Geomorphological evolution of small watershed on the Loess Plateau based on slope shape spectra

doi:10.21203/rs.3.rs-5279046/v1

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Geomorphological evolution of small watershed on the Loess Plateau based on slope shape spectra

https://doi.org/10.21203/rs.3.rs-5279046/v1

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The Loess Plateau, the world's largest loess deposit, is characterized by distinctive landforms and severe soil erosion. Limited long-term observational data has hindered a comprehensive understanding of watershed evolution in this region. This study uses laboratory-based artificial rainfall simulations to investigate the long-term geomorphological evolution of watersheds on the Loess Plateau.Digital elevation models and terrain analysis techniques were employed to analyze geomorphological processes. Through correlation analysis and Sheffield entropy calculations, an optimal combination of terrain factors—including slope, aspect, plan curvature, profile curvature, and elevation—was identified to accurately describe slope characteristics. Multi-resolution image segmentation, Moran's I, and weighted variance were used to optimize segmentation parameters, achieving precise slope unit segmentation. Based on these parameters, we classified slope shapes, constructed slope shape spectra, and analyzed their changes over time. Results indicate that as the watershed evolves, the entropy of slope spectra increases while skewness and kurtosis decrease, indicating greater geomorphological complexity and diversity. The watershed's evolution progresses through three stages: slope shape development, slope shape growth, and slope shape maturity, each marked by distinct geomorphological features. By analyzing landscape indices at both class and landscape levels, we observed significant variation during the early development stage, followed by stabilization during the maturation phase. These findings demonstrate that slope shape spectra effectively capture the dynamic evolution of slopes in the Loess Plateau, offering new insights for understanding its geomorphological evolution and providing a novel method for quantitative geomorphology.

Slope shape spectrum

Segmentation

Evolution

Simulated watershed

Loess landform

The Loess Plateau, the largest loess-covered region in China, is also one of the most severely eroded areas in the world, making it a region of high geoscientific interest (Tang et al. 2017). The ecological environment of this region has long been the focus of loess geomorphology research and has garnered widespread attention from the academic community (Zhao et al. 2013; Liang et al. 2015; Feng et al. 2016; Yang et al. 2024). Over millennia of soil erosion, the Loess Plateau has developed a unique landform. Studying its geomorphological evolution is not only of significant scientific value for understanding the formation and mechanisms of loess landforms but also has practical implications for soil conservation and erosion monitoring in the region.

Current studies on loess watershed evolution often focus on various aspects of watershed morphology. For example, Li analyzed slope changes during watershed evolution using slope spectrum time series (Li et al. 2016), while Lin and Chen applied complex network theory to examine the spatial structure of loess watersheds (Lin and Chen, 2023). Chen employed network graph theory to delineate different stages of simulated watershed development (Chen et al. 2013), and Cao and Zhang used cellular automata models to simulate topographic evolution (Cao et al. 2012; Zhang et al. 2013).Other studies, such as Zhao's investigation into geomorphic entropy provides insights into the complexity of geomorphological processes(Zhao et al. 2023).Yu's research further explores the role of precipitation, vegetation, and land preparation techniques in mitigating soil erosion (Yu et al. 2019). Additionally, Zhu's work offers a detailed analysis of gully nodes༈Zhu et al. 2012༉and gully head (Zhu et al. 2014)morphology in small loess watersheds, contributing valuable knowledge on erosion patterns and watershed evolution.

While these studies provide important insights into watershed evolution, few have specifically explored this process through changes in slope shape. Previous research has confirmed the strong correlation between slope spectrum and geomorphic types on the Loess Plateau (Tang et al. 2008; Li et al. 2016). Given that slope shape is a crucial factor in reflecting the erosion and morphological evolution of slope systems, it can provide a comprehensive view of both macro and micro topographic features. These include elevation, terrain relief, surface roughness, slope, aspect, surface cutting depth,and curvature.

Building on the concept of slope spectrum, this study proposes a new topographic index: the slope shape spectrum. The slope shape spectrum is a statistical representation or mathematical model where the slope shape, determined by quantitative factors of both micro and macro topography, serves as the independent variable, while the proportion of the grid area corresponding to each slope shape in a given region serves as the dependent variable.At different stages of development, loess landforms exhibit significant variation in slope shape due to erosion and sedimentation. As a tool for describing the distribution of slope shapes, the slope shape spectrum offers a unique perspective on the evolution and development of loess landforms. By analyzing the slope shape spectrum, we can better understand the geomorphological complexity of the Loess Plateau.

The primary objective of this study is to analyze the time series of slope shape spectra and investigate their changes as simulated watersheds develop. However, studying long-term geomorphic evolution under natural conditions is challenging. Watersheds serve as fundamental geomorphic units for quantitative analysis of loess evolution (Tang et al. 2009) and play a key role in shaping the diverse landforms of the Loess Plateau (Cai et al. 2014; Zhu et al. 2014). The use of artificial rainfall experiments enables the observation of long-term geomorphic changes within a short time frame, providing a solid foundation for studying the slope shape spectrum.Through artificial rainfall simulations, this study reproduces the evolution of loess watersheds, quantifies the combined characteristics of slope morphology, and introduces qualitative and quantitative indicators to explore the rules governing watershed evolution. This provides new insights for analyzing the geomorphology of the Loess Plateau.

