1. Introduction
At present, nearly half of the total anthropogenic CO
2 emission has been offset by unidentified CO
2 sinks [
1]. Dense observations of column xCO
2 (dry-air mixing ratio of CO
2), supplementing the existing ground-based CO
2 monitoring networks, are urgently needed to provide stronger constraints on inversions of carbon fluxes using the atmospheric inversion technique [
2]. Since 2000, several satellite-based sensors, such as AIRS, SCIAMACHY, and IASI, have been sent into orbit and are able to obtain data of XCO
2 (column-average xCO
2). In recent decades, some dedicated greenhouse gas monitoring missions, including GOSAT, OCO-2, and TanSat, have successfully obtained XCO
2 products with higher accuracy and precision than before. It is widely witnessed that these data are useful in deepening our understanding of the carbon cycle process, despite their susceptibility to aerosols and dependency on sunlight as well as insufficient coverage over high-latitude zones [
3,
4]. Supplementing the passive remote sensing of greenhouse gases, differential absorption LiDAR (DIAL) plays an increasingly critical role in atmospheric CO
2 sensing, owing to the remarkable developments made in both lasers and detectors [
5,
6]. Integrated path differential absorption (IPDA) LiDAR, a special type of DIAL, relies on backscattered signals by hard-targets, thus providing a higher signal-to-noise ratio (SNR) [
7]. Several groups from different countries are dedicated to developing equipment for the active remote sensing of CO
2 equipment based on the integrated-path differential absorption (IPDA) technique, which is expected to obtain CO
2 measurements from space with high accuracy and better coverage at night and at high latitudes [
8,
9,
10]. In recent years, airborne CO
2-IPDA LiDARs have been extensively tested by different research groups funded by the National Aeronautics and Space Administration (NASA) [
5,
11,
12,
13]. In these experiments, results with errors lower than 1% were obtained and researchers are anticipating the realization of spaceborne CO
2-IPDA LiDAR in the near future.
To understand the transportation process of CO
2 in the atmosphere, it is important to identify vertical gradients of xCO
2, especially in the lower troposphere [
14,
15]. Gradients of xCO
2 between the ABL (atmospheric boundary layer) and FT (free troposphere) can be used to estimate the surface net ecosystem exchange [
16]. Vertical gradients of CO
2 would also help us to gain insight into the feedback between the carbon cycle and climate change [
15,
17,
18,
19]. The ground-based dual-wavelength DIAL can measure the range-resolved xCO
2 within 3 km [
18,
20,
21]. However, it relies on aerosol backscattered signals with a relatively lower SNR and thus is unlikely to provide products with accuracies of up to 1 ppm. Moreover, ground-based equipment is difficult to deploy and operate in remote places. The spaceborne CO
2-IPDA LiDAR is a suitable and effective means to obtain measurements of CO
2 with a high accuracy globally. However, most dual-wavelength IPDA systems can only retrieve column-weighted xCO
2 and are incapable of measuring gradients of xCO
2.
To tackle the above-stated problem, we propose a novel multi-wavelength inversion method, which is based on the constrained linear least-squares technique. In the field of IPDA/DIAL, the multiple-wavelength strategy was proposed soon after the dual-wavelength differential absorption inversion algorithm. In 1985, R.E. Warren first presented a generalized DIAL signal model and then proposed an inversion method of the concentration of a target material by using maximum likelihood estimation [
22]. The main goal of R.E. Warren’s method is to retrieve multiple-target materials by means of multiple-wavelength absorption LiDAR. In 1999, Fukuchi presented a multiple wavelength DIAL for measuring range-resolved SO2 and demonstrated that the accuracy can be improved by using averaged signals of a group of on-line/off-line wavelengths instead of a single pair [
23]. Later, in 2002, their group successfully developed a five-wavelength SO2-DIAL and obtained range-resolved SO2 concentrations by means of a curve-fitting technique [
24]. In 2014, Abshire et al. demonstrated airborne measurements of column-weighted xCO
2 using a 30-wavelength CO
2-IPDA LiDAR, which is the first multiple-wavelength CO
2-IPDA LiDAR [
5]. In their inversion method, a layered atmospheric model was used to compute an absorption spectrum for a given path, CO
2 concentrations, and some other parameters. On this basis, the CO
2 concentration was solved by minimizing the error between the measured spectrum and the modeled one. J.R. Chen presented a novel multiple-wavelength IPDA inversion method using symmetrically measured LiDAR soundings, substantially reducing the error from spectral distortion and laser frequency drift in 2014 [
25]. In 2016, Xiang et al. proposed another multiple-wavelength DIAL inversion method which is based on the weighted averaging of observation pairs [
26], and they verified the method using a range-resolved, 11-wavelength CO
2-DIAL in a laboratory environment. This method was further improved via using a better fitting kernel in 2017 [
27]. We can conclude from these previous works that the technical feasibility of multiple-wavelength IPDA LiDAR to measure the CO
2 vertical gradient has been established [
28]. In this work, we aim to discuss the feasibility of obtaining gradients of xCO
2 through our recommended method by multiple-wavelength IPDA LiDAR [
13].
