Rapid, accurate computation of narrow-band sky radiance in the 940 nm gas absorption region using the correlated k-distribution method for sun-photometer observations

Abstract

We developed lookup tables for the correlated k-distribution (CKD) method in the 940 nm water vapor absorption region (WV-CKD), with the aim of rapid and accurate computation of narrow-band radiation around 940 nm (10,000–10,900 ${\mathrm{cm}}^{-1}$) for ground-based angular-scanning 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 sub-bands of $10 {\mathrm{cm}}^{-1}$. Although high-resolution WV-CKD 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 line-by-line approach by our WV-CKD with a resolution of 2 ${\mathrm{cm}}^{-1}$. Furthermore, we confirmed through several simulations that WV-CKD 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 WV-CKD, 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 WV-CKD is < 0.11%. This accuracy is lower than the observational uncertainty of a ground-based angular-scanning 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/amt-13-2635-2020) to SKYNET observations (Chiba, Japan) and compared the results with microwave radiometer values. The precipitable water vapor (PWV) derived with the WV-CKD 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 WV-CKD to actual data analysis, we found that an accurate CKD table is essential for estimating PWV from sky-radiometer 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 satellite-based instruments and estimated using the multi-term 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 angular-scanning 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 satellite-based 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 ground-based angular-scanning radiometer data with high degrees of freedom for control variables. Several RTMs have been used for analysis of ground-based remote sensing data. For example, the AErosol RObotic NETwork (AERONET; Holben et al. 1998), which is an international network of ground-based angular-scanning 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 ground-based angular-scanning 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 Pⁿ-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 ground-based 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 multi-species gas-absorbing broadband radiative flux using the method of Sekiguchi and Nakajima (2008), which is a nonlinearly optimized version of the correlated k-distribution (CKD) method (Lacis and Oinas 1991; Fu and Liou 1992). For narrow-band gas absorption calculation, RSTAR uses the CKD method for which the standard LUT (hereafter, SN-CKD) was designed mechanically with two Gaussian quadrature points, as described in Sect. 2.2. Based on RSTAR with the SN-CKD, 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 multi-term 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 angular-scanning radiometer called sky-radiometer (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 line-by-line (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 long-term observations because a calibration constant can be determined from the PWV data derived from the on-site 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 (WV-CKD), installed the WV-CKD 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 ground-based angular-scanning radiometer observations in the water vapor absorption band. The WV-CKD 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 WV-CKD, and Sect. 3 provides the results of simulations used to estimate the accuracy of the WV-CKD and its impact on PWV estimation from actual SKYNET sky-radiometer observations.

2 Methods/experimental design

Section 2.1 describes the RTM calculation accuracy required to analyze sky-radiometer observations. Section 2.2 examines the challenges facing the currently used LUT of the k distribution (SN-CKD) in RSTAR version 7 (RSTAR7) for narrow-band 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 sky-radiometer 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 sky-radiometer. Section 2.1.2 describes the uncertainty of sky-radiometer observations.

2.1.1 Measurement of sky irradiance

The POM-01 sky-radiometer 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, POM-02, 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 plane-parallel non-refractive atmosphere as follows:

$${F}_{\mathrm{ds}}\left(\lambda \right)=\frac{{F}_{0}\left(\lambda \right)}{{d}^{2}}T\left(\lambda \right),$$

(1a)

$${F}_{\mathrm{df}}\left(\Theta ,\lambda \right)=\frac{{F}_{0}\left(\lambda \right)\mathrm{\Delta \Omega }}{\left|\mu \right|{d}^{2}}L\left(\Theta ,\lambda \right),$$

(1b)

$${F}_{0}\left(\lambda \right)\propto {F}_{\mathrm{sol}}\left(\lambda \right),$$

(1c)

$$T\left(\lambda \right)\equiv \mathrm{exp}\left(-\frac{\tau \left(\lambda \right)}{\left|{\mu }_{0}\right|}\right),$$

(2a)

$$L\left(\Theta ,\lambda \right)\equiv T\left(\lambda \right)\left\{{\int }_{0}^{\tau \left(\uplambda \right)}\mathrm{exp}\left[\left(\tau -t\right)\left(\frac{1}{\left|{\mu }_{0}\right|}-\frac{1}{\left|\mu \right|}\right)\right]{\omega }^{^{\prime}}\left(\lambda ,t\right){P}^{^{\prime}}\left(\Theta ,\uplambda ,t\right)\mathrm{d}t+q\left(\Theta ,\uplambda \right)\right\},$$

(2b)

$$R\left(\Theta ,\lambda \right)\equiv \frac{\left|\mu \right|{F}_{\mathrm{df}}\left(\Theta ;\lambda \right)}{{F}_{\mathrm{ds}}\left(\lambda \right)\mathrm{\Delta \Omega }}=\frac{L\left(\Theta ,\lambda \right)}{T\left(\lambda \right)},$$

