundary profiles of similar landforms have similar morphological and structural features at a certain scale.

Catchment boundary profiles are insignificantly affected by topographic anisotropy, statistical error, and the resolution of digital elevation model (DEM) data; hence, they can be used as the basic data for macro topographic characterization owing to their reliability and stability (^{Farkas et al., 2016}). Catchment boundary profiles have spatially determinate positions and reflect the overall morphological characteristics of catchment landforms to a certain degree. In the present study, the catchment boundary profiles of 14 sample areas on the Loess Plateau were obtained from a 5 m resolution DEM dataset. The topographic relief characteristics of various types of landforms on the Loess Plateau were obtained by analyzing the catchment boundary TRP characteristics using the ensemble empirical mode decomposition (EEMD) spectral analysis method. This study aim to demonstrate the effectiveness of spectral analysis for analyzing digital terrain data and the potential of obtaining different spatial patterns of topographic relief for various landforms on the Loess Plateau.

Materials and methods

Study area

The Loess Plateau extends across a vast area (100°54' E–114°33' E, 33°43' N–41°16' N) that is located to the west of the Taihang Mountains, east of the Riyue-Helan Mountains, north of the Qin Mountains, and south of the Yin Mountains. The 14 sample areas that were selected for this study include Yulin, Jiaxian, Hengshan, Suide, Jingbian, Yan'an, Yichuan, Fuxian, Huanglong, Luochuan, Xunyi, Pucheng, Yongshou, and Qianyang (Table 1). These sample areas are evenly distributed across the northern Shaanxi part of the Loess Plateau and are representative of the wider Loess Plateau.

Data sources

All data used in this study were extracted from DEMs with a resolution of 5 m. After manual correction, data were obtained for the CBP of the 14 study sites. DEM data were provided by the National Bureau of Surveying and Mapping and Geographic Information and used to generate a 1:10000-scale topographic map at 1 m contour intervals. First, the topographic map was drawn via ground surveys and scanned into a computer for geometric correction. Then, the contour lines were digitized and interpolated into a triangulated irregular network (TIN). Finally, the TIN data that were interpolated from grid digital elevation data were manually edited to correct for errors. Extraction was performed in four steps (Fig. 1): 1) extraction of drainage networks; 2) segmentation of rivers; 3) selection of representative drainage catchments; 4) data export for the selected typical catchment boundaries using commands in the ArcToolbox window. Terrain feature points (e.g., catchment outlet, local peaks, and saddles) were identified from catchment boundary profiles (Fig. 2).

Methods

Empirical mode decomposition (EMD) is affected by the frequency of the original signal, and thus has the disadvantages of the edge effect and scale mixture. Therefore, Wu et al. (2009) proposed a new data analysis method based on EMD: ensemble EMD (EEMD). Compared with EMD, EEMD is a time series analysis method with local self-adaptation; it is essentially a noise-aided data analysis technique that works by adding white noise to a signal to generate a noised-signal ensemble as a new signal, which is decomposed to obtain modally-consistent components of intrinsic mode functions (IMFs). The signal consists of components of different scales that are stacked onto a background with a uniform distribution of white noise in the entire time-frequency space. The signal components automatically distribute themselves to appropriate reference scales. Moreover, owing to the zero-mean-value characteristic of white noise, the added white noise can be averaged out after sufficient trials. Thus, the decomposition result for the noise-added signal can be taken as the result for the original signal, which better reflects the characteristics of the original signal.

In this study, EEMD involved an eight step process:

1) An original signal (a data series), x(t), was stacked onto white noise, w(t), at a preset signal-to-noise ratio of 0.2. The resulting signal was then expressed as Eq. (1):

(1)X\left(t\right)=x\left(t\right)+w\left(t\right).

2) EMD was applied to the signal-noise ensemble (the new data series resulting from stacking the original signal onto the white noise), and the resulting IMF components at various levels could then be expressed as Eq. (2):

(2)X\left(t\right)={{\displaystyle \mathrm{\sum }}}_{i=\mathrm{1}}^n{{\displaystyle c}}_i+{{\displaystyle r}}_n.

