Generation of homogeneous 3D Gaussian noise with spherically symmetric covariance

In this paper, an approach for 3D noise generation is presented. The proposed algorithm might be a useful tool for the generation of correlated phase screens. These phase screens can be used for the simulation and modeling of optical wave propagation through atmospheric turbulence. Arbitrary user-defined covariance functions between voxel pairs can be achieved. Correlated 3D noise is formed by superposition of multiple uncorrelated 3D Gaussian noise patterns. These uncorrelated input noise patterns are of different dimensions. They are upsampled to the same target dimensions by linear interpolation. Each input pattern then contributes to total covariance on different spatial scales. The covariances between different voxels are expressed analytically by propagation of error. For a subset of randomly chosen voxels in the entire voxel space, relative deviations between the analytical and user-defined covariances are calculated. A sum of squares of these relative deviations is then minimized by machine learning methods. The optimized parameters are the weighting factors of individual uncorrelated 3D noise patterns. Corresponding covariance functions are numerically evaluated for two current atmospheric turbulence spectra. The first one is the generalized modified atmospheric spectrum. The second one is the generalized modified von Karman spectrum. Based on these covariance functions, optimal superpositions are calculated. Finally, statistical properties of these patterns are validated by ensemble sample covariance analysis.