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Clumped isotope measurements of small carbonate samples
using a high-efficiency dual-reservoir technique
Sierra V. Petersen*, Daniel P. Schrag
Harvard University, Department of Earth and Planetary Sciences, 20
Oxford Street, Cambridge, Massachusetts 02138, USA.
* Author for Correspondence: [email protected]
Supplementary Material
1
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Figure S1: Diagram of 10mL reservoir attached to reference bellows. The MAT253 has a built-in
¼” Swagelok compression fitting as the output to the reference bellows. We attach the 10mL
stainless steel reservoir (Swagelok piece # SS-4CD-TW-10) to the fused silica capillaries (~1m
length, 110m inner diameter, SGE # 0624459) using two converter pieces (1/4” male to 3/8”
female, Swagelok piece # SS-600-R-4, and 3/8” female to 1/16” female, Swagelok piece # SS600-6-1). The capillaries are connected to the 1/16” Swagelok fitting using a graphite-Vespel
composite ferrule (SGE # 072663). On the sample side, the same 2 Swagelok connectors and
10mL reservoir were used to form identical volumes from which the gas bleeds down. However,
the internal volume of the inlet valve adjacent to the sample reservoir is larger than the internal
volume of the MAT 253 valve adjacent to the reference reservoir, so 87 glass beads (borosilicate,
3mm diameter, similar to VWR#26396-630) were placed inside the sample reservoir to balance
this volume difference.
Table S1: Slopes and 1 SE estimates
from data in Figure 4 (48 vs 47),
shown for each sample type regressed
Reference Gas
individually, compared with the slope
RTG (carbonate)
found using the group fit. There is a
CM2 (carbonate)
slight dependence of the slope on the
PUCO2-1000C
average 47 composition of each
sample type, with the least clumped
All together
samples (PUCO2-1000C) having the
steeper slope, and the more clumped (Ref Gas, RTG) having a shallower slope.
Sample Type
# of Data
Points
29
76
108
21
234
Slope in
48 vs 47
0.0477
0.0448
0.0566
0.0798
0.0509
Error
(1SE)
0.0028
0.0025
0.0017
0.0098
0.0013
2
0.5
Comparison of raw and reference-frame corrected D47 vs D48
0.0
-0.5
! 47
CM2 (raw)
RTG (raw)
CM2 (RFAC)
RTG (RFAC)
-1
0
1
2
! 48
3
4
5
Figure S2: 48 vs 47-raw and vs 47-RFAC. The fractionation relationship is preserved through the
correction to the universal reference frame, with essentially no change in slope. The carbonate
data shown here are a subset of data from measurement period #2 which were run at a starting
voltage of m/z 47 = 3300-3800mV. These carbonates were corrected using only heated and
equilibrated gases run within the same voltage range. If the difference in running voltage of the
samples and gas standards were causing the observed fractionation, this reference frame
correction done with similar-sized carbonates and standard gases should remove the fractionation
and flatten out the data in this plot. The fact that the slope is unchanged suggests that differences
in running voltage between samples and standards do not cause the observed fractionation.
Line
CM2
CM2
RTG
RTG
Slope
(47-raw)
(47-RFAC)
(47-raw)
(47-RFAC)
0.0711
0.0700
0.0596
0.0582
Slope error
(1 SE)
0.0076
0.0074
0.0052
0.0051
Intercept
-0.6066
0.2694
-0.2875
0.6290
Intercept
error (1 SE)
0.0112
0.0109
0.0067
0.0066
R2
0.8973
0.8994
0.9632
0.9627
Table S2: Fitted slopes and intercepts for the four lines shown in Figure S2. The slope is
essentially unchanged by the correction to the universal reference frame, and is statistically
different from zero in all cases, suggesting that the reference frame correction does not remove
the fractionation slope, even when the gases and carbonates are run at the same voltage.
3
0.7
0.6
MP1
MP2
MP3
MP4
0.5
! 47RFAC
-0.5
-0.4
-0.3
MP1
MP2
MP3
MP4
0.2
-0.7
0.3
-0.6
! 47raw
CM2 lines over time - RF
0.4
-0.2
CM2 lines over time
0
1
2
3
4
5
! 48
RTG lines over time
6
-1
0
1
2
3
4
5
6
2
3
4
5
6
! 48
RTG lines over time - RF
0.1
-1
0.8
0.7
-0.1
! 47RFAC
0.9
MP1
MP2
MP3
MP4
-0.3
0.6
-0.2
! 47raw
0.0
MP1
MP2
MP3
MP4
-1
0
1
2
3
! 48
4
5
6
-1
0
1
! 48
Figure S3: 48 vs 47-raw and 48 vs 47-RFAC for both CM2 and RTG for the 4 measurement
periods (MP1 = 09/22/13 to 10/03/13; MP2 = 10/07/13 to 12/18/13; MP3 = 01/06/14 to 02/14/14;
MP4 = 02/18/14 to 03/28/14). The PPQ trap material was changed during MP3 and did not have a
large influence on the slope. Horizontal grey lines indicate the published value for each standard.
RTG has few replicates in MP1, resulting in a more divergent slope.
Meas. Samp
Period Type
1
CM2
#
Raw vs
pts Ref. Fr.
10 Raw
Ref. Frame
36 Raw
Ref. Frame
41 Raw
Ref. Frame
29 Raw
Ref. Frame
Slope
2
CM2
3
CM2
4
CM2
1
RTG
5
2
RTG
20
3
RTG
13
4
RTG
40
Raw
Ref. Frame
Raw
Ref. Frame
Raw
Ref. Frame
Raw
Ref. Frame
0.060
0.060
0.057
0.057
0.059
0.061
0.055
0.056
Slope Error
(1 SE)
0.003
0.003
0.004
0.004
0.003
0.003
0.003
0.003
Intercept Intercept
Error (1 SE)
-0.584
0.009
0.260
0.009
-0.592
0.008
0.297
0.008
-0.606
0.011
0.282
0.011
-0.581
0.010
0.273
0.010
0.026
0.026
0.047
0.047
0.044
0.045
0.040
0.041
0.023
0.024
0.003
0.003
0.011
0.011
0.003
0.004
-0.221
0.690
-0.283
0.654
-0.247
0.693
-0.239
0.677
0.029
0.030
0.006
0.006
0.028
0.029
0.011
0.012
Table S3: Slopes and intercepts of the lines shown in Figure S3 (48 vs 47-raw and 48 vs 47-RFAC),
with errors (1 SE). There is a slight noticeable offset between the slopes fitred to CM2 data and
that fitted to RTG.
4
4
6
! 48 (‰)
8
10
-2.2
-2.4
NBS19
0
2
4
6
! 48 (‰)
8
10
4
6
! 48 (‰)
8
10
RTG
0
! 13C (‰)
2
2
-2.30
! 13C (‰)
0
-2.15
10
-4.25
8
2
4
6
! 48 (‰)
8
10
2.00
6
CM2
1.90
4
! 48 (‰)
RTG
0
! 18O (‰)
2
-4.45
! 18O (‰)
0
2.20
2.30
CM2
2.10
! 13C (‰)
-1.8
-2.0
-2.2
! 18O (‰)
Correlations between stable isotopes and D48
NBS19
0
2
4
6
! 48 (‰)
8
10
Figure S4: 18O (left) and 13C (right) values vs 48 for all CM2 (blue), RTG (green), and NBS19
(orange) points measured over 4 different measurement periods. No significant correlation is
observed between 48 and the stable isotope ratios, unlike between 48 and 47. Plots of 13C and
18O values vs 47-RFAC, 47-corr, and 48 values look very similar because of the strong correlation
between those quantities and 48.
