By cross-correlating successive ultrasonic scans of the biological tissue that
is excited by a transient source on the surface, recent experiments show that
the wave displacement during the interval between the two scans can be measured everywhere in the tissue. Assuming we know the displacement history of the
shear wave everywhere, we identify the stiffness change inside the tissue in
terms of the Lame coefficient, $\mu$. 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 recover $\mu$ without directly taking derivatives of the displacement
data.
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