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**Atmospheric Measurement Techniques**
An interactive open-access journal of the European Geosciences Union

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- Abstract
- Introduction
- Overview of line-by-line code and kCompressed database
- kCARTA clear-sky radiative transfer algorithm
- Nonlocal thermodynamic equilibrium computations
- Clear-sky Jacobian algorithm
- Background thermal and temperature variation in a layer
- RTA intercomparisons: kCARTA versus LBLRTM
- Flux computations
- Scattering package included with kCARTA Fortran 90 version
- Conclusions
- Appendix A: UMBC-LBL and kCARTA downloads and auxiliary requirements
- Appendix B: Available spectral regions and f90 kCARTA features
- Appendix C: PCLSAM scattering algorithm
- Code and data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References

**Research article**
30 Jan 2020

**Research article** | 30 Jan 2020

kCARTA: a fast pseudo line-by-line radiative transfer algorithm with analytic Jacobians, fluxes, nonlocal thermodynamic equilibrium, and scattering for the infrared

^{1}JCET, University of Maryland Baltimore County, Baltimore, Maryland, USA^{2}Dept. of Physics, University of Maryland Baltimore County, Baltimore, Maryland, USA^{}deceased

^{1}JCET, University of Maryland Baltimore County, Baltimore, Maryland, USA^{2}Dept. of Physics, University of Maryland Baltimore County, Baltimore, Maryland, USA^{}deceased

**Correspondence**: Sergio DeSouza-Machado (sergio@umbc.edu)

**Correspondence**: Sergio DeSouza-Machado (sergio@umbc.edu)

Abstract

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A fast pseudo-monochromatic radiative transfer package using a
singular value decomposition (SVD) compressed atmospheric
optical depth database has been developed, primarily for simulating
radiances from hyperspectral sounding instruments (resolution ≥0.1 cm^{−1}). The package has been tested extensively for clear-sky
radiative transfer cases, using field campaign data and satellite
instrument data. The current database uses HITRAN 2016 line
parameters and is primed for use in the spectral region spanning 605 to 2830 cm^{−1}. Optical depths for other spectral regions (15–605 and 2830–45 000 cm^{−1}) can also be generated for use by kCARTA. The
clear-sky radiative transfer model computes the background thermal
radiation quickly and accurately using a layer-varying diffusivity
angle at each spectral point; it takes less than 30 s (on a
2.8 GHz core using four threads) to complete a radiance calculation
spanning the infrared. The code can also compute non-local
thermodynamic equilibrium effects for the 4 µm CO_{2} region, as well
as analytic temperature, gas and surface Jacobians. The package also
includes flux and heating rate calculations and an interface to an
infrared scattering model.

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How to cite.

DeSouza-Machado, S., Strow, L. L., Motteler, H., and Hannon, S.: kCARTA: a fast pseudo line-by-line radiative transfer algorithm with analytic Jacobians, fluxes, nonlocal thermodynamic equilibrium, and scattering for the infrared, Atmos. Meas. Tech., 13, 323–339, https://doi.org/10.5194/amt-13-323-2020, 2020.

1 Introduction

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Recent years have seen the launch and routine operation of new-generation infrared sounders on board Earth-orbiting satellites, for
the purposes of providing measurements for data assimilation into
numerical weather prediction (NWP) centers and for monitoring
atmospheric composition. These hyperspectral instruments have low-noise channels with high resolution (≥0.5 cm^{−1}) and provide
gigabytes of data daily, from about 620 to 2800 cm^{−1}. Examples include
the Atmospheric Infrared Sounder (AIRS) (Aumann et al., 2003) on board NASA's
Aqua satellite, the Infrared Atmospheric Sounding Interferometer
(IASI) on board the Metop satellites (Clerbaux et al., 2009), and the Cross
Track Infrared Sounder (CrIS) on board the Suomi and JPSS-1 satellites
(Han et al., 2013).

The radiances measured by these instruments are obtained under all-sky conditions (i.e., clear or cloudy). Publicly available thermodynamic profiles retrieved from these voluminous data are presently performed after cloud-clearing the radiances (Susskind et al., 1998). Monochromatic line-by-line (MNLBL) codes are too slow for use in the operational retrievals from the cloud-cleared radiances. Instead, optical depths (or transmittances) produced by these MNLBL codes are parametrized for use in fast radiative transfer algorithms (RTAs), at the instrument resolution. The accuracy of the retrieved products depends on the accuracy of the fast models, which underlines the importance of the accuracy of line parameters and line shapes used in MNLBL codes, particularly the far-wing effects.

To satisfy the accuracy requirements of convolved optical depths and
radiances used in developing and testing these fast models, the high-altitude (Doppler broadened) lines need to have monochromatic spectral
resolutions of 0.0025 cm^{−1} or better over the almost 2500 cm^{−1} span of a
typical infrared sounder. Using true MNLBL codes to produce optical
depths for training the fast models is computationally intensive, as
accurate line shapes needed to be computed for millions of spectral
points, each at about 100 layers spanning a 0–80 km atmosphere, for
about 40–50 gases; this has to be done for 50 or more profiles. The
acceleration of this part of the process, needed to develop a fast RTA
for the AIRS sounder, was the motivating factor behind the development
of the work presented here. For this we also developed a line-by-line
code (referred to as UMBC-LBL) to produce an accurate precomputed
database of monochromatic atmospheric optical depths. Singular value
decomposition (SVD) was then used to produce a highly compressed
database (referred to as the kCompressed database; Strow et al., 1998)
that is highly accurate, relatively small, and easy to use. When
coupled to an accurate radiative transfer code, this pseudo-line-by-line package can be used as a starting point for developing
tools for atmospheric retrievals (Rodgers, 2000). The key point to note
is that though some optical depth information may be lost due to the
compression and/or resolution of the database, the convolved
radiances are very accurate.

To compute optical depths and radiances at any level for an arbitrary Earth atmospheric thermodynamic + gas profile, we paired together an uncompression algorithm for the kCompressed database with a one-dimensional clear-sky radiative transfer algorithm (RTA). The RTA works for both a down-looking and an up-looking instrument, with geometric ray tracing accounting for the spherical atmospheric layers. The generation of monochromatic transmittances from the compressed database is at least an order of magnitude faster than using a MNLBL code; for the long paths in the atmosphere the computed transmittances are smooth and well behaved and can be used to develop fast forward models. Radiances computed using the compressed database are as accurate as those computed with a MNLBL code as our compression procedure introduces errors well below spectroscopy errors (Strow et al., 1998).

The entire package is called kCARTA, which stands for “kCompressed Atmospheric Radiative Transfer Algorithm”. Although kCARTA is much slower than fast forward models which use effective convolved transmittances, it is much more accurate, and it can be used to generate optical depths and transmittances for developing the faster models. An example is the StandAlone Radiative Transfer Algorithm (SARTA) (Strow et al., 2003) for which kCARTA is the reference forward model; SARTA is used to retrieve level 2 geophysical products from the AIRS (Strow et al., 2003) and CrIS (Gambacorta, 2013) instruments. Other fast forward models for the infrared which parametrize the transmittances of the finite width instrument channels include a principal-component-based radiative transfer model (PCRTM; Liu et al., 2006), Radiative Transfer for TIROS Operational Vertical Sounder (RTTOV; Saunders et al., 1999), and the Jülich Rapid Spectral Simulation Code (JURASSIC; Hoffmann and Alexander, 2009).

kCARTA also includes algorithms to rapidly compute analytic Jacobians
and is available in a Fortran 90 (f90) package. This package (v1.21, April
2019) uses some of the newer Fortran features such as implicit loops and
function overloading and modules, and it includes code for computing
fluxes, heating rates, and the effects of cloud and aerosol scattering
using the Parametrization of Clouds for Longwave Scattering in
Atmospheric Models (PCLSAM) (Chou et al., 1999) algorithm. While kCARTA was
developed for use in the infrared region (605–2830 cm^{−1}), it is trivial
to extend the database out in either direction, to span the far
infrared to the ultra-violet. A clear-sky-only radiance + Jacobian
MATLAB version is also available.

The speed and accuracy plus available run-time options of the code
make it a very attractive alternative to other existing line-by-line
codes. The literature is replete with papers and books describing
spectroscopic calculations and monochromatic radiative transfer and flux
calculations (see for example
Goody and Yung, 1989; Edwards, 1992; Clough et al., 1992; Clough and Iacono, 1995; Tjemkes et al., 2002; Buehler et al., 2011; Schreier et al., 2014; Dudhia, 2017; Vincent and Dudhia, 2017), so
here we chose to emphasize the features (and limitations) of kCARTA that
would interest researchers working in these and related fields, and we
apply kCARTA to quantify how different spectroscopic databases impact
simulated clear-sky top-of-atmosphere (TOA) brightness temperatures.
Focusing on the infrared (605–2830 cm^{−1}) region, this paper begins
with a description of the line-by-line code and the kCompressed
database, followed by a description of the clear-sky radiative
transfer algorithm, together with Jacobians. The paper then discusses
in detail some of the internal machinery of kCARTA, such as a background
thermal computation developed for kCARTA, flux computations, and
scattering packages.

