Estimating uncertainty of the WMO mole fraction scale for carbon dioxide in air

Authors


Abstract

[1] The current WMO CO2 Mole Fraction Scale consists of a set of 15 CO2-in-air primary standard calibration gases ranging in CO2 mole fraction from 250 to 520 μmol mol−1. Since the WMO CO2 Expert Group transferred responsibility for maintaining the WMO Scale from the Scripps Institution of Oceanography (SIO) to the Climate Monitoring and Diagnostics Laboratory (CMDL) in 1995, the 15 WMO primary standards have been calibrated, first at SIO and then at regular intervals, between 1 and 2 years, by the CMDL manometric system. The uncertainty of the 15 primary standards was estimated to be 0.069 μmol mol−1 (one-sigma) in the absolute sense. Manometric calibrations results indicate that there is no evidence of overall drift of the Primaries from 1996 to 2004. In order to lengthen the useful life of the Primary standards, CMDL has always transferred the scale via NDIR analyzers to the secondary standards. The uncertainties arising from the analyzer random error and the propagation error due to the uncertainty of the reference gas mole fraction are discussed. Precision of NDIR transfer calibrations was about 0.014 μmol mol−1 from 1979 to present. Propagation of the uncertainty was calculated theoretically. In the case of interpolation the propagation error was estimated to be between 0.06 and 0.07 μmol mol−1 when the Primaries were used as the reference gases via NDIR transfer calibrations. The CMDL secondary standard calibrations are transferred via NDIR analyzers to the working standards, which are used routinely for measuring atmospheric CO2 mole fraction in the WMO Global Atmosphere Watch monitoring program. The uncertainty of the working standards was estimated to be 0.071 μmol mol−1 in the one-sigma absolute scale. Consistency among the working standards is determined by the random errors of downward transfer calibrations at each level and is about 0.02 μmol mol−1. For comparison with an independent absolute scale, the five gravimetric standards from the National Institute for the Environmental Studies (NIES) in Tsukuba, Japan, ranging in CO2 mole fraction from 350 to 390 μmol mol−1 have been calibrated relative to the CMDL secondary standards. The average and standard deviation of the differences between the NIES gravimetric and CMDL analyzed CO2 mole fraction are 0.004 ± 0.03 μmol mol−1.

1. Introduction

[2] A better understanding of the global carbon cycle is central to resolving uncertainties surrounding the future rate of global change. A major tool in the study of the carbon cycle is provided by the detailed temporal and large-scale spatial patterns of the mole fraction of carbon dioxide (CO2) and its stable isotopic composition in the atmosphere. The atmospheric burden of CO2 has been monitored at the Mauna Loa observatory on Hawaii for nearly half a century, and since then the observing system has been expanded to many sites worldwide [e.g., Keeling et al., 1986; Conway et al., 1988]. In order to use atmospheric transport models for regional-scale CO2 source and sink determination, which is an essential component of the carbon cycle study, it is necessary to combine the measurements made by different laboratories into an international measurement database to expand spatial and temporal coverage. A fundamental requirement for such a program is careful and continuing calibration for CO2 as well as its isotopes. Without a solid and precise link to a common calibration scale, most of the global measurements would be useless in meeting the goals of the carbon budget study.

[3] At present, nondispersive infrared (NDIR) analyzers offer the most robust and precise method of CO2 quantification. However, this technique requires very accurately calibrated standard reference gases. The World Meteorological Organization (WMO) CO2 Expert Meeting in 1975 classified CO2 standards as WMO primary, secondary, and lower standards according to intended use [World Meteorological Organization, 1975]. The primary standards are used to assign accurate CO2 mole fraction values to the secondary standards which are maintained by each laboratory or group of laboratories for further calibration transfer to working standards. The current WMO CO2 Mole Fraction Scale consists of a set of 15 primary CO2-in-air standard calibration gases, ranging in CO2 mole fraction from 250 to 520 μmol mol−1, that were created in 1991. The WMO CO2 Expert Group transferred responsibility for maintaining the WMO Scale from the Scripps Institution of Oceanography (SIO) to the Climate Monitoring and Diagnostics Laboratory (CMDL) in 1995, at which point, the 15 CO2-in-air gas mixtures became the WMO primary standards, then have been calibrated, first at SIO, and then at regular intervals, between 1 and 2 years, by the CMDL manometric system [Zhao et al., 1997]. From 1978 through 1994, the CMDL secondary standards have been calibrated by the SIO against the WMO primary standards approximately every 3 years. After 1995, the CMDL secondary standards are calibrated once a year against WMO Primaries. The secondary standard calibrations are transferred via NDIR analyzers to all other reference gas cylinders, which are used routinely for measuring atmospheric CO2 mole fraction in the WMO Global Atmosphere Watch monitoring program [Tans and Zhao, 2003].

