 Methodology
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Rapid, accurate computation of narrowband sky radiance in the 940 nm gas absorption region using the correlated kdistribution method for sunphotometer observations
Progress in Earth and Planetary Science volume 9, Article number: 10 (2022)
Abstract
We developed lookup tables for the correlated kdistribution (CKD) method in the 940 nm water vapor absorption region (WVCKD), with the aim of rapid and accurate computation of narrowband radiation around 940 nm (10,000–10,900 \({\mathrm{cm}}^{1}\)) for groundbased angularscanning radiometer data analysis. Tables were constructed at three spectral resolutions (2, 5, and 10 \({\mathrm{cm}}^{1}\)) with quadrature values (point and weight) and numbers optimized using simulated sky radiances at ground level, which had accuracies of ≤ 0.5% for subbands of \(10 {\mathrm{cm}}^{1}\). Although highresolution WVCKD requires numerous quadrature points, the number of executions of the radiative transfer model is reduced to approximately 1/46 of the number used in the linebyline approach by our WVCKD with a resolution of 2 \({\mathrm{cm}}^{1}\). Furthermore, we confirmed through several simulations that WVCKD could be used to compute radiances with various vertical profiles. The accuracy of convolved direct solar irradiance and diffuse radiance at a full width at half maximum (FWHM) of 10 nm, computed with the WVCKD, is < 0.3%. In contrast, the accuracy of convolved normalized radiance, which is the ratio of diffuse radiance to direct solar irradiance, at an FWHM of 10 nm computed with the WVCKD is < 0.11%. This accuracy is lower than the observational uncertainty of a groundbased angularscanning radiometer (approximately 0.5%). Finally, we applied the SKYMAP and DSRAD algorithms (Momoi et al. in Atmos Meas Tech 13:2635–2658, 2020. https://doi.org/10.5194/amt1326352020) to SKYNET observations (Chiba, Japan) and compared the results with microwave radiometer values. The precipitable water vapor (PWV) derived with the WVCKD showed better agreement (correlation coefficient γ = 0.995, slope = 1.002) with observations than PWV derived with the previous CKD table (correlation coefficient γ = 0.984, slope = 0.926) by Momoi et al. (Momoi et al., Atmos Meas Tech 13:2635–2658, 2020). Through application of the WVCKD to actual data analysis, we found that an accurate CKD table is essential for estimating PWV from skyradiometer observations.
1 Introduction
Aerosols, clouds, ozone, and water vapor are important parameters for characterizing Earth’s climate and changes therein (e.g., IPCC 2021) and are therefore considered essential climate variables by the World Meteorological Organization (Bojinski et al. 2014). Some of these parameters are measured using ground and satellitebased instruments and estimated using the multiterm least squares method (e.g., Dubovik and King 2000; Rogers 2000), with radiative transfer models (RTMs) as the forward model. For example, Kudo et al. (2021) simultaneously estimated aerosol microphysical and optical properties, as well as ozone and water vapor column concentrations, using the angular distributions of diffuse radiance observed with an angularscanning radiometer (e.g., Holben et al. 1998; Nakajima et al. 2020) proceed with the SKYRAD.pack MRI version 2.
The computational efficiency of the radiance calculation can be improved through use of the lookup table (LUT) method, rather than direct application of the RTM, as the forward model. However, the recent trend in aerosol remote sensing with ground and satellitebased radiometers (e.g., Dubovik et al. 2011; Nakajima et al. 2020; Sinyuk et al. 2020; Kudo et al. 2021) has been toward increasing the number of retrieved parameters, and direct use of RTMs is preferred for analysis of groundbased angularscanning radiometer data with high degrees of freedom for control variables. Several RTMs have been used for analysis of groundbased remote sensing data. For example, the AErosol RObotic NETwork (AERONET; Holben et al. 1998), which is an international network of groundbased angularscanning radiometers, uses a scalar RTM called RSTAR (System for Transfer of Atmospheric Radiation for Radiance calculations; Nakajima and Tanaka 1986, 1988) in version 2 of its inversion algorithm (Dubovik and King 2000; Dubovik et al. 2000, 2006), and a vector RTM called SORD (Successive ORDers of scattering; Korkin et al. 2017) in version 3 (Sinyuk et al. 2020). The SKYNET (Takamura and Nakajima 2004; Nakajima et al. 2007, 2020), another groundbased angularscanning radiometer network, uses RSTAR in most of its analysis packages (Nakajima et al. 1996; Hashimoto et al. 2012; Kudo et al. 2021).
RSTAR is part of the STAR (System for Transfer of Atmospheric Radiation) series, which was developed and distributed by the OpenCLASTR project (http://157.82.240.167/~clastr/) led by some of the authors of this study. In addition to the scalar RTM (RSTAR), the STAR series includes a vector RTM called PSTAR (STAR for Polarized radiance calculations; Ota et al. 2010) and the flux calculation code FSTAR (STAR for Flux calculations; Nakajima et al. 2000), making it a broadly compatible group of packages. In particular, RSTAR and PSTAR introduce efficient calculation methods for IMS (Improved Multiple and Single scattering approximation; Nakajima and Tanaka 1988) and P^{n}IMS (IMS by nth order multiple scattering correction of the forward Peak; Momoi et al. 2022), respectively, enabling accurate reconstruction of sky radiances even in the solar aureole region. Therefore, these RTMs have been widely used in the analysis of satellite and groundbased remote sensing observations, especially by the Japanese research community (e.g., Takenaka et al. 2011; Hashimoto and Nakajima 2017; Sekiguchi et al. 2018; Shi et al. 2019). The STAR series efficiently computes multispecies gasabsorbing broadband radiative flux using the method of Sekiguchi and Nakajima (2008), which is a nonlinearly optimized version of the correlated kdistribution (CKD) method (Lacis and Oinas 1991; Fu and Liou 1992). For narrowband gas absorption calculation, RSTAR uses the CKD method for which the standard LUT (hereafter, SNCKD) was designed mechanically with two Gaussian quadrature points, as described in Sect. 2.2. Based on RSTAR with the SNCKD, Momoi et al. (2020) proposed an approach using the SKYMAP algorithm, which retrieves the concentrations of atmospheric gases, such as water vapor, from the dependence of the angular distribution of diffuse radiance with a multiterm least squares method and calibrates the radiometric sensor using retrieved values, as described in Sect. 3.4.