2 Overview of line-by-line code and kCompressed database

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For an input set of (average temperature, pressure, and gas amount; in molecules per cubic centimeter) parameters, a custom monochromatic line-by-line code (UMBC-LBL) (De Souza-Machado et al., 2002) has been developed in order to accurately compute optical depths. This code defaults to the Van Vleck and Huber line shape (Van Vleck and Huber, 1977; Clough et al., 1980) for almost all molecules, using spectroscopic line parameters from the high-resolution transmission (HITRAN) molecular absorption database.

For each spectral region the UMBC-LBL optical depth computations are
divided into bins that are typically 1 cm^{−1} wide in the infrared. The
optical depth in each of these bins is accumulated in three stages as
shown in Fig. 1. (1) In the fine mesh stage absorption due
to line centers within 1 cm^{−1} of the bin edges is included at a very
high resolution (typically 0.0005 cm^{−1}) and then integrated to the output (typically 0.0025 cm^{−1}) grid using a five-point boxcar; in Fig. 1 these are the red lines within the bin edges at ±0.5 cm^{−1} and the blue lines within 1 cm^{−1} of the same bin edges. (2) In the medium mesh stage absorption from line centers within 1–2 cm^{−1} of the
bin edges is included at 0.1 cm^{−1} resolution, shown in green in the
figure. (3) Finally, in the coarse mesh stage absorption from line
centers within 2–25 cm^{−1} of the bin edges are included at 0.5 cm^{−1}
resolution (none shown in the figure); for (2) and (3) the results are
interpolated to the output grid. The black line is the accumulated
optical depth for that bin.

We note three points here. First, the default kCARTA uses 0.0005 cm^{−1}
resolution between 605 and 880 cm^{−1} and 0.0025 cm^{−1} from 805 to 2830 cm^{−1} (after
the five-point boxcar). Section 7 demonstrates that
convolved radiances computed with these resolutions compare very well
against other RTAs, especially after convolving with a typical
hyperspectral sounder response function. Second, the above
line-by-line computations are very similar to those in other models
(Edwards, 1992; Dudhia, 2017), but we use the “medium” bins and “coarse”
bins for the lines whose centers are within the intervals lying within ±
(1, 2) and ± (2, 25) cm^{−1}, respectively, of the bin edges, instead of
using only coarse bins. Thirdly we note that for most Earth
atmosphere molecules, the line strength–gas amount
combination means the optical depth contribution due to line centers
further than 25 cm^{−1} away from the bin is negligible and can be ignored
(Dudhia, 2017); the exception for the Earth atmosphere are H_{2}O and
CO_{2}, which have countless strong lines further than 25 cm^{−1} away from
bin edges. To speed up the optical depth calculations, the weak but
non-negligible contribution from these “far lines” is added in using
a continuum optical depth contribution which depends on temperature
and gas absorber amount.

The above steps are followed for almost all molecules. Modifications
to the above steps are needed for water vapor (which is separated into
the traditional “basement” plus “continuum” contributions;
Clough et al., 1980, 1989) and CO_{2} in the 4 and 15 µm region, which needs
line-mixing line shapes (Strow and Pine, 1988; Tobin et al., 1996; Niro et al., 2005; Lamouroux et al., 2015). Other
molecules have optical depths that are more easily modeled with the
Van Huber line shape, though recently the infrared absorption due
to CH_{4} has been modeled using line mixing (Tran et al., 2006). The
UMBC-LBL optical depth computation for water vapor should be robust at
all frequencies and allows the addition of water continuum models
such as the recent MT CKD 3.2 coefficients (Mlawer et al., 2012). Spectra from
UMBC-LBL have been extensively compared against optical depths
computed by models such as the Line-by-Line Radiative Transfer Model
(LBLRTM) (Clough et al., 1992, 2005) and the General line-by-line
Atmospheric Transmittance and Radiance model (GENLN2)
(Edwards, 1992).

When applied toward any realistic Earth atmosphere simulation for an observing instrument, the UMBC-LBL calculations described above become impractically slow as they need to be performed for multiple gases in the atmosphere, over ∼100 atmospheric layers and encompassing a wide spectral range.

UMBC-LBL is therefore primarily used to generate an uncompressed database of lookup tables as described below. For each gas other than water vapor, the spectra are computed using the US Standard Atmosphere temperature profile, as well as five temperature offsets (in steps of 10 K) on either side of the temperature profile, for a total of 11 temperatures. Tests using NWP profiles show this is usually sufficient everywhere except for a handful of locations over the winter in Antarctica, which could fall slightly outside the coldest offset (on average by about 3 K) between 600 and 1000 mb; kCARTA handles these extreme cold cases by extrapolating what has been compressed and zero checking the optical depths.

The default infrared database spans 605–880 and 805–2830 cm^{−1},
broken up into 10 000 point intervals that are 5 and 25 cm^{−1} wide,
respectively. Each file contains matrices to compute optical depths
for these 10 000 points at the set resolution. The 100 average
pressure layers used in making the database are from the AIRS Fast
Forward Model. The layers span from 1100 to 0.005 mb (about ground level
to 85 km), and were chosen such that there is less than 0.1 K
brightness temperature (BT) errors in the simulated AIRS
radiances. The layers are about 200 m thick at the bottom of the
atmosphere, gradually getting thicker with height (about 0.65 km at 10 km and 6 km at an altitude of 80 km).

These 10$\phantom{\rule{0.125em}{0ex}}\mathrm{000}\times \mathrm{100}\times \mathrm{11}$ optical depth intervals are then
compressed using singular value decomposition (SVD) to produce the
kCompressed database. Each compressed file will have a matrix of basis
vectors **B**
(size 10 000×*N*) and compressed optical depths
**D**^{′} (size $N\times \mathrm{100}\times \mathrm{11}$), where *N* is the number of
significant singular vectors found. The prime denotes the compression
worked more efficiently when the optical depths were scaled to the
(1∕4)th power (Strow et al., 1998; Rodgers, 2000).

The self broadening of water is accounted for by generating
monochromatic lookup tables for the reference water amount, multiplied
by 0.1, 1.0, 3.3, 6.7, and 10.0 at the 11 temperature profiles
specified above, meaning **D**^{′} for water will have an extra
dimension of length 5. Note that for the infrared we treat the HDO
isotope (HITRAN isotope 4) as a separate gas from the rest of the
water vapor isotopes.

The compressed optical depths **D**^{′} vary smoothly in pressure,
meaning the user is not limited to only using the 100 AIRS layers. For
an arbitrary pressure layering, the lookup tables are uncompressed
using spline or linear interpolation in temperature and pressure and
scaled in gas absorber amount. Temperature interpolation of matrix
**D**^{′} for an AIRS 100-layer atmosphere therefore results in a
matrix *D*^{′′} of size *N*×100, and the final
optical depths (of size 10 000×100) are computed using (*B**D*^{′′})^{4}. Both the spline and linear interpolations allow
easy computation of the analytic temperature derivatives, from which
kCARTA can rapidly compute analytic Jacobians (see Sect. 5).
The cumulative optical depth for each layer in the atmosphere is
obtained by a weighted sum of the individual gas optical depths, with
accuracy limited by that of the compressed database
(Strow et al., 1998). The interested reader is referred to Vincent and Dudhia (2017) for
a further discussion of other RTAs that use compressed databases.

The most recent kCompressed database uses line parameters from the
HITRAN 2016 database (Rothman et al., 2013; Gordon et al., 2017), which together with the
UMBC-LBL line shape models determine the accuracy of the spectral
optical depths in this database. UMBC-LBL CO_{2} line-mixing calculations
use parameters that were derived a few years ago. Newer line-mixing
models exist and we now use optical depths computed using LBLRTM
v12.8 together with the line parameter database file based on HITRAN 2012
(aer_v_3.6) and (a) CO_{2} line mixing by Lamouroux et al., 2010, 2015) and
(b) CH_{4} line mixing by (Tran et al., 2006).

In addition complete kCompressed databases for the IR using optical
depths only from HITRAN 2012, LBLRTM v12.4 code, and GEISA 2015
(Husson et al., 2015) have been generated for comparison purposes. At compile
time we usually point kCARTA to the HITRAN 2016 kCompressed database
made by UMBC-LBL, but at the run time we have switches that easily allow
us to swap in, for example, the CO_{2} and CH_{4} tables generated from
LBLRTM.

