## 3 Analysis of Results
### Comprehensive Assessment of the Extent of Structural Degradation of Red Sandstone  
#### 3.3.1 Overall Planar Ultrasonic Wave Velocity Analysis and Degradation  
Degree Evaluation  
In order to study the changing pattern of the degradation degree of red sandstone under natural environmental erosion, the loss of dynamic elastic modulus is used to define the degradation degree of red sandstone and analyze its change pattern with depth. Figure 9: Projection of the CT grayscale distribution in the X–Z and Y–Z directions at different layers: (**a**) first-layer X–Z plane, (**b**) first-layer Y–Z plane, (**c**) second-layer X–Z plane, (**d**) second-layer Y–Z plane, (**e**) third-layer X–Z plane, (**f**) third-layer Y–Z plane, (**g**) fourth-layer X–Z plane, (**h**) fourth-layer Y–Z plane. According to the relationship between the material dynamic modulus of elasticity \(E_{d}\), density \(\rho\), longitudinal wave speed \(v\), and [PERSON]'s ratio \(\mu\):  
\[E_{d}=\frac{(1+\mu)(1-2\mu)\rho v^{2}}{1-\mu} \tag{15}\]  
Let \(k=\frac{(1+\mu)(1-2\mu)\rho}{1-\mu}\), then the above equation simplifies to  
\[E_{d}=kv^{2} \tag{16}\]  
In this study, we found that there is a certain bias when using the ultrasonic wave velocity in the X or Y direction singularly to define the degradation degree of the overall plane, so it is proposed that a better approach is to characterize the degradation degree of red sandstone from both X and Y, and define the dynamic elastic modulus in X and Y directions, that is:  
\[E_{dx}=kv_{xi}^{2} \tag{17}\]  
\[E_{dy}=kv_{yi}^{2} \tag{18}\]  
and characterize the degree of degradation in two directions, viz. :  
\[df_{x}=\frac{E_{dx0}-E_{dx}}{E_{dx0}},\ df_{y}=\frac{E_{dy0}-E_{dy}}{E_{dy0}} \tag{19}\]  
In order to achieve the calculation of the degradation degree of the overall plane, define a combined degradation degree \(df\), which is expressed as:  
\[df=\sqrt{df_{x}^{2}+df_{y}^{2}} \tag{20}\]  
If the first layer is used as the initial state, i.e., the initial ultrasonic wave velocity values are \(v_{x1}\) and \(v_{y1}\), the final overall degradation degree is calculated by the formula:  
\[df_{i}=\sqrt{\left(\frac{v_{x1}^{2}-v_{x1}^{2}}{v_{x1}^{2}}\right)^{2}+\left( \frac{v_{y1}^{2}-v_{yi}^{2}}{v_{y1}^{2}}\right)^{2}} \tag{21}\]  
of which \(v_{x1}\) and \(v_{y1}\) are the ultrasonic velocities of the \(Li\) layer in the X- and Y-axis directions, respectively. \(df_{xi},\ df_{yi}\) and \(df_{i}\) are the degradation degree of the \(Li\) layer in the X direction, Y direction and the overall plane, respectively. The calculated results are obtained as shown in Figure 10.  
Figure 10 shows that the degree of degradation of red sandstone under natural environmental erosion has a certain positive correlation with the depth from the surface. It can be seen from the figure that the trend of degradation degree is accelerating; the closer to the surface, the faster the trend of degradation degree with depth, i.e., the weathering erosion rate is accelerating. The degradation degree in the X-axis upward is always smaller than that in the Y-axis upward, and the growth rate of the degradation degree in the Y-axis is also larger than that in the X-axis, so it can be judged that the red sandstone specimen is subject to different erosion degrees and erosion rates along different directions in the same plane, so characterizing the degradation degree of the red sandstone from two directions is appropriate. 3.2 Analysis of Surface Degradation Degree of Red Sandstone Based on GM (1, 1) Gray Prediction Model  
In this paper, we use the ultrasonic wave velocity values in two directions to characterize the degree of degradation, but ultrasonic inspection using a probe to collect the ultrasonic wave velocity parameters between the materials, subject to the operational limitations of its probe size, means points cannot be continuously picked, and the top bottom of the red sandstone specimen is a non-parallel surface, which means the ultrasonic wave velocity values at its boundary cannot be accurately detected.