2.1. Isotopic Composition Analysis by Laser Absorption Spectroscopy

According to the Beer-Lambert law of linear absorption, the absorbanceA(v) can be related to the transmitted lightI(v) and incident intensityI₀(v) intensities:

The isotope ratio can thus be determined from the ratio of the integrated areasA_I and the absorption line intensities S of the major and minor isotopic components:

The relative deviation of the isotope ratio in water with respect to the international standard reference known as Vienna Standard Mean Ocean Water (VSMOW), is expressed in terms of the δ-value:

In the present work, the laser instrument was calibrated against the working standards GS-49 (δ¹⁸O = 0.39‰, δ¹⁷O = 0.21‰, and δ²H = 1.7‰, with respect to VSMOW, determined by repeated IRMS analyses at the Center for Isotope Research of the University of Groningen). The accuracy of the laser instrument was evaluated by measurement of another working standard GS-42 with different isotopic composition (δ¹⁸O = −24.62‰, δ¹⁷O = −13.1‰, and δ²H = −187.7‰). Bottled water (Vittel, France) was used as unknown sample material.

2.2. Selection of the Absorption Lines

Selection of suitable absorption lines for water isotopic ratio measurements is one of the most important aspects in instrumental design since the choice of absorption lines has a direct impact on the instrumental performance in terms of measurement sensitivity, precision and selectivity.

Measurements of the isotopic ratios δ¹⁸O, δ¹⁷O, δ²H of the stable isotopologues of water require probing of absorption lines of the four isotopologues H₂¹⁸O, H₂¹⁶O, H₂¹⁷O and HDO within the laser tuning range. It is very desirable that the used absorption lines exhibit similar absorption depths at natural abundance with an absorption intensity as large as possible (for high sensitivity and high precision measurements), are free from interference from the same or other species (high selectivity consideration), and have similar ground state energy (in order to minimize the effects of temperature-dependent line intensities). The spectral region near 2.73 μm, covered by recently commercially available DFB lasers, meets these requirements to a high degree. Parameters of the four molecular lines selected in the present work are summarized inTable 1. Parameters of the previously used lines at 1.4 μm and 6.7 μm are also given for comparison.

The line absorption strength depends on the population of the ground state level and this population in turn depends on the temperature. Temperature drift during measurements may introduce a systematic error in the isotope ratio determination. The temperature coefficients defined as ΔS/S(T₀) for the selected lines are listed inTable 1. A thermal drift of 1 K would lead to a relative variation in the line strength of +1.5‰, +4.6‰, −1.4‰, −3.4‰ for the selected lines of H¹⁸OH, H¹⁶OH, H¹⁷OH and H¹⁶OD respectively. Therefore, realization of a temperature stability better than 0.1 K is essential for high precision measurements.

3.1. Experimental Set-Up

The experimental set-up, mounted on a 50 × 70 cm² optical breadboard, is depicted inFigure 1. The laser source was a room temperature single mode DFB diode laser operating at 2.73 μm (Nanoplus GmbH, Gerbrunn, Germany). It is tunable from 2729 to 2732 nm with an output power of 2 mW. A Thorlabs (Newton, NJ, USA) ITC 502 diode laser controller provided the laser temperature control and laser drive current. The diverging laser beam was collected by an off-axis parabolic mirror PM1 with an effective focal length (EFL) of 25 mm. The laser beam was then transformed into a quasi-parallel beam with a diameter of ∼4 mm by a combination of an antireflection coated CaF₂ lens F1 (f = 200 mm) and an off-axis parabolic mirror PM2 (EFL = 50 mm).

The collimated beam was then divided into two parts. The first part (∼8%) was reflected by a beam splitter (CaF₂) and directed to a homemade Fabry-Perot etalon for spectral metrology. The frequency scale was linearized by means of the interference fringes produced by the etalon consisting of two air-separated uncoated CaF₂ plates with a free spectral range (FSR) of ∼0.0283 cm⁻¹. Positions of the H₂O vapor absorption lines provided by the HITRAN database [29] were used as absolute frequency reference for laser frequency calibration. The main part of the laser beam was coupled into a Herriott cell with an absorption path of 20 m in a 36-pass configuration. The emerging absorption signal was focused with a 50 mm CaF₂ lens F2 onto a LN₂ cooled HgCdTe detector (J15D22-M204-S01M-60). A home-built bridge circuit was employed to realize DC coupling of the HgCdTe detector to a low noise preamplifier (Model 5113, EG&G, Albuquerque, NM, USA).

