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Fernald method is regarded as the standard method for retrieving lidar data, but the retrieval can be performed only when a boundary value is given. Generally, we can select clear atmosphere above the tropopause as a reference to determine the boundary value, but we need to use the slope method to fit the boundary value when the detecting range is lower than the tropopause. The slope method involves significant uncertainty because this algorithm is based on two hypotheses: one is that aerosol vertical distribution is homogeneous, and the other is that either molecule or aerosol components exist in the atmosphere. To reduce the uncertainty, we proposed a new approach, which segments a signal into “uniform” sub-signals to avoid the first hypothesis, and then uses nonlinear two-component fitting to avoid the second one. Compared with the approach based on the slope method, the new approach obtained more accurate boundary values and retrieving results for both of the simulated and real signals. Thus the automatic segmentation algorithm and the two-component fitting method are useful for determining the reference bin and boundary values when the effective detecting range of lidar is lower than the tropopause.
© 2015 Optical Society of America
1. Introduction
Mie lidar is widely used in atmospheric aerosol detection because of its ability to provide atmospheric profiles. Since the 1960s, various methods have been proposed to retrieve atmospheric parameters based on Mie lidar data [1–6]. In the 1980s, two stable retrieval methods based on one-composition model and two-component model of the atmosphere, respectively, were proposed by Klett [2] and Fernald [3]. These two methods are collectively regarded as the standard method and called Klett–Fernald method. These two methods are mathematically equivalent to each other, except that the former separately considers aerosol and molecular components, whereas the latter does not. Considering both aerosol and molecular components are necessary if none of the two components is dominant, the Fernald two-component method is more in accordance with the actual situation and was more widely used in the retrieval than the Klett method. After outstanding contributions of previous studies, three significant error sources were found in the retrieval [2, 7, 8], namely, selection of boundary value, determination of aerosol lidar ratio (the ratio of aerosol extinction to backscattering coefficient), and effect of noise.
Aerosol optical properties can be retrieved by the Klett–Fernald method only when a boundary value is given. We usually need to select a reference height to determine the boundary value by assuming there are few aerosols. Under ideal conditions, the reference height can be set above the tropopause and the ratio of total and molecular backscattering coefficient is empirically known. However, the boundary value is difficult to determine when the detecting range of lidar is lower than the tropopause due to the background is too strong, the laser energy is too weak, the telescope diameter is too small, or a strong attenuation layer exists at the lower height. Thus, various methods have been proposed for determining the boundary value when the detecting range is low. The slope (i.e. one-component fitting) method proposed by Collis can be utilized to fit the boundary value for a homogeneous atmosphere [1]. Furthermore, a method is presented for determining the boundary value in a slant lidar signal through introducing the information of a horizontal lidar signal, which is useful for a scanning lidar [9]. Moreover, a method that replaces the boundary value with the optical transmission along an interested path was proposed, but the transmission is usually unknown in practice [10]. Additionally, the aerosol extinction coefficients (AEC) retrieved by a Raman signal in the reference height can be used for the boundary value of a Mie signal [11]. For the above methods, the slope method is the only method which can be conducted without any additional information. However, the slope method involves significant uncertainty because the method is based on two hypotheses: one is that the atmosphere distribution is strictly homogeneous, and the other is that either molecules or aerosols exist in the atmosphere [12, 13]. A boundary value obtained by the slope method has large errors because vertical distribution of aerosol is changeable even in a small range in the real atmosphere. Moreover, we need to define a relatively clear fitting region, which is usually manually selected according to atmospheric condition [14]. This task is impractical because large difference may be observed between the retrieval results of different researchers, because different methods may empirically define different fitting regions. The two hypotheses limit the application of the slope method in the boundary value selection, and large divergent result may occur in the retrieval.
In this study, we present an automatic segmentation and two-component fitting method to avoid the large uncertainty, which does not require any additional information. We initially search the most clearly segmented region based on the signal segment results to avoid the first hypothesis, and then we use a two-component method to estimate the precise aerosol extinction coefficients as the boundary value to avoid the second hypothesis. We discuss the method in detail, and simulate signals under various conditions to obtain a matrix of fitting accuracy. We search more accurate boundary values to retrieve the AEC with the new approach. Finally, we use simulated and measured signals to verify the accuracy of our approach. The analyses of the comparative experiment demonstrate that the automatic segmentation two-component fitting method is an effective method for selecting the boundary value. The results demonstrate a high consistency between the reference AEC and those retrieved by our approach. In general, the performance of our new approach can refer to the matrix of fitting accuracy.
