Research on Calculation Method of On-Orbit Instrumental Line Shape Function for the Greenhouse Gases Monitoring Instrument on the GaoFen-5B Satellite

Abstract

The Greenhouse Gases Monitoring Instrument is based on the spectroscopic principle of spatial heterodyne spectroscopy technology and has the characteristics of no moving parts, a hyperspectral resolution, and a large luminous flux. The instrumental line shape function is one of the most important parameters characterizing the features of the instrument, and it plays a vital role in the system error analysis of the instrument’s measurements. To accurately obtain the instrumental line shape function of a spatial heterodyne spectrometer during the on-orbit period and improve the accuracy of gas concentration retrieval, this study develops a method to model and characterize the characteristics of the instrumental line shape function, including modulation loss and phase error. This study employs the solar calibration spectrum in the 1.568–1.583 μm bands to conduct iterative calculations of the instrumental line shape function error model. After the instrumental line function is updated, the average relative deviation is reduced from 1.83% to 0.756% between the theoretical and measured solar spectra. Additionally, the average relative deviation is reduced from 7.049% to 2.106% between the GMI nadir and theoretical nadir spectra. The findings demonstrate that updating the instrumental line shape function mitigates the impact of variations in the spectrometer’s instrumental line shape due to alterations in the orbital environment. This study offers a dependable reference for both the enhancement and oversight of a spectrometer’s instrumental line shape function, along with an investigation of shifts in instrument parameters.

1. Introduction

GaoFen-5B (GF-5B) was successfully launched from the Taiyuan Satellite Launch Center on 7 September 2021. It is a fusion application satellite that obtains multiple types of observation data, including hyperspectral, full-spectrum, polarization, multi-angular, and ocean sun glint observation data. It carries out simultaneous hyperspectral observations of the atmosphere and land [1]. GF-5B covers a wide observation spectrum, ranging from ultraviolet to long-wavelength infrared (0.24–13.3 μm). This allows for highly accurate measurements using various detection methods. Consequently, GF-5B achieves comprehensive environmental monitoring of the atmosphere, water resources, and natural ecosystems. The GF-5B satellite carries a powerful suite of seven payloads, including two for Earth imaging and five dedicated to atmospheric measurement. One of the key instruments is the Greenhouse Gases Monitoring Instrument (GMI), developed by the Anhui Institute of Optics and Fine Mechanics of the Chinese Academy of Sciences. This payload leverages spatial heterodyne spectroscopy (SHS) technology, which combines the benefits of grating dispersion and interferogram modulation. This allows the GMI to achieve exceptional performance with a high signal-to-noise ratio (SNR) and hyperspectral resolution, with the added advantage of no moving parts, thus enhancing its reliability.

Since the 1980s, with the rapid development of science and industrial and agricultural technology, the emissions of gases, such as carbon dioxide, methane, nitrous oxide, and halogenated compounds, from industrial and agricultural production and human activities have been increasing. This has led to a continuous increase in the concentration of greenhouse gases (GHGs), causing the global average surface temperature to rise [2]. Among these gases, CO₂ is the most important greenhouse gas in the atmosphere, accounting for about 64% of long-lived greenhouse gases (LLGHGs). Since CO₂ is a trace gas in the atmosphere, its volume fraction is about 3.8 × 10⁻⁴, and the range of its concentration variation is small. The seasonal fluctuation caused by land area and vegetation volume decreases from (12~22) × 10⁻⁶ in the northern hemisphere to (1~2) × 10⁻⁶ in the southern hemisphere [3]. Research by P. J. Rayner and D. M. O’Brien indicates that, for space-based detection, the detection accuracy needs to reach 1% to be effectively used for CO₂ source and sink analyses [4]. To meet the requirements for the high-precision global detection of greenhouse gas concentrations, the GMI payload adopts a solution that is distinct from traditional hyperspectral imaging technologies. It is based on SHS, utilizing a static grating instead of the moving mirror in traditional Fourier transform spectrometers, which has the advantages of a small volume and weight, no moving parts, and a hyperspectral resolution, enabling the acquisition of fine absorption spectra with an extremely high spectral resolution [5,6]. SHS technology differs from the Fourier interferometer scheme employed by Japan’s Greenhouse Gases Observing Satellite (GOSAT) [7] and the grating spectrometer scheme used by the United States’ Orbiting Carbon Observatory-2 (OCO-2) [8]. For the atmospheric detection spectroscopy payload, both radiometric accuracy and spectral precision are critical factors influencing atmospheric retrieval processes [9].

