# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 1 Introduction  
Due to internal material composition, stress environment, sedimentary conditions, weathering degree and burial conditions, natural soil materials show obvious randomness and spatial variability [1; 2; 3; 4; 5; 6]. It is widely acknowledged that these two features make the real distribution of soil parameters more complex and trigger inevitable uncertainties [4; 7; 8; 9; 10]. Ignoring such uncertainties would lead to considerable deviation in geotechnical structure analyses, such as slope reliability. Therefore, it is essential that the accurate simulation of soil parameters with consideration of multisource uncertainties be conducted. At present, the random variable model (RVM) and random field model (RFM) are commonly utilized to simulate spatially variable geotechnical parameters [11; 12; 13; 14]. RVM assumes that each soil parameter in the same layer is homogeneous [11]. Its randomness and spatial variability are mainly characterized by random variables obeying specific probability distribution types, such as truncated normal (TN), lognormal (LN), truncated Gumbel (TG) and Weibull (WB) distributions [9; 15]. However, soil parameters at any space location are supposed to be mutually independent of each other. This obviously does not reflect the real soil state. RFM is known to play an important role in modeling the spatial variability of soil parameters, and was first proposed by [PERSON] in 1977 [13]. Comparedto RVM, it can account for the spatial correlation in the study area and can better describe spatial variability. Presently, great effort has been made in the generation of random fields of soil parameters [16; 17; 18; 19; 20; 21; 22; 23]. For instance, [PERSON] et al. [19] studied rock slope stability by combining RFM with limit equilibrium methods. [PERSON] et al. [20] used the Fourier series method to generate an anisotropic random field. [PERSON] et al. [21] combined the K-L series expansion method, the polynomial chaos expansion method and the Markov Chain Monte Carlo method to simulate spatially variable slopes. [PERSON] et al. [22] simulated the spatially variable soil and carried out slope reliability analysis by the three-dimensional stochastic finite element method. [PERSON] et al. [23] established an effective non-stationary random field model for undrained shear strength parameters. It is worth noting that the correlation of soil parameters is a crucial characteristic of the random field, including autocorrelation and cross-correlation [24]. The former characterizes the correlation of individual soil parameters at different spatial locations, mainly determined by the autocorrelation structure of soil parameter. The latter maintains that soil parameters also tend to be dependent on each other at a given spatial location or in neighborhoods. Both of them act as crucial constraints when simulating random fields. Many studies have claimed that interdependency among soil parameters could affect the distribution of random fields [4; 24]. The challenge then for the geotechnical engineer is to model the spatially variable behavior of soil parameters that exhibit complex correlations. Unfortunately, cross-correlation is seldom considered in the simulation of random fields, which inevitably leads to huge bias in the generation of random fields. Such bias may further develop and expand in subsequent geotechnical structure analysis. These challenges highlight the significance of characterizing the spatial variability of soil profiles accounting for both autocorrelation and cross-correlation. Thus, a need to quantify both spatial variability and cross-correlation of soil parameters has arisen. Fortunately, similar but limited efforts have been undertaken by some scholars [7; 25; 26]. [PERSON] and [PERSON] [7] decomposed correlation matrices of parameters by Cholesky decomposition, and then simulated the cross-correlated random field (CCRF) using lower triangular matrices.