The r-solution and its applications in linearized waveform inversion for a layered background. The IMA volumes in mathematics and its applications

Consider an acoustic half plane with a sound slowness n 2(x, z) close to a given function n 0 2 (Z) (vertically inhomogeneous background). The problem of recovering n 1 2 (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.

[1]  S. Godunov.  Guaranteed Accuracy in Numerical Linear Algebra , 1993 .

[2]  M. Nashed,et al.  Aspects of Generalized Inverses in Analysis and Regularization , 1976 .

[3]  Irshad R. Mufti,et al.  Reversed time migration in spatial frequency domain , 1983 .

[4]  W. Symes,et al.  Uniqueness and continuous dependence for a multidimensional hyperbolic inverse problem , 1985 .

[5]  Albert Tarantola,et al.  Linearized inversion of multioffset seismic reflection data in the ω-k domain: Depth‐dependent reference medium , 1988 .

[6]  Norman Bleistein,et al.  AN EXTENSION OF THE BORN INVERSION METHOD TO A DEPTH DEPENDENT REFERENCE PROFILE , 1984 .

[7]  V. Romanov.  Inverse problems of mathematical physics , 1986 .

[8]  L. Hörmander,et al.  The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients , 1983 .

[9]  On the relation between coefficient and boundary values for solutions of Webster's horn equation , 1986 .

[10]  On the uniquencess of a multidimensional hyperbolic inverse problem , 1988 .

[11]  William W. Symes,et al.  Layered velocity inversion: a model problem from reflection seismology , 1991 .

[12]  L. Kantorovich,et al.  Functional analysis in normed spaces , 1952 .

[13]  A. Tarantola.  LINEARIZED INVERSION OF SEISMIC REFLECTION DATA , 1984 .

[14]  A trace theorem for solutions of the wave equation, and the remote determination of acoustic sources , 1983 .

[15]  V. Cheverda,et al.  R-pseudoinverses for compact operators in Hilbert spaces: existence and stability , 1995 .

[16]  Rakesh.  A Linearised inverse problem for the wave equation , 1988 .

[17]  Fadil Santosa,et al.  An analysis of least-squares velocity inversion , 1989 .

[18]  Jack K. Cohen,et al.  Three‐dimensional Born inversion with an arbitrary reference , 1986 .