Rytov approximation

The Rytov approximation writes a total wave field as $\psi ={\psi }_i{e}^{\varphi }$ and linearizes the equation for the complex logarithmic perturbation $\varphi$. At first order, ${\psi }_i\varphi$ equals the first Born approximation scattered field, soIt can remain useful when phase accumulation is appreciable but local scattering is weak.

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 ${\mu }^2$.

For a centered real scattering potential fluctuation $W$ and the imaginary part $b$ of the incident-field-weighted Green function, $\phi (\mathbf{r})=-\int b(\mathbf{r},{\mathbf{r}}^{\prime })W({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. Its mean vanishes, and its covariance function is the double integral of the two deterministic kernels against the covariance function of $W$. The correlation of the refractive index itself is insufficient when the scattering potential depends quadratically on that index.

The Rytov approximation omits the quadratic gradient source in the logarithmic wave perturbation equation. Its next correction solves ${\mathcal{L}}_i{\chi }_2=-\nabla {\chi }_1\cdot \nabla {\chi }_1$. Control of the propagated correction, weak amplitude fluctuations and avoidance of field zeros are needed. An appreciable smooth accumulated phase can be retained by exponentiation; small total phase is not the same condition as small local phase gradients.

The logarithmic wave perturbation measures relative complex amplitude and phase through $\psi ={\psi }_i{e}^{\chi }$. A continuous branch requires nonzero incident and total fields in the region considered. For the Helmholtz equation with scattering potential convention $L_0\psi =-V\psi$, its equation is $(\Delta +2\nabla \log {\psi }_i\cdot \nabla )\chi +\nabla \chi \cdot \nabla \chi =-V$. The dot product is bilinear, not a squared modulus.

Let ${\psi }_R={\psi }_i{e}^{{\varphi }_1}$ with unit-modulus incidence and ${\varphi }_1={\int }_{D}K(\mathbf{r},{\mathbf{r}}^{\prime })V({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. For a centered real Gaussian random field $V$, the exponential moment of a Gaussian linear functional givesHere $C_V$ is the autocorrelation function of a random field. Both the covariance $\mathbb{E}|{\varphi }_1{|}^2$ and the generally nonzero term $\mathrm{Re}\mathbb{E}{\varphi }_1^2$ enter the exponent; dropping the latter is not justified for an arbitrary finite real random medium.

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 ${\mu }^2{k}_0^2$ and covariance $4{\mu }^2{k}_0^4{C}_W+2{\mu }^4{k}_0^4{C}_W^2$. It is not Gaussian. In the far-field Rytov approximation let $h(y)=2\mathrm{Re}[B{e}^{-iq\cdot y}]$, 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.