Lidar Ratio Regional Transfer Method for Extinction Coefﬁcient Accuracy Improvement in Lidar Networks

: Lidar networks are essential to study the three-dimensional distribution of aerosols on a regional scale. At present, both Mie-scattering lidar (ML) and advanced lidars are being used in lidar networks. The latter can retrieve extinction coefﬁcients without strict assumptions of the lidar ratio, such as Raman lidar (RL) or high-spectral-resolution lidar (HSRL). In order to balance the data quality and instrument costs for the lidar network, the lidar ratio regional transfer method in a lidar network is proposed in this paper. We developed a Lidar Ratio and Aerosol Fraction Non-linear Regression (LR-AFNR) model between the lidar ratio and corresponding absorbing aerosol fraction (this paper studied two types of absorbing aerosols: dust and carbonaceous). The aerosol fraction of the sun photometer retrieval was used as a medium to transfer the lidar ratio of HSRL retrieval to a certain range of MLs. This lidar ratio can be the input parameter for ML retrieval and enables the improvement of the extinction coefﬁcient accuracy. The results show that the LR-APNR model is applicable to atmospheric conditions with high mineral dust or carbonaceous aerosol loading, and the maximum relative error of the ML extinction coefﬁcient can be reduced from 46% (dust) and 64% (carbonaceous aerosol) to 20%.


Introduction
Lidar is widely used in atmospheric aerosol detection because of its high temporal, spatial resolution, and continuous monitoring [1]. A lidar network can be used to observe atmospheric aerosols on a regional scale, which is significant in studying the temporal and spatial distributions of regional atmospheric aerosols and the atmospheric pollution control [2,3]. Over the years, several lidar networks have been established, such as the European Aerosol Research Lidar Network (EARLINET) [4], the Asian Dust and Aerosol Lidar Observation Network (AD-Net) [5], and the NASA Micro-Pulse Lidar Network (MPLNET) [6]. EARLINET mainly uses Raman lidar (RL) mainly, while Mie-scattering lidar (ML) and high-spectral-resolution lidar (HSRL) are applied in AD-Net [4,7]. Considering that HSRL and RL can independently retrieve aerosol backscattering and extinction coefficients, we also developed our HSRL system [8]. However, when establishing a lidar network, the HSRL instrument cost is high and difficult to be deployed on an extensive range. If lower-cost ML instruments are used, the aerosol extinction coefficient retrieval accuracy due to a priori assumption is limited [9,10]. Therefore, we considered whether the lidar ratio retrieved by HSRL can be transferred to ML and used as the input parameter of the Klett-Fernald method for retrieval. Then, the extinction coefficient accuracy of multiple MLs in the region will be improved [11,12]. In this way, this mode of single HSRL with multiple MLs can balance the problem of data quality and instrument cost at the same time.
Since HSRL and MLs are not observed at the same location, it is necessary to ensure that the aerosol detected by HSRL is transferred to the atmosphere above ML detection before Figure 1. The schematic diagram of the lidar ratio regional transfer method and the spatial distribution of the lidar network. (a) The schematic diagram of the lidar ratio regional transfer method. The blue circle in the figure represents the combination of HSRL and the sun photometer (SP) at the same position, and the orange circle represents the combination of ML and the sun photometer (SP) at the same position. The direction of the arrow represents the lidar ratio transfer from HSRL to different MLs using the LR-AFNR. No. 0-n represents the instrument number (n ≥ 1); (b) Schematic of the spatial distribution of the lidar network of HSRL combined with MLs. The blue and orange pentagons correspond to the two combinations of the lidar and sun photometer mentioned in figure (a).
The main process of the lidar ratio regional transfer method includes six steps of data acquisition, data retrieval, data screening, developing the LR-AFNR model, determining the conditions for using the LR-AFNR, and completing the lidar ratio regional transfer ( Figure 2). The specific steps are as follows: First, obtain the relevant data from the HSRL and sun photometer (HSRL and AERONET sites need to be selected in the same location). The data of different lidar ratios and the corresponding aerosol fractions are obtained by retrieving the above two data sources. Then, we proposed the data screening criteria for different lidar ratios and corresponding aerosol fractions and screened the relevant data. The lidar ratio with corresponding aerosol fraction data was selected for the same location and the same hour of observation, and the LR-AFNR model was developed. After that, the conditions of use of LR-AFNR (aerosol fraction and distance conditions) need to be determined. Finally, the aerosol fraction was converted to the lidar ratio based on the conditions of use of LR-AFNR. At the same time, it was used as an input parameter for ML using the Klett-Fernald retrieval method [11]. The main process of the lidar ratio regional transfer method includes six steps of data acquisition, data retrieval, data screening, developing the LR-AFNR model, determining the conditions for using the LR-AFNR, and completing the lidar ratio regional transfer ( Figure 2). The specific steps are as follows: First, obtain the relevant data from the HSRL and sun photometer (HSRL and AERONET sites need to be selected in the same location). The data of different lidar ratios and the corresponding aerosol fractions are obtained by retrieving the above two data sources. Then, we proposed the data screening criteria for different lidar ratios and corresponding aerosol fractions and screened the relevant data. The lidar ratio with corresponding aerosol fraction data was selected for the same location and the same hour of observation, and the LR-AFNR model was developed. After that, the conditions of use of LR-AFNR (aerosol fraction and distance conditions) need to be determined. Finally, the aerosol fraction was converted to the lidar ratio based on the conditions of use of LR-AFNR. At the same time, it was used as an input parameter for ML using the Klett-Fernald retrieval method [11]. Flowchart of the lidar ratio regional transfer method. The flowchart consists of six steps: data acquisition, data retrieval, data screening, developing the LR-AFNR model, determining the conditions for using the LR-AFNR, and completing the lidar ratio regional transfer.