5
2.10
-2.0
2.0
2.5
1.0
1.5
2.0
2.5
RTG
2.0
-2.30
! 13C (‰)
1.5
-2.15
RTG
-4.25
Mass of CaCO3 (mg)
1.0
2.5
1.0
1.5
2.0
2.5
NBS19
NBS19
1.5
2.0
2.5
1.90
-2.2
1.0
2.00
Mass of CaCO3 (mg)
! 13C (‰)
Mass of CaCO3 (mg)
-2.4
! 18O (‰)
1.5
Mass of CaCO3 (mg)
-4.45
! 18O (‰)
1.0
2.25
CM2
! 13C (‰)
-1.8
CM2
-2.2
! 18O (‰)
Correlations between stable isotopes and Sample Size
1.0
Mass of CaCO3 (mg)
1.5
2.0
2.5
Mass of CaCO3 (mg)
Figure S5: 18O (left) and 13C (right) values vs sample size for all CM2 (blue), RTG (green),
and NBS19 (orange) points measured over 4 different measurement periods. There is no
correlation between the stable isotopes and sample size. Plots of 13C and 18O values vs initial
m/z 44 or m/z 47 look very similar because of the strong correlation between those quantities and
sample size.
6
1.0-1.5 mg
1.5-2.0 mg
2.0-2.5 mg
2.5-3.2 mg
0.2
0.3
! 47 (‰)
0.4
0.5
0.6
Relationship between sample mass and size of fractionation
0
2
! 48 (‰)
4
6
8
Relationship between initial V47 and D48
4
0
2
! 48 (‰)
6
1.0-1.5 mg
1.5-2.0 mg
2.0-2.5 mg
2.5-3.2 mg
1000
2000
3000
4000
Initial V47 intensity (mV)
Figure S6: 48 vs 47-RFAC (top) and Initial m/z 47 intensity vs 48 (bottom), separated by sample
size for CM2 runs over all four measurement periods. Smaller sample sizes tend to show higher
48, and therefore 47-RFAC, values, whereas larger samples tend to show lower 48 and 47-RFAC
values, although a few points do not follow this. Grey dashed lines delineate “no fractionation”,
or (48) = 0, covering a range of values for the 4 measurement periods. A wider range of 48
values would be considered uncontaminated using traditional tests (±1.5‰ around dashed lines).
7
4
0
2
! 48 (‰)
6
8
Residual of 'Yield' vs. D48 - Tuning 2
-2e-08
-1e-08
0e+00
1e-08
2e-08
Residual of Increase in Source Vac Pressure rel. to. fitted line
Figure S7: Residual yield (difference between observed yield, measured as increase in source
vacuum gauge pressure, and fitted line shown in Figure 3a) vs 48 for all carbonate data run over
all four measurement periods. There is no correlation between residual yield and 48, indicating
that partial yield is not causing the fractionation. If that were the case, we would expect to see the
highest 48 values occurring either at the highest residual yield (contaminant being added) or the
lowest (fractionation occurring during loss of some gas). Instead we see near-zero residual values
showing the highest 48.
8
0.5
! 48 vs ! 47 for reference gas runs in 3 configurations
0.2
slope=0.047
Full Inlet
No PPQ, Freeze
No PPQ, Expand
-0.1
0.0
0.1
! 47 (‰)
0.3
0.4
slope=0.111
-2
0
2
4
! 48 (‰)
6
8
10
Figure S8: 48 vs 47-raw for runs of reference gas treated as a sample under three configurations:
1) reference gas passed through the full inlet; 2) reference gas frozen into the small U-trap
(bypassing the PPQ trap); 3) reference gas expanded into the small U-trap (bypassing the PPQ
trap). When run through the full inlet, the reference gas shows a similar slope to carbonate
samples and heated gases (~0.05, see Figure 4, and Table S1). In configuration 2 (No PPQ,
Freeze), a strong relationship exists between 48 and 47, but with a slope that is twice as steep
(0.111). Configuration 3 (No PPQ, Expand) shows no significant relationship and the data points
are mostly clumped around the origin, the values that we would expect if there were no
fractionation.
9
8
10
m/z 44 before chopping vs ! 48 for ref. gas run in 3 configurations
! 48 (‰)
-2
0
2
4
6
Full Inlet
No PPQ, Freeze
No PPQ, Expand
5000
10000
Initial m/z 44 before chopping (mV)
15000
10
m/z 47 at start of run vs ! 48 for ref. gas runs in 3 configurations
! 48 (‰)
-2
0
2
4
6
8
Full Inlet
No PPQ, Freeze
No PPQ, Expand
2000
3000
4000
5000
m/z 47 at start of run (mV)
6000
7000
Figure S9: 48 vs m/z 44 before chopping, represents the amount of gas entering the U-trap and
reservoir initially (top). 48 vs m/z 47 at the start of the run, representing the running voltage, or
the size of the sample after chopping (bottom). Data is plotted for the same three run
configurations described in Figure S7. There is no clear relationship between the amount of gas
entering the reservoir initially, or the starting run voltage for the reference gas, and the magnitude
of the fractionation (how far 48 is from the “true 48” of zero), although the smallest aliquots of
reference gas run through the full inlet are not as small as our smallest carbonate samples.
10
4
1
2
3
Samples
HG/EG
0
# samples
5
Measurement Period #1
1000
2000
3000
4000
5000
6000
5000
6000
5000
6000
5000
6000
8 10
6
4
2
0
# samples
Measurement Period #2
1000
2000
3000
4000
8 10
6
4
2
0
# samples
Measurement Period #3
1000
2000
3000
4000
15
10
5
0
# samples
Measurement Period #4
1000
2000
3000
4000
Starting Voltage on mass-47 (mV)
Figure S10: m/z 47 at the start of the run for carbonate samples and calibration gases during the
four measurement periods. During all four periods, our calibration gases were run over a similar
range of ~2000-5000mV initial m/z 47, whereas our samples got smaller over the four
measurement periods. However, looking at just the samples run over the same range (subselecting the range 3300-3800mV during measurement period #2), we still see the fractionation
relationship (Figure S2).
11
0.8
CM2
0.5 0.6
0.4
0.2
0.3
! 47
! 47 (‰)
0.7
uncorrected (! 47-RFAC)
corrected (! 47-corr)
Mean ! 47-RFAC
Mean ! 47-corr
Published Value
1.1
1.0
1.5
2.0
RTG
0.7
0.8
0.9
1.0
uncorrected (! 47-RFAC)
corrected (! 47-corr)
Mean ! 47-RFAC
Mean ! 47-corr
Published Value
0.5
0.6
! 47 (‰)
cap47acr
2.5
Mass of carbonate (mg)
1.5
NBS19
2.0
Mass of carbonate (mg)
2.5
0.3
0.5
uncorrected (! 47-RFAC)
corrected (! 47-corr)
Mean ! 47-RFAC
Mean ! 47-corr
Published Value
0.1
! 47 (‰)
NBS$D47rfac
0.7
1.0
1.0
1.5
2.0
Mass of carbonate (mg)
2.5
Figure S11: 47-RFAC and 47-corr vs sample size for CM2 (top, blue), RTG (middle, green), and
NBS19 (bottom, orange). Filled symbols are the same as open symbols plotted in Figure 6.
Correction for the fractionation relationship brings points and mean values closer to published
values. Error bars account for full error propagation of original 1 SE on 47-raw through all
correction steps carried out (Reference frame for 47-RFAC and Reference frame + 48 correction
for 47-corr). The majority of the increase above the original shot noise error comes from the
Reference frame correction, with a smaller additional increase from the 48 correction.
