Born integral, stationary phase and linearized reflection coefficients in weak anisotropic media

We show that the linearized reflection coefficients for arbitrary anisotropic media embedded in an isotropic background can be derived directly from a Born formalism. Due to rapidly varying phases of the scattered waves from first-order perturbations in density and elastic parameters, the major contributions to the observed wavefield for any source-receiver pair far from the volume of scatterers arise from the stationary points of a scattering integral, called the Born integral. For simple interface models, such integrals can be evaluated analytically using the method of stationary phase. The resulting scattering function relates linearly to the approximate (linearized) reflection coefficient through a scaling factor determined by the angle of incidence and the properties of the background medium. We consider a homogeneous isotropic background to express the approximate reflection coefficients as a sum of an isotropic and an anisotropic reflection coefficient. The isotropic coefficient is a weighted sum of density and isotropic perturbations about the background, whereas the anisotropic coefficient is a weighted sum of anisotropic perturbations where the weights depend on the angles of incidence, the properties of the background medium as well as the azimuth of the plane of reflection with respect to some symmetry plane of the weakly anisotropic medium. We derived expressions for approximate PP and PS reflection coefficients of a weakly isotropic medium, a weakly orthorhombic medium and a weak but arbitrarily anisotropic medium underlying an isotropic medium. Our expressions for PP reflection coefficients are exactly the same as those obtained from first-order perturbation theory which were previously derived by linearization of the exact reflection coefficients. For converted PS waves, our expressions are valid only for small angles of incidence, but have much simpler forms than those obtained by linearization of exact reflection coefficients. We also derive the approximate PP reflection coefficient of a transversely isotropic medium with a tilted axis of symmetry in an explicit form and investigate the effects of dip of the symmetry axis on these reflection coefficients. Numerical results demonstrate that neglecting the dip of a moderately dipping (30-60) symmetry axis of a transversely isotropic medium yields significant errors in determining the weakly anisotropic parameters through an analysis of variations of amplitude with azimuth.