Quantitative Relations between Morphostructural Similarity Degree and Map Scale Change in Contour Clusters in Multi-Scale Map Space

: This paper aims to propose a new approach to calculate the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters for automatic contour generalization. Terrain lines were extracted by pre-processing of unclosed contour lines, and an indirect quantitative expression method of morphostructural similarity relation was proposed based on terrain line hierarchical trees. Thirteen groups of multi-scale contour clusters with different drainage areas of loess geomorphy were employed to explore the changing regularity of morphostructural similarity indices with map scale. Finally, the quantitative relations between mor-phostructural similarity degree and map scale change were calculated using 52 groups of points. The results show that power function is the best ﬁt among the candidate functions, and the quantitative relations between the morphostructural similarity degree and map scale change can be expressed using the same power function, which facilitates the automation of contour generalization.


Introduction
The quantitative expression of spatial relations has been studied by researchers in the cartography [1,2] and geographic information science [3,4] communities for years.Spatial similarity relation [5], including graphic similarity [6], topological similarity [7], morphostructural similarity, and semantic similarity [8], is an essential component of spatial relations.It has been extensively used in human spatial cognition [9,10], pattern recognition [11,12], and spatial data matching [6,[12][13][14].Spatial similarity relation is particularly important in automating map generalization [5,15], which is the process of reducing the level of detail from larger scale maps (e.g., 1:5000) to smaller scale maps (e.g., 1:25,000) [16,17].Contour is an effective means for portraying the three-dimensional topography of the real world on two-dimensional surfaces [18].Its automatic generalization is crucial for downsizing small-scale topographic maps, topographic analysis based on terrain, and construction of multi-scale vector topographic map database.Contour generalization is essentially a kind of spatial similarity transformation between multi-scale contour clusters [5].Cartographers [15,19] contend that the descriptions of spatial relation and geomorphic morphological structure are the key and core of contour generalization.Spatial relation between contours primarily pertains to topological relation [15,20].Existing research [5,7] has manifested that change in topological relations and map scale has a quantitative relation in map generalization.However, the quantitative relation between morphostructural similarity degree and map scale change in multi-scale contours has not been explored, hindering the automation of contour generalization, including automation of generalization algorithms [21,22], automatic control of generalization process [23,24], Research in this field has reached a mature stage.(1) However, most existing algorithms rely on the Töpfer rule, which is based on cartographic experience to determine the selection quantity.(2) The reliability of the generalization results is heavily dependent on the individual experience of the cartographer.(3) Previous research has mainly focused on DEM and lacks exploration based on vector contours.The semi-automatic generalization algorithm falls short in terms of automation, intelligence, and generalization efficiency of the software.
Relevant researches are rare.
(1) Additionally, previous research has primarily concentrated on the multi-source spatial similarity relations of contours generated from raster datasets, such as ASTER GDEM and SRTM.
(2) Only traditional geometric indices of contour line, such as length, sinuosity, and longest common subsequence, etc., were considered, whereas the "clustering nature" and "bending nesting relations" between contour lines were ignored.

Experimental Datasets
Loess geomorphy is the predominant geomorphological type in the Loess Plateau, largely formed through aeolian accumulation and water erosion processes, resulting in unique landforms [33], such as loess tablelands, hills, and ridges, as well as welldeveloped gullies.It is approximately 4~6 km/km 2 , as estimated from 1:50,000 topographic maps [33,34].Therefore, we selected 13 groups of multi-scale contour clusters of loess geomorphy for this study.Each contour cluster covers different drainage areas on the Earth's surface (Table 2).The source map scale of each contour cluster is 1:10,000, and the corresponding generalized contours have target scales of 1:50,000, 1:100,000, and 1:250,000, with contour intervals at 5 m, 20 m, 40 m, and 100 m, respectively.Table 2. Thirteen groups of multi-scale contour clusters used in the study.(A refers to drainage area, N u refers to the number of unclosed contour lines, and N c refers to the number of closed contour lines, respectively).To test and validate the accuracy of terrain line extraction results, three groups of multi-scale contour clusters were randomly selected from the 13 groups.The datasets for these groups were manually generalized by cartographers at multiple scales (1:10,000, 1:50,000, 1:100,000, and 1:250,000) and are available in vector format from the National Geomatics Center of Gansu (NGCG), China.

Pre-Processing of the Dataset
Contours in reality are continuous and closed.However, a topographic map provides only a partial representation of the real world [26], and contour lines are truncated into unclosed contours at the map's border due to being stored according to sheet [27,28].Additionally, unclosed contour lines can cause uncertainty in recognizing curved clusters [27], which may complicate terrain line extraction.Therefore, it is necessary to pre-process these unclosed contours to close them [19,27].
Considering positive topography as an example, Figure 1 illustrates the unclosed contour line closure process, which involves three steps: (i) Constructing an adjacency matrix (R N × N = {R (i,j) }) based on Voronoi polygons of contours.The adjacency relation of contours can be defined by the corresponding Voronoi polygon of contours [35].In addition, there are three types of adjacency relation for any contour line L i : Neighbors contained by L i , neighbors containing Li, and neighbors parallel with L i .In order to avoid repetition, only the latter two are considered; (ii) matching unclosed contours with the elevation field (ID_1) and the adjacency matrix (R (i,j) ), including matching trunk contours and subtree contours; (iii) dissolving unclosed contours according to the matching field (ID_1) into closed contours, and reconstructing the binary adjacency triangular matrix (R M × M (M = max{ID_1} + 1)) of closed contours.

Pre-Processing of the Dataset
Contours in reality are continuous and closed.However, a topographic map provides only a partial representation of the real world [26], and contour lines are truncated into unclosed contours at the map's border due to being stored according to sheet [27,28].Additionally, unclosed contour lines can cause uncertainty in recognizing curved clusters [27], which may complicate terrain line extraction.Therefore, it is necessary to pre-process these unclosed contours to close them [19,27].
Considering positive topography as an example, Figure 1 illustrates the unclosed contour line closure process, which involves three steps: (i) Constructing an adjacency matrix (RN × N = {R(i,j)}) based on Voronoi polygons of contours.The adjacency relation of contours can be defined by the corresponding Voronoi polygon of contours [35].In addition, there are three types of adjacency relation for any contour line Li: Neighbors contained by Li, neighbors containing Li, and neighbors parallel with Li.In order to avoid repetition, only the latter two are considered; (ii) matching unclosed contours with the elevation field (ID_1) and the adjacency matrix (R(i,j)), including matching trunk contours and subtree contours; (iii) dissolving unclosed contours according to the matching field (ID_1) into closed contours, and reconstructing the binary adjacency triangular matrix (RM × M(M = max{ID_1} + 1)) of closed contours.

