Direct and Inverse Problems of Seismic Exploration of Anisotropic and Dispersive Elastic Media on Volume Integral Equations

Abstract: The theory of seismic exploration is based on the theory of elasticity, where an important role 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 dimensionality of 12 × 12 elements. In addition, these parameters can be dispersive, i.e., depend on the frequency. For such a number of parameters, it is difficult to solve the inverse problem using the standard measurement and calculation procedures. A new approach to solve inverse problems based on the development of M.V. Klibanov’s ideas is proposed. 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, based on which the solution of the inverse problem of elasticity theory is obtained. Some examples of the 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. © 2023, Pleiades Publishing, Ltd.