In this paper we suggest and substantiate the approach to the approximation of the generalized inverse (orthogonal generalized inverse, pseudoinverse) operator for a compact operator that acts from the separable Hilbert space X to the separable Hilbert space Υ using the concept of the generalized normal r-solution (solution rank r) which was introduced in [2] for systems of linear equations.We suggest and substantiate the approximation of the r-pseudoinverse operator by the projection method. We have obtained the estimates that characterize the deviation of the r-solution when there are errors in the right-hand side. We also study the behaviour of the r-pseudoinverse when the original operator is perturbed

AbsIracL lhis paper describes a closed cycle of mathematical modelling of wave propagation processes. The half-space z > 0 is assumed to be filled with a verticallyinhomogeneous medium with the wave propagation velacity =(;). A source located on lhe free surface z = 0 causes the wave process U ( z , y , z , t), described by the initial boundary value problem for the wave equation.We consider two main problems: (1) Assuming c.(z) is lmown for all z, we wish to calculate the wave field U ( z , y , z , t ) ; (2) If ~( z )

The efficiency of any inversion method for estimating the medium parameters from seismic data strongly depends on simulation of the wave propagation, i.e., forward modeling. The requirements are that it should be accurate, fast, and computationally efficient. When the inversion is carried out in the frequency domain (FD), e.g., FD full-waveform inversion, only a few monochromatic components are involved in the computations. In this situation, FD forward modeling is an appealing potential alternative to conventional time-domain solvers. Iterative FD solvers, based on a Krylov subspace iterative method, are of interest due to their moderate memory requirements compared with direct solvers. A huge issue preventing their successful use is a very slow convergence. We have developed an iterative solver for the elastic wave propagation in 3D isotropic heterogeneous land models. Its main ingredient is a novel preconditioner, which provides the convergence of the iteration. We have developed and justified a method to invert our preconditioner effectively on the base of the 2D fast Fourier transform and solving a system of linear algebraic equations with a banded matrix. In addition, we determine how to parallelize our solver using the conventional hybrid parallelization (MPI in conjunction with OpenMP) and demonstrate the good scalability for the widespread 3D SEG/EAGE overthrust model. We find that our method has a high potential for low-frequency simulations in land models with moderate lateral variations and arbitrary vertical variations.

One of the most widespread problems in seismology is the necessity to adjust some a priori given medium structure. As a rule, such a priori information is vertically inhomogeneous component of wave propagation velocity. Such theoretical aspects of the problem as its uniqueness estimates of conditional stability are studied rather well. There axe also a variety of algorithms for its numerical solution. Therefore, in this paper the main attention is paid to numerical analysis of resolving ability and information contents of linearized inversion of multi-offset data.*Domaine de Voluceau, The work was supported by RFBR grants No. 96-01-01515 and 97-05-65280; and by the France-Russian Center of Appl. Math, and Inf. grant "Reliable numerical methods of solutions of nonlinear continually correct problems in application to inverse problems of wave propagation theory". Brought to you by | University of Arizona Authenticated Download Date | 5/31/15 6:15 AM * As we shall see from the presentation below, the case of half-space z > 0 does not have serious changes except the calculations which are more tedious. Brought to you by | University of Arizona Authenticated Download Date | 5/31/15 6:15 AM

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