The Effect of a Field of External Origin on Spherical Harmonic Analysis Using Only Internal Coefficients

Abstract

ALLDREDGE and STEARNS (1974)discussed the effect on spherical harmonic analysis of the geomagnetic field of omitting the n=1 terms of external origin (i. e. those associated with the ring current). In a numerical test they added a 100nT uniform field to a field synthesised from a set of (internal) IGRF coefficients, and then reanalysed by least squares fitting to (X, Y, Z) component data at points on a 10X10grid.Using only internal coefficients they were surprised to find that using sin 0 weighting they reproduced the IGRF coefficients almost exactly, but that without this weighting there were significant changes in the zonal (m=0) coefficients of odd degree. Their result is in fact to be expected. With the sin 0 weighting, the least squares' process is to a very good approximation equivalent to determining the coefficients by integrating over the sphere. Now, over the sphere, any harmonic field of external origin is orthogonal to any harmonic field of internal origin (see e. g. JAMES, 1973); similarly the field of any one harmonic is orthogonal to the field of any other harmonic, and each harmonic contributes independently to the mean square vector field (LowES, 1966). So the best the least squares can do is to reproduce the internal coefficients and leave the 100 nT external field unfitted.However without the sin 0 weighting the polar regions are overweighted and there is no longer orthogonality. The least squares will do its best to reduce the (polar weighted) residual field by changing the internal coefficients, and it does in fact manage to reduce the total (polar weighted) residual field to 96nT (vector) r. m. s. by introducing a spurious internal field of 24nT r. m. s. (Properly veraged over the surface the r. m. s. residual field will in fact be (1002+242)1/2 103nT.) That the errors are in the zonal harmonics of odd degree follows from the geometry of the uniform field. It is interesting that when they analysed the total intensity F they obtained the much larger spurious internal fields of 92 and 94nT respectively with and 515

Aliasing and noise in core-surface flow inversions

“…Nevertheless, the effect of ignoring external fields is mitigated by the fact that any harmonic field of internal origin is orthogonal to any harmonic field of external origin when the f y ( t ) in (3) are fixed by integration of magnetic data over the sphere r = a (Lowes 1976). A least-squares determination of the f T ( t ) closely approximates resolution of these terms by integration over the sphere.…”