Vector Errors in Spherical Harmonic Analysis of Scalar Data

TLDR: In this paper, it was shown that spherical harmonic analysis of the geomagnetic scalar intensity F gives a synthesized field having large errors in the vertical component near the equator, where the field is predominantly horizontal.

Abstract: Summary It has recently been noticed that spherical harmonic analysis (SHA) of the geomagnetic scalar intensity F gives a synthesized field having large errors in the vertical component near the equator, where the field is predominantly horizontal. It is now shown that this is just one example of a more general ' perpendicular error ' effect; SHA of a scalar property of a vector field is equivalent to analysing one particular Cartesian component of the vector, and there is a tendency for the resultant synthesized field to have vector errors which are preferentially perpendicular to that component, and which have magnitudes considerably larger than the errors in that component. This paper shows how the (relative) average magnitude of the errors in the SH coefficients obtained from a given type of analysis can be estimated, and how these determine the magnitude and nature of the ' perpendicular error ' effect. It also shows that the effect is not directly related to any lack of uniqueness in the theoretical solution. While the effect can be large for the Earth's magnetic field, it is very small for the gravitational field.