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Abstract
CO2 column-weighted dry-air mixing ratio (XCO2) products with high precision and spatial resolution are essential for inverting CO2 fluxes and promoting our understanding of global climate change. Compared with passive remote sensing methods, IPDA LIDAR, as an active remote sensing technique, offers many advantages in measuring XCO2. However, a significant random error in IPDA LIDAR measurements causes XCO2 values calculated directly from LIDAR signals to be unqualified as the final XCO2 products. Hence, we propose an efficient particle filter-based inversion of CO2 for single observation (EPICSO) algorithm to precisely retrieve the XCO2 of every LIDAR observation while preserving the high spatial resolution of LIDAR measurements. The EPICSO algorithm adopts the sliding average results as the first estimate of the local XCO2; subsequently, it estimates the difference between two adjacent XCO2 points and calculates the posterior probability of XCO2 based on particle filter theory. To evaluate the performance of the EPICSO algorithm numerically, we perform an EPICSO to process pseudo-observation data. The simulation results show that the results retrieved by the EPICSO algorithm satisfy the required high precision and that the algorithm is robust to a significant amount of random errors. In addition, we utilize LIDAR observation data from actual experiments in Hebei, China, to validate the performance of the EPICSO algorithm. The results retrieved by the EPICSO algorithm are more consistent with the actual local XCO2 than those of the conventional method, indicating that the EPICSO algorithm is efficient and practical for retrieving XCO2 with high precision and spatial resolution.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The increasing atmospheric concentration of CO2 is a critical contributor to climate change [1]. Atmospheric inversion is typically performed to estimate CO2 fluxes via the measurement of CO2 concentrations using various techniques [2–4]. Spaceborne remote sensing can provide dense observations of CO2 concentrations globally and can thus supplement ground-based observation networks, such as GLOBALVIEW-CO2, to estimate CO2 fluxes with a finer spatial resolution and reduce the uncertainties of CO2 fluxes in locations where ground-based sites are sparse or unavailable [5,6]. Since 2009, a series of satellites dedicated to monitoring atmospheric CO2 concentrations has been launched [7–9] and have provided unprecedented observations, providing insights into the carbon cycle [10,11]. These missions were intended to improve the estimation of CO2 fluxes at a scale of approximately 1000 km, which imposes the requirement for extremely high accuracy on ${X_{CO2}}$ products [2,12]. Studies have revealed that biases of only a few tenths of parts per million (ppm) in ${X_{CO2}}$ can result in biases in subcontinental CO2 fluxes by up to 109 t of carbon [13–15]. Therefore, temporal–spatial averaging has been widely used to simultaneously suppress errors in ${X_{CO2}}$, which resulted in an extremely low spatial resolution. In recent years, scientists have discovered that ${X_{CO2}}$ products with high spatial resolution can aid in locating and quantifying anthropogenic emissions at the city scale [16–19] as well as at the plant scale [20–22]. The spatial resolution of ${X_{CO2}}$ products is key to the precise estimation of CO2 emissions at the plant and city scales because ${X_{CO2}}$ anomalies or enhancements caused by anthropogenic emissions are invisible under a coarse spatial resolution.
In April 2022, China launched an atmospheric environment monitoring satellite named DQ-1, which carries the first spaceborne LIDAR for CO2 detection [23]. Such a novel spaceborne CO2 monitoring approach is based on the integrated-path differential absorption LIDAR, abbreviated as IPDA LIDAR [24]. Unlike current passive remote sensing missions, such as OCO-2/3 [9] and GOSAT-1/2 [25], IPDA LIDAR does not rely on solar radiation to obtain surface spectra [26]. Therefore, IPDA LIDAR can provide more efficient observations over regions with clouds [27] and high aerosol loading [28] as well as at night and high latitudes [29]. Our recent study demonstrated the significant potential of IPDA LIDAR for the inversion of CO2 fluxes [30]. However, the precise estimation of localized anthropogenic emissions using spaceborne LIDAR is hindered by a significant challenge, namely, the expected insufficient precision of ${X_{CO2}}$ retrievals when echo signals are weak [31,32]. DQ-1 was designed to obtain ${X_{CO2}}$ with a precision of 1 ppm at a spatial resolution of 50 km over land [33,34]. For an orbit altitude of ∼700 km and a repetition frequency of 20 Hz for a single observation, a random error of ∼3.6% is adequate to fulfill this goal. However, such an error is extremely large and hinders the detection of meaningful ${X_{CO2}}$ anomalies for subsequent estimations of localized anthropogenic emissions [35].
