Generation of multivariate (cross-correlated) Gaussian random fields

Input parameters:

• Domain size ($Lx,Ly$)
• Random field data:
• Number of cross-correlated random fields $n$
• Vectors containing the means, standard deviations, correlation lengths ($x$ and $y$ directions)
• Cross correlation coefficient matrix
• Matern smoothness parameter for the adopted Matern kernel

Method:

1. Compute cross- correlation matrix
2. Convert to positive semi definite if it is not
3. Perform eigenvalue decomposition
4. Generate Multivariate random fields via discrete KL expansion

Example

• For the considered application ($n = 8$ cross-correlated ABD matrix components):

$8\times8$ Cross-correlation matrix

• Autocorrelations lie on the diagonal region
• Strong correlations between components are evident (light green/ yellow regions)

Generated multivariate random field realization

• Standard Gaussian random fields are depicted
• Strongly correlated random fields are remarkably similar (e.g., $A_{11}, D_{11}$)