2. Uncertainty of the Primary Standards

[4] The values assigned to the 15 WMO Primaries were fully determined by the SIO calibration data from 1992 to mid-1996, jointly based on the SIO and CMDL measurements from mid-1996 to early 2001, and completely on the CMDL manometric measurements alone from 2001 to present. The WMO primary standard gases were analyzed by the CMDL manometric system in 1996, 1998, 2000, 2001, 2003 and 2004. These manometric calibrations determined the mole fraction of CO2 in each standard gas by measuring the temperature and the pressure of the original dried gas mixture in an approximately 6 liter chamber of the manometric apparatus, then freezing out of the CO2 in two cold traps, and measuring its temperature and pressure in a approximately 10 ml small volume of the manometric apparatus. Because the volume ratio of the small and the large volume is predetermined accurately, the molar ratio of the CO2 in the original dry gas mixture can be calculated with the virial equation of state, taking real gas compressibility into account [Zhao et al., 1997].

2.1. Precision and Accuracy of the Manometric Determination

[5] From September 1996 through January 2005, the WMO 15 primary standard gases have been manomatrically calibrated at intervals of 1 to 2 years. Within each group of calibrations (called a “calibration episode,” of which we now have had six), the standard deviation S of each calibration episode can be calculated by the expression [Laitinen, 1960]:

equation image

where di represents deviation of an individual measurement from the mean of a set; Nt total number of measurements; Ns number of sets. Table 1 lists the standard deviation of each calibration episode calculated by equation (1) from the 15 sets of individual determinations, for each calibration year and the total number of measurements. There are thirteen Primaries in the ambient mole fraction range from 300 to 420 μmol mol−1. Column 4 in Table 1 shows the standard deviation of calibrations for these ambient range Primaries. These standard deviations are measures of the system precision, or reproducibility during a given calibration year, of manometric analysis, mainly arising from short-term imprecision of pressure and temperature transducers, and from variations in gas handling and extraction processing of the CO2 fraction.

Table 1. Standard Deviation of Individual Calibrations During Each Calibration Episode (STDE) of the 15 WMO Primaries
YearNumber of MeasurementsSTDE (All Gases), μmol mol−1STDE (amb_range), μmol mol−1
1996640.120.09
1998580.140.13
2000550.110.10
2001620.090.08
2003620.060.06
2004480.040.04
Overall3490.090.08

[6] The largest scatter in the1998 calibration year was in part caused by pump back stream contaminations during the extraction processing. Instead of the oil pump, a high-vacuum turbomolecular pump (model 969-9170, Varian Vacuum Technologies, Lexington, Massachusetts) was used for the apparatus evacuation and the extraction processing in the two most recent calibration episodes. In addition to prevent back streaming, the high-vacuum dry pump removes residual gases more completely and more quickly. After 2001 there were changes in procedures to evacuate the 10 ml small volume, which is connected with the pressure transducer, because the pure CO2 gas is adsorbed on the walls of the 10 ml small volume and the 1 ml cell of the pressure transducer during the calibration. Experiments show that the previous adsorbed CO2 gas was released very slowly and resulted in a higher CO2 mole fraction measurement. In the last 2 calibration years, the small volume was first evacuated to a residual pressure of 0.1 Pa, and then flushed with the sample air to be measured; the “evacuating-flushing” procedure was repeated at least three times. This new procedure has proved to contribute to the decrease of measurement scatter and to the improvement of the calibration accuracy as well.

[7] As described above, the precision of manometric determinations can be assessed by statistical analysis based on repeated measurements. It represents the short-term random uncertainty. The accuracy of measurements represents an estimate of systematic error which cannot be treated statistically. Because the manometric calibration is an absolute technique, an exhaustive assessment of all potential error components is necessary to determine error bounds. These error components, with their measured (or estimated) magnitudes, are listed in Table 2. Column 1 shows all possible sources of both random and systematic errors. Column 2 and column 3 show the random error of each source and its effect (ΔRi) on the manometric system precision, respectively. Column 4 and column 5 show the potential bias of each source and its effect (ΔBi) on the manometric system accuracy, respectively. In Table 2, each source of error (random and systematic) within each factor is assigned an effect error (column 3 and 5) of the manometric calibrations at the range of ambient CO2 mole fraction of about 375 μmol mol−1 according to the manometric system error analysis [Zhao et al., 1997]. The estimated level of both precision and accuracy (the last row in Table 3) is then calculated by summing each effect error in quadrature (“root-sum-of-squares” method), that is

equation image
equation image

This estimated manometric precision can be compared to the analytical imprecision observed with any particular calibration episode. Observed imprecision greater than that predicted signifies a calibration problem (e.g., in 1998 and 2000 calibration episodes) that can subsequently be identified and corrected. Segregation of random and systematic errors is necessary so that accuracy bounds can be calculated with regard to all systematic errors considered, and secondly, so that an order of influence may be established among random and systematic error sources. Precision and accuracy of the technique may then be improved, when necessary, by minimizing the effects of the most influential error components.