Methodology
Terrain lines not only can effectively express the "clustering nature" and "bending nesting relations" of the contour cluster, but also form a tree structure in itself [27,36] as the "representative" or "substitute" of geomorphologic morphology, since they are easily evaluated [1,20,23].This paper aims to propose an indirect quantitative method of expressing morphostructural similarity between multi-scale contour clusters, based on terrain line extraction results and its corresponding hierarchical tree.Therefore, the extraction of terrain lines and the construction of its hierarchical tree are essential prerequisites for this paper.Furthermore, to explore the change in trends of morphostructural similarity degree with map scale, we present an indirect quantitative expression method of morphostructural similarity relations, which is based on the constructed terrain line hierarchical tree.Figure 2 illustrates the research ideas and technical route.

Methodology
Terrain lines not only can effectively express the "clustering nature" and "bending nesting relations" of the contour cluster, but also form a tree structure in itself [27,36] as the "representative" or "substitute" of geomorphologic morphology, since they are easily evaluated [1,20,23].This paper aims to propose an indirect quantitative method of expressing morphostructural similarity between multi-scale contour clusters, based on terrain line extraction results and its corresponding hierarchical tree.Therefore, the extraction of terrain lines and the construction of its hierarchical tree are essential prerequisites for this paper.Furthermore, to explore the change in trends of morphostructural similarity degree with map scale, we present an indirect quantitative expression method of morphostructural similarity relations, which is based on the constructed terrain line hierarchical tree.Figure 2 illustrates the research ideas and technical route.

Extraction of Terrain Line
Compared with the respective extraction results of constrained Delaunay triangulation (D-TIN) [37] or curvature recognition method [29], a combination of both can avoid their individual deficiency and improve the recognition results of terrain feature points through secondary detection.

Recognition of Contour Curve Cluster
Contour represents surface terrain through closed curves [3,16].Therefore, contour curve cluster recognition is a prerequisite to detect terrain feature points, which involves judging the convexity and concavity of node groups on closed curves.Compared with the D-TIN global recognition method [38] and local feature point segmentation method [39], the vector cross product can merge pseudo bent nodes caused by abnormal jitter through determining the convexity and concavity of adjacent nodes.This allows for the effective recognition of concave and convex segments of contours.Considering positive topography as an example, the nodes on the contour line is assumed to move counterclockwise, concave and convex points correspond to valley line and ridge line points, respectively, while concave and convex segment clusters correspond to catchment area and watershed area, respectively.Its recognition principle is taking three sequential adjacent points {       Terrain lines express the "clustering" relation of contour cluster [17], as they are the structuralization of contours.When the nodes of the valley segment or ridge segment are

Extraction of Terrain Line
Compared with the respective extraction results of constrained Delaunay triangulation (D-TIN) [37] or curvature recognition method [29], a combination of both can avoid their individual deficiency and improve the recognition results of terrain feature points through secondary detection.

Recognition of Contour Curve Cluster
Contour represents surface terrain through closed curves [3,16].Therefore, contour curve cluster recognition is a prerequisite to detect terrain feature points, which involves judging the convexity and concavity of node groups on closed curves.Compared with the D-TIN global recognition method [38] and local feature point segmentation method [39], the vector cross product can merge pseudo bent nodes caused by abnormal jitter through determining the convexity and concavity of adjacent nodes.This allows for the effective recognition of concave and convex segments of contours.Considering positive topography as an example, the nodes on the contour line is assumed to move counterclockwise, concave and convex points correspond to valley line and ridge line points, respectively, while concave and convex segment clusters correspond to catchment area and watershed area, respectively.Its recognition principle is taking three sequential adjacent points {P i−1 (x i−1 , y i−1 ), P i (x i , y i ), P i+1 (x i+1 , y i+1 )} of the contour line and calculating the cross produce of −−→ P i−1 P i and −−→ P i P i+1 .If three adjacent points are collinear, then the curvature of the intermediate node P i is 0, and if −−→ P i−1 P i × −−→ P i P i+1 > 0, then the corresponding middle point P i is the convex point; otherwise, P i is the concave point [20].Therefore, continuous concave and convex nodes form concave and convex segment clusters of the contours, which correspond to catchment and watershed area, respectively.

Detection of Terrain Feature Points
Terrain lines express the "clustering" relation of contour cluster [17], as they are the structuralization of contours.When the nodes of the valley segment or ridge segment are dense, and each bend segment is represented by multiple ridge points or valley points, the terrain feature points corresponding to each concave and convex segment curve can be preliminarily screened by combining extreme points of curvature.Due to the fact that concave and convex points are local feature points calculated using the coordinates of sequential adjacent three points on the contour line, abnormal jitter bending points of contours can be removed by determining the concavity and convexity of adjacent three points.Assuming that the coordinates of three non-collinear adjacent points {P i−1 , P i , P i+1 } on the convex segment contour line are known, and according to the derivation curvature calculation principle, the curvature of the intermediate node P i can be approximated to the reciprocal of the circumscribed circle radius of adjacent three points [20].If the extreme points of contours in a concave or convex segment are not unique, and there are two adjacent concave or convex points with equal extreme values, such as points A and B (red dotted lines) in Figure 2d, then the midpoint of the two points is used as the feature point.

Tracking and Connection of Terrain Line
The tracking and connection of terrain lines involve determining the multi-resolution wedge tracking region, performing a secondary inspection of terrain feature points, and finally, tracking and connection of terrain lines.
Determination of multi-resolution wedge tracking region.Curved clusters corresponding to the secondary valley lines (ridge lines) are constantly simplified during the contour generalization process, and the adjacent ridge (valley) is constantly merged.This results in significant differences in the level of detail on the terrain skeleton lines at different map scales.For contour curve cluster, if their concavity has been marked, the water flows downstream along the descending direction of declivity, and thus forms a wedge-shaped tracking area of the valley line.Otherwise, a wedge-shaped tracking region of the ridge line is formed [21].
Secondary inspection of terrain feature points and tracking and connection of terrain lines.Connection models of terrain line are different corresponding to the different types of constrained D-TIN of contour cluster.Considering Figure 3a as an example, a class I triangle must correspond to one or multiple bends (1 ≤ N ≤ 3) of contour line and at least one class III triangle, and thus the terrain feature point must be a vertex of III class triangle (Figure 3b).Therefore, tracking and connection of models of terrain lines are as follows [21,26]: the terrain feature points corresponding to each concave and convex segment curve can be preliminarily screened by combining extreme points of curvature.Due to the fact that concave and convex points are local feature points calculated using the coordinates of sequential adjacent three points on the contour line, abnormal jitter bending points of contours can be removed by determining the concavity and convexity of adjacent three points.Assuming that the coordinates of three non-collinear adjacent points on the convex segment contour line are known, and according to the derivation curvature calculation principle, the curvature of the intermediate node Pi can be approximated to the reciprocal of the circumscribed circle radius of adjacent three points [20].If the extreme points of contours in a concave or convex segment are not unique, and there are two adjacent concave or convex points with equal extreme values, such as points A and B (red dotted lines) in Figure 2d, then the midpoint of the two points is used as the feature point.

