# Revising the line-shape parameters for air- and self-broadened CO2 lines toward a sub-percent accuracy level

Robab Hashemi, Iouli Gordon, H Tran, Roman Kochanov, Ekaterina Karlovets, Yan Tan, Julien Lamouroux, Ngoc Hoa Ngo, Laurence Rothman

<figure><img src="image_1.png" /></figure>

► To cite this version:

Robab Hashemi, Iouli Gordon, H Tran, Roman Kochanov, Ekaterina Karlovets, et al. Revising the line-shape parameters for air- and self-broadened CO2 lines toward a sub-percent accuracy level. Journal of Quantitative Spectroscopy and Radiative Transfer, 2020, 256, pp.107283. 10.1016/j.jqsrt.2020.107283. hal-03004826

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Submitted on 5 Aug 2025

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<figure><img src="image_2.png" /></figure>
Revising the line-shape parameters for air- and self-
broadened CO₂ lines toward a sub-percent accuracy
level

Robab Hashemi<sup>a,*</sup>, Iouli E. Gordon<sup>a,*</sup>, Ha Tran<sup>b</sup>, Roman V. Kochanov<sup>c,d</sup>,
Ekaterina V. Karlovets<sup>a</sup>, Yan Tan<sup>e</sup>, Julien Lamouroux<sup>f</sup>, Ngoc Hoa Ngo<sup>g</sup>,
Laurence S. Rothman<sup>a</sup>

Abstract

Characterizing and modeling atmospheric CO₂ with a sub-percent accuracy necessitates high-quality spectroscopic line-shape parameters. For this goal, we collected the best experimentally and theoretically determined parameters for self- and air-broadened CO₂ lines to create sets of semi-empirical models for updating all of the carbon dioxide transitions in the HITRAN database using the Voigt and non-Voigt parametrizations.

Based on the available data, we estimated the air- and CO2-broadening half
widths, their associated temperature exponents, and the speed dependence of
the broadening together with its temperature dependence for every transition
in HITRAN. Furthermore, a semi-empirical approach educated by the most
reliable experimental data was used to calculate the line shifts of air- and self-

<footnote><sup>a</sup>Atomic and Molecular Physics Division, Harvard-Smithsonian Center for Astrophysics,
Cambridge, MA, USA.</footnote>

<footnote><sup>b</sup>Laboratoire de Météorologie Dynamique, IPSL, CNRS, Sorbonne Université, Ecole normale supérieure, PSL Research University, Ecole Polytechnique, F-75005 Paris, France.</footnote>

<footnote><sup>c</sup>Laboratory of Quantum Mechanics of Molecules and Radiative Processes, Tomsk State University, 634050 Tomsk, Russia.</footnote>

<footnote><sup>d</sup>V.E. Zuev Institute of Atmospheric Optics, Laboratory of Theoretical Spectroscopy,
Russian Academy of Sciences, 1 Akademichesky Avenue, 634055 Tomsk, Russia.</footnote>

<footnote><sup>e</sup>Hefei National Laboratory for Physical Science at Microscale, University of Science and Technology of China, Hefei, China.</footnote>

<footnote><sup>f</sup>Independent researcher, Paris, France.</footnote>

<footnote><sup>g</sup>Faculty of Physics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam.</footnote>

<footnote>*Corresponding author
Email addresses: robab.hashemi@cfa.harvard.edu (Robab Hashemi),
igordon@cfa.harvard.edu (Iouli E. Gordon)</footnote>
broadened CO₂ absorption lines. The updated line-shape parameters were used for calculating the first-order line-mixing parameters for every transition using the program developed by Lamouroux et al. [JQSRT **151**, 88 (2015)]. The same program was used to calculate absorption coefficients with full line-mixing by using the updated parameters. In order to verify these new data, two different sets of laboratory spectra were used to compare with the absorption coefficient calculated by the line-mixing package program and the HITRAN Application Programming Interface (HAPI) for which necessary functionality was implemented.

The updated data is proposed for the next edition of HITRAN (2020).
Thanks to the relational structure of the HITRAN database, we are able to
provide separate consistent and complete sets of parameters in Voigt and speed-
dependent Voigt parametrizations for every line of carbon dioxide.

Keywords: HITRAN database, CO₂, line-shape parameters, air- and
self-broadening, speed dependence of width, full line-mixing, first-order
line-mixing, Padé approximants

**1. Introduction**

In the Earth's atmosphere, carbon dioxide (CO₂) is a strong absorber of infrared radiation, trapping the heat and causing a potent greenhouse gas effect. The amount of this gas is steadily increasing in the atmosphere from both anthropogenic and natural sources and at the moment the usual sinks of CO₂, including vegetation can not compensate for this efficiently enough. To better understand the sources and sinks of CO₂ and to help make educated political decisions, monitoring CO₂ in the terrestrial atmosphere is the principal goal of many ground-based and space-based observations. The Greenhouse Gases Observing Satellite (GOSAT) [1, 2], observes the concentrations of CO₂ and CH₄ [3, 4]. Additionally, GOSAT-2, launched in October 2018, extends the observation of greenhouse gases [5]. A different space mission is the Orbiting Carbon Observatory-2 (OCO-2) [6, 7], designed to give a global view of CO₂
and attain further data about the location and quantity of the carbon sinks and sources. Furthermore, NASA launched the OCO-3 in May 2019 to collect daily information on carbon dioxide in correspondence with OCO-2 data [8, 9] to know more extensively about human emission and natural fluxes of CO2. The Atmospheric Chemistry Experiment (ACE) is another satellite remote-sensing mission[10] probing from the 2.2 to 13.3 µm region, observing many molecules including CO2 [11] in the limb geometry.

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For investigating CO2 data from these satellites, the intention is to gain the desired level of 0.3 % accuracy on the retrieved CO2 columns[12] by lessening the uncertainties of spectroscopic parameters. Therefore, one needs high-quality line-shape parameters, including the collisional line broadening and pressure shift values with their associated temperature dependences. Moreover, more refined effects including the speed dependence of the line width and line-mixing affect the CO2 retrieval in atmospheric remote sensing. The HITRAN molecular spectroscopic database [13] is a prime resource for reference spectroscopic data for atmospheric applications. Due to its importance, parameters for the carbon dioxide molecule have been included in the database since its inception (see for instance Ref. [14]) and undergo extension or updates in every quadrennial edition of the database.

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The line list of CO2, in the most recent edition of the database (HITRAN2016) [13], covers 0-14076 cm⁻¹ for 12 isotopologues. For each of the CO2 transitions, the line parameters are provided together with line-shape parameters for air and CO2 as perturber gases.

35

In the current HITRAN2016 database [13], except for the 4300-7000 cm⁻¹ wavenumber range, the air-broadening (half width) parameters, their temperature dependence, the air pressure-shift parameters, and the self-broadening half widths of the carbon dioxide lines were based on the semi-empirical calculations by Gamache and Lamouroux [15]. For modeling their vibrational-dependent pressure broadening, the authors have examined Robert-Bonamy formalism and used various experimental results obtained with different line shapes to fit their model. In the 4300-7000 cm⁻¹ region, the parameters were determined

40
from laboratory measurements by Devi et al. [16] using the speed-dependent
Voigt (SDV) profile [17, 18, 19]; however, the speed-dependence parameters are
not provided in HITRAN2016. Unfortunately, as will be discussed in the later
session, different issues have been identified with both experimental and semi-
empirical values used in this approach and therefore the procedure will need
to be revised. Further, the air- and self-shifts from Ref. [15] and effectively in
HITRAN2016 for *P* and *R* branches were the same and did not account for sub-
stantial asymmetric contributions described in Ref. [20]. To moderate this, we
collected the recent experimental results and followed the algorithm proposed
by Hartmann in 2009 [20] to provide the asymmetric air- and self-pressure shifts
for CO₂.

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In sections 2, 3, and 4 of this paper, the models for producing the line-shape
parameters for the non-measured transitions using the Padé approximants are
described.

60

The flexible relational structure [21] of the HITRAN database introduced in
the last edition provides various methods to save and visualize the data when
querying from the internet HITRAN interface at www.hitran.org. In particular,
it allows for providing data beyond the standard 160-character ".par" parameter
sets that was employed in the database since HITRAN2004 [22]. While the
new structure did allow for including the new line-shape parameters, as it was
done for many molecules, this was not the case for the CO₂ line list in the
official release of the HITRAN2016 edition, although in 2019 half widths of CO₂
broadened by water vapor and their temperature dependences were added to
the database [23]. Thanks to the flexible structure of the database, the updated
line list will feature different parameters beyond the Voigt parameters group.

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In this paper, the revised line-shape parameters using the Voigt [24] and
speed-dependent Voigt parameters group in two separate sets are provided. For
example, the SDV parameters are listed in Table 1 and adopted in HAPI using
notation provided in the third column.
<table><caption>Table 1: The HITRAN parametrization of SDV profile in the database.</caption><thead><tr><th>Description</th><th>Symbol</th><th>HITRAN and HAPI notation</th></tr></thead><tbody><tr><td rowspan="2">Broadening half width</td><td>γ<sub>0</sub>-air (cm<sup>-1</sup>atm<sup>-1</sup>)</td><td>gamma_SDV_0_Air_296</td></tr><tr><td>γ<sub>0</sub>-self (cm<sup>-1</sup>atm<sup>-1</sup>)</td><td>gamma_SDV_0_Self_296</td></tr><tr><td rowspan="2">Temp. dep. of half width</td><td>n<sub>γ<sub>0</sub></sub>-air (unitless)</td><td>n_SDV_Air_296</td></tr><tr><td>n<sub>γ<sub>0</sub></sub>-self (unitless)</td><td>n_SDV_Self_296</td></tr><tr><td rowspan="2">Speed dep. of half width</td><td>γ<sub>2</sub>-air (cm<sup>-1</sup>atm<sup>-1</sup>)</td><td>gamma_SDV_2_Air_296</td></tr><tr><td>γ<sub>2</sub>-self (cm<sup>-1</sup>atm<sup>-1</sup>)</td><td>gamma_SDV_2_Self_296</td></tr><tr><td rowspan="2">Temp. dep. of speed dep.</td><td>n<sub>γ<sub>2</sub></sub>-air (unitless)</td><td>ngamma_SDV_2_Air_296</td></tr><tr><td>n<sub>γ<sub>2</sub></sub>-self (unitless)</td><td>n{'gamma_SDV_2_Self_296</td></tr><tr><td rowspan="2">Line shifts</td><td>δ-air (cm<sup>-1</sup>atm<sup>-1</sup>)</td><td>delta_SDV_0_Air_296</td></tr><tr><td>δ-self (cm<sup>-1</sup>atm<sup>-1</sup>)</td><td>delta_SDV_0_Self_296</td></tr><tr><td rowspan="2">The first-order line-mixing</td><td>Y<sub>k</sub>-air (cm<sup>-1</sup>)</td><td>Y_SDV_Air_296</td></tr><tr><td>Y<sub>k</sub>-self (cm<sup>-1</sup>)</td><td>Y_SDV_Self_296</td></tr><tr><td rowspan="2">Temp. dep. of the first-order line-mixing</td><td>n<sub>Y</sub>-air (unitless)</td><td>n{'Y_SDV_Air_296</td></tr><tr><td>n<sub>Y</sub>-self (unitless)</td><td>n{'Y_SDV_Self_296</td></tr></tbody></table>

