2.1 Terminology

The abundance of stable water isotopes in a reservoir is quantified as the concentration ratio of the rare (HD¹⁶O or ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$) to the abundant (${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$) isotopologue. Note that the definition here is referred to as the molecular isotope ratio, as measured by laser spectrometry. Apart from statistical factors, this definition can be shown to be equal, in the first order, to the atomic isotope ratio R determined by mass spectrometry (Mook et al., 2001). The (molecular) isotope ratio of hydrogen in a water reservoir, for example, is then as follows:

Isotope abundance is generally reported as a deviation of the isotope ratio of a sample relative to that of a standard, known as δ value, and is commonly expressed in units of per mil (‰) deviation from the Vienna Standard Mean Ocean Water 2 (VSMOW2) by means of ${}^{\mathrm{2}}{R}_{\text{VSMOW2}}=(\mathrm{155.76}\pm \mathrm{0.05})\times {\mathrm{10}}^{-\mathrm{6}}$ (Hagemann et al., 1970) and ${}^{\mathrm{18}}{R}_{\text{VSMOW2}}=(\mathrm{2005.20}\pm \mathrm{0.45})\times {\mathrm{10}}^{-\mathrm{6}}$ (Baertschi, 1976). The VSMOW2 is distributed by the International Atomic Energy Agency (International Atomic Energy Agency, 2009) and δ is expressed as follows:

The magnitude of the deviation introduced by the mixing ratio dependency is typically at least 1 order of magnitude less than the span of the isotope compositions of the measured standards. To focus on the deviation of the measured isotope compositions at various mixing ratios (δ_meas) from the isotope composition at a reference mixing ratio (δ_cor), we use the Δδ notation, which is defined as follows:

We choose 20 000 ppmv as a reference level within the nominal optimal performance range of the CRDS analyser. The isotope composition at the exact value of 20 000 ppmv is thus obtained from a linear interpolation between the closest measurements above and below 20 000 ppmv (mostly between 19 000 and 21 000 ppmv). Note that all mixing ratios reported in this study are direct (raw) measurements from the CRDS and that the isotope composition–mixing ratio dependency correction is applied to the raw data before calibration.

While Δδ¹⁸O and ΔδD are given directly by the definition above, the deviation for the secondary parameter $d\text{-excess}=\mathit{\delta }\text{D}-\mathrm{8}\cdot {\mathit{\delta }}^{\mathrm{18}}\mathrm{O}$ (Dansgaard, 1964) is obtained from the following calculation: $\mathrm{\Delta }d\text{-excess}=\mathrm{\Delta }\mathit{\delta }\text{D}-\mathrm{8}\cdot \mathrm{\Delta }{\mathit{\delta }}^{\mathrm{18}}\mathrm{O}$.

2.2 Instruments

The instruments investigated in this study include two Picarro L2140-i analysers (serial nos. HKDS2038 and HKDS2039) and one Picarro L2130-i analyser (serial no. HIDS2254; all from Picarro, Inc., USA). Hereafter, we refer to each instrument by their serial number. The instruments record at a data rate of ∼1.25 Hz and with an air flow of ∼35 sccm through the cavity. To minimise instrument drift and errors from the spectral fitting, these CRDS systems control the pressure and temperature of their cavities at (80±0.02) ^∘C and (50±0.1) Torr ((66.66±0.13) hPa) precisely.

For the spectral fitting, the instruments target three absorption lines of water vapour in the region of 7199–7200 cm⁻¹ (Steig et al., 2014). In the CRDS, a laser saturates the measurement cavity at one of the selected absorption wavelengths. After switching the laser off, a photodetector measures the decay (ring down) of photons leaving the cavity through the semi-transparent mirrors (slightly less than 100 % reflectivity). The ring-down time is inversely related to the total optical loss in the cavity. For an empty cavity, the ring-down time is solely determined by the reflectivity of the mirrors. For a cavity containing gas that absorbs light, the ring-down time will be shorter due to the additional absorption from the gas. The absorption intensity at a particular wavelength can be determined by comparing the ring-down time of an empty cavity to the ring-down time of a cavity that contains gas. The absorption intensities at all measured wavelengths generate an optical spectrum in which the height or underlying area of each absorption peak is proportional to the concentration of the molecule that generated the signal. The height or underlying area of each absorption peak is calculated based on the proper fitting of the absorption baseline. At lower water vapour concentrations, the signal-to-noise ratio decreases and the fitting algorithms are affected by various error sources (see Sect. 7).

As a custom modification, the L2130-i (serial no. HIDS2254) operates with two additional lasers that allow for rapid switching between the three target wavelengths, which enables a higher (5 Hz) data acquisition rate and a larger cavity-flow rate than a regular L2130-i. In the present study, we used the flow rates and measurement frequencies as for regular L2130-i analysers. The L2130-i uses peak absorption for the spectral fitting, whereas the L2140-i uses an integrated absorption within the spectral features (Steig et al., 2014). The L2140-i is therefore substantially less sensitive to the pressure broadening and narrowing of the absorption lines due to changes in the matrix gas that can affect older generation analysers, such as the L2120-i (Johnson and Rella, 2017). Manufacturer specifications commonly state a measurement range for vapour from 1000 to 50 000 ppmv. As a custom modification, all instruments used here have been calibrated down to 200 ppmv with an unspecified procedure by the manufacturer.

