Optimization of the Photon Path Length Probability Density Function-Simultaneous (PPDF-S) Method and Evaluation of CO2 Retrieval Performance Under Dense Aerosol Conditions

The photon path length probability density function-simultaneous (PPDF-S) algorithm is effective for retrieving column-averaged concentrations of carbon dioxide (XCO2) and methane (XCH4) from Greenhouse gases Observing Satellite (GOSAT) spectra in Short Wavelength InfraRed (SWIR). Using this method, light-path modification attributable to light reflection/scattering by atmospheric clouds/aerosols is represented by the modification of atmospheric transmittance according to PPDF parameters. We optimized PPDF parameters for a more accurate XCO2 retrieval under aerosol dense conditions based on simulation studies for various aerosol types and surface albedos. We found a more appropriate value of PPDF parameters referring to the vertical profile of CO2 concentration as a measure of a stable solution. The results show that the constraint condition of a PPDF parameter that represents the light reflectance effect by aerosols is sufficiently weak to affect XCO2 adversely. By optimizing the constraint, it was possible to obtain a stable solution of XCO2. The new optimization was applied to retrieval analysis of the GOSAT data measured in Western Siberia. First, we assumed clear sky conditions and retrieved XCO2 from GOSAT data obtained near Yekaterinburg in the target area. The retrieved XCO2 was validated through a comparison with ground-based Fourier Transform Spectrometer (FTS) measurements made at the Yekaterinburg observation site. The validation results showed that the retrieval accuracy was reasonable. Next, we applied the optimized method to dense aerosol conditions when biomass burning was active. The results demonstrated that optimization enabled retrieval, even under smoky conditions, and that the total number of retrieved data increased by about 70%. Furthermore, the results of the simulation studies and the GOSAT data analysis suggest that atmospheric aerosol types that affected CO2 analysis are identifiable by the PPDF parameter value. We expect that we will be able to suggest a further improved algorithm after the atmospheric aerosol types are identified.


Introduction
Among the greenhouse gases emitted by human activities, carbon dioxide (CO 2 ) has the largest radiative forcing as a total effect. Its atmospheric concentration has been increasing since the advent of the Industrial Revolution. The global distribution and temporal variations of the gas have been clarified based on ground-based in situ measurements [1]. However, observation sites are unevenly distributed on a global scale, particularly over the ocean. For this reason, global measurements by satellite have long been recommended.
Based on that background, attempts have been made to retrieve CO 2 concentration data from Thermal InfraRed (TIR) spectra observed by satellite sensors since the 1990s. Chédin et al. [2] first detected a CO 2 signal from TIR spectra, which is sensitive to CO 2 concentration changes in the upper troposphere measured by The High-Resolution Infrared Radiation Sounder (HIRS). Then, they applied the method to demonstrate the usage of the data for CO 2 source/sink analysis [3]. Following these studies, many others have been conducted using sensors of similar types: the Interferometric Monitor for Greenhouse gases (IMG) [4,5], Infrared Atmospheric Sounding Interferometer (IASI) [6], Tropospheric Emissions Spectrometer (TES) [7], Atmospheric Infrared Sounder (AIRS) on the Aqua satellite [8], and Thermal And Near-infrared Sensor for carbon Observation-Fourier Transform Spectrometer (TANSO-FTS) (Band 1, 0.758-0.775 µm; Band 2, 1.56-1.72 µm; Band 3, 1.92-2.08 µm; Band 4, 5.56-14.3 µm; where Band 4 corresponds to the TIR region) on the Greenhouse gases Observing Satellite (GOSAT) [9,10].
However, TIR sounders are, in principle, sensitive to the CO 2 concentration in the troposphere [11], but the sensitivity is low in the planetary boundary layer. This point presents important disadvantages for the study of CO 2 surface fluxes. It is necessary to measure CO 2 concentrations at a low altitude or a columnar concentration for CO 2 budget analysis because most CO 2 sources and sinks are located near the Earth's surface.
By contrast, it is generally preferable to analyze the radiance spectra measured in Short Wavelength InfraRed (SWIR) to analyze the CO 2 concentration rather than in TIR because the SWIR spectra are sensitive to CO 2 concentration changes near the surface. The dry air columnar mixing ratio (XCO 2 ) can be retrieved from the data. The following are examples of satellite sensors that observe spectra in SWIR to measure XCO 2 : Scanning Imaging Absorption Spectrometer for Atmospheric Chartography (SCIAMACHY) [12], TANSO-FTS (Bands 1, 2 and 3) on the GOSAT [13], Orbiting Carbon Observatory-2 (OCO2) [14], Chinese Carbon Dioxide Observation Satellite Mission (TanSat) [15], and Carbon Monitoring Satellite (CarbonSat) [16]. Although the usage of the SWIR band presents the benefit that signals from vertically integrated CO 2 are obtainable, the spectra can be easily affected by multiple light scattering because of clouds and aerosols in the atmosphere. This effect must be examined in order to accurately retrieve XCO 2 from SWIR.
The first satellite dedicated to observing greenhouse gases was GOSAT, which was launched in 2009 to measure greenhouse gases: CO 2 and methane (CH 4 ). Its main sensor, TANSO-FTS, observes atmospheric radiance spectra in both SWIR and TIR ranges.
Many attempts have been undertaken to develop retrieval algorithms to derive XCO 2 from the SWIR band data. Most of these algorithms are based on the so-called 'Full Physics (FP)' method. One is used to process standard products of XCO 2 released from the GOSAT project at the National Institute for Environmental Studies (NIES), which can also be classified as an FP method. In this method, the multiple light scattering processes are explicitly calculated by solving radiative transfer equations accounting for the optical properties of aerosols [17,18]. Many research groups have adopted methods of this type: NIES (Japan) [17,18], Atmospheric CO 2 Observations from Space (ACOS) (NASA JPL, USA) [19,20], Netherlands Institute for Space Research/Karlsruhe Institute of Technology (SRON-KIT) (Netherlands and Germany) [21][22][23], University of Leicester (UK) [14,24,25], and Yonsei University (Korea) [26,27].
A more simplified estimation method, the photon path length probability density function (PPDF)-based method, has been proposed (e.g., [28][29][30][31]). This method can represent the effects of light-path modification without exact radiative transfer calculations because the light-path modification is evaluated statistically for atmospheric transmittance including clouds/aerosols using PPDF parameters. In the new version of the PPDF-based method, the PPDF-simultaneous (PPDF-S) method, gas concentrations are simultaneously retrieved with PPDF parameters [31]. Details of PPDF-based methods are described in Section 2.
Comparing XCO 2 data retrieved using the NIES FP algorithm and those retrieved using the PPDF-S algorithm, the results show that the difference in XCO 2 is large over the middle of Africa and Western Siberia, where many aerosols exist because of biomass burning [32]. The objective of this study is to improve the PPDF-S algorithm and to tune its parameters under high aerosol conditions, as well as to evaluate the retrieval performance of the method through simulation studies. Then, the improved PPDF-S algorithm and tuned parameters are applied to analyses of GOSAT data observed over Western Siberia during the biomass burning season. Additionally, the XCO 2 retrieval accuracy is evaluated.