Table 2. Random and Systematic Error Sources and Magnitudes of the Manometric Calibration at the Range of Ambient CO2 Mole Fraction of About 375 μmol mol−1
Sources of ErrorFactor ImprecisionaEffect on the Manometric Precision ΔR,a μmol mol−1Factor Potential BiasaEffect on the Manometric Accuracy ΔB,a μmol mol−1
  • a

    Method of determination: (1) reported in literature or by manufacturer; (2) estimated according to experimental data.

Pressure measures of the 10 mL volume±0.02 hPa (1)0.023 (1)±0.02 hPa (1)0.023 (1)
Pressure measures of the 6 L volume±0.04 hPa (1)0.014 (1)±0.04 hPa (1)0.014 (1)
Temperature measures of the 10 mL volume±0.02 K (1)0.022 (1)±0.02 K (1)0.022 (1)
Temperature measures of the 6 L volume±0.01 K (1)0.011 (1)±0.01 K (1)0.011 (1)
Volume ratio±0.01% (1)negligible±0.01% (1)0.039 (1)
CO2 sorption on the 10 mL volumeN/A0.01 (2)N/A0.01 (2)
CO2 permeability on the valveN/AnegligibleN/A0.02 (2)
CO2 extractionN/A0.03 (2)N/A0.02 (2)
Virial coefficientsN/Anegligible±0.02 cm3mole (1)0.005 (1)
Estimated uncertainty (∑ΔRi2)1/2 = 0.048 (∑ΔBi2)1/2 = 0.062
Table 3. Transducer Short-Term Stability Measurements by Performing Random Calibrations Using the DeadWeight Tester
DateLinear Coefficient, C1Constant, C0Deviations (20 kPa Range)Deviation (100 kPa Range)
25 June 20040.99978−0.002500.00234−0.00190
26 June 20040.99984−0.00898−0.00292−0.00229
27 June 20040.99979−0.00621−0.00116−0.00458
30 June 20040.99979−0.00726−0.00221−0.00562
2 July 20040.99987−0.00878−0.002010.00148
19 July 20040.99987−0.005360.001290.00429
20 July 20040.99986−0.004730.001850.00460
21 July 20040.99982−0.003640.002010.00103
28 July 20040.99980−0.00578−0.00048−0.00286
7 August 20040.99989−0.005740.001300.00584
Average0.99983−0.005900.000000.00000
Standard deviation0.000040.001950.001880.00383

[8] The accuracy is essential to long-term maintenance of the WMO absolute CO2 mole fraction scale. By inspection, uncertainties in the volume ratio determination and the pressure measurements are the two largest systematic error sources, the former also depends on the uncertainty of pressure measurements. Considerable efforts have been made to reduce systematic errors. The primary pressure standard, a Ruska model 2465-754 absolute deadweight tester, is recertified annually by the manufacturer with an accuracy to 0.0015%. In 2002, the deadweight tester was upgraded with an autofloat controller (model 2465A-200, Ruska Instrument Corporation, Houston, Texas) to generate the desired delivery pressure and to establish the correct piston float position automatically. This upgraded piston gauge reduces workload significantly and eliminates possible artifacts from a manual control of piston float position and rotation, because the error can be produced by accepting a float position that is too high or too low.

[9] In May 1999 the quartz spiral pressure gauge was replaced by a compact, rugged and more stable quartz resonator transducer (model 6000-15A, Paroscientifc, Redmond, Washington) to measure pressure. The transducer uses a vibrating quartz beam with an output frequency which varies with applied pressure. The transducer is regularly calibrated with the standard pressure from 10 kPa to 100 kPa in 10 kPa increments produced by the deadweight tester. Linearity of the transducer across its dynamic range (0–100 kPa) is excellent (r2 = 1.000 000 00). Transducer short-term stability measurements have been made by performing at random intervals calibrations using the deadweight tester. Table 3 shows an example of these measurements. Least squares fits to the calibration data were used to generate the linear coefficient C1 and the constant C2. The pressure to be measured is then calculated by the expression P = C1V + C0 (kPa), where P represents pressure and V the transducer output. Columns 4 and 5 in Table 3 list the pressure deviations from the mean in the small volume pressure range (20 kPa) and in the large volume pressure rang (100 kPa), respectively. During the 2-month period, the offset errors relative to the primary standard due to transducer variations are about 0.004 kPa (0.004%) in the100 kPa range typical of pressure in the 6 L volume and 0.002 kPa (0.01%) in the 20 kPa range typical of pressure in the 10 ml volume, respectively. In June 2003, two more platinum resistance thermometers (PRT) were added to the manometric system to obtain more precise temperature measurements. The PRT probes have been calibrated directly by National Institute of Standards and Technology (NIST). The NIST temperature uncertainty in the bath temperature measurements from 0.5° C to 95° C does not exceed 0.002° C [Taylor and Kuyatt, 1998].

