Abstract
Consider an acoustic half plane with a sound slowness n 2(x, z) close to a given function n 20 (Z) (vertically inhomogeneous background). The problem of recovering n 21 (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 ([1], [2]).
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.
The research described in this publication was made possible in part by Grant N 9300 from the International Science Foundation and Russian Governement.
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Khajdukov, V.G., Kostin, V.I., Tcheverda, V.A. (1997). The r-Solution and Its Applications in Linearized Waveform Inversion for a Layered Background. In: Chavent, G., Sacks, P., Papanicolaou, G., Symes, W.W. (eds) Inverse Problems in Wave Propagation. The IMA Volumes in Mathematics and its Applications, vol 90. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1878-4_13
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