Hence, we propose an efficient particle-filter-based [36,37] inversion of CO2 for a single observation, abbreviated as the EPICSO algorithm, to simultaneously obtain ${X_{CO2}}$ products with high spatial resolution and precision. The remainder of this paper is organized as follows: Section 2 presents the details of the EPICSO algorithm and fundamental materials. In Section 3, an evaluation of the proposed method via simulation experiments is provided. In Section 4, the application of the EPICSO algorithm to process real airborne IPDA LIDAR data is presented. Finally, a summary of this study is presented in Section 5.
2. Methodology
2.1 Fundamentals of IPDA LIDAR
The proposed method can be applied to any typical IPDA LIDAR system. In this study, we selected an atmospheric carbon dioxide LIDAR (ACDL) onboard DQ-1 as our hypothetical equipment. The ACDL emits two lasers with online and offline wavelengths. The online laser is absorbed well through the gas to be measured and attenuated, whereas the offline laser is barely absorbed. The target gas concentration is determined by measuring the differential absorption optical depth. The ACDL adopts the 1.57 µm band as the operating wavelength and deliberately selects its online and offline wavelengths to reduce interferences from water vapor and errors derived from imprecise information regarding the atmospheric temperature and pressure [38]. After optimization, the online and offline wavelengths were determined to be 1572.024 and 1572.084 nm, respectively. Detailed configurations of the ACDL onboard DQ-1 are available in a series of articles published by Chen at the Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences (SIOM) [34,39].
Based on the LIDAR equation [40], the CO2 column-weighted dry air mixing ratio (${X_{CO2}}$) is expressed as shown in (1).
2.2 Description of EPICSO algorithm
The symbols used to describe the EPICSO algorithm are shown in Table 1 and Fig. 1 displays the flow of this algorithm. All raw observations of ${X_{CO2}}$ from the LIDAR measurements are expressed as ${Z_1},{Z_2}, \ldots ,{Z_I}$, where I is the total number of LIDAR measurement points. The formula for calculating ${Z_i}$ is as follows:
Table 1. List of symbols used herein
Herein, i represents the serial number of each point. ${Y_i}$ is a relatively more precise “observation” value of ${X_{CO2}}_i$ that participates in data assimilation in the EPICSO algorithm. If more precise observational data of ${X_{CO2}}_i$ measured by other instruments are available, then the data can be adopted as ${Y_i}$. However, in practice, the useful information typically includes the CO2 concentration of a few locations measured using an in-situ instrument, the LIDAR performance parameters, and the raw observations of ${X_{CO2}}$. As a prior information pertaining to local ${X_{CO2}}$ is not available, we utilized the sliding average method to construct a series of ${Y_i}$ as the first estimate of the local ${X_{CO2}}$.
where $a = \left\{ \begin{smallmatrix} {1\; ,\; i - ({n - 1} )/2 < 1\; }\\ {i - ({n - 1} )/2,i - ({n - 1} )/2 \ge 1\; } \end{smallmatrix},\; b = \left\{ \begin{smallmatrix} {I\; ,i + ({n - 1} )/2 > I}\\ {i + ({n - 1} )/2,\; i + ({n - 1} )/2 \le I\; } \end{smallmatrix} \right. \right.$, and n is the size of the sliding window. The method for setting n is introduced in Section 3.2. The sliding average result ${Y_i}$ cannot be adopted as the retrieval result for ${X_{CO2}}_i$ because it is not sufficiently precise or accurate. However, in contrast with raw observations, the random error of ${Y_i}$ is smaller, which allows it to be used as a relatively more precise “observation” value participating in data assimilation to calculate the posterior probability density of ${X_{CO2}}_i$ using ${Y_i}$.First, we randomly sampled a set of particles denoted as ${X_{CO2}}_1^j$ with weight $w_1^j = \frac{1}{N},$ $({j = 1, \ldots ,N} )$ from the Gaussian distribution of the initial point, whose mean is ${X_{CO2}}_1$, which is estimated using prior information. Next, the transfer equations used to forecast ${X_{CO2}}_i^0$ and ${X_{CO2}}_i^j$ are as follows:
In (4), ${\lambda _{i - 1}}$ ($0 \le {\lambda _{i - 1}} \le 1$) is the acceptability of the estimated difference ${\Delta _{i - 1}}$. As ${\Delta _{i - 1}}$ is not sufficiently accurate, we can only accept it to only some extent. For the random error of every ${X_{CO2}}_i$, we can reasonably assume that all errors obey an identical probability distribution; therefore, the mean values of all the errors are the same. For every two adjacent points, the greater the ${\Delta _{i - 1}}$, which might imply that the actual difference value of the two points is larger, the less it the effect exerted by random error on ${\Delta _{i - 1}}$. Therefore, the acceptability of this ${\Delta _{i - 1}}$ should be higher. Thus, we define ${\lambda _{i - 1}}$ as
where ${\sigma _{{n_m}}}$ is the standard deviation of the random error of ${Y_i}$ (the formula for calculating ${\sigma _{{n_m}}}$ is ${\sigma _{{n_m}}} = {\sigma _{error}}/\sqrt n ,\; $ where ${\sigma _{error}}$ is the standard deviation of the random error of raw observations).Meanwhile, ${Y_{1:i}}$ (${Y_1},{Y_2}, \ldots ,{Y_i}$) is a series of more precise values of ${X_{CO2}}_{1:i}$ that positively affect the retrieval of ${X_{CO2}}$. Therefore, in the data assimilation process, ${Y_{1:i}}$ was adopted as prior information to calculate the posterior density of ${X_{CO2}}_i$, which is a better estimation of ${X_{CO2}}_i$. The posterior density is denoted as $p({X_{CO2}}_i|{Y_{1:i}})$. The final retrieval result for each ${X_{CO2}}_i$ is a mathematical expectation.
