Summary: The class of stochastic non-gaussian positive-definite fields with minimal parameterization proposed by C. Soize [Comput. Methods Appl. Mech. Eng. 195, No. 1–3, 26–64 (2006; Zbl 1093.74065)] to model the elasticity tensor field of a random anisotropic material shows an anisotropy index which grows with the fluctuation level. This property is in contradiction with experimental results in geophysics where the anisotropy index remains limited whatever the fluctuation level. Hence, the main purpose of this paper is to generalize the Soize’s model in order to account independently for the anisotropy index and the fluctuation level. It is then shown that this new model leads to major differences in the wave propagation regimes.