Momoi et al.
Progress in Earth and Planetary Science(2022) 9:10
https://doi.org/10.1186/s40645-022-00467-6

Progress in Earth andPlanetary Science

Open Access

METHODOLOGY

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
Masahiro Momoi1* , Hitoshi Irie1, Miho Sekiguchi2, Teruyuki Nakajima3, Hideaki Takenaka1,
Kazuhiko Miura4,5 and Kazuma Aoki6

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 cm−1) for ground-based angular-scanning radiometer data analysis. Tables were constructed at three spectral
resolutions (2, 5, and 10 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 10cm−1. Although high-resolution
WV-CKD requires numerous quadrature points, the number of executions of the radiative transfer model is reduced toapproximately 1/46 of the number used in the line-by-line approach by our WV-CKD with a resolution of 2 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 withthe 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.
Keywords: Atmospheric gas absorption, Correlated k-distribution, Radiative transfer*Correspondence: m.momoi@biglobe.jp
1Center for Environmental Remote Sensing, Chiba University, 1‑33
Yayoicho, Inage‑ku, Chiba 263‑8522, Japan
Full list of author information is available at the end of the article

1 Introduction
Aerosols, clouds, ozone, and water vapor are important parameters for characterizing Earth’s climate andchanges therein (e.g., IPCC 2021) and are therefore considered essential climate variables by the World Meteorological Organization (Bojinski et al. 2014). Some of theseparameters are measured using ground- and satellitebased instruments and estimated using the multi-term© The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to theoriginal author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images orother third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of thislicence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/.

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

least squares method (e.g., Dubovik and King 2000; Rogers 2000), with radiative transfer models (RTMs) as theforward 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) proceedwith 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 ground-based angular-scanning radiometer datawith 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 aninternational network of ground-based angular-scanningradiometers, uses a scalar RTM called RSTAR (System forTransfer of Atmospheric Radiation for Radiance calculations; Nakajima and Tanaka 1986, 1988) in version 2 ofits 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) inversion 3 (Sinyuk et al. 2020). The SKYNET (Takamura
and Nakajima 2004; Nakajima et al. 2007, 2020), anotherground-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 ofAtmospheric Radiation) series, which was developed anddistributed 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 Polarizedradiance calculations; Ota et al. 2010) and the flux calculation code FSTAR (STAR for Flux calculations; Nakajima et al. 2000), making it a broadly compatible groupof packages. In particular, RSTAR and PSTAR introduce
efficient calculation methods for IMS (Improved Multiple and Single scattering approximation; Nakajima andTanaka 1988) and ­Pn-IMS (IMS by nth order multiple
scattering correction of the forward Peak; Momoi et al.
2022), respectively, enabling accurate reconstruction ofsky 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,Page 2 of 22

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 thecorrelated k-distribution (CKD) method (Lacis andOinas 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 quadraturepoints, 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 andcalibrates 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 andaerosols included in the direct solar irradiance and diffuse
radiance is insufficient due to the large computationalburden. 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). Fromthese 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: lnT̃H2O = −a(mw)b(Bruegge et al. 1992) to define the relationship between
the convolved transmittance of atmospheric water vapor(T̃H2O) 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 lineby-line (LBL) method under a US standard atmosphere,Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

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 ahigh computational cost. Obtaining the convolved transmittance requires regular sensor calibration, in which thesensor output of the extraterrestrial solar irradiance (orcalibration constant) is determined at a specific site (e.g.,
Mauna Loa Observatory), as the convolved transmittanceis the ratio of the sensor output of direct solar irradianceto 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 acalibration constant can be determined from the PWVdata derived from the on-site angular distribution ofdiffuse radiance using the SKYMAP algorithm. Thisapproach requires accurate computations of sky radiancesin 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 methodin the 940 nm water vapor absorption region (WV-CKD),
installed the WV-CKD in RSTAR to enable rapid andaccurate 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 distributionof diffuse radiances, using actual ground-based angularscanning radiometer observations in the water vapor
absorption band. The WV-CKD was constructed throughoptimization 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 theaccuracy of the WV-CKD and its impact on PWV estimation from actual SKYNET sky-radiometer observations.



τ ()

L(�, ) ≡ T ()0Page 3 of 22

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 themethod 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 wasestimated 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 ofsky-radiometer observations.
2.1.1 Measurement of sky irradiance

The POM-01 sky-radiometer measures direct solar anddiffuse irradiances with a finite field of view at sevenwavelengths: 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 describedby a radiative transfer equation in a plane-parallel nonrefractive atmosphere as follows:

Fds () =F0 ()T (),d2

Fdf (�, ) =F0 ()��
L(�, ),|µ|d 2(1a)(1b)

F0 () ∝ Fsol (),

(1c)



τ ()
,
T () ≡ exp −
|µ0 |

(2a)





1
1
′
′
−
exp (τ − t)
ω (, t)P (�, , t)dt + q(�, ) ,|µ0 | |µ|(2b)

Momoi et al. Progress in Earth and Planetary ScienceR(�, ) ≡(2022) 9:10|µ|Fdf (�; )L(�, )=,
Fds ()��
T ()

(2c)

where Fds is the sensor output current of the direct
solar irradiance; Fdf is the sensor output current of diffuse irradiance detected with a finite field of view ();F0 is the calibration constant, which is the sensor output
current for extraterrestrial solar irradiance ( Fsol ) at themean 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);  is wavelength; τ is total optical thickness; m0 is the optical
air mass,
represented
as


′
′
m0 = 1/|cosθ0 | = 1/|µ0 |; P �, , τ ′ and ω , τ ′ are thetotal phase function and total single scattering albedo,′
respectively, at an altitude of τ = τ ; and q is the multiple scattering contribution. Although determination of T
requires F0, R does not require F0 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(�, ) = ωτ P(�, )E(µ, ) + q(�, )T (),E(µ, ) ≡





1
1
|µ0 | − |µ|



(3a)



F̃0 () =

φ()Fdf ()d =��|µ|d 2φ()F0 ()d ∝φ()F0 ()L(�, )d,φ()Fsol ()d.

