Recovery of the potential by using values of the moduleof its gradient
© Cherny F.V., Yakymchuk A.I.
The history of the problem investigation is briefly described. Gravity is characterized. It is proven that no accurate boundary conditions can be deduced for any linear classical problem of the potential theory (including the problem of Stox-Molodensky for the Laplace equation) from the potential gradient module values. It is shown that the error of the transformations of the potential gradient module anomalies as harmonic functions depends on the carvature of the equipotential surfaces of the field, the amplitude of the anomalies and the measure of the domain in which the transformation is made. It is stated that the methods of processing and interpreting gravity and aeromagnetic anomalies developed on the basis of the theory of harmonic functions and successfully used in exploration geophysics may appear to be ineffective in studying the deep regional structure of the Earth. To pass through the crisis we propose a new definition of a non-linear boundary problem of recovering the attraction (or magnetic) potential in an unbounded closed domain from the values of the module of its gradient at the boundary of the domain provided the potential is similar to the given one. A method is proposed to solve the problem in a succession of the solutions of the Neumann’s problems for the Laplace equation which determines the disturbing potential.