2.1 Rainfall experiment of Loess Watershed

Artificial rainfall experiments are widely recognized as an effective method for simulating the long-term evolution of geomorphological features (Borselli et al. 2001; Xue et al. 2007). To investigate the long-term development of the loess watershed, a series of controlled experiments were conducted at the Artificial Rainfall Simulation Laboratory of the Institute of Soil and Water Conservation, Chinese Academy of Sciences. The experiments utilized lowland loess sourced from Yangling, Shaanxi Province, with a unit mass density of 1.39 g/cm³.To ensure that the artificial rainfall conditions closely resembled those of the Loess Plateau, previous studies and expert recommendations were considered (Bryan and Ploey, 1983; Borselli et al. 2001; Wang et al. 2010; Li et al. 2016). These parameters guided the design of the simulated rainfall, including factors such as rainfall duration, intensity, volume, and frequency. The experiment lasted for 72 days, segmented into nine phases, and included a total of 25 artificial rain events, occurring two to five times per week.The loess materials and surface topography were meticulously shaped based on real-world references to ensure that the simulated watershed accurately reflected natural conditions. Figure 1 presents the initial surface of the simulated watershed, while Table 1 details the initial parameters of the watershed, and Table 2 outlines the parameters for the artificial rainfall experiment.

Table 1

geometric characteristics of initial simulated landform in loess watershed
Projected area (m²)	Length (m)	Width (m)	Perimeter (m)	Elevation difference(m)	Mean slope(°)	Channel level	Branch ratio
31.49	9.1	5.8	23.3	2.57	15	2	4

Table 2

The setting parameters of artificial rainfall experiments
Stage	Designed rainfall intensity (mm·min ^− 1)	Actual rainfall intensity (mm·min^− 1)	Total rainfall Period (min)	Total rainfall amount (mm)	Rainfall frequency
Stage1	0	0	0	0	5
Stage2	0.7	0.79	362.9	298.0	2
Stage3	1.5	1.80	76.8	105.3	2
Stage4	0.5	0.58	152.1	91.9	2
Stage5	1.5	1.78	79.1	130.7	3
Stage6	0.5	0.55	185.3	102.1	4
Stage7	1	1.04	185.04	194.8	2
Stage8	2	2.05	64.7	132.1	3
Stage9	0.5	0.56	271.6	152.0	2

2.2 data

In this study, close-range digital photogrammetry was employed to monitor terrain changes in the simulated watershed. A digital elevation model (DEM) with a resolution of 10 mm was generated from stereo images. The process of DEM creation involved four primary steps: control point measurement, close-range photography, stereo camera calibration, and image processing using JX4 software.These DEM images allowed for the reconstruction of the watershed’s evolution, capturing the transformation from an initial straight-slope terrain to a more complex, broken terrain with dense ditches, a result of long-term natural rainfall erosion (Fig. 2). Table 3 presents the topographic attribute data for the watershed at different stages over time.

Table 3

The terrain attributes of loess watershed following the watershed development
Stage	Hypsometric Integral (%)	Slope (degree)	Gully density/ (m·m^–2)	Erosion amount/ kg
Stage1	63.97	16.51	0.5086	0
Stage2	63.20	20.76	0.7404	1355.2
Stage3	62.79	22.73	0.865	1461.3
Stage4	61.22	25.30	0.964	1423.4
Stage5	59.39	28.79	1.1609	1881.7
Stage6	59.02	29.67	1.3551	979.7
Stage7	57.23	31.97	1.847	1597.7
Stage8	56.65	33.24	2.0437	965.4
Stage9	55.58	32.69	1.9488	777.8

3.1 Terrain factor selection for segmentation

Topographic factors derived from Digital Elevation Models (DEMs) are widely recognized as valuable tools for the quantitative description of geomorphology (Ding et al. 2016; Zhao et al. 2017). A combination of multiple terrain factors allows for a comprehensive reflection of the morphological characteristics of landscape entities (Wang et al. 2012; Tang 2014). Based on this, a selection of terrain factors was employed for segmentation.

We pre-selected four macro-topographic factors and four micro-topographic factors as indicators for segmentation. The macro-topographic factors include elevation, topographic relief, surface roughness, and surface cutting depth. Meanwhile, the micro-topographic factors consist of slope, aspect, plane curvature, and profile curvature (Fig. 3).

The expression of topographic and spatial features can be effectively explained by combining various topographic factors from multiple perspectives. However, there is often significant correlation between these factors, which leads to redundant information. To eliminate such redundancy, we applied a correlation analysis method to select a set of low-correlation factors from the eight pre-selected terrain variables. We established a correlation coefficient threshold of 0.8, meaning factors with a correlation coefficient above this threshold are considered highly correlated. In the case of highly correlated factors, our approach is to retain only one and discard the others.To address the varying scales of different dimensions, all terrain factors were normalized, ensuring their values ranged uniformly between 0 and 255. Subsequently, the Sheffield entropy method (Sheffield 1985), commonly used to optimize feature selection (Zhao et al. 2017), was employed to filter out highly correlated terrain factors. This approach ensures the maximization of information provided by the terrain factor combinations. Using this method, we determined an optimal set of terrain factors to serve as input data for image segmentation. The mathematical formula used is presented in Formula 1.

$$\:\text{S}=\frac{\text{n}}{2}+\frac{\text{n}}{2}\cdot\:\text{l}\text{n}\left(2\text{n}\right)+\frac{1}{2}\cdot\:\text{l}\text{n}\left|{\text{M}}_{\text{S}}\right|$$

Where S is the entropy of N-dimensional data subset, n is the dimension, |MS| is the row and column values of the N-dimensional data subset covariance matrix. The larger the value, the greater the entropy of the image, and the more conducive to image classification.

3.2 Optimal segmentation parameters

Satisfactory segmentation results are critical for accurate slope shape classification. Typically, the selection of an optimal segmentation scale relies on subjective trial and error. Baatz and Schäpe (2000) proposed a high-quality optimization method for multi-scale image segmentation, known as Multi-Resolution Image Segmentation (MRS), which has become widely used and is implemented in software like eCognition (Blaschke 2010). MRS is a bottom-up, region-based approach that merges adjacent pixels until the heterogeneity within an object exceeds a user-defined threshold.The MRS algorithm utilizes three user-defined parameters: scale, shape, and compactness. While the local variance method is readily applicable within eCognition (Drăguţ et al. 2010), it primarily focuses on the segmentation scale and does not take into account the other parameters (Drăguţ and Eisank 2012). Thus, a comprehensive approach considering all three parameters is essential for enhancing segmentation quality.