(4b)
(4c)

Hence, the convolved sky intensities (T̃ and L̃) and convolved normalized radiance ( R̃) are defined as follows:F̃ds ()d 2
φ()Fsol ()T ()dT̃ () =,= (5a)φ()Fsol ()dF̃0 ()

L̃(�, ) =

R̃(�, ) ≡

φ()Fsol ()L(�, )d
.
φ()Fsol ()d
φ()Fsol ()L(�, )d,= φ()Fsol ()T ()dT̃ ()

L̃(�, )

Although sky radiance in weak gas absorption regions,such as at 340, 380, 400, 500, 675, 870, and 1020 nm insky-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 lnT̃ andlnR̃, obtained using Eq. (5). The weighted values (σ 2 andT̃σ 2) of the covariance matrix in the algorithms areR̃
expressed under the assumption that F̃ds, F̃df , and F̃0 are
independent of each other, as follows:

σT̃2 = σF̃2 + σF̃2 ,(6a)

σR̃2 = σF̃2 + σF̃2 ,

(6b)

0

ds

df

ds

where σF̃0 is the standard deviation of the sky-radiometer
calibration constant (lnF̃0) or the relative error of F̃0, asT ()µ = µ0−1  



.
τ ()
τ ()
τ ()
exp − |µ| − exp − |µ0 |µ � = µ0

Furthermore, assuming a wideband filter response function (φ ), convolved Fds and Fdf ( F̃ds and F̃df ) with φ can beobtained through convolution of Eq. (1), as follows:
1F̃ds () = φ()Fds ()d = 2φ()F0 ()T ()d,d(4a)F̃df () =Page 4 of 22

(5b)

(5c)

(3b)

∂lnF̃0 ∼ ∂ F̃0 ; σF̃ and σF̃ are the standard deviations of
F̃0

ds

df

sky-radiometer measurements (lnF̃ds and lnF̃df ,respectively).
In this study, we estimated σF̃ from 10 × 10 radiancesds
sampled at 0.1° intervals in a circumsolar domain of ± 1°
in both the zenith and azimuth angle directions. Thismeasurement 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, theuncertainty of sky intensity data is calculated from thedifference (f ) of sky irradiance between observed andGaussian-filtered data, as follows:�f = lnf obs (�) − lnf gf (�),(7)

where f obs and f gf are the observed and Gaussian-filtered sky irradiances, respectively. Because the magnitude of the solar aureole angular distribution differs atlarge scattering angles ( ≥ 0.3o), we used a range of 0.3°.
Therefore, we could estimate the uncertainties of F̃ds
from f , as the solar disk diameter is approximately 0.5°.
Figure 2 shows an example histogram of f at 340, 500,

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

and 940 nm based on measurements taken with the skyradiometer 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 F̃ds and indicates that thestandard deviation of sky-radiometer measurements(σF̃ ) at all wavelengths is less than 5.0 × 10−3. Thus, σF̃dsdfis expected to be larger than σF̃ . This is because F̃ds anddsF̃df are measured using the same detector, but F̃ds ≫ F̃df
by the solar incident beam. Therefore, 5.0 × 10−3 is a reasonable benchmark value for errors in F̃ds and F̃df in thisstudy.
2.2 Challenges regarding the k‑distribution lookup tablein RSTAR​In RSTAR7, gas absorption is considered for ­H2O, ­CO2,­O3, ­N2O, CO, C­ H4, 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 subband. The resolution of the SN-CKD at wavenumber (k )is dlog10 k = 5 × 10−4 (dk ≈ 12.2cm−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 SNCKD because the gas absorption coefficient is small relative to other processes (e.g., Rayleigh scattering andaerosol extinction). However, careful consideration of thequadrature numbers is essential in the water vapor
absorption region around 940 nm due to the complex
absorption characteristics of water vapor. Therefore, wevalidated the radiance calculation with SN-CKD around
940 nm (10,000–10,900 cm−1 [1000–917 nm]) throughPage 5 of 22

the LBL method using simulated band-averaged sky
df (�,)).
intensities for sub-bands ( T = FFds and L = |µ|F
F 0 ()��0
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 subbands 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) withaerosol 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 andprincipal); 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

Fig. 2 Example histogram of the differences at 340, 500, and 940 nmof the sky-radiometer measurements on February 27 and 28, 2020 in
Akiruno, Japan

Table 1 Uncertainty of sky irradiance based on comparison
between f obs and f gf

Fig. 1 Example solar aureole angular distributions of thesky-radiometer at 500 nm obtained through the solar disk scanmethod on February 27, 2020, at Akiruno, Japan. Blue dots aremeasured values and red dots show the angular distribution withnoise reduction using a Gaussian filter

Wavelength

Num. of data

Standard deviation

315 nm127

5.0 × 10−3

340 nm117

3.0 × 10−3

380 nm104

1.9 × 10−3

400 nm122

2.7 × 10−3

500 nm119

2.9 × 10−3

675 nm114

4.1 × 10−3

870 nm135

6.5 × 10−31020 nm105

4.3 × 10−3

940 nm110

4.3 × 10−3

1627 nm56

4.8 × 10−3

2200 nm57

4.4 × 10−3

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 6 of 22

solar irradiance was averaged at the sub-band (dlog10 k )level. Therefore, with the LBL method, T and L areexpressed as:





2
εL,RT
; �, θ0 , w, τa =



LBL

L

ln CKD
L

2

,

(10b)



 
  2  2εImax = max εT2 ,RT  , εL,RT |θ0 = 30(20)70o ; w = 0.5(0.5)6cm; τa = 0.05, 0.2, 1.0 .
T

