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Abstract
The mesopause–lower thermosphere (MLT) region is an important spatial region in the Earth’s atmosphere, making it a valuable area to investigate the temperature variations. Kirchhoff’s law fails with the altitude increase due to the non-local thermal equilibrium effect, resulting in an increase in the error of the method to retrieve the atmospheric temperature in the MLT region using the A-band spectral line intensity. In the non-LTE state, the temperature retrieval method based on the Einstein coefficients is proposed to retrieve atmospheric temperature in the 92–140 km height range using the airglow radiation intensity images obtained from the Michelson Interferometer for global high-resolution thermospheric imaging (MIGHTI) measurements. Results show that the temperature deviation of the two-channel combinations does not exceed 15 K in the altitude range of 92–120 km. This deviation increases up to 45 K when the altitude is in the range of 120–140 km due to the influence of the N2 airglow spectrum. The two-channel combinations self-consistency is increased by 85 K compared with the temperature obtained using the spectral line intensity retrieval. Additionally, the comparison of the retrieval results with the spectral line intensity method and the comparison with the atmospheric chemistry experiment Fourier transform spectrometer (ACE-FTS) temperature measurement data shows that the Einstein coefficient method is significantly more rational and accurate than the spectral line intensity method.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Temperature is one of the most important meteorological parameters that characterizes the earth’s atmospheric environment and spatial physical properties. The transition zone from the top of the mesosphere to the lower thermosphere is the least known but extremely important spatial region. The height of the mesosphere–lower thermosphere (MLT) is 80–140 km [1]. This area is the coldest spatial region in the atmosphere and a key region for the coupling between the atmosphere and the ionosphere. The fluctuation amplitudes of atmospheric circulation, atmospheric tidal waves, and atmospheric acoustic gravity waves in the lower atmosphere rapidly increase as they propagate upward due to the exponential attenuation of atmospheric density, resulting in strong disturbances in the MLT region. The radiation and particle injection from the outer space and the changes in energy and momentum generated by the geomagnetic activities can also cause fluctuations in the various meteorological parameters in the MLT region, thereby affecting the state of the lower atmosphere [2]. The temperature structure and temporal evolution of the MLT are crucial in the dynamic behavior, photochemical reaction efficiency, energy and momentum transfer processes, and atmospheric stability of the atmosphere [3]. The MLT region is also the passage and possible residence zone of the various spacecraft and ultra-high altitude aircraft, and its spatial structure and temporal evolution of the space environment parameters, such as temperature, have significant effects on the accurate orbiting and safe return of the spacecraft. Therefore, the exploration of the MLT region has gradually become a popular research topic around the world.
The two main technical means for temperature detection in the MLT region are as follows: sodium fluorescence lidar and airglow imaging interferometer [4,5]. Sodium fluorescence lidar is the active detection method that adopts the triple-frequency laser technology and locks the laser emission wavelength at three frequency points of the D2 spectral line of sodium atoms to excite the atomic resonance fluorescence of the high altitude sodium layer and uses the Doppler shift information and Doppler spreading information carried by the echo signal to accomplish the retrieval of the atmospheric wind field and temperature field in the sodium layer region [6]. Given the complexity and engineering instability of the sodium lidar system, it can only work in a ground-based detection mode at present, which cannot realize the temperature distribution characteristics of the sodium layer in the global region by spaceborne detection. The airglow imaging interferometer is used to image the radiation signal of the atomic or molecular airglow in the atmosphere by using the spaceborne hyperspectral resolution imager (e.g., Fabry–Perot, Michelson, and DASH), or visible near-infrared cameras equipped with narrow band bandpass filters in the mode of limb viewing and realize the atmospheric temperature profile retrieval by using the phase change caused by the thermal motion of the airglow or the change of intensity ratio of different spectral channels [7–9]. The atomic airglow spectral lines are isolated, and the temperature can only be inferred by using the Doppler spreading information of the atomic airglow sensed by the phase change of the hyperspectral resolution airglow imaging interferometer, which has a large error and can only detect the high altitude space region. Meanwhile, the molecular system, which has a high accuracy and can be extended to a lower airspace, can retrieve the temperature by detecting the intensity ratio of the different vibration–rotation energy levels through multiple spectral channels due to the existence of vibration and rotation energy levels [10]. The multispectral-channel spaceborne airglow imaging interferometer with molecular airglow as the target source can detect the atmospheric temperature in the MLT region with high temporal and spatial resolutions on a global scale. This technology has become a crucial tool for studying the temperature structure and temporal evolution of the MLT [11].
For the MLT region, due to the higher altitude of its coverage, the non-local thermal equilibrium (non-LTE) effect is enhanced, making the temperature retrieval error in this region increase if the local thermal equilibrium (LTE) assumption is still insisted on. This effect is evident for satellite payloads by limb-viewing. In 2011, M. Garćıa-Comas et al. successfully obtained temperature distributions in the 20-105 km altitude range with non-LTE effect corrected by using the IMK/IAA retrieval algorithm for limb-viewing data from the MIPAS [12]. L. Rezac et al. in 2015 proposed a self-consistent algorithm to retrieve atmospheric temperatures in the 60-120 km altitude range using the non-LTE model from SABER observations [13]. In 2018, Marco Matricardi et al. proposed a fast radiative transfer model for non-LTE, which achieves a correction for the effects of non-LTE by retrieving the vibrational temperature based on IASI observations [14]. All of the above work measures and corrects the atmospheric temperature under the influence of a non-LTE effect for vibrational levels. The Michelson interferometer for global high-resolution thermospheric imaging (MIGHTI) on board the Ionospheric Connection Explorer (ICON) satellite launched in the United States in 2019 utilizes five spectral channels to measure the O2 A-band airglow radiation, which is capable of covering the rotational energy levels of the O2 A-band with different rotational quantum numbers, with a view to measuring the atmospheric temperature profile in the region of 90-140 km. Since MIGHTI expects temperature retrieval from spectral band shape measurements, corrections for non-LTE effects under rotation conditions are required. The signal intensity of the target layer is obtained by processing the data of the five channels obtained by MIGHTI under limb-viewing conditions through the “onion peeling” algorithm, and the monotonic relationship between the intensity ratios of the channels and the value of temperature can be utilized to achieve the precise retrieval of the atmospheric temperature. This is because the ratio method eliminates the influence of volume emission rate (VER) on the retrieved temperature, so that the ratio of the O2 airglow signals carries only the temperature-related information.
