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Abstract
Interference data obtained by a spatial heterodyne spectrometer (SHS) is subject to various error factors and suffers from complex phase distortion. Traditional phase correction methods, such as the Amplitude, Merzt, and Forman methods, extract phase distortion in the spectral domain and correct it, which cannot effectively correct spatial phase distortion. Through theoretical derivation and numerical simulation, the spatial phase distortion is firstly determined and corrected in the interference domain. The frequency-dependent phase distortion is then extracted in the spectral domain and corrected. This novel phase distortion correction method named the phase decomposition method was applied to the in-orbit interference data of Greenhouse gases Monitoring Instrument-II (GMI-II). Compared with traditional phase correction methods, the root-mean-square error of the spectrum corrected using the phase decomposition method is reduced by 81.37%.
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1. Introduction
It has become a global consensus to reduce carbon emissions in response to global warming caused by the increase of greenhouse gases such as CO2 and CH4. Passive remote sensing, which uses spectral detection technology to establish a sound carbon monitoring system, is one of the main technologies of greenhouse gas monitoring. The three main technologies for passive remote sensing of greenhouse gases include Michelson interference spectroscopy (MI), grating spectroscopy (GS), and spatial heterodyne spectroscopy (SHS), whose main technical characteristics are shown in Table 1 [1–4].
Table 1. Comparison of spectral detection techniques
Compared with traditional GS and MI, SHS with the advantages of hyperspectrum, high luminous flux, high stability, and small volume and weight, has significant application value in space remote sensing detection, quantitative spectral detection, and other fields [5–9]. Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, developed two Greenhouse gases Monitoring Instruments (GMI) based on SHS for spaceborne payload, successfully launched in 2018 and 2021, known as GMI-I and GMI-II, respectively [10,11]. GMI has a near-infrared detection channel (centred at 0.765 µm, known as the O2 channel) and three short-wave infrared detection channels (centred at 1.58, 1.65, and 2.05 µm, known as CO2-1, CH4, and CO2-2 channels, respectively). The O2 channel adopts a symmetric sampling structure to achieve 0.6 cm-1 spectral resolution providing information on the surface pressure, clouds, and aerosols. CO2-1 and CH4 channels measure essential data regarding the weak absorption bands of CO2 and CH4, providing information on the CO2 and CH4 column concentration. The CO2-2 channel is not sensitive to CO2 column concentration but can be used to correct the effect of cloud and aerosol scattering features on CO2 concentration. These three short-wave infrared channels use an asymmetric sampling structure to increase the maximum optical path difference for improving spectral resolution. The main performance parameters of GMI-II are shown in Table 2.
Table 2. Main performance indicators of GMI-II
Due to the nonuniform deviation of optical elements, such as nonuniform refraction and optical thickness of beam splitter, lens, filter, etc., reading error of electronic elements, and sampling position error, the interference data obtained by SHS have complex phase distortion [12,13]. As shown on the left in Fig. 1, noticeable distortion exists in the interferogram of 760.875 nm monochromatic light obtained by GMI-II in the laboratory. Fourier transform (FT) is directly performed without phase correction on the interference data with distortion on the left of Fig. 1. The recovered spectrum without phase correction has poor symmetry, and even two peaks appear, as shown in the right panel. A highly accurate phase correction is necessary to improve spectral quality.
Fig. 1. Left: 760.875 nm monochromatic interferogram obtained by GMI-II O2 channel, with noticeable distortion due to complex phase distortion. Right: reconstructed spectrum from the interferogram on the left without phase correction, and the magnified image shows the details at the spectral peaks.
Currently, the typical phase correction methods for interference data include Amplitude, Merzt [14], and Forman methods [15]. The Amplitude method requires the collection of symmetric interference data and the modulus calculation in the spectral domain, which will bring the noise of the imaginary part of the spectrum into the correction spectrum resulting in a low SNR. The Forman method realizes phase correction of interferogram in the spatial domain, and there is a convolution operation that requires a large amount of computation [16]. The Merzt method corrects the phase distortion in the spectral domain. For correcting asymmetric interference data, the asymmetric window function has poor apodization, which will reduce the spectral resolution, so the correction effect of Merzt is inferior to that of the Forman method [17]. Additionally, the three traditional phase correction methods extract the phase distortion in the spectral domain, which cannot accurately correct the spatial position-related phase distortion in SHS, so the correction effect of these methods is limited.