In the proposed framework, we sliced the atmosphere into several layers. Then, we constructed simultaneous equations using IPDA equations of multiple wavelengths. On this basis, the simultaneous equations were solved using the constrained linear least-squares technique, obtaining sliced xCO
2. The linear least-squares technique is widely used as a standard approach to approximate the solution of over-determined problems [
29]. However, it searches for a mathematically optimal solution without consideration of physical constraints [
30]. The spatial distribution of CO
2 has its own special features, which can impose physical constraints on inversions [
31]. The ordinary linear least-squares technique would obtain results with a high accuracy in terms of mathematics but with evident errors in terms of physics. We thus consider using a constrained linear least-squares method to obtain results with physical meanings. Such a technique has been used to resolve problems regarding remote sensing [
32] but has not yet been used to deal with data of IPDA LiDAR. The sign of the xCO
2 gradient between the ABL and FT would be auxiliary data to help identify the characteristic of carbon flux (source, neutral, and sink). Moreover, one can retrieve the regional carbon flux using xCO
2 gradients based on the equilibrium boundary layer theory [
33].
The remaining sections are as follows: the principle of the proposed method is presented in
Section 2, as well as relevant methodologies used in the experiments. The performance of the proposed method is evaluated in
Section 3 in terms of the SNR, the number of wavelengths, the number of layers, and the ABLH (the height of the atmospheric boundary layer). Moreover, we tested the ability of the proposed framework to retrieve gradients of xCO
2. In
Section 4, we discuss possible effects of water vapor on retrieving sliced xCO
2 using the proposed method. Finally, the major findings of this work are summarized in
Section 5.
3. Results
3.1. Effect of Signal-to-Noise Ratio
SNRs of received signals of wavelength
can be expressed by Equation (12).
where
Eλ represents the received energy of
λ, M is the internal gain factor of the detector,
R denotes the response of the detector,
B is the electrical bandwidth,
e is the elementary charge,
F is the excess noise factor of the detector (which accounts for additional noise due to the internal amplification statistics), and
is the dark current noise density.
is the energy of solar background radiance.
One can conclude from Equation (12) that
SNRλ is highly related to the hardware configuration. Moreover, some environmental factors can also affect
SNRλ, such as the aerosol optical depth and the surface reflectance which determines
Pλ. Moreover, the laser wavelength is also a critical parameter determining
SNRλ because
Pλ varies significantly with
λ due to CO
2 absorption.
δ(λ,CO2) is the measured optical depth due to CO2 absorption on a certain wavelength of λ, which can be calculated by Equation (13). The relationship between SNRλ and the relative random error of δ(λ,CO2) can be expressed by Equation (14). N is the average number of the shots, which depends on the average time and emitted frequency of the future satellite IPDA. We further defined SNR(T,λ) using Equation (15), where ∆δ(λ,CO2) is the random error of δ(λ,CO2). SNR(T,λ) is very different from SNRλ, The former is used to describe the SNR of measured δ(λ,CO2), which is proportional to XCO2, while the latter is used to describe the SNR of received laser signals. Given the specific configurations of hardware and detection environments, both SNRλ and δ(λ,CO2) are negatively correlated with random errors of retrievals, as shown in Equation (14). However, SNRλ decreases with the increasing δ(λ,CO2). Therefore, it is unwise to establish a relationship between the detection error and SNRλ. Compared with SNRλ, SNR(T,λ) can reflect the accuracy of retrievals more intuitively. Hence, we define SNR(T,λ) herein and correlate it with the detection error in the following experiments.
