The isotopic composition of atmospheric trace gases such as

This paper sets out a practical guide to the calculations required to perform calibrations of isotopologue-specific optical analysers, applicable to both laser and broadband FTIR spectroscopy. Equations to calculate the relevant isotopic and total concentration quantities without approximation are presented, together with worked numerical examples from actual measurements. Potential systematic errors, which may occur when all required isotopic information is not available, or is approximated, are assessed. Fortunately, in most such realistic cases, these systematic errors incurred are acceptably small and within the compatibility limits specified by the World Meteorological Organisation – Global Atmosphere Watch. Isotopologue-based and ratio-based calibration schemes are compared. Calibration based on individual isotopologues is simpler because the analysers fundamentally measure amounts of individual isotopologues, not ratios. Isotopologue calibration does not require a range of isotopic ratios in the reference standards used for the calibration, only a range of concentrations or mole fractions covering the target range. Ratio-based calibration leads to concentration dependence, which must also be characterised.

Until recently, measurements of the amounts of

Non-dispersive infrared (NDIR) analysers have been used for many years as
an instrument of choice for atmospheric trace gas monitoring. NDIR is an
optical technique based on infrared absorption by the target trace gas, and
like any optical or spectroscopic instrument, NDIR instruments have a different
response to different isotopologues of the target species because different
isotopologues have different absorption spectra. Earlier NDIR instruments
such as URAS, UNOR, Siemens and APC employed microphone detectors filled
with the target trace gas that responded selectively to the absorption of
infrared radiation by the target gas in the sample (Griffith, 1982).
The NDIR instrument response depends, in a complex and non-linear way, on the
isotopic composition of the target gas and on the carrier gas. The more
recent LI-COR instruments replaced the microphonic detector with an optical
semiconductor detector that relies on a broad bandpass filter to restrict
the wavelength range from the source to that absorbed by the target gas, for
example, around 4.3

Most recently, laser and Fourier-transform infrared (FTIR) based optical
infrared analysers have taken on a major role in atmospheric trace gas
measurements for many gases, especially the dominant greenhouse gases

Established calibration laboratories using mass spectrometry as the primary
method for isotopic analysis normally provide calibration standards which
specify the total amount and isotopic ratios of a trace gases in an air matrix,
such as

to show that the complete and correct treatment of isotopic composition in calibration calculations is straightforward and that there is no need to invoke some approximations often made in earlier analyses,

to provide a practical guide to isotope-specific calibration calculations, and

to assess the potential errors when all isotopic information is not available and approximations or assumptions must be made.

Using

The most fundamental quantity defining isotopic composition for each element
is the

Standard isotope ratios for relevant reference scales used in atmospheric trace gas analysis.

Isotope ratios are commonly expressed as delta values relative to a standard
or reference material:

For each isotope of an element, the

Similarly, the isotopologue abundances or isotopologue fractions are defined for a molecule; for example, for

Isotopologue fractional abundances and isotopic sums for the
VPDB-

Abundances are taken
from

Abundances of the major and three singly substituted isotopologues and

For a calibration or reference gas,

If the total mole fraction of

Conversely, if a set of calibrated isotopologue mole fractions

In the HITRAN database, tabulated line strengths are normalised by the natural abundance of the relevant isotopologue; the reference isotopologue natural abundances assumed in HITRAN are listed in Table 2. Retrievals from spectra based on HITRAN line parameters thus provide scaled or normalised mole fractions of isotopologues, which are referenced to the isotopic scales assumed by HITRAN. For some purposes it may be convenient to work with these normalised mole fractions directly rather than convert them to absolute mole fractions as in Sect. 2 because the reference isotopologue abundances are inherently included in the normalised amounts. In terms of normalised mole fractions, Eq. (8) becomes

The calculation of

Calibration of an isotopologue-specific analyser can in principle be carried out in two ways: calibrating on either the individual isotopologue amounts or on the derived isotope ratios or delta values. Both methods have been used in published work to date. The former is more fundamental because optical methods actually measure individual isotopologue amounts, not ratios. Ratio- or delta-based calibration leads to the additional complication of concentration dependence in the calibration. A step-by-step method for direct isotopologue calibration is presented in Sect. 4.1 based on the equations of Sect. 2. Ratio or delta calibration is discussed in Sect. 4.2 and the two methods are compared in Sect. 4.3.