To correct the complex phase in SHS, Englert convolved the uncorrected spectrum with the FT of phase distortion at each sampling point extracted from each monochromatic interferogram [18]. Liu et al. used the image distortion model to correct the radial distortion of the interferogram, which can only correct the phase distortion caused by the imaging distortion in SHS [19].
Based on the theory of SHS, this paper analyzes the mechanism of complex phase distortion in SHS, summarizes the deficiency of traditional phase correction methods, and proposes a new phase correction method called the phase decomposition method, which is helpful to improve the accuracy of the correction spectrum. The phase decomposition method is based on a finite number of phase distortion data at a single wavelength, from which two inherent phase distortions are extracted: spatial phase distortion and phase shift. The spatial phase distortion is first corrected in the interference domain, and then the frequency-dependent phase distortion is extracted in the spectral domain and corrected. The novel phase correction method and the three traditional phase correction methods are applied to the in-orbit observation data of GMI-II onboard the Chinese GaoFen-5 satellite-II. The correction result verifies the superiority of the phase decomposition method that is helpful for the high-precision hyperspectral detection of GMI-II.
The structure of this paper is as follows: the second chapter introduces the technical principle of SHS and the traditional phase correction methods, the third chapter expounds the theory of the phase decomposition method, the fourth chapter analyzes the correction results of the in-orbit data of GMI-II, and finally the conclusion is drawn in the fifth chapter.
2. Analysis of phase distortion in SHS
2.1 Principle of SHS
SHS replaces the planar mirrors in MI with a pair of slanted blazing gratings, as shown in Fig. 2. After passing through the collimating system L1, the incident light $B(\sigma )$ 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, two beams of light form a tilt angle ±γ with the optical axis respectively, and 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 2γ, so there are different optical path differences (OPD) at different sampling positions in the spectral dimension direction (the direction of the arrow X in Fig. 2), forming interference fringes. There is an FT relationship between the intensity information of the interference fringes obtained by SHS and the incident spectral information, as shown in Eq. (1).
where, x represents the OPD of the sampling point; $f(\sigma )$ is the spatial frequency, which is linearly related to wavenumber σ and can be obtained by spectral calibration [20,21].Fig. 2. Schematic diagram of SHS. The SHS system consists of a filter, the collimating system L1, a beam splitter, a pair of gratings, the detector optical system L2/L3, and an image detector.
The original interference data obtained by SHS contain various errors, such as phase distortion, blind element point, the nonuniform response of the optical system, etc. Phase distortion is a crucial error that cannot be ignored and affects the accuracy of the recovery spectrum. In order to recover the effective spectral information of the target, phase correction and other data processing work are necessary. Figure 3 shows the spectral reconstruction process of GMI. Firstly, interference data is subjected to detector correction, including dark current deduction, blind element correction, and detector non-uniformity correction [22,23]. Then a flat field correction is carried out, and the baseline is removed [24,25]. After phase correction, apodization is performed to obtain the corrected interference data [26]. The recovered spectra are then obtained by Fourier transform. Finally, spectral calibration and radiometric calibration are performed to obtain accurate spectral information of the target [27].
Fig. 3. Spectral reconstruction process of GMI. Before phase correction, interference data should be preprocessed, including detector correction, flat field correction and baseline correction. After phase correction, the interference data is cut toe first, and then the Fourier transform is used to obtain the recovery spectrum, and finally, the correction spectrum is obtained by spectral calibration and radiometric calibration
2.2 Extraction of phase distortion in SHS
Unlike the traditional MI, SHS imagines the interference data at different OPD of the target to the plane array detector simultaneously. Each interference sampling point corresponds to a small part of the optical system, as shown in the red transparent area in Fig. 2. Therefore, there are more factors affecting phase distortion in SHS, mainly existing in the following three aspects:
- (1) Non-uniformity of system components. For example, the non-uniformity in the grating grooves changes the fringe frequency, the non-uniformity in refractive index and optical thickness of optical components (e.g., beam splitter, lens, filter, etc.) case phase dispersion, electronic devices (e.g., amplifiers, etc.) lead to phase delays, etc.