12
Mass Bin
2.5-2.6mg
2.4-2.5mg
2.3-2.4mg
2.2-2.3mg
2.1-2.2mg
2.0-2.1mg
1.9-2.0mg
1.8-1.9mg
1.7-1.8mg
1.6-1.7mg
1.5-1.6mg
1.4-1.5mg
1.3-1.4mg
1.2-1.3mg
1.1-1.2mg
1.0-1.1mg
All CM2s
2.5-2.6mg
2.4-2.5mg
2.3-2.4mg
2.2-2.3mg
2.1-2.2mg
2.0-2.1mg
1.9-2.0mg
1.8-1.9mg
1.7-1.8mg
1.6-1.7mg
1.5-1.6mg
1.4-1.5mg
1.3-1.4mg
1.2-1.3mg
1.1-1.2mg
1.0-1.1mg
All RTGs
n
5
4
10
11
7
6
7
8
8
6
6
6
5
6
7
6
108
5
4
8
5
4
4
5
4
4
5
5
4
4
4
5
6
76
Mean 47-RFAC (‰)
Mean 47-corr (‰)
Mean 47-corr (‰)
CM2/NBS19 and RTG All 3 carbonates fitted
fitted separately
together
0.352 ± 0.032
0.386 ± 0.040
0.358 ± 0.008
0.369 ± 0.023
0.376 ± 0.029
0.380 ± 0.023
0.417 ± 0.036
0.448 ± 0.047
0.434 ± 0.029
0.428 ± 0.027
0.458 ± 0.048
0.502 ± 0.039
0.558 ± 0.038
0.503 ± 0.035
0.489 ± 0.031
0.504 ± 0.067
0.429 ± 0.010
CM2 samples
0.388 ± 0.011
0.379 ± 0.007
0.392 ± 0.005
0.391 ± 0.010
0.364 ± 0.011
0.369 ± 0.018
0.368 ± 0.011
0.376 ± 0.017
0.358 ± 0.012
0.367 ± 0.010
0.391 ± 0.014
0.385 ± 0.008
0.391 ± 0.007
0.392 ± 0.008
0.382 ± 0.014
0.354 ± 0.015
0.378 ± 0.003
0.384 ± 0.012
0.377 ± 0.007
0.390 ± 0.005
0.389 ± 0.011
0.364 ± 0.011
0.370 ± 0.017
0.370 ± 0.011
0.381 ± 0.019
0.362 ± 0.012
0.371 ± 0.010
0.395 ± 0.015
0.393 ± 0.009
0.402 ± 0.006
0.403 ± 0.009
0.396 ± 0.014
0.376 ± 0.019
0.382 ± 0.003
0.718 ± 0.007
0.742 ± 0.037
0.700 ± 0.024
0.708 ± 0.029
0.689 ± 0.016
0.806 ± 0.033
0.696 ± 0.055
0.755 ± 0.033
0.760 ± 0.014
0.814 ± 0.035
0.814 ± 0.040
0.842 ± 0.019
0.878 ± 0.042
0.841 ± 0.020
0.820 ± 0.025
0.838 ± 0.026
0.773 ± 0.010
RTG samples
0.715 ± 0.010
0.701 ± 0.008
0.708 ± 0.008
0.710 ± 0.005
0.710 ± 0.012
0.703 ± 0.013
0.691 ± 0.012
0.728 ± 0.008
0.729 ± 0.014
0.778 ± 0.019
0.756 ± 0.016
0.742 ± 0.012
0.748 ± 0.023
0.752 ± 0.009
0.717 ± 0.017
0.706 ± 0.020
0.723 ± 0.004
0.717 ± 0.012
0.689 ± 0.020
0.706 ± 0.006
0.710 ± 0.006
0.713 ± 0.012
0.690 ± 0.016
0.690 ± 0.012
0.722 ± 0.002
0.721 ± 0.014
0.770 ± 0.016
0.744 ± 0.013
0.719 ± 0.015
0.725 ± 0.026
0.736 ± 0.008
0.696 ± 0.018
0.682 ± 0.021
0.714 ± 0.004
Table S4: Binned averages (0.1mg bins) for CM2 and RTG, for data uncorrected (47-RFAC) and
corrected (47-corr) for the 48 fractionation using two methods (discussed in later section). Data
shown in Figure 6 comes from the 4th column (CM2/NBS19 and RTG fitted separately).
Comparison of columns 4 and 5 are shown in Figure S20. Column 2 shows the number of
replicates per mass bin. Errors are 1 SE of the samples in each bin.
13
-0.1
-0.3
-0.5
raw ! 47 (‰)
CM2
10
20
! 48 (‰)
30
40
50
30
40
50
-0.15
-0.05
RTG
-0.25
raw ! 47 (‰)
0.05
0
0
10
20
! 48 (‰)
Figure S12: 48 vs 47-raw for measurement period 4 showing contaminated samples compared
with clean samples. Samples that have significantly elevated 48 relative to the amount expected
for a given 47 (in other words, fall far to the right of the fractionation line), are deemed to be
contaminated. The residuals in the x-direction on theses samples are 5-27x bigger than the largest
residual for “clean” samples. These 7 samples were omitted from further calculations. This
method of finding contaminated samples runs the risk of including some “slightly contaminated”
samples in the acceptable data. This would result in lighter 47 values and hotter temperatures.
14
Summary of Corrections from 4 measurement periods:
Measurement Period #1: 09/22/13 to 10/03/14
 20 standard gases
 15 carbonates
Reference Frame Correction:
SlopeEGL = 0.00946 (±0.00123), R2 = 0.9833
SlopeETF = 1.0176 (±0.0239), IntETF = 0.9464 (±0.0125), R2 = 0.9994
Correction for 48 fractionation:
HG/EG: 48 = 0.1718 (±0.0126) * 48 – 0.5552 (±0.2107), R2 = 0.9121
CM2: 48 = 0.9362 (±0.0320) * 48 – 12.6530 (±0.5138), R2 = 0.9899 (for all carbonates)
RTG: 48 = 0.9362 (±0.0320) * 48 – 8.1456 (±0.5533)
SlopeCARB47 = 0.060 (±0.003) for CM2 (no NBS19) and 0.026 (±0.024) for RTG
6
Measurement Period #2: 10/07/13 to 12/18/13
 45 standard gases (50 minus 5 omitted – 4 had anomalously high 48 (residuals
greater than 2 s.d. outside of residuals of good points), 1 had a bad peak center)
 58 carbonates
2
-2
0
! 48
4
Figure S13: 48 vs 48 for heated and
equilibrated gases during measurement
period #2. 4 red Xs represent 4 gases
omitted for anomalously high 48.
Orange star is the gas that was omitted
for bad peak center.
