Chemical Geology 340 (2013) 131–138

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Chemical Geology

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Algorithms for estimating uncertainties in initial radiogenic isotope ratios andmodel ages

Ryan B. Ickert ⁎,1,2

Canadian Centre for Isotopic Microanalysis, Department of Earth and Atmospheric Sciences, University of Alberta, Edmonton, Canada

⁎ Berkeley Geochronology Center, 2455 Ridge Road, BeFax: +1 510 644 9201.

E-mail address: Ryan.Ickert@gmail.com.1 Current affiliation: Berkeley Geochronology Center,2 Current affiliation: Department of Earth and Plan

California, Berkeley, CA, United States.

0009-2541/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.chemgeo.2013.01.001

a b s t r a c t

a r t i c l e i n f o

Article history:Received 5 March 2012Received in revised form 20 November 2012Accepted 2 January 2013Available online 11 January 2013

Editor: L. Reisberg

Keywords:Radiogenic isotopesUncertaintiesError propagationDecay constantCHUR

Initial radiogenic daughter-product isotope ratios such as 143Nd/144Nd, 176Hf/177Hf, 87Sr/86Sr, and 187Os/188Os,their geochemical reservoir normalized equivalents (ε and γ values), andmodel ages are frequently used for geo-chemical inference. These calculated quantities are a function of not only the present day parent–daughter ratiosand radiogenic isotope ratios, but also the compositions of the geochemical reservoirs, decay rates, and indepen-dent geochronological constraints. Here, equations to determine the combined standard uncertainties in thesecalculated quantities that include uncertainties in all input quantities, are derived. The values and uncertaintiesof these model values are reviewed, and it is found that in many cases the uncertainties associated with inputquantities such as decay rates and reservoir compositions are similar to, or exceed analytical uncertainties. For-tunately, most of these uncertainties are highly correlated between samples (i.e., are systematic) and can oftenbe neglectedwhen the objective is to determine the difference between isotope ratios (or other associated quan-tities). A program designed to easily implement these equations is appended.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Long-lived radiogenic isotope systems such as Rb–Sr, Sm–Nd, Re–Os,and Lu–Hf are powerful tools with which the tempo and nature of theevolutionof the solid Earth can be explored. It is often useful –particularlyfor ancient rocks – to use the initial isotope ratio, correcting for radiogenicingrowth to a time of a particular geological event such as a crystallizationage for an igneous rock, or the depositional age for a sedimentary rock.The methods by which to do the calculations are simple and wellestablished, however the calculation and propagation of the attendantuncertainties – both from the measurement of the present day isotopeand chemical ratios, and from system specific parameters such as reser-voir compositions and decay rates – have received only sparse attention(Sambridge and Lambert, 1997).

Here, a contribution is made towards closing this gap in knowledge.The Guide to expression of Uncertainty in Measurement (GUM; JCGM100, 2008) providesmethods on howuncertainties should be combinedin measurement functions when combining quantities. These recom-mendations are followed here. The measurement functions consideredhere include those for calculating initial radiogenic isotope ratios andtheir geochemical-reservoir normalized equivalents (e.g., εNd, εHf and

rkeley, CA 94709, United States.

Berkeley, CA, United States.etary Sciences, University of

rights reserved.

γOs, as defined below) as well as model ages— quantities representingthe time of separation of a geological material from a geochemical res-ervoir, given certain assumptions. These derived quantities requireknowledge of rock ages, reservoir parent–daughter ratios and radiogen-ic isotope ratios, decay rates, as well as the directlymeasured quantitiesof sample parent–daughter and radiogenic isotope ratios.

By employing the recommendations of the GUMon themeasurementfunctions listed above, a set of algorithms – simple equations – are de-rived that can be used to propagate the uncertainty on all input values.These algorithms areuseful because they provide formetrologically trace-able calculated quantities, and can be used to clarify whether measure-ments and calculated quantities are fit-for-purpose. The input valuesthemselves are briefly reviewed, with particular attention to the mannerin which the uncertainties are determined. The implications of these cal-culations and the shortcomings of currently available data are discussed.

2. Measurement functions

Initial isotope ratios are calculated using the following equation:

Ι ¼ D−P eλt−1� �

ð1Þ

where D is the radiogenic isotope ratio, P is the parent–daughter ratio,λ is the decay rate in inverse years (a−1), t is the time – in the past,from present day – at which the initial isotope ratio is calculated inunits of years (Holden et al., 2011), and Ι is the radiogenicdaughter-product isotope ratio at time t.

Table 1Summary of parameters and their uncertainties.

