A method for constructing high-order differential boundary conditions for solving external boundary value problems in geoelectromagnetism

A general approach to the construction of differential boundary conditions for vector fields satisfying the Helmholtz equation is proposed on the basis of the field expansion in multipole series and the application of annihilating operators to them. The resulting differential constraints can be used as boundary conditions in solving external boundary value problems. Examples of their application to the solution of forward geoelectric problems in three-dimensionally inhomogeneous media are examined. Their use at a finite distance from the source of an anomaly is shown to yield more accurate results than those obtained under the assumption that the anomalous field at this distance vanishes. Another effect of their application is a substantial decrease in the dimensions of the modeling domain and therefore in the time required to solve the forward problem. The “safe” distance for using the Dirichlet-type boundary conditions is estimated.