SNR(T,λcentral), where
λcentral is the central wavelength of on-line wavelengths, would be better than 25 based on the technology of the current hardware [
38]. In this group of simulations, the preset vertical profile of xCO
2 is shown in
Table 1. The number of wavelengths used in this group of simulations is 22. If
SNR(T,λcentral) is certain, the
SNRs(T,λi) (
i = 1,2,3…
m,
m = 21) would be calculated by Equation (14).
In order to evaluate the influence of
SNR(T,λcentral) on the final results, we performed 10,000 simulations to evaluate the performance of our recommended method under different
SNR(T,λcentral) (20–26) values. The number of wavelengths is 22 and the assumed ABLH is 1.5 km. Then, the xCO
2 of three sliced layers was retrieved, as shown in
Figure 1. Atmospheric profiles of pressure and temperature were preset using the US standard atmosphere model. The temperature and pressure of the surface are 297 K and 101.32 hPa, respectively. In the subsequent experiments of other subsections, we used the same settings of the atmosphere, unless we have declared otherwise.
As is shown in
Figure 1, the mean error (ME) decreases with the increasing
SNR(T,λcentral). It is worth nothing that MEs would be −0.98 ppm, 0.27 ppm, and −0.15 ppm in ABL, FT, and STA, respectively, when
SNR(T,λcentral) is 25 dB. STD has a negative relationship with the
SNR(T,λcentral) in three layers. However, both ME and STD performance slightly change when
SNR(T,λcentral) is larger than 25 db. Hence, we set
SNR(T,λcentral) to 25 db in the following simulations. Moreover, it is necessary to discuss the feasibility of an
SNR(T,λcentral) of 25 db. We have used the performance simulator and configurations of China’s CO
2-IPDALiDAR described in our previous work [
38] to calculate the SNR and found that it is higher than 25 db. Moreover, according to the flight test reported by Zhu et al., an SNR of 25 db is feasible in real conditions [
39].
3.2. Effect of the Number of Wavelengths and Selecting Modes
A series of experiments was carried out to explore the effect of the number of wavelengths, in which the central wavelength was set to 1572.335 nm. Except for the number of wavelengths, two types of sampling mode were tested to determine whether the strategy of wavelength sampling would exert influences on outcomes. In two different sampling modes, we set the same wavelength step. With such a configuration, we can determine what the dominant factor is, the wavelength number or the wavelength step, in determining the performance of the proposed multi-wavelength method.
Figure 2 demonstrates an example of symmetric sampling with 22 wavelengths (off-line wavelength is not shown), using both solid and dotted lines. An example of unilateral sampling with 12 wavelengths is also shown in this figure, using only the solid lines.
In the next experiment, we wanted to investigate whether different sampling modes would cause differences in the performance of the proposed method. The center wavelength of a CO
2 spectrum would move towards the shorter wavelength with the decreasing pressure. Such a phenomenon is called pressure-induced shift [
40]. Therefore, symmetric sampling in the absolute sense is not possible. This effect has been taken into consideration in the simulations to include possible errors. In our experiments, two sample modes were used: symmetric distribution sampling and unilateral distribution sampling. The
SNR(T,λcentral) was set to 25 dB and the ABLH is 1.5 km. For the simulations in this section, the setting of the number of wavelengths is shown in
Table 2 and the detailed setting of the layers and concentration of CO
2 in each layer is shown in
Table 1.
Figure 2 illustrates two sampling modes in the following simulations. For the symmetric sampling, wavelengths are chosen on both sides of the center wavelength (the black solid line). For the unilateral mode, wavelengths are chosen on one side of the center wavelength (all solid lines or all dotted lines).
