A temperature and concentration reconstruction method based on normalized second harmonic linear model

Abstract

The application provides a temperature and concentration reconstruction method based on a normalized second harmonic linear model, belongs to the technical field of tunable diode laser absorption spectroscopy, and is used for two-dimensional temperature and concentration field reconstruction. The reconstruction system comprises a laser control and generation module, a fiber beam splitter, a sensor, a Mach-Zehnder interferometer, a photodetector, a data acquisition system and a computer. The reconstruction method comprises the following steps: after laser is split, one path is connected to the Mach-Zehnder interferometer and is received by the photodetector, the remaining paths pass through the to-be-detected area after being expanded and are received by a photodetector plate, and normalized second harmonic spectra of transmitted light intensities on each light path are obtained; a normalized second harmonic base matrix is calculated according to discrete temperature and concentration; a linear reconstruction model is established according to the sensor light path arrangement, and an iterative algorithm is used to obtain two-dimensional temperature and concentration distribution. The application effectively utilizes the normalized second harmonic spectrum, increases the number of independent equations, and improves the speed and accuracy of two-dimensional temperature and concentration distribution reconstruction.

Description

Technical Field

[0001] This invention proposes a method for reconstructing temperature and concentration based on a normalized second harmonic linear model, belonging to the field of tunable diode laser absorption spectroscopy technology. This method is used for the simultaneous reconstruction of temperature and concentration distributions in a two-dimensional combustion field. Background Technology

[0002] Tunable Diode Laser Absorption Spectroscopy (TDLAS) is widely used for measuring combustion field temperature and typical molecular concentrations due to its advantages such as non-interference with the flow field, fast response speed, high measurement accuracy, and simple system. TDLAS has two typical implementation methods: Direct Absorption Spectroscopy (DAS) and Wavelength Modulation Spectroscopy (WMS). The DAS method is simple and direct, directly obtaining the absorption spectrum, but it requires a baseline of light intensity when there is no absorption. In field experiments, harsh environmental factors such as high temperature, high pressure, and strong vibration pose significant challenges to obtaining the light baseline, limiting the application of the DAS method in the field. The WMS method uses intensity modulation, which can effectively improve the measurement signal-to-noise ratio and reduce the influence of environmental interference, making it more promising for complex measurement situations compared to the DAS method.

[0003] The Wavelength Modulation (WMS) method uses digital phase-locked loop and low-pass filtering to demodulate the transmitted light intensity, obtaining a harmonic signal containing parameters of the gas being measured. However, the harmonic signal also contains parameters such as the gain of the photoelectric detection system and the laser intensity. Therefore, the WMS method requires prior calibration using known gases, which is very difficult for high-temperature gases. In 2006, Hejie Li et al. published a paper titled "Extension of wavelength-modulation spectroscopy to large modulation depth for diode laser absorption measurements in high-pressure gases" in *Applied Optics*, Vol. 45, No. 5, pp. 1052-1061, which pointed out that under low absorption conditions, normalizing the second harmonic signal using the first harmonic signal can eliminate parameters such as laser intensity and photoelectric gain, eliminating the need to consider experimental parameters in the simulation. This is known as the calibration-free WMS technique. Based on this, using two lasers of different wavelengths, the temperature and concentration of the gas being measured can be determined by colorimetric cancellation of voltage division and path length. In 2007, GB Rieker et al. published a paper titled "A diode laser sensor for rapid, sensitive measurements of gas temperature and watervapour concentration at high temperatures and pressures" in *Measurement Science and Technology*, Volume 18, Issue 5, pp. 1195-1204. This paper used a simulated table of second harmonic ratios with two different spectral characteristics to retrieve the measured second harmonic ratios to invert the temperature, achieving an error of less than 1% in the measured high-pressure shock tube temperature. To avoid the inability to accurately obtain harmonic values ​​at a fixed wavelength due to laser wavelength drift over time, using the wavelength-scanning (WMS) method to obtain the harmonic spectrum is crucial. However, the laser wavenumber model in the wavelength-scanning WMS method is difficult to describe using a single-frequency Fourier series.

