Two-stage Bayesian experimental design optimization for measuring soil–water characteristic curve

Abstract

Direct measurements of soil–water characteristic curve (SWCC) are often costly and time consuming. As a result, a limited number of test data are usually obtained from a single SWCC test, based on which the estimated SWCC is unavoidably associated with uncertainties. It is, hence, prudent to deliberately plan the experimental scheme (i.e., specify the values of the control variable of measuring points) so as to improve the expected value of information of measurement data for reducing the uncertainty in estimated SWCC. This paper developed a Bayesian experimental design optimization (BEDO) approach for SWCC testing based on prior knowledge and information of testing apparatus. The proposed BEDO approach is divided into two stages to respectively identify initial measuring points for controlling the general trajectory of SWCC and determine additional measuring points for further reducing the uncertainty in SWCC. The experimental scheme with the maximum expected utility is identified as the optimal experimental scheme using Subset Simulation optimization in each stage. The proposed approach is illustrated using an experimental design example. Results show that it provides a rational tool to determine the optimal experiment scheme for SWCC testing in terms of the expected utility.

Appendix 1: Derivation of the ψ i, ψ b, ψ r corresponding to VGM

The VGM is expressed as

$${S}_{e}={\left[\frac{1}{1+{\left(\psi /{\alpha }_{vm}\right)}^{{n}_{vm}}}\right]}^{1-1/{n}_{vm}}$$

(22)

At the inflection point, the ${\psi }_{i}$ satisfies (Zhai et al. 2017)

$${\left.\frac{{d}^{2}{S}_{e}\left(\psi \right)}{d{\psi }^{2}}\right|}_{\psi ={\psi }_{i}}\cdot {\psi }_{i}+{\left.\frac{d{S}_{e}\left(\psi \right)}{d\psi }\right|}_{\psi ={\psi }_{i}}=0$$

(23)

where ${\left.\frac{d{S}_{e}\left(\psi \right)}{d\psi }\right|}_{\psi ={\psi }_{i}}$ and ${\left.\frac{{d}^{2}{S}_{e}\left(\psi \right)}{d{\psi }^{2}}\right|}_{\psi ={\psi }_{i}}$ are respectively the first and second derivatives of the VGM evaluated at ${\psi }_{i}$, and they are given by

$$\frac{d{S}_{e}}{d\psi }=\frac{1-{n}_{vm}}{{\alpha }_{vm}}{\left[1+{\left(\psi /{\alpha }_{vm}\right)}^{{n}_{vm}}\right]}^{1/{n}_{vm}-2}{\left(\frac{\psi }{{\alpha }_{vm}}\right)}^{{n}_{vm}-1}$$

(24)

$$\frac{{d}^{2}{S}_{e}}{d{\psi }^{2}}=\frac{1-{n}_{vm}}{{\alpha }_{vm}}\left\{\begin{array}{l}\left(1/{n}_{vm}-2\right){\left[1+{\left(\psi /{\alpha }_{vm}\right)}^{{n}_{vm}}\right]}^{1/{n}_{vm}-3}\cdot {n}_{vm}{\left(\psi /{\alpha }_{vm}\right)}^{{n}_{vm}-1}\cdot \frac{1}{{\alpha }_{vm}}\cdot {\left(\psi /{\alpha }_{vm}\right)}^{{n}_{vm}-1}+\\ {\left[1+{\left(\psi /{\alpha }_{vm}\right)}^{{n}_{vm}}\right]}^{1/{n}_{vm}-2}\left({n}_{vm}-1\right){\left(\frac{\psi }{{\alpha }_{vm}}\right)}^{{n}_{vm}-2}\cdot \left(\frac{1}{{\alpha }_{vm}}\right)\end{array}\right\}$$

(25)

Substituting Eqs. (24) and (25) into Eq. (23) gives

$$\begin{array}{l}\frac{1-{n}_{vm}}{{\alpha }_{vm}}{\left[1+{\left({\psi }_{i}/{\alpha }_{vm}\right)}^{{n}_{vm}}\right]}^{1/{n}_{vm}-2}{\left(\frac{{\psi }_{i}}{{\alpha }_{vm}}\right)}^{{n}_{vm}-1}+\\ {\psi }_{i}\cdot \frac{1-{n}_{vm}}{{\alpha }_{vm}}\left\{\begin{array}{l}\left(1/{n}_{vm}-2\right){\left[1+{\left({\psi }_{i}/{\alpha }_{vm}\right)}^{{n}_{vm}}\right]}^{1/{n}_{vm}-3}\cdot {n}_{vm}{\left({\psi }_{i}/{\alpha }_{vm}\right)}^{{n}_{vm}-1}\cdot \frac{1}{{\alpha }_{vm}}\cdot {\left({\psi }_{i}/{\alpha }_{vm}\right)}^{{n}_{vm}-1}+\\ {\left[1+{\left({\psi }_{i}/{\alpha }_{vm}\right)}^{{n}_{vm}}\right]}^{1/{n}_{vm}-2}\left({n}_{vm}-1\right){\left(\frac{{\psi }_{i}}{{\alpha }_{vm}}\right)}^{{n}_{vm}-2}\cdot \left(\frac{1}{{\alpha }_{vm}}\right)\end{array}\right\}=0\end{array}$$

(26)

The Eq. (26) can be further simplified as

$${n}_{vm}\left[1+{\left({\psi }_{i}/{\alpha }_{vm}\right)}^{{n}_{vm}}\right]+\left(1-2{n}_{vm}\right){\left({\psi }_{i}/{\alpha }_{vm}\right)}^{{n}_{vm}}=0$$

(27)

Using Eq. (27), the ${\psi }_{i}$ can be calculated for a given set of VGM parameters, i.e., ${\alpha }_{vm}$ and ${n}_{vm}$. Moreover, the k₁ (see Fig. 4) can be defined as follows (Zhai and Rahardjo 2012):

$${k}_{1}=\frac{1-{S}_{e,i}}{\mathrm{log}\left({\psi }_{i}\right)-\mathrm{log}\left({\psi }_{b}\right)}$$

(28)

Therefore, the air-entry value ${\psi }_{b}$ is given by

$${\psi }_{b}={\psi }_{i}{0.1}^{\frac{1-{S}_{e,i}}{{k}_{1}}}$$

(29)

In addition, the k₁ and k₂ (see Fig. 4) can be written as (Zhai and Rahardjo 2012)

$${k}_{1}=\frac{{S}_{e,i}-{S}_{e,r}}{\mathrm{log}\left({\psi }_{r}\right)-\mathrm{log}\left({\psi }_{i}\right)}$$

(30)

$${k}_{2}=\frac{{S}_{e,r}-{S}_{e}^{^{\prime}}}{\mathrm{log}\left({\psi }^{^{\prime}}\right)-\mathrm{log}\left({\psi }_{r}\right)}$$

(31)

Substituting Eq. (30) into Eq. (31) gives

$${k}_{2}\left[\mathrm{log}\left({\psi }^{^{\prime}}\right)-\mathrm{log}\left({\psi }_{r}\right)\right]={S}_{e,i}-{S}_{e}^{^{\prime}}-{k}_{1}\left[\mathrm{log}\left({\psi }_{r}\right)-\mathrm{log}\left({\psi }_{i}\right)\right]$$

(32)

Using Eq. (32), the residual suction ${\psi }_{r}$ is written as

$${\psi }_{r}={10}^{\frac{{S}_{e,i}-{S}_{e}^{^{\prime}}+{k}_{1}\mathrm{log}\left({\psi }_{i}\right)-{k}_{2}\mathrm{log}\left({\psi }^{^{\prime}}\right)}{{k}_{1}-{k}_{2}}}$$

(33)

Equations (27), (29), and (33) allow calculating the ${\psi }_{i}$, ${\psi }_{b}$, and ${\psi }_{r}$ for a given set of ${\alpha }_{vm}$ and ${n}_{vm}$, respectively.