Technical Note pubs.acs.org/ac
Uncertainty of Blank Correction in Isotope Ratio Measurement K. E. Anders Ohlsson* Department of Forest Ecology and Management, Swedish University of Agricultural Sciences, SE-90183 Umeå, Sweden S Supporting Information *
ABSTRACT: Blank correction for isotope ratio measurement on small amounts of substances is often limited by presence of a blank, with an apparent isotopic composition different from that of the sample. For isotope ratios, blank correction is commonly performed either by the regression method, which works without the need for estimation of the blank, or by the subtraction method. With the subtraction method, estimation of the amount and isotope delta of the blank is required, and these estimates could be obtained either by direct, semi-indirect, or indirect measurement. Previously given expressions for the standard uncertainties of indirectly measured blank amounts and blank isotope deltas did not account for covariance between input quantities. In the present work, a previously published data set was re-evaluated, with covariance terms properly included in the calculation of uncertainties. It was shown that covariance effects may yield a significant reduction in uncertainty estimates, both for blank quantities and for blank corrected results. For series measurements on a standard material, it was also shown that the distribution of individual corrected isotope delta values around the average value was approximately normal, with its standard deviation equal to the estimated standard uncertainty of the corrected values. In most cases, it was observed that the regression and subtraction methods yield approximately the same blank corrected average values and uncertainties, regardless of method selected for estimation of blank quantities.
T
direct, indirect, and semi-indirect methods for blank estimation? Blank correction for isotope ratio measurements is commonly performed on the basis of an isotope balance.1 Strictly, the isotope balance should be expressed in terms of isotope-amount fractions, and by using this quantity, the following methodology is applicable to cases where the isotopic composition differ widely between sample and blank.14 However, in terminological agreement with most of the previous work, the isotope balance is expressed here as:
he measuring interval for accurate high precision isotope ratio measurement is often limited downward by presence of a blank, with an apparent isotopic composition different from that of sample.1−13 Isotope ratio measurement on small amounts of substances therefore benefit from correction of the systematic error due to blank. For blank correction, two methods were proposed by Gelwicks and Hayes:1 (1) the regression method, where a blank corrected isotope delta value is obtained from a series of measurements on the same sample, and (2) the subtraction method, where the corrected individual value is obtained by subtraction of the blank. With the subtraction method, estimates of the amount and isotope delta of the blank are required. These could be obtained by either of three approaches to measurement: (i) direct, (ii) indirect1 (see below), using two standard series, and (iii) semi-indirect measurement.4 In the semi-indirect method, the blank amount is measured directly, while its isotope delta is estimated from measurement on a series of standards. Despite the importance of blank correction for measurement on small samples, there are very few evaluations of these methods for estimation and correction of blanks, in terms of the uncertainty of the blank corrected values. Furthermore, the existing evaluations provide contradicting evidence for the best methods to use,1,2 and the previously given expressions for blank uncertainties did not account for covariance among input quantities.1 The objective of the present work was therefore to provide answers to the following questions regarding the uncertainty of blank corrected isotope delta values: Is there a significant effect from inclusion of covariance terms in uncertainty expressions? Is there a difference in performance between the regression and subtraction methods for blank correction and between the © XXXX American Chemical Society
n Tδ T = nsδs + nbδ b
(1)
where n is the amount of substance, δ is the isotope delta, and subscripts refer to total amount (T), sample (s), and blank (b), with nT = ns + nb. Basically, there are two approaches to perform the blank correction: In the subtraction approach, values for nb and δb are inserted into the rearranged eq 1 to yield the corrected result as:1 δs = (n Tδ T − nbδ b)/(n T − nb)
(2)
These results, corrected by blank subtraction, thus require estimation of the blank quantities nb and δb, as well as their uncertainties u(nb) and u(δb). In the regression approach, the isotope balance is written instead as:1 δ T = δs + nb(δ b − δs)/n T
(3)
Received: February 5, 2013 Accepted: May 7, 2013