Water vapour measurements with these instruments have a total error budget that involves the uncertainty from the calibration standards projecting onto the VSMOW2 and Standard Light Antarctic Precipitation 2 (SLAP2; collectively VSMOW2–SLAP2) scale and from the time-averaging method employed for the native time resolution data. The Allan deviation quantifies the precision – depending on the averaging time interval. Previous studies found typical Allan deviations of <0.1 ‰ for δ¹⁸O and ∼0.1 ‰ for δD at 15 700 ppmv for averaging times of 1–2 min for the L2130-i (Aemisegger et al., 2012) and similar values for these averaging times for the L2140-i (Steig et al., 2014). Any corrections for the mixing ratio dependency are applied to the raw data at a native time resolution. The uncertainty of any correction is thereby given from a combination of the averaging time of the vapour measurements at a given mixing ratio and the uncertainty of the employed calibration standards.

Table 2Isotope compositions of the standard waters used in this study. The values are reported on the VSMOW2–SLAP2 scale. FARLAB standards are the laboratory standards used at FARLAB, University of Bergen, and UI standards are the laboratory standards used at the isotope laboratory of the University of Iceland. All the waters are at laboratory working standards, except MIX and TAP. MIX is obtained from an even mixing of GSM1 and VATS. TAP is deionised tap water from Bergen, Norway. The isotope compositions of these two waters are calibrated in experiment 9 by using the measured working standards. The σ for FARLAB standards represents the standard deviation of the mean for the six liquid injections, while σ for UI standards is a long-term standard deviation.

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Table 3Summary of the experiment design ordered by the aims and including the dependency on the vapour-generating method, the dependency on the tested instrument, the long-term stability of the dependency behaviour, the influence from the dry gas supply, and the influence of the measuring sequence.

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2.3 Standard waters

To identify the influence of the isotope composition on the mixing ratio dependency, we have used multiple internal standard waters calibrated on the international VSMOW2–SLAP2 scale to characterise the mixing ratio dependency. The standard waters include four laboratory standards in use at the Facility for Advanced Isotopic Research and Monitoring of Weather, Climate and Biogeochemical Cycling (FARLAB) of the University of Bergen and three laboratory standards in use at the isotope laboratory of the University of Iceland (UI; Table 2). For the FARLAB standards, one is obtained from snow in Greenland (GSM1), one is mountain snow from Norway (VATS), one consists of deionised tap water at Bergen (DI), and one is evaporated DI water (EVAP). Besides the four laboratory standards, we have also used an even mixing between GSM1 and VATS (MIX) and the uncalibrated deionised tap water (TAP). For the UI standards, one is from snow at the North Greenland Eemian Ice Drilling (NEEM) site in Greenland, one consists of groundwater in Reykjavik (GV), and one is Milli-Q® purified water based on ocean water from Bermuda (BERM). The isotope compositions of all the waters used span from −33.52 ‰ to 5.03 ‰ for δ¹⁸O and from −262.95 ‰ to 6.26 ‰ for δD (Table 2).

2.4 Vapour-generation methods

We use two methods to generate vapour for characterising the isotope composition–mixing ratio dependency of the three instruments, namely discrete liquid injections and continuous vapour streaming. These are essentially two ways of generating a vapour sample to be analysed by the infrared spectrometers. Both methods involve the injection of a liquid standard water into a heated evaporation chamber in which the injected water is completely evaporated and mixed with a dry matrix gas.

2.5 In situ measurement data for studying the impact of the isotope composition–mixing ratio dependency correction

In order to test the impact of the isotope composition–mixing ratio dependency correction on actual measurements, we applied the proposed correction scheme to two data sets obtained from in situ vapour measurements during the Iceland–Greenland Seas Project (Renfrew et al., 2019) on board a research aircraft (analyser HIDS2254) and a research vessel (analyser HKDS2038) in March 2018 on the Iceland–Greenland seas (∼68^∘ N, 12^∘ W).

The HIDS2254 analyser was installed on board a Twin Otter research aircraft. The instrument was fixed on a rack on the right side of the non-pressurised cabin. A 3.5 m stainless-steel tube with 3∕8 in. diameter, insulated and heated to 50 ^∘C, was led from a backward-facing inlet located behind the right cockpit door to the analyser. A backward-facing inlet was selected to ensure that only vapour (and not particles or droplets) would be sampled. A manifold pump was used to draw the vapour through the inlet at a flow rate of about 8 slpm. The HIDS2254 took a sub-sample through a 0.2 m stainless-steel tubing in low-flow mode at a flow rate of ∼35 sccm. The selected vapour measurements from the aircraft were taken in the lower troposphere above the Iceland Sea during a cold air outbreak (CAO) on 4 March 2018. The particular water vapour measurement segment utilised here was taken during a 9 min long descent of the aircraft from 2900 to 180 m a.s.l. A Greenland blocking associated with northerlies in the Iceland–Greenland seas caused cold atmospheric temperatures, with an average of −12 ^∘C at altitudes below 200 m. Accordingly, mixing ratios at low levels ranged from 2000 to 2700 ppmv. At higher levels, mixing ratios were as low as about 900 ppmv. After applying any correction schemes (see below), the water vapour isotope data from the aircraft are calibrated to the VSMOW2–SLAP2 scale using the long-term average of calibrations, with internal FARLAB laboratory standards on GSM1 and DI, by using the SDM from before and after the flight survey (details described in a forthcoming publication).