- (2) The imaging distortion of the detector optical system causes a change in the fringe frequency on the detector.
- (3) Assembly errors, such as grating tilt and detector shift, introduce phase distortion.
The interference data in SHS is related to the incident light frequency σ and the spatial position x of the sampling point. When the incident light is monochromatic, the phase distortion at each sampling point is denoted as $\varphi (\sigma ,x)$, and the interference expression is
Remove the direct current component in Eq. (2) and convert it into the complex form:
Performing the FT on Eq. (3), filtering out the value of the negative wavenumber in the spectral domain, and then performing an inverse FT, Eq. (3) becomes
The data $I(x)$ becomes plural. According to Eq. (4), the phase distortion $\varphi (\sigma ,x)$ of each interference sampling point at a single wavelength can be obtained.
2.3 Analysis of traditional phase correction methods
Polychromatic light interference with phase distortion $\varphi (\sigma ,x)$ is expressed as
For correcting complex phase distortion $\varphi (\sigma ,x)$ in SHS, Englert proposed to multiply the phase distortion data ${e^{ - i\varphi (\sigma ,x)}}$ directly on both sides of Eq. (6), and then perform FT to get the correction spectrum:
According to Eq. (7), Englert's phase correction method requires obtaining the phase distortion $\varphi (\sigma ,x)$ at each single wavelength according to Eq. (5) or fitting the phase distortion of all wavenumbers using a finite number of monochromatic light phase distortion data based on the assumption that phase distortion slowly changes with frequency [28]. These two operations require a large amount of monochromatic data and a stable instrument response to wavelength. The current phase correction mainly adopts the phase correction methods of the traditional time-modulated interferometer, such as Amplitude, Merzt, and Forman methods.
To simplify the analysis, let $\varphi (\sigma ,x) = \varphi (\sigma ) + \varphi (x)$, then Eq. (6) becomes
FT on both sides of Eq. (8) yields
Since the traditional Amplitude, Merzt and Forman methods do not consider the spatial phase distortion $\varphi (x)$ and assume phase distortion only increases the phase spectrum $\varphi ^{\prime}(\sigma )$, but does not change the amplitude spectrum, i.e., $B(\sigma )\textrm{ = }B^{\prime}(\sigma )$ in Eq. (9). Based on this assumption, these phase correction methods directly solve $\varphi ^{\prime}(\sigma )$ in the spectral domain and correct it. The corrected amplitude spectrum $B^{\prime}(\sigma )$ is then taken as the final correction spectrum, limiting the correction effect. Therefore, correcting the spatial phase distortion is necessary to improve the correction accuracy $\varphi (x)$.
3. Correction principle of the phase decomposition method
3.1 Decomposition of phase distortion
The decomposition process of phase distortion is analyzed using the O2 channel of GMI-II as an example. According to Eq. (5), the phase distortion $\varphi (\sigma ,x)$ at different wavelengths is extracted from single-wavelength spectral calibration data, as shown in Fig. 4.
Fig. 4. Phase distortion data $\varphi (\sigma ,x)$ at different single wavelengths. In the spectrum calibration experiment, the O2 channel of GMI-II took 18 monochromatic interferograms in the ground laboratory. According to Eq. (5), 18 groups of phase distortion $\varphi (\sigma ,x)$ of each sampling point at a single wavelength are extracted from these monochromatic light interferograms.
