Three new ways to calculate average (U Th)/He ages

 Lee Singleton
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1 Avalable onlne at Chemcal Geology 249 (2008) Three new ways to calculate average (U Th)/He ages eter ermeesch School of Earth Scences; Brkbeck, Unversty of London; London WC1E 7HX, Unted Kngdom Receved 15 June 2007; receved n revsed form 8 December 2007; accepted 20 January 2008 Edtor: S.L. Goldsten Abstract Tradtonally the average age of multple (U Th)/He analyses has been calculated as the arthmetc mean age. Ths paper presents three alternatve methods: (a) n analogy wth the fsson track method, the pooled age s calculated by addng the respectve U, Th and He abundances of several grans together, thereby generatng one synthetc multgran measurement; (b) the sochron age s the slope of helum concentraton versus presentday helum producton; (c) the central age s computed from the geometrc mean U Th He composton. Each of these methods s more approprate than the arthmetc mean age n certan applcatons. The pooled age s useful for comparng snglegran wth multgran analyses, whle the sochron age can be used to detect parentless helum. The central age s the most accurate and statstcally robust way to calculate a sample average of several snglegran analyses because U, Th and He form a ternary system and only the central age adequately captures the statstcs of ths compostonal data space. Fortunately, the expected dfference between the arthmetc mean age and the central age s relatvely small, less than 1% f the external age reproducblty s better than 15% (1σ). Fnally, the (U Th)/He age equaton s vsualzed on a ternary dagram to llustrate that the αejecton correcton should be appled before, and not after age calculaton, n order to avod a partal lnearzaton of the age equaton. Includng Sm as a fourth parent element precludes a straghtforward vsualzaton of the age equaton on a twodmensonal plot. Nevertheless, the pooled, sochron and central age methods can be easly generalzed to the case of (U Th Sm)/He datng. To facltate the calculaton of the central age, a webbased calculator s provded at Elsever B.. All rghts reserved. Keywords: (U TH)/He; Thermochronology; Compostonal data analyss 1. Introducton Radogenc helumgeochronology s based on a summed set of dfferental equatons: d½heš dt Xn d½ Š wth : d ½ Š dt dt k ½ Š ð1þ where t=tme, [He]=helum abundance, [ ]=abundance of the th parent nuclde and λ =decay constant of ths nuclde (for 1 n). Despte the smplcty of Eq. (1), there are several ways to solve t, three of whch wll be dscussed n Secton 2. A lnear approxmaton s accurate to better than 1% for ages up to 100 Ma, whch can be consdered satsfactory n comparson wth the external reproducblty of (U Th)/He datng (20 Emal address: 30%; e.g., Stock et al., 2006). Nevertheless, most researchers rghtly decde to calculate an exact age by numercal teraton. Ths paper rases the pont that the accuracy ganed by dong so s easly lost by two common practces: (1) performng the α ejecton after, rather than before age calculaton and (2) usng the arthmetc mean age to summarze a dataset of several snglegran measurements. After Secton 3 presents two smlarly based alternatves to the arthmetc mean age that are approprate for specfc applcatons, Secton 4 ntroduces the central age as the most accurate way to compute average (U Th)/He ages. The accuracy ganed by usng the central age nstead of the arthmetc mean age s comparable to that ganed by teratvely solvng the (U Th)/He age equaton nstead of usng the lnear approxmaton. The only cost of the new procedure s computatonal complexty. To facltate the calculatons, they are mplemented n an onlne calculator (http://pvermees.andropov.org/central) and llustrated on a publshed dataset of nclusonbearng apattes. Fnally, Secton /$  see front matter 2008 Elsever B.. All rghts reserved. do: /j.chemgeo