[0004] In 2013, K Sun et al. published a paper entitled "Analysis of calibration-free wavelength-scanned wavelength modulation spectroscopy for practical gas sensing using tunable diode lasers" in Volume 24, Issue 12 of Measurement Science and Technology. This paper proposed a new method for analyzing wavelength-scanned wavelength modulation (WMS). The method uses measured laser intensity to simulate emitted laser intensity and obtains harmonic signals using digital phase-locked loop and low-pass filtering, thus avoiding the need for Fourier expansion in the classical WMS model. In 2014, Christopher S. Goldenstein et al. published a paper titled "Fitting of calibration-free scanned-wavelength-modulation spectroscopy spectra for determination of gas properties and absorption lineshapes" in *Applied Optics*, Volume 53, Issue 3, pp. 356-367. Building upon this, the paper inferred parameters such as gas temperature and concentration by least-squares fitting of simulated and measured scanned harmonic spectra. However, single-path measurements only yield average temperatures and concentrations along the path, while actual combustion fields are typically non-uniformly distributed.

[0005] By arranging multiple measurement optical paths and combining them with computer tomography (CT) technology, TDLAS technology can effectively measure the gas parameter distribution in a two-dimensional combustion field. The combination of DAS and CT is widely used because the absorption spectrum in DAS is integrable along the optical path. In 2020, the applicant's authorized invention patent, "A Two-Dimensional Temperature and Concentration Reconstruction System and Method Based on Histogram Information" (ZL 201910782513.5), utilized path temperature and concentration histograms to achieve two-dimensional temperature and concentration reconstruction based on the DAS method. However, the DAS method struggles to achieve high-precision imaging in practical measurements with low signal-to-noise ratios. While the WMS method has advantages in complex environments, the lack of suitable integrable parameters in WMS limits its application. In 2014, WeiWei Cai et al. published a paper titled "Multiplexed absorption tomography with calibration-free wavelength modulation spectroscopy" in Applied Physics Letters, Volume 104, Issue 15. This paper proposed combining nonlinear tomography with calibration-free wavelength modulation spectroscopy (WMS). The method uses simulated annealing algorithm to solve for the combustion field temperature and concentration distribution when the simulated and measured normalized second harmonic signals are closest. This successfully achieved two-dimensional imaging based on the WMS method. However, the complex and extensive calculations made the method too time-consuming. In 2020, Wenshuai Zhao et al. published a paper titled "AWMS Based TDLAS Tomographic System for Distribution Retrievals of Both Gas Concentration and Temperature in Dynamic Flames" in IEEE Sensors Journal, Volume 20, Issue 8, pp. 4179-4188. The paper pointed out that under weak absorption, the normalized second harmonic peak value is a parameter that can be integrated along the path. By combining it with CT technology, two-dimensional temperature and gas concentration distribution can be reconstructed. Compared with nonlinear models, this effectively reduces the computation. However, in a single optical path, only the peak value of the normalized second harmonic is used as projection data for reconstruction, resulting in a serious underdeterminacy in the reconstruction problem.

[0006] Based on the above background, this paper proposes a method for reconstructing temperature and concentration using a normalized second harmonic linear model. By establishing a linear calculation model of the normalized second harmonic spectrum on a single optical path using pre-discrete temperature and concentration pairs, and organically integrating it with CT technology, a two-dimensional imaging method for temperature and gas molecule concentration distribution in a combustion field is achieved. This method has the following advantages: it inherits the anti-interference advantages of the WMS method, making it more suitable for measurements in harsh environments; it effectively utilizes the shape information of the normalized second harmonic spectrum, increasing the number of independent equations, which effectively reduces the underdeterminacy of the reconstruction problem, improves the reconstruction quality of temperature and concentration distribution, and the established normalized second harmonic linear model requires less computation and has a faster reconstruction speed. Summary of the Invention

[0007] To address the temperature and concentration distribution of a two-dimensional combustion field, this paper proposes a method for reconstructing temperature and concentration based on a normalized second harmonic linear model. This method establishes a linear model of the normalized second harmonic spectrum under non-uniform distribution, enabling simultaneous reconstruction of two-dimensional temperature and concentration distribution.

