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.

Shear-wave velocities (V5 \ which are widely used for earthquake ground-motion site characterization studies, are now largely obtained using borehole methods. Drilling holes, however, is expensive. Surface methods are less expensive for obtaining V5 information, but not many comparisons with direct borehole measurements have been made. Because different assumptions are used in data interpretation of each surface method, and because public safety is involved in site characterization for engineering structures, it is important to validate the surface methods by additional comparisons with borehole measurements. We compare results obtained from a particular surface method (array measurement of surface waves associated with microtremor) with results obtained from borehole methods. Using a ten-element nested-triangular array of 100-m aperture, we measured surface-wave phase velocities at two California sites, Garner Valley near Hemet and Hollister Municipal Airport. The Garner Valley site is located at an ancient lake bed where water-saturated sediment overlies decomposed granite on top of granite bedrock. Our array was deployed at a location where seismic velocities had been determined to a depth of 500 m by borehole methods. At Hollister, where the near-surface sediment consists of clay, sand, and gravel, we determined phase velocities using an array located close to a 60-m deep borehole where downhole velocity logs already exist. Because we want to assess the measurements uncomplicated by uncertainties introduced by the inversion process, we compare our phase-velocity results with the borehole V5 depth profile by calculating fundamental-mode Rayleigh-wave phase velocities from an earth model constructed from the borehole data. For wavelengths <~2 times of the array aperture at Garner Valley, phase-velocity results from array measurements agree with the calculated Rayleigh-wave velocities to better than 11%. Measurement errors become larger for wavelengths >2 times of the array aperture. At Hollister, the measured phase velocity at 3.9 Hz (near the upper edge of the microtremor frequency band) is within 20% of the calculated Rayleigh-wave velocity. Because shear-wave velocity is the predominant factor controlling Rayleigh-wave phase velocities, these comparisons suggest that this non-intrusive method can provide Vs information adequate for ground motion estimation provided two conditions are met. These conditions are: (1) the site velocity structure can be approximated by a horizontally-layered structure at least on the size of the seismic array, and (2) when the surface \\ avelength is <~2 times of the array aperture.

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Attenuation of seismic waves indicates that the earth is not perfectly elastic. Dispersion accompanying absorption gives frequency-dependent "elastic" moduli, a fact that must be taken into account when inverting seismic data. Normal mode data are reinverted after correcting for absorption. The correction removes the discrepancy between body wave and free oscillation interpretations of earth structure.

A detailed evaluation on the method of internal friction measurement by the stress‐strain hysteresis loop method from 0.01 to 1 Hz at 10−8 to 10−7 strain amplitude and 23.9°C is presented. Significant systematic errors in relative phase measurement can result from convex end surfaces of the sample and stress sensor and from end surface irregularities such as nicks and asperities. Preparation of concave end surfaces polished to optical smoothness having a radius of curvature >3.6×104 cm reduces the systematic error in relative phase measurements to <(5.5±2.2)×10−4 radians. The values of QE−1 (internal friction under uniaxial compression) determined from the relative phase measurements are |QE−1–Qs−1|< 2.8×10−3 for the tool steel sample and |QE−1–Qs−1|< 2.2×10−3 for the Westerly granite sample, where Qs−1 is the internal friction of the fused quartz stress sensor under uniaxial compression. These values are consistent with those inferred from the relative modulus dispersion data also presented in this paper. The polymethyl methacrylate (PMM, trade name Plexiglass) sample shows high values of internal friction (QE−1 ≅5×10−2) with strong frequency dependence and with a maximum in QE−1 at ≅0.4 Hz.