Reference

Rb–SrΛ 1.3968±27·10−11 a−1a Rotenberg et al. (2012)87Rb/86SrCHUR 0.084±8 DePaolo and Wasserburg (1976a)87Sr/86SrCHUR 0.7045±5 DePaolo and Wasserburg (1976a)

Lu–HfΛ 1.867±8·10−11 a−1 Söderlund et al. (2004)176Lu/177HfCHUR 0.0336±1 Bouvier et al. (2008)176Hf/177HfCHUR 0.282785±11 Bouvier et al. (2008)rLu/Hf, 176Hf/177Hf 0.897 Bouvier et al. (2008) and this work

Sm–NdΛ 6.539±99·10−12 a−1 Lugmair and Marti (1978)147Sm/144NdCHUR 0.1960±4 Bouvier et al. (2008)143Nd/144NdCHUR 0.512630±11 Bouvier et al. (2008)rSm/Nd,

143Sm/

144Nd 0.829 Bouvier et al. (2008) and this work

147Sm/144NdDM 0.2137 Goldstein et al. (1984)143Nd/144NdDM 0.513143 Goldstein et al. (1984)

Re–OsΛ 1.6668±34·10−11 a−1 Selby et al. (2007)187Re/188OsMantle 0.422±50 Walker et al. (2002)187Os/188OsMantle 0.1283±33 Walker et al. (2002)rRe/Os, 187Os/188Os 0.557 Walker et al. (2002) and this work

All uncertainties are at a coverage factor of 2 (approximately 95% confidence limits).The values of r (Pearson's r) are the correlation between the parent–daughter ratiosand radiogenic daughter isotope ratios.

a The uncertainty given by Rotenberg et al. (2012) is +0.0027–0.0018, but a symmetric

uncertainty is required for the equations here.

132 R.B. Ickert / Chemical Geology 340 (2013) 131–138

Geochemical reservoir normalized values are either reported as ε(for Sm–Nd, Lu–Hf, and less commonly Rb–Sr; Papanastassiou andWasserburg, 1968) or γ values (for Re–Os; Walker et al., 1989) bythe equation:

ε;γ ¼ θΙsΙc−1

� �ð2Þ

where θ is either 104 or 102, depending on whether ε or γ units, re-spectively, are used. The subscripts s and c refer to the sample andthe geochemical reservoir, respectively.

Unfortunately, geochemical-reservoir normalized values are used torepresent similar but slightly different quantities. Being able to differen-tiate between these uses is critical to an accurate use of the uncertaintypropagation equations. A common use of these values is as a linear trans-formation of a radiogenic isotope ratio to a quantity with a smallernumber of significant digits. This is done to present a value that is easierto either read or speak aloud. For example, two rocks with present day143Nd/144Nd of 0.5126813±0.000051 and 0.5127325±0.000051 aremore easily distinguished when represented as the present day εNdvalues 1.0±0.1 and 2.0±0.1. For this purpose, the choice of normalizingvalue is arbitrary— they are deviations in parts per ten thousand from anumber in the range of typical natural 143Nd/144Nd. A geochemical reser-voir is used by convention.

A second, also common, use of geochemical-reservoir normalizedquantities is to cast initial radiogenic isotope ratios as the relative dif-ference between a sample and a geochemical reservoir at a giventime. In this second case, the calculated quantities are dependent onthe choice of normalizing value for the reservoir. A related use ofthese calculated quantities is to normalize data taken in differentlaboratories that are corrected relative to different stable isotoperatios.

Related to ε and γ quantities aremodel ages: Apparent age of extrac-tion from a geochemical reservoir. These include intercepts with chon-dritic or bulk silicate earth evolution models (ICE-ages, Lugmair et al.,1975, 1976; or TCHUR DePaolo and Wasserburg, 1976a), variationsfrom depleted mantle reservoirs (TDM; DePaolo, 1981), or rhenium de-pletion ages (model ages assuming a present-day parent–daughterratio of zero; TRD, Walker et al., 1989).

TCHUR, TMa and TDM ages are calculated using the following rela-tionship:

TCHUR; TMa; TDM ¼ T model ¼1λln

Dc−Ds

Pc−Psþ 1

� �ð3Þ

In the case of TDM ages, the reservoir represented by Dc and Pc arenot the chondritic parameters, but those describing a model depletedmantle reservoir.

Rhenium depletion ages (TRD) involve a two step calculation, wherethe sample is corrected for radiogenic decay to some point (often aneruption age of a mantle xenolith, which is the t parameter in the equa-tions below), and the intersection between the reservoir and the sam-ple is taken assuming a Ps of zero. Some workers have used similartwo step calculations for other isotope systems (e.g., Sm–Nd; Liewand Hofmann, 1988). The equation for TRD is:

TRD ¼ 1λln

Dc−ΙsPc

þ 1� �

ð4Þ

2.1. Model parameters

The decay rates and geochemical reservoir compositions, whichare input values in the equations introduced above, are brieflydiscussed below (Table 1). The object is not to provide a comprehen-sive, critical review nor to recommend particular values to the

geochemical community, but to document their mean values, uncer-tainties, and correlations, particularly where they have not beforebeen tabulated. Nevertheless, the summaries below are constructedsuch that the values presented below are as consistent with canonicalvalues as possible and that the associated uncertainties and correla-tions are an accurate portrayal of the data distributions.

All uncertainties here are expressed as expanded standard uncer-tainties with a coverage factor (k) of two (JCGM 200, 2008), approx-imately equivalent to 95% confidence limits. Correlations are Pearsonproduct–moment correlation coefficients, or Pearson's r, defined as rin the equations.