The steps described here are consistent with those recently published by
Flores et al. (2017) and Tans et al. (2017). Griffith et al. (2012)
previously described the same methods but used a minor approximation in
accounting for the sum of all multiply substituted isotopologues in the
calculation of

There are two parts to the calibration and unknown measurement procedure:
(1) determination of the reference isotopologue amounts and the calibration
equation for each isotopologue in a calibration gas, and (2) measurement of
the isotopologue amounts in an unknown sample and calculation of its total
trace gas amount and delta quantities. As above,

From reference standard tank data provided by the calibration laboratory

Calculate the calibrated amount of each isotopologue

Measure uncalibrated analyser responses or raw isotopologue amounts of each
standard

Derive the calibration equation for each isotopologue, for example, for a
linear calibration:

Measure the sample with the analyser and determine the analyser responses or raw isotopologue amounts.

Apply the inverted calibration determined in step 4 (Eq. 14) above for each isotopologue to determine calibrated isotopologue amounts.

Calculate

Calculate

Calculate total

With this scheme, for complete calibration of the analyser for the total

Spectroscopic analysers fundamentally determine the amounts of individual isotopologues, and the isotopologue-based analysis as described in the preceding section is the natural choice as a basis for calibration. Historically, however, isotope ratio mass spectrometry (IRMS) has been the method of choice for isotopic analysis because many sources of noise cancel in calculating the ratio. Traditional IRMS calibration schemes are based on standards over a range of isotope ratios or delta values directly, rather than on isotopologue amounts. Ratio or delta calibration schemes have thus, perhaps inevitably, flowed through to optical techniques. Ratio calibration schemes use calibration standards which cover a range of delta values and derive calibration equations analogous to Eq. (14) directly in terms of delta values rather than isotopologue amounts. The raw measured delta values are calculated from the uncalibrated isotopologue amounts. However, as shown in the following, this method inevitably leads to a concentration dependence of the calibration equations, which must be characterised as part of (and which significantly complicates) the calibration procedure.

Several groups have reported on ratio calibration schemes and the consequent
concentration dependence (e.g. Griffith et al., 2012; Wen et al., 2013;
Rella et al., 2015; Pang et al., 2016; Braden-Behrens et al., 2017; Flores
et al., 2017). The concentration dependence inevitably follows if the actual
calibration relationships between measured and true amounts of individual
isotopologues (Sect. 4.1, Eq. 14) have a
non-zero

Figure 1 illustrates this concentration dependence
with a typical

Example of

The concentration dependence is a function of the isotopologue calibration coefficients, and thus in principle for best accuracy it should be redetermined for every calibration, complicating the calibration procedure. The Thermo Fisher Delta Ray isotope analyser, for example, takes this approach in a prescribed sequence of measurements using several reference standards; however, Braden-Behrens et al. (2017) and Flores et al. (2017) found this procedure not to be sufficiently accurate or stable and invoked separate a posteriori calibration schemes. Rella et al. (2015) and Picarro (2017) similarly describe a calibration procedure for Picarro analysers to take concentration dependence into account.

Worked data for calibration of an FTIR analyser using four
reference standards in

As an example, assume a calibration laboratory provides calibrated reference
gases with an absolute accuracy of 0.05 ppm for total

The isotopologue scheme does not require calibration gases spanning a range of delta values; it is sufficient to span the range of total amount fractions of interest. This simplifies the preparation of reference gases for calibration laboratories.

The ratio scheme has an unavoidable

Errors are discussed further in Sect. 6.

This section presents a worked example of the calibration of an optical
analyser using reference gases of given total

The calibration data were collected in the laboratory at the University of
Wollongong on 27 September 2017. Four reference tanks were sourced from CSIRO, with total

Worked calibration of sample data in Fig. 3 at four times with varying

Calibration plots for three

Figure 3 shows an example of 1 day of calibrated
1 min measurements from the same FTIR analyser collected at a rural site
in SE Australia on 23 and 24 January 2018. Table 4
illustrates the worked calibration of the raw data at four times of
differing

Actual isotopologue amounts and

Calibrated total

Table 5 shows examples of actual isotopologue
amounts for samples with total CO

Keeling plot of data shown in Fig. 3.

Details of isotopologues of common atmospheric species.

These potential errors in computation of delta values should also be viewed
in the context of experimental measurement errors. Flores et al. (2017) formally evaluated the uncertainty budget for their
particular FTIR measurements of

Three assumptions, previously mentioned and summarised here, have negligible impact on the calculations of Sect. 2 and Table 5.

The relative amounts of multiply substituted minor isotopologues are assumed to be in statistical relative abundance, i.e. there is no isotope clumping. Clumping refers to the case where the enrichment (or depletion) of two or more isotopes in a multiply substituted isotopologue are correlated, rather than each following their statistical amounts independently. Clumping effects are normally much less than 1 ‰, and according to Table 5 are therefore insignificant.