The original lookup tables for the thermal infrared occupy hundreds of
gigabytes, while the compressed monochromatic absorption coefficients
are a much more manageable 824 megabytes (218 megabytes (water + HDO) + 76 megabytes (CO_{2}) + 530 megabytes (about 40 other molecular and 30
cross-section gases)). A general overview of some of the factors
involved in compressing lookup tables for use in speeding up
line-by-line codes is found in Vincent and Dudhia (2017), while more details
about the detailed testing and generation of the kCARTA SVD compressed
database are found in Strow et al. (1998). Appendix B discusses
the extension of the database to span 15 to 44 000 cm^{−1}, though we
note that kCARTA lacks built-in accurate scattering calculations in the
shorter wavelengths. In order to resolve the narrow Doppler lines at
the top of the atmosphere, the resolution *δ**ν* of the spectral
bands in Appendix B is adjusted according to $\mathit{\delta}\mathit{\nu}\sim {\mathit{\nu}}_{\mathrm{0}}\sqrt{({k}_{\mathrm{b}}T/m)}/c$, where *ν*_{0} is the band center,
and *T* and *m* are the temperature and mass of the molecule, respectively,
while *k*_{b} and *c* are Boltzmann's constant and speed of light.

The default kCARTA mode is to use the first 42 molecular gases in the HITRAN database, together with about 30 cross-section gases, for which we have reference profiles. If the user does not provide the profiles for any of these gases, kCARTA uses the US standard profile for that gas. The user can also choose to only use a selected number of specified gases. While running kCARTA, the user can then define different sets of mixed paths, where some of the gases are either turned off or the entire profile is multiplied by a constant number, which is very useful when for example we want to include only certain gases when we parametrize optical depths for SARTA.

3 kCARTA clear-sky radiative transfer algorithm

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As a stream of radiation propagates through a layer, the change in diffuse beam
intensity *R*(*ν*) in a plane-parallel medium is given by the standard
Schwarzschild equation (Liou, 1980; Goody and Yung, 1989; Edwards, 1992):

$$\begin{array}{}\text{(1)}& \mathit{\mu}{\displaystyle \frac{\mathrm{d}R\left(\mathit{\nu}\right)}{\mathrm{d}{k}_{\mathrm{e}}}}=-R\left(\mathit{\nu}\right)+J\left(\mathit{\nu}\right),\end{array}$$

where *μ* is the cosine of the viewing angle, *k*_{e} is the
extinction optical depth, *ν* is the wavenumber, and *J*(*ν*) is the
source function. For a non-scattering clear sky, the source
function is usually the Planck emission *B*(*ν*,*T*) at the layer
temperature *T*, leading to an equation that can easily be solved for
an individual layer. The general solution for a down-looking instrument
measuring radiation propagating up through a clear-sky atmosphere can be
written in terms of four components:

$$\begin{array}{}\text{(2)}& R\left(\mathit{\nu}\right)={R}_{\mathrm{s}}\left(\mathit{\nu}\right)+{R}_{\text{lay}}\left(\mathit{\nu}\right)+{R}_{\mathrm{th}}\left(\mathit{\nu}\right)+{R}_{\text{solar}}\left(\mathit{\nu}\right),\end{array}$$

which are the surface, layer emissions, and downward thermal and solar terms, respectively. In terms of integrals the expressions can be written as (see e.g., Liou, 1980; Dudhia, 2017)

$$\begin{array}{}\text{(3)}& \begin{array}{rl}& R(\mathit{\nu},\mathit{\theta})={\mathit{\u03f5}}_{\mathrm{s}}\left(\mathit{\nu}\right)B(\mathit{\nu},{T}_{\mathrm{s}}){\mathit{\tau}}_{\text{atm}}(\mathit{\nu},\mathit{\theta})\\ & +\underset{\text{surface}}{\overset{\text{TOA}}{\int}}B(\mathit{\nu},T(z\left)\right){\displaystyle \frac{\partial \mathit{\tau}(\mathit{\nu},\mathit{\theta})}{\partial s}}\mathrm{d}s+{\displaystyle \frac{\mathrm{1}-{\mathit{\u03f5}}_{\mathrm{s}}\left(\mathit{\nu}\right)}{\mathit{\pi}}}{\mathit{\tau}}_{\text{atm}}\left(\mathit{\nu}\right)\\ & \int \mathrm{d}{\mathrm{\Omega}}^{+}\underset{\text{TOA}}{\overset{\text{surface}}{\int}}B(\mathit{\nu},T(s\left)\right){\displaystyle \frac{\partial \mathit{\tau}\left(\mathrm{\Omega}\right)}{\partial s}}\mathrm{cos}\left(\mathit{\theta}\right)\mathrm{d}s\\ & +{\mathit{\rho}}_{\mathrm{s}}\left(\mathit{\nu}\right){B}_{\u22a1}\left(\mathit{\nu}\right)\mathrm{cos}\left({\mathit{\theta}}_{\u22a1}\right){\mathit{\tau}}_{\text{atm}}(\mathit{\nu},{\mathit{\theta}}_{\u22a1}){\mathit{\tau}}_{\text{atm}}(\mathit{\nu},\mathit{\theta})\end{array},\end{array}$$

where *B*(*ν*,*T*) is the Planck radiance at temperature *T*, *T*_{s} is
the skin surface temperature, *ϵ*_{s} and *ρ*_{s} are the surface
emissivity and reflectivity, *B*_{⊡}(*ν*) is the solar radiance
at TOA, *θ*_{⊡} is the solar zenith angle, *θ* is the
satellite viewing angle, *τ*(*ν*,*θ*) is the transmission at
angle *θ*, and *τ*_{atm} is the total atmospheric
transmission. The dΩ^{+} in the middle term indicates
integration over the upper hemisphere.

In what follows we discretize Eq. (2) so that
layer *i*=1 is the bottom and *i*=*N* (=100) the uppermost layer,
schematically shown in Fig. 2 for a clear-sky
four-layer atmosphere, with O being the center of the
Earth. A is the satellite while S is the satellite
sub-point directly below it. Point P is the ground scene being
observed by the satellite (slightly away from nadir), and N is the local
normal at P. ∠SAP is the satellite scan angle while ∠APN is
the satellite zenith angle *θ*; ∠NPI is the solar zenith
angle *θ*_{⊡}. Note that as the radiation propagates
through the pressure layers from P to H_{1} to H_{2} to H_{3}
to H_{4} to A, the local angle (between the radiation ray and
the local normal at any of the concentric circles) keeps changing due
to the spherical geometry of the layers (refraction effects can also be included).

The default mode of kCARTA (f90 version) assumes linear variation in layer temperature with optical depth, uses a background thermal diffusive angle that varies with the layer-to-ground optical depth (instead of a constant value typically assumed to be cos${}^{-\mathrm{1}}(\mathrm{3}/\mathrm{5})$), and performs ray tracing to account for the spherical atmospheric layers (but with no density effects). The f90 version of kCARTA also allows the user to choose constant layer temperature and to choose alternate ways of computing the background term, which will be discussed in Sect. 6.

Here we describe radiative transfer for the constant layer temperature
case; see Sect. 6.2 for a short discussion
when using the linear-in-tau option. For an arbitrary layer *i* with
(nadir) optical depth *k*_{i}(*ν*), the transmittance of a beam
passing from the bottom to the top of the layer at angle *θ* is
given by ${\mathit{\tau}}_{i}(\mathit{\nu},\mathit{\theta})={e}^{-{k}_{i}\left(\mathit{\nu}\right)/\mathrm{cos}\left(\mathit{\theta}\right)}$. The
transmittance from the top of layer *i* to space is then the product
of the individual transmittances of the layers above *i*

$$\begin{array}{}\text{(4)}& {\displaystyle}{\mathit{\tau}}_{i+\mathrm{1}\to \text{TOA}}(\mathit{\nu},\mathit{\theta})={\mathit{\pi}}_{j=i+\mathrm{1}}^{N}{\mathit{\tau}}_{j}(\mathit{\nu},\mathit{\theta}),\end{array}$$

with the special case of transmission from ground to space (*i*=0) involving all *N* layers.

The individual contributions to the upwelling radiance are then computed as follows.

The kCARTA surface emission is given by

$$\begin{array}{}\text{(5)}& {R}_{\mathrm{s}}\left(\mathit{\nu}\right)={\mathit{\u03f5}}_{\mathrm{s}}\left(\mathit{\nu}\right)B(\mathit{\nu},{T}_{\mathrm{s}}){\mathit{\tau}}_{\text{GND}\to \text{TOA}}(\mathit{\nu},\mathit{\theta}),\end{array}$$

where *ϵ*(*n**u*) is the user-supplied emissivity.