2.1.1. Decay ratesThere is a broad consensus that the decay rate of 147Sm is close to

6.54·10−12 a−1, the value proposed by Lugmair and Marti (1978).Kossert et al. (2009) provides a review of data published up to 2009,not including the more recent measurements of Su et al. (2010). Withthe exception of the measurements of Kinoshita et al. (2003), allpost-1978 measurements are consistent with the canonical Lugmairand Marti value. Therefore that value, 6.539±99·10−12 a−1 is usedhere.

The decay rate of 176Lu is not well determined (Begemann et al.,2001; Scherer et al., 2001; Söderlund et al., 2004; Amelin, 2005;Amelin and Davis, 2005). Direct counting experiments scatter widelyoutside of their reported uncertainties, implying the presence ofunresolved systematic errors (Amelin and Davis, 2005). Arguably, themost precise and accurate results are those derived from directly cali-brating the 176Lu decay rate to the U–Pb decay system. There are twocommonly cited terrestrial determinations, from Scherer et al. (2001;1.865±15·10−11a−1) and from Söderlund et al. (2004; 1.867±8·10−11a−1) and a meteoritic determination from Amelin (2005;1.864±16·10−11a−1 and 1.832±29·10−11a−1). All of the 176Ludecay rates determined by comparison between the Lu–Hf and U–Pbsystems are in good agreement according to the uncertainties ascribedto them by the original authors, although the methods used to derivethem are not consistent — Scherer et al. use the standard deviation of

133R.B. Ickert / Chemical Geology 340 (2013) 131–138

their measurements, whereas Söderlund et al. propagate the uncer-tainties of each individual Lu–Hf U–Pb pair. It should be noted that av-eraging different results, although in principle providing a moreprecise and perhapsmore accurate estimate, is complicated by complexuncertainty correlations due to the use of different spike solutions andto the correlated uncertainty in the uranium decay rates. Here, themost recently determined and nominally most precise terrestrialvalue of Söderlund et al. (2004) value is used.

Measurements of the decay rate of 187Re by direct counting, accu-mulation and geological comparison do not all agree (Begemann et al.,2001). Two geological age comparison experiments yield consistentdecay rates. Smoliar et al. (1996) compared the Re–Os systematics ofiron meteorites to their U–Pb ages and calculated a decay rate of1.666±17·10−11a−1 (or ±5·10−11a−1 if spike uncertainties areignored, see Selby et al. (2007)). More recently, Selby et al. (2007)published an identical, but far more precise estimate of 1.6668±34·10−11a−1 (relative to the U decay rates of Jaffey et al., 1971)based on the comparison of Re–Os dates from molybdenite formed inmagmatic ore deposits with the U–Pb zircon ages of their associatedmagmas. The uncertainty on the Selby et al. value is underestimated,however, because the uncertainties in the decay rates of 235U and238U were added to the U–Pb ages of each Re–Os/U–Pb pair prior tothe calculation of an inverse variance weighted mean and subsequentpropagation of uncertainty onto the final result. This treats the uncer-tainties in the uranium decay rates as independent from sample-to-sample and consequently demagnifies their effects. Given that theuncertainties for each Re–Os/U–Pb pair contain a large component de-rived from the U–Pb age, (28%–81%; Selby et al., 2007), this may havea substantial effect on the final uncertainty. The Selby et al. value isused here, but the uncertainty in the decay rate should be used withcaution.

Rotenberg et al. (2012) have reviewed recent direct counting, agecomparison, and accumulation experiments (e.g., Kossert, 2003;Nebel et al., 2011; Rotenberg et al., 2012), and based on good agree-ment between the methods, used the accumulation experiments toa 87Rb decay rate of 1.3968+27

−18·10−11a−1, which is used here.A symmetric uncertainty is required for the uncertainty propagationequations, and therefore a value of ±27·10−11a−1 is adopted. Thesmall increase in the lower bound on the decay rate makes only asmall difference to the calculations.

2.1.2. Geochemical reservoirsAs noted above, the parent–daughter ratios and radiogenic isotope

ratios of reference geochemical reservoirs have the dual purpose ofproviding a convention for linear transformation of isotope ratiosinto numbers with less significant digits (e.g., Papanastassiou andWasserburg, 1968; DePaolo and Wasserburg, 1976b), and for com-paring the isotope ratio with that of a geochemical reservoir at apoint in time. For the purpose of the former, the uncertainty in thereservoir composition is not required, but for the purpose of the lat-ter, it is.

The two quantities used to characterize a geochemical reser-voir, the parent–daughter ratio and the radiogenic isotope ratio,are usually correlated, either due to radiogenic ingrowth (forthose parameters directly measured in meteorites) or due to afunctional dependence, as with the determination of Re/Os fromthe inferred 187Os/188Os of the primitive upper mantle. This corre-lation has an important effect on the propagation of the uncer-tainty when a correction for radiogenic ingrowth is required.Therefore a correlation coefficient (Pearson's r, abbreviated as r)is reported along with the values and their uncertainties. Thechoice of the statistic to describe the uncertainty depends on howthe parent and daughter nuclides were distributed in the earlysolar system, added to the early Earth, and subsequently mixed.

