Robust modelling of the Earth's magnetic field

We consider the consequences of a priori assumptions made about the probability density function used for historical geomagnetic data modelling. We describe a method, based on penalized maximum likelihood estimation, that can model data under any (for p greater than or equal to 1) p-norm measure of misfit, although only p = 1 and p = 2 are considered here. For p = 2 it is implicitly assumed that the errors in the data originate from a Gaussian distribution, whereas p = 1 assumes that the errors arise from a double exponential (or Laplace) distribution. We show that the geomagnetic main field models are consistent (in the sense that their residuals give excellent agreement with the assumed error distribution) only when modelled under the 1-norm measure of misfit. Least-squares (2-norm) methods that reject data at a pre-assigned residual level depend critically on the correct assignment of a priori error estimates, which are almost never known accurately. We demonstrate that main field models of the Earth's magnetic field generated by 1-norm modelling are robust, and therefore the nn hoc data rejection procedure, required by 2-norm modelling, should not be implemented.