Solving the nonlinear Riccati equation for the 1D plane-wave reflection response from a velocity gradient interface by a Heun to hypergeometric reduction

For 1D acoustic wave propagation in the frequency domain, the complex nonlinear Riccati differential equation describes the dependence of the plane-wave reflection response in depth, defined as the upgoing part of the wavefield divided by the down -going part of the wavefield at that depth. In the case that the model is a velocity gradient interface expressed in terms of a smooth Heaviside function defined by the Fermi-Dirac function, the Riccati equation is shown to have an analytical solution for the reflection response. The solution accounts for all wave phe-nomena in the gradient zone. The solution of the Riccati equation for the reflection response is obtained by introducing the Riccati equation for the reciprocal of the reflection response which can be related to a second-order self-adjoint linear differential equation. The two latter equations obey the same radiation condition. Fur-thermore, a connection between the second-order linear equation and the Heun equation is obtained, both being characterized by having four regular singular points. By a change of variables, the Heun equation can be transformed into Gauss' hypergeometric equation, whereby the solution is expressed as the sum of two linearly independent functions containing hypergeometric func-tions. The advantage of applying the Heun-to-hypergeometric reduction to the nonlinear Riccati equation is that solutions to the hypergeometric equation are well known. The two functions allow us to analytically determine the Riccati solution because the ratio of the linearly independent functions at infinity is constant. In contrast to the classic frequency-independent reflection coef-ficient from two layers in welded contact, the reflection coeffi-cient at the surface associated with the gradient interface depends on frequency. In the limit of zero frequency, the reflection coef-ficient approaches the classic one. We have implemented the iter-ative solution of the integral form of the Riccati equation and found that the solution builds up nonlinearly to the correct sol-ution. The model is verified by a numerical example valid for 1D wave propagation.