Basis of PPDF-S Retrieval
Most sensors designed to measure gaseous constituents in the atmosphere, such as TANSO-FTS onboard GOSAT, detect the solar radiation in the SWIR region reflected by the earth's surface or scattered by the atmosphere. When we analyze CO 2 concentrations from the measured spectral radiation, scattering effects by clouds/aerosols in the atmosphere make the analysis very difficult because they shorten and lengthen the light-path length from the geometrical length. If these properties are estimated inappropriately, then large errors in the estimation of CO 2 concentrations can easily occur.
In the FP method adopted for GOSAT data analyses, the scattering processes in the atmosphere including aerosols are represented using the Discrete Ordinate Method (DOM) for accurate XCO 2 retrieval [17,18]. However, the PPDF-based method does not solve the radiative transfer equation directly for the representation of scattering effects. In the method, the light-path modification is described by effective transmittance using the following basic PPDF parameters: the altitude of the cloud/aerosol layer h; the relative reflection of the cloud/aerosol layer α, which represents the relative fraction of the photons scattered singly to the detector; ρ, which characterizes the relative mean path length between the layer and ground surface; and an adjustment parameter γ accounting for multiple scattering (e.g., [28][29][30][31]). A diagram of the definitions of α and ρ is presented in Figure 1.
Oshchepkov et al. [31] proposed the use of effective transmittance within a three-layered PPDF model using eight PPDF parameters to characterize the light-path modification. Four parameters represent the Rayleigh scattering effects: h r , α r , ρ r , and γ r . The remaining four parameters represent the aerosol reflection/scattering effects: h a , α a , ρ a , and γ a . Using these parameters, the effective transmittance can be represented by the following equations: where where θ and θ 0 respectively denote the solar and satellite zenith angles; k(h) is the gas absorption coefficient at altitude h; and h TOA is the altitude of the top of the atmosphere. When the gas concentrations are simultaneously retrieved with the PPDF parameters described above, the method is designated as the PPDF-simultaneous (PPDF-S) method: the newest version among PPDF-based methods applied to GOSAT measurements. The history of the development of the PPDF-S method and an explanation of its performance are described in the following section.

Basis of PPDF-S Retrieval
Most sensors designed to measure gaseous constituents in the atmosphere, such as TANSO-FTS onboard GOSAT, detect the solar radiation in the SWIR region reflected by the earth's surface or scattered by the atmosphere. When we analyze CO2 concentrations from the measured spectral radiation, scattering effects by clouds/aerosols in the atmosphere make the analysis very difficult because they shorten and lengthen the light-path length from the geometrical length. If these properties are estimated inappropriately, then large errors in the estimation of CO2 concentrations can easily occur.
In the FP method adopted for GOSAT data analyses, the scattering processes in the atmosphere including aerosols are represented using the Discrete Ordinate Method (DOM) for accurate XCO2 retrieval [17,18]. However, the PPDF-based method does not solve the radiative transfer equation directly for the representation of scattering effects. In the method, the light-path modification is described by effective transmittance using the following basic PPDF parameters: the altitude of the cloud/aerosol layer h; the relative reflection of the cloud/aerosol layer α, which represents the relative fraction of the photons scattered singly to the detector; ρ, which characterizes the relative mean path length between the layer and ground surface; and an adjustment parameter γ accounting for multiple scattering (e.g., [28,29,30,31]). A diagram of the definitions of α and ρ is presented in Figure 1.

History and Performance of PPDF-S
Actually, NIES has started to develop a PPDF-based retrieval method for application to GOSAT SWIR measurements. First, based on the forward Monte Carlo simulation for the analysis of photon trajectory statistics by Bril et al. [28], four PPDF parameters were introduced: h, α, ρ, and γ. In the next step, Oshchepkov et al. [29] used PPDF parameters for XCO 2 retrieval and established the PPDF-based retrieval method. After Bril et al. [33] improved the method for application to the atmosphere including desert dust aerosols, Oshchepkov et al. [30] executed an additional improvement defining two sets of PPDF parameters. Those parameters, h c , α c , ρ c , and γ c , and h a , α a , ρ a , and γ a , were introduced respectively to distinguish the scattering process by clouds and that by aerosols. The parameters were applied to the three-layered atmosphere. The method was used for GOSAT data processing. Global characteristics of the retrieved XCO 2 and PPDF parameters were subsequently investigated by Oshchepkov et al. [34].
To evaluate the retrieval accuracy of the improved PPDF-based method, the retrieved XCO 2 was validated using ground-based remote sensing data provided by the Total Column Observing Network (TCCON [35]) by Oshchepkov et al. [36]. PPDF-based retrieval results and validation results were cross-compared with those of other XCO 2 results retrieved using various retrieval algorithms based on the FP method in Oshchepkov et al. [37]. According to Oshchepkov et al. [36], the validation results in comparison with TCCON 12 sites data showed that PPDF-based XCO 2 had a bias of −0.43 ppm and a standard deviation of 1.80 ppm. The paper also showed that the detected effects of light-path modification represented by the PPDF parameters were physically interpreted by the seasonal trends of the aerosol optical depth derived from an aerosol transport model, as well as by the time series of cirrus optical depth derived from space-based lidar measurements. By comparing the results of these retrieval analyses of XCO 2 of different types, Oshchepkov et al. [37] demonstrated that PPDF-based XCO 2 was as accurate as the retrieval results obtained using other algorithms based on the FP method.
In the earlier version of the PPDF-based algorithm, the PPDF parameters were used as a pre-screening test to identify GOSAT observation data that are not distinctly affected by atmospheric light scattering. The near clear sky conditions are recognized by low values of PPDF parameters retrieved from radiance spectra in the O 2 A-band (0.76 µm). After the prescreening test, the gas concentration retrieval is executed in the near-infrared CO 2 bands, assuming that light-path modification is nearly zero. The retrieval scheme is based on Differential Optical Absorption Spectroscopy (DOAS). Therefore, the method is designated as a PPDF-D method. However, the observation data that are used for gas retrieval with the method are limited. For the PPDF-D method, further improvement was executed to retrieve PPDF parameters and CO 2 concentration simultaneously, which enables an increase of the amount of observation data that are applicable for the retrieval. First, Bril et al. [38] reported pre-results of the simultaneous retrieval of XCO 2 and PPDF parameters. From the retrieval, only two PPDF parameters were derived from GOSAT data measured over ocean areas: the effective altitude of the aerosol layer (h a ) and the relative layer reflectivity (α a ). Subsequently, Oshchepkov et al. [31] further improved the method for the simultaneous retrieval of XCO 2 and eight PPDF parameters: h r , α r , ρ r , γ r for Rayleigh scattering by atmospheric molecules; and h a , α a , ρ a , and γ a for scattering by aerosol. The paper established the current PPDF-S method. The retrieved XCO 2 was validated in comparison with TCCON data at 12 sites. Reportedly, the accuracy of the study was a bias of 0.08 ppm, with a standard deviation of 1.90 ppm.
Iwasaki et al. [32] also validated the PPDF-S-based XCO 2 , as well as XCH 4 , by expanding the time period of validation and describing their global characteristics, which demonstrated that the bias and standard deviation of XCO 2 over the land are, respectively, 0.73 and 1.83 ppm. The paper also presented validation of the Level 2 data as standard products for General Users (i.e., GU products) and only for Research Announcement principal investigators (RA products) retrieved using the FP method and released from NIES in comparison with the PPDF-S-based products. GU products were selected from RA products using screening tests referring to atmospheric parameters, such as aerosol optical thickness (AOT), surface conditions, and radiometric parameters.
To investigate the uncertainty of the retrieved results of the GOSAT data observed under high-AOT conditions, the validation results were compared for the XCO 2 of the PPDF-S product and the data removed from the RA products through screening tests, which were mainly removed because of the high-AOT values. From the analysis, it was pointed out that the XCO 2 bias and its standard deviation in the PPDF-S product are smaller than those of the removed RA products.
However, comparing the global distribution of PPDF-S products to those of the GU products, large differences in both the XCO 2 and XCH 4 were found over the middle of Africa and Western Siberia, where biomass burning occurs and where large amounts of aerosols exist. The different retrieval performances of PPDF-S and FP methods might result from the poor representativity of the light-path modification by the PPDF-S method under high-AOT conditions. As a conclusion of this study, the importance of improving the PPDF-S retrieval algorithm to be more accurate under denser aerosol conditions is indicated. The improvement method used for this study is described in Section 3.

Basic Equations of Retrieval
The PPDF-S retrieval algorithm is based on optimal estimation to minimize a cost-function J(x), the weighted least-squares difference [39], which is defined by four components: measured radiance spectra R*; simulated radiance spectra R ; the a priori data x a for the target state vector x; the state vector about the amount of gases in each atmospheric layer x gas ; and other state vectors x nongas , including h r , α r , ρ r , γ r , h a , α a , ρ a , and γ a . J(x) is formatted as where x = x gas x nongas , S a = S a,gas 0 0 S a,nongas , and f (x) is the forward model based on PPDF parameters. Additionally, S y and S a are covariance matrices of measurement errors and a priori data. Following the Gauss-Newton method, the solution is obtained in an iterative manner as where K i is the Jacobian matrix, calculated with respect to x = x i as Hereinafter, each element of x a is described as x a . The square root values of the diagonal element of S y and S a are respectively described as σ y and σ a for this study.

