A new paleointensity determination protocol (the IZZI method) was recently proposed. The IZZI technique combines the Aitken (in-field, zero-field; IZ) and Coe (zero-field, in-field; ZI) methods. The IZZI protocol of paleointensity method was experimentally tested, showing a strong angular dependence resulting from the undemagnetized portions of partial thermoremanent magnetization (pTRM) tails. The IZZI method is better than the conventional techniques (Aitken, Coe, and Thellier) in three respects: (1) it can easily detect the angular dependence; (2) it provides a quantitative estimate for the consistency of the outcome between IZ and ZI step; and (3) it is quicker because the extra pTRM tail check step is unnecessary.
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 The geomagnetic field is a vector quantity, so that both magnitude and direction are necessary to fully describe it. For the past five decades, paleomagnetists have concentrated on determining the direction of the Earth's magnetic field at the time a rock formation was magnetized. Compared to directional studies, the intensity variation of the ancient geomagnetic field has been less studied. Deciphering the variation of ancient geomagnetic field intensity is no less important and may have broad applications from geochronology [e.g., Carlut and Kent, 2000] to discussions regarding the growth of the inner core and possibly the evolution of core-mantle boundary [e.g., Hale, 1987; Labrosse et al., 2001].
 Despite common use of the so-called Thellier-type techniques in paleointensity determination, a detailed understanding of these experiments is still lacking. Recently, reviews of various Thellier-type techniques were given by Selkin and Tauxe  and Valet .
 In the original “Thellier-Thellier” method [Thellier and Thellier, 1959], the specimen is heated twice at each temperature step, cooling first in a laboratory field (H2), then inverting the specimen and cooling in −H2. At each temperature, the natural remanent magnetization (NRM) lost can be estimated through vector subtraction.
 In practice, the most commonly used technique is the “Coe” method [Coe, 1967], in which we first heat the specimen to Ti and cool it in zero field to determine the NRM lost directly. Then we heat the specimen again to Ti and cool it in a laboratory field (H2) to determine the partial thermoremanent magnetization (pTRM) gained. As the temperature is increased in a stepwise fashion, the NRM is progressively replaced by pTRMs. This we will call the “zero-field/in-field (ZI)” method in the following.
 In order to compensate for what was called the “zero field memory effect,” Aitken et al.  modified the Coe method by reversing the order of the double heatings. In the “in-field/zero-field (IZ)” method, we impart the pTRM before carrying out zero-field heating [see also Valet et al., 1998; Biggin and Bohnel, 2003].
 In addition to the double heating, an in-field step at a lower temperature is repeated to check the capacity of the specimen to acquire pTRM (the pTRM checks). Furthermore, in order to check whether the pTRM acquired in the in-field step is completely removed by reheating to Ti, it has now become fashionable to insert a second zero-field step after the in-field step in the Coe method [e.g., Riisager et al., 2000; Riisager and Riisager, 2001]. This “pTRM tail check” is unavailable in either the Aitken method or the classical Thellier technique.
 Therefore there are several widely used paleointensity determination techniques that carry out stepwise double heatings to test reciprocity and reproducibility of pTRMs. In practice, these techniques are generally considered to be functionally interchangeable, producing equally reliable paleointensity estimates. However, on the basis of a simple quantitative model, Yu et al.  found that both IZ and ZI methods show a strong angular dependence on (H2), while the Thellier method is independent of the H2 direction. Contrary to common intuition, each method yields quite different outcomes as the reciprocity of thermal blocking and unblocking is violated, even with marginal (10%) tails of partial thermoremanence [Yu et al., 2004]. According to their simulation, the ZI method with the H2 antiparallel to the NRM is the most robust paleointensity determination technique when the intensity of the laboratory-induced field H2 is smaller than ancient field H1. The IZ method with the laboratory field parallel to the NRM is the optimum approach when the intensity of the laboratory-induced field H2 is larger than the ancient field H1. By far the most efficient method, however, is the IZZI method [e.g., Tauxe and Staudigel, 2004; Yu et al., 2004]. This method easily detects the angular dependence on H2 resulting from pTRM tails. In the present study, we will experimentally test the IZZI protocol by using well-defined samples to experimentally confirm the angular dependence of IZZI method.
2. IZZI Method
2.1. Some Preliminaries
 In the simplest case, there are two different types of pTRMs. To produce a pTRM↓(T1, T0), we heat a sample in zero field to the Curie point (Tc) and cool it in zero field to the upper end of the blocking temperature spectrum (T1) at which point the field is turned on and the specimen cooled to room temperature, T0. To produce a pTRM↑(T1, T0), we heat the specimen in zero field to T1 and cool it in a laboratory field to T0. In both cases, the portions that survive reheating above T1 (so-called high unblocking tails) are pTRM tail t↓(T1 < Tub < Tc) and t↑(T1 < Tub < Tc), respectively.