2 340. ermeesch / Chemcal Geology 249 (2008) presents a generalzed method to calculate central ages for datasets that also nclude a fourth radoactve parent, 147 Sm. 2. Calculatng snglegran ages: many ways to skn a cat Ten naturallyoccurng longlved αemttng radonucldes exst on Earth: 144 Nd, 147 Sm, 148 Sm, 152 Gd, 174 Hf, 186 Os, 190 t, 232 Th, 235 U and 238 U. For the purpose of helumthermochronology, all but the heavest three of these nucldes can often be neglected because of ther low abundance and low helumyeld. For example, only one αpartcle s produced per 147 Sm, whereas sx to eght are formed n the Th and U decay seres. Further smplfcaton s possble because the presentday 238 U/ 235 Urato s constant n the solar system (=137.88; Steger and Jäger, 1977). Therefore, the ngrowth of helum wth tme (t) can be wrtten as a functon of the elemental U, Th and He abundances or concentratons: ½HeŠ 8 137:88 138:88 ek 238t 7 1 þ 138:88 ek 235t 1 ½UŠþ6 e k232t 1 ½ThŠ ð2þ the average has been estmated by the arthmetc mean of the snglegran ages. Ths secton wll ntroduce two alternatve methods for calculatng average ages, and the next secton wll add a thrd. Each of these new methods s more approprate than the arthmetc mean age n specfc applcatons The pooled age Helum can be extracted from the host gran ether n a resstance furnace or by laserheatng n a mcrooven (House et al., 2000). In the former, but sometmes also n the latter case, t may be necessary to analyze multple mneral grans together (e.g., ersano et al., 2007). oolng several grans boosts the sgnal strength and sometmes averages out αejecton correcton errors caused by zonng and mneral nclusons. ermeesch et al. (2007) ntroduced the pooled age as the best way to compare multple snglegran ages wth one or more multgran ages, or to compare two sets of multgran ages wth each other (Fg. 1). The pooled age s calculated by addng the respectve U, Th and He abundances (n moles) of several measurements together, thereby generatng one synthetc mult wth λ 232, λ 235 and λ 238 the decay constants of 232 Th, 235 U and 238 U, respectvely. Eq. (2) has no analytcal soluton but s easy to solve teratvely. However, for young ages (t 1/λ 235 ), a reasonably accurate lnear approxmaton also exsts: t ½HeŠ ð3þ wth the presentday helum producton rate: 8 137:88 138:88 k 238 þ 7 138:88 k 235 ½UŠþ6k 232 ½ThŠ ð4þ The accuracy of ths soluton wll be dscussed n Secton 4. Besdes beng easy to mplement, the lnear age equaton s useful for llustratve purposes and opens up some new applcatons whch wll be dscussed n Secton 3. Meesters and Duna (2005) ntroduced an alternatve drect soluton to the (U Th)/He age equaton: t 1 ln 1 þ k wm k wm ½HeŠ ð5þ wth λ wm the weghted mean decay constant: k wm 8 137:88 138:88 k2 238 þ 7 138:88 k2 235 ½UŠþ6k ½ThŠ ð6þ As shown by Meesters and Duna (2005) and n Secton 4, ths soluton s remarkably accurate for all practcal applcatons. 3. Multgran ages Eqs. (2 6) can be used to calculate (U Th)/He ages from ndvdual U, Th and He measurements, but do not explan how to calculate the average value of multple analyses. Tradtonally, Fg. 1. Lnearzed (U Th)/He dagram wth a subset of the HFtreated nclusonbearng apatte data of ermeesch et al. (2007). 30% uncertanty (2σ) was added to the helum abundances to account for αejecton correcton nduced scatter (ermeesch et al., 2007). (a) Accordng to the lnear age equaton, (U Th)/He ages are gven by the slopes of lnes connectng each (,[He])pont wth the orgn; (b) the pooled age s a synthetc multgran age calculated from the summed producton rates and helum abundances of all the measurements. The box n 1.b marks the outlne of 1.a. The pooled age of the sample s 11.28±0.14 Ma.