2. Principle and method
Mie lidar is commonly used because of its relatively simple operational design. A single-scattering equation for Mie lidar is written as follows [3, 6]:
where P(r) represents the lidar signal and r is the range (altitude). C is a constant that is called the lidar constant. G(r) is the overlap factor caused by the misalignment of the laser beam and the telescope [15]. β1(r) and β2(r) are the molecular and aerosol backscattering coefficients, respectively. α1(r) and α2(r) are the aerosol and molecule extinction coefficients, respectively. e(r) is the noise, which we regard as Gaussian.Figure 1 shows the flow chart of the retrieval of lidar signal based on the segmentation algorithm, the two-component fitting method, and the Fernald method. First, we read the Mie scattering lidar data, and then select the data in a far range where we consider that the signal does not contain any information about atmosphere but contains background noise. Second, we remove the background noise by subtracting the average of the background signal from the entire signal, and then we calculate the standard deviation of the background signal for subsequent segmentation and retrieval. Third, if the valid detecting range is larger than the tropopause, then the boundary value is determined by the empirical backscattering ratio. Otherwise, we segment the signal into “uniform” sub-signals, and then we search the most clearly segmented region and use the two-component fitting method to estimate the AEC as the boundary value to reduce the uncertainty of the slope method. Fourth, the lidar ratio is empirically given or provided in real time by combining auxiliary sensors, such as the sun photometer, Raman scattering lidar data, and CALIPSO (Cloud-aerosol Lidar And Infrared Pathfinder Satellite Observations) [16]. Finally, the Fernald method is utilized to retrieve the aerosol optical properties based on the lidar ratio, boundary value, and American standard atmosphere.
Fig. 1 Flow chart of the lidar data retrieval based on the segmentation algorithm, two-component fitting method, and Fernald method.
2.1 Automatic segmentation algorithm
Segmenting a signal or curve into a few “uniform” sub-signals or sub-curves is an important approach to facilitate signal processing and application. This approach has been maturely applied in various research fields, such as map generalization [17] and data mining [18]. To the best of our knowledge, the initial form of the automatic segment approach for lidar data processing was first introduced by Wei. Gong, et al. [19] and subsequently improved [20]. The lidar signal segmentation algorithm segments the signal into “uniform” sub-signals in which any backscatter contributions are considered to originate from “uniform” objects. The lidar range-corrected signal X(r) can be described as follows:
where S(r) is the pure signal, and the noise level of X(r) is equivalent to e(r)r2. Starting with a range-corrected signal with n range bins X(r1), X(r2), ... and X(rn), Xseg(r) is defined as the signal derived by linear interpolation with X(r) of the beginning bin X(r1) and ending bin X(rn). The absolute intensity difference of the two signals can be written as follows:After finding the maximum difference at range rm, we select the corresponding range bin as a break-bin if d(rm) exceeds a given threshold value dthr(rm). Theoretically, the corresponding envelope curve of the noise e(r)r2 is ± 3σ·r2, where σ is the standard deviation of the background signal. Thus, a threshold array dthr(r) can be defined as 6σ·r2. Recursive segmenting calculations are conducted based on the two sub-signals segmented by this break-bin until the value no longer exceeds that of the corresponding threshold. Based on the preceding discussion, the recursive algorithm can be outlined as follows:
- Xseg(r1:rn) is obtained by interpolating X(r1) and X(rn) linearly, and then the absolute difference between X(r1:rn) and Xseg(r1:rn) is obtained.
- The threshold dthr(rm), which is equal to 6σ·rm2, are calculated.
- dmax and its corresponding range bin X(rm) are determined.
- If d(rm) exceeds the threshold value dthr(rm), then the left and right parts are segmented recursively; Otherwise, the break bins are the first and end bins of X(r1:rn).
- The first and end bins of every segmentation are combined as final break bins.
