We derive a lithospheric magnetic field model up to equivalent spherical harmonic degree 1000 over southern Africa. We rely on a joint inversion of satellite, near-surface, and ground magnetic field data. The input data set consists of magnetic field vector measurements from the CHAMP satellite, across-track magnetic field differences from the Swarm mission, the World Digital Magnetic Anomaly Map, and magnetic field measurements from repeat stations and three local INTERMAGNET observatories. For the inversion scheme, we use the revised spherical cap harmonic analysis (R-SCHA), a regional analysis technique able to deal with magnetic field measurements obtained at different altitudes. The model is carefully assessed and displayed at different altitudes and its spectral content is compared to high-resolution global lithospheric field models. By comparing the shape of its spectrum to a statistical power spectrum of Earth's lithospheric magnetic field, we infer the mean magnetic thickness and the mean magnetization over southern Africa.

The increasing availability of magnetic field measurements, both at satellite
altitude and near or at the Earth's surface, makes it possible to improve
models of the lithospheric magnetic field periodically, both in terms of
accuracy and spatial resolution. Marine and aeromagnetic measurements,
because of their proximity to lithospheric sources, are capable of
capturing the small-scale features of the lithospheric magnetic field well.
However, its large-scale contributions are not accessible by means of these
measurements because of their limited spatial extent. Low Earth orbiting
magnetic satellite missions, such as CHAMP (

Satellite measurements and a coordinated international effort to collect and
merge all publicly available near-surface measurements led to the first
global grid of lithospheric field anomalies, the World Digital Magnetic
Anomaly Map (WDMAM), which was published in 2007 (

These global models represent the first attempt to merge the information
content of satellite and near-surface measurements in the form of SH models.
However, this merging is not carried out by a joint inversion of satellite
and near-surface measurements, mainly because of the existence of wide data
gaps that lead to numerical instabilities. The high inhomogeneity in data
availability calls for regional modeling approaches that are flexible in
adapting their spatial resolution to the available data coverage. Over
regions well covered by magnetic field measurements, regional models can
achieve higher accuracy and spatial resolution than global models. A review
of available regional modeling techniques in the framework of geomagnetism
is given by

In this study, we focus on the southern part of Africa, which is well covered
by near-surface measurements, and we opt for the revised spherical cap
harmonic analysis (R-SCHA,

The combined use of magnetic field measurements taken at different altitudes
requires careful data processing in order to maximize their compatibility
(see, e.g.,

We consider magnetic field CHAMP satellite measurements that were selected
and processed following the procedure described by

The selection and correction procedure of Swarm magnetic field measurements
closely follows the one described in

Since 1970, coordinated efforts have been made by African countries and
collaborators towards the aeromagnetic mapping of southern Africa. This
African magnetic mapping project (

Our ground data set comprises magnetic measurements from southern African
repeat stations and geomagnetic observatories as processed by

The input data set is summarized in Table

The amount of input data per data type.

Histograms of the different types of magnetic field data included in our model as a function of their altitude above the Earth's mean reference radius (according to the geocentric reference frame).

Revised spherical cap harmonic analysis (

This problem can be written as a linear system of equations:

The data are inverted in an iteratively re-weighted least square sense with
Huber weighting (see, e.g.,

In order to estimate the spectral content of the regional model, we compute
its R-SCHA surface power spectrum according to

This power spectrum is equivalent to an SH power spectrum (

The residuals of each data type after the inversion are shown in Figs.

Residuals between our model and the input data for

The residuals between the model and the WDMAM, shown in Fig.

Figure

The root mean square deviation (RMSD) for each data type and each component
is
summarized in Table

The root mean square deviation (RMSD) and the correlation coefficient between our model and the input data.

The predictions of our model over the ground measurement points (red
circles) and the mean of the measurements (blue squares) for the

Figure

The

Figure

The power spectrum of our regional lithospheric field model (blue curve) and the power spectra of the global WDMAMv2-SH800 and EMM2017 calculated over the region of interest (red and green curves, respectively). The solid line is used for the bandwidth over which the comparison between the spectra of the regional and global models is valid (see text for details).

We use the power spectrum of our model to estimate the mean magnetic
thickness and the mean magnetization over southern Africa. Traditionally,
such studies rely on Fourier power spectra (see, e.g.,

The misfit functional between the observational and the statistical power
spectra is defined as follows (

We calculate the misfit functional

The regional field model derived in this study offers insights into several
aspects of lithospheric field modeling. The first concerns the contamination
of lithospheric field models by external field sources (see

The second aspect concerns the flexibility of regional modeling in the use of
regularization. As shown in Fig.

The third aspect concerns the estimation of the magnetic thickness and
magnetization through spectral analysis of lithospheric field models.

Finally, the high-resolution vectorial maps of the magnetic field model
displayed at different altitudes offer important insights concerning the
sources of the observed anomalies. Although a thorough description and
interpretation of these maps is beyond the scope of our study, we point out
some striking features of the regional lithospheric field. Some of the
prominent anomalies visible in the

In this study we jointly modeled satellite, aeromagnetic, marine, and ground
magnetic field measurements and derived a lithospheric magnetic field model
over southern Africa. We showed that all different data sets are highly
compatible with each other, with the horizontal satellite components being
the least compatible with the rest of the data set due to external field
contamination. We showed that Swarm across-track differences at midlatitudes
efficiently eliminate the offsets between adjacent satellite tracks due to
the rapid variations of the ring current (see Fig.

By means of the R-SCHA power spectrum we showed that our model carries more
energy over the bandwidth constrained by near-surface measurements than
global high-resolution lithospheric field models over the same region, as it
does not require regularization (see Fig.

A wealth of information about the sources of lithospheric field anomalies can
be extracted from our regional, high-resolution, vector lithospheric field
model that can be accurately upward and downward continued (see Figs.

Through a spectral analysis of our model we inferred that the mean magnetic
thickness and mean magnetization over southern Africa lies between 11 and
22 km and 0.6 and 0.9 A m

Grids of our lithospheric field model at various altitudes are available in the Supplement.

FV and ET designed the study and prepared the necessary code. ET provided the processed Swarm magnetic field data. MK provided the processed surface magnetic field data. FV ran the calculations. FV, ET, and MK analyzed the results. FV wrote the paper with input from ET and MK.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Dynamics and interaction of processes in the Earth and its space environment: the perspective from low Earth orbiting satellites and beyond”. It is not associated with a conference.

The authors wish to thank Vincent Lesur for making available the processed CHAMP satellite data set and the South African National Space Agency (SANSA) Space Science in Hermanus for the collaborative effort to obtain and process the southern African repeat station data. Foteini Vervelidou was partly funded by Région Île-de-France and partly by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Schwerpunktprogramm 1788 “DynamicEarth” under the grant LE2477/7-1. This study was partly funded by CNES in the framework of the Swarm project. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Nicolas Gillet Reviewed by: Dhananjay Ravat and one anonymous referee