# Mapping of CH4 atmospheric plumes from point sources by open-path TDLAS and AI-assisted tomographic reconstruction: numerical proof of concept

Claire Birot, Gautier Vilmart, Aurélien Plyer, Frédéric Champagnat, Christophe Brossard, Grégoire Broquet, Pramod Kumar, Myriam Raybaut

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Claire Birot, Gautier Vilmart, Aurélien Plyer, Frédéric Champagnat, Christophe Brossard, et al. Mapping of CH4 atmospheric plumes from point sources by open-path TDLAS and AI-assisted tomographic reconstruction: numerical proof of concept. SPIE Sensors + Imaging 2025, Sep 2025, Madrid, Spain. 10.1117/12.3069656 . hal-05391712

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# Mapping of CH4 atmospheric plumes from point sources by open-path TDLAS and AI-assisted tomographic reconstruction: numerical proof of concept

Claire Birot<sup>a</sup>, Gautier Vilmart<sup>a</sup>, Aurélien Plyer<sup>b</sup>, Frédéric Champagnat<sup>b</sup>, Christophe Brossard<sup>a</sup>, Grégoire Broquet<sup>c</sup>, Pramod Kumar<sup>c</sup>, and Myriam Raybaut<sup>a</sup>

<sup>a</sup>DPHY, ONERA, Université Paris-Saclay, 91120, Palaiseau, France

<sup>b</sup>DTIS, ONERA, Université Paris-Saclay, 91120, Palaiseau, France

<sup>c</sup>Laboratoire des Sciences du Climat et de l'Environnement (LSCE/IPSL), CEA-CNRS-UVSQ, Université Paris Saclay, 91191 Gif-sur-Yvette, France

## ABSTRACT

Optical diagnostics, such as Tunable Diode Laser Absorption Spectroscopy (TDLAS), are commonly used to measure concentration in a non-intrusive way. One possible application is the detection and quantification of gas into the atmosphere, such as CH<sub>4</sub> leaks from industrial sites and facilities. However, the TDLAS technique does not allow the retrieval of the spatial distribution of the corresponding atmospheric plume. With the aim to recover an atmospheric concentration map of the corresponding atmospheric plume, we study the feasibility and relevance of developing a tomography reconstruction algorithm. The principle is to use multiple laser beam paths in a plane, in order to probe the gas absorption. We encode the prior information about the plume shape into a neural network. Tomography is studied on synthetic data generated from Gaussian plume models and Large Eddy Simulations (LES).

**Keywords:** Absorption spectroscopy, tomographic algorithm, autoencoder

## 1. INTRODUCTION

Methane (CH<sub>4</sub>) is the second most important greenhouse gas contribution to climate change following carbon dioxide. Its global potential is much higher than that of carbon dioxide (CO<sub>2</sub>), at 28 times over a 100-year period.<sup>1</sup> Industrial sites are a common source of methane emissions. Monitoring industrial sites is thus necessary to assess local environmental impact and evaluate solutions to reduce its effects. In particular, the detection of leaks is of primary concern. Many gas detection sensors exist, based on different fields of physics such as optics, calorimetry, pyro-electricity, electrochemistry and semiconductors.<sup>1</sup> Spectroscopic methods, in particular, offer a high sensitivity and both high repetition rates and broad dynamic ranges. Spectroscopic measurements are commonly made using in situ sensors based on absorption spectrometry on static stations or mobile systems.<sup>2</sup> Several absorption-based techniques have been implemented, such as Fourier Transform Infrared spectroscopy (FTIR),<sup>3</sup> Cavity Ring-Down Spectroscopy, (CRDS, e.g., Picarro systems),<sup>4</sup> or Dual Comb Spectroscopy (DCS).<sup>5</sup> In presence of a leak, it is usually desirable to detect its point source. Because the measurements are local, an additional step is needed to convert them into maps that can pinpoint the source of the leak. One possibility is to use a tomographic reconstruction. Different tomographic algorithm can be used. For instance, a Bayesian algorithm as been used for the reconstruction and quantification of methane, carbon dioxide emissions and has been applied in various methane or carbon dioxide measurement campaigns<sup>4,5</sup>.