The atmospheric absorption and re-emission is modeled as

$$\begin{array}{}\text{(6)}& {R}_{\text{lay}}=\sum _{i=\mathrm{1}}^{i=N}B(\mathit{\nu},{T}_{i})(\mathrm{1.0}-{\mathit{\tau}}_{i}(\mathit{\nu}\left)\right){\mathit{\tau}}_{i+\mathrm{1}\to \text{TOA}}(\mathit{\nu},\mathit{\theta}).\end{array}$$

Layers with negligible absorption (*τ*_{i}→1) contribute
negligibly to the overall radiance, while those with large optical depths
(*τ*_{i}→0) “black” out radiation from below.
(1.0−*τ*_{i}(*ν*)) is the emissivity of the layer while
$(\mathrm{1.0}-{\mathit{\tau}}_{i}(\mathit{\nu}\left)\right){\mathit{\tau}}_{i+\mathrm{1}\to \text{TOA}}(\mathit{\nu},\mathit{\theta})$ is the weighting
function *W*_{i} of the layer.

The atmosphere also emits radiation downward, at all angles, in a manner analogous to the upward layer emission just discussed. The total background thermal radiance at the surface is an integral over all (zenith and azimuth) radiance streams propagating from the top of the atmosphere (set to 2.7 K) to the surface. This is time consuming to compute using quadrature, and one approximation is to use a single effective (or diffusivity) angle of ${\mathit{\theta}}_{\text{diff}}={\mathrm{cos}}^{-\mathrm{1}}(\mathrm{3}/\mathrm{5})$ at all layers and wavenumbers:

$$\begin{array}{}\text{(7)}& \begin{array}{rl}& {R}_{\text{th}}^{\text{surface}}\left(\mathit{\nu}\right)=\mathit{\pi}{\mathit{\rho}}_{\mathrm{s}}\sum _{i=N}^{i=\mathrm{1}}B\left({T}_{i}\right)\left[{\mathit{\tau}}_{i-\mathrm{1}\to \text{ground}}(\mathit{\nu},{\mathit{\theta}}_{\text{diff}})\right.\\ & -{\mathit{\tau}}_{i\to \text{ground}}(\mathit{\nu},{\mathit{\theta}}_{\text{diff}})\left.\right].\end{array}\end{array}$$

The summation is from the top of the atmosphere to the ground, and *ρ*_{s} is the
surface reflectivity discussed above. Current sounders have channel
radiance accuracy better than 0.2 K, so while the above term is much
smaller than the surface or upwelling atmospheric emission
contributions, it has to be computed accurately. Section 6 includes a detailed discussion of how kCARTA
improves the accuracy of this background term by using a lookup table
to rapidly compute a spectrally and layer-varying diffusive angle.

Letting the surface reflectivity be denoted by ${\mathit{\rho}}_{\mathrm{s}}(\mathit{\nu},\mathit{\theta},\mathit{\varphi})$, then the solar contribution to the TOA radiance is given by

$$\begin{array}{}\text{(8)}& \begin{array}{rl}& {R}_{\u22a1}\left(\mathit{\nu}\right)={\mathit{\rho}}_{\mathrm{s}}(\mathit{\nu},\mathit{\theta},\mathit{\varphi}){B}_{\u22a1}\left(\mathit{\nu}\right)\mathrm{cos}\left({\mathit{\theta}}_{\u22a1}\right)\\ & \times {\mathit{\tau}}_{N\to \text{ground}}(\mathit{\nu},{\mathit{\theta}}_{\u22a1})\left){\mathit{\tau}}_{\text{ground}\to \text{TOA}}\right(\mathit{\nu},\mathit{\theta}\left)\right){\mathrm{\Omega}}_{\u22a1},\end{array}\end{array}$$

where *B*_{⊡}(*ν*) is the solar radiation at the top of
the atmosphere and accounts for the solar disk. Over ocean, if the wind
speed and solar and satellite azimuth angles are known, the
reflectivity can be precomputed using the bidirectional reflectance
distribution function (BRDF) and input to kCARTA; see for example
Appendix C in Nalli et al., 2016. It is not easy to compute the BRDF over
land, and the reflectivity could be simply modeled as ${\mathit{\rho}}_{\mathrm{s}}\left(\mathit{\nu}\right)=\frac{\mathrm{1}-{\mathit{\u03f5}}_{\mathrm{s}}\left(\mathit{\nu}\right)}{\mathit{\pi}}$.

${\mathrm{\Omega}}_{\u22a1}=\mathit{\pi}({r}_{\mathrm{s}}/{d}_{\mathrm{se}}{)}^{\mathrm{2}}$ is the solid angle subtended
at the Earth by the sun, where *r*_{e} is the radius of the sun and
*d*_{se} is the Earth–sun distance. The solar radiation incident at
the TOA *B*_{⊡}(*ν*) comes from data files related to the ATMOS
mission (Farmer et al., 1987; Farmer and Norton, 1989) and is modulated by the angle the sun makes with the
vertical, cos (*θ*_{⊡}) (day-of-year effects are not included in the
Earth–sun distance).

4 Nonlocal thermodynamic equilibrium computations

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During the daytime, incident solar radiation is preferentially
absorbed by some CO_{2} and O_{3} infrared bands, whose kinetic
temperature then differs from the rest of the bands or molecules. This
leads to enhanced emission by the lines in these bands.

Limb sounders detect NLTE effects in the 15 µm CO_{2} bands (and in
other molecular bands, for example O_{3}) due to the extremely long
paths involved, but these are not modeled in the package as kCARTA is
designed for nadir sounders.

For a nadir sounder, the most important effects are seen in the CO_{2} 4 µm (*ν*_{3}) band. kCARTA includes a computationally intensive
line-by-line nonlocal thermodynamic equilibrium (NLTE) model to
calculate the effects for this CO_{2} band. The model requires the
kinetic temperature profile and NLTE vibrational temperatures of the
strong bands in this region to compute the optical depths and Planck
modifiers for the strong NLTE bands and the weaker LTE bands
(Edwards et al., 1993, 1998; Lopez-Puertas and Taylor, 2001; Zorn et al., 2002), which are then used to compute a
monochromatic top-of-atmosphere nadir radiance.

AIRS provided the first high-resolution nadir data of NLTE
in the 4 µm CO_{2} band. Using the kCARTA NLTE line-by-line model, a
fast NLTE model (De Souza-Machado et al., 2007) for sounders has already been
developed, which is used in the NASA AIRS L2 operational product.

5 Clear-sky Jacobian algorithm

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Retrievals of atmospheric profiles (temperature, humidity, and trace gases) minimize the differences between observations and calculations, by adjusting the profiles using the linear derivatives (or Jacobians) of the radiance with respect to the atmospheric parameters. This section describes the computation of analytic Jacobians by kCARTA. Note that kCARTA currently computes Jacobians and weighting functions using a constant layer temperature assumption. For a downward-looking instrument, for simplicity consider only the upwelling terms in the radiance equation (atmospheric layer emission and the surface terms). Assuming a nadir satellite viewing angle, the solution to Eq. (1) is

$$\begin{array}{}\text{(9)}& \begin{array}{rl}& R\left(\mathit{\nu}\right)={\mathit{\u03f5}}_{\mathrm{s}}B({T}_{\mathrm{s}},\mathit{\nu}){\mathit{\tau}}_{\mathrm{1}\to \text{TOA}}\left(\mathit{\nu}\right)\\ & +{\mathrm{\Sigma}}_{i=\mathrm{1}}^{i=N}B({T}_{i},\mathit{\nu})(\mathrm{1.0}-{\mathit{\tau}}_{i}(\mathit{\nu}\left)\right){\mathit{\tau}}_{i+\mathrm{1}\to \text{TOA}}\left(\mathit{\nu}\right).\end{array}\end{array}$$

Differentiation with respect to the *m*-layer variable *s*_{m} (gas amount or layer temperature ${s}_{m}={q}_{m\left(g\right)},{T}_{m}$) yields

$$\begin{array}{}\text{(10)}& \begin{array}{rl}& {\displaystyle \frac{\partial R\left(\mathit{\nu}\right)}{\partial {s}_{m}}}={\mathit{\u03f5}}_{\mathrm{s}}B\left({T}_{\mathrm{s}}\right){\displaystyle \frac{\partial {\mathit{\tau}}_{\mathrm{1}\to \text{TOA}}\left(\mathit{\nu}\right)}{\partial {s}_{m}}}\\ & +\sum _{i=\mathrm{1}}^{N}B({T}_{i},\mathit{\nu})(\mathrm{1.0}-{\mathit{\tau}}_{i}(\mathit{\nu}\left)\right){\displaystyle \frac{\partial {\mathit{\tau}}_{i+\mathrm{1}}\left(\mathit{\nu}\right)}{\partial {s}_{m}}}\\ & +\sum _{i=\mathrm{1}}^{N}{\mathit{\tau}}_{i+\mathrm{1}\to \text{TOA}}\left(\mathit{\nu}\right){\displaystyle \frac{\partial}{\partial {s}_{m}}}\left[B({T}_{i},\mathit{\nu})(\mathrm{1.0}-{\mathit{\tau}}_{i}(\mathit{\nu}\left)\right)\right],\end{array}\end{array}$$

