Data gaps in ®nite-dimensional boundary value problems for satellite gradiometry A. Albertella, F. Migliaccio, F. Sanso DIIAR, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy e-mail: alberta@ipmtf4.topo.polimi.it; Tel.: +39-02-2399-6504; Fax: +39-02-2399-6530 Received: 14 August 2000 / Accepted: 12 April 2001 Abstract. In the framework of a boundary value prob- lem BVP), when areas on the boundary are void of data the solution of the problem becomes undetermined and clearly more dicult. 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 ®nite-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 con®rm the theoretical result. Key words: Polar gap ± Finite-dimensional model 1 De®nition of the problem A boundary value problem BVP) has from the abstract point of view the same outlook as the problem of intersecting two straight lines in a plane or, a little more generally, as intersecting two complementary linear manifolds in R n . Basically this has the form: ®nd x 2 R n such that Ax a Bx b 1 where A and B are matrices; a and b are vectors with the same number of rows as A and B respectively. If each of Eqs. 1) does have solutions, i.e. there do exist two vectors x and e x such that A x a; Be x b then, after setting x x n; x e x g Equations 1) can be transformed into: ®nd n; g such that n g e x x 2 An 0 Bg 0 3 Basically Eqs. 2) and 3) mean that we want to decompose the known vector e x x into two compo- nents n; g such that n belongs to the null space of A and g to that of B, i.e. e x x n g n 2 N A ; g 2 N B 4 As we know, such a decomposition always exists and is unique if we have simultaneously N A N B R n N A \ N B f0g 5 If we relax one of the constraints of Eqs. 4), e.g. if we cancel one of the rows of Bx b, then N B gets larger and of course the ®rst of Eqs. 5) is still valid i.e. we guarantee the existence of the solution) but the second no longer needs to be true i.e. we loose the uniqueness). The opposite happens if we increase the number of constraints. In the framework of a BVP, Eqs. 1) can be rein- terpreted as described below. The variable x is an un- known element belonging to some to make it simple) Hilbert space H of functions de®ned on a possibly regular) open set X, typically a relatively compact set or, as in the geodetic case, the complement of a com- pact set. The operator A is a dierential operator sending H onto another Hilbert space L, to which a belongs. The Correspondence to: A. Albertella Journal of Geodesy 2001) 75: 641±646