Numerical evaluation of the scatter of principal magnetic susceptibilities from the uncertainties in experimental measurements

Abstract

Explicit equations are derived which specify the total eigenvector uncertainties for a tensor of second rank in terms of the matrix of experimental measurements and the estimated measurement errors appropriate to the experiments. Only the minimum but sufficient number of measurements necessary to specify the representative ellipsoid are considered in this analysis. The specific application in mind is the measurement of magnetic anisotropy of rocks, and the calculation of the statistical scatter of the total susceptibility ellipsoids. The result of the calculations provide direct proof that the errors in the triad of principal axes calculated from a meaned set of measurements for a sample should be less than the errors in the triad of axes calculated from a single measurement of the sample. This provides the investigator with the option of using a formal statistical approach to the analysis of anisotropy data based directly upon the raw measurements and estimates of instrumental accuracy, rather than upon the scatter of spatial orientations of the total susceptibility ellipsoids estimated from the measurements. The occasionally poor agreement of a set of individual specimen measurements with matrix theory is likely due to uncorrelated errors in the raw measurements and not necessarily to operational error, instrumental defects, or the specimen itself.