Potential for the measurement of mesosphere and lower thermosphere (MLT) wind, temperature, density and geomagnetic field with Superconducting Submillimeter-Wave Limb-Emission Sounder 2 (SMILES-2)

Abstract

Submillimeter-Wave Limb-Emission Sounder 2 (SMILES-2) is a satellite mission proposed in Japan to probe the middle and upper atmosphere (20–160 km). The main instrument is composed of 4 K cooled radiometers operating near 0.7 and 2 THz. It could measure the diurnal changes of the horizontal wind above 30 km, temperature above 20 km, ground-state atomic oxygen above 90 km and atmospheric density near the mesopause, as well as abundance of about 15 chemical species. In this study we have conducted simulations to assess the wind, temperature and density retrieval performance in the mesosphere and lower thermosphere (60–110 km) using the radiometer at 760 GHz. It contains lines of water vapor (${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$), molecular oxygen (${\mathrm{O}}_{\mathrm{2}}$) and nitric oxide (NO) that are the strongest signals measured with SMILES-2 at these altitudes. The Zeeman effect on the ${\mathrm{O}}_{\mathrm{2}}$ line due to the geomagnetic field ($\mathit{B}$) is considered; otherwise, the retrieval errors would be underestimated by a factor of 2 above 90 km. The optimal configuration for the radiometer’s polarization is found to be vertical linear. Considering a retrieval vertical resolution of 2.5 km, the line-of-sight wind is retrieved with a precision of 2–5 m s${}^{-\mathrm{1}}$ up to 90 km and 30 m s${}^{-\mathrm{1}}$ at 110 km. Temperature and atmospheric density are retrieved with a precision better than 5 K and 7 % up to 90 km (30 K and 20 % at 110 km). Errors induced by uncertainties on the vector $\mathit{B}$ are mitigated by retrieving it. The retrieval of $\mathit{B}$ is described as a side-product of the mission. At high latitudes, precisions of 30–100 nT on the vertical component and 100–300 nT on the horizontal one could be obtained at 85 and 105 km (vertical resolution of 20 km). SMILES-2 could therefore provide the first measurements of $\mathit{B}$ close to the electrojets' altitude, and the precision is enough to measure variations induced by solar storms in the auroral regions.

1 Introduction

The mesosphere and lower thermosphere (MLT) is a transitional region (60–110 km) between atmospheric layers with very different characteristics, namely the stratosphere (15–60 km) and the thermosphere (90–400 km) . In the stratosphere, ${\mathrm{O}}_{\mathrm{3}}$ controls the chemical and radiative processes; hence, it also regulates the temperature and the dynamics. In the thermosphere, the chemistry and the radiative balance are mainly controlled by the oxygen atoms. In this region, wind and temperature exhibit large diurnal variations and are strongly influenced by tides generated in the lower atmosphere. The thermosphere is also the region of interactions between the ionized (plasma) and neutral atmosphere.

The mean physical characteristics of the MLT (wind, temperature and density) are primarily established by energy transferred from the troposphere via small-scale gravity waves (GWs) . Hence, the MLT state deviates significantly from the radiative equilibrium, as illustrated by the occurrence of the coldest point of the Earth system ($\approx \mathrm{150}$ K) in the summer polar mesopause. Waves with planetary scales also contribute to the upper atmosphere climate (general circulation) through their momentum and energy transport/deposition . In particular, tides that are mainly driven by diurnally varying diabatic heating in the troposphere and the stratosphere propagate upward, with their amplitude reaching a maximum in the MLT . Hence, the MLT plays a key role in connecting the lower and upper atmosphere and also in linking both hemispheres . Furthermore, the increase in anthropogenic ${\mathrm{CO}}_{\mathrm{2}}$ is responsible for a cooling of 1–3 K decade${}^{-\mathrm{1}}$ in the MLT that has been measured since the early 1990s .

The processes behind these phenomena are still not well quantified. The difficulty arises from the nonlinear interactions between the GWs, tides, planetary waves, the background wind and the electromagnetic field . The system is further complicated by the interconnections between the dynamics and highly variable chemical species, as well as the very different temporal and spatial scales of these processes. Observations of the MLT, in particular of wind, temperature and density, are therefore essential to further our understanding of this region .

Continuous measurements of temperature and wind are performed from ground-based stations using lidars , radars   and, up to 70 km, millimeter radiometers . Density was recently monitored using meteor radars , but measurements remain scarce. Satellite observations of the MLT have also been performed for several decades. The missions currently in operation and capable of measuring at these altitudes are listed in Table . Temperature is measured with various techniques and spectral domains , but discrepancies larger than 10 K can be found between these measurements above 80 km . and described the past and current wind measurements from space. Currently only TIDI and MLS (and soon MIGHTI) are capable of measuring MLT winds but with a poor sensitivity below 80 km , and MLS, which is equipped with a single antenna, can only measure one component of the wind vector (it was not designed for wind measurement).

Table 1

Current and future satellites and instruments capable of measuring the MLT (60–110 km).

Satellites in operation	Launch year
Odin	2001
Thermosphere Ionosphere Mesosphere Energetics and Dynamics, TIMED	2001
Atmospheric Chemistry Experiment, ACE	2004
Aura	2004
Ionospheric Connection Explorer, ICON	Second quarter of 2019
Instruments	Main products
Sub-millimeter limb sounder, SMR (Odin)	$T$, ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$, $\mathrm{NO}$
Optical Spectrograph and InfraRed Imaging System, OSIRIS (Odin)	$T$, airglow
Sounding of the Atmosphere using Broadband Emission Radiometry, SABER (TIMED)	$T$, $\mathrm{O}$
TIMED Doppler Interferometer, TIDI (TIMED)	$W$, $T$, airglow
Microwave Limb Sounder, MLS (Aura)	$T$, ($W$); $<\mathrm{90}$ km
Michelson Interferometer for Global High-resolution Thermospheric Imaging, MIGHTI (ICON)	$W$, $T$, airglow; $>\mathrm{90}$ km
Planned missions
Mesospheric Airglow/Aerosol Tomography Spectroscopy, MATS (Innosat), 2019	GWs, $T$, NLC; 90–100 km
Stratospheric Inferred Winds, SIW (Innosat), 2023	$W$, $T$; $<\mathrm{80}$ km

$W$, $T$ and NLC denote wind, temperature and noctilucent cloud.

In the future, we clearly risk a lack of satellite observations since all the current missions (except ICON) have already exceeded their theoretical lifetime. Sweden is preparing two Innosat-based missions that are of interest for the study of the MLT (Table ). The MATS mission aims at characterizing the 3-D structure of the GWs near 90–100 km using the oxygen A-band emission and the ultraviolet light scattered by noctilucent clouds . Information on temperature will also be retrieved. The other mission is SIW, a sub-millimeter limb sounder that will measure horizontal wind, temperature and trace gases up to about 80 km . The MATS and SIW missions will be operational for 2 years between 2019 and 2021 and between 2023 and 2025, respectively. Other projects have not been selected yet and remain uncertain. For example, proposed a THz limb sounder (TLS) to measure the atomic oxygen line at 2 THz. Such an instrument could fly together with a new version of SABER . The European Space Agency (ESA) is studying a limb sounder operating between 0.8 and 4 THz for the retrieval of the abundance of chemical species such as atomic oxygen (O) or the hydroxyl radical (OH) . TALIS, a limb sounder using similar spectral bands to Aura MLS, is being studied in China . described a concept for a limb sounder on board a CubeSat to measure temperature with high horizontal resolution using the molecular oxygen (${\mathrm{O}}_{\mathrm{2}}$) A-band infra-red emission.