Here, we study the theoretical feasibility of such monitoring by implementing the TDLAT technique (*Tunable Diode Laser Absorption Tomography*), which is based on TDLAS (*Tunable Diode Laser Absorption Spectroscopy*) measurements combined with a tomographic reconstruction algorithm. TDLAS is a well-known absorption

<footnote>Further author information: (Send correspondence to Claire Birot)</footnote>

<footnote>Claire Birot: E-mail: claire.birot@onera.fr, Telephone: +33 1 80 38 61 93</footnote>

<footnote>Gautier Vilmart: E-mail: gautier.vilmart@onera.fr</footnote>
technique widely used to measure the concentration of gas species quantitatively. In contrast to using FTIR spectrometer or a Picarro system, which are limited to local measurements, TDLAS can cover a wide spatial region using freely propagating laser beams, thereby simplifying, in some cases, the design and deployment of the experimental setup. However, the resulting spectra, which are path-integrated along the optical beams, are not spatially resolved and therefore require tomographic reconstruction algorithms to recover the local distribution of the measured quantities in the same manner as for local measurements. In Sec. 2 we first present a method for generating concentration maps from integrated measurements using a reconstruction algorithm that relies on an autoencoder. Then, Sec. 3 introduces the plume model, the training of the concentration map, the laser beam geometry and the spectral domain. The algorithm is first tested with a Gaussian plume model, then with Large Eddy Simulation (LES) data. Finally, the robustness of the algorithm is tested by modifying the laser beam geometry.

## 2. CONCENTRATION MAP RECONSTRUCTION FROM INTEGRATED MEASUREMENTS

### 2.1 Reconstruction problem

Absorption spectroscopy allows the measurement of integrated concentration and temperature of the medium of interest. This method, based on the Beer-Lambert law (Eq. 1), consists in propagating through the investigated medium a laser beam emitted by a diode laser tunable in wavelength. The measured transmission spectrum depends on the spatial distribution of temperature, concentration and pressure following the Beer-Lambert equation⁶ (Eq. 1 and Eq. 2).

$$ \frac{I(\sigma)}{I_0(\sigma)} = \exp(-\alpha(\sigma)) \qquad (1) $$

with $\alpha$ the absorbance of the medium for the case of a single absorption line defined by

$$ \alpha(T, P, N, \sigma) = \int_{0}^{L} S(T(l)) \phi(T(l), P, \sigma) N(l) dl. \qquad (2) $$

where $S$ is the line strength, $\phi$ is the spectral lineshape function, $\sigma$ is the wavenumber, $T$ is the temperature, $P$ is the pressure and $N$ is the number density of the absorbing species, $L$ is the absorption length. Fitting an experimental spectrum covering one or several absorption lines with the Beer-Lambert law allows the determination of the line-of-sight integrated values of concentration of the probed species at known temperature and pressure. Based on homogeneous distributions hypothesis for temperature, pressure and concentration, the TDLAS technique allows the inference of absolute concentration with a relative ease of implementation at a repetition rate of 10 kHz or more.⁷ However, the main limitation of TDLAS is the lack of spatial resolution along beam propagation. A solution is the development of TDLAT in order to obtain a 2D mapping of concentration from multiple beam paths and associated integrated measurements. The well-known algorithms that are used in the biomedical domain cannot be used for the TDLAT because the number of projections is too low. A different and more suitable algorithm has to be developed. Tomography is an ill-posed problem because of the high number of free parameters; therefore, the reliability of the reconstruction is strongly dependent on the number and the geometry of beam paths.

Two categories of algorithms among others are used in TDLAT: algebraic algorithms and algorithms based on neural networks. Algebraic algorithms, such as Tikhonov or Landweber algorithms⁸, are based on minimising a loss function which contains a regularisation term:

$$ L(N) = \sum (\|\widehat{SP}_i - SP(\theta_i, r_i, N)\|^2) + R(N) \qquad (3) $$

where $L(N)$ is the loss function, $\widehat{SP}_i$ is the experimental absorption spectrum ($I/I_0$) of the $i^{th}$ beam, $SP(\theta_i, r_i, N)$ is the calculated spectrum, $R(N)$ the regularisation term and $(\theta_i, r_i)$ the coordinates of the $i^{th}$ beam.
beam. The difficulty in this type of algorithm is to carefully select the regularisation term, to obtain a stable and reproducible result.