 The detailed mathematical model of various Thellier-type techniques has been provided elsewhere [Yu et al., 2004]; hence we include only the short summary. In the Thellier-type double heating experiments, we replace the NRM with successive pTRM↑s produced in H2 where H2 is generally not parallel to H1 or of equal magnitude. The ratio of the two magnitudes ∣H2∣/∣H1∣ is p and the angles relating the two fields are ϕ (in the horizontal plane) and θ (in the vertical plane).
2.2. New Protocol: IZZI
 In recent studies, the IZZI method combines the Aitken (in-field, zero-field; IZ), Coe (zero-field, in-field; ZI), pTRM checks, and pTRM tail check methods [e.g., Riisager et al., 2000; Riisager and Riisager, 2001; see also Tauxe and Staudigel, 2004; Yu et al., 2004]. In the IZZI method, we carry out quadruple heatings (as in the pTRM tail check method) at every other temperature step, while carrying out double heatings (as in the Aitken method) at the intervening temperature steps. On the basis of the mathematical model [see also Yu et al., 2004], we can predict the outcome of IZZI paleointensity determinations. Results in the Arai diagrams [Nagata et al., 1963] should be zigzagged if there are significant pTRM tails (Figure 1). This zigzag pattern is predicted to be prominent as the amount of t↑ increases and as H2 deviates from the NRM direction (Figure 1). As a result, data points from the entire IZ steps and those from entire ZI steps will form two different trend lines (Figure 1).
 If the two steps zigzag, the question arises as to which trend line, the ZI or the IZ results represent the more reliable paleointensity? The answer depends on the direction of H2 and the magnitude of p.
 In general, the ZI data set yields more reliable paleointensities than that of the IZ set. For example, as H2 deviates from the NRM, a more pronounced concave down feature develops in the IZ (Aitken) set than in ZI (Coe) set (Figure 1). On the other hand, the IZ set is superior when θ is zero (Figure 1) and when p > 1 [e.g., Yu et al., 2004, Figure 6].
3. Testing the IZZI Method
 In order to bridge from a mathematical approach to experiment, it is necessary to test the IZZI method using well-defined samples. On the basis of unpublished results (study of Israeli copper mine slags), we have selected samples that had shown the least sign of alteration. We chose a set of 23 samples including not only the best results but also samples rejected because of their nonlinear (“zigzagged”) Arai plots.
 A thermoremanent magnetization (TRM) was produced by cooling from 600°C in a laboratory field of H1 = 40 μT along the z axis (parallel) for 11 samples, x axis (perpendicular) for 6 samples, and −z axis (antiparallel) for 6 samples. We then carried out the IZZI experiment. During the in-field step, we applied a field H2 = 40 μT (p = 1) along the z axis of the samples. Double heatings were carried out for IZ steps at 350, 425, 470, 490, 510, 530, 550, and 570°C and quadruple heatings (including pTRM checks and pTRM tail check) for ZI steps at 400, 450, 480, 500, 520, 540, 560, 580°C.
 IZZI experiments (Figures 2a, 2c, and 2e) and associated vector projections (Figures 2b, 2d, and 2f) for three representative samples are shown. These three samples were previously rejected for paleointensity work because of their nonlinear Arai plots. For clarity, we have omitted pTRM checks in Arai diagrams. (But the pTRM checks were satisfied within 5% in all samples). When H2 is parallel to H1 (Figure 2a), the IZ and ZI sets are indistinguishable, agreeing well with the prediction of Yu et al. . As H2 deviates from H1, two distinctively different behaviors are expressed (Figures 2c and 2e). Except for the parallel case (Figure 2a), the ZI sets always fall closer to the ideal line and hence would be less biased in real paleointensity determinations (Figures 2c and 2e). In particular, the vector projection of NRM steps for the perpendicular case shows a slight zigzagging behavior (Figure 2d).
 As predicted [Yu et al., 2004], the pTRM tails also show an angular dependence (Figure 2). For example, pTRM tails are zero for the parallel test (Figure 2a), positive for the perpendicular test (Figure 2c), and negative for the antiparallel test (Figure 2e). This observation holds throughout the entire temperature range (Figure 2).
 We have plotted pTRM tail checks for all 25 samples as a function of temperature (Figure 3). Depending on the angular difference between H1 and H2, each type of experiment forms distinctive envelopes that barely overlap with one another (Figure 3). For a given sample, the ratio of (pTRM tail check)/TRM remains nearly constant over the entire temperature range (Figure 3). Note that individual samples have different unblocking temperature spectra. In particular, on average, pTRM tails were null when H2 was parallel to H1 (Figure 3), in excellent agreement with previous predictions and testing [Yu and Dunlop, 2003; Yu et al., 2004].
