Maximum likelihood solution for inclination-only data in paleomagnetism

Abstract

S U M M A R YWe have developed a new robust maximum likelihood method for estimating the unbiased mean inclination from inclination-only data. In paleomagnetic analysis, the arithmetic mean of inclination-only data is known to introduce a shallowing bias. Several methods have been introduced to estimate the unbiased mean inclination of inclination-only data together with measures of the dispersion. Some inclination-only methods were designed to maximize the likelihood function of the marginal Fisher distribution. However, the exact analytical form of the maximum likelihood function is fairly complicated, and all the methods require various assumptions and approximations that are often inappropriate. For some steep and dispersed data sets, these methods provide estimates that are significantly displaced from the peak of the likelihood function to systematically shallower inclination. The problem locating the maximum of the likelihood function is partly due to difficulties in accurately evaluating the function for all values of interest, because some elements of the likelihood function increase exponentially as precision parameters increase, leading to numerical instabilities. In this study, we succeeded in analytically cancelling exponential elements from the log-likelihood function, and we are now able to calculate its value anywhere in the parameter space and for any inclination-only data set. Furthermore, we can now calculate the partial derivatives of the log-likelihood function with desired accuracy, and locate the maximum likelihood without the assumptions required by previous methods. To assess the reliability and accuracy of our method, we generated large numbers of random Fisher-distributed data sets, for which we calculated mean inclinations and precision parameters. The comparisons show that our new robust Arason-Levi maximum likelihood method is the most reliable, and the mean inclination estimates are the least biased towards shallow values.

Fast Estimation of Covariance Parameters in Least-Squares Collocation by Fisher Scoring with Levenberg–Marquardt Optimization

“…He presents different levels of noise in ice-core data determinable by the LLF extremum. Arason and Levi (2010), although involving a few numerical techniques of LLF maximum approximation in the paleomagnetic studies, finally verify these estimates by the use of LLF contours. Many of the implemented methods fail in their investigation of two covariance parameters.…”

“…The examples in geodesy include horizontal deformations, gravity data modeling or adjustment of the geodetic networks (Grodecki 1999(Grodecki , 2001Jarmołowski 2016;Kubik 1970;Yu 1996). ML has been used in geomagnetism for paleomagnetic studies, identification of magnetostratigraphic polarity zones and dating of magnetostratigraphic sections (Arason and Levi 2010;Man 2008Man , 2011. The hydrologic sciences provide the applications related with floods and soil hydraulic properties (Chowdhary and Singh 2010;Hoeksema and Kitanidis 1985;Sveinsson et al 2003;Tsai and Li 2008;Ye et al 2004;Zimmermann et al 2008).…”

Site U1373

“…Discrete samples were obtained from only 9 of 10 in situ lava flows. The archive-half measurements provide the most comprehensive view of inclination variations in these flows, and we calculated average inclinations using inclination-only statistics (Arason and Levi, 2010) for each lithologic unit in two ways (Fig. F59C-F59E) (see "Paleomagnetism" in the "Methods" chapter [Expedition 330 Scientists, 2012a]).…”

Methods

“…For azimuthally unoriented cores, the simple arithmetic mean of inclination data will be biased to shallower values (e.g., Kono, 1980;McFadden and Reid, 1982;Arason and Levi, 2010). To compensate for this bias, we have used the inclination-only statistics of Arason and Levi (2010) to calculate the overall mean inclination for the cored interval and appropriate subintervals.…”