A
dx.doi.org/10.1021/ac4003968 | Anal. Chem. XXXX, XXX, XXX−XXX
Analytical Chemistry
Technical Note
Table 1. Estimates of Blank Quantities and the Input Quantities Required for Use of Indirect and Semi-Indirect Methodsa input
m1
s
m2
s
δ1
s
nitrogen carbon
−2195 −219.4 direct
19 16.6
−421.4 17.74 direct
17.9 8.03
20.76 −10.84 semi
0.15 0.24
δ2
blank
nb
s
δb
s
δb
s
nb
s
δb
s
nitrogen carbon
83.08 17.13
0.31 0.59
−5.11 −27.76
0.02 0.22
−5.66 −23.65
0.14 0.89
86.21 12.74
0.55 0.84
−4.70 −28.06
0.11 0.44
0.18 −29.46 indirect
s
r (δ1, m1)
0.13 0.19
−0.925 −0.804 indirect
r (δ2, m2) −0.972 −0.802
Isotope delta values are given as ‰ vs air and vs VPDB for N and C, respectively. The unit for nb is nmol and V·s for N and C, respectively. The standard uncertainty of each quantity is given as the standard deviation s. Calculations were based on data set from Polissar et al.2 Standards 1 and 2 are as in Table 2. a
For multiple analyses of the same sample, the measured δT values are regressed on corresponding 1/nT values, and the corrected result δs is obtained as the intercept of the regression line at 1/nT = 0. Its standard uncertainty u(δs) is estimated from the standard error of the intercept. Note that blank correction, using the regression procedure, works without estimation of blank quantities. In some cases, the blank quantities nb and δb could be estimated by direct measurement on the blank. In other cases,1,3,4,8,11,13 where the direct method yields poor precision, there are two alternative methods available: First, nb and δb could be estimated using the indirect method described by Gelwicks and Hayes,1 where multiple measurement results for two different standard samples give two regression lines (eq 3), with intercepts δ1 and δ2, and slopes m1 = vb(δb − δ1) and m1 = nb(δb − δ2). From these data, the blank estimates are obtained as: nb = (m1 − m2)/(δ2 − δ1) δ b = (m2δ1 − m1δ2)/(m2 − m1)
r(b0 , b1) = −X̅ (n/∑ Xi 2)1/2
where X̅ is the average of the n observations Xi of the x variable. The previously given analytical expressions for u2(nb) and u2(δb) (their eqs 12 and 13)1 are here complemented by adding the last two terms of eq 7, which reads respectively: 2M [u(m1)u(δ1)r(m1 , δ1) + u(m2)u(δ2)r(m2 , δ2)] D3
−
2D [m2 2u(m1)u(δ1)r(m1 , δ1) + m12u(m2)u(δ2)r(m2 , δ2)] M3
(9)
(10)
⎛ ∂f ⎞2 2 ⎛ ∂f ⎞2 2 ⎛ ∂f ⎞2 2 u (δ b) = ⎜ ⎟ u ( n b) + ⎜ ⎟ u (m1) + ⎜ ⎟ u (δ1) ⎝ ∂δ1 ⎠ ⎝ ∂m1 ⎠ ⎝ ∂nb ⎠ 2
(4)
∂f ∂f m2 u(m1)u(δ1)r(m1 , δ1) = 14 u 2(nb) ∂m1 ∂δ1 nb 1 2 2 + 2 u (m1) + u 2(δ1) + u(m1)u(δ1)r(m1 , δ1) nb nb +2
(5)
■
(11)
COMPUTATIONS For the direct method, nb and δb were estimated as the average value of n blank measurements (nb and δb), with their uncertainty u obtained by type A evaluation as (s/(n)1/2), where s = standard deviation of either quantity.15 For the indirect and semi-indirect methods, u(nb) and u(δb) were estimated by applying the corresponding eqs 7 and 11. Finally, the uncertainty of the blank corrected sample values, u(δs), was obtained from eq 14 of ref 1, or as the standard deviation of the intercept, for the subtraction and regression methods, respectively. All calculations on propagation of uncertainties were performed by applying the numerical procedure given by Kragten,17 using the Microsoft Office Excel spreadsheet program. With the Kragten procedure, the functional relations for δs, nb, and δb were obtained from eqs 2, 4, and 5 (or 6), respectively. The computations were based on the same data set as used previously by Polissar et al.2 in elemental analyzer−isotope ratio mass spectrometry measurements of δ13C and δ15N (see Supporting Information). For nitrogen, the data for nT and nb are calibrated and presented in units of nmol of N. For carbon, however, the available complete data set includes values of the indication quantity I only, where I is the integrated ion current in units of V·s. Estimation of isotope deltas for corrected sample or blank could be performed using the indication
(6)
The expressions, given in previous work, for calculation of the blank standard uncertainties u(nb ) and u(δb ) are incomplete, in that they do not include terms for the covariance between intercept and slope estimates for the regression lines (their eqs 11−13).1 Consequently, blank uncertainties are potentially overestimated, and the uncertainty of δs may appear to be higher than it actually is. With covariance terms included, the uncertainties u(nb) and u(δb) are expressed as:15 ⎛ ∂f ⎞2 2 ⎛ ∂f ⎞2 2 ⎛ ∂f ⎞2 2 u (z ) = ⎜ ⎟ u (m2) + ⎜ ⎟ u (δ1) ⎟ u (m1) + ⎜ ⎝ ∂δ1 ⎠ ⎝ ∂m2 ⎠ ⎝ ∂m1 ⎠ 2
⎛ ∂f ⎞2 2 ∂f ∂f u(m1)u(δ1)r(m1 , δ1) +⎜ ⎟ u (δ 2 ) + 2 ∂m1 ∂δ1 ⎝ ∂δ2 ⎠ ∂f ∂f u(m2)u(δ2)r(m2 , δ2) +2 ∂m2 ∂δ2
−
where M = m1 − m2 and D = δ1 − δ2. With the semi-indirect method (eq 6), the complete expression for u(δb) is as follows:
Second, there exists also the semi-indirect method, where nb is estimated from direct measurement, while δb is obtained by the indirect method, but using only regression parameters (δ1 and m1) from a single standard.4,7,8 In this case, δb is estimated as: δ b = m1/nb + δ1
