In view of approximating the fractional Brownian process in the plane B. Mandelbrot examined the superimposition of rectilinear faults, centered on the axis of a Poisson process with arbitrary large rate, the profiles of which being of the type \(Q\cdot sgn(x)| x|^{\alpha}\), \(| \alpha | <1/2\) and Q a real valued random variable. In the following, we derive from general hypotheses a formula which characterises any random profile leading to a Gaussian process of given type and thus providing explicit examples of profiles, thinkably less contrived than the former; some results on the quality of convergence are given.