2.2 Two-component fitting method
For a retrieval that lidar detected the tropopause and aerosol can be ignored above the tropopause, the reference height rc can be set at tropopause and β1(rc)/β2(rc) can be assumed to be a constant value. However, we usually have to select a clear region in the lidar signal when the detecting range of lidar is lower than the tropopause and fit the AEC as the boundary value by the slope method. Vertically, the molecular backscatter and extinction coefficients exponentially decreases, which are proportional to the air density and can be calculated from the Rayleigh-theory based on the standard atmosphere [21]. The characteristic of exponentially decreases does not meet the prerequisite of the slope method that the atmospheric extinction coefficient is constant in a clear region, which cause large errors. In a region between r1 and r2 where the overlap factor is a constant, the lidar equation can be written as follows:
The two parts of the preceding equation are represented by a and b, as follows:
Assuming the aerosol and molecular backscatter ratio as constant is reasonable when the aerosol is stable as we assumed for that of the atmosphere above tropopause. Thus, We assume that β1(r)/β2(r) is constant as well as S1(r). Therefore, a and b are constant. The Eq. (4) can be written as follows:
We can obtain a and b by fitting each segment using a nonlinear least squares method, which minimizes the fitting error Δ:
After obtaining the optimal estimated values of a and b, we can determine the AEC according to the definition of b as follows:
The two-component fitting method requires a clear signal for retrieval. One of the segmented clear regions obtained by the aforementioned automatic segmentation algorithm serves as a fitting region. Several segmented regions can usually be regarded as clear regions for selecting the boundary value. To select the optimum clear region, we assume that fitting accuracy W(R, n) mainly depends on the noise level R and the number of range bins n for a region. We define W(R, n) as follows:
where f(R) and h(n) are functions of noise level and the number of data bins. R is simply defined as the ratio of the central signal in the fitting region to the standard deviation of the background noise. In this study, we simulated multi-group signals with different noise levels and number of bins to fit the boundary value according to the above Eqs. (8) and (9). We compared the fitted boundary value with the true value and obtained fitting precision matrix W(R,n). The fitting precision matrix W(R,n) was regarded as a criterion to determine the optimal region.2.3 Fernald method
The Fernald method is commonly used in the lidar data retrieval, because separately considers aerosol and molecular components. The Fernald two-components algorithm contains forward and backward parts, which can be given as follows [3]:
Backward part:
Forward part:
where A(i) is equal to β2(i) is the backscatter coefficient that can be obtained from the standard atmosphere model [21], and S2(i) = α2(i)/β2(i) = 8π/3 sr is the lidar ratio of atmospheric molecules. Before the retrieval, two parameters should be given, namely, the lidar ratio of aerosol and the boundary value. The lidar ratio of aerosol varies with the chemical and physical properties of the aerosol. However, according to the preliminary observation of an atmospheric background period, we can assume that S1(i) is a constant and is equal to 50 sr [22]. Reference height rc is generally set at approximately 10 km, and the β(rc)/β2(rc) is typically assumed to be 1.01~1.05 for retrieving the aerosol backscatter and extinction coefficients.3. Results and discussion
This section reports the findings of the retrieval approach applied to a variety of signals as described previously. The approach is tested by complex real and simulated data sets and compared with the slope method.
3.1 Simulated signal experiment
To verify the automatic segmentation and two-component fitting method of boundary value selection, we simulated a single backscatter signal of a ground-based lidar with a wavelength of 532 nm under standard atmosphere [Fig. 2(a)]. The lidar ratio of aerosol was set to be 50 sr. We also simulated an optical thick layer and a planetary boundary layer at 4-5 km and 0-2 km, respectively. The simulated signal composed of true value and Gaussian noise is illustrated in Fig. 2(a). The breakpoints produced by the automatic segmentation algorithm show that the segment tends to combine as many bins as possible in every segment, which is beneficial for providing robust fittings by the slope method and two-component fitting method.
Fig. 2 Simulated signal experiment: (a) Simulated lidar signals and breakpoints produced by the segmentation algorithm. (b) True value (True) and fitted AEC by the slope (i.e. one-component fitting) method (Fit 1) and two-component fitting method (Fit 2), as well as AEC retrieved by Fernald method with the boundary values obtained based on the two-component fitting method (Fernald 1) and the two-component fitting method (Fernald 2), respectively.
