Comment on "Attenuative Dispersion of P Waves in and near the New Madrid Seismic Zone" by L. Cong, J. Mejia, and B. J. Mitchell

Abstract

Manuscript received 15 February 2001. Cong et al. (2000) used P -wave dispersion from small earthquakes in the New Madrid Seismic Zone to estimate Q P through the use of a continuous-relaxation model. This technique was used as an alternative to the usual spectral-slope techniques and employed an innovative step of comparing group delays relative to those at a reference frequency in order to reduce the bias in the dispersion due to errors in the assumed distance and velocity. The resultant Q m values showed an increasing trend with distance, which they interpreted as due to an increase of crustal Q with depth and the portion of the wave propagation through an active seismic zone. The method used by Cong et al. (2000) suffers, however, from two important problems. First, their equation for estimating the relative group delay requires a correction that changes their derived relaxation parameters. Second, their use of short signal-duration windows, just the first cycle of the P -wave arrival, to define signal dispersion at periods much greater than the window duration is wrong. These problems lead to several difficulties in the interpretation of their results. A number of causal Q models have been presented in the literature (Ben-Menahem and Singh, 1981). The continuous-relaxation model is constructed as a superposition of relaxation mechanisms such that Q is constant over a passband. For reference, this operator is (Ben-Menahem and Singh, 1981, equations 10.285 and 10.287) \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[ Q^{-1}(f)=\frac{2}{{\pi}}Q\_{m}^{-1}\mathrm{tan}^{-1}\left[\frac{{\omega}({\tau}\_{1}-{\tau}\_{2})}{1+{\omega}^{2}{\tau}\_{1}{\tau}_{2}}\right],\] \end{document}1 where ω is the angular frequency, τ 1 and τ 2 are the relaxation times, and Q m is approximately Q(f) at intermediate frequencies. An example of this function is plotted in Figure 1. The causal frequency-dependent phase velocity C ( ω ) is given by \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[ \frac{C({\omega})}{C\_{{\infty}}}=\left\{1+\frac{1}{{\pi}Q\_{m}}\left[\mathrm{ln}\frac{{\tau}\_{1}}{{\tau}\_{2}}+\frac{1}{2}\mathrm{ln}\left(\frac{1+{\omega}^{2}{ }{\tau}\_{2}^{2}}{1+{\omega}^{2}{ }{\tau}\_{1}^{2}}\right)\right]\right\}^{-1}\] \end{document}2 Figure 1. Comparison of the Q -1( f ) obtained using the published and corrected dispersion relation. The …