3) Steps 1) and 2) were repeated n times, with new white noise of the same amplitude, w_j(t), added to the original signal, x(t), each time. The resulting signal-noise ensemble for the j-th trial could then be expressed as Eq. (3):

(3){{\displaystyle X}}_j\left(t\right)={{\displaystyle \mathrm{\sum }}}_{i=\mathrm{1}}^n{{\displaystyle c}}_{ji}+{{\displaystyle r}}_{jn}.

4) The IMF component for the jth trial, c_n(t), could then be expressed as Eq. (4):

(4){{\displaystyle c}}_n\left(t\right)={\displaystyle \frac{\mathrm{1}}{n}}{{\displaystyle \mathrm{\sum }}}_{j=\mathrm{1}}^N{{\displaystyle c}}_{j,n}\left(t\right).

5) The real mode was approximated through multiple averaging operations. The final decomposition result could then be expressed as Eq. (5):

(5)x\left(t\right)={{\displaystyle \mathrm{\sum }}}_{n=\mathrm{1}}^m{{\displaystyle c}}_n\left(t\right)+{{\displaystyle r}}_m\left(t\right).

6) A Monte Carlo significance test was then performed on the resultant IMF components, and those with a desired significance level were selected for the subsequent analyses. The energy spectrum density of the kth IMF component could then be expressed as Eq. (6):

(6){{\displaystyle E}}_k={\displaystyle \frac{\mathrm{1}}{n}}\mathrm{\sum }_{j=\mathrm{1}}^N{\left|{{\displaystyle I}}_k\left(j\right)\right|}^{\mathrm{2}}.

where N is the length of an IMF component and I_k(j) is the kth IMF component.

7) The approximate relationship between the mean energy spectrum density, {{\displaystyle \overline{E}}}_k, and the mean period, {{\displaystyle \overline{T}}}_k, of the k-th IMF component was obtained by performing a Monte Carlo significance test to the white noise as defined by Eq. (7):

(7)\ln {{\displaystyle \overline{E}}}_k+\ln {{\displaystyle \left\{{{\displaystyle \overline{T}}}_k\right\}}}_{\mathrm{a}}=\mathrm{0.}

Equation (7) is represented as a line with a slope of-1 in a coordinate system with {\left\{{\overline{T}}_k\right\}}_a and ln{\overline{E}}_k as the horizontal and vertical axes, respectively. Theoretically, the IMF of the white noise falls on the line. However, IMFs obtained during real applications may deviate from the line. Thus, a confidence interval for the energy spectrum distribution of the white noise was defined as Eq. (8):

(8)\ln {{\displaystyle \overline{E}}}_k=\ln {{\displaystyle \left\{{{\displaystyle \overline{T}}}_k\right\}}}_{\mathrm{a}}\pm \alpha {{\displaystyle \sqrt{\mathrm{2}/Ne}}}^{\ln \left({{\displaystyle \left\{{{\displaystyle \overline{T}}}_k\right\}}}_a/\mathrm{2}\right)},

where α is the significance level.

Results and discussion

EEMD and testing of CBP

The EEMD results of the catchment boundary profiles of the 14 samples areas are shown in Table 2. Owing to the self-adaption of EEMD, the decomposition result of each sample area depended entirely on its CBP. Among the 14 sample areas, four (Hengshan, Suide, Jingbian, and Pucheng) had ten IMF components and a residual error term, and ten (Yulin, Jiaxian, Yan’an, Yichuan, Huanglong, Luochuan, Xunyi, Yongshou, and Qianyang) had eleven IMF components and a residual error term. The EEMD results of the 14 samples areas showed that as the decomposition period increased, the frequency gradually decreased. This was because the topography of the Loess Plateau is heterogeneous, nonlinear, and nonstationary. As multiple EEMD trials of IMF components result in an approximation of the real mode, the last term of the decomposition result, the residual error, can be taken as the overall topographic variation trend.