(8)
(7)
where the function z = f(m1,m2,δ1,δ2) is given by eq 4 for nb and eqs 5 or 6 for δb, respectively: The r(qi,qj) is the correlation coefficient for quantities qi and qj. With the straight line equation y = b0 + b1x obtained from regression, the correlation between b0 and b1 is estimated as:16 B
dx.doi.org/10.1021/ac4003968 | Anal. Chem. XXXX, XXX, XXX−XXX
Analytical Chemistry
Technical Note
Table 2. Blank Corrected Values of δ15N (‰ vs air) and δ13C (‰ vs VPDB), Obtained from Elemental Analyzer−Isotope Ratio Mass Spectrometry Measurementsa nitrogen
IAEA-N2b
s
IAEA-N1c
s
peru mud
s
caffeine
s
intercept direct semi indirect ΔS−B n sres carbon
20.76 20.46
0.15 0.18
0.18 0.43 1.02
0.13 0.10 0.14
7.17 6.93 7.46 6.97 12.28 12 0.14 s
0.17 0.12 0.13 0.13
1.68 1.80 2.43 1.60 6.79 4 0.22
0.52 0.27 0.29 0.29
intercept direct semi indirect ΔS−B n sres
25.87 9 0.17 IAEA-CH-6b
5.30 4 0.06 s
NBS22c
s
peru mud
−10.84 −10.07
0.24 0.18
−29.46 −29.65 −30.66
0.19 0.16 0.18
−20.56 −20.38 −21.15 −20.73 7.20 17 0.25
16.92 17 0.59
−1.70 14 0.43
0.10 0.08 0.09 0.08
ABS 1
caffeine
s
ABS 1
s
−37.68 −37.65 −38.19 −37.23 −9.92 7 0.33
0.18 0.15 0.16 0.15
−25.64 −25.59 −26.20 −25.65 2.12 12 0.42
0.17 0.15 0.16 0.15
−1.18 −1.40 0.09 −2.18 3.93 8 0.10 NBS19 1.53 2.52 1.88 1.40 29.29 10 0.55
s 0.26 0.18 0.26 0.23
s 0.30 0.22 0.22 0.22
a
Calculations were based on the data set from Polissar et al.2 For the regression method, the blank corrected value equals the intercept of the regression line δs (row denoted “intercept”). Table rows denoted “direct”, “semi”, and “indirect” give results for the corresponding methods of blank estimation, when used with the blank subtraction method. For the subtraction method, the weighted average of blank corrected isotope delta values δs from measurements on the series of standard samples were given, with weights proportional to 1/u2(δs), and normalized to set the sum of weights equal to unity. ΔS−B is the difference in isotope delta value between standard and blank. For calculation of u(δs), the standard deviation of residuals (sres; after regression) was used as estimate of u(δT), while u(nT) was selected as 2% of nT. bUsed as standard 1 in indirect and semi-indirect blank estimation. cSelected as standard 2 in indirect blank estimation.
quantity I as well. This is because I and nT are related through the calibration curve: I = k·nT, where k is the calibration constant, and the fact that k cancels out in the relevant eqs 2, 5, and 6. Note that the use of I values for estimation of the blank makes it unnecessary to accurately weigh the standard samples or to determine the calibration constant k. Results from measurements on several standard material series were included in the data sets. Two of these standard series were selected to be used for blank estimation, with the indirect and semi-indirect methods, while the remaining standard series were employed as independent data used for validation of the blank correction procedure.
■
clearly seen in Figure 1. The difference between estimates of u(δs), obtained with or without covariance terms present,
RESULTS AND DISCUSSION
Table 1 shows the results from estimation of the amount and isotope delta of blanks, including their standard deviations, using the direct, indirect, and semi-indirect methods. Despite significant differences between the blank estimates, they yielded in most cases the same weighted average and standard deviation of blank corrected values, obtained from measurement on standard materials (cf. Table 2). This was, in most cases, also true for blank corrected results obtained with the subtraction and regression methods, respectively. A closer look into the normalized differences between 210 individual δs values (δ13C and δ15N) and the corresponding weighted average value for each standard, (δs − δs)/u(δs), revealed that 15 and 5 of these differences were outside the ±2 and ±3 limits, respectively (all values for peru mud, caffeine, ABS1, and NBS19). This agreed with what could be approximately expected from a normal distribution. For the subtraction method, the conclusion from these data was therefore that blank corrected isotope delta values were normally distributed, with a standard deviation σ given by u(δs), regardless of method of estimation of blank quantities. The importance of properly including covariance terms in the expressions for the uncertainty in blank estimates was
Figure 1. Measurement uncertainty u(δs) for blank corrected δ15N values as a function of total amount of nitrogen nT. The indirect method was used for estimation of blank quantities, with or without covariance terms included in uncertainty expressions (filled and open markers, respectively). Standard materials were peru mud (circles), caffeine (triangles), and ABS1 (squares).