The signal background (corresponding to “laser off”), determined by the dark current level of the detector, was found to be fluctuating by about 1% over a 1 h time interval. This probably was due to variation of the detector temperature or its DC power supply. In order to correctly retrieve absorption spectra, a beam shutter (Thorlabs, SH05) was placed before the detector for background level acquisition at the beginning of each absorption spectral scan.

The laser frequency was periodically scanned at a rate of 10 Hz across the absorption lines of the water isotopologues H¹⁶OH, H¹⁷OH, H¹⁸OH, and HDO by means of a triangular wave voltage. The output signal from the preamplifier was digitized with a laptop using a 16-bit analogue/digital data acquisition card (DAQ Card-6036E, National Instruments, (Austin, TX, USA) controlled with a Lab Windows-based program.

3.2. Experimental Protocol

Careful attention has been paid in the present work to sample mass effects and sample memory effects that affect the determination of the isotopic ratios, as discussed in detail by, among others, Kerstelet al. [15] and more recently Liset al. [21]. Liquid water samples of 12 μL were injected into the pre-evacuated gas cell through a silicon membrane using a syringe resulting in a water vapor pressure of ∼4.5 mbar inside the gas cell (the saturated vapor pressure is ∼42 mbar), which corresponds to a H₂O molecule number density of 1.1 × 10¹⁷ mol/cm³ in the cell. Besides temperature fluctuations, variations in sample pressure inside the cell, resulting from sample injection by a syringe through a silicon membrane, will have a strong effect on the precision and accuracy of the measurements. Although constant volume was used for both sample and reference standard injections in our experiments, variances in the H₂O number density in the cell may occur due to H₂O leakage through the silicon membrane, over-tightening or under-tightening of the silicon membrane. In order to minimize the impact of this effect on the final results, only those injections resulting in a gas cell pressure within ±0.1 mbar of 4.5 mbar were accepted. Furthermore, in order to avoid sample memory effects due to the “stickiness” of water on the gas cell wall, the first three injections were discarded for each isotope ratio determination.

The temperature of the gas cell was actively controlled to 30 °C by the use of a heater band, and maintained constant within ±0.1 °C by a PID controller. The cell temperature was monitored with calibrated platinum resistors (Pt100) with an accuracy of 0.03 °C and a precision of 0.01 °C. No temperature gradient along the cell axis was observed within the measurement precision of the temperature sensors.

In our experiment, a slight drift of the laser wavelength with time has been observed. In order to minimize the effects of the instrumental drift on the determination ofA_I, only 10 laser scans were co-added, resulting in a raw δ-value at 1Hz averaged data acquisition rate. Further improvement in precision has been achieved through the use of Kalman filtering technique. As can be seen in the results presented in Section 5, the use of Kalman filtering approach allows minimizing the effect of laser wavelength drift since it filters the measurement data at 1 s time intervals.

4.1. Spectral Fitting Algorithm

In order to determine the integrated absorbanceA_I (the area under the absorption line profile), absorption spectra were fitted to Voigt [32] and Galatry [33] profiles, using a Levenberg-Marquardt multi-line fitting algorithm. In the fit procedure, the baseline approximated with a 4th-order polynomial was experimentally found to well represent laser power ramp, allowing good removal of the variation of laser power in the fitted residual. Apart from the four selected lines of H¹⁸OH, H¹⁶OH, H¹⁷OH and H¹⁶OD, two H¹⁶OD lines at 3663.2250 cm⁻¹ and 3663.3879 cm⁻¹ on either side of the H¹⁷OH line were also taken into account in the spectral fitting program. As indicated by the fit residuals shown inFigure 3, the Galatry profile (b) fit resulted in a better residual than using the Voigt profile (a). In fact, the Voigt profile does not take into account correlations between molecular velocities and collisional processes. Corrections for velocity-changing collisions are included in the soft-collision (Galatry) model [33]. Speed dependence of the relaxation rates can be accounted for with the speed-dependent (SD) Voigt profile [34]. Both Galatry and SD-Voigt profiles are reported for recovering the experimentally observed lineshapes, and Galatry profile essentially behaves like the speed-dependent Voigt model [35]. However, Galatry profile and SD-Voigt profile are based on completely different processes. As can be seen inFigure 3c, d, relatively big residuals around absorption peaks are still evidenced even using Galatry model, which leads to a bigger uncertainty in the determination of the integrated area A_I. In order to significantly minimize the uncertainty in the determination of the integrated area under the absorption features, a more sophisticated model could be adopted in the future (for instance, the speed-dependent Galatry profile), to take into account the narrowing due to the speed-dependence of relaxation rates as well as the averaging effect of velocity-changing collisions.