The symbols listed in Table 1 are presented in a functional form in section 2.2. The temperature dependence of the line-shift parameter is not presented in the table nor discussed throughout this paper because there were not many available measurements for this parameter in the literature. The existing measured values were obtained with large uncertainty and do not improve the results. The double power-law [25] would justify the temperature dependence of the shift parameter better in the case of having sufficient laboratory measurements in a wide range of temperatures.

After obtaining the above-mentioned line-shape parameters, we used them to update the CO₂ line-mixing package developed by Lamouroux et al. [26] to compute the absorption coefficients with full line-mixing. Furthermore, first-order line-mixing values (section 5) are obtained and will be provided with every CO₂ line in HITRAN.
## 2. Collisional spectral broadening and the corresponding temperature dependence for CO₂ transitions perturbed by air and CO₂

In the following subsections, the simple semi-empirical models are described for producing the broadening parameters and their temperature dependence for transitions of CO₂ to provide an appropriate data set for the HITRAN database. One challenge is to choose a particular set of precise measurements/calculation to fit the models, acknowledging various measurements and calculations performed for the line-shape study of CO₂. The selected data set should satisfy the demand concerning completeness in terms of having sufficient coverage of the rotational transitions. Additionally, it is beneficial that different line-shape parameters are obtained in a variety of temperatures and pressures. Furthermore, the desired results should be complemented by several other measurements.

A problem that might arise when using parameters from diverse data sources is the possibility of combining various systematic errors of different experimental measurements or numerical errors in calculations. This can be problematic in achieving a sub-percent accuracy level. Therefore, it is essential to verify the compatibility of the results from various sources. For instance, to predict the half widths using the complex Robert-Bonamy (CRB) algorithm in Ref. [15], the authors educated their model by a large amount of data in different vibrational bands from different measurements that employed diverse techniques and different line-shape profiles for fitting the data. The potentials used for the CRB calculations were determined by fitting to the experimental measurements. The data for transitions with the same *J* (but from different vibrational bands) were fit to Equation (5a) of Ref. [15] to determine the γ₀ (Lorentzian component) for each of these rotational transitions. A set of three coefficients for every rotational transition were obtained. Table 2 of Ref. [15] lists 600 coefficients provided for *P* branch lines with *J* up to 200. While the approach discussed in Ref. [15] has a sound physical basis, the fits yielded a much stronger vibrational dependence than one would expect for the γ₀ values of transitions in a linear molecule. This is largely due to systematic offsets in some of the data
and different line-shape models used in different studies. It is well known that the employment of different profiles yields $\gamma_0$ values that can be quite different and for CO$_2$ deviate by more than 5 % (see, for instance, Refs [27, 28, 29]). Therefore, in this paper, we use only the recent experiments that report extensive coverage of rotational transitions. This is not at all to say that there are no good measurements beyond the ones that we compared and eventually used.

It is important to stress again that we treat the data derived from the Voigt and SDV profiles as a separate set to avoid correlation due to co-mixture of two different line profiles.

## 2.1. The air-broadening half widths obtained using the Voigt profile

The $\gamma_0$-air half widths of CO$_2$ lines have been retrieved using the Voigt profile (VP) in various studies. Here we present the well-known Voigt profile formalism [24]:

$$I(x, y) = \frac{1}{\sqrt{\pi}} \mathrm{Re} \left[ \frac{i}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-t^2}}{x - t + iy} dt \right]. \qquad (1)$$

The parameter $x$ is given by the following equation,

$$x = \left( \frac{\nu - \nu_0 - \delta}{\sigma} \right). \qquad (2)$$

where $\nu_0$ is the central line position and $\delta$ is the collisional shift. The term $\sigma = \nu_0 \sqrt{\frac{2k_B T}{Mc^2}}$ is the Doppler broadening at $1/e$ intensity. $c$ is the speed of light, and $k_B$ is the Boltzmann constant. The molecular mass is shown with $M$. The unit-less $y$ parameter in Eq. 1 is given by

$$y = \frac{\gamma}{\sigma}, \qquad (3)$$

with $\gamma$ being the collisional width.

Figure 1 exhibits the comparison of results for collisional $\gamma_0$-air versus $|m|$, where $m = -J$ for the $P$ branch lines and $m = J+1$ for the $R$ branch. The filled blue circles show the $\gamma_0$-air obtained using a semi-empirical fit of selected pre-1992 experimental data by Rothman et al. [30] assuming that the broadening
of CO₂ lines is vibrationally-independent; this fit was used for the HITRAN editions from 1992 to 2004 [31, 32, 33, 22] and for the majority of transitions in the 2008 edition [34], while the purple triangles represent those measured by Predoi-Cross et al. [35] for the 30012-00001 band. The experimental results by Gulidova et al. [36] in the 00011-00001 band, which cover an extended range of *J*, appear to be lower than other measurements. The diagram also presents the measured line broadening by Hashemi et al. [37] in the laser band 00011-00001 band (the green stars). Among these studies, the largest reported γ₀-air values are reported by Tanaka et al. [38]. The authors have obtained γ₀-N₂ and γ₀-O₂ to combine them for calculating the γ₀-air in the 11101-00001 band (shown by the half-filled gray circles), which are in poor agreement with the other studies. It is noted that the results by Tanaka et al. [38] were used by Ref. [15] in the CRB prediction and also by the GOSAT mission.
<figure><img src="image_3.png" /><figcaption>Figure 1: The comparison of air-broadening half widths of CO₂ lines at 296 K plotted versus *m* in unit of cm⁻¹atm⁻¹. The results obtained by Rothman et al. [30] are shown with the filled blue circles. The measured values by Predoi-Cross et al. [35] are presented with the purple triangles, the ones from Gulidova et al. [36] presented by the open red rhombus. Further studies are measurement by Hashemi et al. [37] (the green star), and Tanaka et al. [38] (by the half-filled gray circles). Each measurement data is fitted using Padé approximants (shown by lines) except for Tanaka et al. [38].</figcaption></figure>

To extend the experimental results to the higher *J* transitions, Padé approximants have been used to fit the data with the benefit of producing more reasonable asymptotic results when *J* is large. It can be advantageous for the HITEMP database [39] as well. The following 3rd/4th order equation is used:

$$ \gamma_0 = \frac{(c_0 + c_1 \times |m| + c_2 \times |m|^2 + c_3 \times |m|^3)}{(1 + d_1 \times |m| + d_2 \times |m|^2 + d_3 \times |m|^3 + d_4 \times |m|^4)} \quad (4) $$

The fit coefficients in this equation can be found in Table 2. Among all these
measured $\gamma_0$-air, we intended to keep the data set which results in smaller residuals in modeling the experimental spectra.

<table><caption>Table 2: Fitting coefficients (Eq. 4) for calculating air- and CO₂-broadening of CO₂ at room temperature (in cm⁻¹·atm⁻¹) for the VP and SDV line-shape profiles. The reference number in each column refers to the selected studies used to fit Eq. 4.</caption><thead><tr><th>Coef.</th><th>γ₀-air (VP) [35, 37]</th><th>γ₀-air (SDV)</th><th>γ₀-self (VP) [37]</th><th>γ₀-self (SDV) [35]</th></tr></thead><tbody><tr><td>c₀</td><td>1.906 × 10⁻¹</td><td>3.459 × 10⁻¹</td><td>1.336 × 10⁻¹</td><td>1.268 × 10⁻¹</td></tr><tr><td>c₁</td><td>0.1268 × 10¹</td><td>0.459 × 10¹</td><td>5.164 × 10⁻²</td><td>-3.980 × 10⁻³</td></tr><tr><td>c₂</td><td>-1.799 × 10⁻²</td><td>-1.384 × 10⁻¹</td><td>-1.510 × 10⁻³</td><td>3.745 × 10⁻⁵</td></tr><tr><td>c₃</td><td>4.027 × 10⁻⁴</td><td>3.280 × 10⁻³</td><td>2.784 × 10⁻⁵</td><td>-6.604 × 10⁻⁸</td></tr><tr><td>d₁</td><td>1.415 × 10¹</td><td>5.191 × 10¹</td><td>5.122 × 10⁻¹</td><td>-1.635 × 10⁻²</td></tr><tr><td>d₂</td><td>2.731 × 10⁻²</td><td>-9.999 × 10⁻¹</td><td>-1.406 × 10⁻²</td><td>-2.043 × 10⁻⁴</td></tr><tr><td>d₃</td><td>-9.286 × 10⁻⁴</td><td>2.328 × 10⁻²</td><td>3.062 × 10⁻⁴</td><td>4.825 × 10⁻⁶</td></tr><tr><td>d₄</td><td>6.254 × 10⁻⁵</td><td>2.209 × 10⁻⁴</td><td>9.557 × 10⁻⁷</td><td>-1.547 × 10⁻⁸</td></tr></tbody></table>

Based on Figure 1, the proposed semi-empirical models reproduce the rota-tional dependence of broadening parameters very well. Another valid conclusion would be that the vibrational dependence of the broadening of the CO₂ spectral lines is very weak. Hence, we can use the properly-determined values of one band, and extend them to populate the HITRAN database for the entire bands. This does not appear to influence the accuracy, but allows the use of just eight coefficients in Table 1 as opposed to 600 coefficients in Ref. [15] that were used to generate the HITRAN2012 and 2016 values. The estimated uncertainty of our model is assumed to be the same as the experimental error.

