A theoretical analysis of the scattering of High-Frequency (HF), electromagnetic waves from a rough surface such as the ocean surface is proposed. This is required for an interpretation of the radar signature when using an HF radar for the remote sensing of ocean surface parameters. The analysis is based on Walsh's (1980b) formulation in the spatial Fourier transform domain for the scattering from a general time invariant surface. Initially, a two dimensional spatially periodic surface with a high refractive index is considered. For this surface, a series solution is derived for the surface electric field in the spatial transform domain maintaining the choice of any finite source. The source then considered is an elementary vertical dipole excited by a pulsed sinusoidal current. It is assumed that the surface slopes are small compared to unity. For this source, zero first, and second order approximations of the vertical component of the surface field are inverse spatially transformed in an asymptotic sense. These solutions are in the form of ground waves with modified surface impedances. By assuming a narrow beam receiving antenna, the inverse transforms for the first and second order solutions that involve spatial convolution integrals, are evaluated asymptotically for the two orders of the backscattered surface field. -- By modelling the ocean surface as a three dimensional periodic surface in space and time with random Fourier coefficients, the above two orders of the backscattered field are suitably modified to include the time dependency and the statistical variation of the surface. Based on this model an average first and second order backscattered Doppler spectra and consequently the two orders of the cross section are derived. The first order cross section is the same as that derived by Barrick (1972b, 1977a) using the Rice perturbation technique. The second order result contains three parts. The first part is almost the same as that obtained by Barrick. The two additional parts result from surface and incident field interaction along the path from transmitter to the primary ocean scattering patch and off-patch double scatter respectively. The effect of the former may be very pronounced, particularly at the higher Doppler frequencies when the radar is situated in the middle of the ocean.