All three parameters—scale, shape, and compactness—are addressed in the study by Johnson and Xie (2011), who argue that optimal image segmentation is determined through an unsupervised evaluation method that considers both global intra-segment homogeneity and inter-segment heterogeneity measurements. To assess these criteria, this study employs Moran's I (MI) and weighted variance (WV) (Johnson and Xie 2011; Schlögel et al. 2018; Alvioli et al. 2020). MI quantifies inter-segment heterogeneity, while WV measures intra-segment homogeneity. Consequently, lower values of MI and WV (see Formulas 2 and 3) indicate better segmentation quality. The definitions of MI and WV are as follows:

$$\:Ml=\frac{n\sum\:_{i=1}^{n}\sum\:_{j=1}^{n}{w}_{ij}\left({x}_{i}-\overline{x}\right)\left({x}_{j}-\overline{x}\right)}{\sum\:_{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}\left(\sum\:_{i\ne\:j}{w}_{ij}\right)}$$

$$\:WV=\frac{\sum\:_{i=1}^{n}{a}_{i}\cdot\:{v}_{i}}{\sum\:_{i=1}^{n}{a}_{i}}$$

Where n is the number of segments,x_i and x_j are the eigenvalues of segments i and j, respectively, and x represents the average value of the image.Wij is a spatial weight that measures the spatial adjacency between segments i and j. In this study, only adjacent regions (i.e. regions sharing borders) are considered for the calculation of the MI, so if regions xi and xj are adjacent, W_ij=1 otherwise W_ij=0.Lower MI indicates higher intersegment heterogeneity.Where a_i and v_i represent the area and standard deviation of segment i, respectively.WV is chosen as a measure of intra-segment goodness because a segment with a low variance should be relatively homogeneous.

In the context of image segmentation, the goodness measures of plane curvature and section curvature are computed independently. To ensure that the intra-segment and inter-segment goodness measures are considered on equal footing, we employ the normalization formula (Formula 4) to re-scale both measures to a common range of 0 to 1.

$$\:\left(\begin{array}{c}X-{X}_{\text{m}\text{i}\text{n}}\end{array}\right)/\left(\begin{array}{c}{X}_{\text{m}\text{a}\text{x}}-{X}_{\text{m}\text{i}\text{n}}\end{array}\right)$$

Low values of Moran's I and weighted variance are considered ideal for achieving optimal image segmentation. To evaluate segmentation results, Johnson and Xie (2011) introduced the Global Score (GS), which combines normalized Moran's I (MI_norm) and normalized weighted variance (V_norm) (Formula 5). Earlier studies, such as Espindola et al. (2006), primarily utilized the near-infrared (NIR) band for segmentation quality assessment. However, Johnson and Xie (2011) expanded this approach by incorporating the mean global scores from the NIR, red, and green bands, providing a more comprehensive evaluation of segmentation performance.

$$\:GS={V}_{\text{n}\text{o}\text{r}\text{m}}+M{I}_{\text{n}\text{o}\text{r}\text{m}}$$

In this study, recognizing that the slope shape classification is influenced by curvature segmentation effects, we utilize the average global scores of plane curvature and profile curvature to determine optimal segmentation parameters, as shown in Formula 6.

$$\:{GS}_{mean}=（{GS}_{\text{p}\text{l}\text{a}\text{n}\text{e}\:\text{c}\text{u}\text{r}\text{v}\text{a}\text{t}\text{u}\text{r}\text{e}\:}+{\text{G}\text{S}}_{\text{p}\text{r}\text{o}\text{f}\text{i}\text{l}\text{e}\:\text{c}\text{u}\text{r}\text{v}\text{a}\text{t}\text{u}\text{r}\text{e}}）/\:2$$

The optimization parameters based on GS are identified through the following process: first, we establish the possible range for each parameter via rough trials and visual assessments. In this research, the scale parameters range from 5 to 20, shape parameters from 0.3 to 0.7, and compactness from 0.4 to 0.6. Segmentation results are generated using various combinations of these parameters, and the combination yielding the lowest mean GS (GS_mean) is selected as the optimal parameter set.

3.3 Classification of slope shape element and construction of slope shape spectrum

The calculation of curvature for slope shape elements is crucial for their classification. In this study, we employ a fitting equation for surface expression proposed by Zevenbergen and Thorne (1987) to calculate the plane curvature and profile curvature of slope shape elements. Optimal segmentation parameters are then applied to segment and classify these elements within the study area.Based on previous research (Drăguţ and Blaschke 2006), the slope classification rules are defined according to slope shape elements as follows:

1.A slope unit with an average gradient of less than 3° is classified as a gentle slope.

2.A slope unit with an average gradient greater than 45° is classified as a steep slope.

3.For unclassified objects, they are further categorized according to the profile curvature and plan curvature of the slope unit into: straight-straight, convex-convex, concave-concave, straight-convex, straight-concave, convex-straight, concave-straight, concave-convex, and convex-concave slope shapes.

The detailed classification rules for slope shapes are presented in Table 4.

Table 4

Classification rules of slope shape
Slope unit	Plan curvature	Profile curvature	Slope (°)
SS slope	0	0	ND
VV slope	> 0	< 0	ND
CC slope	< 0	> 0	ND
SV slope	> 0	0	ND
SC slope	< 0	0	ND
VS slope	0	< 0	ND
CS slope	0	> 0	ND
VC slope	> 0	> 0	ND
CV slope	< 0	< 0	ND
Flat slope	ND	ND	< 3
Steep slope	ND	ND	> 45
(Notes:S = straight,V = convex,C = concave,ND = Not Defined;The priority of flat slope and steep slope is higher than the other nine slope shapes;Among them, the former letter represents the profile shape, and the latter letter represents the plane shape)

In our analysis, the slope shape classification results serve as independent variables, while the area proportion of each slope shape functions as the dependent variable. This relationship is utilized to establish the slope shape spectrum, which graphically represents the area proportions of slope shape elements within a unit area.

