## 1. Introduction

For many decades, hysteresis measurements have been an essential tool for rock magnetic research and, more broadly, for magnetic research in physics and materials science [e.g., *Ewing*, 1885; *Stacey and Banerjee*, 1974; *Dunlop and Özdemir*, 1997]. Simple qualitative interpretation of mineralogy and/or grain sizes based on loop shapes and intercept values can be quite valuable, but increasingly sophisticated quantitative methods have provided powerful new ways of analyzing and “unmixing” the complex magnetic assemblages in natural materials [e.g., *Jackson et al.*, 1990; *Roberts et al.*, 1995; *Tauxe et al.*, 1996; *von Dobeneck*, 1996; *Fabian and von Dobeneck*, 1997; *Dunlop*, 2002a, 2002b; *Fabian*, 2003; *Yu and Tauxe*, 2005; *Fabian*, 2006; *Leonhardt*, 2006].

Curiously, however, little attention has been paid to quantifying the signal/noise ratio in hysteresis measurements and to the uncertainties in the parameters and the magnetization curves derived from hysteresis loops. In comparison, related magnetic unmixing approaches based on remanent magnetization curves [e.g., *Dunlop*, 1972; *Robertson and France*, 1994; *Heslop and Dillon*, 2007] have been subjected to much more rigorous scrutiny [e.g., *Egli*, 2003, 2004a, 2004b]. Published hysteresis data are most commonly reported only in terms of summary parameters (saturation magnetization *M*_{S}, saturation remanent magnetization *M*_{RS}, coercivity *H*_{C} and remanent coercivity *H*_{CR}) and ratios thereof, typically in graphical form on “Day plots” [*Day et al.*, 1977; *Dunlop*, 2002a, 2002b] or “squareness-coercivity plots” [*Nagata*, 1961; *Tauxe et al.*, 2002; *Wang and Van der Voo*, 2004], with at most a few full loops shown as representative examples. This makes it difficult to assess data quality and to fully appreciate the significance of data points located in any particular region of the Day plot.

In the laboratory database of the Institute for Rock Magnetism are more than 35,000 hysteresis loops measured on two Princeton MicroMag vibrating sample magnetometers (VSMs), by a large number of investigators, on a wide variety of materials, and over a broad range of temperatures. Signal/noise ratios vary enormously, as do the proportions of ferromagnetic, ferrimagnetic, paramagnetic, diamagnetic and antiferromagnetic contributions to the total signal (Figure 1). Recurring issues that arise in the processing and interpretation of these loops have led us to define some simple quantitative measures of data quality, which we describe in this paper.

Conventional loop processing (as in the MicroMag software, for example) involves several basic operations: (1) fitting the high-field data (the portion of the loop with *H*/*H*_{max} above a specified threshold, where the ferromagnetic (in a broad sense) moment is assumed to be saturated) with a line, whose intercept is taken as *M*_{S} and whose slope is taken to represent the (field-independent) susceptibility *χ*_{HF} of paramagnetic, diamagnetic and/or antiferromagnetic materials; (2) subtraction of the linear contribution *χ*_{HF}*H* from the measured *M*(*H*) to obtain a slope-corrected loop, assumed to represent the hysteresis behavior of ferromagnetic material in the specimen; and (3) finding the intercepts of the slope-corrected loop on the vertical and horizontal axes (*M*_{RS} and *H*_{C}, respectively). An optional additional step in the MicroMag software is the computation of, and correction for, vertical and/or horizontal displacements of the loop, based on inequality of the positive and negative axis intercepts, to correct for sensor offsets. More advanced processing techniques include: (4) decomposition of the loop into “reversible” (or “induced hysteretic”) and “irreversible” (or “remanent hysteretic”) components *M*_{ih}(*H*) and *M*_{rh}(*H*) (see Figures 1 and 6 for examples) [e.g., *von Dobeneck*, 1996; *Fabian and von Dobeneck*, 1997]; (5) decomposition into analytical basis functions and isolation of “partial loops” of distinct coercivity classes [*von Dobeneck*, 1996]; (6) “unmixing” in terms of canonical empirical and/or theoretical functions [*Carter*-*Stiglitz et al.*, 2001; *Dunlop and Carter*-*Stiglitz*, 2006]; and (7) nonlinear high-field *M*(*H*) fitting to characterize the approach to saturation of hard magnetic phases and obtain improved estimates of *M*_{S} and other parameters [e.g., *Fabian*, 2006]. An even wider variety of parameter calculations opens up when the loop is supplemented with additional data such as the initial magnetization curve starting from *M*_{RS} (the *M*_{si} curve of *Fabian and von Dobeneck* [1997], called the “ZFORC” by *Yu and Tauxe* [2005]); derived quantities include the *M*_{BF} (*H*) curve and its field axis intercept *H*_{CR} [*Fabian and von Dobeneck* 1997], transient energy dissipation (“TED”) [*Fabian*, 2003], and the “transient hysteresis” [*Yu and Tauxe*, 2005].

All of these calculations and the parameters derived from them have accompanying uncertainties related to inevitable experimental imperfections (noise, drift, etc.) and subtle mathematical complexities (ill-conditioned inversion). Our primary goal in this paper is to take some important initial steps toward a longer-term goal of developing rigorous estimates of uncertainty in all of the properties determined by processing of hysteresis data. We begin with the essential step of quantifying the uncertainty in the individual magnetic moment measurements in the loop, and defining a simple, objective, quantitative measure of the signal/noise ratio. Although this can be done most directly by measuring multiple loops and calculating a mean and standard deviation for each point, we focus here instead on developing a method that can be applied to the measurements in a single loop, because (1) there exists an enormous amount of valuable previously measured hysteresis data for which replicate measurements are unavailable but which can be reanalyzed using suitable methods and (2) in certain circumstances it may be impractical to make replicate measurements. The quantified signal/noise ratio provides a criterion that can be used in deciding whether filtering of the loop data is likely to result in more reliable estimates of fundamental properties. A second significant step that we take here is to define some simple statistical tests that use the individual measurement uncertainties to evaluate certain models by testing for lack of fit to the data set. In particular, we develop tests for linearity of *M*(*H*), in selected high-field intervals or for the whole loop, and for “unmixed” models using analytical basis functions. We apply these tests in addressing the important and difficult problem of robust quantification of parameters describing the nonlinear high-field *M*(*H*) behavior of unsaturated materials.

The organization of this paper essentially follows a general processing protocol for hysteresis loops, which inevitably contain both random noise and systematic errors related to instrument drift and sensor offsets. We begin with the concept of loop symmetry, which is essential to our development of self-contained single-loop noise quantification. We develop a robust method for quantifying loop shifts (offsets of the center of symmetry from the origin), and we discuss some critical limitations on correcting for them. Noise quantification follows directly from these considerations. Before we can address the question of high-field *M*(*H*) fitting, it is necessary for us to discuss prerequisites including drift correction and filtering of loop data. Finally we evaluate different methods of analyzing the approach to saturation [e.g., *Fabian*, 2006], and we show that robust estimation of related model parameters is far more difficult than is generally appreciated.