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The fourth moment plays an important role in wave propagation and scattering in random media. However, in the strong scattering regimes, it remains to be solved although a rich variety of methods have been proposed. The fourth moment equation is solved using the modified Gaussian solution method proposed in this paper. The fourth moment can be considered as a sum of the Gaussian solution and non-Gaussian correction term. The Gaussian solution is the fourth moment in the saturated regime, and can be obtained as a sum of products of the second-order moments. The derived equation of the non-Gaussian correction term is solved using the Rytov approximation and the Green's function. The fourth moment solution obtained is general for without restrictions on incident wave forms. Specifically, the solution for a plane wave is derived and applied to study the amplitude scintillation and intensity correlation function of trans ionospheric radio signals. The theoretical calculation on the scintillation index agrees well with the experimental results, which confirms the fourth moment solution to some extent.