3.4 Slope shape spectrum indices

3.4.1 Quantity characteristic index of slope shape spectrum

Based on histograms, the slope shape spectrum can visually depict the frequency and variations of different slope shapes. Applying the slope shape spectrum to digital terrain analysis involves a critical step of extracting a series of quantitative changes. These variables can reveal the evolution of slope shapes. Through systematic analysis and comparison, three quantitative characteristic indicators were selected: slope shape spectrum entropy, skewness, and kurtosis, to quantify different aspects of the slope shape spectrum.

Over the past several decades, the concept of information entropy has been extensively applied in geosciences (Cao 2007; Zhao et al. 2023). From the perspective of information theory, this paper adopts the concept of slope shape spectrum information entropy (H) to describe the uniformity of slope shape distribution across a sample area and the variation in frequency data across different groups in the slope shape spectrum. The formula for calculating this entropy is provided as Eq. 7.

$$\:H=-\sum\:_{i=1}^{m}{P}_{i}\text{ln}{P}_{i}$$

In the formula, m denotes the number of slope shape classes, and P_i represents the frequency of each slope shape class. In this study, 11 slope shape classes are identified, thus m is set to 11. The information entropy of the slope shape spectrum serves as a metric to quantify the complexity of the spectral model. Geomorphologically, regions with gentle surface fluctuations and simpler slope shapes exhibit lower information entropy. Conversely, regions characterized by significant surface relief variations, such as those with complex gullies and diverse slope shapes distributions, display higher entropy values.

Skewness (S) and Kurtosis (K) are statistical measures used to describe the shape of the distribution. Skewness quantifies the degree and direction of asymmetry in a distribution, reflecting how data is distributed around the mean. Kurtosis measures the sharpness or flatness of the probability density distribution around the mean, indicating how peaked or flat the data is. The formulas for skewness (Formula 8) and kurtosis (Formula 9) are defined as follows:

$$\:s=\frac{m}{\left(m-1\right)\left(m-2\right)}\sum\:{\left(\frac{{p}_{i}-\overline{p}}{\delta\:}\right)}^{3}$$

$$\:K=E\left[{\left(\frac{X-\mu\:}{\sigma\:}\right)}^{4}\right]-3$$

Where m represents the number of slope shape categories, pi represents the frequency of each category, p represents the average frequency, δ represents the standard deviation, x represents the slope shape area, and µ represents the average value of all slope shape areas.When S < 0, the data distribution is more dispersed on the right side of the mean, whereas S > 0 means that the data distribution is more dispersed on the left side. K < 0 indicates that less extreme data is distributed at both ends, and K > 0 indicates that more data is distributed at both ends.

3.4.2 Spatial structure characteristic index of slope shape spectrum

The quantitative metrics of the slope shape spectrum, such as information entropy, skewness, and kurtosis, are effective for analyzing the frequency distribution of slope shapes. However, these metrics fall short in describing the spatial structure of slope shape distribution. To address this gap, landscape pattern concepts have been increasingly applied in geoscience studies(Coco and Buccolini 2016). The landscape pattern approach focuses on understanding how ecological processes are related to spatial differentiation and scale by analyzing the type, quantity, and spatial positioning of landscape units, such as patches, matrices, and corridors.

In this study, slope shape elements are abstracted as spatial units within a landscape ecology framework. By treating the slope shape distribution of the entire sample area as a cohesive landscape unit, landscape ecology methods are applied to quantitatively assess the spatial structure of the slope shape spectrum. Specifically, eight landscape indices (Table 5) are employed to quantify the spatial characteristics of the slope shape landscape, facilitating a more comprehensive analysis of its spatial structure.

