An analytical method has been developed to study the steady-state diffraction and scattering of mono-chromatic Rayleigh waves by shallow circular canyons. The shape of the canyon can vary from very shallow to semi-circular. The method is be sed on reprecentation of the scattered waves in series of cylindrical wave functions and on approximation of the half-epace surface by a cylindrical surface of very large radius. Along the canyon bottom the stresses of the free-fleld motion are approximated by finite Fourier series to satisfy the sero-stress boundary condition tiere. The horisontal and vertical displacement amplitudes are illustrated for canyon motion excited by Rayleigh waves with different wave lengths.