where, as usual, ${\mathit{\tau}}_{m}\left(\mathit{\nu}\right)={e}^{-{k}_{m}\left(\mathit{\nu}\right)}$ and ${\mathit{\tau}}_{m\to \text{TOA}}\left(\mathit{\nu}\right)={\mathrm{\Pi}}_{j=m}^{N}{e}^{-{k}_{j}\left(\mathit{\nu}\right)}$. The differentiation yields

$$\begin{array}{}\text{(11)}& \begin{array}{rl}& {\displaystyle \frac{\partial R\left(\mathit{\nu}\right)}{\partial {s}_{m}}}=\left[{\mathit{\u03f5}}_{\mathrm{s}}B\left({T}_{\mathrm{s}}\right){\mathit{\tau}}_{\mathrm{1}\to \text{TOA}}\right](-\mathrm{1}){\displaystyle \frac{\partial {k}_{m}\left(\mathit{\nu}\right)}{\partial {s}_{m}}}\\ & +\left[\sum _{i=\mathrm{1}}^{m-\mathrm{1}}(\mathrm{1.0}-{\mathit{\tau}}_{i}(\mathit{\nu}\left)\right){B}_{i}\left(\mathit{\nu}\right)\mathit{\tau}(\mathit{\nu}{)}_{i+\mathrm{1}\to \text{TOA}}\right](-\mathrm{1}){\displaystyle \frac{\partial {k}_{m}\left(\mathit{\nu}\right)}{\partial {s}_{m}}}\\ & +\left[(\mathrm{1.0}-{\mathit{\tau}}_{m}(\mathit{\nu}\left)\right){\displaystyle \frac{\partial {B}_{m}\left(\mathit{\nu}\right)}{\partial {s}_{m}}}-B({T}_{m},\mathit{\nu}){\displaystyle \frac{\partial {\mathit{\tau}}_{m}\left(\mathit{\nu}\right)}{\partial {s}_{m}}}\right]\\ & {\mathit{\tau}}_{m+\mathrm{1}\to \text{TOA}}\left(\mathit{\nu}\right).\end{array}\end{array}$$

The individual Jacobian terms $\frac{\partial {k}_{m}}{\partial {s}_{m\left(g\right)}}$ are rapidly computed by kCARTA, as follows. The gas amount derivative is simply $\frac{\partial {k}_{m}}{\partial {q}_{m\left(g\right)}}=\frac{{k}_{m}}{{q}_{m\left(g\right)}}$ (with added complexity for water, to account for self-broadening), and the temperature derivative $\frac{\partial {k}_{m}}{\partial T}$ is cumulatively obtained while kCARTA is performing the temperature interpolations during the individual gas database uncompression.

The solar and background thermal contributions are also included in the Jacobian calculations. The thermal background Jacobians are computed at ${\mathrm{cos}}^{-\mathrm{1}}(\mathrm{3}/\mathrm{5})$ at all levels, for speed. This would lead to slight differences when comparing the Jacobians computed as above to those obtained using finite differences. kCARTA also computes the weighting functions and Jacobians with respect to the surface temperature and surface emissivity.

6 Background thermal and temperature variation in a layer

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In this section we take a closer look at the computation of downwelling background thermal radiation and layer temperature variation.

The contribution of downwelling background thermal to
top-of-atmosphere upwelling radiances is negligible in regions that
are blacked out as the instrument cannot see surface-leaving emissions.
Similarly in layers/spectral regions where there is very little
absorption and re-emission, the contribution is negligible as the
effective layer emissivity (denoted by Δ*τ*_{i}(*ν*) below)
goes to zero. The background contribution thus needs to be performed most
accurately in the window regions (low but finite optical depths);
depending on the surface emissivity (and hence reflectivity) in the
window regions, in terms of BT this term contributes as much as 4 K of
the total radiance when reflected back up to the top of the
atmosphere.
The contribution at the surface by a downwelling radiance stream
propagating at angle (*θ*,*ϕ*) through layer *i* is given by

$$\begin{array}{}\text{(12)}& \begin{array}{rl}& \mathrm{\Delta}{R}_{i}(\mathit{\nu},\mathit{\theta},\mathit{\varphi})\\ & =B(\mathit{\nu},{T}_{i})(\mathrm{1.0}-{\mathit{\tau}}_{i}(\mathit{\nu},\mathit{\theta},\mathit{\varphi}\left)\right){\mathit{\tau}}_{i-\mathrm{1}\to \text{ground}}(\mathit{\nu},\mathit{\theta},\mathit{\varphi})\\ & =B(\mathit{\nu},{T}_{i})\left({\mathit{\tau}}_{i-\mathrm{1}\to \text{ground}}\right(\mathit{\nu},\mathit{\theta},\mathit{\varphi})-{\mathit{\tau}}_{i\to \text{ground}}(\mathit{\nu},\mathit{\theta},\mathit{\varphi}\left)\right),\end{array}\end{array}$$

where *θ* is the zenith and *ϕ* is the azimuth angle, and
*τ*_{i→ground} represents the layer-to-ground transmittances,
derived from layer-to-ground optical depths *x*. This equation can be
rewritten as

$$\begin{array}{}\text{(13)}& \mathrm{\Delta}{R}_{i}(\mathit{\nu},\mathit{\theta},\mathit{\varphi})=B(\mathit{\nu},{T}_{i})\times \mathrm{\Delta}{\mathit{\tau}}_{i}(\mathit{\nu},\mathit{\theta},\mathit{\varphi}).\end{array}$$

An integral over (*θ*,*ϕ*) would give the contribution from the
layer. The total downwelling spectral radiance at the surface would be a sum over
all *i* layers (and the downwelling flux at the surface would be the integral
over all wavenumbers).

The integral over the azimuth is straightforward (assuming isotropic radiation), but the integral over the zenith is more complex. Since the reflected background term is much smaller than the surface or atmospheric terms, a single stream at the effective angle ${\mathit{\theta}}_{\text{diff}}={\mathrm{cos}}^{-\mathrm{1}}(\mathrm{3}/\mathrm{5})$ (Liou, 1980) is often used as an approximation, at all layers and wavenumbers.

We have refined the computation as follows. Recall that Δ*R*(*ν*) in Eq. (13) depends on the layer-to-ground
optical depth *x*, letting *μ*=cos *θ* the integral over the
zenith (${\int}_{\mathrm{0}}^{\mathrm{1}}{e}^{-x/\mathit{\mu}}\mathit{\mu}d\mathit{\mu}={E}_{\mathrm{3}}\left(x\right)$, more commonly known
as the exponential integral of the third kind). The area under the
*E*_{3}(*x*) curve would be the total flux coming from all optical depths ($\mathrm{0}\le x\le \mathrm{\infty}$); over 77 % of this area comes from the range $\mathrm{0}\le x\le \mathrm{1}$.

Applying the mean value theorem for
integrals (MVTI) to *E*_{3}(*x*), we can write Eq. (13) in
terms of two effective diffusive angles ${\mathit{\theta}}_{\mathrm{d}}^{i},{\mathit{\theta}}_{\mathrm{d}}^{i-\mathrm{1}}$ at each layer *i*:

$$\begin{array}{}\text{(14)}& \begin{array}{rl}& \mathrm{\Delta}\mathit{\tau}(i,i-\mathrm{1})=\mathit{\tau}\left(i-\mathrm{1}\to \text{ground},{\mathit{\theta}}_{\mathrm{d}}^{i-\mathrm{1}},\mathit{\nu}\right)\\ & -\mathit{\tau}\left(i\to \text{ground},{\mathit{\theta}}_{\mathrm{d}}^{i},\mathit{\nu}\right){R}_{\text{th}}^{\text{surface}}\left(\mathit{\nu}\right)\\ & =\mathrm{2}\mathit{\pi}{\mathit{\rho}}_{\mathrm{s}}\sum _{i=N}^{i=\mathrm{1}}B(\mathit{\nu},{T}_{i})\mathrm{\Delta}\mathit{\tau}(i,i-\mathrm{1})\end{array},\end{array}$$

with the effective angles varying as a function of the layer-to-ground space optical depth of that layer and the layer immediately
below it. Numerical solutions to the MVTI show that when *x*→0 then *μ*_{d}→0.5 (or ${\mathit{\theta}}_{\mathrm{d}}\to \mathrm{60}{}^{\circ}$). Similarly, as *x*→∞ then ${\mathit{\theta}}_{\mathrm{d}}\to \mathrm{0}{}^{\circ}$, but this optically thick atmosphere means an
instrument observing from the TOA cannot see the surface, so we use a
lower limit (of 30^{∘}) for the diffusive angle. Finally when *x*=1.00 we find the special case ${\mathit{\mu}}_{\mathrm{d}}=\mathrm{0.59274}\simeq (\mathrm{3}/\mathrm{5})$. For
“optically thin” regions, the layers closest to the ground
contribute most to *R*_{th}(*ν*).

