Then, the two-step MPSTD algorithm formulated above is employed to determine the scattering of a buried cylinder below each of the rough surfaces generated.
The second example is devoted to validate the two-step MPSTD approach.
This demonstrates that the length of the period of the random periodic surface considered in the computation is sufficiently large so that the interaction between the buried cylinder and the other periods of the rough surface (x < -5 and x > 5) can indeed be neglected and the total length of the rough surface included in the finite computation domain is adequate; hence verifies that the two-step MPSTD approach presented in Section 3 works effectively for a random periodic rough surface with a period of sufficiently large length.
From the limited CPU time increase, it is expected that the two-step MPSTD method can also effectively work well for larger L.
16(b), the results obtained by the two-step approach for the flat interface is exactly the same as that got from the "regular" Monte-Carlo MPSTD algorithm presented in  as expected.
The two-step Monte-Carlo MPSTD numerical technique developed in this work can be employed for determining the scattering of a cylinder of arbitrary shape buried below a random periodic rough surface.
Different from the previously published MPSTD analysis, which mainly analyzed the scattering of buried objects under flat or undulated surfaces, this work and the authors' previous one  involve a random rough surface.
But in a MPSTD subdomain that is partially bounded by the rough surface, the profile [y.sub.mapped] is a function of [x.sub.mapped], which are related to the CGL points in the ([xi], [eta], [zeta]) coordinates by the coordinate transformation.
As the first step of the 3D MPSTD formulation, the computation domain is divided into 216 non-overlapping hexahedral subdomains.
Most recently, a Monte-Carlo MPSTD algorithm is developed for the analysis of scattering from a 2D cylinder buried below a random rough surface .
To validate the MPSTD algorithm developed, we introduce a virtual random rough surface placed along y = 0 and the half spaces above and below it are both set to be free space.
At two observation points, one is below and the other is above the virtual rough surface, the MPSTD results of the total field [E.sub.z] are compared with the free-space results as well those obtained using the FDTD method.