[0008] The reconstruction system includes a laser control and generation module, an optical fiber beam splitter, sensors, a Mach-Zehnder interferometer, a photodetector, a data acquisition system, and a computer; the sensors consist of a laser input module, a Powell prism, and a photodetector plate. The reconstruction method first measures and calculates the normalized second harmonic signal on the sensor's optical path, then numerically calculates the normalized second harmonic basis matrix based on the reconstruction range of the measured temperature and concentration fields, and finally uses a linear reconstruction model to achieve two-dimensional temperature and concentration field imaging. Specifically, it includes the following steps:

[0009] Step 1: Obtain the normalized second harmonic signal of the laser transmitted light intensity on the sensor optical path; the laser control and generation module uses a low-frequency scanning signal superimposed with high-frequency sinusoidal modulation as the injection current to drive the laser. The output laser passes through an optical fiber beam splitter, one path is connected to a Mach-Zehnder interferometer, and then to a photodetector. After acquisition, the change of laser output wavenumber v(t) over time is calculated.

[0010]

[0011] in, Where is the laser center wavenumber, a is the modulation depth, and ω = 2πf is the modulation angular frequency;

[0012] The model for the incident laser intensity I0(t) is as follows:

[0013]

[0014] in, wave number The laser intensity at the location, i0 and i2 are the linear and nonlinear intensity modulation amplitudes, respectively. and This corresponds to the phase shift between intensity modulation and frequency modulation.

[0015] The remaining optical paths of the fiber optic beam splitter are connected to the laser input modules of the sensor, and expanded by a Powell prism. After passing through the target gas, the laser beam is received by the photoelectric detection plate on the sensor. For any laser optical path of the sensor, after the incident laser passes through the target gas, the transmission intensity I... t (t) decays to

[0016]

[0017] Where τ is the transmission coefficient and α is the absorptivity;

[0018] For the case of low absorption of the target gas, i.e., α(v) < 0.05, the transmittance can be approximately calculated as follows:

[0019]

[0020] The transmission coefficient τ[v(t)], being periodic, can be expanded into a Fourier series.

[0021]

[0022] Among them, Fourier coefficients The calculation formula is:

[0023]

[0024] Using digital phase-locked loop and low-pass filter to extract transmitted light intensity I t Extracting the first and second harmonic signals from (t): Multiply the transmitted light intensity by the reference signals cos(ωt) and sin(ωt) respectively, and then obtain the X component X of the first harmonic signal 1f through low-pass filtering. 1f and Y component Y 1f Then, the amplitude R of the first harmonic signal is calculated. 1f Similarly, the transmitted light intensity is multiplied by the reference signals cos(2ωt) and sin(2ωt) respectively, and then the X component of the second harmonic signal 2f is obtained by low-pass filtering. 2f and Y component Y 2f The amplitude R of the second harmonic signal was calculated. 2f For linear intensity modulation with a phase shift of π, i.e. And when i2 = 0, the normalized second harmonic signal S 2f / 1f Represented as

[0025]

[0026] Step 2: Construct the normalized second harmonic basis matrix; based on a rough estimate of the temperature and concentration range of the target gas, discretize the gas parameters into M temperature and concentration pairs: {T1,X1},{T2,X2},…,{T… M ,X M The absorption rates β1(v), β2(v), ..., β of these M sets of gas parameters per unit path were numerically calculated. M (v); The total absorptivity along the laser path can be expressed as the sum of the absorptions of the gas in each state.

[0027]

[0028] Where L1, L2, ..., L M Let M be the path lengths occupied by each of the M gas states; the Fourier coefficients of the M gas parameters under unit length absorption can be obtained from equation (7). According to equation (9), the Fourier coefficients of the total path absorption can be obtained.