Reconstruction algorithms based on neural networks integrate this regularisation in their training⁹. The training database has to be large and also represent correctly the shape of the medium of interest.

## 2.2 Structure of the reconstruction algorithm

The proposed reconstruction algorithm is inspired by Bora et al.¹⁰ and builds a data-based mapping $N(z)$ between a lower-dimensional latent space $\mathcal{Z}$ and the space of concentration maps $\mathcal{N}$. The mapping $N(z)$ is a CNN trained on a concentration data set generated from the Gaussian plume model,¹¹ see details in Section 3.1 and Section 3.2. Once trained, the mapping $N$ is plugged into a loss that measures the discrepancy between predicted and measured spectra:

$$ \mathcal{L}(z) = \sum_{i} \| \widetilde{SP}_{i} - SP(\theta_{i}, r_{i}, N(z)) \|^{2}. \qquad (4) $$

Note that the regularization is embedded in the mapping $N$. The loss $\mathcal{L}$ is optimized with rmsprop of the optax optimization library for JAX. Once the optimal latent space parameter $\hat{z}$ has been found, the best concentration map is defined as $N(\hat{z})$.

<figure><img src="image_2.png" /><figcaption>Figure 1: Reconstruction tomographic algorithm based on an autoencoder to generate concentration maps from its latent space; the gradient descent is made from the loss function calculated using Eq. 4.</figcaption></figure>

# 3. MAIN FEATURES OF THE METHOD

We intend to demonstrate on simulated data the feasibility of TDLAT using an autoencoder to reconstruct the concentration map of a methane release. The training is made using the Gaussian model and is tested on a synthetic leak simulated with the Gaussian model and Large Eddy Simulations. The assumptions taken are explained in the following parts.

## 3.1 Gaussian plume model dataset

The plume is often simulated with a Gaussian model when tomography is performed, like in the studies of L. Cartwright et al.⁴ or S. Coburn et al.⁵ This model is used to describe the dispersion from continuous sources¹¹ with the following assumptions: the wind is uniform and all variables are averaged over about 10 minutes. The Gaussian model formulation of the mass concentration $C$ is the following¹²:

$$ C(X, Y, Z) = \frac{Q_s}{2\pi\sigma_Y\sigma_Z U_e} \exp -\frac{Y^2}{2\sigma_Y^2} \times \left( \exp -\frac{(Z - z_e)^2}{2\sigma_Z^2} + \exp -\frac{(Z + z_e)^2}{2\sigma_Z^2} \right), \qquad (5) $$
where $Q_s$ is the emission rate of the point source underlying the plume, $z_e$ the effective release height above the ground surface, $U_e$ the effective mean wind speed at the height of the release. The dispersion coefficients in, respectively, lateral and vertical directions, $\sigma_Y$ and $\sigma_Z$ are defined by

$$ \sigma_Y = \sigma_v t \left( 1 + \sqrt{\frac{t}{2T_Y}} \right)^{-1}, \qquad (6a) $$

$$ \sigma_Z = \sigma_w t \left( 1 + \sqrt{\frac{t}{2T_Z}} \right)^{-1}, \qquad (6b) $$

where $t = X/U_e$ is the travel time from origin to X, $T_Y$ and $T_Z$ are the Lagrangian timescales in lateral and vertical directions, respectively, $\sigma_v$ and $\sigma_w$ are the velocity fluctuations in, respectively, lateral and vertical directions.

The mass concentration $C$ ($g/m^3$) is proportional to density $N$ (molecules/cm$^3$):

$$ N = \kappa C, \text{ where } \kappa = \frac{N_A}{M_{CH_4}} \times 10^{-6}. \qquad (7) $$

$N_A$ is the Avogadro constant (mol$^{-1}$) and $M_{CH_4}$ is the methane molar mass (g/mol).

