Abstract
For the weakly inhomogeneous acoustic medium in Ω={(x, y, z:z>0}, we consider the inverse problem of determining the density function p(x, y). The inversion input for our inverse problem is the wave field given on a line. We get an integral equation for the 2-D density perturbation from the linearization. By virtue of the integral transform, we prove the uniqueness and the instability of the solution to the integral equation. The degree of ill-posedness for this problem is also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Colton, D.L., Monk, P.: A comparison of two methods for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. J. Comput. Appl. Math., 42(1): 5–16 (1992)
Courant, R., Hilbert, D.: Methods of mathematical physics, Vol.2, John Wiley, New York, 1962
Devaney, A.J.: Geophysical diffraction tomography. IEEE Trans. Geosci. Remote Sensing, GE-22,3–12 (1984)
Friedlander, F.G.: Sound pulse. Cambridge University Press, Cambridge, 1958
Kress, R.: Linear integral equations. Springer-Verlag, Berlin, 1989
Newton, R.G.: Inverse scattering (2), three dimensions. J. Maths. Phys., 21: 1698–1715 (1980)
Sacks, P.: A velocity inversion problem involving an unknown source. SIAM J. Appl. Math., 50(3): 931–941 (1990)
Sacks, P.: The inverse problem for a weakly inhomogeneous two-dimensional acoustic medium. SIAM J. Appl. Math., 48(5): 1167–1197 (1988)
Sacks P., Symes, W.: Uniqueness and continuous dependance for a multidimensional hyperbolic inverse problem. Comm. PDE, 10(6): 635–676 (1985)
Symes, W.: Linearization stability for an inverse problem in several dimensional wave propogation. SIAM, J. Math. Anal., 1986, 17(1): 132–151
Tarantola, A.: The Seismic Reflection Inverse Problem. Proceeding of the conference on inverse problem of acoustic and elastic waves, Santosa F. et al eds., SIAM, Philadelphin, PA, 1984
Xie, Ganquan, Li, Jianhua: Nonlinear integral equation of inverse scattering problems of wave equation and iteration. J. Comput. Math., 8(3): 289–297 (1990)
Yagle, A.E., Levy, B.C.: Lay-stripping solutions of multidimensional inverse scattering problems. J. Math. Phys., 27(6): 1701–1710 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Science Foundation of Southeast University (No.9207011148)
Rights and permissions
About this article
Cite this article
Liu, Jj. Linearization Ill-Posedness for 2.5-D Wave Equation Inversion Model. Acta Mathematicae Applicatae Sinica, English Series 18, 219–230 (2002). https://doi.org/10.1007/s102550200021
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s102550200021