1 Introduction

Carbon dioxide (CO2) is considered an important anthropogenic greenhouse gas that causes global climate change [1]. The concentration of atmospheric CO2 has increased from approximately 280–380 ppm (parts per million) over the past century. The present level of CO2 should be much higher than what we have observed, suggesting the existence of some unknown carbon sink [2]. Therefore, several instruments have been or are currently being developed for mapping the global atmospheric CO2 column concentration over large areas with high precision. In comparison with passive remote-sensing techniques such as the orbiting carbon observatory (OCO) and the greenhouse gases observing satellite (GOSAT), differential absorption lidar (DIAL) provides many advantages, including daytime coverage, availability of all latitudes, less interference from clouds and aerosol scattering, and precise column height determination [3–5].

Airborne or satelliteborne lidar measurements of atmospheric CO2 using the IPDA technique, a special case of the DIAL technique, are promising approaches for retrieving global CO2 column concentrations [6–8]. Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS) and Advanced Space Carbon and Climate Observation of Planet Earth (A-SCOPE) missions, aiming at global CO2 column concentration measuring, had been proposed and researched by NASA and ESA, respectively. Both of them have not been selected as the spaceborne mission yet because of missing technology readiness which asks for continuous effort on greenhouse gas lidar technologies. With precise measurements of the CO2 global distribution, one can determine the carbon sink/source using different models and develop a better understanding of the processes that regulate atmospheric CO2 and its role in the carbon cycle [9]. Multi-satellite monitoring of atmospheric CO2 is a powerful tool for use in global climate and environment research. Although this method of monitoring has existed and will continue to exist, there is still a need for simultaneous observations of atmospheric CO2 with high vertical and temporal resolutions as well as long-term observations. Gibert [10] indicated that a vertical profile would be ideal for the detection of concentration changes in atmospheric CO2. Kawa [11] concluded that diurnal differences in CO2 column abundance, indicative of plant photosynthesis and respiration fluxes, are difficult to detect using a satelliteborne lidar. Therefore, ground-based DIAL systems are of great significance because they are able to provide atmospheric CO2 measurements that are range-resolved, highly accurate, and have high temporal resolution to complement airborne and satelliteborne measurements.

Our group focused on developing measuring equipment for sensing atmospheric CO2 using DIAL techniques. The instrument used in this study, a ground-based system using pulsed lasers, was designed to simultaneously perform range-resolved atmospheric CO2 measurements with a high temporal resolution. Eventually, the apparatus will be installed in Yangbajing, a town in Tibet, as a part of the atmospheric profiling synthetic observation system (APSOS) [12, 13]. Ehret analyzed the relationship between the relative inversion error and the frequency offset from the line center [7]. For a small displacement in either direction, the relative error sharply increases to a maximum value. It was suggested via simulation analysis that a frequency stability of up to 0.3 MHz would be ideal for on-line wavelength locking of CO2 soundings at 1.6 µm. For all systems reported in [6, 8, 14–16], the on-line wavelength stabilization was achieved by the stabilization of a continuous-wave (CW) injection seed laser under the assumption that the wavelength of the output laser can be precisely controlled by the injection seed process [17]. Unlike these systems, a pulsed dye laser was adopted in our system for wavelength modulation. DIAL using a pulsed dye as a wavelength modulation unit is very common in atmospheric water vapor and ozone measuring [18–20]. Hence, such technique may be more mature and stable. However, spectrum of CO2 (0.26 nm) is much narrower than that of O3 (90 nm) and the desired accuracy (0.25–1 %, 1–4 ppm) is better than that of H2O (5–10 %), resulting that requested accuracy of wavelength calibration and stabilization is much higher. The existing methodology of wavelength calibration for DIAL using dye is incapable of providing results with adequate accuracy because the wavelength control and stabilization have to be performed with respect to a pulsed laser which is more challenge than similar technique work well with a CW laser [21–23]. The low signal to noise ratio (SNR) resulting from the inevitable energy fluctuations of the pulsed laser is a huge obstacle for the precise calibration of a desired wavelength, namely the so-called on-line wavelength used in the DIAL technique. To our knowledge, there is no wavelength calibration method which is capable of providing results of requested high accuracy for pulsed laser of ~1,570 nm which is necessary and significant for achieving the ultimate goal of mixing ratio measurement accuracy of 1 ppm.

