A perturbation method for treating the scattering of plane waves by small surface imperfections on an elastic half space is presented. The solution to the first order approximation is given as convolution integrals of the surface imperfection with kernel functions defined by Fourier inversion integrals. The evaluation of these integrals is discussed and their asymptotic representations determined. The far field scattered displacements are explicitly obtained for arbitrary imperfections. The scattered field consists of a Rayleigh surface wave and four body phases which at the free surface travel with the speed of dilational or distortional waves. Numerical examples are given. In particular the error in the apparent angle of emergence due to the scattered waves is obtained. The body phases exhibit the familiar 3/2 geometric attenuation, but still may make a significant contribution at moderately long distances. A strong dependence of the magnitude of the error on the angle of incidence is demonstrated.

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