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Guidelines for assessing the reliability of 40Ar/39Ar plateau ages: Application
to ages relevant to hotspot tracks

Ajoy K. Baksi

Department of Geology & Geophysics, Louisiana State University, Baton Rouge, LA 70803, USA

akbaksi@yahoo.com

“Therefore O students study mathematics and do not build without foundations”

Leonardo da Vinci, ca. 1500

Abstract

The literature contains numerous 40Ar/39Ar “ages” that, based on statistical tests, have no geological significance. Herein, I set out some simple guidelines to permit readers to assess the reliability of published ages. The emphasis is on 40Ar/39Ar data on whole-rock material, though the methods are applicable to all published plateau data. I illustrate the use of the techniques by looking at published age data for hotspot tracks in the Atlantic Ocean (the Walvis Ridge), as well as newly published ages for the British Tertiary Igneous Province.

Introduction

The proliferation of 40Ar/39Ar dating laboratories has led to a large number of papers reporting age data. However, in many instances, the published “ages” have no merit, as they fail the simple statistical tests that should be applied to all such data. Herein, I consider results of  40Ar/39Ar step-heating studies only. In these experiments, a sample is heated in steps of increasing laboratory extraction temperature, until all the argon is released. The argon released in each step is measured to calculate a “step age” with an associated analytical error. At the end of the series of experiments, the step ages (± one sigma errors are quoted herein) are plotted against the cumulative amount of 39Ar released. The resulting figure is called an age spectrum (e.g., Figure 1). This approach is designed to look at the gas released from sites of increasing argon retentivity. When a reasonable number of consecutive steps, carrying a substantial amount of the total argon released, give the same age, the resulting average value carries geological significance. For unmetamorphosed igneous rocks, the latter would normally represent the crystallization age.

An adjunct method of evaluation is to present the relevant data on an isotopic ratio plot and look for straight-line segments through three or more consecutive heating steps. This is the isochron technique (see York, 1969; Roddick, 1978; Dalrymple et al., 1988 for details). Whether the data are evaluated in spectrum or isochron form, they must pass relevant statistical tests for “reliability”. These tests are outlined herein. In earlier work (Baksi, 1999, 2003), I set out guidelines for assessing the reliability of 40Ar/39Ar ages based on critical evaluation of isochron diagrams. This work followed the first efforts (Brooks et al., 1972) to distinguish isochrons from “errorchrons” – the latter occuring where “geological error” was present and negating the use of the age for geological purposes. In evaluating 40Ar/39Ar data on isochron diagrams, the data must be subject to straight-line fitting (e.g., York, 1969). The resultant “goodness of fit” parameter, in tandem with the number of points on the line, must be evaluated for statistical reliability (95% confidence level generally) using Chi Square Tables (e.g., deVor et al., 1992). If the data scatter badly around the best-fitting straight line, the resulting “age” should be rejected as being not reliable.

An easier method is to look at the data on age-spectrum plots and assess the reliability of “plateau” sections. Though definitions of what constitutes a plateau vary, a general guideline is that such sections of the age spectra should contain at least three consecutive steps, carrying > 50% of the total 39Ar released, whose ages “overlap”. It on this last issue that I shall focus. Two steps can never define a plateau, and such data cannot be evaluated on an isochron diagram. Any two points in the universe lie on a straight line! In evaluating isochron “goodness of fit parameters”, the number of degrees of freedom for Chi Square Tables is N-2, and for two steps, this is zero.

I look primarily at age spectra and note that plateau sections represent steps that “overlap” using (40Ar/36Ar)initial = 295.5 (the atmospheric argon value) for data reduction. I note that cases where:

  1. (40Ar/36Ar) initial > 295.5, (excess 40Ar argon is present), are taken care of by isochron plots, and
  2. (40Ar/36Ar) initial < 295.5, are “disturbed samples”, e.g., partial loss of (radiogenic) 40Ar* (formed by in situ decay of 40K) (Lanphere & Dalrymple, 1978), and the resultant “age” has no geological significance.