To tackle the above problem, a novel wavelength calibration procedure is proposed and tested in this work. The wavelength of the output pulsed laser was first stepped around the peak area by precisely controlling the stepping motor of the dye laser. Subsequently, a two-stage wavelength calibration strategy based on the Voigt fitting program was used to search for the on-line wavelength hidden by noise after preliminary energy normalization was performed within the appropriate time boundaries. It is worth mentioning that the absolute accuracy of the on-line wavelength calibrations cannot be accessed directly, because wavemeters with very high accuracy (<10 MHz) are unavailable for pulsed lasers of ~1,570 nm to date. Therefore, the Voigt fitting/simulation program is also an indirect means for performance evaluation when laser intensity variations are known or measured. Through simulations, it is observed that the absorption peak can be accurately determined despite the large intensity variations of the pulsed laser. On this basis, we tested the proposed procedure in our wavelength control unit by analyzing measurement data acquired by a 16-m gas absorption cell and found it to be an effective method to calibrate the on-line wavelength for our DIAL system. Meanwhile, we also found that the proposed procedure could deal with simulated signals with very low SNR. Our experimental results showed that the on-line wavelength can be accurately identified in such extreme cases by using the proposed procedure. This is a promising method for simplifying wavelength control units to be installed in remote locations that are difficult to access. Furthermore, it enables wavelength optimization for a multi-wavelength DIAL system to retrieve CO2 concentrations accurately at different altitudes.

2 System and principle

2.1 Lidar setup

Figure 1 shows the configuration of our ground-based DIAL setup. A 16-m gas absorption cell filled with pure CO2 at a pressure of 1 atm at room temperature is employed to obtain a CO2 spectrum from which the absorption peak can be determined. Preliminary energy normalization is applied using two detectors prior to further processing. Despite that, intensity fluctuations are still too large to accurately determine the desired on-line wavelength using traditional methods. We omit the details of other parts of the setup in Fig. 1 and illustrate the schematic diagram of the transmitter in Fig. 2 because wavelength calibration is the focus of this work. The dye system acts as a tunable frequency converter to generate desired wavelengths. The target wavelengths are generated by difference frequency mixing between the fundamental of the Nd:YAG laser(Spitlight 2000 NdYAG) and the dye laser which is pumped by the second harmonic of the Nd:YAG. As a consequence, the wavelength of the output laser can be changed by tuning the red laser of which wavelength is changed by a stepper motor. Therefore, traditional wavelength calibration and stabilization techniques which were proved to provide ideal results with the injection-seeded CW laser are not feasible for this setup. Thus, a novel method capable of highly accurate pulsed laser wavelength calibration is necessary. Further, the assumption that the wavelength of the output laser can be precisely controlled by the injection seed process is no longer the foundation of our wavelength control unit, which requires the application of wavelength calibration and direct control over the final output laser.

Fig. 1
figure 1

Configurations of our ground-based CO2 DIAL system

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Fig. 2
figure 2

Schematic diagram of the output laser

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2.2 Voigt function

Spectral lines in the discrete absorption or emission spectra are never strictly monochromatic [24]. An excited atom can emit its excitation energy as spontaneous radiation. However, because the amplitude of the oscillation decreases gradually, the frequency of the emitted radiation is no longer monochromatic as it would be for an oscillation of constant amplitude. The frequency distribution can be described by the Lorentzian profile. Generally, the Lorentzian line profile with the natural line width \(\delta v_{n}\) cannot be observed without the use of special techniques as it is completely concealed by other broadening effects. One of the major contributions to the spectral line width in gases at low pressures is the Doppler width, which is due to the thermal motion of the absorbing or emitting molecules. A Doppler-broadened spectral line can be represented by a Gaussian profile.

In spectroscopy, the Voigt profile is a line profile resulting from the convolution of two broadening mechanisms, one of which would produce a Gaussian profile as a result of the Doppler broadening and the other would produce a Lorentzian profile [25]. Furthermore, both Dicke effect (i.e., molecular confinement or diffusion due to velocity changing collisions) and speed dependence of molecular relaxation lead to absorption lineshapes that are narrower than the usual Voigt profile with a higher absorption peak and that may be asymmetric [26]. To address this problem, Galatry and speed-dependent Voigt profiles have been used as alternatives to achieve better performance of fitting [27]. These profiles are superior to usual Voigt because they account for the influences of pressure on the broadening coefficient. However, in this work, what we concern is the position but not the intensity of the absorption peak. Besides, Galatry and speed-dependent Voigt profiles require more parameters. Hence, the Voigt profile is adequate and convenient to fulfill the task.