LBL  

LBL 

L

 =

1
dκ
�κ κ 2
1�,  =  dκ�κ κ 2

TA ()TR ()TH2O ()Tcont ()
�κ

dκ
,
κ2

(8a)
TA ()TR ()TH2O ()Tcont ()R(�, )
�κ

dκ
,
κ2

(8b)
where TA and TR are the monochromatic transmittancesof aerosol extinction and Rayleigh scattering, respectively; TH2O is the monochromatic transmittance of water
vapor absorption (line and self-continuum); and Tcont is
the monochromatic transmittance of the ­O2 and ­O3 continuum absorption. Because the differential interval of
the numerical integral is too small (dκ = 0.01cm−1) for
use with the LBL method, TH2O is obtained using Beer–
Lambert’s law as follows: zln[TH2O ()] = −m0 (θ0 )σH2O,line (K , p, ) + σH2O,cont (K , ) dz,
0(9)where σH2O,line and σH2O,cont are the absorption coefficients [/m] of the water vapor line and self-continuum
absorptions, respectively; K is the temperature; p is thepressure; and z is the geometric thickness of the atmosphere [m]. Figure 3 shows the maximum errors (εImax ) of
T and L obtained with the SN-CKD compared with the
LBL method, which were determined as:


LBL 2


T
2
(10a)
εT ,RT ; θ0 , w, τa = ln CKD ,
T

(10c)

The root mean square errors of εImax for CA-DB were
8.24 × 10−1 (approximately 82.4% error) at 10,000–
10,902 cm−1, and 1.16 (approximately 116% error) at
10,411–10,864 cm−1 [961–920 nm]. This error propagates to the convolved sky intensities. Thus, the errorof 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 σFds. 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 lackof optimization, which leads large error under the US
atmosphere.
2.3 Method used to create the new k‑distribution lookup
table

In this study, a new LUT of the k distribution at 10-cm−1
intervals (�κ ) from 10,000 cm−1 to 10,900 cm−1 (hereafter WV-CKD-10) was constructed through optimizationof the quadrature values (point and weight) and numbers
through the LBL method. Radiation around 940 nm is
attenuated mainly through aerosol extinction, Rayleighscattering, 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 ­O2and ­O3 continuums. Those LUTs were created from

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)

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

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 forE(µ, ) under six AFGL standard atmospheres (US standard, tropical, mid-latitude summer, mid-latitude winter,high-latitude summer, and high-latitude winter)in
2
RSTAR7. In this study, the maximum error ( εE ) of
E(µ, ) for �κ = 10cm−1 is achieved at values less than
5.0 × 10−3. The quadrature values, consisting of a pair ofpoint 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 thevalidation dataset of the SN-CKD.

Page 7 of 22

Nch

i=1

 and L)In the CKD method, convolved sky intensities (Tare determined with the stepwise filter response function
φ , as follows: () =
T

 
φ  ≡

i=1


    L�,  = T A  T R  T cont 
Nch
 
(i)
ξi TH2O,ckd  R(i) (�, ),CKD φ · F sol · L · �κ 

Ns2

φ · F sol · �κ dκ�κ Fsol () κ 2
,
2
�κ 

2

.

(13b)

(13c)

dκ
�κ φ() κ 2
,
2
�κ 

(13d)

�
� �
2 2
CKD
Ns
φ · F sol · T· �κ 
ln �
2
LBL
Ns
φ · F sol · T· �κ 
� ��
�
 ��
2
2 2
CKD
LBL
Ns
Ns
φ · F sol · T· �κ  −φ · F sol · T· �κ 

∼
�Ns
2
LBL
φ · F sol · T· �κ 
�
�Ns �
2 2
LBL
φ · F sol · T· εT ,RT · �κ 
=
,
��
�
2 2
LBL
Ns
φ · F sol · T· �κ 

εT2� ,RT =

(14a)

2
ε�L,RT
(�) =

�

�Ns

ln �

Ns

φ · F sol · L

CKD
LBL

(�) · �κ 

2

2

�2

φ · F sol · L (�) · �κ 
� ��
�
 ��
2
2 2
CKD
LBL
Ns
Ns
φ · F sol · L· �κ  −φ · F sol · L· �κ 

∼
�Ns
2
LBL
φ · F sol · L· �κ 
�
�Ns �
2 2
LBL
φ · F sol · L (�) · εL,RT · �κ 
,
=
��
�
2 2
LBL
Ns
φ · F sol · L (�) · �κ 

(11a)

(11b)

i=1

where T A and T R are the band-averaged values of TA andTR, respectively; Nch is the number of quadrature points;(i)
TH2O,ckd and R(i) are the ith transmittance and normalized radiance of the quadrature values of the k distribution at �κ , respectively; and ξi is the ith quadrature
weight of the k distribution at �κ , which is normalized to:



(13a)

,

2

φ · F sol · �κ 

Ns


2

where F sol is the band-averaged extraterrestrial solar
 andirradiance and Ns is the number of sub-bands. TT̃ in Eq. (5a) ( L and L̃ in Eq. (5b)) are not entirely synonymous, but are generally equivalent based on theassumption that the extraterrestrial solar irradiance andfilter response function being nearly constant across sub and bands. The residual errors (εT ,RT and εL,RT) of T
L for
the CKD method are obtained as follows:    Nch (i)
 = T A  T R  T cont 
ξi TH2O,ckd  ,CKD  

φ · F sol · T · �κ 

Ns

 
F sol  ≡3.1 Optimized k‑distribution lookup table

T

Ns


L(�, ) =3 Results and discussion
Figure 4a shows the quadrature
numbers and Fig. 4b2shows the maximum error ( εImax ), which was satisfied
at values less than 5.0 × 10−3 for 10,000–10,900 cm−1.
The quadrature number falls into the range of 3–15 at
10,000–10,900 cm−1 and 4–15 at 10,410–10,870 cm−1
[961–920 nm]. The median quadrature number is 7 at
10,000–10,900 cm−1 and 8 at 10,410–10,870 cm−1 [961–920 nm], as the water vapor line absorption around
940 nm is complex and requires numerous quadraturepoints 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 cm−1 must be computed 1000 times with the LBL
method (= 10cm−1 /0.01cm−1) in RSTAR7 (Table 2).
The equations for band-averaged sky intensities differ
CKD
markedly between the LBL and CKD methods. T
and
CKD, derived using the WV-CKD, are as follows:L(12)

ξi = 1.


