Direct and inverse problems of seismic exploration of anisotropic and dispersive elastic media based on volume integral equations

The theory of seismic exploration is based on the theory of elasticity, where one of the important roles is played by material equations - Hooke's law. The equations of elasticity theory include the density of the medium. In the general case, at each point of the medium, it is necessary to determine a matrix of parameters with a dimension of 12$\times$12 elements. In addition, these parameters can be dispersive, i.e. depend on the frequency. For such a number of parameters, the solution of the inverse problem, using standard measurement and calculation procedures, is difficult.
A new approach to solving inverse problems based on the development of M.V. Klibanov. The balance of elastic energy is obtained based on the vector representation of the equations of the theory of elasticity and integral equations for studying the reciprocity principle. Volumetric integral equations are derived, on the basis of which the solution of the inverse problem of elasticity theory is obtained. Some examples of numerical implementation of the solution of direct and inverse problems of the theory of elasticity in three-dimensionally inhomogeneous anisotropic models of the geological environment are considered.