Equations 5 and 7 is used to create the database for the autoencoder training. The target area taken as an example here is a square of dimension $20 \times 20$ m$^2$, at $Z = 2$ m and $z_e = 0.01$ m. Four parameters are randomly drawn in the following ranges: $Q_s$ from 0.3 to 150 g/s, $U_e$ from 1.2 to 3.4 m/s, $\sigma_w$ and $\sigma_v$ from both 0.2 and 0.8 m/s. Moreover, a random rotation of the wind is applied from -10° to 10°. The database contains $M = 10000$ images denoted $\{N_m\}_{m=1}^M$, 8000 images 64 by 64 pixels are used for training and the remaining images are used for testing.

## 3.2 Concentration map $N(\cdot)$ training

The mapping $N(z)$ is trained using an autoencoder framework.13 The encoder part $E(N)$ maps a concentration map to a reduced dimension latent space $\mathcal{Z}$. The decoder is the mapping $N(\cdot)$, it regenerates the concentration maps from the values of the latent space coefficients $z$. The encoder part is made up of five convolution layers with 32 features. The kernel size is (3,3) and the stride is 2. The activation function is a Gaussian Error Linear Unit (GELU). The parameter set of the encoder is denoted $\theta_e$. The data are reshaped to be a single feature vector. The decoder is made up of five deconvolution layers. The kernel size is (3,3), and the strides are (2,2). The parameter set of the decoder is denoted $\theta_d$. The autoencoder is trained on the Gaussian plume dataset, the training loss is defined as

$$ \mathcal{L}_{ae}(\theta_e, \theta_d) = \sum_{m=1}^{8000} \| N_{\theta_d}(E_{\theta_e}(N_m)) - N_m \|^2. \qquad (8) $$

Training is performed with the Adam optimiser from optax with a batch size of 128, 500 epochs and a learning rate of $1.10^{-5}$.

## 3.3 Geometry of the optical paths

An important parameter is the geometry of the laser system. It is indeed desirable to cover all the studied area with the most lines of sight as possible. However, from an experimental point of view, more beams lead to a more complex and expensive system. A compromise should be made between the number of beams and the complexity and feasibility from an experimental point of view. The reconstruction algorithm is tested with three different numbers of beams: 50, 30 and 10 (Fig. 2). We consider here the use of corner cube retroreflectors to reflect the laser beams back to their origin in order to double the absorption length of the optical paths. Retroreflectors reflect the incoming beams with a spatial shift of a few millimeters; the two directions are considered the same due to the small size of the optical beam compared to the pixel size. In this study, the beam configuration is chosen to best cover the spatial extent of the leak, whose overall direction is known. This represents a best-case scenario and will serve as a baseline. The case where the configuration is not adequately chosen is discussed in Section 4.3.
<figure><img src="image_3.png" /><figcaption>Figure 2: Three optical beams arrangements used : a: 10 optical beams; b: 30 optical beams; c: 50 optical beams.</figcaption></figure>

## 3.4 Spectral domain for methane absorption spectroscopy

The spectral domain around $1.65\mu m$ is often used$^{14}$ for methane measurements over a few meters, with low concentrations. Diode laser sources$^{15}$ can be used to target this domain. Here, we study the absorption line at $6096.2cm^{-1}$. This transition has a higher line strength and low interference with atmospheric lines of $H_2O$ or $CO_2$ (as can be seen in Fig. 3 for an absorption length of 40m).

<figure><img src="image_4.png" /><figcaption>Figure 3: Absorption spectrum of water (2%), carbon dioxide (0.04%) and methane (0.000002%) at 6096.2 $cm^{-1}$, for a 40m abortion length. The concentrations used are typical in the atmosphere.</figcaption></figure>

# 4. RESULTS

The reconstruction algorithm is tested with known distributions : first with a Gaussian plume and then with LES data. To validate the results, three parameters are monitored: the direction of the release, the dispersion coefficient in the lateral direction $\sigma_Y$, and the mean concentration of the methane released. In order to measure the direction of release, for each $X$ value, the $Y$ coordinate corresponding to the maximum concentration is noted. The orientation of the release is then represented by the plot of this peak location with respect to $X$. The dispersion coefficient corresponds to the full width at half maximum (FWHM) of these profiles.