During the launch and on-orbit periods, the optical, structural, and electronic components of the spectrometer may undergo performance changes, resulting in laboratory calibration results that cannot be directly applied to the on-orbit spectrum. Thus, it is necessary to correct such changes during the on-orbit period, requiring on-orbit calibration, including both spectral and radiometric calibration. Spectral calibration includes determining the relationship between the spatial frequency of interference fringes and the incident wavelength, as well as the instrumental line shape function (ILSF). Spectral calibration is the primary task carried out to ensure the genuine and effective acquisition of target spectral information. Before the launch, the GMI payload completed high-precision laboratory spectral calibration using a tunable laser combined with a speckle integrating sphere system. The uncertainty of spectral calibration reached 0.015 cm⁻¹ [10], and the ILSF was determined through instrument measurements of monochromatic light data [11]. However, during satellite launch, factors such as instrument vibrations, changes in on-orbit thermal conditions, outer space gravity and radiation, instrument aging, and differences between on-orbit and laboratory experimental environmental conditions can lead to subtle variations in wavelength and the spectrometer’s ILSF during the on-orbit period. Therefore, it is essential to monitor and update the wavelength and ILSF changes of the spectrometer during the on-orbit period [12]. During the on-orbit period of the satellite, on-orbit wavelength calibration was performed using the measured high-resolution solar Fraunhofer as the characteristic spectral lines. After calibration, the spectral line deviation reduced from 0.133 cm⁻¹ to 0.009 cm⁻¹ [13]. To further improve the accuracy of on-orbit spectral calibration, accurately obtain the fine absorption spectral information of greenhouse gases, and ensure the precision of atmospheric greenhouse gas retrieval, studying the ILSF of the spectrometer during the on-orbit period is of great significance, especially for quantitative analyses of hyperspectral remote sensing data.

For the retrieval of high-precision greenhouse gases in the atmosphere, there are three main factors that constitute the primary constraints: atmospheric radiative transfer calculation, instrument models, and retrieval algorithms. The most important parameter in the instrument model is the ILSF, which determines the final spectral form of the radiation entering the instrument. It represents the instrumental spectral resolution and the distribution of energy across spectral lines. Therefore, it is important to accurately obtain the ILSF of the spectrometer to precisely simulate the theoretical spectrum and quantitative retrieval [9,14]. When a Fourier transform spectrometer performs spectroscopy through spatial modulation, it is assumed that the theoretical ILSF for each spectral band is identical. In ground-based laboratories, methods such as gas absorption [15,16] and adjustable lasers [11,17] are often utilized to determine the ILSF. This involves measuring the ILSF by selecting prominent gas absorption lines or laser wavelengths within the working spectral range. For on-orbit ILSF determination, laser diodes are commonly chosen as light sources [18,19,20], sharing the same principles as tunable lasers. Another option is to use the solar spectrum, although adjustments to parameters such as the distance from the Sun to the Earth and solar activity cycles are necessary [21].

The GMI payload employs a static interferogram technology system and lacks experimental equipment for directly measuring the ILSF using monochromatic light or gas absorption cells. Therefore, it is impossible to directly measure the ILSF in a way similar to that used in ground-based laboratory environments. Currently, there is limited research both domestically and internationally on updating the ILSF of space-borne spectrometers during the on-orbit period. This study, based on the principles of SHS technology and the processing of interferograms, derives theoretical mathematical expressions for the ILSF. Utilizing the GMI payload as the carrier and combining the laboratory measurement results of the ILSF and the analysis of instrument parameters, this study utilizes the on-orbit solar observation calibration spectrum of the GMI payload, which is not affected by the atmosphere and features an independent solar Fraunhofer line. The model establishes the ILSF concerning modulation loss and phase error, using the stable Kurucz solar spectrum as the theoretical reference spectrum. A model function for the ILSF concerning modulation loss and phase error is established. Through residual iterative optimization with the solar calibration spectrum measured by the GMI payload, the parameters of the ILSF model are ultimately obtained. This achieves the update of the GMI measurement of the on-orbit ILSF, with an ILSF on-orbit monitoring error of less than 1%. This study provides a reference for subsequent on-orbit calibration, monitoring of the ILSF, and analyses of instrumental parameter changes for spectrometers.

2. GMI Payload Overview and Ground-Based ILSF Testing

2.1. Overview of the GMI Payload

The GMI adopts the principle of SHS technology, as illustrated in Figure 1. It consists of a collimating front lens (L1), a spatial heterodyne interferometer unit (beam splitter and two-arm grating), imaging lenses (L2 and L3), and a detector.

According to the interference principle of SHS technology, the spectrum of the measured target, obtained through this technology, appears as an interferogram. After passing through collimating system L1, the incident light spectrum $B\left(\sigma \right)$ is directed parallel to the beam splitter and divided into two beams of light, which are directed towards the gratings in each arm. After diffraction by the grating, the two beams of light form a tilt angle $\pm \theta$ with the optical axis. Then, the two diffractive beams return to the beam splitter again and are finally projected onto the detector through the detector optical system L2/L3. Among them, the wavefront of the two coherent beams produces an inclination angle of $2\theta$, so there are different optical path differences at different sampling positions in the spectral dimension direction (the direction of arrow X in Figure 1), forming interference fringes. There is a Fourier transform (FT) relationship between the intensity information of the interference fringes obtained by SHS and the incident spectral information, as shown in Equation (1) [22]:

$$I\left(x\right)={\displaystyle {\int }_0^{\infty }B\left(\sigma \right)}\left[1+\cos \left(2\pi \cdot f\left(\sigma \right)x\right)\right]d\left(\sigma \right)$$

(1)

Here, $x$ represents the optical path differences (OPD) of the sampling point; $f\left(\sigma \right)=4\left(\sigma -{\sigma }_0\right)\tan \theta$ represents the spatial frequency of the interference fringes generated when a light beam with wavenumber sigma enters the spatial heterodyne spectrometer; ${\sigma }_0$ and $\theta$ are the Littrow wavenumber and angle; and $B\left(\sigma \right)d\left(\sigma \right)$ is the intensity at wavenumber $\sigma$.