Table 5

Quantifying the landscape index of slope shape spectrum
Quantitative index	Quantitative index	Remark
Mean patch area (AREA_MN)	$\:\text{A}\text{R}\text{E}{\text{A}}_{-}\text{M}\text{N}=\sum\:_{\text{j}=1}^{\text{n}}\frac{{\text{a}}_{\text{i}\text{j}}}{{\text{n}}_{\text{i}}}$	a_ij is the area of the patch, and n_i represents the number of patches included in the Class i slope shape. AREA_MN represents the complexity of slope shape patches in the landscape.
Perimeter-Area Fractal Dimension (PAFRAC)	$\:PAFRAC=\frac{\frac{2}{\left[{n}_{i}\times\:\sum\:_{j=1}^{n}\left(\text{l}\text{n}{P}_{ij}\times\:\text{l}\text{n}{a}_{ij}\right)\right]-\left[\left(\sum\:_{j=1}^{n}\text{l}\text{n}{P}_{ij}\right)\times\:\left(\sum\:_{j=1}^{n}\text{l}\text{n}{a}_{ij}\right)\right]}}{\left({n}_{i}\times\:\sum\:_{j=1}^{n}\text{l}\text{n}{P}_{ij}^{2}\right)-{\left(\sum\:_{j=1}^{n}\text{l}\text{n}{P}_{ij}\right)}^{2}}$	a_ij is the patch area, p_ij is the proportion of this type of patch in the study sample area, and ni represents the number of patches included in the Class i slope shape. PAFRAC represents the complexity of the slope shape patches of the overall landscape.
Contagion Index (CONTAG)	$\:CONTAG=\left[1+\frac{\sum\:_{i=1}^{m}\sum\:_{k=1}^{m}\left[\left({P}_{i}\right)\left(\frac{{g}_{ik}}{\sum\:_{k=1}^{m}{g}_{ik}}\right)\right]\times\:\left(\text{l}\text{n}\left({P}_{i}\right)\left(\frac{{g}_{ik}}{\sum\:_{k=1}^{m}{g}_{ik}}\right)\right)}{2\text{l}\text{n}\left(m\right)}\right]\times\:100$	Where p_i is the percentage of area occupied by type i patches, g_ik is the number of adjacent type i patches and type k patches, and m is the total number of patches types in the landscape. CONTAG represents the degree of aggregation of slope shape patches in the landscape.
Interspersion and Juxtaposition Index (IJI)	$\:IJI=\left[1+\frac{-\sum\:_{k=1}^{m}\left[\left(\frac{{e}_{ik}}{\sum\:_{k=1}^{m}{e}_{ik}}\right)\text{l}\text{n}\left(\frac{{e}_{ik}}{\sum\:_{k=1}^{m}{e}_{ik}}\right)\right]}{\text{l}\text{n}\left(m-1\right)}\right]\times\:100$	m refers to the total number of patch types in the landscape, and e_ik is the side length of patch i adjacent to the patch of patch type k. IJI represents the degree of aggregation and separation of slope shape patches in landscape. When the corresponding plaque type was adjacent to only one other plaque type, the IJI was close to 0. If this plaque type is adjacent to all other plaques, IJI = 100.
Shannon's diversity index (SHDI)	$\:\text{S}\text{H}\text{D}\text{I}=-\sum\:_{\text{i}=1}^{\text{m}}\left[{\text{P}}_{\text{i}}\text{l}\text{n}\left({\text{P}}_{\text{i}}\right)\right]$	m refers to the total number of patch types in the landscape, and P_i refers to the area ratio of patch type i to the entire landscape. SHDI represents the type richness of slope shape patches in landscape.
Shannon's evenness index (SHEI)	$\:SHEI=\frac{-\sum\:_{i=1}^{m}\left[{P}_{i}\text{l}\text{n}\left({P}_{i}\right)\right]}{\text{l}\text{n}m}$	m refers to the total number of patch types in the landscape, and P_i refers to the area ratio of patch type i to the entire landscape. SHEI represents the uniformity of slope shape patches in landscape area distribution.
Landscape shape index (LSI)	$\:LSI=\frac{0.25E}{\sqrt{A}}$	E is the edge length of the patch, and A is the area of the patch. LSI represents the complexity of slope shape patches in landscape.
Patch density (PD)	$\:PD=\frac{NP}{\text{A}}$	NP refers to the total number of patches in the landscape, and A is the total area of the landscape or patch. PD indicates the fragmentation degree of a certain slope shape patch in the landscape

4.1 Selection of terrain factors for segmentation

In the process of segmentation, selecting the appropriate topographic factors is essential to minimize redundancy and maximize the information gained. Table 6 reveals high correlations between the terrain factors of slope, terrain relief, surface roughness, and surface cutting depth. To reduce redundancy, one highly correlated factor is selected, along with four lower-correlation factors, for further analysis. Four combinations of these factors were tested, and their entropy values were recorded as 23.339, 21.792, 21.987, and 21.794.

Among these combinations, the one including slope yielded the highest entropy (Table 7), indicating that slope combination provides the most information. Therefore, the final selected combination of topographic factors consists of slope, aspect, plane curvature, profile curvature, and elevation.

Table 6

Correlation coefficient matrix of terrain factors
Terrain factors	Elevation	Slope	Aspect	Plane C	Profile C	TR	SR	SCD
Elevation	1	-	-	-	-	-	-	-
Slope	−0.127	1	-	-	-	-	-	-
Aspect	0.063	0.038	1	-	-	-	-	-
Plane C	0.015	−0.002	0.004	1	-	-	-	-
Profile C	−0.116	−0.007	0.004	−0.281	1	-	-	-
TR	−0.047	0.909	0.038	−0.021	−0.008	1	-	-
SR	−0.015	0.830	0.028	−0.031	−0.009	0.974	1	-
SCD	−0.037	0.904	0.037	−0.018	−0.085	0.996	0.970	1
(Notes: Plane C = Plane curvature, Profile C = Profile curvature, SR = Surface roughness, TR = Terrain relief;SCD = Surface cutting depth)

Table 7

Terrain factor combinations and the corresponding entropy values
Combination	Entropy value
Slope,Aspect,plane C,Profile C,Elevation	23.339
TR,Aspect,plane C,Profile C,Elevation	21.792
SR,Aspect,plane C,Profile C,Elevation	21.987
SCD,Aspect,plane C,Profile C,Elevation	21.794
(Notes: Plane C = Plane curvature, Profile C = Profile curvature, SR = Surface roughness, TR = Terrain relief;SCD = Surface cutting depth)

4.2 Segmentation optimization

In Section 3.2, 240 different parameter settings were tested for optimal segmentation: scale (5 to 20, intervals of 1), shape (0.3 to 0.7, intervals of 0.1), and compactness (0.4 to 0.6, intervals of 0.1). The GSmean (mean Goodness of Segmentation) values for these parameter combinations showed a trend where GSmean initially decreases and then increases as the scale parameter rises.Table 8 shows the partial results of this analysis, indicating that the smallest GSmean, which corresponds to the optimal segmentation, occurs at a scale of 16, with a shape value of 0.6 and a compactness value of 0.5. These parameter values were selected as the best combination for achieving the most precise segmentation.