With today's high-speed computers, kCARTA uses an effective diffusive
angle *θ*_{d} tabulated as a function of layer to ground optical
depth *x*, as follows. For each 25 cm^{−1} interval spanning the infrared range
the layer *L* above which ${\mathrm{cos}}^{-\mathrm{1}}(\mathrm{3}/\mathrm{5})$ can be safely used was
determined; below this layer, the lookup table is used. The table has
higher resolution for *x*≤0.1 and becomes more coarse as *x*
increases, with the effective diffusive angle cutoff at 30^{∘} when
the optical depths are larger than about 15.

We have tested this method of computing the background thermal against
both the 20-point Gauss–Legendre quadrature and the three-point exponential
Gauss quadrature (used by LBLRTM flux computations), and we found the
method to be very accurate and fast, in terms of both the downwelling
flux at the surface and also the final TOA computed radiance, even
when the emissivity is as low as 0.8 (which means a significant
contribution from the reflected thermal). At this low emissivity value,
the constant ${\mathrm{cos}}^{-\mathrm{1}}(\mathrm{3}/\mathrm{5})$ diffusivity angle model produces a final
TOA BT which differs from the Gauss–Legendre model by as much as 1.3 K
(for the tropical profile) at for example 900 cm^{−1}, while the
exponential quadrature and our model have errors smaller than 0.005 K.

LBLRTM (Clough et al., 1992, 2005) has been extensively tested and shown to
be very accurate in its computation of optical depths, radiances, and
fluxes. In the computation of radiances, both kCARTA and LBLRTM codes
use a “linear in *τ*” layer temperature variation; the former uses
a higher-order expansion (accurate to O (*τ*^{5}) for small *τ*)
and also has an option to use “constant” layer temperature. Here we
summarize the relevant equations. For an individual layer, with lower
and upper boundary temperatures *T*_{L} and *T*_{U}, the linear in *τ*
approximation leads to the following expression for the radiance at
the top of the layer (rewritten from Eq. (13) in Clough et al., 1992):

$$\begin{array}{}\text{(15)}& \begin{array}{rl}& I\left(\mathit{\nu}\right)={I}_{\mathrm{0}}\left(\mathit{\nu}\right)T+(\mathrm{1}-T)\mathit{\left\{}{B}_{\mathrm{av}}\right(\mathit{\nu})+({B}_{\mathrm{u}}\left(\mathit{\nu}\right)-{B}_{\mathrm{av}}\left(\mathit{\nu}\right))\\ & \left(\mathrm{1}-\mathrm{2}\left({\displaystyle \frac{\mathrm{1}}{\mathit{\tau}}}-{\displaystyle \frac{T}{\mathrm{1}-T}}\right)\right)\mathit{\}},\end{array}\end{array}$$

where the optical depth *τ* includes the view angle $\mathit{\tau}={\mathit{\tau}}_{\text{layer}}/\mathrm{cos}\left(\mathit{\theta}\right)$ and transmission $T=\mathrm{exp}(-\mathit{\tau})$. *I*_{0}(*ν*) is the radiation incident at the bottom of the
layer, *B*_{av}(*ν*) is the Planck radiance corresponding to the
average layer temperature, while *B*_{u}(*ν*) is the Planck radiance
corresponding to the upper boundary. For large *τ*, *T*→0 and *I*(*ν*)→*B*_{u}(*ν*). For small *τ*→0
the expression can be further expanded as follows:

$$\begin{array}{}\text{(16)}& \begin{array}{rl}& I\left(\mathit{\nu}\right)={I}_{\mathrm{0}}\left(\mathit{\nu}\right)T+(\mathrm{1}-T)\mathit{\left\{}{B}_{\mathrm{av}}\right(\mathit{\nu})+({B}_{\mathrm{u}}\left(\mathit{\nu}\right)-{B}_{\mathrm{av}}\left(\mathit{\nu}\right))\\ & \left({\displaystyle \frac{\mathit{\tau}}{\mathrm{6}}}-{\displaystyle \frac{{\mathit{\tau}}^{\mathrm{3}}}{\mathrm{360}}}+{\displaystyle \frac{{\mathit{\tau}}^{\mathrm{5}}}{\mathrm{15}\phantom{\rule{0.125em}{0ex}}\mathrm{120}}}\right)\mathit{\}}.\end{array}\end{array}$$

Comparing to the top of layer radiance in the constant in *τ* model,

$$\begin{array}{}\text{(17)}& I\left(\mathit{\nu}\right)={I}_{\mathrm{0}}\left(\mathit{\nu}\right)T+(\mathrm{1}-T){B}_{\mathrm{av}}\left(\mathit{\nu}\right),\end{array}$$

one sees the expressions are identical if there is no temperature
variation, i.e., (*B*_{u}(*ν*)=*B*_{av}(*ν*)). The default kCARTA model
layers are approximately 0.25 km thick (or a temperature spread of
about 1.5 K for a 6 K km^{−1} lapse rate) at the bottom of the atmosphere
and about 2 km thick in the stratosphere (a temperature difference of
10 K). The gaseous absorption in these upper layers is
typically negligible, except deep inside the strongly absorbing 15
and 4 µm CO_{2} bands, which is where one would expect the largest differences
between a linear-in-tau versus a constant-in-tau temperature model.

7 RTA intercomparisons: kCARTA versus LBLRTM

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In this section we describe brightness temperature differences
(ΔBT) between kCARTA and LBLRTM. As kCARTA is designed to be
accurate for typical hyperspectral sounders, we show that after
convolution the kCARTA spectral radiances compare very well against
similarly convolved LBLRTM radiances; for completeness we also
discuss how the monochromatic brightness temperature differences (ΔBT) change as a function of
resolution in the 10–15 µm O_{3} and temperature sounding regions.

For these runs we use our 49 regression profiles, with emissivity = 1
and reflectivity = 0 to exclude differences due to reflected thermal
contributions. To exercise the kCARTA ray tracing, we use a ground
satellite zenith angle of 24.5^{∘}, which becomes a TOA satellite
scan angle of 22^{∘} (typical average sounder scan angle). The tests
are run at various kCARTA database resolutions. Note that when results
are stated for a particular resolution, this means the kCARTA database
(after five-point boxcar integration) was at this resolution; similarly
the internal LBLRTM radiances were output at a resolution such that
we can apply the same five-point boxcar for the radiance comparisons.

The comparisons are divided into two sets. For the first set of
comparisons, we use an atmosphere consisting only of
H_{2}O, CO_{2}, and O_{3}, with the optical depths generated using LBLRTM
v12.8. This is done to assess the linear-in-tau radiative transfer
while limiting differences due to spectroscopy, especially in the
high-altitude 15 µm CO_{2} and 10 µm O_{3} sounding regions. For these
tests, three resolutions spanning 605–1205 cm^{−1} were used: 0.0025, 0.0005, and 0.0002 cm^{−1}.

At low resolution (0.0025 cm^{−1}), the mean differences right on top of
the high-altitude temperature sounding lines in the 630–700 cm^{−1} region
are large (≥10 K). However, these differences drop significantly
as we increase the resolution, to within 4 K ± 1.2 K at 0.0005 cm^{−1}
(default kCARTA resolution) and 1 K ± 0.4 K at 0.0002 cm^{−1}. The
default kCARTA 0.0005 cm^{−1} resolution results are shown in Fig. 3a, b. Panel (a) is the mean while the panel (b) is the standard
deviation. Note that in the 10 µm O_{3} sounding region (where the
Doppler-broadened width of the high-altitude lines would be wider than
in the 15 µm region), the differences are consistently much smaller:
$-\mathrm{0.3}\pm \mathrm{0.1}$ K at 0.0005 cm^{−1} resolution, dropping to $-\mathrm{0.1}\pm \mathrm{0.05}$ K at 0.0002 cm^{−1} resolution.

Figure 3c, d show a zoom-in of a typical unit
wavenumber interval deep in the 15 µm region and show the
differences are zeros away from lines and largest around the peaks of
the high sounding lines, each encompassing a very narrow spectral
range of less than ∼0.005 cm^{−1}. These would be expected to
contribute minimally to the convolutions using typical sounder
spectral response functions, as will be shown below.

Taken together these mean that the kCARTA RTA is working as expected: in
the very long wavelength 15 µm CO_{2} region the differences decrease as
we increase the spectral resolution while at 10 µm the differences
remain quite small. We conjecture the remaining differences between
kCARTA and LBLRTM are due to (a) algorithms: we use Eq. (16) to the 5th order while LBLRTM may use a
Padé approximation and/or Eq. (16)
to the 1st order; (b) there may be some very slight broadening effects
right on top of the high-altitude CO_{2} lines that we have not captured
when generating the compressed database.

