Efficient simulation of multivariate three-dimensional cross-correlated random fields conditioning on non-lattice measurement data

Abstract

It is challenging to simulate large-scale or fine-resolution multivariate three-dimensional (3D) cross-correlated conditional random fields because of computational issues such as inverting, storing or Cholesky decomposition of large correlation matrices. Recently, an efficient univariate 3D conditional random field simulation method was developed based on the separability assumption of the autocorrelation functions in the vertical and horizontal directions. The developed simulation method allows for Kronecker-product derivations of the large correlation matrices and thus does not need to invert and store large matrices. Moreover, it can handle univariate non-lattice data (e.g., all soundings measure the data of one soil property and there exists missing data at some depths at some soundings). It may be more common to see multivariate non-lattice data (e.g., all soundings measure the data of multiple soil properties and there exists missing data of some properties at some depths at some soundings) in practical site investigations. However, the proposed method is not applicable to multivariate non-lattice data because it cannot directly account for the cross-correlation among different variables The purpose of the current paper is to extend the previous method to accommodate the multivariate non-lattice data. The extended method still takes advantage of the Kronecker-product derivations to avoid the mathematical operation of the large correlation matrices. A simulated example is adopted to illustrate the effectiveness of the extended method.

Introduction

Spatial variability of soil properties has a significant influence on the failure mechanism and reliability of geotechnical structures such as foundations (e.g., [1], [2], [3]), tunnels (e.g., [4], [5], [6]), retaining wall (e.g., [7], [8], [9]) and slopes (e.g., [10], [11], [12], [13], [14], [15]). It is important to account for the spatial variability of soil properties when performing the geotechnical reliability and risk analysis. The conventional geotechnical reliability analysis often simplifies the spatial variability of soil property as a one-dimensional (1D) [16], [17] or two-dimensional (2D) problems [18], [19], [20] although soil properties generally exhibit a three-dimensional (3D) spatial variability. Some recent studies have recognized that the sophisticated 3D spatial variability cannot be fully represented by the simplified 1D or 2D spatial variability (e.g., [14], [21], [22], [23], [24]). To effectively capture the failure mechanism and accurately estimate the failure risk of geotechnical structure, it entails the modeling of the 3D spatial variability, which can be represented by a 3D random field. Moreover, site investigation for many realistic sites is generally performed by various test methods such as cone penetration test (CPT), standard penetration test (SPT), vane shear test (VST), and/or laboratory triaxial compression test. The associated sites often contain observed/measured data for multiple soil property parameters. It is also vital to simulate multivariate cross-correlated 3D random fields conditioning on the observed multivariate data.

