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) on board NASA's Aqua satellite, the Infrared Atmospheric Sounding Interferometer (IASI) on board the Metop satellites , and the Cross Track Infrared Sounder (CrIS) on board the Suomi and JPSS-1 satellites .

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 . 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; ) 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 . 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 .

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) for which kCARTA is the reference forward model; SARTA is used to retrieve level 2 geophysical products from the AIRS and CrIS 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; ), Radiative Transfer for TIROS Operational Vertical Sounder (RTTOV; ), and the Jülich Rapid Spectral Simulation Code (JURASSIC; ).

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) 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 ), 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.

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 : 1μdR(ν)dke=-R(ν)+J(ν), where μ is the cosine of the viewing angle, ke 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: 2R(ν)=Rs(ν)+Rlay(ν)+Rth(ν)+Rsolar(ν), 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., ) 3R(ν,θ)=ϵs(ν)B(ν,Ts)τatm(ν,θ)+∫surfaceTOAB(ν,T(z))∂τ(ν,θ)∂sds+1-ϵs(ν)πτatm(ν)∫dΩ+∫TOAsurfaceB(ν,T(s))∂τ(Ω)∂scos⁡(θ)ds+ρs(ν)B⊡(ν)cos⁡(θ⊡)τatm(ν,θ⊡)τatm(ν,θ), where B(ν,T) is the Planck radiance at temperature T, Ts is the skin surface temperature, ϵs and ρs 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 τatm is the total atmospheric transmission. The dΩ+ in the middle term indicates integration over the upper hemisphere.

In what follows we discretize Eq. () so that layer i=1 is the bottom and i=N (=100) the uppermost layer, schematically shown in Fig.  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 H1 to H2 to H3 to H4 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-1(3/5)), 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. .

Here we describe radiative transfer for the constant layer temperature case; see Sect.  for a short discussion when using the linear-in-tau option. For an arbitrary layer i with (nadir) optical depth ki(ν), the transmittance of a beam passing from the bottom to the top of the layer at angle θ is given by τi(ν,θ)=e-ki(ν)/cos⁡(θ). The transmittance from the top of layer i to space is then the product of the individual transmittances of the layers above i 4τi+1→TOA(ν,θ)=πj=i+1Nτj(ν,θ), 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.

During the daytime, incident solar radiation is preferentially absorbed by some CO2 and O3 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 CO2 bands (and in other molecular bands, for example O3) 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 CO2 4 µm (ν3) band. kCARTA includes a computationally intensive line-by-line nonlocal thermodynamic equilibrium (NLTE) model to calculate the effects for this CO2 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 , 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 CO2 band. Using the kCARTA NLTE line-by-line model, a fast NLTE model for sounders has already been developed, which is used in the NASA AIRS L2 operational product.

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. () is 9R(ν)=ϵsB(Ts,ν)τ1→TOA(ν)+Σi=1i=NB(Ti,ν)(1.0-τi(ν))τi+1→TOA(ν).

Differentiation with respect to the m-layer variable sm (gas amount or layer temperature sm=qm(g),Tm) yields 10∂R(ν)∂sm=ϵsB(Ts)∂τ1→TOA(ν)∂sm+∑i=1NB(Ti,ν)(1.0-τi(ν))∂τi+1(ν)∂sm+∑i=1Nτi+1→TOA(ν)∂∂smB(Ti,ν)(1.0-τi(ν)), where, as usual, τm(ν)=e-km(ν) and τm→TOA(ν)=Πj=mNe-kj(ν). The differentiation yields 11∂R(ν)∂sm=ϵsB(Ts)τ1→TOA(-1)∂km(ν)∂sm+∑i=1m-1(1.0-τi(ν))Bi(ν)τ(ν)i+1→TOA(-1)∂km(ν)∂sm+(1.0-τm(ν))∂Bm(ν)∂sm-B(Tm,ν)∂τm(ν)∂smτm+1→TOA(ν).

The individual Jacobian terms ∂km∂sm(g) are rapidly computed by kCARTA, as follows. The gas amount derivative is simply ∂km∂qm(g)=kmqm(g) (with added complexity for water, to account for self-broadening), and the temperature derivative ∂km∂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 cos⁡-1(3/5) 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.

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

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 O3 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 H2O, CO2, and O3, 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 CO2 and 10 µm O3 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. a, b. Panel (a) is the mean while the panel (b) is the standard deviation. Note that in the 10 µm O3 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: -0.3±0.1 K at 0.0005 cm-1 resolution, dropping to -0.1±0.05 K at 0.0002 cm-1 resolution.

Figure c, 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 CO2 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. () to the 5th order while LBLRTM may use a Padé approximation and/or Eq. () to the 1st order; (b) there may be some very slight broadening effects right on top of the high-altitude CO2 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 CO2 and CH4 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. a, 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 O3 lines they could be as large as 5 K. Instead we show the differences after convolution with AIRS SRFs. Figure a, 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, H2O, and O3) 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.

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. () 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 ). As in Sect.  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 CO2 band. We use the Rapid Radiative Transfer (Longwave) model (RRTM-LW) 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 (δν 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 H2O, CO2, and O3 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  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.  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.

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 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.

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 N2 and O2 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 CO2 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.