If the nuclides were distributed heterogeneously and were notcompletely homogenized (either during accretion or afterwards by

convective stirring), then scatter in the parent–daughter ratios and ra-diogenic isotope ratios will be mirrored by scatter in the “true” compo-sition of the geochemical reservoir. If scatter (in excess of analyticaluncertainty) is present in measurements, interpreted as nebular het-erogeneity (as opposed to redistribution duringmetamorphism, for ex-ample), and there is reason to suspect that objects with thesecompositions accreted to form Earth but did not homogenize, then thestandard deviation correctly captures the uncertainty of the reservoircomposition. In contrast, a homogenized Earth effectively has apoint-value for a geochemical reservoir and the standard error will cor-rectly capture the uncertainty of the reservoir composition.

Convention is followed here, where the uncertainties in the Sm–Ndand Lu–Hf systems are described by standard errors, and followingWalker et al. (2002), the Re–Os system is described by standarddeviations.

For the Sm–Nd and Lu–Hf isotopic systems, the normalizing geo-chemical reservoirs are directly derived from measurements of ra-diogenic isotope ratios and parent–daughter ratios of primitivemeteorites. This reservoir is commonly referred to as CHUR, orCHondritic Uniform Reservoir (DePaolo and Wasserburg, 1976b). Itis an isotopic reference frame defined by the closed system isotopicevolution of a reservoir with chondritic chemical and isotopic abun-dances of refractory lithophile elements. Usually it is meant to repre-sent 143Nd/144Nd or 176Hf/177Hf isotopic composition of the bulkEarth at a particular time from 4.56 Ga to the present.

Although originally conceived as genuinely capturing the trueisotopic composition of the bulk earth, it is currently unclear as tothe degree to which this assumption continues to hold. For example,workers have identified clear isotopic differences between refractory el-ements (including Nd) in Earth and primitive meteorites (e.g., Boyetand Carlson, 2005, 2006), leading some to propose a non-chondriticEarth (e.g., Campbell andO'Neill, 2012). Nevertheless, the difference be-tween the radiogenic isotope compositions of measured samples andputative cosmochemical reservoirs are still of great interest.

The determination of CHUR parameters has a long history (Bouvier etal., 2008 provides a useful summary), however, they are nowmainly de-finedby the average values of Sm/Nd, 143Nd/144Nd, Lu/Hf, and 176Hf/177Hfof a large number of bulk measurements of chondritic meteorites. Themost frequently used present day values for Sm–Nd are those pub-lished by Jacobsen and Wasserburg (1980) and are 147Sm/144Nd=0.1966±4 143Nd/144Nd=0.512638±25 (after correcting the data to146Nd/144Nd=0.7219; Hamilton et al., 1983; Amelin and Rotenberg,2004). Bouvier et al. (2008) measured the Lu–Hf and Sm–Nd systematicsof a large number of chondritic meteorites and determined that usefulLu–Hf data is best derived from unequilibrated chondrites, due to Lu–Hfredistribution due to growth ofmetamorphicminerals. Based on adatasetmainly comprised of values determined in their lab, they recommendedusing CHUR parameters of 176Lu/177Hf=0.0336±1 and 176Hf/177Hf =0.282785±11 (r=0.897). Based on an expanded dataset, they deter-mined values for CHUR of 147Sm/144Nd=0.1960±4 143Nd/144Nd=0.512630±11 (r=0.829). This value is within uncertainty of both theJacobsen and Wasserburg (1980) and Amelin and Rotenberg values(2004). Because they are based on datasets that, in part, consist of Hf–Nd analyses on the same aliquots, the values for CHUR reported byBouvier et al. (2008), alongwith their associated uncertainties and covari-ances, are used here.

There are a number of formulations of the Nd depleted mantle forTDM (e.g., Nägler and Stille, 1993). The first parameterization(DePaolo, 1981) was a second order polynomial of εNd as a functionof time, but Goldstein et al. (1984) subsequently provided a simpler,linear model. Nagler and Kramers (1998) provides a more sophisti-cated examination linking it to a forward model of crustal recyclingthrough time and results in a third order polynomial. The depletedmantle preferred here is that of Goldstein et al. (1984) as it is numer-ically simpler and has more-radiogenic 143Nd/144Nd at a given timethan the other models and may better account for sampling bias.

134 R.B. Ickert / Chemical Geology 340 (2013) 131–138

Geochemical reservoir normalization is not usually undertakenwhen reporting or using Rb–Sr isotope data. This is, in part, due tothe lack of knowledge regarding the Rb/Sr and 87Sr/86Sr of Earth.Estimations from primitive meteorites are hindered by the variabilityof Rb/Sr in these objects due to the high relative volatility of Rb.DePaolo and Wasserburg (1976a) recommended a pair of values thatmay represent the bulk Earth. They noted that the initial 87Sr/86Sr andinitial 143Nd/144Nd in basalts were correlated, and inferred that the87Sr/86Sr at εNd=0, approximately 0.7045, was the present day bulkEarth 87Sr/86Sr. With this information, and assuming an initial 87Sr/86Srof about 0.698976 (e.g., Papanastassiou andWasserburg, 1968) they de-termined present day approximations of the bulk Earth of 87Rb/86Sr=0.084±8 and 87Sr/86Sr=0.7045±5. However, it is now well knownthat correlations between isotope ratios in basalts have a more complexrelationship (e.g., Zindler and Hart, 1986), and these values are probablynot accurate.

