The reconstruction of Mercury's internal magnetic field enables us to take a look into the inner heart of Mercury. In view of the BepiColombo mission, Mercury's magnetosphere is simulated using a hybrid plasma code, and the multipoles of the internal magnetic field are estimated from the virtual spacecraft data using three distinct reconstruction methods: the truncated singular value decomposition, the Tikhonov regularization and Capon's minimum variance projection. The study shows that a precise determination of Mercury's internal field beyond the octupole up to the dotriacontapole is possible and that Capon's method provides the same high performance as the Tikhonov regularization, which is superior to the performance of the truncated singular value decomposition.

The in-depth analysis of planetary magnetic fields is a key element to understand the structure and dynamics of planetary interiors

Besides the parametrization of the magnetic field, the application of a robust inversion method for separating the magnetic field contributions is required. Several inversion methods have successfully been applied to the reconstruction of planetary magnetic fields. For example, the Tikhonov regularization

As mentioned in

In preparation for the analysis of the BepiColombo magnetic field data, the goal of the present study is the comparison of Capon's method with the truncated singular value decomposition and the Tikhonov regularization in application to simulated Mercury magnetic field data. Although the simulated data have to be regarded as a proxy for the MPO data that are not yet available, the great advantage of the application to simulated data lies in the fact that the exact solution of the inversion problem is known from the input of the simulation, which enables the comparison of the estimators with the ideal solution. The plasma interaction of Mercury with the solar wind is simulated using a hybrid plasma code. The resulting magnetic field data are parametrized by making use of a combination of the Gauss representation

The fluxgate magnetometer on board the Mercury Planetary Orbiter (MPO)

Since both the internal and external fields

In general, the shell includes current-carrying regions, and therefore, the contributions

It should be noted that the internal field is canonically described in a Mercury body-fixed (MBF) corotating coordinate system, whereas the external field is canonically described in a Mercury solar-orbital system (MSO), with the

In general, the number of data points from an orbital mission is much larger than the number of wanted model coefficients. Thus, the resulting shape matrix

For the reconstruction of Mercury's internal magnetic field, various kinds of data inversion techniques are available, such as the least-squares fit method, the singular value decomposition, the Tikhonov regularization and Capon's minimum variance method (e.g., Haykin, 2014). The construction of these inversion techniques is reviewed along with merits and demerits in this section.

The most prominent inversion method for linear inversion problems is the least-squares fit method (LSF) (e.g., Haykin, 2014). The method minimizes the quadratic deviation

The singular value decomposition (SVD) generalizes the LSF method (e.g., Haykin, 2014). The method is based on the decomposition of the shape matrix

One of the most commonly used techniques for reducing the condition number is the low-rank approximation or truncated singular value decomposition (TSVD)

On the one hand, the decreased condition number improves the solvability of the inversion problem. On the other hand, it should be noted that the elimination of singular values causes a lack of information, which can decrease the performance of the data analysis. This lack of information can be quantitatively estimated via the model resolution matrix

Within the application of the LSF method and the singular value decomposition, the wanted model coefficients are required to satisfy the highly overdetermined system of linear equations

Considering the analysis of Mercury's internal magnetic field, the wanted model coefficients describe the amplitude of the magnetic field. Thus, it is obvious that the currents flowing within the magnetosphere as well as inside of Mercury generate magnetic fields with a minimal energy. The energy spectrum

Inserting the singular value decomposition of the matrix

Comparison of the SVD estimator

In analogy to the TSVD, the modification of the shape matrix results in a nontrivial resolution matrix

Capon's minimum variance projection is broadly established in the analysis of seismic and plasma waves

Due to the complexity of Mercury's magnetosphere, the entire parametrization of the magnetic field contributions, generated by currents flowing within the magnetosphere, is unrealizable. Thus, it is useful to decompose the magnetic field

In the case of small singular values of the shape matrix

In the following, the above-presented inversion methods are applied to simulated Mercury magnetic field data for reconstructing Mercury's internal multipole field up to the dotriacontapole term. The internal field is modeled as being generated by the dynamo field (crustal fields are not considered here), and the multipole spectrum model is taken from the MESSENGER results

The magnetic field resulting from the plasma interaction of Mercury with the solar wind is simulated with the hybrid code AIKEF

Simulated magnitude of the magnetic field

The geometry of Mercury's magnetosphere is mainly dominated by the internal dipole field. Also the internal quadrupole field in terms of the apparently northward shifted dipole field is visible. The influence of the octupole, hexadecapole and dotriacontapole fields is not visually noticeable within the figure.