Screening of Different Lidar Ratio Data
Taking advantage of the broad spectrum of Cabannes-Brillouin scattering from atmospheric molecules, the HSRL technique employs a narrow spectral filter to reject the aerosol Mie-scattering component in the lidar return signals. Therefore, the HSRL can directly obtain the aerosol extinction coefficient and backscatter coefficient [19,20]. The lidar Figure 2. Flowchart of the lidar ratio regional transfer method. The flowchart consists of six steps: data acquisition, data retrieval, data screening, developing the LR-AFNR model, determining the conditions for using the LR-AFNR, and completing the lidar ratio regional transfer.

Screening of Different Lidar Ratio Data
Taking advantage of the broad spectrum of Cabannes-Brillouin scattering from atmospheric molecules, the HSRL technique employs a narrow spectral filter to reject the aerosol Mie-scattering component in the lidar return signals. Therefore, the HSRL can directly obtain the aerosol extinction coefficient and backscatter coefficient [19,20]. The lidar ratio can be calculated from these two parameters. Considering a mathematical relationship between the fraction of dust or carbonaceous aerosols with the lidar ratio [14], the lidar ratio obtained by HSRL is first screened. This paper selects the data of the University of Wisconsin HSRL (UW-HSRL) [21], and the three sites selected are shown in Table 1. The first site is named northern Great Plains (SGP) and is located on the northern plains of Oklahoma. This system (BagoHSRL) has ceased observation (belongs to the archived site). We downloaded observations data for this system from January 2015-October 2017. The second site is named KORUS and is located at Yonsei University in Seoul, Korea. This system (AHSRL) has ceased observation (belongs to the archived site). We downloaded observations data from this system from January 2016-December 2018. The third site is named Madison, WI, located at the University of Wisconsin-Madison in Wisconsin. This system (BagoHSRL) maintains routine observations (belongs to the active site). We downloaded the observation data of the system from November 2012-June 2019. The wavelength and altitude ranges of the data are 532 nm and 0-10 km, respectively. In addition, the spatial and temporal resolutions of the data are 7.5 m and 10 min, and the download formats and file modes of the data are NetCDF4 and day, respectively. Details can be found at http://hsrl.ssec.wisc.edu (last accessed on 26 April 2021).
In this paper, the parameter retrieval using the UW-HSRL included the aerosol depolarization ratio δ a , aerosol backscatter coefficient β a , aerosol extinction coefficient α a , and lidar ratio S a = α a /β a . In order to make a more reasonable comparison with the sun photometer data, this paper needs to make an assumption about column properties: the aerosol is well-mixed in the entire boundary layer [22]. Considering the existence of a near-surface layer of more than 4 km [23][24][25], the near-surface layer of this paper is selected to be 4 km, i.e., the altitude range of HSRL data selection is 0.5-4 km (removal of the part affected by the overlap factor below 0.5 km). Based on [26], the authors found pure dust δ a = 0.3 and pure carbonaceous aerosol δ a = 0.05. When dust and carbonaceous aerosols are uniformly mixed, as well as the dust fraction of total backscatter (called the dust absorbing mixing ratio) in the range of 50-100%, dust is considered to be the primary absorbing aerosol with δ a values ranging from 0.15-0.3. It should be noted that the dust (carbonaceous aerosol) absorbing mixing ratio refers to the fraction of dust (carbonaceous aerosol) to the total sum of dust and carbonaceous aerosols. On the contrary, the carbonaceous aerosol is considered the main one with δ a values ranging from 0.05 to 0.15, while the carbonaceous aerosol absorbing mixing ratio range is 50-100%. In order to distinguish between dust and carbonaceous aerosols, we also need to use the lidar ratio. According to relevant historical documents, the polluted dust lidar ratio range is 30-60 Sr, and the polluted carbonaceous aerosol lidar ratio range is 40-100 Sr [27][28][29][30][31][32][33]. In addition, to eliminate the influence of cloud and low signal-to-noise data, the scattering ratio R = (β m + β a )/β m is set in the range of 1.2-10 [8,34]. β m is the backscatter coefficient of atmospheric molecules, which can be obtained through the 1976 U.S. Standard Atmosphere Model. Considering that aerosols may still occur above 4 km, Remote Sens. 2022, 14, 626 5 of 22 this paper adopts the method of [8] to set a threshold value of the scattering ratio R to screen data with aerosols. We established the theoretical scattering ratio threshold T t and the practical scattering ratio threshold T p . The expressions are where σβ a represents the uncertainty of aerosol backscatter, and the specific calculation method can be found in Equations (3)-(5), (14) and (17)-(21) mentioned in the paper of [8].
When the altitude is greater than 4 km and R > T p , we consider the presence of aerosols and discard the lidar ratio profile data. Therefore, the lidar ratio screening criteria ( Table 2) are proposed for dust and carbonaceous aerosols. The lidar ratio data outside the reasonable range of the criteria are set to the NaN value. It should be noted that the dust-dominated and carbonaceous aerosol-dominated screening criteria are only different in the aerosol depolarization ratio and lidar ratio.   The lidar ratio data retrieved from the three HSRL sites was screened with dust-dom inated data and carbonaceous aerosol-dominated data through screening criteria. T data mentioned above were averaged at a spatial resolution of 0.5-4 km and a time perio of one hour.

