Explicit form of the random field spectral representation and some applications

We present here an explicit form of the random spectral measure element, what allows us to express a stationary random field as a stochastic integral explicitly depending on its power spectrum and a spectral tensor if the field is a vector one. It has been shown here that convergence mechanism of such integral is significantly different from the one of the Fourier transform and that the traditional formalism is a partial limiting case of the one presented here. The fact that there is an explicit expression of a random field makes calculation of higher order statistics of it much more straightforward (see for example Chepurnov et al. 2020). For a vector field such expression contains a projection of an isotropically distributed random vector by a spectral tensor, what makes geometrical interpretation of harmonics behavior possible, simplifying its analysis (see Sect. 2). This spectral representation also makes straightforward numerical generation of a random field, what is extensively used by Chepurnov et al. 2020. We also present here some practical applications of this formalism.