A significance test was performed on the decomposition results for the 14 sample areas. Components with desired significance levels were then used for further investigation of the spatial variation patterns in the topography of the 14 samples areas. To analyze the amount of information with actual physical meaning in the IMF components, the variance contribution rate, period, and confidence level of the components were computed, as shown in Table 2. We note that in order to maintain the total energy of the signal at a desirable level, some IMF components that fell in the area of a significant white noise but failed the significance test were included in the computation of the variance contribution rate.

Figure 3(a) clearly shows that the second IMF component failed the significance test (i.e., it fell below the 95% confidence line), and thus contained relatively less information of actual physical meaning. All of the other IMF components fell above the 99% confidence line and thus contained more information of actual physical meaning. Inspection of Fig. 3(a) and Table 2 reveals that among the IMF components of the CBP from the EEMD for Yulin (excluding the residual, RES), IMF9 had the highest variance contribution rate (2.1906). Thus, the principal period of the catchment boundary topographic relief for Yulin was IMF9. The variation period of IMF9 was then computed and resulted in a mean period of the catchment boundary topographic relief of 1331.25 m for Yulin.

Figure 3(b) and Table 2 show that for the CBP based on the EEMD for Jiaxian, IMF2 and IMF3 failed the significance test; of the IMF components that passed the significance test (excluding the RES), IMF8 had the highest variance contribution rate (2.4140) and the highest confidence level. Thus, the principal period of the catchment boundary topographic relief for Jiaxian was IMF8. The variation period of IMF8 was then computed and yielded a mean period of the catchment boundary topographic relief of 1153.25 m for Jiaxian.

It can been seen from Fig. 4(a) and Table 2 that among the IMF components of the CBP from the EEMD for Hengshan, IMF2 had a confidence level of<95%; of all the IMF components with confidence levels>99% (excluding the RES), IMF8 had the highest variance contribution rate (5.8280) and was thus most representative of the topographic features of this sample area. Thus, the principal period of the CBP topographic relief for Hengshan was IMF8. The variation period of IMF8 was computed and resulted in a mean period of the catchment boundary topographic relief of 742.20 m for Hengshan.

Figure 4(b) and Table 2 reveal that among the IMF components of the CBP based on the EEMD for Suide, IMF2 failed the significance test; of the IMF components that passed the significance test (excluding the RES), IMF8 had the highest variance contribution rate (3.0651) and the highest confidence level. Thus, the principal period of the catchment boundary topographic relief for Suide was IMF8. The variation period of IMF8 was computed using the equation for the mean period computation, and yielded a mean period of the CBP topographic relief of 956.25 m for Suide.

It can be seen from Fig. 5(a) and Table 2 that among the ten IMF components of the CBP from the EEMD for Jingbian, IMF2 and IMF 3 fell below the 95% confidence line and failed the significance test, while the other IMF components fell above the 99% confidence line; among the significant IMF components (excluding the RES), IMF8 had the highest variance contribution rate (51.0685). Thus, the principal period of the catchment boundary topographic relief for Jingbian was IMF8. The variation period of IMF8 was computed and yielded a mean period of the catchment boundary topographic relief of 1707 m for Jingbian.

Figure 5(b) and Table 2 show that among the IMF components of the CBP based on the EEMD for Yan’an, IMF2 failed the significance test; of the IMF components that passed the significance test (excluding the RES), IMF8 had the highest variance contribution rate (21.8382). Thus, the principal period of the catchment boundary topographic relief for Yan’an was IMF8. The variation period of IMF8 was computed and resulted in a mean period of the catchment boundary topographic relief of 1114.25 m for Yan’an.

It can be seen from Fig. 6(a) and Table 2 that among the 11 IMF components of the CBP from the EEMD for Yichuan, only IMF2 failed the significance test and thus contained relatively less information of actual physical meaning; of the other IMF components (excluding the RES), IMF9 had the highest variance contribution rate (2.0095). Thus, the principal period of the catchment boundary topographic relief for Yichuan was IMF9. The variation period of IMF9 was computed and yielded a mean period of the catchment boundary topographic relief of 1129.25 m for Yichuan.

