= Gaussian intensity in the first Rytov approximation
{c}
Let $\psi_R=\psi_i e^{\phi_1}$ with unit-modulus incidence and $\phi_1=\int_DK(\mathbf r,\mathbf r')V(\mathbf r')\,d\mathbf r'$. For a centered real <Gaussian random field> $V$, the <exponential moment of a Gaussian linear functional> gives
$$
\mathbb E|\psi_R|^2=\exp\left[\frac12\int_D\int_D a(\mathbf r')a(\mathbf r'')C_V(\mathbf r'-\mathbf r'')\,d\mathbf r'\,d\mathbf r''\right],\qquad a=2\operatorname{Re}K.
$$
Here $C_V$ is the <autocorrelation function of a random field>. Both the covariance $\mathbb E|\phi_1|^2$ and the generally nonzero term $\operatorname{Re}\mathbb E\phi_1^2$ enter the exponent; dropping the latter is not justified for an arbitrary finite real random medium.
Back to article page