Tracking and Connection of Terrain Line
The tracking and connection of terrain lines involve determining the multi-resolution wedge tracking region, performing a secondary inspection of terrain feature points, and finally, tracking and connection of terrain lines.
Determination of multi-resolution wedge tracking region.Curved clusters corresponding to the secondary valley lines (ridge lines) are constantly simplified during the contour generalization process, and the adjacent ridge (valley) is constantly merged.This results in significant differences in the level of detail on the terrain skeleton lines at different map scales.For contour curve cluster, if their concavity has been marked, the water flows downstream along the descending direction of declivity, and thus forms a wedge-shaped tracking area of the valley line.Otherwise, a wedge-shaped tracking region of the ridge line is formed [21].
Secondary inspection of terrain feature points and tracking and connection of terrain lines.Connection models of terrain line are different corresponding to the different types of constrained D-TIN of contour cluster.Considering Figure 3a as an example, a class Ⅰ triangle must correspond to one or multiple bends (1 ≤ N ≤ 3) of contour line and at least one class Ⅲ triangle, and thus the terrain feature point must be a vertex of Ⅲ class triangle (Figure 3b).Therefore, tracking and connection of models of terrain lines are as follows [21,26]: Rule 1: If a class Ⅰ triangle corresponds to two or three bends, the terrain lines should be orthogonal to the ridge or valley line [40] (Figure 3a).The tracking and connection models of terrain lines are shown in Figure 3a,b.
Rule 2: According to the principle that the motion or slope direction of a particle located on the terrain surface is the fastest in the direction of elevation decline [20], when a contour line passes through a ridge or a valley, the connection line of the midpoint of a class Ⅱ triangle is the dividing line of the catchment or watershed area [40] (Figure 3c).Rule 1: If a class I triangle corresponds to two or three bends, the terrain lines should be orthogonal to the ridge or valley line [40] (Figure 3a).The tracking and connection models of terrain lines are shown in Figure 3a,b.
Rule 2: According to the principle that the motion or slope direction of a particle located on the terrain surface is the fastest in the direction of elevation decline [20], when a contour line passes through a ridge or a valley, the connection line of the midpoint of a class II triangle is the dividing line of the catchment or watershed area [40] (Figure 3c).
Rule 3: If there are only class II triangles between two adjacent contour lines, then the contour lines are dense.The tracking and connection models of terrain lines are shown in Figure 3d.If the extreme values of adjacent concave or convex points are equal, for example, A and B in Figure 3d, then the feature point is the midpoint of A and B.

Construction of Hierarchical Tree of Terrain Lines
Terrain lines serve as a "representative" or "substitute" of geomorphic morphology, establishing a "clustering" relationship between contours [27].For example, a valley line represents a group of contour curve clusters that correspond to the valley.Therefore, the evaluation and treatment of geomorphic structures can be transformed into the processing of their "substitute".Additionally, terrain lines are tree structures.As a result, to evaluate and process terrain lines as a whole, it is necessary to organize them organically.The primary way to achieve this is through the construction of a corresponding hierarchical tree.
Considering positive topography as an example, the principle of contour generalization involves eliminating secondary valleys and expanding corresponding ridges [27].The elimination of valleys indicates the merger of adjacent ridges (black bold dotted line in Figure 4a).Therefore, the processing of valley lines entails the modification of corresponding ridge lines.Therefore, geomorphological generalization can be viewed as a generalization of valley information [27].According to the principles of hydrology, the hierarchical tree of terrain lines reflects the inheritance relations between trunk and branch valley lines (Figure 4b).The more branches a valley line has, the higher the hierarchical encoding of its trunk valley line [41].Contour curve cluster corresponding to the more highly encoded valley lines is more likely to be retained during the process of contour generalization; otherwise, they are simplified to emphasize the main geomorphological morphology [28].However, the hierarchical encoding of terrain lines reflects their relative levels (Figure 4b).

Construction of Hierarchical Tree of Terrain Lines
Terrain lines serve as a "representative" or "substitute" of geomorphic morphology, establishing a "clustering" relationship between contours [27].For example, a valley line represents a group of contour curve clusters that correspond to the valley.Therefore, the evaluation and treatment of geomorphic structures can be transformed into the processing of their "substitute".Additionally, terrain lines are tree structures.As a result, to evaluate and process terrain lines as a whole, it is necessary to organize them organically.The primary way to achieve this is through the construction of a corresponding hierarchical tree.
Considering positive topography as an example, the principle of contour generalization involves eliminating secondary valleys and expanding corresponding ridges [27].The elimination of valleys indicates the merger of adjacent ridges (black bold dotted line in Figure 4a).Therefore, the processing of valley lines entails the modification of corresponding ridge lines.Therefore, geomorphological generalization can be viewed as a generalization of valley information [27].According to the principles of hydrology, the hierarchical tree of terrain lines reflects the inheritance relations between trunk and branch valley lines (Figure 4b).The more branches a valley line has, the higher the hierarchical encoding of its trunk valley line [41].Contour curve cluster corresponding to the more highly encoded valley lines is more likely to be retained during the process of contour generalization; otherwise, they are simplified to emphasize the main geomorphological morphology [28].However, the hierarchical encoding of terrain lines reflects their relative levels (Figure 4b).Compared with Horton, Strahler, and Shreve [42][43][44], the Branch rule [45] is the most common encoding method of tree structure.This is due to the fact that it reflects the number of incoming tributaries along the entire river from its source to its mouth, and the difference in the number of tributaries per unit distance [30,45].Therefore, this study uses the Branch rule to build the hierarchical tree of terrain line.Considering loess geomorphy as an example, Figure 5 shows the Branch codes of valley line hierarchical trees corresponding to multi-scale contour clusters.Compared with Horton, Strahler, and Shreve [42][43][44], the Branch rule [45] is the most common encoding method of tree structure.This is due to the fact that it reflects the number of incoming tributaries along the entire river from its source to its mouth, and the difference in the number of tributaries per unit distance [30,45].Therefore, this study uses the Branch rule to build the hierarchical tree of terrain line.Considering loess geomorphy as an example, Figure 5 shows the Branch codes of valley line hierarchical trees corresponding to multi-scale contour clusters.

