Viscoacoustic wave equations for the power-law dependence of Q on frequency

Abstract

The power-law dependence of the quality factor Q on frequency (i.e. Q = Q 0 | f / f 0 | γ ), which is called the power-law frequency-dependent Q for brevity, is a common phenomenological description of seismic wave attenuation in the interior of the Earth. The wave equation in differential form is essential to forward and inverse modelling of dissipative seismic waveforms in an accurate and efficient manner. However, all existing methods are seemingly unable to explicitly incorporate the exponent parameter γ into the wave equation in differential form. This drawback apparently limits the development of gradient-based inverse methods for spatially varying γ via the wave equation. In this paper, we use a weighting function method to derive the viscoacoustic wave equations for the power-law frequency-dependent Q , which explicitly involve the parameters γ and Q 0 . A critical step in this method is to construct a dissipative model, for which the complex modulus is expressed as an N -order series in terms of the weighting function. Numerical examples are used to illustrate the accuracy of the dissipative model, the effect of γ on waveforms, and the application of the newly proposed wave equations in viscoacoustic wavefield modelling for the power-law frequency-dependent Q .