[0029]

[0030] Based on the measured incident laser intensity I0(t) and laser output wavenumber v(t), the transmitted light intensity per unit length after absorption for the M gas parameters is obtained using equation (3). Then, the normalized second harmonic signal is obtained using the digital phase-locked loop and low-pass filtering method in step one. The normalized second harmonic signal corresponding to the i-th gas parameter is expressed as R. 2f / 1f,i ;

[0031] Based on the linear relationship in equation (8) and equation (10), the normalized second harmonic signal on the total path satisfies the normalized second harmonic signal of the unit absorption of the M gas parameters.

[0032] Normalized second harmonic R of M gas parameters 2f / 1f,i The harmonic basis matrix R is formed by arranging the elements row by row. M×K Where K is the number of data points of the normalized second harmonic signal;

[0033] Step 3: Establishing a linear reconstruction model and imaging two-dimensional temperature and concentration distribution; measuring the transmitted light intensity of the Q laser paths on the sensor and calculating the corresponding normalized second harmonic signals according to Step 1, arranging them row by row to obtain matrix S. Q×K ;

[0034] The two-dimensional region of interest is discretized into D grids, and the sensitivity matrix W is calculated based on the laser optical path layout of the sensor. Q×D The element w in the i-th row and j-th column i,j It is the optical path length of the i-th laser path of the sensor passing through the j-th grid;

[0035] Using a 0-1 binary matrix Y D×M The matrix Y characterizes the temperature and concentration distribution of the target gas field. D×M Each row represents a grid, containing one element (1) and all other elements (0). The temperature and concentration values ​​at that location are obtained based on the column containing element 1.

[0036]

[0037] In the formula, and These represent the temperature and concentration distributions of the D grids, respectively, with the superscript R indicating the reconstructed distribution;

[0038] Histogram matrix L Q×M Let M represent the path lengths occupied by the M gas states on the Q laser paths. Equation (11) satisfied by the Q laser paths can be written in matrix form as follows:

[0039] S Q×K =L Q×M ·R M×K (13)

[0040] According to the definition of the sensitivity matrix, we have

[0041] W Q×D ·Y D×M =L Q×M (14)

[0042] Furthermore, the histograms along each laser path satisfy the prior condition that the sum of their lengths equals the total length of the laser path, subject to the following constraints.

[0043] L Q×M ·e M×1 =W Q×D ·e D×1 (15)

[0044] In the formula, column vector e M×1 and e D×1 All elements in the array are 1;

[0045] The comprehensive models (13), (14), and (15) have reconstruction models.

[0046] W Q×D ·Y D×M ·(R M×K e M×1 )=(S Q×K W Q×D ·e D×1 (16)

[0047] The matrix Y in equation (16) is solved using the Simultaneous Algebraic Reconstruction Technique (SART). D×M Then, the two-dimensional temperature and concentration distribution are calculated according to equation (12). Attached Figure Description

[0048] Figure 1 This is a flowchart of the reconstruction method.

[0049] Figure 2 The reconstruction system structure diagram consists of the following parts: laser control and generation module (101), fiber beam splitter (102), Powell prism (103), photoelectric detection board (104), Mach-Zehnder interferometer (105), photoelectric detector (106), data acquisition system (107) and computer (108).

[0050] Figure 3 The two-dimensional temperature distribution (a) and concentration distribution (b) of the flame obtained from the simulation under a 120×120 grid are used as the objects for simulation reconstruction.

[0051] Figure 4 The temperature distribution (a) and concentration distribution (b) are obtained by simulation reconstruction under a 40×40 mesh. Detailed Implementation

[0052] The invention will be further illustrated below with reference to examples.