All these extrapolated results, shown in Figure 1, have been tested to determine the residuals (experiment-calculations) in modeling the laboratory spectra. To extrapolate the results for the lines with large *J*, the values from Ref. [35] (43 < *J* < 58) were combined with the measurements of Ref. [37] (*J* < 43). We also tested the provided values of Ref. [36] up to *J* = 83 to model the spectra. The results of Ref. [38] are not included in our fitting procedure because of the
systematic large discrepancy with other measurements.

Much effort has been given over the years to find the best broadening solutions for the 2.06 µm band employed by the OCO-2 mission [6]. The Lorentzian parameters distributed in the HITRAN2016 ".par" files are those determined in Devi et al. [40] using SDV profile and as discussed in Lamouroux et al. [26] need to be scaled to be used with Voigt profile. However a single scaling factor was suggested by Lamouroux et al., and after the release of the HITRAN2016 database it was realized that it was better to supply the package with Voigt parameters from 1992 algorithm [30]. Currently the line-mixing package distributed with HITRAN2016 uses these values for 2.06 µm band in contrast with the ".par" file, which definitely adds to the confusion for the users of the database. Figure 2 illustrates the residuals only for a couple of lines of CO₂-air mixture in this spectral region (these lines are selected just for the purpose of better representation). Based on the RMS values (which were determined for the 4700-5100 cm⁻¹ spectral range, much wider than that presented in Figure 2), the use of γ₀-air of Ref. [36] results in significant residuals. Except for a few transitions (shown in Figure 2), the residuals for Ref. [35] and Ref. [30] were close. There are transitions where the residuals in blue (Ref. [35]) are smaller than those in red ([30]). Visually, the residuals are best minimized when using the results of Ref. [37] combined with Ref. [35] for the higher *J* lines as the minimum RMS value confirms. Therefore, to update the HITRAN γ₀-air, the combined results of [37, 35] will be used and the fitting coefficients are provided in the second column of Table 2. This approximation works efficiently without adding complexity in calculation, despite ignoring the vibrational dependence of the line widths. Based on Figure 2, the residuals were obtained by adjusting the half widths. This figure shows how the use of different line widths affects the residuals. All the residuals are generated using LM Fortran code (available at HITRANonline) and only the Voigt profile is used to calculate the transmission and the residuals. We will see in section 5 that taking line-mixing into account will improve the spectral residuals with respect to the use of the Voigt profile only.
<figure><img src="image_4.png" /></figure>

Figure 2: The Fourier Transform (FT) transmission spectrum of CO₂ diluted by air at T=294.1 K and P=598.9 Torr measured by Sung, JPL (see section 6). The CO₂ mixing ratio is 0.0145. The bottom panel shows the comparison of the residuals when using different air-broadening half widths of CO₂ measured using the VP. The residuals for Rothman et al. [30] are shown by the red line, the ones for Predoi-Cross et al. [35] are presented by the blue line, the ones for Gulidova et al. [36] displayed with the black line, and the residuals for combined Hashemi et al. [37] and Predoi-Cross et al. [35] are shown by the green line.

## 2.2. The air-broadening half width obtained using the speed-dependent Voigt profile

Determining the pressure dependence of line profiles, solely with pressure broadening and shift parameters, is known to be inadequate. More refined collisional effects such as the speed dependences of the line broadening and shifting, and the collision-induced velocity changes must be considered [41]. Therefore, it is essential to take into account the broadening values retrieved using refined line-shape profiles such as the speed-dependent Voigt profile (SDV) and the speed-dependent Nelkin-Gatak profile (SDNG) [42].

Based on several measurements with sophisticated SDV and SDNG profiles,
we can generate a separate data table of line parameters for these profiles, which are missing in the database right now. Ideally, to populate the HITRAN database with the parameter group of the SDV and SDNG profiles, it is preferred to use one comprehensive study with all the required line-shape parameters measured (or calculated) to avoid the possible inconsistencies and systematic biases between data from different sources.

The study by Nguyen et al. [43] for analysis of CO₂-air spectra meets this criterion in which the line-shape parameters have been determined from re-quantized classical molecular dynamics simulations (rCMDS) using the SDNG profile. In this study we only concentrate on using the VP and SDV profiles because these profiles are implemented in some of the atmospheric retrieval codes and also the CO₂ line-mixing software [26] that will be discussed in section 5.

The quadratic formulation of the SDV line shape [44, 17, 45] which includes the dependence of relaxation rates $\gamma$ on the absorber speed, $v_a$, is as follows,

$$ \gamma(v_a) = \gamma_0 + \gamma_2\left[\left(\frac{v_a}{v_{a0}}\right)^2 - \frac{3}{2}\right]. \qquad (5) $$

The parameter $\gamma_0$ is the half width of SDV line shape (gamma\_SDV\_0\_air\_296) which corresponds to the mean relaxation rate ($\gamma_0 = <\gamma(v_a)>$), while the speed dependence of the relaxation rate is presented with $\gamma_2$ (gamma\_SDV\_2\_air\_296). Similarly for the collisional shifts we will have,

$$ \delta(v_a) = \delta_0 + \delta_2\left[\left(\frac{v_a}{v_{a0}}\right)^2 - \frac{3}{2}\right]. \qquad (6) $$

where $\delta_2$ presents the dependence of shift values on the speed, $v_a$. Here, to present the functional form of the SDV profile, we repeat the Hartmann–Tran profile (HTP) as provided in Refs [46, 47, 48],

$$ I_{HTP}(\nu) = \frac{1}{\pi} \mathrm{Re} \left\{ \frac{A(\nu)}{1 - \left[\nu_{vc} - \eta(N_0 - \frac{3N_2}{2})\right]A(\nu) + \left(\frac{\eta N_2}{v_{a0}^2}\right)B(\nu)} \right\}. \qquad (7) $$

The $A(\nu)$ and $B(\nu)$ can be written as

$$ A(\nu) = \frac{\sqrt{\pi c}}{\nu_0 v_{a0}} [w(iZ_-) - w(iZ_+)], \qquad (8) $$
$$B(\nu) = \frac{v_{a0}^2}{N_3} \left[ -1 + \frac{\sqrt{\pi}}{2\sqrt{Y}}(1 - Z_-^2)w(iZ_-) - \frac{\sqrt{\pi}}{2\sqrt{Y}}(1 - Z_+^2)w(iZ_+) \right] \quad (9)$$

with $w(z)$ defined as

$$w(z) = \frac{i}{\pi} \int_{-\infty}^{+\infty} \frac{e^{-t^2}}{z-t} dt = e^{-z^2} \text{errfunc}(-iz) \quad (10)$$

where errfunc is the Gauss error function. The terms $Z_{\pm} = \sqrt{X+Y} \pm \sqrt{Y}$, with $X = \frac{-i(\nu_0-\nu)+N_4}{N_3}$ and $Y = \left(\frac{\nu_0 v_{a0}}{2cN_3}\right)^2$ where $N_3 = (1-\eta)N_2$ and $N_4 = (1-\eta)(N_0 - \frac{3N_2}{2}) + \nu_{vc}$. We note that $N_0 = \gamma_0 + i\delta_0$ and $N_2 = \gamma_2 + i\delta_2$. Eq. (7) reduces to the SDV profile when $\nu_{vc} = \eta = \delta_2 = 0$.

The line-shape parameters have been estimated from rCMDS using the SDV profile and can be found as Table 1 in the supplementary files. Figure 3 presents the values of air-broadening of CO$_2$ obtained using the SDV profiles. Because of the non-sufficient signal-to-noise ratio, the results are reported for $m$ up to 50 only. Also, the uncertainties of the rCMDS values [43] are not reported due to the complexity of determining them accurately. By comparing the retrieved $\gamma_0$-air from these analyses with the measured results by Long et al. [27], Bui et al. [29], and Devi et al. [40], it can be seen that the original simulated broadening values from rCMDS calculations are generally smaller than the experimental values, with systematic underestimation of the line broadening at lower $J$.

Now, if we tend to use the rCMDS $\gamma_0$-air, the predicted line broadening needs to be scaled based on the experimental broadening values. For a logical comparison, we used the previous study of Nguyen et al. [49] for CO$_2$-N$_2$ and compared it to the very precise experimental measurement by Wilzewski et al. [50] for the same system. The scale factor is obtained to correct all the simulated values with the ratio of $\gamma_0(\text{CO}_2\text{-N}_2)$ of Ref. [50]/$\gamma_0(\text{CO}_2\text{-N}_2)$ of Ref. [49]. The line widths were scaled up using a scale factor of 1.003 (the average value over $m$). The fitting parameters for Eq. (4) are obtained for the re-scaled air-broadening half widths, and can be found in Table 2.
<figure><img src="image_5.png" /><figcaption>Figure 3: The air-broadening half widths of CO₂ deduced from the rescaled rCMDS-calculated results modeled by the SDV profile and fitted using Padé approximants. The results are compared with measurements of Long et al. [27] pCSDNG profile, and those of Devi et al. [40] measured using the SDV profile. The two transitions measured by Bui et al. [29] are also presented with the pink stars.</figcaption></figure>

## 2.3. The self-broadening half widths obtained using the Voigt and speed-dependent Voigt profile

With a similar fitting approach, we produced the results for the γ₀-self pa- rameter using the Voigt profile. As for the γ₀-air, the measurement of γ₀-self half widths by Hashemi et al. [37] was used for extrapolating the results for the transitions with high *J*. The coefficients for the Padé approximants are shown in Table 2. Although the SDV profile was used in Ref. [37], the speed-dependent parameters were not provided [37], and the measurement by Predoi-Cross et al. [51] was used for the γ₀-self deduced with the SDV line-shape model. The measured self-broadening half widths were fitted using Padé approximants and the fitting constants are presented in Table 2.