The ratio method has been applied in several technical fields to measure the temperature of gas molecules and has achieved a high measurement accuracy. Tunable diode laser absorption spectroscopy uses a dual-wavelength laser to scan two isolated spectral lines of a target molecule and achieves the measurement of temperature information in high-temperature combustion environments through the ratio of the intensity of the spectral lines as a function of temperature [15]. Pure rotational Raman Lidar uses two adjacent narrow-band bandpass filters to acquire pure rotational Raman signals with opposite temperature sensitivities of N2 and O2 in the atmosphere and retrieve the temperature information of the middle and lower atmosphere by making a ratio [16,17]. Ground-based Airglow Imaging Interferometer uses Fabry–Perot or Michelson interferometer to image different spectral lines of the O2 or OH airglow in the upper atmosphere and retrieve the temperature information of the upper atmosphere by doing the ratio method [18]. The physical principle of the above-mentioned three techniques using the ratio method is that the molecular energy level population follows the Boltzmann distribution, and the relationship between the spectral intensity of each molecular energy level and the temperature is determined by the partition function. The ICON team designed and developed the MIGHTI to enable atmospheric temperature detection, which takes advantage of the physical law that molecular energy levels’ population follows the Boltzmann distribution [19]. However, a prerequisite exists for the Boltzmann distribution law to hold, which is the assumption of the non-LTE. The gas density in the middle and low atmosphere is large, and the collision frequency between molecules is much larger than the energy level transition frequency accompanying the molecular radiation, thus in the LTE effect [20]. However, the atmospheric density exponentially decays with the increase in altitude, and the collisions between the gas molecules rapidly weaken. Meanwhile, the energy level transitioning processes caused by the airglow radiation and photochemical reactions gradually dominate. At this time, the atmosphere is in non-LTE, and the blackbody radiation law, Boltzmann distribution law, Kirchhoff’s law, and other radiative transfer theories are no longer applicable [21]. According to Kirchhoff’s law, the emission rate of any object at a given temperature is numerically equal to the absorption rate of this object. Therefore, the temperature in LTE can be retrieved by the ratio of intensity of the spectral line associated with the absorption rate. However, Kirchhoff’s law fails in the non-LTE, and the emission rate and absorption rate are no longer equal. At this point, the retrieval of the temperature using the ratio of the spectral line intensities will inevitably result in systematic errors [22].
In view of this situation, this work proposes to retrieve the atmospheric temperature by processing the signal measured by the MIGHTI using the radiation-dependent Einstein coefficients. This work also compares the results with those of the ICON team based on the spectral line intensity to analyze the specific influence of the non-LTE effects on the temperature detection in the middle and upper atmosphere. The first part of this work ${O_2}({b^1}\sum _g^{})$ introduces the photochemical reaction of and the mechanism of its generation. The second part describes the rotational spectral distribution of ${O_2}({b^1}\sum _g^{})$ based on the spectral intensity and Einstein coefficient. The third part describes the specific principle of the MIGHTI temperature measurement. Finally, the retrieval outcomes of the middle and upper atmospheric temperatures using the Einstein coefficients is compared with the retrieval results of the ICON team using the spectral intensity and the NRSMSIS 2.0 model and the atmospheric chemistry experiment Fourier transform spectrometer (ACE-FTS) measurements. Furthermore, this work verifies that the Einstein coefficient method is more accurate and reasonable compared with the spectral line intensity method.
2. ${O_2}({b^1}\sum _g^ + )$ airglow production mechanisms
Airglow is a self-luminous phenomenon produced by atoms or molecular gases in the atmosphere that are excited to high energy levels by photochemical reactions and transition to lower energy levels. The O2 A-band airglow at 762 nm is the most intense visible airglow, which is produced by the transition of the excited state ${O_2}({b^1}\sum _g^ + )$ to the ground state ${O_2}({X^3}\Sigma )$ and can cover basically the entire near space during the daytime (the airglow is typically distributed in the altitude range of 30–150 km). Given the band shape and spectral line shape of its airglow radiation carrying information, such as Boltzmann population, Doppler frequency shift, and broadening, this phenomenon has become an important tracer for remote sensing of atmospheric meteorological parameters (such as wind and temperature fields) [23].
The radiation source of the O2 A-band airglow at 762 nm is the excited state ${O_2}({b^1}\sum _g^ + )$, and its generation mechanism mainly includes three types: (1) atmospheric resonance absorption, (2) ground state O2 colliding with the O(1D) atoms produced by the photolysis of O2 and O3, and (3) Bass chemical reaction. The specific mechanism is shown in Fig. 1 [24].
