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

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 (Padé) approximation to the relaxation spectrum 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 D. Zhang (B) · M. P. Lamoureux Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada e-mail: [email protected] M. P. Lamoureux e-mail: [email protected] G. F. Margarve Department of Geoscience, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada e-mail: [email protected] E. Cherkaev Department of Mathematics, University of Utah, 155 S 1400 E, JWB 233, Salt Lake City, UT 84112-0090, USA e-mail: [email protected] (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 Padé approximation can be used for determination of the internal memory variables in time-domain finite difference numerical simulation of viscoelastic wave propagation.