Generally, the multivariate cross-correlated 3D conditional random field simulation might be subject to two difficulties: cross-correlated random field simulation and conditional random field simulation. In terms of the former, various cross-correlated random field simulation methods have been proposed. The conventional random field simulation method such as Cholesky decomposition, local average subdivision (e.g., [25]), Karhunen-Loève expansion (e.g., [26]) and spectral representation (e.g., [27]) methods can be readily extended to simulate the cross-correlated random field by taking the cross-correlated matrix into consideration during the random field generation process. For instance, Robin et al. [28] proposed a Fourier transform-based cross-correlated random field simulation method. Vořechovský [29] proposed a series expansion-based cross-correlated random field simulation method. Zhao and Wang [30] proposed a Bayesian compressive sampling and Karhunen–Loève (KL) expansion-based cross-correlated random field simulation method. These methods are frequently used for simulating the 2D cross-correlated random fields in geotechnical engineering (e.g., [10], [31], [32]). These methods generally are efficient when simulating 3D cross-correlated unconditional random fields. However, they might be highly computationally intensive for simulating 3D cross-correlated conditional random fields especially for cases with large-scale domains. In terms of the latter, various conditional random field simulation methods such as Kriging methods (e.g., [33], [34], [35]), patching algorithm-based method [36], Bayesian methods (e.g., [37], [38]) and some non-parametric methods (e.g., [39], [40], [41]) and have been proposed. These methods are commonly utilized to simulate a 2D conditional random field although they might be extended to simulate a 3D conditional random field for a domain with a relatively small scale or with a low resolution (e.g., [37], [42], [43], [44]). Nevertheless, they are not applicable to the 3D conditional random field simulation for a domain with a large scale or with a fine resolution. In fact, due to the significant computational cost required in 3D problem, even the simulation of a 3D unconditional random field over a large domain with a high resolution is challenging with the existing random field simulation methods such as Cholesky decomposition, Karhunen-Loève expansion (e.g., [26]) and spectral representation (e.g., [27]). The local average subdivision (e.g., [25]) is computational efficiency for 3D problem; however, it requires a uniform grid to simulate the random field. Recently, this computational challenge of 3D unconditional random field simulation is addressed by Cheng et al. [5] and Li et al. [45]. Cheng et al. [5] suggested simulating each sub-domain at the 3D space sequentially by conditioning on neighboring sub-domains with distances less than a few times of the scale of fluctuation (SOF). This method still requires considerable computational cost if SOF is large. Li et al. [45] proposed a stepwise covariance matrix decomposition (CMDC) method with the separability assumption of the autocorrelation function in the horizontal and vertical directions. The CMDC method requires the 3D random field to be simulated follows a 3D lattice structure so that it can take advantages of the Kronecker-product derivations to decompose the correlation matrices. As a result, the CMDC does not need to invert and store large matrices and its computational cost is independent of the SOF. Noted that the above-mentioned studies [5], [45] did not address the simulation of a 3D conditional random field. Zhao and Wang [40], [41] developed non-parametric methods for simulating 3D conditional random field samples from measurement data. Although this method bypasses the usage of parametric correlation model, it is computationally inefficient. Ching et al. [46] further proposed a method to address the computational inefficiency faced by the 3D conditional random field based on two assumptions of (a) the separable auto-correlation and (b) the regular-3D-lattice-distributed sounding data. The first assumption might be not strong while the second assumption is a little strong because the measured sounding data in geotechnical practice generally have a non-lattice 3D structure. To relax the second assumption, Yang and Ching [47] further proposed an efficient univariate 3D conditional random field simulation method. Their method is admitted by the 3D non-lattice sounding data. However, both Zhao and Wang [40], [41], Ching et al. [46] and Yang and Ching [47] did not address the simulation of multivariate 3D cross-correlated conditional random fields.

In practice, it is not uncommon that the measured sounding data are multivariate and not with a regular 3D lattice structure. Consider a schematic diagram for sounding data at a typical site shown in Fig. 1. The site investigations are performed by various tests including VST, SPT, CPT and laboratory triaxial compression test. As a result, multiple soil property parameters are measured at this site. If the random fields of multiple soil strength parameters such as undrained shear strength $s_{\mathrm{u}}$, cohesion $c$, friction angle $\varphi$ and elastic modulus $E$ are concerned, the measured multiple soil property data by VST, SPT, CPT and laboratory triaxial compression might be converted to the soil strength parameters through certain transformation relationships (e.g., [48], [49], [50]). For illustration, it is supposed that the cross-correlated 3D conditional random fields of two soil property parameters $X_1$ and $X_2$ at this typical site are to be simulated. $X_1$ and $X_2$ can be the measured and/or transformed soil strength data based on the measured sounding data obtained from the VST, SPT, CPT or laboratory triaxial compression tests. Suppose that the sounding S1, S2, (S3, S4) and (S5, S6) contain the data collected from VST, laboratory triaxial compression test, CPT test and SPT test, respectively. It is not uncommon that different test methods (e.g., SPT, VST, CPT and laboratory triaxial compression tests) have different sampling depth intervals. Moreover, the same test method (e.g., CPT) may also have different sampling depths at different soundings (e.g., S3 and S4). When these non-lattice test data are converted to $X_1$ and $X_2$ data, $X_1$ and $X_2$ data are also not regularly distributed. That is to say, some $X_1$ and $X_2$ data are missing or have unequal spacing. For example, the missing data that cause the non-lattice structure of $X_1$ and $X_2$ are shown as open circles and open triangles in Fig. 1. This non-lattice structure hinders the application of the Kronecker product in decomposing the auto-covariance matrix, hence prohibits an effective simulation of multivariate 3D cross-correlated conditional random fields.

This paper aims at extending the Yang and Ching [47]’s method to simulate the multivariate 3D cross-correlated random field conditioning on irregularly distributed (non-lattice) sounding data. The extended method does not need to invert and store large matrices, either. This paper starts with the problem statement, followed by a brief review of the univariate 3D conditional random field simulation method proposed in [47]. The modified method is then illustrated with the typical site data shown in Fig. 1. A simulated example is employed to demonstrate the effectiveness of the modified method.