All errors herein are given at the 1-sigma level. For plateau ages, I list the standard error on the mean (the SEM).

Methodology

The basic principle involved is to evaluate critically how much the step ages on the plateau differ from one another.  This is done by calculating the relevant Mean Square Weighted Deviate (MSWD = F). This parameter measures the scatter of the individual step ages, with their associated errors (ti ± dti), from the mean.

where

is the mean (the “plateau” age) and N contiguous steps are used.

It is important to understand that the MSWD value obtained either from the age-spectrum plot or the isochron must be used along with the number of steps for assessing reliability using Chi Square Tables. Two parameters  are used in such a test. The first is the number of “degrees of freedom” of the data set (nu) which is equal to N-1 for plateau calculations and N-2 for isochron (straight line fitting) purposes.  The second is the Chi Square value, which is equal to MSWD multiplied by nu. When the relevant Chi Square Tables (e.g., deVor et al., 1992) indicate that the probability of occurence is < 0.05 (95% confidence level), the “age” (plateau or isochron) should be rejected. Significant error is then present, rendering the number (the “age”) and its associated error estimate geologically meaningless. As will be demonstrated herein, many “ages” have been presented in the literature wherein the relevant statistical tests demonstrate that the probability of having obtained a statistically meaningful age is not only < 0.01 (1%), but often < 0.000001 (10-6).

I begin by noting that Brooks et al. (1972), in their treatment of geochronological data, suggested rejection of straight lines as isochrons when MSWD > 2.5. For data with thenumber of degrees of freedom (nu = N – 2) = 1, 3, 5, 10 the corresponding probability values are 0.11, 0.06, 0.03 and 0.005, respectively.

Note that the same value of MSWD (= 2.5), makes an isochron statistically acceptable for up to 5 points (nu = 3), but unacceptable when seven or more points are used (nu > 5). This is of particular importance today, when age spectra often contain 10 – 20 steps. Clearly each case must be evaluated individually for “statistical reliability”.

I begin by looking at a hypothetrical data set. Consider a case of five steps carrying equal quantities of gas giving rise to step ages of 99.0, 101.0, 100.0, 101.0 and 99.0 Ma, each with an error of ± 0.5 m.y. The plateau age of 100.00 Ma, with a standard error on the mean:

of ± 0.22 Ma is found to be unacceptable, with p ~ 0.003 (Figure 1).

Visually, there appear to be fairly minor “bumps” on the plateau section, but no proper estimate of the crystallization age can be made from this age spectrum. The crystallization age may lie between ~ 99 and 101 m.yr. However, its exact value, and equally important its error estimate, cannot be stated with statistical confidence. When other age spectra show steps deviating markedly from each other i.e., much more severe “bumps”, clearly no plateau section has been developed. This will be considered in detail for two specimens below.

Figure 1: Age spectrum plot for hypothetical data set

In the above instance (Figure 1), if each step age error were ± 1.0 m.y., the corresponding p value would be ~ 0.41, making the plateau age (100.00 ± 0.45 Ma) acceptable as a crystallization value. Step age errors must be estimated/calculated properly. Errors in isotopic ratio measurements must be used correctly (see Dalrymple et al., 1981) to calculate step age errors. In particular, sigma-j, the estimated error in determination of the irradiation parameter, must NOT be used in (each) step age error calculation.

It has been pointed out by Roddick (1978) that in instances where the MSWD > 1, the calculated error in the plateau should be multiplied by (MSWD)1/2. I will follow this procedure only when the plateau is statistically acceptable (p > 0.05). In cases where MSWD is very large (e.g., > 10), such multiplication of the plateau age error, would make the age “less precise” and may be thought to be functionally usable. However, if p < 0.05, the probability of the steps defining a proper plateau are unacceptable. Estimating its age with a less precise “number” is to be avoided.

In evaluating age spectra sections thought to make up plateau sections, nu = N-1. I illustrate the use of this technique for data taken from the literature. I presume that the relevant data sets (step ages with associated errors, or preferably measured isotopic ratios with associated errors) are either available in tables within the paper, at a supplementary site, or will be supplied by the authors. Since it is not desirable to reinterpret all published data, I offer some guidelines for data sets that are worth closer inspection.