Table 8

GS—measure metrics (partial) with different scale, shape, and compactness parameters
Scale	Shape	Compactness	GS_{Plan curvature}	GS_{profile curvature}	GS_mean
5	0.5	0.5	0.834	1.000	0.917
5	0.6	0.5	0.927	0.907	0.917
10	0.5	0.5	0.559	0.370	0.465
10	0.6	0.5	0.393	0.280	0.337
15	0.5	0.5	0.109	0.218	0.164
15	0.6	0.5	0.372	0.203	0.288
16	0.5	0.5	0.120	0.207	0.163
16	0.6	0.4	0.044	0.103	0.074
16	0.6	0.5	0.056	0.088	0.072
17	0.5	0.5	0.126	0.208	0.167
17	0.6	0.5	0.062	0.112	0.087
20	0.5	0.5	0.150	0.108	0.129
20	0.6	0.5	0.071	0.159	0.115

4.3 Slope shape classification results and slope shape spectrum

In Section 3.3, slope shapes were classified based on two curvature metrics: plane curvature and section curvature. The classification was visualized and analyzed using a hill shading map (Fig. 2), allowing a comparison between slope classification results and overlapping areas in a randomly sampled part of the study area. The results demonstrated that the object-oriented classification method is robust and reliable for this task.

Subsequently, the classification results were fused to create a comprehensive spatial distribution of slope shape units over nine time periods, as illustrated in Fig. 4. This method provided a clear representation of slope dynamics and geomorphological changes over time, emphasizing the effectiveness of object-oriented approaches in terrain analysis.

4.4 Time variation of slope shape spectrum

Using the slope shape classification results, we calculated the proportion of different slope shape elements for each stage of the simulated watershed. These proportions were used to establish the slope shape spectrum, where the slope shape category was treated as the independent variable, and the area proportion of each slope shape as the dependent variable. The slope shape spectra, illustrating the evolution of slope shapes over different stages of the simulated watershed, are presented in Fig. 5.

From the analysis of Figs. 4 and 5, it is evident that the slope shapes in different stages of the simulated watershed vary significantly. The slope shape spectrum entropy (Fig. 6a) shows an increasing trend, while the skewness and kurtosis of the slope shape spectrum (Figs. 6b, 6c) exhibit a decreasing trend. This is because, at the initial stage of the simulated watershed (Fig. 4, stage 1), the surface undulations are relatively gentle, and the slope shapes are relatively uniform, primarily consisting of double straight slopes. As a result, the slope shape spectrum entropy is low, while its skewness and kurtosis are high. As the simulated watershed evolves through different stages (Fig. 4, stages 2–9), a variety of slope shapes begin to emerge and increase in number, with surface undulations becoming more pronounced and the slope shape distribution becoming more complex. Consequently, the slope shape spectrum entropy shows an increasing trend, while the skewness and kurtosis decrease. Therefore, there is a certain spatial coupling between slope shape spectrum entropy and the development stages of the simulated watershed, indirectly reflecting the development process of the Loess landform. The overall development and evolution of the Loess Plateau geomorphological system correspond to an entropy increase process, which is consistent with the conclusion that geomorphological systems exhibit ordered evolutionary characteristics (Chen et al. 2006).

According to the change rate of information entropy of slope shape spectrum (Fig. 6,a), the simulated watershed evolution was divided into three stages.

The first stage is the slope shape development stage (Figs. 4 and 5, stage 1-stage 2), during which slope shape changes are highly pronounced. This is due to the initial surface experiencing 298 mm of artificial rainfall (Table 3), far exceeding the rainfall in other periods.Under the continuous impact of rain and erosion, the original double-straight slopes gradually transformed into other slope shapes.As a result, the slope shape spectrum entropy rapidly increased, reaching its highest rate of change in this stage.The second stage is the slope shape growth stage (Figs. 4 and 5, stage 3-stage 5), characterized by active slope changes.With ongoing rainfall erosion and sediment deposition, different slope shapes gradually evolved.The proportion of double-straight slopes decreased, while steep slopes increased, and other slope shapes appeared more frequently.Consequently, the slope shape spectrum entropy continued to increase, with a relatively high rate of change.The third stage is the slope shape maturation stage (Figs. 4 and 5, stage 6-stage 9), where slope changes became more stable. In this phase, the slope system was dominated by headward erosion, with slow surface incision, and slope shapes gradient changes gradually leveled out.Steep slopes tended to transform into double-concave, double-convex, concave-convex, and convex-concave slope shapes.Under sedimentation, concave-convex and convex-concave slope shapes frequently transitioned into concave-straight, convex-straight, straight-concave, and straight-convex slope shapes. The double-straight slope both transformed and was transformed, resulting in minimal overall change, while flat slopes appeared sporadically within the double-straight slopes. Therefore, in this stage, the slope shape spectrum entropy increased steadily, with the lowest rate of change.

During the dynamic evolution of slope systems, a series of complex natural processes, including erosion, cutting, merging, and siltation, are at play. These processes interact to facilitate the gradual transformation of the watershed's terrain from its initial double straight slope morphology into a diverse array of slope shape structures. Over time, these varied slope shapes undergo reorganization and adjustment at their bases. Ultimately, this leads to the formation of new double straight slopes or flat slope shapes, thereby completing a comprehensive cycle of slope state transformation.

4.5 Spatial variation of slope shape spectrum

4.5.1 Class level indices

Figure 7 illustrates the class level index across various stages of watershed evolution, offering crucial insights for analyzing the spatial structural changes of slope shape patches throughout this process.

The PD (patch density) of various slope shape patches (Fig. 7,a) exhibited a clear upward trend throughout the evolution of the watershed. This demonstrates how erosion contributes to an increase in the distribution density of slope shape patches, intensifying landscape fragmentation.Notably, during the slope shape development phase, erosion is particularly pronounced, causing a sharp rise in the distribution density and a rapid increase in PD.As the watershed evolves into the slope shape growth and maturity stages, the combined effects of erosion and sedimentation result in a more complex and fluctuating upward trend in PD.This model captures both the intricate influence of erosion and sedimentation on patch distribution density and the dynamic changes in landscape fragmentation. The alterations in slope shape patches suggest that initial rainfall erosion transforms the relatively smooth Loess Plateau surface into a more complex terrain, marked by increased surface roughness and expanding gullies, accelerating slope shape evolution.Therefore, the greater the watershed's development, the more severe the erosion, leading to higher fragmentation of slope shape patches. Conversely, when erosion is weak, the fragmentation degree is lower.

