Recovery of the Lamé parameter μ in biological tissues

By cross-correlating successive ultrasonic scans of the biological tissue that is excited by a transient source on the surface, recent experiments show that elastic wave displacement during the interval between the two scans can be measured everywhere in the tissue. Assuming that we know the displacement history, w(t, x), of a propagating shear wave everywhere, we identify the stiffness change inside the tissue in terms of the Lamé coefficient, μ. So, the goal is: find μ from w. In the experiment, we consider that there is a dominant frequency, τ , called a central frequency. We then take the Fourier transform of w (in time) and evaluate this transform at τ . Using this transform of the data, our method starts with the asymptotic expansion of geometrical optics. By tracing the amplitude change of the displacement along the geometrical rays as they travel into the medium, we are able to recoverμ without directly taking derivatives of the displacement data.