Estimation of Q and phase velocity Estimation of Q and phase velocity using the stress-strain relaxation spectrum Dali Zhang, Michael P. Lamoureux, and Gary F. Margrave ABSTRACT 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. Math- ematically the problem is formulated as an inverse spectral problem for reconstruction of spectral measure in the analytic Stieltjes representation of the complex modulus using ra- tional approximation. A rational (Padé) 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 stan- dard 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 Padé approximation can be used for determination of the internal memory variables in TDFD numerical simulation of viscoelastic wave propagation. INTRODUCTION We present a method to recover relaxation spectrum of the medium given measure- ments of complex velocity or complex viscoelastic modulus, and to further estimate the quality Q-factor and phase velocity. We formulate the problem as an approximation to the spectral measure in the Stieltjes representation of the complex modulus using rational (Padé) approximation. The method of construction of Padé approximation is based on con- strained least squares minimization algorithm, regularized by the constraints derived from the analytic Stieltjes representation of the complex modulus. Solution of the constrained minimization problem gives us coefficients of a rational approximation to the spectral mea- sure of the medium. This rational approximation is transformed into Padé approximation by partial fraction decomposition. The method can use as data the values of measured, or simulated (or desired) complex modulus or complex velocity in certain interval of frequen- cies. The recovered lower order rational ([p, q ]-Padé) approximation can be used for deter- mination of the internal memory variables in TDFD numerical simulation of viscoelastic wave propagation. The developed technique together with finite difference modeling may eventually lead to an alternative formulation for numerical simulation of viscoelastic wave propagation. The present approach may suggest a new simultaneous inversion technique for estimation of the frequency dependent complex velocities, Q-factors and phase veloc- ities in anelastic attenuating media from vertical seismic profile (VSP) data in geophysics prospecting. CREWES Research Report — Volume 21 (2009) 1