Soil-water characteristic curve (SWCC) represents the variation of volumetric water content (or effective saturation) with the matrix suction, which is significant to estimate unsaturated soil parameters (e.g., unsaturated shear strength and permeability coefficient) [1] - [3] ). Only a limited number of SWCC measuring data can be obtained considering that the direct measurement of SWCC is often costly and time-consuming through in-situ or laboratory tests according to some prescribed experimental schemes (i.e., the number of measuring points and their corresponding values of the control variable). The uncertainty of estimating SWCC based on limited data is inevitable, which depends on the data obtained from prescribed experimental schemes and affects the estimation of unsaturated soil parameters and geotechnical reliability analysis [4] . Determining an optimal experimental scheme is vital for reducing the uncertainty in SWCC estimated from a limited number of data points.

Experimental design optimization (EDO) provides a rational vehicle to determine the optimal experimental scheme for acquisition of measuring data in a cost-effective way [5] . Several EDO methods have been developed in the literature, including conventional experimental design optimization (CEDO) methods based on classical statistics [6] and Bayesian experimental design optimization (BEDO) methods based on Bayesian inference and/or information theory [7] [8] . Compared with CEDO, the BEDO has an advantage of quantifying various uncertainties, which has been recently applied in geotechnical and geological engineering to design in-situ instrumentation [9] and site investigation programs [10] . Despite of these previous studies on in-situ monitoring and sampling design, research on applying BEDO to design geotechnical laboratory tests that can be troublesome and time-consuming, e.g., SWCC test, is rare. Ding et al. (2022) [8] proposed a BEDO approach for SWCC testing, which, however, requires to implement the optimization procedure twice at two stages of the experimental design for determining control and additional measuring points, respectively.

This paper presents a one-stage Bayesian experimental design optimization (OBEDO) method for SWCC testing based on Fredlund and Xing (1994) (FX) model, which determines the optimal experimental scheme by implementing a single run of optimization procedure. The proposed method adopts expected utility to quantify the expected value of information provided by SWCC testing. The ancestral sampling and Bayesian method are used to generate simulated data to evaluate the effect of uncertainty on soil parameters. The optimal scheme with maximal expected utility is searched out with Subset Simulation Optimization (SSO), which improves the efficiency of determining the optimal scheme in the design space. This paper starts with a description of the proposed OBEDO framework based on FX model, followed by quantifying the expected utility of candidate experimental schemes and optimizing the experimental scheme by maximizing the expected utility using SSO [11] . Then, the proposed approach is illustrated using a SWCC experimental design example.

As shown in Figure 1 , the proposed OBEDO framework starts with collecting available prior knowledge (i.e., prevailing SWCC models and typical ranges of its model parameters) before testing on the SWCC of soils concerned and the information of testing apparatus and technique, which are used to determine the design space of candidate experiment schemes. The proposed OBEDO framework is comprised of three steps: determination of the candidate experimental schemes, calculation of the expected utility, U€, of a possible experimental scheme E that is specified by the number, n, of measuring points, and optimization of the experimental scheme performed by SSO to maximize the U€. Details of the three steps of the proposed OBEDO framework are provided in the following three sections.

The trajectory of SWCC can be generally controlled by characteristic matric suction values (such as the air-entry value ${\psi }_b$, the matric suction at the inflection point ${\psi }_i$, and the matric suction corresponding to the residual water content ${\psi }_r$) and their corresponding degrees of saturation. For a given SWCC parametric model, the ${\psi }_b$, ${\psi }_i$, and ${\psi }_r$ divide the SWCC into four partitions. There are, at least, four control measuring points selected within the ranges of the matric suction, i.e., $\left(0,{\psi }_b\right]$, $\left({\psi }_b,{\psi }_i\right]$, $\left({\psi }_i,{\psi }_r\right]$, $\left({\psi }_r,{\psi }_m\right)$ to capture the general trajectory of the estimated SWCC and a certain number of additional points selected within the ranges of the matric suction, i.e., $\left(0,{\psi }_m\right)$ to reduce its associated uncertainty. Let n denote the total number of measuring points. Each candidate experimental scheme, E, of SWCC testing is comprised of four control points (i.e., A₁, A₂, A₃, A₄) and (n-4) additional points (i.e., B₁-B_n-4), as shown in Figure 2 .