Figure 2(b) shows that the AEC fitted by the slope (Fit 1) and two-component fitting methods (Fit 2) are significantly larger than the true value (True) when the signal rapidly declines at the planetary boundary layer (1-2 km) and the range between the peak and top of the optical thick layer (4.5-5 km). Meanwhile, the retrieved AEC is negative at the region with rapidly rising signals between the base and peak of the optical thick layer (4-4.5 km). Therefore, both the slope and two-component fitting methods are highly sensitive to the changes of layers. Thus, we should avoid selecting the boundary value in layers. In the clear regions, the two-component fitting result is very close to the true value. However, the fitting result of the slope method and the true value differs by one order of magnitude. This phenomenon indicates that even under clear conditions, the variation of the air molecules along with the change of height must be considered during the retrieval. Actually, the signal decrease is mainly caused by the decline of backscatter coefficient under clear conditions, but the slope method miss-interprets that the signal decrease is mainly caused by attenuation of larger constant AEC. Moreover, we employed the Fernald method to retrieve the AEC based on the fitted AEC of the two-component fitting as the boundary value at 6 km, which is the center of the clear region between 5 km and 7 km. Subsequently, accurate AEC (Fernald 2) is obtained, which agrees well with the true value at the entire detecting range as shown in Fig. 2 (b). However, the AEC retrieved by the Fernald method with the boundary value obtained based on the fitted AEC of the slope method (Fernald 1) is with significantly larger error.
In the retrieval, more than one segmentation can usually be regarded as clear and be used as fitting regions for selecting the boundary value, such as 2.5-4 km and 5-7 km in Fig. 2(a). Thus, we have to establish a criterion to evaluate these segmentations and use the optimal one. We simulated 1000 group signals with various noise levels and number of bins and then obtained the AEC by the two-component fitting method. The standard deviation of the relative error was calculated and regarded as the criterion for selecting the optimal fitting region. The fitting accuracy matrix rapidly varies with SNR and number of range bins (Fig. 3). In applications, we need to find the corresponding standard deviation depending on the number of data bins and SNR for each segmented region, and consider the region with the smallest standard deviation as the optimum region. Thus, we select the center bin of the region as the reference bin and its fitted AEC as the boundary value.
Fig. 3 Fitting accuracy matrix W(R, n) varies with noise level and number of range bins, which is calculated based on 1000 simulations.
3.2 Measured signal experiment
We further verify the performance of the retrieval approach by using real signals, which were measured by a self-developed ground-based lidar with a wavelength of 532 nm at Wuhan University (30.5° N, 114.3° E) on November 30, 2008. The vertical and temporal resolutions of the measured signal are 7.5 m and 1 min, respectively.
Figure 4(a) shows one of the real signals and break-bins produced by automatic segmentation algorithm. The breakpoints in Fig. 4(a) demonstrated that the automatic segmentation algorithm has good applicability in segmenting the signal to “uniform” sub-signals. Figure 4(b) shows the AEC retrieved by the Fernald method with calibration at 10 km where is almost aerosol free, which we consider as reference value for the other retrievals. In the following, we only use the signals below 7 km for further testing to imitate the situations when lidar detection capability is less than 7 km or the signal above 7 km is fully attenuated by a lower thick layer. Figure 4(b) shows the fitted AEC of the slope method (Fit 1) and the two-component fitting method (Fit 2), which are usually negative or significantly larger than the reference value below 4 km where the planetary boundary layer has influence [23]. This phenomenon indicates that the two fitting methods are highly sensitive to change in a layer. Thus, we may only use the signal above the planetary boundary layer for the boundary value selection when processing a real signal. We determined clear regions and boundary values based on the slope method and the two-component fitting method below 7 km, then retrieved the AEC by the Fernald method assuming the backscatter ratio and the aerosol lidar ratio as empirical value of 1.05 and 50 sr, respectively. Figure 4(b) shows that the retrieved AEC (Fernald 1) based on the boundary value fitted by the slope method has a large difference with the reference value. However, the retrieved AEC (Fernald 2) based on the boundary value fitted by the two-component fitting method is in agreement with the reference value very well.
Fig. 4 Measured signal experiment: (a) Measured lidar signals and break bins generated by automatic segmentation algorithm. (b) Aerosol extinction coefficient fitted by the slope method (Fit 1) and segmented two-component fitting method (Fit 2), of which the negative part is not shown. Three profiles of aerosol extinction coefficients retrieved by the Fernald method were based on the boundary value determined by calibrating at 10 km (Ref.), fitted by the slope method (Fernald 1) and the two-component fitting method (Fernald 2), respectively.