Rhenium and Os are highly siderophile elements, and there-fore during the early stages of planetary differentiation they werestrongly partitioned into the core (Ringwood, 1966). However,Re/Os and 187Os/188Os measurements of upper mantle rocks havebeen used to show that Re–Os systematics of the primitive uppermantle are close to that of enstatite or ordinary chondritic meteor-ites (Meisel et al., 1996, 2001; Walker et al., 2002; Becker et al.,2006), probably because of a post-accretion fertilization with chon-dritic meteorites (Chou, 1978). The Re–Os isotopic systematics cantherefore either be derived from modeling the depletion history ofsamples of upper mantle peridotites (e.g., Meisel et al., 1996) orby assuming that the close agreement of enstatite and ordinarychondrites and the present day upper mantle reflects a geneticlink, allowing use of the chondritic values. Neither estimate isideal, but the similarity between the two estimates lends credibilityto both. The value derived from primitive meteorites is the moststraightforward, because the 187Re/188Os and 187Os/188Os can bedetermined on the same samples and the uncertainties on thevalues are simpler to derive and more intuitive than those derivedfrom regressions against proxies for depletion, which are minimumestimates (e.g., Meisel et al., 1996, 2001).

Following Walker et al. (2002), the enstatite and ordinary chondriteRe–Os values, and by inference the upper mantle, are 187Re/188Os=0.422±50 and 187Os/188Os=0.1283±33 with r=0.557. From Meiselet al. (2001) and Becker et al. (2006), the upper mantle Re–Os systemat-ics derived from xenoliths are 187Re/188Os 0.431±90 and 187Os/188Os=0.1296±23. The two values are nearly perfectly correlated with r=0.997.

3. Uncertainty propagation

The GUM advocates the use of a linear approximation of the mea-surement function (f) to combine the standard uncertainties (u) ofthe input quantities (x), which they have termed the law of propaga-tion of uncertainties. The uncertainty in the quantity calculated by themeasurement function is referred to as a combined standard uncer-tainty (uc). The law of propagation of uncertainties is a first-orderTaylor expansion about the expectation values of the input quantities.The equation is

u2c ¼

Xni¼1

∂f∂xi

uxi

!2

ð5Þ

For n=2 this expands to

u2c ¼ ∂f

∂x1

� �2

u2x1þ ∂f

∂x2

� �2

u2x2þ 2ux1

ux2rx1 ;x2

∂f∂x1

∂f∂x2

ð6Þ

The value r (Pearson's product–moment correlation coefficient)represents the tendency of a pair of input quantities to be correlated.For example, a large positive rx1 ;x2 means that if quantity x1 is high, itis likely that x2 will be high as well. The term that incorporates r iszero if the uncertainties in two input quantities are uncorrelated.This is the case for most input quantities, with the notable exceptionof the radiogenic isotope ratios and parent–daughter ratios of geo-chemical reservoirs.

3.1. Calculation results

By substituting Eq. (1) into Eq. (2) (where subscripts s and c referto the sample and the geochemical reservoir, respectively) we find:

ε;γ ¼ θDs−Ps eλt−1

� �Dc−Pc eλt−1

� �−1

0@

1A

Using the law of propagation of uncertainty, the combined stan-dard uncertainty for an ε or γ quantity is the following (for simplicity,ε only is used):

u2c ¼ ∂εt

∂DsuDs

� �2

þ ∂εt∂Dc

uDc

� �2

þ ∂εt∂Ps

uPs

� �2

þ ∂εt∂Pc

uPc

� �2

þ∂εt∂t ut

� �2

þ ∂εt∂λ uλ

� �2

þ 2rPcDcuPc

uDc

∂εt∂Dc

∂εt∂Pc

ð7Þ

∂εt∂Ds

¼ θΙc

ð8Þ

∂εt∂Ps

¼θ eλt−1� �

Ιcð9Þ

∂εt∂Dc

¼ −1ð ÞθΙsΙ2c

ð10Þ

∂εt∂Pc

¼θ eλt−1� �

Ιs

Ι2cð11Þ

∂εt∂t ¼ θPsλe

λt

Ιc

" #− θΙsPcλe

λt

Ι2c

" #ð12Þ

∂εt∂λ ¼ θPste

λt

Ιc

" #− θΙsPcte

λt

Ι2c

" #ð13Þ

Note that the covariance term is zero for every pair of input quan-tities except for the parent–daughter ratio and radiogenic isotoperatio of the normalizing geochemical reservoirs, Pc and Dc.

Both t and λ are present in the calculations of initial ratios in thebulk earth and sample ratios (this is why there are two terms inEqs. (12) and (13)). This has an important effect on propagating uλand ut onto uεt . For a sample with a parent–daughter ratio withinthe range of most natural values, uλ and ut are highly correlated forthe sample and CHUR initial ratios, and consequently the uncer-tainties nearly cancel each other out. Geometrically, this is becausean εHf value, for example, is simply the “distance” (in isotope-ratiospace) between the 176Hf/177Hf of a sample and CHUR, at a giventime on a plot of time vs. 176Hf/177Hf. The magnitude of the correla-tion between uλ and ut depends on how close Ps/Pc is to unity, but italways serves to shrink the size of uλ and ut relative to their effecton an initial isotope ratio alone.