Figure 6(b) and Table 2 show that among the IMF components of the CBP based on the EEMD for Fuxian, IMF2 also failed the significance test; of the IMF components that passed the significance test (excluding the RES), IMF9 had the highest variance contribution rate (8.2622). Thus, the principal period of the catchment boundary topographic relief for Fuxian was IMF9. The variation period of IMF9 was computed and resulted in a mean period of the catchment boundary topographic relief of 1670.33 m for Fuxian.

It can be seen from Fig. 7(a) and Table 2 that among the 11 IMF components of the CPB from the EEMD for Huanglong, IMF2 failed the significance test; of all the components that passed the significance test (excluding the RES), IMF8 had the highest variance contribution rate (33.7958). Thus, the principal period of the catchment boundary topographic relief for Huanglong was IMF8. The variation period of IMF8 was computed and yielded a mean period of the catchment boundary topographic relief of 1069.75 m for Huanglong.

Figure 7(b) and Table 2 reveal that among the IMF components of the CBP based on the EEMD for Luochuan, IMF2 had a confidence level of<95%, while the other components had confidence levels>99%; of all the components with confidence levels>99% (excluding the RES), IMF9 had the highest variance contribution rate (2.6711). Thus, the principal period of the catchment boundary topographic relief for Luochuan was IMF9. The variation period of IMF9 was computed and yielded a mean period of the catchment boundary topographic relief of 2154.50 m for Luochuan.

Figure 8(a) and Table 2 show that among the IMF components of the CBP from the EEMD for Xunyi, IMF2 failed the significance test; of the components that passed the significance test (excluding the RES), IMF9 had the highest variance contribution rate (3.3380). Thus, the principal period of the catchment boundary topographic relief for Xunyi was IMF9. The variation period of IMF9 was computed and resulted in a mean period of the catchment boundary topographic relief of 2125.50 m for Xunyi.

It can be seen from Fig. 8(b) and Table 2 that among the ten IMF components of the CBP based on the EEMD for Pucheng, IMF2 also failed the significance test and thus contained relatively less information of actual physical meaning; of the components that passed the significance test (excluding the RES), IMF9 had the highest variance contribution rate (1.6306). Thus, the principal period of the catchment boundary topographic relief for Pucheng was IMF9. The variation period of IMF9 was computed and yielded a mean period of the catchment boundary topographic relief of 2008.50 m for Pucheng.

Figure 9(a) and Table 2 reveal that among the IMF components of the CBP from the EEMD for Yongshou, IMF2 failed the significance test; of the components that passed the significance test (excluding the RES), IMF8 had the highest variance contribution rate (4.2091). Thus, the principal period of the catchment boundary topographic relief for Yongshou was IMF8. The variation period of IMF8 was computed and resulted in a mean period of the catchment boundary topographic relief of 973.80 m for Yongshou.

Finally, Fig. 9(b) and Table 2 show that among the IMF components of the CBP based on the EEMD for Qianyang, IMF2 and IMF3 failed the significance test; of the components that passed the significance test (excluding the RES), IMF9 had the highest variance contribution rate (7.5288). Thus, the principal period of the catchment boundary topographic relief for Qianyang was IMF9. The variation period of IMF9 was computed and yielded a mean period of the catchment boundary topographic relief of 2310.50 m for Qianyang.

