Online generation of temporal and spatial fractal red noise

It is relatively easy to generate fields with fractal noise using Fourier, wavelet or cascade algorithms (see also my page on cloud generators). However, these noise fields have to be calculated fully before their first use. This can be problematic in case a 2D or 3D spatial field is needed as input to a dynamical model that should also have a temporal fractal structure. Often the number of temporal time steps is so large, that the noise field become impractically large. Therefore, this essay introduces an online method to generate fractal red noise fields, where the new field is calculated from the previous field without the need to know all previous fields; only the current and the new field have to be stored in memory.
The algorithm involves two steps. The spatially correlated red noise is calculated from a white noise field using Fourier filtering. The white noise field evolves temporally correlated by addition of Gaussian noise. In other words, every pixel of the white noise field represents a fractal Brownian noise time series. Fortunately, the spatially correlated noise field retains the fractal nature of the temporal structure of the white noise field.