-10
0
10
! 48
20
30
40
50
Reference Frame Correction:
SlopeEGL = 0.00691 (±0.00047), R2 = 0.9912
SlopeETF = 1.0091 (±0.0109), IntETF = 0.9392 (±0.0057), R2 = 0.9999
Correction for 48 fractionation:
HG/EG: 48 = 0.0998 (±0.0059) * 48 + 0.0760 (±0.1289), R2 = 0.8697
CM2: 48 = 0.9676 (±0.0098) * 48 -13.2081 (±0.1494), R2 = 0.9946 (for all carbonates)
RTG: 48 = 0.9676 (±0.0098) * 48 – 8.4263 (±0.1600)
NBS19: 48 = 0.9676 (±0.0098) * 48 – 12.3481 (±0.1685)
SlopeCARB47 = 0.057 (±0.004) for CM2/NBS19 and 0.047 (±0.003) for RTG
15
Measurement Period #3: 01/06/14 to 02/14/14
 27 standard gases (31 minus 4 omitted – 1 from partial blockage of water trap, 3
from a bad batch of CO2 blue gases)
 56 carbonates (57 minus 1 omitted for a bad peak center)
Reference Frame Correction:
SlopeEGL = 0.00599 (±0.00103), R2 = 0.9816
SlopeETF = 1.0308 (±0.0396), IntETF = 0.9376 (±0.0203), R2 = 0.9985
Correction for 48 fractionation:
HG/EG: 48 = 0.0988 (±0.0067) * 48 – 0.0339 (±0.2006), R2 = 0.8976
CM2: 48 = 0.9620 (±0.0104) * 48 -13.1040 (±0.1745), R2 = 0.9945 (for all carbonates)
RTG: 48 = 0.9620 (±0.0104) * 48 – 8.3941 (±0.1876)
NBS19: 48 = 0.9620 (±0.0104) * 48 – 12.3315 (±0.1967)
SlopeCARB47 = 0.059 (±0.003) for CM2/NBS19 and 0.045 (±0.011) for RTG
10
Measurement Period #4: 02/18/14 to 03/28/14
 33 standard gases (39 minus 6 omitted – 4 from bad batch of CO2 blue gases, 2
with anomalously high 48 (residual greater than 2 s.d. outside residuals of good
points))
 74 carbonates (87 minus 13 omitted – 7 from contamination at the end of the acid
bath (see Fig. S12), 1 from bad yield, 5 from obvious fractionation (negative 48,
and 18O and 13C values fractionated by 0.3-0.7‰))
4
-2
0
2
! 48
6
8
Figure S14: 48 vs 48 for heated and
equilibrated gases during
measurement period #4. 2 red Xs
represent 4 gases omitted for
anomalously high 48. Turquoise stars
are the gases that were from a bad
batch of CO2 blue gases.
-20
0
20
! 48
40
60
80
Reference Frame Correction:
SlopeEGL = 0.00750 (±0.00079), R2 = 0.9751
SlopeETF = 1.0335 (±0.0387), IntETF = 0.9237 (±0.0197), R2 = 0.9986
Correction for 48 fractionation:
HG/EG: 48 = 0.0935 (±0.0043) * 48 + 0.3054 (±0.1534), R2 = 0.9389
CM2: 48 = 0.9432 (±0.0099) * 48 -12.6916 (±0.1673), R2 = 0.9923 (for all carbonates)
RTG: 48 = 0.9432 (±0.0099) * 48 – 8.1828 (±0.1785)
NBS19: 48 = 0.9432 (±0.0099) * 48 – 12.0647 (±0.1846)
SlopeCARB47 = 0.055 (±0.003) for CM2/NBS19 and 0.041 (±0.003) for RTG
16
Discussion of shot noise limit with decreasing beam intensity:
Shot noise was nicely defined by Merritt & Haye [1] and the shot noise limit (),
or the smallest standard error that can be reached based on counting statistics, can be
calculated with the following equation:
2 = 2 x 106 * (R/R)2
(S1)
This can be rewritten as: 2 = 2 x 106 * (1+R)/R * (44*qe)/(t*V44)
(S2)
Figure S15: Mass of reacted
carbonate vs calculated (line) and
observed (points) shot noise limit.
0.005
shot noise limit (permil)
0.010
0.015
0.020
where R is the ratio of the number of ions collected per time unit (current) in the mass of
interest (47) relative to the most common mass (44) (=i47/i44 = 4.8x10-5), 44 is the
resistor on m/z 44 (=3x107 ohms), and qe is the charge on each ion (=1.6x10-19 C).
In the traditional dual-bellows measurement configuration, V44 is the target
voltage for the m/z 44 beam at which every cycle is measured. The quantity V44*t/qe,
where t is the total number of seconds of integration time over all cycles and acquisitions,
gives an estimate of the total number of ions collected over the whole measurement
period.
In our dual-reservoir measurement configuration, the beam intensity (and
therefore V44) is constantly changing throughout the run. In order to quantify the total
number of ions collected over the measurement period, we “integrate” the beam strength
over time. We take V44 for each cycle, multiply by 26 seconds, the integration time of a
single cycle, and then sum all the cycles. This is computationally equivalent to taking the
average V44 value over the whole run and inputting that in place of the target voltage V44
in the equation above.
We observe a linear relationship between the average V44 and the initial V47,
which we can relate to sample size by the following equations:
V44-average = 0.4197* V47-initial + 50.2
V47-initial = 2231*(Mass of CaCO3) -1714
These equations are defined based on the sensitivity levels observed during this study
(Smaller U-trap, tuning 2), and would change with changes in sensitivity. We can use
Sample Size vs. Shot noise limit (for 7acquisitions x 14 cycles)
these equations to plot the shot
noise limit against sample size,
assuming that all samples were run
Calculated from equations
for the same amount of time (7
Measured 1sigma SE on D47
acquisitions x 14 cycles x 26
second integration time). We see
our measured standard error
increasing in line with the
predicted shot noise limit at small
sample sizes.
1.0
17
1.5
2.0
mass of reacted carbonate (mg)
2.5
Discussion of Group Fit vs Individual Fit
When correcting carbonate data for the 48 correction, there are two steps in the
correction process for which you can choose a group vs individual fit: 1) solving for the
slope and intercepts of carbonate data in 48 vs 48 space and 2) solving for the slope of
carbonate data in 48 vs 47 space. In this case, we have many replicates of CM2 and
RTG in each measurement period, giving us more than enough points to get a robust fit
for each of these sample types using the individual fit method. In contrast, we have many
fewer replicates of NBS19. In many cases where we would be measuring unknowns, we
might only have 6 or fewer replicates of each unknown during a single measurement
period (like NBS19 in this study). Depending on the spread of the data, it might be
difficult to fit an individual line to so few points that accurately captures the slope of the
48 vs 47 relationship or the 48 vs 48 relationship.
In the first group vs individual fit decision (48 vs 48), the group fit is suggested
for all data sets, regardless of the number of replicates. SlopeCARB48 values from
individual fits to CM2 and RTG agree within error in 3 out of 4 measurement periods (the
exception being measurement period #3 where RTG has a steeper slope) (see Table S5).
With enough replicates, the slopes of the two individual fits converge, implying they are
representing the same true slope, and that a group fit is appropriate. For NBS19, which
has so few replicates per measurement period, the slopes are similar, but do not agree
within error. If enough NBS19 replicates were run in a single measurement period, the
individual fit would probably converge to the same slope as the other two carbonates.
Other unknown samples that we have measured show the same slope. Performing the
group fit in this case allows NBS19, and other carbonates with few replicates, to be
corrected using the true slope, as constrained by the other carbonate standards. In
addition, the error on the group fit slopes and intercepts are smaller, especially for NBS19,
and this improves the error on the calculated “true 48” value. In future studies, this step
will be conducted using the group fit method, which accurately captures the slope for all
samples, regardless of the number of replicates.
In the second group vs individual fit decision step (48 vs 47), the group vs
individual fit becomes even more important for samples with few replicates, due to
increased scatter in the relationship. In the case of a small number of replicates, the
individual fit is therefore more likely to misrepresent the true slope of the relationship.
Although the SlopeCARB47 is fairly similar (~0.05) among all sample types and fairly
constant through time (Fig. S3), we do observe a slight but consistent difference between
the individual fits of the CM2 and RTG data in 48 vs 47 (see Table S3). The slope for
RTG is ~0.04-0.045, whereas the slope for CM2 is ~0.057-0.061. This difference may be
driven by differences in their average 47 values. Gases with more clumping tend to have
a shallower slope, whereas gases closer to the stochastic distribution have a steeper slope.
In Table S1, PUCO2_1000C (a heated gas, 47-raw ~ -0.95) has the steepest slope, and the
slopes decrease as the 47 values increase (CM2, 47-raw ~ -0.5 steeper than RTG, 47-raw ~
-0.3 and Reference gas, 47-raw near zero). Therefore, when correcting unknowns, it is
prudent to choose the standard that is most similar in 47. For most biogenic carbonates,
this would be RTG, whose 47 value corresponds to earth-surface temperatures. For
NBS19, CM2 would be the most similar.
