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Simulation evaluation of a single-photon laser methane remote sensor for leakage rate monitoring

Shouzheng Zhu, Shijie Liu, Guoliang Tang, Xin He, Hao Zhou, Senyuan Wang, Shicheng Yang, Pujiang Huang, Wenhang Yang, Bangjian Zhao, Chunlai Li, and Jianyu Wang

Open Access Open Access

Abstract

We propose a novel methane leakage rate remote sensor that combines a single-photon avalanche diode detector with a near-infrared 1653.7 nm low-power laser. The proposed M sequence and triangle wave signal modulation method simultaneously realizes the detection of methane leakage and target point clouds. Innovatively, the sensor’s methane concentration and leakage rate quantification ability were simulated by combining the Gaussian plume diffusion model and the Risley prism. The effects of the prism rotation ratio, wind speed, leakage rate, atmospheric stability (AS), target reflectivity, signal averaging period, and concentration spatial interpolation method on leakage rate are discussed. When plume methane concentrations reduce from 10,000 to 500 ppm·m, the relative concentration bias rise from 1% to 30%, the absolute concentration bias is approximately 100 ppm·m. Two spatial concentration interpolation methods introduced leakage rate bias ranging from 6%-25%. For a low AS, the leakage rate bias under the cubic interpolation method was small (approximately 1.6%). In addition, when the initial leakage rate increased from 100 to 1,000 mg/s, the leakage rate bias was approximately 20% smaller.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Achieving significant methane (CH4) reductions would substantially affect the atmospheric warming potential [1,2]. Methane is emitted from various anthropogenic sources, such as landfills, oil and natural gas systems, agricultural activities, coal mining, stationary and mobile combustion, wastewater treatment, and industrial processes [3]. Estimates of methane emissions are subject to a high degree of uncertainty in these fields. The oil and gas industry is one of the major sources of methane, accounting for an estimated 24% of global anthropogenic emissions, and contributes to energy wastage and uncertainty in climate assessments [4]. Maintaining the security and integrity of a natural gas system is a continuous process involving searching, locating, and repairing leakages. However, meeting the criteria for large-scale deployment in an industry driven by cost reduction and practicality is challenging. To this end, there is a drive across the industry to identify economical and feasible methods for leakage detection and repair (LDAR) operations [5].

The standard optical imaging approach employs a portable optical gas imaging camera that operates in the mid-wave (3-5 µm) to long-wave (7-14 µm) infrared to locate gas leakage [6,7]. However, the radiative background and poor distinguish of the gas significantly affect device performance, altering the signal-to-noise ratio and necessitating expert analysis [8]. Although approaches to gas concentration simulations are being developed, their quantification remains unreliable [9]. Satellites that observe global atmospheric methane concentrations can be applied to detect and quantify high-intensity point sources and characterize emissions at the regional and national levels [10]. However, they cannot meet the spatial resolution and timeliness detection requirements to monitor methane leakage in factories. Moreover, the current satellite-based methane concentration quantification algorithm is immature [11]. Excepting for the above passive techniques, laser absorption spectroscopy has been utilized in laboratories for many years and is now employed in industries with an open laser beam path and environment [12,13]. Tunable diode laser absorption spectroscopy (TDLAS) measures the precise concentration of a specific gas in a laser path by quickly modulating lasers over a target gas absorption line. TDLAS devices perform gas imaging by detecting the diffusive scattering from noncooperative objects in the environment [14,15]. However, they cannot simultaneously detect the distance and gas concentration, lack gas three-dimensional distribution information, and are restricted by the signal strength, resulting in the short measurement distance. Other active remote detection techniques include differential absorption LIDAR (DIAL) and IPDA LIDAR [16,17], which provide high-fidelity quantification and localization but are large, expensive, and nonportable. For example, the NPL's DIAL equipment is installed in an articulated truck and requires skilled operation, posing many logistical difficulties when employed in regular surveys [18]. Moreover, its high cost and size make it unsuitable for continuous monitoring of individual emissions.

Therefore, developing an efficient, convenient, real-time, and high-precision methane industrial leakage remote sensor is crucial for improving methane leakage monitoring [19,20]. Compared to the passive optical scheme, the active laser scheme can quantitatively detect methane leakage effectively. However, achieving concentration detection with a high distance resolution using traditional pulsed IPDA detection is challenging. The higher the distance resolution, the narrower the pulse width, higher the repetition frequency, and lower the pulse energy. Based on the above investigation of the methane quantitative monitoring requirements and the shortcomings of the current quantitative detection methods, we use near-infrared single-photon avalanche diode (SPAD) to reduce the receiving optical size of the system and realize equipment miniaturization. Furthermore, rapid continuous laser wavelength tuning improves the time resolution of gas concentration. Combined with pseudo-random binary sequence (PRBS) modulation, the sensor achieved high-range resolution. The signal modulation method can distinguishes the return signal from the background light and random noise detected by the sensor [21].