3 . ermeesch / Chemcal Geology 249 (2008) gran measurement. The age of the pooled measurement can then be calculated usng any of the equatons gven n Secton 2. An obvous dsadvantage of poolng compostonal data s that the resultng age s based to the hgh U, Th or He compostons. Because such bas can be assocated wth anomalous grans affected by radaton damage or mplanted helum, the pooled age may be wrong by effectvely gvng extra weght to outlers. However, these objectons are also true for standard multgran analyses, whch cannot be avoded when datng small, young, or U Thpoor grans (e.g., ersano et al., 2007). In short, the pooled age must be used for and only for averagng multgran alquots The (U Th)/He sochron The prevous secton showed that (U Th)/He data can be vsualzed on a twodmensonal plot of helum abundance or concentraton versusproducton (Fg. 1). To calculate a pooled age, t s mportant that [U], [Th] and [He] are elemental abundances, expressed n moles. If the data are recast n unts of concentraton, some of the bas towards hgh U Thgrans dsappears and the He dagram can be used to defne a (U Th)/He sochron. Ths s an unconstraned lnear ft through a seres of snglegran (,[He]) measurements. For an applcaton of the sochron method, consder the U Th rch mneral nclusons n apatte whch are often held responsble for erroneously old (U Th)/He ages, because they produce parentless He. Ths problem can be detected wth the (U Th)/ He sochron. In the absence of mneral nclusons, the sochron goes through the orgn (=[He]=0). However, n the presence of αemttng nclusons, the sochron s ether not defned or does not go through the orgn. For example, consder the worstcase scenaro of an αemttng zrcon ncluson contaned n an apatte wthout U and Th. The ncluson ejects He nto the surroundng apatte that s measured followng degassng by heatng wth a laser or n a resstance furnace. However, the zrcon ncluson wll not dssolve n the concentrated HNO 3 that s commonly used to dgest apattes pror to U Th analyss. Therefore, the apparent (U Th)producton of such a sample s zero, and ts sochron does not go through the orgn of the [He] dagram. ermeesch et al. (2007) solved the parentless helum problem by dssoluton of the apatte and ts nclusons n hot HF. The effectveness of ths technque s llustrated by comparng an nclusonrch sample from Naxos usng the tradtonal HNO 3 method wth an HFtreated alquot of the same sample. The latter defnes a wellconstraned (U Th)/He sochron wth zero ntercept, whereas the former does not (Fg. 2). Calculaton of the sochron age, ncludng error propagaton, can easly be done usng the Isoplot Excel addn (Ludwg, 2003). form a ternary system, can be plotted on a ternary dagram, and are subject to the pecular mathematcs of the ternary dataspace. In a threecomponent system (A+B+C=1), ncreasng one component (e.g., A) causes a decrease n the two other components (B and C). Another consequence of socalled data closure s that the arthmetc mean of compostonal data has no physcal meanng (Weltje, 2002) lottng the (U Th)/He age equaton on ternary dagrams Followng the nomenclature of Atchson (1986), the ternary dagram s a 2smplex (Δ 2 ). The very fact that t s possble to plot ternary data on a twodmensonal sheet of paper tells us that the sample space really has only two, and not three dmensons. As a soluton to the compostonal data problem, Atchson (1986) suggested to transform the data from Δ 2 to R 2 usng the lograto transformaton. After performng the desred ( tradtonal ) statstcal analyss on the transformed data n R 2, the results can be transformed back to Δ 2 usng the nverse lograto transformaton (Fg. 3). Implementaton detals about the lograto transformaton wll be gven n Secton 4.3. Ternary dagrams and lograto plots are useful tools for vsualzng U Th He data and the (U Th)/He age equaton. 4. U Th He as a ternary system The ageeqs. (2), (3) and (5) do not specfy the measurement unts of [U], [Th] and [He]. These can be expressed n moles or moles/g, but they can also be nondmensonalzed by normalzaton to a constant sum: [U ] [U]/([U]+[Th]+[He]), [Th ] [Th]/([U]+[Th]+[He]) and [He ] [He]/([U]+[Th] +[He]) so that [U ]+[Th ]+[He ]=1. Therefore, U, Th and He Fg. 2. (U Th)/He sochron plots for nclusonbearng apattes from Naxos, Greece (ermeesch et al., 2007). (a) HNO 3 treated apattes do not plot on a lne, ndcatng parentless helum caused by undssolved mneral nclusons contanng mssng U and Th; (b) HFtreated apattes from the same sample do form a welldefned sochron ntersectng the orgn, ndcatng that all parent and daughter nucldes are accounted for. The sochron age s 12.0±4.2 Ma.