(2c)

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:

$$L\left(\Theta ,\lambda \right)=\omega \tau P\left(\Theta ,\uplambda \right)E\left(\mu ,\lambda \right)+q\left(\Theta ,\uplambda \right)T\left(\lambda \right),$$

(3a)

$$E\left(\mu ,\lambda \right)\equiv \left\{\begin{array}{cc}T\left(\lambda \right)& \mu ={\mu }_{0}\\ {\left[\left(\frac{1}{\left|{\mu }_{0}\right|}-\frac{1}{\left|\mu \right|}\right)\tau \left(\lambda \right)\right]}^{-1}\left[\mathrm{exp}\left(-\frac{\tau \left(\lambda \right)}{\left|\mu \right|}\right)-\mathrm{exp}\left(-\frac{\tau \left(\lambda \right)}{\left|{\mu }_{0}\right|}\right)\right]& \mu \ne {\mu }_{0}\end{array}\right..$$

(3b)

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:

$${\tilde{F }}_{\mathrm{ds}}\left(\lambda \right)=\int \phi \left(\lambda \right){F}_{\mathrm{ds}}\left(\lambda \right)\mathrm{d}\lambda =\frac{1}{{d}^{2}}\int \phi \left(\lambda \right){F}_{0}\left(\lambda \right)T\left(\lambda \right)\mathrm{d}\lambda ,$$

(4a)

$${\tilde{F }}_{\mathrm{df}}\left(\lambda \right)=\int \phi \left(\lambda \right){F}_{\mathrm{df}}\left(\lambda \right)\mathrm{d}\lambda =\frac{\mathrm{\Delta \Omega }}{\left|\mu \right|{d}^{2}}\int \phi \left(\lambda \right){F}_{0}\left(\lambda \right)L\left(\Theta ,\lambda \right)\mathrm{d}\lambda ,$$

(4b)

$${\tilde{F }}_{0}\left(\lambda \right)=\int \phi \left(\lambda \right){F}_{0}\left(\lambda \right)\mathrm{d}\lambda \propto \int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)\mathrm{d}\lambda .$$

(4c)

Hence, the convolved sky intensities ($\tilde{T }$ and $\tilde{L }$) and convolved normalized radiance ($\tilde{R }$) are defined as follows:

$$\tilde{T }\left(\lambda \right)=\frac{{\tilde{F }}_{\mathrm{ds}}\left(\lambda \right){d}^{2}}{{\tilde{F }}_{0}\left(\lambda \right)}=\frac{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)T\left(\lambda \right)\mathrm{d}\lambda }{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)\mathrm{d}\lambda },$$

(5a)

$$\tilde{L }\left(\Theta ,\lambda \right)=\frac{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)L\left(\Theta ,\lambda \right)\mathrm{d}\lambda }{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)\mathrm{d}\lambda }.$$

(5b)

$$\tilde{R }\left(\Theta ,\lambda \right)\equiv \frac{\tilde{L }\left(\Theta ,\lambda \right)}{\tilde{T }\left(\lambda \right)}=\frac{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)L\left(\Theta ,\lambda \right)\mathrm{d}\lambda }{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)T\left(\lambda \right)\mathrm{d}\lambda },$$

(5c)

Although sky radiance in weak gas absorption regions, such as at 340, 380, 400, 500, 675, 870, and 1020 nm in sky-radiometer observations, can be assumed to represent narrow spectral bands (Eq. [2]), sky radiance in gas absorption regions, such as the 940 nm band in sky-radiometer observations, requires convolution (Eq. [5]).

2.1.2 Uncertainty of sky-radiometer observations

In the SKYNET analysis algorithms (e.g., Momoi et al. 2020; Kudo et al. 2021) used with sky-radiometer 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:

$${\sigma }_{\tilde{T }}^{2}={\sigma }_{{\tilde{F }}_{\mathrm{ds}}}^{2}+{\sigma }_{{\tilde{F }}_{0}}^{2},$$

(6a)

$${\sigma }_{\tilde{R }}^{2}={\sigma }_{{\tilde{F }}_{\mathrm{df}}}^{2}+{\sigma }_{{\tilde{F }}_{\mathrm{ds}}}^{2},$$

(6b)

where ${\sigma }_{{\tilde{F }}_{0}}$ is the standard deviation of the sky-radiometer 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 sky-radiometer 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 Gaussian-filtered data, as follows:

$${\Delta }_{f}=\mathrm{ln}{f}^{\mathrm{obs}}\left(\Theta \right)-\mathrm{ln}{f}^{\mathrm{gf}}\left(\Theta \right),$$

(7)

where ${f}^{\mathrm{obs}}$ and ${f}^{\mathrm{gf}}$ are the observed and Gaussian-filtered 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 sky-radiometer POM-02 (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 sky-radiometer 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.