Nevertheless, during the experimental design stage, the ${\psi }_b$, ${\psi }_i$, and ${\psi }_r$ values corresponding to the prescribed SWCC model are unknown. The expected value (i.e., ${\bar{\psi }}_i$, ${\bar{\psi }}_b$, and ${\bar{\psi }}_r$) of ${\psi }_i$, ${\psi }_b$, ${\psi }_r$ is adopted to constrain the matric suction range of control point (i.e., A₁, A₂, A₃, A₄), which can be determined using Monte Carlo simulation based on the prior knowledge of SWCC model parameters. Consider, for example, the FX model given below:

$S_e=\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}\text{=}\left[1-\frac{\ln \left(1+\psi /{\psi }_r\right)}{\ln \left(1+{10}^6/{\psi }_r\right)}\right]{\left[\frac{1}{\ln \left(e+{\left(\psi /a_{fx}\right)}^{n_{fx}}\right)}\right]}^{m_{fx}}$ (1)

The values of ${\psi }_i$, ${\psi }_b$, ${\psi }_r$ corresponding to the FX model satisfy Eqs. (2)-(4) [12] .

$\begin{array}{l}\left[1-\ln \left(1+{\psi }_i/C_r\right)/\ln \left(1+{10}^6/C_r\right)\right]\frac{m_{fx}{n}_{fx}}{a_{fx}}\frac{1}{\ln \left(e+\left({\psi }_i/a_{fx}\right)\right)}\frac{1}{e+{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}}\\ \times {\left({\psi }_i/a_{fx}\right)}^{n_{fx}-1}\{\frac{\left(m_{fx}+1\right)n_{fx}}{\ln \left(e+{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}\right)}\frac{1}{e+{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}}{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}\\ +\frac{n_{fx}}{e+{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}}{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}-n_{fx}\}+\frac{{\psi }_i}{\ln \left(1+{10}^6/C_r\right)}{\left(\frac{1}{C_r+{\psi }_i}\right)}^2\\ 2\frac{m_{fx}{n}_{fx}}{\ln \left(1+{10}^6/C_r\right)}\frac{1}{C_r+{\psi }_i}\frac{1}{\ln \left(e+{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}\right)}\frac{1}{e+{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}}{\left({\psi }_i/a_{fx}\right)}^{n_{fx}}\\ -\frac{1}{\ln \left(1+{10}^6/C_r\right)}\frac{1}{C_r+{\psi }_i}=0\end{array}$ (2)

${\psi }_b={\psi }_i{0.1}^{\frac{1-S_{e,i}}{k_1}}$ (3)

${\psi }_r={10}^{\frac{S_{e,i}-{S^{\prime }}_e+k_1\log \left({\psi }_i\right)-k_2\log \left({\psi }^{\prime }\right)}{k_1-k_2}}$ (4)

where $a_{fx}$, $n_{fx}$ and $m_{fx}$ are the model fitting parameters of FX model; $S_{e,i}$ is an effective degree of saturation corresponding to ${\psi }_i$; k₁ is the slope at the inflection point; ${\psi }^{\prime }$ is the matric suction where the SWCC starts to drop linearly; ${S^{\prime }}_e$ is an effective degree of saturation corresponding to ${\psi }^{\prime }$; and k₂ is the slope at the point $\left({\psi }^{\prime },{S^{\prime }}_e\right)$. These symbols are illustrated in Figure 3 . N_p estimates of ${\psi }_i$, ${\psi }_b$, and ${\psi }_r$ can be obtained with the number, N_p, of random samples of $a_{fx}$, $n_{fx}$ and $m_{fx}$ simulated from their uniform prior distribution through Monte Carlo simulation. Based on the N_p estimates of ${\psi }_i$, ${\psi }_b$, and ${\psi }_r$, their respective mean values (i.e., ${\bar{\psi }}_i$, ${\bar{\psi }}_b$, and ${\bar{\psi }}_r$) are evaluated, with which the matric suction values (i.e., ${\psi }_{{\text{A}}_1},{\psi }_{{\text{A}}_2},{\psi }_{{\text{A}}_3}$, and ${\psi }_{{\text{A}}_4}$) of the four control measuring points A₁, A₂, A₃, and A₄ are, respectively, assigned within the matric suction ranges $\left(0,{\bar{\psi }}_b\right]$, $\left({\bar{\psi }}_b,{\bar{\psi }}_i\right]$, $\left({\bar{\psi }}_i,{\bar{\psi }}_r\right]$, and $\left({\bar{\psi }}_r,{\psi }_m\right)$. The matric suction values (i.e., ${\psi }_{{\text{B}}_1}$- ${\psi }_{{\text{B}}_{n-4}}$) of n-4 additional measuring points B₁-B_n-4 belong to the range, $\left(0,{\psi }_m\right)$, but should not be equal to any values of the four control measuring points A₁, A₂, A₃, A₄.

