 Height-correlation dependence of coherent reflection

The quadratic coherent correction weights the power spectrum of surface height by $\beta (q)$, so equal root mean square heights can give different mean fields. Only an additional long-spatial-scale approximation, with spectrum concentrated at $|q|\ll k$, reduces it to $2k^2{\sigma }^2$. Second-order surface scattering therefore contains information absent from first-order coherent reflection.