Minimum variance distortionless projection, the so-called Capon method, serves as a powerful and robust data analysis tool when working on various kinds of ill-posed inverse problems. The method has not only successfully been applied to multipoint wave and turbulence studies in the context of space plasma physics, but it is also currently being considered as a technique to perform the multipole expansion of planetary magnetic fields from a limited data set, such as Mercury's magnetic field analysis. The practical application and limits of the Capon method are discussed in a rigorous fashion by formulating its linear algebraic derivation in view of planetary magnetic field studies. Furthermore, the optimization of Capon's method by making use of diagonal loading is considered.

Nonlinear and adaptive filter techniques have a wide range of applications in geophysical and space science studies to find the most likely parameter set describing the measurement data or to decompose the data into a set of signals and noise. Above all, the minimum variance distortionless projection introduced by

From a theoretical point of view there are several origins for the derivation of the method. The first derivation of Capon's method

The analysis of planetary magnetic fields is of great interest and one of the main tasks in space science. Here we pay special attention to the analysis of Mercury's internal magnetic field, which is one of the primary goals of the BepiColombo mission

For example, when only data in current-free regions are analyzed, the planetary magnetic field is non-rotational and can be parameterized via the Gauss representation

Summarizing the magnetic field measurements for all

Within Eq. (

Since the method does not require the orthogonality of the basis functions, it has a wider range of applications when decomposing the measured data into a set of superposed signals, especially when the number of data points is limited. For example, when the magnetic field data are measured on a dense grid in the vicinity of the planet, the Gauss coefficients can be estimated via integration of the data. But in the case of a limited data set those integrals cannot be evaluated.

The following derivation of Capon's method is based on the linear algebraic formulation

As illustrated in the previous section, the magnetic field

For every data point

Because

Despite its simplicity it is obviously incorrect. As

For the inversion of Eq. (

Here,

If the non-invertibility of the matrix

In contrast to the estimator, the true coefficient vector is a theoretical given vector that is not affected by the averaging (

In the limit of vanishing errors

For the further evaluation of Eq. (

Taking into account that

Since Eq. (

Applying the filter matrix to the average of the non-parameterized parts of the field in Eq. (

This equation is one of the important constraints for the construction of the wanted filter matrix

Referring to Eq. (

Using Eq. (

Therefore, the output power

Multiplication of Eq. (

Because

Due to the ensemble averaging the matrix

By means of Eq. (

Regarding the expensive derivation, this compact formula for Capon's estimator is surprising.

The filter matrix

The filter matrix is applied to the disturbed data for estimating the output power that is related to Capon's estimator. Thus, the output signal–noise ratio (resulting from the filtering) can be expressed as

The input signal–noise ratio is given by the data and the underlying model, and therefore

It should be noted that the matrix

Referring to the previous section and considering the additional quadratic constraint, the filter matrix can be calculated by solving

In Fig.

Sketch of the trace of the filter matrix with respect to

Since

In the literature there are several methods for determining the optimal diagonal loading parameter

The additional quadratic constraint (Eq.

Sketch of the

When Capon's method is applied to the reconstruction of Mercury's internal magnetic field

Capon's estimator inherits the matrix operation structure of the least square fit estimator, which is given by

The least square fit method minimizes the deviation between the disturbed measurements

Referring to the abovementioned substitutions, Capon's method can be interpreted as measuring the deviation in Eq. (

Thus, Capon's method can be regarded as a special case of the least square fit method or more precisely of the weighted least square fit method, whereby the data and the model are measured and weighted with the inverse data covariance matrix. This property is useful for the practical application of Capon's method. In contrast to the computationally burdensome matrix inversions in Eq. (

To illustrate the mathematical foundations presented above, Capon's method is applied to simulated magnetic field data in order to reconstruct Mercury's internal magnetic field. As a proof of concept, the underlying model is restricted to the internal dipole and quadrupole contributions to the magnetic field as discussed in Sect.

For the reconstruction of Mercury's internal dipole and quadrupole field the internal Gauss coefficients

Implemented and reconstructed Gauss coefficients for the internal dipole and quadrupole field.

The deviation

Capon's method has been previously applied to the analysis of waves. Thus, the existing derivations treat the non-parameterized parts of the field

As already mentioned in the Introduction (Sect.

Capon's method is a robust and useful tool for various kinds of ill-posed inverse problems, such as Mercury's planetary magnetic field analysis. The derivation of the method can be regarded from different mathematical perspectives. Here we revisited the linear algebraic matrix formulation of the method and extended the derivation for Mercury's magnetic field analysis. Capon's method becomes even more robust by incorporating the diagonal loading technique. Thereby, the construction of a filter matrix is vital to the derivation of Capon's estimator.

The trace of the filter matrix in particular determines the array gain, which is defined as the ratio of the output and the input signal–noise ratio. If the trace is large, the output signal–noise ratio can decrease, resulting in signal elimination, and thus the performance of Capon's estimator degrades. Bounding the trace of the filter matrix results in diagonal loading of the data covariance matrix, which improves the robustness of the method. The main problem of the diagonal loading technique is that in general it is not clear how to choose the optimal diagonal loading parameter. Since for the analysis of planetary magnetic fields we prefer measurements that are stationary up to the Gaussian noise, estimators for the diagonal loading parameter that are related to eigenvalues of the data covariance matrix corresponding to interference and noise cannot be applied. Making use of the

For the calculation of Capon's estimator several computationally burdensome matrix inversions are necessary. Interpretation of Capon's method as a special case of the least square fit method enables the usage of numerically more stable and fewer burden minimization algorithms, e.g., the gradient descent or conjugate gradient method, for calculating Capon's estimator.

It should be noted that the parameterization of Mercury's internal magnetic field via the Gauss representation, as mentioned in Sect.

The influence of the averaging on the determinant of the data covariance matrix

The data covariance matrix is defined as

The raw data supporting the conclusions of this article will be made available by the authors without undue reservation.

All authors contributed to the conception and design of the study; ST, YN and UM wrote the first draft of the paper; all authors contributed to paper revision as well as reading and approving the submitted version.

The authors declare that they have no conflict of interest.

The authors are grateful for stimulating discussions and helpful suggestions by Karl-Heinz Glassmeier and Alexander Schwenke.

We acknowledge support by the German Research Foundation and the Open Access Publication Funds of the Technische Universität Braunschweig. The work by Y. Narita is supported by the Austrian Space Applications Programme at the Austrian Research Promotion Agency under contract 865967. D. Heyner was supported by the German Ministerium für Wirtschaft und Energie and the German Zentrum für Luft- und Raumfahrt under contract 50 QW1501.

This paper was edited by Lev Eppelbaum and reviewed by two anonymous referees.