Despite the progress described above, detailed assessment of the information contents of the water vapor and aerosols included in the direct solar irradiance and diffuse radiance is insufficient due to the large computational burden. The main instrument of the SKYNET is an angularscanning radiometer called skyradiometer (Prede, Tokyo, Japan) that measures direct solar and diffuses irradiances with a finite field of view at multiple wavelengths, including a water vapor absorption band (940 nm). From these measurements, we retrieve aerosol properties (e.g., size distribution, refractive index, and particle shape), cloud properties (e.g., effective radius and cloud optical thickness; Khatri et al. 2019), and ozone and water vapor amounts (Khatri et al. 2014; Uchiyama et al. 2014). In general, the column total atmospheric water vapor (precipitable water vapor [PWV]) is estimated from atmospheric transmittance in water vapor absorption regions, such as 940 nm, using observations taken with a photometer (Fowle 1912, 1915; Bruegge et al. 1992; Schmid et al. 1996, 2001; Halthore et al. 1997; Holben et al. 1998; Campanelli et al. 2014, 2018; Uchiyama et al. 2014, 2019). Campanelli et al. (2014, 2018) and Uchiyama et al. (2014, 2019) used the empirical equation: \(\mathrm{ln}{\tilde{T }}_{\mathrm{H}2\mathrm{O}}=a{\left(mw\right)}^{b}\) (Bruegge et al. 1992) to define the relationship between the convolved transmittance of atmospheric water vapor (\({\tilde{T }}_{\mathrm{H}2\mathrm{O}}\)) and PWV (\(w\)). In this equation, \(a\) and \(b\) are adjustment parameters, which are affected by the filter response function of the radiometer and can be determined using several approaches, including comparison with other instruments (Campanelli et al. 2014, 2018) and theoretical calculations (Uchiyama et al. 2014; Giles et al. 2019). For determining the parameters through theoretical calculations, the AERONET (Giles et al. 2019) uses the linebyline (LBL) method under a US standard atmosphere, while Uchiyama et al. (2014) used the CKD method under Air Force Geophysics Laboratory (AFGL) standard atmospheres. However, Campanelli et al. (2014, 2018) reported that the parameters vary seasonally and spatially due to differences in the vertical profiles of water vapor, temperature, and pressure. Therefore, the parameters should be estimated seasonally and spatially, but implementation of this approach using the LBL method has a high computational cost. Obtaining the convolved transmittance requires regular sensor calibration, in which the sensor output of the extraterrestrial solar irradiance (or calibration constant) is determined at a specific site (e.g., Mauna Loa Observatory), as the convolved transmittance is the ratio of the sensor output of direct solar irradiance to the calibration constant. Recently, Momoi et al. (2020) reported that the angular distribution of diffuse radiances for the water vapor absorption band in the almucantar plane is affected by PWV and proposed another PWV retrieval method based on this relationship. This method is suitable for us with longterm observations because a calibration constant can be determined from the PWV data derived from the onsite angular distribution of diffuse radiance using the SKYMAP algorithm. This approach requires accurate computations of sky radiances in the water vapor absorption band by the RTM. Furthermore, new algorithms for simultaneous retrieval of water vapor and aerosols and assessment of the retrieval using the water vapor absorption band are needed, as the diffuse radiances at 940 nm in parts of the sky other than the almucantar plane contain information about aerosol vertical inhomogeneity (Momoi et al. 2020).
In this study, we developed LUTs for the CKD method in the 940 nm water vapor absorption region (WVCKD), installed the WVCKD in RSTAR to enable rapid and accurate computation of the 940 nm band for retrieval of PWV, and assessed the impact of the introduction of the LUTs on PWV estimations from the angular distribution of diffuse radiances, using actual groundbased angularscanning radiometer observations in the water vapor absorption band. The WVCKD was constructed through optimization of the quadrature values (point and weight) and numbers. Section 2 describes the methods and experimental setup used to create the WVCKD, and Sect. 3 provides the results of simulations used to estimate the accuracy of the WVCKD and its impact on PWV estimation from actual SKYNET skyradiometer observations.
2 Methods/experimental design
Section 2.1 describes the RTM calculation accuracy required to analyze skyradiometer observations. Section 2.2 examines the challenges facing the currently used LUT of the k distribution (SNCKD) in RSTAR version 7 (RSTAR7) for narrowband sky radiance computation around the 940 nm region. Section 2.3 describes the method used to create a new LUT of the k distribution.
2.1 RTM calculation accuracy requirements
The skyradiometer measures direct solar and diffuse irradiances. In previous studies (e.g., Uchiyama et al. 2014; Momoi et al. 2020; Kudo et al. 2021), PWV was estimated from direct solar irradiance and/or “normalized radiance,” which is defined as the ratio of diffuse radiance to direct solar irradiance (Nakajima et al. 1996). Section 2.1.1 describes these values measured by the skyradiometer. Section 2.1.2 describes the uncertainty of skyradiometer observations.
2.1.1 Measurement of sky irradiance
The POM01 skyradiometer measures direct solar and diffuse irradiances with a finite field of view at seven wavelengths: 315, 400, 500, 675, 870, 940, and 1020 nm. The latest model, POM02, measures irradiance at 340, 380, 1627, and 2200 nm in addition to those wavelengths. Diffuse irradiance is measured at scattering angles of 2, 3, 4, 5, 7, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, and 160° in the almucantar and principal planes.
Assuming a narrow spectral band filter response function, direct solar and diffuse irradiances can be described by a radiative transfer equation in a planeparallel nonrefractive atmosphere as follows:
where \({F}_{\mathrm{ds}}\) is the sensor output current of the direct solar irradiance; \({F}_{\mathrm{df}}\) is the sensor output current of diffuse irradiance detected with a finite field of view (\(\mathrm{\Delta \Omega }\)); \({F}_{0}\) is the calibration constant, which is the sensor output current for extraterrestrial solar irradiance (\({F}_{\mathrm{sol}}\)) at the mean distance between earth and the sun; \(T\) is transmittance; \(L\) is diffuse intensity defined as Eqs. (1b) and (2b); \(R\) is normalized radiance; d is the distance between Earth and the sun (AU); \(\lambda\) is wavelength; \(\tau\) is total optical thickness; \({m}_{0}\) is the optical air mass, represented as \({m}_{0}=1/\left\mathrm{cos}{\theta }_{0}\right=1/\left{\mu }_{0}\right\); \({P}^{^{\prime}}\left(\Theta ,\uplambda ,\tau ^{\prime}\right)\) and \({\omega }^{^{\prime}}\left(\lambda ,\tau ^{\prime}\right)\) are the total phase function and total single scattering albedo, respectively, at an altitude of \(\tau ={\tau }^{^{\prime}}\); and \(q\) is the multiple scattering contribution. Although determination of \(T\) requires \({F}_{0}\), \(R\) does not require \({F}_{0}\) due to the normalization. Hereafter, \(T\) and \(L\) are referred to as sky intensities. Assuming a single homogeneous layer, \(L\) can be simplified as follows:
Furthermore, assuming a wideband filter response function (\(\phi\)), convolved \({F}_{\mathrm{ds}}\) and \({F}_{\mathrm{df}}\) (\({\tilde{F }}_{\mathrm{ds}}\) and \({\tilde{F }}_{\mathrm{df}}\)) with \(\phi\) can be obtained through convolution of Eq. (1), as follows:
Hence, the convolved sky intensities (\(\tilde{T }\) and \(\tilde{L }\)) and convolved normalized radiance (\(\tilde{R }\)) are defined as follows:
Although sky radiance in weak gas absorption regions, such as at 340, 380, 400, 500, 675, 870, and 1020 nm in skyradiometer observations, can be assumed to represent narrow spectral bands (Eq. [2]), sky radiance in gas absorption regions, such as the 940 nm band in skyradiometer observations, requires convolution (Eq. [5]).
2.1.2 Uncertainty of skyradiometer observations
In the SKYNET analysis algorithms (e.g., Momoi et al. 2020; Kudo et al. 2021) used with skyradiometer observations, the measurement values are treated as \(\mathrm{ln}\tilde{T }\) and \(\mathrm{ln}\tilde{R }\), obtained using Eq. (5). The weighted values (\({\sigma }_{\tilde{T }}^{2}\) and \({\sigma }_{\tilde{R }}^{2}\)) of the covariance matrix in the algorithms are expressed under the assumption that \({\tilde{F }}_{\mathrm{ds}}\), \({\tilde{F }}_{\mathrm{df}}\), and \({\tilde{F }}_{0}\) are independent of each other, as follows:
where \({\sigma }_{{\tilde{F }}_{0}}\) is the standard deviation of the skyradiometer calibration constant (\(\mathrm{ln}{\tilde{F }}_{0}\)) or the relative error of \({\tilde{F }}_{0}\), as \(\partial \mathrm{ln}{\tilde{F }}_{0}\sim \frac{\partial {\tilde{F }}_{0}}{{\tilde{F }}_{0}}\); \({\sigma }_{{\tilde{F }}_{\mathrm{ds}}}\) and \({\sigma }_{{\tilde{F }}_{\mathrm{df}}}\) are the standard deviations of skyradiometer measurements (\(\mathrm{ln}{\tilde{F }}_{\mathrm{ds}}\) and \(\mathrm{ln}{\tilde{F }}_{\mathrm{df}}\), respectively).
In this study, we estimated \({\sigma }_{{\tilde{F }}_{\mathrm{ds}}}\) from 10 × 10 radiances sampled at 0.1° intervals in a circumsolar domain of ± 1° in both the zenith and azimuth angle directions. This measurement protocol is mainly used for calibration of the solid view angle through the solar disk scan method (Nakajima et al. 1996, 2020; Boi et al. 1999; Uchiyama et al. 2018). The solar disk scan method provides a solar aureole angular distribution (Fig. 1), which reduces observational noise with a Gaussian filter. Therefore, the uncertainty of sky intensity data is calculated from the difference (\({\Delta }_{f}\)) of sky irradiance between observed and Gaussianfiltered data, as follows:
where \({f}^{\mathrm{obs}}\) and \({f}^{\mathrm{gf}}\) are the observed and Gaussianfiltered sky irradiances, respectively. Because the magnitude of the solar aureole angular distribution differs at large scattering angles (\(\Theta \ge {0.3}^{\mathrm{o}}\)), we used a range of 0.3°. Therefore, we could estimate the uncertainties of \({\tilde{F }}_{\mathrm{ds}}\) from \({\Delta }_{f}\), as the solar disk diameter is approximately 0.5°. Figure 2 shows an example histogram of \({\Delta }_{f}\) at 340, 500, and 940 nm based on measurements taken with the skyradiometer POM02 (serial no. PS2501401) on February 27 and 28, 2020, in Akiruno, Tokyo (35.751°N, 139.323°E). Table 1 lists the uncertainties of \({\tilde{F }}_{\mathrm{ds}}\) and indicates that the standard deviation of skyradiometer measurements (\({\sigma }_{{\tilde{F }}_{\mathrm{ds}}}\)) at all wavelengths is less than \(5.0\times {10}^{3}\). Thus, \({\sigma }_{{\tilde{F }}_{\mathrm{df}}}\) is expected to be larger than \({\sigma }_{{\tilde{F }}_{\mathrm{ds}}}\). This is because \({\tilde{F }}_{\mathrm{ds}}\) and \({\tilde{F }}_{\mathrm{df}}\) are measured using the same detector, but \({\tilde{F }}_{\mathrm{ds}}\gg {\tilde{F }}_{\mathrm{df}}\) by the solar incident beam. Therefore, \(5.0\times {10}^{3}\) is a reasonable benchmark value for errors in \({\tilde{F }}_{\mathrm{ds}}\) and \({\tilde{F }}_{\mathrm{df}}\) in this study.
2.2 Challenges regarding the kdistribution lookup table in RSTAR
In RSTAR7, gas absorption is considered for H_{2}O, CO_{2}, O_{3}, N_{2}O, CO, CH_{4}, and O_{2} using HITRAN 2004 database (Rothman et al. 2005) and MT_CKD version 1 (Mlawer et al. 2012) for continuum absorptions. Gas absorption is calculated using the CKD method with a LUT (ckd.g.ch_2_2e3; SNCKD), which generates two Gaussian quadrature points without optimization in each subband. The resolution of the SNCKD at wavenumber (\(k\)) is \({\mathrm{dlog}}_{10}k=5\times {10}^{4}\) (\(\mathrm{d}k\approx 12.2 {\mathrm{cm}}^{1}\) at 940 nm). In weak gas absorption regions (e.g., skyradiometer measurements at 340, 380, 400, 500, 675, 870, and 1020 nm), the radiance can be accurately computed using the SNCKD because the gas absorption coefficient is small relative to other processes (e.g., Rayleigh scattering and aerosol extinction). However, careful consideration of the quadrature numbers is essential in the water vapor absorption region around 940 nm due to the complex absorption characteristics of water vapor. Therefore, we validated the radiance calculation with SNCKD around 940 nm (10,000–10,900 \({\mathrm{cm}}^{1}\) [1000–917 nm]) through the LBL method using simulated bandaveraged sky intensities for subbands (\(\overline{T }=\frac{{\overline{F} }_{\mathrm{ds}}}{{\overline{F} }_{0}}\) and \(\overline{L }=\frac{\left\mu \right{\overline{F} }_{\mathrm{df}}\left(\Theta ,\lambda \right)}{{\overline{F} }_{0}\left(\lambda \right)\mathrm{\Delta \Omega }}\)). The LBL method is based on the line absorption calculation from HITRAN 2012 (Rothman et al. 2013) and continuum absorption from MT_CKD version 3.2 (Mlawer et al. 2012). The bandaveraged sky intensities for subbands were computed using RSTAR7 with the IMS method (Nakajima and Tanaka 1988). The validation dataset (CADB) consisted of bandaveraged sky intensities at ground level, which were calculated from continental averaged aerosol conditions (Hess et al. 1999) with aerosol optical thickness (AOT) of 0.05, 0.20, and 1.00 at 940 nm; solar zenith angles of 30, 50, and 70 degrees in two skyradiometer observation planes (almucantar and principal); and PWV from 0.5 to 6 cm at an interval of 0.5 cm. The vertical atmospheric profile is the US standard atmosphere employed in RSTAR7. Extraterrestrial solar irradiance was averaged at the subband (\({\mathrm{dlog}}_{10}k\)) level. Therefore, with the LBL method, \(\overline{T }\) and \(\overline{L }\) are expressed as:
where \({T}_{\mathrm{A}}\) and \({T}_{\mathrm{R}}\) are the monochromatic transmittances of aerosol extinction and Rayleigh scattering, respectively; \({T}_{\mathrm{H}2\mathrm{O}}\) is the monochromatic transmittance of water vapor absorption (line and selfcontinuum); and \({T}_{\mathrm{cont}}\) is the monochromatic transmittance of the O_{2} and O_{3} continuum absorption. Because the differential interval of the numerical integral is too small (\(\mathrm{d}\kappa =0.01 {\mathrm{cm}}^{1}\)) for use with the LBL method, \({T}_{\mathrm{H}2\mathrm{O}}\) is obtained using Beer–Lambert’s law as follows:
where \({\sigma }_{\mathrm{H}2\mathrm{O},\mathrm{line}}\) and \({\sigma }_{\mathrm{H}2\mathrm{O},\mathrm{cont}}\) are the absorption coefficients [/m] of the water vapor line and selfcontinuum absorptions, respectively; \(K\) is the temperature; \(p\) is the pressure; and \(z\) is the geometric thickness of the atmosphere [m]. Figure 3 shows the maximum errors (\({\varepsilon }_{\overline{I}\mathrm{max} }\)) of \(\overline{T }\) and \(\overline{L }\) obtained with the SNCKD compared with the LBL method, which were determined as:
The root mean square errors of \({\varepsilon }_{\overline{I}\mathrm{max} }\) for CADB were \(8.24\times {10}^{1}\) (approximately 82.4% error) at 10,000–10,902 \({\mathrm{cm}}^{1}\), and \(1.16\) (approximately 116% error) at 10,411–10,864 \({\mathrm{cm}}^{1}\) [961–920 nm]. This error propagates to the convolved sky intensities. Thus, the error of convolved sky intensities with a filter response function of Gaussian shape at a full width at half maximum (FWHM) of 10 nm, corresponding to the FWHM of the skyradiometer’s filter, reaches 22%, as discussed in Sect. 3.2. This residual error is larger than \({\sigma }_{{F}_{\mathrm{ds}}}\). One reason for this large error is an update of the absorption database from HITRAN 2004 to HITRAN 2012 where the number of water vapor absorption lines in this band increased more than fourfold. Another reason is the lack of optimization, which leads large error under the US atmosphere.
2.3 Method used to create the new kdistribution lookup table
In this study, a new LUT of the k distribution at 10\({\mathrm{cm}}^{1}\) intervals (\({\Delta }_{\kappa }\)) from 10,000 \({\mathrm{cm}}^{1}\) to 10,900 \({\mathrm{cm}}^{1}\) (hereafter WVCKD10) was constructed through optimization of the quadrature values (point and weight) and numbers through the LBL method. Radiation around 940 nm is attenuated mainly through aerosol extinction, Rayleigh scattering, and gas absorption including O_{2} continuum, O_{3} continuum, and water vapor line and selfcontinuum absorptions. Thus, WVCKD10 consists of three LUTs: quadrature weights, water vapor k distribution, and O_{2} and O_{3} continuums. Those LUTs were created from HITRAN 2012 (Rothman et al. 2013) and MT_CKD version 3.2 (Mlawer et al. 2012). Because computation of the RTM requires the number of quadrature values, WVCKD optimally contains the minimum quadrature numbers in addition to quadrature values optimized for \(\overline{E }\left(\mu ,\lambda \right)\) under six AFGL standard atmospheres (US standard, tropical, midlatitude summer, midlatitude winter, highlatitude summer, and highlatitude winter) in RSTAR7. In this study, the maximum error (\(\sqrt{{\varepsilon }_{E}^{2}}\)) of \(\overline{E }\left(\mu ,\lambda \right)\) for \({\Delta }_{\kappa }=10 {\mathrm{cm}}^{1}\) is achieved at values less than \(5.0\times {10}^{3}\). The quadrature values, consisting of a pair of point and weight values, were optimized with the Gauss–Newton method, and the quadrature numbers were determined through a linear search from 2 to 64. After construction of the WVCKD10, we validated it with CADB, as described in the previous subsection for the validation dataset of the SNCKD.
3 Results and discussion
3.1 Optimized kdistribution lookup table
Figure 4a shows the quadrature numbers and Fig. 4b shows the maximum error (\(\sqrt{{\varepsilon }_{\overline{I}\mathrm{max} }^{2}}\)), which was satisfied at values less than \(5.0\times {10}^{3}\) for 10,000–10,900 \({\mathrm{cm}}^{1}\). The quadrature number falls into the range of 3–15 at 10,000–10,900 \({\mathrm{cm}}^{1}\) and 4–15 at 10,410–10,870 \({\mathrm{cm}}^{1}\) [961–920 nm]. The median quadrature number is 7 at 10,000–10,900 \({\mathrm{cm}}^{1}\) and 8 at 10,410–10,870 \({\mathrm{cm}}^{1}\) [961–920 nm], as the water vapor line absorption around 940 nm is complex and requires numerous quadrature points to maintain accuracy. In conclusion, using the WVCKD10 reduces the number of executions of the RTM to approximately 1/100 of the number needed for the LBL method, as the bandaveraged sky intensities at 10 \({\mathrm{cm}}^{1}\) must be computed 1000 times with the LBL method (\(=10 {\mathrm{cm}}^{1}/0.01 {\mathrm{cm}}^{1}\)) in RSTAR7 (Table 2).
The equations for bandaveraged sky intensities differ markedly between the LBL and CKD methods. \({\overline{T} }^{\mathrm{CKD}}\) and \({\overline{L} }^{\mathrm{CKD}}\), derived using the WVCKD, are as follows:
where \({\overline{T} }_{A}\) and \({\overline{T} }_{R}\) are the bandaveraged values of \({T}_{A}\) and \({T}_{R}\), respectively; \({N}_{\mathrm{ch}}\) is the number of quadrature points; \({T}_{\mathrm{H}2\mathrm{O},\mathrm{ckd}}^{\left(i\right)}\) and \({R}^{(i)}\) are the ith transmittance and normalized radiance of the quadrature values of the k distribution at \({\Delta }_{\kappa }\), respectively; and \({\xi }_{i}\) is the ith quadrature weight of the k distribution at \({\Delta }_{\kappa }\), which is normalized to:
In the CKD method, convolved sky intensities (\(\widehat{T}\) and \(\widehat{L}\)) are determined with the stepwise filter response function \(\overline{\phi }\), as follows:
where \({\overline{F} }_{\mathrm{sol}}\) is the bandaveraged extraterrestrial solar irradiance and \({N}_{\mathrm{s}}\) is the number of subbands. \(\widehat{T}\) and \(\tilde{T }\) in Eq. (5a) (\(\widehat{L}\) and \(\tilde{L }\) in Eq. (5b)) are not entirely synonymous, but are generally equivalent based on the assumption that the extraterrestrial solar irradiance and filter response function being nearly constant across subbands. The residual errors (\({\varepsilon }_{\widehat{T},\mathrm{RT}}\) and \({\varepsilon }_{\widehat{L},\mathrm{RT}}\)) of \(\widehat{T}\) and \(\widehat{L}\) for the CKD method are obtained as follows:
If we assume that \({\overline{T} }_{\mathrm{H}2\mathrm{O}}\) is roughly randomly distributed in the range of \({\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{min}}=0\) to \({\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{max}}=1\), \({\varepsilon }_{\widehat{T},\mathrm{RT}}^{2}\) and \({\varepsilon }_{\widehat{L},\mathrm{RT}}^{2}\) are obtained as follows:
where \(<>\) indicates an averaging operation;
In Eq. (15), \({\varepsilon }_{\overline{T },\mathrm{RT}}\sim {\varepsilon }_{\overline{L },\mathrm{RT}}\sim <{\varepsilon }_{\overline{I}\mathrm{max} }^{2}>\). The second term on the righthand side of Eq. (16a) is a rough assumption because of the biased probability distribution of \({\overline{T} }^{\mathrm{LBL}}\). For example, \({D}^{2}\) is 1.18, 1.38, and 1.82 at 10,410–10,870 \({\mathrm{cm}}^{1}\) [961–920 nm] under the US standard atmosphere with a respective PWV of 0.7, 1.4, and 2.8 cm and a solar zenith angle of 70 degrees. The second expression on the righthand side of Eq. (16b) is an estimate for the situation in which the FWHM is 10 nm and the central wavelength is 940 nm using the extraterrestrial solar irradiance reported by Coddington et al. (2021). Moreover, the residual error (\({\varepsilon }_{\widehat{R},\mathrm{RT}}\)) of the convolved normalized radiance (\(\widehat{R}\)) with the stepwise filter response function is obtained as follows:
Using Eqs. (10) and (15), \({\varepsilon }_{\widehat{I}\mathrm{max}}^{2}\) is estimated as \({\varepsilon }_{\widehat{I}\mathrm{max}}^{2}\sim {D}^{2}{\Phi }^{2}<{\varepsilon }_{\overline{I}\mathrm{max} }^{2}>\sim {\left(D\Phi {\upvarepsilon }_{E}\right)}^{2}\) and the expected residual errors of radiances simulated with the WVCKD are estimated as follows:
3.2 Evaluation of the WVCKD
To evaluate sky radiance at ground level in detail, we constructed a finescale LUT of the k distribution (\({\Delta }_{\kappa }=2, 5 {\mathrm{cm}}^{1}\); hereinafter, WVCKD2 and WVCKD5, respectively) in the manner described in Sec. 2.3; in this case, \(\sqrt{{\varepsilon }_{E}^{2}}\) was satisfied at values below \(11.2\times {10}^{3}\) and \(7.1\times {10}^{3}\) to maintain accuracy of the \(10 {\mathrm{cm}}^{1}\) bandaveraged sky intensities below \(5.0\times {10}^{3}\) (equal to that of the WVCKD10). The details of the WVCKD2 and WVCKD5 are summarized in Table 2. The quadrature numbers of the WVCKD2 and WVCKD5 fall into the ranges of 1–9 and 2–11, respectively. The median quadrature numbers of the WVCKD2 and WVCKD5 are 4 and 6, respectively. Therefore, using all of the WVCKD developed here allows for ≥ 46fold more rapid calculation than the LBL method. The simulation (using a dataset consisting of sky intensities at ground level under several aerosol conditions from Dubovik et al. 2000; hereinafter, DUDB) was conducted for the two aerosol types reported by Dubovik et al. (2000) and two atmospheric profiles, described in Table 3. The atmospheric profiles used represented the SKYNET Chiba site (35.63°N, 140.10°E) and were obtained from National Centers for Environmental Prediction (NCEP) reanalysis 1 data for 2018 (Fig. 5). The sky intensities were convolved using filter response functions for three Gaussian shapes (FWHM: 5, 10, 15 nm) at two central wavelengths of 936 and 940 nm (Fig. 6).
3.2.1 Comparison with convolved sky intensities obtained using the stepwise filter response function
\({\widehat{T}}^{\mathrm{LBL}}\), \({\widehat{L}}^{\mathrm{LBL}}\), and \({\widehat{R}}^{\mathrm{LBL}}\) assume that the extraterrestrial solar irradiance and filter response function are nearly constant in each subband. Therefore, this section aims to evaluate whether the WVCKD can be used for aerosol and atmospheric vertical profiles other than the six AFGL standard atmospheres with continental averaged aerosols.
Table 4 summarizes the maximum residuals obtained between the LBL and CKD methods. The expected values (\(D\Phi \sqrt{{\varepsilon }_{E}^{2}}\)) estimated using Eq. (15) in each simulation are also presented in Table 4. The \(\sqrt{{\varepsilon }_{\widehat{I}\mathrm{max}}^{2}}\) values obtained using all WVCKD were smaller than \(1.64\times {10}^{3}\) and similar to the expected residual errors (\(D\Phi \sqrt{{\varepsilon }_{E}^{2}}=2.0\times {10}^{3},1.4\times {10}^{3}\) and \(1.1\times {10}^{3}\) at FWHM of 5, 10, and 15 nm, respectively). In some cases, the residual errors exceed \(D\Phi \sqrt{{\varepsilon }_{E}^{2}}\), but fall within two standard deviations (\(2D\Phi \sqrt{{\varepsilon }_{E}^{2}}\)) due to fluctuations in transmittance [Eq. (16a)]. This suggests that the WVCKD can be used under conditions other than the six AFGL standard atmospheres with continental averaged aerosols; moreover, the expected residual error (\(D\Phi \sqrt{{\varepsilon }_{E}^{2}}\)) is a useful benchmark of \({\widehat{T}}^{\mathrm{CKD}}\) and \({\widehat{L}}^{\mathrm{CKD}}\). Additionally, the \(\sqrt{{\varepsilon }_{\widehat{R}\mathrm{max}}^{2}}\) values were smaller than \(3.97\times {10}^{4}\) and much smaller than \(\sqrt{{\varepsilon }_{\widehat{I}\mathrm{max}}^{2}}\). This difference arises because the normalized radiance cancels the residual error of \({\widehat{T}}^{\mathrm{CKD}}\) and \({\widehat{L}}^{\mathrm{CKD}}\), as shown in Eq. (17a). In contrast, the \(\sqrt{{\varepsilon }_{\widehat{I}\mathrm{max}}^{2}}\) values obtained using the SNCKD were much larger, ranging from \(7.53\times {10}^{3}\) to \(2.21\times {10}^{1}\), and were not negligible relative to the uncertainty of skyradiometer observations. Thus, the \(\sqrt{{\varepsilon }_{\widehat{R}\mathrm{max}}^{2}}\) of the SNCKD is better than \(\sqrt{{\varepsilon }_{\widehat{I}\mathrm{max}}^{2}}\), but reaches \(6.71\times {10}^{2}\) at the FWHM of 5 nm and \(3.89\times {10}^{2}\) at an FWHM of 10 nm.
3.2.2 Comparison with convolved sky intensities obtained using a smooth filter response function
