Computation of differential seismograms for point and plane scatterers in layered media

The first-order scattering effects of elastic waves generated by point sources in layered media are computed using two different geometries and formulations. We first consider point diffractors and demonstrate for elastic scattering Trorey's (1970) result of the equivalence between a specular reflection from a plane interface and a "Kirchhoff" summation of point diffractors in a planar array. We then propose a fast and convenient formulation to calculate the first-order scattering of an infinite slab within a layered elastic medium. Our study shows that the first-order scattering theory is remarkably accurate at angles of incidence not exceeding 35 degrees, and that the formulation used allows us to model density or velocity perturbations of up to 10%. These conclusions are in close agreement with results previously obtained in the plane-wave domain. The methods and expressions presented have implications for Kirchhoff back-projection imaging methods and may be useful in assessing the sensitivity of the seismic response of depth-dependent elastic media to various combinations of elastic parameters. They are also directly applicable to linearized inversion algorithms of seismograms using gradient techniques.