This is a useful property when true comparisons with CHUR are ofinterest as it minimizes the uncertainty associated with decay ratesand dates. However, if the initial isotope ratio alone is of interest

135R.B. Ickert / Chemical Geology 340 (2013) 131–138

(not the relative isotope ratios of the sample and the reservoir), butnormalization is employed simply to present clearer numerical

values, the right hand terms of Eqs. (12) and (13) (e.g., θΙsPcλeλt

Ι2c

h iand

θΙsPcteλt

Ι2c

h i) can be set to zero.

The distinction between the two different uses of ε or γ values isimportant for calculating a combined standard uncertainty – in thefirst case, the value can be thought of as the difference betweentwo isotopic compositions – the sample and the geochemical reser-voir. In the second case, the normalization is linear transformationof the data to recast the isotope ratio into a number with fewer sig-nificant digits (for example to be more intelligible on a figure or ina body of text) and the choice of normalizing reservoir is set byconvention.

Here, the name Model-1 is given to a combined standard uncer-tainty that compares a sample to a geochemical reservoir, incorporat-ing the correlations between the sample and geochemical reservoirassociated with uλ and ut. A Model-2 type combined standard uncer-tainty is identical to an initial isotope ratio, transformed into unitsof ε or γ. OnlyModel-2 type uncertainties are appropriate when calcu-lating an initial isotope ratio rather than an ε or γ value.

When an initial isotope ratio, IS is of interest, rather than an ε or aγ value, the uncertainties in the geochemical reservoir composition,uDc and uPc , are set to zero and a Model-2 type uncertainty is used.The following identity can be employed to recover the combinedstandard uncertainty in the isotope ratio from the uncertainty in theε or a γ value. The identity is constructed by rearranging Eq. (2) tosolve for Ιc and then combining this with Eq. (5).

uΙs ¼Ιcθuεt ð14Þ

The combined standard uncertainty for a model age is describedby:

Tmodel2u ¼ ∂T model

∂DsuDs

� �2

þ ∂Tmodel

∂DcuDc

� �2

þ ∂Tmodel

∂PsuPs

� �2

þ ∂Tmodel

∂PcuPc

� �2

þ ∂Tmodel

∂λ uλ

� �2

þ 2rPcDcuPc

uDc

∂Tmodel

∂Dc

∂Tmodel

∂Pc

ð15Þ

∂Tmodel

∂λ ¼ −1λ2 ln

Dc−Ds

Pc−Psþ 1

� �ð16Þ

∂Tmodel

∂Ds¼ 1

λΨð17Þ

∂Tmodel

∂Dc¼ −1

λΨð18Þ

∂Tmodel

∂Ps¼ Dc−Ds

λΨ Pc−Psð Þ ð19Þ

∂T model

∂Pc¼ Dc−Ds

λΨ Ps−Pcð Þ ð20Þ

Where

Ψ ¼ Dc−Ps þ Pc−Ds ð21Þ

The combined standard uncertainty for a Re-depletion age is de-scribed by:

u2TRD

¼ ∂TRD

∂DsuDs

� �2

þ ∂TRD

∂DcuDc

� �2

þ ∂TRD

∂PsuPs

� �2

þ ∂TRD

∂PcuPc

� �2

þ ∂TRD

∂λ uλ

� �2

þ 2rPcDcuPc

uDc

∂TRD

∂Dc

∂TRD

∂Pc

ð22Þ

By expanding Eq. (23) using Eq. (23), the terms in Eq. (23) are:

∂TRD

∂λ ¼ 1λ2

λPsteλt

Ω

!þ ln

ΩPc

� � !ð23Þ

∂TRD

∂Ds¼ −1

λΩð24Þ

∂TRD

∂Dc¼ 1

λΩð25Þ

∂TRD

∂Ps¼

eλt−1� �

λΩð26Þ

∂TRD

∂Pc¼ Pc−Ω

λPcΩð27Þ

∂TRD

∂t ¼ Pseλt

PsΩð28Þ

where

Ω ¼ Dc þ Pc−Ιs ð29Þ

4. Discussion

The influence of these input quantities on the combined standarduncertainty is evaluated here. The uncertainties are cast in terms ofgeochemical reservoir normalized units and as a Model-2 type. As de-scribed above, Model-2 uncertainties are appropriate when the reser-voir normalization is used simply as a convention for lineartransformation of the data. All uncertainties are reported at a cover-age factor of 2 (e.g., at approximately the 95% confidence level). Theuncertainties are calculated for parent–daughter ratios identical totheir respective geochemical reservoir (e.g., CHUR or PUM), thereforethe contributions of decay rates will be larger than those reportedhere if the parent–daughter ratios are higher, and smaller if the par-ent–daughter ratios are lower.