Fig. 1

Example solar aureole angular distributions of the sky-radiometer at 500 nm obtained through the solar disk scan method on February 27, 2020, at Akiruno, Japan. Blue dots are measured values and red dots show the angular distribution with noise reduction using a Gaussian filter

Fig. 2

Example histogram of the differences at 340, 500, and 940 nm of the sky-radiometer measurements on February 27 and 28, 2020 in Akiruno, Japan

Table 1 Uncertainty of sky irradiance based on comparison between $f^{{{\text{obs}}}}$ and $f^{{{\text{gf}}}}$

2.2 Challenges regarding the k-distribution lookup table in RSTAR

In RSTAR7, gas absorption is considered for H₂O, CO₂, O₃, N₂O, CO, CH₄, and O₂ 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; SN-CKD), which generates two Gaussian quadrature points without optimization in each sub-band. The resolution of the SN-CKD 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., sky-radiometer measurements at 340, 380, 400, 500, 675, 870, and 1020 nm), the radiance can be accurately computed using the SN-CKD 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 SN-CKD around 940 nm (10,000–10,900 ${\mathrm{cm}}^{-1}$ [1000–917 nm]) through the LBL method using simulated band-averaged sky intensities for sub-bands ($\overline{T }=\frac{{\overline{F} }_{\mathrm{ds}}}{{\overline{F} }_{0}}$ 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 band-averaged sky intensities for sub-bands were computed using RSTAR7 with the IMS method (Nakajima and Tanaka 1988). The validation dataset (CA-DB) consisted of band-averaged 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 sky-radiometer 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 sub-band (${\mathrm{dlog}}_{10}k$) level. Therefore, with the LBL method, $\overline{T }$ and $\overline{L }$ are expressed as:

$${\overline{T} }^{\mathrm{LBL}}\left(\overline{\lambda }\right)=\frac{1}{{\int }_{{\Delta }_{\kappa }}\frac{\mathrm{d}\kappa }{{\kappa }^{2}}}{\int }_{{\Delta }_{\kappa }}{T}_{\mathrm{A}}\left(\lambda \right){T}_{\mathrm{R}}\left(\lambda \right){T}_{\mathrm{H}2\mathrm{O}}\left(\lambda \right){T}_{\mathrm{cont}}\left(\lambda \right)\frac{\mathrm{d}\kappa }{{\kappa }^{2}},$$

(8a)

$${\overline{L} }^{\mathrm{LBL}}\left(\Theta ,\overline{\lambda }\right)=\frac{1}{{\int }_{{\Delta }_{\kappa }}\frac{\mathrm{d}\kappa }{{\kappa }^{2}}}{\int }_{{\Delta }_{\kappa }}{T}_{\mathrm{A}}\left(\lambda \right){T}_{\mathrm{R}}\left(\lambda \right){T}_{\mathrm{H}2\mathrm{O}}\left(\lambda \right){T}_{\mathrm{cont}}\left(\lambda \right)R\left(\Theta ,\lambda \right)\frac{\mathrm{d}\kappa }{{\kappa }^{2}},$$

(8b)

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 self-continuum); and ${T}_{\mathrm{cont}}$ is the monochromatic transmittance of the O₂ and O₃ 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:

$$\mathrm{ln}\left[{T}_{\mathrm{H}2\mathrm{O}}\left(\lambda \right)\right]=-{m}_{0}\left({\theta }_{0}\right){\int }_{0}^{z}\left[{\sigma }_{\mathrm{H}2\mathrm{O},\mathrm{line}}\left(K,p,\lambda \right)+{\sigma }_{\mathrm{H}2\mathrm{O},\mathrm{cont}}\left(K,\lambda \right)\right]dz,$$

(9)

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 self-continuum 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 SN-CKD compared with the LBL method, which were determined as:

$${\varepsilon }_{\overline{T },\mathrm{RT}}^{2}\left(\overline{\lambda };{\theta }_{0}, w,{\tau }_{a}\right)={\left(\mathrm{ln}\frac{{\overline{T} }^{\mathrm{LBL}}}{{\overline{T} }^{\mathrm{CKD}}}\right)}^{2},$$

(10a)

$${\varepsilon }_{\overline{L },\mathrm{RT}}^{2}\left(\overline{\lambda };\Theta ,{\theta }_{0}, w,{\tau }_{a}\right)={\left(\mathrm{ln}\frac{{\overline{L} }^{\mathrm{LBL}}}{{\overline{L} }^{\mathrm{CKD}}}\right)}^{2},$$

(10b)

$${\varepsilon }_{\overline{I}\mathrm{max} }^{2}\left(\overline{\lambda }\right)=\mathrm{max}\left\{{\varepsilon }_{\overline{T },\mathrm{RT}}^{2}\left(\overline{\lambda }\right),{\varepsilon }_{\overline{L },\mathrm{RT}}^{2}\left(\overline{\lambda }\right)|{\theta }_{0}=30 \left(20\right) {70}^{\mathrm{o}};w=0.5 \left(0.5 \right) 6 \mathrm{cm};{\tau }_{a}=0.05, 0.2, 1.0\right\}.$$

(10c)

The root mean square errors of ${\varepsilon }_{\overline{I}\mathrm{max} }$ for CA-DB 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 sky-radiometer’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.

