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The concept of a relaxation spectrum is used to compute the absorption and dispersion of a linear anelastic solid. The Boltzmann aftereffect equation is solved for a solid having a linear relationship between stress and strain and their first time derivatives, the 'standard linear solid', and having a distribution of relaxation times. The distribution function is chosen to give a nearly constant Q over the seismic frequency range. Both discrete and continuous relaxation spectra are considered. The resulting linear solid has a broad absorption band which can be interpreted in terms of a superposition of absorption peaks of individual relaxation mechanisms.The accompanying phase and group velocity dispersion imply that one cannot directly compare body wave, surface wave, and free oscillation data or laboratory and seismic data without correcting for absorption. The necessary formalism for making these corrections is given. In the constant Q regions the correction is the same as that implied in the theories of Futterman, Lomnitz, Strick and Kolsky.

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Comparison of seismic velocities in mantle minerals, under mantle conditions, with seismic data is a first step toward constraining mantle chemistry. The calculation, however, is uncertain due to lack of data on certain physical properties. "Global" systematics have not proved very useful in estimating these properties, particularly for the shear parameters. A new approach to elasticity estimation is used in this study to produce estimates of unknown quantities, primarily pressure and temperature derivatives of elastic moduli, from the structural and chemical trends evident in the large amount of elasticity data now available. These trends suggest that the derivatives of unmeasured high-pressure phases can be estimated from "analogous" low-pressure phases. Using these predictions and the best available measurements, seismic velocities are computed along high-temperature adiabats for a set of mantle minerals using third-order finite strain theory. The calculation of density and moduli at high temperature, to initiate the adiabat, must be done with care since parameters such as thermal expansion are not independent of temperature. Both compressional and shear seismic profiles are well-matched by a mineralogy dominated by clinopyroxene and garnet and with an olivine content of approximately 40% by volume. Between 670 and 1000 km, perovskite alone provides a good fit to the seismic velocities. Combining seismic velocities with recent phase equilibria data for a hypothetical pure olivine mantle suggests that a mineralogy with a maximum of 35% olivine (shear profile) or 40-53% olivine (compressional profile) by volume can satisfy the constraint imposed by the 400-km discontinuity. Other features of the upper mantle can then be matched by appropriate combinations of pyroxenes, garnets, and their high-pressure equivalents. While mantle models with a substantially larger fraction of olivine cannot be ruled out, they are acceptable only if the derivatives of the spine! phases are substantially different from olivine and deviate from trends in the larger data set.

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