Simulations of sensor are performed based on their parameters to reduce errors and analyze their sources [22]. Hu et al. investigated energy monitoring methods for IPDA LIDAR systems and used ground-glass diffusers to reduce the speckle effects during energy monitoring [23]. Zhu et al. analyzed the influence of laser fluctuations on inversion results and investigated a set of systematic errors in detection devices to improve system performance [24]. Ye et al. simulated a gas leakage localization algorithm and verified their feasibility for concentration localization [25]. Thus, the simulation of proposed sensor provides significant support for evaluating the system parameters to guide subsequent experiments and algorithm optimization.

The remaining paper is organized as follows: Section 2 introduces the novel methane-remote system model framework and sensor structures. The gas concentration, leakage rate inversion algorithm and system simulation model are described in Section 3. Furthermore, Section 4 presents the investigation of the quantitative impact of various errors on the leakage rate and concentration, comprehensive analyses of the sensor quantitative ability under different factors. Finally, Section 5 summarizes the performance of the proposed sensor and its ability to determine the methane concentration and leakage rate quantitatively.

2. Theory and sensor system

2.1 Basic system model framework

Similar to IPDA LIDAR, the proposed sensor estimates the path-integrated column concentration (PICC) of methane released into the atmosphere through measuring the back scattered echoes from hard target. Firstly, methane absorption spectroscopy is performed using the approximate MHz TDLAS modulation of continuously sweeping laser [26]. Next, the PRBS modulation and time correlation single photon counting (TSCPC) were performed to discern the target [27,28]. The laser modulation plays a pivotal role in the gas concentration and range measurements, encompassing two modulation modes. One is continuous laser triangle wave sweeping, which alters the laser emission power and wavelength by adjusting the laser current. The relationship between the wavelength and laser current can be written as

$${\lambda _i} = {\lambda _0} + {I_i} \ast {\alpha _1}$$
where λi is the wavelength, i is the number of wavelengths, λ0 is the initial modulation wavelength, Ii is the modulated current, and α1 is the wavelength modulation coefficient. Similarly, the relationship between the emission power and wavelength can be written as
$$P({{\lambda_i}} )= {\lambda _0} \ast {\alpha _2} + {P_0}$$
where P(λi) is the ramp sweeping emission power, P0 denotes the initial modulation power, and α2 represents the power modulation coefficient.

The PRBS owns various types, the Maximal Length Sequence (M sequence) is one of the most common forms, which can effectively ensure superior ranging accuracy and signal-to-noise ratio compared with other coding schemes. The fundamental M-sequence, along with the cross-correlation of an N-bit PRBS, where a[i]∈ [0,1], and its bipolar sequence can be described by a'[i] = 2a[i]-1∈ {-1,1}, exhibits significant values only at zero shifts and near zero for other positions. The M-sequence modulation and continuous laser sweeping signals were superimposed onto the emitted laser. The emitted laser signal can be written as

$${P_l}(i) = P \ast a[i ],\textrm{ }i \in [{1:N} ]$$
where Pl(i) represents the emission power at wavelength i, i denotes the time bin corresponding to the wavelength sequence. The emitted signals with average power P(λi) in different wavelengths are coded with M-sequence a[i]∈ [0,1]. The received signals are dominated by the target types, which encounters a time delay τ and are weakened by the aerosol and gas absorption. The received echo photon counts per bit nλi are given by [29]
$${n_{{\lambda _i}}} = {P_l}({\lambda _i}){\eta _l}{\eta _r}{\eta _q}{T_c}\rho \frac{{{\lambda _i}}}{{hc}}\frac{{{A_r}}}{{\pi {R^2}}}\exp ( - 2OD({\lambda _i}))$$
where Pl(λi) is the initial laser power emitted at various wavelengths, ηl and ηr are the emission and receiving optical efficiency, respectively, ηq is the quantum efficiency, TC is the bit time (chip time) of M-sequence, ρ represents the reflectivity of the target (approximate Lambertian apparatus), λi denotes the sweeping wavelength values, h is the Planck's constant, c is the speed of light, Ar is the receiving optical unit area, R is the detection distance, τ is the flight time, and OD(λi) denotes the total column optical depth in different wavelengths. It can be written as
$$OD({{\lambda_i}} )= \int_0^R {[{\alpha _a}({\lambda _i}) + {\sigma _{{\lambda _i}}}{N_{C{H_4}}}]} \textrm{ }dr$$
where αa (λi) is the aerosol extinction coefficient, σλi denotes the methane absorption cross-section at different wavelengths, and NCH4 is the methane gas molecules along the laser integration path. The photoelectron nb detected due to ambient light is expressed as
$${n_b} = \rho {A_r}{\eta _r}{\eta _q}{T_c}{L_s}\varDelta \lambda \frac{\lambda }{{hc}}\frac{{\theta _r^2}}{4}{e ^{ - \int_o^R {{\alpha _a}({{\lambda_i}} )dr} }}$$
where Ls denotes the Nadir solar spectral radiance, Δλ represents the bandwidth of the optical filter at the detector, and θr is the receiver field-of-view. The photoelectron nd generated by the dark current of the detector is expressed as
$${n_d} = {k_{dc}}{T_c}$$
where kdc denotes the SPAD dark count rate. The total echo photons is expressed as
$${n_s} = {n_{{\lambda _i}}} + {n_b} + {n_d}.$$