Let ${\psi }_h$ denote the feasible discrete matric suction value, and a set of possible value of ${\psi }_h$ can be expressed as ${\Omega }_{\text{O}}=\left(\text{0:}\Delta {\psi }_1:{\bar{\psi }}_b\right]\cup \left({\bar{\psi }}_b\text{:}\Delta {\psi }_2:{\bar{\psi }}_i\right]\cup \left({\bar{\psi }}_i\text{:}\Delta {\psi }_3:{\bar{\psi }}_r\right]\cup \left({\bar{\psi }}_r\text{:}\Delta {\psi }_4:{\psi }_m\right]$, where $\Delta {\psi }_1$, $\Delta {\psi }_2$, $\Delta {\psi }_3$, $\Delta {\psi }_4$ are discrete intervals (e.g., the minimum increment of the matric suction that can be applied by the testing apparatus). The above discre- tization procedure of the matrix suction results in a total of $N_{\text{O}}$ possible values of ${\psi }_h$. Assume that $N_{{\text{A}}_{\text{1}}}$, $N_{{\text{A}}_{\text{2}}}$, $N_{{\text{A}}_{\text{3}}}$, and $N_{{\text{A}}_{\text{4}}}$ values of ${\psi }_h$ in ${\Omega }_{\text{O}}$ fall within $\left(0,{\bar{\psi }}_b\right]$, $\left({\bar{\psi }}_b,{\bar{\psi }}_i\right]$, $\left({\bar{\psi }}_i,{\bar{\psi }}_r\right]$, and $\left({\bar{\psi }}_r,{\psi }_m\right)$, respectively, which constitute the set ${\Omega }_{{\text{A}}_{\text{1}}}$, ${\Omega }_{{\text{A}}_{\text{2}}}$, ${\Omega }_{{\text{A}}_{\text{3}}}$, ${\Omega }_{{\text{A}}_{\text{4}}}$. The matric suction values (i.e., ${\psi }_{{\text{A}}_1},{\psi }_{{\text{A}}_2},{\psi }_{{\text{A}}_3}$, and ${\psi }_{{\text{A}}_4}$) of the control measuring points A₁, A₂, A₃, and A₄ satisfied ${\psi }_{{\text{A}}_{\text{1}}}\in {\Omega }_{{\text{A}}_{\text{1}}}$, ${\psi }_{{\text{A}}_2}\in {\Omega }_{{\text{A}}_2}$, ${\psi }_{{\text{A}}_3}\in {\Omega }_{{\text{A}}_3}$, and ${\psi }_{{\text{A}}_4}\in {\Omega }_{{\text{A}}_4}$, respectively. Let ${\Omega }_{\text{B|A}}$ denote the set of possible values of the matric suction (i.e., ${\psi }_{{\text{B}}_j}$) of each additional measuring point B_j ( $j=1,2,\cdots ,n-4$), which can be written as a set ${\Omega }_{\text{B|A}}=\left\{{\psi }_{{\text{B}}_j}|{\psi }_{{\text{B}}_j}\in {\Omega }_{\text{O}}\text{and}{\psi }_{{\text{B}}_j}\notin {\psi }_{{\text{A}}_i}\right\}$( $i=1,2,3,4$; $j=1,2,\cdots ,n-4$). Each set of possible values of ${\psi }_{{\text{A}}_1},{\psi }_{{\text{A}}_2},{\psi }_{{\text{A}}_3},{\psi }_{{\text{A}}_4}$ and ${\psi }_{{\text{B}}_j}$( $j=1,2,\cdots ,n-4$) constitute a candidate experimental scheme E, which can be expressed as Eq.(5)