Retrieval Method of Aerosol Fraction and Data Screening
AERONET is a globally distributed network of automated sun and sky radiomete The lidar ratio data retrieved from the three HSRL sites was screened with dustdominated data and carbonaceous aerosol-dominated data through screening criteria. The Remote Sens. 2022, 14, 626 6 of 22 data mentioned above were averaged at a spatial resolution of 0.5-4 km and a time period of one hour.

Retrieval Method of Aerosol Fraction and Data Screening
AERONET is a globally distributed network of automated sun and sky radiometers, which measure the aerosol optical depth (AOD) and absorption aerosol optical depth (AAOD) at four wavelengths (440, 675, 870, and 1020 nm), as well as other properties [35]. This paper uses the Level 1.5 and Version 3 data from AERONET. Details can be found at https://aeronet.gsfc.nasa.gov (last accessed on 23 March 2021) [36]. Since this paper needs to use single-scattering albedo (SSA) data with AOD < 0.4, and level 2.0 data only have SSA data when AOD > 0.4, the data of level 1.5 were used in this paper [35]. Three AERONET sites were selected that overlap with the HSRL sites (Table 3) in the previous section. Based on [15], the authors proposed a method combining the absorption Angstrom Exponent (AAE) in visible and near-UV spectra to separate the total AAOD observed by the sun photometer into BC, BrC, and dust, where the AAE value of each absorbing component is assumed known and constant; the expression is where there are two inherent assumptions in Equation (3): (1) The AAE value of each absorbing component is an intrinsic property that does not depend on the mixing state and (2) the measured AAOD value represents a well-mixed sample of these species. Based on the two assumptions mentioned above, the Equation (3) of AAOD for different absorbing components can be developed at three different wavelengths λ i (440, 675, and 870 nm). The Equation (3) can provide three equations with three unknowns. AAE values for λ 1 = 440 nm and λ 2 = 675 nm are denoted as AAE1, and AAE values for λ 1 = 675 nm and λ 2 = 870 nm are denoted as AAE2. Based on [15], the authors showed that AAE1 values of BC, BrC, and dust are calculated to be 0.55 ± 0.24, 4.55 ± 2.01, and 2.20 ± 0.50, respectively, and AAE2 values of BC and dust are calculated to be 0.85 ± 0.40 and 1.15 ± 0.50, respectively. It should be noted that the AAE2 of BrC is not defined because BrC has minimal absorption in near-infrared spectra, and it is assumed that BrC does not absorb at 870 nm. Since the wavelength of UW-HSRL data used in this paper is 532 nm, the data of this wavelength are not available in AERONET. Therefore, it is necessary to convert the extinction Angstrom Exponent (EAE), AAE, AOD, and AAOD between different wavelengths to calculate the correlation result for λ re f = 532 nm. The conversion equation is as follows [15,22]: where λ 1 and λ 2 represent different reference wavelengths (λ 1 = 440 nm, λ 2 = 532 nm).
Equations (4) and (5)  and AAOD(532), respectively. The corresponding EAE and AAE values can be obtained directly from the AERONET Version 3 data product. In addition, the relationship between AOD, SSA, and AAOD is also satisfied Based on [15], SSA values of BC, BrC, and dust at 550 nm were found to be 0.225 ± 0.075, 0.9 ± 0.05, and 0.925 ± 0.025, respectively. For both AAE and SSA above, we chose the average value to be used in the Equation (6) for calculation. One-hour average AOD values of the three absorbing aerosols of BC, BrC, and dust in the corresponding observation period were calculated by the Equation (6). Then, the values were normalized to obtain the percentage of BC, BrC, dust, and other aerosols at each of the three sites at the corresponding observation period (take the sum of BC and BrC as carbonaceous aerosols). Since the South Korean region is greatly affected by the dust transfer from the Gobi in spring [37], the average monthly fraction of dust in March and April each year at the Yonsei University site was selected for the study. In the United States, a large number of carbonaceous aerosols are produced in the summer and autumn due to agricultural waste and wildfire burning [38]. Thus, the monthly average carbonaceous aerosol fraction from July to October at the Cart site was selected for the study ( Figure 4).  The average dust fraction at the Yonsei University site in March and April of year during the period selected in this paper was calculated as 21.1%. The ave carbonaceous aerosol fraction at the Cart site from July to October was calculated as 16 For convenience, the data of dust and carbonaceous aerosol fractions exceeding 20% 15% were selected for the following study.

