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Write $\psi =e^{ikx}E(x,\mathbf{s})$, $\mathbf{s}=(y,z)$. In the weak-contrast approximation $n^2\simeq 1+2\mu W$, neglecting $E_{xx}$ relative to $k{E}_x$ gives the parabolic wave equationFor the mutual coherence $\Gamma (x;{\mathbf{s}}_1,{\mathbf{s}}_2)=\mathbb{E}[E_1{E}_2^{\ast }]$, differentiating gives the exact moment equation within this wave approximation,This is an evolution equation but not a closed one. Even a jointly Gaussian random field does not make $W$ independent of $E$, which depends on its history. Transverse statistical homogeneity alone therefore does not close the second moment. A covariance or longitudinal Markov approximation is additional information.

Under the white-in-$x$ assumption introduced in the next part, interpret $Wdx=d{\mathcal{B}}_x$, a longitudinal Brownian field with transverse covariance $C$, and use the Stratonovich integral as the smooth-medium limit. Conversion to the Itô integral yieldsThe product rule with quadratic variation gives the closed mutual coherence in a white-noise parabolic medium equation,The two attenuation drifts contribute $-k^2{\mu }^{2}C(0)$ and the cross variation contributes $+k^2{\mu }^{2}C({\mathbf{s}}_1-{\mathbf{s}}_2)$. The coefficient $C$ is the white-noise covariance intensity; it replaces, rather than literally retains, the original finite pointwise variance of $W$.