= Weak-fluctuation memory equation for the coherent field
For a <parabolic wave equation> with $n=1+\mu W$, a zero-mean stationary real field $W$ of unit variance, and $L_0=i\partial_z^2/(2k)$, finite-correlation perturbation theory gives
$$
\begin{aligned}
m_x&=L_0m+\frac{ik\mu^2}{2}m\\
&\quad-k^2\mu^2\int_{x_0}^x\int_{\mathbb R}C(x-s,z-\zeta)G_{x-s}(z-\zeta)m(s,\zeta)\,d\zeta\,ds+O(\mu^3).
\end{aligned}
$$
Here $m$ is the <coherent field>, $C$ is the <autocorrelation function of a random field> and $G$ is the one-coordinate <Fresnel propagator>. Expand the random solution once using the <Duhamel principle>, multiply by $W$ and average to obtain the memory term. The local phase term comes from the $\mu^2W^2$ term in $n^2$. The expansion is for fixed propagation distances with suitable covariance regularity and moment bounds. A <Markov approximation> is an additional scale assumption that can turn this integral equation into a local attenuation equation.
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