However, because $p({X_{CO2}}_i|{Y_{1:i}})$ is unknown, directly determining the abovementioned expectation is difficult. Hence, we adopted a method based on particle filter theory [36,37], where particles $\{{{X_{CO2}}_i^j} \}_{j = 1}^N$ were sampled from a known distribution $q({X_{CO2}}_i|{Y_{1:i}})$. Thus, the expectation can be expressed as
In addition, because
andThe recursive form of the weight is written as [37]
As a known distribution, $q({{X_{CO2}}_i^j\textrm{|}{X_{CO2}}_{1:i - 1}^j,{Y_{1:i}}} )$ is typically set as
However, as the number of iterations increases, degeneracy occurs [41,42], in which the weights of most particles become extremely small and are thus negligible. This implies that only a few particles with large weights affect retrieval, resulting in the degradation of retrieval performance. Generally, the number of effective particles is used to express the degree of particle degeneracy as follows [36]:
If (17) is smaller than a preset threshold, then certain measures must be adopted. To solve the degeneracy problem, the following two approaches are typically adopted:
In this study, we adopted the second approach, i.e., we replicated every particle, where the number of replications are based on the weight of every particle. The larger the weight of the particle, the more times the replications should be. After replication, the weights of all the particles in the same state are set to $1/N$, and the set of particles is expressed as $\{{\widehat {{X_{CO2}}}_i^j} \}_{j = 1}^N$. Therefore, the final retrieval result for every ${X_{CO2}}_i$ is
The flow of the EPICSO algorithm is as follows:
- 1. Use the raw observations ${Z_1},{Z_2}, \ldots ,{Z_I}$ as the input data, set ${X_{CO2}}_1^0 = {X_{CO2}}_1$, initialize the particles as $i = 2$, and generate particles $\left\{ {{X_{CO2}}_1^j,\; w_1^j = \frac{1}{N}} \right\}_{j = 1}^N$ from the prior distribution $p({{X_{CO2}}_1} )$;
- 2. For $i = 2,\; 3 \ldots \ldots $, repeat the following steps:
① Calculate ${Y_i}$, ${\Delta _{i - 1}}$, ${\lambda _{i - 1}}$ and ${X_{CO2}}_i^0$. Using (4), sample particles $\{{{X_{CO2}}_i^j} \}_{j = 1}^N$ from the Gaussian distribution, whose mean is ${X_{CO2}}_{i - 1}^j + {\lambda _{i - 1}}{\Delta _{i - 1}}$ and standard deviation is ${\sigma _{{n_t}}}$. Next, using (16), calculate the $w_i^j$ of every particle. Subsequently, normalize all weights.
② If ${N_{eff}} < threshold$, then resample $\{{{X_{CO2}}_i^j,w_i^j} \}_{j = 1}^N$ and the set of particles becomes $\left\{ {\widehat {{X_{CO2}}}_i^j,w_i^j = \frac{1}{N}} \right\}_{j = 1}^N$; otherwise, proceed to step ③ directly.
③ Calculate the retrieval result of ${X_{CO2}}_i$: ${X_{CO2}}_i^{final} = \mathop \sum \limits_{j = 1}^N \widehat {{X_{CO2}}}_i^jw_i^j$. Subsequently, set $i = i + 1$.