Fig. 3

The maximum error of the SN-CKD with CA-DB. SN-CKD is the standard lookup table for the CKD method in the radiative transfer model RSTAR. CA-DB is dataset consisting of radiances at ground level under continental averaged aerosol condition of Hess et al. (1999)

2.3 Method used to create the new k-distribution 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 WV-CKD-10) 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 O₂ continuum, O₃ continuum, and water vapor line and self-continuum absorptions. Thus, WV-CKD-10 consists of three LUTs: quadrature weights, water vapor k distribution, and O₂ and O₃ 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, WV-CKD 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, mid-latitude summer, mid-latitude winter, high-latitude summer, and high-latitude 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 WV-CKD-10, we validated it with CA-DB, as described in the previous subsection for the validation dataset of the SN-CKD.

3 Results and discussion

3.1 Optimized k-distribution 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 WV-CKD-10 reduces the number of executions of the RTM to approximately 1/100 of the number needed for the LBL method, as the band-averaged 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).

Fig. 4

Architecture of the WV-CKD-10. The top panel (a) shows the number of quadrature points in the sub-band. The bottom panel (b) shows the maximum residual error relative to the LBL method using CA-DB. WV-CKD-10 is a lookup table for the CKD method in the water vapor absorption region of 940 nm at 10-${\mathrm{cm}}^{-1}$ intervals. CA-DB is dataset consisting of radiances at ground level under continental averaged aerosol condition of Hess et al. (1999)

Table 2 Description of the WV-CKDs. WV-CKD is a series of lookup tables for the CKD method in the water vapor absorption region of 940 nm at 2-, 5-, and 10-${\text{cm}}^{ - 1}$ intervals

The equations for band-averaged sky intensities differ markedly between the LBL and CKD methods. ${\overline{T} }^{\mathrm{CKD}}$ and ${\overline{L} }^{\mathrm{CKD}}$, derived using the WV-CKD, are as follows:

$${\overline{T} }^{\mathrm{CKD}}\left(\overline{\lambda }\right)={\overline{T} }_{\mathrm{A}}\left(\overline{\lambda }\right){\overline{T} }_{\mathrm{R}}\left(\overline{\lambda }\right){\overline{T} }_{\mathrm{cont}}\left(\overline{\lambda }\right){\sum }_{i=1}^{{N}_{\mathrm{ch}}}{\xi }_{i}{T}_{\mathrm{H}2\mathrm{O},\mathrm{ckd}}^{\left(i\right)}\left(\overline{\lambda }\right),$$

(11a)

$${\overline{L} }^{\mathrm{CKD}}\left(\Theta ,\overline{\lambda }\right)={\overline{T} }_{\mathrm{A}}\left(\overline{\lambda }\right){\overline{T} }_{\mathrm{R}}\left(\overline{\lambda }\right){\overline{T} }_{\mathrm{cont}}\left(\overline{\lambda }\right){\sum }_{i=1}^{{N}_{\mathrm{ch}}}{\xi }_{i}{T}_{\mathrm{H}2\mathrm{O},\mathrm{ckd}}^{\left(i\right)}\left(\overline{\lambda }\right){R}^{(i)}\left(\Theta ,\lambda \right),$$

(11b)

where ${\overline{T} }_{A}$ and ${\overline{T} }_{R}$ are the band-averaged 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:

$${\sum }_{i=1}^{{N}_{\mathrm{ch}}}{\xi }_{i}=1.$$

(12)

In the CKD method, convolved sky intensities ($\widehat{T}$ and $\widehat{L}$) are determined with the stepwise filter response function $\overline{\phi }$, as follows:

$$\widehat{T}\left(\lambda \right)=\frac{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot \overline{T}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}},$$

(13a)

$$\widehat{L}\left(\Theta ,\lambda \right)=\frac{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot \overline{L}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}.$$

(13b)

$${\overline{F} }_{\mathrm{sol}}\left(\overline{\lambda }\right)\equiv \frac{{\int }_{{\Delta }_{\kappa }}{F}_{\mathrm{sol}}\left(\lambda \right)\frac{\mathrm{d}\kappa }{{\kappa }^{2}}}{{\Delta }_{\kappa }{\overline{\lambda }}^{2}},$$