CBP topographic relief characteristics

The sequence of the 14 sample sites in terms of topographic relief were ranked from high to low as: Qianyang, Luochuan, Xunyi, Pucheng, Jingbian, Fuxian, Yulin, Jiaxian, Yichuan, Yan’an, Huanglong, Yongshou, Suide, and Hengshan. The major topographic relief characteristics of the Loess Plateau were determined to be in the north–south direction, the period of topographic relief increases, while the spatial frequency of topographic relief decreases. Four sample areas located in loess tablelands of the southern Loess Plateau—Qianyang, Luochuan, Xunyi, and Pucheng—were found to have the lowest spatial frequency of topographic relief. This is because the tablelands have a relatively integrated land surface, with fewer complicated gullies than other areas of the Loess Plateau. Six sample areas—Jingbian, Fuxian, Yulin, Jiaxian, Yichuan, and Yan’an—were determined to have slightly higher spatial frequencies of topographic relief compared with the aforementioned four samples areas. Among these, Yulin and Jiaxian are located in desert-loess transition areas of the northern Loess Plateau, which is characterized by relatively low precipitation, less severe water erosion, and a lower density of gully than other areas of the Loess Plateau. Jingbian, Fuxian, Yichuan, and Yan'an, on the other hand, are located in hill-gully areas of the southern Loess Plateau, which is characterized by relatively a low density of gullies compared with the loess tablelands. The remaining sample areas—Huanglong, Yongshou, Suide, and Hengshan—were found to have the highest spatial frequencies of topographic relief. These four sample areas are located in typical loess hill-gully areas, which have many hills and rugged landforms with significant topographic relief, and are therefore characterized by a low stability of surface matter and a high possibility of gully erosion.

Topographic relief is a geographical field with a strong self-affinity and self-similarity (^{Bertassello et al., 2018}). The topographic relief of different types of loess landforms exhibits great heterogeneity (^{Li et al., 2015}). As an important theory and method of geomorphology (^{Mark and Aronson, 1984}), fractals are of great significance to the development and evolution of surface topography (^{Czirók et al., 1994}), type division (^{Andrle, 1992}), and change simulation and prediction (^{Cao et al., 2015}; ^{Luo et al., 2018}). Some parameters of the multifractal spectrum can describe and express the similarities and differences between objects from a local and global perspective (^{Ariza-Villaverde et al., 2015}; ^{Dutta, 2017}). For a relatively large width in the multifractal spectrum, the multifractal structure of the CBP data series presented a large global variation. Conversely, for a relatively small multifractal spectrum width, the multifractal structure of the CBP topographic relief data series showed a small global fluctuation. The multifractal spectrum has a long left-tail when the CBP topographic relief data series has a multifractal structure that is sensitive to local fluctuations with large magnitudes (^{Ihlen, 2012}). In contrast, the multifractal spectrum has a long right-tail when the CBP topographic relief data series has a multifractal structure that is sensitive to local fluctuations with small magnitudes (^{Kantelhardt et al., 2002}). Thus, the width and shape of the multifractal spectrum are able to classify a wide range of different scale invariant structures of the CBP topographic relief data series (^{Kantelhardt et al., 2003}; ^{Cao et al., 2017}; ^{Cao et al., 2018}). It can be seen from Figs. 10(c), 10(f), 10(i), and 10(m) that the multifractal spectrum curve exhibits a distinct left-bias curve with a long right-tail, and that the spectral width of the multifractal spectrum curve is 1.1111. This means that the CBP topographic relief of Hengshan, Yan’an, Huanglong, and Yongshou is dominated by local small fluctuations, and the global variation is mainly based on large fluctuations. In Figs. 10(a), 10(d), 10(g), 10(j), 10(k), and 10(n), the shape of the multifractal spectrum curve exhibits a hook-shaped curve with a long right-tail, and the multifractal spectrum width of these sampling sites are>1.2192. This indicates that the topographic relief features of the CBP in Yulin, Suide, Yichuan, Luochuan, Xunyi, and Qianyang are characterized by smaller local fluctuations and larger global amplitude fluctuations. In Figs. 10(b), 10(e), and 10(l), the shape of the multifractal spectrum curve reflects an approximate left and right symmetric bell-curve with a larger width (1.2579). This indicates that the topographic relief feature of the CBP in Jiaxian, Jingbian, and Pucheng are characterized by i) local fluctuations that have smaller magnitudes that are intersected by local fluctuations with larger magnitudes, and that they are proportionally the same, and ii) global fluctuations with larger amplitudes. In Fig. 10(h), the shape of the multifractal spectrum curve is also a left-bias curve with a long right-tail, but its multifractal spectrum has the largest width (1.277). This indicates that the topographic relief characteristics of the CBP for the Fuxian sampling sites is also dominated by local fluctuations with small magnitudes, and that the global fluctuations are dominated by the largest amplitudes.