Development of a Non-Linear Regression Model
A non-linear regression model of two lidar ratio and corresponding aerosol fract was developed ( Figure 5). It should be noted that the fraction of dust (carbonac aerosol) represented by the horizontal coordinate in Figure 5 refers to the proportio dust (carbonaceous aerosol) in the total aerosols. In addition to dust and carbonac aerosols, total aerosols also include other absorbing and non-absorbing aerosols. The (carbonaceous aerosols) absorbing mixing ratio refers to the fraction of (carbonaceous aerosols) to the total sum of dust and carbonaceous aerosols. Referrin [14], the non-linear regression model is fitted using only simple quadratic polyno Based on Figure 5, the R 2 of fitting curves between the two absorbing lidar ra retrieved by HSRL and the corresponding percentage retrieved by the sun photom were both greater than 0.5, which can be considered to be highly correlated [39]. The average dust fraction at the Yonsei University site in March and April of each year during the period selected in this paper was calculated as 21.1%. The average carbonaceous aerosol fraction at the Cart site from July to October was calculated as 16.1%. For convenience, the data of dust and carbonaceous aerosol fractions exceeding 20% and 15% were selected for the following study.

Development of a Non-Linear Regression Model
A non-linear regression model of two lidar ratio and corresponding aerosol fractions was developed ( Figure 5). It should be noted that the fraction of dust (carbonaceous aerosol) represented by the horizontal coordinate in Figure 5 refers to the proportion of dust (carbonaceous aerosol) in the total aerosols. In addition to dust and carbonaceous aerosols, total aerosols also include other absorbing and non-absorbing aerosols. The dust (carbonaceous aerosols) absorbing mixing ratio refers to the fraction of dust (carbonaceous aerosols) to the total sum of dust and carbonaceous aerosols. Referring to [14], the non-linear regression model is fitted using only simple quadratic polynomial functions (i.e., y = ax 2 + bx + c).

Screening of AERONET Sites
After developing the LR-AFNR model, it is necessary to clarify the usage conditions of this model. This paper uses AERONET data from different sites to carry out the study on the usage conditions considering that dust in Asia and carbonaceous aerosols in Europe and the United States are more typical [7]. The AERONET sites in the regions mentioned above are relatively concentrated, and there are numerous downloadable data. Based on Figure 5, the R 2 of fitting curves between the two absorbing lidar ratios retrieved by HSRL and the corresponding percentage retrieved by the sun photometer were both greater than 0.5, which can be considered to be highly correlated [39]. The fitting lidar ratio in Figure 5a-c decreased gradually with an increase in the dust fraction. Because the three sites selected in this paper are all located inland, with the lack of a mixture of aerosols with a low lidar ratio similar to sea salt aerosols, it can be considered that the dust studied in this paper is mainly polluted dust. With the increase in the dust fraction, the results approach the lidar ratio of pure dust, which is generally 35-50 Sr [30]. Further, the trends of the dust-dominated lidar ratio fitting curves are similar to the results of [14]. The fitting lidar ratio in Figure 5d-f increases gradually with the increase in the carbonaceous aerosol fraction. We consider that this part of the carbonaceous aerosol may contain a mixture of smoke with relatively high lidar ratio [31]. We used the average of the fitting curves in Figures 5a-c and 5d-f as the LR-AFNR model for dust and carbonaceous aerosols, respectively.

Screening of AERONET Sites
After developing the LR-AFNR model, it is necessary to clarify the usage conditions of this model. This paper uses AERONET data from different sites to carry out the study on the usage conditions considering that dust in Asia and carbonaceous aerosols in Europe and the United States are more typical [7]. The AERONET sites in the regions mentioned above are relatively concentrated, and there are numerous downloadable data. Thus, this paper selected the data of 63 sites in six regions to carry out the study. The distribution of sites can be found in Figure A1 of Appendix A. In addition, the site name, location, and data selection period information can be found in Table A1 of Appendix A (the altitude of all selected sites was less than 0.5 km).
The six AERONET sites of the 1 MD Science Center, 2 Mainz, 3 Carpentras, 4 Beijing, 5 Taihu, and 6 Yonsei University were taken in the six regions mentioned above as the center, and the data of a total of 63 sites were selected within 500 km around the central sites to study the usage conditions of the LR-AFNR [36].