2.2. Combined Standard Uncertainty of the WMO Scale

[10] Table 4 shows the manometric calibration data of all six CMDL manometric calibration episodes for all Primary gas mixtures. Calibration precision, shown with each CO2 number in Table 4 is indicated by the standard deviations associated with the determined average CO2 mole fractions of the individual measurements of each cylinder during the calibration episode. The last row in Table 4 shows the difference of the mean of all 15 Primaries in each episode from the mean of all episodes. The last column in Table 4 shows the standard error of the mean (denoted by σm) of each primary cylinder for all manometric determinations. The standard error of the mean (σm) is simply the standard deviation (σ) of the individual measurement episodes of each Primary divided by the square root of the total number (N) of episodes, that is σm = σN−1/2. In fact, standard deviation (σ) characterizes the average uncertainty of the individual measurement episodes from which it is calculated, it represents the uncertainty in a single episode. It essentially describes the random error of the manometric calibration system. Since the current WMO CO2 Mole Fraction Scale is defined by the average of all six CMDL manometric calibration episodes for each Primary gas mixture, therefore, the standard error of the mean (σm) should be considered as the best estimate of the uncertainty of the CO2 primaries arising from the random error in the manometric calibrations. In consideration of both standard error of the mean (σm = 0.030 μmol mol−1, an average of the last column in Table 4) and systematic error (0.062 μmol mol−1 in Table 2), the combined standard uncertainty of the WMO Primaries is given by [Taylor and Kuyatt, 1994]

equation image
Table 4. NOAA/CMDL Manometric Calibration Data for 15 WMO CO2-in-Air Primary Standards From 1996 Through 2004
Tank Number1996, μmol mol−11998, μmol mol−12000, μmol mol−12001, μmol mol−12003, μmol mol−12004, μmol mol−1Average, μmol mol−1σm,a μmol mol−1
  • a

    See text for description of variable.

110246.57 ± 0.01246.60 ± 0.03246.80 ± 0.02246.73 ± 0.03246.63 ± 0.01246.65 ± 0.04246.66 ± 0.0890.036
102304.32 ± 0.08304.42 ± 0.12304.42 ± 0.07304.40 ± 0.03304.31 ± 0.06304.33 ± 0.04304.37 ± 0.0510.021
111323.97 ± 0.12323.94 ± 0.14324.16 ± 0.12324.01 ± 0.09323.99 ± 0.06323.98 ± 0.04324.01 ± 0.0770.032
130337.18 ± 0.03337.35 ± 0.11337.27 ± 0.02337.29 ± 0.08337.22 ± 0.06337.32 ± 0.02337.27 ± 0.0630.026
121349.35 ± 0.09349.36 ± 0.01349.47 ± 0.04349.39 ± 0.03349.38 ± 0.05349.38 ± 0.01349.39 ± 0.0420.017
103353.30 ± 0.04353.45 ± 0.11353.35 ± 0.06353.26 ± 0.04353.20 ± 0.04353.22 ± 0.05353.30 ± 0.0940.038
139360.93 ± 0.06360.90 ± 0.01360.87 ± 0.04360.90 ± 0.04360.90 ± 0.01360.88 ± 0.04360.90 ± 0.0220.009
105369.37 ± 0.06369.32 ± 0.13369.39 ± 0.14369.40 ± 0.11369.37 ± 0.03369.38 ± 0.03369.37 ± 0.0270.011
136381.21 ± 0.10381.43 ± 0.15381.35 ± 0.07381.37 ± 0.01381.30 ± 0.02381.31 ± 0.05381.33 ± 0.0770.031
146389.48 ± 0.04389.53 ± 0.19389.55 ± 0.09389.59 ± 0.01389.56 ± 0.04389.62 ± 0.02389.56 ± 0.0480.020
101396.20 ± 0.11396.47 ± 0.12396.40 ± 0.04396.37 ± 0.09396.23 ± 0.06396.34 ± 0.02396.33 ± 0.1030.042
106411.97 ± 0.07412.07 ± 0.15412.21 ± 0.05412.12 ± 0.11412.02 ± 0.03412.00 ± 0.01412.06 ± 0.0900.037
123422.95 ± 0.09423.11 ± 0.14423.25 ± 0.11422.99 ± 0.08423.03 ± 0.07423.14 ± 0.02423.08 ± 0.1100.045
107453.23 ± 0.22453.20 ± 0.06453.09 ± 0.14452.98 ± 0.08452.98 ± 0.05453.02 ± 0.01453.08 ± 0.1210.049
132521.26 ± 0.07521.35 ± 0.18521.51 ± 0.13521.46 ± 0.10521.46 ± 0.06521.41 ± 0.05521.41 ± 0.0920.037
Mean374.75 ± 0.12374.83 ± 0.14374.87 ± 0.11374.81 ± 0.09374.77 ± 0.06374.80 ± 0.04374.81 ± 0.0740.030
Differencea−0.0550.0280.0650.006−0.036−0.0080.000 