4 342. ermeesch / Chemcal Geology 249 (2008) Fg. 3. Ternary U Th He data can be mapped to twodmensonal logratospace wthout loss of nformaton. Thus, t can be shown that the lnear age equaton s accurate to better than 1% for ages up to 100 Ma (Fg. 4.a) whereas the equaton of Meesters and Duna (2005) reaches the same accuracy at 1 Ga (Fg. 4.b). Fg. 4.c represents a warnng aganst applyng the αejecton correcton after, rather than before the age calculaton. Ths causes a partal lnearzaton of the age equaton and results n a loss of accuracy. For example, dvdng an uncorrected (U Th)/He age by an αretenton factor F t of 0.7 results n a msft that s 30% of the lnear age equaton msft. To take full advantage of the accuracy of the exact age equaton, one must dvde [He] by F t before calculatng the (U Th)/He age The central age The lograto transformaton s useful for more than just the purpose of vsualzaton. It provdes a fourth and arguably best way to calculate the average age of a populaton of snglegran (U Th)/He measurements. The central age s calculated from the average U Th He composton of the dataset, where average s defned as the geometrc mean of the snglegran U, Th and He measurements. The geometrc mean of compostonal data equals the arthmetc mean of lograto transformed data. How mportant s the dfference between the arthmetc mean age and the central age? To smplfy ths queston, consder the specal case of a sample wth only one radoactve parent, say Th. Assume that W = ln([he]/[th]) s normally dstrbuted wth mean µ and standard devaton σ. Usng the lnearzed age equaton for clarty, the central age t c s gven by: t c Ce A ð7þ wth C=1/(6λ 232 ) for Th. Usng the frst raw moment of the lognormal dstrbuton (Atchson and Brown, 1957), the arthmetc mean age t m s: t m Ce Aþr2 =2 so that the relatve dfference between t m and t c s: t m t c t c e r2 =2 1 ð9þ ð8þ Usng the second central moment of the lognormal dstrbuton (Atchson and Brown, 1957), the varance of the snglegran ages s gven by: r 2 t C 2 e r2 1 e 2Aþr2 ð10þ lottng (t m t c )/t c versus σ t /t c reveals that the central age s systematcally younger than the mean age. Fortunately, the dfference s small. For example, for a typcal external reproducblty of ~25% (e.g. σ t /t =11% for Stock et al., 2006), the expected dfference s b1% (Fg. 5). Fnally, t s nterestng to note that the geometrc mean of the lognormal dstrbuton equals ts medan. Therefore, the central age asymptotcally converges to the medan age. However, typcal numbers of replcate analyses are not suffcent for ths approach to be truly benefcal Applcaton to HFtreated Naxos apattes We now return to the sample of HFtreated nclusonbearng apattes from Naxos that was prevously used to llustrate the pooled and sochron age (Fgs. 1 and 2). The raw data and the dfferent steps of the central age calculaton are gven n Table 1. We wll now walk through the dfferent parts (labeled a, b and c) of ths table. (a) The upper left part of Table 1 lsts the U, Th and He abundances of 11 snglegran analyses. Ther respectve snglegran ages (t) were calculated usng the exact age equaton, even though the lnear age approxmaton (Eq. (3)) s accurate to better than 0.1% for such young ages (Fg. 4. a). The pooled U, Th and He abundances are obtaned by smple summaton of the consttuent grans. Note that the helum abundances are corrected for αejecton pror to beng pooled. A nomnal σ = 15% statstcal uncertanty s assocated wth F t, assumng randomly dstrbuted mneral nclusons (ermeesch et al., 2007). The pooled abundances were normalzed to unty to facltate comparson wth the geometrc mean composton (see below).
5 . ermeesch / Chemcal Geology 249 (2008) (b) To calculate the sochron age, the abundances are frst rescaled to unts of concentraton (e.g. n nmol/g). Ths removes the bas towards large grans, whch can domnate the pooled age calculaton. The αproducton rate s gven by Eq. (4). The lnear regresson (Fg. 2) was done Fg. 5. Expected dfference between the mean age t c and the central age t m plotted aganst the relatve spread of the snglegran ages. usng Isochron 3.0 (Ludwg, 2003), yeldng a slope of 12.0 ±4.2 Ma wth an ntercept of 0.05±0.45 nmol/g He. (c) Central ages are somewhat more complcated to calculate than arthmetc mean ages, pooled ages or sochron ages. Therefore, these calculatons