CO 2 retrieval Based on Simulation
To investigate the accuracy of the PPDF-S retrieval, simulation studies have been conducted. In the analysis, measured radiance described as Y* in the equation for the cost function is simulated by an atmospheric Radiative Transfer Model (RTM), assuming various atmospheric conditions and surface characteristics. The CO 2 amount is retrieved from the simulated radiance. The retrieval accuracy is defined in this study as the XCO 2 bias, which is calculated by subtracting the assumed XCO 2 in the simulation from the retrieved XCO 2 . The RTM parameter settings and datasets used for the calculation of radiance are shown in Table 1. In the calculation, a clear sky condition is also presumed by setting AOT = 0.0. On the other hand, F(x) is calculated using PPDF parameters in the retrieval process as a forward calculation step. The solar irradiance model, zenith angles, surface albedo, and atmospheric profiles, which are used for the forward calculation, are the same as those used for the Pstar3 simulation, except for the aerosol models. In the PPDF-based forward model, light-path modification is explained as the change of effective transmittance described by PPDF parameters for any aerosol model. The a priori and variance of each parameter included in the state vector x are shown in Table 2. The values of σ y in the retrieval are shown below. σ y for Band 1: 4.0 × 10 −7 /SNR [W/m 2 /str/cm −1 ] σ y for Band 2: 3.5 × 10 −7 /SNR [W/m 2 /str/cm −1 ] σ y for Band 3: 2.5 × 10 −7 /SNR [W/m 2 /str/cm −1 ] signal-to-noise ratio (SNR) = 400 × β ρ × ScaleFactor ρ , 3 γ = exp −β 2 γ × ScaleFactor γ , (ScaleFactor x : Adjustment parameter for representing the dependence of parameter x on wavelength). 4 Gain H: Case of 'High' gain setting of GOSAT sensor. 5 Gain M: Case of 'Middle' gain setting of GOSAT sensor.

Optimization of PPDF Parameter Settings for More Adequate XCO 2 Retrieval
For the accurate retrieval of XCO 2 , it is generally important to derive a stable CO 2 profile. To achieve this, PPDF parameters should also be derived appropriately because the state vector x gas for the CO 2 amount in each layer and x nongas for PPDF parameters are complementary, and because an inappropriate setting of variation of the former parameters can engender instability in the CO 2 profile.
However, it is difficult to determine the appropriate value and variance of PPDF parameters a priori because PPDF parameters must represent light reflectance/scattering situations of many types with a limited number or parameters and because radiation propagation processes are complicated. If very large values are set to S a when describing the wide range of dispersion a priori, then the possibility exists that an inappropriate value of retrieved PPDF parameters from correct answers will engender an undesirable value of XCO 2 .
One possible means of finding the best value or at least a value better than the current value of the variance of state vector (σ a ), as well as the priori value (x a ), is to find parameters with unnecessarily large values and then make them smaller than the current values. The spectral radiance corresponding to each variance can be a good measure to evaluate the effects of the variation of each PPDF parameter on XCO 2 retrieval. The radiance can be evaluated through the radiative transfer equation Y = K·σ a . Based on the idea that a larger value of K·σ a more effectively influences the parameter affecting XCO 2 retrieval, we found the PPDF parameter with unnecessarily large variance and made it smaller to optimize the parameter: we constrained the parameter to its better a priori value. We determined the more appropriate value of σ a by referring to the shape of the retrieved vertical profile of CO 2 concentration as a measure of a stable solution. The optimization was aimed at improving the accuracy of CO 2 concentration for various atmospheric conditions, irrespective of existence or non-existence of the aerosol or aerosol type.

PPDF Parameter Optimization
We compared the value of K·σ a for each PPDF parameter and evaluated the effect of its variation on XCO 2 retrieval. The formulation of the Jacobian K for each parameter is based on the procedure reported by Oshchepkov et al. [30]. Examples of spectral K·σ a for α a , ρ a , and γ a in Band 2 are shown in Figure 2. Results of comparisons showed that the K·σ a of α a representing the light reflectance by aerosols is greater by about two orders of magnitude among the PPDF parameters (α r , ρ r , γ r , ρ a , and γ a ) in all GOSAT measurement bands. Consequently, the variation of α a might be too large to cause instability of the retrieved CO 2 profile. Therefore, we tried to optimize the σ a of α a , as well as its a priori value, based on the results retrieved from various simulations using the radiative transfer code, Pstar3. variation of each PPDF parameter on XCO2 retrieval. The radiance can be evaluated through the radiative transfer equation Y = K·σa. Based on the idea that a larger value of K·σa more effectively influences the parameter affecting XCO2 retrieval, we found the PPDF parameter with unnecessarily large variance and made it smaller to optimize the parameter: we constrained the parameter to its better a priori value. We determined the more appropriate value of σa by referring to the shape of the retrieved vertical profile of CO2 concentration as a measure of a stable solution. The optimization was aimed at improving the accuracy of CO2 concentration for various atmospheric conditions, irrespective of existence or non-existence of the aerosol or aerosol type.

PPDF Parameter Optimization
We compared the value of K·σa for each PPDF parameter and evaluated the effect of its variation on XCO2 retrieval. The formulation of the Jacobian K for each parameter is based on the procedure reported by Oshchepkov et al. [30]. Examples of spectral K·σa for αa, ρa, and γa in Band 2 are shown in Figure 2. Results of comparisons showed that the K·σa of αa representing the light reflectance by aerosols is greater by about two orders of magnitude among the PPDF parameters (αr, ρr, γr, ρa, and γa) in all GOSAT measurement bands. Consequently, the variation of αa might be too large to cause instability of the retrieved CO2 profile. Therefore, we tried to optimize the σa of αa, as well as its a priori value, based on the results retrieved from various simulations using the radiative transfer code, Pstar3. As a method of optimizing the deviation value, we tested various magnification values, which were multiplied by the original value of the σa from 1 to 1/100. Furthermore, we investigated the retrieval accuracy of XCO2. Referring to the shape of the retrieved CO2 vertical profile, the results showed that 1/20 is the most adequate magnitude for σa. We also found that the value of 1/200 instead of 1/20 used in the equation for the a priori value of αa shown in Table 2 for βαa was effective As a method of optimizing the deviation value, we tested various magnification values, which were multiplied by the original value of the σ a from 1 to 1/100. Furthermore, we investigated the retrieval accuracy of XCO 2 . Referring to the shape of the retrieved CO 2 vertical profile, the results showed that 1/20 is the most adequate magnitude for σ a . We also found that the value of 1/200 instead of 1/20 used in the equation for the a priori value of α a shown in Table 2 for β αa was effective at reducing XCO 2 bias with the smaller σ a . The retrieval results obtained using the original settings are shown in Table 2. Results based on the optimization explained above are compared in relation to both a clear sky condition and atmosphere including aerosols of various types.

CO 2 Retrieval Results
For the retrieval simulations, the true value of XCO 2 is assumed to be 390 ppm. The a priori value of XCO 2 is set as 385 ppm, as described in Section 3.2. 'XCO 2 bias' is defined in this subsection as the value resulting from subtraction of the true XCO 2 of 390 ppm from the retrieved XCO 2 .

Clear Sky Condition
For synthesizing the measured radiance under a clear sky condition, surface albedo is assumed to be 0.2 for all spectral bands of the GOSAT sensor. The averaging kernel which represents the vertical resolution of the retrieval analysis and the column averaging kernel which represents the sensitivity profile to the total column value of the retrieved parameter are almost identical when comparing the original method and the optimized method for various atmospheric conditions. As examples, the vertical profile of the retrieved CO 2 concentration, averaging kernel, and column averaging kernel are shown in Figure 3a-c. The scale on the horizontal axis of Figure 3a is set to be the same as that of the results for the atmosphere including aerosols, as depicted in Figure 4. In Figure 3a, the retrieved CO 2 profiles assuming SNR = 400 and 2400 are depicted by red and blue lines, respectively. The vertically constant profile of 385 ppm is the a priori profile (pink line) and that of 390 ppm is the true profile (gray line) in the figure. The retrieved CO 2 profile for SNR = 400 exists between the a priori and true profiles, but its shape is not vertically uniform, with a peak at a height in the middle of the troposphere corresponding to the sensitivity peak of the retrieval system. If the SNR of the measurement was ideally very high, for example, 2400, then the retrieved profile was closer to the true profile, but it remained not perfectly uniform. This point of caution is discussed in Section 6. Figure 3b,c present the results obtained when assuming SNR = 400.
both a clear sky condition and atmosphere including aerosols of various types.