The LSI (Landscape Shape Index) shown in Fig. 7b also exhibits an upward trend alongside the watershed's evolution.The rapid increase in the LSI during the slope shape development phase indicates intense erosion and a corresponding rise in the edge complexity of slope shape patches.his complexity arises from continuous erosion within the loess watershed, leading to the formation of new tributary gullies as existing tributary ditches are further incised.Following the slope shape growth and maturity stages, the fluctuations in the LSI stabilize, suggesting that the edge complexity of slope shape patches is becoming increasingly stable.In the late stages of watershed evolution, the spatial structure of the basin tends to stabilize, resulting in fewer new tributary gullies;however, the LSI remains elevated due to the intricate topographic features created by earlier erosion.It is noteworthy that the LSI for flat slopes exhibits minimal change, which is attributable to their flat terrain characteristics, yielding more regular and simple patch edges.Thus, it can be concluded that as the watershed matures, both the topography and the complexity of slope shape patches become more stable.

4.5.2 Landscape level indices

Table 9 presents the landscape-level indices at various stages of watershed evolution, highlighting distinct trends in each index. The analysis reveals significant differences in trend patterns across the evolutionary stages of the simulated watershed. Notably, two indices, AREA_MN and CONTAG, exhibited downward trends. AREA_MN decreased sharply from 1.762 to 0.154, while CONTAG fell from 98.202 to 77.788. These declines indicate that during the slope shape development phase, the effects of artificial rainfall resulted in intense surface erosion and numerous alterations in slope shapes. Consequently, this led to increased fragmentation and a reduction in the average patch area, as well as decreased aggregation among similar slope shapes. As the watershed progresses into the growth and maturity stages, the transformation rate among slope shapes slows down, resulting in a gentler decline in the corresponding indices.

Simultaneously, four indices exhibited an upward trend: PAFRAC, IJI, SHDI, and SHEI. During the slope shape development phase, slope shape types evolved from homogeneous to diverse, and the boundaries transitioned from regular to complex, leading to a significant increase in SHDI and PAFRAC. Additionally, the distribution of various slope shapes became more uniform across the watershed, resulting in a marked increase in SHEI and IJI. As the watershed entered the stages of growth and maturity, the growth rates of the IJI, SHDI, and SHEI indices gradually slowed, eventually leveling off, although they continued to exhibit an upward trend. Notably, PAFRAC stabilized around 1.4 during these stages, suggesting that the watershed is maturing, the terrain is stabilizing, the impacts of erosion and sedimentation are diminishing, and the complexity of basin boundaries has reached a relatively stable state

Through the combination of slope shape spectrum and landscape index, it can be found that all indices change with the evolution of the simulated watershed, and it has the ability to describe and simulate the topographic characteristics and evolution stages of the loess watershed.

Table 9

Landscape level indices at different stages of watershed evolution
Stage	AREA_MN/m²	PAFRAC	CONTAG/%	IJI/%	SHDI	SHEI
Stage1	1.762	1.082	98.202	51.458	0.065	0.031
Stage2	0.154	1.431	77.788	77.915	0.928	0.387
Stage3	0.104	1.397	71.443	80.893	1.186	0.494
Stage4	0.094	1.451	66.965	79.504	1.371	0.572
Stage5	0.075	1.414	62.832	79.964	1.535	0.640
Stage6	0.068	1.433	60.770	79.598	1.606	0.670
Stage7	0.069	1.435	59.186	77.680	1.672	0.697
Stage8	0.061	1.428	57.262	79.734	1.744	0.727
Stage9	0.051	1.404	54.553	81.480	1.841	0.768

This study simulated the long-term evolution of watersheds in the Loess Plateau through indoor artificial rainfall experiments. The slope shape spectrum was then applied to analyze and model the basin's evolution. A series of quantitative indicators were introduced to reveal and examine changes in slope morphology during this process. The key findings and conclusions of the study are as follows:

Optimization of Terrain Factors and Image Segmentation Parameters: By conducting correlation analysis and entropy calculations, this study identified the optimal combination of terrain factors—slope, aspect, plan curvature, profile curvature, and elevation—that most accurately describe slope shape classification. The optimal combination of scale, shape, and compactness parameters for the multi-resolution image segmentation algorithm was also determined, ensuring precise slope shape classification.
Construction of the Slope Shape Spectrum and Division of Watershed Evolution Stages: Using the optimal segmentation parameters, the slope shapes of the simulated watershed were classified, and a slope shape spectrum was constructed.The analysis revealed that as the watershed evolved, the information entropy of the slope shape spectrum gradually increased, while skewness and kurtosis progressively decreased.This indicates growing geomorphological complexity and a smoother watershed morphology.Based on the rate of change in information entropy, the evolution of the simulated watershed was divided into three stages: the slope shape development stage, the slope shape growth stage, and the slope shape maturity stage, each characterized by distinct geomorphic evolution features.
Spatial Variation Analysis of Landscape Index: Through class-level and landscape-level landscape index analysis, this study revealed the dynamic spatial changes of slope shape patches as the simulated watershed evolved. These changes included increased density of slope patches, greater edge complexity, and heightened landscape fragmentation. The findings suggest that as watershed development progresses, erosion becomes more intense, leading to higher fragmentation of slope patches. In the later stages of evolution, the watershed tends to stabilize, and the complexity of the basin's boundaries reaches a relatively stable state.

In summary, this study introduces the concept of the slope shape spectrum, offering a comprehensive research framework for investigating geomorphic evolution on the Loess Plateau. By integrating topographic analysis, image segmentation techniques, and landscape ecology methods, this framework provides a valuable reference for both research and practical applications in related fields.