wll be dscussed n more detal. Frst, transform each of the n snglegran analyses to logratospace (Fg. 6): ½U Š ½Th Š ln ; W ln ð11þ ½He Š ½He Š Fg. 4. (a) Relatve msft of the lnear approxmaton (Eq. (3)) to the exact age equaton (Eq. (2)); (b) msft of the drect soluton of Meesters and Duna (2005) (Eq. (5)); (c) msft caused by makng the αejecton correcton after the age calculaton (wth F t =0.7). For = 1,...,n. Note that ths transformaton can be done rrespectve of whether the U, Th and He measurements are expressed n abundance unts or n unts of concentraton. Followng standard error propagaton, the (co)varances of these quanttes are estmated by: ru U 2 ; He W rth Th 2 ; cov ;W rhe 2 He He ð12þ Next, calculate the arthmetc mean of the lograto transformed data: 1 ; W 1 W ; n n Wth the followng (co)varances: 1 n 2 ; W 1 n 2 W ; cov ;W 1 n 2 cov ;W ð13þ ð14þ Note that Eq. (14) only propagates the nternal (.e. analytcal) uncertanty, and not the external error. Snglegran (U Th)/He ages tend to suffer from overdsperson wth respect to the formal analytcal precson for a number of reasons (Ftzgerald et al., 2006; ermeesch et al., 2007). Therefore, t may be better to use an alternatve equaton propagatng the external error: n 2 n ; W W W 2 ; ð15þ nn ð 1Þ nn ð 1Þ n cov ;W W W nn ð 1Þ
6 Table 1 Stepbystep data reducton of the Naxos dataset of 11 nclusonbearng apattes (wth an arthmetc mean age of 11.58±0.20 Ma) 344. ermeesch / Chemcal Geology 249 (2008) Ths table should be read from a to c and from left to rght: (a) U, Th and He abundances and pooled age (11.28±0.14Ma).[He ]stheαejecton corrected Heabundance ([He ] [He]/F t ); (b) The U, Th and He concentratons, requred for the calculaton of a (U Th)/He sochron. The sochron age s 12.0±4.2 Ma. (c) lograto transformed data (,W) and the (dmensonless) geometrc mean composton, resultng n the central age of 11.38±0.20 Ma.
7 . ermeesch / Chemcal Geology 249 (2008) Errorweghtng can be done by trval generalzatons of Eqs. (13) (15), whch are mplemented n the webcalculator. The geometrc mean composton s gven by the nverse lograto transformaton (Atchson, 1986; Weltje, 2002): ½U Š e e þ e W þ 1 ; ½ThŠ e W e þ e W þ 1 ; ½HeŠ 1 e þ e W þ 1 ð16þ Wth varances: U a 2 b 2 A b 2 c 2 2bc d 2 e 2 2de He W cov ;W 1 A where e e W þ 1 e þ W a 2 b 2 e þ e W þ 1 e þ e W þ 1 e W e þ 1 e c 2 d 2 e þ e W þ 1 e þ e W þ 1 e W e 2 e þ e W þ 1 ð17þ The central age s then smply calculated by pluggng [U ], [Th ] and [He ] and ther uncertantes nto Eqs. (2), (3) or (5). As predcted (Fg. 5), the arthmetc mean age s older than the central age. There s less than 2% dsagreement between the arthmetc mean age (~11.58 Ma) and the central age (~ Ma), and 7% dfference between the pooled age (~11.28 Ma) and the sochron age (~12.0 Ma). 5. Generalzed equatons for (U Th Sm)/He datng For reasons gven n the Introducton, 147 Sm s often neglected n helum thermochronometry. However, n rare cases t does happen that apatte contans hgh abundances of Sm, affectng the helum age on the percent level. Ths secton wll explan how to add a fourth radoactve parent to the methods descrbed above. The exact age equaton (Eq. (2)) and the presentday helum producton rate (Eq. (4)) can easly be generalzed to nclude Sm: ½HeŠ 8 137:88 138:88 ek 238t 7 1 þ 138:88 ek 235t 1 ½UŠþ6 e k232t 1 ½ThŠþ0:1499 e k 147 t 1 ½SmŠ ð18þ and 8 137:88 138:88 k 238 þ 7 138:88 k 235 ½UŠþ6k 232 ½ThŠþ0:1499k 147 ½SmŠ ð19þ Wth λ 147 the decay constant of 147 Sm and all other parameters as n Eqs. (2) and (4). Usng Eq. (19), calculatng an sochron age for (U Th Sm)/He proceeds n exactly the same way as for the ordnary (U Th)/He method, and the same s true for the pooled age (Fg. 6). Calculatng (U Th Sm)/He central ages s also very smlar, although the equatons are a bt longer. In addton to and W (Eq. (11)), we defne a thrd lograto varable X (1 n): ½U Š ½Th Š ½Sm Š ln ; W ln ; X ln ½He Š ½He Š ½He Š ð20þ Because there are three nstead of two lograto varables, the (U Th Sm)/He age equaton cannot be vsualzed on a straghtforward bvarate dagram, but forms a set of hypersurfaces n trvarate logratospace (Fg. 7). Lkewse, (U Th Sm)/He data do not form a ternary, but a tetrahedral system n Fg. 6. (a) Ternary dagram of the Naxos data (Table 1). (b) The same data plotted n logratospace. Error ellpes are 2σ.