CO2 Retrieval Results
For the retrieval simulations, the true value of XCO2 is assumed to be 390 ppm. The a priori value of XCO2 is set as 385 ppm, as described in Section 3.2. 'XCO2 bias' is defined in this subsection as the value resulting from subtraction of the true XCO2 of 390 ppm from the retrieved XCO2.

Clear Sky Condition
For synthesizing the measured radiance under a clear sky condition, surface albedo is assumed to be 0.2 for all spectral bands of the GOSAT sensor. The averaging kernel which represents the vertical resolution of the retrieval analysis and the column averaging kernel which represents the sensitivity profile to the total column value of the retrieved parameter are almost identical when comparing the original method and the optimized method for various atmospheric conditions. As examples, the vertical profile of the retrieved CO2 concentration, averaging kernel, and column averaging kernel are shown in Figure 3a-c. The scale on the horizontal axis of Figure 3a is set to be the same as that of the results for the atmosphere including aerosols, as depicted in Figure 4. In Figure 3a, the retrieved CO2 profiles assuming SNR = 400 and 2400 are depicted by red and blue lines, respectively. The vertically constant profile of 385 ppm is the a priori profile (pink line) and that of 390 ppm is the true profile (gray line) in the figure. The retrieved CO2 profile for SNR = 400 exists between the a priori and true profiles, but its shape is not vertically uniform, with a peak at a height in the middle of the troposphere corresponding to the sensitivity peak of the retrieval system. If the SNR of the measurement was ideally very high, for example, 2400, then the retrieved profile was closer to the true profile, but it remained not perfectly uniform. This point of caution is discussed in Section 6. Figure 3b,c present the results obtained when assuming SNR = 400.
The XCO2 biases of the results retrieved using the original method and the optimized method are −0.47 ppm and −0.63 ppm, respectively. The accuracy of XCO2 becomes slightly worse for the clear sky condition after the optimization of αa, but the value is still sufficiently small compared to the variation in CO2 concentration of natural phenomena.   In the simulation for the atmosphere including aerosols, the measured radiance is synthesized assuming various values of surface albedo, aerosol types, and AOT, as shown in Table 1. In Figure 4, examples show the CO2 profile retrieved using the original method (left) and the optimized method The XCO 2 biases of the results retrieved using the original method and the optimized method are −0.47 ppm and −0.63 ppm, respectively. The accuracy of XCO 2 becomes slightly worse for the clear sky condition after the optimization of α a , but the value is still sufficiently small compared to the variation in CO 2 concentration of natural phenomena.

Atmosphere Including Aerosols of Various Types
In the simulation for the atmosphere including aerosols, the measured radiance is synthesized assuming various values of surface albedo, aerosol types, and AOT, as shown in Table 1. In Figure 4, examples show the CO 2 profile retrieved using the original method (left) and the optimized method (right) in the cases of AOT of 1.0 at 0.55 µm for aerosols of four types: Rural (blue), Urban (green), Soot (red), and Dust-like (black). For these calculations, the surface albedos are set as 0.2 for all GOSAT spectral bands. Because the averaging kernel and column averaging kernel are almost identical to those under the clear sky condition, they are not shown here.
In the original method, the retrieval results obtained under the condition that includes Urban-type aerosols are not provided in the figure because the solution related to x does not converge on a value in the retrieval iteration analysis based on a Gauss-Newton method. However, the retrieval is completed to obtain solutions by the optimized method of all conditions. Although the variance of the CO 2 profile retrieved by the optimized method for Dust-like aerosols becomes smaller after the optimization of α a , it is still larger than those found for aerosols of other types. XCO 2 biases for the aerosols of four types are evaluated as explained below. For synthetic analyses of the retrieval accuracy after the optimization for α a , XCO 2 biases are calculated for various combinations of surface albedo and AOT. A contour plot of the results is portrayed in Figure 5. They show that the bias (in parts per million: ppm) depends on the aerosol type, surface albedo, and AOT. Especially when the surface albedo is small, the dependence of the XCO 2 bias on AOT is large for aerosols of all types. Each result has a line representing zero bias, except for Soot, which shows a negative bias in all cases. For Dust-like aerosols, the bias exceeds ±5 ppm. It is larger than those for other types of aerosols. Urban aerosols comprise Rural and Soot aerosols. Therefore, the XCO 2 bias for the Urban type apparently has an intermediate property of those for the Rural and Soot aerosols.

Application of the Optimized Method to GOSAT Data Observed for Western Siberia
As described in Section 5.2, aerosol densities in Western Siberia can be high when biomass burning occurs there. Therefore, improving the retrieval accuracy for GOSAT data from the area is an interesting topic. This chapter describes the application of the optimized method to GOSAT data for Western Siberia considering both clear sky conditions and aerosol dense conditions.

Application to Clear Ssky Conditions in Western Siberia and Validation of the Retrieved XCO2 Using Ground-Based FTS at Yekaterinburg
At Yekaterinburg (57.038° N and 59.545° E) in Western Siberia, XCO2 has been measured using a ground-based Fourier Transform Spectrometer (FTS) by the Ural Federal University. The XCO2 data accuracy was reported by Nikita et al. [44]. They are sufficiently accurate to be used for GOSAT data validation. We applied the optimized method to GOSAT data observed near Yekaterinburg during 2010-2014 and retrieved XCO2. Then, the retrieved XCO2 was validated using data observed at Yekaterinburg. The match-up criterion between GOSAT and ground-based FTS is ±2 × 2 degrees in latitude and longitude, and within a ±1.5 h observational time. In addition, Yekaterinburg is a site of the Aerosol Robotic NETwork (AERONET); AOT has been measured using a sun spectral photometer set. Actually, the AOTs observed at Yekaterinburg for the cases used in the validation

Application of the Optimized Method to GOSAT Data Observed for Western Siberia
As described in Section 5.2, aerosol densities in Western Siberia can be high when biomass burning occurs there. Therefore, improving the retrieval accuracy for GOSAT data from the area is an interesting topic. This chapter describes the application of the optimized method to GOSAT data for Western Siberia considering both clear sky conditions and aerosol dense conditions.

Application to Clear Ssky Conditions in Western Siberia and Validation of the Retrieved XCO 2 Using Ground-Based FTS at Yekaterinburg
At Yekaterinburg (57.038 • N and 59.545 • E) in Western Siberia, XCO 2 has been measured using a ground-based Fourier Transform Spectrometer (FTS) by the Ural Federal University. The XCO 2 data accuracy was reported by Nikita et al. [44]. They are sufficiently accurate to be used for GOSAT data validation. We applied the optimized method to GOSAT data observed near Yekaterinburg during 2010-2014 and retrieved XCO 2 . Then, the retrieved XCO 2 was validated using data observed at Yekaterinburg. The match-up criterion between GOSAT and ground-based FTS is ±2 × 2 degrees in latitude and longitude, and within a ±1.5 h observational time. In addition, Yekaterinburg is a site of the Aerosol Robotic NETwork (AERONET); AOT has been measured using a sun spectral photometer set. Actually, the AOTs observed at Yekaterinburg for the cases used in the validation were nearly 0.1. Considering the low AOTs and the restriction that ground-based FTS can only take measurements under clear sky conditions, the validation was executed presuming that the atmosphere included no cloud or aerosol.
A comparison of PPDF-S data and ground-based FTS data at Yekaterinburg is presented in Figure 6. The results obtained using the original parameter setting in the PPDF-S method and those obtained using the optimized setting are colored as gray and red, respectively. The numbers of GOSAT data shown in the figure are three and five, respectively, for the original and optimized settings. To represent the retrieval accuracy, the bias and standard deviation of XCO 2 are calculated by subtracting ground-based data from the GOSAT data. The results show that the bias and its standard deviation of XCO 2 are, respectively, 0.75 and 0.57 ppm for the original setting, and 0.69 and 1.79 ppm for the optimized data. were nearly 0.1. Considering the low AOTs and the restriction that ground-based FTS can only take measurements under clear sky conditions, the validation was executed presuming that the atmosphere included no cloud or aerosol. A comparison of PPDF-S data and ground-based FTS data at Yekaterinburg is presented in Figure 6. The results obtained using the original parameter setting in the PPDF-S method and those obtained using the optimized setting are colored as gray and red, respectively. The numbers of GOSAT data shown in the figure are three and five, respectively, for the original and optimized settings. To represent the retrieval accuracy, the bias and standard deviation of XCO2 are calculated by subtracting ground-based data from the GOSAT data. The results show that the bias and its standard deviation of XCO2 are, respectively, 0.75 and 0.57 ppm for the original setting, and 0.69 and 1.79 ppm for the optimized data.