Figure 4 shows that the extracted phase distortion correlates significantly with the wavenumber σ and the spatial position x of the sampling point. The complex phase distortion can be decomposed to extract two inherent phase distortions in the system: (1) the phase shift distortion $\varphi (\sigma )$ that is only related to the frequency and (2) the spatial phase distortion $\varphi (x)$ that is independent of the frequency. That is, phase distortion $\varphi (\sigma ,x)$ can be expressed as:
where, $\varphi ^{\prime}(\sigma ,x)$ is a weak residual distortion term that cannot be decomposed.3.1.1 Phase shift $\varphi (\sigma )$
There is a fixed sampling deviation error $\Delta x$ between the actual and theoretical zero OPD points because of alignment errors. This offset of the zero OPD point produces a linear phase distortion, defined as a phase shift $\varphi (\sigma )$:
Equation (11) shows that the phase shift $\varphi (\sigma )$ is only related to the wavenumber σ of monochromatic light and has nothing to do with the spatial position x of the sampling point.
A piece of bilateral interference data is intercepted near the zero OPD, and the phase shift $\varphi (\sigma )$ of each monochromatic optical interference data in Fig. 4 are calculated according to (12):
In Fig. 5, the blue asterisk marks the phase shift $\varphi (\sigma )$ extracted from the O2 channel at different monochromatic lights, and the solid red line is the linear fitting result of $\varphi (\sigma )$. The fitting residual of phase shift (black curve) is minimal and controlled within ±0.02 rad, which indirectly verifies the effectiveness of the phase shift extraction algorithm.
Fig. 5. The linear fitting result of extracted phase shift. The blue asterisk marks the phase shift $\varphi (\sigma )$ extracted from the O2 channel of GMI-II at different monochromatic lights, the solid red line is the linear fitting of $\varphi (\sigma )$, and the black curve is the fitting residual.
3.1.2 Spatial phase distortion $\varphi (x)$
For SHS, each interference sampling point is a small part of the optical system. The actual interference data of each sampling point and the theoretical interference data are biased because of the non-uniformity of the optical system. Therefore, there is a frequency-independent phase distortion component at each sampling point, defined as the spatial phase distortion $\varphi (x)$.
The phase shift $\varphi (\sigma )$ in Fig. 5 is subtracted from the phase distortion $\varphi (\sigma ,x)$ in Fig. 4, whose result is shown in Fig. 6. The phase distortion after deducting the phase shift is significantly correlated with sampling points, but poorly correlated with the frequency. Therefore, the mean fitting of phase distortion data in Fig. 6 is carried out to obtain the spatial phase distortion $\varphi (x)$, as shown by the thick red solid line in Fig. 6.
Fig. 6. Extraction result of spatial phase distortion $\varphi (x)$. The thin solid line represents the phase distortion after deducting the phase shift under different wavenumbers, and the thick red solid line is the spatial phase distortion $\varphi (x)$ of the mean fitting.
3.1.3 Residual phase distortion $\varphi ^{\prime}(\sigma ,x)$
In the above analysis, the phase distortion term independent of spatial position, i.e., phase shift $\varphi (\sigma )$, and phase distortion term independent of frequency, i.e., spatial phase distortion $\varphi (x)$, are extracted from the complex phase distortion. In the actual optical system, there is residual unresolvable phase distortion that is related to the frequency and spatial position, defined as the residual phase distortion $\varphi ^{\prime}(\sigma ,x)\textrm{ = }\varphi (\sigma ,x)\textrm{ - }\varphi (\sigma )\textrm{ - }\varphi (x)$.
As shown in Fig. 7, the residual phase distortion is not significantly correlated with frequency and spatial position. 99% of residual phase distortion is controlled within ±0.15 rad.
Fig. 7. Residual phase distortion $\varphi ^{\prime}(\sigma ,x)$ that cannot be further decomposed. The left figure shows the residual phase distortion at each sampling point for different monochromatic lights, and the right figure shows the histogram of the data in the left figure.
3.2 Principle of correction of phase distortion
The interference expression of polychromatic light with phase distortion is
Multiply ${e^{ - i\varphi (x)}}$ directly on the original interferogram to obtain:
- (2) Correction of frequency-dependent phase distortion $\varphi (\sigma )$ and $\varphi ^{\prime}(\sigma ,x)$
Because the residual phase distortion $\varphi ^{\prime}(\sigma ,x)$ is controlled within ±0.15 rad and is not significantly correlated with spatial position, its effect on the amplitude spectrum can be neglected, and only the phase spectrum is affected, as shown in Eq. (15).