22εImax= max εT2 ,RT , εL,RT
(�) .

(14b)
(14c)

If we assume that T H2O is roughly randomly distributedin the range of T H2O,min = 0 to T H2O,max = 1, ε2
and
T ,RT
ε2 are obtained as follows:
L,RTMomoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 8 of 22

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-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-cm−1 intervals

Resolution
 1/2
Maximum error of E(µ, ): εE2

WV-CKD-2

WV-CKD-5

WV-CKD-10

LBL

2 cm−15 cm−1

10 cm−1

0.01 cm−1

11.2 × 10−3

7.1 × 10−3

5.0 × 10−3–

Median quadrature numbers (10,000–10,900 cm−1)

4

6

7

–

Median quadrature numbers (10,410–10,870 cm−1)

5

6

8

–

Mean computation times (10,000–10,900 cm−1)

2.18/cm−1

1.18/cm−1

0.734/cm−1

100/cm−1

Mean computation times (10,410–10,870 cm−1)

2.44/cm−1

1.33/cm−1

0.848/cm−1

100/cm−1

D of Eq. 16 (FWHM = 5 nm)

0.176

0.280

0.394

–

0.124

0.198

0.278

–

D of Eq. 16 (FWHM = 15 nm)

0.102

0.162

0.228

–

D of Eq. 16 (FWHM = 10 nm)

22∼ D2 �2 < εImax
εT2 ,RT ∼ εL,RT
>,

(15)

where <> indicates an averaging operation;��� T H2O,max 2
LBL 2
x dx
>
< T
T H2O,min∼
D2 ≡ �
�2
LBL
T−T
H2O,max
H2O,min<T>2
� T H2O,max
xdx
4
T H2O,min/ = ≈ 1.33,
3
T H2O,max − T H2O,min

(16a)

Ns 

�2 ≡ 

Ns



2 2

φ · F sol · �κ 2
 ∼ (0.242) .

2 2

φ · F sol · �κ 

(16b)

2
>. The second term
In Eq. (15), εT ,RT ∼ εL,RT ∼< εImax
on the right-hand side of Eq. (16a) is a rough assumption
LBL
because of the biased probability distribution of T .
2
For example, D is 1.18, 1.38, and 1.82 at 10,410–10,870
cm−1 [961–920 nm] under the US standard atmospherewith a respective PWV of 0.7, 1.4, and 2.8 cm and aMomoi et al. Progress in Earth and Planetary Science(2022) 9:10

solar zenith angle of 70 degrees. The second expression
on the right-hand side of Eq. (16b) is an estimate for thesituation 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 (εR,RT) of the convolved normalized ) with the stepwise filter response function isradiance ( Robtained as follows:CKD (�) ≡
R


LCKD

 CKD (�)T(�)
LBL (�)eεR,RT=R

LBL (�)eεL,RT (�)−εT ,RT ,
=R



2
2
εRmax
= max εR,RT
(�) .



(17a)

(17b)

Using Eqs. (10) and (15), ε2
is
Imax
2
>∼ (D�εE )2 andε2∼ D2 �2 < εImax
Imax

estimated as
the expected
residual errors of radiances simulated with the WV-CKD
are estimated as follows:22≤ εImax
εT2 ,RT ∼ εL,RT∼ (D�εE )2 ∼ (0.278)2

2
× 5.0 × 10−3 ≪ σF2ds ∼ σF2df .

(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(�κ = 2, 5cm−1; hereinafter, WV-CKD-2 and WVCKD-5, respectively)in the manner described in Sec. 2.3;
in this case, εE2 was satisfied at values below 11.2 × 10−3
and 7.1 × 10−3 to maintain accuracy of the 10cm−1 bandaveraged sky intensities below 5.0 × 10−3 (equal to that of
the WV-CKD-10). The details of the WV-CKD-2 andWV-CKD-5 are summarized in Table 2. The quadrature
numbers of the WV-CKD-2 and WV-CKD-5 fall into theranges 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 atmosphericprofiles used represented the SKYNET Chiba site
(35.63°N, 140.10°E) and were obtained from National
Centers for Environmental Prediction (NCEP) reanalysis

Page 9 of 22

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 wavelengthsof 936 and 940 nm (Fig. 6).
3.2.1 Comparison with convolved sky intensities obtainedusing the stepwise filter response function

LBL assume that the extraterrestrial solar
 LBL, T
LLBL, and R
irradiance and filter response function are nearly constantin 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� εE2 ) estimated using Eq. (15) in each simulation are
also presented in Table 4. The

ε2

Imax

values obtainedusing all WV-CKD were smaller than 1.64 × 10−3 andsimilartotheexpected
residualerrors−3−3−32( D� εE = 2.0 × 10 , 1.4 × 10
and 1.1 × 10
at
FWHM of 5, 10, and 15 nm, respectively).
In
somecases,
D� εE2 , but fall within two
the residual errors exceed standard deviations (2D� εE2 ) due to fluctuations intransmittance [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� εE2 ) CKD and 
LCKD. Additionally,
is a 
useful benchmark of T
the ε2
values were smaller than 3.97 × 10−4 andRmaxmuch smaller than ε2 . This difference arises because
Imax

the normalized radiance cancels the residual error of
 CKD and 
T
LCKD, as shown in Eq. (17a). In contrast, theε2values obtained using the SN-CKD were muchImaxlarger, ranging from 7.53 × 10−3 to 2.21 × 10−1, and were
not negligible relative to the 
uncertainty of sky-radiometer observations. Thus, the ε2of the SN-CKD is
Rmax2ε , but reaches 6.71 × 10−2 at thebetter thanImax