increased rapidly when nT decreased toward nb. For carbon data, the effect of omitting covariance terms was smaller in comparison to the effects observed for nitrogen results. Figure 2 shows a comparison of u(δs) estimates obtained with the direct, indirect, and semi-indirect methods for estimation of blank quantities (see also S-13, S-14, and S-15, Supporting Information). A significant variation was observed for u(δs) values, at any fixed level of nT. A minor part of this variation was due to the different methods applied for blank estimation, while the major part of the variation was between the different standard materials. The Kragten procedure for propagation of uncertainties showed that the dominant source C
dx.doi.org/10.1021/ac4003968 | Anal. Chem. XXXX, XXX, XXX−XXX
Analytical Chemistry
Technical Note
(12) Ziolkowski, L. A.; Druffel, E. R. M. Anal. Chem. 2009, 81, 10156−10161. (13) Lang, S. Q.; Bernasconi, S. M.; Früh-Green, G. L. Rapid Commun. Mass Spectrom. 2012, 26, 9−16. (14) Coplen, T. B. Rapid Commun. Mass Spectrom. 2011, 25, 2538− 2560. (15) ISO Evaluation of measurement data - Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections), JCGM 100:2008, 1st ed.; BIPM: Geneva, 2008. (16) Draper, N. R.; Smith, H. Applied Regression Analysis, 3rd ed.; Wiley: New York, 1998. (17) Kragten, J. Analyst 1994, 119, 2161−2165. Figure 2. Measurement uncertainty u(δs) for blank corrected δ13C values as a function of total amount of carbon nT. Standard samples included were peru mud, caffeine, ABS1, and NBS19. Blank quantities were estimated directly (dots), indirectly (open circles), and semiindirectly (cross).
of u(δs) was the u(δT) component, which was estimated as the standard deviation of residuals about the regression for each standard series. Variation in u(δT) may be caused by differences in sample homogeneity, which become more pronounced with decreasing sample amounts.
■
ASSOCIATED CONTENT
S Supporting Information *
An example of the procedure for computation of blank corrected values and their uncertainties. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*Phone: +46-90-7868369. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS Pratigya Polissar is gratefully acknowledged for providing me with the data set from ref 2 in digital form. I also owe a great thanks to two anonymous reviewers for their useful and encouraging comments.
■
REFERENCES
(1) Gelwicks, J. T.; Hayes, J. M. Anal. Chem. 1990, 62, 535−539. (2) Polissar, P. J.; Fulton, J. M.; Junium, C. K.; Turich, C. C.; Freeman, K. H. Anal. Chem. 2009, 81, 755−763. (3) Komatsu, D. D.; Tsunogai, U.; Kamimura, K.; Konno, U.; Ishimura, T.; Nakagawa, F. Rapid Commun. Mass Spectrom. 2011, 25, 3351−3359. (4) Fry, B.; Brand, W.; Mersch, F. J.; Tholke, K.; Garritt, R. Anal. Chem. 1992, 64, 288−291. (5) Kornfeld, A.; Horton, T. W.; Yakir, D.; Turnbull, M. H. Rapid Commun. Mass Spectrom. 2012, 26, 460−468. (6) Polissar, P. J.; Freeman, K. H.; Rowley, D. B.; McInerney, F. A.; Currie, B. S. Earth Planet. Sci. Lett. 2009, 287, 64−76. (7) Hagopian, W. M.; Jahren, A. H. Rapid Commun. Mass Spectrom. 2010, 24, 2542−2546. (8) Popp, B. N.; Hayes, J. M.; Boreham, C. J. Energy Fuels 1993, 7, 185−190. (9) Sessions, A. L.; Sylva, S. P.; Hayes, J. M. Anal. Chem. 2005, 77, 6519−6527. (10) Hwang, J.; Druffel, E. R. M. Radiocarbon 2005, 47, 75−87. (11) Panetta, R. J.; Ibrahim, M.; Gélinas, Y. Anal. Chem. 2008, 80, 5232−5239. D
dx.doi.org/10.1021/ac4003968 | Anal. Chem. XXXX, XXX, XXX−XXX