The Voigt profile is common in many branches of spectroscopy and diffraction and defined as follows:

$$y = y_{0} + A \cdot \frac{2\ln 2}{{\pi^{3/2} }} \cdot \frac{{\omega_{L} }}{{\omega_{G}^{2} }} \cdot \int_{ - \infty }^{\infty } {\frac{{e^{{ - t^{2} }} }}{{\left( {\sqrt {\ln 2} \cdot \frac{{\omega_{L} }}{{\omega_{G} }}} \right)^{2} + \left( {2\sqrt {\ln 2} \cdot \frac{{x - x_{c} }}{{\omega_{G} }} - t} \right)}}} {\text{d}}t$$
(1)

where \(x_{c}\) is the center frequency, \(\omega_{L}\) is the Lorentzian width, \(\omega_{G}\) is the Doppler width, \(A\) is the area under the line, and \(y_{0}\) is the offset. In our application, \(x_{c}\) corresponds to the on-line wavelength and \(\omega_{L}\) and \(\omega_{G}\) can be calculated with respect to temperature and pressure by means of relevant equations [28] and parameters from HITRAN 2012 [29]. On that basis, the desired on-line wavelength could be identified from a sequence data of transmission by using the Voigt function. Owing to its physical implications, we believe that the Voigt function is superior to other functions such as polynomial fittings for the on-line wavelength calibration in CO2 DIAL measuring.

Furthermore, the half-width of the CO2 absorption spectrum is so narrow that the requirements of on-line wavelength calibration accuracy are extremely high. However, there are no available wavemeters with a desired accuracy for use with pulsed lasers of 1,570 nm to enable on-line wavelength calibration. The Voigt function helps us to generate simulation data by setting function parameters in the light of HITRAN 2012 and different noise levels. We then in turn retrieve x c and compare it with the initial parameter to calculate biases. As a result, performances are evaluated, in theory, under different noise level conditions. Using these results, the corresponding accuracy can be determined by measuring the noise levels in practice measurements. It is worth noting that three parameters, including the central wavelength, Lorentzian width, and Doppler width, can be retrieved according to the Eq. 1 by using the Voigt fitting. However, the intention of this work aims at on-line wavelength calibration. Moreover, it is found that Lorentzian width and Doppler width cannot be retrieved as accurately as the central wavelength through experiments. Hence, they are not involved in the following experiments.

2.3 Processing procedures

Because sequences of measurement data are required for the fitting process, a key problem here is to determinate the appropriate number of points required. More points are expected to result in better results. However, the computing power necessary to handle a large number of sample points would be impractical in most laboratory settings, making a two-stage wavelength calibration procedure indispensable. In the first stage, larger steps are adopted for coarse calibration. Our goal in this stage is to set the wavelength to the desired value with accuracy of ~5 pm as quickly as possible. Next, a fine calibration will be applied in the second stage. Finally, we search for the absorption peak in the area given in the first stage to the greatest degree of accuracy possible. Besides, it is noting that the issue of the computational load associated with the use of the Voigt profile is not discussed in this work. Because the elapsed time associated with the use of the Voigt profile is less than one second. Moreover, the elapsed time of computational load is evidently smaller than that of samples acquisition.

3 Simulation analysis

3.1 Assignments of requirements on wavelength calibration

With regarding to the issue of frequency uncertainty, the fact whether the on-line should be stabilized on the line center or wing of the respective absorption line is of great importance. In general, the requirement of locking the on-line wavelength to a point on the wing is more stringent than that of locking to the peak because the center of an absorption line is more “flat” than its wings. The advantage of tuning wavelength from the center line is to optimize optical depth [15]. Bruneau concluded that the lowest possible DIAL error is obtained for a signal–noise limit direct detection with an optical thickness of 1.28 and on-line-to-off-line energy ratio of 3.6 [30]. However, for a vertical DIAL using aerosol-distributed target, optimizing of optical depth is not a major concern as aerosols become very rare beyond atmospheric boundary layer height (ABLH). Given the ABLH is 1,000 m and the atmospheric CO2 concentration is 400 ppm, the optical depth owing to absorption is about 1.15 <1.28. Therefore, there is no need to tune the on-line wavelength from the center line for a ground-based DIAL using aerosol-distributed target for optimizing optical depth.

The bandwidth and the spectral purity are two significant factors to achieve retrievals with high precision and accuracy for a DIAL. However, this work is focus on the issue of wavelength calibration which aims at reducing the frequency uncertainty. The bandwidth and the spectral purity will evidently affect the intensity of the detected spectrum but hardly affect the shape of the detected spectrum which is the very foundation of wavelength calibration. Besides, they are parameters in regarding to different laser sources. For these sakes, they are not involved in this work.