(13c)

$$\overline{\phi }\left(\overline{\lambda }\right)\equiv \frac{{\int }_{{\Delta }_{\kappa }}\phi \left(\lambda \right)\frac{\mathrm{d}\kappa }{{\kappa }^{2}}}{{\Delta }_{\kappa }{\overline{\lambda }}^{2}},$$

(13d)

where ${\overline{F} }_{\mathrm{sol}}$ is the band-averaged extraterrestrial solar irradiance and ${N}_{\mathrm{s}}$ is the number of sub-bands. $\widehat{T}$ 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 sub-bands. 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:

$${\varepsilon }_{\widehat{T},\mathrm{RT}}^{2}={\left(\mathrm{ln}\frac{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{CKD}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{LBL}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}\right)}^{2}\sim {\left[\frac{\left({\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{CKD}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right)-\left({\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{LBL}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right)}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{LBL}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}\right]}^{2}=\frac{{\sum }^{{N}_{\mathrm{s}}}{\left[\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{LBL}}\cdot {\varepsilon }_{\overline{T },\mathrm{RT}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right]}^{2}}{{\left[{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{LBL}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right]}^{2}},$$

(14a)

$${\varepsilon }_{\widehat{L},\mathrm{RT}}^{2}\left(\Theta \right)={\left(\mathrm{ln}\frac{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{CKD}}\left(\Theta \right)\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{LBL}}\left(\Theta \right)\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}\right)}^{2}\sim {\left[\frac{\left({\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{CKD}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right)-\left({\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{LBL}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right)}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{LBL}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}\right]}^{2}=\frac{{\sum }^{{N}_{\mathrm{s}}}{\left[\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{LBL}}\left(\Theta \right)\cdot {\varepsilon }_{\overline{L },\mathrm{RT}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right]}^{2}}{{\left[{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{LBL}}\left(\Theta \right)\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right]}^{2}},$$

(14b)

$${\varepsilon }_{\widehat{I}\mathrm{max}}^{2}=\mathrm{max}\left\{{\varepsilon }_{\widehat{T},\mathrm{RT}}^{2},{\varepsilon }_{\widehat{L},\mathrm{RT}}^{2}\left(\Theta \right)\right\}.$$

(14c)

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:

$${\varepsilon }_{\widehat{T},\mathrm{RT}}^{2}\sim {\varepsilon }_{\widehat{L},\mathrm{RT}}^{2}\sim {D}^{2}{\Phi }^{2}<{\varepsilon }_{\overline{I}\mathrm{max} }^{2}>,$$

(15)

where $<>$ indicates an averaging operation;

$${D}^{2}\equiv \frac{<{\left({\overline{T} }^{\mathrm{LBL}}\right)}^{2}>}{{\left(<{\overline{T} }^{\mathrm{LBL}}>\right)}^{2}}\sim \frac{{\int }_{{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{min}}}^{{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{max}}}{x}^{2}\mathrm{d}x}{{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{max}}-{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{min}}}/{\left(\frac{{\int }_{{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{min}}}^{{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{max}}}x\mathrm{d}x}{{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{max}}-{\overline{T} }_{\mathrm{H}2\mathrm{O},\mathrm{min}}}\right)}^{2}=\frac{4}{3}\approx 1.33,$$

(16a)

$${\Phi }^{2}\equiv \frac{{\sum }^{{N}_{\mathrm{s}}}{\left[\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right]}^{2}}{{\left[{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}\right]}^{2}}\sim {\left(0.242\right)}^{2}.$$

(16b)

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 right-hand 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 right-hand 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:

$${\widehat{R}}^{\mathrm{CKD}}\left(\Theta \right)\equiv \frac{{\widehat{L}}^{\mathrm{CKD}}}{{\widehat{T}}^{\mathrm{CKD}}\left(\Theta \right)}={\widehat{R}}^{\mathrm{LBL}}\left(\Theta \right){e}^{{\varepsilon }_{\widehat{R},\mathrm{RT}}\left(\Theta \right)}={\widehat{R}}^{\mathrm{LBL}}\left(\Theta \right){e}^{{\varepsilon }_{\widehat{L},\mathrm{RT}}\left(\Theta \right)-{\varepsilon }_{\widehat{T},\mathrm{RT}}},$$

(17a)

$${\varepsilon }_{\widehat{R}\mathrm{max}}^{2}=\mathrm{max}\left\{{\varepsilon }_{\widehat{R},\mathrm{RT}}^{2}\left(\Theta \right)\right\}.$$

(17b)