For the vapour measurements on board the research vessel (R/V Alliance), the HKDS2038 analyser was installed inside a heated measurement container that was placed on the crew deck at about 6 m above the water's surface. The ambient air was drawn into the container with a flow rate of around 8 slpm by a manifold pump through a 5 m long 1∕4 in. stainless-steel tube and was heated to about 50 ^∘C with self-regulating heating tape. The tube inlet was mounted 4 m above the deck and was protected from precipitation with a downward-facing tin can. The selected time period from the research vessel was acquired during a CAO event between 14 and 16 March 2018. At the beginning of the event, the mixing ratio dropped from 6000 to below 3000 ppmv within 2 h and then stayed at around 3000 ppmv for about 24 h before it increased to 8000 ppmv again.

Figure 1Mixing ratio dependency of uncalibrated measurements for (a) δ¹⁸O, (b) δD, and (c) d-excess for five standard waters (GSM1, MIX, VATS, DI, and EVAP) on instrument HIDS2254 (Picarro L2130-i) with discrete liquid injections via an autosampler (experiment 1). Mixing ratio dependency is expressed as the deviation Δ of the measured isotope composition at each mixing ratio with respect to the reference mixing ratio of 20 000 ppmv. The symbol and error bar represents the mean and the standard deviation of the mean for the last three of a total of four injections at each mixing ratio. Solid lines are fits with the function $f\left(x\right)=\frac{a}{x}+bx+c$. Dashed lines are the 95 % confidence interval for the corresponding fit. Measurements and fits for d-excess are calculated with $\mathrm{\Delta }d\text{-excess}=\mathrm{\Delta }\mathit{\delta }\text{D}-\mathrm{8}\cdot \mathrm{\Delta }\mathit{\delta }{}^{\mathrm{18}}\mathrm{O}$. The typical 1 standard deviation of a single injection at the corresponding mixing ratio is indicated with grey error bars. Two outliers (at about 4300 and 8900 ppmv) are removed from the GSM1 measurements, and one outlier (at about 9000 ppmv) is removed from the VATS measurements.

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5.1 General formulation

1. We obtain the mixing ratio dependency for each water standard as raw (uncorrected, uncalibrated) measurements of the isotope compositions. The water standards thereby cover a wide range of isotope compositions and different mixing ratios, particularly also at low mixing ratios.

2. We express the mixing ratio dependency for each water standard as the deviation of the raw measurements to the reference value at 20 000 ppmv (Eq. 3). The reference value is obtained from a linear interpolation between the closest measurements above and below 20 000 ppmv. These deviations are denoted as Δδ¹⁸O, ΔδD, and Δd-excess as described in Sect. 2.1.

3. A suitable fitting function is fitted to the mixing ratio dependency of each standard water. Here we used fitting functions with the following form:

   $$\begin{array}{}\text{(4)}& f_{\mathit{\delta }}\left(x\right)={\displaystyle \frac{a_{\mathit{\delta }}}{x}}+b_{\mathit{\delta }}x+c_{\mathit{\delta }},\end{array}$$

   where x is the mixing ratio, δ indicates the isotope composition of the standard waters, and a_δ, b_δ, and c_δ are fitting coefficients for each water standard and isotope species.

4. We express the obtained fitting coefficients as a function of the isotope composition as a(δ), b(δ), and c(δ) for all the standard waters (Fig. 2, symbols). This reveals a dependency of the fitting coefficients on the isotope composition of the water standard. We now fit a suitable function to this dependency by using the following quadratic polynomial:

   $$\begin{array}{}\text{(5)}& \left\{\begin{array}{l}a\left(\mathit{\delta }\right)=m_a(\mathit{\delta }-n_a{)}^{\mathrm{2}}+k_a,\\ b\left(\mathit{\delta }\right)=m_b(\mathit{\delta }-n_b{)}^{\mathrm{2}}+k_b,\\ c\left(\mathit{\delta }\right)=m_c(\mathit{\delta }-n_c{)}^{\mathrm{2}}+k_c,\end{array}\right.\end{array}$$

   where δ is the isotope composition and m, n, and k are the respective fitting coefficients of the quadratic polynomials.