The GMI payload is equipped with four spectral channels, namely, O₂ (0.765 μm), CO₂-1 (1.575 μm), CO₂-2 (2.05 μm), and CH₄ (1.65 μm). The 0.765 μm band is utilized to mainly provide information on surface pressure, clouds, and aerosols. The 1.575 μm and 1.65 μm bands are used to estimate the atmospheric concentrations of CO₂ and CH₄, respectively. The 2.05 μm band mainly provides information on aerosols and CO₂. The main technical specifications of the GMI payload are shown in

Table 1. Main technical indicators of the GMI.

Parameters	Technical Indicators
	O2	CO2-1	CH4	CO2-2
Central Wavelength/μm	0.765	1.575	1.65	2.05
Spectral Range/μm	0.765~0.769	1.568~1.583	1.642~1.658	2.043~2.058
Spectral Resolution/cm−1	0.6	0.27	0.27	0.27
SNR (Albedo = 0.3; Sun Elevation = 30°)	300	250
Radiometric Calibration	Absolute Accuracy: 5%; Relative Accuracy: 2%
Field of View (FOV)	14.6 mrad (10.3 km@705 km)
Observation Mode	Nadir Observation Mode: 1, 5, 7, 9 Points (Default 5 Points) Solar Observation Mode: Calibration Observation

.

2.2. The ILSF Model and Laboratory Testing

2.2.1. Ideal Measurement Model

The ILSF characterizes the spectral response of a spectrometer to a monochromatic light source at a given wavelength, serving as a crucial parameter for the high-precision spectral calibration of spectrometers [23]. The input signal is typically set as an impulse function $\delta \left(\sigma -{\sigma }_0\right)$, where the frequency domain of the impulse function is a constant, i.e., $F\left[\delta \left(\sigma -{\sigma }_0\right)\right]=1$. For incident light with the Littrow wavenumber, the relationship between the OPD $u$ of the two beams upon re-encountering after dispersion and the distance from the grating dispersion direction to optical axis $x$ can be expressed as follows:

$$u=4x\tan \theta$$

(2)

The input spectra in the SHS system can be symmetrized by defining the following:

$$B_e\left({\sigma }^{\prime }\right)=\frac{1}{2}\left(B\left({\sigma }^{\prime }\right)+B\left(-{\sigma }^{\prime }\right)\right)$$

(3)

Here, ${\sigma }^{\prime }=\sigma -{\sigma }_0$. This is a generation of the relationship $B_e\left(\sigma \right)=\left(B\left(\sigma \right)+B\left(-{\sigma }^{\prime }\right)\right)/2$, which is often applied in the analysis of FTS systems and accounts for the symmetry in the recovered transform. After making these changes, Equation (1) can be rewritten as follows:

$$I\left(u\right)={\displaystyle {\int }_{-\infty }^{\infty }B\left({\sigma }^{\prime }\right)}\left[1+\cos \left(2\pi u{\sigma }^{\prime }\right)\right]d\left({\sigma }^{\prime }\right)$$

(4)

By setting $B\left({\sigma }^{\prime }\right)=\delta \left({\sigma }_1^{\prime }-{\sigma }^{\prime }\right)$ in Equation (3), the response of the basic SHS systems given by Equation (4) is as follows:

$$I\left(u\right)=\cos \left(2\pi u{\sigma }_1^{\prime }\right)$$

(5)

Here, ${\sigma }_1^{\prime }={\sigma }_1-{\sigma }_0$, and the constant term is removed. Since $I\left(u\right)$ is measured only over a finite range of $u\left(u<\left|L\right|=u_{\max }\right)$, in the simplest case, the transform of Equation (5) can be written as follows:

$$B_I\left({\sigma }^{\prime }\right)=2L\left\{\sin c\left(2\pi L\left({\sigma }_1^{\prime }+{\sigma }^{\prime }\right)\right)+\sin c\left(2\pi L\left({\sigma }^{\prime }-{\sigma }_1^{\prime }\right)\right)\right\}$$

(6)

As indicated by Equation (6), under ideal conditions, spectrum $B_I\left({\sigma }^{\prime }\right)$ reconstructed from monochromatic light interference data is a $\sin c$ function, oscillating with its peaks centered around zero and gradually decaying towards zero. This is commonly referred to as the instrumental line shape function. The full width at half maximum (FWHM) of the ILSF reflects the broadening influence of the instrument on the spectrum, and its size can determine the spectral resolution of the instrument. Under ideal conditions, the ILSF of the instrument is only affected by modulation loss [24]. Figure 2 presents typical theoretical results for the ILSF.