Look for the following:

  1. where the relevant MSWD value is not quoted,
  2. where the figures are not drawn to an ideal scale (see example below), and
  3. where step ages often differ by more than the relevant 95% confidence value amount (see Baksi, 1999).

The significance of i. above is self-evident. If the MSWD value is not quoted, it can easily be calculated and used in conjunction with Chi Square Tables to assess reliability. Point ii. is illustrated in Figure 2. The apparent age may be drawn on a far-from-ideal scale, precluding quick visual evaluation of the “plateau” (Figures 2a, b). The same data sets are redrawn to a proper scale in Figures 2c and 2d. Visual inspection suggests a plateau may be developed for the RT sample (Baksi, 2001), but not for LB1 San (Chambers & Pringle, 2001), wherein the age spectrum shows severe bumps. Proper statistical evaluation confirms this. Note that in Figure 2c the last four or five steps may be taken to define a plateau. The two sets of plateau ages (see Figure 2c) overlap statistically. Four steps were used to define the plateau age by Baksi (2001), noting slight loss of 40Ar* (lower ages) at both the lowest and highest temeprature steps.

Figure 2: Examples of acceptable (left) and unacceptable (right) age spectra where relevant details of the plateau sections are obscured by use of inappropriate apparent age scales (top panels).

Point iii. above forms the main focus of this webpage and is illustrated below using examples taken from the literature. I briefly consider age spectra where low temperature steps yield older ages than those at higher temperatures. The former and latter derive gas from low and high argon rentention sites respectively. Thus low-temperature steps normally exhibit ages lower than, or equal to, those of the higher temerature steps. “Descending staircase” type age spectra may result from two totally different phenomena. The first is due to the presence of excess 40Ar in the rock, which is often released preferentially to 40Ar* at relatively low temperatures. The second explanation involves 39Ar recoil within the sample. Prior to laboratory heating of the rock, during irradiation of the sample in a nuclear reactor, a fast neutron enters the 39K atom, converting it to 39Ar, as a proton is released (fired out). Like the recoil of a gun when a bullet is fired, the 39Ar formed by the reaction, recoils. In this process, it may move from one mineral phase to a contiguous one, creating a “disturbed” age spectrum. Examples of both phenomena (excess 40Ar and 39Ar recoil) are given in Baksi (1994) and references cited therein.

Case Histories

(1) Examination of ages reported for whole-rock basalts from the Walvis Ridge, Atlantic Ocean.

O’Connor & Duncan (1990) reported a number of “plateau ages” to temporally define the track of the Tristan da Cunha hotspot. These ages were subsequently used by Muller et al. (1993) to derive plate motions relative to hotspots. Based on evaluation of isochron plots for each specimen Baksi (1999) showed that almost all the ages were untenable as crystallization values. In this web page, I take a similar approach (but one that is somewhat easier to visualize) to evaluate these ages based on their age spectra.

In Figures 3 – 6 plateau ages as reported by O’Connor & Duncan (1990) are shown in italics, and the proper statistical evaluation is shown in bold (F = MSWD value, p = probability of occurrence based on Chi Square Tables, using the appropriate number of degrees of freedom). All the samples dated by O’Connor & Duncan (1990) were whole-rock basalts.

DSDP 528-40-5/525A-57-5, AII-93-14-9: Age spectra are shown in Figure 3. The first sample shows a descending-staircase type of age spectrum. Based on only two steps O’Connor & Duncan (1990) reported a plateau age of 77.6 ± 0.5 Ma. This is by definition untenable. Similar two-step plateau ages of 79.4 ± 0.4 Ma and 61.5 ± 0.3 Ma (O’Connor & Duncan, 1990) for specimens DSDP 525A-57-5 and AII-93-14-9, are also rejected. Baksi (1999) showed that the isochrons (ages) reported by O’Connor & Duncan (1990) for the two DSDP specimens show MSWD > 100; p <10-10. For the third specimen, the isochron (for all six steps) used by O’Connor & Duncan (1990), yields MSWD > 1000 (Baksi, 1999) and p < 10-20. None of these specimens yields an age (plateau or isochron) sufficiently accurate to be utilized for tracing hotspot tracks.