2.2 Proposed sensor structure

The structure of the methane remote sensor is illustrated in Fig. 1. The optical modules include a polarizing beam splitter (PBS), Risley prism, focusing lens, filter lens, and SPAD. The electrical modules comprise laser emission unit, temperature, current controller of the laser, signal generation unit, prism motion control unit, and data acquisition and analysis unit. The near-infrared SPAD is sensitive to single photons of light between 1-1.7 µm, making it well-suited for the 1.65 µm methane absorption. A distributed feedback (DFB) continuous-wave laser with a central wavelength of 1653.7 nm was chosen as the dedicated light source. The DFB laser is driven and controlled by the temperature and current control units. The signal generator (Tektronix, AWG 31252) generates the laser triangle sweeping and encoded PRBS amplitude modulation signals. The encoded PRBS signal modulates the emitted laser signal using a commercial electro-optical modulator (iXblue, MXER-LN-10). The modulated laser is collimated to a parallel light beam using a fiber port collimator (Thorlabs). The half-waveplate adjusts the polarization direction of the emitted light to match the PBS. A visible laser is coaligned with the infrared laser through a beam filter to determine the beam position. The optical design employs a single coaxial optical transmit and receive design integrated with a PBS to align the transmit and receive path over long ranges. The polarization selection characteristic of the PBS protects the SPAD from laser reflections inside the device. Additionally, the mechanically controlled Risley prism facilitates rapid scanning of the transmitted laser beam across the detection field. The received signal is captured via the optical path and focused onto the SPAD and processed by photon-counting module. Subsequently, the developed algorithm calculates the target distances and gas concentrations in the field. The primary performance parameters of the system are listed in Table S1.

 figure: Fig. 1.

Fig. 1. Structure of the proposed sensor. Dichroic mirror (combined mirror); EOM, electro-optical modulator; PBS, polarizing beam splitter; SPAD, Single Photon Avalanche Diode.

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3. Simulation processing and methods

3.1 Signal generation and parameters inversion

In the methane sensor, we perform signal generation, signal acquisition, wavelength modulation, and data analysis by the signal generator and photon counting acquisition unit. A high-frequency triangle wave (∼1 MHz) and a M sequence is generated and then applied separately to the laser diode and electro-optical modulator (EOM) for wavelength modulation. The M-sequence chip time Tc of 10 ns, corresponding to a bit rate of f = 100 MHz and range resolution Δd = c/2f = 1.5 m. The length of the M sequence is N = 27-1 = 127, the unambiguous range is d = N*c/2f = 190.5 m. The continuous laser covers the methane absorption line, which is simultaneously encoded by M sequence, the emitted and received laser signal are shown in Fig. S1. The period of M sequence is 127*10 ns = 1.27 µs, corresponding to the frequency of 787.4 kHz. Because the M sequence and laser sweeping were performed simultaneously, the sweep period of the continuous laser was set to 787.4 kHz. During the laser sweeping process, current value is associated with specific wavelengths, and there exists a correspondence between the M-sequence time bin and the current value with respect to the wavelength. When calculating the gas absorption line, the time dimension is converted to the wavelength dimension, i.e., λi = f(τi).

The M sequence possesses inherent autocorrelation properties, the cross-correlation result between the received photon signal n[i] and bipolar sequence described as a'[i] = 2a[i]-1∈ {-1.1} is written as

$${C_i}[\tau /{T_c}] = {n_s} \otimes {a^{\prime}}[i] = \sum\limits_{k = 1}^{N - 1} {{n_s}(k)\cdot } {a^{\prime}}[N + k].$$

The maximum value position of the cross-correlation coefficient is k=τ/Tc, where τ is the signal flight time, the detection distance can be calculated as

$$R = {{c\tau } / 2}.$$

The methane concentration calculation process is illustrated in Fig. 2.

 figure: Fig. 2.

Fig. 2. Simulation flow chart with system parameter modeling, concentration and distance module, and leakage rate calculation.

