Calculation of Expanded Uncertainties for CO_{2}
Last update: February 2017 (Brad Hall)
The general method for estimating the expanded (total) uncertainty of the X2007 CO_{2} mole fraction scale is outlined here.
Definitions
ppm = µmol mol^{-1}primary = CO_{2} primary standard (modified natural air)
standard uncertainty = approx. 1-sigma, (68% confidence level)
expanded uncertainty = approx. 2-sigma, (95% confidence level)
X = mole fraction in dry air
Φ = volume ratio (large volume relative to small volume)
P = pressure
T = temperature
uξ = standard uncertainty estimated for the quantity ξ (T, P, Φ, etc)
Measurement Equation:
The mole fraction of CO_{2} in a sample of dry air is calculated as:
where R is the gas constant, β is the second Virial coefficient, Φ is the volume ratio, and X_{N2O} is the mole fraction of N_{2}O.
Estimating the total uncertainty involves the following steps:
- Estimate uT and uP applicable to a volume ratio experiment. a. Here we use the “metRology” software developed by NIST, which calculates partial derivatives and performs the error propagation for a given mean value and standard uncertainty of each variable in a measurement equation. The software accounts for correlations among variables, which were determined separately following JCGM (2008). An alternate software tool, available at http://uncertainty.nist.gov/, gives similar results.
- Calculate uΦ.
- Estimate the uncertainties of pressure (uP) and temperature (uT) measurement during a manometer run.
- Calculate uCO_{2} for a single manometer run and then include typical repeatability of manometer runs of the same primary standard. This gives the standard uncertainty for analysis of a primary standard with mole fraction XCO_{2}.
- Combine results from 4 with an estimate of the scale transfer uncertainty (reproducibility).
Steps 1-2
Uncertainties associated with pressure measurement are estimated from pressure calibrations performed using a piston gauge (primary pressure standard), manufacturer’s specifications, and variability of the pressure zero reading at vacuum.
Uncertainties on temperature measurement are estimated from calibration certificates obtained by accredited calibration facilities, and variability among different sensors placed at the same location inside a temperature-controlled oven. The repeatability associated with temperature measurement using a single probe is small and is ignored.
The volume ratio (VR) is calculated from successive expansions of air in four steps, bridging the difference between the large and small volumes.
Φ = r1*r2*r3*r4 - r1*r2*r3 + r1 (2)
Where r1, r2, r3, and r4 are intermediate volume ratios determined at each expansion step. Typical central values (mean) and standard uncertainties are shown in Table 1.
Table 1: Parameters of a typical series of volume expansions, P1 ==> P2, used to determine the volume ratio.P1 (kPa) | uP1 (kPa) | P2 (kPa) | uP2 (kPa) | ri | uri |
---|---|---|---|---|---|
80 | 0.0020 | 20 | 0.0015 | 4.15 | 0.00036 |
80 | 0.0020 | 10 | 0.0012 | 7.95 | 0.00109 |
80 | 0.0020 | 18 | 0.0015 | 4.53 | 0.00043 |
80 | 0.0020 | 12 | 0.0012 | 6.85 | 0.00074 |
Using the “metRology” software we calculate the standard uncertainty on the volume ratio to be 0.206 for a volume ratio of 880.10, or ~0.023%.
Step 3
For a manometer run, we estimate uT_air = 0.009 deg C, and uT_CO_{2} = 0.015 deg C. uT_CO_{2} is larger than uT_air because the oven temperature is disturbed upon freezing the CO_{2} sample at liq-N_{2} temperature, and takes some time to equilibrate. T_air is determined from the average of 3 PRTs and one thermistor, while T_CO_{2} is determined from the average of 2 PRTs. For pressure, we estimate uP_air = 0.0025 kPa and uP_CO_{2} = 0.0022 kPa. There is a slight difference here because P_air is determined near the upper range of the pressure calibration, while P_CO_{2} is closer to midrange. These uP are likely overestimated based on current run procedures, but we use them here because the CO_{2} scale is based on several years of data, so we want to estimate “typical” values from the past decade for this calculation. We also include uncertainty in the second Virial coefficient, β, taken from Zhao and Tans (1997).
Step 4
Using central values and standard uncertainties for T, P, VR, and β for typical measurement conditions (again, using the “metRology” software), we estimate the standard uncertainty for a single manometer run as a function of CO_{2} mole fraction. We then combine this estimate with the repeatability of manometer results (8 episodes over 20 years), and the analytical reproducibility.
Table 2: Uncertainty Budget at nominal 400 µmol/molVariable | Typical value | Standard uncertainty | unit | Contribution to uncertainty | Relative contribution to uncertainty |
---|---|---|---|---|---|
P_{air} | 83 | 0.0025 | kPa | 1.198E-08 | 7% |
P_{CO2} | 30 | 0.0022 | kPa | 3.018E-08 | 17% |
T_{air} | 310 | 0.009 | K | 1.152E-08 | 6% |
T_{CO2} | 310 | 0.015 | K | 1.920E-08 | 11% |
Φ (vol. ratio) | 880.10 | 0.206 | dimensionless | 9.400E-08 | 53% |
Β_{air} | -5.87 | 0.2 | cm^{3}/mol | 8.940E-14 | 0% |
Β_{CO2} | -112.8 | 0.2 | cm^{3}/mol | 2.560E-13 | 0% |
X_{N2O} | 0.325 | 0.0005 | nmol/mol | 2.441E-12 | 0% |
repeatability, N=8 | 0.035 | μmol/mol | 7.813E-09 | 5% | |
analytical reproducibility* | 0.01 | µmol/mol | 1.2500E-09 | <1% | |
* based on less than one year of data |
Step 5
Combining uncertainties from Table 2, we obtain an estimate for the standard uncertainty. Repeating this exercise at different mole fraction we obtain the results shown in Table 3, and then multiply by coverage factor k=2 to obtain the expanded uncertainty (Figure 1).
CO_{2} (ppm) | Standard uncertainty (ppm) |
---|---|
250 | 0.079 |
300 | 0.090 |
350 | 0.100 |
400 | 0.110 |
450 | 0.120 |
500 | 0.132 |
550 | 0.142 |
References
- JCGM 100:2008 Evaluation of Measurement Data – Guide to the Expression of Uncertainty in Measurement (ISO GUM 1995 with minor corrections), Joint Committee for Guides in Metrology (2008); http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf
- Tans et al (2015), “History of WMO CO_{2} X2007 scale: long-term reproducibility”, presented at the WMO/IAEA GGMT Meeting, Sept. 13-17, 2015; La Jolla, CA.
- Zhao, C., P.P. Tans, and K.W. Thoning (1997), A high precision manometric system for absolute calibrations of CO_{2} in dry air, J. Geophys. Res. 102, 5885-5894.
- Zhao, C., and P.P. Tans (2006), Estimating uncertainty of the WMO Mole Fraction Scale for carbon dioxide in air, J. Geophys. Res. 111, D08S09, doi: 10.1029/2005JD006003.