Figure 4 shows the overlaid results of γ₀-self for both the VP and SDV profiles. For the γ₀-self values of CO₂ deduced using the VP, two sets of mea-
surements from Casa et al. [52] and Predoi-Cross et al. [51] are plotted for comparison. The results of Ref. [51] from the VP and SDV are very close, especially for $20 < |m| < 60$ and, compared to the values reported in Ref. [37], there is 4 % difference in the same region. This relatively large discrepancy between the retrieved $\gamma_0$-self using the two profiles is normal for the higher $m$.

<figure><img src="image_6.png" /><figcaption>Figure 4: The plot shows the self-broadening half widths for CO₂ transitions at 296 K based on the fitting results from both experiments using the VP and the SDV profiles. The fitted $\gamma_0$-self are shown here with solid lines of different colors which are valid up to $|m| = 150$. The values are deduced from the data by Predoi-Cross et al. [51] (open red circles) fitted to Padé approximants (solid red line). For the $\gamma_0$-self retrieved using the VP, the measurement of Ref. [37] is used to fit with Eq. 4 and compared with measurements of Refs [51, 52] obtained using the VP.</figcaption></figure>

The comparison of the results obtained by using Padé approximants with HITRAN2016 is shown for a couple of bands in Figure 5. For the higher $J$ values, the use of Padé approximants smooths the results and agrees very well with the experiment by Predoi-Cross et al. [51]. There was about a maximum
of 2 % difference in the broadening half width for the stronger lines at lower J.

<figure><img src="image_7.png" /><figcaption>Figure 5: The air and self-broadening half widths of CO₂ lines, deduced from the data by fitting to Padé approximants and compared with the current values in the HITRAN2016 edition [13] obtained using the SDV profile.</figcaption></figure>

## 2.4. The temperature dependence exponents of the line half widths

It is important to provide accurate temperature dependence exponents of the line broadening in the HITRAN database to model CO₂ absorption cross-sections for performing correct atmospheric retrievals. The following power-law function has been used [32],

$$ \gamma_0(T) = \gamma_0(T_{ref}) (T_{ref}/T)^{n_{\gamma_0}} \quad (11) $$

where $T_{ref}$ is the reference temperature (296 K) and $n_{\gamma_0}$ is the temperature-dependent exponent of the broadening parameter. Note that in order to cover wider temperature ranges (i.e. that include temperatures much lower or much higher than those encountered in the terrestrial atmosphere) this equation that represents a single power-law would not yield the most accurate conversion and a
double power-law as described in Ref. [25] should be used. However, there is not enough experimental data for CO₂ to provide the double power-law parameters for every line, and the focus of this study is to primarily aid remote-sensing retrievals of the terrestrial atmosphere.

The graph of temperature-dependence exponents versus $|m|$ (for example, Fig. 6) shows a reduction in the values up to $J = 10$ and then a rising trend up to $J = 50$ confirmed with the experimental data sets presented (Refs[53, 50]). For the temperature dependence of the air-broadening, the experimental data of Devi et al. [40] is used because the authors have measured the temperature dependence of broadening for a broad range of $J$. For the $n_{\gamma_0}$ (n\_SDV\_air\_296) of rCMDS predictions, the rotational dependence is small (at $m$ around 30–50 our predicted values seem to underestimate experimental data [53]). Hence, the agreement between the rCMDS predicted results for $n_{\gamma_0}$ and the measured values in Ref. [40] is not good especially after $m = 20$ as confirmed by the comparison between rCMDS results [49] and those of Ref. [50] for CO₂-N₂. The difference can lead to a difference up to 2-3 % for line widths calculated at 200 K, for instance, which is not negligible. The reason for the discrepancy observed for rCMDS predicted line broadenings and their temperature dependences is, as explained in Ref. [54], probably due to the simple classical treatment for the rotation of the molecules in the rCMDS. It can be observed from Fig. (6) that the data for $n_{\gamma_0}$-air by Ghysels et al. [55] do not agree well with the other measurements. Equation 4 is used to fit $n_{\gamma_0}$ results of Ref. [40]. In Table 3, the fitting coefficients can be found (column 2 for the SDV profile results).
<figure><img src="image_8.png" /></figure>

Figure 6: The temperature exponents of air-broadening half widths of CO₂, deduced from measured data by Devi et al. [40] using SDV profile and fitted to Eq. 4, are in good agreement with the results by Refs [50, 53] for CO₂-N₂ obtained using SDNG profile. The temperature exponent measured in Ref. [55] is presented with the grey triangle symbol. Also, the measurement by Ma et al. [56] is presented. The blue line presents the results of $n_{\gamma_0}$ for the VP using Eq. (4).

In Figure 6, the results of $n_{\gamma_0}$ deduced using VP by Ma et al. [56] are presented, but only for a few transitions. Due to a lack of data, the measured $n_{\gamma_0}$ for the SDV profile and the results by Ma et al. [56] can only be used to obtain a scale factor as the ratio of $n_{\gamma_0}$ (SDV)/$n_{\gamma_0}$ (VP). Using the scale factor of 1.05, the results in Figure (6) are calculated for the VP and plotted with the solid blue line. Then by fitting to the Padé approximants (Eq. 4), the temperature dependence of CO₂-air lines used with the VP is generated. We also compared them to the values in the HITRAN2016 obtained using the VP; the rescaled temperature dependences are in agreement with the ones in HITRAN except for some of the transitions (with $|m| < 7$ and $35 < |m| < 42$).

For updating the temperature exponents of self-broadening half widths of
<table><caption>Table 3: Fitted coefficients (from Eq.(4)) for estimating the temperature exponent of air and CO₂-broadening of CO₂ lines. Each column represents a Padé approximants fit to the experimental data for n<sub>γ₀</sub>-air (VP) [56], n<sub>γ₀</sub>-air (SDV) [40], n<sub>γ₀</sub>-self (VP) [57, 58], and n<sub>γ₀</sub>-self (SDV) [59].</caption><thead><tr><th>Coefs.</th><th>n<sub>γ₀</sub>-air (VP)</th><th>n<sub>γ₀</sub>-air (SDV)</th><th>n<sub>γ₀</sub>-self (VP)</th><th>n<sub>γ₀</sub>-self (SDV)</th></tr></thead><tbody><tr><td>c<sub>0</sub></td><td>0.0223 × 10<sup>1</sup></td><td>-2.265 × 10<sup>3</sup></td><td>7.926 × 10<sup>-1</sup></td><td>-0.2262 × 10<sup>1</sup></td></tr><tr><td>c<sub>1</sub></td><td>0.1365 × 10<sup>1</sup></td><td>6.409 × 10<sup>3</sup></td><td>-5.339 × 10<sup>-2</sup></td><td>3.311 × 10<sup>1</sup></td></tr><tr><td>c<sub>2</sub></td><td>-0.2273 × 10<sup>-1</sup></td><td>1.017 × 10<sup>3</sup></td><td>5.805 × 10<sup>-5</sup></td><td>-0.1530 × 10<sup>1</sup></td></tr><tr><td>c<sub>3</sub></td><td>3.010 × 10<sup>-3</sup></td><td>2.390 × 10<sup>1</sup></td><td>6.916 × 10<sup>-5</sup></td><td>0.2445 × 10<sup>1</sup></td></tr><tr><td>d<sub>1</sub></td><td>1.585 × 10<sup>1</sup></td><td>6.186 × 10<sup>3</sup></td><td>-4.258 × 10<sup>-2</sup></td><td>3.959 × 10<sup>1</sup></td></tr><tr><td>d<sub>2</sub></td><td>-2.008 × 10<sup>-1</sup></td><td>1.987 × 10<sup>3</sup></td><td>-2.530 × 10<sup>-3</sup></td><td>-0.10 × 10<sup>1</sup></td></tr><tr><td>d<sub>3</sub></td><td>0.4239 × 10<sup>-1</sup></td><td>6.790 × 10<sup>-13</sup></td><td>1.644 × 10<sup>-4</sup></td><td>0.3034 × 10<sup>1</sup></td></tr><tr><td>d<sub>4</sub></td><td>4.837 × 10<sup>-5</sup></td><td>3.627 × 10<sup>-1</sup></td><td>-1.619 × 10<sup>-7</sup></td><td>2.131 × 10<sup>-2</sup></td></tr></tbody></table>

CO₂, the theoretical values obtained by Predoi-Cross et al. [60] were used. Figure 7 shows that close to J = 20 as the most populated state of the perturbing molecule for CO₂-CO₂, there is a bump for the extrapolated results (following the Robert-Bonamy theory). The fitting coefficients are presented in Table 3 and the extrapolated results are plotted versus *m* in Figure 7 (shown by the solid red line). The comparison of different measurements are shown in Figure 7. The measured data by Suarez et al. [57] and Devi et al. [58] in the ν₃ band, deduced using the VP are presented and it can be observed that the measurements of Ref.[58] are larger than the other presented values. The results in Ref.[56] are also plotted which are in agreement with the measurement of Ref. [60] (except for two transitions). Another recent set of data available for comparison is the n<sub>γ₀</sub>-self measured by Li et al. [61]. The agreement between the semi-empirically generated lines with the measured values in Ref.[61] is not satisfactory; as can be seen their results are scattered with large uncertainties.
<figure><img src="image_9.png" /><figcaption>Figure 7: The extrapolated temperature exponents of self-broadening half widths of CO₂, deduced from theoretical data by Predoi-Cross et al. [60] and compared with the measured data in the same study using the VP profile. The measurements of Refs [56, 57, 58, 61] are presented for comparison.</figcaption></figure>

Also, in the Figure 8, the results by Devi et al. [59] are presented with the brown circles obtained using the SDV profile. These temperature-dependence exponents do not show the peak value which appeared for the $n_{\gamma_0}$ (VP) results versus $|m|$. The difference might be due to the uncertainty of the measured values. We used the measurement from Ref. [59] and extrapolated them with the same function (the solid red line) and the fitting constants can be found in Table 3. The measured temperature exponents of Refs [53, 60] have shown a resealable agreement with the extrapolated values.
<figure><img src="image_10.png" /><figcaption>Figure 8: The temperature exponents of self-broadening half widths of CO₂, deduced from measured data by Devi et al. [59] using the SDV profile and the comparison with those by Li et al. [61]. Also, the results by Benner et al. [53] and Predoi-Cross. [60] are presented for the SDV profile set which are in agreement for the lower *m* numbers.</figcaption></figure>

**3. The speed dependence of the line width**

As discussed in section 2.2, a complete set of calculations of line-shape pa-
rameters using the SDV profile was deduced from rCMDS and the results are
provided as a supplementary table. The speed dependence of line width pre-
dicted from the rCMDS, is shown by $a_w$ as the ratio of $\gamma_2/\gamma_0$ and plotted by the
filled brown circles in Fig. (9) versus $m$. The experimentally-measured values of
Ref. [40], using the same line-shape model, are also checked. As can be noticed,
the agreement is quite poor between the predicted and the experimental val-
ues. The results of Ref. [40] show a sharp rotational dependence. The authors
have argued that this overestimation might be as a result of including a large
number of low temperature spectra in their analysis. In Ref. [40] it was also assumed that the speed dependence is independent of the perturber gas and the temperature of the gas sample which can lead to this discrepancy. Besides, the measured speed-dependent parameters of Ref. [40] may be correlated with intensities of the lines.

