3.1.1. Concept of Terrain Openness

3.1.2. Basic Process for Extracting Karst Depressions Using the Topographic Openness Index Method

(1) 

Determining optimal analysis radius: The mean change point method can be applied to determine the optimal analysis radius. This method determines whether a mutation point occurs using Eqs. (3) and (4) [31]. Specifically, we set the ordered series ${\mathit{X}}_{\mathit{i}},\mathit{i}$ = 1, 2, 3, …, $\mathit{N},\mathbf{where} \mathit{N}$ represents the number of samples and ${\mathit{X}}_{\mathit{i}}$ represents the boundary. The data are divided into two segments, and the sum of the squared deviations of each sample segment (S_i) and the sum of the squared deviations of the whole sample (S) are calculated. The existence of a change point will increase the difference between the sum of the squared deviations ${\mathit{S}}_{\mathit{i}}$ of the sample and the sum of the squared deviations $\mathit{S}$ of the segmented sample. The point corresponding to the point when the difference between ${\mathit{S}}_{\mathit{i}}$ and $\mathit{S}$ reaches the maximum is the change point. The analysis radius corresponding to this change point is the optimal analysis radius.

$${\mathit{S}}_{\mathit{i}}=\sum _{{\mathit{t}}_{\mathit{i}-1}}^{\mathit{i}-1}{\left({\mathit{X}}_{{\mathit{t}}_1}-{\overline{\mathit{X}}}_{{\mathit{i}}_1}\right)}^2+\sum _{{\mathit{t}}_{2=\mathit{i}}}^{\mathit{N}}{\left({\mathit{X}}_{{\mathit{t}}_2}-{\overline{\mathit{X}}}_{{\mathit{i}}_2}\right)}^2$$

(3)

$$\mathit{S}=\sum _{\mathit{i}=1}^{\mathit{N}}{\left({\mathit{X}}_{\mathit{i}}-\overline{\mathit{X}}\right)}^2$$

(4)

In Eqs. (3) and (4), $\mathit{i}$ is the sequence number, ${\mathit{t}}_{\mathit{i}}$ = 1, 2, 3, …, $\mathit{i}-1$; ${\mathit{t}}_2$ =1, i + 1, i + 2, …, $\mathit{N}$；$\overline{\mathit{X}}$ is the arithmetic mean of the overall samples; $\mathit{N}$ is the number of total samples; $\mathit{S}$ is the total sum of the squares of departures; and ${\mathit{S}}_{\mathit{i}}$ is the difference of the sum of the squares of the departures of the samples of the two segments.

(2) 

Detecting saddle points: Saddle points are important terrain control points that can be obtained by extracting ridge and valley lines and finding their intersections [32].

(3) 

Obtaining the opening difference graph: Based on the determined optimal analysis radius, positive and negative openness maps of the study area are obtained. The obtained positive and negative openness images are differenced to obtain openness difference maps reflecting the more dominant up-convex or down-convex of each image element.

(4) 

Optimal segmentation threshold determination: Based on the saddle points obtained in Step (2), the saddle point opening difference values in the opening difference graph in Step (3) are extracted, and a statistical map presenting an approximate normal distribution is obtained after statistics are performed. Combining the principle of normal distribution σ and the meaning of karst depression, the optimal segmentation thresholds of the openness difference map are determined.

(5) 

Closed depression detection: In the open difference biplot obtained based on the optimal segmentation thresholds, there are a few broken and discontinuous regions, which can be eliminated by applying opening and closing operations in the mathematical morphology method [33](Figure 3).

3.3.1. Why Is the Terrain Openness Index Method Improved Based on Slope Mutation Points?

3.3.2. Technical Routes for ROBSMP Extraction of Karst Depressions

4.2.1. Optimal Segmentation Thresholds Determined Using the Topographic Openness Index Method

4.2.2. Optimal Segmentation Thresholds Determined Using ROBSMPs