For the second set of monochromatic tests, kCARTA and LBLRTM used 42 molecular gases and 13 cross-section gases, using the current kCARTA
default resolution of 0.0005 cm^{−1} and 0.0025 cm^{−1} for the spectral ranges
605–880 cm^{−1} and 805–2830 cm^{−1}; the overlap region allows us to convolve the
resulting radiances with AIRS spectral response functions (SRFs). Note
that we used the default optical depths for kCARTA (currently HITRAN
2016, except for CO_{2} and CH_{4} which come from LBLRTM v12.8),
while the LBLRTM v12.8 line file is based on HITRAN 2012. We only
briefly summarize the monochromatic differences: deep in the 15 µm
they are the same as Fig. 3a, b, but are noticeably different in other
regions because of differences in underlying spectroscopy and (for
high-altitude lines) possibly also resolution; for example on top of
the 10 µm O_{3} lines they could be as large as 5 K. Instead we show
the differences after convolution with AIRS SRFs. Figure 4a, b show the biases and
standard deviations of these differences; as described the noticeable
differences at 10 and 6.7 µm arise primarily because of
spectroscopy. For completeness, panels (c) and (d) show the
mean BT spectra for the 49 regression panels (panels a and c) and the
variation in computed BT (panels b and d), which are due to profile
differences (temperature, H_{2}O, and O_{3}) as well as surface
temperatures. Any user interested in reducing the monochromatic
differences could easily do so by generating and using higher
resolution compressed databases.

8 Flux computations

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Longwave fluxes at the top and bottom of the atmosphere, as well as
the heating and cooling rates, are computed by integrating spectral
radiances from Eq. (2) over all angles and over
the infrared spectral region: kCARTA is limited to the spectral range 15–3000 cm^{−1} spanned by the different bands of kCARTA (see Appendix B). As in Sect. 7 the limitation of kCARTA
for flux calculations is the spectral (infrared) resolution at every
layer, compared to the varying-with-height resolution employed by
other models such as LBLRTM. This impacts the high-altitude longwave
cooling in the 15 µm CO_{2} band.
We use the Rapid Radiative Transfer (Longwave) model (RRTM-LW) (Mlawer et al., 2012) as
our reference model for flux and heating rate comparisons in a clear-sky atmosphere. This fast model computes fluxes and heating rates in 16
bands spanning from 10 to 3000 cm^{−1} and was developed using LBLRTM;
the latter uses a varying spectral resolution at each layer (*δ**ν* equal to four points per half-width in each layer), which means the
spectra for the upper atmosphere layers have very high resolution. kCARTA
uses the same approach as RRTM-LW and LBLRTM to compute fluxes and
heating rates: the angular integration uses an exponential
Gauss–Legendre with three or four terms, with a linear in *τ* layer
temperature variation.

The accuracy of the flux and heating rate algorithm in kCARTA at the
various resolutions was assessed by comparing fluxes and heating
rates in the dominant 15 to 10 µm bands (fourth to eighth RRTM-LW
bands, spanning 630–1180 cm^{−1}) computed using RRTM-LW and kCARTA, using the 49 regression profile set.

At 0.0025 cm^{−1} resolution the kCARTA and RRTM-LW heating rates differ by
less than 0.2 K d^{−1} on average for altitudes below 40 km, but at
higher altitudes the differences were much larger and could be 1.5 K d^{−1}. Switching to the 605–1205 cm^{−1} H_{2}O,
CO_{2}, and O_{3} test atmosphere database at 0.0005 cm^{−1} significantly
improves the results, with heating rate differences dropping to about
0.2 K d^{−1} almost everywhere.

Figure 5 shows the heating rate differences between
kCARTA and RRTM-LW. Panel (a) shows differences between kCARTA
and RRTM-LW, with the mean and standard deviation being solid
and dashed, respectively; panel (b) shows mean calculations as a
function of height. The blue curves were calculated at 0.0025 cm^{−1}
resolution while the red curves were calculated at higher than 0.0005 cm^{−1}
resolution. While the agreement is better than 0.05 K d^{−1} in the
lowest 30 km, Fig. 5 shows the heating rates using
the low resolution begin to differ noticeably above 45 km (blue curve);
conversely the high-resolution heating rates (red curves) are within
0.2 K d^{−1} till about 65 km.

9 Scattering package included with kCARTA Fortran 90 version

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The daily coverage of hyperspectral sounders provides us with
information pertaining to the effects of cloud contamination on
measured radiances. Ignoring these effects can negatively impact
retrievals used for weather forecasting and climate modeling. A
scattering package based on the PCLSAM (Parametrization of
Cloud Longwave Scattering for use in Atmospheric Models) scheme
(Chou et al., 1999) has been interfaced into the f90 version of
kCARTA (see Appendix C). The
implementation allows kCARTA to compute radiances very quickly in the
presence of scattering media such as clouds or aerosol. For a given
scattering species and assumed particle shape and distribution, the
extinction coefficients, single-scattering albedo, and asymmetry
parameters needed by the scattering code are stored in tables as a
function of wavenumber and effective particle size (for a particle
amount of 1 g m^{−2}). The PCLSAM package is optimized for use
in the thermal infrared, away from regions where solar contributions
are important. As kCARTA currently does not handle Rayleigh
scattering, one can easily use kCARTA to output monochromatic optical
depths that can be imported into well-known scattering packages. More
details about PCLSAM and our cloud representation models are
found in Appendix C.

10 Conclusions

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We have described the details of a very fast and accurate pseudo-monochromatic code, optimized for the thermal infrared spectral region used by operational weather sounders for thermodynamic retrievals. It is much faster than line-by-line codes, and the accuracy of its spectroscopic database has been extensively compared to GENLN2 and more recently to LBLRTM. Updating the spectroscopy in a selected wavenumber region for a specified gas is as simple as updating the relevant file(s) in the database: for example, our custom UMBC-LBL enables us to rebuild entire databases within weeks of the latest HITRAN release.

The computed clear-sky radiances includes a fast, accurate estimate of
the background thermal radiation. Analytic temperature and gas amount
Jacobians can be rapidly computed. Early in the AIRS mission,
comparisons of AIRS observations against kCARTA simulations allowed for
the quick implementation of modifications to gas optical depths: our
modifications to the CKD2.4 and MT-CKD 1.0 continuum versions are very
similar to what is now in the MT-CKD2.5 version. We now use the MT-CKD
3.2 continuum, together with the N_{2} and O_{2} continuum
contributions bundled with that same version. We use two resolutions
in the infrared: 0.0005 cm^{−1} for the 605–880 cm^{−1} region (to accurately
resolve the high-altitude CO_{2} lines) and 0.0025 cm^{−1} elsewhere; a user
can easily switch to an alternate resolution by generating the
appropriate compressed databases for use with kCARTA, though this could
be at the expense of speed (at these current resolutions kCARTA takes 30 s to compute TOA spectra from 605 to 2830 cm^{−1} while LBLRTM takes
over 3 min). Tests show that brightness temperature
differences between kCARTA and for example LBLRTM are largest right on
top of a small number of high-altitude temperature sounding lines in
the 15 µm region (and close to zero elsewhere); these differences
decrease as the resolution is increased. Since the disparity is right at
the peaks of the lines, the differences after convolution with a
typical sounder SRF such as AIRS are on average much smaller than the
NeDT.

kCARTA is fast enough to be used in optimal estimation retrievals for
instruments spanning a reasonably small wavenumber range. The kCARTA
database has been extended to include 15–44 000 cm^{−1}, which eventually
needs to be updated to HITRAN 2016 (see the Appendix). In the future
we plan to augment the optical depth calculations performed by
UMBC-LBL by using speed-dependent line shapes as parameters become
available.

Appendix A: UMBC-LBL and kCARTA downloads and auxiliary requirements

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The kCompressed database is supplied in Fortran little-endian binary files
that contain the optical depths for a specific gas. Each file contains
optical depths at 10 000 spectral points and average pressures
corresponding to the 100 AIRS layers. Links to the
(605–2830 cm^{−1}) compressed database can be found at
http://asl.umbc.edu/pub/packages/kCompressedDatabase.html (last access: 2 January 2020).

We also supply the US Standard Profile for all gases in the database
and kLAYERS, a program that takes in a point profile (from
sondes and NWPs) and outputs an AIRS 100-layer path-averaged profile
(molecules cm^{−2}). kLAYERS needs our supplied HDF file
implementation (RTP) source code.

The MATLAB version should work with R2012+ while the compiler for the Fortran version must support structures, such as Absoft, ifort, and PGF. As the RTP file contains the atmospheric profile and scan geometry, both the MATLAB and f90 kCARTA only need a simple additional (name list) file to drive either code. The f90 version of kCARTA outputs binary files, which typically have header information such as kCARTA version number, number of layers and gases, and parameter setting values, followed by panels, each 10 000 points long, containing the optical depths, radiances, Jacobians or fluxes computed, and output by kCARTA. A number of MATLAB-based readers can then be used to further process the kCARTA output as needed. More information is found at http://asl.umbc.edu/pub/packages/ kcarta.html (last access: 12 January 2020).