<table><caption>Table 4: Fitted coefficients to Eq. (4) for estimating the air- and CO₂-speed dependence of half widths ($a_w$-air and $a_w$-self) of CO₂ at room temperature and the temperature dependence of $\gamma_2$-air ($n_{\gamma_2}$-air). The results are valid for $1 < |m| < 150$.</caption><thead><tr><th>Coefficients</th><th>a<sub>w</sub>-air (rCMDS)</th><th>a<sub>w</sub>-self [62]</th><th>n<sub>γ₂</sub>-air (rCMDS)</th></tr></thead><tbody><tr><td>c<sub>0</sub></td><td>-1.235 × 10<sup>1</sup></td><td>-2.265 × 10<sup>3</sup></td><td>3.341 × 10<sup>2</sup></td></tr><tr><td>c<sub>1</sub></td><td>1.230 × 10<sup>1</sup></td><td>6.409 × 10<sup>3</sup></td><td>1.662 × 10<sup>2</sup></td></tr><tr><td>c<sub>2</sub></td><td>1.422 × 10<sup>-1</sup></td><td>1.017 × 10<sup>3</sup></td><td>1.072 × 10<sup>1</sup></td></tr><tr><td>c<sub>3</sub></td><td>0.1650 × 10<sup>1</sup></td><td>2.390 × 10<sup>1</sup></td><td>0.1061 × 10<sup>1</sup></td></tr><tr><td>d<sub>1</sub></td><td>-0.100 × 10<sup>1</sup></td><td>6.186 × 10<sup>3</sup></td><td>4.784 × 10<sup>2</sup></td></tr><tr><td>d<sub>2</sub></td><td>5.419 × 10<sup>1</sup></td><td>1.988 × 10<sup>3</sup></td><td>1.173 × 10<sup>1</sup></td></tr><tr><td>d<sub>3</sub></td><td>1.179 × 10<sup>1</sup></td><td>6.79 × 10<sup>-13</sup></td><td>-0.100 × 10<sup>1</sup></td></tr><tr><td>d<sub>4</sub></td><td>1.185 × 10<sup>-1</sup></td><td>3.627 × 10<sup>-1</sup></td><td>1.827 × 10<sup>-1</sup></td></tr></tbody></table>

365

We also examined the results of Nguyen et al. [43] for the CO₂-air system using the SDNG profile (the blue rhombus in Figure (9)). Not surprisingly, the $a_w$-air from the SDNG profile agrees well with the experimental result by Bui et al. [29] for the CO₂-air and Wilzewski et al. [50] for the CO₂-N₂ system analyzed using the same line-shape model. There is an average difference of about 20 % between the results of rCMDS for $a_w$-air obtained using the i) SDNG profile and the ii) SDV profile, which is indicating the difference in using these two speed-dependent profiles. The results show that the ratios $a_w$ obtained by the SDV model are larger than the ones obtained with the SDNG profile due to neglecting the Dicke narrowing effect [63]. By employing Eq. (4) to the rCMDS results and assuming there is no vibrational dependence for the speed dependence parameter, the extrapolated results (the coefficients of Eq. (4) are

370

375
presented in Table 4) can be used for all the bands of CO₂.

<figure><img src="image_11.png" /><figcaption>Figure 9: The speed dependence of air-broadening half widths of CO₂; $a_w$ deduced from rCMDS prediction (the filled brown circles) and extrapolated using Eq. 4 (solid red line) versus $|m|$. The measured $a_w$ by Ref. [40] are plotted for comparison (the pink triangles). These two data set do not agree well. The rCMDS calculated $a_w$-air obtained using SDNG profile are presented with the blue rhombus which are very similar to the measured speed dependence by Bui et al. [29] for CO₂ in air (shown by the orange squares) and Wilzewski et al. [50] for the CO₂-N₂ system using the same line-shape model (shown by the green stars in the graph).</figcaption></figure>

Similarly, for modeling the $a_w$-self half widths, the measurement by Danesh-var et al. [62] was employed and the coefficients are given in the third column of Table 4.

The temperature exponent of $\gamma_2$ ($n_{\gamma_2}$), predicted by the rCMDS model, are also presented.

The temperature dependence of $\gamma_2$ is modelled using the following power-law equation:

$$ \gamma_2(T) = \gamma_2(T_{ref}) (T_{ref}/T)^{n_{\gamma_2}} \quad (12) $$

Figure 10 shows the smoothed temperature dependence of speed dependence
calculated with the rCMDS model, presented with the filled blue circle, and by
fitting them to Eq. (4), we can obtain the best fit line. The results are compared
to the measured values of Ref. [50] for the CO₂-N₂ system. The experimentally-
measured values of Ref. [50] confirm that the temperature dependence of γ₂ is
in fact different than γ₀, (i.e. nγ₀≠nγ₂) as the presented nγ₀ values shown by the
purple line (obtained in this study) are larger than the nγ₂ values. Although the
perturber gas is different, the experimental results of Ref.[50] are in moderately
good agreement with the rCMDS predictions. The large uncertainties most
probably are as a result of numerical errors because of insufficient signal/noise
in their data. The γ₂-air of Ref. [43] obtained from rCMDS prediction using the
SDNG profile, presented by the red squares, show close agreement with both
experimentally-measured values of Ref. [50] and the extrapolated SDV results
of this study.
<figure><img src="image_12.png" /></figure>

Figure 10: The smoothed temperature dependence of γ₂-air of CO₂ lines, deduced from rCMDS prediction obtained with the SDV profile (the filled blue circles) and fitted using Padé approximants (see Table 4). In the absence of measurement of the temperature dependence of speed dependence, we used the study by Wilzewski et al. [50] for the CO₂-N₂ and with the SDNG profile for comparison (the green scattered stars). The deduced γ₂-air from rCMDS prediction obtained with the SDNG profile of Ref. [43] are displayed as well. The reported uncertainties are quite large, confirming the difficulty of retrieving this parameter, particularly for the higher |m|. We also displayed the n<sub>γ₀</sub> values obtained in this study by the solid purple line.

The temperature dependence of the half widths is displayed in Figure 10 as well. As shown in the figure, the agreement between n<sub>γ₀</sub> and n<sub>γ₂</sub> values is poor and they behave differently. We investigated how the use of n<sub>γ₀</sub> may affect the speed dependent parameter and eventually the calculated transmission of a sample spectrum at 170 K. The results showed that no improvement was achieved when using the same temperature exponent for γ₂. We also employed the method suggested in Ref. [55] and computed the temperature exponent. The use of these values did not improve the residuals. In section 6, we provide a sample of transmission and the residuals created for testing the estimated n<sub>γ₂</sub>
values.

## 4. Calculating the pressure shift of air- and self-broadened CO₂ lines

In this section, an empirical approach proposed by Hartmann [20] is used to obtain the collisional air- and self-shifts of lines for the vibrational bands of CO₂. To summarise Hartmann’s algorithm, which is based on Robert-Bonamy formalism [64], the shifts of each transition for all the bands can be calculated from the experimentally-measured shift values for a specific band assuming the contribution of symmetric and asymmetric functions. The detailed equations can be found in section 2 of Ref. [20]. The asymmetric term depends on the rotational quantum number. The symmetric term is vibrational dependent and is a function of *m*. Repeating the equations (11) to (15) derived by Hartmann [20], and knowing that the symmetric term is proportionate to the changes in quantum numbers ($Δν_j = ν'_j - ν''_j$), one can obtain

$$ \begin{aligned} \delta[(\nu_1 + \Delta\nu_1, \nu_2 + \Delta\nu_2, \nu_3 + \Delta\nu_3, J_f) \leftarrow (\nu_1, \nu_2, \nu_3, J_i)] = \\ (J_i - J_f)\delta_{\text{rot}}(|m|) + (a_1\Delta\nu_1 + a_2\Delta\nu_2 + a_3\Delta\nu_3)\delta_{\text{vib}}(|m|). \end{aligned} \quad (13) $$

The terms $δ_{Rot}(|m|)$ as half-difference and $δ_{Vib}(|m|)$ as half-sum can be calculated by the following equations using measured values for *P* and *R* branches for the same $|m|$ in any band,

$$ \delta_{Rot}(|m|) = \frac{\delta[P(-|m|)] - \delta[R(+|m|)]}{2}, \quad (14) $$

$$ \delta_{Vib}(|m|) \propto \frac{\delta[P(-|m|)] + \delta[R(+|m|)]}{2}. \quad (15) $$

We should interpolate the measured shifts for the *R* branch to calculate the shifts for even values of *m* in a way that they correspond to the same $|m|$ value in the *P* branch. Then half sum and half differences of the *P* and *R* branch shifts for the same $|m|$ can be calculated. The important point in obtaining the $δ_{Rot}(|m|)$ is that the half-difference should not depend on any considered vibrational band. Furthermore, it should go toward zero for the larger values of *m*. For a given
$|m|$, to calculate the $\delta_{Vib}(|m|)$, it is possible to represent their value by a linear combination of the $\Delta\nu_i$. For instance, for the $(\nu_i^1, \nu_i^2, \nu_i^3) \rightarrow (\nu_f^1, \nu_f^2, \nu_f^3)$ band, one should have

$$[\delta(m) + \delta(-m)]/2 = A(m) \times [a_1(\nu_f^1 - \nu_i^1) + a_2(\nu_f^2 - \nu_i^2) + a_3(\nu_f^3 - \nu_i^3)]. \quad (16)$$

