## 4 Results and Discussions
### Information Spread Estimation  
Information spread is a method to make up for the lack of information by optimizing the sample information to carry out set-valued fuzzy processing on samples. The method can transform single-valued samples into set-valued samples and improve the accuracy of population distribution [35].  
In the information spread theoretical model, let \(X\) be the sample set of actual observed values:  
\[X=\{x_{1},\ x_{2},\ x_{3},\cdots,\ x_{m}\} \tag{7}\]  
where \(x_{i}\) is the sample observed value, and \(m\) is the number of observed samples.  
Assumed \(U\) is the information spread range of \(x_{i}\) in \(X\) set:  
\[U=\{u_{1},\ u_{2},\ u_{3},\cdots,\ u_{n}\} \tag{8}\]  
where \(u_{j}\) represents the discrete real values in the interval [\(u_{1}\),\(u_{n}\)] after discretization at a fixed interval, and \(n\) is the number of discrete values.  
In general, the information carried by each single-valued observation sample \(x_{i}\) in the set \(X\) can be spread to all points in the index domain \(U\) according to normal distributions:  
\[f_{i}\big{(}u_{j}\big{)}=\frac{1}{h\sqrt{2\pi}}\exp\frac{\big{(}x_{i}-u_{j} \big{)}^{2}}{2h^{2}} \tag{9}\]  
where \(f_{i}\big{(}u_{j}\big{)}\) is the information spread function of normal distribution, and \(h\) is the information spread coefficient, i.e.,  
\[h=\frac{\alpha(v_{\max}-v_{\min})}{n-1} \tag{10}\]  
where \(v_{\max}\) is the maximum value of sample observations, \(v_{\min}\) is the minimum value, and \(\alpha=1.420693101\) when \(n\geq 17\).  
\[C_{i}=\sum_{j=1}^{n}f_{i}\big{(}u_{j}\big{)},\ i=1,2,\cdots,\ m \tag{11}\]  
Figure 13: CEI results of the normal, lognormal, and Gamma distributions of the average \(V_{D<0.1\ \mu\text{m}}\).  
where, \(C_{i}\) is the accumulation of normal spread information of the observed sample \(x_{i}\). Then, the normalized information distribution \(\mu_{xi}\big{(}u_{j}\big{)}\) of sample point \(x_{i}\) can be expressed by  
\[\mu_{xi}\big{(}u_{j}\big{)}=\frac{f_{i}\big{(}u_{j}\big{)}}{C_{i}},\ i=1,2, \cdots,m;\ j=1,2,\cdots,n \tag{12}\]  
Figures 14 and 15 show the residual plot of microstructure parameters estimated by the normal distribution and information spread estimation (from left to right: the plots of 1 to 8, 9 to 16, 17 to 24, and 25 to 32 were from silty clays with NS contents of 0, 0.25%, 1%, and 3%, respectively). From Figure 14a, the 20 th residual plot of normal distribution is singularity, and the residuals of normal distribution are greater than those of the information spread estimation in Figure 15. Therefore, the information spread estimation has a better fitting effect than the normal distribution for the NS-modified silty clay, because it has combined the normal distribution information of each observation. Meanwhile, the information spread estimation fits with all observations with the nonlinear regression method.  
Figure 14: The residual plot of (**a**) normal and (**b**) information spread distributions for the \(V_{D<0.1\mu\)m.  
As a result, even for the unsmooth parts of the observations, the normal diffusion estimation in combination with the statistical features has a better prediction than the normal distribution. However, the normal diffusion estimation still has many limits because of the complex expression and operating process, especially in practical applications [36]. Furthermore, the superiority of the normal diffusion estimation is not shown in large samples. In general, for the Qinghai-Tibet Plateau silty clay, the pore size distribution of soil samples is regular whether nanomaterials are added or not. The Gaussian distribution is more representative than other statistical distributions.