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Step 1: According to the cross-correlation results of the received signal and bipolar sequence, we obtained the flight time τ and corresponding signal shift bit k. As shown in Fig. 2, the echo signals at the corresponding wavelengths are reduced due to methane absorption. The echo signals are shifted by bit k to the wavelength sequence corresponding to the emission laser, the shift recovery signal is expressed as

$${n_s}\_re(i )= [{{n_s}({k + 1:end} ),{n_s}({1:k} )} ]$$
where ns_re represents the shifted signal, with the original emission wavelength sequence denoted as λi. Each signal maintains a one-to-one correspondence.

Step 2: Original signal extraction - time bin τi and wavelength λi have one-to-one correspondence. The signal is extracted at the corresponding wavelength according to the sampling interval and is expressed as ne(λi).

Step 3: Interpolate wavelength sequence λi and extracted signal ne(λi). The interpolated wavelengths λinterp and signals ninterp(λi) are then baseline-fitted, the baseline-fitted signal is nbase.

Step 4: The corresponding interpolated data at different wavelengths ninterp are divided from the baseline data nbase to obtain the transmittance data Tsignal = ninterp / nbase.

Step 5: Fit λinterp and Tsignal to Lorentz curve - λinterp and Tsignal are one-dimensional data.

Step 6: Lookup table establishment - First, according to the theoretical absorption model, the theoretical methane absorption curve is obtained, and a data search matrix of wavelength, transmittance, and concentration is established.

Lookup Table (LUT) establishment steps:

  • (1) An initial methane concentration sequences were established and the theoretical absorption cross-section lines were calculated according to the atmospheric temperature and pressure data.
  • (2) The ideal methane transmission lines Ts at different concentrations were calculated based on the gas parameters.
  • (3) The sum of squares error (SSE) between Ts and Tsignal is calculated as
    $$SSE = {\sum\limits_{j = 1}^m {({{T_{signal}}(j )- {T_s}(j )} )} ^2}.$$
    m is the number of data points after interpolation.
  • (4) The initial concentration was adjusted such that the residual in Eq. (12) is minimized.
  • (5) The corresponding initial concentration is the ideal concentration.

Step 7: The reversed plume methane concentrations were spatially interpolated, the leakage rate was calculated based on the mass balance algorithm (MBA), and the bias between the theoretical leakage rate and reversed leakage rate was calculated.

3.2 Gas detection simulation model

Based on the sensor equipment, we constructed a methane detection model for a specific scenario, assuming the deployment of the sensor for remote methane leakage detection. As is shown in Fig. 3, the sensor was installed on a high tower to observe methane leakage. The gas concentration and background target range were detected using the received backscatter signal. In this model, the gas plume diffusion model was applied to simulate methane leakage, the plume PICC of z axis is also calculated [30]. And the MBA was used to calculate leakage rate because of it’s convenient and offers superior real-time performance [31–33].

 figure: Fig. 3.

Fig. 3. Schematic diagram of the device observation of methane leakage and the leakage rate calculation for the mass balance algorithm.

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4. Results and analysis

4.1 Gas diffusion characteristic and concentration localization

The Risley prism are used as the spatial point detection scheme of this sensor. The different rotation speed ratios of the two prisms correspond to different numbers and locations of sampling points. The rotation speed of one prism was fixed at 0.9 r/s, and the other was configured with three relative rotation speed ratios (-1.65, -2.2, and -2.7). The distribution characteristics of the systematic sampling points were analyzed under three different rotation speeds and three gas plumes. Furthermore, we aligned the scanning imaging with the location of the leakage source to avoid scanning blind spots. The scanning results at three relative rotation speed ratios for gas plume 1 (leakage rate: 100 mg/s, wind speed: 3 m/s, and AS: F) are shown in Fig. 4 (a-c). Similarly, Fig. 4 (d-f) and (g-i) correspond to the scanning results for gas plumes 2 (leakage rate: 500 mg/s, wind speed: 5 m/s, AS: A) and 3 (leakage rate: 1000 mg/s, wind speed: 1 m/s, AS: D), respectively. The gas plume distribution in both axis directions significantly influences the number of concentration points and values sampled by the Risley prism at the same rotation speed ratio. In gas plume 1, smaller concentration values and narrower concentration distribution in the y-axis increased concentration sample value errors and fewer sampling points. In gas plume 2, the broader y-axis distribution partially compensated for the limited concentration points. Risley prism are better suited for later quantitative calculations of the concentration and leakage rates because of their higher concentration values within gas plume 3 and the wider distribution of sampling points. For the same gas plume, the distribution of scanning points within the plume is not uniform at a rotation speed ratio of -2.2, significantly reducing the number of points within the plume. A broader distribution of scanning points within the gas plume enhanced the quantification of the leakage rate. Therefore, the selection of rotation speed ratio aims to satisfy more scanning points and a larger spatial range of sample points within the plume.

 figure: Fig. 4.