After obtaining $\delta_{Rot}(|m|)$ and $\delta_{Vib}(|m|)$, the parameters $a_1, a_2$, and $a_3$ can be obtained from experimental measurements of various bands since the requirement is to have the shift of both P and R branches and to have the shifts for as many values of $\Delta\nu_1, \Delta\nu_2, \Delta\nu_3$ as possible, where $\Delta\nu_i$ is the differences between the number of quanta of the vibrational mode $\nu_i$ in the upper and lower states. While the $a_1, a_2$, and $a_3$ parameters are the same for all the bands and lines, $A(m)$ depends on $m$ but not on the vibrational states. Then one can write

$$(a_1\Delta\nu_1 + a_2\Delta\nu_2 + a_3\Delta\nu_3)\delta_{Vib}(|m| = J) = \frac{1}{2}\delta[(\nu_1 + \Delta\nu_1, \nu_2 + \Delta\nu_2, \nu_3 + \Delta\nu_3, J) \leftarrow (\nu_1, \nu_2, \nu_3, J-1)] \quad (17) \\ + \delta[(\nu_1 + \Delta\nu_1, \nu_2 + \Delta\nu_2, \nu_3 + \Delta\nu_3, J-1) \leftarrow (\nu_1, \nu_2, \nu_3, J)].$$

From the available shift measurements, the $\delta_{Rot}(|m|)$ and $\delta_{Vib}(|m|)$ functions are modeled using the empirical formulas presented as Eq. (15) of Ref. [20]. Correcting the typographical error in Eq. (15) of Ref. [20], we rewrite this equation as follows,

$$\begin{aligned} \delta_{Vib}(|m|) &= \alpha_1^V + \alpha_2^V e^{-|m|\times\beta_2^V} + \alpha_3^V e^{-|m|\times\beta_3^V} \\ \delta_{Rot}(|m|) &= \alpha_1^R + \alpha_2^R e^{-|m|\times\beta_2^R} + \alpha_3^R e^{-|m|\times\beta_3^R}. \end{aligned} \quad (18)$$

The coefficients $\alpha^R, \alpha^V, \beta^R$, and $\beta^V$ are taken from Table 1 of ref. [20]. Also, $a_1, a_2$, and $a_3$ parameters are taken from the same table to calculate the $\delta$-air values. Figure 11 presents these values for several bands and compared to HITRAN2016. There is good agreement between the experimental values and the calculated shifts using this approach compared with HITRAN2016 values, produced by the routine explained in Ref. [15]. The asymmetry of shifts in the
<figure><img src="image_13.png" /><figcaption>Figure 11: The air-induced line-shift values estimated at room temperature for the present study (PS) calculated using Hartmann’s approach (the solid blue line) [20], the experimental measurement at the same band (the filled blue circles) [36], and compared to those in HITRAN2016 (the open green circles) [13]. The same comparison is performed for another band and compared to the measured δ-air of Ref. [61] (the filled brown circles).</figcaption></figure>

P and R branches are well produced using this approach. For the Q-branch lines, where $m = J_i = J_f$, we used the same equations to calculate the pressure shifts.

For calculating the self-shifts of CO₂ lines, similar steps were followed with slight modification in modeling $\delta_{Rot}(|m|)$ and $\delta_{Vib}(|m|)$. The following empirical functions were used,

$$ \begin{aligned} \delta_{Vib}(|m|) &= -\alpha_1^V - \alpha_2^V e^{-|m| \times \beta_1^V} \cos(|m| \times \beta_2^V + \alpha_3^V) \\ \delta_{Rot}(|m|) &= -\alpha_1^R - \alpha_2^R e^{-|m| \times \beta_1^R} \cos(|m| \times \beta_2^R + \alpha_3^R). \end{aligned} \quad (19) $$

The adoption of these functions was optional to include sufficient coefficients for performing better quality fits. To obtain the coefficients $\alpha^R, \alpha^V$ and $\beta^R, \beta^V$, the experimental shifts such as measured values in Refs [62, 65] were used to optimize the fit. The best fit was obtained using the $\nu_1 + 5\nu_3$ band measured
by Lu et al. [65]. The same measurement was used to estimate the $a_1$, $a_2$, and $a_3$ constants. The transitions of the $\nu_1 + 5\nu_3$ band showed the most promising fit because it is more complete in terms of having sufficient $m$ values.

For obtaining the $a_1$, $a_2$, and $a_3$ parameters, the experimental values measured by Ref. [65] for the $\nu_1 + 5\nu_3$ band were used. By fitting Eq. (17) to the measured pressure shifts in this band and minimizing the difference, we obtained the coefficients $a_1$ and $a_3$ ($a_1 = 0.77$ and $a_3 = 1.77$). We also assumed $a_2 = a_1/2$ to calculate $a_2$ and then tested the fit using the measured values for $\nu_1 + 2\nu_2$ by Daneshvar et al. [62] as shown in Figure 12. The calculated coefficients presented in Table 5 can be used in Eq. (13) to obtain the $\delta$-self for different bands.

<table><caption>Table 5: The coefficients obtained when fitting the experimental $\delta$-self values of Ref. [65] to Eq. (19).</caption><thead><tr><th>Coefficients ($\delta_{vib}$)</th><th>Coefficients ($\delta_{Rot}$)</th></tr></thead><tbody><tr><td>$\alpha_1^V = 0.2672 \times 10^{-1}$</td><td>$\alpha_1^R = 0.1321 \times 10^{-3}$</td></tr><tr><td>$\alpha_2^V = 0.6122 \times 10^{-1}$</td><td>$\alpha_2^R = 0.2209 \times 10^{-3}$</td></tr><tr><td>$\alpha_3^V = 0.1876 \times 10^{-1}$</td><td>$\alpha_3^R = 0.1999 \times 10^{-10}$</td></tr><tr><td>$\beta_1^V = 0.2073 \times 10^{-1}$</td><td>$\beta_1^R = 1.411 \times 10^{-35}$</td></tr><tr><td>$\beta_2^V = 0.1782 \times 10^{-3}$</td><td>$\beta_1^R = 0.8869 \times 10^{-1}$</td></tr></tbody></table>

Figure 12 shows that the best agreement is obtained for the $\nu_1 + 5\nu_3$ band since these data were used to retrieve the empirical functions. For the rest of the bands, the $\delta_{Rot}(|m|)$ and $\delta_{Vib}(|m|)$ functions were fixed.
<figure><img src="image_14.png" /><figcaption>Figure 12: Measured δ-self at 296 K versus *m* shown by symbols and the calculated ones using the semi-empirical approach using Table 5 and Eq. (13) in the present study (PS) shown by the lines at 296 K. The green filled circles are the measured values in the $ν_1 + 2ν_2$ band [62]. The red filled circles present the δ-self measured in the $3ν_1 + ν_3$ band [51], and the blue filled triangles show the measured δ-self for CO₂ in Ref. [65] for $ν_1 + 5ν_3$.</figcaption></figure>

The δ-self for all the bands are produced using this method.

**5. First-order and full line-mixing**

For isolated lines, having the line-shape parameters for each transition of CO₂ enables the modeling of spectra from the line center to the relatively near wings of the line. However, for regions with closely-spaced lines (*Q* branch, for instance), the line-mixing effect needs to be included to accurately model the atmospheric spectra [66]. Because there are not many experimental measurements of line-mixing available [67, 68] (and references therein), we used the approach by Lamouroux et al. [26] which is based on the Energy-Corrected Sudden (ECS) approximation to predict the line-mixing in all bands and regions for the first 10 isotopologues of CO₂ (ordered by the natural terrestrial abundance).
dance assumed by the HITRAN database). The software is a Fortran code with the functions for reading the CO₂ band information, reading spectroscopic data tables, constructing the relaxation matrix and calculating the absorption coefficient [66]. The instruction for using this package is provided in HITRANonline (https://hitran.org/supplementary/).

By taking into account the line-mixing effect, the absorption coefficient for the mixture of CO₂-air at specific temperature T, pressure P, volume mixing ratios of xCO₂, and neglecting the Doppler effect, (considering the binary collision and impact approximations) is [67]

$$ \alpha^{LM}(\nu, x_{CO_2}, P, T) = \frac{8\pi^2}{3hc}\nu[1 - \exp(-hc\nu/k_B T)](x_{CO_2}P)] \times \sum_k \sum_{k'} \rho_k(T)d_k d_{k'} \mathrm{Im}\{\langle k' | [\Sigma - L_0 - iPW_{CO_2-air}(T)]^{-1} | k \rangle\}. \quad (20) $$

where h is the Planck constant and ρk presents the population of the rotational level of line and dk presents the dipole moment matrix components. The terms Σ and L₀ in the imaginary part present the diagonal operators which are linked with the position ν and νk of the lines which are not perturbed. The term WCO₂-air is called the relaxation operator which carries the impact of collisions on the shape of spectral lines. This operator depends on the temperature, the band of interest, and on the broadener (we use air as example perturber here). The diagonal terms of the matrix are associated with the line width (γ) and shift (δ) parameters,

$$ \langle k' | W_{CO_2-air}(T) | k \rangle = \gamma^{CO_2-air}(T) - i\delta^{CO_2-air}(T). \quad (21) $$

By a few assumptions which are described in Ref. [67], the first-order line mixing (Yk(T)) [69] can be obtained to be added as an asymmetry to the line profile. The Yk(T) parameter is defined as

$$ Y_k^{CO_2-X}(T) = 2 \sum_{k'=k} \frac{d_{k'}}{d_k} \frac{\langle k' | W_{CO_2-X}(T) | k \rangle}{\nu_k - \nu_{k'}}. \quad (22) $$
The line-mixing is proportionate to the pressure and hence the first-order approximation is effective if the following condition is true for the pressure

$$P|\langle k|W_{CO_2-air}(T)|k'\rangle| \ll |\nu_k - \nu_{k'}|(k \neq k') \quad (23)$$

Also, the sum rules should be satisfied for the $Y_k$, (e.g. the sum over the lines of a given band of the $Y_k$ multiplied by the line intensities should be zero).