[0053] This example uses a regular heptagonal sensor, with 12 photoelectric detection plates evenly distributed along each side. A center wavenumber of 7185.56 cm⁻¹ was selected. -1 and 7444.34cm -1 The H2O absorption spectrum was obtained, and the laser wavenumber model was based on a low-frequency scanning sawtooth wave superimposed with a high-frequency sinusoidal modulation. For the given temperature and concentration distribution in the simulation, the normalized second harmonic signal along all light rays on the sensor was theoretically calculated to simulate the experimental measurement and calculation results. Then, the established linear model was used to reconstruct the two-dimensional temperature and concentration distribution; the reconstruction process is as follows... Figure 1 As shown, it includes the following steps:

[0054] Step 1: Theoretically calculate the normalized second harmonic signal of the reconstructed object along all light rays under the current sensor settings. The regular heptagonal sensor structure used in this example is as follows: Figure 2As shown in the heptagonal structure, the light source located at the seven vertices of the heptagon is expanded into a fan-shaped beam by a Powell prism. The divergence angle of the Powell prism is approximately 90 degrees, so a single laser beam can illuminate 44 detectors directly opposite the light source. The entire sensor has a total of 308 light rays. The selected absorption spectral line has a center wavenumber of 7185.56 cm⁻¹. -1 and 7444.34cm -1 The wavenumber models were all based on 1kHz scanning sawtooth waves superimposed with 100kHz sinusoidal modulation, with a modulation depth of 0.1. The wavenumber v(t) was calculated based on the Mach-Zehnder interferometer signal. The linear intensity modulation amplitude of the laser input light intensity was 0.1, the phase shift was π, and the measured value was I0.

[0055] The bimodal temperature and concentration distributions were selected as the objects for simulation reconstruction. The temperature and concentration distributions are as follows: Figure 3 As shown in (a) and (b), when calculating the projected absorptivity, the two-dimensional test field is uniformly divided into a 120×120 grid, assuming that each grid has a uniform temperature and gas concentration. The heptagonal sensor contains 6092 grids, and the absorption spectrum per unit length within each grid is calculated. Then, based on the 120×120 grid division, the matrix of the lengths of 308 laser rays passing through each grid is calculated. This matrix is ​​multiplied by the matrix composed of the absorption spectra per unit length within all grids to obtain the projected absorptivity α along all rays. 308×4000 .

[0056] The transmitted light intensity after the laser passes through the grid on each optical path is calculated based on the projected absorptivity and the laser input intensity I0. The transmitted light intensity of the i-th optical path is...

[0057] I t =I0·exp(-α) i (17)

[0058] The transmitted light intensity of each optical path is amplified by lock-in and low-pass filtered to obtain the first and second harmonic signals. Dividing these signals yields the normalized second harmonic spectrum for each optical path. Subtracting the normalized second harmonic spectrum outside the region of interest yields the projection measurement data for the reconstructed region. 136 sampling points near the peak of each spectral line are selected, and the two spectral lines form a measurement data matrix S for 308 optical paths. 308×272 .

[0059] Step 2: Set the temperature and concentration values ​​for reconstruction and calculate the normalized second harmonic basis matrix. Based on the reconstruction range of temperature and concentration, select 8 temperature values ​​(300K, 471K, 643K, 814K, 986K, 1157K, 1329K, and 1500K) and 8 concentration values ​​(0.002, 0.0231, 0.0443, 0.0654, 0.0866, 0.1077, 0.1289, and 0.15) for reconstruction. Generally, the temperature and concentration distributions in the combustion field are correlated; higher temperatures result in higher product concentrations, and lower temperatures result in lower product concentrations. Eight temperature-concentration combinations were generated using a one-to-one correspondence: (300K, 0.002), (471K, 0.0231), (643K, 0.0443), (814K, 0.0654), (986K, 0.0866), (1157K, 0.1077), (1329K, 0.1289), and (1500K, 0.15). The wavenumber points for both spectral lines were selected at 2000 points each, for a total of 4000 points. The absorbance β per unit length was calculated for each temperature-concentration combination. 8×4000 .

