Estimation of Q and phase velocity using the stress-strain relaxation spectrum

Dali Zhang, Michael P. Lamoureux, Gary F. Margrave

The article presents a numerical inversion method for estimation of Q-factor and phase velocity in linear, viscoelastic, isotropic media using reconstruction of relaxation spectrum from measured or computed complex velocity or complex modulus of the medium. Mathematically the problem is formulated as an inverse spectral problem for reconstruction of spectral measure in the analytic Stieltjes representation of the complex modulus using rational approximation. A rational (Pad) approximation to the spectral measure 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 Pad pproximation can be used for determination of the internal memory variables in TDFD numerical simulation of viscoelastic wave propagation.