The internal Gauss coefficients implemented in the simulation code represent the ideal solution

Synthetically generated measurement positions (red dots) in the

The optimal regularization parameter for the application of the Tikhonov regularization results in

Implemented and reconstructed internal Gauss coefficients for the dipole, quadrupole, octupole, hexadecapole and dotriacontapole field. The implemented value

The deviations between the reconstructed and the implemented coefficients result in

Implemented and reconstructed internal Gauss coefficients for the dipole, quadrupole, octupole, hexadecapole and dotriacontapole field. The implemented multipole spectrum is taken from the MESSENGER results

In the present study, simulated magnetic field data and synthetically generated measurement positions are evaluated. Within the practical application to in situ spacecraft data, it is expectable that the measurement positions as well as the measurements will be determined defectively, resulting in estimation errors (e.g.,

Implemented and reconstructed internal Gauss coefficients for the dipole, quadrupole, octupole, hexadecapole and dotriacontapole field. For the implemented quadrupole and octupole coefficients, the lower boundary from the MESSENGER results

The analysis of MESSENGER magnetic field data provided a value of

The optimal regularization parameter results in

Implemented and reconstructed internal Gauss coefficients for the dipole, quadrupole, octupole, hexadecapole and dotriacontapole field. The implemented axisymmetric coefficients are taken from the MESSENGER results

The relative errors result in

Besides the uncertainty concerning the internal coefficient

Although Mercury's internal magnetic field is dominated by an axisymmetric geometry

As newly established, Capon's method provides the same performance as the Tikhonov regularization. For the comparison of the two methods, a more general comment is appropriate. Both the methods incorporate the constraint of a minimum norm solution and, therefore, deliver superior results than the TSVD method. Within the derivation of the Tikhonov estimator

The detailed characterization of Mercury's internal magnetic field is expected to play an important role in understanding the origin of the field. Due to the interference of the internal parts with the magnetospheric field contributions, on the one hand, each part of the field has to be parametrized properly. On the other hand, a robust inversion method for reconstructing the wanted model coefficients is required.

In preparation for the analysis of the magnetic field measurements provided by the magnetometer on board the MPO, the plasma interaction of Mercury with the solar wind is simulated numerically. The resulting magnetic field data are parametrized by a combination of the Gauss representation with the Mie representation, called the Gauss–Mie representation, and the corresponding expansion coefficients are reconstructed from the data using the truncated singular value decomposition and the Tikhonov regularization as well as Capon's method. The reconstructed internal Gauss coefficients of the dipole, quadrupole, octupole, hexadecapole and dotriacontapole fields are in very good agreement with the coefficients implemented into the simulation code, and thus, a high-precision determination of Mercury's internal magnetic field up to the dotriacontapole is expectable. The quality of the reconstructed internal coefficients depends on the magnitude of the values. For example, in the case of an internal dotriacontapole coefficient of

The comparison of the inversion methods shows that Capon's method and the Tikhonov method provide a comparative performance. Since both the methods incorporate the constraint of a minimum norm solution, which is equivalent to minimum energy, Capon's estimator as well as the Tikhonov estimator deliver superior results than the truncated singular value decomposition. It should be noted that the constraint of a minimum norm solution is included synthetically within the derivation of the Tikhonov estimator. Since Capon's method is based on the minimization of the output power, which corresponds to the norm of the estimator, the constraint of a minimum norm solution is naturally implemented in the method. Furthermore, Capon's method weights the data adaptively, since the weighting is determined by the measurement positions and the measurements themselves, whereas the Tikhonov method weights all data equally. Besides the constraint of a minimum norm solution, further physically based constraints, for example, at the core–mantle boundary

Within the analyses presented here, simulated stationary magnetic field data resulting from the plasma interaction of Mercury with the solar wind under constant external conditions, i.e., constant solar wind density, velocity and interplanetary magnetic field orientation, are evaluated. Due to the fast temporal variability of the Hermean environment (e.g.,

Simulation data can be provided upon request.

All authors contributed to the conception and design of the study. ST and UM wrote the first draft of the manuscript. All authors contributed to manuscript revision and read and approved the submitted version.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful for stimulating discussions and helpful suggestions by Alexander Schwenke. Simon Toepfer and Uwe Motschmann acknowledge the North-German Supercomputing Alliance (HLRN) for providing HPC resources that have contributed to the research results reported in this paper.

This research has been supported by the Österreichische Forschungsförderungsgesellschaft (grant no. 865967) and the Bundesministerium für Wirtschaft und Energie (grant no. 50QW1501).This open-access publication was funded by Technische Universität Braunschweig.

This paper was edited by Anna Milillo and reviewed by two anonymous referees.