Distribution of the Aerosol Fraction and Distance Usage Conditions
According to Equations (3)-(6), the dust and the carbonaceous aerosol fraction of 63 sites with a time resolution of one hour were calculated. We compared the probability distribution of dust and carbonaceous aerosol fractions in the six regions in Figure 6. The fraction probability exceeded 0.4 concentrated in a range of 20-40% for dust. For carbonaceous aerosol (the horizontal coordinate (10-20% of the scale) actually included only 15-20% of the fraction), the fraction probability exceeded 0.4 concentrated in a range of 15-20%. According to the calculation results, it was found that the probability of the carbonaceous aerosol fraction exceeding 60% in all six regions was less than 0.02. Thus, the condition that the carbonaceous aerosol fraction exceeded 60% was not considered. This paper defines the dust fraction in the range of 20-40% as light dust and that in the range of 40-100% as heavy dust. At the same time, the carbonaceous aerosol fraction in the range of 15-20% or 20-60% is considered light carbonaceous aerosol or heavy carbonaceous aerosol, respectively.
Based on the four types of absorbing aerosol fractions mentioned above, the distribution of the lidar ratio relative error with distance was studied using data from different sun photometer sites. The fractions of dust and carbonaceous aerosols obtained by any two sun photometer sites (assumed names A and B) were used as input parameters into the LR-AFNR and converted into the lidar ratio. Then, the average lidar ratio for the same hour at sites A and B was calculated within the period selected in Table A1. The lidar ratio obtained in the atmosphere over site A was taken as the actual value of site A, and the lidar ratio obtained at site B was taken as the reference value of site A. The relative distance between sites of A and B, as well as the average of the lidar ratio relative error for a temporal resolution of one hour within the selected period was calculated (here, the relative error was taken as the absolute value). It should be noted that the relative distance between the selected two sites should be less than 500 km, and the total number of data points used to calculate the average lidar ratio relative error between the two sites should be greater than 50. tribution of dust and carbonaceous aerosol fractions in the six regions in Figure 6. The fraction probability exceeded 0.4 concentrated in a range of 20-40% for dust. For carbonaceous aerosol (the horizontal coordinate (10-20% of the scale) actually included only 15-20% of the fraction), the fraction probability exceeded 0.4 concentrated in a range of 15-20%. According to the calculation results, it was found that the probability of the carbonaceous aerosol fraction exceeding 60% in all six regions was less than 0.02. Thus, the condition that the carbonaceous aerosol fraction exceeded 60% was not considered. This paper defines the dust fraction in the range of 20-40% as light dust and that in the range of 40-100% as heavy dust. At the same time, the carbonaceous aerosol fraction in the range of 15-20% or 20-60% is considered light carbonaceous aerosol or heavy carbonaceous aerosol, respectively. Based on the four types of absorbing aerosol fractions mentioned above, the distribution of the lidar ratio relative error with distance was studied using data from different The data from any two sun photometer sites in the six regions mentioned above were compared to obtain a series of the relationship of the relative distance and average relative error of the lidar ratio for the four types of light/heavy dust, light/heavy carbonaceous aerosols (Figure 7). Based on the results shown in the scatter of Figure 7, we used the inverse tangent function (i.e., y = a · arctan(bx)) for the fit.
Through the screening conditions mentioned above, the final effective data points under the four different aerosol absorbing conditions are 100, 88, 70, and 98, respectively. Figure 7 illustrates that when the relative distance between AERONET sites increases, the relative error of the lidar ratio also becomes larger but finally tends to a stable value. Moreover, under the conditions of heavy dust and heavy carbonaceous aerosols, the final tendency of the lidar ratio relative error in the six regions is much larger than that under the conditions of light dust and light carbonaceous aerosols. The main reason is that the maximum absorbing aerosol fraction does not exceed 100%, limiting the relative error of the lidar ratio. Since the lidar ratio directly affects the retrieval accuracy of α a using the Klett-Fernald method, the effect of the LR-AFNR on the relative error of α a is used to study the range of usage distance. The results mentioned above will be discussed and analyzed in more detail in the next section (the selection of points A, B in Figure 7 are introduced in the next section). average lidar ratio relative error between the two sites should be greater than 50.
The data from any two sun photometer sites in the six regions mentioned above we compared to obtain a series of the relationship of the relative distance and average relati error of the lidar ratio for the four types of light/heavy dust, light/heavy carbonaceo aerosols (Figure 7). Based on the results shown in the scatter of Figure 7, we used t inverse tangent function (i.e., arctan( ) y a bx = ⋅ ) for the fit.