The errors induced by a possible frequency uncertainty are illustrated in Fig. 3. In this figure, the relative error in the two-way transmission is plotted as a function of the frequency offset from the center line. This error can be directly translated into a relative measurement error for the optical depth. All spectroscopic parameters involved in this simulation are from HITRAN 2012 [29]. The temperature and pressure are set to be 288.15 K and 1 atm, respectively. The optical length is set to be 500 m, and atmospheric CO2 is set to be 400 ppm. Here, the frequency uncertainty consists of calibration uncertainty and stabilization uncertainty. The accuracy of frequency stabilization for a pulsed laser is influenced by the energy fluctuation of the transmitted beam. As such fluctuations are inevitable and are much more evident than that of the CW laser, the accuracy of frequency stabilization may not be very high. Hence, it is of great significance to improve the accuracy of wavelength calibration to keep the total frequency uncertainty at an acceptable level. Figure 3 confirms that high frequency uncertainty is very harmful for accurate CO2 measurements when the on-line wavelength is on the wing of the respective absorption line. Given maximum permissive transmission error is 0.1 %, the frequency uncertainty should be restricted to lower than 5 MHz for the side-line tunable strategy. However, if the center line was adopted as the on-line wavelength, the maximum permissive frequency uncertainty would be much higher. Figure 4 demonstrates the relationship between transmission errors and the frequency uncertainty for center wavelength strategy. In this case, the optical depth owing to CO2 absorption is always underestimated as shown in Fig. 4. A frequency uncertainty better 75 MHz would guarantee errors no more than 0.1 % in the light of Fig. 6. Also A. Amediek [16] indicated that an on-line stability better than 70 MHz@1,572 nm is needed to achieve error values below 0.1 %.

Fig. 3
figure 3

Errors derived from frequency uncertainty when different on-line wavelength is adopted

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Fig. 4
figure 4

Relationship between errors and frequency uncertainty when the center line is selected as the on-line wavelength. The dotted line depicts an error budget of 0.1 %

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Besides, it is also worth noting that the proposed calibration method would be used in two ways. On the one hand, we can utilize this method to calibrate the wavelength of pulsed laser as the equivalent of the lock-in amplifier technique for CW laser. On the other hand, we can carry out a coarse calibration (by using either the proposed method or a wavemeter of high performance) at first. Subsequently, it is intended to continuously scan over the peak area. At last, we identify the desired on-line wavelength in sequences of backscattered signals of different wavelength by using the proposed method. The major advantage of this strategy is simplification of wavelength control unit. The shortcoming is fewer retrieval than the traditional strategy in a certain time period. Meanwhile, the noise of backscattered signal is much higher than that of the signal acquired by the gas cell. Hence, a high noise level is indeed necessary to take into consideration in the following tests.

3.2 Coarse calibration

For coarse calibration, wavemeters are available for both CW and pulsed lasers. Here, we still use the proposed method to meet our goal of verifying continuity and comprehensiveness. Here, we also take system simplification and economic viability into consideration.

The absorption peak is easily recognized in a series of data under ideal conditions. Difficulties appear only if noise is present in the measuring data as illustrated in Fig. 5. Consequently, noise levels as well as the number of points are taken into consideration in our simulations. All relevant spectral parameters used in the simulations of this study are taken from the HITRAN database [31–33]. Other general parameters are set as follows: temperature = 296 K, pressure = 1 atm, optical path = 1.3 m, natural abundance = 98.42 %, and isotopologues of O16C12O16.

Fig. 5
figure 5

Comparison of ideal signal (left panel) and contaminated signal with noise level of (right panel)

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In this study, accuracy refers to the absolute error, which is the difference between the measured or inferred value of a quantity x o and its actual value x. In other words, a higher value of accuracy indicates a worse performance and vice versa. Equation 2 describes the meaning of the accuracy denoted as \(\Delta x\).

$$\Delta x = \left| {x_{0} - x} \right|$$
(2)

Five levels of noise, 2, 5, 10, 15, and 20 %, were chosen as representatives of all possible noise levels. In practice, the noise level of laboratory data acquired by using a long-path gas cell and a pulsed laser is less than 5 % in most cases, according to our observations. The potential of the proposed strategy to deal with backscatter signals which contain more noise than the laboratory data. In order to do this, we used simulated contaminated signals with noise levels of up to 20 %. However, Fig. 5 shows that the data quality for these measurements is often too low for further processing measures to be applied. Such contaminated data can hardly be adopted to retrieve carbon dioxide concentrations even if the absorption peak can be accurately recognized and, therefore, simulations with high levels of noise are excluded.

Before simulation analysis and verifications, a term “step size” is clarified here. In the numerical analysis, the original “step size” refers to the discretization of the simulated spectra. For consistency, we have already transformed the original “step size” into the wavelength step. Besides, the stepper motor alters the wavelength of the red beam (~634 nm) directly. But the wavelength increment of the red beam is not equal to that of the infrared beam (~1,572 nm). Consequently, we have also transformed the step size of the stepper motor into wavelength increment of the infrared beam. Consequently, in the whole manuscript, the “step size” consistently refers to the wavelength step of the infrared laser (~1,572 nm).

Figure 6 demonstrates the relationships between the accuracy of wavelength calibration with respect to step size (left panel) and the noise level (right panel). Each dot on the left (right) panel denotes the mean accuracy of the results obtained from different noise level settings (step). The dotted line with circles represents the results obtained using the ordinary method which takes the absorption peak as the maximum value, and the solid line with squares denotes the results obtained by means of Voigt fitting. Overall, we found that the mean accuracy deteriorates when the step or noise level increased for either method. However, Voigt fitting was evidently superior to the ordinary method in searching for the absorption peak regardless of the size of the step and noise levels, as shown in Fig. 6.

