Wave Processes in Inhomogeneous Media

Nowadays, a theory and methods of wavefields simulation and inversion is one of the leading edges of modern science and technology. Their succesful application in advanced methods of oil and gas prospecting, computer tomography, non-destructive analysis, etc. (the list can be lengthened) is well known. However, they continue to grow and develop very actively and, apparently, it can be explained and understood taking into account the inner logic of historical development of sciences, enlarging of practical requirements and great opportunities provided by modern computing technologies.

In this project main attention is concentrated to inverse problems of wave propagation theory. The inversion hypothesis suggests that a useful way to interpret reflection seismograms and other geophysical data is to create a detailed mechanical model of the earth which predicts them. Then the contents of the model provide a more immediate basis for interpretation than do the data themselves. This approach has become thinkable in recent years as computers grow powerful enough to simulate seismic waves in believably complex multidimensional models. Judjing the merit of the inversion hypothesis will require both a collection of functional algorithms to invert geophysical data, and a database of experience in using the resulting model based interpretations in an industrial setting. The latter is impossible without the former, and up to now the former has been lacking.

The inverse problems of wave propagation theory are usually non-linear, large scaled, complicated by their ill-posedness and can be resolved only numerically. Each of these items brings its own specific difficulties, so, in order to get solution in acceptable cost some simplifications are usually assumed. In their own turn such simplifications should be considered as perturbations of an initial setting of the problem and, consequently, as a source of errors in the solution.

The main goal of the project is study and development of reliable and efficient numerical methods to solve inverse problems of the wave propagation theory. We believe that several important obstacles standing on the way of this goal are essentially mathematical and computational in nature, and that we are well positioned to grapple with these. We also believe that contact with field data, and with people who apply seismology to solve industrial problems, is essential to keep our mathematical and computational work properly focused.

In a framework of the project a range of questions is investigated both theoretically and numerically. Below a breaf list of such questions is given:

• Choice and analysis of mathematical model for a physical phenomenon under investigation. It means to clear up crucial physical pecularities for a specific process of wave propagation, to derive corresponding equations and initial and boundary conditions and to study well posedness of the problem. As a rule a simulation with acceptable quality of any physical process of wave propagation phenomena presupposes to reach at a compromise between degree of detalization and efficiency of its implementation. A reliable conclusion on the quality of simulation can be made only on the base of real data. As an example of such analysis the detailed investigation of attenuation and derivation of visco-elastic model presented in could be mentioned.
• Construction of an operator of the inverse problem which maps unknowm medium parameters into data of measurements. This operator is implicitly defined by the initial boundary value problem that simulates the wave process under consideration (the ''direct problem'') and the map of its solution to available data (input information for inverse problem) and should be studied carefully. Namely, its domain and range, injectivity, smoothness, existence and pecularities of its Frechet derivative. When the spatial dimension is one or parameters of a medium depend on depth only these questions are fairly well studied. But for multidimensional statements very little is known. The difficulties are essentially due to the ill-posed nature of the timelike hyperbolic Cauchy problem. Apparently, study of this kind of problems needs in modifications of the standard integral energy method. Generalizations of the well known Limiting Absorption Principle seem to be helpful in progress here too;
• Study and clearing up possibly most features of the operator of the inverse problem are important because on their base one can clarify a notion of a solution, and propose and justify a method to search for it. It seems to be almost natural that an inverse of the operator would be discontinuous and any solving procedure needs in some regularization in order to provide stability with respect to incoming errors. This leads to necessity of comparison of regularized and exact solutions in a case when the last one exists. Another trouble may appear while an iterative method is proposed to be developed to solve a non-linear problem because the operator may posess a derivative with an unbounded inverse and, moreover, the derivative itself may be unbounded as well. Apparently, a regularization method of selecting of stable subspaces in the model space could be appropriate when it is realized on the base of SVD analysis.
• Well-founded finite dimensional approximation of an operator equation of the inverse problem is a necessary step to implement any numerical approach to solve it. This includes the approximation of the equations and boundary conditions simulating the process of wave propagation and of the operator of the inverse problem and its derivative. Any finite dimensional approximation should be consistent with the choosen method of regularization. In order to make approximation effective and reliable it is necessary to investigate relationships between linear spans of bases in functional spaces which can be efficiently constructed (polynomials, trigonometric, ''wavelets'' and so on) and mentioned above stable subspaces in order to provide nearness of these subspaces. This can essentially reduce the computing ''cost'' comparing with the full SVD analysis.
• Developement of reliable and effective methods to solve specific problems of linear algebra appearing in the process of studying and resolution of inverse problems. This means, for example, that in investigations of well posedness of boundary value problems for Partial Differential Equations (including Absorbing Boundary Conditions and of stability analysis of Finite Difference Schemes modern algorithms of Unsymmetric Eigenvalue Problem of Linear Algebra are going to be applied. Solution of systems of linear algebraic equations arising in implementation of iterative processes for inversion of the non-linear operator will be founded on application of reliable direct and iterative schemes.