5. By replacing the parameters a_δ, b_δ, and c_δ in Eq. (4) with their parametric expressions in Eq. (5), we obtain the generalised fitting for the mixing ratio dependency, which is a function of both the mixing ratio x and the isotope composition of the measured water δ as follows:

   $$\begin{array}{}\text{(6)}& f(x,\mathit{\delta })={\displaystyle \frac{a\left(\mathit{\delta }\right)}{x}}+b\left(\mathit{\delta }\right)x+c\left(\mathit{\delta }\right).\end{array}$$

6. Using Eq. (6), we can now correct the measured isotope compositions to a reference mixing ratio at 20 000 ppmv when given any measured raw mixing ratio and isotope composition within the range investigated here. Thus, the isotope composition at 20 000 ppmv (δ_cor) is the unknown; its analytical solution is found by solving the following equation:

   $$\begin{array}{}\text{(7)}& {\mathit{\delta }}_{\text{meas}}-{\mathit{\delta }}_{\text{cor}}={\displaystyle \frac{a\left({\mathit{\delta }}_{\text{cor}}\right)}{h}}+b\left({\mathit{\delta }}_{\text{cor}}\right)\cdot h+c\left({\mathit{\delta }}_{\text{cor}}\right),\end{array}$$

   where h is the measured raw mixing ratio and δ_meas is the measured isotope composition at that mixing ratio. The right-hand side of the equation is the isotopic deviation determined from Eq. (6). The coefficients a(δ_cor), b(δ_cor), and c(δ_cor) are determined from Eq. (5). Equation (7) is a quadratic function; the procedure to obtain its analytical solution is given in Appendix C.

5.2 Correction function for analyser HIDS2254

We now exemplify the general steps given previously for the HIDS2254 analyser. The results from step 1 and 2 for HIDS2254 are presented in Sect. 3. Here we use a range from −33.07 ‰ to 5.03 ‰ for δ¹⁸O and from −262.95 ‰ to 4.75 ‰ for δD, with mixing ratios between 500 and 25 000 ppmv (experiment 1 in Table 1).

Table 4Fitting coefficients for the mixing ratio dependency behaviour of the five standard waters measured with HIDS2254. Coefficients a, b, and c are calculated with respect to the fitting function $f_{\mathit{\delta }}\left(x\right)=\frac{a_{\mathit{\delta }}}{x}+b_{\mathit{\delta }}\cdot x+c_{\mathit{\delta }}$. The reported uncertainty is 1 standard deviation. The fitting lines are shown in Fig. 1.

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Table 5Fitting coefficients for the isotope composition dependency of the mixing ratio dependency coefficients a,b, and c in Table 4. Coefficients m,n, and k are with respect to the fitting function $f_j\left(\mathit{\delta }\right)=m_j(\mathit{\delta }-n_j{)}^{\mathrm{2}}+k_j$. The reported uncertainty is 1 standard deviation. The fitting lines are shown in Fig. 2.

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The coefficients a_δ, b_δ, and c_δ obtained in step 3 from Eq. (4) for the five standard waters measured on instrument HIDS2254 are given in Table 4. While the magnitude differs between the coefficients, scaling analysis shows that each of the terms ($\frac{a}{x}$, bx, and c) on the right-hand side of Eq. (4) contributes similarly to the isotope composition–mixing ratio dependency (not shown). The fitting results from step 4 are shown in Fig. 1 (solid colour line). The choice of this type of function captures the behaviour of the isotope composition–mixing ratio dependency for both Δδ¹⁸O and ΔδD of each standard water. Thus, the fit for Δd-excess is calculated from the fit of Δδ by $\mathrm{\Delta }d{\text{-excess}}_{\text{fit}}=\mathrm{\Delta }\mathit{\delta }\text{D}-\mathrm{8}\cdot \mathrm{\Delta }{\mathit{\delta }}^{\text{18}}\mathrm{O}$.

The coefficients m, n, and k obtained in step 4 in Eq. (5) are given in Table 5. The fitting results (solid line) with the fitting uncertainty (95 % confidence interval; black dotted line) are shown in Fig. 2. Since we only fit 5 data points, the fitting uncertainty is large, which results in a relatively large standard deviation for the isotope composition deviations Δδ. This large standard deviation for Δδ can be reduced by using a bootstrapping approach (Efron, 1979) to estimate the fitting uncertainty in Eq. (5; Appendix B).

Figure 3Surface function of the isotopic deviations for (a) δ¹⁸O and (b) δD based on the isotope composition–mixing ratio dependency of instrument HIDS2254 (Picarro L2130-i). The x axis is the raw mixing ratio, and the y axis shows the raw isotope composition at 20 000 ppmv. Contours with numbers indicate the isotopic deviation of Δδ. Symbols show the isotope measurements over the Iceland Sea: measurements averaged to 1 min from an aircraft in the lower troposphere (red crosses) and measurements at a 10 min resolution from a research vessel (blue dots). The measurements from the aircraft were done with instrument HIDS2254 (Picarro L2130-i), and measurements on the research vessel were done with instrument HKDS2038 (Picarro L2140-i).

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Following step 5, this results in a two-dimensional correction surface for each isotopologue as shown in Fig. 3 (black contours). For illustration purposes, some contours below 4000 ppmv are omitted for both δ¹⁸O and δD. The isotope composition–mixing ratio dependency for both δ¹⁸O and δD increases substantially at low mixing ratios. For δ¹⁸O, the deviation changes from positive to negative as the mixing ratio decreases below ∼4000 ppmv. For δD, the deviation increases below 10 000 ppmv and splits into both positive and negative contributions – depending on the isotope composition. The uncertainty (1 standard deviation) for the deviation Δδ is typically 1 order of magnitude smaller than the Δδ values at the corresponding position.

