= Mean-potential correction to weak Rytov intensity
{title2=$\log I=\int h\overline V+\tfrac12\iint hhC_V+O(\mu^4)$}
For $n=1+\mu W$ with jointly centered unit-variance <Gaussian random field> $W$, the <scattering potential> $V=k_0^2(n^2-1)$ has mean $\mu^2k_0^2$ and covariance $4\mu^2k_0^4C_W+2\mu^4k_0^4C_W^2$. It is not Gaussian. In the far-field <Rytov approximation> let $h(y)=2\operatorname{Re}[B e^{-iq\cdot y}]$, so intensity is $\mathbb E\exp(\int hV)$. Expansion through order $\mu^2$ includes both the mean term and the covariance term shown. The remainder starts at order $\mu^4$ under suitable exponential integrability. Keeping a covariance correction while omitting the same-order mean term is inconsistent. The exact quadratic model uses the <Gaussian quadratic exponential moment>.
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