The plane-wave primary reflection response from an impedance gradient interface

A weak scattering model that allows prediction of the one-dimensional acoustic plane wave primary reflection response from an impedance gradient interface is described. The velocity and density gradient profiles are represented by a smooth approximation to the Heaviside function of the Fermi-Dirac distribution type. The profiles are described by the velocities and densities at minus and plus infinity, the reference depth of the gradient interface, and its smoothness. The primary response is derived by using the Bremmer series to reduce a generally complex reflection problem to the simpler one of the primary reflections which is a valid solution for small impedance contrasts. The reflection response can be expressed in terms of the Appellian hypergeometric functions of two variables of the first kind and Gaussian hypergeometric functions. When the reflection response is evaluated at sufficiently large distance above the reference depth, the Appellian functions are reduced to Gaussian hypergeometric functions. In the Born approximation, the reflection response simplifies. In the limit of zero frequency, the reflection coefficient in the small impedance contrast approximation can be related to the classic reflection coefficient for two impedance layers in welded contact.