Application to Biomass Burning Area in Western Siberia
Biomass burning is a generic term for the burning of forests, grassland, and crop residues. It is known to release aerosols as black carbon and organic carbon into the atmosphere along with greenhouse gases. Western Siberia is a large biomass burning area [45] for which the main land cover type is boreal coniferous forests and croplands. In the southern part of the area, the combustion of spring wheat residues is performed as an activity of farm operations. If the agricultural burning is not managed well, then it can cause uncontrolled wildfires.
To investigate the retrieval accuracy of the optimized PPDF-S method under denser aerosol conditions, we applied the method to the retrieval of XCO2 from GOSAT data of the selected Western Siberia area: within 45° N to 65° N latitude and 30° E to 75° E longitude. This area includes an observation site of the Ural Federal University at Yekaterinburg, where XCO2 has been measured using ground-based FTS. A land type map of the target area and the location of the Yekaterinburg site are shown in Figure 7a. During the summer of 2013, numerous fires and smoke plumes emitted from biomass burning were detected using a MODerate resolution Imaging Spectroradiometer (MODIS). Furthermore, high AOTs were detected at Yekaterinburg as an AERONET site. For this reason, we specifically undertook the analysis of GOSAT data during the summer (June-August) in 2013 over the Western Siberia area described above. As a measure of biomass burning activity in this area, accumulated counts of thermal anomalies detected by MODIS for the period are shown in Figure 7b.

Application to Biomass Burning Area in Western Siberia
Biomass burning is a generic term for the burning of forests, grassland, and crop residues. It is known to release aerosols as black carbon and organic carbon into the atmosphere along with greenhouse gases. Western Siberia is a large biomass burning area [45] for which the main land cover type is boreal coniferous forests and croplands. In the southern part of the area, the combustion of spring wheat residues is performed as an activity of farm operations. If the agricultural burning is not managed well, then it can cause uncontrolled wildfires.
To investigate the retrieval accuracy of the optimized PPDF-S method under denser aerosol conditions, we applied the method to the retrieval of XCO 2 from GOSAT data of the selected Western Siberia area: within 45 • N to 65 • N latitude and 30 • E to 75 • E longitude. This area includes an observation site of the Ural Federal University at Yekaterinburg, where XCO 2 has been measured using ground-based FTS. A land type map of the target area and the location of the Yekaterinburg site are shown in Figure 7a. During the summer of 2013, numerous fires and smoke plumes emitted from biomass burning were detected using a MODerate resolution Imaging Spectroradiometer (MODIS). Furthermore, high AOTs were detected at Yekaterinburg as an AERONET site. For this reason, we specifically undertook the analysis of GOSAT data during the summer (June-August) in 2013 over the Western Siberia area described above. As a measure of biomass burning activity in this area, accumulated counts of thermal anomalies detected by MODIS for the period are shown in Figure 7b. It is pointed out that the amount of XCO2 data retrieved using the original PPDF-S method is less than that of the NIES RA product [32], which is derived using the FP method and only provided for the Research Announcement principal investigators of the GOSAT project. However, after PPDF parameter optimization, the amount of data retrieved using the PPDF-S method increased from the original number for the target area in Western Siberia during the analysis period.
For retrieval analysis of GOSAT measurement data, the atmospheric transmittance of gas species was calculated at the high spectral resolution of 0.01 cm −1 using v3.0 ACOS/OCO-2 based on the absorption coefficient (ABSCO) look-up tables (LUTs) provided by the Absorption Coefficient (ABSCO) Team instead of line-by-line (LBL) calculation using the HITRAN database [43]. These LUTs record molecular absorption cross-sections in a three-dimensional space of wavelengths, temperatures, and pressures for O2, H2O, and CO2, including radiative effects of various kinds, such as non-Voigt line shapes, speed dependence, line mixing, and collision-induced absorption [46].

Comparison of Results Retrieved Using Optimized PPDF-S Method and Full Physics Method
The distribution of XCO2, which is newly retrieved using an optimized method for the target dataset, is shown in Figure 8a. The amount of data retrieved using the PPDF-S method increased by about 70%. In some cases, XCO2 can be retrieved by the optimized method even over smoke from biomass burning. An example of those scenes observed on 10 August 2013 is shown in Figure 8b. A background photograph of the figure is a visible composite image of MODIS. As an overall feature of XCO2 distribution, smaller values of XCO2 were observed in the northern part of the boundary at about 55° N, with larger values in the southern part. To compare the results of the optimized PPDF-S retrieval and the NIES RA product, the XCO2 values analyzed using the PPDF-S method are shown against the NIES RA product in Figure 9. The color scale in the figure represents the averaged surface albedo in the spectral band of the GOSAT sensor at around 1.6 μm. It shows fairly good interrelation among these data, but some data show that the XCO2 by the optimized PPDF-S was smaller than the NIES RA product, which ranges from 380 ppm to 370 ppm.  It is pointed out that the amount of XCO 2 data retrieved using the original PPDF-S method is less than that of the NIES RA product [32], which is derived using the FP method and only provided for the Research Announcement principal investigators of the GOSAT project. However, after PPDF parameter optimization, the amount of data retrieved using the PPDF-S method increased from the original number for the target area in Western Siberia during the analysis period.
For retrieval analysis of GOSAT measurement data, the atmospheric transmittance of gas species was calculated at the high spectral resolution of 0.01 cm −1 using v3.0 ACOS/OCO-2 based on the absorption coefficient (ABSCO) look-up tables (LUTs) provided by the Absorption Coefficient (ABSCO) Team instead of line-by-line (LBL) calculation using the HITRAN database [43]. These LUTs record molecular absorption cross-sections in a three-dimensional space of wavelengths, temperatures, and pressures for O 2 , H 2 O, and CO 2 , including radiative effects of various kinds, such as non-Voigt line shapes, speed dependence, line mixing, and collision-induced absorption [46].