${O_2}({b^1}\sum _g^ + )$ in the excited state is highly unstable (their energy level lifetime is approximately 12 s) and prone to loss and degeneration to the ground state ${O_2}({X^3}\Sigma )$ due to the principle of lowest energy [25]. The loss process of the excited ${O_2}({b^1}\sum _g^ + )$ includes two mechanisms: spontaneous emission and annihilation caused by collisions with neutral gas molecules in the atmosphere (such as N2 and O2), which are expressed as follows:
where ${A_\Sigma} = 0.085\textrm{ }{s^{ - 1}}$ is the spontaneous emission coefficient of the O2 airglow in the 762 nm band [26].The transition of O2 from the excited state to the ground state is necessarily accompanied by the release of energy. During the collision–annihilation, the energy released by the energy level transition is converted into kinetic energy of O2, while the energy level transition during the spontaneous radiation releases photons followed by airglow radiation. In the low altitude region, the smaller the average molecular free path of the gas, the higher the frequency of intermolecular collisions because the atmospheric density exponentially decays with altitude, and the collision–annihilation effect dominates. In the middle and high altitude regions, the transition rate of the excited state molecules is greater than the collision rate, and the spontaneous radiation dominates. Therefore, the radiation efficiency of airglow generally increases with the increase in altitude.
The contributions of the different physical mechanisms to the A-band VER profile of O2 at each height level are shown in Fig. 2 [27]. The black dotted line indicates the contribution of O(1D) generated by O2 under ultraviolet (UV) photolysis in the Lyman-α band and Schumann–Runge band colliding with the ground state O2 to produce 762 nm ${O_2}({b^1}\sum _g^ + )$ airglow through energy exchange. The solid purple line indicates the VER contribution of the O2 A-band airglow produced by the collision of O(1D) generated by the UV photolysis of the O3 in the Hartley band with the ground state O2. The green dotted line and the blue solid line indicate the contributions of the three-body chemical reactions and atmospheric resonance absorption effects to the VER of the ${O_2}({b^1}\sum _g^ + )$ airglow, respectively. The red solid line indicates the total VER contribution of the ${O_2}({b^1}\sum _g^ + )$ airglow from all mechanisms together. The excited ${O_2}({b^1}\sum _g^ + )$ generated by the collisional energy exchange excitation between O(1D) produced by the UV photolysis of O3 and O2 and ground state ${O_2}({X^3}\Sigma )$ mainly contributes to the A-band airglow radiation of 762 nm O2 molecules below 80 km and above 100 km, respectively. In addition, the resonance absorption effect of O2 below 100 km dominates the generation of the ${O_2}({b^1}\sum _g^ + )$ airglow. When the altitude is higher than 100 km, its contribution to the O2 A-band airglow rapidly decreases due to the sharp decrease of the O2 molecular number density.
Fig. 2. Contribution of different physical mechanisms to the O2 A-band airglow VER at each altitude level.
3. Rotational spectral distribution of the ${O_2}({b^1}\sum _g^ + )$ airglow
The absorption or emission of photons by molecules is achieved through their inherent energy level transitions. Unlike the atomic energy level that has only an electronic state energy level transition, the energy level structure of molecules includes the electronic state energy level, vibrational state energy level, and rotational state energy level. Accordingly, the molecular spectrum shows a band structure. The O2 A-band airglow is generated by the transition from its second excited state $({b^1}\sum _g^ + )$ to the ground state $({X^3}\Sigma _g^ - )$. Given that the vibrational state transition rate of the ${O_2}({{b^1}\Sigma _g^ + ,v^{\prime} \ge 1} )$ state quenching to the ${O_2}({{b^1}\Sigma _g^ + ,v^{\prime} = 0} )$ state is much faster than the electronic state transition rate of the ${O_2}({{b^1}\Sigma _g^ + ,v^{\prime} = 0} )$ state radiation to the ${O_2}({{X^3}\Sigma _g^ - ,v^{\prime\prime} = 0} )$ state, the ${O_2}({b^1}\sum _g^ + )$ airglow radiation is dominated by the fundamental frequency vibrational state transition, and its spectral distribution is around at 762 nm. During the fundamental frequency vibrational transition of ${O_2}({{b^1}\Sigma _g^ + ,v^{\prime} = 0} )\to {O_2}({{X^3}\Sigma _g^ - ,v^{\prime\prime} = 0} )$, it is accompanied by a rotational transition. According to the vibrational–rotational transition selection rule, the transitions in which the upper and lower energy levels are Σ states belong to parallel band transitions, and only transitions of $\Delta J ={\pm} 1$ are allowed for such spectral bands. Accordingly, the O2 airglow spectrum has only the P-branch (the sub-band corresponding to the $\Delta J ={-} 1$ transition, located at the long-wave position of the spectral band) and the R-branch (the sub-band corresponding to the $\Delta J ={+} 1$ transition, located at the short-wave position of the spectral band), but not the Q-branch (the sub-band corresponding to the $\Delta J = 0$ transition located at the middle of the spectral band). The rotational spectral distribution of the ${O_2}({b^1}\sum _g^ + )$ airglow is shown in Fig. 3 [28], where the red line is the spectral intensity of each rotational line, which corresponds to the absorption transition, and the blue line is the Einstein coefficient, which corresponds to the emission transition, both of which are represented by the left and right vertical coordinates, respectively. The rotational spectral distribution of the ${O_2}({b^1}\sum _g^ + )$ airglow depicted by the spectral line intensities and Einstein coefficients greatly varies in the spectral band shape.
