A proposal for a consistent parametrization of earth models

Abstract

SUMMARY The current way to parametrize earth models in terms of real-valued seismic velocities and quality factors is incomplete as it does not specify how complex-valued viscoelastic moduli or complex velocities should be computed from them. Various ways to do this can be found in the literature. Depending on the context they may specify (1) the real part of the viscoelastic modulus, (2) the absolute value of the viscoelastic modulus, (3) the real part of complex velocity or (4) the phase velocity of a propagating plane wave. We propose here to exclusively use the first alternative because it is the only one which allows both a flexible choice of elastic parameters and a mathematically rigorous evaluation of the complex-valued viscoelastic moduli. The other definitions only permit an evaluation of viscoelastic moduli if the tabulated quality factors are directly associated with the listed velocities. Ignoring the subtle differences between the three definitions leads to variations in viscoelastic moduli which are second order in 1/Q where Q is a quality factor. This may be the reason why the topic has never been discussed in the literature. In case of shallow seismic media, however, where quality factors may assume values of less than 10, the subtle differences become noticeable in synthetic seismograms. It is then essential to use the same definition in all algorithms to make results comparable. Matters become worse for anisotropic media, which are commonly specified in terms of real elastic moduli and quality factors for effective isotropic moduli. In that case, the complex-valued viscoelastic moduli cannot be determined uniquely. However, interpreting the tabulated constants as the real parts of the complex-valued viscoelastic moduli at least allows a consistent definition, which respects the relative magnitude of the anelastic and anisotropic parts compared to the elastic parts. It should be noted that all these considerations apply to complex-valued viscoelastic moduli at a fixed frequency. Specifying the frequency dependence of the viscoelastic moduli requires additional knowledge about the rheology of the material.