We consider a complexified Green function for the 3D Helmholtz operator. This function, G* = exp (ikR*)/R*, where , describes a Gaussian beam propagating along the z-axis. It satisfies a certain inhomogeneous Helmholtz equation in . The corresponding source distribution is a generalized function supported by a 2D surface, , which depends on the branch cut of the square root defining R*. We discuss various choices of the branch cut, and show that the corresponding surface can have a complicated structure. In order to describe the source distribution, we first choose the cut and the corresponding branch of the square root, and then regularize a certain integral with a power-type singularity. The paraxial and far-field asymptotic representations are presented for G*.
An inverse problem of wave propagation into a weakly laterally inhomogeneous medium occupying a half-space is considered in the acoustic approximation. The half-space consists of an upper layer and a semi-infinite bottom separated with an interface. An assumption of a weak lateral inhomogeneity means that the velocity of wave propagation and the shape of the interface depend weakly on the horizontal coordinates, x = (x1, x2), in comparison with the strong dependence on the vertical coordinate, z, giving rise to a small parameter ε 1. Expanding the velocity in power series with respect to ε, we obtain a recurrent system of 1D inverse problems. We provide algorithms to solve these problems for the zero and first-order approximations. In the zero-order approximation, the corresponding 1D inverse problem is reduced to a system of non-linear Volterra-type integral equations. In the first-order approximation, the corresponding 1D inverse problem is reduced to a system of coupled linear Volterra integral equations. These equations are used for the numerical reconstruction of the velocity in both layers and the interface up to O(ε 2 ).
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