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The coupled time-domain integral equations (TDIE), involving surface equivalent electric and magnetic currents, for both conducting and dielectric bodies is developed. Besides, composite transient electromagnetic scattering from a 3-D conducting object above a 2-D dielectric Gaussian random rough surface is analyzed with the proposed method. The spatial and temporal bases are RWG basis functions and weighted Laguerre polynomials, respectively. Numerical results are given to verify the theory and codes.