The first-order line-mixing parameters are deduced from the calculation of the relaxation matrix as shown in Eq. (22). Yet the difference of full line-mixing and the first-order approximation is within the concept of individual transitions; the use of full line-mixing changes the shape of lines in a way that the conception of single lines vanishes [70]. However, with the first-order approximation each transition has particular line parameters. Another difference between the two approaches is apparent when the absorption lines are too close, especially for some Q branch lines (where the wavenumber difference between the lines is less than the Lorentzian half width). The first-order approximation fails in modeling the absorption since it supposes that the spectral lines are not too close to each other and their interaction is weak in this case. Therefore, $Y_k$ parameters must be used very cautiously in clusters of closely spaced and strongly overlapping lines.

It should be pointed out that the full line-mixing impacts other line parameters. The line-shape parameters along with the first-order line-mixing parameter differ with the change of pressure and temperature meaning that each single line in the calculation of the absorption coefficient has non-constant intensity.

To perform the calculation of full and the first-order line-mixing, the Lamouroux et al. [26] routine was updated to overcome a previously-existing problem of over-writing the HITRAN line widths. The original version of the code (which is based on a preliminary version explained in Ref. [15]), when generating the line-mixing data tables, overwrites HITRAN widths with an alternative semi-empirical procedure yielding different results in some bands. This was justified for some of the bands including 2.06 µm band but was yielding worse results in others. Now the line-mixing package is completely consistent
with parametrizations provided in HITRAN. Another useful feature, added to
the update of the line-mixing package, is regarding the possibility of the pres-
ence of humidity to reach a sub-percent accuracy in atmospheric retrieval of
CO2. Therefore, the water-broadening half widths of the VP were used along
with the temperature dependence of the water-broadening (obtained in Refs
[23, 71, 72, 73]) added to the spectroscopic input files. The contribution of
collisions with water is taken into account only in the diagonal elements of the
relaxation matrix (through the line broadening) but not for the off-diagonal
elements. The water-broadening half width for the SDV profile were produced
using the average ratio of air-broadening by the Voigt to SDV air-broadening.
Also, the spectroscopic data tables were constructed with the appropriate line-
shape parameters. The updated line-shape parameters, obtained in this paper,
were used for the calculation of line-mixing. For example, the air-broadening
half widths were replaced with the results of Refs [37, 35] to make sure that the
well-isolated lines are adequately produced.

The HITRAN2016 line list [13] of CO₂ below 8000 cm⁻¹ is mainly a com-
bination of the Carbon Dioxide Spectroscopic Database (CDSD) line positions
[74] and the ab initio line intensities of the University College London (UCL)
[75, 76, 77]. The CDSD provides a computed line list based on the global mod-
eling of the line positions and intensities within the framework of the method
of effective operators [78, 79]. The UCL variational approach provides semi-
empirical line positions based on the fitted potential energy surface and line
intensities computed with an ab initio dipole moment surface. A number of is-
sues for the line positions and intensities have been identified in new laboratory
and atmospheric spectra below 8000 cm⁻¹ (mainly from the Kitt Peak National
Observatory, MkIV balloon and Cavity Ring Down spectra, Total Carbon Col-
umn Observing Network (TCCON)). A general review of the bands affected by
these issues in the HITRAN2016 database and the necessary updates for the
line positions and intensities based on Refs [80, 81, 82] were performed. The
changes made to the line positions and intensities for the upcoming 2020 version
of the HITRAN spectroscopic database were implemented here but will be de-
scribed in the separate publication. The partition functions were also updated using TIPS2017 [83], enabling the computation for the twelve isotopologues involved in the HITRAN2016 edition. The updated package will be available in HITRANonline (https://hitran.org/supplementary/).

To examine how the modified line-mixing (LM) package works beyond the first-order, the absorption coefficients are produced using the VP plus full line-mixing for three sets of parameters: i) the updated Voigt parameters obtained in this study; ii) the former version of Lamouroux's code (where there was the problem of over-writing HITRAN line-width); and iii) the HITRAN 2016 parameters. The laboratory spectra measured at room temperature were used to obtain the residuals. Figure 13 presents the magnified residuals for one transition. As can be seen, the minimum residuals were achieved when using the updated parameters in the present study.
<figure><img src="image_15.png" /></figure>

Figure 13: The FT transmission spectra of CO₂-air at T=294.1 K, total P=598.9 Torr and $P_{CO_2}$ = 8.68 Torr (see section 6 for details on the laboratory spectra). The bottom panel shows the magnified residuals for three types of calculations: i) the residuals for the updated Voigt parameters, ii) line-shape parameters from [30] supplied in HITRAN 2016 line mixing package, and iii) the residuals for the former version of Lamouroux's package. All three include full line-mixing obtained using Lamouroux's LM package [26] in the 4700-5100 cm⁻¹ region.

Besides calculating the absorption coefficient with full line-mixing, the line-mixing code is adopted in such a way that it calculates the first-order line-mixing parameters ($Y_k$) for all the allowed transitions. Therefore, using the line-mixing package, the absorption coefficient using full line-mixing or the first-order line-mixing parameter can be calculated. The new edition of the HITRAN database will include these calculated $Y_k$ values for each of the lines.

The calculated $Y_k$s for one band is presented in Figure 14 and compared to the results of Ref. [35] obtained based on the exponential power gap law (EPG) formalism for the same band. The program is capable of calculating the absorption coefficient in four ways, first by the VP without including line-mixing, then using the VP including the first-order line-mixing, third, by using
<figure><img src="image_16.png" /><figcaption>Figure 14: The first-order line-mixing of CO₂ calculated from the line-mixing code developed by Lamouroux et al. [26] and comparison with those measured by Predoi-Cross et al. [35]. The presented Yₖ values are calculated at 296 K and 1 atm. The CO₂ mixing ratio was 1.5 × 10⁻²</figcaption></figure>

VP accounting for full diagonalization line-mixing, and last by the SDV profile including first-order line-mixing (see section 6).

To include the temperature dependence of the first-order line-mixing ($n_Y$), we used the results predicted by rCMDS. Figure (15) presents the $n_Y$ and their agreement with the measured values of Ref. [50]. A power-law model was used to represent the temperature dependence of the first-order line-mixing parameter. To simplify the calculations with HAPI, two average values over $m$ were obtained for the higher and lower $m$ values (i.e. for the sake of simplicity in calculation using HAPI, the average values are used for all the bands with the following functional form $Y_k(T) = Y_k(T_{ref}) \times (T_{ref}/T)^{1.6}$ for $m \ge 16$ and $n_Y = 0.41$ for the $m < 16$ for the transition of interest $k$). The preliminary examinations showed that the use of the average values provide satisfactory results. However, it is not implying that there is no difference in the calculated
605 spectra using the individual $n_Y$ from rCMDS and the average value.

Although the self-broadening of CO₂ lines is not particularly significant for the atmosphere of Earth, we updated them to calculate the Yₖ-self.

<figure><img src="image_17.png" /><figcaption>Figure 15: The temperature exponent of first-order line-mixing for the CO₂-air system, deduced using rCMDS and their agreement with the measured values of Ref. [50].</figcaption></figure>

## 6. Validation of the results using laboratory spectra

Finally, for validating the updated line-shape parameters, laboratory spectra were measured by Sung at the Jet Propulsion Laboratory (JPL) by using a Bruker 125HR Fourier transform spectrometer in the 2.1 µm region (GOSAT and OCO-2 missions range) [53]. The second set, measured by Vander Auwera at the Université Libre de Bruxelles (ULB) using a Bruker IFS 125h Fourier transform spectrometer in the 4.3 µm region (ACE-FTS mission range) [26]. These measurements were used to compare with the calculation of the absorption coefficient/transmission by HAPI [84] and line-mixing package which are accessible in the HITRAN web page.

With the numerous functionalities that HAPI holds, the user can select the desired line-shape profile for calculating the absorption coefficients and download the parameter groups for each profile. Then by entering an appropriate
broadener and molecule (CO₂-air), the proper line list can be fetched. Then by calling the defined line-shape functions, the spectra can be produced. Furthermore, the first-order line-mixing is now implemented in the formulation of HAPI and can be adopted with different line-shape models [84].

Figure 16 shows the transmission of CO₂ perturbed by air at room temperature and a pressure of 598.9 Torr recorded between 4700 and 5100 cm⁻¹ at JPL.

<figure><img src="image_18.png" /></figure>

Figure 16: The FT transmission spectra of CO₂-air at T=294.1 K, total P=598.9 Torr and P<sub>CO₂</sub> = 8.68 Torr (the top panel). The second panel shows the magnified residuals for two types of calculations using the Fortran package developed by Lamouroux et al. [26] and HAPI: 1) The residuals for the updated Voigt parameters, 2) the HITRAN2016 line-shape parameters modeled with the VP, 3) the residuals for the updated SDV parameters accounting for first-order line-mixing using HAPI, 4) the same residuals (SDV+Y<sub>k</sub>) using the Fortran code, and 5) the residuals for the updated Voigt parameters including full line-mixing using the Fortran code.

Various sets of residuals were computed to compare the laboratory spectra with the calculations made by the Fortran LM code [26] and HAPI. In the second panel, the transmission spectra was calculated for the .par (.par is the
default line transition format in HITRAN) files updated using the Voigt line-
shape parameters obtained in this paper (γ₀-air, γ₀-self, nγ₀ – air, δ-air, and
Yₖ-air and self). In addition, the HITRAN 2016 line-shape parameters are used
as they are and the transmission is calculated with the VP. The third and the
forth sets are the SDV line-shape parameters including the Yₖ parameters (γ₀-
air, γ₀-self, nγ₀ – air, γ₂ – air, nγ₂ – air, δ-air, and Yₖ-air and self) using HAPI
and the Fortran code package developed by Lamouroux et al. [26]. The fifth
set uses the updated Voigt input file with full line-mixing. Including the full
line-mixing in the calculation of absorption cross-section using the SDV profile
is difficult and not cost-effective since it involves diagonalizing the relaxation
matrix and then inverting it for each value of the molecular speed.

