## 2 Sampling and Methodology
### Quantitative Analysis of Loess Microstructure in Different Regions under Natural Conditions
The correlation coefficient R\({}^{2}\) of the fractal dimension of porosity distribution and the proportion of pore area are 0.916 and 0.950, respectively. This finding indicates that the porosity area ratio has a greater relationship with collapsibility. Quantitative Analysis of Loess Microstructure and Microstructural Changes under Uniform Initial Water Content across Different Regions  
To enhance the accuracy of the analysis, loess samples from all four regions were subjected to testing under a controlled and uniform initial water content. Given that the general trends in loess collapsibility under identical initial water content conditions were largely consistent across the different regions, a group of samples exhibiting the most pronounced differences in collapsibility was selected for further investigation. Table 3 presents the quantitative microstructural parameters of loess samples with an initial water content of 6.79% before and after immersion. Figure 12 illustrates the changes in these micro-quantitative parameters for loess from different regions before and after soaking. Figure 11: Relationship between the quantitative factors of microstructure and collapsibility coefficient. ((**(a)**) average form factor, (**b**) probability entropy, (**c**) porosity distribution dimension, (**d**) pore area ratio). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sampling Site**} & **State of Soil** & **Average form** & **Probability** & **Porosity** & **Pore Area Ratio** \\  & **Sample** & **Factor** & **Entropy** & **Practal Dimension** & **Pore Area Ratio** \\ \hline \multirow{2}{*}{Shanzhou} & Before soaking & 0.398 & 0.994 & 3.4 & 45.79\% \\  & After soaking & 0.371 & 0.990 & 3.2 & 30.28\% \\ \hline \multirow{2}{*}{Mianchi} & Before soaking & 0.364 & 0.989 & 2.9 & 41.77\% \\  & After soaking & 0.345 & 0.986 & 2.5 & 27.78\% \\ \hline \multirow{2}{*}{Gongyi} & Before soaking & 0.358 & 0.987 & 2.6 & 17.52\% \\  & After soaking & 0.332 & 0.983 & 2.3 & 8.77\% \\ \hline \multirow{2}{*}{Xingyang} & Before soaking & 0.342 & 0.982 & 2.3 & 16.21\% \\  & After soaking & 0.332 & 0.979 & 2.0 & 7.52\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Quantitative microstructural parameters of loess in different regions (\(\omega\) = 6.79%). 1. Average form factor. Despite the uniform initial moisture content across the four regions, variations in soil particle morphology are still evident. Notably, the pore shape in Shanzhou appears smoother compared to Xingyang. Following immersion, the average shape coefficient of loess in all regions decreases, with the most rapid decline observed in Shanzhou. The trend aligns with the changes observed in the average shape coefficient of loess under natural water content conditions. 2. Probability entropy. Under the same initial water content, the probability entropy of the loess shows a gradual decrease from Shanzhou to Xingyang. This observation suggests a progressive increase in pore orderliness and a more stable arrangement distribution. 3. Porosity distribution fractal dimension. The fractal dimension of pore distribution in loess across the four regions remains largely consistent with that observed in the natural state, albeit with minor variations. The lower fractal dimension values in Xingyang and Gongyi indicate a more concentrated distribution of soil particles, suggestive of a condensed state. 4. Pore area ratio. The pore area ratio of loess in each region manifests a gradual decrease both before and after soaking and collapsibility, implying a corresponding increase in the area ratio occupied by particles. Xingyang loess displays the smallest pore area ratio, both before and after soaking compared to other regions, further supporting the notion of its denser soil structure. Figure 12: Variation curves of quantitative microstructural parameters in different regions. (**(a)** Average form factor, **(b)** Probability entropy, **(c)** Porosity distribution dimension, **(d)** Pore area ratio). 5.