[11] When the means of each cylinder for each calibration episode are calculated, it is seen that those means differ from one another by more than what would be expected by calculating the calibration standard error of the mean (σm) of each cylinder for each calibration episode. The larger than expected differences imply that there were slight variations in calibrations of pressure, volume ratio, and temperature, or variations in procedure between episodes mentioned in section 2.1, and that the best statistic to characterize variance of the measurements over many years should be obtained by using the means of each Primary for each episode. Slight variations in transducer calibrations or in operating procedure are also suggested by calculating how the mean of all 15 Primaries differs between episodes, as show in the last row of Table 4. There is no evidence of the mean mole fraction drift of the WMO Primaries or of the system from 1996 to 2004.

3. Uncertainty of Standard Transfer

[12] In order to maximize the lifetime of its highest-level standards, CMDL has always transferred the scale via a set of Secondary standards, which in turn are used to calibrate all other standard gases. The Secondary standards are calibrated once or twice a year against the WMO Primaries. The Secondaries last from 3–5 years. All calibration transfers from a higher level to a lower level are done with four higher-level standards always bracketing the mole fractions of the two or four lower-level standards that are being calibrated. All of these transfer calibrations have been done with NDIR analyzers, but with a succession of different instruments. In practice, the standard transfer involves two distinct steps, of direct measurements followed by calculation. There are two inevitable errors in the standard transfer process: the NDIR measurement error arising from instrument random effects and the propagation error arising in the calculation of the CO2 value due to the uncertainty of the reference gases used in the NDIR calibration. The random error can be evaluated statistically; the propagation effect can be calculated theoretically.

3.1. Uncertainty of the NDIR Analyzer

[13] The CO2 measurements by using a differential NDIR analyzer in CMDL are based on the differences in absorption of infrared radiation passing through two gas sampling cells. Infrared radiation is transmitted through both cell paths, and the output of the analyzer is proportional to the difference in absorption between the two cells. The differential instrument random errors are mainly arising from the intrinsic noise and gain fluctuations. In order to reduce the random error caused by gain fluctuations the reference cell of the analyzer is flushed with a nonzero gas containing ≈350 μmol mol−1 CO2 in air at a flow rate of 10 ml min−1. The standard gases and the gases to be calibrated flow through the sample cell at 300 ml min−1.

[14] Procedures to perform NDIR calibrations in CMDL have been followed closely since 1979. Operation is essentially automatic. Four standard gases of known mole fractions are used to determine the CO2 mole fractions for either two or four unknowns, the unknown gases are computed from a least squares quadratic fit to the standard gases. The standard gases, named L, M, H and Q, and the unknown gases to be calibrated, named W1, and W2, each flow through the sample cell of the NDIR analyzer alternately for 90 s flushing followed by 30 s measuring in a pyramidal sequence: L-W1-M-W2-H-Q-Q-H-W2-M-W1-L at 300 ml min−1. These pyramidal calibrations are usually repeated five times. This procedure eliminates most of the systematic drift of the zero and gain. The mean and standard deviations are then computed from the five values as a final result [Komhyr et al., 1985]. Typical data output of calibrations of unknown gases W1, W2, W3 and W4 with standards L, M, H and Q is shown in Figure 1a. Calibration precision, as indicated by the standard deviation related to measured mean CO2 mole fractions, is typically a few hundredths of a part per million (ppm). Figure 1b presents that, in this calibration example, the residual error from the quadratic fits (assigned value minus fitted value, indicated by a triangle) and the residual deviation of each unknown's measurement from the mean of five calibrations (indicated by a square) are less than 0.01 μmol mol−1. On the right vertical coordinate, a red line obtained from the least squares quadratic fits to four standard gases plots the NDIR outputs in volts versus CO2 mole fractions in μmol mol−1.

Figure 1a.

Typical output data obtained from CO2 standard transfer calibration system.

Figure 1b.