Funding Statement:

None

Author Contribution

J:Hongtao Jiang ；C：Nan Chen ；W：Chen Wang ； L：Sijia Li ; O：Mengyao OUJ finished the experiment, C guided J to write the paper, and W, L and O helped to revise the paper. All the authors reviewed the manuscript.

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Geomorphological evolution of small watershed on the Loess Plateau based on slope shape spectra

Status:

Version 1

Abstract

Figures

1 Introduction

2. Experiment and data

2.1 Rainfall experiment of Loess Watershed

2.2 data

3. Method

3.1 Terrain factor selection for segmentation

3.2 Optimal segmentation parameters

3.3 Classification of slope shape element and construction of slope shape spectrum

2.A slope unit with an average gradient greater than 45° is classified as a steep slope.

3.4 Slope shape spectrum indices

3.4.1 Quantity characteristic index of slope shape spectrum

3.4.2 Spatial structure characteristic index of slope shape spectrum

4 Results and discussion

4.1 Selection of terrain factors for segmentation

4.2 Segmentation optimization

4.3 Slope shape classification results and slope shape spectrum

4.4 Time variation of slope shape spectrum

5 Conclusions

Declarations

Funding Statement:

Author Contribution

References

Additional Declarations

Status:

Version 1

Quantitative index	Quantitative index	Remark
Mean patch area (AREA_MN)	\(\:\text{A}\text{R}\text{E}{\text{A}}_{-}\text{M}\text{N}=\sum\:_{\text{j}=1}^{\text{n}}\frac{{\text{a}}_{\text{i}\text{j}}}{{\text{n}}_{\text{i}}}\)	a_ij is the area of the patch, and n_i represents the number of patches included in the Class i slope shape. AREA_MN represents the complexity of slope shape patches in the landscape.
Perimeter-Area Fractal Dimension (PAFRAC)	\(\:PAFRAC=\frac{\frac{2}{\left[{n}_{i}\times\:\sum\:_{j=1}^{n}\left(\text{l}\text{n}{P}_{ij}\times\:\text{l}\text{n}{a}_{ij}\right)\right]-\left[\left(\sum\:_{j=1}^{n}\text{l}\text{n}{P}_{ij}\right)\times\:\left(\sum\:_{j=1}^{n}\text{l}\text{n}{a}_{ij}\right)\right]}}{\left({n}_{i}\times\:\sum\:_{j=1}^{n}\text{l}\text{n}{P}_{ij}^{2}\right)-{\left(\sum\:_{j=1}^{n}\text{l}\text{n}{P}_{ij}\right)}^{2}}\)	a_ij is the patch area, p_ij is the proportion of this type of patch in the study sample area, and ni represents the number of patches included in the Class i slope shape. PAFRAC represents the complexity of the slope shape patches of the overall landscape.
Contagion Index (CONTAG)	\(\:CONTAG=\left[1+\frac{\sum\:_{i=1}^{m}\sum\:_{k=1}^{m}\left[\left({P}_{i}\right)\left(\frac{{g}_{ik}}{\sum\:_{k=1}^{m}{g}_{ik}}\right)\right]\times\:\left(\text{l}\text{n}\left({P}_{i}\right)\left(\frac{{g}_{ik}}{\sum\:_{k=1}^{m}{g}_{ik}}\right)\right)}{2\text{l}\text{n}\left(m\right)}\right]\times\:100\)	Where p_i is the percentage of area occupied by type i patches, g_ik is the number of adjacent type i patches and type k patches, and m is the total number of patches types in the landscape. CONTAG represents the degree of aggregation of slope shape patches in the landscape.
Interspersion and Juxtaposition Index (IJI)	\(\:IJI=\left[1+\frac{-\sum\:_{k=1}^{m}\left[\left(\frac{{e}_{ik}}{\sum\:_{k=1}^{m}{e}_{ik}}\right)\text{l}\text{n}\left(\frac{{e}_{ik}}{\sum\:_{k=1}^{m}{e}_{ik}}\right)\right]}{\text{l}\text{n}\left(m-1\right)}\right]\times\:100\)	m refers to the total number of patch types in the landscape, and e_ik is the side length of patch i adjacent to the patch of patch type k. IJI represents the degree of aggregation and separation of slope shape patches in landscape. When the corresponding plaque type was adjacent to only one other plaque type, the IJI was close to 0. If this plaque type is adjacent to all other plaques, IJI = 100.
Shannon's diversity index (SHDI)	\(\:\text{S}\text{H}\text{D}\text{I}=-\sum\:_{\text{i}=1}^{\text{m}}\left[{\text{P}}_{\text{i}}\text{l}\text{n}\left({\text{P}}_{\text{i}}\right)\right]\)	m refers to the total number of patch types in the landscape, and P_i refers to the area ratio of patch type i to the entire landscape. SHDI represents the type richness of slope shape patches in landscape.
Shannon's evenness index (SHEI)	\(\:SHEI=\frac{-\sum\:_{i=1}^{m}\left[{P}_{i}\text{l}\text{n}\left({P}_{i}\right)\right]}{\text{l}\text{n}m}\)	m refers to the total number of patch types in the landscape, and P_i refers to the area ratio of patch type i to the entire landscape. SHEI represents the uniformity of slope shape patches in landscape area distribution.
Landscape shape index (LSI)	\(\:LSI=\frac{0.25E}{\sqrt{A}}\)	E is the edge length of the patch, and A is the area of the patch. LSI represents the complexity of slope shape patches in landscape.
Patch density (PD)	\(\:PD=\frac{NP}{\text{A}}\)	NP refers to the total number of patches in the landscape, and A is the total area of the landscape or patch. PD indicates the fragmentation degree of a certain slope shape patch in the landscape