FWHM of 5 nm and 3.89 × 10−2 at an FWHM of 10 nm.
3.2.2 Comparison with convolved sky intensities obtainedusing a smooth filter response functionT̃ LBL, L̃LBL, and R̃LBL are calculated using the LBL method
and convolved with a smooth filter response function andthe high-resolution (0.01cm−1) extraterrestrial solar irradiance data from Coddington et al. (2021). This section CKD, LCKD,provides a comprehensive assessment of T
CKD in addition to the general performance of gas
and Rabsorption discussed in Sect. 3.2.1. The residual errors of
the convolved sky intensities are defined as follows:Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 10 of 22

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

VariableType 1

AerosolComponent (Dubovik et al. 2000)

Water-soluble

Biomass burning

Vertical profile

exp (−z/H), H = 2 km

exp (−z/H), H = 4 km

Optical depth at 500 nm
Atmosphere
GeometryεT̃2 =

εL̃2 (�) =ln

εR̃2 (�) =0.2Type 3

Type 4

0.4

Vertical profile

Chiba Jan

Chiba-Apr

Chiba-Jul

Chiba-Oct

PWV

0.78 cm

1.79 cm

4.64 cm

2.58 cm

Solar zenith angle

60 degree CKD
TT̃ LBLln

Type 2

2

2
 
2 
CKD
Ns
φ · F sol · T· �κ 
φ()Fsol ()T LBL ()d∼ εT2 ,RT +εT̃2 ,FRF ,
/
= ln
Ns
2
φ()F
()d
sol
φ · F sol · �κ 

2

2
 
2 
CKD
Ns
φ · F sol · L
φ()Fsol ()LLBL (�, )d(�) · �κ 22
∼ εL,RT+εL̃,FRF, (19b)
/
= ln
Ns
2
φ()F
()d
sol
φ · F sol · �κ 


LCKD
L̃LBL

 

40 degree
LCKD L̃LBL/
ln
 CKD T̃ LBL
T

2

2
 
2 
CKD
Ns
φ · F sol · L
φ()Fsol ()LLBL (�, )d(�) · �κ 
,
/ 
= ln
Ns
2
CKD
φ()Fsol ()T LBL ()dφ · F sol · T· �κ 

2
where ε2and εL̃,FRF
are the residual errors arising
T̃ ,FRF
from the assumptions of the extraterrestrial solar irradiance and filter response function. The maximum residuals can be obtained as follows:
2= max εT̃2 , εL̃2 (�) ,εĨmax
2= max εR̃2 (�) .
εR̃max

(19a)

(19c)

(20a)
(20b)

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

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 11 of 22

Fig. 6 Filter response functions (FWHM: 5, 10, and 15 nm). Black lines represent true filter response functions at central wavelengths of 936 and940 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-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

Figure 7 shows the angular distribution of the convolved
normalized radiances simulated for type 2 DU-DB.
Although the convolved R̃ value obtained with the SNCKD had large errors in backward scattering and thezenith region (approximately 1.5%), convolved R̃ values
from the WV-CKDs showed better performance (< 0.1%).
2
Table 5 summarizes εĨmaxand ε2for the simulationR̃maxusing DU-DB. Although a finer �κ 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 affectperformance 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 × 10−3. In
the case of an FWHM of 5 nm, the residual errors fromthe WV-CKD-10 are significantly large (> 1 × 10−2) due
to the assumption of the stepwise function. In contrast,CKD was less strongly affected than T
 CKD and 
R
LCKD


( ε2

R̃max

≤ 1.1 × 10−3), in accordance with the relation-

ship between ε2and ε2 described in Sect. 3.2.1. This
Imax
Rmax
finding indicates an advantage of sky-radiometer observations, as Momoi et al. (2020) proposed estimation of PWVfrom the angular distribution of R̃. With an FWHM of
10 nm,
 corresponding to the sky-radiometer specification,2the εĨmax
values of the WV-CKD-2, WV-CKD-5, and

WV-CKD-10 are less than 3.2 × 10−3, 1.3 × 10−3, and
4.0 × 10−3, respectively.
These values are significantly2εĨmax obtained using the SN-CKD
smaller than
−1values of all(≤ 1.5 × 10 ). In comparison, the ε2
R̃max

WV-CKDs reach 1.1 × 10−3. In conclusion, this simulation suggests that the WV-CKD-5 is useful for computation of sky radiances (direct solar irradiances andnormalized radiances) based on sky-radiometer observations. Moreover, this process is approximately 75-foldmore rapid than the LBL method.

(ALM)ε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax −3εR̂max (×10 )ε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax −3εR̂max (×10 )
0.177

