On the Observability of the Time-Variable Lithospheric Signal in Satellite Magnetic Data

Abstract

The lithospheric magnetic field, which is one of the main objectives of ESA’s mission Swarm, is slowly varying in time due to an induced component. This variation is small (usually it is omitted in the lithospheric modelling) but recent advances in processing strategies and still-growing amount of satellite data open questions whether such an effect should be considered in the development of the lithospheric models—when using data from missions like CHAMP and Swarm. This effect can now be estimated over a period of 17 years (since the launch of CHAMP), and it is shown how the satellite measurements (over the observable part of the spectrum) can be referenced to one common epoch. For this purpose, we first inverted the magnetic field vector from CHAOS-6 over degrees 21–120, after subtraction of a remanent model, to a vertically integrated susceptibility map. Using this susceptibility distribution and taking into account the evolving core fields from the CHAOS-6 model, the time-varying lithospheric signal is computed. The results depend on the time span and the altitude considered, e.g., an altitude of 400 km and a span of 17 years can produce more than 0.5 nT variations resulting in a peak-to-peak value of nearly 1 nT. The vertically integrated quantities appear to be a useful choice for parameterising lithospheric time variations, also for providing data corrections at the satellite altitude. The effect of the choice of the core field, which enters the inversion, on the lithospheric time variation is also studied—this effect is found less important even for core fields 20 years apart.

Appendix: Tile-Wise Strategy for Estimating a Vertically Integrated Susceptibility

Ideally, a solution to the linear inverse problem associated with Eq. (1) is based upon a complete (global) integration domain. Although satellite lithospheric geomagnetic fields do not typically exceed the spherical harmonic degree 180, global inversions at a corresponding spatial resolution may be too demanding for ordinary PCs. Here, we outline an iterative approach that has very low hardware requirements, but it is capable of providing a global seamless solution. The problem is solved by splitting the integration domain into tiles, and then by taking care of the edge effects that are iteratively reduced. This approach is also applicable to other integral equations in the geopotential field modelling (e.g., Poisson integral equation).

Fig. 8

Tiling of the integration domain and the meaning of the margins used in the inversion for obtaining a global VIS map

The design of the iterative procedure is as follows:

• (1) The global integration domain is divided into tiles according to Fig. 8, where each tile is further extended with a (user-defined) margin.

• (2) Magnetic data using Eq. (1) (after it is adapted for the magnetic field vector) are inverted in such an extended area (a tile with red margins from Fig. 8), and VIS is obtained for each tile. For the inversion, the variance component estimation technique is used following Koch and Kusche (2002); Kusche (2003a, b) and Xu et al. (2006); Shen et al. (2012).

• (3) A tile by tile, global map of the VIS is constructed—however, only the innermost area of each tile is put into this map (green in Fig. 8). In this way, the largest edge effects accumulated in the margins are eliminated.

• (4) With the global VIS, a lithospheric signal at the upper altitude is computed—now, the integration domain excludes the extended tile so that the forward calculation estimates the edge effects for a given tile at the upper altitude. The edge effects are not estimated in the first iteration, as there is no VIS available.

• (5) The edge effects are then subtracted from the original data in each extended tile, and one continues with the inversion from step (2). The whole procedure is repeated until a chosen criterion is met, and the seamless global solution of VIS is obtained.

The use of the margins helps the procedure to converge. This is because a VIS composed from the innermost areas estimates the edge effects better than a VIS composed from the tiles containing also the margins deteriorated by the local inversion. Note the tiling scheme is independent of the inversion so that different methods can be employed in step (2). To stop the iteration procedure, we use a fixed number of iterations although there might be more sophisticated criteria considered (e.g., data residuals or differences between two last solutions, etc.). In our case, the results do not change much with more than 40 iterations ($<<0.1\%$). Each iteration (on a 1 arc-deg global grid) took about 10 minutes with the tiles 30 × 30 arc-deg large and the margins thick 5 and 10 arc-deg in latitude and longitude, respectively.

Fig. 9

Difference of the nth and $(n-1)$th estimate of the magnetic susceptibility in the iterative inversion procedure with $n=40$. The left and right panel displays the situation with and without the margins involved, respectively

The magnetic susceptibility obtained with and without the margins is shown in Fig. 9. When the margins are not involved (right panel), the tile-wise structure of the integration domain is clearly visible since the procedure does not converge. Conversely, using the margins we can find quite a seamless solution with some minor effects ($<|1|\,\mathrm {m}\cdot \mathrm {SI}$)—these appear close to the magnetic equator where the spherical equiangular discretization does not coincide with polarity changes of the magnetic field vector used in the inversion. Also note that each polar area is resolved within a single tile for mitigating a singularity problem with this type of grid.