The emission rate $\varepsilon ( J^{\prime},J^{\prime\prime})$ of each rotational spectral line is calculated by multiplying the Einstein A coefficient ${A_{J^{\prime}}}_{,J^{\prime\prime}}$ of the ${O_2}({{b^1}\Sigma _g^ + ,v^{\prime} = 0} )\to {O_2}({{X^3}\Sigma _g^ - ,v^{\prime\prime} = 0} )$ transition by the relative particle number of the ${O_2}({{b^1}\Sigma _g^ + ,v^{\prime} = 0} )$ level [29]:
$Q_{tot}^{up}(T)$ is the vibrational–rotational state partition function of the upper energy level at temperature T, which can be expressed as follows [30]:
The spectral line intensity $S(J^{\prime\prime},T)$ is related to the absorption rate and is determined by the Einstein B coefficient ${B_{J^{\prime},J^{\prime\prime}}}$, which can be expressed as follows:
$Q_{tot}^{low}(T)$ is the vibrational–rotational state partition function of the lower energy level at temperature T, which can be expressed as follows [30]:
The relationship between Einstein’s A and B coefficients can be expressed as follows:
Therefore, the ratio between the emission rate $\varepsilon ( J^{\prime},J^{\prime\prime})$ and the spectral line intensity $S(J^{\prime\prime},T)$ for the rotational state transition $J^{\prime} \to J^{\prime\prime}$ should be:
The variation of the ratio of the emission rate to the spectral lines intensity in the A band with respect to the wavelength at the temperatures of 150, 250, 350, 450, and 550 K at which O2 is located is shown in Fig. 4. The trend of the ratio with wavelength becomes evident with the decrease in temperature. Therefore, the difference in the rotational spectral distribution of the ${O_2}({b^1}\sum _g^ + )$ airglow calculated from the emission rate and the spectral line intensity gradually increases with the temperature decrease.
Fig. 4. Ratio of the emission lines to the absorption lines as a function of wavelength at different temperatures.
4. Temperature measurement method
4.1 MIGHTI temperature measurement principle
Five discrete spectral channels with different wavelengths are used by MIGHTI to measure the A-band spectral shape of O2. Three spectral channels with central wavelengths of 760.0, 762.8, and 765.2 nm are more sensitive to the atmospheric temperature changes and are temperature sample channels with a channel bandwidth of 4 nm and the intensity ratio of the two adjacent channels, which can retrieve the atmospheric temperature. MIGHTI uses two spectral channels with central wavelengths of 754.1 and 780.1 nm as the background channels to reduce the influence of the background radiation signal on the accuracy of atmospheric temperature retrieval and the temperature measurement error by subtracting the background intensity from the signal channels. The filter transmittance of the three temperature sample spectral channels is shown in Fig. 5. In combination with Fig. 3, channels B and D sample the two wings of the ${O_2}({b^1}\sum _g^ + )$ airglow spectral band (i.e., the airglow spectral lines with high rotational quantum numbers), while channel C samples the middle region of the ${O_2}({b^1}\sum _g^ + )$ airglow spectral band (i.e., the airglow spectral lines with low rotational quantum numbers). According to Boltzmann distribution theory and Eq. (3) or Eq. (5), the molecular population of the rotational states with high quantum number increases with increasing temperature. Meanwhile, the molecular population of the rotational states with low quantum number decreases with increasing temperature. Consequently, the signal intensities of channels B and D increase with increasing temperature, while channel C decreases. The different temperature sensitivities of the three channels are the key to the retrieval of the atmospheric temperature by MIGHTI using the shape of the airglow spectrum.
Figure 6(a) and 6(b) show the relative intensity of each signal channel as a function of temperature using Eqs. (3) and (5), respectively, in combination with the transmittance functions of channels B, C, and D. Figure 6(a) shows the relative intensity obtained based on the spectral line intensity, and Fig. 6(b) shows the relative intensity obtained based on the Einstein coefficient. The relative intensity of all three channels show a monotonic exponential-like variation with temperature. The relative intensities of channels B and D increase with increasing temperature, and channel C decreases with increasing temperature. Moreover, the temperature sensitivities (i.e., slope of the curve) of channels B, C, and D decrease with increasing temperature.
Fig. 6. Intensities of three signal channels concerning with absorption and emission as functions of temperature. (a) Absorption line as a function of temperature; (b) emission line as a function of temperature.
The signal intensity of each channel is mainly determined by the VER of the ${O_2}({b^1}\sum _g^ + )$ airglow, in addition to being influenced by temperature. The ratio of the signal intensity of two adjacent channels with opposite temperature sensitivity is only temperature-dependent and is not affected by the VER. Figure 7 shows the relative intensity ratios of the B-to-C and D-to-C channel combinations as functions of temperature, where the red line represents the ratio of relative intensity calculated from the spectral line intensity, and the blue line indicates the ratio based on the Einstein coefficient. The channel ratios monotonically vary with temperature, in which the relative intensity ratio of the B-to-C channel combination varies monotonically with temperature, while the D-to-C channel combination shows exponential-like characteristics. The retrieval of the atmospheric temperature in the MLT region can be achieved by using either the B-to-C or D-to-C channel combination. The integration of the two combinations can greatly improve the accuracy of the temperature retrieval. The ratios obtained from the spectral line intensity and the Einstein coefficients are different as a function of temperature, and some differences are expected in the retrieval results when the atmospheric temperature is retrieved as a function of the two.
Fig. 7. Intensity ratios varying with temperature for channel combinations B-to-C and D-to-C concerning with absorption and emission, respectively.