Figure 3: Age spectra for disturbed samples from the Tristan da Cunha hotspot track (the Walvis ridge).

Figure 4: Age spectra showing older apparent ages for low-temperature steps than for high-temperature steps.

V29-9-1, AII-93-3-25/93-14-1/93-11-8: Age spectra are shown in Figure 4. The plateau ages reported by O’Connor & Duncan (1990), are untenable because of very high MSWD values. The corresponding p values are all < 0.0005. These descending staircase age spectra may indicate the presence of excess 40Ar. For this reason (see Introduction above) the corresponding isochron plots need to be critically examined. Baksi (1999) showed that the MSWD values for these straight lines are 36, 5.5, 6.2 and 60 respectively. The corresponding probability values are < 10-6, 0.004, 0.012 and <10-20, respectively. These four specimens gave no plateau/isochron ages that can be utilized for delineating hotspot tracks in the South Atlantic Ocean.

AII-93-5-3/93-3-1/93-8-11: Age spectra are shown in Figure 5. The plateau ages reported by O’Connor & Duncan (1990) are untenable because of very high MSWD values, corresponding to p values < 10-10 in each case. The isochron plots used by O’Connor & Duncan (1990) showed MSWD values of 174, 30 and 2.5 respectively (Baksi, 1999). The first two yield p < 10-10. For AII-93-8-11, the probability value is 0.08. However, the isochron is derived from the first four steps, two of which show very high atmospheric argon (36Ar) contents. The gas is derived from altered sites and hence the corresponding age is suspect (Baksi, 1999). These three specimens gave no plateau/isochron ages that can be utilized for tracking hotspot tracks in the South Atlantic Ocean.

AII-93-10-11: A plateau age of 46.2 ± 0.3 Ma (Figure 6) was reported by O’Connor & Duncan (1990). The three-step plateau yields an acceptable MSWD value (p ~ 0.29). The corresponding isochron (Baksi, 1999) yields a low precision age of 49.8 ± 4.7 Ma, MSWD = 6.2 and p ~ 0.01. This rock could be said to yield a somewhat imprecise but usable age for evaluating the hotspot track trace.

Figure 5: Age spectra yielding unacceptable ages from the Tristan da Cunha hotspot track.

Figure 6: Age spectrum yielding an imprecise but usable age for evaluating the Tristan da Cunha hotspot track.

“Plateau” ages were also reported by Duncan (1978, 1984) for the track of the Kerguelen Hotspot in the Indian Ocean and the Great Meteor Hotspot in the North Atlantic Ocean, respectively. Based on statistical evaluation of the isochron plots Baksi (1999) showed that each of these was untenable. Elsewhere it will be shown that this conclusion is fully supported by critical examination of the individual age spectra.

The plate motion model derived by Muller et al. (1993) was based solely on the ages referred to above. Only a very few of these purported ages can be used to date time of crystallization. The plate motion model of Muller et al. (1993) is unsubstantiated since it is not based on a sufficient number of crystallization ages for hotspot tracks (Baksi, 1999).

(2) Recent ages for rocks from the British Tertiary Igneous Province (BTIP).

Chambers & Pringle (2001) reported a number of plateau/isochron ages for both rocks and mineral separates from the BTIP. These authors used these “ages” for defining the time of initiation and duration of flood-basalt volcanism in this province and to refine details (cryptochrons) of the geomagnetic polarity time scale from ~ 60 – 55 Ma. In their Table 1 Chambers & Pringle (2001) list the MSWD values for their “plateaux” and isochron plots, and state “of the 10 Mull samples analyzed, eight met all of the reliability criteria” (Chambers & Pringle, 2001, p. 334). Critical scrutiny of their data shows that almost all of their plateau/isochron ages are unacceptable on statistical grounds. Furthermore, a cursory glance at the age spectra drawn to an appropriate scale, shows few if any acceptable ages were recovered (Figures 7 – 9). I examine their results for rocks from the Isle of Mull.