$E=\left\{{\psi }_{{\text{A}}_1},{\psi }_{{\text{A}}_2},{\psi }_{{\text{A}}_3},{\psi }_{{\text{A}}_4},{\psi }_{{\text{B}}_{\text{1}}},{\psi }_{{\text{B}}_{\text{2}}},{\psi }_{{\text{B}}_{\text{3}}},\cdots ,{\psi }_{{\text{B}}_{n\text{-4}}}\right\}$ (5)

The optimal experimental scheme is determined by maximizing the expected data worth (i.e., the expected utility U(E)) of the SWCC test performed according to candidate experimental schemes using SSO. Calculations of the U(E) of each candidate experimental scheme, E, and its optimization through SSO are provided in the following two sections, respectively.

Consider, for example, a candidate experimental scheme E. The data worth of SWCC test can be quantified by the relative entropy, $R\left(E\right)$, that indicates the statistical difference between the updated distribution, $p\left(\Theta |S_e,E\right)$, of SWCC model parameters, $\Theta$, given a set of newly-obtained data (e.g., values of effective degree of saturation, $S_e$), obtained according to E and the prior distribution, $p\left(\Theta |E\right)$, of $\Theta$. $R\left(E\right)$ can be written as Eq. (6) [5]

$R\left(E\right)={\displaystyle \int p\left(\Theta |S_e,E\right)\ln \left[p\left(\Theta |S_e,E\right)/p\left(\Theta |E\right)\right]\text{d}\Theta }$ (6)

Without the real measurement data at the experimental design stage, the expected utility, U(E), of SWCC measurement data corresponding to E is adopted to quantify the expected worth of data, which is evaluated as Eq. (7) [13] :

$U\left(E\right)={\displaystyle \int R\left(E\right)p\left(S_e|E\right)\text{d}{S}_e}$ (7)

where $p\left(S_e|E\right)$ is the probability density function (PDF) of $S_e$ corresponding to E. Substituting Eq. (6) into Eq. (7) gives

$U\left(E\right)={\displaystyle \iint \ln \left[\frac{p\left(\Theta |S_e,E\right)}{p\left(\Theta |E\right)}\right]p\left(S_e,\Theta |E\right)\text{d}\Theta \text{d}{S}_e}$ (8)

Using the $\Theta$ and $S_e$ samples, Eq. (8) is re-written as Eq. (9) [8] :

$U\left(E\right)\approx \frac{1}{N_r}{\displaystyle \underset{r=1}{\overset{N_r}{\sum }}\left\{\ln \left[p\left({\Theta }_r|S_{e,r},E\right)\right]-\ln \left[p\left({\Theta }_r|E\right)\right]\right\}}$ (9)

where $p\left({\Theta }_r|S_{e,r},E\right)$ is the posterior distribution of $\Theta$ evaluated at ${\Theta }_r$ given $S_{e,r}$.

For a given number of measuring points, the optimal experimental scheme $E^{*}$ is taken as the scheme with the maximum U(E) among candidate experimental schemes, i.e.,

$E^{*}=\arg \max U\left(E\right)$ (10)

The next section makes uses of SSO to identify the $E^{*}$ among candidate experimental schemes.