## 4.1 Reconstruction for a Gaussian plume

In order to validate the reconstruction algorithm, we first use a Gaussian plume. The concentration map serving as the ground truth is presented Fig 4.a, along with the positions of the 50 beams. Fig 4.b shows the fit after 1000 iterations of spectrum # 25 whose position is shown in Fig 4.a. The reconstruction algorithm converges
to minimize the loss function. The error between the synthetic "experimental" spectrum and the fitted one is about 1% at the main peak. Surprisingly, as we will see, this relatively large error does not lead to a substantial deviation in the mean concentration.

<figure><img src="image_5.png" /><figcaption>Figure 4: Comparison between the synthetic "experimental" spectrum calculated from the propagation of beam # 25 and the fitted spectrum using the Gaussian plume model as ground truth.</figcaption></figure>

The reconstruction map obtained after optimization is represented in Fig. 5. In this figure, as will also be the case for the following figures, the mean concentration is expressed in ppm (mole fraction with respect to air in the atmospheric environment) and calculated over the whole map by the equation:

$$ mean \ concentration = k \times \frac{1}{n_{pixel}} \sum_{i}^{n_{pixel}} N(i). \qquad (9) $$

where $N(i)$ is the local concentration of pixel $i$, $k$ is the conversion factor between ppm and molecules/cm³.

The mean concentration error is 1.5%. The reconstructed map has a similar shape as the Gaussian plume model, as expected.

<figure><img src="image_6.png" /><figcaption>Figure 5: Reconstructed concentration map for a Gaussian plume ground truth; the mean concentration is indicated in the box. For the error map (c) the mean absolute concentration error is indicated in the box.</figcaption></figure>

The direction of the release is very well reproduced as shown in Fig. 6. As the plume expands due to diffusion, the dispersion coefficient increases with distance from the leak location, as can be seen. The error in reconstruction also increases with distance from the source, reaching up to 2 m at the plume's end.

Although there are errors in the dispersion coefficient at the end of the plume, the release is correctly reconstructed. Moreover, the overall mean concentration is correctly estimated. This test case allows to verify and validate the algorithm performance since it is trained with Gaussian data. The neural network is properly trained and the tomographic inversion results in coherent concentration maps. We can now test the reconstruction algorithm for a ground truth closer to reality, using LES data by making some assumptions.
<figure><img src="image_7.png" /><figcaption>Figure 6: Direction of the plume and dispersion coefficient in the lateral direction for a Gaussian plume model as ground truth.</figcaption></figure>

## 4.2 Reconstruction for a LES simulation

### 4.2.1 Assumptions for the reconstruction based on LES data

The ground truth used is the result of a Large Eddy Simulation (LES), as seen in Fig. 15a, calculated with the PALM model,¹⁶ a solver for the Navier-Stokes equations. The plume is sampled at one minute intervals in all three dimensions (X, Y, Z). A user-defined power law vertical wind speed profile is used with a reference wind speed of 2.76 m/s measured at 5 m height above the ground surface. The release rate ($Q_s$) is 10 g/s and the release height is 1.5 m. The target area, $20 \times 20$ m², is chosen so that the source point is close to that on the training data. Indeed, there is not much variation in the training set in terms of the location of the source point.

As previously indicated, the Gaussian plume model reproduces well the dispersion from continuous sources after averaging the plume concentration over 10 minutes. However, the LES simulations are time-resolved. To compare the two models, and because the autoencoder is trained on the Gaussian model and thus is expected to work best with plume similar in shape, the ground truth concentration map is generated by averaging the LES simulation over 10 minutes. The temperature and the pressure are constant. So, the logarithm of the absorption is proportional to the concentration, which allows us to average. Synthetic experimental spectra are calculated from this ground truth. In the context of this feasibility study, noise is neglected. This assumption is realistic for the high frequency noise since the sampling rate is expected to be in the $kHz$ range, therefore averaging over several minutes provides a consistent decrease of the noise power.

