The radiation patterns available in the literature for seismic surface sources are restricted to far-field, low-frequency solutions for undamped half-space models. A theoretical study of the radiation pattern of a circular baseplate vibrating torsionally on the surface of an N-layered anelastic medium demonstrates that the patterns in the literature do not reasonably represent the radiated field of a source on a damped layered medium.The radiation pattern of a source is a measure of the strength of the output signal as a function of direction and is determined by calculating the displacements at points along the arc of a circle at a specified radial distance from the source. The solution for the displacement due to a vibrator or a plane-layered anelastic medium is obtained by solving the elastodynamic wave equation using Fourier and Hankel transforms. The displacement is expressed in terms of an inverse Hankel transform which is performed numerically.The shape of the torsional radiation pattern in a half-space is a function of frequency, baseplate radius, shear-wave velocity, and radial distance between the source and the observation points. As frequency increases, the source beams more energy in a near-vertical direction, and less in directions near the horizontal. Vertical beaming also increases as baseplate radius increases and as shear-wave velocity decreases. The radiated field approaches an asymptotic far-field radiation pattern at radial distances in excess of 5 to 8 baseplate radii.In a layered medium, constructive and destructive interference of direct, reflected, and refracted waves results in a characteristic lobate structure in the radiated field. A minimum occurs just below the layer interface due to the combined effects of reflections and refractions. As frequency increases, a greater proportion of energy penetrates the layer interface and enters the underlying medium. At frequencies typically used in exploration seismology the majority of the signal from a torsional source is trapped in the surface layer.The shape of the radiation pattern at a given distance from the source is a function of the density, velocity, and Q contrasts between layers. For a given acoustic impedance contrast, more energy is trapped in the surface layer if the contrast is caused primarily by a velocity rather than a density change since the velocity contrast results not only in reflections but also in refractions and Love waves. A variation in Q between layers further distorts the radiated field. The pattern for a layered model does not asymptote to a far-field pattern until the radial distance is much greater than the total thickness of the layers in the model.Radiation pattern modeling for layered media may help resolve the discrepancies between theoretical patterns and field measurements and may provide a means of optimizing the radiated field of a surface source or source array in terms of its directivity and the partitioning of energy between various wave types.