18
Below we show two examples of the group fit vs the individual fit in the case of
having 5 replicates or 2 replicates of an unknown (in this case NBS19) in a given
measurement period. We will compare three fitting methods: 1) NBS19 alone; 2) All 3
fitted together; and 3) NBS19 and CM2 fitted together, RTG fitted separately.
Meas. Sample
Per
Type
N
SlopeCARB48
(48 vs 48)
Slope Err.
(1SE)
IntCARB48
Int. Err. (1SE)
1
2
3
4
CM2
CM2
CM2
CM2
10
36
41
29
0.932
0.973
0.956
0.940
0.038
0.014
0.012
0.019
-12.587
-13.292
-13.008
-12.642
0.611
0.220
0.201
9.309
1
2
3
4
RTG
RTG
RTG
RTG
5
20
13
40
1.000
0.964
0.994
0.954
0.079
0.013
0.019
0.009
-8.781
-8.387
-8.755
-8.311
0.790
0.136
0.214
0.112
1
2
3
4
NBS19
NBS19
NBS19
NBS19
0
2
2
5
NA
0.829
1.142
0.816
NA
0
0
0.061
NA
-10.482
-14.829
-10.124
NA
0
0
0.931
1
Group fit
(no
NBS19)
Group fit
(all 3)
15
0.936
0.032
-12.653
-8.146
0.514
0.553
58
0.968
0.010
3
Group fit
(all 3)
56
0.962
0.010
4
Group fit
(all 3)
74
0.943
0.010
-13.208
-8.426
-12.348
-13.104
-8.394
-12.332
-12.692
-8.183
-12.065
0.149
0.160
0.168
0.174
0.187
0.196
0.167
0.178
0.184
2
Table S5: Slopes and intercepts of individually-fitted 48 vs 48 lines for CM2, RTG, and NBS19
during each measurement period, compared with the group-fit slopes and intercepts (bottom).
1SE errors are shown on slopes and intercepts. Errors on slope and intercept are zero when there
are only 2 points.
Measurement period #4: An unknown (NBS19) with a typical number of replicates (5)
A typical number of replicates of an unknown in one measurement period could
be ~4-6. Here we look at the case of NBS19 during measurement period #4, in which
there were 5 replicates run. For the first step (48 vs 48), there tends to be little scatter
around the carbonate lines, meaning that the individual fits are very close to the group fit,
even with few replicates (Table S5 shows the group vs individual fit slopes for this step).
In this example, we compare two scenarios: 1) NBS19 fitted alone; and 2) All three
19
carbonates fitted together. We do not separate RTG and CM2 in this case because their
SlopeCARB48 values are the same within error (Table S5).
Individual: 48 = 0.8163 (±0.061) * 48 – 10.1239 (±0.931), R2 = 0.9837
Group with CM2+RTG: 48 = 0.9432 (±0.010) * 48 – 12.0647 (±0.185), R2 = 0.9923
4
2
NBS19 replicates
Group Fit
Individual Fit
Heated + Equil. Gases
-4
-2
0
! 48 (‰)
6
8
10
48 vs 48 lines:
-20
0
20
! 48 (‰)
40
60
80
Figure S15: 48 vs 48 for NBS19 run during measurement period #4 showing two fitted lines –
one from an individual fit of just NBS19 data points, and one from the group fit including all
CM2 and RTG from measurement period #4. Black points show the heated and equilibrated gases
run during this period.
These fits in 48 vs 48 space are needed to calculate the “true 48” value, also
known as the intersection with the heated and equilibrated gas line in 48 vs 48 space
(HG/EG: 48 = 0.0919 * 48 + 0.2690). Although the fits are different, the important quantity
((48) = 0, the intersection point with the HG/EG data) is nearly identical for the two
cases. Each fit corresponds to its own intersection point, but the “true 48” values only
differ by 0.012‰. However, the error on the slope and intercept in the group fit is
reduced relative to the individual fit, which results in a smaller error on the “true 48”
value for the group fit (1SE = 0.241 vs 0.336). This difference is propagated through the
subsequent 48 correction step, resulting in larger errors on the final corrected data.
Intersection points:
Individual fit: (true 48, true 48) = (14.429, 1.655 ± 0.336)
Group fit with CM2+RTG: (true 48, true 48) = (14.559, 1.667 ± 0.241)
In the second step, where we solve for the slope in 48 vs 47 space (SlopeCARB47),
the group fit is more important than in the first step due to the increased scatter around
this relationship. Here we compare three scenarios: 1) NBS19 alone; 2) NBS19 group fit
with CM2 and RTG; and 3) NBS19 group fit with CM2 only.
48 vs 47 lines:
Individual: SlopeCARB47 = 0.056 ± 0.009
Group fit with CM2+RTG: SlopeCARB47 = 0.047 ± 0.002
Group fit with CM2 only: SlopeCARB47 = 0.055 ± 0.003
20
0.50
0.45
0.40
! 47 (‰)
0.35
NBS19 replicates
Group Fit
Individual Fit
1.0
1.5
2.0
2.5
! 48 (‰)
3.0
3.5
4.0
Figure S16: 48 vs 47 for NBS19 run during measurement period #4 showing two fitted lines,
from the individual and group fits (with CM2+RTG).
In measurement period #4 the slopes from the individual fits for CM2 and RTG
are 0.055 and 0.040, and there are 29 and 40 replicates, respectively. The group fit with
all three sample types therefore has a slope intermediate between these (0.047), driven by
the relative weighting of the sample types. This slope is shallower than the NBS19 or
CM2 individual fit slopes (0.056 and 0.055), or their combined fit (0.055). The combined
CM2/NBS19 slope is very similar to each sample type’s individual fit due to their similar
47 values, but the error is much larger on the individual fit. This larger error would be
propagated through the correction, resulting in a final error 0-0.006‰ higher on the
individual-fit points. The similarity between the fits of NBS19 and CM2 suggests that the
5 replicates of NBS19 are enough to properly capture the slope of the 48 vs 47
relationship. This is aided by the 3‰ spread in 48. A group of 5 replicates with a very
narrow range in 48 would be less likely to properly represent the slope.
We can use these different slopes to correct the NBS19 data and compare the
results.
48
47-RFAC
47-corr
(Individual Fit)
47-corr (Group
Fit- CM2+RTG)
47-corr (Group
Fit- CM2 only)
SlopeCARB47 #1
1.022
0.328
0.056
0.364
#2
1.502
0.378
0.387
#3
2.431
0.385
0.342
#4
2.729
0.414
0.355
#5
Mean 1 SE
4.151
0.517 0.404 0.031
0.378 0.365 0.008
0.047
0.358 0.386 0.349
0.364
0.400 0.371
0.009
0.055
0.363 0.387 0.343
0.356
0.380 0.365
0.008
Table S6: Comparison of raw NBS19 data (48 and 47-RFAC) with data corrected with each of the
three fit methods (47-corr) for measurement period #4.
All three methods of correction result in a mean value closer to the published
value of 0.373 ± 0.007‰ [Dennis et al., 2011], and a large reduction in the error of the
mean (0.031 down to 0.008-0.009) relative to the uncorrected data. The individual fit and
21
the group fit with CM2 only result in the same mean and SE, due to the very similar
SlopeCARB47 values between these two fits. This shows that performing a group fit with a
standard of similar 47 does not introduce any biases. When the shallower slope of the
CM2/RTG/NBS19 group fit is used, the resulting mean is higher. This shows that if an
incorrect SlopeCARB47 is used, such as one calculated from a carbonate standard of
different 47 composition, the resulting corrected data can be skewed away from the true
value. The biggest difference between the two methods is seen in points with the largest
magnitude of fractionation (ie. 48 farthest from the “true 48”).