2.2.2. The Laboratory ILSF Experiment

When measuring the ILSF, it is essential that the wavelength uncertainty of the light source is minimal, ensuring good stability. Additionally, there should be multiple lines uniformly distributed within the spectral range of the instrument, and the spectral resolution of these lines should be significantly smaller than the spectral resolution of the instrument. The line intensity should also meet the SNR requirements of the instrument. The GMI payload belongs to super-resolution spectral spectrophotometry technology. The spectral calibration scheme selected involves a SANTEC TSL-510C tunable laser from SANTEC Corporation combined with a speckle integrating sphere system [10]. This instrument can output continuous tunable monochromatic light (1 pm intervals) within the spectral range of the instrument, with a linewidth smaller than 1 pm. The wavelength absolute accuracy is better than 5 pm, and the output power meets the requirements of the spatial heterodyne spectrometer. The light output from the laser is introduced into an integrating sphere to provide a uniform surface light source that fills the FOV of the instrument. The laser spot on the inner wall of the integrating sphere directly illuminates a small area, and the method of using a rotating diffuse reflection plate is employed to eliminate the speckle caused by the spatial coherence of the laser [25,26]. The interferogram is restored to the spectrum after preprocessing and phase correction. To ensure the accurate peak positioning of the spectral data, the original 500 valid data points are zero-filled to 2¹⁴. After zero-filling, the spectral resolution remains unchanged, and the spectral curve becomes smoother. A preprocessed interferogram at an incident wavenumber of 6349.21 cm⁻¹ is shown in Figure 3a, and restored spectral data after zero-filling are shown in Figure 3b.

When a Fourier spectrometer performs spectral splitting through spatial modulation, the ILSF is the same for each spectral band. Therefore, it is not necessary to determine the ILSF for each incident wavenumber. In actual processing, the spectrum of individual monochromatic lights is normalized and shifted to the same position based on the location of their maxima. The result is shown in Figure 4a. Because of the narrow spectral range of the instrument, the ILSF variation is minimal across the spectrum. Therefore, averaging the ILSF at various wavenumbers provides the overall ILSF for the entire spectral range. The FWHM of the instrument is 0.268 cm⁻¹, as depicted in Figure 4b.

2.2.3. The ILSF Absorption Cell Test Validation

Before the GMI payload launched, absorption spectrum testing experiments were conducted using specialized equipment, such as hyperspectral resolution calibration equipment and an atmospheric environment simulation calibration chamber [27,28]. The experimental equipment is shown in Figure 5. Firstly, the atmospheric environment simulation calibration chamber is evacuated to a high vacuum state with air extraction equipment to measure the background spectrum; then, gas distribution equipment is employed to proportion greenhouse gases at standard concentrations. Temperature control equipment is utilized for temperature regulation, and once the temperature and pressure parameters stabilize, the payload conducts absorption spectrum measurements. The measured transmittance spectrum is obtained based on the target and background spectrum.

According to the Beer–Lambert law [29], the theoretical transmittance calculation formula for radiant energy passing through the target gas is as follows:

$$\tau \left(\lambda \right)=e^{-\alpha \left(\lambda \right)cL}$$

(7)

Here, $\lambda$ is the wavelength, $\alpha$ is the gas absorption cross-section, $c$ is the target gas concentration, and $L$ is the absorption path length. The target gas concentration is obtained by combining the gas pressure and temperature with the ideal gas equation [30]. The calculation of the atmospheric absorption cross-section $\alpha$ is under the condition of the absorption cell state when the payload measures the actual transmittance. The absorption cross-section for each channel is calculated based on the HITRAN database [31] and relevant formula $\sigma =n_0\alpha \cdot P{T}_0/P_{0}T$, where $\sigma$ is the absorption coefficient, $n_0$ is the standard Lorentzian constant, $P$ is the actual pressure, $T$ is the actual temperature, and $P_0$ and $T_0$ are the pressure and temperature in the standard atmospheric state.

The calculated transmittance spectrum is convolved with the ILSF of the corresponding channel, resulting in the theoretical transmittance for each channel. Figure 6 illustrates a comparison between the measured (blue curve) and theoretical transmittance (red curve) of the CO₂-1 bands observed by the GMI payload. This comparison was carried out in a controlled environment, where 100 kPa of CO₂ gas was introduced into the simulated calibration chamber, with the gas temperature rigorously maintained at 20 ± 0.5 °C. A comparative analysis between the measured transmittance of the absorption cell and the theoretical transmittance revealed that the absorption peak positions are accurate, confirming the precision of the wavelength calibration results. According to Equation (8), the average relative deviation of the spectral lines was 0.818%, demonstrating good agreement within the depth and width of the absorption peaks and confirming the accuracy of the ILSF calibration.

$$RMD=\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{\left|x_i-X_i\right|}{X_i}}\cdot 100\%$$

(8)

Here, $x$ is the measured spectrum, $X$ is the reference spectrum, and $i$ is the number of spectral points.