- 3. To achieve stable results, repeat the EPICSO algorithm multiple times and use the mean values of the results as the final results.
3. Algorithm evaluation and sensitivity testing via simulation experiments
3.1 Data for simulation experiments
Figure 2 shows data for simulation experiments. These three types of data were used to simulate the true values of ${X_{CO2}}$ with varying degrees of fluctuation. Each type of data was used to simulate the true values of 550 single observations of ${X_{CO2}}$. Data 2 were obtained from a snippet of actual observations in one day. By enlarging and shrinking humps A, B, and C of Data 2, we generated Data 3 and Data 1, which represent high and low fluctuations, respectively. Before conducting the simulation experiments, a random error was applied to the data, as shown in Fig. 3. Notably, we did not add additional bias to the data because in the actual scenario, the bias of LIDAR is insignificant. Every data point with random error is a pseudo-observation for simulating a single observation of ${X_{CO2}}$ from the LIDAR, which features a large random error. Subsequently, we used the EPICSO algorithm to process the pseudo-observations and evaluated its effectiveness.
Fig. 2. Three types of data for simulating true values of ${{\boldsymbol X}_{{\boldsymbol CO}2}}$ with various amplitudes of fluctuation in actual situations. Data 1, 2 and 3 represent true values of ${{\boldsymbol X}_{{\boldsymbol CO}2}}$ with low, medium, and high fluctuations, respectively.
Fig. 3. Pseudo-observations for simulating ${Z_{1:I}}$; raw observations of ${X_{CO2}}$ with large random error and low bias (based on Data 3 for instance, where the standard deviation of random error applied is 2 ppm). The black line is the true values of ${X_{CO2}}$ in the high fluctuation case based on Data 3.
3.2 Method for setting $n$
The symbols used in this section are shown in Table 2. In (3), a sliding window size n is introduced to acquire ${Y_{1:I}}$, which is a series of more precise observation values of ${X_{CO2}}$ for calculating the posterior possibility. An appropriate value of n renders ${Y_{1:I}}$ suitable and improves the estimation of ${X_{CO2}}$. Therefore, a relatively appropriate n must be selected based on prior information such that the final retrieval results of the EPICSO algorithm is more precise. Herein, we propose a method to determine the appropriate value of n.
Table 2. List of symbols used in Section 3.2
Naturally, the variance of ${Y_{1:I}}$ decreases compared to that of the raw data ${Z_{1:I}}$ after using the sliding average. For convenience, the variance of ${Y_{1:I}}$ when the sliding window size is n is denoted as $va{r_{{Y_{1:I}}}}(n )$. Clearly, $va{r_{{Y_{1:I}}}}(1 )= va{r_{{Z_{1:I}}}} = va{r_{True}} + va{r_{error}}$, i.e., without the sliding average, the variance of ${Y_{1:I}}$ is merely the variance of ${Z_{1:I}}$, which equals the sum of the variances of the true values and the random error. Meanwhile, $va{r_{{Y_{1:I}}}}({2I - 1} )= 0$ because, in this case, every ${Y_i}$ is the mean of ${Z_{1:I}}$. As n increases from $1$ to $2I - 1$, $va{r_{{Y_{1:I}}}}(n )$ decreases from $va{r_{{Z_{1:I}}}}$ to 0. When $va{r_{{Y_{1:I}}}}(n )> va{r_{True}}$, a large random error occurs in ${Y_{1:I}}$, whereas when $va{r_{{Y_{1:I}}}}(n )< va{r_{True}}$, ${Y_{1:I}}$ is extremely flat and does not reflect a reasonable amount of fluctuation in the true values. Therefore, when $va{r_{{Y_{1:I}}}}(n )= va{r_{True}}$, n is the most appropriate.
In this study, we used Data 2 with different random errors as an example to examine the relationship between $va{r_{{Y_{1:I}}}}(n )$ and n.