Figure 3 shows that both ME and STD decrease with the increasing number of wavelengths in three layers. We also found that the number of wavelengths, rather than the wavelength step, is the dominant factor in determining the performance of inversions. Two sampling modes yield very similar performance with the same number of wavelengths. With 22 wavelengths, MEs of retrievals are −0.98 ppm, 0.27 ppm, and −0.15 ppm in ABL, FT, and STA, respectively. We think that 22 wavelengths are adequate and appropriate to obtain retrievals of each layer with an error of less than 1 ppm. Consequently, we set the number of wavelengths to 22 in the following simulations.
3.3. Effect of the Number of Layers
In order to determine the optimal number of sliced layers, we discussed different options for the number of slicing layers, including two, three, and four layers, into the inversion. In this experiment, we changed the default setting of the CO
2 profile into a four-layer profile, as shown in
Table 3. The simulated signals were then generated on this basis and were used to obtain sliced xCO
2 of two-layer and three-layer. We also calculated the pseudo-truth of sliced xCO
2 using weighted averaging in terms of sums of atmospheric pressures in different layers for two-layer and three-layer inversions. Then, the mean error was calculated as the difference between retrieved XCO
2 and the corresponding pseudo-truths. The results of the simulations are shown in
Table 3. In this section, the
SNR(T,λcentral) was set to 25 dB, the setting of the number of wavelengths was 22, and the detailed setting of the layers and concentration of CO
2 in each layer shown in
Table 1.
ABLB represents the layer below ABLH, ABLU represents the layer upper ABLH, LFT represents the lower free troposphere, and UFT represents the upper free troposphere. The column, xCO2, represents preset xCO2 in each layer.
Table 3 shows that the number of sliced layers have evident influence on the performance of our retrieval model. Both the accuracy and the precision of retrievals decrease with the increasing number of sliced layers. The scheme of two layers yields results with the highest accuracy and precision, in which MEs are −0.46 ppm and 0.11 ppm, respectively, whose corresponding STDs are also the lowest among the three schemes. Moreover, these results exhibit that both the accuracy and the precision are related to the range of each layer. The proposed method can obtain xCO
2 with errors of less than 1 ppm for each layer when the number of sliced layers is less than three.
More sliced layers help us to gain more information on the vertical mixing of CO2 but lead to lower accuracy and precision. Hence, there is a compromise in terms of the selection of an appropriate number of sliced layers. Near-surface CO2 has a strong relationship with human activity and ecosystems. The CO2 gradient around PBL is meaningful for us to quantitatively calculate carbon fluxes. If we sliced the atmosphere into two layers, the concentration of CO2 in ABL would be acquired with high accuracy, but the range of ABLU layer is too large compared with the range of ABLB. It is thus not reasonable to calculate the CO2 gradient around the ABL. If we sliced the atmosphere into four layers, the accuracy of retrievals in the ABL and FT would be insufficient to obtain accurate estimates of CO2 gradients between ABL and FT. Therefore, three sliced layers would be the most appropriate.
3.4. Effect of the Atmospheric Boundary Layer Height (ABLH)
The ABL is the lowest layer of the atmosphere, which is directly affected by the Earth’s surface and responds to radiative forcing. LiDAR is capable of measuring the ABLH with high accuracy. ABLH has a relationship with the location, the landform, and meteorological conditions. To evaluate the performance of the proposed retrieval method with respect to different heights of PBL, we carried out inversion simulations under different settings of the PBLH, namely 1.0 km, 1.5 km, and 2.0 km. In this section, the SNR(T,λcentral) was set to 25 dB, and the setting of the number of wavelengths was 22.
Table 4 shows that the accuracy of the retrieved xCO
2 in ABL would increase with the ascending ABLH. Especially for the xCO
2 in ABL, the mean error of xCO
2 in ABL was −1.76 ppm when ABLH was set to 1 km. Along with the ME, STD also falls with the increasing ABLH, implying that the precision of xCO
2 is also affected by the ABLH. We believe that the mechanism by which the ABLH affects the performance of the proposed method can be explained by Equation (9). A thicker ABL means a longer optical path, which leads to a higher DAOD. This would produce better adjustment in the ABL layer to promise the accuracy for xCO
2 in ABL during the process of our retrieval model.
