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Revisiting the eccentric dipole approximation of the geomagnetic field

Abstract

**Abstract**. In this paper we approach the problem of describing the Earth’s magnetic ﬁeld with a computationally non-intensive model by ﬁtting parameters of a multivariate eccentric dipole (ED) model to values produced by IGRF, as opposed to computing it using the first eight Gauss coefficients of the spherical harmonic expansion. We discuss the differences between these approaches and the potential benefits of the former one. We also demonstrate the validity of our argument by comparing the measured and predicted positions of R1 to the estimated position of the magnetic center in the GSM system, as well as performing the same comparison for other magnetospheric parameters.

A common representation of the magnetic field of the Earth’s internal sources is its approximation by the magnetic field of an eccentric dipole with its shift relative to the Earth’s center represented by a (dx, dy, dz) vector in Cartesian geographic coordinate system (GEO). This approach was first popularized in (citation not found: Cole_1963); a good review of its evolution and modifications can be found in (citation not found: Frazer-Smith-1987). However, currently a representation based on spherical harmonic analysis (SHA) of the internal source magnetic field potential is usually preferred in order to account for inhomogeneities, local deviations from a dipole field and other artifacts and high-order harmonics (citation not found: igrf_11).

Still, as convincingly argued in (citation not found: Campbell_2004), the eccentricity of the dipole component of the geomagnetic field is probably real and should be treated as such. This phenomenon would manifest itself in displacement of the van Allen belts, which would be centered on the dipole instead of the geometric center of the Earth, among other effects. Assuming that the dipole is located in the center of the Earth inevitably introduces high-order harmonics in the SHA representation of the geomagnetic field. Furthermore, secular variations of the eccentric magnetic dipole’s position and orientation directly influence the estimated position of geomagnetic poles; ignoring the dipole’s eccentricity may lead to significant discrepancies between the observed magnetic dip pole and geomagnetic pole positions and velocities (citation not found: Newitt_2009), and thus its contribution should not be discounted lightly.

This argument may further be developed by taking into account the fact that the total number of observatories used in building the IGRF model is finite and it is indeed possible to compute polynomials that will produce a trivial solution; thus, a grid based solely on measurements by observatories and/or other sources cannot be considered orthonormal.

We decided to revisit the possibility of representing the geomagnetic field with an eccentric dipole in order to remove the aforementioned controversies and to locate the origin of the geocentric solar-magnetospheric coordinate system, which should in our opinion rather be dipole-centric. Also, it can be easily concluded that magnetospheric sources will not be geocentric and will be centered on the ED as the main contributing factor of their formation.

Subsequently, since in the expression for expansion of V in spherical harmonics

\(V(r,\theta,\phi)=R_E \sum\limits_{n=1}^\infty \sum\limits_{m=0}^n (\frac{R_E}{r})^{n+1} (g_n^m cos m\phi + h_n^m sin m\phi )P_n^m(cos\theta) + R_E \sum\limits_{n=1}^\infty \sum\limits_{m=0}^n (\frac{r}{R_E})^{n} (G_n^m cos m\phi + H_n^m sin m\phi )P_n^m(cos\theta) \)

we need to use a single coordinate system for both internal and external sources, we would rather place the origin at the location of the dipole in order to remove the dependence on time that arises from the consideration that the ED is engaged in a daily motion around the Earth’s center. If we now decompose the contributions of both internal and external sources in the dipole-centric coordinate system \(r', \theta', \phi'\), the only dependence on time will be expressed in secular variations, and in such a coordinate system it will be more convenient to study them.

In order to reliably determine the position of the eccentric dipole we use the following algorithm. First, we randomly choose a number of points on the surface of the Earth (100 points in our case). For each of these points we compute the reference field value components using IGRFv11 and store them in an array. Then we iterate over x, y, z coordinates in some specified ranges (in our case, from \(-0.1R_E\) to \(0.1R_E\)); in each iteration we place the dipole in the currently evaluated coordinates (dx, dy, dz) and compute the magnetic field value in the same set of points as the reference values:

\( x = sin\theta_0 \cdot cos \phi_0 - dx \\ y = sin\theta_0 \cdot sin \phi_0 - dy \\ z = cos\theta_0 - dz \\ \theta = acos(\frac{z}{\sqrt[]{x^2 + y^2 + z^2}}) \\ \phi = atan2(y, x) \\ B_r = -2 \cdot B_0 / R^3 \cdot cos\theta \\ B_\theta = -B_0 / R^3 \cdot sin\theta \\ \\ \left( \begin{array}{c} B_x \\ B_y \\ B_z \end{array} \right) = \left( \begin{array}{ccc} sin\theta \cdot cos\phi & cos\theta \cdot cos\phi & -sin\phi \\ sin\theta \cdot sin\phi & cos\theta \cdot sin\phi & cos\phi \\ cos\theta & -sin\theta & 0 \end{array} \right) \times \left( \begin{array}{c} B_r \\ B_\theta \\ 0 \end{array} \right) \)

We now fit these values to the reference ones and calculate the correlation coefficient and the standard deviation. If the standard deviation appears to be the lowest seen so far, we save the current shift values as the result candidate.

As soon as every single combination of x,y,z shifts has been evaluated and the best-fitting one selected, we fit it to IGRF values calculated in the coordinates of active magnetometer observatories (citation not found: kyoto_list) where, by definition, the values produced by IGRF correspond to experimentally observed ones. This cross-validation produces a correlation coefficient and a standard deviation which can now safely be used in further calculations. This procedure is necessary to avoid overfitting to the initial set of random points.

Finally, we repeat the whole process a sufficiently large number of times on different sets of random points in order to avoid hitting local minima and to choose the best estimate of dipole position. The shift vector produced after this step is considered to be the final result of the current version of our model.

We now briefly turn to comparison of our results to values produced by the method initially proposed in (citation not found: Schmidt_34) and further popularized by (citation not found: Bartels_36). The essence of this method lies in minimization of second-order terms of the SHA of the magnetic field potential. The eccentric dipole produced by this procedure is identical to the initial centered dipole in both magnetic moment and orientation, but is located in coordinates whose components can be derived from the Gauss coefficients. A more detailed description of this method can also be found at http://www.spenvis.oma.be/help/background/magfield/cd.html.

The ED position computed with our method gives a comparable but somewhat better correlation for examined dates. For instance, in case of Jan 01, 2000 the correlation coefficient of the ED found with our method is 0.97, while the Bartels method produces an ED with a 0.96 correlation coefficient.

Lastly, we should mention efforts of other researchers in this direction, namely, (citation not found: Cain_85), in which secular drift of ED poles is analyzed, (citation not found: Sano_90), which applies the same least-squares method to obtain extra dipoles that would account for previously neglected higher order components, and (citation not found: Ladynin_2008), in which ED orientation is also computed.

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