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##### On the non-uniqueness of main geomagnetic field determined by surface intensity measurements - the Backus problem

**Authors**: P. Alberto, O. Oliveira, M. A. Pais

**Ref.**: Geophys. J. Int. **159**, 548-554 (2004)

**Abstract**: We revisit the problem of non-uniqueness of harmonic magnetic ﬁeld models in a region outside a sphere containing the ﬁeld sources, when only intensity values on the sphere surface are known. Using the angular momentum algebra and the Clebsch-Gordan coefﬁcients, we are able to treat different aspects of this non-uniqueness following a uniﬁed line of reasoning. In this new framework, we ﬁrst recover two Backus results, namely the proof of uniqueness in the case of a ﬁeld generated by a ﬁnite number of harmonics and the recurrence relation that deﬁnes the well-known Backus series. This formalism allows us to extend previous studies in two ways: ﬁrstly, we show how to produce an harmonic series orthogonal on the sphere to some other arbitrary harmonic series; secondly, we outline a new method for computing magnetic ﬁeld models starting from scalar intensity values alone.