The effect of a ~0.4% uncertainty in λ176Lu is modest, extendinglinearly to ±0.47 ε units at 4.5 Ga (Fig. 1a). The uncertainties on CHURare at a maximum of±0.39 ε units at 0 Ma, and decreasemonotonicallyto ±0.18 ε units by 4.5 Ga. It is useful to examine the effect of the corre-lation between the CHUR 176Lu/177Hf and 176Hf/177Hf estimates on theuncertainty in CHUR -without it the uncertainty in CHURwould increaseto ±0.50 ε units. The uncertainties on the decay rate and CHUR, whencombined have a minimum of ±0.34 at ~1.6 Ga. An alternative tousing the covariance term was presented by Blichert-Toft and Albarède(1997) who used the conditional standard deviation u(1−r2). Usingthismethod on the Bouvier et al. (2008) CHUR parameters (and ignoringthe covariance term in Eq. (6)), the calculated uncertainties on CHUR as afunction of time underestimate the true uncertainties at all times exceptfor very ancient samples, and ranges ±0.07 to ±0.10 ε units from 0 to4.5 Ga.

Radiogenic Hf isotope ratios can be determined to better than ±0.5 ε units (e.g., Weis et al., 2007) and Lu/Hf can be determined tobetter than 0.2% (Vervoort et al., 2004), which typically contributes

0 1 2 3 4time (Ga)

0

0.2

0.4

0.6

0.8

1.0

1.2

± ε

Hf

(a) Hf

0 1 2 3 4time (Ga)

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

± ε

Nd

(b) Nd

0 1 2 3 4time (Ga)

0

1

2

3

4

5

6

± γ

Os

0

0.01

0.02

0.03

0.04

0.05

0.06

λ ± γO

s

0.07

0.08

(d) Os

0 1 2 3 4time (Ga)

0

2

4

6

8

10

12

± ε

Sr

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

λ ± εS

r (2σ)

(c) Sr

0 1 2 3 4time (Ga)

0

0.10

0.20

0.30

0.40

0.50

± G

a

(e) TRD

λ (decay rate)

CHUR or Mantle

λ+ (CHUR or Mantle)

CHUR or Mantle, ρ=0

Fig. 1. The uncertainty in the Sr, Nd or Hf isotopic composition of CHUR, the Os isotopic composition of the upper mantle, and TRD as a function of time. The red curve is the sum ofthe uncertainties from the decay rate, the parent–daughter ratio of the model reservoir, and the radiogenic daughter isotope ratio of the model reservoir. The black curve is onlyCHUR uncertainties, the green curve is only decay rate uncertainties, and the dashed grey curve is identical to the black curve but shows the effect of neglecting the covarianceterm between the parent–daughter ratio and radiogenic isotope ratio. For (c) and (d), the uncertainty in the decay rate is small and it is depicted on a different scale (on theright), and the CHUR is effectively the entire uncertainty. (e) The uncertainty on TRD ages as a consequence of the uncertainty in the CHUR parameters. The effect of the decay con-stant uncertainty is incorporated, but not graphed, as it is insignificant.

136 R.B. Ickert / Chemical Geology 340 (2013) 131–138

an uncertainty of less than ±0.2 ε units during correction for radio-genic ingrowth, even for very old rocks. Therefore the uncertaintiesin the decay rates and the CHUR values are not negligible in size rel-ative to the uncertainties associated with sample measurements.

The ~1.5% uncertainty on λ147Sm has a substantial effect on theuncertainty in the calculated isotopic composition of the bulk mantleas a function of time, rising to ±1.8 ε units at 4.5 Ga (Fig. 1b). The un-certainty in CHUR is not as large, changing monotonically from±0.21ε units to ±0.13 ε units. The effect of the covariance term reduces theapparent uncertainty by more than a factor of two. When uncer-tainties on CHUR and the decay rate are considered simultaneously,the decay rate dominates after ~1.5 Ga.

Analytical uncertainties on radiogenic Nd isotope ratios and Sm/Ndare similar to those of Hf — typically better than ±0.5 ε units and

0.2%, respectively. The contribution of the decay rate to the total com-bined uncertainty can dominate the uncertainty for either very ancientrocks or for rocks with high Sm/Nd (e.g., those needing large decay cor-rections). The uncertainty in CHUR is mainly negligible.

Uncertainties on the mantle Re–Os systematics are large. Uncer-tainties in the primitive upper mantle composition, derived frommeasurements of meteorites, are ±2.5 γ units at 0 Ga, have a mini-mum of ±2.4 γ units at ~1.5 Ga and rise to ±3.6 γ units by 4.5 Ga(Fig. 1d). The uncertainty in the decay rate, as reported by Selby etal. (2007), has a corresponding uncertainty that rises to ±0.07 γunits at 4.5 Ga. Uncertainties in TRD ages (for zero age samples)range from ±0.47 Ga for a TRD=0 Ga, have a minimum at TRD=2.3 Ga of ±0.37 Ga, and increases to ±0.45 Ga at TRD=4.5 Ga(Fig. 1e).

137R.B. Ickert / Chemical Geology 340 (2013) 131–138

Given the high variability in Re/Os and Os abundances in naturalsamples, it is difficult to generalize the effects of these uncertainties.However, radiogenic Os isotope ratios can be measured to betterthan 0.1%, which means that the uncertainty on the PUM value isthe same order of magnitude, but the nominal uncertainty on thedecay rate is negligible. However, as described above, that uncer-tainty should be viewed with caution, and its magnitude may berevised.