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 WV-CKD are estimated as follows:

$${\varepsilon }_{\widehat{T},\mathrm{RT}}^{2}\sim {\varepsilon }_{\widehat{L},\mathrm{RT}}^{2}\le {\varepsilon }_{\widehat{I}\mathrm{max}}^{2}\sim {\left(D\Phi {\upvarepsilon }_{E}\right)}^{2}\sim {\left(0.278\right)}^{2}\times {\left(5.0\times {10}^{-3}\right)}^{2}\ll {\sigma }_{{F}_{\mathrm{ds}}}^{2}\sim {\sigma }_{{F}_{\mathrm{df}}}^{2}.$$

(18)

3.2 Evaluation of the WV-CKD

To evaluate sky radiance at ground level in detail, we constructed a fine-scale LUT of the k distribution (${\Delta }_{\kappa }=2, 5 {\mathrm{cm}}^{-1}$; hereinafter, WV-CKD-2 and WV-CKD-5, 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 $10 {\mathrm{cm}}^{-1}$ band-averaged sky intensities below $5.0\times {10}^{-3}$ (equal to that of the WV-CKD-10). The details of the WV-CKD-2 and WV-CKD-5 are summarized in Table 2. The quadrature numbers of the WV-CKD-2 and WV-CKD-5 fall into the ranges of 1–9 and 2–11, respectively. The median quadrature numbers of the WV-CKD-2 and WV-CKD-5 are 4 and 6, respectively. Therefore, using all of the WV-CKD developed here allows for ≥ 46-fold 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, DU-DB) 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).

Table 3 Description of the simulation data used in Sec 3.2 (DU-DB)

Fig. 5

Monthly mean atmospheric vertical profiles at the SKYNET Chiba site in 2018 obtained from NCEP reanalysis 1 data. Left and center panels show vertical profiles of pressure and temperature, respectively. Right panel is the vertical profile of water vapor normalized to PWV of 2 cm

Fig. 6

Filter response functions (FWHM: 5, 10, and 15 nm). Black lines represent true filter response functions at central wavelengths of 936 and 940 nm. Blue, red, and green lines show stepwise functions integrated at the sub-band level for WV-CKD-2, -5, and -10, respectively. WV-CKD-2, -5, and -10 are lookup tables for the CKD method in the water vapor absorption region of 940 nm at 2-, 5-, and 10-${\mathrm{cm}}^{-1}$ intervals, respectively. FWHM of (a) 5 nm, (b) 10 nm, and (c) 15 nm at a central wavelength of 936 nm; FWHM of (d) 5 nm, (e) 10 nm, and (f) 15 nm at a central wavelength of 940 nm

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 sub-band. Therefore, this section aims to evaluate whether the WV-CKD 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 WV-CKD 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 WV-CKD 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 SN-CKD 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 sky-radiometer observations. Thus, the $\sqrt{{\varepsilon }_{\widehat{R}\mathrm{max}}^{2}}$ of the SN-CKD 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.

Table 4 Maximum residual errors of the convolved radiances assuming sub-band-averaged extraterrestrial solar irradiance and a filter response function. Residual errors lager than the expected residual values are underlined

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 high-resolution ($0.01 {\mathrm{cm}}^{-1}$) 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:

$${\varepsilon }_{\tilde{T }}^{2}={\left(\mathrm{ln}\frac{{\widehat{T}}^{\mathrm{CKD}}}{{\tilde{T }}^{\mathrm{LBL}}}\right)}^{2}={\left[\mathrm{ln}\left(\frac{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{CKD}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}/\frac{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right){T}^{\mathrm{LBL}}\left(\lambda \right)\mathrm{d}\lambda }{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)\mathrm{d}\lambda }\right)\right]}^{2}\sim {\varepsilon }_{\widehat{T},\mathrm{RT}}^{2}+{\varepsilon }_{\tilde{T },\mathrm{FRF}}^{2},$$

(19a)

$${\varepsilon }_{\tilde{L }}^{2}\left(\Theta \right)={\left(\mathrm{ln}\frac{{\widehat{L}}^{\mathrm{CKD}}}{{\tilde{L }}^{\mathrm{LBL}}}\right)}^{2}={\left[\mathrm{ln}\left(\frac{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{CKD}}\left(\Theta \right)\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}/\frac{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right){L}^{\mathrm{LBL}}\left(\Theta ,\lambda \right)\mathrm{d}\lambda }{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right)\mathrm{d}\lambda }\right)\right]}^{2}\sim {\varepsilon }_{\widehat{L},\mathrm{RT}}^{2}+{\varepsilon }_{\tilde{L },\mathrm{FRF}}^{2},$$