 Because pTRM tails have a pronounced angular dependence, the Aitken, Coe, and Thellier techniques are not interchangeable unless samples truly satisfy the reciprocity law. It is interesting that the angular dependence is absent in the classical Thellier technique because the tails in the two in-field steps compensate for the angular dependence [see Yu et al., 2004]. But the TRM tails cannot be checked in this method, a severe drawback.
 Contrary to common intuition, the pTRM tail check method detects the vectorial difference between the high-temperature tails of pTRM↑ and the preexisting high-temperature tails t↓ that are embedded in TRM [e.g., Shcherbakov et al., 2001; Yu and Dunlop, 2003; Yu et al., 2004; this study]. In other words, relying solely on the linearity of the Arai plot, pTRM checks, and even the conventional pTRM tail checks is insufficient to guarantee a reliable paleointensity result. In fact, the pTRM tail check can be used as a selection criterion only when p as well as θ are available.
 The best way to diagnose the existence of angular dependence of pTRM tails is to carry out the IZZI protocol with H2 nonparallel to H1. For samples with small tails, the zigzagging behavior of IZZI data is subdued. In principle, the zigzagging of IZZI data can be used as a sample selection criterion. The question thus arises: what quantitative measure can represent the zigzagging effect in Arai and vector plots?
 The zigzag effect appears both on Arai plot and vector projections (Figure 2). Because of its vectorial nature, the NRM zigzag is visible for H2 ⊥ H1 (Figure 2d) while the NRM zigzag for H2 // H1 and H2 \\ H1 is unresolvable (Figures 2b and 2f). The zigzag feature in the Arai plot is nearly absent for H2 // H1 (Figure 2a), and increased as the angle between H2 and H1 increased (Figures 2c and 2e). Apparently, H2 // H1 is the least sensitive for detecting zigzagging both in Arai plot and vector projections (Figures 2a and 2b).
 In order to assess the quality of the IZZI paleointensity results, we used the most fundamental relationship in the paleointensity determination. At a given temperature, the ratio of b (=NRM loss)/(pTRM gained) must remain constant to yield a reliable paleointensity. Note that the b ratio uses an NRM loss that is a difference between NRM and NRM remaining at each temperature. The results in Figure 2 are replotted to represent the temperature dependence of b (Figure 4), showing three interesting aspects. First, in a given sample, the scatter of data points in IZ and ZI set shows a strong angular dependence (Figure 4). For I067, the ZI set is slightly more scattered than the IZ set (Figure 4a), agreeing well with the prediction (Figure 1). A recently observed slight superiority of the Aitken method over Coe method [e.g., Biggin and Bohnel, 2003] is indeed true in the special circumstance when H2 is parallel to H1 and p = 1/2 (Figure 1c) [see also Yu et al., 2004, Figure 6c]. However, the ZI data set shows less scatter than the IZ data set as H2 deviates from the H1 (Figures 4b and 4c). Second, the magnitude of scatter increased as H2 deviated from H1 (Figure 4). The maximum scatter was 40%, 70%, and 140% for parallel, perpendicular, and antiparallel setting (Figure 4). Third, the observed peaks near 500°C in Figure 5 represent the nonlinear feature of Arai plots, illustrating that more remanence was demagnetized during the zero-field step than was magnetized during the in-field step. In all samples, the b ratio was not constant for T < 540°C (Figure 4).
 Three additional examples are shown in Figure 5. Sample I084 (parallel) shows a perfect linear Arai plot (Figure 5a) with a uniform b ratio (Figure 5b). Two other examples are also acceptable because they pass all the conventional selection criteria, but they show a slight zigzagging in Arai plots (Figures 5c and 5e) associated with a nonuniform b ratio below 500°C (Figures 5d and 5f). Although substantial peaks occur apparently at <500°C (Figures 5d and 5f), the corresponding pTRMs represent <10% of TRM (Figures 5c and 5e). On the other hand, the last 6–8 data points, covering a significant fraction of TRM, show a rather constant b ratio (Figures 5d and 5f). Somewhat varying values for the ideal line reflect the anisotropic feature of the sample that shows a different capacity in acquiring TRM along the three composite axes (Figures 4 and 5).
 A simple comparison between I067 (Figures 2a and 4a) and I069 (Figures 5c and 5d) or I070 (Figures 5e and 5f) clearly demonstrates the difficulty in using the b ratio as a simple selection criterion. Despite its nonlinear features in Arai plots, a much more uniform b ratio was observed for I067 (Figures 2a and 4a). This paradox results from the fact that b is controlled by three competing factors: 1) the amount of tail t↑TRM, 2) the angular dependence of H2 and H1, and 3) the corresponding pTRM fraction (=pTRM/TRM). As a result, the b ratio alone fails to diagnose the quality of paleointensity determination.