Superconducting Submillimeter-Wave Limb-Emission Sounder 2 (SMILES-2) is a middle and upper atmospheric satellite mission proposed to the Japan Aerospace Exploration Agency (JAXA) . If selected, it will be launched around 2026 on a JAXA M-class satellite. The objectives are to provide geophysical information with unprecedented precision and altitude coverage such as the temperature between 15 and 160 km, horizontal wind between 30 and 160 km, atmospheric density up to 110 km, ground state of atomic oxygen between 90 and 160 km, and more than 15 trace gases' abundance . The proposed satellite will be equipped with two antennas for the limb measurement of horizontal winds, and three radiometers near 0.7 and 2 THz cooled at 4 K, a technology successfully tested with JEM/SMILES . With a precessing orbit and the high receiver precision, it will be possible to retrieve diurnal variations of very weak signals, as demonstrated with JEM/SMILES .

In this study we discuss the potential for SMILES-2 to measure the main characteristics of the neutral MLT, namely wind, temperature and atmospheric density. An essential source of information is the ${\mathrm{O}}_{\mathrm{2}}$ transition at 773.8 GHz. As a magnetic dipole, ${\mathrm{O}}_{\mathrm{2}}$ is subject to the Zeeman effect induced by the Earth's magnetic field ($\mathit{B}$). Special care is taken to properly include this effect in the simulations in order to correctly assess the measurement performance. Retrieval errors induced by uncertainties on $\mathit{B}$ are mitigated by retrieving its three components simultaneously with other atmospheric parameters. The scientific interest of the retrieval of $\mathit{B}$ is also discussed. In Sect. , the characteristics and principle of the observations are presented in details. Sections  and  describe the Zeeman model and the retrieval setting, respectively. The retrieval errors are discussed in Sect. . Finally, we summarize the results and discuss future analysis for SMILES-2.

Figure 1

SMILES-2 orbit over the Northern Hemisphere. The red and purple lines show the forward and aftward lines of sight (LOSs). The circles show the tangent-point footprints .

[Figure omitted. See PDF]

2 Measurement principle

2.1 Observation method

The observation characteristics are summarized in Table . The atmospheric limb is scanned from about 20 to 180 km. Scans are performed alternatively with two antennas looking at perpendicular directions to each other. Both antennas can probe the same atmospheric column with a 7 min delay (Fig. ), allowing us to derive the 2-D horizontal winds. The same method will be used for SIW and more information is given in . The limb geometry provides a high vertical resolution of 2–3 km, and the zonal and meridional samplings at the Equator are about 20${}^{\circ }$ (2200 km) and 6${}^{\circ }$ (650 km), respectively.

The orbit precesses with a period of about 3 months. The satellite orientation is reversed after every half precession cycle in order to keep the solar panels properly illuminated and the radiative-cooling panels in the shadow side. The latitude coverage is between 50${}^{\circ }$ S and 80${}^{\circ }$ N or 80${}^{\circ }$ S and 50${}^{\circ }$ N depending on the satellite orientation. At low and middle latitudes, the same latitude is observed twice per orbit, with LT differences close to 12 h. Hence, gathering the observations between each maneuver allows us to piece together the complete diurnal cycle of the retrieved parameters.

Table 2

SMILES2 observation characteristics.

Satellite altitude	550 km
Latitude range	50${}^{\circ }$ S–80${}^{\circ }$ N or 80${}^{\circ }$ S–50${}^{\circ }$ N
Scan altitude	20–200 km (4.27 ${}^{\circ }$)
Scan velocity	0.1${}^{\circ }$ s${}^{-\mathrm{1}}$ (43 s scan${}^{\text{a}}$)
Spectrum integration time	0.25 s (1.1 km at 80 km${}^{\text{b}}$)
Antenna diameter	75 cm
Frequency $\iff$ velocity at 760 GHz	1 m s${}^{-\mathrm{1}}$ $\iff$ 2.5 kHz

${}^{\text{a}}$ Calibration measurements will be performed over the upper range. ${}^{\text{b}}$ Tangent point vertical displacement during the integration time.

2.2 Spectral bands

Three spectral bands near 638, 763 GHz and 2 THz are measured simultaneously . The band at 638 GHz contains a strong stratospheric and lower mesospheric signal from ozone (${\mathrm{O}}_{\mathrm{3}}$). This band is the same as that selected for SIW and its main characteristics are described in . Two THz bands are measured alternatively: one contains $\mathrm{OH}$ lines and the second one an $\mathrm{O}$ line . The $\mathrm{O}$ line is used to retrieve between 90 and 160 km, the abundance of $\mathrm{O}$ in its ground state, wind and temperature .

Table 3

The 763 GHz spectral band.

Local oscillator	Lower sideband	Upper sideband	DSB Tsys	Vertical
($\mathrm{GHz}$)	($\mathrm{GHz}$)	($\mathrm{GHz}$)	Resolution	resolution
763.5	750.0–756.0	771.0–777.0	180 K	0.0366${}^{\circ }$
	${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$ at 752 GHz	${\mathrm{O}}_{\mathrm{2}}$ at 773 GHz	0.5 $\mathrm{MHz}$	1.9 km${}^{\ast }$

${}^{\ast }$ Estimated for a tangent height of 80 km including the antenna field of view and the scan velocity.

Figure 2

(a) Spectra at the Equator in daytime for tangent heights of 50, 70 and 90 km. The $x$ coordinates are the intermediate frequency (IF). The yellow dashed lines indicate the noise standard deviation (2$\mathit{\sigma }$). The grey area shows the frequency range of 200 MHz in which the Zeeman radiative transfer model is used (only for the upper-side band). (b) Same as (a) but for 80${}^{\circ }$ N and nighttime winter conditions. The red labels indicate molecular lines in the upper-side band. The LO frequency is 763.5 GHz.

[Figure omitted. See PDF]

The 763 GHz band (Table ) is the band considered in this study. It contains lines of water vapor (${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$) at 752.03 GHz and ${\mathrm{O}}_{\mathrm{2}}$ at 773.84 GHz (Fig. ) that provide a strong signal in the MLT. It also contains other molecular lines, weaker but still suitable for our study: nitric oxide ($\mathrm{NO}$, 751.67–752.00 and 773.02–773.05 GHz), ${\mathrm{O}}_{\mathrm{3}}$ (754.46 and 776.66 GHz) and carbon monoxide isotopologue (${}^{\mathrm{13}}\mathrm{CO}$) at 771.183 GHz. The bands have changed compared to those originally described by , a change motivated to reduce the power consumption. In the new setting, the $\mathrm{CO}$ line is about 50 times weaker than that previously selected.

2.3 Qualitative description of the information content

Most of the lines in the spectral bands are emitted by chemical species in their ground state under local thermodynamic equilibrium. The molecular abundance and the temperature are retrieved from the amplitude of the lines. Their Doppler shift (2.5 kHz for 1 m s${}^{-\mathrm{1}}$) is used to retrieve the line-of-sight (LOS) wind. The atmospheric density is derived from the ${\mathrm{O}}_{\mathrm{2}}$ abundance considering that the volume mixing ratio of ${\mathrm{O}}_{\mathrm{2}}$ is well known below 110 km .

Above about 70 km, the lines are broadened by the random molecular motions, i.e., Doppler broadening, and they do not carry direct information on the pressure (Appendix A). Consequently, the density of the molecule can be retrieved and not the volume-mixing ratio (VMR) as in the lower altitudes.

2.4 LOS altitude

In this study, we consider LOS tangent heights between 60 and 110 km. They are provided as input for the inversion algorithm; therefore they must be known before inverting the spectra. Height registration for a complete scan is calculated differently in the lower part of the scan and in the range of interest (between $\sim \mathrm{20}$ and 60 and between 60 and 110 km, respectively).