In this example, the individual fit shows good agreement with the two group fit
methods, showing that 5 replicates can be enough to be fit reliably with the individual fit
method.
NBS19 during measurement period #3:
An unknown with a very small number of replicates (2)
4
2
NBS19 replicates
Group Fit
Individual Fit
Heated + Equil. Gases
-4
-2
0
! 48 (‰)
6
8
10
Typical clumped isotope measurements have 3-6 replicates of an unknown.
However, sometimes not all of these replicates will be conducted in the same
measurement period. In this case, we would need to correct a small number of replicates
with a given heated gas line. Here we show an example of a case where only 2 replicates
of NBS19 were measured in a given measurement period.
-20
0
20
! 48 (‰)
40
60
Figure S17: 48 vs. 48 for NBS19 run during measurement period #3 showing two fitted lines –
one from an individual fit of just NBS19 data points, and one from the group fit including all
CM2 and RTG from measurement period #3. Black points show the heated and equilibrated gases
run during this period.
Although there are only two points, the lines fitted in 48 vs. 48 space using the
group and individual fits are fairly similar. The error on the slope and intercept of the
individual fit line is zero because there are only 2 points to fit.
48 vs. 48 lines: Individual: 48 = 1.142 (±0) * 48 – 14.829 (±0), R2 = 1
Group fit with CM2 + RTG: 48 = 0.9620 (±0.010) * 48 – 12.3315 (±0.169), R2 = 0.9941
22
Calculating the intersection points with the heated and equilibrated gas line (HG/EG: 48 =
0.0988 * 48 – 0.0339), we see that the intersection points agree within error.
Intersection points:
Individual fit: (true 48, true 48) = (14.186, 1.368 ± 0.320)
Group fit with CM2 + RTG: (true 48, true 48) = (14.225, 1.374 ± 0.328)
0.340
0.335
! 47 (‰)
0.345
In this case the error for the individual fit is smaller because there is no error on
the slope and intercept of the carbonate line. In general, the group fit would have a
smaller error because of the larger number of points used to fit. The agreement between
the “true 48” calculated by the two fits is aided by both replicates being very close to the
heated gas line, despite the slopes being more different than in the first example.
0.330
NBS19 replicates
Group Fit
Individual Fit
0.8
1.0
! 48 (‰)
1.2
1.4
Figure S18: 48 vs. 47 for NBS19 run during measurement period #3 showing two fitted lines,
from the individual and group fits with CM2+RTG.
The 48 vs. 47 fit in this case demonstrates how the individual fit to a small
number of replicates can wildly misrepresent the slope of the relationship. In this case,
we can see that the two NBS19 points do not form a line that reflects the fractionation we
know occurs (Slope = ~0.05). The slope of the individual fit is -0.024 – the wrong
direction entirely! In measurement period #3 the slopes from the individual fits for CM2
and RTG are 0.059 and 0.044, and there are 41 and 13 replicates, respectively. The group
fit with all three is dominated by CM2, resulting in a slope very close to the individual
CM2 fit. The group fit with CM2 alone has a slope identical to the CM2 individual slope,
due to the miniscule influence of the 2 NBS19 points on the 41 CM2 points. The wild
divergence between the NBS19 individual fit and the known slope of ~0.05, or with the
slope calculated by either of the group fits shows that two points is not enough to
accurately capture the slope of the 48 vs. 47 relationship, given the scatter observed
around that line.
48 vs. 47 lines:
Individual: SlopeCARB47 = -0.024 ± 0.000
Group fit with CM2+RTG: SlopeCARB47 = 0.056 ± 0.003
Group fit with CM2 only: SlopeCARB47 = 0.059 ± 0.003
We can use these different slopes to correct the NBS19 data and compare the
results.
23
SlopeCARB47
48
47-RFAC
47-corr (Individual Fit) -0.024
0.056
47-corr (Group Fit –
CM2+RTG)
0.059
47-corr (Group Fit –
CM2 only)
#1
0.658
0.348
0.331
0.388
#2
1.407
0.329
0.330
0.327
Mean
1 SE
0.339
0.330
0.358
0.010
0.000
0.031
0.390
0.327
0.359
0.032
Table S7: Comparison of raw NBS19 data (48 and 47-RFAC) with data corrected with each of the
two fits (47-corr) for measurement period #3.
Both group fits brings the mean value of these two replicates much closer to the
published value (0.373 ± 0.007‰[2]). The individual fit in this case actually shifts the
mean farther away from the published value. This shows that for a very small number of
replicates, the group fit does substantially better than the individual fit. It is possible that
you could have a case where two replicates form a line that perfectly matches the true
slope but, given the scatter that we see around this relationship, it is unlikely that such a
small number of replicates will properly capture the slope on their own. The two group
fits have very similar results because CM2 replicates dominate the group fit with all three
sample types.
We can also calculate an individual fit for NBS19 replicates measured during
measurement period #2, which also had 2 replicates. Combining all 9 replicates of
NBS19 over three measurement periods, we get a mean value of 0.357 ± 0.007‰ for the
individual fits compared with a mean of 0.366 ± 0.007‰ for the group fit with CM2 only
and a mean of 0.368 ± 0.007‰ for the group fit with CM2 and RTG. Both the group fits
are closer to the published value of 0.373 ± 0.007‰, showing the importance of the
group fit for samples with a small number of replicates.
For all future unknowns that have less than 5-6 replicates in a single measurement
period, or if the spread of data points does not define a clear slope, a group fit of some
kind should be performed. In all cases, the relationship between 48 and 47 should be
independently assessed for unknown sample types before choosing the appropriate groupfit pairings. Ideally, each sample would have enough replicates to fit independently.
Another possibility is to do a group fit of all unknowns and no carbonate standards. This
could work well if there are a few unknowns with more replicates (>5) that can drive the
slope to correct other unknowns with fewer replicates. If it is necessary to include a
carbonate standard, the group fit should be performed with the standard of nearest 47
value (CM2 in this case, RTG in the case of low-temperature samples).
Samples with many replicates: Group fit vs Individual fit?
Performing a group fit on multiple types of carbonates implies that they are all
following the same slope. In 48 vs 48 space, the slopes of the CM2 and RTG agree
24
within error (Table S5), justifying the group fit in this first step. However, we observe a
slight but consistent difference between the individual fits of the CM2 and RTG data in
48 vs 47 space (see Table S3). The slope for RTG is ~0.044, whereas the slope for CM2
is ~0.057. In this section we compare CM2 and RTG data processed using two different
fitting methods: 1) CM2 and RTG fitted independently (no NBS19) vs 2) all three fitted
together. As seen above, samples with a small number of replicates benefit greatly from
the group fit to define the slope, especially in 48 vs 47 space, but what happens when
you group-fit two data sets that both have many replicates and have disparate individual
slopes?
Table S8 shows a comparison of the 48 vs 47 slopes for the 2 fitting methods.
CM2 and RTG show consistently different individual-fit slopes, with RTG always having
a shallower slope. The group-fit slope is intermediate between the two individual slopes
and the magnitude of the group-fit slope is driven by the relative number of replicates of
each sample type in a given measurement period (shown in parentheses) and by the
magnitude of the fractionation of those points (the leverage).
Fit Method
CM2 alone
3 together
RTG alone
MP1
Slope (# samples)
0.060 (10)
MP2
Slope (# samples)
0.057 (36)
MP3
Slope (# samples)
0.059 (41)
MP4
Slope (# samples)
0.055 (29)
±0.003
±0.004
±0.003
±0.003
0.060 ±0.005
0.026 (5)
0.052 ±0.003
0.047 (20)
0.056 ±0.003
0.045 (13)
0.047 ±0.002
0.041 (40)
±0.024
±0.003
±0.011
±0.003
Table S8: Slopes and 1SE errors for the group fit (all 3 together) vs the RTG and CM2 individual
fits. Number of replicates per measurement period are shown in parentheses.