SO33, BM67, BM64, B-1: (All whole-rock samples; see Figure 7). The first (center lava), shows an age spectrum with numerous ups and downs. Seven steps were used to arrive at a “plateau” age of 58.94 ± 0.30 Ma. The corresponding MSWD value is ~ 19, and p < 10-20. Clearly this is no plateau. BM67 and 64 (Ben More lavas) were said to yield plateau ages of 58.66 ± 0.25 Ma and 58.19 ± 0.26 Ma, respectively, although many step ages obviously do not overlap in each case. The MSWD ~ 26 and ~ 18, yield p < 10-20 in each case. Clearly these ages are unacceptable. Thus what are we to make of the plateau section for BM64, as defined by Chambers & Pringle (2001)? Step ages “stagger around” from ~ 61 – 58 Ma, with each step error being of the order of 0.20 m.yr. How can one claim, as do Chambers & Pringle ( 2001), that the statistically significant mean value is 58.19 ± 0.26 Ma? For BI (Lower lava) a descending age spectrum was used to arrive at a plateau age of 60.56 ± 0.29 Ma. The MSWD value of ~ 15 (p < 10-14) makes this clearly unacceptable. The corresponding isochron plots for these four rocks (Chambers & Pringle, 2001, their Table 1) yield MSWD values of 26, 21, 14 and 11, respectively. Corresponding p values are < 10-10 in all cases. None of these four rocks yielded a proper plateau and/or isochron age.

Figure 7: Unacceptable age spectra for samples from the BTIP.

SO17 (whole-rock – center lava): The age spectrum (Figure 8) shows that the rock contains excess argon, giving a saddle-shaped age spectrum (cf. Lanphere & Dalrymple, 1976). Nine steps define were used by Chambers & Pringle (2001) to define a plateau age of 61.54 ± 0.24 Ma. MSWD = 2.02 and p = 0.04. The corresponding isochron (Chambers & Pringle, 2001, their Table 1), shows an initial argon ratio of 316 ± 6 (excess argon is present), an age of 59.66 ± 0.64 and MSWD = 1.28. For the latter, p = 0.25, making the isochron but not the plateau age acceptable. Chambers & Pringle (2001) reject this age on statistical grounds but accept others with much worse statistical credentials.

Figure 8: Age spectrum yielding an unacceptable age. The isochron age for this sample was 59.66 ± 0.64 and is acceptable.

Figure 9: Age spectra that are suggestive of excess argon. Only sample MD3 (lowermost panel) yields an acceptable age.

LB1 (Sanidine – Loch Ba ring dyke): This mineral has been used very successfully to date rocks and generally yields good plateau ages. Chambers & Pringle (2001; their Figure 2) presented the age spectrum inadequately, the ordinate running from 40 to 80 Ma (see Figure 2b herein), When replotted for proper visual inspection (Figure 2d), the age spectrum shows numerous ups and downs that are much more serious than the hypothetical case shown in Figure 1. Chambers & Pringle (2001) used 12 steps to arrive at a “plateau age” of 58.48 ± 0.18 Ma. Their MSWD value for these 12 steps is 3.05, for nu (degrees of freedom = N-1) = 11. The corresponding Chi Square value is 33.55 (MSWD x nu), and Chi Square Tables tell us that the probability of this occurrence is p ~ 0.0004. The age of Chambers & Pringle (2001) must thus be rejected on these grounds. The lowest section of the age spectrum (Figure 2d) shows numerous steps of ~ 58 – 59 Ma. The mineral may have a crystallization age in this range, but it contains excess argon (higher step ages on either side of the “plateau”). There is no justification for quoting an age of 58.48 Ma with a very small error margin of ±0.18 m.yr. Evaluation of the corresponding isochron plot gave MSWD = 3.9 (Chambers & Pringle, 2001; their Table 1). This yields p ~ 0.00002.
Both the plateau and isochron approach statistics are unequivocal – a large amount of excess scatter is present in the data sets (“geological error”). This mineral separate fails to  yield either a plateau or an isochron age. Note that almost all the other cases considered in here yield much lower probability values, often < 10-6 and in some instances < 10-20! Clearly in such such cases, the quoted plateau/isochron ages are totally spurious.