Between 20 and 60 km, an approach similar to that used for Aura MLS  can be used. The LOS tangent pressure and atmospheric temperature would be retrieved simultaneously from the ${\mathrm{O}}_{\mathrm{2}}$ line near 763 GHz and from ${\mathrm{O}}_{\mathrm{3}}$ lines in the 638 GHz band. The height of the pressure levels would then be derived from the hydrostatic equilibrium equation. The resulting precisions are estimated to be better than 1 % and 75 m for the LOS tangent pressure and height, respectively .

In the altitude range of interest ($>\mathrm{60}$ km), the LOS tangent heights are inferred from the extrapolation of those calculated previously for the lower altitudes and attitude data from the star-trackers and GPS on board the satellite. Based on JEM/SMILES results, the expected precision on the retrieved LOS tangent heights will be 100 m or better .

3 Zeeman effect modeling

The Zeeman effect on atmospheric molecular-oxygen lines has been extensively studied . In this study, we describe the polarized radiance with Stokes vectors as in (e.g., Eq. 1.32), and . The magnetic field characteristics (amplitude and orientation angles with respect to the LOS) are defined at the LOS tangent height (Fig. ) and are assumed to be constant over the LOS. This approximation is the same as that used by , and it is justified since most of the retrieved information comes from a thin altitude range around the tangent point.

3.1 Absorption matrix

The interaction between the radiation and the atmosphere are described by the $\mathrm{4}\times \mathrm{4}$ absorption matrix $\mathbf{K}$:

1

Zeeman transitions characteristics for the ${\mathrm{O}}_{\mathrm{2}}$ line at 773.84 GHz ($J:\mathrm{4}\to \mathrm{4}$ and $=\mathrm{1}$). The relative strengths are normalized such as $\sum s_{\mathit{\pi }}=\sum s_{{\mathit{\sigma }}^{+}}=\sum s_{{\mathit{\sigma }}^{-}}=\mathrm{1}$ (Table 3.1 in ). The frequency shift factors ${\mathit{\beta }}_{\mathrm{m}}$ are from Eq. ().

[Figure omitted. See PDF]

The coefficients of ${\mathbf{K}}_{\mathrm{o}}$ are derived from  (Eq. 5.36): 6$$\begin{array}{rl}& q={\displaystyle \frac{{\sin }^{\mathrm{2}}\left(\mathit{\theta }\right)\cos \left(\mathrm{2}\mathit{\varphi }\right)}{\mathrm{2}}}\\ & (\sum _{\mathit{\pi }}\left[s_{\mathit{\pi }}F(\mathit{\nu },{\mathit{\nu }}_{\mathit{\pi }},{\mathrm{\Gamma }}_{x,z})\right]-\sum _{{\mathit{\sigma }}^{\pm }}\left[{\displaystyle \frac{s_{\mathit{\sigma }}}{\mathrm{2}}}F(\mathit{\nu },{\mathit{\nu }}_{\mathit{\sigma }},{\mathrm{\Gamma }}_{{\mathrm{O}}_{\mathrm{2}},z})\right]),\\ & u={\displaystyle \frac{{\sin }^{\mathrm{2}}\left(\mathit{\theta }\right)\sin \left(\mathrm{2}\mathit{\varphi }\right)}{\mathrm{2}}}\\ & (\sum _{\mathit{\pi }}\left[s_{\mathit{\pi }}F(\mathit{\nu },{\mathit{\nu }}_{\mathit{\pi }},{\mathrm{\Gamma }}_{x,z})\right]-\sum _{{\mathit{\sigma }}^{\pm }}\left[{\displaystyle \frac{s_{\mathit{\sigma }}}{\mathrm{2}}}F(\mathit{\nu },{\mathit{\nu }}_{\mathit{\sigma }},{\mathrm{\Gamma }}_{{\mathrm{O}}_{\mathrm{2}},z})\right]),\\ & v=\cos \left(\mathit{\theta }\right)\\ & (\sum _{{\mathit{\sigma }}^{\pm }}\pm {\displaystyle \frac{s_{{\mathit{\sigma }}^{\pm }}}{\mathrm{2}}}F(\mathit{\nu },{\mathit{\nu }}_{{\mathit{\sigma }}^{\pm }},{\mathrm{\Gamma }}_{{\mathrm{O}}_{\mathrm{2}},z}\left)\right).\end{array}$$ The parameters $u^{\prime }$, $v^{\prime }$ and $q^{\prime }$ are computed by replacing the term $F$ with $F^{\prime }$, the dispersive part of the complex Voigt function (see Appendix  and  ).

3.2 Radiative transfer

The LOS is divided in narrow ranges of size d$s$ (typically 5 km long) in which the atmospheric parameters are considered constant. The change of the polarized radiance passing through an homogeneous range is derived from a matrix equation which is similar to the scalar radiative transfer one used for a nonpolarized radiation :

7$${\mathit{b}}_{\mathrm{a}}(s+\mathrm{d}s)=(\mathbf{I}-\mathbf{\Lambda }(s,s+\mathrm{d}s\left)\right)\cdot {\mathit{b}}_{\mathrm{p}}\left(s\right)+\mathbf{\Lambda }(s,s+\mathrm{d}s)\cdot {\mathit{b}}_{\mathrm{a}}\left(s\right),$$ where ${\mathit{b}}_{\mathrm{a}}\left(s\right)$ is the Stokes vector at the position $s$ on the LOS (the frequency dependence is omitted), “$\cdot$” is the matrix multiplication operator, ${\mathit{b}}_{\mathrm{p}}\left(s\right)=\left[P\right(s),\mathrm{0},\mathrm{0},\mathrm{0}{]}^T$ describes the nonpolarized source function between $s$ and $s+\mathrm{d}s$, $P\left(s\right)$ is the Planck function, and $\mathbf{\Lambda }(s,s+\mathrm{d}s)$ is $\mathrm{4}\times \mathrm{4}$ evolution operator matrix defined as follows: 8$$\mathbf{\Lambda }(s,s+\mathrm{d}s)=\exp (-\mathbf{K}(s\left)\mathrm{d}s\right).$$

Figure 5

Upper panels: ${\mathrm{O}}_{\mathrm{2}}$ lines simulated for antenna-1 (blue) and antenna-2 (yellow) at 80${}^{\circ }$ N over the ascending orbit. Panels from left to right show the results for a detector with horizontal, $+\mathrm{45}{}^{\circ }$, vertical, $-\mathrm{45}{}^{\circ }$ and right-circular polarization. The dashed and full yellow thin lines are spectra calculated with an angular tilt of the antenna-2 detector of $-\mathrm{20}$ and $+\mathrm{20}{}^{\circ }$. The black dashed lines are the measurement noise STD $\times$ 10 for antenna-1. Lower panels: same as upper panels but for the Equator.