3.5. Ability to Retrieve Gradients of xCO2
In this subsection, experiments are designed to explore the ability of the proposed multi-wavelength method to retrieve gradients of CO
2 under different circumstances. The ultimate goal of any space-based CO
2 sensor is to map the dynamic distribution of carbon sources and sinks. The vertical gradient of CO
2 is a critical parameter for distinguishing carbon sources and sinks. With enough accuracy, the vertical gradient of CO
2 can provide the potential ability to estimate carbon fluxes quantitatively. Three typical CO
2 profiles are assumed, as shown in
Table 5, to simulate the possible characteristics of different carbon fluxes. In this series of experiments, we assumed three scenarios according to differences between
XCO
2 in ABL and FT and named them as shown in
Table 5. It is worth noting that the vertical profiles of CO
2 are determined by many factors. In some extreme cases, a lower xCO
2 in ABL with respect to FT could be witnessed for CO
2 source regions and vice versa. Herein, we utilized the potential source, sink, and neutral to name the three scenarios, purely for simplification. In this group of experiments, the number of wavelengths was set to 22, the
SNR(T,λcentral) was set to 25 dB, and the wavelength range was set to 1572.305–1572.365 nm.
Figure 4 shows the ME and STD of retrievals in different layers for three scenarios. Each subfigure shows the performance of the proposed method in a specific layer and different colors are used to represent different characteristics of CO
2 vertical profiles, namely a potential source, a potential sink, and a potential neutral area. The MEs of xCO
2 retrievals in ABL are −0.98 ppm, −0.31 ppm, and −0.11 ppm for a potential carbon source, a potential sink, and a potential neutral region, respectively. Compared with retrievals in ABL, retrievals in the other two layers are much more accurate, yielding an ME of no more than 0.34 ppm regardless of carbon characteristics. The STD never exceeds 0.73 ppm in any case. These results indicate that the proposed method can obtain accurate and precise results regardless of carbon characteristics.
The mean errors of xCO2 in ABL and xCO2 in FT would be anti-correlated (one is positive while the other negative) for the source and sink cases because, in the retrieval model, it would adjust the value of DAOD in each layer according to the weight matrix W in the manuscript. Additionally, it would promise that the total DAOD is certain. The DAOD in STA is larger than that of ABL and FT. Hence, the main adjustments of DAOD occur in ABL and FT, and mean errors in DAODs in ABL and FT would have a neutralized relationship. Meanwhile, it is also worth noting that the performance of the proposed method varies with different characteristics of CO2 vertical profiles.
Figure 5 shows the ME and STD of retrieved xCO
2 gradients, including the xCO
2 gradient between ABL and FT and between FT and STA. For a potential carbon source, the ME and STD of the retrieved xCO
2 gradient (xCO
2ABL- xCO
2FT) are −1.25 ppm and 1.05 ppm, xCO
2ABL is xCO
2 in ABL, and xCO
2FT is xCO
2 in FT. For a potential carbon sink, the ME and STD of the retrieved xCO
2 gradient (xCO
2ABL- xCO
2FT) are -1.03 ppm and 1.02 ppm. For a potential neutral region, the ME and STD of the retrieved xCO
2 gradient (xCO
2ABL- xCO
2FT) are −0.17 ppm and 0.99 ppm. Consequently, the accuracy and precision of the retrieved xCO
2 gradients are also influenced by the characteristics of the vertical CO
2 profiles. Considering that the true carbon gradient (xCO
2ABL- xCO
2FT) is set to 8 and −12 ppm, its accuracy would be acceptable for its largest uncertainty of only around 16%.
As for the xCO
2 gradient between FT and STA, the uncertainty of the retrieved xCO
2 gradients is less than 7.5% in all three scenarios. Moreover, both the accuracy and precision of the retrieved xCO
2 gradients between FT and STA are higher than the carbon gradients between ABL and FT. Referring to
Figure 4, we believe that the lower accuracy and precision of the xCO
2 retrievals in ABL are responsible for such a phenomenon.