### 数学代写|概率论代写Probability theory代考|MATHS 2103

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|概率论代写Probability theory代考|Fredlund and Xing Equation for SWCC

Based on the assumption that the shape of the SWCC depends on the pore-size distribution of the soil, Fredlund and Xing [7] proposed a three-parameter equation for the $\mathrm{SWCC}$, with flexibility to fit a wide variety of soils:
$$\theta=\frac{\theta_{\mathrm{s}}}{\left{\ln \left[\exp (1)+(\varphi / a)^{k}\right]\right}^{m}}\left[1-\frac{\ln \left(1+\varphi / \varphi_{\mathrm{r}}\right)}{\ln \left(1+10^{6} / \varphi_{\mathrm{r}}\right)}\right]$$
where $\theta_{s}$ is the saturated volumetric water content; $a$ is the fitting parameter related to the air-entry value for the soil; $k$ is the fitting parameter related to the maximum slope of the curve; $m$ is the fitting parameter related to the curvature of the curve; $\varphi$ is the soil suction; $\varphi_{\mathrm{r}}$ represents the soil suction related to the residual volumetric water content. In order to reduce the complexity of the equation, it was suggested that the residual suction $\varphi_{\mathrm{r}}$ takes the value of $3000 \mathrm{kPa}$, regardless of the soil types [38]. In addition, the normalized volumetric water content, that is, $S=\theta / \theta_{s}$, is introduced and Eq. (2.1) can be re-written as:
$$S=\frac{1}{\left{\ln \left[\exp (1)+(\varphi / a)^{k}\right]\right}^{m}}\left[1-\frac{\ln \left(1+\varphi / \varphi_{\mathrm{r}}\right)}{\ln \left(1+10^{6} / \varphi_{\mathrm{r}}\right)}\right]$$
The fitting parameters, that is, $a, k$ and $m$, can be obtained by nonlinear leastsquares methods. Although the Fredlund and Xing equation is capable of describing the SWCC by fitting the measured data, it cannot distinguish the SWCCs for the same soil under different initial dry densities. It is because the same set of fitting parameters $(a, k, m)$ is used in the Fredlund and Xing equation to describe their SWCCs. In other words, for the same soils with different initial dry densities, their SWCCs from Eq. (2.2) are identical. This induces larger level of prediction uncertainty in the estimation of unsaturated soil properties as the soil initial dry density has a significant influence on the SWCC, permeability and shear strength.

## 数学代写|概率论代写Probability theory代考|Effect of Initial Dry Density and Proposed Estimation Method

Soil is a porous medium with different particle sizes and complex texture. Granular soils at different compaction states are associated with different dry densities and porosities. The water in the porous media with different pore size exhibits different tension due to capillary effect. In the drying process, water is more difficult to flow out from soil with larger dry density, and hence the soil has different water retentions under different densities. Generally speaking, granular soil with larger dry density has larger air-entry value and larger residual water content. As a result, its SWCC will associate with smaller maximum slope $[12,17]$. It should be noted that the contribution of the capillary mechanisms and associated pore structure to water retention is more important under relatively low suction. At high suction, the water retention is dominated by the surface adsorption mechanism. Although the clayey soil basically has this feature, the volume shrinkage of samples in the SWCC test is not negligible. For this reason, only granular soil is considered in this chapter.

From the above discussion about the effect of initial dry density on the SWCC of granular soil, a new estimation model is proposed. Sheng and Zhou [18] found that the modified suction can reflect the influence of porosity $(n)$ on the hydraulic relationship between the water content and suction:
$$\varphi^{}=n \cdot \varphi$$ In this study, the volume change during the desorption process is ignored. Inspired by the concept of modified suction, the initial porosity ( $n_{0}$ ) is used in the estimation to consider the effect of initial dry density on the SWCC. After substantial investigations, the fitting parameters are proposed to be modified as follows: $$a=a^{} / n_{0}, k=k^{} \cdot n_{0}, m=m^{} \cdot n_{0}$$
where $n_{0}$ is the measured initial porosity which is introduced to the set of fitting parameters $\left(a^{}, k^{}, m^{}\right)$ to consider the effect of initial dry density on soil-water characteristic curve. In particular, $a=a^{} / n_{0}$ reflects the influence of the initial porosity on the air-entry value, and the air-entry value decreases as the initial porosity $n_{0}$ increases; $k=k^{} \cdot n_{0}$ and $m=m^{} \cdot n_{0}$ reflect the effect of initial porosity on the maximum slope and curvature of the SWCC. The water in the soil with larger $n_{0}$ is easier to flow out, so its SWCC is associated with larger maximum slope and curvature. Thus, the new estimation model is:
$$S=\frac{1}{\left{\ln \left[\exp (1)+\left(\varphi \cdot n_{0} / a^{}\right)^{k^{} \cdot n_{0}}\right]\right}^{m^{} \cdot n_{0}}}\left[1-\frac{\ln \left(1+\varphi / \varphi_{\mathrm{r}}\right)}{\ln \left(1+10^{6} / \varphi_{\mathrm{r}}\right)}\right]$$ The effect of $n_{0}$ is further shown in Fig. 2.1. The three curves in this figure represent the $\mathrm{SWCCs}$ of the same soil $\left(a^{}=1, k^{}=15, m^{}=2\right)$ with different initial porosities.