Comparison of Results Retrieved Using Optimized PPDF-S Method and Full Physics Method
The distribution of XCO 2 , which is newly retrieved using an optimized method for the target dataset, is shown in Figure 8a. The amount of data retrieved using the PPDF-S method increased by about 70%. In some cases, XCO 2 can be retrieved by the optimized method even over smoke from biomass burning. An example of those scenes observed on 10 August 2013 is shown in Figure 8b. A background photograph of the figure is a visible composite image of MODIS. As an overall feature of XCO 2 distribution, smaller values of XCO 2 were observed in the northern part of the boundary at about 55 • N, with larger values in the southern part. To compare the results of the optimized PPDF-S retrieval and the NIES RA product, the XCO 2 values analyzed using the PPDF-S method are shown against the NIES RA product in Figure 9. The color scale in the figure represents the averaged surface albedo in the spectral band of the GOSAT sensor at around 1.6 µm. It shows fairly good interrelation among these data, but some data show that the XCO 2 by the optimized PPDF-S was smaller than the NIES RA product, which ranges from 380 ppm to 370 ppm. about 55° N, with larger values in the southern part. To compare the results of the optimized PPDF-S retrieval and the NIES RA product, the XCO2 values analyzed using the PPDF-S method are shown against the NIES RA product in Figure 9. The color scale in the figure represents the averaged surface albedo in the spectral band of the GOSAT sensor at around 1.6 μm. It shows fairly good interrelation among these data, but some data show that the XCO2 by the optimized PPDF-S was smaller than the NIES RA product, which ranges from 380 ppm to 370 ppm.   Figure 9 shows that the XCO2 of the optimized PPDF-S retrieval became large when the albedo at 1.6 μm was large. This result implies that the retrieved XCO2 is strongly related to the surface albedo corresponding to the land type in the scene. The distributions of the averaged surface albedo in the GOSAT spectral band at around 1.6 μm and 2.0 μm are shown in Figures 10a and 10b, respectively. The albedo was evaluated in relation to the retrieval procedure for XCO2. In these figures, the surface albedo values for both measurement bands are small, below 0.1, in areas with a latitude higher than 55° N. Larger values were found for lower-latitude areas. The comparison of the results presented in Figure 9; Figure 10 shows that XCO2 values retrieved using the PPDF-S method are correlated with the surface albedo. That dependency, which is similar to what Oshchepkov et al. [34] reported, is also investigated through a comparison of the retrieved PPDF parameters. The horizontal distributions of the retrieved αa, ρa, and αa/ρa are shown in Figure 11a-11c, respectively. The αa/ρa parameter can represent the ratio of the shortening effect for the light-path length caused by light refraction (αa) to lengthen the effects of the light-path length because of light scattering (ρa). The dependence of αa and αa/ρa on the surface albedo corresponding to the land surface type is presented in Figure 11. Discussions of these relations among PPDF parameters are presented in Section 6.  Figure 9 shows that the XCO 2 of the optimized PPDF-S retrieval became large when the albedo at 1.6 µm was large. This result implies that the retrieved XCO 2 is strongly related to the surface albedo corresponding to the land type in the scene. The distributions of the averaged surface albedo in the GOSAT spectral band at around 1.6 µm and 2.0 µm are shown in Figure 10a,b, respectively. The albedo was evaluated in relation to the retrieval procedure for XCO 2 . In these figures, the surface albedo values for both measurement bands are small, below 0.1, in areas with a latitude higher than 55 • N. Larger values were found for lower-latitude areas. The comparison of the results presented in Figure 9; Figure 10 shows that XCO 2 values retrieved using the PPDF-S method are correlated with the surface albedo. That dependency, which is similar to what Oshchepkov et al. [34] reported, is also investigated through a comparison of the retrieved PPDF parameters. The horizontal distributions of the retrieved α a , ρ a , and α a /ρ a are shown in Figure 11a-c, respectively. The α a /ρ a parameter can represent the ratio of the shortening effect for the light-path length caused by light refraction (α a ) to lengthen the effects of the light-path length because of light scattering (ρ a ). The dependence of α a and α a /ρ a on the surface albedo corresponding to the land surface type is presented in Figure 11. Discussions of these relations among PPDF parameters are presented in Section 6.
investigated through a comparison of the retrieved PPDF parameters. The horizontal distributions of the retrieved αa, ρa, and αa/ρa are shown in Figure 11a-11c, respectively. The αa/ρa parameter can represent the ratio of the shortening effect for the light-path length caused by light refraction (αa) to lengthen the effects of the light-path length because of light scattering (ρa). The dependence of αa and αa/ρa on the surface albedo corresponding to the land surface type is presented in Figure 11. Discussions of these relations among PPDF parameters are presented in Section 6. (a) (b) (c) Figure 11. Distribution of the retrieved PPDF parameters: αa (a), ρa (b), and αa/ρa (c).

Identification of Atmospheric Aerosol Types Using PPDF Parameters
As described in Section 5.2, αa/ρa can be used as a measure to represent the ratio of the shortened light-path length caused by light refraction (αa) to lengthen the light-path length because of light scattering (ρa). These parameters vary, depending on surface reflectance and on the characteristics of aerosols in the atmosphere. Each type of aerosol has its own characteristics of light reflectance/scattering corresponding to its own refractive index and particle size distribution. However, surface reflectance and its wavelength dependence differ among land cover types. We investigated the dependence of αa and αa/ρa on surface albedo for each land cover type. In the analysis, the surface albedo was related to the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) spectral library. Figure 12 shows the surface albedo at CO2 bands (1.6 μm and 2.0 μm) for each land cover type. A diagram of the correlation between αa and αa/ρa retrieved from simulations is shown in Figure 13. In the simulations, surface albedos of two types were assumed: (1) surface albedo varied as 0.05-0.5, but it was the same for all GOSAT spectral bands changing the AOT from 0.05 to 1.0; and (2) the surface albedo had more realistic values when presuming 19 land cover types related to the ASTER spectral library with a fixed AOT of 0.5. As shown in Figure 13, the result from simulation (1) is depicted by gray plots, but only for the Soot type aerosols as an example. The spread of plots aligned in a row results from the variation of AOT. For simulation (2), results are shown for aerosols of all types. The numbers adjacent to each dot represent the land cover type. The axis range is restricted to the same range as that for the results from the GOSAT data shown in Figure 14. For land cover types of '0, water', '17, water bodies', '15, snow and ice', and '18, tundra', values of αa or αa/ρa are so large that they exceed the axis range and they are not shown in Figure 13. As the results for '9, savanna', '10, grasslands', and '12, croplands' are overlapping with '8, woody savanna', only a label for '8' is plotted on the figure. We also investigated the correlation between αa and αa/ρa for the GOSAT data over the target area in Western Siberia. The result is shown in Figure 14. The color range represents the retrieved XCO2. Data are spread along a linear line with simulations for the type of aerosol. Its slope and absolute values are almost identical to those for the Soot-type aerosols in the simulation. The tendency shown in Figure 14 is similar to that for Soot shown in Figure 13, which might imply that Soot aerosols were included in the atmosphere over the target area during the analysis period. The implication is reasonable in light of the fact that the main aerosol types emitted from biomass burning are organic carbon and black carbon that are the same kind of Soot [47]. This also corresponds to the AERONET observation at Yekaterinburg. The observation showed a high value of absorption AOT, which means that atmospheric aerosol types can be Soot. We conclude that the PPDF parameters chart, as Figure 11. Distribution of the retrieved PPDF parameters: α a (a), ρ a (b), and α a /ρ a (c).

Identification of Atmospheric Aerosol Types Using PPDF Parameters
As described in Section 5.2, α a /ρ a can be used as a measure to represent the ratio of the shortened light-path length caused by light refraction (α a ) to lengthen the light-path length because of light scattering (ρ a ). These parameters vary, depending on surface reflectance and on the characteristics of aerosols in the atmosphere. Each type of aerosol has its own characteristics of light reflectance/scattering corresponding to its own refractive index and particle size distribution. However, surface reflectance and its wavelength dependence differ among land cover types. We investigated the dependence of α a and α a /ρ a on surface albedo for each land cover type. In the analysis, the surface albedo was related to the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) spectral library. Figure 12 shows the surface albedo at CO 2 bands (1.6 µm and 2.0 µm) for each land cover type. A diagram of the correlation between α a and α a /ρ a retrieved from simulations is shown in Figure 13. In the simulations, surface albedos of two types were assumed: (1) surface albedo varied as 0.05-0.5, but it was the same for all GOSAT spectral bands changing the AOT from 0.05 to 1.0; and (2) the surface albedo had more realistic values when presuming 19 land cover types related to the ASTER spectral library with a fixed AOT of 0.5. As shown in Figure 13, the result from simulation (1) is depicted by gray plots, but only for the Soot type aerosols as an example. The spread of plots aligned in a row results from the variation of AOT. For simulation (2), results are shown for aerosols of all types. The numbers adjacent to each dot represent the land cover type. The axis range is restricted to the same range as that for the results from the GOSAT data shown in Figure 14. For land cover types of '0, water', '17, water bodies', '15, snow and ice', and '18, tundra', values of α a or α a /ρ a are so large that they exceed the axis range and they are not shown in Figure 13. As the results for '9, savanna', '10, grasslands', and '12, croplands' are overlapping with '8, woody savanna', only a label for '8' is plotted on the figure. We also investigated the correlation between α a and α a /ρ a for the GOSAT data over the target area in Western Siberia. The result is shown in Figure 14. The color range represents the retrieved XCO 2 . Data are spread along a linear line with simulations for the type of aerosol. Its slope and absolute values are almost identical to those for the Soot-type aerosols in the simulation. The tendency shown in Figure 14 is similar to that for Soot shown in Figure 13, which might imply that Soot aerosols were included in the atmosphere over the target area during the analysis period. The implication is reasonable in light of the fact that the main aerosol types emitted from biomass burning are organic carbon and black carbon that are the same kind of Soot [47]. This also corresponds to the AERONET observation at Yekaterinburg. The observation showed a high value of absorption AOT, which means that atmospheric aerosol types can be Soot. We conclude that the PPDF parameters chart, as shown in Figures 13 and 14, presents the possibility of identifying atmospheric aerosol types.     '15, snow and ice', and '18, tundra', values of α a or α a /ρ a are so large that they exceed the axis range. They are not shown in Figure 13. As the results for '9, savanna', '10, grasslands', and '12, croplands' are overlapping with '8, woody savanna', only a label for '8' is plotted on the figure.