3. On-Orbit ILSF Status Analysis

3.1. On-Orbit Calibration Scheme

The GMI payload on-orbit calibration scheme uses highly stable sunlight as the calibration light source. It establishes a space-level large-area radiation standard source through a solar illumination calibrator and fills its aperture of observation and FOV at the front end of the payload optical path to achieve entire end-to-end optical path calibration. During the on-orbit calibration process, the payload’s two-dimensional scanning mechanism shifts the nadir observation mode to the on-orbit calibration mode; a schematic diagram of the calibration optical path is shown in Figure 7b. This action allows the calibration light source, sunlight, to enter the payload telescope, initiating data collection in calibration mode. The structure of the calibrator is shown in Figure 7a. The on-orbit calibrator consists of a solar diffuser, a light trap, a radiometer, a calibration door, a driving mechanism, and a calibration cabin. It achieves entire aperture, FOV, and optical path calibration for the payload. The calibrators serve distinct functions within the instrument system. Firstly, the solar diffuser plate, constructed from polytetrafluoroethylene, functions as a standard light source by harnessing on-orbit solar illumination. Secondly, the radiometer is tasked with tracking and monitoring the decay of the diffuser plate over time. Additionally, the light trap plays a crucial role in providing a dark background reference for instrument calibration and conducting dark current measurements. Furthermore, the calibration cabin and space diaphragm are employed to eliminate stray light and safeguard the space environment. Finally, the door and driving mechanism work in conjunction with the two-dimensional rotation of the scanning mirror to execute calibration procedures; the door-blocking structure is shown in Figure 7c. These calibration routines are conducted monthly to ensure the precision and accuracy of the instrument measurements.

The on-orbit calibration of the GMI payload includes spectral calibration and radiometric calibration, both of which use unified on-orbit calibration equipment, and the calibration time is selected near the Antarctic region of the satellite. The spectral calibration time must be selected so that the sunlight can be incident on the calibration equipment, the sunlight can pass through the upper atmosphere, and the height range is 10–100 km. During the radiometric calibration, it is crucial to select a calibration time for when sunlight can illuminate the calibration device without being affected by atmospheric absorption.

3.2. Theoretical Spectral Processing

The solar spectrum is an absolute value of solar irradiance in a certain wavelength band. The solar spectrum, which is usually determined during the solar minimum or low-solar-activity periods, has important implications for climate science and sensor radiometric calibration. The Kurucz solar spectrum is often used as the standard solar spectrum to compare and calibrate observations. The Kurucz solar spectrum, developed by Robert Kurucz, is a theoretical representation of the solar spectrum, calculated using specific physical assumptions and mathematical models. The Kurucz solar spectrum spans a broad range of wavelengths, from ultraviolet to infrared, offering detailed information on the sun’s radiation intensity and spectral characteristics at various wavelengths [32].

The solar spectrum observed by the GMI payload can clearly distinguish the solar Fraunhofer absorption lines formed by the absorption of the outer atmospheric elements of the sun. The spectral resolution of the Kurucz solar spectrum is higher than that of the GMI payload. Therefore, the Kurucz solar spectrum with a high spectral resolution can be used as the theoretical reference spectrum for the on-orbit spectral calibration of the GMI payload. The Kurucz solar spectrum is derived from theoretical and empirical model calculations, covering a spectral range of 0.2 to 200 μm. The uncertainty of its standard value in the CO₂-1 bands can reach 3% [8,33,34]. The data can be accessed from a website (http://Kurucz.harvard.edu/sun, accessed on 12 December 2023). Figure 8 shows the spectral line distribution of the Kurucz solar spectrum in the CO₂-1 bands. The Kurucz solar spectrum is denoted as the input spectrum $S\left(\sigma \right)$, and the ILSF signifies the response of a single detector element to the optical wavelength of monochromatic light on this pixel, set to $f\left(\sigma \right)$; the theoretical reference solar spectrum $S^{\prime }\left(\sigma \right)$ is equivalent to the input spectrum, and the convolution of the ILSF of the spectrometer is [35].

$$S^{\prime }\left(\sigma \right)=S\left(\sigma \right)\ast f\left(\sigma \right)$$

(9)

Here, $\sigma$ is the corresponding wavenumber of the spectrum.

According to the principle of selecting characteristic lines, appropriate solar spectral lines are selected [36], as shown in Figure 8, where the blue curve is the theoretical reference solar spectrum after convolution, and the red curve is the selected reference solar characteristic line.

3.3. Solar Calibration Spectrum Processing

The on-orbit spectral calibration of the GMI payload must first carry out the preprocessing of the interferogram [37]. In contrast with traditional interferometer spectrometers, the GMI payload obtains an interferogram through heterodyne modulation, making it susceptible to various factors such as detector noise, machining, and assembly errors in optical components, as well as background interference. Therefore, the preprocessing of the interferogram includes dark current (DC) noise subtraction, abnormal pixel correction, detector corresponding error correction (uniformity correction), interferogram baseline correction, phase error correction, and Fourier transform (FFT). Considering the rise in abnormal pixels during the GMI payload on-orbit period, the algorithm for detecting and correcting abnormal pixels is enhanced. For trend items influenced by factors such as light source uniformity, optical path stability, and processing and assembly errors during the collection process, the variational mode decomposition (VMD) method is employed, effectively removing trend terms [13]. The on-orbit calibration data processing of the GMI payload is shown in Figure 9. The preprocessed interferogram of the calibration of the first solar observation in the CO₂-1 bands after the GMI payload entered orbit (11 October 2021) is shown in Figure 10.

