Relation between the connected diagram and smoothing methods for rough surface scattering

In previous work by the author on connected diagram expansion methods for the problem of scattering from a random rough surface a stochastic Lippmann–Schwinger integral equation in Fourier transform space for the scattered part of the Green’s function was derived. Averaging techniques using homogeneous statistics and a statistical cluster decomposition on the surface interaction function yielded a connected diagram expansion for the coherent and incoherent Green’s functions. Here it is demonstrated that the smoothing method applied to this stochastic integral equation yields a result that agrees with the connected diagram expansion only to second order in the surface interaction. For third- and higher-order interactions, the smoothing method does not yield connected terms.