The individual fit RTG slope is always shallower than the group fit, and the CM2
slope is always steeper. This results in the group fit skewing RTG data too light and CM2
data too shallow (Fig. S19). The difference between the 47 values resulting from the two
methods scales with 1) the difference between the group and individual fit slopes in that
measurement period, and 2) the magnitude of the fractionation (difference between 48
and “true 48”). In Fig. S19, the group and individual fit correction arrows diverge as the
magnitude of the correction gets larger.
Correction for one sample:
47-corr = 47-RF/AC – (48- true48) * SlopeCARB47
Difference between two methods:
47-corr(Indv) - 47-corr(Grp) = (48- true48) * (SlopeCARB47(Indv) - SlopeCARB47(Grp))
25
1(47), "&8 9: &<<<<<<<<<<<<<<<<<<<<<<<<<<&7, 0/(, "&8 9: &
8 9: &
' #"&! "# 3&4"#$%&5)&&
4(/, 6&1(47), "&8 9: &
! "#$%&'()&*(+, &
-+. (/(. $01&'()&*(+, &2#"&! "# $
-+. (/(. $01&'()&*(+, &2#"&%& ' &
! "#$%&'()&*(+, &
' #"&%& ' 3&4"#$%&5)&&
4(/, 6&7, 0/(, "&8 9: &
8 9; &
MP1
MP2
MP3
MP4
0.30
CM2
0.35 0.40 0.45
Group Fit
RTG and CM2 fit separately
0.65
0.70
0.75
0.80
RTG and CM2 fit separately
0.30
0.35
0.40
0.45
0.50
Figure S19: Schematic of group fit vs individual fit slopes, and the resulting corrected values for
CM2 and RTG. This direction of change (steeper slope = lighter 47) is applicable to all sample
types.
0.50
MP1
MP2
MP3
MP4
0.65
RTG
0.70
0.75
Group Fit
0.80
Figure S20: Cross-plot of 47-corr, calculated using the group fit vs the individual fit for CM2
(left) and RTG (right), separated by measurement period. A 1:1 line is plotted for reference.
Figure S20 shows a point-by-point comparison for CM2 and RTG data fitted in
two ways, separated by measurement period. The largest differences occur during
measurement period #4. In this period, the individual slopes for CM2 and RTG are the
most divergent (Table S8), resulting in the largest shifts between the two methods. In
addition, many of the smallest samples (<1.5mg) were run during this measurement
period. Smaller samples tend to have higher 48 values (ie. magnitude of fractionation),
which contributes to the larger difference observed in this measurement period.
Using the group fit instead of the individual fit shifts the mean value for RTG
from 0.723 ± 0.004‰ to 0.714 ± 0.004‰. For CM2, the group fit shifts the mean value
26
from 0.378 ± 0.003‰ to 0.382 ± 0.003‰. This is in line with the schematic shown in
Figure S19, which predicted lighter values for RTG and heavier values of CM2 in the
group fit.
Figure S21 shows the mass-binned averages for each of the two methods. The
largest shifts are seen in the smallest mass bins because 1) many of these smallest
samples have higher 48 and require larger corrections, so differences in SlopeCARB47 are
magnified and 2) most of the smallest samples were run in measurement period #4 where
the difference in SlopeCARB47 values was greatest. We cannot separate the influences of
these two effects.
This comparison shows the influence of group fitting with a standard of different
slope. In future studies of unknowns, all effort should be made to identify the true slope
of the unknown sample type individually, through samples with a larger number of
replicates. Sensitivity tests should be performed to determine the influence of differing
SlopeCARB47 values for various fitting methods. To account for possibly getting the slope
wrong when measuring unknowns, the error on SlopeCARB47 could be artificially
increased in error propagation calculations. In this study, the errors on calculated
SlopeCARB47 values obtained using the RTG and CM2/NBS19 fits were ~0.002-0.005‰
over the four measurement periods. This is similar to the mean difference between group
and individual fit slopes (0.006‰, excluding RTG in measurement period #1). Increasing
the error on SlopeCARB47 by a factor of two increases the uncertainty on each point by an
average of 0.001‰ (0.002‰ for samples smaller than 1.5mg). Whenever possible,
individual fits are preferred, but only when enough replicates are present to reliably fit a
slope.
0.42
0.38
group fit
individual fit
0.34
! 47-corr (‰)
CM2
0.72
0.76
group fit
individual fit
0.68
! 47-corr (‰)
RTG
1.0
1.5
2.0
2.5
Mass of carbonate (mg)
Figure S21: Comparison of binned averages for the CM2 and RTG data from all four
measurement periods calculated using either the group fit or for the case where RTG and CM2
were fitted separately. Group fit data for this plot are shown in Table S4. Individual fit data for
RTG is also in Table S4. The individual fit CM2 data are very similar to the CM2/NBS19 data
shown in Table S4 (the few NBS19 replicates in that group fit do not have much influence).
27
Error Propagation in the Clumped Isotope Correction:
The correction from a raw 47 value to a fully-corrected 47 value is made up of a few
steps. This includes 3 steps to get from the raw value to a value in the absolute reference
frame[2], as well as 2 steps to correct for the “unknown fractionation” with 48 observed
in our samples. All the data and regression outputs that go into these corrections have
errors that need to be propagated.
To get from the raw value to the absolute reference frame value takes 3 steps:
1. 47-SGvsWG0 = 47-raw - 47raw* SlopeEGL
(S1)
2. 47-RF = 47-SGvsWG0 * SlopeETF + IntETF
(S2)
3. 47-RF/AC = 47-RF + Acid Frac. Factor
(S3)
To correct the absolute reference frame value for the 48 fractionation takes 2 steps:
4. true48 = (IntCARB48 * SlopeHG/EG – IntHG/EG * SlopeCARB48) * (SlopeHG/EG –
SlopeCARB48)
(S4)
5. 47-corr = 47-RF/AC – (48- true48) * SlopeCARB47
(S5)
We begin with the raw carbonate data points, containing the following values and errors:
47 +/- se47
47 +/- se47
48 +/- se48
48 +/- se48
where the delta values and errors represent the mean and standard error of all cycles and
acquisitions in one sample run (7acq x 14 cyc = 98 points to average per sample).
We also have the outputs of two regressions, the equilibrium gas lines (EGLs) and the
empirical transfer function (ETF). A statistical program such as R will output the
standard error of the slope and intercept estimates with the regression information. We
also take the published value and error for the acid fractionation factor of your choice
(based on reaction temperature).
SlopeEGL+/- seSlpEGL
SlopeETF +/- seSlpETF
IntETF +/- seIntETF
Acid Fractionation Factor +/- erAcidFr
For the 48 correction, we also need the slope and intercept of the heated gas (HG/EG)
and carbonate (CARB48) data in 48 vs 48 space, as well as the regression of 48 vs 47RF/AC for carbonates (CARB47). These give us the following additional values:
SlopeHG/EG+/- seSlpHGEG
IntHG/EG+/- seIntHGEG
SlopeCARB48+/- seSlpCARB48
IntCARB48+/- seIntCARB48
SlopeCARB47+/- seSlpCARB47
To propagate the errors I will use the two following formulae, which are basic definitions
of error propagation.
For z = x + y
For z = x*y
dz = SQRT(dx^2 + dy^2)
dz = x*y*SQRT((dx/x)^2 + (dy/y)^2)
28
(S6)
(S7)
Step 1: Linearity correction
47-SGvsWG0 = 47-raw - 47raw* SlopeEGL
(S1)
To propagate the errors in this set of operations, we combine Eqns. S6 and S7 in the
correct order to get:
err47-SGvsWG0 = SQRT[ se472 + (47raw* SlopeEGL)2 *( (se47/47raw)2 +
(seSlpEGL/SlopeEGL)2) ]
(S8)
Step 2: ETF correction
47-RF = 47-SGvsWG0 * SlopeETF + IntETF
(S2)
We combine Eqns. S6 and S7 in the correct order, and input the error calculated in Eqn.