The surface function exhibits the same features as those determined by the experimental results, underlining that the fitting procedure reflects the main characteristics of the isotope composition–mixing ratio dependency from the experimental data. At 20 000 ppmv, the correction function is not exactly 0, as the fit that is based on all the measurements is not constrained to the point [20 000, 0] ppmv. This deficiency could be addressed by a modified fitting procedure. Below 500 ppmv, the correction function has larger uncertainties due to the lack of measurements at this mixing ratio range. Note that the fitting functions used in Eqs. (4) and (5) are purely phenomenological and do not result from a particular physical model. Still, we recommend the proposed parameterisation to characterise individual instruments.

6.1 Impact on the aircraft measurements

Low water vapour mixing ratios and a relatively wide range of (depleted) isotope compositions make water vapour isotope measurements from a research aircraft particularly suitable for demonstrating the impact of the new isotope composition–mixing ratio dependency correction. Figure 4 shows a vertical profile of 1 min averaged water vapour isotope measurements above the Iceland Sea (Sect. 2.5). During the descent of the aircraft from 2900 m a.s.l. to the minimum safe altitude, the water vapour mixing ratio gradually increases from about 800 ppmv at the top to 2300 ppmv near the surface (Fig. 4d). The stable isotope profiles (Fig. 4a–c) show three main characteristics. Above about 2000 m a.s.l., δ¹⁸O and δD are depleted ($\sim -\mathrm{42}$ ‰ and −320 ‰) with d-excess between 10 ‰ and 20 ‰. Between 2000 and 1400 m a.s.l., there is a transition where δ¹⁸O and δD increase to around −30 ‰ and −240 ‰, respectively, and d-excess decreases to $\sim -\mathrm{5}$ ‰. Below 1400 m a.s.l., δ¹⁸O, δD, and d-excess gradually increase until reaching about −22 ‰, −170 ‰, and 8 ‰, respectively, near the surface. The uncertainty (1 standard deviation) of the profile is obtained using the uncertainty propagation law, including the uncertainty (1 standard deviation) of the 1 min averaged data set (here the dominating source of uncertainty) and the uncertainty of the correction scheme.

First we investigate the impact of the new correction scheme introduced here. This correction, abbreviated as iso-hum-corr, modifies the uncorrected data set in the region above 2000 m a.s.l. by about −0.4 ‰ and −13.3 ‰ for δ¹⁸O and δD, respectively, resulting in a change of about −10.4 ‰ for d-excess (Fig. 4; red circles vs. black dots). The impact of applying the new correction scheme to the aircraft measurements can be understood by examining where the data sets align in the correction surface function (Fig. 3; red crosses). For both δ¹⁸O and δD, the aircraft data set is characterised by a low mixing ratio and depleted isotope compositions, and it is clustered in the bottom-left corner of the surface function. This is the most sensitive area of the correction, thus causing the largest deviations in the surface function.

To assess the benefit of the new isotope composition–mixing ratio dependency correction, we take this new correction scheme as the reference scheme and compare its impact to three other correction schemes. The first scheme only corrects for the mixing ratio dependency based on standard DI (hum-corr-DI, Fig. 4; orange crosses). The second correction follows the same procedure but is based on standard GSM1 (hum-corr-GSM1, Fig. 4; blue squares), and the third correction follows an approach proposed by Bonne et al. (2014) for in situ vapour measurements in southern Greenland. Instead of correcting the mixing ratio dependency for the vapour measurements, the Bonne et al. (2014) approach corrects the mixing ratio dependency for the calibration standards. Thus, by assuming that the mixing ratio dependencies of the two employed standards remain stable during the measurement period, the measured isotope compositions of these two standards are corrected by using the mixing ratio dependency function to the ambient air mixing ratio of each single vapour measurement. Then, the linear regression computed from these two corrected standard measurements against their certified values is applied to calibrate the vapour measurement to the VSMOW2–SLAP2 scale. This scheme is hereafter referred to as 2-std-hum-corr (Fig. 4; green triangles).

The different correction schemes modify the uncorrected data set differently. The δ¹⁸O profile is only marginally affected by the correction below 1400 m a.s.l. (Fig. 4a). For the measurements above 1400 m a.s.l., differences become more pronounced but are masked by the large uncertainty as the aircraft was descending through a strong mixing-ratio gradient. All corrections show clear deviations at elevations above 2000 m a.s.l., and we focus our comparison on this region. The hum-corr-DI stands out, with a positive correction of about 0.8 ‰, while the other three schemes induce a negative correction of between −0.3 ‰ and −0.7 ‰. The δD profile exhibits a similar pattern but with more apparent differences between the correction schemes (Fig. 4b). This results in an even more pronounced correction in the d-excess profile (Fig. 4c).

