Summary
This paper is concerned with an inverse problem for two-dimensional elastic solids. It seeks to recover the subsurface density profile based on the measurements obtained at the boundary. The method considers a temporal interval for which time dependent measurements are provided. It formulates an optimal estimation problem which seeks to minimize the error difference between the given data and the response from the system. It uses a boundary regularization term to stabilize the inversion. The method leads to an iterative algorithm which, at every iteration, requires the solution to a two-point boundary value problem. Several numerical results are presented which indicate that a close estimate of the unknown density function can be obtained based on the boundary measurements only.
Similar content being viewed by others
References
Beck, J. V., Blackwell, B., St. Clair, C. R.: Inverse heat conduction: ill-posed problems. New York: Wiley 1985.
Gottfried, A.: Inverse problem in differential equations. New York: Plenum Press 1990.
Tikhonov, A. N., Arsenin, V.: Solutions of ill-posed problems. New York: Wiley 1977.
Parker, R. L.: Geophysical inverse theory. Princeton: Princeton University Press 1994.
Blakemore, M., Georgious, G. A.: Mathematical modelling in non-destructive testing. New York: Oxford 1988.
Stoffa, P. L., Sen, M. K.: Nonlinear multiparameter optimization using genetic algorithms: inversion of plane-wave seismograms. Geophysics56, 1794–1810 (1991).
Mosegaard, K., Verstergaard, P. D.: A simulated annealing approach to seismic model optimization with sparse prior information. Geophys. Prospect.89, 599–611 (1991).
Jim Yeh, T. C., Zhang, J.: A geostatistical inverse method for variably saturated flow in the vadose zone. Water Resour. Res.32, 2757–2766 (1996).
Woodburry, A.: Minimum relative entropy inversion: theory and application to recovering the release history of a groundwater contaminant. Water Resour. Res.32, 2671–81 (1996).
Chew, W., Wang, Y.: Reconstruction of two-dimensional permittivity distribution using the distorted born iterative method. IEEE Trans. Medical Imaging9, 218–25 (1990).
Keys, R.: An application of Marquardt's procedure to the seismic inversion problem. Proc. of the IEEE74, 476 (1986).
Tadi, M.: Explicit method for inverse wave scattering in solids. Inverse Problems13, 509–521 (1997).
Graff, K. F.: Wave motion in elastic solids. London: Oxford University Press 1975.
Tadi, M.: Variational method for the solution of tpbvp arising in mechanics. Comp. Mech.20, 468–473 (1997).
Anderson, B. D. O., Moore, J. B.: Optimal control, linear quadratic methods. Englewood Cliffs: Prentice Hall 1990.
Zienkiewicz, O. C.: The finite element method. Berkshire: McGraw-Hill 1977.
Richtmyer, R. D., Morton, K. W.: Difference methods for initial-value problems. New York: Wiley 1967.
Bond, L. J., Punjani, M., Saffari, N.: Ultrasonic wave propagation and scattering using explicit finite differencing methods. In: Mathematical modelling in non-destructive testing (Blackmore, M., Georgious, G., eds.), pp. 81–124. New York: Oxford 1988.
Ilan, A., Ungar, A., Alterman, Z.: An improved representation of boundary conditions in finite difference schemes for seismological problem. Geophys. J. Royal Astron. Soc.43, 727–745 (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tadi, M. Inverse wave scattering in 2-D elastic solids. Acta Mechanica 136, 1–15 (1999). https://doi.org/10.1007/BF01292294
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01292294