The multifractal form based on the partition function only provides a global description of the data series singularity and does not give local information. However, in practical applications, it is often more important to reveal the local singularity information of the data series. The singularity strength and the shape of the multifractal can accurately distinguish the variation characteristics of the data series of study objects from a local and global perspective (^{Greiner et al., 1998}; ^{Struzik, 2000}; ^{Stephen and Dixon, 2011}; ^{Chhabra and Jensen, 1989}). Figure 11 illustrates that according to the singularity strength of the CBP topographic relief variation of different sampling sites, 14 sampling sites could be divided into four types. The singularity strength of sampling sites 3 (Hengshan), 6 (Yan’an), 9 (Huanglong), and 13 (Yongshou) were exactly the same, and belong to a type that involves mainly small fluctuations in the local and global topographic relief. Sampling sites 1 (Yulin), 4 (Suide), 7 (Yichuan), 10 (Luochuan), 11 (Xunyi), and 14 (Qianyang) belong to a type that is characterized by small local fluctuations and large global fluctuations. Sampling sites 2 (Jiaxian), 5 (Jingbian), and 12 (Pucheng) belong to a type that is characterized by both small and large local fluctuations of equal proportions as well as large global fluctuations. Sampling site 8 (Fuxian) was found to be a separate type that is characterized by the smallest local fluctuations and the largest global fluctuations.

Conclusions

In this study, the topographic relief of the Loess Plateau was investigated by performing an EEMD of the CBPs of 14 sample areas. A Monte Carlo significance test was performed on the IMF components determined from the EEMD. An in-depth analysis of the variance contribution rate and mean period of the IMF components revealed that components with higher significance levels contained relatively more information of actual physical meaning, and thus better reflect the catchment boundary topographic characteristics. Moreover, this method effectively revealed that areas with higher densities of gullies had smaller TRPs.

The spatial frequency of the topographic relief of the 14 samples areas was then investigated, and revealed that the sequence of the 14 samples areas in terms of topographic relief could be ranked from low to high as: Qianyang (loess tablelands, ridges, and loess-covered medium-elevation hills), Luochuan (medium-relief, high-slope, and medium-elevation hills), Xunyi (medium-altitude loess ridges and tablelands, and low-altitude loess tablelands), Pucheng (loess tablelands, river terraces, and alluvial plains), Jingbian (loess ridges and hills with high shallow gullies), Fuxian (residual loess tablelands), Yulin (wind-deposited sand dunes), Jiaxian (loess hills and ridges with high shallow gullies), Yichuan (loess hills and ridges with high deep gullies), Yan’an (loess-covered, low-slope, and medium-elevation hills), Huanglong (loess tablelands and hills), Yongshou (medium-relief and medium-elevation hills), Suide (loess hills and ridges), and Hengshan (loess hills with high deep gullies).

Of the 14 sample areas, Qianyang was found to have the lowest spatial density of gullies, thus indicating a low topographic relief. Hengshan, followed by Suide, had the highest spatial frequency of gullies, thus indicating a high topographic relief and severe erosion. In comparison with other data analysis methods, the EEMD of the CBPs in combination with the Monte Carlo significance test is more appropriate for investigating the topographic relief characteristics of the Loess Plateau. Through the analysis of the multifractal spectrum and the singularity strength of the fluctuation characteristics of the CBP for the 14 sampling sites, the topographic relief of the Loess Plateau can be divided into four types, the features of which exhibit obvious spatial differences.

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