## 数学代写|概率论代写Probability theory代考|Verification and Discussion

The lab test data of SWCCs for three types of sands, namely Berlin coarse sand, Berlin medium sand and Wagram sand, with different initial porosities were collected from the UNSODA [37] database. According to the database manual [37], the tensiometry and gamma-ray attenuation were used to measure the suction and volumetric water content of Berlin coarse and Berlin medium sands, respectively. The pressure outflow method was used to determine the water retention of Wagram sand. The relevant soil

properties and texture contents are summarized in Table 2.1. Figure $2.2$ shows the process of data partitioning, curve-fitting and verification. The data from the same soil are divided into two groups randomly and repeatedly according to proportion of $2: 1$ : the first group of two-thirds of the data points for obtaining the fitting parameters and the rest of the one-third for verification. In the fitting process, different sets of fitting parameters, that is, $(a, k, m)$ for the Fredlund and Xing equation and $\left(a^{}, k^{}\right.$, $\left.m^{*}\right)$ for the proposed estimation method, are obtained for different data sets.

It was found that the calculated fitting parameters could depend heavily on the partitioning of the two groups of data for fitting and verification due to the limited number of data points. Therefore, to assure the reliability of the data fitting, a simple criterion is imposed for data partitioning. According to the process of Fig. 2.2, 2250 sets of data are randomly selected from each soil type and 2250 sets of fitting parameters are computed by both Fredlund and Xing equation and the proposed method. Sorting the values of each fitting parameter, their values at $97.5$ and $2.5$ percentiles are obtained. Thus, the corresponding $95 \%$ confidence interval (CI) for each parameter can be constructed. 1000 sets of test data are randomly selected to analyze statistically if their fitting parameters, $(a, k, m)$ and $\left(a^{}, k^{}, m^{*}\right)$, are all located in their corresponding $95 \%$ CIs. Figure $2.3$ shows an example of frequency distribution about the relevant fitting parameters by Fredlund and Xing equation and the proposed method. Finally, a random data set is used as Group I data, and then the rest of data points are categorized into Group II data for verification.

## 数学代写|概率论代写Probability theory代考|Fredlund and Xing Equation for SWCC

\theta=\frac{\theta_{\mathrm{s}}}{\left{\ln \left[\exp (1)+(\varphi / a)^{k}\right]\right}^{m }}\left[1-\frac{\ln \left(1+\varphi / \varphi_{\mathrm{r}}\right)}{\ln \left(1+10^{6} / \varphi_{ \mathrm{r}}\right)}\right]\theta=\frac{\theta_{\mathrm{s}}}{\left{\ln \left[\exp (1)+(\varphi / a)^{k}\right]\right}^{m }}\left[1-\frac{\ln \left(1+\varphi / \varphi_{\mathrm{r}}\right)}{\ln \left(1+10^{6} / \varphi_{ \mathrm{r}}\right)}\right]

S=\frac{1}{\left{\ln \left[\exp (1)+(\varphi / a)^{k}\right]\right}^{m}}\left[1-\frac {\ln \left(1+\varphi / \varphi_{\mathrm{r}}\right)}{\ln \left(1+10^{6} / \varphi_{\mathrm{r}}\right) }\正确的]S=\frac{1}{\left{\ln \left[\exp (1)+(\varphi / a)^{k}\right]\right}^{m}}\left[1-\frac {\ln \left(1+\varphi / \varphi_{\mathrm{r}}\right)}{\ln \left(1+10^{6} / \varphi_{\mathrm{r}}\right) }\正确的]

## 数学代写|概率论代写Probability theory代考|Effect of Initial Dry Density and Proposed Estimation Method

S=\frac{1}{\left{\ln \left[\exp (1)+\left(\varphi \cdot n_{0} / a^{}\right)^{k^{} \cdot n_ {0}}\right]\right}^{m^{} \cdot n_{0}}}\left[1-\frac{\ln \left(1+\varphi / \varphi_{\mathrm{r} }\right)}{\ln \left(1+10^{6} / \varphi_{\mathrm{r}}\right)}\right]S=\frac{1}{\left{\ln \left[\exp (1)+\left(\varphi \cdot n_{0} / a^{}\right)^{k^{} \cdot n_ {0}}\right]\right}^{m^{} \cdot n_{0}}}\left[1-\frac{\ln \left(1+\varphi / \varphi_{\mathrm{r} }\right)}{\ln \left(1+10^{6} / \varphi_{\mathrm{r}}\right)}\right]的效果n0进一步如图 2.1 所示。该图中的三条曲线代表小号在CCs相同的土壤(一个=1,ķ=15,米=2)具有不同的初始孔隙率。

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## MATLAB代写

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