Figure 14.
Correlation diagram between αa and αa/ρa for GOSAT data observed over the target area in Western Siberia. The color represents retrieved XCO2.

Discussion
As described in Section 4.2.1, simulation studies show that the retrieved CO2 profile is not vertically uniform, even under a clear sky condition. The non-uniformity might result from the difference in the characteristics of radiative transfer models between Pstar3 used for synthesizing the spectral radiance and the model with PPDF parameters used for simulating a priori radiance. The former is Pstar3, which is based on the Discrete Ordinate Method (DOM), whereas the latter is an original code in which the light-path length modification is represented by the atmospheric transmittance and PPDF parameters. The difference of radiance corresponding to Rayleigh light scattering at Band 1 (0.76 μm) between these two models is about 0.25%.
Simulation studies also show that the XCO2 bias for the atmosphere including Dust-like aerosols is very large compared with that for the other types of aerosols (Rural, Urban, and Soot). This result might derive from the strong effects of light scattering by Dust-like aerosols with AOT: they are four times stronger than those of the other aerosols in the main spectral band (1.6 μm) used for CO2 retrieval. This might be attributable to the same order of the particle size of Dust-like aerosols as the wavelength. From the retrieval results of the simulation study, we found that the derived XCO2 in the first step of the iterative calculation to minimize the cost function J(x) in the PPDF-S retrieval is closest to the true value for Dust-like aerosols. Although the result shows that the possibility of another xa is more appropriate for Dust-like aerosols, adoption of the derived XCO2 in the first step as a retrieval result is a good countermeasure against the large XCO2 bias for Dust-like aerosols.
For Soot-type aerosols, XCO2 bias is negative under most AOT and surface reflectance conditions. It can result from underestimation of the effects of a shortened light-path length on the PPDF-S retrieval. The effect of light absorption by Soot at 1.6 μm is very strong: the value of the imaginary part of the complex index of refraction of Soot at the wavelength is larger than that of the other aerosols (Rural, Urban, and Dust-like) by about two orders of magnitude. The property also affects the efficiency of light scattering. The size parameter of Soot is about 0.5-1.0 at 1.6 μm. For the size parameter, if the value of the imaginary part of the complex index of refraction is large, then the effect of light scattering will be greater [48]. This means that relative reflection of the aerosol layer (αa) will be greater. The light scattering also lengthens the light path in the Soot layer by multiple scattering of photons. However, at the same time, the Soot absorbs the photons in the layer. As a result, the light-path length in the layer is shortened. Then, the relative mean path length between the layer and ground surface (ρa) will be shorter. Therefore, the total effect of light absorption and scattering by Soot can be effective to make the total light-path length short. The Figure 14. Correlation diagram between α a and α a /ρ a for GOSAT data observed over the target area in Western Siberia. The color represents retrieved XCO 2 .