[Figure omitted. See PDF]

The integration over the LOS is performed by applying the scalar equation given by to Stokes parameters: 9$$\begin{array}{rl}& {\mathit{b}}_{\mathrm{a}}\left(\mathrm{sat}\right)=\sum _{i=\mathrm{0}}^{N-\mathrm{1}}\mathbf{\Lambda }(i+\mathrm{1},\mathrm{sat})\cdot (\mathbf{I}-\mathbf{\Lambda }(\mathrm{0},i+\mathrm{1})\cdot \mathbf{\Lambda }(\mathrm{0},i+\mathrm{1}\left)\right)\\ & \cdot \left({\mathit{b}}_{\mathrm{p}}\right(i)-{\mathit{b}}_{\mathrm{p}}(i+\mathrm{1}\left)\right)+(\mathbf{I}-\mathbf{\Lambda }(\mathrm{0},\mathrm{sat})\cdot \mathbf{\Lambda }(\mathrm{0},\mathrm{sat}\left)\right)\\ & \cdot {\mathit{b}}_{\mathrm{p}}\left(N\right),\end{array}$$ where ${\mathit{b}}_{\mathrm{a}}\left(\mathrm{sat}\right)$ is the Stokes vector representing the radiation state at the antenna position, $i$ is the index of the level at $s_i$ ($i=\mathrm{0}$ for the tangent point) and $N$ is the number of levels above the tangent point. The cosmic background radiation is neglected. We use the relationship $\mathbf{\Lambda }(i,j)=\mathbf{\Lambda }(k,j)\cdot \mathbf{\Lambda }(i,k)$ with $i<k<j$ (the two matrices on the right-hand side of the equality do not commute).

4 Measurement and retrieval setting

4.1 Measured radiance

The measured radiance for antenna $a$ ($a=\mathrm{1}$ or 2) at the elevation angle $\mathit{\theta }$ and the IF $\mathit{\nu }$ is as follows:

10$$y_{\mathit{\theta },\mathit{\nu }}^a={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\mathit{R}}_{\mathit{\theta },\mathit{\nu }}^{\text{u}}\colorbox[rgb]{}{$$}{\mathit{y}}^{\text{a,u}}({\mathit{\nu }}_{\text{LO}}+\mathit{\nu })+{\mathit{R}}_{\mathit{\theta },\mathit{\nu }}^{\text{l}}\colorbox[rgb]{}{$$}{\mathit{y}}^{\text{a,l}}({\mathit{\nu }}_{\text{LO}}-\mathit{\nu })\right),$$ where ${\mathit{y}}^{\text{a,u}}$ and ${\mathit{y}}^{\text{a,l}}$ are the atmospheric specific intensities in the upper and lower sidebands around the local oscillator frequency ${\mathit{\nu }}_{\text{LO}}$, ${\mathit{R}}_{\mathit{\theta },\mathit{\nu }}$ represents the antenna and spectrometer functions, and is the convolution operator . A simple case with a constant upper and lower sideband ratio is considered. The Zeeman model is only used within a bandwidth of 200 MHz encompassing the ${\mathrm{O}}_{\mathrm{2}}$ line (upper sideband). Outside this range, the nonpolarized radiative transfer model described in is used. In order to transform the Stokes vector (Eq. ) to the specific intensity associated with the radiometer's polarization, we first rotate the vector from the atmospheric frame to the detector frame as follows: 11$${\mathit{b}}_{\mathrm{d}}={\mathrm{M}}_{\mathrm{r}}\left({\mathit{\alpha }}_{\mathrm{d}}\right)\cdot {\mathit{b}}_{\mathrm{a}},$$ where ${\mathit{b}}_{\mathrm{d}}$ is the Stokes vector in the instrument frame and ${\mathrm{M}}_{\mathrm{r}}\left({\mathit{\alpha }}_{\mathrm{d}}\right)$ is the Mueller matrix for a rotation ${\mathit{\alpha }}_{\mathrm{d}}$: 12$${\mathrm{M}}_{\mathrm{r}}\left({\mathit{\alpha }}_{\mathrm{d}}\right)=\left[\begin{array}{cccc}\mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \cos \left(\mathrm{2}{\mathit{\alpha }}_{\mathrm{d}}\right)& \sin \left(\mathrm{2}{\mathit{\alpha }}_{\mathrm{d}}\right)& \mathrm{0}\\ \mathrm{0}& -\sin \left(\mathrm{2}{\mathit{\alpha }}_{\mathrm{d}}\right)& \cos \left(\mathrm{2}{\mathit{\alpha }}_{\mathrm{d}}\right)& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right].$$ The specific intensity $y$ corresponding to the detector polarization is 13$$y={\mathit{b}}_{\mathrm{d}}\left[\mathrm{1}\right]+m{\mathit{b}}_{\mathrm{d}}\left[n\right],$$ where ${\mathit{b}}_{\mathrm{d}}\left[n\right]$ is the $n$th component of the Stokes vector and $(m,n)$ is $(-\mathrm{1},\mathrm{1})$, $(\mathrm{1},\mathrm{1})$, $(-\mathrm{1},\mathrm{2})$, $(\mathrm{1},\mathrm{2})$, $(-\mathrm{1},\mathrm{3})$ and $(\mathrm{1},\mathrm{3})$ for horizontal, vertical, $+\mathrm{45}$, $-\mathrm{45}{}^{\circ }$, right and left circular polarizations, respectively.

Figure 6

Magnetic field (${\mathit{\theta }}_{\mathrm{1},\mathrm{2}}$, ${\mathit{\varphi }}_{\mathrm{1},\mathrm{2}}$,$\left|\mathit{B}\right|$ and ${\mathrm{\Phi }}_{\mathrm{b}}$) and LOS (${\mathrm{\Phi }}_{\mathrm{1},\mathrm{2}}$) parameters (Fig. ) with respect to latitudes. The blue (yellow) lines are for ANT1 (ANT2) data. The circle–dashed (square–full) lines are data on the descending (ascending) orbit branch. The grey arrows in panels 2 and 5 indicate the direction of the satellite motion.

[Figure omitted. See PDF]

Figure  shows simulated spectra of the ${\mathrm{O}}_{\mathrm{2}}$ line over the Equator and at 80${}^{\circ }$ N when the satellite is moving toward north (ascending orbit branch). The tangent height is 100 km and the atmospheric conditions are representative of the Northern Hemisphere in wintertime . The magnetic field characteristics are zonal means inferred from a quiet solar day (Fig. ). Spectra are shown for different radiometer's polarizations. Over the Equator, $\mathit{B}$ is along the meridional direction and clear differences are seen between the radiances measured with both antennas, except if the detector has a vertical polarization. In that case, the radiometer detects only the ${\mathit{\sigma }}^{\pm }$ lines independently of the LOS orientation. The antenna-1 spectrum measured with a radiometer with a horizontal polarization is sensitive to the $\mathit{\pi }$ components which gives the visible double line shape. A receiver with a right-circular polarization measures mainly the ${\mathit{\sigma }}^{+}$ components since the antenna-1 is nearly aligned with the magnetic field (${\mathit{\theta }}_{\mathrm{1}}=\mathrm{20}{}^{\circ }$ in Fig. ). The spectrum looks like a single line with a frequency shift of $\mathrm{\Delta }\mathit{\nu }\approx \mathrm{420}({\mathit{\beta }}_{\mathrm{m}=-\mathrm{1}}+{\mathit{\beta }}_{\mathrm{m}=\mathrm{0}})=-\mathrm{21}$ kHz (Eq. ), equivalent to a LOS wind of 8 m s${}^{-\mathrm{1}}$.

Over the polar region, the spectra measured by both antennas are very similar since the vector $\mathit{B}$ is almost vertical and perpendicular to both LOSs (Fig. ). Only the Zeeman components $\mathit{\pi }$ are detected with the receiver with vertical polarization, while the horizontally polarized one detects ${\mathit{\sigma }}^{\pm }$ components (Fig. ).

4.2 Retrieval setting

The geomagnetic field may exhibit rapid temporal and spatial variations that can be as large as hundreds of nanoteslas . Such variations will be difficult to take into account when processing the data and may lead to retrieval errors with the same magnitude as those induced by the measurement noise.