[0060] The transmitted light intensity is calculated based on the unit length absorptivity and laser input intensity for each of the eight combinations. Lock-in amplification and low-pass filtering are performed on each transmitted light intensity to obtain the first and second harmonic signals for each of the eight combinations, thus generating the normalized second harmonic spectrum for the eight combinations. This forms the normalized second harmonic basis matrix. Similarly, 136 points near the peak of each spectral line are selected, and the basis matrix R is obtained from the two spectral lines. 8×272 .

[0061] Step 3: Establish a linear reconstruction model and use an iterative algorithm to solve for the two-dimensional temperature and concentration distribution. During reconstruction, the two-dimensional field to be measured is uniformly divided into a 40×40 grid, with 1020 grid points within the region of interest. Calculate the matrix W representing the length of 308 laser rays passing through 1020 grid points under the 40×40 grid division. 308×1020 A two-dimensional linear reconstruction model of temperature and concentration distribution was obtained.

[0062] W 308×1020 ·Y 1020×8 ·(R 8×272 e 8×1 )=(S 308×272 W 308×1020 ·e 1020×1 (18)

[0063] In the formula e 8×1 and e 1020×1 Both are column vectors with all elements equal to 1, Y 1020×8 It is the matrix to be reconstructed.

[0064] For Y 1020×8 Each column is solved using the SART algorithm to obtain Y. 1020×8 The two-dimensional temperature distribution is then obtained by multiplying it with the discrete value matrix of temperature and concentration. and concentration distribution

[0065] The two-dimensional temperature and concentration distributions within the region of interest of the field to be measured are obtained by rearranging the corresponding positions, as shown below. Figure 4 As shown in (a) and (b).

[0066] The above description of the present invention and its embodiments is not limited thereto, and the accompanying drawings are only one embodiment of the present invention. Any structure or embodiment similar to this technical solution designed without departing from the spirit of the present invention shall fall within the protection scope of the present invention.