86.7

0.180

32.5

0.329

26.7
11.0

0.011

1.639

0.295

0.472

11.5

85.3

0.472
0.033

11.5

80.0

0.347

72.8
24.0

0.007

72.8
1.66

0.018

0.996

176

2.0

0.020

14.4

0.996

1.470

93.2
144

0.005

14.4

1.470

0.397

67.1
93.2

1.038

194

1.038
0.019

23.3

127

0.057

0.451

23.2

221

0.451
0.016

221
12.7

0.205

0.053

0.335

0.278

0.506

0.044

0.228

0.112

0.260

0.046

0.261

0.058

0.578

0.011

0.520

0.035

0.541

0.035

0.512

0.183

0.619

0.013

0.437

0.126

0.331

0.053

0.063

0.018

0.087

0.283

0.185

0.039

0.185

0.073

0.719

0.060

0.646

0.026

0.205

0.023

0.205

0.084

0.214

0.002

0.214

0.046

0.123

0.032

0.123

0.024

0.063

0.024

145

12.5

44.2

14.8

140

14.8

127

28.2

136

3.71

136

1.4

13.4

128

11.7

114

38.9

43.2

11.9

7.58

28.8

136

17.4

107

21.2

145

8.13

0.165

0.007

1.486

0.203

0.631

0.019

0.631

0.230

0.391

0.005

0.166

0.139

0.611

0.006

0.611

0.075

1.213

0.043

1.213

0.265

0.624

0.002

0.624

0.216

0.375

0.005

0.022

0.040

0.407

0.103

0.378

0.024

0.275

0.052

0.099

0.029

0.099

0.056

0.557

0.028

0.501

0.094

0.405

0.048

0.311

0.109

0.550

0.038

0.441

0.074

0.071

0.019

0.140

0.022

0.169

0.088

0.216

0.029

0.157

0.208

0.407

0.051

0.199

0.037

0.254

0.037

0.254

0.085

0.218

0.037

0.218

0.065

0.239

0.054

0.239

0.183

0.140

0.062

110

11.6

35.7

15.3

130

15.3

116

25.6

128

4.77

128

1.1

10.1

98.7

10.1

91.3

24.0

41.8

8.19

17.9

18.0

105

14.0

86.8

18.2

110

7.00

0.173

0.026

1.136

0.159

0.573

0.002

0.573

0.253

0.416

0.006

0.169

0.094

0.573

0.015

0.573

0.067

1.006

0.067

1.000

0.169

0.573

0.026

0.573

0.249

0.421

0.002

0.117

0.045

0.342

0.033

0.356

0.028

0.324

0.047

0.037

0.018

0.028

0.052

0.517

0.033

0.465

0.080

0.329

0.065

0.249

0.089

0.525

0.045

0.436

0.055

0.169

0.014

0.193

0.040

0.101

0.033

0.235

0.033

0.236

0.263

0.217

0.051

0.097

0.043

0.293

0.033

0.293

0.052

0.178

0.052

0.178

0.105

0.296

0.042

0.296

0.265

0.193

0.071

(2022) 9:10

Type 3

(PPL)

(ALM)

Type 2

(PPL)

(ALM)

Type 1(b) 940 nm

Expected residual error(×10−3) 1/2D� εE2

(PPL)

(ALM)

Type 4

(PPL)

(ALM)

Type 3

(PPL)

(ALM)

Type 2

(PPL)

(ALM)

Type 1(a) 936 nm

SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10

15

Database

10

5

FWHM (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

Momoi et al. Progress in Earth and Planetary Science
Page 12 of 22

Expected residual error(×10−3) 1/2D� εE2

(PPL)

(ALM)

Type 4

(PPL)

0.465
0.150

81.9
7.72

2.0

0.033

0.465

0.170

1.639

7.72

80.3

20.3

47.0

0.123

0.497

0.030

0.374

0.119

0.455

0.129

0.121

0.021

0.013

0.018

0.087

1.4

9.73

136

9.73

130

31.4

75.6

0.102

0.638

0.020

0.638

0.104

1.486

0.051

0.382

0.016

0.331

0.040

0.445

0.039

0.300

0.016

0.277

0.022

0.169

1.1

10.3

126

10.3

119

29.8

65.5

0.083

0.573

0.006

0.573

0.039

1.136

0.024

0.369

0.018

0.346

0.048

0.389

0.022

0.296

0.022

0.296

0.040

0.101

SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10ε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax ε(×10−3)
 R̂maxε(×10−3)
 Îmax −3εR̂max (×10 )15

Database

10

5

FWHM (nm)Table 4 (continued)

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10
Page 13 of 22

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 14 of 22

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-cm−1 intervals. Note that the lines in the top panels overlap

3.3 Relationship between convolved normalized radiances
around 940 nm and PWV

Momoi et al. (2020) investigated the relationships amongR̃ around 940 nm, PWV, and the vertical profiles of aerosols and reported two major findings. First, R̃ depends
on PWV in both almucantar and principal planes, withR̃ in the principal plane being more strongly dependent
on PWV. Second, R̃ in the principal plane depends onthe vertical aerosol profile, whereas R̃ in the almucantar
plane is nearly independent of the vertical aerosol profile. Because Momoi et al. (2020) used the SN-CKD, werepeated 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 differedfrom the assumption used in Momoi et al. (2020). The
spheroid particles used here are the kernels developedby 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 thefilter response function also differed from the function
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 describedby Momoi et al. (2020). Although the magnitude of R̃ differed from the value reported by Momoi et al. (2020),
the relationship between R̃ around 940 nm and PWV
was consistent with their findings. Other sensitivity tests (Figs. 4–6 in Momoi et al. 2020) had the samecharacteristic.
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 T̃ derived from the angular distribution of R̃,



−3
ε

Ĩmax (×10 )ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)