S8 (err47-SGvsWG0). We thus get:
err47-RF = SQRT[ seIntETF2 + (47-SGvsWG0 * SlopeETF)2 * ( (err47-SGvsWG0/47-SGvsWG0)2
+ (seSlpETF/SlopeETF)2 ) ]
(S9)
Step 3: Acid fractionation correction
47-RF/AC = 47-RF + Acid Frac. Factor
(S3)
We combine Eqns. S6 and S7 in the correct order, and input the error calculated in Eqn.
S9 (err47-RF ). We thus get:
err47-RF/AC = SQRT [ err47-RF2 + errAcidFr2 ]
(S10)
Combining Eqns. S8, S9, and S10, we can create one equation for the error on 47-RF/AC
that contains only known quantities. All the inputs to this equation should be known from
original data, regression outputs, or literature values (erAcidFr).
err47-RF/AC = SQRT [ ( seIntETF2 + (47-SGvsWG0 * SlopeETF)2 * ( ( se472 + (47raw*
SlopeEGL)2 *( (se47/47raw)2 + (seSlpEGL/SlopeEGL)2) )/47-SGvsWG02 +
(seSlpETF/SlopeETF)2 ) ) + errAcidFr2 ]
(S11)
Step 4: Calculate true48
true48 = (IntCARB48 * SlopeHG/EG – IntHG/EG * SlopeCARB48) * (SlopeHG/EG – SlopeCARB48)
(S4)
The value true48 is the y-coordinate (48) of the intersection between the heated gas and
carbonate data in 48 vs 48 space. Because this is the intersection of two lines with errors
on their slopes and intercepts, it is a more complicated calculation to get errTrue48, and
this is discussed below.
29
Step 5: Correction for 48 fractionation
47-corr = 47-RF/AC – (48-raw- true48) * SlopeCARB47
(S5)
We combine Eqns. S6 and S7 in the correct order, and input the error calculated in Eqns.
S10 or S11 (err47-RF/AC). We thus get:
err47-corr = SQRT [ err47-RF/AC2 + ((48-raw- true48) * SlopeCARB47)2 * ( (se482 +
errTrue482)/(48-raw- true48)2 + (seSlpCARB47/SlopeCARB47)2 ) ]
(S12)
Combining e Eqns. S11 and S12, we can create an equation for the error on 47-corr, the
fully corrected value that contains only known quantities. Everything in this equation
should be one of the given values at the start, except for errTrue48.
err47-corr = SQRT [ ( ( seIntETF2 + ((47-raw - 47raw* SlopeEGL)* SlopeETF)2 * ( ( se472 +
(47raw* SlopeEGL)2 *( (se47/47)2 + (seEGL/SlopeEGL)2) )/(47-raw - 47raw* SlopeEGL)2 +
(seSlpETF/SlopeETF)2 ) ) + errAcidFr2 ) + ((48-raw- (IntCARB48 * SlopeHG/EG – IntHG/EG *
SlopeCARB48) * (SlopeHG/EG – SlopeCARB48)) * SlopeCARB47)2 * ( (se482 + errTrue482)/(482
raw- (IntCARB48 * SlopeHG/EG – IntHG/EG * SlopeCARB48) * (SlopeHG/EG – SlopeCARB48)) +
2
(seSlpCARB47/SlopeCARB47) ) ]
(S13)
Error in the intersection of two fitted lines
We have two lines in 48 vs 48 space, one for heated and equilibrated gases (HG/EG) and
one for carbonate (CARB48). These are normal regressions, and we get the standard error
on the slope and intercept from the regression output (for example from the summary
function in R):
SlopeHG/EG+/- seSlpHGEG
IntHG/EG+/- seIntHGEG
SlopeCARB48+/- seSlpCARB48
IntCARB48+/- seIntCARB48
For now, let us make the notation simpler and say that we have two lines:
Line 1: y = a*x + c
Line 2: y = b*x + d
These lines intersect at the point (X,Y), such that X = (d-c)/(a-b) and Y = (a*d-b*c)/(a-b).
The lines are both linear regression fits with errors on the slope and intercept: a, b, c,
d. Each fit also has a covariance coefficient, which is between 0 and 1 and can be
calculated as follows:
r = cov(x,y)/ [ sd(x) * sd(y) ]
(S14)
48
where x and y in Eqn. S14 are the vectors of data ( and 48) for either HG/EG or the
carbonates which were used for the regression. This r is the same as the square root of the
R2 value given by the regression output. We can calculate this for each of the two lines
and call them r1 and r2.
We can then write the error matrix (covariance matrix) for the intersection point (X,Y)
30
é s 2
X
ê
ër * sXsY
r * sXsY ù
ú= N
sY 2 û
(S15)
We can write the covariance matrix comparing a, c, b, and d. Note: We are assuming that
the two lines are independent. Since the lines come from different sample runs (gas
standards or carbonates), this is a fairly good assumption.
é s2
ù
r1* sa * sc
0
0
a
ê
ú
sc 2
0
0
êr1* sa * sc
ú=M
(S16)
2
ê
0
0
sb
r2 * sb * sd ú
ê
ú
0
0
r2 * sb * sd
sd 2
ë
û
We also need a transformation matrix, which calculates the partial derivatives of x and y
with each of the parameters a, b, c, d.
é ¶x ¶y ù
ê ¶a ¶a ú
é-k -b* k ù
ê ¶x ¶y ú
ú
ê
ú æ 1 öê -1
-b ú
¶
c
¶
c
ê
ú =ç
T= ê
÷
a* k ú
ê ¶x ¶y ú è a - bøê k
ê
ú
ê ¶b ¶b ú
a û
ë +1
ê ¶x ¶y ú
êë¶d ¶d úû
where k = (d-c)/(a-b), for simplicity.
(S17)
We can then use the matrix multiplication TTMT = N to solve for x and y. They have
the following solution, after matrix multiplication and algebra…
X2 = 1/(a-b)2 * [k2 * (a2 + b2) + 2*k*(r1*a*c + r2*b*d) + c2 + d2)
(S18)
Y2 = 1/(a-b)2 * [k2 * (b2*a2 + a2*b2) + 2*k*(r1*b2*a*c + r2*a2*b*d) + b2*c2 +
a2* d2)
(S19)
If we now translate this back into our clumped isotope calculation, we only really care
about Y, which is equal to errTrue48, or the error on the y-value at the point of
intersection. X represents the error in the x-value (48) at the point of intersection.
Letting Line 1 be the HG/EG line and Line 2 be the CARB48 line, we can make the
following substitutions:
a = SlopeHG/EG
a = seSlpHGEG
c = IntHG/EG
c = seIntHGEG
b = SlopeCARB48
b = seSlpCARB48
d = IntCARB48
d = seIntCARB48
Finally, we just plug these values into Eqn. S19 and solve for Y. This is then plugged
into Eqn. S13 along with other known quantities to solve for our final product, err47-corr,
or the error on the fully corrected 47 value (47-corr).
31
References
1 D. A. Merritt, J. M. Hayes. Factors controlling precision and accuracy in isotope-ratiomonitoring mass spectrometry. Anal. Chem. 1994, 66, 2336.
2 K. J. Dennis, H. P. Affek, B. H. Passey, D. P. Schrag, J. M. Eiler. Defining an absolute
reference frame for ‘clumped’ isotope studies of CO2. Geochim. Cosmochim. Acta 2011,
75, 7117.
32