In Fig. 4, diagrams (a), (b), and (c) show the relationship between $va{r_{{Y_{1:I}}}}(n )$ and n, where the standard deviations of the random error applied to Data 2 were 2, 6, and 18 ppm, respectively, to simulate the effects of various random error magnitudes. After calculating all $va{r_{{Y_{1:I}}}}(n )$, we selected the power function $va{r_{{Y_{1:I}}}}(n )= a{n^b} + c$ to fit these points and recorded the parameters, as shown in Table. 3. The table shows the standard deviations of the random error, the parameters of the fitting function, the R-square of the fitting results, and ${n_0}$, which may be the most appropriate value of n derived from $va{r_{True}} = a{n^b} + c$. In the three cases, all the R-square values exceeded 97.8%, and those of (b) and (c) exceeded 99%, indicating that $va{r_{{Y_{1:I}}}}(n )= a{n^b} + c$ can express the relationship between $va{r_{{Y_{1:I}}}}(n )$ and n accurately. We set $va{r_{{Y_{1:I}}}}(n )= va{r_{True}}$ to calculate ${n_0}$, which is the theoretically most appropriate n. Subsequently, we varied n from $1$ to $2I - 1$ again, and for each n, we executed the EPICSO algorithm to calculate the RMSE of the final results to validate that ${n_0}$ was the best. (In this study, the RMSE of certain data expresses the root mean square error between these data and the true values to represent the magnitude of the random error.)
Table 3. Parameters used for describing relationship between $va{r_{{Y_{1:I}}}}(n )$ and $ n$ in Fig. 4
Fig. 4. Relationship between $va{r_{{Y_{1:I}}}}(n )$ and n. We applied three types of random error to Data 2, where the standard deviations of random error are 2, 6 and 18 ppm, separately. Subsequently, we varied n from $1$ to $2I - 1$ ($where\; 2I - 1 = 1099.$) and calculate the corresponding $va{r_{{Y_{1:I}}}}(n )$. We identified a power function that can express this relation accurately. The blue lines in each diagram express the positions of $va{r_{True}}$ and ${n_0}$.
We used the results of the EPICSO algorithm and Data 2 to calculate the RMSE of the former results. Fig. 5 shows that in all cases, as n increased, the RMSE of the EPICSO results decreased initially and then increased when it passed an inflection point. The n at the inflection point was ${n_{actual}}$, which is the most appropriate value. Table 4 shows that ${n_{actual}}$ is similar to ${n_0}$ and that the RMSEs of the two n values are almost identical and small compared with the standard deviations of the random error, which verifies that the ${n_0}$ estimated previously can be adopted as an appropriate n to allow the EPICSO algorithm to achieve the best results. In addition, the RMSE–n diagrams indicate that when n is ahead of the inflection point, the EPICSO algorithm is more sensitive to the values of n and that an appropriate n can significantly reduce the RMSE or random error as well as improve the performance of the EPICSO algorithm.
Table 4. Parameters pertaining to Fig. 5. (Here, we list the data for comparing $n_0$ and ${n_{actual}}$; clearly, $n_0$ and ${n_{actual}}$ are extremely similar.)
Fig. 5. RMSE of results yielded by EPICSO algorithm for different n. ${n_{actual}}$ is the n value when the RMSE is the minimum, which is the actual most appropriate n. The blue lines on each diagram express the positions of the minimum RMSEs and ${n_{actual}}$, and the green lines express the positions of ${n_0}$ and the RMSE when $n = {n_0}$.
After validating the appropriateness of ${n_0}$, we summarized the method for setting n. In practice, because the relationship between $va{r_{{Y_{1:I}}}}(n )$ and n satisfies the power function $va{r_{{Y_{1:I}}}}(n )= a{n^b} + c$, we require only three ($n,\; va{r_{{Y_{1:I}}}}(n )$) points to solve the three parameters $a,b,and\; c$. Two points, i.e., ($1,\; va{r_{{Y_{1:I}}}}(1 )= va{r_{{Z_{1:I}}}}$) and ($2I - 1,\; va{r_{{Y_{1:I}}}}({2I - 1} )= 0$), are known. Therefore, we only require another point to calculate $a,b,and\; c$ as well as ($I,\; va{r_{{Y_{1:I}}}}(I )$). The method for setting n is as follows:
- (1) Set $n = I$; calculate ${Y_{1:I}}$ and $va{r_{{Y_{1:I}}}}(I )$.
- (2) Use three points, i.e., ($1,\; va{r_{{Z_{1:I}}}}$), ($2I - 1,\; 0$), and ($I,\; va{r_{{Y_{1:I}}}}(I )$) to fit the curve $va{r_{{Y_{1:I}}}}(n )= a{n^b} + c$; subsequently, obtain the three parameters $a,b,\; and\; c$.
- (3) Set $va{r_{True}} = va{r_{{Z_{1:I}}}} - va{r_{error}}$ and calculate n from $va{r_{True}} = a{n^b} + c$.
- (4) The n obtained is the most appropriate one.
