First-Rytov coherent mean ($\mathbb{E}[{\psi }_R]={\psi }_i\exp ({\mu }^2\mathbb{E}[{\chi }_1^2]/2)$)

If the first logarithmic perturbation ${\chi }_1$ is a centered linear functional of a Gaussian random field, the complex Gaussian exponential moment gives the displayed mean of the first Rytov approximation ${\psi }_R={\psi }_i{e}^{\mu {\chi }_1}$. This is the mean of the truncated logarithmic model. A nonzero mean of the omitted second logarithmic perturbation can change the true field mean at order ${\mu }^2$.