The linear phase shift $\varphi (\sigma )$ and residual phase distortion can be regarded as a frequency-dependent phase distortion $\varphi ^{\prime}(\sigma )$ that is extracted from the spectrum of the interferogram after the correction of spatial phase distortion:
The traditional Forman or Merzt methods can be used for phase correction of $\varphi ^{\prime}(\sigma )$. These correction methods are too well established to be described in detail here. The complete correction process for the phase decomposition method is shown in Fig. 8.
Fig. 8. Phase decomposition method correction process. Before phase correction, spatial phase distortion is extracted from monochromatic light interference data at different wavelengths. In the phase correction process, the spatial phase distortion is first corrected in the interferometric domain, and then the frequency-dependent phase distortion is corrected using the Merzt and Forman methods to obtain the phase correction spectrum.
For the O2 channel with a symmetrical sampling structure to achieve 0.6 cm-1 spectral detection capability, the Merzt method is used to correct the frequency-dependent phase distortion. The other three infrared channels, which are the main channels for CO2 and CH4 detection, use the asymmetric sampling design to increase the maximum OPD for improving spectral resolution [29]. If the asymmetric interference data are corrected by the Merzt method, the toe-cutting effect is poor, and the spectral resolution will be reduced. Therefore, the Forman method corrects frequency-dependent phase distortion in the CO2-1, CH4, and CO2-2 channels.
4. Analysis of correction results of GMI-II observation data
Phase correction analysis is performed on the atmospheric observation data of GMI-II, which is derived from the observation data of GF-5 satellite-II on 29 October 2021.
The O2 channel interference data of GMI-II were used to gradually analyze the phase decomposition method's correction process. Firstly, the spatial phase distortion $\varphi (x)$ is corrected, and the interference diagram before and after phase correction is shown in Fig. 9. It can be noticed that the interference data before the correction have absolute equivalent maximum and minimum values. However, after correction $\varphi (x)$, the minima are suppressed, the maximum value increase and the symmetry of the interference data is greatly improved. The improvement of corrected interference data in Fig. 9 fully demonstrates the correctness of spatial phase distortion extraction and the necessity of correction.
Fig. 9. Correction effect of spatial phase distortion. The blue solid line is the original interference data before phase distortion. The red dotted line is the interference data after correcting spatial phase distortion.
Continue to correct the phase distortion associated with the frequency, and the corrected interference data is shown in Fig. 10. The traditional Merzt method and the phase decomposition method proposed in this paper can effectively solve the asymmetric problem of the original interference data. Moreover, the maximum and minimum values of interference data corrected by the two methods overlap. Nevertheless, the fluctuation in the interference data is entirely different, which inevitably leads to differences in the spectrum of recovery in Fig. 11.
Fig. 10. Interference data after correction of phase distortion. The blue solid line is the interference data corrected by the traditional Merzt method, and the red dotted line is the interference data corrected by the phase decomposition method.
Fig. 11. Comparison of the corrected spectrum with the theoretical spectrum. (a) Comparison of the corrected spectrum with the spectrum corrected by phase decomposition method; (b) Comparison of the corrected spectrum with the spectrum corrected by traditional Merzt method; (c) Comparison of the corrected spectrum with the spectrum corrected by traditional Forman method; (d) Comparison of the corrected spectrum with the spectrum corrected by traditional amplitude method.
Taking root-mean-square error (RMSE) of the corrected spectrum and theoretical spectrum as the evaluation index of corrected effect:
where, M is the number of spectral channels.The RMSE of the correction spectrum of the Forman, Merzt, Amplitude, and phase decomposition methods are 5.603 × 10−8, 5.660 × 10−8, 5.660 × 10−8 and 1.044 × 10−8 respectively. The RMSE of the spectrum corrected using the phase decomposition method is significantly smaller than that of the traditional phase correction methods. The coefficient R in Eq. (18) represents the improvement of spectral quality corrected using the phase decomposition method. The phase decomposition method can improve the spectral quality by up to 81.37%.
where, RMSEtra is the RMSE corresponding to the best correction spectrum of the three traditional phase correction methods, and RMSEde is the RMSE of the correction spectrum of the phase decomposition method.Figure 12 shows a comparison between the correction spectrum and the theoretical spectrum of the other three observation channels. The correction spectrum RMSE of the four phase correction methods for each GMI-II channel are listed in Table 3. According to Table 3, compared with the traditional phase correction methods, spectral quality corrected using the phase decomposition method is significantly improved, which proves the superiority of the phase decomposition method.