The impact of applying the correction scheme using single standard water (thus accounting for only mixing ratio dependency) relies on the choice of the used standard water. Using the hum-corr-DI correction introduces the largest deviation (1.1 ‰, 18.3 ‰, and 9.5 ‰ for δ¹⁸O, δD, and d-excess respectively), while using the hum-corr-GSM1 correction produces results much closer to that of the reference scheme (with an offset of 0.1 ‰ and 4.2 ‰ for δ¹⁸O and δD, respectively, and 3.4 ‰ for d-excess). For this specific aircraft measurement (where the surface condition is already quite dry during the cold air outbreak event) the isotope composition of standard GSM1 happens to closely resemble the average isotope composition of the measurement. However, in the case of a previously unknown range of isotope compositions or strongly varying conditions, a comprehensive characterisation of mixing ratio dependency with multiple standard waters can provide advantages and should be preferred. Unknown ranges are particularly likely for atmospheric measurements of vertical profiles in humid regions (e.g. the tropics and subtropics) or over a wide area from moving platforms.

Finally, applying the alternative calibration approach (2-std-hum-corr) used in Bonne et al. (2014) results in only slightly more depleted isotope values than the reference, with an offset of about −0.3 ‰ and −0.5 ‰ for δ¹⁸O and δD, respectively, resulting in a change of 1.7 ‰ for d-excess. The small discrepancy between the 2-std-hum-corr and iso-hum-corr is mainly due to three factors. First, depending on the number of the used standard waters, the interpolation scheme for the isotopic deviations in between those of the used standard waters can be different. The 2-std-hum-corr makes use of the mixing ratio dependency functions of only two standard waters. In this way, the deviations can only be linearly interpolated between the two standard waters. In contrast, the reference scheme is able to account for non-linearities during interpolation by measuring five standard waters. Second, 2-std-hum-corr corrects the two standard waters to the mixing ratio of the measurement while the reference scheme corrects the measurement to the mixing ratio of the two standard waters (i.e. the reference mixing ratio). Based on the mixing ratio dependency feature of the two standard waters, the choice of correcting the two standard waters will result in a higher slope for the VSMOW2–SLAP2 calibration line. Consequently, the measurements after calibration are stretched to two ends, i.e. the measurements with the isotope composition close to that of standard DI become more enriched and those close to that of standard GSM1 become more depleted. Finally, the mixing ratio dependency functions for GSM1 and DI in 2-std-hum-corr (using the individual fit for GSM1 or DI respectively) are not exactly identical to those used in the reference scheme (from the surface function determined by the measurements of five standard waters). Despite the small discrepancy, the consistent results of 2-std-hum-corr with that of the reference scheme indicate that a correction scheme using the mixing ratio dependency functions of only two standard waters covering the measured isotope composition range can work sufficiently well in certain situations.

6.2 Impact on the ship-based measurements

Applying the four different correction schemes to the ship-based measurement data has a much weaker impact on the corrected series of vapour measurements (not shown). After applying our isotope composition–mixing ratio dependency correction scheme, the uncorrected data set changes to the order of 0.06 ‰ and 0.15 ‰ for δ¹⁸O and δD, respectively, leading to a change to the order of −0.5 ‰ for d-excess. This is mainly because these ship measurements were carried out at the ocean surface, with relatively high mixing ratios (from 2500 to 8000 ppmv) and less-depleted isotope compositions (−23 ‰ to −12 ‰ for δ¹⁸O and −160 ‰ to −100 ‰ for δD) compared to the aircraft measurements. As shown in Fig. 3, the ship data (blue dots) are coincidentally located in an area with low sensitivity at the correction surface. A linear interpolation between two standards may not capture such a saddle point correctly. This indicates that measurements are not sensitive to the correction of the isotope composition–mixing ratio dependency under all conditions. Ultimately, however, the certainty about a reliable correction will only be achieved with a complete characterisation of the isotope composition and mixing ratios covered by the measurements.

7.1 Artefact from mixing

If we assume the dependency is as a result of mixing with remnant water, then there would mainly be two candidates for the background moisture source: (1) the remaining water vapour in the dry gas supply and (2) the remaining water vapour from previous measurements in the analyser. By changing the dry gas supply from the ambient air dried through Drierite to synthetic air or N₂ from cylinder, which typically provides a dry gas with a mixing ratio below 10 ppmv, we can exclude the possible influence of the background moisture in the dry gas supply. In order to quantify the amount and the isotope compositions of the remaining water vapour from previous measurements, we have applied several successive, so-called empty injections via the autosampler. Thus, no liquid is injected and only dry gas is flushed into the vaporiser. Results from these empty injections show that the remaining water vapour in the system typically has a mixing ratio of about 60–80 ppmv, with its isotope composition closely following those of the previous injections. If the mixing ratio dependency was a result of the mixing between the injected water and the remaining water vapour from previous measurement, then we would expect a mixing of two water vapour masses of the same isotope compositions at different mixing ratios when injecting the same standard water during a characterisation run. As a consequence, we would expect the mixed vapour to have the same isotope composition, which is not the case. Finally, the shape of the isotope composition–mixing ratio dependency with a maximum between 2000 and 6000 ppmv (Fig. 1a) is not consistent with the expectation of a memory effect that would monotonously increase with the decreasing mixing ratio. The slight hysteresis observed during the upward/downward calibration runs indicates that there may be contributions from remnant water on walls or filter surfaces in the analyser, for example, that only exchange once a sufficiently humid air mass is introduced into the analyser. Such contributions do, however, appear to be of second order when compared to the substantial changes of the mixing ratio dependency with the isotope composition.