By using Equation (9), the Kurucz solar spectrum in the CO₂-1 bands is convolved with the ILSF. Figure 11 shows the comparison results between the measured spectrum and the reference spectrum in the CO₂-1 bands and the relative deviation results of the two spectra. This spectrum is derived from processing the GMI payload data following the calibration data processing flowchart shown in Figure 11. The figure shows that the spectral lines of the theoretical solar spectrum and the measured spectrum closely match, although there are variations in the depth of the absorption peak. The relative average deviation of the spectral lines across the entire band is approximately 1.83%. The differences may be attributed to significant distinctions between the on-orbit operating environment, conditions, and laboratory environment during satellite mission execution. Additionally, changes in equipment parameters may contribute to errors when directly applying the ILSF obtained in the laboratory to on-orbit observational data.

4. On-Orbit ILSF Iterative Model Algorithm

After obtaining the processing results of the GMI payload on-orbit solar observation spectrum in Section 3, the laboratory ILSF measurement results cannot be directly applied. To effectively monitor the changes in instrument performance and obtain an accurate spectrum, it is necessary to adjust the ILSF after the on-orbit GMI payload. However, because of the static interferogram technology of the GMI payload and the absence of measuring equipment for monochromatic light, it is not possible to directly measure the ILSF of the instrument. Additionally, the factors influencing the ILSF result from a complex interplay of effects, and it is difficult to accurately derive a mathematical expression for its function directly from the perspective of a physical model. Luo et al. [38] analyzed the modulation efficiency of the interferograms of a spatial heterodyne spectrometer during the engineering implementation process. They focused on factors such as the splitting characteristics of the beam splitter, aberrations of the optical system [39,40] (including the installation accuracy of the collimation system [41] and the exit wavefront error, interferometer component surface shape errors, imaging system defocus, and MTF curve degradation), and the influence of the extended light source, as well as detector sampling density and spacing.

The ILSF determines the degree of modulation of the instrument on incident radiation, that is, the spectral resolution of the instrument and the spectral line energy distribution. The ILSF can be divided into two parts. The first part describes the modulation loss caused by the inherent self-apodization of the instrument. It can be used to calculate the optical thickness of the spectrometer and the FOV. The second part is caused by the uncalibrated and optical phase difference of the instrument, characterized by modulation efficiency amplitude and phase error. The modulation efficiency amplitude characterizes the width of the ILSF, and the phase error quantifies the degree of ILSF symmetry. Therefore, it is effective to correct the ILSF function by constructing models with respect to modulation loss and phase error [42].

The calibration data of the solar spectrum measured by the orbiting high-resolution spectrometer are not affected by aerosols and have the characteristics of independent solar Fraunhofer lines. The stable Kurucz solar spectrum is used as the reference spectrum [43] to establish the ILSF modulation term error and phase error model. By iteratively optimizing the ILSF model parameters, it is possible to achieve the best agreement between the theoretical spectrum and the measured solar spectrum in the CO₂-1 bands.

4.1. Error Function Model

The modulation term error of the instrument is an inherent property of the instrument, and the Gaussian line shape is the curve most used to describe the shape of peaks in a spectrum [44]. The modulation loss and phase error are randomly generated using a Gaussian function, which has the following 1D form:

$$f\left(x\right)=a{e}^{\frac{-{\left(x-\mu \right)}^2}{2{\delta }^2}}$$

(10)

Here, $a$ is the height of the peak; $\mu$ is the central coordinate of the peak; $\delta$ is the width of the bell-shaped curve, also known as the standard deviation; and $x$ is the coordinate position.

According to the above formula, the Gaussian function selected for the simulation generation of modulation loss and phase error curve is as follows:

$$f_{ki}\left(y\right)={\displaystyle \sum _{j=0}^{n-1}{a}_j{e}^{\frac{-{\left(y-\mu \right)}^2}{2{\delta }_j^2}}}$$

(11)

Here, $a$ denotes the error function parameter, $\mu$ is the mean, and ${\delta }^2$ is the variance.

