## 5 Analysis and Discussion
### Area and Volume Evaluation at Different Spatial Scales  
As shown in Figure 6, the evaluations of the area at different spatial scales were obtained. Figure 6a-c presents the area evolution results of specimens S2, S3, and S5, respectively.  
Figure 5: The spatial characterization results of specimen S3. Black dots mark the microcracks, the red ellipsoid represents the space formed by MVEE, the black ellipsoid represents the TDPS space, where the three red line segments are the three axes of TDPS, and the yellow irregular geometry is formed by 3D convex hull.  
Figure 6: Area evaluation of three methods. (**a**) Area evaluation of Specim S2. (**b**) Area evaluation of Specim S3. (**c**) Area evaluation of Specim S5.  
In Figure 6a, the theoretical result at the 100% position is relatively large. The rest of the values show potential regularity. It is possible that the number of microcracks monitored in this specimen is relatively small and the distribution is sparse. Therefore, the variation is significantly on a larger base when the range of spatial scales is reduced. It can also be interpreted as a similar variation at 100% for the other two specimens, as shown in Figure 6b,c. However, due to a greater number of microcracks, the density of microcracks in the other two specimens is relatively greater. Therefore, the differences were lower compared to specimen S2, and even a similar fall was found in specimens S3 and S5.  
Among the three methods, it is found that the area evaluation results from the theoretical distribution pattern space are the largest, and the 3D convex hull result has the smallest values. As the scale of selected intervals decreases, the values of the three methods gradually become closer. In general, the values in the MVEE method are closer to the TDPS method results than the 3D convex hull results. It is possible that both the TDPS method and the MVEE method are based on an ellipsoid space, which can degenerate into a sphere space, the area values are closer, and the MVEE results are lower than those of the theoretical distribution pattern space because of the minimum volume target. Therefore, this relationship can be used to find the optimal characterization interval of the theoretical distribution pattern space.  
In spite of the differences in microcrack density and the area variation from the results of three specimens, the values of the TDPS and MVEE method are closer to each other in the interval below 85%. The position is illustrated by the distance between the circle and triangle symbols in the figure.  
As shown in Figure 7, the volume evaluation results at different spatial scales in three methods were obtained, and a similar trend to that of area variation was exhibited. However, the variation is greater at the 100% scale, compared to the area evaluation from Figure 6. The crack space formed at the 100% scale based on the TDPS method is significantly larger than the specimen volume since the volume of the test specimen is only \(5.4\times 10^{5}\) cm\({}^{3}\). The fact that the theoretical volume is greater than the actual volume of the test specimen does not mean that it is an error. The specimen is limited in size and the damage can only be confined and exhibited within the specimen. If the specimen boundary is expanded virtually, then the crack space has the potential to continue to expand. In this way, the results of the TDPS are well interpreted, demonstrating the ability to find potential crack spaces based on existing microcracks.  
Figure 7: Volume evaluation of three methods. (**a**) Volume evaluation of Specim S2. (**b**) Volume evaluation of Specim S3. (**c**) Volume evaluation of Specim S5.  
Similar to the trend in area evolution, the results of the theoretical distribution pattern space and the MVEE method are gradually matched in volume values starting at 85%.