 We therefore need a parameter that compensates for the effect of 3 but represents the effects of 1 and 2. We define the “zigzag” parameter Z as follows:
where ri is the pTRM fraction (=pTRM↑TRM(Ti, T0)/TRM). For a uniform application to all samples, it was necessary to correct the effect of magnetic anisotropy (from the results of 3-axis TRM). We then normalized all the values of observed magnetization to TRM. The first term, (bi − p), reflects a scatter between the observed b and the ideal reference value (=pTRM/NRM). For anisotropic samples, anisotropy correction is necessary prior to Z estimation. In practice, p is unknown, the ideal value can be replaced with the estimated ratio of ∣H2∣/∣H1∣. The second term, ri, represents a weighting factor, in order to account for the increasing amount of pTRM fraction in the high-temperature range. As a sum, the Z parameter represents the cumulative scatter of b ratios at all temperature ranges.
 In the IZZI experiment, two separate values of Z can be determined for the ZI (ZZI) and IZ (ZIZ) sets. In addition to the 23 samples analyzed in section 3, we have added 5 results where the IZZI method was applied to the NRMs, for a total of 28 samples. Note that the IZZI method was applied to TRM for 23 samples in section 2.1. A total of 11 samples yielded acceptable results while 17 results show severe nonlinearity in Arai plots (Table 1).
We calculated the Z parameter for the IZ and ZI data sets separately. The angles relating H1 and H2 are θ.
 Both ZZI and ZIZ from Table 1 are plotted as a function of θ (Figures 6a and 6b). The accepted results show nearly uniform values of ZZI < 0.06 (Figure 6a). On the other hand, ZIZ shows an angular dependence, with increasing ZIZ as the θ increases (Figure 6b).
 We also examined the possible correlation between ZZI and ZIZ, showing a roughly linear relation (Figure 6c). Note that the results are biased toward ZZI for the parallel case and toward ZIZ for nonparallel θ. This observation also validates our mathematical model, where the Aitken method (IZ step) yields a slightly better result when θ is zero (Figure 1a).
 All the 11 accepted results form a tight cluster in a shaded area, bounded by ZZI = 0.10 and ZIZ = 0.15 (Figure 6c). Can we suggest the shaded area as a selection criterion for IZZI protocol? Although no accepted result was observed for θ = 180°, we can tentatively suggest that ZZI = 0.10 and ZIZ = 0.15 as a selection criteria because all the accepted results for θ = 0–136° fall in the shaded area (Figure 6c). However, it requires a caution due to the angular dependence effect. For example, a tighter cluster at small values of Z was observed for θ = 0° (red diamonds in Figure 6c). Thus, in the case of θ = 0, a much more stringent bound of ZZI = 0.07 and ZIZ = 0.10 would be helpful.
 In practice, it is common to apply a fixed sample selection criterion. Unfortunately, we may discard what are otherwise acceptable results. We therefore need a flexible criterion that can incorporate the angular dependence of pTRM tails. We tentatively suggest that allowing a less stringent criteria for the antiparallel case (H2 \\ H1) than for the parallel case (H2 // H1). For example, a tentative criterion of pTRM (tail) check would be 2%, 3%, and 4% (or 4%, 6%, and 8%) for parallel, perpendicular, and antiparallel cases.
 Why is the “IZZI” protocol better than conventional method? In principle, the weighted pTRM tail can be as useful as the Z parameter as long as the p and angular relation between H1 and H2 are available. The IZZI method is far superior because it can easily detect the angular dependence and it provides a quantitative estimate for the consistency of the outcome between IZ and ZI data sets. It is also quicker experimentally, by not requiring the extra pTRM tail check step.
 (1) The IZZI protocol of the paleointensity method was tested, confirming a strong angular dependence of pTRM and pTRM tails. (2) The conventional paleointensity sample selection criteria require revision to incorporate the effect of angular dependence. For example, a much more stringent sample selection criterion is required for H1 parallel to H2. (3) The IZZI method is superior than the conventional paleointensity techniques (Aitken, Coe, and Thellier) because it can easily detect the angular dependence of pTRM and it provides a quantitative estimate for the consistency of the outcome between IZ and ZI steps. In addition, it is quicker because the extra pTRM tail check step is unnecessary.
 This research was supported by NSF grant EAR0229498 to L. Tauxe. Peter Riisager, Dennis Kent, and an anonymous reviewer provided helpful comments. We benefited from fruitful discussions with Peter Selkin and Agnès Genevey. We thank Jason Steindorf for help with the measurements.