Consider an acoustic half plane with a sound slowness n²(x, z) close to a given function n²₀ (z) (vertically inhomogeneous background). The problem of recovering n²₁(x, z)(local lateral variations) using as data a series of point sources responses measured at the line {z=0} is studied. By means of formal linearization and Fourier transformation with respect to time, lateral coordinate and source position this problem is reduced to a splitting family of 1D linear integral equations of the first kind in L₂ spaces. To solve these equations a notion of r-solution is used. The r-solution of a linear equation with a compact operator in Hilbert spaces is the generalized normal solution of an equation with finite-dimensional operator being a restriction of the initial operator onto the span of its r largest singular vectors. The main features of this solution are its stability with respect to perturbations and existence of numerical algorithms for its reliable computing. Results of a numerical analysis of a problem are presented and discussed including singular value decomposition of mentioned above 1D integral operators and r-solutions for different values of r.

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