Quantitative Expression Method of Multi-Scale Contour Cluster Morphostructural Similarity
Hydrological geomorphic characteristic parameters include topological and geometrical characteristic parameters [46,47].The former focuses on stream bifurcation and stream order, while the latter includes stream length, stream length law, chain length, inlet angle, river density, and stream gradient, etc. Stream length law and chain length essentially describe stream length, and stream gradient describes three-dimensional characteristics, which can be excluded.Stream inlet angles are generally unchanged after generalization [30].Therefore, based on the parameters mentioned above [46,47] and previous research on generalizing drainage networks [30,45], stream bifurcation, river density, stream length, and stream order have been selected.Therefore, the degree of morphostructural similarity between multi-scale contour clusters can be indirectly measured from the following four aspects.

Topological Similarity
The topological relation of a tree-like network mainly refers to the descendent relation [30], with stream bifurcation describing the descendant relations between the main stream and its tributaries in hydrology.Therefore, the valley line bifurcation ratio can be used to indirectly express the topological similarity between multi-scale contour clusters quantitatively.Assuming that the contour cluster M1 at scale S1 is generalized to contour cluster M2 at scale S2, then the formula for calculating the topological similarity degree (

S
) is as follows: where 1 2 g , g refers to the level of valley line before and after generalization, respectively.

Quantitative Expression Method of Multi-Scale Contour Cluster Morphostructural Similarity
Hydrological geomorphic characteristic parameters include topological and geometrical characteristic parameters [46,47].The former focuses on stream bifurcation and stream order, while the latter includes stream length, stream length law, chain length, inlet angle, river density, and stream gradient, etc. Stream length law and chain length essentially describe stream length, and stream gradient describes three-dimensional characteristics, which can be excluded.Stream inlet angles are generally unchanged after generalization [30].Therefore, based on the parameters mentioned above [46,47] and previous research on generalizing drainage networks [30,45], stream bifurcation, river density, stream length, and stream order have been selected.Therefore, the degree of morphostructural similarity between multi-scale contour clusters can be indirectly measured from the following four aspects.

Topological Similarity
The topological relation of a tree-like network mainly refers to the descendent relation [30], with stream bifurcation describing the descendant relations between the main stream and its tributaries in hydrology.Therefore, the valley line bifurcation ratio can be used to indirectly express the topological similarity between multi-scale contour clusters quantitatively.Assuming that the contour cluster M 1 at scale S 1 is generalized to contour cluster M 2 at scale S 2 , then the formula for calculating the topological similarity degree (S Topo S 2 S 1 ) is as follows: where g 1 , g 2 refers to the level of valley line before and after generalization, respectively.N represent the level of valley line and bifurcation number between the higher level (g i +1)(g 1 = 1, 2, . . ., m; g 2 = 1, 2, . . ., n) valley line and g i level valley line of contour cluster at scale S 1 , S 2 , respectively.Meanwhile, m, n are the total numbers of valley line levels and R Considering Figure 5 as an example, the Branch codes of first-level valley lines corresponding to contour clusters at map scales of 1:10,000, 1:50,000, 1:100,000, and 1:250,000 are 173, 45, 22, and 9, respectively.This indicates that the total numbers of valley line bifurcations are 172, 44, 21, and 8, respectively.Therefore, the topological similarity (S Topo S 2 S 1 ) of valley line hierarchical trees corresponding to multi-scale contour clusters is 0.2558, 0.1221, and 0.0465, when the contour clusters are generalized from 1:10,000 to 1:50,000, 1:100,000, and 1:250,000, respectively.

Directional Similarity
In contrast to the discrete settlement groups [5], the direction of which is changed after generalization, the contours on topographic maps are spatial targets that have high positional accuracy and are often used for spatial positioning.In map generalization processing, it is generally not allowed to move contour lines on maps [2].Therefore, the directional relationships between contour clusters are not changed after map generalization, and the corresponding directional similarity degree (S Dir S 2 S 1 ) should be viewed as 1.

Distance Similarity
Gully density refers to the lengths of gullies per unit area, and the greater its value, the more the watershed cuts into the geomorphology.Therefore, gully density is one of the most important indicators to represent regional geomorphic features [33], and can be used to measure the distance similarity between valley line hierarchical trees.Due to the fact that the drainage area remains unchanged after contour generalization; therefore, the distance similarity degree (S Dis S 2 S 1 ) can be calculated as follows: where S Dis S 2 S 1 ∈ (0, 1], g 1 , g 2 represent the level of valley line; t 1 , t 2 are the total numbers of valley lines; and p, q (p = 1, 2, . . ., t 1 ; q = 1, 2, . . ., t 2 ) refers to the p-th or q-th valley line of g 1 or g 2 level before and after generalization, respectively.

Attribute Similarity
During the process of contour structural generalization, contour curve clusters corresponding to the main valley lines are retained, but those corresponding to the secondary valley lines, such as ditches, gullies, etc., are gradually "eroded" [23].Therefore, attribute similarity (S Att S 2 S 1 ) can be expressed by Formula (3): (3) where i refers to the i-th valley line; t 1 , t 2 refer to the total numbers of valley line before and after generalization, respectively; A S 2 i refers to the Branch code of the i-th valley line in the generalized contour cluster; A S 1 i refers to the Branch code of the i-th valley line in the original contour cluster, and exists in the generalized contour cluster, S Att S 2 S 1 ∈ (0, 1].For example, when the contour cluster is generalized from 1:10,000 to 1:100,000, the sum of valley line Branch codes in the generalized contour cluster is 51 (Figure 5c), and the sum of valley line Branch codes in the original contour cluster as well as that existing in the generalized contour cluster (red valley lines in Figure 5a) is 310.Therefore, the multi-scale attribute similarity degree between multi-scale contour clusters corresponding to the valley line hierarchical trees is 0.1645.

Multi-Scale Morphostructural Similarity Degree
The analysis in Section 3.2 shows that the morphostructural similarity relation between multi-scale contour clusters can be indirectly and quantitatively expressed by the corresponding multi-scale valley line hierarchical trees.Therefore, the multi-scale contour cluster morphostructural similarity degree (S S 2 S 1 ) can be calculated by Formula (4): where w 1 , w 2 , w 3 , w 4 are 0.22, 0.25, 0.31, and 0.22, respectively, and these weight parameters are obtained through a large number of spatial psychological tests conducted by our team [5].

Results Analysis of Pre-Processing
The pre-processing of unclosed contour lines, which includes topology checking, matching, and closure of unclosed contour lines, is the prerequisite for extraction terrain lines from contour cluster.Tables 3 and 4 show the elevation and code matching results of unclosed contour lines in Figure 6.multi-scale attribute similarity degree between multi-scale contour clusters ing to the valley line hierarchical trees is 0.1645.