Fig. 6
figure 6

Influences of noise level and step size on the accuracy of coarse wavelength calibration. Two panels share the same scale. The accuracy is demonstrated in pm on the left axis and in Mhz on the right axis. In the left panel, each dot on a step represents the average accuracy of results obtained with noise levels ranging from 2 to 20 %. In the right panel, each dot on a noise level represents the average accuracy of results obtained with steps ranging from 1 to 27.5 pm

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The left panel of Fig. 6 shows that the accuracy fluctuated and slightly increased as step size was decreased for the maximum method for steps smaller than 15 pm. In other words, it is unlikely that performance could be further improved even if smaller steps were adopted. In contrast, the growth of the accuracy with decrease in step size accelerated when the step was smaller than 5 pm for the Voigt fitting method. This trend suggests that further improvements in performance are possible. Similar trends were also observed in the right panel of Fig. 6. In conclusion, better performance and greater potential suggest that Voigt fitting should be substituted for the ordinary method for accurate wavelength calibration of the on-line wavelength. Meanwhile, the performance curves demonstrate that Voigt fitting fulfilled our demand for coarse calibration. Fewer points are required to finish the wavelength calibration with an accuracy of up to 5 pm by using the proposed method even in the case of unfavorable noise conditions. In the light of our experiments, a step size of 10 pm is sufficient to ensure a result with an accuracy of at least 5 pm regardless of a noise level of 20 %. Taking into consideration the full width at half maximum (FWHM) of the carbon dioxide peak around 1,572 nm, steps of 10 pm are equivalent to 25 sample points. The total time period of a wavelength calibration process is determined by the number of sample points and the elapsed time of a single Voigt calculation. When 25 sample points are collected, the time consumption will be 10–62.5 s in terms of the laser frequency. The elapsed time due to a Voigt fitting is less than 1 s. Consequently, such consumption is affordable for practical applications.

3.3 Fine calibration

After course calibration, it is assumed that the bias between the desired wavelength and the preliminary result is within 5 pm. For fine calibrations, further performance improvements are expected.

In the DIAL technique, the on-line wavelength (\(\lambda_{\text{on}}\)) is set around or exactly at the absorption peak, while the off-line (\(\lambda_{\text{off}}\)) wavelength is set far away from the absorption peak. For CO2 detection, the on-line and off-line wavelengths are so close that the atmospheric backscatter coefficient together with the atmospheric extinction coefficient can be considered to be equivalent at two slightly different wavelengths. However, water vapor is taken into account when the accuracy requirement is extremely stringent. Given that \(P_{0} \left( {\lambda_{\text{on}} ,R} \right) = P_{0} \left( {\lambda_{\text{off}} ,R} \right),\) the average concentration of the target trace gas at range R with depth \(\Delta R = R_{\text{top}} - R_{\text{bottom}}\) can be described using expression (3). There are several candidates lines for CO2 DIAL at both 1.6 and 2.0 µm. Concerning selection of on-line and off-line wavelength, it is strongly recommended to keep away from some strong absorption lines of H2O. With cautious selection of on-line and off-line wavelength, the minuend in (3) could be neglected. For instance, the ratio of the absorption cross sections of water vapor and carbon dioxide is only 0.02 when \(\lambda_{\text{on}}\) and \(\lambda_{\text{off}}\) are 1573.332 and 1573.200 nm, respectively, as reported in [34]. Assuming that \(\sigma_{{{\text{co}}_{ 2} }} (\lambda_{\text{on}} )\) and \(\sigma_{{{\text{co}}_{2} }} (\lambda_{\text{off}} )\) were calculated properly, the remaining errors are then derived from \(P\left( {\lambda_{\text{on}} ,R} \right)\) and \(P\left( {\lambda_{\text{off}} ,R} \right)\), which represents the accuracy of the wavelength calibration. Because \(\sigma_{{{\text{co}}_{2} }} (\lambda_{\text{on}} )\) is much greater than \(\sigma_{{{\text{co}}_{2} }} (\lambda_{\text{off}} )\) the accurate calibration of on-line wavelength then plays a crucial role in retrieving carbon dioxide concentrations.