4.2. Suppression of Oscillation Structure on the Baseline

A periodic oscillatory structure on the baseline was observed in both the Voigt (Figure 3a) and the Galatry fit residuals (Figure 3b). This kind of sinewave-like undulation, usually assumed to be fringes resulting from optical interference, was, however, also observed with laser turned off. This means that the periodic oscillatory structure was caused by an electrical perturbation. In order to further improve the measurement precision, we applied a Fourier filtering approach [18] to suppress this oscillatory baseline structure. For this purpose, Fourier transformation of the residuals of the fit was performed in order to determine the oscillation frequencies. As can be seen in the noise power spectral density graph (Figure 4), the noise at frequencies higher than 0.001 Hz was associated with the Voigt fit residual as it has been almost completely removed in the Galatry fit. We determined that the noise peaks at frequencies lower than 0.001 Hz were associated with the baseline oscillatory structure. In order to take this undulation structure into account, the baseline was then modeled as a composition of a 4th-order polynomial and a Fourier series (FS) function of sine and cosine waves (for simulation of the sinewave-like oscillatory structure of the baseline). The transmitted power signal recorded by experiment was thus fitted to the following function:

To obtain the integrated absorbanceA_I, we fitted the F_fit function (Equation (6)) to the experimentally recorded data with fixed isotope-dependent Doppler widthγ_D (scaled with the square root of the isotopologue mass), while the following parameters were determined in the fit: the integrated absorbanceA_I, the line center positionν₀, the Lorentzian pressure broadening widthγ_L, the Dicke narrowing parameterβ_C, and the baseline modeling parameters.

Figure 3c and the black lines inFigure 4 show the results after suppression of the periodic undulation structure by fitting the spectrum toEquation (6). As can be observed inFigure 5, compared to the results obtained with a simple 4th-order polynomial as baseline model (left panel inFigure 5), baseline correction using a composition of a 4th-order polynomial and a Fourier series function (central panel inFigure 5) leads to a reduction of up to 3-fold in the standard deviation (SD) in the isotope ratio determination.

Though application of Fourier filtering can improve the measurement precision by removing oscillatory structure from the baseline, it is hard to completely account for the exact baseline structure with only three Fourier components. In fact, the residuals ofFigure 3c still show signs of a residual periodic structure. We therefore proceeded to eliminate the source of the electrical perturbation, which was determined to be related to an electronic component failure in the preamplifier. In the case of no electrical perturbation, as shown inFigure 3d, a 3.7-fold improvement in the measurement precision was obtained (right panel inFigure 5), in comparison with the results achieved using a Fourier filtering (middle panel inFigure 5).

4.3. Allan Variance

The measurement precision is usually affected by the instrument instability and measurement errors related to sample handing and injection (e.g., incomplete evacuation of the gas cell between two consecutive measurements will lead to memory effects and thus affect the measurement precision). Another important limiting factor to achieving high precision is the signal-to-noise ratio (SNR) of the spectral data. In LAS, high SNR can be obtained by enhancing the signal (by selecting stronger absorption lines and using long absorption path length or cavity enhanced spectroscopy) and reducing the noise (by averaging N laser scans and using modulation or/and balanced-beam detection techniques). With the signal averaging approach, the optimal averaging number N, limited by the stability of the instrument, can be determined by an Allan variance analysis [36]. Alternatively, the individual spectral scans can be processed, and the resulting isotope ratios may be further averaged (within the system's stability time) to obtain the desired precision level. In the present work, due to laser wavelength drift, the maximum averaging number of the laser scans was limited at about N = 10, corresponding to an acquisition time of 1 s. The 1 s raw data were processed to provide 1 s raw δ values with a 1σ precision of 7.8‰ for δ¹⁸O, 6.6‰ for δ¹⁷O, and 8.0‰ for δ²H of a bottled water (Vittel, France).

In order to further improve the measurement precision, the 1 s raw δ values are further averaged. Allan variance analysis has been performed to determine the optimum averaging number. As can be seen inFigure 6 (lower panel), the optimum averaging time for the present instrument was ∼30 s. We then used conventional averaging of 30 measured δ -values corresponding to the optimum averaging time of 30 s. Herewith the precision was improved to 1.4‰ for δ¹⁸O, 1.1‰ for δ¹⁷O, and 1.5‰ for δ²H, respectively.