Fig. 4. Risley prism scanning point and gas plume coincidence results. Distribution under different rotation speed ratios for (a-c) gas plume1; (d-f) gas plume 2; (g-i) gas plume 3.

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Later, the concentration values corresponding to the coordinates of the plumes were assigned to the scanning point of the coordinate prism. The x and y-axis coordinates of the scanning points and corresponding concentration values were obtained. The corresponding data matrices were integrated into the sensor system to derive the concentration values and subsequent calculations. In Fig. 5, the concentration values at the prism coordinates corresponding to different air masses and rotation speed ratios are determined and represented using different colors.

 figure: Fig. 5.

Fig. 5. Scanning points corresponding to the concentration of neighboring plumes. Distribution under different rotation speed ratios for (a-c) gas plume 1; (d-f) gas plume 2 (g-i) gas plume 3.

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4.2 System performance evaluation

In Section 4.1, we discuss the coordination and correlation between prisms and gas plumes. The acquired data matrices contain points and concentration values for various rotation speed ratios and plumes. These data matrices were subsequently used for systematic concentration inversion. The emitted laser energy signal of the sensor is shown in Fig. 6 (a). The emitted linear triangular wave energy was modulated by the M sequence, as described in Eq. (3). A gas absorption peak exists in the rising and falling segments of the triangular wave, and two absorption curves exist during one sweep period. The concentrations are expressed in Eq. (4) for the echo signal calculation. As shown in Fig. 6 (b), the on1 and on2 positions correspond to the two peak absorption positions, and the corresponding signal amplitudes decrease, indicating signal attenuation at a certain concentration.

 figure: Fig. 6.

Fig. 6. (a) Emission-coded and absorbed cross-section superimposed signals. (b) Echo signal superimposed on concentration absorption signal.

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The extracted noise-containing gas absorption signals were processed, which included signal interpolation, baseline fitting, filtering, and Lorentzian fitting to obtain the gas transmittance lines for calculating the corresponding concentration values. The results are shown in Fig. S2. The 1000- and 100000-times averaged signals were selected for signal processing to observe the inversion effects of the concentration values. As is shown in Table 1, the relative and absolute errors of the concentration inversion for several concentration values were calculated for different target reflectivities. The concentration values are 10296, 6618, 1885, and 575 ppm·m. As the concentration decreases, the corresponding relative error increases as a whole. At 10296 ppm·m, the error is within 1%, while at 575 ppm·m, the concentration bias exhibits more fluctuations, with a maximum bias of 30.51%. The inversion errors are mostly contained within 5%, reaching a maximum of 7.98% at reduced reflectivity at 1885ppm·m.

Tables Icon

Table 1. Relative and absolute errors of inversion methane concentration at typical target reflectivity

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4.3 Rotation speed error analysis

Previous analysis demonstrated the influence of the rotation speed ratios of the Risley prism on the distribution of the sampling points. The rotation speed causes jitters during scanning, which reduces the scanning accuracy. A rotation speed error with a standard bias of 0.01 r/s was added to each of the Risley prism. The distributions of the sampling points before and after the addition of errors were studied, as shown in Fig. S3. Four rotation speed ratios were chosen, in which the scanning points before and after adding errors at -1.65 and 1.95 ratios are more uniformly distributed across the field of view. For -2.2 and -2.7 rotation ratios, the distribution becomes uniform throughout the field of view after increasing the error compared to that before the error. Furthermore, the distribution characteristics of the scanning points within different plumes before and after adding errors at different rotation speeds were investigated.

Figure 7 shows the points distribution characteristics before and after introducing errors at ratios of -1.65, -1.95, and -2.7. This consistency is observed in the number of scattering points near the leakage source, the distribution of scattering points in the other regions is also uniform. Specifically, at -2.2 rotation ratio, the distribution area of the sampling points increases after introducing the error. The overall distribution of the sampling points broadens and significantly differs from that before the error. Further studies on the reversed gas concentration values before and after the error and quantification of the leakage rate will be conducted.

 figure: Fig. 7.

Fig. 7. Results of the coordinates of the scanning points falling into different gas plumes before and after the prism error. Distribution under different rotation speed ratios for (a-d) gas plume 1; (e-h) gas plume 2; (i-l) gas plume 3.