$$N_{{{\text{CO}}_{ 2} }} = \frac{1}{{2[\sigma_{{{\text{CO}}_{ 2} }} (\lambda_{\text{on}} ) - \sigma_{{{\text{CO}}_{ 2} }} (\lambda_{\text{off}} )\Delta R]}}\times\ln \frac{{P(\lambda_{\text{off}} ,R_{\text{top}} ) \cdot P(\lambda_{\text{on}} ,R_{\text{bottom}} )}}{{P(\lambda_{\text{on}} ,R_{\text{top}} ) \cdot P(\lambda_{\text{off}} ,R_{\text{bottom}} )}} - N_{{{\text{H}}_{ 2} {\text{O}}}} \frac{{\sigma_{{{\text{H}}_{ 2} {\text{O}}}} (\lambda_{\text{on}} ) - \sigma_{{{\text{H}}_{ 2} {\text{O}}}} (\lambda_{\text{off}} )}}{{\sigma_{{{\text{CO}}_{ 2} }} (\lambda_{\text{on}} ) - \sigma_{{{\text{co}}_{ 2} }} (\lambda_{\text{off}} )}}$$
(3)

The nature of the Voigt function, which is the product of the convolution of the Gaussian and Lorentzian functions, results in non-convergence. The functional form for the Voigt profile allows the Gaussian and Lorentzian portions to have different line widths. Thus, the solved parameters are non-uniform when repeated experiments are carried out. Because of this, the standard deviation was calculated for each case to evaluate the stability of the algorithm. Such variations are negligible for coarse determinations because of the greater tolerance allowed at this stage. This is why the standard deviation was not calculated in previous experiments.

Similar to the results presented in Fig. 6, the left and right panels show the influences of step size and noise level, respectively, on the performance. Because the performances of the Voigt fitting were much better than that of the ordinary method, a secondary axis (the right axis of each panel) was used to show the performance curves of the ordinary method. The two curves in the right panel shared a similar trend, and accuracies were found to be inversely proportional to noise level. Furthermore, the standard deviation was found to increase with the noise level. It is inferred from such phenomena that the noise level degraded not only the accuracy but also the stability of the proposed method. In addition, the ordinary method was completely incapable of reaching the targets previously described. A mean accuracy of 0.4 pm was obtained with the maximum method even though the noise level was only 2 %. In contrast, the accuracy was better than 0.2 pm for a noise level of 20 % with a standard deviation of approximately 0.04 pm. From this result, we conclude that the proposed method can provide the desired accuracy in wavelength calibration by using long-path gas cells in which the noise level of the measuring data is <5 % under normal circumstances.

Figure 7 indicates that smaller steps lead to better results when the Voigt fitting method was applied. In addition, the stability was also improved when smaller steps were used. A step of 0.2 pm seems to be sufficient to obtain high-quality results. There was no significant relationship between step size and performance in the case of the maximum method. The best accuracy achieved for the maximum method was 0.47 pm, which is insufficient for fine calibration.

Fig. 7
figure 7

Influences of noise level and step size on the accuracy of fine wavelength calibration. Results of Voigt fitting and maximum are shown in left and right axes, respectively. Two left axes share the same scale but in different units (left panel for pm and right panel for MHz). So do the right axes

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Using simulations, the accuracies of the proposed method for cases of different step sizes and noise levels were calculated and listed in Table 1. In the actual measurements, step size is a parameter set by the user. If the noise level can be measured, the accuracy of the wavelength modulation can be determined by querying from Table 1.

Table 1 Accuracy matrix
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In conclusion, the proposed method is superior to the ordinary method with respect to wavelength calibration. Furthermore, simulations revealed that accurate wavelength calibration can be achieved by using the Voigt fitting with the pulsed laser when long-path gas cells are involved. Furthermore, the Voigt fitting has the potential to deal with the backscatter signal which contains much more noise. Hence, the long-path gas cell may not be indispensable in a ground-based CO2 DIAL system. In the following section, these conclusions from simulation will be verified using real data.

4 Experiments with real signal

4.1 Noise analysis

For performance evaluation in this proposed method, the noise level is the key parameter to be solved for while step size is the known parameter. In this section, we analyzed the formation of total noise and then provided a method for noise level measurement.

Here, the total measured noises were classified into the baseline fluctuation and the random noise. The baseline fluctuation, a periodic fluctuation, is mainly due to the presence of interference fringes caused by parasitic reflections in the gas cell and is very common in absorption measurements taken with sources having a long coherence length. The baseline fluctuation is related to the design of the gas cell and the wavelength of the beam. Therefore, special design of a long-path gas cell will cancel out the baseline fluctuation. However, in applications, such special designed gas cell is not adopted because of its large volume. To our knowledge, the size of a well-designed gas cell is more than 10 times of an ordinary one. Fortunately, such baseline fluctuation is a function of the wavelength but not of the time when the gas cell is fixed. Hence, one can measure the baseline fluctuation before the atmospheric sounding experiment and can subtract it from the received signals, while noises originating from intensity fluctuations of the pulsed laser and unknown environmental factors are categorized as the random noise. In the previous simulations, the noise level refers to the level of random noise. Hence, it is necessary to subtract the baseline fluctuation from the total measured noise; otherwise, the error evaluation matrix will cease to be effective. More samples at each node are helpful to suppress random noise. However, large samples lower down the scanning velocity in turn. In this work, each node represents average of 50 samples because we found that larger samples at each note improve the performance inconspicuously, while fewer samples lead to significant deterioration. The time period depends on the number of scanning notes and frequency of the transmitted laser after determination of samples at each note. Data shown in Figs. 8 and 9 were acquired in about 8.4 and 3.4 min, respectively.