MD – 1/2/3 (Felsic groundmass separated from dykes): All three age spectra are suggestive of the presence of excess argon (Figure 9). Chambers & Pringle (2001) reported plateau ages of 58.07 ± 0.23 Ma and 58.33 ± 0.27 Ma for MD 1 and 2 respectively. In both cases MSWD values are very high (~ 10 – 15), with p values correspondingly low (< 10-8 in each case). These ages are rejected. The corresponding isochron plots also have high MSWD values (10 and 19), and the associated p values are extremely low (both < 10-9). Chambers & Pringle (2001) listed a plateau age of 58.04 ± 0.20 Ma for MD3. In this instance the MSWD and p values (1.68 and 0.09 respectively) are acceptable. The corresponding isochron plot (Chambers & Pringle, 2001, their Table 1) shows MSWD = 2.12 (p ~ 0.04) and verges on being acceptable. The age of this sample can be taken to be 58.1 ± 0.3 Ma.

Summary for Isle of Mull ages

Of the eight specimens said by Chambers & Pringle (2001) to yield fully acceptable plateau/isochron ages, on critical examination, only two pass the relevant statistical tests – SO17 (isochron age) and MD3. The claim that “The duration of activity that produced the Isle of Mull Tertiary igneous center has been constrained to 2.52 ± 0.36 million years” (Chambers & Pringle, 2001, p. 333) is thus without scientific basis. These two acceptable ages, together with one from the Isle of Skye, do not help to resolve details of the geomagnetic polarity time scale for the period ~ 61–56 Ma, and the models shown by Chambers & Pringle (2001, Figure 5), are without supporting geochronological data.

Samples from the British Tertiary Igneous Province have proved more difficult to date accurately than whole-rock basalts from other flood basalt provinces. This may be due to the fact that many sections of the exposed BTIP rocks have undergone low-grade metamorphism (Mussett, 1986). In my estimation, the best ages for this province are those of Mussett (1986) wherein the data were scrupulously evaluated by proper statistical criteria. These ages are 60.0 ± 0.5 Ma for the base of the Mull lava pile and 56.5 ± 1.0 Ma for the Loch Ba dyke (see Chambers & Pringle, 2001, p. 334).

Concluding Remarks.

All 40Ar/39Ar age data must be rigorously examined for conformity to plateau and isochron ages. Figures must be presented to permit quick but accurate visual inspection of results. MSWD values must be reported for both plateau and isochron plots and/or raw data should be included in Tables/Supplementary Data Sets or made readily available on request. Proper ages must always be based on at least three consecutive steps carrying > 50% of the total 39Ar released. MSWD values for both plateau/isochron data must yield p values > 0.05 (95% confidence level). It may be argued that cases with (say) p > 0.01 but < 0.05 should be considered “valid”. In such cases, where should the line be drawn in the sand? A 95% confidence test is almost universally accepted as the first “cut”. Certainly, ages where the corresponding p values are < 0.001 are anathema. 

Good examples of proper statistical care being exercised in evaluating 40Ar/39Ar step-heating data, are given, amongst others, in the work of Mussett (1986) and Knight et al. (2003). In the absence of proper care being exercised in these matters by authors/editors/reviewers, it falls on the reader to evaluate 40Ar/39Ar ages prior to making use of them. “Always verify your references” (East, 1998). Failure to adhere to this adage has led to numerous incorrect ages becoming entrenched in the literature, following repeated citation by unsuspecting/uncritical scientists.

Acknowledgments

I thank Hetu Sheth and especially Gillian Foulger for making this effort possible.

References

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