Multi-Scale Morphostructural Similarity Degree
The analysis in Section 3.2 shows that the morphostructural similarity tween multi-scale contour clusters can be indirectly and quantitatively expr corresponding multi-scale valley line hierarchical trees.Therefore, the multicluster morphostructural similarity degree ( , , , w w w w are 0.22, 0.25, 0.31, and 0.22, respectively, and these w eters are obtained through a large number of spatial psychological tests cond team [5].

Results Analysis of Pre-Processing
The pre-processing of unclosed contour lines, which includes topolo matching, and closure of unclosed contour lines, is the prerequisite for extra lines from contour cluster.Tables 3 and 4 show the elevation and code mat of unclosed contour lines in Figure 6.Tables 3 and 4 reveal that the 44 unclosed contour lines in Figure 6 are formed 22 closed contour lines after pre-processing, which include 19 trunk  Tables 3 and 4 reveal that the 44 unclosed contour lines in Figure 6 are matched and formed 22 closed contour lines after pre-processing, which include 19 trunk contour lines (Table 3) and three branch contour lines (Table 4).The closure results of 13 groups of unclosed multi-scale contours are shown in Table 2.

Extraction Results Accuracy Evaluation of Terrain Lines
Typically, 10-20% of experimental datasets are used to evaluate the accuracy of extraction results.Therefore, to evaluate the geometry and position accuracy of three groups of valley line extraction results, three sets of ranked samples of multi-scale contour clusters at map scale 1:10,000, 1:50,000 are selected [48].Table 5 shows the geometry and position accuracy errors of the valley line extraction results of the three groups of samples, compared with the standard river dataset of the corresponding map scale.Table 5 indicates that compared with the standard river dataset of 1:10,000 and 1:50,000, the maximum errors ratio is only {0.0314,0.1363},{0.0805,0.2007},respectively.Overall, the local position accuracy (PE) of valley line extraction results corresponding to the 1:10,000 contour clusters is higher than that of 1:50,000, but the overall position accuracy (SMD) of the latter is higher than the former, with a maximum overall position accuracy error of 0.8945.
Considering sample 1 as an example, Figure 7 shows the overlap of the valley line extraction results with the standard river data of the corresponding map scale.Figure 7 demonstrates that the position accuracy SMD, PE [48,49], and geometric accuracy LR, SD [48] of the valley line extraction results are {0.6532,0.0024, 1.0150, 0.9852}, {0.4028, 0.0102, 0.9727, 1.0279}, respectively (Table 5), compared with standard river data of 1:10,000, 1:50,000.The geometric accuracy errors ΔLR, ΔSD that tend toward zero are only {0.0150, 0.0148}, {0.0276, 0.0279}, respectively, and the geometric accuracy is higher.The local position accuracy of the valley line extraction results corresponding to 1:10,000 Figure 7 demonstrates that the position accuracy SMD, PE [48,49], and geometric accuracy LR, SD [48] of the valley line extraction results are {0.6532,0.0024, 1.0150, 0.9852}, {0.4028, 0.0102, 0.9727, 1.0279}, respectively (Table 5), compared with standard river data of 1:10,000, 1:50,000.The geometric accuracy errors ∆LR, ∆SD that tend toward zero are only {0.0150, 0.0148}, {0.0276, 0.0279}, respectively, and the geometric accuracy is higher.The local position accuracy of the valley line extraction results corresponding to 1:10,000 contour cluster is higher than that of 1:50,000 (Figure 7a), but the overall position accuracy of the latter is higher than the former (Figure 7b), with a smaller position error margin.In conclusion, the accuracy of the valley line extraction results can be used to express the morphostructural similarity between multi-scale contour clusters quantitatively.Figure 7 demonstrates that the position accuracy SMD, PE [48,49], and geometric accuracy LR, SD [48] of the valley line extraction results are {0.6532,0.0024, 1.0150, 0.9852}, {0.4028, 0.0102, 0.9727, 1.0279}, respectively (Table 5), compared with standard river data of 1:10,000, 1:50,000.The geometric accuracy errors ΔLR, ΔSD that tend toward zero are only {0.0150, 0.0148}, {0.0276, 0.0279}, respectively, and the geometric accuracy is higher.The local position accuracy of the valley line extraction results corresponding to 1:10,000 contour cluster is higher than that of 1:50,000 (Figure 7a), but the overall position accuracy of the latter is higher than the former (Figure 7b), with a smaller position error margin.In conclusion, the accuracy of the valley line extraction results can be used to express the morphostructural similarity between multi-scale contour clusters quantitatively.

Neighbor Scale Morphostructural Indices
Based on the analysis in Section 3.3, the morphostructural similarity relation between multi-scale contour clusters can be indirectly and quantitatively evaluated through their corresponding valley line hierarchical trees.The measurement indices that relate to the quantitative expression of morphostructural similarity relations of multi-scale contour clusters mainly include the length of valley line, bifurcation number, and Branch codes of valley lines, etc. Figure 8 displays the variation trends of the neighbor scale morphostructural indices of 13 groups of contour clusters of loess geomorphy.Some insight can be gained from Figure 8.Some insight can be gained from Figure 8.
(1) In contrast to the bifurcation ratio and length ratio of neighbor scale valley lines, the average of closed contour lines number ratio is equal to the contour interval ratio (CI) of neighbor scale contour clusters.However, the number ratio of neighbor scale closed contour lines is not a constant; therefore, it fluctuates around the average of neighbor scale contour interval ratio due to differences in surface relief, steep slope, and surface fragmentation [7].For instance, as the contour cluster is generalized from 1:100,000 to 1:250,000, the neighbor scale closed contour lines number ratio fluctuates around 2.5 (Figure 8a).(2) Although the length ratio and bifurcation ratio of neighbor scale valley lines both fluctuate around their average, the fluctuation degree of the latter among different samples is greater than the former (Figure 8c).The maximum fluctuation range of the latter is [1.1, 8.0588] with a difference of 6.9588 (Figure 8b).There is no specific proportional relationship between the neighbor scale morphostructural indices and the corresponding contour interval ratio or contour interval distance ratio of neighbor scale contour clusters.Consequently, the following sections further investigate the changing trends of multi-scale morphostructural similarity indices with map scale.