4.2 “Onion peeling” algorithm
The MIGHTI observes the atmosphere in the mode of limb viewing to obtain high vertical resolution wind and temperature profile information, where each line of its CCD image is derived from the airglow radiation of the atmosphere at the corresponding limb viewing altitude. The line contains contributions from the target layer and the spatial altitude atmosphere above it due to the effect of the observation geometry. The limb viewing geometry is shown in Fig. 8. To simplify the forward simulation and retrieval algorithms, the atmospheric layers in each path segment are assumed to have the same absorption and emission properties, and their optical path are represented by equivalent uniform paths.
As previously mentioned, MIGHTI samples the O2 A-band airglow intensity through five discrete spectral channels, the spectral shapes measured in the middle three spectral channels (760.0, 762.8, and 765.2 nm) are used to retrieve the atmospheric temperature, and the channels on both sides (754.1 and 780.1 nm) are used to determine the background intensity to reduce retrieval errors. The images of the O2 A-band airglow radiation signal measured by the MIGHTI are shown in Fig. 9. The five image regions, namely, A, B, C, D, and E, correspond to the five spectral channels, and the vertical direction of the image represents the different limb viewing heights [31].
Assuming that the Earth and the atmosphere are spherical, the relative signal intensity $I_{k,m}^{obs}$ observed in the field of view (FOV) of the mth row of MIGHTI’s CCD can be written as the integral of the airglow radiation signal at each point along the line of sight of the line of view [32]:
Under the assumption of sphericity, the dependence of the atmospheric state on latitude and longitude in the observed region within the same altitude layer at LOS is negligible. Hence, the integral form of Eq. (5) can be transformed into an algebraic summation form by discretizing [32]:
The solution of the nonlinear equations in Eq. (6) can be obtained by the reverse replacement process using the “onion peeling” algorithm. The mathematical processing process of the “onion peeling” algorithm is to start solving from the top LOS, and the results of the upper layer are used as input to the next layer and sequentially solved downward. Since there is no contribution from other heights, the signal intensity of the target layer at the top layer can be obtained directly. Then, the contribution of the top height is removed from the second layer measurements to obtain the signal intensity of the second layer target segment. The signal intensity information of the target layer at each limb height can be obtained. The mathematical representation of the “onion peeling” algorithm can be written as follows [32]:
The background intensity images measured in channels A and E will be used as reference and will be deducted from channels B, C, and D (Fig. 9). Then, the “onion peeling” algorithm will be used to process the emission intensity signals measured in channels B, C, and D and obtain the relative radiation intensity of the target layer of each signal channel, as shown in Fig. 10. The relative radiation intensity profiles of the target layers of channels B, C, and D do not vary with height, and their relative differences imply the atmospheric temperature information. The atmospheric temperature information of the MLT region can be obtained by retrieval of the channel intensity ratio as a function of temperature shown in Fig. 10.
4.3 Error analysis
The signal intensity observed by the near infrared channel of MIGHTI consists of three components: the radiated signal of the O2 A-band airglow, the atmospheric background signal, and the noise. Among them, the atmospheric background signal can be corrected through the A and E channels to eliminate its influence on temperature measurement, while the temperature measurement error generated by noise cannot be directly eliminated. The sources of error in the MIGHTI temperature measurement channel include shot noise (SN), readout noise and dark noise, in which SN is caused by the statistical properties of photons, obeys the Poisson distribution and is the main source of noise. SN is propotional to the square root of the signal intensity of the airglow radiation.
In order to analysis the self-consistency of temperature retrieval, two combinations of B-to-C and D-to-C channels were used to obtain the temperature profiles, respectively. The specific equation for the temperature error of the two channel combinations can be expressed as:
The retrieval error for temperature observation can be reduced though cooperative inversion of the two channel combinations of B-to-C and D-to-C, and the combined error of the cooperative inversion method can be expressed as:
5. Comparison and discussion
To assess the rationality and reliability of temperature retrieval using Einstein coefficients, this section calculates the measurement error of MIGHTI based on the consideration of its SNR, and verifies the retrieval results of MIGHTI data from different perspectives: the NRSMSIS 2.0 atmospheric model, the MIGHTI temperature self-consistency, and the actual data from the ACE-FTS.
5.1 Comparison with atmosphere mode
We use the Einstein coefficients to calculate the variation of temperature with height at different latitudes and atmospheric model 2.0 simulation results for comparison to verify the rationality of the Einstein coefficient method for retrieval of temperature on a global scale. The data measured by MIGHTI on January 5, 2022, from 08:00 to 17:00 local time are selected for Fig. 11(a) to plot the results of the temperature distribution in latitude at 5° intervals, 9°S to 40°N, and 92–140 km altitude region. Figure 11(b) shows the atmospheric model 2.0 [33] temperature results for the corresponding time period. The atmospheric temperature distribution obtained by the Einstein coefficient method retrieval shows similar latitude and altitude trends with the atmospheric model 2.0, verifying the high rationality of using the Einstein coefficient method to retrieve the atmospheric temperature in the MLT region.
Fig. 11. Comparison of the temperature distribution obtained by the Einstein coefficient method and the atmospheric model at 9°S to 40°N. (a) Einstein coefficient method; (b) atmospheric model.