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We have acquired the coordinates and concentrations of the scanning points for different gas plumes and different rotation speed ratios before and after the error. The acquired concentration values were incorporated into a sensor system for simulation, and the discrepancy between the inverted and theoretical concentrations was examined. Subsequently, the correlation between the concentrations before and after introducing the error and the relative and absolute bias was explored. Concentration correlation results for various conditions are shown in Fig. 8. Because the distribution characteristics at -1.65 and -1.95 are similar, only the ratios of -1.65, -2.2, and -2.7 are discussed. Gas plume 3 has a leakage rate of 1000 mg/s, and the correlation of the concentration values is better, with a mean absolute percentage bias of less than 20%, which is the smallest bias among the three gas plumes. The low leakage rates of plumes 1 and 2 increases the bias. The ratio of -1.65 corresponds to the smallest bias and the best correlation for all three plumes, therefore, its effect is minimized for different plumes. For different plumes, the concentration bias before and after the rotation speed error is approximately 8% or less. In addition, the results of the differences in the concentration values before and after the speed error for different plumes are shown in Fig. 9. The concentration bias of gas plume 1 is within ±100 ppm·m, and the mean concentrations absolute bias is within 50 ppm·m. The jittery range of the bias values of plume 2 increased after the adding speed error, but the average bias values are more consistent. For plume 3, the range of concentration bias before and after introducing the error is relatively consistent with the individual extremes. However, the overall average concentration bias is consistent, and the bias is the smallest compared to the other plumes. The leakage rate significantly affects the concentration bias. The average absolute bias of the concentration inversions from the theoretical values is within 60 ppm·m.

 figure: Fig. 8.

Fig. 8. Correlation and mean absolute percentage error between the simulation concentration values and theoretical concentration values before and after the prisms error. Results under different rotation speed ratios for (a-c) gas plume 1; (d-f) gas plume 2; (g-i) gas plume 3.

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 figure: Fig. 9.

Fig. 9. Concentration absolute error between the simulated and theoretical concentration values before and after prisms error. (a-b) gas plume 1 results. (c-d) gas plume 2 results. (e-f) gas plume 3 results.

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4.4 Leakage rate estimation

We have obtained the coordinates of the scanning points and associated concentrations, which were subsequently used to calculate leakage rate. First, because the scanning points are sparsely distributed, interpolating the spatial gas concentration is crucial for compensating the absence of spatial data points. Two interpolation methods were selected: cubic and Kriging interpolation. The interpolated raw data include the prism coordinates, reversed concentrations of the three plumes, and rotation speed ratios.

The interpolation results are shown in Fig. 10, each subplot includes the results of the concentration distribution mapping obtained using both interpolation methods. In gas plume 1, cubic interpolation reveals high concentration in low-value areas. The characteristics of the two methods in relation to the high-value areas of concentration interpolation are approximate. The interpolation characteristics of the high-concentration areas are more consistent for plumes 2 and 3. Nevertheless, compared to cubic interpolation, Kriging interpolation exhibits small areas of concentration outliers and lower spatial interpolation uniformity. In plumes 1 and 2, persistent high values are observed within the low-value regions in the cubic interpolation. This phenomenon is primarily attributed to concentration outliers in the inversion and low concentration values, resulting in regional variations in the interpolation. Therefore, in leakage rate calculation, the main consideration is the cross-section of the leakage rate calculation within the effective distribution interval, and outliers are ignored. Furthermore, for uniform gas plume conditions, the concentration distributions associated with the different rotation speeds do not align consistently. The distribution of the prism scanning points within the high-value concentration zone near the leakage source primarily determines this discrepancy. The interpolation errors are large when the high-concentration value points are few. Therefore, the prism rotation ratio critically influences the scanning of high-value areas within the plume.

 figure: Fig. 10.

Fig. 10. Cubic and Kriging interpolation of statistical concentration values. (a-c) gas plume 1, ratio: -1.65, -2.2, -2.7. (d-f) gas plume 2, ratio: -1.65, -2.2, -2.7. (g-i) gas plume 3, ratio: -1.65, -2.2, -2.7.

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Leakage rate was calculated using the interpolated concentration mapping derived from the inverted concentration values. The leakage rate was determined by applying the MBA. The plume and rotation ratios were aligned with prior measurements, and comparisons were conducted between the surface reflectivity, signal averaging periods, interpolation method, and impact on quantifying the leakage rate before and after adding prism rotation error. Table 2 and 3 show the leakage rate calculation results for the target reflectivity of 0.2 and 0.4 sr-1 to analyze the effect of target reflectivity on the leakage rate quantification. The deviation in the corresponding leakage rate under different reflectivities is small in plume 1, indicating that under 0.2 and 0.4 sr-1 reflectivity, the system exhibits a relatively small sensitivity to leakage rate quantization and is robust against the influence of reflectivity. This characteristic makes it suitable for accurately detecting leakage rates. Except for individual anomalies, increasing the number of averaging periods reduces the leakage rate error by 1% to 8% for high-averaging periods. In addition, the difference in the quantification of the leakage rate of plume 1 before and after adding the error of prisms is small, and both can effectively quantify the leakage rate of plume 1. Regarding the interpolation method, the error in the cubic interpolation method is relatively small for plume 1. The difference in the leakage rates between the two interpolation methods is approximately 6%.

