Spherical harmonic analysis of paleomagnetic data: The case of linear mapping

We consider the problem of describing the ancient magnetic field as completely as possible using paleomagnetic data. One of the best ways to do this is to represent the field by the use of Gauss coefficients. Unlike ordinary cases, however, spherical harmonic analysis of paleomagnetic data is a complicated problem because (1) the data cannot be considered as contemporaneous and (2) they are mostly available only in the form of directions. If the mapping between the model parameters (Gauss coefficients) and the data is linear (e.g., the three components of the field), there is one-to-one correspondence between the means of the parameters and the means of the data. Conventional least squares techniques can be applied to the means as the relations between the data and parameters are the same as in the instantaneous case. One-to-one correspondence also exists between the variances of the parameters and the variances of the data. However, the functions appearing in these relations are not orthogonal to each other, and the matrix to be inverted is quite ill conditioned. For magnetic field directions that are nonlinearly related to Gauss coefficients, one-to-one correspondence is completely absent between the means (variances) of the parameters and those of the data. This means that inversion becomes quite complicated because the mean of such data contains information on both the mean and fluctuations of the parameters. We applied inversion to the paleointensity data of the past 5 Myr and obtained the means and variances of Gauss coefficients for that period. The means are well determined and show that the time-averaged field is dominated by the axial dipole component. It is more axisymmetric than the recent field, suggesting that equatorial dipole components are largely averaged out as assumed by paleomagnetic dipole hypothesis. Inversion of the variances gave less well determined results. Among the variances of Gauss coefficients that are significantly different from zero that of the axial dipole is much larger than those of other coefficients, indicating that the observed variation in paleointensity is largely due to the fluctuation of the magnitude of the dipole moment.

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