Giant Gaussian process models of geomagnetic palaeosecular variation: a directional outlook

Abstract

SUMMARY Assessment of long-term palaeosecular variation (PSV) of the geomagnetic field is frequently based on simplified versions of a class of statistical models known as giant Gaussian processes (GGP) used to represent temporal variations in spherical harmonic descriptions of the field. Here we propose a new type of analysis to assess the shape and dispersion of the directional distributions caused by PSV. The quantities analysed in this study are equal-area coordinates of rotated distributions of palaeomagnetic directions, ${x_E}$ (east−west) and ${x_N}\ $(north−south) and their standard deviations (${\sigma _E}$ and ${\sigma _N}$). These are easy to determine, and can readily be numerically predicted for any GGP model, avoiding the need for the numerous simulations generally used to determine the scatter and/or elongation of directional distributions. Mean predictions of $\overline {{x_N}} $ for a simplified GGP model are different from the expected geocentric axial dipole (GAD) directions, in agreement with inclination differences noted in previous studies. The best estimates for palaeomagnetic inclination are the expected directions from the mean of virtual geomagnetic poles (VGPs) calculated using an iterative angular cut-off process. Predictions of ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}$ vary with latitude and are symmetric about the Equator. The N–S direction (${\sigma _{\rm N}}$) is always larger than E–W (${\sigma _{\rm E}}$), but the difference decreases from a maximum at the Equator to the poles, where ${\sigma _{\rm N}} = \ {\sigma _{\rm E}}$. A simplified GGP model is used to show that the parameter α (affecting variances in all Gauss coefficients) is positively correlated with ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}\ $ while the β parameter, the ratio of dipole to quadrupole family standard deviations, modifies the latitudinal dependence of ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}$. Experimental error in ${\sigma _{\rm E}}$ and ${\sigma _{\rm N}}$ can be accommodated using the common statistical parameters found in palaeomagnetic data sets, as ${\alpha _{95}}$ from site-mean directions. Predictions of simplified GGP models are compared with both numerical simulations and real data spanning the last 10 Ma. The latitudinal dependence of the proposed measures of PSV (${\sigma _{\rm E}}$ and ${\sigma _N}$) provide useful diagnostics for testing the validity of a GGP model. For the past 10 Ma the best-fitting GGP model with a mean GAD field set to $g_1^0 = \ - 18\ \mu T$ has α = 6.7 µT and β = 4.2. These new directional diagnostics will be used to investigate changes in overall geomagnetic field behaviour over other geological time intervals.