## 2 Sampling and Methodology
### Quantitative Analysis of Loess Microstructure in Different Regions under Natural Conditions
Values closer to 0 indicate a more ordered and regular arrangement, while values approaching 1 signify greater disorder and irregularity. When the pore and particle distribution is more irregular, the soil is unstable and prone to collapsibility, and the probability entropy is larger. This parameter is calculated using the following formula: \[H_{m}=-\sum_{i=1}^{n}\frac{m_{i}}{M}\times\frac{\ln(\,m_{i}\times M)}{\ln n}\] where \(m_{i}\) represents the number of pores in the ith location of the n locations within the 0-180\({}^{\circ}\) range, \(M\) denotes the total number of pores. Following immersion, the probability entropy of loess across all four regions manifests an upward trend. Notably, Shanzhou loess possesses the lowest probability entropy prior to soaking, whereas Xingyang loess displays the highest. This finding implies a less ordered pore arrangement within Shanzhou loess compared to Xingyang loess, suggesting a greater degree of stability in the pore structure of the latter. 3. Porosity distribution fractal dimension [37]. The distribution fractal dimension reflects the spatial distribution of objects. A smaller fractal dimension implies a more concentrated distribution of particles or pores. The pore distribution is more concentrated, which indicates that the particles are arranged closely, the intergranular pores increase,and the number of large pores decreases. The box-counting method is employed for calculation:  
\[D_{V}=-\underset{r\to 0}{\lim}\frac{\ln N(r)}{\ln r}=-K\]  
where \(r\) represents the side length of the square box, \(N(r)\) denotes the number of boxes containing objects, and \(K\) indicates the linear slope. Analysis of the fractal dimension reveals that the loess in Gongyi and Xingyang manifests a lower value compared to Minachi and Shanzhou. Furthermore, all four regions demonstrate a decrease in fractal dimension following collapse. This observation suggests a weaker tendency towards dispersed and collectivized pore distribution in Xingyang and Gongyi loess. Conversely, Malan loess in Minachi and Shanzhou displays a more concentrated pore distribution. (4) Pore area ratio. This represents the proportion of pores within the soil as observed in the microscopic image. The value ranges from 0 to 1, with higher values indicating a greater abundance of pores. It provides enough space for the occurrence of loess collapsibility, and thus the collapsibility of the loess increases. Under the influence of external loads and water, the pore area of loess in all regions manifests a decreasing trend. Notably, Shanzou loess possesses a higher initial pore area ratio compared to the other three regions, and it also experiences the most rapid decline following immersion and external load application. This observation suggests that loess in the eastern part of western Henan has a denser skeletal particle arrangement compared to the western part. In conjunction with previous findings, it can be inferred that the naturally high water content of Malan loess in Gongyi and Xingyang contributes to the destruction of large pores and overhead pores between soil particles. Consequently, the reduction in pore area ratio after soaking is less pronounced in these areas compared to Minachi and Shanzhou. Interestingly, the pore area ratio of Minachi and Shanzhou loess after collapsibility remains higher than that of Gongyi and Xingyang loess before collapsibility. The relationship between the loess collapsibility coefficient and the quantitative parameters of microstructure was compared by statistical linear regression analysis (Figure 11). It is found that there is a certain linear relationship between the four quantitative parameters and loess collapsibility.With the increase of average form factor, probability entropy, fractal dimension of porosity distribution and Pore area ratio, the collapsibility of loess increases linearly. The correlation coefficients R\({}^{2}\) of average form factor and probability entropy are 0.937 and 0.938, respectively.