4.2. Construction of an Error Model for the Modulation Term

The expression of the interferogram collected by the theoretical spatial heterodyne spectrometer indicates that $B\left(\sigma \right)$ is the optical radiation intensity and that $x$ is the position of the detector pixel. When $\sigma <0$ and $B\left(\sigma \right)=B\left(-\sigma \right)$, the interferogram can be represented as follows:

$$I\left(x\right)={\displaystyle {\int }_{-\infty }^{\infty }B\left({\sigma }^{\prime }\right)}\cos \left(2\pi {\sigma }^{\prime }u\right)d\left({\sigma }^{\prime }\right)={\displaystyle {\int }_{-\infty }^{\infty }B\left({\sigma }^{\prime }\right)}{e}^{i2\pi {\sigma }^{\prime }u}d\left({\sigma }^{\prime }\right)$$

(12)

When the spectrometer operates in a non-ideal state, it introduces modulation errors to the interferogram. In this case, the interferogram is expressed as follows:

$$I\left(x\right)={\displaystyle {\int }_{-\infty }^{\infty }A\left({\sigma }^{\prime },x\right)B\left({\sigma }^{\prime }\right)}{e}^{i2\pi {\sigma }^{\prime }u}{e}^{i\phi \left({\sigma }^{\prime },x\right)}d\left({\sigma }^{\prime }\right)$$

(13)

Here, $A\left({\sigma }^{\prime },x\right)$ is the modulation loss of the interferogram, and $\phi \left({\sigma }^{\prime },x\right)$ is the phase error.

The interferogram of the monochromatic light modulation term error with frequency is ${\sigma }_0^{\prime }$, which can be expressed as follows:

$$I_o\left(x\right)=A\left({\sigma }_0^{\prime },x\right)B_o\left({\sigma }_0^{\prime }\right)\left(e^{i\left[{\sigma }_0^{\prime }x+\phi \left({\sigma }_0^{\prime },x\right)\right]}+e^{-i\left[{\sigma }_0^{\prime }x+\phi \left({\sigma }_0^{\prime },x\right)\right]}\right)$$

(14)

Since the interferogram has no practical significance in the negative direction, the modulation error interferogram can also be expressed as follows:

$$I_o\left(x\right)=A\left({\sigma }_0^{\prime },x\right)B_o\left({\sigma }_0^{\prime }\right)e^{i\left[{\sigma }_0^{\prime }x+\phi \left({\sigma }_0^{\prime },x\right)\right]}$$

(15)

The theoretical spectrum $B\left({\sigma }_0^{\prime }\right)$ of monochromatic light can be obtained as follows:

$$B\left({\sigma }_0^{\prime }\right)=I_o\left(x\right)\frac{1}{A\left({\sigma }_0^{\prime },x\right)}{e}^{i{\sigma }_0^{\prime }x}{e}^{-i\phi \left({\sigma }_0^{\prime },x\right)}=B_0\left({\sigma }_0^{\prime }\right)\otimes D\left({\sigma }_0^{\prime }\right)$$

(16)

Here, $A\left({\sigma }_0^{\prime },x\right)={\displaystyle \sum _{j=0}^{n-1}{a}_{aj}{e}^{\frac{-{\left(x-{\mu }_{aj}\right)}^2}{2{\delta }_{aj}^2}}}$ is the modulation loss term, $\phi \left({\sigma }_0^{\prime },x\right)={\displaystyle \sum _{j=0}^{n-1}{a}_{pj}{e}^{\frac{-{\left(x-{\mu }_{pj}\right)}^2}{2{\delta }_{pj}^2}}}$ is the modulation term error, $B_o\left({\sigma }_0^{\prime }\right)$ is the measured ILSF in the laboratory, and $D\left({\sigma }_0^{\prime }\right)$ is the Fourier transform of the modulation term error.

4.3. The ILSF Modulation Term Error Correction

A flowchart of the ILSF characterization is presented in Figure 12. Different characterization and modeling algorithms are presented in this section. First, the solar spectrum is measured with the GMI payload to characterize its properties. To produce the reference spectrum, the same radiance can be either modeled or measured at a high resolution so that absorption or emission lines are resolved to their natural shape. Once these two sets of data are obtained, the characterization becomes an iterative optimization process on the ILSF model parameters. Once the distorted spectrum is close to the measured spectrum up to a certain threshold value, the ILSF parameters are stored to feed the correction algorithm.

The main steps are as follows:

(1) Obtain the solar spectrum measured by the GMI payload and amplitude brightness value of the characteristic spectral lines.

(2) Select the high-precision Kurucz solar spectrum of the corresponding band as the reference spectrum.

(3) Acquire the ILSF data measured using laboratory tunable lasers.

(4) Construct a GMI payload instrument modulation term error model to obtain the new ILSF.

(5) Calculate the residual of the measured solar spectrum characteristic lines and the reference spectrum characteristic lines after convolving the ILSF with Kurucz.

(6) Iteratively optimize the modulation term error model parameters.

(7) Calculate the corresponding model parameters $a,\delta ,\mu$ when the evaluation function ${\chi }^2$ is the smallest. Then, obtain the final ILSF data according to Equation (16).

The average relative deviation is used as the evaluation function, as shown in Equation (17):

$${\chi }^2\left(a,\delta ,\mu \right)=\frac{1}{N}{{\displaystyle \sum _{j=1}^N\left[\frac{G\left(j\right)-S\left(i,a,\delta ,\mu \right)}{G\left(j\right)}\right]}}^2$$

(17)

Here, $G\left(j\right)$ is the radiation value of the spectrometer at wavenumber $WN\left(j\right)$, N is the total number of spectral points selected for evaluation, and $S\left(i,a,\delta ,\mu \right)$ is the radiation value of the reference spectrum after convolution with the ILSF at wavenumber $WN\left(j\right)$.

