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The inverse problems of geo-exploration implies investigation of domains inside the earth crust which contain gas, oil or other minerals fields as well as determination of physical properties of these fields such as density, porosity, pressure, geometry and so on. The most important problem is determination of the boundaries of the fields which gives information about its location, volume and geometry and makes possible to predict costs and outcomes in exploitation.In practice, in e.g. oil exploration and seismology this inverse problem is mainly solved by means of themap migration method. The use of this method assumes an a priori knowledge (or, better say, a good a priori guess)of the velocity profiles in each particular field constituting the considered part of the earth. Then the method is numerically efficient and sufficiently robust. However, a good a priori guess is often difficult to achieve, especially taking into account large scales of geophysics/oil exploration.Recently, A.Blagovestchenskii, Y.Kurylev and V.Zalipaev suggested another method, using dynamic data, to simultaneously determine velocity (and density) profiles and location of interfaces. It is based on the Blagovestchenskii method of paired Volterra equationsfor one-dimensional inverse problems and is valid for weakly lateral heterogeneous media, i.e. the case when parameters depend mainly onthe depth. The method, which employs a relatively low-frequency part of the response signals, is rather efficient numerically partly circumventing the notorious ill-posedness of multidimensional inverse problems. However, to achieve better reconstruction of the interfaces which,as we have already mentioned, is the main goal in seismology/oil exploration, the accuracy obtained by this method may be not enough. However, it provides a good starting point for methods based on high-frequency asymptotics in wave propagation,in particular the method of non-stationary Gaussian beams. The use, on this stage, of non-stationary Gaussian beams may have an additional advantage, in comparison with the map migration techniques, being stable with respect to singularitiesof the ray field. The aim of the proposed project is to develop, theoretically and numerically, a combined method which would utilize a relatively low and middle frequency part of the response signal to construct an approximationto the velocity/density profiles and location of interfaces and then the higher-frequency part to improve the reconstruction of interfaces.Section 1-3
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