Using these new parameters led to improvements in calculating the spectra
when including the line-mixing effects, and the desired accuracy of less than
0.3 % was achieved for most of the lines in the 4700-4900 cm⁻¹ region as it is
displayed for two transitions in Figure 16 (the RMS values are shown for each
calculation). For the tested region (with the *P* and *R* branches), very small dif-
ferences in residuals can be observed when the transmission spectra are modeled
using the VP plus full line-mixing compared to the transmission obtained using
the SDV profile with first-order line-mixing. However, it must be emphasized
that the first-order line-mixing does not work effectively when the coupling is
excessively strong. In the case of "perpendicular" bands with the densely packed
*Q* branches or wherever the lines are strongly affected by the line-mixing, the
use of full line-mixing is strongly recommended. Similar results were obtained
using HAPI in calculating the residuals. The highest amplitudes of the residuals
obtained when using HITRAN2016 line-shape parameters. Comparison of the
residuals for the entire region shows that the improvement achieved by the new
parameters and the inclusion of line-mixing is better than 0.5 % compared to
HITRAN2016.

Another validation test is to compare the actual transmission calculated by
the different sets of line parameters for the VP and SDV profiles. Because
in fitting the spectra, typically the retrieved half widths and intensities are
correlated, the unaccounted effects (of broadening) might impact the retrieved intensity for achieving optimized fit.

<figure><img src="image_19.png" /><figcaption>Figure 17: The FT transmission spectra of CO₂-air measured by Sung at JPL (see section 6), at T=294.1 K and P=598.9 Torr with 0.0145 as the mixing ratio of CO₂ (shown with the blue line). Various transmissions are calculated for the HITRAN2016 parameters with the VP (the red line), the updated VP parameters (the black line), the updated VP parameters accounting for Yₖ (the red line), and the updated SDV parameters (the green line).</figcaption></figure>

Figure 17 presents the observed and calculated transmission spectra when using the VP and SDV parameters group. Ideally, the intensity of lines which is the area under curve should be the same for all the used line-shape profiles. However, the unaccounted broadening effect may cause a slight difference. The magnified transmission for the transition near 4863.4 cm⁻¹ shows the discrepancy of HITRAN2016 fitted with the VP with the experimental spectrum. It should be noted that the half widths in HITRAN2016 are based on the measured data of Ref. [16] which are retrieved using the SDV profile. This observed difference verifies that the width of VP should not be substituted with those of the SDV line widths.

For verifying the temperature dependence of γ₂, the transmission spectrum measured at T=170.1 K was tested to see how the residuals change when the SDV profile is used. In Figure 18, the transmission spectrum and the resid-
uals for the VP and the SDV profiles are shown. Applying the temperature
dependence of the speed dependence improved the obtained residuals slightly.
For most of the transitions, the best residuals were obtained when using the
SDV profile either with or without the first-order line-mixing (RMS=0.0054
and 0.0055, respectively), and next by using the VP and full line-mixing effect
(RMS=0.0064). Furthermore, we checked the use of the temperature depen-
dence of γ₀ for γ₂ as suggested by Ghysels et al. [55] and no improvement was
achieved in the residuals.

680

<figure><img src="image_20.png" /><figcaption>Figure 18: The top panel shows the FT transmission spectra of CO₂-air measured by Sung at JPL (see section 6), at T=170.1 K and P=400.4 Torr with 0.03 as the mixing ratio of CO₂. The bottom panel displays the residuals multiplied by 100. The blue line shows the difference of experimental and calculated spectra when the HITRAN2016 VP parameters are used in calculations, the green line shows the new VP parameters, the red line presents the new Voigt parameters when including the full line-mixing calculated using Fortran code, the cyan line shows the residuals using SDV parameters including the first-order line-mixing and the purple line presents the same residual calculated without line-mixing.</figcaption></figure>

685

This verification does not necessarily imply that the predicted temperature
dependence values are the best. More measurement data and further examina-
tions are needed to confirm or refine the results.
We also validated the results in a different spectral region as presented in Figure 19 using the line-mixing package by Lamouroux et al. [26]. The transmission file for CO₂-air was measured by Vander Auwera at ULB at T=295 K and P = 818.3 Torr, and a mixing ratio of XCO₂ = 0.0047, The second panel shows that the use of the VP and SDV profiles generates comparable residuals when including the first-order line-mixing (the RMS value for the VP+Yk is 0.00202 and for the SDV+Yk, it is 0.00201). The use of the VP accounting for full line-mixing led to slight improvement in the residuals (the RMS value is 0.002). Using the Fortran LM package, the difference obtained between the full line-mixing and first-order line mixing calculations for this pressure and temperature condition is well below 0.1 % as shown. In the third panel, the HITRAN2016 residuals are compared with the new Voigt parameters when including and not including the first-order line-mixing effect. The residuals for the new VP are improved by 0.5 % compared to HITRAN2016 CO₂ parameters for most of the lines. For the 4.3 µm region, the remaining residuals are due to the errors of line intensities and CO₂ amount in the experimental samples but not to the line-shape parameters as explained in Ref. [26]. It should be mentioned that before testing for line-mixing, we tried to resolve the line intensities and sample amount issue. The line intensities were enhanced by 4 % for all CO₂ isotopologues whereas for the second most abundant CO₂ isotopologue the line intensities were lowered by 2 %.
<figure><img src="image_21.png" /></figure>

Figure 19: The FT transmission spectra of CO₂-air at T = 295 K, total P = 818.3 Torr and XₐCO₂ = 0.0047. The second panel shows the magnified residuals for the calculation of transmission when using: 1) the new SDV + first-order line-mixing, 2) HITRAN2016 VP + full line-mixing, 3) the new VP + full line-mixing. For the third panel: 1) the HITRAN 2016 VP with no line-mixing, 2) the new VP with no line-mixing, 3) the HITRAN 2016 VP + the first-order line-mixing and 4) the new VP + first-order line-mixing.

All the validation tests verify that including line-mixing parameters in modeling the spectra either as full line-mixing or the first-order line-mixing improves the residuals. The best results are achieved when using the updated Voigt parameter groups when accounting for the full line-mixing and very closely by modeling using the SDV parameter group with the first-order line-mixing parameters. It is worth noting that for the regions where the lines are dense and strongly affected by line-mixing, the line-mixing effect is predominant compared to the speed dependence effect. Therefore, the absorption coefficient most reliably can be produced by using the Voigt profile, including the full line-mixing effect. The benefit of the first-order approximation is that for each transition,
the line mixing can be calculated quite fast and we should emphasize that the
difference between the full and the first-order line-mixing is that when the sep-
aration between the mixed lines is less than a few Lorentz line broadening, the
first-order approximation is not capable of fitting the spectra correctly [70].

7. Future work

The future goal involves the extension of the HITRAN database by including
the CO₂ line-shape parameters obtained using the Hartmann-Tran (HT) profile
[85] and using the semi-empirical models similar to those described in this paper
to generate the HT parameters for all the transitions. This will be challenging
regarding the extended number of parameters needed for the HT profile and
concerning their correlation. Therefore, it is necessary to acquire high signal-
to-noise spectra measured in a wide range of pressures.

Feasibly, there are measurements and calculations available for this pur-
pose, including the rCMDS-calculated results by Nguyen et al. [43] shown in
the present study. Since the infrastructure for adding HT parameters to the
database has already been created [19, 21, 84], HAPI can be modified to ac-
count for the first-order line-mixing in the HT profile likewise.

As it was emphasized in previous sections, the use of the first order line-
mixing in bands with Q-branches can result in nonphysical results including
negative absorption, especially further from the line centers. The use of full
line-mixing is strongly advised, which can be realized by pre-calculating look up
tables for remote-sensing missions. However, some missions require calculating
absorption in real time and calculating full line-mixing can be too slow for
these applications. It is desirable to create a model that would be able to
renormalize Y-coefficients. However, this is a very non-trivial task and will
likely be very band-dependent. Additionally, remote-sensing experiments often
require an inclusion of the so-called far wing effect. For instance, in the version
4 [86] retrievals for the ACE-FTS, far-wing parametrizations of Ref. [87] are
used.
## 8. Conclusion

This work presents an update and extension of the HITRAN CO₂ spectral parameters, which includes updates on the CO₂-air and CO₂-CO₂ line-shape parameters together with the speed dependence of the broadening parameters, their temperature dependent exponents, the full and first-order line-mixing as well as their temperature dependencies.

Furthermore, a semi-empirical approach was examined for calculating the line shifts of air- and CO₂ broadened CO₂ absorption lines. This method yielded reliable estimates of pressure shifts for every transition. The rotational and vibrational dependence of the line shifts were reproduced well by the suggested process. For most of the bands, the average difference of the new shifts with those in HITRAN2016 is about 7 %, which might lead to observable improvements in radiative-transfer codes.

Different sets of spectra in two wavenumber regions were used for validating the updated line-shape parameters. Significant improvements were achieved for both regions examined in this study and, compared to HITRAN2016, 0.5 % improvement was achieved which will be advantageous for the CO₂ retrievals for the OCO-2 mission, for example. In particular, near 2.1 µm the residuals were obtained at a desired level of 0.3 % accuracy at room temperature when including the line-mixing effect. For lower temperature when testing the SDV profile, the fit quality was not as notable as at room temperature, probably due to the large uncertainties on the temperature dependence of the speed dependence parameter. Therefore, additional examinations of experimental laboratory spectra and also atmospheric spectra are desired to assess the achievement of these new parameters.

The new data is proposed for the HITRAN2020 edition and can be tested using the HAPI program and the line-mixing package available in https://hitran.org/supplementary/. The extrapolation approach described in this work is extendable to estimate air and CO₂ line-shape parameters for lines beyond the scope of HITRAN and can be used for the CO₂ HITEMP database
as well.

## 9. Acknowledgements

The authors are thankful for the support and experimental data received from Keeyoon Sung at the Jet Propulsion Laboratory (JPL) and Jean Vander Auwera at the Université Libre de Bruxelles (ULB). Furthermore, we are very appreciative to Jean-Michel Hartmann for his excellent guidance through the calculation of CO₂ line shifts. Robert Gamache is thanked for helpful discussions. We also would like to thank Sergio De Souza-Machado for finding deficiencies in previous implementations of line-shape parameters in HITRAN and for stimulating discussions.

Development of the HITRAN database is supported through the NASA Aura grant NNX17AI78G.

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