Values

Similar considerations apply to other molecular species, see
Table 6. For

For

Several commercial manufacturers offer isotopologue-specific optical analysers based on laser (Campbell Scientific, Picarro, Los Gatos Research, Aerodyne Research, Thermo Fisher Scientific) or FTIR (Ecotech) spectroscopy that analyse sampled air for one or more specific isotopologues. These instruments report results in a variety of ways, as isotopologue mole fractions and/or as total mole fractions and isotopic delta values, both calibrated and uncalibrated. In most cases the scheme by which total mole fractions and delta values are calculated from the raw measured data is not fully described, although some details are available in user manuals and published works. In most cases some level of approximation is used in accounting for the full molecular isotopic composition when converting between isotopologue amounts and total amounts and delta values. As shown in Sect. 6, these approximations are fortunately in most cases acceptably small, but it is nevertheless recommended that they be assessed and documented if the full computation scheme is not used or measurement, and calibration data for all isotopologues are not available.

GAW reports on Carbon Dioxide, other Greenhouse Gases and Related Tracers Measurement Techniques since 2011 (WMO-GAW, 2012) recommend that the computational scheme for isotopic quantities derived from all commercial and non-commercial analysers be published and fully transparent to the user to avoid the potential for biases and inaccuracies stemming from different calibration and calculation schemes. Potential errors and calibration biases due to inconsistent isotopic calculations and the empirical determination of concentration dependences can be avoided if only the raw output isotopologue amounts from the analyser(s) are used and calibrated and isotopic quantities are calculated a posteriori following consistent calculation schemes, such as those described here and in Flores et al. (2017) and Tans et al. (2017).

Optical trace gas analysers based on laser or FTIR spectroscopy measure the
concentrations or mole fractions of individual isotopologues of a trace gas
rather than the total amount of all possible isotopologues of the target
gas. This leads to potential calibration inaccuracies in relating the
individual isotopologue measurements made by the analyser to the more usual
quantities of total amount and isotopic ratios or delta values. This paper
reviews previous studies addressing isotopic calibration of optical
analysers and presents a practical guide to the calculations required to
completely and rigorously account for the isotopic composition of a trace
gas when determining its total concentration with an isotopologue-specific
optical analyser. Although most previous work has made some level of
approximation in accounting for the full isotopic composition, this paper
shows that such approximations are not required and save little effort – the
complete calculations are relatively straightforward. The approach described
here is consistent with those of Flores et al. (2017) and Tans
et al. (2017); for

Potential errors which may arise when making sometimes-unavoidable
approximations in the calculations are assessed and, in most cases,
fortunately found to be small and often negligible. However, significant
errors can arise when the isotopic composition of an air sample is very
different from that used to calibrated the analyser. Two common cases where
this may occur in practice are in the production of synthetic reference
standards using highly depleted

Provided the appropriate calibration standards are available, this paper recommends that the calibration of optical analysers be carried out via direct measurement of the amounts of individual isotopologues, from which the total trace gas amount and isotopic composition can then be calculated completely and accurately. It recommends against ratio or delta-based calibration because this approach leads inevitably to concentration dependences in the calibration that must be characterised. Direct isotopologue calibration avoids concentration dependence and requires only reference standards spanning the range of concentrations to be measured and of known isotopic composition. There is no requirement for the reference gases to span the range of expected delta values; they can all be produced from the same source of trace gas and all have the same isotopic composition.

Optical FTIR and laser methods do not currently meet GAW requirements for
repeatability of

Data in the paper are only illustrative of the calculations. There are no original or published data.

The author is a consultant to Ecotech Pty Ltd., manufacturer of the Spectronus trace gas analyser under licence to the University of Wollongong.

This article is part of the special issue “The 10th International Carbon Dioxide Conference (ICDC10) and the 19th WMO/IAEA Meeting on Carbon Dioxide, other Greenhouse Gases and Related Tracer Measurement Techniques (GGMT-2017) (AMT/ACP/BG/CP/ESD inter-journal SI)”. It is a result of the 19th WMO/IAEA Meeting on Carbon Dioxide, Other Greenhouse Gases, and Related Tracer Measurement Techniques (GGMT-2017), Empa Dübendorf, Switzerland, 27–31 August 2017.

I would like to thank the GAW-GGMT community for the many discussions on this topic, and especially Edgar Flores, Joelle Viallon, Camille Yver, Grant Forster, Kentaro Ishijima and Jessica Conolly who provided comments on the manuscript and checked the calculations. Edited by: Hubertus Fischer Reviewed by: two anonymous referees