0.570

87.0

0.363

33.4

0.333

1.077

60.6

1.034
0.049

11.3

40.6

0.047

98.0
11.6

0.048

11.6

0.317

0.509

25.6
93.5

0.878

83.6

0.369
0.075

1.40

83.6

1.378

163

2.0

0.108

14.2

1.378

2.897

108
130

0.093

13.9

2.897

0.808

68.8
108

1.569

182

1.569
0.162

23.0

113

0.169

0.247

21.4

211

0.247
0.108

211
12.8

0.961

1.417

0.114

1.417

0.857

0.724

0.148

0.724

0.504

0.695

0.210

0.190

0.113

1.807

0.113

1.807

0.185

2.402

0.119

2.402

0.178

1.905

0.162

1.905

0.221

0.961

0.090

5.043

19.19

0.134

19.19

2.323

16.77

0.378

16.77

2.463

11.72

0.761

11.72

0.327

4.375

0.128

4.176

0.597

3.330

0.168

2.901

0.379

4.588

0.061

4.271

0.422

5.038

0.053

147

80.0

12.5

48.8

14.7

144

14.7

131

28.7

139

3.62

139

13.1

131

11.5

118

38.5

47.2

11.6

8.69

28.1

139

17.1

111

21.6

147

7.94

1.740

0.648

0.171

0.648

0.182

0.942

0.182

0.942

0.394

0.767

0.160

0.767

1.4

0.435

2.339

0.435

2.339

0.468

3.153

0.468

3.153

0.523

2.706

0.523

2.706

0.531

1.740

0.471

0.398

1.022

0.173

0.936

0.217

1.369

0.164

1.152

0.142

1.133

0.142

1.078

0.342

0.353

0.342

0.353

0.325

1.197

0.325

1.197

0.395

0.631

0.371

0.631

0.355

0.398

0.355

0.919

3.572

0.213

3.572

0.505

2.875

0.242

2.875

0.685

1.333

0.305

1.333

0.345

2.521

0.319

2.521

0.514

3.982

0.160

3.982

1.058

2.850

0.390

2.850

0.602

0.919

0.506

112

67.1

11.5

37.4

15.2

132

15.2

117

25.8

130

4.71

130

10.1

100

10.1

92.9

23.9

43.5

8.20

19.6

17.8

106

14.0

88.5

18.4

112

6.99

1.521

1.848

0.256

1.848

0.277

1.855

0.277

1.855

0.468

1.494

0.242

1.494

1.1

0.211

1.643

0.215

1.647

0.161

1.728

0.161

1.728

0.251

1.871

0.251

1.871

0.492

1.521

0.309

0.246

0.566

0.190

0.536

0.205

0.808

0.205

0.686

0.184

0.597

0.184

0.549

0.130

0.655

0.171

0.655

0.073

0.555

0.110

0.555

0.114

0.610

0.114

0.523

0.189

0.295

0.189

0.772

1.113

0.202

1.113

0.233

1.019

0.227

1.019

0.403

0.357

0.262

0.357

0.124

1.159

0.154

1.189

0.024

1.104

0.077

1.158

0.359

1.286

0.119

1.286

0.417

0.772

0.308

(2022) 9:10

(ALM)

Type 3

(PPL)

(ALM)

Type 2

(PPL)

(ALM)

Type 1(b) 940 nm

Ĩmax



ε
(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax −3εR̃max (×10 )Expected residual error (×10−3) 1/2D� εE2

(PPL)

(ALM)

Type 4

(PPL)

(ALM)

Type 3

(PPL)

(ALM)

Type 2

(PPL)

(ALM)

Type 1(a) 936 nm

SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10

15

Database

10

5

FWHM (nm)Table 5 As described for Table 4, but the reference radiances were computed with a smooth filter response function and the high-resolution (0.01cm−1) extraterrestrial solar
irradiance data

Momoi et al. Progress in Earth and Planetary Science
Page 15 of 22

Expected residual error (×10−3) 1/2D� εE2

(PPL)

(ALM)

Type 4

(PPL)

R̃max

2.0

0.017

0.389

7.82

0.017

94.6

0.391

0.049

7.82

93.5

20.1

0.357

0.466

0.116

0.466

0.339

0.938

16.35

0.279

16.35

0.807

9.72

140

9.72

134

31.2

1.4

0.117

0.754

0.130

0.767

0.171

0.124

1.305

0.140

1.265

0.173

0.194

2.827

0.206

2.839

0.213

10.2

127

10.2

120

29.7

1.1

0.205

1.634

0.218

1.647

0.256

0.167

0.809

0.187

0.809

0.190

0.190

0.945

0.211

0.966

0.202

SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10 SN-CKD WV-CKD-2 WV-CKD-5 WV-CKD-10ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)
 R̃maxε(×10−3)
 Ĩmax ε(×10−3)

15

Database

10

5

FWHM (nm)Table 5 (continued)

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10
Page 16 of 22

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 17 of 22

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

Transported dust

Components

Size distribution

Refractive index
at 940 nm

Relative weight in total
optical thickness at
500 nm

Vertical profileModeradius(μm)

Mode width

Real

ImaginaryWater-soluble

0.18

0.81

1.43

Soot

0.05

0.69

1.75

0.0074

0.90

0.44

0.07

exp (−z/H), H = 8 km

Insoluble

5.98

0.92

1.52

0.008

0.03

Dust

3.23

0.79

1.53

0.004

0.25

Water-soluble
Soot

0.18

0.81

1.43

0.0074

0.67

0.05

0.69

1.75

0.44

0.05

Insoluble

5.98

0.92

1.52

0.008

0.03

exp (−z/H), H = 4 km

exp (−z/H), H = 2 km


c)
√1
exp − (z−z, zc = 3.5 km
22σ2πσσ = 0.4 km
exp (−z/H), H = 8 km
exp (−z/H), H = 4 km

exp (−z/H), H = 2 km

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 = 0cm). 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 alookup table for the CKD method in the water vapor absorption region of 940 nm at 2-cm−1 intervals

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 theangular distribution of R̃ in the range of 340–1020 nm,
except at 940 nm; T̃ at 940 nm is estimated from theMomoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 18 of 22

Fig. 9 Analytical flow of Momoi et al. (2020). SDF, CR, and NS ratio are aerosol volume size distribution, complex refractive index, and non-sphericalratio, respectively. Aerosol channels denote weak gas absorption bands (i.e., 340, 380, 400, 500, 675, 870, and 1020 nm). SKYMAP is a new on-sitemethod 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

angular distribution of R̃ at 940 nm using aerosol properties interpolated from aerosol optical properties at 870
and 1020 nm; finally, F̃0 at 940 nm is determined from T̃
at 940 nm. After calibration, PWV can be estimated fromF̃ds using F̃0 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.
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 thesame location. The MWR measured the zenith brightness
temperature in the 22–30 GHz region at 1-min temporal resolution, and P­ WVMWR 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 verticalstructures 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 theold sky-radiometer model POM-01. In the case of the SKYMAP with WV-CKD-2, the annual mean F̃0 at 940 nm was
retrieved as 2.079 × 10−4 A, which is 7.5% greater than thevalue determined with the SN-CKD (1.933 × 10−4 A). Using
F̃0, the PWV was estimated with the DSRAD, as shown in
Fig. 10. Figure 11a illustrates that ­PWVSNCKD derived using
the SN-CKD was underestimated relative to P
­WVMWR.
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,­PWVWVCKD derived using the WV-CKD-2 showed good
agreement with ­PWVMWR (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 throughinterpolation 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Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