\({\tilde{T }}^{\mathrm{LBL}}\), \({\tilde{L }}^{\mathrm{LBL}}\), and \({\tilde{R }}^{\mathrm{LBL}}\) are calculated using the LBL method and convolved with a smooth filter response function and the highresolution (\(0.01 {\mathrm{cm}}^{1}\)) extraterrestrial solar irradiance data from Coddington et al. (2021). This section provides a comprehensive assessment of \({\widehat{T}}^{\mathrm{CKD}}\), \({\widehat{L}}^{\mathrm{CKD}}\), and \({\widehat{R}}^{\mathrm{CKD}}\) in addition to the general performance of gas absorption discussed in Sect. 3.2.1. The residual errors of the convolved sky intensities are defined as follows:
where \({\varepsilon }_{\tilde{T },\mathrm{FRF}}^{2}\) and \({\varepsilon }_{\tilde{L },\mathrm{FRF}}^{2}\) are the residual errors arising from the assumptions of the extraterrestrial solar irradiance and filter response function. The maximum residuals can be obtained as follows:
Figure 7 shows the angular distribution of the convolved normalized radiances simulated for type 2 DUDB. Although the convolved \(\tilde{R }\) value obtained with the SNCKD had large errors in backward scattering and the zenith region (approximately 1.5%), convolved \(\tilde{R }\) values from the WVCKDs showed better performance (< 0.1%). Table 5 summarizes \({\varepsilon }_{\tilde{I }\mathrm{max}}^{2}\) and \({\varepsilon }_{\tilde{R }\mathrm{max}}^{2}\) for the simulation using DUDB. Although a finer \({\Delta }_{\kappa }\) is effective for assessing the shape of the filter response function, Table 5 does not follow this trend. This discrepancy arises because extraterrestrial solar irradiance has strong wavelength dependence (Coddington et al. 2021), so it might affect performance more strongly than the shape of the response function. In that case, the residual errors obtained with the WVCKD2 and WVCKD5 are within \(3\times {10}^{3}\). In the case of an FWHM of 5 nm, the residual errors from the WVCKD10 are significantly large (> \(1\times {10}^{2}\)) due to the assumption of the stepwise function. In contrast, \({\widehat{R}}^{\mathrm{CKD}}\) was less strongly affected than \({\widehat{T}}^{\mathrm{CKD}}\) and \({\widehat{L}}^{\mathrm{CKD}}\) (\(\sqrt{{\varepsilon }_{\tilde{R }\mathrm{max}}^{2}}\le 1.1\times {10}^{3}\)), in accordance with the relationship between \({\varepsilon }_{\widehat{R}\mathrm{max}}^{2}\) and \({\varepsilon }_{\widehat{I}\mathrm{max}}^{2}\) described in Sect. 3.2.1. This finding indicates an advantage of skyradiometer observations, as Momoi et al. (2020) proposed estimation of PWV from the angular distribution of \(\tilde{R }\). With an FWHM of 10 nm, corresponding to the skyradiometer specification, the \(\sqrt{{\varepsilon }_{\tilde{I }\mathrm{max}}^{2}}\) values of the WVCKD2, WVCKD5, and WVCKD10 are less than \(3.2\times {10}^{3}\), \(1.3\times {10}^{3}\), and \(4.0\times {10}^{3}\), respectively. These values are significantly smaller than \(\sqrt{{\varepsilon }_{\tilde{I }\mathrm{max}}^{2}}\) obtained using the SNCKD (\(\le 1.5\times {10}^{1}\)). In comparison, the \(\sqrt{{\varepsilon }_{\tilde{R }\mathrm{max}}^{2}}\) values of all WVCKDs reach \(1.1\times {10}^{3}\). In conclusion, this simulation suggests that the WVCKD5 is useful for computation of sky radiances (direct solar irradiances and normalized radiances) based on skyradiometer observations. Moreover, this process is approximately 75fold more rapid than the LBL method.
3.3 Relationship between convolved normalized radiances around 940 nm and PWV
Momoi et al. (2020) investigated the relationships among \(\tilde{R }\) around 940 nm, PWV, and the vertical profiles of aerosols and reported two major findings. First, \(\tilde{R }\) depends on PWV in both almucantar and principal planes, with \(\tilde{R }\) in the principal plane being more strongly dependent on PWV. Second, \(\tilde{R }\) in the principal plane depends on the vertical aerosol profile, whereas \(\tilde{R }\) in the almucantar plane is nearly independent of the vertical aerosol profile. Because Momoi et al. (2020) used the SNCKD, we repeated their analysis with the WVCKD2. In this section, we describe sensitivity tests conducted under two aerosol conditions with US standard atmospheres, as described by Momoi et al. (2020): the continental average and the continental average + transported dust in the upper atmosphere (Table 6). All continental average aerosols were assumed to be spherical, and the dust aerosols were assumed to be spheroid, which differed from the assumption used in Momoi et al. (2020). The spheroid particles used here are the kernels developed by Dubovik et al. (2006) with an aspect ratio set to 0.6, representing the yellow sand particles reported by Nakajima et al. (1989). The radiances were convolved with a filter response function of Gaussian shape with an FWHM of 10 nm and central wavelength of 940 nm. Note that the filter response function also differed from the function described by Momoi et al. (2020).
Figure 8 shows the results of sensitivity tests conducted with the SNCKD and WVCKD2. The results obtained with the SNCKD are similar to those described by Momoi et al. (2020). Although the magnitude of \(\tilde{R }\) differed from the value reported by Momoi et al. (2020), the relationship between \(\tilde{R }\) around 940 nm and PWV was consistent with their findings. Other sensitivity tests (Figs. 4–6 in Momoi et al. 2020) had the same characteristic.
3.4 PWV estimation from skyradiometer observations
3.4.1 Brief outline of the method proposed by Momoi et al. (2020)
According to Momoi et al. (2020), the calibration constants for the water vapor band (940 nm) can be determined using \(\tilde{T }\) derived from the angular distribution of \(\tilde{R }\), which is referred to as the SKYMAP algorithm. The SKYMAP algorithm includes the following three steps (Fig. 9). Aerosol properties (aerosol size distribution, complex refractive index, and sphericity) are estimated from the angular distribution of \(\tilde{R }\) in the range of 340–1020 nm, except at 940 nm; \(\tilde{T }\) at 940 nm is estimated from the angular distribution of \(\tilde{R }\) at 940 nm using aerosol properties interpolated from aerosol optical properties at 870 and 1020 nm; finally, \({\tilde{F }}_{0}\) at 940 nm is determined from \(\tilde{T }\) at 940 nm. After calibration, PWV can be estimated from \({\tilde{F }}_{\mathrm{ds}}\) using \({\tilde{F }}_{0}\) through a physicsbased algorithm (DSRAD; Momoi et al. 2020; Fig. 9), rather than through an empirical equation (Bruegge et al. 1992). In this section, we assessed the influence of the LUT of the k distribution on PWV estimation from actual skyradiometer observations through the SKYMAP/DSRAD algorithm using the WVCKD2.
3.4.2 PWV estimation from actual skyradiometer observations