An example of the NDIR calibration by using Siemens Ultramat-3 analyzer. A triangle indicates the residual error of each standard gas from the quadratic fits (assigned value minus fitted value). A square indicates the residual deviation of each unknown's measurement from the mean. On the right vertical coordinate, a red line shows the NDIR outputs versus CO2 mole fractions obtained from the least squares quadratic fits to four standard gases, and a diamond indicates the NDIR output of the standard gas.

[15] From September 1979 to December 2004 a total of 9423 CO2 standard transfer calibrations has been made via the NDIR analyzers in CMDL. Figure 2 shows calibration precision (1σ standard deviation) for each unknown determination by a plus sign. The solid lines show the distribution of the NDIR calibration precision. The two vertical dashed lines divide all transfer calibration data into three sections according to different type of the NDIR analyzer used. Table 5 summarizes the statistical results of the calibrations in the three periods. The number in the parentheses in Table 5 shows the 1σ standard deviation associated with the mean. Column 6 shows the average calibrating precision in the CO2 mole fraction range from 300 to 420 μmol mol−1.

Figure 2.

NDIR precision of all the CMDL standard transfer calibrations from 1979 to 2004. A plus indicates the 1σ standard deviation of each unknown determination associated with the mean. The solid lines show the NDIR calibration precision versus the percentage of the total calibrations indicated on the top horizontal axis. The vertical dashed lines indicate the place where the NDIR analyzers were changed.

Table 5. Summary of NDIR Calibrations in CMDL From 1979 to 2004
PeriodsAnalyzersNumber of CalibrationsMean CO2 of Unknowns, μmol mol−1Precision of All Calibrations, μmol mol−1Precision of Ambient Range, μmol mol−1
1979–1986UNOR-2914340.84 (±13.13)0.035 (±0.029)0.035 (±0.029)
1986–2000Ultramat-35136357.30 (±35.67)0.012 (±0.023)0.009 (±0.019)
2000–2004Licor-62523373384.40 (±117.40)0.015 (±0.018)0.013 (±0.020)
Overall 9423365.38 (±76.50)0.016 (±0.023)0.014 (±0.022)

3.2. Propagation of Uncertainty

[16] The determination of CO2 value by an NDIR analyzer is a relative measurement against reference gases of known CO2 mole fractions. The uncertainty of reference gases will “propagate” through the calculations to produce an uncertainty in the final results. In the simplest case, two independent reference gases C1 ± ΔC1 and C2 ± ΔC2 are used to determine an unknown gas mole fraction C by

equation image

where C represents CO2 mole fractions and A the output of the analyzer (absorption), a0 and a1 are constants defined by the response of the analyzer to the two reference gases. The propagation of the uncertainty Up can be calculated by the expression [Taylor, 1982]

equation image

where ∂C/∂C1 and ∂C/∂C2 are partial derivatives, ΔC1 and ΔC2 represent the uncertainty of the independent reference gas C1 and C2, respectively. Combined (3) and (4) will produce

equation image

where A1 and A2 represent the absorptions of the reference gas C1 and C2, respectively, A is the absorption of the unknown gas C, assuming two reference gases have a identical and uncorrelated uncertainty ΔC, that is, ΔC1 = ΔC2 = ΔC, then

equation image

Usually the reference gases are always chosen to bracket the mole fractions of unknowns. If the mole fraction of the unknowns is at the midpoint of two independent references, equation (6) will give the minimum propagation error as

equation image

In the case of the linear interpolation to determine the unknown gas mole fraction, the propagation error Up calculated from (6) is between 0.707 ΔC and ΔC, that is

equation image

where ΔC represents the uncertainty of independent reference gases, γ the propagation error coefficient. In the case of the quadratic fit to three independent reference gases, equations (3) and (4) turn to

equation image

and

equation image

respectively, where ∂C/∂C1, ∂C/∂C2 and ∂C/∂C3 are partial derivatives, and ΔC1, ΔC2 and ΔC3 represent the uncertainty of the independent reference gas C1, C2 and C3, respectively. If we again assume that all reference gases have an identical uncorrelated uncertainty ΔC, the propagation error Up of calculating from (8) is then between 0.77 ΔC and ΔC (0.77 ≤ γ ≤1). Figure 3 shows the propagation error coefficient γ versus variations of the CO2 mole fraction (NDIR output) by using Siemens Ultramat-6F analyzer data. In the case of interpolation to determine the unknowns (references are bracketing unknowns), the propagation error coefficient γ is less or equal to one. However, the use of an extrapolating method results in a larger propagation error, as is clearly illustrated in Figure 3.

Figure 3.

Propagation error coefficient γ versus variations of the CO2 mole fraction (NDIR output) by using Siemens Ultramat-6F analyzer data. The vertical dashed lines indicate three standard reference gases in CO2 values to be 350, 355, and 360 ppm. The line labeled (a) indicates variations of the propagation error while using a quadratic fit to three reference gases; the line labeled (b) indicates variations of the propagation error while using a linear fit to two reference gases (350 and 360 ppm).