Fig. 8
figure 8

Measurement of noise signals. It shows the ratio of signals from the absorption and the reference path

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Fig. 9
figure 9

Decompositions of noise signals. Two noise curves mark the right vertical axis, while original and de-noise curves mark the left one. The simulation curve has no unit, and it marks the left axis just for a better comparison

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In our system, dual-path measurements were adopted to suppress noise. As shown in Fig. 1, 1 % of the output laser is sent to the wavelength control unit for wavelength calibration and stabilization. And that beam is split into two beams again by 50 and 50 %.The reference absorption path is used to monitor the intrinsic fluctuation of the laser, and the absorption path includes a gas cell filled with CO2 at a certain temperature and pressure. The temperature and pressure will be set to some special values in terms of the detection range in the future. In this experiment, they are set to 300 K and 1 atm, respectively, just for conveniences of maintenance. Though dual-path measurements were adopted, noise is still present even when improvements in the setup are made, demonstrating why the traditional method cannot be used with our system. Figure 8 shows scanning data when the gas cells are filled with air. The absorption due to carbon dioxide is neglected in this case because, to our knowledge, the 16-m optical path cannot provide the necessary optical thickness for the measurement of carbon dioxide concentrations of only 400 ppm. The curve in Fig. 8 should theoretically be a horizontal line if no noise is present.

Figure 8 illustrates that the total noise fluctuates regularly within a range of 20 % with the wavelength. Such regular variations can be seen as the baseline fluctuation. Furthermore, there are several ‘burrs’ on the curve, which can be classified as random noise. A fitting program was applied to the scanning data to separate noises into the baseline fluctuation and random noises. Several fitting functions were tested, and the Gaussian function was found to be suitable for the task. An adjusted R 2 value of 0.937 indicated that the noises are well separated. The fitting curve was then seen as the baseline fluctuation. On that basis, we extracted the baseline fluctuation from the measurements and then applied the Voigt fitting to the processed data. Biases of approximately 0.2 pm were observed between the results obtained from the original and the processed data. Meanwhile, the root-mean-square errors (RMSEs) for the original and processed data were roughly the same with values of 0.052 and 0.053, respectively. In summation, significant differences between the fitting results were observed, while only small differences between the fitting precision results were found. Therefore, we draw the conclusion that the baseline fluctuation exerts a linear influence upon the final results. Actually, the carbon dioxide spectrum is quite narrow and the peak area which contains the wanted on-line wavelength is definitely smaller than 10 pm. As illustrated in Fig. 8, the periodic baseline fluctuation shows the monotonicity over such a tiny area. This explains the phenomenon that the RMSEs of the original data and the de-noised data are roughly the same.

Figure 9 illustrates the composition of the measured data and the different kinds of noise. The original line, which marks the left axis as well as the de-noised line, denotes the original measuring data without any processing. The de-noised line represents the data from which the systematic noise has already been subtracted. B-noise and R-noise lines, which share the right axis, denote the baseline fluctuation and the random noise, respectively. The simulation line is a synthetic curve with random noise of 5 %. The ‘intensity’ here refers to the relative optical depth but not the exact optical depth. In addition, there is no physical meaning attributed to the simulation line because the optical length, concentrations of CO2, and line intensity are not appointed according to conditions of experiments. Hence, its value is incommensurate with that of the other curves. Figure 9 demonstrates that the baseline fluctuation is significantly greater than the random noise, accounting for nearly 15–30 % of the total signal. Fortunately, such periodic noise, which can be subtracted from the signal, is measurable and regular.

As the intensity of the original data varies with the wavelength (owing to the absorption of carbon dioxide), the percentage noise level is never a constant even if the amount of noise is. Assuming that the amount of noise is a fixed value, the corresponding noise level would be higher on the wings but smaller at the peak. In this case, the original data contained a random noise of 1 % at the peak which increased to 5 % on wings. The purple line represents a simulated curve with a constant noise level of 4 %. As we have seen, the shape of this line is quite similar to that of the measured data on the wings. However, the original as well as the de-noised line is smoother at the peak. Conversely, the simulated curve remained highly volatile at the peak area. Consequently, random noise accounts for 1–4 % of the total signal for our system and we think a mean noise level of 2 % may be appropriate to evaluate the accuracy of wavelength determination. Further, we have come to the conclusion that the baseline fluctuation must be measured well and excluded from the original data for performance improvement according to our noise analysis.

