Current loops fitted to geomagnetic model spherical harmonic coefficients.

Abstract

One hundred-sixty circular current loops with radial axes were fitted by least squares to the 899 spherical harmonic coefficients of a 29th degree model. In the first case, the parameters that were fitted for each loop were the normalized magnetic moment, the distance from the center of the Earth to the current element, the colatitude and E. longitude of the loop axis, and one-half of the central apex angle of the loop. For this case, two of the loops converged near the inner-core outer-core boundary. They accounted for most of the Bipolar field. Twenty of the loops , all with much smaller magnetic moments than the two deep loops, converged in the distance range of 0.42 to 0.67 Earth's radius from the center of the Earth. The other 138 loops , after many iterations, were located at distances between 0.81 and 1.0 Earth's radius from the center of the Earth. The loops with radial distances between 0 .21 and 0.67 Earth's radius from the center of the Earth are referred to as "core" loops and those at distances greater than 0.81 Earth's radius as "crustal" loops. The spherical harmonic coefficients from these 160 loops, when subtraced from the 899 coefficients of the original model, left a root-mean-square residual of only 0.2 nT. A second case was tried which constrained the 138 "crustal" loops to be at 0.996 Earth's radius (25.5 km depth). In this case, the root-mean-square residual of the spherical harmonic coefficients from the original model was 0.9 nT.

The Geomagnetic Main Field and the Geodynamo

“…Other interpretations of the field include representations of the field in terms of dipole sources [ Alldredge 1987c; Benton and Alldredge 1987].…”

An alternate representation of the geomagnetic core field obtained using machine learning

“…There is a long history of other previous works searching for current loop representations of the geomagnetic field with a similar goal in mind, such as Alldredge (1987), Rong et al (2021) and references therein and Peddie (1979). All these attempted to fit either one or more magnetic dipoles or current loops to some representation of the Earth's magnetic field or a dataset that contains measured magnetic field values.…”

“…In Alldredge (1987) the author first inverted the parameters of two loops in axial position. Then based on the residual magnetic field, judiciously set the initial parameters for 5 additional current loops and carried out the inversion for the newly added loop parameters.…”

A New Technique to Diagnose the Geomagnetic Field Based on a Single Circular Current Loop Model

“…Because optimal fitting is strongly dependent on initial input parameters, no general technique exists for determining whether a particular minimum error corresponds to the best solution. The more loops that are involved, the more parameters that are required and the more time‐consuming calculation become (without additional constraints and assumptions) (Alldredge, 1987; Demina & Farafonova, 2016; Peddie, 1979; Zidarov, 1985). Peddie (1979) tried to avoid this problem by making 20 preliminary computer runs using random initial parameters for his loop models.…”

“…Zidarov (1985) chose a suitable initial "point" of loop parameters to iterate the calculation in his multiple loops model. In the study of Alldredge (1987), the initial values were chosen by referring to results obtained by Benton and Alldredge (1987). Demina and Farafonova (2016) selected 23 starting points to invert the seven parameters of a single current loop.…”