Geomagnetic field analysis -- I. Stochastic inversion

Most of the Earth's magnetic field and its secular change originate in the core. Provided the mantle can be treated as an electrical insulator, stochastic inversion enables surface observations to be analysed for the core field. A priori information about the variation of the field at the core boundary leads to very stringent conditions at the Earth's surface. The field models are identical with those derived from the method of harmonic splines (Shure, Parker & Backus) provided the a priori information is specified appropriately. The method is applied to secular variation data from 106 magnetic observatories. Model predictions for fields at the Earth's surface have error estimates associated with them that appear realistic. For plausible choices of a priori information the error of the field at the core is unbounded, but integrals over patches of the core surface can have finite errors. The hypothesis that magnetic fields are frozen to the core fluid implies that certain integrals of the secular variation vanish. This idea is tested by computing the integrals and their standard and maximum errors. Most of the integrals are within one standard deviation of zero, but those over the large patches to the north and south of the magnetic equator are many times their standard error, because of the dominating influence of the decaying dipole. All integrals are well within their maximum error, indicating that it will be possible to construct core fields, consistent with frozen flux, that satisfy the observations.