Claims

1. A method for reconstructing temperature and concentration based on a normalized second harmonic linear model, the reconstruction system comprising a laser control and generation module, an optical fiber beam splitter, a sensor, a Mach-Zehnder interferometer, a photodetector, a data acquisition system, and a computer; wherein the sensor consists of a laser input module, a Powell prism, and a photodetector plate; the reconstruction method first measures and calculates the normalized second harmonic signal of the laser transmitted light intensity on the sensor optical path, then constructs the normalized second harmonic basis matrix according to the reconstruction range of the measured temperature and concentration field, and finally uses the linear reconstruction model to realize two-dimensional temperature and concentration field imaging; Specifically, the following steps are included: Step 1: Obtain the normalized second harmonic signal of the laser transmitted light intensity on the sensor optical path; the laser control and generation module uses a low-frequency scanning signal superimposed with high-frequency sinusoidal modulation as the injection current to drive the laser. The output laser passes through an optical fiber beam splitter, one path is connected to a Mach-Zehnder interferometer, and then to a photodetector. After acquisition, the change of laser output wavenumber v(t) over time is calculated. in, Where is the laser center wavenumber, a is the modulation depth, and ω = 2πf is the modulation angular frequency; The model for the incident laser intensity I0(t) is as follows: in, wave number The laser intensity at the location, i0 and i2 are the linear and nonlinear intensity modulation amplitudes, respectively. and This corresponds to the phase shift between intensity modulation and frequency modulation. The remaining optical paths of the fiber optic beam splitter are connected to the laser input modules of the sensor, and expanded by a Powell prism. After passing through the target gas, the laser beam is received by the photoelectric detection plate on the sensor. For any laser optical path of the sensor, after the incident laser passes through the target gas, the transmission intensity I... t (t) decays to Where τ is the transmission coefficient and α is the absorptivity; For the case of low absorption of the target gas, i.e., α(v) < 0.05, the transmittance can be approximately calculated as follows: The transmission coefficient τ[v(t)], being periodic, can be expanded into a Fourier series. Among them, Fourier coefficients The calculation formula is: Using digital phase-locked loop and low-pass filter to extract transmitted light intensity I t Extracting the first and second harmonic signals from (t): Multiply the transmitted light intensity by the reference signals cos(ωt) and sin(ωt) respectively, and then obtain the X component X of the first harmonic signal 1f through low-pass filtering. 1f and Y component Y 1f Then, the amplitude R of the first harmonic signal is calculated. 1f Similarly, the transmitted light intensity is multiplied by the reference signals cos(2ωt) and sin(2ωt) respectively, and then the X component of the second harmonic signal 2f is obtained by low-pass filtering. 2f and Y component Y 2f The amplitude R of the second harmonic signal was calculated. 2f For linear intensity modulation with a phase shift of π, i.e. And when i2 = 0, the normalized second harmonic signal S 2f / 1f Represented as Step 2: Construct the normalized second harmonic basis matrix; based on a rough estimate of the temperature and concentration range of the target gas, discretize the gas parameters into M temperature and concentration pairs: {T1,X1},{T2,X2},…,{T… M ,X M The absorption rates β1(v), β2(v), ..., β of these M sets of gas parameters per unit path were numerically calculated. M (v); The total absorptivity along the laser path can be expressed as the sum of the absorptions of the gas in each state. Where L1, L2, ..., L M Let M be the path lengths occupied by each of the M gas states; the Fourier coefficients of the M gas parameters under unit length absorption can be obtained from equation (7). According to equation (9), the Fourier coefficients of the total path absorption can be obtained. Based on the measured incident laser intensity I0(t) and laser output wavenumber v(t), the transmitted light intensity per unit length after absorption for the M gas parameters is obtained using equation (3). Then, the normalized second harmonic signal is obtained using the digital phase-locked loop and low-pass filtering method in step one. The normalized second harmonic signal corresponding to the i-th gas parameter is expressed as R. 2f / 1f,i ; Based on the linear relationship in equation (8) and equation (10), the normalized second harmonic signal on the total path satisfies the normalized second harmonic signal of the unit absorption of the M gas parameters. Normalized second harmonic R of M gas parameters 2f / 1f,i The harmonic basis matrix R is formed by arranging the elements row by row. M×K Where K is the number of data points of the normalized second harmonic signal; Step 3: Establishing a linear reconstruction model and imaging two-dimensional temperature and concentration distribution; measuring the transmitted light intensity of the Q laser paths on the sensor and calculating the corresponding normalized second harmonic signals according to Step 1, arranging them row by row to obtain matrix S. Q×K ; The two-dimensional region of interest is discretized into D grids, and the sensitivity matrix W is calculated based on the laser optical path layout of the sensor. Q×D The element w in the i-th row and j-th column i,j It is the optical path length of the i-th laser path of the sensor passing through the j-th grid; Using a 0-1 binary matrix Y D×M The matrix Y characterizes the temperature and concentration distribution of the target gas field. D×M Each row represents a grid, containing one element (1) and all other elements (0). The temperature and concentration values ​​at that location are obtained based on the column containing element 1. In the formula, and These represent the temperature and concentration distributions of the D grids, respectively, with the superscript R indicating the reconstructed distribution; Histogram matrix L Q×M Let M represent the path lengths occupied by the M gas states on the Q laser paths. Equation (11) satisfied by the Q laser paths can be written in matrix form as follows: S Q×K =L Q×M ·R M×K , (13) According to the definition of the sensitivity matrix, we have W Q×D ·Y D×M =L Q×M (14) Furthermore, the histograms along each laser path satisfy the prior condition that the sum of their lengths equals the total length of the laser path, subject to the following constraints. L Q×M ·have been M×1 =W Q×D ·have been D×1 , (15) In the formula, column vector e M×1 and e D×1 All elements in the array are 1; The comprehensive models (13), (14), and (15) have reconstruction models. W Q×D ·Y D×M ·(R M×K e M×1 )=(S Q×K W Q×D ·e D×1 ), (16) The matrix Y in equation (16) is solved using the Simultaneous Algebraic Reconstruction Technique (SART). D×M Then, the two-dimensional temperature and concentration distribution are calculated according to equation (12).

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