Multi-Scale Morphostructural Indices
By analyzing the distribution characteristics of morphostructural indices, including average value ( − X), standard deviation (δ), and variation coefficient (C V ), etc., the variation trends of these indices can be discovered.Standard deviation reflects the dispersion of a set of statistics relative to its average, and the variation coefficient represents the relative fluctuation of statistical data [50].Table 6 shows the statistical indices of 13 groups of multi-scale morphostructural similarity indices with the same map scale change in loess geomorphy.The following three conclusions can be drawn from Table 6 and Figure 9.
(1) Multi-scale morphostructural similarity indices of contour cluster gradually decrease with the increasing map scale change, regardless of multi-scale morphostructural similarity degree or other measures, such as distance similarity degree.For example, the average of multi-scale morphostructural similarity degree decreases from 0.5271 to 0.3849 as the map scale change increases from 5 to 25 (Figure 9a).(2) Compared with the same index of different map scale changes, the standard deviation of multi-scale morphostructural similarity indices decreases with the increasing map scale change.For example, the multi-scale morphostructural similarity degree decreases from 0.0666 to 0.0321 as the map scale change increases from 5 to 25.However, the variation coefficients of other indices decrease at first and then increase, except for topological similarity degree (Table 6).This indicates that the dispersion degree between multi-scale morphostructural similarity degree and its average gradually decreases with the increase in map scale change (black line > red > gray) (Figure 9).However, the degree of relative fluctuation among different samples decreases from 0.1265 to 0.0884 at first, and then increases from 0.0884 to 0.0923 (Table 6).Therefore, when the map scale change (C) is 10, the fluctuation degree of morphostructural similarity degree is the gentlest, which is confirmed by Figure 9.
(3) Compared with different indices of the same map scale change, the standard deviation and variation coefficient of multi-scale morphostructural similarity degree are both minimum (Table 6).This indicates that the agglomeration degree of multi-scale morphostructural similarity degree is the largest, and all fluctuate around its average.For example, when the map scale change is 10, all of the multi-scale morphostructural similarity degree fluctuates around 0.3927 (Figure 9a).This conclusion further indicates that the relations between morphostructural similarity degree and map scale change between multi-scale contour clusters can be expressed quantitatively by the same function.Therefore, the following contents further formulate the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters.Based on the above analysis, the functions that conform to the trend can be listed as follows: Figure 10 shows the fitting results between the multi-scale morphostructural similarity degree and map scale change, considering 52 pairs of <C, S> fitting points of loess geomorphy as an example.It is observed in Figure 10 that although the quadratic function has the highest fitting accuracy among the candidate functions, it has two inflection points and is not always monotonic in the change range of map scale, and thus it can be excluded.Consequently, the power function (Formula ( 6)) provides the best fit for the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters of loess geomorphy, R 2 = 0.9235.
Figure 10 also indicates that the morphostructural similarity degree of multi-scale contour clusters decreases sharply when the map scale change is less than 5, and then levels off gradually with an increase in map scale change.To investigate whether the quantitative relations between morphostructural similarity degree and map scale change It is observed in Figure 10 that although the quadratic function has the highest fitting accuracy among the candidate functions, it has two inflection points and is not always monotonic in the change range of map scale, and thus it can be excluded.Consequently, the power function (Formula ( 6)) provides the best fit for the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters of loess geomorphy, R 2 = 0.9235.
Figure 10 also indicates that the morphostructural similarity degree of multi-scale contour clusters decreases sharply when the map scale change is less than 5, and then levels off gradually with an increase in map scale change.To investigate whether the quantitative relations between morphostructural similarity degree and map scale change in 13 groups of multi-scale contour clusters can be expressed quantitatively by the same power function, the following contents further explore the influence of sample size on the types and precisions of the fitting function.clusters of loess geomorphy, R 2 = 0.9235.
Figure 10 also indicates that the morphostructural similarity degree of multi-scale contour clusters decreases sharply when the map scale change is less than 5, and then levels off gradually with an increase in map scale change.To investigate whether the quantitative relations between morphostructural similarity degree and map scale change in 13 groups of multi-scale contour clusters can be expressed quantitatively by the same power function, the following contents further explore the influence of sample size on the types and precisions of the fitting function.Figure 11 also demonstrates that the fitting accuracy of the power function is always the highest among the candidate functions with the increase in sample sizes.Therefore, the power function is always the best fit for the quantitative relation between multi-scale morphostructural similarity degree and map scale change.Their fitting accuracies (R 2 ) are not less than 0.8986, and the maximum fitting accuracy is up to 0.9401.This experimental result further confirms that the power function (Formula ( 6)) is the best to express the quantitative relation between morphostructural similarity degree and map scale change in multi-scale contour clusters.
Theoretically, a complete fit is R 2 = 1, but with the increase in sample size, R 2 decreases and eventually converges to a certain value.Therefore, considering the contour cluster with different contour interval changes as an example, Figure 12 shows the influence of sample sizes on the accuracy and coefficients of fitting results of power function.Figure 11 also demonstrates that the fitting accuracy of the power function is always the highest among the candidate functions with the increase in sample sizes.Therefore, the power function is always the best fit for the quantitative relation between multi-scale morphostructural similarity degree and map scale change.Their fitting accuracies (R 2 ) are not less than 0.8986, and the maximum fitting accuracy is up to 0.9401.This experimental result further confirms that the power function (Formula 6) is the best to express the quantitative relation between morphostructural similarity degree and map scale change in multi-scale contour clusters.
Theoretically, a complete fit is R 2 = 1, but with the increase in sample size, R 2 decreases and eventually converges to a certain value.Therefore, considering the contour cluster with different contour interval changes as an example, Figure 12 shows the influence of sample sizes on the accuracy and coefficients of fitting results of power function.As shown in Figure 12, with the increase in sample sizes, the fitting precision (R 2 ) of the power function increased at first and then decreased, and eventually stabilized.The fitting accuracy (R 2 ) of the power function converges to 0.93 (Figure 12a).The fitting coefficient "a" basically remains unchanged with the increase in sample size, while the fitting As shown in Figure 12, with the increase in sample sizes, the fitting precision (R 2 ) of the power function increased at first and then decreased, and eventually stabilized.The fitting accuracy (R 2 ) of the power function converges to 0.93 (Figure 12a).The fitting coefficient "a" basically remains unchanged with the increase in sample size, while the fitting coefficient "b" decreases initially and then tends to stabilize.The fitting coefficients "a" and "b" converge to 0.95 and −0.35, respectively (Figure 12b,c).S = 0.95C −0.34 R 2 = 0.93 (7) The above analysis further indicates that the quantitative relation between morphostructural similarity degree and map scale change in 13 groups of samples of multi-scale contour clusters with different drainage areas of loess geomorphy can be expressed quantitatively by the same power function (Formula ( 7)).This conclusion demonstrates that it is reasonable and feasible for contour cluster to realize the automation of map generalization based on the multi-scale spatial similarity relations.

