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

Abstract

S U M M A R YWe 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.

Waveform inversion for orthorhombic anisotropy with P waves: feasibility and resolution

“…2(b)), which prompted 8 Vladimir Kazei & Tariq Alkhalifah the development of linearized reflection coefficients (Rüger 1997, e.g. ) and, more recently, reflection-based radiation patterns (Shaw & Sen 2004;Gholami et al 2013b;Alkhalifah & Plessix 2014). The relation between the reflection patterns and linearized coefficients is well explained by Shaw & Sen (2004).…”

“…For instance, it is also valid for the perturbations within thin low-contrast layers and nearnormal wave incidence angles (Shaw & Sen 2004). To obtain a simplified expression for the scattering of plane waves on an arbitrary perturbation, we first manipulate the right hand side of equation 1.…”

“…Namely, the scattered wavefield from a point perturbation δm = δ(x)(δρ, δC) T in the firstorder Born approximation is proportional to the scattering function (Beylkin & Burridge 1990;Eaton & Stewart 1994;Shaw & Sen 2004):…”

“…The frequency domain Born approximation for the scattered wavefield δU in an anisotropic elastic medium is (e.g. Hudson & Heritage 1981;Beylkin & Burridge 1990;Shaw & Sen 2004) …”

On the Resolution of Inversion for Orthorhombic Anisotropy

“…For the case of a point scatterer δρ(K) = const, δĉ i jkl (K) = const, equation 2 provides the radiation pattern or scattering function of these scatterers (Eaton and Stewart, 1994;de Hoop et al, 1999;Shaw and Sen, 2004). Under the plane-wave and Born applicability assumptions, equation 2 generalizes the point scatterer radiation patterns to arbitrary perturbations.…”