8 346. ermeesch / Chemcal Geology 249 (2008) Fg. 7. The lograto transformaton can easly be generalzed to the case of 3 radoactve parents but ths precludes a straghtforward vsualzaton on a twodmensonal dagram. (a) the (U Th Sm)/He age equaton n compostonal dataspace. (b) two hypersurfaces representng two ages n logratospace. compostonal dataspace (Fg. 7). Generalzng the (co)varances of Eq. (12): ru 2 ; rˆ W 2 U He rth 2 ; Th He X rsm 2 ð21þ Sm He cov ;W cov ;X cov W ;X rhe 2 He Calculatng the arthmetc logratomeans: 1 n ; W 1 n W ; X 1 n X ð22þ The (co)varances of the logratomeans, propagatng only the nternal error: 1 n 2 cov ;W 1 n 2 cov W ;X 1 n 2 ; W 1 n 2 W ; X 1 n 2 cov ;W ; cov ;X 1 n 2 cov W ;X cov ;X ; X ð23þ The (co)varances of the logratomeans, propagatng the external error: n 2 n ; rˆ2 W W W 2 nn ð 1Þ nn ð 1Þ n cov ;W W W ;cov nn ð 1Þ ;X n cov W ; X W W X X nn ð 1Þ n ; X X X 2 nn ð 1Þ n ð nn ð 1Þ Þ X X ð24þ The nverse lograto transformaton: e ½U Š e þ e W þ e X þ 1 ; ½ThŠ e þ e W þ e X þ 1 e ½SmŠ X e þ e W þ e X þ 1 ; ½HeŠ 1 e þ e W þ e X þ 1 ð25þ Fnally, calculatng the standard error propagaton of the geometrc mean compostons: rˆ U a rˆ 2 2 b 2 c 2 2ab 2ac 2bc Th C Sm A b 2 d 2 e 2 W 2bd 2be 2de B c 2 e 2 f 2 X 2ce 2cf 2ef A cov B ;W C rˆ 2 g 2 h 2 2 2gh 2g cov A He ;X cov W ;X wth e e W þ e X þ 1 e þ W a 2 ; b 2 e þ e W þ e X þ 1 e þ e W þ e X þ 1 e þ X e W e þ e X þ 1 c 2 ; d 2 e þ e W þ e X þ 1 e þ e W þ e X þ 1 e W þ X e X e þ e W þ 1 e 2 ; f 2 e þ e W þ e X þ 1 e þ e W þ e X þ 1 e g 2 ; h 2 e þ e W þ e X þ 1 e þ e W þ e X þ 1 e X 2 e þ e W þ e X þ 1 e W e W ð26þ An example of a wellbehaved (U Th Sm)/He dataset from the Fsh Lake alley apatte standard (provded by rof. Danel Stockl, Unversty of Kansas) s gven n the webcalculator (http://pvermees.andropov.org/central). The arthmetc mean of 28 snglegran ages s 6.36±0.11 Ma, the pooled age 6.43±
9 . ermeesch / Chemcal Geology 249 (2008) Ma, the sochron ages 6.44±0.67 Ma (wth an ntercept of 0.005±0.056 fmol/µg, and the central age 6.41±0.14 Ma. Note that the central age s older and not younger than the arthmetc mean age. Ths ndcates that random varatons exceed the very small systematc dfference between the arthmetc and geometrc mean compostons. However, the central age probably stll s more accurate than the arthmetc mean age because t s less senstve to outlers. 6. Conclusons Ths paper compared three ways to calculate an age from a sngle set of U, Th and He measurements and four ways to calculate the average of several alquots of the same sample. U, Th and He form a ternary system, and the ternary dagram was ntroduced as an elegant way to make such a comparson. Ths reveals that the accuracy ganed by the exact soluton of the (U Th)/He equaton s easly lost f the average age of replcate measurements s calculated by the arthmetc mean. As a better alternatve, the central age s calculated from the geometrc mean composton of a dataset. In addton to the central age, the paper also ntroduced the pooled age and the sochron age as valuable alternatves to the arthmetc mean age n certan applcatons. The pooled age s calculated by addng the U, Th, (Sm) and He contents of several sngleand/or multgran alquots of the same sample. ooled ages are based to hgh U Thgrans whch may be affected by radaton damage, but are the only sensble way to average multgran alquots. The sochron age s gven by the slope of a lnear ft of a dagram that plots helum content aganst presentday helum producton rate. In order to reduce the bas towards large grans, t s a good dea to transform the nput data to unts of concentraton, whch can be done by dvdng the atomc abundances by the estmated volume or mass of the component