Such errors are mitigated by retrieving the three components of $\mathit{B}$ simultaneously with other atmospheric parameters. It is done by using the scans of the same atmospheric column measured with the two antennas (Fig. ). The measurement vector $\mathit{y}$ is defined accordingly as follows:

14$${\mathit{y}}^T=\left[{\mathit{y}}^{a_{\mathrm{1}}},{\mathit{y}}^{a_{\mathrm{2}}}\right],$$ where the superscripts $a_{\mathrm{1}}$ and $a_{\mathrm{2}}$ denote that the parameters are associated with the antennas 1 and 2, respectively. The vector $\mathit{x}$ describing the retrieved parameters contains the profiles of the chemical species having the most significant features in the MLT spectra, namely ${\mathrm{O}}_{\mathrm{2}}$, ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$, ${\mathrm{O}}_{\mathrm{3}}$, $\mathrm{NO}$ and $\mathrm{HDO}$ (Fig. ). It also includes the profiles of temperature $T$, LOS winds (LW) and the three components of $\mathit{B}$. It is defined as follows: 15$$\begin{array}{rl}{\mathit{x}}^T=& \left[{\mathit{x}}_{{\mathrm{O}}_{\mathrm{2}}}^{a_{\mathrm{1}}},\mathrm{\dots },{\mathit{x}}_{\mathrm{T}}^{a_{\mathrm{1}}},{\mathit{x}}_{\mathrm{LW}}^{a_{\mathrm{1}}},\right.\\ & {\mathit{x}}_{{\mathrm{O}}_{\mathrm{2}}}^{a_{\mathrm{2}}},\mathrm{\dots },{\mathit{x}}_{\mathrm{T}}^{a_{\mathrm{2}}},{\mathit{x}}_{\mathrm{LW}}^{a_{\mathrm{2}}},\\ & \left.{\mathit{x}}_{\mathrm{Bw}},{\mathit{x}}_{\mathrm{Bu}},{\mathit{x}}_{\mathrm{Bv}}\right],\end{array}$$ where ${\mathit{x}}_{\mathrm{Bw}}$, ${\mathit{x}}_{\mathrm{Bu}}$ and ${\mathit{x}}_{\mathrm{Bv}}$ are the profiles of the vertical, zonal and meridional components of $\mathit{B}$. The abundance and temperature profiles are retrieved for each antenna in order to account for differences between both scan locations. This is a similar approach to that used by for the measurement of winds with the ground-based radiometer WIRA.

The retrieval error induced by the measurement noise is  16$${\mathit{ϵ}}_n^{\mathrm{2}}=\mathrm{diag}\left\{{\left({\mathbf{K}}^T{\mathbf{S}}_y^{-\mathrm{1}}\mathbf{K}+{\mathbf{U}}^{-\mathrm{1}}\right)}^{-\mathrm{1}}\right\},$$ where $\mathbf{K}=\frac{\mathrm{d}\mathit{y}}{\mathrm{d}\mathit{x}}$ is the Jacobian matrix of the retrieved parameters $\mathit{x}$ and $\mathbf{U}$ is a diagonal matrix to ensure a stable inversion but with values large enough to allow us to neglect its effects in the altitude range where the retrievals are relevant . The matrix ${\mathbf{S}}_y$ is the diagonal covariance matrix associated with the measurement noise: 17$${\mathbf{S}}_{y,i,i}={\displaystyle \frac{{\left(T_{\text{sys}}+y_i\right)}^{\mathrm{2}}}{\mathit{\delta }\mathit{\nu }\mathit{\delta }t}},$$ and ${\mathbf{S}}_{y,i,i}$ is the noise induced variance on the $i$th component of the measurement vector $\mathit{y}$, $T_{\text{sys}}$ is the system temperature (Table ), $\mathit{\delta }\mathit{\nu }$ is the frequency resolution (0.5 MHz) and $\mathit{\delta }t$ is the spectrum integration time (0.25 s).

The radiative transfer model computes the Jacobian ${\mathbf{K}}_B=\partial {\mathit{y}}^{a_i}/\partial {\mathit{x}}_B$ with respect to antenna-$i$ frame ($\mathit{\{}{\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{i}},{\mathit{z}}_{\mathit{i}}\mathit{\}}$ in Fig. a). The matrix ${\mathbf{K}}_B$ is then computed in the atmospheric frame (Fig. ): 18$${\displaystyle \frac{\partial {\mathit{y}}^{a_i}}{\partial {\mathit{B}}_q}}=\sum _{k=\mathit{\{}{x}_i,y_i,z_i\mathit{\}}}{\displaystyle \frac{\partial {\mathit{y}}^{a_i}}{\partial {\mathit{B}}_k}}{\displaystyle \frac{\partial {\mathit{B}}_k}{\partial {\mathit{B}}_q}},$$ where $q=\mathit{\{}\mathit{u},\mathit{v},\mathit{w}\mathit{\}}$ denote the atmospheric frame axes, and 19$$\begin{array}{rl}& {\mathit{B}}_{x_i}={\mathit{B}}_w,\\ & {\mathit{B}}_{y_i}=\cos \left({\mathrm{\Phi }}_i\right){\mathit{B}}_u+\sin \left({\mathrm{\Phi }}_i\right){\mathit{B}}_v,\\ & {\mathit{B}}_{z_i}=-\sin \left({\mathrm{\Phi }}_i\right){\mathit{B}}_u+\cos \left({\mathrm{\Phi }}_i\right){\mathit{B}}_v,\end{array}$$ where ${\mathrm{\Phi }}_i$ is the angle between the antenna-$i$ LOS and the meridional direction (Fig. a).

Figure 7

Vertical profiles of atmospheric number density, temperature and VMRs of ${\mathrm{O}}_{\mathrm{2}}$, ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$, ${\mathrm{O}}_{\mathrm{3}}$ and $\mathrm{NO}$. They are representative of a Northern Hemisphere winter period (DJF), during daytime at 50${}^{\circ }$ S and the Equator (dashed-grey and red lines, respectively), and polar night at 80${}^{\circ }$ N (blue lines).

[Figure omitted. See PDF]

5 Retrieval errors

Figure  shows the retrieval errors on the atmospheric density, temperature, LOS wind and the main chemical species at three latitudes (50${}^{\circ }$ S, Equator, 80${}^{\circ }$ N). For the instrumental setting, we considered a radiometer with a linear vertical polarization and the forward-looking antenna (antenna-1). The vertical resolution of the retrieved profiles is 2.5 km for the main parameters (temperature, LOS wind, ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$ and ${\mathrm{O}}_{\mathrm{3}}$), 5 km for $\mathrm{NO}$ and 20 km for the components of $\mathit{B}$. Errors are computed for the same winter (DJF) climatology described in the previous section. The corresponding atmospheric state includes a stable polar vortex and does not show any $\mathrm{NO}$ enhancement due to energetic particle precipitation. The results at the Equator and the Southern Hemisphere (SH) mid-latitudes (50${}^{\circ }$ S) are for daytime conditions, while the Northern Hemisphere (NH) results at 80${}^{\circ }$ N are representative of the polar night. We did not find significant differences between daytime and nighttime except for the relative error in ${\mathrm{O}}_{\mathrm{3}}$ retrieval, which is photo-dissociated between 60 and 80 km.

Figure 9

Retrieval errors on the temperature profile induced by the measurement noise, for a radiometer with horizontal (H), 45${}^{\circ }$ (45), vertical (V) and $-\mathrm{45}{}^{\circ }$ ($-\mathrm{45}$) linear polarizations, and right circular (RC) polarization. The colored dashed lines are results for the descending orbit branch while the full lines ones are those for the ascending branch. The blue (yellow) lines show the results for antenna-1 (antenna-2). The black dashed lines show the errors that occur if the Zeeman effect is not considered. Vertical resolution of the retrieved profiles is 2.5 km.

