Spherical harmonic analysis of the geomagnetic field: an example of a linear inverse problem

Abstract

The geomagnetic field at the Earth's surface may be represented in terms of spherical harmonic coefficients. The determination of the coefficients from observed values of the field, and the evaluation of a field value from a set of coefficients, may be regarded as linear inverse problems of discrete and continuous types respectively. Well-known procedures of inverse theory are found to provide new insight into some of the hazards of conventional methods of analysis, as well as suggesting some novel methods of approach. The methods are applied to a dataset of measurements of all three components of the field from 80 magnetic observations. Determining coefficients from observations is usually effected by least-squares analysis after truncating the harmonic series at some degree and order. Inverse methods all seem to have their counterpart in least-squares theory, but they do not help decide the best truncation level. An alternative approach, Parker's method of linear inference, has been developed to apply to spherical harmonic series. The results are similar to least squares but the error bounds are more pessimistic. In evaluating the field components from a finite set of coefficients it is only possible to find a weighted average of the field value over a certain area of a spherical surface. The averaging function appropriate to the usual spherical harmonic sum formula has oscillations and is not a well-localized average. Alternative formulae are suggested which give a better, localized average. The methods are applied to fields downward continued to the core-mantle boundary. Most striking results are obtained when the covariance matrix for the spherical harmonic coefficients is used to damp the field determination. This procedure retains the necessary information about the original distribution of the data and it is found to eliminate some of the features of magnetic charts in the oceans where data coverage is poor, showing that these anomalies are not real.