7.2 Spectroscopic effect

Now we explore the second hypothesis, namely that the isotope composition–mixing ratio dependency is an instrument behaviour resulting from spectroscopic effects. The manufacturer recommends a procedure for water vapour dependency calibration using their SDM, or similar, device (Picarro Inc., 2017), which is similar to what we have employed. While the first-order effect can be removed from a linear fit, there are second-order, non-linear components that become more apparent the more the water concentration changes from the recommended range of operation (5000–25 000 ppmv). In the following paragraphs, we discuss the potential reasons for the origin of the water vapour and isotope dependency from a spectroscopic standpoint that is based on the published literature (Steig et al., 2014; Rella et al., 2015; Johnson and Rella, 2017).

Figure 5Absorption spectrum of ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$, ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$, and HD¹⁶O in the frequency range targeted by the laser of Picarro models L2130-i and L2140-i. Simulations with two water concentrations, i.e. 25 000 (black line) and 5000 ppmv (orange line), were performed by using http://spectraplot.com (last access: 19 February 2020) with the HITRAN–HITEMP database (Goldenstein et al., 2017). Simulation parameters are set according to the cavity conditions of the Picarro analyser as follows: T=80 ^∘C, P=50 Torr, and L=1 cm. Panel (a) is an enlarged version of the shaded area in panel (b).

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The two modules of the CRDS analyser used in our experiments (i.e. Picarro L2140-i and L2140-i) target three absorption lines of water vapour in the region 7199–7200 cm⁻¹, namely near 7199.960, 7200.135, and 7200.305 cm⁻¹ for ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$, ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$, and HD¹⁶O respectively (Fig. 5). These absorption lines broaden or narrow, depending on the partial pressure of the gas mixture in the cavity, and can be affected by changes in their baseline due to other strong absorption lines nearby. A fitting algorithm then fits the measured absorption spectrum to an expected model spectrum and adjusts the model parameters in order to minimise the residual error. Broadening/narrowing of lines due to changing gas mixture and baseline shifts are particular challenges to the fitting algorithm (Johnson and Rella, 2017) because this causes residuals that induce instrument error during the fitting procedure.

The L2130-i and earlier spectrometers use the absorption peak as a free parameter in the fitting algorithm. The peak shape and, thus, the peak amplitude can suffer from the above-mentioned broadening/narrowing effect, which introduces potential error under conditions of varying concentration or matrix gases. The fitting algorithms of the L2140-i spectrometers, in contrast, have a higher number of ring downs due to a different strategy for obtaining laser resonance and use a different laser stabilisation scheme. This allows us to fit the integrated absorption, rather than the peak amplitude, of each absorption line. Since the integrated absorption is a constant independent of pressure, the fitting is expected to be more accurate, with a low sensitivity to broadening/narrowing effects arising from changes of water concentrations and background gas compositions (Steig et al., 2014). One part of the retrieval algorithm is the removal of the baseline from the ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$ spectrum. To this end, changes in the baseline from nearby strong absorption lines as a result of concentration changes or cross-interference from other gas species is a possible source of error for either fitting algorithm. Other possible sources of error can be absorption loss non-linearities due to small imperfections in the instrument, such as the non-zero shut-off time of the laser and the response time of the ring-down detector (Rella et al., 2015). Unless fitting algorithms take the actual line shapes into consideration directly, some residual effects are likely to persist.

The retrieval of H₂O concentration and the stable isotope compositions of δ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$ and δHD¹⁶O (identical to δ¹⁸O and δD) are implemented in a manner that is similar to the procedure described for CH₄ and δ¹³CH₄ (Rella et al., 2015). Considering a linear dependency of absorption to concentration (which is not always true), where the mole fraction of ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$ (c₁₈) is related to the absorption peak height α₁₈ with a proportionality constant k₁₈ and an error offset ϵ₁₈, we have the following equation:

Note that the expressions above should apply to both the L2130-i and L2140-i spectrometers, with the only difference being that the absorption peak height is replaced by the integrated absorption (Steig et al., 2014).

Based on the molecular definition of a δ value with respect to the VSMOW2, an isotope ratio of the sample ${}^{\mathrm{18}}R=\frac{\left[{\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}\right]}{\left[{\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}\right]}$, and of VSMOW2 ${}^{\mathrm{18}}{R}_{\text{VSMOW2}}=(\mathrm{2005.20}\pm \mathrm{0.45})\times {\mathrm{10}}^{-\mathrm{6}}$ (Baertschi, 1976), we have the following equation:

The retrieval can then be formulated as follows:

For an ideal spectrometer, the calibration coefficients are constants (i.e. k₁₈=κ₁₈ and k₁₆=κ₁₆), and the calibration offsets are 0 (i.e. ${\mathit{ϵ}}_{\mathrm{18}}={\mathit{ϵ}}_{\mathrm{16}}=\mathrm{0}$). These assumptions lead to the expected retrieval for a spectrometer as follows:

The actual spectrometer is not ideal, but in most situations it has a highly linear and stable performance (Rella et al., 2015). Nevertheless, it can be calibrated based on the linear dependency of $\mathit{\delta }{\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$ to $\frac{{\mathit{\alpha }}_{\mathrm{18}}}{{\mathit{\alpha }}_{\mathrm{16}}}$ by using a linear expression with the following form:

where the calibration constants A and B can be determined based on the measurable quantities $\mathit{\delta }{\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$ and the $\frac{{\mathit{\alpha }}_{\mathrm{18}}}{{\mathit{\alpha }}_{\mathrm{16}}}$ from a reference instrument in the factory. These determined calibration constants deviate slightly from the expected values in Eq. (11). They are then transferred from the reference instrument to each new instrument of the same type (Rella et al., 2015).