The reconstruction process is initialized by a concentration map generated from random parameters in the latent space of the autoencoder. The influence of the initialization is not studied here. The number of iterations is 1000 and the learning rate is $2 \times 10^{-2}$.

As for the Gaussian model, to validate the results, three parameters are monitored: the direction of the release, the dispersion coefficient in the lateral direction $\sigma_Y$, and the mean concentration of the methane released.

### 4.2.2 Results

Fig 7.b and c presents for two lines of sight the experimental and reconstructed spectra whose position are shown in Fig 7.a. The algorithm converges to minimise the loss function as defined. Spectrum # 6 is very well reproduced, when spectrum # 7 shows an error of about 10%.

Fig 8 represents the reconstructed concentration map obtained for the three different arrangements represented in Fig. 2. In this figure, LES data averaged from 0 to 10 minutes are used as ground truth. Similar maps are obtained for 30 and 50 beams, with a shape of the release closer to the ground truth than with 10 beams. The error between the ground truth and the mean estimated concentration is 0.4% and 1.1% for the 30 and 50 beams cases respectively, and only 0.3% for the 10 beams case. This shows that the mean concentration is recovered very well, and that having 10 beams can be sufficient to recover the amount of methane released. The origin of the release is not located correctly. The reconstruction with 30 and 50 are very similar, and the error actually increases for the configuration with more beams.
<figure><img src="image_8.png" /><figcaption>(a) Concentration map with the optical paths</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) Spectrum # 6</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) Spectrum # 7</figcaption></figure>

Figure 7: Result of the fitting between the experimental and simulated spectra in transmission for spectra number 6 and 7 based on the LES plume simulation averaged from 0 to 10 minutes.

<figure><img src="image_11.png" /><figcaption>(a) Ground truth</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) 10 optical paths</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) 30 optical paths</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(d) 50 optical paths</figcaption></figure>

Figure 8: Reconstructed concentration map for different numbers of beams for LES data averaged from 0 to 10 minutes as ground truth; the mean concentration is indicated in the box.

Concentration error maps are displayed on Fig. 9 in ppm. The errors range from -20 ppm to 50 ppm. The region of origin of the plume exhibits the highest errors. The highest values are observed for the geometry with 10 beams, as expected. The error distribution of the two cases with 30 and 50 beams are almost identical. The amplitude of the error distribution with 10 beams is about twice as large as in the other two cases. The mean absolute concentration error equals 3.55 ppm for 10 beams, 2.98 ppm for 30 beams and 3.08 ppm for 50 beams. Using more than 10 beams allows to have lower local error and to detect higher spatial frequencies, i.e. to reduce the smoothing effect. In all, since adding 20 or 40 beams is experimentally more complex, 10 beams may be sufficient for this application, depending on the required precision. Still, using 30 beams could be preferable for more accurately locating the leak source, as it offers better positioning.

<figure><img src="image_15.png" /></figure>

Figure 9: Concentration error map for the different number of beams for LES data averaged from 0 to 10 minutes as ground truth; the mean error is indicated in the box.

The direction of the leak relative to the horizontal direction is determined as described in Sect. 4.2.1. For all
optical configurations, the direction of the leak is well reproduced as seen in Fig. 10. The width of the plume in the lateral direction is better reproduced by the optical configuration with fewer beams; for the two other configurations, the width compares well with the ground truth until 17 m, where there is a discrepancy. This gap can be explained by the fact that the beam density is lower at the end of the plume than in the middle. However, the error is less than 1.5 m. These two physical criteria support the idea that the algorithm provides coherent results compared to the ground truth.

<figure><img src="image_16.png" /><figcaption>Figure 10: Direction of the plume and dispersion coefficient in the lateral direction for LES data averaged from 0 to 10 minutes as ground truth and for the three beams arrangements.</figcaption></figure>

Other ground truths are tested to assess if the same performance can be achieved. For the next reconstructions, we consider only the geometry with 30 beams. The following reconstructions are made from the ground truth of the same LES data but by averaging over different time intervals (Fig. 11 and Fig. 12). The errors in the mean concentration obtained are 0.8% and 1.6%, with mean errors of 2.44 ppm and 1.61 ppm, respectively;

<figure><img src="image_17.png" /><figcaption>Figure 11: Reconstruction concentrated map from ground truth LES data averaged for the 30 beams from 20 to 30 minutes; the mean concentration is indicated in the box.</figcaption></figure>

For both cases, the direction of the methane release is well reproduced. The release expansions follow the same trends as the ground truths. We observe a higher error at the end of the plume for the reconstruction corresponding to Fig. 12. Overall, the end region of the plume is not as well reconstructed probably because the beam density is lower than in the centre of the image.