Tables Icon

Table 2. Relative error of inversion leakage rate before and after prism error, gas plume 1, reflectivity: 0.2

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Table 3. Relative error of inversion leakage rate before and after prism error, gas plume 1, reflectivity: 0.4

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Table 4 presents the results of the leakage rate quantification for gas plume 2. The overall leakage rate error increases, with the error calculated from the signals averaged over 100,000 periods, ranging from approximately 8% to 20%. The difference between the two averaged periods ranges from approximately 5% to 30%, and increasing the number of periods reduces the leakage rate error significantly. The cubic method results in a smaller bias in the calculated leakage rate, with only a 2% difference between the two methods, which is consistent with the results for gas plume 1. Table 5 shows the quantitative results for the leakage rate of gas plume 3. The bias in the leakage rate obtained by Kriging interpolation is smaller than that calculated by the cubic method, with a difference of 5%-25% between the two methods. The Kriging interpolation calculates a lower bias in the leakage rate and is better suited for situations involving low leakage rates. However, owing to the lower leakage rates and higher wind speeds, the overall bias in the leakage rate is greater than that of the previous two gas plumes, with bias ranging from 30%-40%.

Tables Icon

Table 4. Relative error of inversion leakage rate before and after prism error, gas plume 2, reflectivity: 0.4

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Table 5. Relative error of inversion leakage rate before and after prism error, gas plume 3, reflectivity: 0.4

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The effects of wind speed, leakage rate, and AS on the quantification of the leakage rate under different interpolation methods were individually analyzed, with 100000 signals averaging periods. Table S2 illustrates the impact of the wind speed on the leakage rate. The absolute bias gradually increases with increasing wind speed for the Kriging interpolation method, whereas the opposite trend is observed for cubic interpolation. The Kriging method is suitable for cases with relatively large and concentrated concentration values at the leakage point, whereas cubic interpolation results in larger errors in such situations. Furthermore, the cubic interpolation method is suitable for cases with high wind speeds, whereas the Kriging method is more appropriate for scenarios with low wind speeds. Specifically, the Kriging interpolation method exhibited a bias of approximately 2% at low wind speeds (1 m/s), whereas the cubic interpolation method showed a bias of approximately 5% at high wind speeds (5 m/s). The results of the bias of the simulated leakage rates from the theoretical values are listed in Table S3. The Kriging interpolation method exhibits a lower bias for the low theoretical leakage rates of 100 and 500 mg/s, whereas the cubic interpolation method exhibited a small bias of approximately 5% when the theoretical leakage rate is 1000 mg/s. Table S4 shows the bias of the simulated leakage rates from the theoretical values for different AS. For the low AS, the leakage rate bias under cubic interpolation method is small, approximately 1.6%. As AS increases, the bias of the leakage rate calculated by the Kriging and cubic interpolation methods are similar. Moreover, the cubic and Kriging methods fits low- and high-AS conditions, respectively.

5. Conclusions

We propose a novel miniaturized sensor for continuous methane monitoring and quantitative calculation of the leakage rate that overcomes the challenges of traditional large-scale methane monitoring systems. We conducted a leakage rate assessment by incorporating various gas plumes and utilizing Risley prism to validate the methane leakage monitoring capabilities of the sensor. In addition, we investigated the effects of the prism rotation speed error, wind speed, leakage rate, AS, and concentration spatial interpolation algorithm on the sensor’s performance. We find that methane concentrations ranged from 10,000 to 500 ppm·m, with relative concentration bias varying from 1% to 30% and absolute concentration error of approximately 100 ppm·m. The concentration bias before and after the rotation speed error is within 8% and the average absolute bias is within 60 ppm·m.

Furthermore, we selected three different plumes to examine the effects of different interpolation methods, signal averaging periods, and rotation speeds of the prisms on leakage rate errors. The cubic and Kriging interpolation methods can measure small bias in the calculated leakage rate, and is suited for low and high AS cases, respectively. Overall, the leakage rate bias varies from 1%-40% across the three plumes, and the two concentration interpolation methods introduce errors ranging from 6%-25%. The effects of wind speed, leakage rate, and AS on the inverted concentrations and leakage rates are discussed separately. As wind speed increases, the concentration bias increases and the concentration correlation deteriorates; the Kriging and cubic interpolation methods have a bias of approximately 2% and 5% for the low (1 m/s) and high wind speed (5 m/s), respectively. In addition, an increase in the initial leakage rate can reduce the inversion error from 100 to 1000 mg/s, the bias is reduced by approximately 20%. For a low AS, the leakage rate bias under the cubic interpolation method was small (approximately 1.6%). This suggests that the cubic interpolation method fits scenarios with low AS.

