Uncertainty estimates in geomagnetic field modeling

Abstract

Geomagnetic field models usually consist of coefficients of a truncated spherical harmonic analysis derived by a weighted least squares analysis. The data are typically assumed to be uncorrelated, and an estimated, diagonal, data covariance matrix is incorporated as an inverse weighting function. Accuracy estimates on the derived coefficients are taken to be the usual output covariance matrix. This procedure does not take into account the field from the truncated terms, the presence of crustal fields, or the presence of external fields, such as Sq. The resulting accuracy estimates are believed to be too low by factors of 2–10. A formalism is presented within which approximate account can be taken of these neglected fields. This formalism includes a weight matrix in which the observations are suitably correlated. A priori statistics describing the truncated main field, the crustal field, and the Sq field are required. An estimate of the properties of the crustal field is made which is consistent with both surface and satellite data. The properties of the Sq field are inferred from the model of Malin [1973]. Given these estimates, computation of the elements of the weight matrix is straightforward. For surface data the weight matrix is adequately approximated as diagonal for moderate sized data sets and, in those cases, is easily incorporated into the least squares formalism. However, for satellite data and for large quantities of surface data, the matrix is “full,” and an approximation is adopted to make the analysis tractible. Models are derived using a selected subset of Magsat data, both with and without the correlated weight matrix. Inclusion of the correlated weight matrix increases the error estimates of the coefficients by a factor of up to 7.5. Calculation of such models using no approximations in the correlated weight matrix are limited by computer size to observation sets of about 1500 points or less. However, using the approximation, much larger data sets can be accommodated, and it becomes practical to perform the calculations on satellite data sets. The GSFC(12/83) model, based on Magsat data, was rederived using this method. The coefficient uncertainty estimates increased by factors of 4–70, while the coefficient values remained essentially the same.

Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

“…A few published models of the geomagnetic field [viz. the NASA Goddard Space Flight Center Models: GSFC (12/83);GSFC 1985S;GSFC 1985DS;GSFC 1990S;GSFC 1990DS] give numerical values for both the spherical harmonic coefficients and their standard errors (Langel et al, 1989. The standard error in each spherical harmonic coefficient is approximately comparable with the magnitude of the corresponding annual variation, at least for the candidate models for DGRF 1985and IGRF 1990(Barraclough et al, 1992Langel et al, 1992).…”

“…These models also take account of neglected higher-degree fields (truncation), crustal fields and S q . It is believed that the resulting (standard) error estimates are a good indicator of actual coefficient accuracy , at least to within a factor of 2 (Langel et al, 1989). Hopefully, future research will result in even more realistic estimates of the errors in the spherical harmonic coefficients, based on a more rigorous treatment of both systematic and stochastic errors.…”

“…Consequently, these earlier uncertainties underestimated the actual uncertainties by a factor of between 2 and 10. The model published by Langel et al (1989) represents the first attempt to estimate realistic coefficient uncertainties. In their model, which is essentially a revision of an earlier model, approximate allowance is made for the magnetic field arising from neglected higher-degree fields (truncation), the presence of crustal magnetic fields and the existence of external magnetic fields, particularly S q (which represents the solar diurnal variation of ionospheric currents under quiet geomagnetic conditions).…”

“…1, are usually calculated from a weighted-least-squares fitting procedure that uses all available magnetic data (for a particular epoch) on a global scale, from observatories, satellites and repeat stations, as well as from various aeromagnetic, shipborne and land surveys (Langel et al, 1989Barraclough, 1990). To perform the least-squares analysis, the infinite summation in Eq.…”

“…DGRF or IGRF). Characteristic uncertainties in field-line tracing in the magnetosphere are estimated using one of the few published models of the geomagnetic field that presents both the spherical harmonic coefficients and their standard errors (Langel et al, 1989. Uncertainties in the specification of the external sources of magnetic field in the magnetosphere (viz.…”