If Eq. (12) is used for calibration of the water analysers (not reported in the published literature), then there are two potential sources of error. First, the dependencies on the isotope ratio may not be entirely linear (even when assuming a linear relationship the coefficient is not necessarily a constant and the offset is not necessarily 0) and remain as residuals. Second, the change of this relationship with the differing mixing ratio may remain unexplored. Furthermore, manufacturing tolerances will induce deviations from the reference instrument on which such an initial calibration has been carried out, and instruments therefore have to be calibrated individually to obtain suitable post-processing methods. The initial instrument calibration procedure may therefore be one potential origin for the isotope composition–mixing ratio dependency identified here.

Deviations from an ideal spectrometer stem from the potential spectrometer errors due to small imperfections of the instrument. One possible error is the absorption-loss offset that could occur when the baseline loss is not reproduced well by the fitting algorithm. This absorption offset then leads to a mole-fraction offset, namely ϵ₁₈ in Eq. (8). The measured $\mathit{\delta }{\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$, including the effect of absorption offset, can be formulated explicitly by following Rella et al. (2015) – and their Appendix S1.1 – as follows:

where α₀ is the net absorption loss parameter that should, to the first order, be independent of water concentration and isotope ratio. Comparing this to Eq. (11) for an ideal spectrometer, the additional term of $\frac{{\mathit{\alpha }}_{\mathrm{0}}}{{\mathit{\alpha }}_{\mathrm{16}}}$ in Eq. (13) creates an inverse relationship with the water concentration and should be responsible for deviations from the ideal spectrometer that are mostly evident at low mixing ratios.

Another possible spectrometer error is the so-called absorption loss non-linearity, which describes effects due to a shorter or longer ring-down time than expected in the optimal range of operations. These effects can be included as additional terms, again by following Rella et al. (2015) – and their Appendix S1.2 – with a non-linear dependency of α₁₈ on α₁₆ (i.e. ${\mathit{\alpha }}_{\mathrm{18}}\Rightarrow {\mathit{\alpha }}_{\mathrm{18}}+\mathit{\beta }{\mathit{\alpha }}_{\mathrm{16}}+\mathit{\gamma }{\mathit{\alpha }}_{\mathrm{16}}^{\mathrm{2}}$) as follows:

The calibration coefficients are thus assumed constant. The non-linearities from spectral crosstalk between ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$ and ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$ or imperfections in the baseline removal of the ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$ absorption spectrum are present in several terms; such effects require the calibration of each individual instrument to be accounted for. When written in this explicit form, it appears consistent with expectations that a systematic isotope composition–mixing ratio dependency may be detected from careful analyser characterisation. The equation can be further simplified as follows:

The difference between Eq. (15) and Eq. (12) represents the deviation from an ideal spectrometer due to non-linearities from imperfections in the baseline removal and spectral crosstalk between ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$ and ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$ and can be denoted as follows:

The dependency on water concentration (c₁₆) in Eq. (16) appears consistent with the mixing ratio dependency function (Eq. 7) identified in our systematic investigation of three analysers, supporting the hypothesis of a spectrometric origin for the isotope composition–mixing ratio dependency.

A similar form of mixing ratio correction is applied to the ¹⁷O measurements using the L2140-i analyser in the study of Steig et al. (2014), and their Eq. 22, in which the integrated absorption area, instead of the peak amplitude, is used to calculate the absorption loss, and the crosstalk between ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{18}}\mathrm{O}$ and ${\mathrm{H}}_{\mathrm{2}}^{\mathrm{16}}\mathrm{O}$ is modelled with a bilinear relationship. Steig et al. (2014) note that the introduction of an integrated absorption detection leads to a substantially improved behaviour for the mixing ratio dependency over the peak amplitude detection for δ¹⁸O but not for δD, with the reason remaining unclear. It is also worth noting that their instrument has not been evaluated for the low mixing ratio range, which is the focus of this paper. It may be possible that part of the identified isotope composition dependency of the mixing ratio dependency stems from the, thus far, lacking systematic analysis of the low mixing ratio range of the analyser for this effect.

One aspect that is not addressed here, but may be valuable for further consideration in the future, is the availability of analysers with higher flow rates of above 300 sccm, for example, for use in research aircraft (Sodemann et al., 2017). Given recent indications that the flow rate affects the isotope composition–mixing ratio dependency (Thurnherr et al., 2020), forthcoming studies should explore the flow rate as an additional parameter. This requires the availability of suitable methods to generate standard vapour at these higher flow rates.