As mentioned in Section 3 entitled “Candidate experimental schemes based on FX model”, the number of candidate experimental schemes is equal to $N_{{\text{A}}_{\text{1}}}\cdot N_{{\text{A}}_2}\cdot N_{{\text{A}}_3}\cdot N_{{\text{A}}_4}\cdot C_{N_o-4}^{n-4}$. Identifying the $E^{*}$ among candidate experimental schemes can be formulated as an optimization problem below:

$\begin{array}{l}\underset{E}{\max }U\left(E\right)\\ \text{s}\text{.t}.E=\left\{{\psi }_{{\text{A}}_1},{\psi }_{{\text{A}}_2},{\psi }_{{\text{A}}_3},{\psi }_{{\text{A}}_4},{\psi }_{{\text{B}}_{\text{1}}},{\psi }_{{\text{B}}_{\text{2}}},{\psi }_{{\text{B}}_{\text{3}}},\cdots ,{\psi }_{{\text{B}}_{n\text{-4}}}\right\};\\ {\psi }_{{\text{A}}_i}\in {\Omega }_{{\text{A}}_i}\left(i=1,2,3,4\right);{\psi }_{{\text{B}}_j}\in {\Omega }_{\text{D|}C}\left(j=1,2,\cdots ,n-4\right)\end{array}$ (11)

where the feasible domains (i.e., ${\Omega }_{{\text{A}}_i}$ and ${\Omega }_{\text{B|A}}$) of ${\psi }_{{\text{A}}_i}$ ( $i=1,2,3,4$) and ${\psi }_{B_j}$ ( $j=1,2,\cdots ,n-4$) are defined previously in Section 3. In this study, SSO is used to search the E^* in the design space. SSO is a global optimization algorithm that was originally developed from Subset simulation [11] . The proposed OBEDO approach makes use of SSO to identify the optimal experimental scheme E^* according to the expected utility, where only one-stage optimization is involved, and returns the one with the maximum expected utility as the E^*, which contains the optimal control and additional measuring points.

Within the SSO framework, E^* can be found among the candidate schemes by solving the following reliability analysis problem [14] :

$P\left(F\right)=P\left(U\left(E\right)>U\left(E^{*}\right)\right)$ (12)

where $F=\left\{U\left(E\right)>U\left(E^{*}\right)\right\}$ is an auxiliary failure event. $P\left(F\right)$ represents the probability that event F occurs, which becomes zero as scheme E is equal to E^*. A number of conditional samples of a series of nested intermediate failure events satisfying $F_1\supset F_2\supset F_3\supset \cdots F_{N_s}=F$ is generated with SSO, with which $P\left(F\right)$ is expressed as Eq. (13):

$P\left(F\right)=P\left(F_{N_s}\right)=P\left(F_1\right){\displaystyle \underset{m=2}{\overset{N_s}{\prod }}P\left(F_m|F_{m-1}\right)}$ (13)

where $F_m=\left\{U\left(E\right)>U_m\left(E\right)\right\}$, $m=1,2,3,\cdots ,N_s$. $P\left(F_1\right)$ is equal to $P\left(U\left(E\right)>U_1\left(E\right)\right)$; $U_1\left(E\right)<U_2\left(E\right)<\cdots <U_{N_s}\left(E\right)=U\left(E^{*}\right)$ are an increasing sequence of N_s intermediate threshold values, which are determined adaptively with simulated samples so that the sample estimates of $P\left(F_1\right)$ and $P\left(F_m|F_{m-1}\right)$ are always equivalent to a specific value of conditional probability p₀ (e.g., 0.1) [15] . The implementation of SSO can refer to Li and Au (2010) [14] and Ding et al. (2022) [8] .

This paper developed a one-stage Bayesian experimental design optimization (OBEDO) approach for determining the optimal experimental scheme of SWCC test using the prior knowledge and the information of testing apparatus. The candidate experimental scheme with the maximum expected utility is identified as the optimal experimental scheme using Subset Simulation optimization (SSO).

1) The proposed OBEDO approach was illustrated using a design example. It was shown that the expected utility of the optimal experimental scheme improves by adding more measurements. Such an improvement becomes marginal as the number of measuring points is sufficiently large (e.g., 17 in the illustrative example). Hence, the optimal number of measuring points can be determined as a trade-off between the improvement of data worth and the commitment involved in testing.

2) The proposed approach was also verified using real loess data. Results showed that the arbitrary arrangement of measuring points of SWCC test is hardly to give the optimal experiment scheme in terms of the expected utility (or values of information). The proposed OBEDO approach provides a rational tool to optimize the arrangement of measuring points of SWCC test based on prior knowledge and the information of testing apparatus so as to obtain SWCC measurement data with relatively high value of information for uncertainty reduction.