[17] The overall uncertainty of the CMDL secondary standards can be calculated by

equation image

where Undir = 0.014 μmol mol−1 represents uncertainty arising from the random error (1σ) of the NDIR instruments (see Table 5), and Up the propagation error arising from the uncertainty of the Primaries. Choosing the maximum propagation coefficient (γ = 1) for a conservative estimate, then Up = γΔC = 1 × 0.069 = 0.069 μmol mol−1, thus

equation image

However, the uncertainty of the Primaries is common to all Secondaries, thus the consistency among the Secondaries is about 0.014 μmol mol−1 (equal to NDIR random error). Similarly in the one-sigma absolute scale, the uncertainty of the working standards, which are used routinely for measuring atmospheric CO2 mole fractions in the CMDL sampling network and at the CMDL observatories, can be calculated by

equation image

Consistency among the working standards is determined by the random components of the measurement errors at each level, that is (0.0142 + 0.0142)1/2 = 0.02 μmol mol−1.

3.3. Comparison With Gravimetric CO2 Standards

[18] In addition to manometric technique, there is an approach to making the high-accuracy CO2 standards by the gravimetric technique. The preparation of the standards using the gravimetric technique is conducted by filling pure CO2 of a specified quantity into a standard cylinder and then diluting it with a CO2-free gas mixture according to the required CO2 concentration. The masses of the component gases (CO2 and the carrier gas) are accurately weighed by a balance. The CO2 mole fraction of the standards is then obtained by converting the mass ratio to the molar ratio. The accuracy of the gravimetric standards depends on the CO2 purity and the balance accuracy as well as the precisions of the temperature, pressure and humidity measurements in the standards preparation environment [Tanaka et al., 1983].

[19] In April 2005, CMDL received five CO2-in-air gravimetric standard cylinders from the National Institute for the Environmental Studies (NIES) in Tsukuba of Japan for comparison between the WMO CO2 mole fraction scale and the NIES gravimetric CO2 mole fraction scale. These five high-pressure (10 MPa) cylinders ranging in CO2 mole fraction from 350 to 390 μmol mol−1 have been calibrated three times by using Licor-6252 NDIR analyzer against the CMDL secondary standards. The results of these calibrations are listed in Table 6. The average and standard deviation (1σ) of the differences (in absolute values) between the NIES gravimetric and CMDL analyzed CO2 mole fraction (CMDL-NIES) are 0.004 ± 0.03 μmol mol−1. The standard gas, CPB30089, has the largest deviation of 0.06 μmol mol−1. Without this gas, the average and standard deviation of the differences are 0.01 ± 0.01 μmol mol−1, which is about the uncertainty of Licor-6252 NDIR analyzer. The result of these comparisons illustrates an excellent agreement between the two independent methods, manometric and gravimetric, to determine the CO2 mole fraction in dry air. The total uncertainty of the NIES gravimetric CO2 standards is estimated to be 0.04 μmol mol−1 in one-sigma absolute sense (Y. Tohjima and C. Zhao, personal communications, 2005). This comparison result is essentially an independent confirmation of the absolute accuracy of the WMO CO2 mole fraction scale.

Table 6. CMDL Calibrations of NIES Gravimetric CO2-in-Air Standard Cylinders
DateCylinder NumberCMDL NDIR CalibrationsaCMDL Meana,bNIES Assigned Valuea,cDifference (CMDL-NIES)a
  • a

    Values are mole fractions given in μmol mol−1.

  • b

    Uncertainty values for CMDL calibrations are one standard deviation associated with the mean.

  • c

    Uncertainty values for NIES gravimetric standards are expressed in the one-sigma absolute scale (Y. Tohjima and C. Zhao, personal communications, 2005).

4 May 2005CPB30089350.20   
9 May 2005CPB30089350.21   
10 May 2005CPB30089350.20350.20 ± 0.01350.14 ± 0.040.06
4 May 2005CPB30091350.03   
9 May 2005CPB30091350.03   
10 May 2005CPB30091350.04350.03 ± 0.01350.02 ± 0.040.01
4 May 2005CPB30092390.09   
10 May 2005CPB30092390.08   
11 May 2005CPB30092390.09390.09 ± 0.01390.11 ± 0.04−0.02
4 May 2005CPB30093390.09   
10 May 2005CPB30093390.10   
11 May 2005CPB30093390.09390.09 ± 0.01390.11 ± 0.04−0.02
4 May 2005CPB30094389.02   
10 May 2005CPB30094389.02   
11 May 2005CPB30094389.02389.02 ± 0.00389.03 ± 0.04−0.01
Average 373.886373.886 ± 0.01373.882 ± 0.040.004