Given the chance of retrieval results for single date data, we choose the MIGHTI temperature retrieval results using the Einstein coefficient method at 08:00-17:00 local time from January 1 to January 8, 2022, to compare with the atmospheric model 2.0 temperature profile for the corresponding time and geographic location. The selected data were calculated at 5° and 12 km intervals in latitude and altitude, respectively, and the mean values of the temperature at different altitudes in the same interval from 9°S–40°N were obtained, as shown in Fig. 12. The red and blue lines show the trends with latitude of the temperature retrieval results of MIGHTI and the temperature simulation results with the NRSMSIS 2.0 atmospheric model in the corresponding altitude region, respectively, and the error bars on the red line represent the measurement errors of MIGHTI in that altitude region. When the altitude is low, the temperature retrieved from the Einstein coefficient method retrieval is basically consistent with the temperature distribution of the atmospheric model 2.0, with the maximum difference being only 33 K. The temperature difference between the two at altitudes of 116–127 km increases, and the maximum value of their difference rises to 70 K. The reason for this increasing is that the accuracy of the atmospheric model decreases with increasing altitude. When the altitude is between 128 km and 140 km, the difference in temperature distribution between the two is further increased, with a maximum difference of up to 120 K. In addition to the further reduction in the accuracy of the atmospheric model, the difference is mainly due to the existence of N2 1Pg 3-1 band airglow radiation in the region above 130 km, which overlaps with the airglow spectral region in the O2 A-band, resulting in the pollution of the spectral signal and the reduction in the accuracy of the temperature retrieval results in this region [34]. As also can be seen from Fig. 12, the MIGHTI observation error is only 1 K in the lowest altitude region and increases with altitude, and increases to 12 K in the highest altitude region. Although there are some discrepancies in the altitude range above 130 km, the relative difference between the retrieval result and the atmospheric model is in the range of 2% to 10% over most of the MLT region where the measurement results are less affected by N2 1Pg 3-1 band airglow. It proves that the retrieval result and the atmospheric model are in good agreement, and validates the rationality of the Einstein coefficient method for retrieving temperature at the global scale.
Fig. 12. Comparison of the temperature fluctuations obtained by the Einstein coefficient method and the atmospheric model at different height regions.
5.2 Self-consistency analysis
MIGHTI combines the intensity ratio of the two channels as a function of temperature to retrieve the atmospheric temperature in the MLT region. Accordingly, the self-consistency of the temperature of the two channel combinations is important for corroborating the reliability of the MIGHTI temperature measurement results. We selected the MIGHTI observation data on January 5, 2022, and plotted the atmospheric temperature profile of the MLT region retrieved using the Einstein coefficients and spectral line intensity and compared it with the NRSMSIS 2.0 atmospheric model, as shown in Fig. 13. Figure 13(a) and 13(b) show the atmospheric temperature profiles retrieved using the Einstein coefficient and spectral line intensity methods, respectively. The blue and red solid lines represent the atmospheric temperature obtained by the B-to-C and D-to-C channels of the two methods, respectively, and the error bars denote the uncertainty of the MIGHTI temperature observations. The right panel illustrates the deviation rate, defined as the ratio of the temperature difference obtained from the combined retrieval of the two channels to its temperature mean, which reflects the trend of the temperature deviation of the combined two channels with height and can more intuitively represent the self-consistency of the two retrieval methods. The larger the absolute value of the deviation rate, the worse the temperature self-consistency. The atmospheric temperature profile obtained from the combination of two channels through spectral line intensity retrieval has a significant deviation, and the retrieval result deviation increases with increasing altitude. When the altitude is at 120 km, the deviation increases to a maximum of 130 K. Subsequently, the deviation of the retrieved temperature gradually decreases with increasing altitude. When the altitude is at 100 km, the deviation rate of the spectral line intensity method can reach 44.6%, and the deviation rate decreases with the altitude, with a minimum of approximately 18.5%, but always higher than the Einstein coefficient method. By contrast, the atmospheric temperature profiles of the two-channel combination retrieved by the Einstein coefficient have less deviation and better self-consistency. Below 120 km, the temperature profiles of the two-channel combination almost coincide, and the deviation does not exceed 15 K. When the altitude rises above 120 km, the deviation slightly increases and reaches up to 45 K. This phenomenon is due to the signal intensity of the sampled channel that decreases when the altitude is higher than 120 km. When the relative radiation intensity of the target layer is retrieved using the “onion peeling” algorithm, its relative radiation intensity decreases, resulting in a lower signal-to-noise ratio and a higher deviation in the channel. In addition, the right panel reflects the small deviation rate of the Einstein coefficient method with a maximum of 9.8%. Meanwhile, we added the atmospheric model 2.0 for comparison, and the temperature profiles of both channel combinations of the Einstein coefficient method are in better consistency with the atmospheric model compared with the spectral line intensity method. The comparison results of the deviation rate and atmospheric model verify that the Einstein coefficient method has better self-consistency.
Fig. 13. Comparison of the atmospheric temperature profiles obtained by Einstein coefficient and spectral line intensity and their difference rates with the atmospheric model as a reference. (a) Atmospheric temperature profiles retrieved from Einstein coefficient and difference rates; (b) atmospheric temperature profiles retrieved from the spectral line intensity and difference rates.