Discussion
Insights can be obtained from the experimental results presented above.First, the ratio of neighbor scale contour line number to the corresponding neighbor scale contour interval distance ratio is constant.The reason is that the process of contour generalization includes two parts: Thinning and simplification [27].Contour lines are thinned according to a certain contour interval in the generalization process, which is in essence a generalization of contours in quantity [28].However, the number ratio of neighbor scale closed contour lines fluctuates around the average of the neighbor scale contour interval ratio, due to variations in geomorphological characteristics, such as surface relief, steep slopes, and surface fragmentation degree.For instance, large mountainous and broken hilly topographies have distinct morphological characteristics.Similarly, this is true for contour clusters with identical map scale and geomorphological type [7].
Second, the fluctuation degree of the bifurcation ratio of valley lines diminishes with the decrease in map scale change.Contour structure generalization involves the constant deletion of secondary valley lines in the large map scale, the continuous merging of adjacent ridge, and the preservation of main valley lines with the decrease in map scale.When the map scale reduces to a certain extent, the valley lines, which highlight the primary geomorphological features in the contour generalization process, remain unchanged.Consequently, the corresponding valley line bifurcation ratio tends to be more stable with the increase in map scale change in Figure 8b.These findings align with the principle of contour structural generalization and can provide guidance for improving spatial cognition, spatial reasoning, and map design.
Third, the results of valley line extraction, which only considered curvature, reveal that numerous feature points are extracted in the U-shaped region of gentle bending, increasing the complexity of valley line tracking and being significantly affected by the threshold value [27].The geometric accuracy and position accuracy of valley line extracted through the combination of constrained Delaunay triangulation and curvature are generally higher.The primary source of error is that the theoretically extracted valley lines are the connecting lines of the midpoint of constrained Delaunay triangulation, while rivers in reality are naturally formed catchment lines.Consequently, the error of local position accuracy between valley line extraction results and rivers is more significant in areas with larger river bends, as demonstrated in Figure 7.
Fourth, the method proposed in this paper can also pertain to the contour cluster extracted from raster data.However, there are significant differences between the simulated contour cluster extracted from raster data using existing software modules, such as "Contour" module tool in ArcMap, and the vector contour cluster generated by a cartographer.Contour cluster extracted from corresponding raster data with different resolutions by existing software modules only refer to rarefaction of contour lines, and ignore the process of simplification, the merging of secondary positional hilltops, and displacement, etc.These deficiencies can lead to mistakes in the extraction of terrain lines, which will future lead to unreliable calculation results of multi-scale morphostructural similarity.Currently, our research team mainly focuses on the research of spatial similarity relations in multi-scale map spaces and its application in automatic map generalization.Our future research direction is to study the spatial similarity between raster dataset, such as DEM with different resolutions.
Finally, the power function is the best to express the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters.When the sample size increases to a certain amount, the fitting accuracy and coefficients of power function tend to be stable.Therefore, the quantitative relations between morphostructural similarity degree and map scale change in multi-scale contour clusters with different drainage areas of loess geomorphy can be expressed by the same power function.
This conclusion has significance in the automation of contour generalization.First, it demonstrates the feasibility of realizing the automatic generalization of contour cluster based on multi-scale spatial similarity relations.Second, Formula (7) can be used in automatic contour generalization, including automation of contour generalization algorithm, automatic control of generalization process, and automatic evaluation of contour generalization results.For example, in automatic control of contour structural generalization process, if a large scale contour topographic map is generalized to any small scale contour topographic map, and the map scale change (C) in contour clusters is given, the theoretical morphostructural similarity degree S T between the original contour cluster and generalized one can be calculated by Formula (7).The practical morphostructural similarity degree S i between the original contour cluster and the intermediate result generalized by the contour generalization algorithm can also be calculated using the proposed model in Section 3.3.When |S i − S T | → min , the intermediate result of contour generalization can be regarded as the target scale topographic map, and the generalization process of contour generalization algorithm should be stopped.This may facilitate the automation of map generalization software.

Conclusions
Obtaining morphostructural similarity degrees among the same contour clusters on multi-scale maps is important in automatic map generalization.However, the quantitative expression of the morphostructural similarity relationship between contour clusters in multi-scale map spaces is a precondition for exploring the trends between morphostructural similarity degree and map scale change in multi-scale contour cluster.This paper proposes an indirect and quantitative expression model of morphostructural similarity relations of multi-scale contour clusters by constructing terrain line hierarchical trees.On this basis, the different quantitative trends of morphostructural similarity degrees of multiscale contour clusters with the change in map scale were explored.It can be concluded that power function (S = aC −b (a > 0, b > 0, )) is the best fit to express the quantitative relations between morphostructural similarity degree and map scale change in multiscale contour clusters.The quantitative relations of 13 groups of multi-scale contour clusters with different drainage areas of loess geomorphy can be expressed using the same power function.This conclusion indicates that it is reasonable and feasible to realize the automation of contour generalization based on multi-scale spatial similarity relations, which may facilitate the automation of map generalization software.

Future Works
Currently, only four kinds of map scale and multi-scale vector contour clusters were considered in this study.However, experiments indicate that the morphostructural similarity of multi-scale contour clusters changes drop drastically at small map scale change (C ≤ 5).Additionally, morphostructural similarity relations of multi-scale contour cluster have not been well applied to the processing of contour generalization.Therefore, our future study will increase the sample sizes of large scale datasets of different geomorphologic morphology types, such as mountainous topography, Aeolian landform, and fluvial landform, etc.We will also apply multi-scale contour spatial similarity relations to the processing of contour generalization, which will provide a foundation for human spatial cognition, matching of spatial data, and pattern recognition, especially the automation of map generalization for contour clusters.In addition, the spatial similarity relations between multi-resolution raster DEMs are a direction worth studying.
Future research on this issue may focus on the following areas: (1) Conducting more experiments to improve the accuracy and adaptability of the proposed models and formulas.Scale changes are an inductive behavior of spatial thinking, and it is necessary to obtain universal, unified, and high-level spatial knowledge from a large number of specific, individual, and underlying data through granularity filtering, hierarchical merging, and linguistic generalization [51].Therefore, it would be more useful to perform the analysis on more map samples.Additionally, it is essential to explore the extent to which the preservation of geomorphological characteristics at smaller scales is possible, as the nature of maps is changing from topographic to general geographic.(2) Is it possible to design one model that can calculate the spatial similarity degree of individual point objects, and/or object groups, such as point clouds, parallel line clusters, intersected line networks, tree-like networks discrete polygon groups, by taking map scale change as an independent variable?Previous studies on models for spatial similarity relations in multi-scale map spaces indicated that there is a quantitative relationship between map scale change and spatial similarity degree.However, this quantitative relationship has not been determined, and some studies consider it to be a power function, while others consider it to be an exponential function.
(3) It is worth exploring approaches for automatically obtaining the parameters used in the algorithms and operators that are not parameter free with the help of the models and formulae of multi-scale spatial similarity relations.Progress in this area will lay a good foundation for full automation of map generalization.