grans. Dong so wll translate the dataponts along a straght lne through the orgn of the sochron plot and mproves ts power for detectng parentless helum. If there s no parentless helum, data must plot on a sngle lne gong through the orgn. But f, nstead, the data do not defne a lne, or ths lne does not go through the orgn of the sochron dagram, parentless helum or a smlar problem may be present. The sochron age s less well suted for ages older than 100 Ma because t uses the lnearzed age equaton (Eq. (3)). Although most (U Th)/He geochronologsts are probably already aware that some accuracy s lost by calculatng the α ejecton correcton by smply dvdng the uncorrected (U Th)/ He age by the αretenton factor F t, t bears repeatng that nstead, the measured helum concentraton should be dvded by F t before the age calculaton. The effects dscussed n ths paper are relatvely mnor, affectng the calculated ages by at most a few percent. Nevertheless, the added computatonal cost of followng the above recommendatons pales n comparson wth the cost of collectng, separatng and analyzng samples. Therefore, there s no reason why not to gan the extra percent of accuracy. To facltate the calculaton of the central age, a webbased calculator s provded at central. It mplements the calculatons of central ages wth or wthout Sm, and offers several optons for propagatng nternal or external uncertantes. The webcalculator also allows the calculaton of errorweghted central ages, and ncludes two dataset for testng purposes: the nclusonbearng (U Th)/He data from Naxos whch s also summarzed n Table 1, and (U Th Sm)/He data from a Fsh Lake alley apatte lab standard. Acknowledgments I would lke to thank Jeremy Hourgan and an anonymous revewer for the constructve comments, and Danny Stockl for sharng hs Fsh Lake alley (U Th Sm)/He data. Ths work was done whle the author was a Mare Cure postdoctoral fellow at ETHZürch n the framework of the CRONUSEU ntatve (RTN project reference ). References Atchson, J., The Statstcal Analyss of Compostonal Data. Chapman and Hall, London. 416 pp. Atchson, J., Brown, J.A.C., The Lognormal Dstrbuton. Cambrdge Unversty ress, London. 176 pp. Ftzgerald,.G., Baldwn, S.L., Webb, L.E., O Sullvan,.B., Interpretaton of (U Th)/He sngle gran ages from slowly cooled crustal terranes: a case study from the Transantarctc Mountans of southern ctora Land. Chemcal Geology 225, House, M.A., Farley, K.A., Stockl, D., Helum chronometry of apatte and ttante usng NdYAG laser heatng. Earth and lanetary Scence Letters 183 (3 4), Ludwg, K.R., Isoplot 3.00 user manual, Berkeley Geochronology Center Specal ublcaton, vol. 4. Meesters, A.G.C.A., Duna, T.J., A nonteratve soluton of the (U Th)/ He age equaton. Geochemstry, Geophyscs, Geosystems 6 (4). ersano, C., Barfod, D.N., Stuart, F.M., Bshop,., Constrants on early Cenozoc underplatngdrven uplft and denudaton of western Scotland from low temperature thermochronometry. Earth and lanetary Scence Letters 263, Steger, R.H., Jäger, E., Subcommsson on geochronology: conventon on the use of decay constants n geoand cosmochronology. Earth and lanetary Scence Letters 36, Stock, G.M., Ehlers, T.A., Farley, K.A., Where does sedment come from? Quantfyng catchment eroson wth detrtal apatte (U Th)/He thermochronometry. Geology 34, ermeesch,., Seward, D., Latkoczy, C., Wpf, M., Guenther, D., Baur, H., Alphaemttng mneral nclusons n apatte, ther effect on (U Th)/ He ages, and how to reduce t. Geochmca et Cosmochmca Acta 71, Weltje, G.J., Quanttatve analyss of detrtal modes; statstcally rgorous confdence regons n ternary dagrams and ther use n sedmentary petrology. EarthScence Revews 57 (3 4),
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