5. Experiment and Discussion

The spectral calibration data selected for analysis were derived from solar observations within the CO₂-1 bands following the deployment of the GMI payload on orbit. The Kurucz solar spectrum’s corresponding band was convolved with the ILSF to create a reference spectrum. Figure 13 shows the solar calibration data obtained after the deployment of the GMI payload on orbit. The theoretical spectrum (the blue curve) and the measured spectrum (the red curve) show a substantial consistency, with relative deviations being large relative to the baseline near the solar Fraunhofer lines, with overall relative deviations in the ±2% range and relative mean deviations of about 0.83%. Six calibration datasets—specifically from 11 October 2021, 24 November 2021, 27 December 2021, 25 January 2022, 24 February 2022, and 26 March 2022—were selected for ILSF calculation and verification.

The error model of the modulation term was iterated, and Figure 14 shows the ILSF measured in the laboratory using a monochromatic laser before the GMI load launch (the blue curve) and the adjusted ILSF obtained using the iterative error function model after the GMI load on orbit (the green curve). The resolution is reduced after the ILSF correction, and the asymmetric side lobe oscillations indicate that the corrected ILSF enhances the consistency between the theoretical solar spectrum and the measured solar spectrum in the corresponding band spectral line.

Figure 15 shows a comparison between the corrected CO₂-1 bands’ measured solar spectrum and the theoretical spectrum after ILSF correction, along with the relative residual distribution. From the comparison, it is evident that the spectra are highly consistent. The relative residual distribution indicates that the overall relative deviation is within the ±2% range, with larger deviations observed in the bands with dense solar Fraunhofer lines. The average relative deviations for the six measured solar spectra are 0.83%, 0.667%, 0.636%, 0.725%, 0.858%, and 0.817%, all of which are around 0.8%. This also demonstrates that the instrument maintains relative stability over this period after ILSF correction. By averaging these six results, it is found that the ILSF accuracy error is 0.756%, with deviations within 1%. This indicates that the ILSF correction method has a good corrective effect.

To further analyze the impact of on-orbit updates and adjustments to the ILSF on the GMI payload’s instrumental performance, the nadir observation spectrum of the CO₂-1 bands on 11 November 2021 was selected. The results are shown in Figure 16. Figure 16a, b compares the simulated and measured spectra before and after the ILSF correction, exhibiting consistent distribution characteristics. Figure 16c shows the distribution of relative residuals, indicating larger residuals in bands with dense absorption peaks. Figure 16d shows that, after the ILSF correction, the average relative deviation between the measured and simulated spectra decreases from 7.049% to 2.106%. The residual values indicate that the discrepancy between the theoretical solar spectrum and the measured solar spectrum improves after updating the ILSF. This demonstrates that the changes in the ILSF caused by the on-orbit environment can be effectively corrected.

According to the algorithm in Section 4, the other three bands are corrected. As shown in the Figure 17, the ILSF oscillation of the O₂ bands is reduced, but not obviously so, and the residuals before and after correction are also relatively close. Figure 18 shows the ILSF correction results for CH₄ bands, which are relatively close to those of the CO₂-1 bands. Figure 19 shows the ILSF correction results for the CO₂-2 bands, where the ILSF changes are more pronounced, with a trend similar to the CO₂-1 and CH₄ bands. The ILSF of CO₂-2 bands is broadened and the oscillation is reduced. However, the correction effect of the residuals before and after the ILSF correction for the CO₂-2 bands is not significant. This may be due to the detector for these bands being greatly affected by temperature, which impacts the solar calibration data, as well as the inapplicability of the radiometric calibration coefficients. Currently, on-orbit absolute radiometric calibration is ongoing, and after its completion, the CO₂-2 bands will be the focus of the analysis.

6. Conclusions

Based on the GMI, this study proposes a method to construct an ILSF error model function using Gaussian line shapes in order to monitor and update the ILSF. This addresses the challenges arising from instrument degradation and performance changes during the on-orbit period, preventing spectra from being measured by tunable lasers or standard gas with absorption characteristics. By combining the two main characteristics (modulation loss and phase error) of the ILSF, an ILSF functional model is constructed. The solar spectrum is not affected by aerosols and has the characteristics of independent solar Fraunhofer lines. The model parameters of the on-orbit ILSF are obtained by iterating the residuals of the measured solar spectrum and the theoretical reference solar spectrum of the spectrometer, and the update and supervision of the ILSF variations are completed to further improve the measurement accuracy of the spectrometer on orbit, thus verifying that the use of Gaussian line shapes to construct the ILSF error function has a good effect.

During the on-orbit period, the monitoring of changes in the updated ILSF provides a basis for subsequent research on the on-orbit parameter changes of the instrument. Additionally, the ILSF correction in this study was conducted under the assumption that the solar diffuser plate of the calibrator was not degraded, without considering the influence of the radiometric calibration results. Subsequent research on on-orbit radiometric calibration will further elaborate on the correction of ILSF variations.