Error Analysis
In the previous section, we analyzed the relationship between the lidar ratio relative error and the different relative distances of the instruments using the LR-AFNR under four different aerosol conditions (light/heavy dust, light/heavy carbonaceous aerosols). However, we did not obtain the most suitable range for using the LR-AFNR model. According to [40], the use of α a as a parameter for the retrieval of aerosol microphysical properties is meaningful only when its uncertainty is less than 20%. We therefore limited the maximum allowed relative uncertainty of the extinction coefficient to 20%. Next, the lidar ratio relative error was derived from the extinction coefficient relative error. Then, the corresponding relative distance was determined from the lidar ratio relative error. Based on [41], we performed a correlation error study using the lidar simulation signal. HSRL system parameters as well as basic parameters for lidar signal simulation were adopted from [42]. Specific information can be found in Table A2 of Appendix A. The aerosol loading was set to the bottom exponential aerosol type [43][44][45], and the boundary layer height was set at 3 km. In addition, the average values of AOD were calculated for 63 sites with a dust fraction greater than 20% and carbonaceous aerosol fraction greater than 15%, which were 0.36 and 0.31, respectively. Therefore, the simulation set AOD = 0.36 for the dust-dominated condition and AOD = 0.31 for the carbonaceous aerosol-dominated condition. The actual lidar ratio was calculated by LR-AFNR, ranging from 42.4-56.8 Sr and 54.3-87.9 Sr under dust and carbonaceous aerosol conditions, respectively. Furthermore, the reference lidar ratio was the initial lidar ratio when the Klett-Fernald method was used, set in the range of 30-60 Sr and 40-100 Sr for dust and carbonaceous aerosol conditions, respectively. The values of β m , α m (atmospheric molecular extinction coefficient) were calculated by the 1976 U.S. Standard Atmosphere Model. The background aerosol scattering ratio was set to 1.02 [46]. In this study, Poisson noise was added as the main error source [47,48]. Through the above-mentioned basic parameter setting, a lidar simulation signal was simulated.
It should be noted that the lidar ratio was set to a constant, and each simulated signal was averaged 500 times. The retrieval used the Klett-Fernald method [11] and started from the boundary layer at 3 km, and the initial reference β a used the actual value of the simulation. The results (Figure 8) were retrieved using 50 simulated signal profiles. Based on Figure 8, the larger the relative error of the lidar ratio, the larger the relative error of the a α retrieval. The simulated signal had a relative error greater than 40% for the extinction coefficient due to the lidar ratio relative error.
The aerosol actual lidar ratio and the reference lidar ratio were taken as the horizontal and vertical coordinates, respectively, and each corresponding coordinate calculated a relative error profile of a α , similar to Figure 8b. The average value of this profile was calculated (as the absolute value). At the same time, the relative error of the lidar ratio was calculated (as the absolute value). The above-mentioned two average values were used as the third coordinate to obtain the results (Figure 9).
Combined with Figure 7, the relative error of a α varied with the distance between the two AERONET sites. Since the horizontal and vertical coordinates of the relative error of a α and the lidar ratio had the same value, the relative error of the dust and carbonaceous aerosol lidar ratio was calculated according to the limit of a α , i.e., a relative error of 20%. As shown by the black dotted lines in Figure 9b,d, the relative error of the two aerosol lidar ratios was limited to 23.7% and 22.9% (the average of the values of the upper Figure 8. Take dust as an example, the Klett-Fernald method was used to retrieve the α a , as well as the relative error distribution of α a . (a) The results of α a retrieval using 50 simulated signal profiles in the 3 km boundary layer. The dotted black line ( 1 ), the solid blue line ( 2 ), the solid purple line ( 3 ), and the solid orange line ( 4 ) represent the simulated actual α a (the lidar ratio was 56.8 Sr) and the average of α a retrieval after substituting three reference lidar ratios (lidar ratios of 60, 45, and 30 Sr). The three error bars represent the standard deviation of α a retrieval using the three reference lidar ratios (b) Relative error profile of α a retrieval using three reference lidar ratios.
Based on Figure 8, the larger the relative error of the lidar ratio, the larger the relative error of the α a retrieval. The simulated signal had a relative error greater than 40% for the extinction coefficient due to the lidar ratio relative error.
The aerosol actual lidar ratio and the reference lidar ratio were taken as the horizontal and vertical coordinates, respectively, and each corresponding coordinate calculated a relative error profile of α a , similar to Figure 8b. The average value of this profile was calculated (as the absolute value). At the same time, the relative error of the lidar ratio was calculated (as the absolute value). The above-mentioned two average values were used as the third coordinate to obtain the results (Figure 9).
Combined with Figure 7, the relative error of α a varied with the distance between the two AERONET sites. Since the horizontal and vertical coordinates of the relative error of α a and the lidar ratio had the same value, the relative error of the dust and carbonaceous aerosol lidar ratio was calculated according to the limit of α a , i.e., a relative error of 20%.
As shown by the black dotted lines in Figure 9b,d, the relative error of the two aerosol lidar ratios was limited to 23.7% and 22.9% (the average of the values of the upper and lower black solid lines). Based on a comparison with the results of Figure 7, the abovementioned relative error in Figure 7a,c corresponded to 108 km (point A) and 85 km (point B), respectively. In other words, after using LR-AFNR to obtain the lidar ratio, the range of the relative error of dust α a and carbonaceous aerosol α a was 4-20% when the distance between ML and HSRL was within 108 km and 85 km (4% was the minimum relative error of the signal simulation). According to the calculated results, the range of the dust α a relative error in Figure 9a was 4-43% (including the grey part), and the range of the carbonaceous aerosol α a relative error in Figure 9c was 4-64% (including the grey part), which indicated that after using the LR-AFNR, HSRL had a significant improvement in the accuracy of ML retrieving α a for the same range, and the maximum relative error of dust and carbonaceous α a decreased from 43% and 64% to 20%.
Nevertheless, the relative error of the lidar ratio in Figure 7b,d was smaller, and the relative error of α a did not exceed 20%. The relative error of α a was not caused primarily by the relative error of the aerosol fraction under the condition of light dust and light carbonaceous aerosols. Nevertheless, the relative error of the lidar ratio in Figure 7b,d was smaller, and the relative error of a α did not exceed 20%. The relative error of a α was not caused primarily by the relative error of the aerosol fraction under the condition of light dust and light carbonaceous aerosols.

The Influence of the LR-AFNR Model on the Lidar Ratio
This section will discuss the influence of different aerosol fractions on the lidar ratio after using LR-AFNR. Here, only dust is taken as an example. Assuming that the actual value of the dust fraction of site A is X %, the reference value of site B should be 20-100%, Figure 9. The relationship between the relative error of α a and the lidar ratio, as well as the actual lidar ratio and the reference lidar ratio. (a,b) represent the relative error of α a and the lidar ratio under the dust condition; (c,d) represent the relative error of α a and the lidar ratio under the carbonaceous aerosol condition. The pink dotted line represents the position where the relative error of α a was 20%, and the black dotted line represents the relative error of the lidar ratio at the same position (same horizontal and vertical coordinates) as the pink dotted line in the figure. The grey area represents the part that exceeded the upper limit of the color bar.

