Bayesian inference in geomagnetism

Abstract

In the existence half of a geophysical inverse problem (finding a model to fit the data), any method of regularization is acceptable, and the damping parameter λ should be made as large as still permits a reasonable model to fit the data adequately. In the uniqueness half of the inverse problem (finding all reasonable models that fit the data) two common methods for regularizing are stochastic inversion (SI) and Bayesian inference (BI). In both methods λ is determined by the observer's prior beliefs. If the errors and the prior model distribution are both gaussian, SI and BI lead to the same calculations, but are interpreted differently. In Gubbins and Bloxham's (G & B's) recent use of surface and satellite magnetic data to find the radial magnetic field at the core-mantle boundary (CMB), their choice of BI seems appropriate. However, their method of choosing λ is suited to the existence problem rather than the uniqueness problem and overestimates the resolution which the data provide on the CMB. As a prior belief, the heat flow bound at the CMB would call for a λ 6000 times smaller than G & B's smallest λ. How this change would affect G & B's conclusions cannot be ascertained without repeating their calculations with the smaller λ, but recent work by Shure, Parker & Langel (1985) suggests that the data cannot determine the Gauss coefficients of the core for degrees above 10.