[Figure omitted. See PDF]

5.1 Atmospheric density, temperature and LOS wind

The achieved precision of the atmospheric density (or ${\mathrm{O}}_{\mathrm{2}}$) profile is better than 5 % up to about 95 km at all latitudes. Above 90 km, the signal intensity drops significantly and errors quickly increase, up to 20 % at 110 km. Outside of the 70–90 km range, there are significant differences between the error profiles calculated for the full- and narrow-band inversions. This shows that spectral lines from other molecular species also have an impact on the ${\mathrm{O}}_{\mathrm{2}}$ retrievals. This impact probably occurs through the temperature retrieval. For instance, over the winter polar region, the strong $\mathrm{NO}$ signal significantly improves the temperature retrievals and thus indirectly improves the ${\mathrm{O}}_{\mathrm{2}}$ abundance retrieval. Similarly, including ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}$ and ${\mathrm{O}}_{\mathrm{3}}$ lines leads to an improvement of the ${\mathrm{O}}_{\mathrm{2}}$ retrieval quality below 70 km.

For all latitudes, the temperature retrieval error is better than 5 K below 90 km and 30 K at 110 km. The ${\mathrm{O}}_{\mathrm{2}}$ line is the main source of information on the temperature near 90 km.

Figure 11

Same as Fig.  but for the geomagnetic field components and altitudes between 85 and 105 km. The retrieval vertical resolution is 20 km.

[Figure omitted. See PDF]

5.2 Geomagnetic field

Figure  shows the retrieval errors on the three components of $\mathit{B}$ at 85 km and 105 km (vertical resolution of 20 km). The results strongly depend on the radiometer's polarization. Best performance is achieved with a $\pm \mathrm{45}{}^{\circ }$ linear polarization. Errors are clearly smaller when the retrieved component is aligned with the background magnetic field: the error on ${\mathit{B}}_v$ is smallest at the Equator where $\mathit{B}$ is horizontal and in the meridional plane, and the error on ${\mathit{B}}_w$ minimizes at high latitudes where $\mathit{B}$ is nearly vertical. The best sensitivity is found at 85 km where the precision is better than 400 nT for all components and at all latitudes, except for the zonal component ($B_u$) in the tropics. At high latitudes, errors are between 50 and 100 nT for the vertical component ($B_w$) and between 100 and 200 nT for the horizontal ones ($B_u$ and $B_v$). At 105 km, errors increase, for example to 80–500 nT outside the tropics.

Contrary to the results shown in Sect.  where it was the optimal configuration, the linear vertical polarization yields a worse retrieval performance for $\mathit{B}$. In this case, the retrieval errors on ${\mathit{B}}_w$ and ${\mathit{B}}_u$ over the tropics are much larger than those found with a slant polarization. Only the meridional component (${\mathit{B}}_v$) can still be retrieved with a reasonable precision of 100–400 nT. At middle and high latitudes, best precision is found for ${\mathit{B}}_w$ (30–50 nT at 85 km and 50–70 nT at 105 km). At 85 km, the error on ${\mathit{B}}_v$ and ${\mathit{B}}_u$ are between 200 and 300 and between 300 and 2000 nT, respectively. Large errors in ${\mathit{B}}_u$ are found at 40${}^{\circ }$S and 70${}^{\circ }$ N where the LOS is aligned with the $U$ and $V$ axes (${\mathrm{\Phi }}_{\mathrm{1}}$ or ${\mathrm{\Phi }}_{\mathrm{2}}=\mathrm{0}$, Fig. ).

Our results show that the sensitivity of the SMILES-2 instrument is high enough to potentially measure the electrojet-induced variations of $\mathit{B}$ at high latitudes even under quiet sun conditions, provided that the data are properly averaged. used the Zeeman effect on the AURA/MLS ${\mathrm{O}}_{\mathrm{2}}$ line to derive variations of 100–200 nT in the intensity of $\mathit{B}$. During solar storms, the amplitude of the perturbations in the auroral regions could be considerably larger (several hundreds of nanoteslas)  and could be detected with single measurements along the vertical and at least one horizontal component of $\mathit{B}$. Hence, SMILES-2 could allow us to infer information on the 3-D variations of the auroral electrojet.

Perturbations of the geomagnetic field near the Equator (30 and 80 nT for the surface vertical and horizontal components of $\mathit{B}$) are much smaller than the retrieval precision . Therefore, extracting interesting information on the equatorial jet will be more challenging, and a receiver with a slant polarization could be necessary.

6 Conclusions

This analysis demonstrates the potential of SMILES-2 for the measurement of the temperature, atmospheric density and LOS wind in the MLT (60–110 km). The retrieval precision was assessed, focusing on the SMILES-2 band at 760 GHz, the most suitable for such measurements. Special care was taken to properly include the Zeeman effect on the ${\mathrm{O}}_{\mathrm{2}}$ line. Our results showed that neglecting it could lead to underestimating the retrieval errors by a factor of up to 2 above 90 km. Because the ${\mathrm{O}}_{\mathrm{2}}$ line is polarized, the radiometer's polarization configuration had to be investigated. We found that the optimal configuration was vertically linear. The LOS wind is retrieved with a precision of 2–5 m s${}^{-\mathrm{1}}$ up to 90 km (30 m s${}^{-\mathrm{1}}$ at 110 km) and a vertical resolution of 2.5 km. Temperature and atmospheric density are retrieved with a precision better than 5 K (30 K) and 7 % (20 %) up to 90 km (110 km), respectively. The achieved precision of the wind measurements, a key product for SMILES-2, is comparable to the requirements for the new ICON mission . However, unlike optical sensors, SMILES-2 can acquire high-precision measurements during day and night, and at all latitudes, even during auroral events. The low noise level achieved by the 4 K super-cooled radiometers is essential to achieve good performance above 90 km, where sensitivity becomes critical due to significantly weaker signals.

The retrieval of the geomagnetic field using the ${\mathrm{O}}_{\mathrm{2}}$ line was also discussed. We showed that valuable information on the horizontal and vertical components of $\mathit{B}$ could be determined directly near the E-region auroral electrojets. highlighted the need for such observations since, currently, only measurements from the ground or from low-orbit satellites near 400 km are available. proposed a CubeSat constellation, with the purpose of measuring the ${\mathrm{O}}_{\mathrm{2}}$ line at 119 GHz to produce high spatial and temporal observations of $\mathit{B}$ perturbations. It is worth mentioning that this methodology, based on ${\mathrm{O}}_{\mathrm{2}}$ spectral lines, has also been proposed to measure the Martian residual magnetic field . Further analyses should be conducted, to characterize more precisely the potential of SMILES-2 for the study of the 3-D ionospheric electrojets.

The final instrumental setup is still under discussion. In terms of possible instrumental developments, the spectral bandwidth of the 763 GHz band might be reduced in the definitive configuration of SMILES-2. Narrowing the bandwidth by a factor of 2 (while ensuring a correct adjustment of the LO frequency) would cause minimal degradation of the measurement performance, limited to altitudes below about 40 km.

Future work to improve MLT retrievals will include the two other SMILES-2 bands. Indeed, the atomic oxygen line at 2 THz contains temperature and wind information above 100 km. This line can help us to improve the wind retrieval precision to 10 m s${}^{-\mathrm{1}}$ at 110 km . In the 638 GHz band, a strong signal from ${\mathrm{O}}_{\mathrm{3}}$ will be measured below about 70 km in daytime and 90 km in nighttime. Furthermore, new parameters for the Zeeman model became recently available . Applying the updated parameters should induce a change of the ${\mathrm{O}}_{\mathrm{2}}$ and $\mathrm{O}$ line intensities, of up to a few percent. The Zeeman effect on other spectral lines, $\mathrm{OH}$, $\mathrm{NO}$ and $\mathrm{ClO}$, should also be studied.