The sensor system was built in the laboratory to perform some of the experimental tests. We validated the system parameters, providing a reference for subsequent experimental tests and algorithmic research. In addition, the proposed methane remote detection method provides a reference for other researchers and gas-leakage industrial application manufacturers. In the future, our group will conduct further experimental tests and joint observations using passive gas detection equipment to better validate and optimize the methane quantification parameters of the sensor. In addition, further research on data filtering, concentration error control algorithms, and optimization of prism rotation speed ratios will be conducted to promote quantitative methane precision.

Funding

National Defense Pre-Research Foundation of China during the 14th Five-Year Plan Period (D040107); 173 Key Projects of Basic Research (2021-JCJO-ZD-025-11); High Resolution Imaging Spectrometer (HRIS) technology and equipment (2023C03012); Development of a high-precision CO2 online analyzer based on cavity-enhanced absorption spectroscopy (B02006C019019); Research on active detection technology of extreme sensitivity gas based on mid-infrared band (B02006C019001).

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.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Supplementary materials of “Simulation evaluation of a single-photon laser methane remote sensor for leakage rate monitoring”

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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Figures (10)
Fig. 1.
Fig. 1. Structure of the proposed sensor. Dichroic mirror (combined mirror); EOM, electro-optical modulator; PBS, polarizing beam splitter; SPAD, Single Photon Avalanche Diode.

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Fig. 2.
Fig. 2. Simulation flow chart with system parameter modeling, concentration and distance module, and leakage rate calculation.

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Fig. 3.
Fig. 3. Schematic diagram of the device observation of methane leakage and the leakage rate calculation for the mass balance algorithm.

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Fig. 4.
Fig. 4. Risley prism scanning point and gas plume coincidence results. Distribution under different rotation speed ratios for (a-c) gas plume1; (d-f) gas plume 2; (g-i) gas plume 3.

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Fig. 5.
Fig. 5. Scanning points corresponding to the concentration of neighboring plumes. Distribution under different rotation speed ratios for (a-c) gas plume 1; (d-f) gas plume 2 (g-i) gas plume 3.

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Fig. 6.
Fig. 6. (a) Emission-coded and absorbed cross-section superimposed signals. (b) Echo signal superimposed on concentration absorption signal.

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Fig. 7.
Fig. 7. Results of the coordinates of the scanning points falling into different gas plumes before and after the prism error. Distribution under different rotation speed ratios for (a-d) gas plume 1; (e-h) gas plume 2; (i-l) gas plume 3.

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Fig. 8.
Fig. 8. Correlation and mean absolute percentage error between the simulation concentration values and theoretical concentration values before and after the prisms error. Results under different rotation speed ratios for (a-c) gas plume 1; (d-f) gas plume 2; (g-i) gas plume 3.

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Fig. 9.
Fig. 9. Concentration absolute error between the simulated and theoretical concentration values before and after prisms error. (a-b) gas plume 1 results. (c-d) gas plume 2 results. (e-f) gas plume 3 results.

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Fig. 10.
Fig. 10. Cubic and Kriging interpolation of statistical concentration values. (a-c) gas plume 1, ratio: -1.65, -2.2, -2.7. (d-f) gas plume 2, ratio: -1.65, -2.2, -2.7. (g-i) gas plume 3, ratio: -1.65, -2.2, -2.7.

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Tables (5)

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Table 1. Relative and absolute errors of inversion methane concentration at typical target reflectivity

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Table 2. Relative error of inversion leakage rate before and after prism error, gas plume 1, reflectivity: 0.2

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Table 3. Relative error of inversion leakage rate before and after prism error, gas plume 1, reflectivity: 0.4

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Table 4. Relative error of inversion leakage rate before and after prism error, gas plume 2, reflectivity: 0.4

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Table 5. Relative error of inversion leakage rate before and after prism error, gas plume 3, reflectivity: 0.4

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Equations (12)

Equations on this page are rendered with MathJax. Learn more.

λ i = λ 0 + I i ∗ α 1
P ( λ i ) = λ 0 ∗ α 2 + P 0
P l ( i ) = P ∗ a [ i ] ,   i ∈ [ 1 : N ]
n λ i = P l ( λ i ) η l η r η q T c ρ λ i h c A r π R 2 exp ⁡ ( − 2 O D ( λ i ) )
O D ( λ i ) = ∫ 0 R [ α a ( λ i ) + σ λ i N C H 4 ]   d r
n b = ρ A r η r η q T c L s Δ λ λ h c θ r 2 4 e − ∫ o R α a ( λ i ) d r
n d = k d c T c
n s = n λ i + n b + n d .
C i [ τ / T c ] = n s ⊗ a ′ [ i ] = ∑ k = 1 N − 1 n s ( k ) ⋅ a ′ [ N + k ] .
R = c τ / 2 .
n s _ r e ( i ) = [ n s ( k + 1 : e n d ) , n s ( 1 : k ) ]
S S E = ∑ j = 1 m ( T s i g n a l ( j ) − T s ( j ) ) 2 .
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