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SUMMARY The sensitivity of Rayleigh wave amplitude to Earth structure has applications to seismic tomography, both in cases where amplitude information is used to supplement phase velocity data to improve images of elastic parameters, and to correct amplitudes for local Earth structure in attenuation tomography. We review the theoretical basis of the ray theoretical approximation, in which the wave amplitudes are controlled by a combination of geometrical spreading and local changes in energy density due to Earth structure. We focus mainly on the latter effect, which we term the constant energy flux approximation. We investigate the ray theoretical basis for this approximation, test it against a full waveform simulation that verifies its accuracy and show how it can be used to compute the sensitivity of amplitude to elastic moduli and density. We investigate how perturbing these parameters in a set of simple Earth models affects Rayleigh wave amplitudes, and demonstrate that a slow velocity heterogeneity can cause either increased or reduced amplitudes, depending upon the depth of the heterogeneity and the observation frequency. Consequently, amplitude sensitivity can be either positive or negative, and its magnitude can vary significantly with frequency. Although an added complication, the very different behaviour of phase velocity and amplitudes to changes in Earth structure implies that the two types of data are complementary and suggest the effectiveness of using both in Rayleigh wave tomography.