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 astandard lookup table for the CKD method in the radiative transfer
model RSTAR, and WV-CKD-2 is a lookup table for the CKD methodin the water vapor absorption region of 940 nm at 2-cm−1 intervals.
DSRAD estimates AOT and PWV from the calibrated direct solar
irradiance from sky-radiometer observations

AERONET photometer reached 0.01, corresponding to an
absolute PWV error of about 0.08 cm calculated using theequation of Momoi et al. (2020). In contrast, the ­PWVSNCKD
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. Thissuggests that SKYMAP/DSRAD calculation with the WVCKD is useful for accurate estimation of PWV and determination of the calibration constant F̃0 for the water vapor
absorption band of sky-radiometer observations.

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

Page 19 of 22

a LUT (WV-CKD). We found the challenges facing thecurrently used LUT (SN-CKD) 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 SN-CKD, the root mean square error of the
maximum error in sub-bands obtained with the SNCKD is 116% across the range of 10,411–10,864 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(�κ = 2, 5, 10cm−1 in WV-CKD-2, WV-CKD-5, and
WV-CKD-10, respectively). The quadrature values andnumbers 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-bandsof 10cm−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, and7, 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 thatdiffered from the conditions used for optimization of
the tables. The residual errors of convolved sky inten CKD and 
LCKD) were similar to the expected
sities (Terrors 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 normalCKD) was less strongly affected by residized radiance ( R CKD andual errors obtained using the WV-CKD than T
CKD
CKD

 CKD

L
, as Rcancels the residual errors of both T
CKD
L
and 
. 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 R̃ and measurement uncertainty (approximately 0.5%). Finally, weapplied the SKYMAP and DSRAD algorithms (Momoi
et al. 2020) to SKYNET observations and compared theresults with those of the microwave radiometer. PWVshows better agreement when derived with the WVCKD (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 tableis essential for estimating PWV from sky-radiometer
observations.

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

Page 20 of 22

Fig. 11 Comparison of PWV observed at Chiba in 2019 between the microwave radiometer (­ PWVMWR) and sky-radiometer ­(PWVSKYR) with (a)SN-CKD and (b) WV-CKD-2. The bottom value in the legend is the mean and standard deviation of the difference (­ PWVSKYR—PWVMWR). γ is thecorrelation 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-cm−1 intervals

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
k-distribution; SN-CKD: Standard lookup table for the correlated k-distribution
method in RSTAR​; RSTAR7: RSTAR version 7; PSTAR4: PSTAR version 4; PWV:
Precipitable water vapor; AERONET: Aerosol robotic network; LBL: Line-by-line;
AFGL: Air Force Geophysics Laboratory; WV-CKD: Lookup table for the correlated k-distribution method in the water vapor absorption region of 940 nm;
CA-DB: Dataset consisting of radiances at ground level under continental
averaged aerosol condition; AOT: aerosol optical thickness, FWHM: Full widthat half maximum; DU-DB: Dataset consisting of the radiances at ground level
under several aerosol conditions from Dubovik et al. (2000); MWR: Microwave
radiometer.
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 wereobtained 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/ (lastaccessed: Oct. 2021).
Authors’ contributions
This study was designed by MM and HI. The gas absorption calculation modules for RSTAR7 and the WV-CKDs were developed by MM, TN, and MS. The
sky-radiometer data were analyzed by MM, HI, KM, and KA. The sky-radiometer(S/N PS2501401) was maintained by MM, KM, and KA. The sky-radiometer (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.
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, 2013-2014), and JAXA Contract research (“Fundamental
development of radiative transfer codes for advanced utilization of earth
observation satellites,” FY2021).
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(WV-CKDs) and gas absorption calculation codes for RSTAR7 and PSTAR4 areavailable on request from the first author. The SKYMAP and DSRAD algorithms
are also available on request from the first author. The sky-radiometer data are
available from the SKYNET website (http://​www.​skynet-​isdc.​org/, last accessed:
Oct. 2021). The MWR data are available from CEReS, Chiba University (http://​atmos3.​cr.​chiba-u.​jp/​skynet/, last accessed: Oct. 2021). The AERONET sun–sky
radiometer data are available from the AERONET website (https://​aeron​et.​gsfc.​
nasa.​gov/, last access: Oct. 2021).

Declarations
Competing interests
The authors declare that they have no competing interest.
Author details
1
Center for Environmental Remote Sensing, Chiba University, 1‑33 Yayoicho,
Inage‑ku, Chiba 263‑8522, Japan. 2 Faculty of Marine Technology, Tokyo
University of Marine Science and Technology, 2‑1‑6 Etchujima, Koto‑ku,
Tokyo 135‑8533, Japan. 3 National Institute for Environmental Studies, 16‑2
Onogawa, Tsukuba, Ibaraki 305‑8506, Japan. 4 Department of Physics, Facultyof Science Division I, Tokyo University of Science, 1‑3 Kagurazaka, Shinjuku‑ku,
Tokyo 162‑8601, Japan. 5 Laboratory for Environmental Research at Mount Fuji,
2‑5‑5 Okubo, Shinjuku‑ku, Tokyo 169‑0072, Japan. 6 Graduate School of Science and Engineering, University of Toyama, 3190 Gofuku, Toyama, Toyama
930‑8555, Japan.
Received: 12 August 2021 Accepted: 7 January 2022

Momoi et al. Progress in Earth and Planetary Science(2022) 9:10

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