4.4. Kalman Filtering

Though signal averaging enables one to improve the measurement precision [1,11], such “post-processing” results in a slow temporal response of the system. For certain specific applications, such as the on-line monitoring of exhaled breath [37], it is highly desirable to be able to perform real-time measurements with high sensitivity and precision, while maintaining a fast system response. Kalman filtering, being an adaptive filtering technique uses a recursive procedure for “true value” prediction based on the previously determined values. It can efficiently remove the shot-to-shot variability related to the real-time noise in the measured data with minimal deformation of the physical quantity to be measured. Kalman filtering has been successfully applied before to real-time trace gas concentration measurements [38,39]. We recently introduced this technique to the field of isotope ratio measurements [28] in order to perform fast and high precision measurements. In this section, we describe in detail the theoretical consideration of a Kalman filter model for application to isotope ratio measurements.

Using a linear stochastic difference model, the true isotope ratioδ̂_k+1 at timek+1 is evolved from the valueδ̂_k given atk according to:

At timek the measured isotope ratioz_k of the true valueδ̂_k can be expressed as follows:

The time update procedure projects forward in time current isotope ratio and error variance estimates to obtaina priori estimates for the next time step. The prediction equations can be expressed as:

The measurement update incorporates a new measurement into thea priori isotope ratio estimate to obtain an improveda posteriori estimate as filtered isotope ratio value, δ. The filtering equations involved in the measurement update are:

In practical applications, the measurement noise σ²_v and the true δ variability σ²_w (related to real isotope abundance variation and real-time drifts resulting from laser frequency shift, thermal fluctuation, pressure variation,etc.) should be well defined.

Information on the measurement noise variance σ²_v is usually known, because it depends on the quality of the measurement instrument, while the variance σ²_w of the true isotopic ratio variability is quite subjective. Whereas both σ²_v and σ²_w vary with real variations of the isotope abundance, once the measurement system has reached equilibrium, the ratio of σ²_v to σ²_w should be constant. We therefore define:

The parameter q is known as the filter tuning parameter. The choice of q depends on the particular instrument and its application environment. For a larger q value, the system's response time is longer to follow real-time variation in isotopic composition. Conversely, the filtering is less efficient in removing shot-to-shot real-time noise when a smaller q value is used.Figure 7a shows the 1σ precision (standard deviations) in isotope ratios determination of a bottled water (Vittel) as a function of q. As shown inFigure 7a, the improvement in the precision is clearly evident, as the standard deviation decreases exponentially with increased q value. The decreasing trend for δ¹⁸O is toward a steady state value for q > 800. For δ¹⁷O and δ²H, the decreasing trend slowed down for q > 800, while toward a steady state value for q > 2000. This may be explained by small drift in δ¹⁷O, δ²H resulting a Gaussian distribution, high q value is needed to get steady state, because the larger q value, the more efficient is the filtering δ-value fluctuations due to random noise. In the present work, a value of q = 800 was chosen as a compromise between fast temporal response and high filtering efficiency (thus high measurement precision) for the developed laser instrument. In the present work, a value of q = 800 was chosen as a compromise between fast temporal response and high filtering efficiency (thus high measurement precision) for the developed laser instrument. In our experiment, σ²_v was determined by the δ-value variance deduced from the first 10 raw measurements, and σ²_w was calculated by dividing σ²_v by q.

The effects of the Kalman filtering on the measurement accuracy were investigated in the present work. For this purpose, the working standard GS-42 with well known isotope ratios was used. The relative deviation in water (GS-42) isotope ratio measurement accuracy, (δ −δ̄);/δ̄, is plotted in function of the q-parameter inFigure 7b. As can be seen, the relative deviation for δ¹⁸O and δ²H are almost constant in the range of ±1% with increased q-value. Though the relative deviation for δ¹⁷O varies evidently with q-value, the values are under a range of ±0.7%. The Kalman filtering is more effective to remove fluctuations due to random white noise. However, apart from the measurement random noise, temperature fluctuations of the gaseous sample and baseline drift may result in drift over time in δ values, which cannot be removed by the Kalman filtering. Because of slow drift in δ¹⁷O values, the fluctuation of δ¹⁷O mean for different q-value is almost two times larger than δ¹⁸O and δ²H. The maximal differences in relative deviation for the measurement accuracy are 1.6 × 10⁻³ for δ¹⁸O, 1 × 10⁻² for δ¹⁷O, and 5 × 10⁻⁴ for δ²H inFigure 7b.