Appendix A Spectroscopic parameters

Table A1

Parameters of ${\mathrm{O}}_{\mathrm{2}}$ lines in the ground electronic and vibrational levels between 100 and 1000 GHz. Values are taken from the HITRAN 2008.

Frequency	Strength	$E_{\mathrm{L}}$,	$N$:	$J$:
MHz	at 296 K cm${}^{-\mathrm{1}}$	cm${}^{-\mathrm{1}}$	lower,	lower,
	molecule${}^{-\mathrm{1}}$ cm${}^{\mathrm{2}}$		upper	upper
118 750.3408	$\mathrm{9.956}\times {\mathrm{10}}^{-\mathrm{26}}$	0.0000	1, 1	0, 1
368 498.3839	$\mathrm{2.213}\times {\mathrm{10}}^{-\mathrm{26}}$	3.9611	1, 3	1, 2
424 763.1626	$\mathrm{2.414}\times {\mathrm{10}}^{-\mathrm{25}}$	2.0843	1, 3	2, 2
487 249.4142	$\mathrm{1.031}\times {\mathrm{10}}^{-\mathrm{25}}$	2.0843	1, 3	2, 3
715 393.1236	$\mathrm{6.246}\times {\mathrm{10}}^{-\mathrm{26}}$	18.3372	3, 5	3, 4
773 839.7019	$\mathrm{3.943}\times {\mathrm{10}}^{-\mathrm{25}}$	16.3876	3, 5	4, 4
834 145.7729	$\mathrm{1.359}\times {\mathrm{10}}^{-\mathrm{25}}$	16.3876	3, 5	4, 5

The spectroscopic parameters are taken from the HITRAN database . The line strength at the temperature $T$ is as follows: A1$$\begin{array}{rl}& S\left(T\right)={\displaystyle \frac{C_{\mathrm{H}}}{r_{\mathrm{iso}}}}{S}_{\mathrm{H}}\left(T_{\mathrm{0}}\right){\displaystyle \frac{e^{-C_{\mathrm{E}}{E}_{\mathrm{L}}/k_{\mathrm{b}}T}}{e^{-C_{\mathrm{E}}{E}_{\mathrm{L}}/k_{\mathrm{b}}{T}_{\mathrm{0}}}}}\\ & \left({\displaystyle \frac{\mathrm{1}-e^{-C_{\mathrm{E}}{\stackrel{\mathrm{‾}}{\mathit{\nu }}}_{\mathrm{0}}/k_{\mathrm{b}}T}}{\mathrm{1}-e^{-C_{\mathrm{E}}{\stackrel{\mathrm{‾}}{\mathit{\nu }}}_{\mathrm{0}}/k_{\mathrm{b}}{T}_{\mathrm{0}}}}}\right){\displaystyle \frac{Q\left(T_{\mathrm{0}}\right)}{Q\left(T\right)}}\left(\mathrm{Hz}{\mathrm{m}}^{\mathrm{2}}{\mathrm{molecule}}^{-\mathrm{1}}\right),\end{array}$$ where $k_{\mathrm{b}}=\mathrm{1.380662}\times {\mathrm{10}}^{-\mathrm{23}}$ J K${}^{-\mathrm{1}}$ is the Boltzmann constant, ${\stackrel{\mathrm{‾}}{\mathit{\nu }}}_{\mathrm{0}}$ (cm${}^{-\mathrm{1}}$) is the transition wavenumber, $S_{\mathrm{H}}\left(T_{\mathrm{0}}\right)$ is the HITRAN line strength (cm${}^{-\mathrm{1}}$ cm${}^{\mathrm{2}}$ molecule${}^{-\mathrm{1}}$), $T_{\mathrm{0}}=\mathrm{296}$ K and $E_{\mathrm{L}}$ (cm${}^{-\mathrm{1}}$) is the lowest energy of the transition. The partition function $Q$ is calculated from tabulated values between 120 and 500 K, a range that encompasses the temperatures found between 50 and 130 km ($Q\left(\mathrm{296}\right)=\mathrm{215.77}$). The constants $C_{\mathrm{E}}={\mathrm{10}}^{\mathrm{2}}{h}_{\mathrm{p}}c$ and $C_{\mathrm{H}}={\mathrm{10}}^{-\mathrm{2}}c$ allow the conversion of the HITRAN units to the International System (SI) ones. The isotopic ratio $r_{\mathrm{iso}}$ is taken away from $S_{\mathrm{H}}$ and added to the density profile instead. The Table  shows parameters of the main ${\mathrm{O}}_{\mathrm{2}}$ millimeter lines.

Above the altitude of about 70 km, the real part of the Voigt function $F$ (Eq. ) is close to the Gauss function that describes lines broadened by random molecular velocities (Doppler broadening): A2$$F\left(\mathit{\nu }\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{d}}}}{\left({\displaystyle \frac{\ln \mathrm{2}}{\mathit{\pi }}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}{e}^{-\ln \mathrm{2}{\left(\frac{\mathit{\nu }-{\mathit{\nu }}_{\mathrm{0}}}{\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{d}}}\right)}^{\mathrm{2}}}\left({\text{Hz}}^{-\mathrm{1}}\right)$$ with A3$$\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{d}}={\displaystyle \frac{{\mathit{\nu }}_{\mathrm{0}}}{c}}{\left({\displaystyle \frac{\mathrm{2}\ln \mathrm{2}RT}{M}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}\left(\text{Hz}\right),$$ and $\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{d}}$ is the Doppler broadening parameter (i.e., the half-width at half-maximum, HWHM) of $F$, ${\mathit{\nu }}_{\mathrm{0}}$ is the frequency of the transition, $c=\mathrm{2.997924}\times {\mathrm{10}}^{\mathrm{8}}$ m s${}^{-\mathrm{1}}$ is the speed of light in vacuum, $R=\mathrm{8.31446}$ J K${}^{-\mathrm{1}}$ mol${}^{-\mathrm{1}}$ is the gas constant and $M$ is the molar mass (0.031980 kg mol${}^{-\mathrm{1}}$ for ${\mathrm{O}}_{\mathrm{2}}$). At 80 km, $\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{d}}$ is about 0.6–0.7 MHz for the ${\mathrm{O}}_{\mathrm{2}}$ line at 773 GHz, while the pressure broadening HWHM is only 0.01–0.02 MHz.

The dispersion profile used for the calculation of the coefficient $q^{\prime }$, $u^{\prime }$ and $v^{\prime }$ (Eq. ) is given by the following: A4$$F^{\prime }\left(\mathit{\nu }\right)=\sqrt{\mathrm{2}}F\left(\mathit{\nu }\right)\mathrm{erfi}\left(v\right)\left({\mathrm{Hz}}^{-\mathrm{1}}\right),$$ with $v=\ln \mathrm{2}\left(\frac{\mathit{\nu }-{\mathit{\nu }}_{\mathrm{0}}}{\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{d}}}\right)$ and $\mathrm{erfi}\left(v\right)=\sqrt{\mathrm{2}/\mathit{\pi }}{\int }_{\mathrm{0}}^v\exp \left(t^{\mathrm{2}}\right)d$t the imaginary error function (Eq. 5.54 in ).

