Importance of physical dispersion in surface wave and free oscillation problems: Review

Abstract

Physical dispersion resulting from anelasticity is investigated from the point of view of linear viscoelastic models and causality relations. It is concluded that inasmuch as Q in the earth's mantle is nearly independent of frequency, at least in the seismic frequency band, a dispersion relation in the form of C(ω) = C(ωr)[1 + (1/πQm) In (ω/ωr)] must be used for correcting the effect of physical dispersion arising from anelasticity. (Here C(ω) is the phase velocity of either body waves, surface waves, or free oscillations, ω is the angular frequency, ωr is the reference angular frequency, and Qm is the path average Q for body waves or Q of a surface wave or a mode of angular frequency ω; for surface waves and free oscillations, C(ωr) should be understood as the phase velocity at ω computed by using the elastic moduli at ω = ωr.) The values of Q outside the seismic frequency band affect mainly the absolute value of the phase velocity but do not affect significantly the relative dispersion within the seismic frequency band. Even if the microscopic mechanism of attenuation is nonlinear, this dispersion relation can be used if departure from elasticity is relatively small, so that the signal can be approximated by a superposition of propagating harmonic waves. Since surface wave and free oscillation Q is 100–500 for fundamental modes, a correction of 0.5–1.5% must be made for joint interpretation of body wave and surface wave data. This correction is nearly 1 order of magnitude larger than the uncertainties associated with these data and are therefore very significant. When this correction is made, the discrepancy between the observed surface wave phase velocities and free oscillation periods and those predicted by the Jeffreys or Gutenberg model becomes much smaller than has previously been considered.