Rational approximation for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media

Abstract

The article presents a numerical inversion method for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media using reconstruction of relaxation spectrum from measured or computed complex velocity or modulus of the medium. Mathematically, the problem is formulated as an inverse problem for reconstruction of relaxation spectrum in the analytic Stieltjes representation of the complex modulus using rational approximation. A rational (Pade) approximation to the relaxation spec trum is derived from a constrained least squares minimization problem with regularization. The recovered stress-strain relaxation spectrum is applied to numerical calculation of frequency-dependent Q factor and frequency-dependent phase velocity for known analytical models of a standard linear viscoelastic solid (Zener) model as well as a nearly constant-Q model which has a continuous spectrum. Numerical results for these analytic models show good agreement between theoretical and predicted values and demonstrate the validity of the algorithm. The proposed method can be used for evaluating relaxation mechanisms in seismic wavefield simulation of viscoelastic media. The constructed lower order Pade approximation can be used for determination of the internal memory variables in time-domain finite difference numerical simulation of viscoelastic wave propagation.