Troyan, V. N.; Ryzhikov, G. A. Diffraction tomography: Construction and interpretation of tomographic functionals. (English. Russian original) Zbl 0973.74587 J. Math. Sci., New York 86, No. 3, 2773-2786 (1997); translation from Zap. Nauchn. Semin. POMI 218, 176-196 (1994). Summary: On the basis of a linearized model of propagation of seismic wave fields, we introduce the notion of tomographic functionals. The physical interpretation of tomographic functionals is that their integral kernels are spatial functions of the influence of variations of medium parameters on particular measurements of the wave field of sound signal. The norm of a tomographic functional is determined by the intensity of the influence function related to the interaction operator. The field is generated by a “source” with the dependence on time determined by the apparatus function of seismic channel. The analysis of tomographic functionals makes it possible to design mathematically tomographic experiments for monitoring active seismic zones by controlling the parameters of tomographic functionals. The richness in content of a tomographic experiment is determined not only by the norms of tomographic functionals, but also by the region where their supports overlap. Finally, we analyze tomographic functionals for the wave and Lamé equations. Cited in 4 Documents MSC: 74J25 Inverse problems for waves in solid mechanics 86A22 Inverse problems in geophysics 86A15 Seismology (including tsunami modeling), earthquakes Keywords:linearized model of propagation of seismic wave; tomographic functionals; norm; influence function; seismic channel; Lamé equations Citations:Zbl 0924.00023 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Belishev, M. I., The Gel’fand-Levitan equations in the multidimensional problem for the wave equation, Zap. Nauchn. Semin. LOMI, 156, 15-20 (1987) · Zbl 0695.35111 [2] Blagoveshenskii, A. S., On a local method for solving a nonstationary inverse problem for an inhomogeneous medium, Tr. Mat. Inst. Akad. Nauk SSSR, 115, 28-38 (1971) · Zbl 0231.35044 [3] Blagoveshenskii, A. S., Inverse problems of the theory of propagation of elastic waves, Izv. Akad. Nauk SSSR, Ser. Fiz. Zemli, 12, 16-29 (1978) [4] A. S. Blagoveshenskii, “On an inverse problem of the theory of propagation of seismic waves,” in:Problems of Mathematical Physics [in Russian], Leningrad (1966), pp. 18-31. [5] V. G. Romanov,Inverse Problems of Mathematical Physics [in Russian], Moscow (1984). · Zbl 0576.35001 [6] V. G. Romanov,Some Inverse Problems for the Hyperbolic-Type Equations [in Russian], Novosibirsk, (1972). [7] L. D. Faddeev, “Inverse problems in the quantum theory of scattering,” in:Modern Problems of Mathematics (Itogi Nauki Techniki) [in Russian], Vol. 3, Moscow, (1974), pp. 93-181. [8] A. L. Buchgeim,Introduction to the Theory of Inverse Problems [in Russian], Novosibirsk (1988). · Zbl 0653.35001 [9] A. N. Tichnov and V. Ya. Arsenin,Methods of Solving Incorrect Problems [in Russian], Moscow (1986). [10] V. N. Troyan and Yu. M. Sokolov,Methods for Approximating Geophysical Data on Computers [in Russian] Leningrad (1989). [11] G. A. Ryzhikov and V. N. Troyan,Tomography and Inverse Problems of Remote Sensing [in Russian], St. Petersburg (1994). · Zbl 0973.74587 [12] Ryzhikov, G. A.; Troyan, V. N., Tomographic functionals in interpretation problems of the elastic waves sounding, Vopr. Dinam. Teor. Raspr. Seismich. Voln, 28, 87-90 (1989) [13] G. A. Ryzhikov and V. N. Troyan, “Diffraction tomography and backprojection,”Proc. of the 9th International Seminar on Model Optimization in Exploration Geophysics, Berlin, Vieweg, 47-52 (1991). [14] G. A. Ryzhikov and V. N. Troyan, “On regularization methods in 3-D tomography,”Proc. of the 9th International Seminar on Model Optimization in Exploration Geophysics, Berlin, Vieweg, 53-61 (1991). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.