Bina, C. R., and G. R. Helffrich, Calculation of elastic properties from thermodynamic equation of state principles, Annual Reviews of Earth and Planetary Sciences, 20, 527-552, doi:10.1146/annurev.ea.20.050192.002523, 1992.
Introduction. The analysis of seismic velocities at depth within the earth has a long history and attracts an increasingly broad spectrum of researchers. Almost as soon as seismic travel times were available, Williamson & Adams  used them to obtain velocity profiles and check their consistency with the densities that would result from self-compression of a uniform material. Birch  addressed the same question of homogeneity and composition by incorporating the effect of pressure in his extrapolation of the shallow earth's density and wave speeds to mantle depths. This traditional use of seismological data to investigate mantle properties and phenomena continues since seismic waves (in a broad sense that includes normal modes as well) constitute our primary probe of earth structure. For example, core-diffracted P and S waves [Wysession & Okal, 1988; Wysession & Okal, 1989] carry information concerning the properties of the D'' layer at the core-mantle boundary [Bullen, 1949; Lay & Helmberger, 1983; Lay, 1986] that can be related to temperature and composition [Wysession, 1991]. Slabs of subducted oceanic lithosphere influence the travel times and waveforms of seismic waves [Oliver & Isacks, 1967; Toksöz et al., 1971; Sleep, 1973; Engdahl et al., 1977; Jordan, 1977; Suyehiro & Sacks, 1979; Creager & Jordan, 1984; Creager & Jordan, 1986; Silver & Chan, 1986; Vidale, 1987; Fischer et al., 1988; Cormier, 1989] which can be related to their thermal structure and composition. Seismic waves reflected and refracted at the slab-mantle interface also provide constraints on the properties of this interface that can be interpreted by computing seismic velocities in slab mineralogies [Helffrich et al., 1989]. In a similar fashion, the earth's velocity structure and discontinuities provide information concerning the composition of the mantle [Birch, 1952; Lees et al., 1983; Anderson & Bass, 1984; Bass & Anderson, 1984; Bina & Wood, 1984; Weidner, 1985; Bina & Wood, 1987; Akaogi et al., 1989; Duffy & Anderson, 1989]. Finally, the issue of the composition of the lower mantle can be addressed by comparing computed densities of the candidate mineralogies with density profiles of the lower mantle [Watt & O'Connell, 1978; Jackson, 1983; Anderson, 1987; Chopelas & Boehler, 1989; Bina & Silver, 1990; Fei et al., 1991].
The variety of approaches involved in these studies demonstrates the need for a review of the methodology used to compute elastic wave speeds, in the spirit of an earlier review by Anderson et al. . Three broad areas bear on this topic: thermodynamic analysis, continuum mechanics, and solid state physics. Experiment provides the raw data and some important rules of thumb that make it possible to compute elastic velocities inside the earth. Our intent is to combine theory and data into a practical method that will facilitate calculations of this type and to indicate some of the failings and alternative approaches that may be pursued.
The elastic properties of the Earth's interior - i.e., density and elastic wave velocities - are functions of composition, mineralogy, pressure, and temperature. The dependence of the elastic properties on these factors has been measured for numerous minerals over a range of conditions. Compositional variations traditionally have been treated through empirical systematics based on the mineral structure and the mean atomic weight of a mineral [e.g., Birch, 1961; Anderson & Nafe, 1965; Anderson, 1967; Anderson, 1987]. On the other hand, temperature and pressure effects have been investigated through laboratory measurements [e.g., Christensen, 1984]. These results have given rise to the view that pressure and temperature derivatives of wave speeds, assumed to be constant and applied to speeds measured at room conditions, yield elastic wave speeds under mantle conditions. This process requires extensive extrapolations, by factors of up to ~20 in temperature and 100 in pressure. In general, however, these derivatives are not constant throughout the extrapolation interval, and this assumption is likely to lead to incorrect results. When large extrapolations away from measured properties are necessary, they should be guided by a theoretical framework that controls the functional form. The purpose of this paper is to review the relations which provide a conceptual basis for bridging the gap between experimental and mantle conditions. We deal first with the relevant thermodynamic relations and then briefly with the continuum mechanical and lattice dynamical theories that bear on elastic properties at mantle conditions.Copyright © 1992 Annual Reviews, Inc.