We performed a statistical comparison of the temperature retrieval results from the two methods using the scatter plot shown in Fig. 14. A straight line is also fitted as a reference because the temperature distribution data set obtained by the two methods is linearly correlated. Figure 14(a), 14(b), 14(c), and 14(d) represent the data sets at altitudes of 95, 105, 115, and 125 km, respectively. The horizontal axis is the temperature of the B-to-C channel combination, and the vertical axis is the temperature of the D-to-C channel combination. The red line is fitted by the Einstein coefficient method, the blue line is fitted by the spectral line intensity method, and the proportional function $y = x$ indicates that the temperature of the two-channel combinations is exactly identical. When the fitted straight line is closer to the proportional function $y = x$ (i.e., the slope of the fitted equation k = 1 and the intercept b = 0), the temperature self-consistency of the two-channel combination is better. The red line intercepts are all smaller than the blue line intercepts. The blue line has a large intercept at all altitudes, while the red line has a small intercept at lower altitudes, but the intercept increases when the altitude is at 125 km. As previously mentioned, this phenomenon occurred because the channel signal-to-noise ratio decreases when the altitude is above 120 km, resulting in an increase in the retrieval deviation of the two channels. Furthermore, the red line is closer to the straight line $y = x$ than the blue line at different heights, verifying that the self-consistency of the Einstein coefficient method for retrieving temperature is better than that of the spectral line intensity method.
Fig. 14. Comparison of the retrieval accuracy of the two methods at different heights. Data from January 5, 2022 and January 17, 2022. (a) Fit equations of the Einstein coefficient method and spectral line intensity method at 95 km; (b) fit equations of the Einstein coefficient method and spectral line intensity method at 105 km; (c) fit equations of the Einstein coefficient method and spectral line intensity method at 115 km; (d) fit equations of the Einstein coefficient method and spectral line intensity method at 125 km.
In addition, we retrieved the temperature data on January 5, 2022, at 1 km intervals in the spatial range of 92–140 km. Slope k and intercept b of each fitted line are calculated in the above form, and their relationship with altitude is shown in Fig. 15. Figure 15(a) shows the profile of slope variation with height, and Fig. 15(b) presents the profile of the intercept variation with height, where the blue and red lines indicate the fitting results of the two methods of Einstein coefficient and spectral line intensity, respectively, and the black lines in the two subplots indicate the ideal fitting results with a slope of one and intercept of zero for comparative analysis. Theoretically, the closer the slope is to one and the closer the intercept is to zero, the better the self-consistency is. Slope k1 of the equation fitted by the spectral line intensity method is closer to the reference value one than slope k2 of the equation fitted by the Einstein coefficient method when the altitude is in the range of 99–111 km, and intercept b1 of the equation fitted by the spectral line intensity method is closer to the reference value zero than intercept b2 of the equation fitted by the Einstein coefficient method when the altitude is in the range of 123–128 km. Therefore, the relationship between the temperature self-consistency of the two methods is difficult to directly verify based only on the variation of a single parameter with height.
Fig. 15. Comparison of the intercept and slope of the fitted equations of the two retrieval methods at different heights. (a) Comparison of slope profiles between two methods; (b) comparison of the intercept profiles between two methods.
As previously mentioned, combining the two parameters organically provides more informative results than to evaluate the self-consistency of the two methods’ retrieval temperatures in isolation by a single parameter. Consequently, the fitting results were further processed using the following method: The maximum deviation is only 12 K because the mean values of the atmospheric temperature of the B-to-C and D-to-C channels at different altitudes obtained by the two methods of retrieval are basically the same. Accordingly, the average value of the atmospheric temperature of the two channels retrieved by the Einstein coefficient method is taken as the “true value” of the atmospheric temperature and is denoted as $\bar{x}$. Then, slopes k1 and k2 and intercepts b1 and b2 for each height obtained by fitting the spectral line intensity method and the Einstein coefficient method with the previous method are again fitted to the equations in the following form:
Fitted temperatures y1 and y2 are used as the “measured values” retrieved by the two methods, and the “true values” are subtracted from the “measured values” to obtain:
where $\Delta {y_i}$ reflects the MIGHTI temperature retrieval self-consistency, and the smaller its absolute value, the better the tempe $\Delta {y_1} = {y_1} - \bar{x}$ rature retrieval self-consistency. This result allows for a comparison of the self-consistency of the temperature retrieval results of both two methods. Figure 16 plots the profile of the $\Delta {y_i}$ variation with height for each height layer. The aforementioned figure shows that $\Delta {y_1}$ is larger in value and increases with increasing altitude, reaching a maximum value of 147 K at 112 km. Thereafter, the overall trend decreases according to the increasing altitude. Meanwhile, $\Delta {y_2}$ is smaller in value and does not exceed 20 K when the altitude is below 125 km, after which it reaches a maximum value of 72 K at 134 km with increasing altitude, which is only half of the maximum value of $\Delta {y_1}$. The absolute value of $\Delta {y_2}$ is smaller than that of $\Delta {y_1}$ at different heights, which confirms the higher temperature self-consistency of the Einstein coefficient method compared with the spectral line intensity method while avoiding chance, further demonstrating the reliability of the Einstein coefficient method for retrieving the temperature of the middle and upper atmosphere.5.3 Comparison with ACE-FTS
Comparing data products from another independent satellite with different measurement principles is also an effective way to verify the accuracy of the retrieval results. The Atmospheric Chemistry Experiment (ACE) satellite, launched in the United States in 2003, carries the Fourier Transform Spectrometer (FTS), which utilizes occultation observations to measure the atmospheric composition changes with an accuracy of ±4 K in the MLT region. The atmospheric temperatures obtained by the Einstein coefficient method and the spectral line intensity method of retrieval are compared using satellite’s temperature data. The MIGHTI data on April 19, 2022 were selected to retrieve the atmospheric temperature in the 92–140 km region by using two methods. The average temperature of the two-channel combinations was taken as the retrieval result to plot the atmospheric temperature profile, and the temperature data from ACE-FTS on the same date were selected for comparison. Moreover, the data that fit the latitude and longitude were selected for comparison to avoid the influence of geographical differences on the temperature measurement results. The local solar times of the MIGHTI data selected were all located between 08:00 and 11:00 to ensure the accuracy of the retrieval results. Three sets of atmospheric temperature profiles are plotted in Fig. 17. The red and blue lines represent the atmospheric temperature profiles retrieved by the Einstein coefficient method and the spectral line intensity one, the green line is the temperature data measured by ACE-FTS, and the MIGHTI observation error is shown in the right panel. The atmospheric temperature profile retrieved by the spectral line intensity method has a certain deviation from the ACE-FTS satellite temperature profile, which is especially evident when located above 120 km. This situation occurs because Kirchhoff’s law fails in the non-LTE state, resulting in large errors caused by using the spectral line intensity to retrieve the gas temperature. Furthermore, the trend of the atmospheric temperature profile retrieved by the Einstein coefficient method is consistent with the temperature profiles of the ACE-FTS, reflecting the rationality of the temperature retrieved by the Einstein coefficient method.
