On Numerical Reconstruction of a Function from Incomplete Data on Arc Means in Seismic Tomography

• Autorzy: Denisiuk A.
• Języki publikacji: EN

Abstrakty

EN

Consider the problem of reconstruction of a small perturbation of the acoustic wave speed field from traveltime data with linear background slowness. Mathematically, the problem is equivalent to reconstruction of a function from the data of integrals along the circle arcs. The data is limited, in the sense that the base points belong to a compact set. We propose and numerically test a new approach, based on reduction of the problem to the inverse problem for the Radon transform. The data completion procedure is considered as well.

Słowa kluczowe

EN

radon transform; seismic tomography; inverse kinematic problem; spherical mean transform; interpolation of band-limited function

• Wydawca: Institute of Information Technology of the Warsaw University of Life Sciences – SGGW
• Czasopismo: Machine Graphics and Vision
• Rocznik: 2011
• Tom: Vol. 20, No. 4
• Strony: 413--437
• Opis fizyczny: Bibliogr. 21 poz., il., rys.

Twórcy

autor

Denisiuk A.

• University of Warmia and Mazury in Olsztyn

Bibliografia

• [1] Lavrent’ev M., Romanov V., and Vasiliev V. Multidimensional inverse problems for differential equations. Lecture Notes in Mathematics, 167, 1970.
• [2] Harger R. O. Synthetic Aperture Radar Systems. Academic Press, New-York, 1970.
• [3] Shepp L. A. and Logan B. F. The fourier reconstruction of a head section. IEEE Trans. Nucl. Sci., NS-21: 21-43, 1974.
• [4] Romanov V. G. Integral Geometry and Inverse Problems for Hyperbolic Equations. Springer, Berlin, 1974.
• [5] Herman G. T. Image Reconstruction from Projections. The Fundamentals of Computerized Tomography. Academic Press, New-York, 1980.
• [6] Fawsett J. A. Inversion of n-dimensional spherical averages. SIAM J. Appl. Math., 45: 336-341, 1985.
• [7] Tarantola A. Inversion of travel times and seismic waveforms. In G. Nolet, editor, Seismic Tomography, pages 135-157. Reidel, 1987.
• [8] Hellsten H. and Andersson L. E. An inverse method for the processing of synthetic aperture radar data. Inverse Problems, 3: 111-124, 1987.
• [9] Firbas P. Tomography from seismic profiles. In G. Nolet, editor, Seismic Tomography, pages 189-202. Reidel, 1987.
• [10] Denisjuk A. and Palamodov V. Inversion de la transformation de radon d’après des données incomplètes. C. R. Acad. Sci. Paris, 307, Serie 1: 181-183, 1988.
• [11] Alekseev A. S., Lavrent’ev M. M., Romanov M. E., and Romanov V. G. Theoretical and computational aspects of seismic tomography, volume 11, pages 395-409. Kluver Academic Publishers, Novosibirsk, 1990.
• [12] Palamodov V. P. Inversion formulas for the three-dimensional ray transform. Lecture Notes in Math, 1499: 53-62, 1991.
• [13] Palamodov V. P. On reliability of reconstruction a velocity field from hodograph. In Teriya i praktika issledovaniya litosfery, pages 63-71, Petropavlovsk-Kamchatskii, 1991.
• [14] Denisjuk A. Integral geometry on the family of semi-spheres. Fractional Calculus and Applied Analysis, 2: 42-59, 1999.
• [15] Denisjuk A. On reconstruction of a stable part of band-limited function by interpolation. Proceedings of the Mathematical Institute of Belarus Nat. Acad. Sci., 5: 60-62, 2000.
• [16] Denisjuk A. On two approaches to the problem of reconstruction from the arc means with incomplete data, volume 2, pages 11-15. BrSU, 2000.
• [17] Palamodov V. P. Reconstruction from limited data of arc means. Fourier analysis and applications, 6: 25-42, 2000.
• [18] Natterer F. The mathematics of computerized tomography. SIAM, 2001.
• [19] Nolet G. A Breviary of Seismic Tomography: Imaging the Interior of the Earth and Sun. Cambridge University Press, New-York, 2008.
• [20] Kuchment P. and Kunyansky L. Mathematics of thermoacoustic tomography. Euro. Jnl of Applied Mathematics, 19: 191-224, 2008.
• [21] Agranovsky M, Finch D., and Kuchment P. Range conditions for a spherical mean transform. Inverse Problems and Imaging, 3: 373-382, 2009.