3.3 Results of simulation experiments
We performed simulation experiments to evaluate the performance of the EPICSO algorithm. In the simulation experiments, three types of random errors with standard deviations of 2, 6, and 18 ppm, separately, were applied to data points 1, 2, and 3 to simulate the effects of various random error magnitudes on ${X_{CO2}}$ with different amplitudes of fluctuation. For every case, we adopted the method introduced in Section 3.2 to set the parameter n and calculate the sliding average result ${Y_{1:I}}$ for the subsequent calculation using the EPICSO algorithm. In addition, we regarded the average values of the results of 10 repeated experiments as the final retrieval results and calculated the ME and RMSE of the EPICSO results. The ME of a certain data point is the mean error between the data point and its true value, thus reflecting the bias in the data point. The simulation results are presented as follows.
The true values in the same row are the same, and the standard deviations of the random error in the same column are the same. Three random error amplitudes were applied to each true value. The EPICSO results and true values were used to calculate the ME and RMSE of the retrieval results, which are shown in each figure. By comparing the raw data (cyan), EPICSO results (red), and true values (black), we can intuitively observe that the error has been eliminated considerably after using the EPICSO algorithm, and that results yielded by the EPICSO algorithm retain a reasonable fluctuation of true values. Even in the case with large random errors (18 ppm), the EPICSO results retained the small humps of the true values.
Fig. 6 (c) shows the simulation results for an extremely high-error case of spaceborne IPDA LIDAR measurements. Although the error applied was significant (18 ppm), the EPICSO results retained small humps of Data 1, meanwhile, the ME (within 0.1 ppm) and RMSE (within 1 ppm) of the retrieval results were both low, which implies that the EPICSO algorithm effectively retrieved the single observation of spaceborne IPDA LIDAR.
Fig. 6. (a-i) Final retrieval results of ${X_{CO2}}$ by applying EPICSO algorithm. True values of (a)(b)(c), (d)(e)(f), and (g)(h)(i) are those from Data 1, Data 2, and Data 3, respectively. The standard deviations of the random error of the first, second, and third columns are 2, 6 and 18 ppm, respectively. The cyan line shows the raw data (pseudo observations), the yellow line shows the sliding average results ${Y_{1:I}}$ when n is the most appropriate, the red line shows the results of the EPICSO algorithm, and the black line shows the true value data. The standard deviations of random error, ME, and RMSE of the EPICSO results are shown on every figure.
Table 5 lists the ME and RMSE values of the EPICSO results. The ME was adopted to express the bias and the RMSE was used to reflect the random error of the results. Compared with the raw data, the random error of the EPICSO results declined significantly, whereas the ME of the results changed slightly, which implies during the retrieval, the EPICSO algorithm did not magnify the bias but caused it to decrease slightly. In most cases, the MEs and RMSEs of the results were within 0.1 and 1 ppm, respectively, which indicates that the results retrieved by the EPICSO algorithm with high precision are adequate to be ${X_{CO2}}$ products. Some ME values exceeded 0.1 ppm (e.g., Figs. 6(e) and (f)) because the bias of the raw data was high. However, in practice, the bias of LIDAR measurements is relatively low, which does not pose an issue. Because of point-by-point retrieval, the EPICSO results retained a high spatial resolution. Based on Fig. 6 (c), which shows the simulation results for the extremely large-error case of the spaceborne IPDA LIDAR, the RMSE of the EPICSO results reduced by 95.07% compared with the raw data, which indicates the favorable performance of the EPICSO algorithm. For the same true-value data, the RMSE of the final retrieval results increased with the error. However, when the design error was similar, the greater the fluctuation of the true values, the higher was the RMSE of the final results. This is attributable to the following reasons: (1) the EPICSO algorithm has a low-pass property, which is more suitable for the lower-fluctuation data than for the higher-fluctuation data. (2) In the simulation experiments, when we set the same standard deviation of error, owing to the randomness of error generation, a larger error may be generated, which consequently increases the RMSE. For example, as shown in Table 5, when the designed standard deviations of the random error were the same, the actual random error of Data 3 was larger than those of Data 1 and Data 2.
Table 5. ME and RMSE in each case. (We calculated the ME and RMSE of noisy data and the sliding average results ${{\boldsymbol Y}_{1:{\boldsymbol I}}}$ for comparison.)
