About the book
To study complex dynamic processes and their monitoring, a synchronous technique for observing fields from various sources is already required. In this case, in contrast to previously solved problems related to the Geo-mapping of 3D layered - block media, when the unity of the complex interpretation model was achieved by superimposing separate models obtained by independent interpretation of data from various geophysical fields: gravitational, magnetic, seismic, electromagnetic, thermal, it is necessary take into account the thermodynamic relationship of the material parameters of the environment. Hence, the interpretation process no longer represents an independent solution of inverse problems for each field separately. The operator of the inverse problem in a complex interpretation is written out in the form of a system of equations for fields related by thermodynamic material relations. Examples include algorithms for determining convection mechanisms in the Earth’s mantle, interpretation of natural field anomalies associated with electrokinetic phenomena.
The theory of interpretation of geophysical fields is directly related to the solution of ill-posed problems and the theory of regularization created by the works of A.N. Tikhonov, V.K.Ivanov, M.M. Lavrentiev and their schools. As a result of its application, important results have been achieved in constructing models of local, regional, and global distributions of physical properties in the Earth and other celestial bodies, on which geophysical studies were carried out, in the framework of piecewise homogeneous models and linear approximation for describing the propagation of fields.
Further development of geophysical methods in the direction of constructing more and more adequate models of media and phenomena necessarily leads to more and more complex problems of mathematical geophysics, for which not only inverse, but also direct problems become significantly incorrect. In this regard, it is necessary to develop a new concept of regularization for simultaneously solving a system of heterogeneous operator equations.
Currently, the study of processes associated not only with geophysics and astrophysics but also with biology and medicine requires even more complication of interpretation models from non-linear and heterogeneous to hierarchical. This book will be devoted to the creation of new mathematical theories for solving ill-posed problems for complicated models.