On the S-approximation of the Earth's gravity field

The linear integral representation method has been developed from the Backus and Gilbert (Backus, G. and Gilbert, F., 1967, Numerical application of formalism for geophysical inverse problems. Geophysical Journal of the Royal Astronomical Society, 13, 247-276; Backus, G. and Gilbert, F., 1968, The resolving power of gross Earth data. Geophysical Journal of the Royal Astronomical Society, 16, 169-205) method for solving problems, in which the number of data is sufficiently large; at the same time the spatial distribution of some physical properties is represented by a large number of the parameters, or even is an unknown function of spatial coordinates. The linear integral representation method enables us to obtain the solution using discrete and approximately given field data. Elements of the external gravity field are represented by a sum of simple and double layer potentials, which are harmonic outside the field sources. The field sources are distributed on one or several planes lying under the Earth surface. We show that this approximation (we call it S-approximation) gives reasonable results in the synthesis problems, since various gravity field elements can be obtained from the S-approximation of only one field element by elementary operations.