4.2 Wavelength calibration experiments

Ten groups of wavelength calibration experiments using a gas cell with a step size of 6 pm were carried out. About 0.25 nm (30 GHz) is adequate for a single absorption line scan. In three-lines scanning experiments, wavelength span is 0.8–0.9 nm (110 GHz). It costs 250 s to sample 100 different wavelengths in the span range for each group. It is worth noting that we collected so many samples of different wavelength for a better visualization. Ten samples of different wavelength are adequate for the practical use purpose.

According to HITRAN 2012, there are three absorption peaks in this range: R14, R16, and R18 in the 30012 ← 00001 band of carbon dioxide. Hence, the distance from one peak to another, which would not be affected by inaccurate wavelength calibration using a wavemeter(Bristol 821-B) with insufficient accuracy, can be compared with the theoretical values after the on-line wavelength was determined. Table 2 demonstrates the results of the wavelength calibration experiments with a step size of 6 pm.

Table 2 Performance evaluations using real signals
Full size table

Given that our wavemeter’s accuracy is approximately 5 pm, all three peaks were erroneously determined by the maximum value method. In contrast, they are all accurately determined by the Voigt fitting method. Moreover, the bias between the results calculated using the maximum value method and the theoretical value exceeds 30 and 7 pm for R14–R16 and R16–R18, respectively. Such bias between the Voigt fitting method and the theoretical value remains below 0.06 and 1.5 pm for R14–R16 and R16–R18, respectively. As analyzed previously, the random noise is 1–2 % at the peak for this experiment. According to Table 1, the accuracy is superior to 0.164 pm. Hence, these results are consistent with those of the simulation experiments. Moreover, in such noise level, a step size of 3 pm is adequate to obtain results with accuracy superior to 0.1 pm. Then, we calibrate line R16 with step size of 3 pm.

Figure 10 demonstrates ten groups of wavelength calibration experiments with a step size of 3 pm. The difference between results obtained by the two methods is approximately 2 pm. Based on previous simulations and real signal experiments, we suppose that the result of the Voigt fitting is more accurate, namely that they are closer to the desire peak. In addition, the stability of the results obtained by the Voigt fitting method is superior to those obtained using the traditional method because the variance of results obtained by the Voigt fitting method is evidently smaller.

Fig. 10
figure 10

Results of on-line wavelength determination

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Furthermore, six groups of experiments with a step size of 1 pm were performed. Similar with the previous experiments, 100 samples of different wavelength were collected for each group. The mean on-line wavelength is 1572.53583 nm. We also calibrated the wavelength after resampling the step size to 3 pm to compare the results to those obtained from the original data. The resultant on-line wavelength using the resampled data is 1572.53575 nm. The difference was 0.08 pm, <0.1 pm. This is a little bigger than the difference calculated by simulations. But it does not exceed the calculated difference of 5 % noise level (0.769 pm). Consequently, it is concluded that the accuracy matrix listed in Table 1 is reliable.

According to Table 1, it was determined that a step size of 3 pm is adequate for wavelength calibration using a gas cell when the proposed strategy is applied. A smaller step size is not necessary unless signals with a high level of noise are used.

If the retrieval error budget of the wavelength uncertainty is set to 0.1 %, corresponding to roughly 0.4 ppm given a mean CO2 concentration of 400 ppm, a wavelength drift of 0.3 pm is allowed for a wavelength calibration of ~1,572 nm. Based on previous simulation analysis and real signal experiments, we believe that the proposed strategy is capable of accurate wavelength calibration for pulsed lasers using a gas cell.

5 Conclusion

In summary, a two-stage wavelength calibration strategy based on Voigt fitting was proposed for accurate wavelength calibration of a pulsed laser after briefly introduction of a ground-based DIAL aimed at profiling vertical atmospheric CO2 concentrations. Simulation analysis demonstrated that the traditional method cannot provide sufficiently accurate wavelength calibration results for retrieving CO2 concentrations. In contrast, our simulations showed that the wavelength calibration bias could be suppressed to less than 0.2 pm by using the proposed strategy, despite the high noise levels in the simulated data. Several groups of real signal experiments verified the results of the simulation analysis. We confirmed that the scanning step of 1–3 pm is adequate for wavelength calibration of a pulsed laser using a gas cell. Furthermore, we speculate that the proposed strategy may work well with the scattering signals, which propagate through the atmosphere and are absorbed by atmospheric CO2, under conditions that accurate profiles of the ambient temperate and pressure are known and aerosol-distributed targets provide range-resolved signals with noise level of <20 %. Once this inference is proved by future works, it would be a promising feature for developing a DIAL system without wavelength calibration and stabilization units.