The viscoacoustic Green's function for the Helmholtz equation in a velocity gradient interface model

In viscoacoustically stratified media where the density is constant, 3-D wave propagation in the frequency–radial wavenumber domain is governed by the Helmholtz equation. In the case that the model is a velocity gradient interface where the squared velocity in depth is represented by a smooth Heaviside function of the Fermi–Dirac distribution type, the Helmholtz equation for a point source at arbitrary location is shown to have analytical solution for the Green's function. The velocity depth profile, which is a modification of the Epstein profile which has been thoroughly studied in different branches of physics, is described by four parameters: the velocities at minus and plus infinity, the reference depth of the gradient interface, and its smoothness. The Helmholtz equation is first solved in a model where the point source is absent. The solution to the source-free equation has four unknown constants that must be determined. The radiation conditions at minus and plus infinity and two conditions on the Green's function at the source depth allow the constants to be found. The Green's function solution can be represented in two mathematically equivalent algebraic forms involving ordinary hypergeometric functions. The first form allows a numerically stable implementation over all wavenumber components. The second form allows a physical, intuitive interpretation and is expressed mathematically as the sum of two terms. Each of the terms contains the product of constant-velocity reference phase shift functions and hypergeometric functions which take the role to adjust the amplitude and phase shift calculated by the reference phase shift functions to account for the depth varying velocity profile. Inverse Fourier transforms take the Green's function from the frequency–wavenumber domain to the frequency–space domain or time–space domain. The Green's function solution is valid for any sharpness of the interface. Selected numerical results are presented for the 1-D and 2-D Helmholtz equation to demonstrate the influence of the velocity gradient zone on the wavefield. The 1-D solution in an acoustic model is compared in time domain to the classical finite-difference wave propagation solution. For the purpose of interpretation of seismograms, we model for comparison the wavefield response in a model of two half-spaces in welded contact. For brevity, the latter model is referenced as the HS model.