A new compressed database (spanning the infrared 500–2830 cm^{−1} region)
is generated for kCARTA every 4 years, roughly within a few months of
a HITRAN database release. The current f90 version described in this
paper is identified on GitHub as SRCv1.21_f90 and currently uses
HITRAN 2016 line parameters for all gases except CO_{2} and CH_{4}
where we used LBLRTM v12.8 optical depths, together with the MT-CKD3.2
continuum. These were used to generate the most recent SARTA v2.01 fast
model coefficients; earlier SARTA versions were developed using kCARTA
v1.07 and v1.18 (with HITRAN databases updated as they became
available).

Appendix B: Available spectral regions and f90 kCARTA features

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Asterisks indicate the region spanned by hyperspectral sounders, which kCARTA focuses on for testing and validation.

The UMBC-LBL line-by-line code has been used to generate optical
depths in the spectral regions seen in Table B1.
The asterisks mark the 605–2830 cm^{−1} spectral region spanned by the current hyperspectral sounders, where we focus our validation of spectroscopy and radiative transfer. The current database in this spectral
region uses line shape parameters from HITRAN 2016. The Van Vleck and
Huber line shape is used for all HITRAN molecules from ozone onward;
water vapor uses the “without basement” plus MT-CKD 3.2, and
CO_{2}, and CH_{4} use line-mixing optical depths generated from LBLRTM
v12.8. Note that in the important 4.3 µm
temperature sounding region, the f90 version can also include the
N_{2}∕H_{2}O and N_{2}∕CO_{2} collision-induced absorption (CIA) effects modeled in
Hartmann et al. (2018) and Tran et al. (2018), which depend on CO_{2}, H_{2}O, and N_{2} absorber amounts.

A clear-sky radiance calculation in the infrared takes about 30 s, using a 2.8 GHz 32-core multi-threading Intel machine. The
run time goes to 120 s if Jacobians are also computed (for nine gases). A full radiance calculation from 15 to 44 000 cm^{−1} takes less
than 5 min.

Table B2 lists a number of the features of kCARTA, with the ones marked by an asterisk only available in the f90 version. Note that the tables default to describing the spectroscopy for the infrared region.

The spline versus linear temperature interpolation differences, as tested on 49 regression profiles, are 0.0004±0.0040 K, with a maximum absolute difference of 0.342 K (in the 15 µm region).

Appendix C: PCLSAM scattering algorithm

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The PCLSAM scattering algorithm for longwave radiances has
applications ranging from dust retrievals (De Souza-Machado et al., 2010) to modeling
the effects of clouds on sounder data (Matricardi, 2005; Vidot et al., 2015). This
scattering model changes the extinction optical depth from *k*(*ν*) to
a parametrized number ${k}_{\text{eff. extinction}}^{\text{scatterer}}\left(\mathit{\nu}\right)$
(Chou et al., 1999) and is designed for cases of the single-scattering
albedo *ω* being much less than 1, such as in the thermal
infrared, where *ω* for cirrus and water droplets and aerosols is
typically on the order of 0.5.

Since ${k}_{\text{eff. extinction}}^{\text{scatterer}}\left(\mathit{\nu}\right)$ is now effectively the
absorption due to the cloud or aerosol, for each layer *i* that
contains scatterers we replace the gas absorption optical depth with
the total absorption optical depth:

$$\begin{array}{}\text{(C1)}& {k}_{\text{total}}\left(\mathit{\nu}\right)={k}_{\text{atm}}^{\text{gases}}\left(\mathit{\nu}\right)+{k}_{\text{eff. extinction}}^{\text{scatterer}}\left(\mathit{\nu}\right),\end{array}$$

where (Chou et al., 1999) ${k}_{\text{eff. extinction}}^{\text{scatterer}}\left(\mathit{\nu}\right)={k}_{\text{extinction}}^{\text{scatterer}}\left(\mathit{\nu}\right)\times (\mathrm{1}-\mathit{\omega}(\mathit{\nu}\left)\right)(\mathrm{1}-b(\mathit{\nu}\left)\right)$ and
the backscatter $b\left(\mathit{\nu}\right)=(\mathrm{1}-g(\mathit{\nu}\left)\right)/\mathrm{2}$, with *g*(*ν*) being the
asymmetry factor. Using this for every layer containing scatterers,
the radiative transfer algorithm is now the same as clear-sky
radiative transfer, with very little speed penalty.

kCARTA is capable of using a TwoSlab (De Souza-Machado et al., 2018) cloud representation scheme for use with PCLSAM. This allows for non-unity fractions for up to two clouds, so that radiative transfer then assumes the total radiance is a sum of four radiance streams (clear, cloud 1, cloud 2, and the cloud overlap) weighted appropriately:

$$\begin{array}{}\text{(C2)}& r\left(\mathit{\nu}\right)={c}_{\text{overlap}}{r}^{\left(\mathrm{12}\right)}\left(\mathit{\nu}\right)+{c}_{\mathrm{1}}{r}^{\left(\mathrm{1}\right)}\left(\mathit{\nu}\right)+{c}_{\mathrm{2}}{r}^{\left(\mathrm{2}\right)}\left(\mathit{\nu}\right)+{f}_{\text{clr}}{r}^{\text{clr}}\left(\mathit{\nu}\right).\end{array}$$

With this model kCARTA allows the user to specify up to two types of
scatterers in the atmosphere (ice–water, ice–dust, and water–dust or even
ice–ice, water–water, and dust–dust); the two scatterers are placed in
separate “slabs” which occupy complete AIRS layers and are specified
by cloud top/bottom pressure (millibars), cloud amount (g m^{−2}),
cloud effective particle diameter (µm). After the computations are
done, all five radiances are output when two clouds are defined
(overlap, two clouds separately, clear, and the weighted sum), and
three radiances if only one cloud is defined (one cloud, clear,
weighted sum).

Analytic Jacobians for temperature, gas amounts, and cloud microphysical parameters (effective size and loading) can also be computed, as can be fluxes and associated heating rates, though the slab boundaries could introduce spikes in the heating rate profiles.

kCARTA does not have built-in multiple-scattering capabilities to handle, for example, Rayleigh scattering in the ultra-violet. To handle this we have written MATLAB routines to read in kCARTA optical depths and pipe them into LBLDIS (Turner et al., 2003; Turner, 2005), a code that merges optical depths and scattering using the extensively tested discrete ordinate radiative transfer (DISORT) (Stamnes et al., 1988) algorithm.

Code and data availability

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Code and data availability.

The UMBC-LBL code has been developed in MATLAB, with extensive use of Mex files to speed up loops. The package is available at https://github.com/ sergio66/UMBC_LBL (last access: 12 January 2020) (De Souza-Machado and Strow, 2000) and is fully described in De Souza-Machado et al. (2002). The compression code is available upon request.

The f90 and MATLAB versions of kCARTA can be cloned from https://github.com/ sergio66/kcarta_gen (last access: 12 January 2020) (De Souza-Machado et al., 2019) and https://github.com/strow/kcarta-matlab (last access: 12 January 2020) (De Souza-Machado et al., 2012), respectively.

Author contributions

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Author contributions.

SDM prepared the paper with contributions from all (living) co-authors. The initial compressed database coding and testing was done by LS, HM, and SH. Following this deS-M wrote the Fortran and MATLAB wrapper codes for clear-sky radiative transfer and Jacobians, which were tested and validated by the other authors. Scattering and flux capabilities were added and tested by deS-M.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank the reviewers whose comments/suggestions helped
improve this paper. This research has been supported by NASA (award 80NSSC18K0618). David Tobin of the University of Wisconsin-Madison helped with UMBC-LBL CO_{2} line-mixing code and
modifying the water continuum coefficients. David Edwards of the
National Center for Atmospheric Research provided the GENLN2
line-by-line code to compare kCARTA against. Both David Edwards and
Manuel Lopez-Puertas of the Instituto de Astrofisica de Andalucia
(Spain) contributed to the NLTE portions of the code. Optical depth
and flux comparisons against LBLRTM were facilitated by Eli Mlawer
(Atmospheric and Environmental Research, Lexington, MA), while Guido
Masiello (University of Basilicata, Italy) helped with the radiance
intercomparisons.

Financial support

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Financial support.

This research has been supported by NASA (award 80NSSC18K0618).

Review statement

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Review statement.

This paper was edited by Lars Hoffmann and reviewed by J.M. Blaisdell and two anonymous referees.

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Short summary

The current instruments being used for weather forecasting and climate require accurate radiative transfer codes to process the acquired data. In addition the codes are becoming more realistic, as they can now account for the effects of cloud and aerosols, rather than only simulating radiances for a clear sky. We describe a fast, accurate, and general purpose code that we have developed to help model data from these instruments.

The current instruments being used for weather forecasting and climate require accurate...

Atmospheric Measurement Techniques

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