Comments on “Field-Strength Dependence of Directional Dispersion in Magnetizations of Sediment Samples: A Statistical Model and Paleomagnetic Evidence” by Hyodo and Itota

Abstract

Hyodo and Itota (1993) asserted that the relative paleointensity can be estimated from the directional dispersion of specimen magnetizations of sediment, based on a model that the directional dispersion of particle magnetic moments is dependent on the magnetic field intensity. In addition, they presented paleomagnetic data claimed to support their idea. However their statistical treatment is only qualitative and misleading by not taking account of experimental error. An alternative interpretation is shown for their paleomagnetic data. Hyodo and Itota (1993) introduced the Fisher distribution (Fisher, 1953) to describe the directions of particle moments in sediment. Contrary to their qualitative treatment, I show a quantitative treatment based on the statistics on Fisher distribution. According to Fisher et al. (1987), the Fisher distribution for the directions of particle moments can be denoted by F{a, tcp}, where A denotes the mean direction and Kp is the concentration parameter (usually referred to as the precision parameter). A larger value of ,cp means a higher degree of directional concentration of particle moments. Now it is assumed that the strength of particle moment is uniformly m and each specimen contains a large number of N particles, then the magnetization intensity of a specimen has a distribution N(NmL(np), m(1 -2L(tcp)/tep -(L(ip))2)) (Mardis, 1972, p. 265), where N(p, 02) denotes a normal distribution with a mean of µ and a variance of a2 and L(a) expresses the Langevin function given by L(a) = coth a -1/a. We can find the mean intensity of specimen magnetizations NmL(np) directly dependent on the directional dispersion of particle moments rp. Unfortunately, there is at present no satisfactory theory to relate the field intensity to the magnetization intensity or the dispersion of particle moments. If we accept a paramagnetic gas theory for the alignment of non-interacting particle moments in sediment (Collinson, 1965), the directions of particle moments exactly follow the Fisher distribution with tcp = mB/kBT, where B is the field intensity, kB is the Boltzmann constant, and T is the absolute temperature (Watson, 1983), although this theory is not directly applicable to sediment especially for postdepositional realignment process (e.g. Barton et al., 1980). By using an analogy to the central limit theorem in a linear distribution, Hyodo and Itota (1993) showed that the directional dispersion of particle moments is positively correlated to the observed dispersion of specimen magnetizations in a single horizon. Without using such an analogy, it can be shown that the correlation is not necessarily true by applying the distribution theory on Fisher distribution. We can regard the direction of a specimen magnetization as a mean direction of random samples extracted from a population of particle moment directions following F{A, tcp}. Then the directional distribution of specimen magnetization follows F{a, tc,q}, where KS is given by tcpR and R follows N(NL(tcp), (1 -2L(Kp)/Kp -(L(tcp))2)) (Mardia, 1...