Due to the small uncertainties in the 87Rb decay rate, its contribu-tion to the combined standard uncertainty of an initial 87Sr/86Sr issmall for an 87Rb/86Sr of b0.1, however the 87Rb/86Sr of rocks likegranites can range up to 10 (e.g., McCulloch and Chappell, 1982).For a 87Rb/86Sr of 3, for example, it contributes ±5.6 ε units at 4.5 Ga.

Although the contribution to the combined uncertainties due tothe decay rates can be large, for many applications the difference be-tween initial radiogenic isotope ratios is more important than theabsolute values. For these cases, the propagated uncertainties due todecay rates are highly correlated between samples and therefore con-tribute negligibly to the difference between them – in these casesthey can be safely ignored. Nevertheless, it is important to acknowl-edge that uncertainties of these magnitudes are present in the de-rived quantities as published, regardless of whether they arerelevant to the solution of any particular geochemical problem.

4.1. Example

As an example, the total combined uncertainty is calculatedfor sample JEH-2007-23 from Hoffmann et al. (2011), in which theLu–Hf and Sm–Nd systematics are determined for a ca. 3.2 Gametabasaltic rock. Two sets of uncertainties are calculated for eachisotope system. One that includes the uncertainty in CHUR and theModel-1 type uncertainties—where the magnitude of the uncertain-ty components in the age and decay rate are small due to the corre-lation between the isotopic composition of CHUR and themetabasalt. The other is a Model-2 type uncertainty, appropriatewhen the ε notation is used only as a linear transformation — it isequivalent to an initial isotope ratio. For the purpose of the uncer-tainty propagation, the age correction is assumed to have an uncer-tainty of 1%. All uncertainties are in units of parts per ten thousand(equivalent to ε units) and at a coverage factor of two (approximate95% confidence level).

Considering only the uncertainties in the radiogenic isotope ratiosand the parent–daughter ratios (as reported in the table of the originalwork), the uncertainties on initial εNd values are ±0.23, which break-down into ±0.14 from the 143Nd/144Nd and ±0.18 from 147Sm/144Nd.The corresponding uncertainty in the initial εHf values is ±0.28 withcontributions of ±0.21 from the 176Hf/177Hf, and ±0.18 from the176Lu/177Hf.

AModel-1 type uncertainty yields a combined standard uncertain-ty on the initial εNd of ±0.27. It is only a small increase due to thecorrelation of the decay rate and age uncertainties (contributing ±0.04 and ±0.03, respectively) and the magnitude of the uncertaintyin CHUR (±0.12 ε). For εHf, the total uncertainty increases to ±0.42, in part because of the relatively large component due to the un-certainty in the age, ±0.23. The uncertainties in the decay rate of176Lu and of CHUR contribute only ±0.10 and ±0.23 to the combinedstandard uncertainty, respectively.

The Model-2 type calculations (equivalent to initial isotoperatios) yield much larger combined standard uncertainties. For theinitial εNd they are ±1.71, primarily due to the large uncertaintyon the 147Sm decay rate and age (±1.41 and ±0.93, respectively).For εHf, the total combined uncertainty rises to ±0.77, mainly asso-ciated with the large propagated uncertainty in the age (±0.66) withthe uncertainty on the 176Lu decay rate contributing only a smallamount (±0.28).

5. Conclusions

Simple equations have been presented that incorporate the uncer-tainties in measured radiogenic isotope ratios and parent–daughterratios, quantity of time required for radiogenic ingrowth correction,decay rates and model geochemical reservoirs on initial isotope ra-tios, reservoir normalized values, and model ages.

Most usefully, these algorithms will allow workers to estimate thecombined standard uncertainties for initial isotope ratios and modelages by correctly weighting the uncertainties in age, radiogenic iso-tope ratio, and parent–daughter ratios. This allows for an evaluationof whether a measurement is fit for purpose. These quantities are typ-ically uncorrelated between samples and therefore are important inmost cases. Fortunately, when differences between isotope ratios,rather than absolute isotope ratios or geochemical-reservoir normal-ized values, are of interest, the uncertainties in the decay rates arehighly correlated and therefore become unimportant.

Notwithstanding the usefulness in solving geochemical problems,the full set of algorithms provides the means by which to account forevery input quantity in these measurement functions. This is a pre-requisite for providing full traceability in measurement — a goal ofmetrologically sound measurements.

The equations are straightforward to implement in a spreadsheetor statistical software package, and are incorporated into a MicrosoftExcel spreadsheet that is available as part of the Supplementary on-line material.

Acknowledgments

RBI thanks Graham Pearson and Yuri Amelin for discussion. Twoanonymous reviewers and the editor are thanked for their constructivecriticism that contributed to the clarity of this manuscript. This workwas supported by NSERC PGS-M and PGS-D scholarships, an IzaakWalton Killam Memorial Postdoctoral Fellowship and an NSERC Dis-covery grant to Thomas Stachel.

Appendix A. Supplementary data

The electronic supplementary material includes a Microsoft Excelspreadsheet implementing the algorithms of this paper, an exampleworksheet, and a brief description of the functions and argumentsof the algorithms in the spreadsheet. Supplementary data to this arti-cle can be found online at http://dx.doi.org/10.1016/j.chemgeo.2013.01.001.

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