Fig. 17. Comparison of the atmospheric temperature profiles of MIGHTI obtained by Einstein coefficient and spectral line intensity with observation results of ACE-FTS as a reference, the right pane is the temperature uncertainty of MIGHTI.
To visualize the difference between the retrieval results of the two methods and the ACE-FTS temperature measurement data, the root-mean-square (RMS) of the Einstein coefficient method and the spectral line intensity method were separately calculated:
where ${X_i}$ is the ith set of MIGHTI data, $\overline F$ is the average of all ACE-FTS temperature data at each altitude, and N is the number of MIGHTI data used. The smaller the RMS value, the smaller the deviation between the corresponding method and ACE-FTS temperature measurement results. The results of the RMS calculation are shown in Fig. 18.Fig. 18. RMS between observation results of ACE-FTS and temperature profiles of MIGHTI obtained by Einstein coefficient and spectral line intensity.
The figure clearly demonstrates that the RMS values for the spectral line intensity method and Einstein coefficient method are lower in the altitude range of 92–104 km, and the maximum values does not exceed 7 K. The RMS of the Einstein coefficient method are smaller than those of the spectral line intensity method for the remaining heights in the MLT region, verifying the higher accuracy of the Einstein coefficient method compared with the spectral line intensity method for temperature retrieval. When the altitude is above 125 km, the RMS of the two methods rapidly increases and finally reaches a maximum in the region around 130 km, with the Einstein coefficient method reaching a maximum of 65 K at 130 km and the spectral line intensity method attaining a maximum of 94 K at 133 km. The main reason for this phenomenon is the spectral contamination of the airglow radiation in the N2 1Pg 3-1 band in the space region around 130 km, resulting in a large retrieval error in this region [34].
6. Conclusion
This paper presents a new method of retrieving the atmospheric temperature profiles based on Einstein coefficients, aiming at correcting the influence of non-LTE effect and reducing the temperature retrieval error. We retrieve the O2 A-band airglow signal measured by MIGHTI using the “onion peeling” algorithm of the limb viewing mode based on the emission spectrum of the Einstein coefficients and obtain an atmospheric temperature profile of 92–140 km, which can effectively cover the entire MLT region.
Based on the comparison of the retrieved atmospheric temperature through the Einstein coefficient method with simulation data from the NRSMSIS 2.0 atmospheric model and the measurement result from ACE-FTS, good agreement is achieved, which verifies the rationality and accuracy of the new method.
The result of this study shows that the atmospheric temperature in the MLT region obtained by the Einstein coefficient method has better self-consistency than that of the spectral line intensity method under the influence of non-LTE effect. The maximum deviation of the inverted temperature from t two different channel combinations is not more than 45 K for the Einstein coefficient method, which is better than that of the spectral line intensity method with a maximum deviation of 130 K. Moreover, we also compared the temperature retrieval results of the two methods with the measured data of ACE-FTS. The result reflects that the Einstein coefficient method has better consistency with the data of ACE-FTS, and the maximum RMS of temperature profiles between MIGHTI and ACE-FTS is reduced from 94 K to 65 K with non-LTE effect corrected. This further verifies the accuracy of the Einstein coefficient method for temperature inversion in the MLT region. The result also shows that the temperature deviation between the two different channel combinations of MIGHTI increases with altitude and reaches a maximum above 130 km, which is due to the influence of spectral interference from the N2 1Pg 3-1 band airglow.
The Einstein coefficient method proposed in this work greatly improves the self-consistency of the atmospheric temperature retrieval, and its superior performance in the middle and upper atmospheric temperature retrieval reflects great scientific research value and engineering application prospects. The method contributes to further studies of temperature remote sensing in the MLT region and helps in promoting the understanding and exploration of the coupling mechanism in the region, favorably contributing to the middle and upper atmosphere temperature retrieval. Predictably, this new method may become a standardized method for temperature retrieval of the middle and upper atmosphere in the foreseeable future and will be widely applied to temperature data processing for other satellite payloads instead of being limited to MIGHTI. In future endeavors, it is expected to do a more in-depth study on the correction of the non-LTE effect to further improve the precision of temperature retrieval in the MLT region though eliminating the spectral interference of N2 1PG 3-1 band airglow in the 130 km region.
Funding
National Natural Science Foundation of China (41704178, 41975039, 62305283); Natural Science Foundation of Shandong Province (ZR2021QD088); Youth Innovation Technology Project of Higher School in Shandong Province (2021KJ008); Graduate Innovation Foundation of Yantai University (GGIFYTU2313).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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