On the quantitative analysis and evaluation of magnetic hysteresis data

Magnetic hysteresis data are centrally important in pure and applied rock magnetism, but to date, no objective quantitative methods have been developed for assessment of data quality and of the uncertainty in parameters calculated from imperfect data. We propose several initial steps toward such assessment, using loop symmetry as an important key. With a few notable exceptions (e. g., related to field cooling and exchange bias), magnetic hysteresis loops possess a high degree of inversion symmetry (M(H) = -M(-H)). This property enables us to treat the upper and lower half-loops as replicate measurements for quantification of random noise, drift, and offsets. This, in turn, makes it possible to evaluate the statistical significance of nonlinearity, either in the high-field region (due to nonsaturation of the ferromagnetic moment) or over the complete range of applied fields (due to nonnegligible contribution of ferromagnetic phases to the total magnetic signal). It also allows us to quantify the significance of fitting errors for model loops constructed from analytical basis functions. When a statistically significant high-field nonlinearity is found, magnetic parameters must be calculated by approach-to-saturation fitting, e. g., by a model of the form M(H) = M-s + chi H-HF + alpha H-beta. This nonlinear high-field inverse modeling problem is strongly ill conditioned, resulting in large and strongly covariant uncertainties in the fitted parameters, which we characterize through bootstrap analyses. For a variety of materials, including ferrihydrite and mid-ocean ridge basalts, measured in applied fields up to about 1.5 T, we find that the calculated value of the exponent beta is extremely sensitive to small differences in the data or in the method of processing and that the overall uncertainty exceeds the range of physically reasonable values. The unknowability of beta is accompanied by relatively large uncertainties in the other parameters, which can be characterized, if not rigorously quantified, through the bootstrapped distribution of best fit models. Nevertheless, approach-to-saturation fitting yields much more accurate estimates of important parameters like M-s than those obtained by linear M(H) fitting and should be used when maximum available fields are insufficient to reach saturation.