Figure 5 is for investigating the ability of our new approach to automatically retrieve more than 500 profiles when their effective range is lower than the tropopause. Figure 5(a) is the reference AEC retrieved by Fernald method with calibrating at 10 km where is almost aerosol free, whereas Fig. 5(b) and 5(c) are AECs respectively retrieved based on the boundary value fitted by the slope method and the two-component fitting method by only using the signals below 7 km to imitate the situations when the detection range is less than 7 km. Figure 5 (d) shows large difference with a mean value of 8.8 × 10−5 m−1 between the AEC retrieved based on the slope method [as shown in Fig. 5 (c)] and the reference value [as shown in Fig. 5 (a)]. Figure 5 (e) is the same as (d), but with a smaller mean value of 2.9 × 10−5 m−1 for the difference of the reference AEC and the AEC retrieved based on the boundary value fitted by the two-component method. Figs. 5(c) and 5(e) show high consistency between the AEC retrieved by our new approach and the reference value. The comparison primarily indicates that the two-component fitting method has better applicability than that of the slope method.
Fig. 5 Aerosol extinction coefficient retrieved by Fernald method and their differences: (a)-(c) are the aerosol extinction coefficient retrieved by the Fernald method based on the boundary value determined by calibrating at 10 km, fitted by the slope method and the two-component fitting method, respectively.(d) is the difference of the retrieved extinction based on the slope method and the reference value. (e) is the same as (d), but for the two-component fitting method.
In general, we can automatically determine the boundary values and retrieve the entire profile based on the segmentation of two-component fitting when lidar is unable to detect the tropopause. The performance of the two-component fitting method is notably better than that of the slope method because we use a more reasonable hypothesis based on a priori information according to the US standard atmosphere [21]. However, large error may be caused, if we use the two-component fitting method to a range that is affected by the planetary boundary layer where the aerosol may be dominant, because it does not meet the assumption that the extinction is exponentially decreasing. Thus, strictly a priori information should be given and the two-component fitting equation should be reformulated correspondingly, otherwise, the solution may produce unrealistic results. Furthermore, under ideal conditions and when only range-independent white noise is considered, the performance of our new method can refer to the matrix of fitting accuracy as shown in Fig. 3.
4. Summary
Previous studies have made outstanding contributions to lidar data retrieval. However, the retrieval accuracy depends on boundary value selection, which may cause large errors. To reduce the errors, we propose a two-component fitting method to fit an accurate AEC as the boundary value based on the optimal segmentation, which is searched from the results of the segment algorithm. Before retrieval, a statistical matrix of fitting accuracy is obtained by repeated simulation with various noise level and range bins. In practical application, we searched the corresponding standard deviation according to the SNR and the number of range bins for each segmentation. We considered the region with the smallest standard deviation as the best region, then selected the middle bin and fitted AEC of the best region as the reference bin and boundary value, respectively. The AEC retrieved by the Fernald method with calibration at 10 km is considered as reference value for the other retrievals. Both the real and simulated cases show that the AEC retrieved based on the boundary value fitted by the two-component fitting method is significantly closer to the reference value than that of the slope method. Therefore, we suggest determining the reference bin and boundary values based on the two-component fitting method and automatic segmentation algorithm when the detecting range of lidar is lower than the tropopause.
Future work should consider the potential use of the approach in various applications of lidar data processing because this approach to select the boundary value is a new exploration. Improving and proposing new automatic segmentation and fitting algorithms are necessary. For example, the two-component fitting method can be improved based on statistical prior information even in the low altitude. Furthermore, the multiple scattering effects should be considered in the retrieval under optical thick atmospheric conditions. Further statistical analysis on fitting the AEC should be conducted in the future.
Acknowledgment
This work was supported by National Natural Science Foundation of China (Grant No.41127901), Program for Innovative Research Team in University of Ministry of Education of China(Grant No.IRT1278), Major Project of Hubei Collaborative Innovation Center for High-efficiency Utilization of Solar Energy (Grant No.HBSZD2014002). Part of this research was also supported by Strategic Priority Research Program of the Chinese Academy of Sciences (XDA05040300), National High Technology Research and Development Program of China (2011AA12A104), and by Science and Technology Research Foundation of SGCC contract (DZB17201200260).
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