Compared with the sliding average results ${Y_{1:I}}$, the RMSEs of the EPICSO results were generally less than the RMSEs of ${Y_{1:I}}$. The magnitude of this improvement was 0.1 to 0.3 ppm and when the random error was larger, the RMSEs of the EPICSO results improved more significantly in general. As shown in Fig. 6, in the lower-error case (2 ppm), the improvement by the EPICSO algorithm in contrast with ${Y_{1:I}}$ is not significant; however, in the larger-error case (6 and 18 ppm), the EPICSO results clearly maintained the fluctuations of the true values (“humps”) more accurately than ${Y_{1:I}}$. In our opinion, this occurred because in the higher random error cases, the window size n was large, and it is easy for the sliding average method to make the data too flat and lose details of the small data fluctuation, although we have optimized the sliding window size n. Therefore, in an actual measuring scenario where the random error is high, the EPICSO algorithm is more adequate for retrieving precise ${X_{CO2}}$ products.
Additionally, we investigated the robustness of the EPICSO algorithm. As an example, we applied different random errors to Data 2 and calculated the ME and RMSE of the raw data and EPICSO results. We conducted 100 replicates of the EPICSO retrieval process using the mean value of the 100 MEs and the mean of the 100 RMSEs as the ME and RMSE of the EPICSO results, respectively. The following figure shows the MEs and RMSEs for various standard deviations of random error.
Based on Fig. 7, as the error increased, the RMSE of the EPICSO results increased; however, in contrast to the RMSE of the raw data, the growth rate of the above was extremely low, which indicates that even if the standard deviation of the error is high, the performance of the EPICSO algorithm remains favorable. Therefore, the EPICSO algorithm is robust against relatively large random errors. No clear relationship was indicated between the ME and standard error because the bias and random error were not correlated. Based on a comparison with the raw data, the ME of the EPICSO results changed slightly, which implies that the EPICSO algorithm does not magnify the bias.
Fig. 7. MEs and RMSEs for various standard deviations of random error. The blue and red points show the absolute values of MEs and RMSEs, respectively.
In summary, results from the simulation experiments show that the EPICSO algorithm can successfully retrieve ${X_{CO2}}$ with high precision and spatial resolution from raw observations yet perform effectively in cases involving large random errors. Next, we validate the performance of the algorithm based on actual observations.
4. Algorithm validation based on actual observations
4.1 Data from experiment on March 2019 in Hebei
Northern China, particularly the Beijing–Tianjin–Hebei region, has the world’s largest population and the highest man-made emissions. Accurate and precise measurements of atmospheric CO2 in this region are critical [39]. The first flight experiment was conducted in March 2019. In this experiment, a Yun-8 transport plane was used to transport the IPDA LIDAR. To detect the atmospheric CO2 concentrations on various surface types, flight paths were designed to pass over the sea, urban areas, and mountains. The plane carried an IPDA LIDAR, which measured the ${X_{CO2}}$ along the flight path. An aircraft integrated meteorological measurement system [39] was installed on this plane to measure meteorological data, which were then used to calculate the integral weight function. To validate the retrieval results, an in-situ gas measuring instrument was used in the plane to measure the CO2 concentration.
After eliminating some unqualified observations (e.g., when the number of pulse signals was not four, the ranges measured by the LIDAR were incorrect, and the roll angle of the plane was extremely large), we performed calculations based on the raw observations of ${X_{CO2}}$.
Fig. 8 shows that random errors were present in the raw observations of ${X_{CO2}}$. Therefore, the EPICSO algorithm was adopted to obtain more precise results of ${X_{CO2}}$.
Fig. 8. Raw observations of ${X_{CO2}}$ in Beijing–Tianjin–Hebei region after eliminating unqualified observations. Random error remained in the results. The points are used as ${Z_{1:I}}$ and input to the EPICSO algorithm to retrieve a more precise ${X_{CO2}}$.
4.2 Final retrieval results of ${X_{CO2}}$
We used raw observations of ${X_{CO2}}$ (as shown in Fig. 8) as the input data for the EPICSO algorithm. The relative random errors (RRE) of this LIDAR were approximately 7.56% (above sea) and 2.5% (above land), which were obtained via simulation [31]. We multiplied the mean value of the raw observations by the RRE to calculate the standard deviation of the random error (approximately 31 and 10 ppm above sea and land levels, respectively). Subsequently, we set n using the setting method introduced in Section 3.2 and continued executing the EPICSO algorithm. The final results are as follows.