The Influence of the LR-AFNR Model on the Lidar Ratio
This section will discuss the influence of different aerosol fractions on the lidar ratio after using LR-AFNR. Here, only dust is taken as an example. Assuming that the actual value of the dust fraction of site A is X%, the reference value of site B should be 20-100%, and the absolute error of the dust fraction can be in the range of (20 − X %)-(100 − X %). The above-mentioned actual value of site A and the reference value of site B were input parameters into the LR-AFNR, and the relative error of the corresponding lidar ratio was calculated. According to this calculation method, the actual value of the dust fraction was the horizontal coordinate, and the absolute error range of the dust fraction was the vertical coordinate. The relative error of the lidar ratio was the third coordinate. After changing the value of X from 20-100%, we obtained the results shown in Figure 10a. In Figure 10a, under the condition of heavy dust, the range of the dust fraction was 40-100%. When two AERONET sites were relatively close to each other, the difference between the actual value of the dust fraction and the reference value was small, and the relative error of the lidar ratio value fell in the area within 20% of the dotted line of Figure  10a. However, due to the upper limit of the dust fraction, the relative error of the lidar ratio did not continue to increase with the relative distance and eventually stabilized. Thus, the value fell in the dotted line area between 20-30% in Figure 10a. Under the condition of light dust, the range of the dust fraction was 20-40% because the fraction itself was relatively small. Therefore, the relative error value of the lidar ratio basically fell within the dark blue area in Figure 10a, and the relative error of the lidar ratio did not continue increasing with the relative distance and eventually stabilized. The abovementioned analysis also explains the scatter plot distribution of the lidar ratio relative error and the relative distance in Figure 7.

The Influence of Different Types of Aerosols on the LR-AFNR Model
When we used Equations (3)-(6) to calculate the BC, BrC, and dust fraction, the AAE and SSA values used were the average of the reference values. However, in fact, the AAE and SSA values of different types of carbonaceous aerosols are also different [49,50]. According to [51], we found that the AAE and SSA of fresh smoke in carbonaceous aerosol was lower than the corresponding value of aged smoke (where fresh refers to smoke formed within 5 min and aged refers to smoke from one hour to several days old). To facilitate comparing the influence of different aerosol types (i.e., different AAE and SSA Figure 10. The relationship between the two absorbing aerosol fractions and the lidar ratio relative error. The horizontal and vertical coordinates represent the actual value of the two absorbing aerosol fractions and the absolute error of the reference value, respectively. The color bar represents the relative error of the lidar ratio calculated after the two absorbing aerosol fractions were input into the LR-AFNR. The white part in the figure represents no data. (a) Under the condition of dust, the sky blue, black, and white dotted lines represent the positions where the relative error of the lidar ratio was 30%, 20%, and 10%, respectively; (b) Under the condition of carbonaceous aerosol, the sky blue, black and white dotted lines represent the positions where the relative error of the lidar ratio was 40%, 20%, and 10%, respectively.
In Figure 10a, under the condition of heavy dust, the range of the dust fraction was 40-100%. When two AERONET sites were relatively close to each other, the difference between the actual value of the dust fraction and the reference value was small, and the relative error of the lidar ratio value fell in the area within 20% of the dotted line of Figure 10a. However, due to the upper limit of the dust fraction, the relative error of the lidar ratio did not continue to increase with the relative distance and eventually stabilized. Thus, the value fell in the dotted line area between 20-30% in Figure 10a. Under the condition of light dust, the range of the dust fraction was 20-40% because the fraction itself was relatively small. Therefore, the relative error value of the lidar ratio basically fell within the dark blue area in Figure 10a, and the relative error of the lidar ratio did not continue increasing with the relative distance and eventually stabilized. The above-mentioned analysis also explains the scatter plot distribution of the lidar ratio relative error and the relative distance in Figure 7.

