Data gaps in finite-dimensional boundary value problems for satellite gradiometry

In the framework of a boundary value problem (BVP), when areas on the boundary are void of data the solution of the problem becomes undetermined and clearly more difficult. Physically. this could be the situation in which a gradiometer on a satellite on a perfectly circular orbit covers a sphere with measured second radial derivatives: if the satellite orbit is not polar, there are caps at satellite altitude which are not covered by data. A solution is presented based on an iterative algorithm, under the hypothesis of using a finite-dimensional model as is usually done in the time-wise approach. The convergence of the iterative solution is proved and a numerical example is shown to confirm the theoretical result.