(19b)

$${\varepsilon }_{\tilde{R }}^{2}\left(\Theta \right)={\left[\mathrm{ln}\left(\frac{{\widehat{L}}^{\mathrm{CKD}}}{{\widehat{T}}^{\mathrm{CKD}}}/\frac{{\tilde{L }}^{\mathrm{LBL}}}{{\tilde{T }}^{\mathrm{LBL}}}\right)\right]}^{2}={\left[\mathrm{ln}\left(\frac{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{L} }^{\mathrm{CKD}}\left(\Theta \right)\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}{{\sum }^{{N}_{\mathrm{s}}}\overline{\phi }\cdot {\overline{F} }_{\mathrm{sol}}\cdot {\overline{T} }^{\mathrm{CKD}}\cdot {\Delta }_{\kappa }{\overline{\lambda }}^{2}}/\frac{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right){L}^{\mathrm{LBL}}\left(\Theta ,\lambda \right)\mathrm{d}\lambda }{\int \phi \left(\lambda \right){F}_{\mathrm{sol}}\left(\lambda \right){T}^{\mathrm{LBL}}\left(\lambda \right)\mathrm{d}\lambda }\right)\right]}^{2},$$

(19c)

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:

$${\varepsilon }_{\tilde{I }\mathrm{max}}^{2}=\mathrm{max}\left\{{\varepsilon }_{\tilde{T }}^{2},{\varepsilon }_{\tilde{L }}^{2}\left(\Theta \right)\right\},$$

(20a)

$${\varepsilon }_{\tilde{R }\mathrm{max}}^{2}=\mathrm{max}\left\{{\varepsilon }_{\tilde{R }}^{2}\left(\Theta \right)\right\}.$$

(20b)

Figure 7 shows the angular distribution of the convolved normalized radiances simulated for type 2 DU-DB. Although the convolved $\tilde{R }$ value obtained with the SN-CKD had large errors in backward scattering and the zenith region (approximately 1.5%), convolved $\tilde{R }$ values from the WV-CKDs 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 DU-DB. 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 WV-CKD-2 and WV-CKD-5 are within $3\times {10}^{-3}$. In the case of an FWHM of 5 nm, the residual errors from the WV-CKD-10 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 sky-radiometer 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 sky-radiometer specification, the $\sqrt{{\varepsilon }_{\tilde{I }\mathrm{max}}^{2}}$ values of the WV-CKD-2, WV-CKD-5, and WV-CKD-10 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 SN-CKD ($\le 1.5\times {10}^{-1}$). In comparison, the $\sqrt{{\varepsilon }_{\tilde{R }\mathrm{max}}^{2}}$ values of all WV-CKDs reach $1.1\times {10}^{-3}$. In conclusion, this simulation suggests that the WV-CKD-5 is useful for computation of sky radiances (direct solar irradiances and normalized radiances) based on sky-radiometer observations. Moreover, this process is approximately 75-fold more rapid than the LBL method.

Fig. 7

Angular distributions of simulated normalized radiances simulated for type 2 (Table 3) in the almucantar and principal planes with an FWHM of 10 nm and a central wavelength of 940 nm. a, b Convolved normalized radiance, $R$. c, d Relative errors of the simulated radiance with the SN-CKD and WV-CKDs. SN-CKD is a standard lookup table for the CKD method in the radiative transfer model RSTAR. WV-CKD is a series of lookup tables for the CKD method in the water vapor absorption region of 940 nm at 2-, 5-, and 10-${\mathrm{cm}}^{-1}$ intervals. Note that the lines in the top panels overlap

Table 5 As described for Table 4, but the reference radiances were computed with a smooth filter response function and the high-resolution ($0.01\user2{ }{\mathbf{cm}}^{ - 1}$) extraterrestrial solar irradiance data

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 SN-CKD, we repeated their analysis with the WV-CKD-2. 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).

Table 6 Microphysical and optical properties and vertical profiles of aerosols used in sensitivity testing, in accordance with the approach described by Momoi et al. (2020)

Figure 8 shows the results of sensitivity tests conducted with the SN-CKD and WV-CKD-2. The results obtained with the SN-CKD 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.

Fig. 8

Angular distributions of the simulated normalized radiances for the continental average aerosol (Table 6) in the almucantar plane with AOT of 0.02 and 0.20 at 940 nm. Simulations were conducted for a solar zenith angle of 70 degrees and PWV ($w$) = 0, 1, 2, 3, 4, and 5 cm. a, b Convolved normalized radiance, $R$. c, d Ratio of $R(w)$ to $R(w = 0 \mathrm{cm})$. Solid lines show radiances obtained with the SN-CKD and dashed lines show radiances obtained with the WV-CKD-2. SN-CKD is a standard lookup table for the CKD method in the radiative transfer model RSTAR, and WV-CKD-2 is a lookup table for the CKD method in the water vapor absorption region of 940 nm at 2-${\mathrm{cm}}^{-1}$ intervals

3.4 PWV estimation from sky-radiometer 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 physics-based 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 sky-radiometer observations through the SKYMAP/DSRAD algorithm using the WV-CKD-2.

