The response to seismic excitation of a three-dimensional rigid foundation embedded into an elastic half space is analyzed. The analysis is separated into two physically meaningful subproblems: the first deals with the determination of the restraining forces acting on the foundation; the second deals with the evaluation of driving forces induced by seismic waves on the fixed foundation. This analysis is applied to the case of a hemispherical foundation embedded in a homogeneous elastic half space. The exact series solutions of the mixed boundary value problem are presented. The results are compared with those of a circular flat foundation. The comparison indicates that the embedment may have a marked influence of foundation response. The impedance components of the embedded hemispherical foundation are generally larger than those of the disc foundation. For incident plane P, SV and SH waves, scattering due to embedment reduces the foundation translational motion at high frequencies, but leads to significant rocking and torsional motions.