Fig. 12. Comparison of the theoretical and correction spectrum for each GMI-II channel: (a) CO2-1 channel; (b) CH4 channel; (c) CO2-2 channel. In each subgraph, the correction spectrum of the phase decomposition method was compared with that of the Forman method that has the best phase correction effect among the three traditional phase correction methods.
Table 3. RMSE of the correction spectrum for each GMI-II channel
It is worth noting that the phase decomposition method shows a qualitative improvement in the correction spectrum of the O2 channel, but a slight improvement in the corrected spectral quality of the CO2-1 channel. The essential difference between the phase decomposition and traditional phase correction methods is that the spatial phase distortion is extracted and corrected separately. Therefore, the relationship between the spatial phase distortion of the four channels and the correction effect is analyzed.
Figure 13 shows the monochromatic light interferogram and the extracted spatial phase distortion $\varphi (x)$ of GMI-II channels. The mean square error (MSE) of phase distortion in the O2, CO2-1, CH4, and CO2-2 channels is 3.753, 0.00506, 0.854, and 0.6837, respectively. It can be found that the spatial distortion of the O2 channel is the largest and most complex, so that the phase decomposition method can improve the correction spectrum quality best. The spatial distortion of the CO2-1 channel is smaller than that of other channels, and the corresponding correction effect is limited.
Fig. 13. The extracted spatial phase distortion of the four channels of GMI-II. (a) O2 channel: the MSE of the spatial phase distortion is 3.753. (b) CO2-1 channel: the MSE of the spatial phase distortion is 0.0506. (c) CH4 channel: the MSE of the spatial phase distortion is 0.8574. (d) CO2-2 channel: the MSE of the spatial phase distortion is 0.6837.
Further analysis revealed that the improvement R of the phase decomposition method to the correction spectrum is positively correlated with the MSE of spatial phase distortion, as shown in Fig. 14. This conclusion thoroughly verifies the necessity of the spatial phase distortion correction and reflects the superiority of the phase decomposition correction method.
Fig. 14. The relationship between the spectral improvement of the phase decomposition method with spatial phase distortion $\varphi (x)$. “*” in the figure is the actual data, and the red curve is the fitting curve. In the absence of spatial phase distortion, the phase decomposition method is no different from the traditional phase correction methods, so the fitting curve will pass through the point (0,0).
5. Conclusion
SHS is a spatial modulated interferometric spectroscopy whose interferogram distortion is easily affected by the non-uniformity of the optical system. Traditional phase correction methods cannot effectively correct the spatial phase distortion, and the phase correction effect is limited. In this paper, the spatial phase distortion is extracted from the phase distortion data of monochromatic light and corrected separately, and then the frequency-dependent phase distortion is corrected to achieve high precision correction of complex phase distortion in SHS.
The correction results of GMI-II's in-orbit observation data show that the spectral quality of the phase decomposition method proposed in this paper can be improved by up to 81.37% compared with traditional phase correction methods, which thoroughly verifies the superiority of the phase decomposition method. The spectrum correction results show that independent correction of spatial phase distortion is necessary to improve the quality of the spectrum reconstructed from interferograms. In the case of large spatial phase distortion, the phase decomposition method has absolute advantages over the traditional phase correction method.
Funding
National Natural Science Foundation of China (41975033, 61975212); National Key Research and Development Program of China (2021YFB3901000, 2021YFB3901004); Key Research Program of the Chinese Academy of Sciences (JCPYJJ-22010); HFIPS Director’s Fund (YZJJ202210-TS).
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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