Normal vector models of the geomagnetic secular variation during late Tertiary time

During late Tertiary time, the secular variation of the geomagnetic field vector at a number of widely separated sites can be modelled as the sum of the field vector of a randomly sampled isotropic normal dipole moment and a non-dipole field vector that is a function of the assumed source geometry. Here, the non-dipole field vector distribution is calculated in the limit for an infinite number of radial dipole sources on the core surface with a possible latitudinal bias in geographic distribution and a normal moment distribution that is invariant with respect to geographic location. The model therefore consists of four degrees of freedom, and for the usual case of unit vector data the number reduces to three because the dipole variance the and non-dipole source moment variance can be specified only as their ratios to the mean dipole moment. The resultant non-dipole field vectors are non-isotropically normal with zero mean (if and only if the mean of the source moment distribution is everywhere zero). For the assumed source geometry, the dipole-non-dipole sum is normally distributed with mean and covariance as functions of latitude. For direct comparison with the available directional data, the normal distribution is integrated over all possible vector magnitudes to yield the associated unconditional (unit vector) distribution. In most statistical models of the distribution of the angular variance of the palaeomagnetic field vector as a function of time, the field vector is assumed to belong to a unit vector population whose variance in magnitude can be ignored. Owing to experimental difficulty and the mineralogical complexity of most rocks, estimates of the palaeomagnetic field direction generally do not include reliable estimates of the vector magnitude. In most cases, therefore, a description of the data in terms of their unit vector directions alone seems appropriate and sufficient. A major difficulty inherent in unit vector analysis is the fact that even though the true (threedimensional) vector distribution may be conceptually simple in form, the distribution of the directions of its constituents may be analytically intractable. For example, rotational symmetry about the mean vector direction (provided that it exists) is usually regarded as a necessary, but not sufficient, condition for analytical simplicity of a unit vector distribution; yet it is possible to construct many simple vector sets whose unit vectors do not possess such symmetry. The two directional distributions that have been analysed in most detail are those of Fisher (1953) and Bingham (i964, 1974), including its special cases, the Dimroth-Watson distributions. Both Fisher's and Bingham's distributions may be defined as conditional distributions of the three-dimensional normal distribution. Fisher's distribution is the subset of the isotropic normal conditional upon a given vector length (Downs 1966; Downs & Gould I967), whereas Bingham's distribution is the subset of a non-isotropic normal with zero mean conditional upon a given vector length (Bingham I964). Fisher's distribution has the additional remarkable '3 Vol. 306. A~~~~~~~~~~~~ym