## 4.3 Reconstruction for a global rotation of the laser beams arrangement

As previously discussed, the beam geometry has been designed for a release in a privileged direction (in the X direction), assuming a constant wind direction during the measurement. To showcase the importance of the
<figure><img src="image_18.png" /><figcaption>Figure 12: Reconstructed concentration map from ground truth LES data averaged for the 30 beams from 30 to 40 minutes; the mean concentration is indicated in the box.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 13: Direction of the plume and dispersion coefficient in the lateral direction for two different ground truth cases and for the 30 beams arrangement.</figcaption></figure>

optical configuration, and to test the tomographic algorithm in a less ideal configuration, a new reconstruction is conducted, applying a rotation of 90 degrees on the laser beam geometry while retaining the 30-beam configuration (Fig. 14).

<figure><img src="image_20.png" /><figcaption>Figure 14: Laser beam geometry after a rotation of 90 degrees.</figcaption></figure>

As can be seen in Fig. 15, the reconstruction algorithm still provides a coherent concentration map. Although the reconstructed map does not really resemble a Gaussian plume, this shows that the reconstruction is not very sensitive to the laser beam geometry.
<figure><img src="image_21.png" /><figcaption>Figure 15: Reconstructed concentration map obtained with a 90° rotated beam geometry, for LES data averaged from 0 to 10 minutes as ground truth, the mean concentration is indicated in the box.</figcaption></figure>

## 4.4 Discussion

It is quite remarkable that the autoencoder, trained on a simple model, still manages to perform well for concentration distributions that are closer to reality. This can be explained by the fact that the LES simulations are averaged in a way that makes them resemble the Gaussian model; however, the high frequencies that remain present do not disturb it. Another noteworthy point is that relatively few beams seem to be sufficient to recover a distribution that could be deemed acceptable. It should be kept in mind that the reconstruction map may be dependent on the training dataset. In particular, the training dataset includes privileged direction of a plume model up to ten degrees variation. In this study, the direction of the release for the maps to be reconstructed is globally known (towards increasing X) and does not change during the measurement. We constructed the geometry of the beam paths on this basis. In real cases, the wind direction can change during the data recording. To be more sensitive to the change of the wind direction, the autoencoder could be trained with more variability of the release direction.

# 5. CONCLUSION AND PERSPECTIVES

In this preliminary work, we developed a reconstruction algorithm for absorption tomography to map a methane release. The reconstruction is based on a reduced order model build from a synthetic dataset using an autoencoder framework. The developed algorithm has been tested on an hyperspectral TDLAS measurement model, but could also be used on simpler two-frequencies setups. We have shown that with only 10 beams a reconstruction with less than 1% error on the mean concentration can be obtained; the local concentration on a pixel of size has an mean error of less than 35%. Furthermore, the direction of the leak is estimated with a very good accuracy. The increase of the dispersion coefficient with the distance from the leak source location is well recovered up to 10 m, with an error of less than 1 m. The errors at the end of the image are higher probably because of the lower beam density. For future work, the database for autoencoder training can be augmented with data more representative of reality such as with the LES simulations. Since these are resolved in time, it would also be possible to detect the leak more quickly, without having to average over 10 minutes. In addition, the source height is fixed in the present training database, the database can be easily augmented with a range of source heights. The present study assumes spatially homogeneous temperature and pressure, with known values. In such a context, the logarithm of the absorption is proportional to the concentration up to a known spectrum profile. This feature can be exploited to drastically compress the input spectrum measurement data and yield simpler and cheaper algorithms.

# ACKNOWLEDGMENTS

The authors thank the French Defense Innovation Agency (AID) for supporting this work.
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