Figure 1 .
Figure 1.Matching and closure of unclosed contour lines.

Figure 1 .
Figure 1.Matching and closure of unclosed contour lines.

Figure 2 .
Figure 2. Research ideas and technical route.

Rule 3 :
If there are only class Ⅱ triangles between two adjacent contour lines, then the contour lines are dense.The tracking and connection models of terrain lines are shown in Figure 3d.If the extreme values of adjacent concave or convex points are equal, for example, A and B in Figure 3d, then the feature point is the midpoint of A and B.

Figure 3 .
Figure 3. Tracking and connection models of terrain lines.((a) Tracking and connection model 1 of the class I triangle, (b) Tracking and connection model 2 of the class I triangle, (c) Connection model of the class III triangle, and terrain feature point must be a vertex of the class III triangle, (d) connection model of the class II triangle when the contour lines are dense, and feature point is the midpoint of A and B).

20 Figure 3 .
Figure 3. Tracking and connection models of terrain lines.((a) Tracking and connection model 1 of the class Ⅰ triangle, (b) Tracking and connection model 2 of the class Ⅰ triangle, (c) Connection model of the class Ⅲ triangle, and terrain feature point must be a vertex of the class Ⅲ triangle, (d) connection model of the class Ⅱ triangle when the contour lines are dense, and feature point is the midpoint of A and B).

Figure 4 .
Figure 4. Bifurcation structure of valley lines and its Branch hierarchical encoding.((a) Valley lines corresponding to contour cluster, (b) valley line hierarchical tree and its Branch encoding).

Figure 4 .
Figure 4. Bifurcation structure of valley lines and its Branch hierarchical encoding.((a) Valley lines corresponding to contour cluster, (b) valley line hierarchical tree and its Branch encoding).

g
level valley line of con- tour cluster at scale S1, S2, respectively.Meanwhile, m, n are the total numbers of valley line levels and represent the total numbers of valley line bifurcation before and after generalization, respectively.According to the Branch encoding principle[47], , N refers to the branch code of the first-level valley line at map scale Si,

S 1 b 2 S 1 ∈
, R S 2 b represent the total numbers of valley line bifurcation before and after generalization, respectively.According to the Branch encoding principle[47], R S i b = N − 1, N refers to the branch code of the first-level valley line at map scale S i , S Topo S (0, 1].

4. 3 .
Quantitative Trends of Multi-Scale Morphostructural Indices 4.3.1.Neighbor Scale Morphostructural Indices Based on the analysis in Section 3.3, the morphostructural similarity relation between multi-scale contour clusters can be indirectly and quantitatively evaluated through their corresponding valley line hierarchical trees.The measurement indices that relate to the quantitative expression of morphostructural similarity relations of multi-scale contour clusters mainly include the length of valley line, bifurcation number, and Branch codes of valley lines, etc. Figure 8 displays the variation trends of the neighbor scale morphostructural indices of 13 groups of contour clusters of loess geomorphy.

Figure 8 .
Figure 8. Variation trends of neighbor scale morphostructural indices of different groups of samples.( y , CI refer to the average value of the corresponding indicator and contour interval ratio, respectively.(a), (b), and (c) refer to number ratio variation trends of neighbor scale closed contour lines, variation trends of neighbor scale valley line bifurcation ratio, and variation trends of neighbor scale valley line length ratio of different groups of samples, respectively).

Figure 8 .
Figure 8. Variation trends of neighbor scale morphostructural indices of different groups of samples.(y, CI refer to the average value of the corresponding indicator and contour interval ratio, respectively.(a-c) refer to number ratio variation trends of neighbor scale closed contour lines, variation trends of neighbor scale valley line bifurcation ratio, and variation trends of neighbor scale valley line length ratio of different groups of samples, respectively).

Figure 9
Figure 9 shows the variation trends of morphostructural similarity indices of multiscale contour clusters at the same or different map scale change(s) in loess geomorphy.

20 Figure 10 .
Figure 10.Fitting results between multi-scale morphostructural similarity degree and map scale change.

Figure 10 .
Figure 10.Fitting results between multi-scale morphostructural similarity degree and map scale change.

4. 4 . 2 .
Influence of Sample Size on the Types and Precisions of Fitting FunctionIt is still unknown whether the types of optimal fitting function change constantly with the increase in sample sizes.Therefore, this study considers multi-scale contour clusters with different drainage areas of loess geomorphy as research objects.Figure11shows the influence of sample size on the types of quantitative function relations between multi-scale morphostructural similarity and map scale change.

4. 4 . 2 .
Influence of Sample Size on the Types and Precisions of Fitting FunctionIt is still unknown whether the types of optimal fitting function change constantly with the increase in sample sizes.Therefore, this study considers multi-scale contour clusters with different drainage areas of loess geomorphy as research objects.Figure11shows the influence of sample size on the types of quantitative function relations between multiscale morphostructural similarity and map scale change.

Figure 11 .
Figure 11.Influence of sample size on the types and accuracies of function fitting results.

20 Figure 11 .
Figure 11.Influence of sample size on the types and accuracies of function fitting results.

Figure 12 .
Figure 12.Variation trends of fitting accuracy and coefficient of power function with the increase in sample sizes.((a) Fitting accuracy R 2 , (b) fitting coefficient a, (c) fitting coefficient b).

Figure 12 .
Figure 12.Variation trends of fitting accuracy and coefficient of power function with the increase in sample sizes.((a) Fitting accuracy R 2 , (b) fitting coefficient a, (c) fitting coefficient b).

Table 3 .
Elevation and code matching of unclosed trunk contour lines.("Elev" and "ID_1" refer to elevation and matching field, respectively).

Table 6 .
Statistical indices of multi-scale morphostructural similarity indices of contour cluster.
Author Contributions: Conceptualization, Haowen Yan and Rong Wang; methodology, Rong Wang; validation, Rong Wang and Xiaomin Lu; data curation, Xiaomin Lu; writing-original draft preparation, Rong Wang; writing-review and editing, Haowen Yan; funding acquisition, Haowen Yan and Rong Wang.All authors have read and agreed to the published version of the manuscript.This research was funded by National Science Foundation Committee of China grant number 41930101 and The APC was funded by Science and Technology Project of Gansu Province, grant number 22JR11RE190. Funding: