Characterization of inherent spatial variability of loess deposit properties in Shaanxi Province, China

Abstract

Purpose

Inherent spatial variability of properties of loess deposit along depth direction plays important roles in soil reclamation, agricultural irrigation, and risk assessment and management of geo-hazards on the Loess Plateau. This study aims to develop a database for loess deposit properties in Shaanxi Province first, followed by evaluating and reporting comprehensively the abovementioned spatial variability along depth.

Materials and methods

A comprehensive literature review and numerous laboratory tests were conducted for 37, 81, and 177 loess profiles respectively for sandy, silty, and clayey loess to examine their properties in Shaanxi Province, China. For each property collected in the database, the means, coefficients of variation (COV), most suitable probability distribution functions, and spatial correlation lengths along the depth direction were evaluated and reported.

Results and discussion

The physical and index properties and strength properties of the three loess soil generally exhibit relatively low variabilities with a mean COV of less than 20.0% in most cases. In contrast, the deformation properties exhibit medium to high variabilities with a mean COV varying from roughly 27.0 to 85.0%. For all of these properties mentioned above, the spatial correlation length along depth was estimated to be approximately 3 ~ 8 m.

Conclusions

(1) The COV of loess properties at one site can be significantly different from that at other sites. Thus, the site-specific variabilities of loess deposit properties shall be considered in geological design and analysis. (2) When site-specific data of properties for loess deposits are not available for on-going geological/geotechnical projects, the typical ranges of loess properties obtained in this study, including their statistics and correlation lengths, can be used as guidelines for approximation.

Appendix

A Bayesian framework was adopted to estimate the spatial auto-correlation length λ of loess properties. In a Bayesian paradigm, updated information on parameters of interest was quantitatively characterized by its posterior probability density function (PDF), which is expressed as (e.g., Ang and Tang 2007; Zhao et al. 2023).

$$P(\lambda |Data)=\frac{P(Data|\lambda )P(\lambda )}{P(Data)}$$

(A1)

where Data = {(x₁, y₁), …, (x_n, y_n)} represents measurements at different depths; P(λ) is the prior PDF of λ, representing the prior knowledge in the absence of measurements; P(Data|λ) is the likelihood function, expressing the plausibility of observing “Data” given a particular λ; P(Data) is a normalizing constant, ensuring integration of P(λ|Data) with respect to λ equaling to one. Due to the theorem of central limits, loess properties tend to be normally or log-normally distributed, which is similar to that of other soils (Lacasse and Nadim 1996). In this case, the likelihood function is expressed as

$$P(Data\vert\lambda)=\frac1{\sqrt{\det(2\pi\Sigma)}}\exp\left(-\frac{\left(y-\mu\right)^{\mathrm T}\Sigma^{-1}\left(y-\mu\right)}2\right)$$

(A2)

where y represents y₁(x₁), y₂(x₂), …, y_n(x_n) or its logarithm in a column vector format; μ is the mean of loess property of interest, which has been computed in Eq. (1); $\Sigma \text{ = }{\sigma }^{2}R$, where $\sigma$ is obtained from Eq. (1), and R_i,j is the correlation coefficient calculated using Eq. (3) in the main text. In addition to the likelihood function in Eq. (A2), the prior PDF of λ is constructed as below:

$$P(\lambda)=\left\{\begin{array}{cc}\frac1{\lambda_{\max}-\lambda_{\min}}&\lambda\in\left[\lambda_{\min},\lambda_{\max}\right]\\0&otherwise\end{array}\right.$$

(A3)

where λ_min and λ_max are respectively the lower and upper bounds of correlation length for a particular loess property of interest, which can be obtained from literature (Phoon and Kulhawy 1999a, b; Cao et al. 2016). In this study, λ_min and λ_max are taken as 0.5 m and 9.5 m, respectively, for all loess properties, which are consistent with the typical ranges of correlation length reported in Phoon and Kulhawy (1999a, b). Given the prior PDF in Eq. (A3), and likelihood function in Eq. (A2), the posterior PDF of λ can be characterized by Eq. (A1) using the Bayesian theorem. Because the prior PDF in Eq. (A3) is often not conjugate to the likelihood function in Eq. (A2), the posterior PDF obtained is intractable. In this case, numerical integration or MCMC method may be used to depict the posterior PDF of λ (e.g., Ching and Chen 2007; Straub and Papaioannou 2015; Tian et al. 2016), from which its mean and SD can be obtained (e.g., Ang and Tang 2007). Note that the presented method above has been tested and validated before using it for estimating correlation length of various loess properties in the manuscript. Besides, it is worth noting that de-trending shall be carried out first to measurements when obvious trend functions are clearly identified in the measurements. In this paper, Bayesian evidence was computed to select the optimal trend function (in terms of the first or second order polynomial function) when probabilistically estimating λ. Details for evidence computation is referred to Yan et al. (2009), Cao and Wang (2013), Schöniger et al. (2014), among others.