In this subsection, we describe the impact of introducing the new CKD table (WVCKD) into the analysis of actual observation data obtained with the skyradiometer (POM02, serial no. PS2501417) at Chiba University (35.63°N, 140.10°E) in 2019. PWV reference values were obtained with a microwave radiometer (MWR; MP1500; Radiometrix) at the same location. The MWR measured the zenith brightness temperature in the 22–30 GHz region at 1min temporal resolution, and PWV_{MWR} was estimated using the default software. Using the SKYMAP and DSRAD algorithms, aerosol optical properties and PWV were estimated from the data at 400, 500, 675, 870, 940, and 1020 nm, along with the vertical structures of temperature, pressure, and water vapor from NCEP reanalysis 1 data. Sky intensities at 340 and 380 nm were also measured, but were not used in the present analysis for the sake of simplicity, as these data do not significantly affect PWV retrieval. Note that the 6 wavelengths used in the present analysis are those implemented in the old skyradiometer model POM01. In the case of the SKYMAP with WVCKD2, the annual mean \({\tilde{F }}_{0}\) at 940 nm was retrieved as \(2.079\times {10}^{4}\) A, which is 7.5% greater than the value determined with the SNCKD (\(1.933\times {10}^{4}\) A). Using \({\tilde{F }}_{0}\), the PWV was estimated with the DSRAD, as shown in Fig. 10. Figure 11a illustrates that PWV_{SNCKD} derived using the SNCKD was underestimated relative to PWV_{MWR}. This underestimation was significant, as bias of approximately –0.3 cm occurred in July and August (Fig. 10b), when PWV was higher (4–6 cm) than in other seasons, as shown in Fig. 10a. This result indicates a similar error tendency to the findings of Momoi et al. (2020). In contrast, PWV_{WVCKD} derived using the WVCKD2 showed good agreement with PWV_{MWR} (correlation coefficient γ = 0.995, slope = 1.002; Fig. 11b), even in July and August (Fig. 10b). Momoi et al. (2020) attributed the large bias error in the PWV to error in the AOT at 940 nm propagated through interpolation of AOT at 870 and 1020 nm. Figure 10c indicates that the uncertainties in observed AOT values at 675, 870, and 1020 nm from the collocated skyradiometer and AERONET photometer reached 0.01, corresponding to an absolute PWV error of about 0.08 cm calculated using the equation of Momoi et al. (2020). In contrast, the PWV_{SNCKD} errors larger than 0.08 cm shown in Fig. 10b are considered to represent the errors in PWV arising from the SNCKD, rather than from the uncertainty in AOT estimation. This suggests that SKYMAP/DSRAD calculation with the WVCKD is useful for accurate estimation of PWV and determination of the calibration constant \({\tilde{F }}_{0}\) for the water vapor absorption band of skyradiometer observations.
4 Conclusions
To compute direct solar irradiance and diffuse radiance at ground level around 940 nm with narrowband RTMs, we developed a rapid calculation module using a LUT (WVCKD). We found the challenges facing the currently used LUT (SNCKD) in RSTAR7 for narrowband sky radiance computation around 940 nm region. As shown by comparison of the sky intensities of subbands obtained with the LBL method and CKD method using the SNCKD, the root mean square error of the maximum error in subbands obtained with the SNCKD is 116% across the range of 10,411–10,864 \({\mathrm{cm}}^{1}\) [961–920 nm]. This large error may arise from the updated database and lack of optimization. Our WVCKDs were created at three different spectral resolutions (\({\Delta }_{\kappa }=2, 5, 10 {\mathrm{cm}}^{1}\) in WVCKD2, WVCKD5, and WVCKD10, respectively). The quadrature values and numbers of these WVCKDs were optimized using sky intensities based on the singlescattering approximation at ground level, with an accuracy of ≤ 0.5% for subbands of \(10 {\mathrm{cm}}^{1}\). The quadrature numbers affected computational efficiency. The median quadrature numbers of WVCKD2, WVCKD5, and WVCKD10 are 4, 6, and 7, respectively; their calculations were ≥ 46fold more rapid than the LBL method.
Radiance calculation with the WVCKD was evaluated for two aerosol types and four vertical profiles that differed from the conditions used for optimization of the tables. The residual errors of convolved sky intensities (\({\widehat{T}}^{\mathrm{CKD}}\) and \({\widehat{L}}^{\mathrm{CKD}}\)) were similar to the expected errors observed during optimization of quadrature values and numbers. This finding suggests that the WVCKD has sufficient versatility for application under actual atmospheric conditions. The convolved normalized radiance (\({\widehat{R}}^{\mathrm{CKD}}\)) was less strongly affected by residual errors obtained using the WVCKD than \({\widehat{T}}^{\mathrm{CKD}}\) and \({\widehat{L}}^{\mathrm{CKD}}\), as \({\widehat{R}}^{\mathrm{CKD}}\) cancels the residual errors of both \({\widehat{T}}^{\mathrm{CKD}}\) and \({\widehat{L}}^{\mathrm{CKD}}\). Additionally, while the error of convolved sky intensities obtained with the SNCKD is within 15%, use of WVCKD provides convolved sky intensities with an accuracy of ≤ 0.3% at an FWHM of 10 nm, equal to the FWHM of the skyradiometer. This accuracy is lower than the PWV dependence on \(\tilde{R }\) and measurement uncertainty (approximately 0.5%). Finally, we applied the SKYMAP and DSRAD algorithms (Momoi et al. 2020) to SKYNET observations and compared the results with those of the microwave radiometer. PWV shows better agreement when derived with the WVCKD (correlation coefficient γ = 0.995, slope = 1.002) than with the SNCKD (correlation coefficient γ = 0.984, slope = 0.926), as used by Momoi et al. (2020). Therefore, through application of the WVCKD to actual data analysis, we demonstrated that an accurate CKD table is essential for estimating PWV from skyradiometer observations.
Availability of data and materials
RSTAR and PSTAR are available from the OpenCLASTR project website (http://157.82.240.167/~clastr/, last accessed: Oct. 2021). The LUTs around 940 nm (WVCKDs) and gas absorption calculation codes for RSTAR7 and PSTAR4 are available on request from the first author. The SKYMAP and DSRAD algorithms are also available on request from the first author. The skyradiometer data are available from the SKYNET website (http://www.skynetisdc.org/, last accessed: Oct. 2021). The MWR data are available from CEReS, Chiba University (http://atmos3.cr.chibau.jp/skynet/, last accessed: Oct. 2021). The AERONET sun–sky radiometer data are available from the AERONET website (https://aeronet.gsfc.nasa.gov/, last access: Oct. 2021).
Abbreviations
 RTM:

Radiative transfer model
 LUT:

Lookup table
 RSTAR:

System for Transfer of Atmospheric Radiation for Radiance calculations
 PSTAR:

System for Transfer of Atmospheric Radiation for Polarized radiance calculations
 CKD:

Correlated kdistribution
 SNCKD:

Standard lookup table for the correlated kdistribution method in RSTAR
 RSTAR7:

RSTAR version 7
 PSTAR4:

PSTAR version 4
 PWV:

Precipitable water vapor
 AERONET:

Aerosol robotic network
 LBL:

Linebyline
 AFGL:

Air Force Geophysics Laboratory
 WVCKD:

Lookup table for the correlated kdistribution method in the water vapor absorption region of 940 nm
 CADB:

Dataset consisting of radiances at ground level under continental averaged aerosol condition
 AOT: aerosol optical thickness, FWHM:

Full width at half maximum
 DUDB:

Dataset consisting of the radiances at ground level under several aerosol conditions from Dubovik et al. (2000)
 MWR:

Microwave radiometer
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Acknowledgements
We are grateful to the OpenCLASTR project (http://157.82.240.167/~clastr/, last accessed: Oct. 2021) for allowing us to use SKYRAD.pack version 4.2 (skyradiometer analysis package) and RSTAR7 (System for Transfer of Atmospheric Radiation for Radiance calculations) in this study. NCEP reanalysis 1 data were obtained from the website of the National Oceanic & Atmospheric Administration (NOAA) Earth System Research Laboratory (ESRL) Physical Sciences Division (PSD; Boulder, CO, USA) at http://www.esrl.noaa.gov/psd/ (last accessed: Oct. 2021).
Funding
This study was partly supported by the Environment Research and Technology Development Fund (JPMEERF20192001 and JPMEERF20215005) of the Environmental Restoration and Conservation Agency of Japan, JSPS KAKENHI (Grant Numbers JP19H04235, JP20H04320, and JP21K12227), the JAXA 2nd research announcement on the Earth Observations (Grant Number 19RT000351), a Tokyo University of Science Joint Research Grant (Representative Kazuhiko Miura, 20132014), and JAXA Contract research (“Fundamental development of radiative transfer codes for advanced utilization of earth observation satellites,” FY2021).
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This study was designed by MM and HI. The gas absorption calculation modules for RSTAR7 and the WVCKDs were developed by MM, TN, and MS. The skyradiometer data were analyzed by MM, HI, KM, and KA. The skyradiometer (S/N PS2501401) was maintained by MM, KM, and KA. The skyradiometer (S/N PS2501417) and microwave radiometer were maintained by HI. The highperformance computers used for analysis were operated and maintained by MM and HT. The manuscript was written by MM, and all authors contributed to its editing and revision. All authors read and approved the final manuscript.
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Momoi, M., Irie, H., Sekiguchi, M. et al. Rapid, accurate computation of narrowband sky radiance in the 940 nm gas absorption region using the correlated kdistribution method for sunphotometer observations. Prog Earth Planet Sci 9, 10 (2022). https://doi.org/10.1186/s40645022004676
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DOI: https://doi.org/10.1186/s40645022004676