We introduce a formalism for estimating local spatial averages of the core-mantle boundary (CMB) radial magnetic field and its time derivatives, based on magnetic field observations collected by low-Earth-orbit satellites. This provides a useful alternative to conventional core field modelling based on global spherical harmonic basis functions, where noise in the polar regions maps into all harmonics, and model regularization and spectral truncation are required. A powerful perspective offered by the proposed technique is formal appraisal of the spatial resolution and variance of the resulting field averages. We use the Green’s functions for the Neumann boundary value problem to link the satellite observations to the radial magnetic field on the CMB and estimate field averages using a modified Backus-Gilbert inversion approach. Our approach builds on the Subtractive Optimally Localized Averages (SOLA) method developed in helioseismology, that seeks averaging kernels as close as possible to a chosen target kernel. We are able to account for both internal and external field sources and can easily incorporate data error covariance information, for example describing along-track serial error correlation. As a proof of concept we present a global map collecting local estimates of the radial main field (MF) constructed on a grid at the CMB with one degree spacing in latitude and longitude, derived from one month of three component vector magnetic field data collected by the Swarm satellite trio, using data from dark and geomagnetically quiet times. Using sums and differences of the field components taken along track and in the east-west direction we obtain estimates with spatial resolution kernel widths varying between 18 and 54 degrees depending on the latitude, and a standard deviation of approximately 10μT (i.e. 5% of the mean CMB field amplitude). The morphology of our CMB radial field map agrees well with results from conventional spherical harmonic field models. In a second application, we determine local estimates of the average rate of change, or secular variation (SV), of the radial field at the CMB, initially considering two year time windows, and performing the analysis on data collected by either the Swarm or CHAMP satellites. We obtain stable local estimates of the SV at the CMB, and present maps of estimates with averaging kernel widths of approximately 42, 33 and 30 degrees on the equator, with corresponding standard derivations of 0.25, 2.5 and 5 μT/yr. By subtracting SV estimates constructed at different epochs we are able to calculate the local aggregated secular acceleration (SA) and to study its time changes. Differencing SV estimates 2 years apart, and considering an averaging kernel width of 42 degrees on the equator, we obtain SA maps very similar to those found in the CHAOS-6-x7 field model truncated at SH degree 10. Using our approach we are able to directly control the width of the spatial averaging kernel and the length of the time window, enabling us to directly study the robustness of the inferred SA. Pushing to higher resolution in time, considering one year differences of SV estimates constructed using one year windows, we are able to track the evolution of coherent SA structures in time-longitude plots at the equator. At 25° W in mid 2007 we find a distinctive SA ’cross-over’ event, with strong, oppositely signed and adjacent, SA features rapidly changing sign within a year. Our method is well suited for studying such spatio-temporally localized SA events at high resolution; there will be further opportunities for such investigations as the time series of data provided by the Swarm mission lengthens.
- Secular variation
- Secular acceleration