The intraday retrieval results of ${X_{CO2}}$ yielded by the EPICSO algorithm are shown in Fig. 9 (a), and a comparison among the EPICSO results, the results retrieved by Zhu et al. [43], and the sliding average results (the sliding window size is the most appropriate) is shown in Fig. 9 (b). In Fig. 9 (b), the measurement locations of the three types of results used for comparison were above the sea, where the random error was the highest. The carbon sources and sinks of the sea were similar in a small spatial range, which implies that the true values of ${X_{CO2}}$ in this small area can be regarded as constant, and the standard deviation of the data in this area can be adopted as an approximated random error. Therefore, for every 20 points of the results that were spatially extremely close (within 100 m), we calculated their standard deviation as the random error in this area and used the mean value of all standard deviations as a precision indicator to validate the precision advantage of the EPICSO algorithm. The mean standard deviations of the three results are listed in Table. 6.
Table 6. Mean standard deviations of three results shown in Fig. 9(b)
Fig. 9. (a-b) Final retrieval results of ${X_{CO2}}$ after using EPICSO algorithm. (a) EPICSO results during the entire measuring time. As an example, we selected the EPICSO results above the sea in the red box in (a) and zoomed them out to (b) to compare them with ${X_{CO2}}$ results retrieved by Zhu et al. [43] and the sliding average results in the same period.
As shown in Table 6, the standard deviations decreased significantly after applying the various retrieval methods. Compared with the other two methods, the EPICSO algorithm yielded the lowest standard deviation, which indicates that in large-error cases, the EPICSO algorithm is more precise than the sliding average and the method presented in [43]. The precision indicator for the EPICSO algorithm was 0.380 ppm.
The accuracy of the EPICSO algorithm must be further validated. Owing to space limitations, details regarding the further validation are provided in the Supplemental Material. The accuracy indicator of the EPICSO algorithm was 0.337 ppm.
The results retrieved by Zhu et al. differed from those of the EPICSO algorithm. The validation in the Supplemental Material shows that the relatively true ${X_{CO2}}$ of the sea was 411–413 ppm, which implies that the results retrieved by Zhu et al. (415–430 ppm) were biased.
Fig. 10 shows all the EPICSO results on a map. The spatial distribution of local ${X_{CO2}}$ is shown, where the values of ${X_{CO2}}$ in the southeast are higher than those in the west, based on the northwest wind in the area.
Fig. 10. EPICSO retrieval results during entire measuring time (shown on the map). Gaps appeared in the flight path on the map because the observations at the gaps were unqualified and have been eliminated. The unqualified observations at the track corners were due to turns made by the plane, which resulted in overly large roll angles of the plane.
After validating the reliability of the EPICSO algorithm, we used it to retrieve local ${X_{CO2}}$ data on another day. The results obtained are as follows.
In terms of the data shown in Fig. 11, the methods used to calculate the accuracy and precision were the same as those used earlier. The accuracy of the EPICSO algorithm was 0.289 ppm, and its precision was 0.346 ppm. Fig. 11 (a) shows that along the 326.42 km flight path, the number of the result points was 28688; consequently, the spatial resolution of the EPICSO results was 11.4 m. This result is much better than that of the conventional temporal-spatial averaging method because the EPICSO algorithm retrieves ${X_{CO2}}$ for every single observation. Clearly, the ${X_{CO2}}$ near the urban area is higher than that in the north, which can be explained by the fact that the area in the north is mountainous and has more vegetation for absorbing more CO2.
5. Conclusion and future studies
In this study, a retrieval algorithm for ${X_{CO2}}$, named the EPICSO algorithm, was proposed to retrieve more precise results. By performing simulation experiments, we identified a method for setting an essential parameter n in this algorithm and discovered that when n was ahead of the inflection point, EPICSO algorithm was sensitive to the values of n and that an appropriate n can significantly reduce the RMSE and improve the performance of the EPICSO algorithm. Additionally, the simulation experiments showed that EPICSO results exhibited high precision and retained reasonable fluctuations in the true values; therefore, the EPICSO algorithm is adequate for obtaining ${X_{CO2}}$ products. In addition, the EPICSO algorithm was robust to significant errors. Data from experiments were used to validate the proposed algorithm. The results indicated that the EPICSO algorithm is more effective for the precise retrieval of single observation data as it can preserve the distribution characteristics and high spatial resolution of the data in the local area.
However, we did not use more data from actual experiments to evaluate the EPICSO algorithm. In the future, we shall utilize the data measured from DQ-1, which was developed by the SIOM and launched on April 16, 2022, to investigate the effects of the EPICSO algorithm in various situations (e.g., cloud- and aerosol-covered areas and high-latitude areas).
Funding
Hubei Luojia Laboratory; National Natural Science Foundation of China (41971283).
Acknowledgment
The authors would like to thank the National Natural Science Foundation of China and Hubei Luojia Laboratory for the financial support.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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