The Influence of Different Types of Aerosols on the LR-AFNR Model
When we used Equations (3)-(6) to calculate the BC, BrC, and dust fraction, the AAE and SSA values used were the average of the reference values. However, in fact, the AAE and SSA values of different types of carbonaceous aerosols are also different [49,50]. According to [51], we found that the AAE and SSA of fresh smoke in carbonaceous aerosol was lower than the corresponding value of aged smoke (where fresh refers to smoke formed within 5 min and aged refers to smoke from one hour to several days old). To facilitate comparing the influence of different aerosol types (i.e., different AAE and SSA values) on the LR-AFNR, we assumed that the lower AAE and SSA values were for carbonaceous aerosols dominated by fresh smoke. On the contrary, they are carbonaceous aerosols dominated by aged smoke. We set 0.9 and 1.1 times the average value of AAE in Section 2.3 (both within the range of reference values given by [15]) under the condition of fresh smoke and aged smoke, respectively, and the lower and upper limits of the SSA reference values were set under the condition of fresh smoke and aged smoke, respectively (Table 4). In addition, the AAE and SSA values of dust remained unchanged. We chose Figure 5b,f, which were dominated by dust and carbonaceous aerosols, as examples for our study. Moreover, we still used the HSRL and sun photometer data corresponding to the scattered points in Figure 5. The different values of AAE and SSA in Table 4 were reused for the corresponding observational data, and a comparison curve ( Figure 11) with Figure 5b,f (the original fitting curve) was obtained.
A comparison of Figure 11a,b, indicated that under the dust-dominated condition, the fitting curve obtained from fresh smoke with lower AAE and SSA values has smaller values than the original fitting curve. Moreover, the fitting curve obtained from aged smoke with larger AAE and SSA values had larger values than the original fitting curve because the dust fraction calculated by the lower AAE and SSA values was also smaller. A comparison of Figures 5b and 11a showed that the dust fraction of the scatter points in Figure 11a was concentrated in the range of 10-20%, which led to the fitting curve being smaller than the original fitting curve. For the aged smoke of carbonaceous aerosols, by comparing Figures 5b and 11b, it was found that the dust fraction of the scatter points in Figure 11b was concentrated in the range of 80-100%, which led to the fitting curve being larger than the original fitting curve. However, under the carbonaceous aerosol-dominated condition, since the lidar ratio increased with increasing aerosol fraction, Figure 11c,d yielded almost opposite conclusions to Figure 11a,b. Through the above comparison, we found that different types of aerosols affect the accuracy of the LR-AFNR, but at present, we have not further analyzed the influence of different types of aerosols on the LR-AFNR. We believe that a more refined classification of aerosol types is needed in the future to obtain a more credible LR-AFNR model, which is a worthy direction to work on. We chose Figure 5b,f, which were dominated by dust and carbonaceous aerosols, as examples for our study. Moreover, we still used the HSRL and sun photometer data corresponding to the scattered points in Figure 5. The different values of AAE and SSA in Table 4 were reused for the corresponding observational data, and a comparison curve ( Figure 11) with Figure 5b,f (the original fitting curve) was obtained.   Figure 5f. The solid black line represents the original fitting curve, and the blue dotted lines represent the fitting curves of fresh and aged smoke. N and R 2 represent the number of fitting points and the coefficient of determination, respectively. It should be noted that the solid red line corresponds to the dust fraction range of 20-100% in Figure 5b, and the solid black line corresponds to the carbonaceous aerosol fraction range of 15-60% in Figure 5f, so only part of the curve is shown in this Figure.

Conclusions
Based on the data of AERONET sites and University of Wisconsin HSRL sites in the same location, this paper developed a non-linear regression relationship between the absorbing aerosol (dust and carbonaceous aerosols) lidar ratio and corresponding fraction. The results show that the R 2 values were all higher than 0.5, which can be considered highly correlated. Then, the paper compared the fitting relationship of the two absorbing aerosols at each of the three sites and found similarities, and the LR-AFNR model was developed in this paper. A total of 63 AERONET sites in six regions were selected to study the model usage conditions. It was found that the model can control the relative error of α a within 20% under the conditions of heavy dust and heavy carbonaceous aerosols when the usage range does not exceed 108 km and 85 km, and the maximum relative error of dust and carbonaceous α a were reduced from 43% and 64% to 20%. According to the study results, this paper proposes a lidar network composed of HSRL and ML: put the HSRL in the center and deploy multiple MLs around it. All lidars are deployed in locations where sun photometers exist, such as in meteorological bureaus and universities. When the relative distance between the HSRL and the ML is within 85 km, the HSRL can be transferred to the lidar ratio as the input parameters of the Klett-Fernald method retrieval for the ML through the sun photometer as the medium. In this way, this method implements the indirect transfer of the lidar ratio from HSRL to MLs.
However, this paper only discusses the model for dust and carbonaceous aerosols, and there is a lack of more in-depth research on the classification of refined aerosols. Through our research, we found that different types of aerosols would cause different AAE and SSA values, which would influence the accuracy of the LR-AFNR model. In addition, the accuracy of the LR-AFNR model was also affected by the different regions and other constraints. Therefore, it is necessary to promote the HSRL system in the future. More of the above-mentioned lidar networks should be deployed. Moreover, we should continuously improve the LR-AFNR model by using the lidar ratio obtained by HSRL and the relevant data obtained by sun photometers (or other relevant instruments) to develop the relationship between the relevant aerosol fraction and lidar ratio in different regions. This lidar network mode can also improve the detection accuracy of the internal MLs while controlling the cost. Combining the advantages of both lidars is of great significance for accurate monitoring research of the three-dimensional distribution of aerosols on a regional scale.

Appendix A
This paper selected data from 63 sites in six regions for research. The location info mation can be found in Figure A1, and the specific site name, location, and data selecti period time information can be found in Table A1. Figure A1. The six regional sites of AERONET are marked on the map. The regions ①-⑥ rep sent eastern U.S., northern Europe, northern Europe, northern China, eastern China, and South K rea, respectively. The six figures around the world map represent the detailed distribution map the selected AERONET site locations in each region. The numbers in the figure correspond to t information of each corresponding numbered site in Table A1; where the color bar represents a tude, it should be noted that the figure omits six detailed figures of the color bar with an altitu range of 0~2 km. Figure A1. The six regional sites of AERONET are marked on the map. The regions 1 -6 represent eastern U.S., northern Europe, northern Europe, northern China, eastern China, and South Korea, respectively. The six figures around the world map represent the detailed distribution map of the selected AERONET site locations in each region. The numbers in the figure correspond to the information of each corresponding numbered site in Table A1; where the color bar represents altitude, it should be noted that the figure omits six detailed figures of the color bar with an altitude range of 0~2 km.