Fig. 9

Analytical flow of Momoi et al. (2020). SDF, CR, and NS ratio are aerosol volume size distribution, complex refractive index, and non-spherical ratio, respectively. Aerosol channels denote weak gas absorption bands (i.e., 340, 380, 400, 500, 675, 870, and 1020 nm). SKYMAP is a new on-site method for self-calibrating the water vapor channel of the sky-radiometer using diffuse radiances normalized by direct solar irradiance. DSRAD estimates AOT and PWV from the calibrated direct solar irradiance from sky-radiometer observations

3.4.2 PWV estimation from actual sky-radiometer observations

In this subsection, we describe the impact of introducing the new CKD table (WV-CKD) into the analysis of actual observation data obtained with the sky-radiometer (POM-02, serial no. PS2501417) at Chiba University (35.63°N, 140.10°E) in 2019. PWV reference values were obtained with a microwave radiometer (MWR; MP-1500; Radiometrix) at the same location. The MWR measured the zenith brightness temperature in the 22–30 GHz region at 1-min 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 sky-radiometer model POM-01. In the case of the SKYMAP with WV-CKD-2, 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 SN-CKD ($1.933\times {10}^{-4}$ 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 SN-CKD 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 WV-CKD-2 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 sky-radiometer 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 SN-CKD, rather than from the uncertainty in AOT estimation. This suggests that SKYMAP/DSRAD calculation with the WV-CKD is useful for accurate estimation of PWV and determination of the calibration constant ${\tilde{F }}_{0}$ for the water vapor absorption band of sky-radiometer observations.

Fig. 10

The top row (a) shows time-series PWV from sky-radiometer (SKYR) observations derived using the SN-CKD (red) and WV-CKD-2 (blue) in 2019 at Chiba. The middle row (b) is the difference in PWV between the sky-radiometer and microwave radiometer. The bottom row (c) shows differences in AOT at 675 (red), 870 (blue), and 1020 (black) nm between the DSRAD algorithm and AERONET retrieval results. Circles and error bars in the middle and bottom rows represent means and standard deviations, respectively. SN-CKD is a standard lookup table for the CKD method in the radiative transfer model RSTAR, and WV-CKD-2 is a lookup table for the CKD method in the water vapor absorption region of 940 nm at 2-${\mathrm{cm}}^{-1}$ intervals. DSRAD estimates AOT and PWV from the calibrated direct solar irradiance from sky-radiometer observations

Fig. 11

Comparison of PWV observed at Chiba in 2019 between the microwave radiometer (PWV_MWR) and sky-radiometer (PWV_SKYR) with (a) SN-CKD and (b) WV-CKD-2. The bottom value in the legend is the mean and standard deviation of the difference (PWV_SKYR—PWV_MWR). γ is the correlation coefficient. SN-CKD is a standard lookup table for the CKD method in the radiative transfer model RSTAR, and WV-CKD-2 is a lookup table for the CKD method in the water vapor absorption region of 940 nm at 2-${\mathrm{cm}}^{-1}$ intervals

4 Conclusions

To compute direct solar irradiance and diffuse radiance at ground level around 940 nm with narrow-band RTMs, we developed a rapid calculation module using a LUT (WV-CKD). We found the challenges facing the currently used LUT (SN-CKD) in RSTAR7 for narrow-band sky radiance computation around 940 nm region. As shown by comparison of the sky intensities of sub-bands obtained with the LBL method and CKD method using the SN-CKD, the root mean square error of the maximum error in sub-bands obtained with the SN-CKD 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 WV-CKDs were created at three different spectral resolutions (${\Delta }_{\kappa }=2, 5, 10 {\mathrm{cm}}^{-1}$ in WV-CKD-2, WV-CKD-5, and WV-CKD-10, respectively). The quadrature values and numbers of these WV-CKDs were optimized using sky intensities based on the single-scattering approximation at ground level, with an accuracy of ≤ 0.5% for sub-bands of $10 {\mathrm{cm}}^{-1}$. The quadrature numbers affected computational efficiency. The median quadrature numbers of WV-CKD-2, WV-CKD-5, and WV-CKD-10 are 4, 6, and 7, respectively; their calculations were ≥ 46-fold more rapid than the LBL method.

Radiance calculation with the WV-CKD 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 WV-CKD 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 WV-CKD 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 SN-CKD is within 15%, use of WV-CKD provides convolved sky intensities with an accuracy of ≤ 0.3% at an FWHM of 10 nm, equal to the FWHM of the sky-radiometer. 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 WV-CKD (correlation coefficient γ = 0.995, slope = 1.002) than with the SN-CKD (correlation coefficient γ = 0.984, slope = 0.926), as used by Momoi et al. (2020). Therefore, through application of the WV-CKD to actual data analysis, we demonstrated that an accurate CKD table is essential for estimating PWV from sky-radiometer observations.