# Exploration of Kastification and Characterization Based on Borehole Image
## 3 Identification of the Karst Pore Structure  
### Eccentric Image Acquisition Model  
Since the actual geological borehole is usually in a non-absolute vertical state, the actual state of the hole panoramic probe without the added probe centering device is mostly an eccentric acquisition image state, resulting in the collected borehole image being prone to uneven illumination phenomena. The more the probe favors the borehole wall, the stronger the light received at the wall and the clearer the wall image becomes. The farther away from the probe, the less light the wall receives, and the more blurred the image. The schematic representation of the eccentric image is demonstrated in Figure 2. Figure 2a presents a schematic diagram of the relative position of the probe with respect to the borehole. Figure 2b illustrates the expanded image of the corresponding borehole wall. The brighter the color, the stronger the light intensity, the darker the color, and the weaker the light intensity. Position A in Figure 2 represents the closest position to the probe and receives the strongest light intensity, which more visibly presents the pore characteristics on the pore wall at this position. Position C in Figure 2 is the furthest point from the probe and receives the weakest light intensity, making it difficult to present the pore characteristics on the pore wall at this position. In order to adapt to the actual test state, uneven hole wall illumination caused by probe eccentricity is avoided. This paper aims to establish an image acquisition model based on the eccentric image acquisition model. As illustrated in Figure 3a, the central origin of the probe is coincident with point \(O\), the origin of the rectangular coordinate system. In such a system, the horizontal and vertical directions are set as the \(x\)-axis and the \(y\)-axis, respectively. Figure 2: Schematic representation of the light intensity at eccentric position: (**a**) schematic diagram of position relationship and (**b**) schematic diagram of expanding the light intensity. Let us set the radius of the probe, radius of the hole, and coordinates of the center point \(O^{\prime}\) equal to \(r\), \(R\), and (\(x_{0}\), \(y_{0}\)), respectively. Then, the circular equation of the circle outside the probe and the wall of the borehole hole could be, respectively provided by:  
\[\left\{ \begin{array}{l}x^{2}+y^{2}=r^{2}\\ (x-x_{0})^{2}+(y-y_{0})^{2}=R^{2}\end{array} \right.. \tag{1}\]  
Suppose a ray with an angle of \(\alpha\) along the central point \(O\) intersects the probe circle at point \(P_{1}\) and the hole wall circle at point \(P_{2}\). Using Equation (1), the relations corresponding to point \(P_{1}\) and point \(P_{2}\) are stated by:  
\[\left\{ \begin{array}{l}\left(r\cos\alpha\right)^{2}+\left(r\sin\alpha\right) ^{2}=r^{2}\\ \left(\left(r+l\right)\cos\alpha-x_{0}\right)^{2}+\left(\left(r+l\right)\sin \alpha-y_{0}\right)^{2}=R^{2}\end{array} \right.. \tag{2}\]  
Among them, parameter \(l\) is the shortest distance between the probe and the hole wall, and the light intensity is evaluated by the size of its value. In other words, the smaller the \(l\), the stronger the light, and the larger the \(l\), the weaker the light. To this end, the corresponding valid solution to Equation (2) is sought as follows:  
\[