Appendix B Matrix exponential

The computation of the matrix exponential in Eq. () is the performance bottleneck in our implementation of the radiative transfer solver if we use a general algorithm. A significantly faster algorithm has been implemented using the symmetry in ${\mathbf{K}}_{\mathrm{o}}$ (Eq. ). The evolution operator $\mathbf{\Lambda }$ (Eq. ) is written as $\exp (-k_{\mathrm{a}}\mathrm{d}s)\exp \left({\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}\right),$ with $k_{\mathrm{a}}$ the scalar absorption coefficient (Eq. ) and ${\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}=-{\mathbf{K}}_{\mathrm{o}}\mathrm{d}s$. The Cayley–Hamilton theorem is used to compute $\exp \left({\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}\right)$: B1$$\exp \left({\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}\right)=\sum _{k=\mathrm{0}}^{\mathrm{3}}{\mathit{\kappa }}_k{\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}^k,$$ where ${\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}^{\mathrm{0}}$ is the identity matrix. The coefficient ${\mathit{\kappa }}_k$ are derived using the four eigenvalues of ${\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}$: $$\left\{\begin{array}{l}{\exp }^{{\mathit{\lambda }}_{\mathrm{1}}}={\mathit{\kappa }}_{\mathrm{0}}+{\mathit{\kappa }}_{\mathrm{1}}{\mathit{\lambda }}_{\mathrm{1}}+{\mathit{\kappa }}_{\mathrm{2}}{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\kappa }}_{\mathrm{3}}{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{3}},\\ {\exp }^{-{\mathit{\lambda }}_{\mathrm{1}}}={\mathit{\kappa }}_{\mathrm{0}}-{\mathit{\kappa }}_{\mathrm{1}}{\mathit{\lambda }}_{\mathrm{1}}+{\mathit{\kappa }}_{\mathrm{2}}{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}-{\mathit{\kappa }}_{\mathrm{3}}{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{3}},\\ {\exp }^{j{\mathit{\lambda }}_{\mathrm{2}}}={\mathit{\kappa }}_{\mathrm{0}}+j{\mathit{\kappa }}_{\mathrm{1}}{\mathit{\lambda }}_{\mathrm{2}}-{\mathit{\kappa }}_{\mathrm{2}}{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}-j{\mathit{\kappa }}_{\mathrm{3}}{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{3}},\\ {\exp }^{-j{\mathit{\lambda }}_{\mathrm{2}}}={\mathit{\kappa }}_{\mathrm{0}}-j{\mathit{\kappa }}_{\mathrm{1}}{\mathit{\lambda }}_{\mathrm{2}}-{\mathit{\kappa }}_{\mathrm{2}}{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}+j{\mathit{\kappa }}_{\mathrm{3}}{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{3}},\end{array}\right.$$ where ${\mathit{\lambda }}_{\mathrm{1},\mathrm{2}}$ is positive real-valued numbers that determine the four eigenvalues $\pm {\mathit{\lambda }}_{\mathrm{1}}$ and $\pm j{\mathit{\lambda }}_{\mathrm{2}}$ of ${\stackrel{\mathrm{̃}}{\mathbf{K}}}_{\mathrm{o}}$. This gives the following: B2$$\begin{array}{rl}& {\mathit{\kappa }}_{\mathrm{0}}={\displaystyle \frac{{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}\cosh \left({\mathit{\lambda }}_{\mathrm{1}}\right)+{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}\cos \left({\mathit{\lambda }}_{\mathrm{2}}\right)}{{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}}},\\ & {\mathit{\kappa }}_{\mathrm{1}}={\displaystyle \frac{{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}\sinh \left({\mathit{\lambda }}_{\mathrm{1}}\right)/{\mathit{\lambda }}_{\mathrm{1}}+{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}\sin \left({\mathit{\lambda }}_{\mathrm{2}}\right)/{\mathit{\lambda }}_{\mathrm{2}}}{{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}}},\\ & {\mathit{\kappa }}_{\mathrm{2}}={\displaystyle \frac{\cosh \left({\mathit{\lambda }}_{\mathrm{1}}\right)-\cos \left({\mathit{\lambda }}_{\mathrm{2}}\right)}{{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}}},\\ & {\mathit{\kappa }}_{\mathrm{3}}={\displaystyle \frac{\sinh \left({\mathit{\lambda }}_{\mathrm{1}}\right)/{\mathit{\lambda }}_{\mathrm{1}}-\sin \left({\mathit{\lambda }}_{\mathrm{2}}\right)/{\mathit{\lambda }}_{\mathrm{2}}}{{\mathit{\lambda }}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\lambda }}_{\mathrm{2}}^{\mathrm{2}}}}.\end{array}$$ The eigenvalue parameters are ${\mathit{\lambda }}_{\mathrm{1}}=-\mathrm{d}s\sqrt{(A+B)/\mathrm{2}}$ and ${\mathit{\lambda }}_{\mathrm{2}}=-\mathrm{d}s\sqrt{(A-B)/\mathrm{2}}$, where $$\begin{array}{rl}& A=\left[\right.\mathrm{8}(q{q}^{\prime }v{v}^{\prime }+u{u}^{\prime }q{q}^{\prime }+u{u}^{\prime }v{v}^{\prime })+q^{\mathrm{4}}+u^{\mathrm{4}}+v^{\mathrm{4}}\\ & +{q^{\prime }}^{\mathrm{4}}+{u^{\prime }}^{\mathrm{4}}+{v^{\prime }}^{\mathrm{4}}+\mathrm{2}\left(\right.q^{\mathrm{2}}(u^{\mathrm{2}}-{u^{\prime }}^{\mathrm{2}}+v^{\mathrm{2}}-{v^{\prime }}^{\mathrm{2}}+{q^{\prime }}^{\mathrm{2}})\\ & +u^{\mathrm{2}}({u^{\prime }}^{\mathrm{2}}+v^{\mathrm{2}}-{v^{\prime }}^{\mathrm{2}}-{q^{\prime }}^{\mathrm{2}})\\ & +v^{\mathrm{2}}(-{u^{\prime }}^{\mathrm{2}}+{v^{\prime }}^{\mathrm{2}}-{q^{\prime }}^{\mathrm{2}})\\ & +{v^{\prime }}^{\mathrm{2}}({u^{\prime }}^{\mathrm{2}}+{q^{\prime }}^{\mathrm{2}})\\ & +{u^{\prime }}^{\mathrm{2}}{q^{\prime }}^{\mathrm{2}}){}^{\frac{\mathrm{1}}{\mathrm{2}}}\end{array}$$ and $$B=q^{\mathrm{2}}-{q^{\prime }}^{\mathrm{2}}+u^{\mathrm{2}}-{u^{\prime }}^{\mathrm{2}}+v^{\mathrm{2}}-{v^{\prime }}^{\mathrm{2}}.$$

Code availability

Model is available upon request.

Author contributions

PB designed the study, performed the simulations and wrote the paper. All co-authors provided valuable information. MS is the PI of the mission proposal.

Competing interests

The authors declare that they have no conflict of interest.

Acknowledgements

Philippe Baron would like to thank Franz Schreier (German Aerospace Center, DLR) for providing the python implementation of the complex Voigt function used in GARLIC (see reference given for Eq. 3). We would like to thank Hugh C. Pumphrey and the two anonymous referees for their valuable comments that helped us to improve the paper.

Financial support

SMILES-2 studies are supported by the strategic development research fund from the Institute of Space and Astronautical Science (ISAS)/JAXA. Huixin Liu acknowledges support by JSPS KAKENHI (grants no. 18H01270, 18H04446 and 17KK0095).

Review statement

This paper was edited by Christian von Savigny and reviewed by Hugh C. Pumphrey and two anonymous referees.