Harmonic splines for geomagnetic modelling

Which features of a geomagnetic field model on the surface of the core are really necessary in order to fit, within observational error, the field observations at and above the Earth's surface? To approach this question, we define ‘roughness’ in various ways as a norm on an appropriate Hilbert space of field models which is small when the field is smooth on the core surface. Then, we calculate the model with least norm (the smoothest model) which fits the data, sources outside the core being treated as noise. Sample calculations illustrate the effects of noise, of the choice of norm and of an uneven distribution of observing stations.