A Kirchhoff Migration scheme for elastic obstacle identification

Kirchhoff Migration (KM), sometimes called Arrival (or Travel) Time Imaging, is a basic and popular imaging technique based on the arrival time of waves from given sources to given sensors. It is commonly used in the fields of underwater acoustics and solid earth geophysics, for both subsurface structure analysis and for identifying unknown local obstacles (scatterers) in the medium. The present paper concentrates on the latter application. For acoustics, the KM algorithm is extremely simple and efficient, although it usually produces a rather crude image, which is the reason for its use as the method of choice when high resolution is not needed, or as a fast technique to produce an initial guess for a more sophisticated imaging method. For elasticity, KM is much more involved, as the arrival-time algorithm is not obvious, mainly since there is more than one wave speed at each spatial point. In this paper, a new KM scheme is proposed for obstacle identification in an isotropic piecewise-homogeneous elastic medium. The scheme is based on measuring two quantities that are second-order operators of the displacement field, which are related to P and S waves, and applying the acoustic KM algorithm to each of them, with the appropriate wave speed. It is demonstrated numerically that the operator related to S waves results in very good identification in many cases. The fact that measurements based on the S-related operator are preferred over those based on the P-related operator is an empirical observation, and awaits full analysis, although a partial explanation is given here.