4. Conclusion and Discussion

[20] The current WMO Mole Fraction Scale is defined by the average of all six CMDL calibration episodes for all Primary gas mixtures. The combined standard uncertainty of the 15 primary standards was estimated to be 0.069 μmol mol−1 in the one-sigma absolute scale. Manometric calibrations results indicated that there is no evidence of overall drift of the Primaries from 1996 to 2004. The standard deviation of differences between the means for each calibration episode is 0.07 μmol mol−1 on average for all Primaries (see the second row from the bottom of column 8 in Table 4). With current procedures, any potential drift of a primary standard will have to be evaluated in light of a basic “noise” level of about 0.07 μmol mol−1. The CMDL secondary standards are calibrated once a year against WMO Primaries. The secondary standard calibrations are transferred via NDIR analyzers to the working standards, which are used routinely for measuring atmospheric CO2 mole fraction in the WMO Global Atmosphere Watch monitoring program. The uncertainty of the working standards was estimated to be 0.071 μmol mol−1 in the one-sigma absolute scale. Consistency among the working standards is determined by the random errors of downward transfer calibrations at each level, and is about 0.02 μmol mol−1. The one-sigma absolute accuracy of CMDL's manometric determinations was estimated to be 0.062 μmol mol−1. For comparison with an independent absolute scale, the five NIES gravimetric standards ranging in CO2 mole fraction from 350 to 390 μmol mol−1 have been calibrated relative to the CMDL secondary standards. The average and standard deviation of the differences between the NIES gravimetric and CMDL analyzed CO2 mole fraction are 0.004 ± 0.03 μmol mol−1.

[21] There are some issues pertinent to the transfer calibrations that need to be discussed here. A calibration scale, relating instrument response to mole fraction of the gas being measured, is defined by a curve fit to a set of well-defined reference gas mixtures. Ideally one would like the scale to be uniquely defined by the set of reference gases because that allows the scale to be transferred between laboratories. There is a limit to the accuracy with which this can be done. We have found that, when the same gases are run though three NDIR analyzers (Siemens Ultramat-6, Licor-6252, and Licor-7000) in direct sequence, the residuals of such a curve fit are different for each analyzer at the level of 0.01–0.03 μmol mol−1. The differences remain when the experiment is repeated at a later time. Especially, in the case of measuring CO2-in-synthetic-air as opposed to atmospheric air gas mixtures the bias can be as large as 0.07 μmol mol−1 between Siemens Ultramat-6 and Licor-6252. They are thus a property of the analyzer, and do not transfer when a set of reference gases is sent to another laboratory.

[22] As described in section 2, the past assignments of the WMO Primaries were based on the original values received from Scripps. In late 2001, we decided to base the WMO Scale entirely on the CMDL manometric calibrations alone. The average difference of assigned values for each primary gas between WMO current scale and the Scripps X99A revision listed in Table 7 is only 0.02 μmol mol−1 (exclusive of the two highest mole fraction cylinders). It should be noted, however, that the original assignments received from Scripps differ from the revision X99A which we received in early 2002. Inconsistencies between successive Scripps calibrations and between Scripps and CMDL translated only gradually and partially into working level standards. These inconsistencies are still being evaluated and will be reported on at a later time.

Table 7. Comparison of Assigned Values for 15 WMO CO2-in-Air Primary Standards Between the Current WMO Mole Fraction Scale and the SIO X99A Scale
Cylinder Serial NumberWMO Scale, μmol mol−1SIO X99A, μmol mol−1WMO-SIOX99, μmol mol−1
110246.66 ± 0.02246.59 ± 0.080.07
102304.37 ± 0.07304.35 ± 0.090.02
111324.01 ± 0.10324.01 ± 0.050.00
130337.27 ± 0.05337.27 ± 0.020.00
121349.39 ± 0.04349.36 ± 0.010.03
103353.30 ± 0.06353.20 ± 0.030.10
139360.90 ± 0.03360.87 ± 0.060.03
105369.37 ± 0.09369.40 ± 0.06−0.03
136381.33 ± 0.07381.34 ± 0.08−0.01
146389.56 ± 0.07389.60 ± 0.08−0.04
101396.32 ± 0.07396.30 ± 0.070.02
106412.06 ± 0.07412.08 ± 0.12−0.02
123423.08 ± 0.09423.05 ± 0.130.03
107453.09 ± 0.09452.96 ± 0.160.13
132521.42 ± 0.10521.07 ± 0.640.35

Acknowledgments

[23] We thank Yasunori Tohjima of the National Institute for the Environmental Studies of Japan for providing their five CO2 gravimetric standards. Discussions with Jason M. Zhao of Johns Hopkins University were very helpful for estimating propagation of uncertainty.

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