In order to provide reliability of the final results each of mentioned above subproblems should be analyzed carefully in order to get constructive errors estimates. Nowadays, for some of these subproblems such analysis has become a general tool. In particular, error and dispersive analysis of finite difference schemes for partial differential equations and problems of linear algebra are traditionally used to get an quantitative information about relation between an exact and approximate solutions of a specific problem. Unfortunately this analysis is usually essentially labour consumed and this could be a reason why it is not a common tool. However, as far as to problems of linear algebra are concerned, it should be recalled that a well known notion of matrix condition and its generalizations are appropriate in order to evaluate error estimates of numerical solutions. Moreover, there exists widely spread software (e.g. NAG, LAPACK) that provide a possibility to compute all necessary parameters of a problem under consideration.

Attainment the formulated goals of the project is inconceivable without of active use of modern computing technologies and development of software based on modern reliable and efficient algorithms. At the moment it is impossible to predict what specific software will be necessary, but we can name a few topics to which some efforts will be certainly devoted:

• finite-difference simulators with improved accuracy, absorbing boundary conditions;
• bases generators (in particular, wavelets) for numerical implementation of Galerkin type methods;
• computing of multidimensional integrals;
• Newton-like methods for non-linear problems;
• some problems of linear algebra, in particular, adaptation of SVD for a specific circumstances and unsymmetric eigenvalue problem to investigate well-posedness of boundary value problems for partial differential equations.

It should be underlined especially, that essential efforts will be goaled to the software development for parallel computations, as numerical investigation and resolution of the listed above problems for realistic models is very time and RAM consuming.

In such a way the combined research team presents the specialists as in wave propagation theory as in parallel computing too. To speed up our investigations we are going to implement numerical experiments together with the theoretical study in order to develop a suitable technological complex providing applications of new algorithms in wavefields inversion and simulation.

At present our investigations are supported by:

• Grant of Russian Foundation of Basic Research 96-01-01515 ''Resolving ability and stability of inverse problems of wave propagation for multicoverage systems'';
• Institut Franco-Russie de Matematique Appliquee within the framework of the theme ''Numerical methods of wavefields inversion''.

Recent selective publications

1. Kostin V.I. Tcheverda V.A. 1995 r-pseudoinverses for compact operators in Hilbert spaces: existence and stability. Journal of Inverse and Ill-Posed Problems, 1995, v.3, n.2.
2. Khajdukov V.G., Kostin V.I., Tcheverda V.A. The $r$-solution and its applications in linearized waveform inversion for a layered background. Accepted to IMA volume "Inverse Problems of Wave Propagation", Springer Verlag, New York.
3. Khajdukov V.G., Kostin V.I., Tcheverda V.A., Clement F., Chavent G., Numerical comparison of SVD and Propagator/Reflectivity decomposition for the acoustic wave equation. INRIA Research Report n. 2888, May 1996 (co-authors Khajdukov V.G., Kostin V.I., Clement F., Chavent G.)