Discussion
As described in Section 4.2.1, simulation studies show that the retrieved CO 2 profile is not vertically uniform, even under a clear sky condition. The non-uniformity might result from the difference in the characteristics of radiative transfer models between Pstar3 used for synthesizing the spectral radiance and the model with PPDF parameters used for simulating a priori radiance. The former is Pstar3, which is based on the Discrete Ordinate Method (DOM), whereas the latter is an original code in which the light-path length modification is represented by the atmospheric transmittance and PPDF parameters. The difference of radiance corresponding to Rayleigh light scattering at Band 1 (0.76 µm) between these two models is about 0.25%.
Simulation studies also show that the XCO 2 bias for the atmosphere including Dust-like aerosols is very large compared with that for the other types of aerosols (Rural, Urban, and Soot). This result might derive from the strong effects of light scattering by Dust-like aerosols with AOT: they are four times stronger than those of the other aerosols in the main spectral band (1.6 µm) used for CO 2 retrieval. This might be attributable to the same order of the particle size of Dust-like aerosols as the wavelength. From the retrieval results of the simulation study, we found that the derived XCO 2 in the first step of the iterative calculation to minimize the cost function J(x) in the PPDF-S retrieval is closest to the true value for Dust-like aerosols. Although the result shows that the possibility of another x a is more appropriate for Dust-like aerosols, adoption of the derived XCO 2 in the first step as a retrieval result is a good countermeasure against the large XCO 2 bias for Dust-like aerosols.
For Soot-type aerosols, XCO 2 bias is negative under most AOT and surface reflectance conditions. It can result from underestimation of the effects of a shortened light-path length on the PPDF-S retrieval. The effect of light absorption by Soot at 1.6 µm is very strong: the value of the imaginary part of the complex index of refraction of Soot at the wavelength is larger than that of the other aerosols (Rural, Urban, and Dust-like) by about two orders of magnitude. The property also affects the efficiency of light scattering. The size parameter of Soot is about 0.5-1.0 at 1.6 µm. For the size parameter, if the value of the imaginary part of the complex index of refraction is large, then the effect of light scattering will be greater [48]. This means that relative reflection of the aerosol layer (α a ) will be greater. The light scattering also lengthens the light path in the Soot layer by multiple scattering of photons. However, at the same time, the Soot absorbs the photons in the layer. As a result, the light-path length in the layer is shortened. Then, the relative mean path length between the layer and ground surface (ρ a ) will be shorter. Therefore, the total effect of light absorption and scattering by Soot can be effective to make the total light-path length short. The effect is confirmed by the difference of atmospheric transmittance at 1.6 µm between clear sky and the atmosphere including each type of aerosol. The difference is represented by the ratio (Tr ratio ), which is calculated by dividing the transmittance for the atmosphere containing aerosols (Tr aerosol ) by that for clear sky (Tr clear ); Tr ratio = Tr aerosol /Tr clear . The difference between the Tr ratio on the CO 2 absorption line and that on the baseline outside the absorption line can be a measure of the magnitude of light-path modification by aerosols. A smaller Tr ratio on the CO 2 absorption line represents that light has been absorbed effectively by CO 2 because of the lengthened light-path length by aerosols. However, larger Tr ratio on the absorption line represents a shortened light-path length by aerosols. Figure 15 shows Tr ratio for Rural, Urban, Soot, and Dust-like with AOT of 0.5 at 0.55 µm for the surface albedo of 0.2, which is independent of wavelength. The results show that the Tr ratio on the absorption line is smaller than that on the baseline for Rural, Urban, and Dust-like; it is larger for Soot. The result shows that the former three aerosol types work to increase the light-path length at the 1.6 µm band, whereas Soot decreases the light-path length. According to the optical effect by Soot described above and the definition of α a and ρ a , a larger a priori value of α a than the original value can represent the shortened light-path length and can reduce the XCO 2 bias for Soot. A value of 36/20 instead of 1/20 used in the equation for the a priori value of α a shown in Table 2 for β αa is most effective at reducing the XCO 2 bias. effect is confirmed by the difference of atmospheric transmittance at 1.6 μm between clear sky and the atmosphere including each type of aerosol. The difference is represented by the ratio (Trratio), which is calculated by dividing the transmittance for the atmosphere containing aerosols (Traerosol) by that for clear sky (Trclear); Trratio = Traerosol/Trclear. The difference between the Trratio on the CO2 absorption line and that on the baseline outside the absorption line can be a measure of the magnitude of light-path modification by aerosols. A smaller Trratio on the CO2 absorption line represents that light has been absorbed effectively by CO2 because of the lengthened light-path length by aerosols. However, larger Trratio on the absorption line represents a shortened light-path length by aerosols. Figure 15 shows Trratio for Rural, Urban, Soot, and Dust-like with AOT of 0.5 at 0.55 μm for the surface albedo of 0.2, which is independent of wavelength. The results show that the Trratio on the absorption line is smaller than that on the baseline for Rural, Urban, and Dust-like; it is larger for Soot. The result shows that the former three aerosol types work to increase the light-path length at the 1.6 μm band, whereas Soot decreases the light-path length. According to the optical effect by Soot described above and the definition of αa and ρa, a larger a priori value of αa than the original value can represent the shortened light-path length and can reduce the XCO2 bias for Soot. A value of 36/20 instead of 1/20 used in the equation for the a priori value of αa shown in Table 2 for βαa is most effective at reducing the XCO2 bias. Results for retrieval from GOSAT data measured in Western Siberia show that about 10% of the retrieved XCO2 take small values below 380 ppm, as shown in Figure 9. Although the retrieval result accuracy cannot be evaluated because no verification data exist for the target area during the analysis period, the value is extremely small compared with the background level of XCO2 for the year in Western Siberia. Therefore, the possibility that the airmass considered in the PPDF-S is larger than the true one can cause the overestimation of XCO2. That possibility was investigated. Corresponding to the airmass, we compared the surface pressure for the PPDF-S method and that for the FP method. For the comparison, the a priori value of surface pressure used in the PPDF-S method was used because surface pressure is not derived in the method. However, surface pressure Results for retrieval from GOSAT data measured in Western Siberia show that about 10% of the retrieved XCO 2 take small values below 380 ppm, as shown in Figure 9. Although the retrieval result accuracy cannot be evaluated because no verification data exist for the target area during the analysis period, the value is extremely small compared with the background level of XCO 2 for the year in Western Siberia. Therefore, the possibility that the airmass considered in the PPDF-S is larger than the true one can cause the overestimation of XCO 2 . That possibility was investigated. Corresponding to the airmass, we compared the surface pressure for the PPDF-S method and that for the FP method.
For the comparison, the a priori value of surface pressure used in the PPDF-S method was used because surface pressure is not derived in the method. However, surface pressure was retrieved along with the RA product of XCO 2 in the FP method. The retrieved surface pressure was used in the comparison. As a result, the relation of surface pressure between the two methods for the cases in which XCO 2 retrieved using the PPDF-S method was below 380 ppm was identical to that for other cases in which small XCO 2 was not retrieved using the PPDF-S method. The a priori value of surface pressure used for the PPDF-S method was not so wrong that it could cause the underestimation of XCO 2 . On the other hand, as described in Section 5.3, Figures 13 and 14 show that the consideration derives from the inference that Soot might be included in the atmosphere. Then, we also considered the possibility that the effect of the shortened light-path length by Soot is underestimated in the PPDF-S retrieval. If the value of 36/20 was used for the equation for the a priori value of α a shown in Table 2 for β αa , then the value of the retrieved XCO 2 was increased by about 5 ppm. Because the retrieved XCO 2 value was more realistic, the identification of Soot from the α a -α a /ρ a chart might be appropriate. The AOT values were 0.1-0.4 for the cases of small XCO 2 . Although the value is not so high, the underestimation of XCO 2 can be caused by negative XCO 2 biases that can occur even for low AOT, as shown in Figure 5.
We investigated the dependence of retrieved XCO 2 and PPDF parameters on the altitude of the aerosol layer because light scattering/absorption properties depend on the layer altitude. We executed the PPDF-S retrieval from simulated radiance spectra considering aerosols of four types with the volume mixing ratio at 0-3 km, or at 0-2 km (as shown in Table 1) in Pstar3. Although the difference in XCO 2 caused by a change in the altitude was as great as 2.05 ppm, the difference in PPDF parameters was small; those for Rural, Urban, Soot, and Dust-like were, respectively, about 5%, 2.5%, 0.01%, and 3%. The difference was so small that one might infer that the dependence of the retrieval results on the aerosol layer altitude is negligible. The result also indicates that the identification of aerosol types using the α a -α a /ρ a chart is applicable to various situations of the atmosphere, including aerosols.
From the possibility of the identification of aerosol types using the α a -α a /ρ a chart and the countermeasures against the large XCO 2 bias for Dust-like aerosols and against the negative XCO 2 bias for Soot, a more improved PPDF-S retrieval flowchart can be inferred, as shown in Figure 16. In the figure, the improved part is shown in red letters, in addition to the original flowchart.

Conclusions
We have improved the PPDF-S retrieval method for XCO2 retrieval from GOSAT measurement spectra in SWIR to retrieve XCO2 more accurately under dense aerosol conditions. In the improved version, the constraint of a PPDF parameter that represents the light reflectance effect by aerosols (αa) was optimized based on simulation studies of various types of aerosols and surface albedos, referring to the vertical profile of CO2 concentration as a measure of a stable solution. The new method was applied to retrieval analysis of GOSAT data measured in Western Siberia, where biomass burning is very active. By optimizing the PPDF parameter, the retrieval came to be executable, even under smoky conditions, and the total number of retrieved data increased by about 70%. A comparison of the XCO2 of the data provided by NIES as an RA product and the XCO2 values of newly retrieved data showed no marked inconsistency. The results show that the PPDF parameters retrieved from GOSAT data measured for Western Siberia are consistent with those from the simulations for Soot. The result might imply that Soot aerosols were included in the atmosphere over the target area during the analysis period, which is consistent with real phenomena and ground-based observations. Therefore, we conclude that the PPDF parameters have the possibility of identifying atmospheric aerosol types. The results also show that the effect of shortening the light-path length by Soot can be underestimated by PPDF-S retrieval. According to the simulation studies, the result was solved by setting the a priori value of αa as larger to represent the shortened light-path length. This countermeasure for Soot was also effective for retrieving more realistic XCO2 from GOSAT data for Western Siberia. From these results, we suggest the use of a further improved algorithm after the atmospheric aerosol type can be identified. The algorithm is anticipated for Figure 16. Improved PPDF-S retrieval flowchart.

Conclusions
We have improved the PPDF-S retrieval method for XCO 2 retrieval from GOSAT measurement spectra in SWIR to retrieve XCO 2 more accurately under dense aerosol conditions. In the improved version, the constraint of a PPDF parameter that represents the light reflectance effect by aerosols (α a ) was optimized based on simulation studies of various types of aerosols and surface albedos, referring to the vertical profile of CO 2 concentration as a measure of a stable solution. The new method was applied to retrieval analysis of GOSAT data measured in Western Siberia, where biomass burning is very active. By optimizing the PPDF parameter, the retrieval came to be executable, even under smoky conditions, and the total number of retrieved data increased by about 70%. A comparison of the XCO 2 of the data provided by NIES as an RA product and the XCO 2 values of newly retrieved data showed no marked inconsistency. The results show that the PPDF parameters retrieved from GOSAT data measured for Western Siberia are consistent with those from the simulations for Soot. The result might imply that Soot aerosols were included in the atmosphere over the target area during the analysis period, which is consistent with real phenomena and ground-based observations. Therefore, we conclude that the PPDF parameters have the possibility of identifying atmospheric aerosol types. The results also show that the effect of shortening the light-path length by Soot can be underestimated by PPDF-S retrieval. According to the simulation studies, the result was solved by setting the a priori value of α a as larger to represent the shortened light-path length. This countermeasure for Soot was also effective for retrieving more realistic XCO 2 from GOSAT data for Western Siberia. From these results, we suggest the use of a further improved algorithm after the atmospheric aerosol type can be identified. The algorithm is anticipated for application to the analysis of carbon monoxide (CO), which is newly observed by the successor of GOSAT: GOSAT-2. The results of simultaneous observations and analyses of CO 2 , CO, and aerosols using the method are expected to contribute to a better capability of distinguishing anthropogenic CO 2 emissions from the total emissions of urban areas or biomass burning regions where gases and aerosols are released simultaneously.