The method presented here is designed for high-resolution hyperspectral satellites. The images we work with are Level 1 (L1) images from PRISMA and EnMAP . Both of these satellites provide hyperspectral images with a spatial resolution of 30 m. The methane absorption bands are located inside the 1500–2450 nm range, in the shortwave infrared (SWIR) range. This range is covered by the spectral channels of both PRISMA and EnMAP. In the SWIR, the spectral resolution of PRISMA varies between 9 and 15 nm, and the spectral resolution of EnMAP is approximately 10 nm.

The deep learning models presented here were trained on simulated plumes. To train our networks, we inserted those plumes in true EnMAP L1 images to reproduce plumes with real background noise. The result of the LES procedure is an enhancement map providing the enhancement in methane concentration for each pixel. From this simulated enhancement map, we compute the plume transmittance at each wavelength by using a radiative transfer model and methane absorption cross-sections from the HITRAN data base . The plume transmittance is then convolved with the instrument's spectral response function to obtain a spectrum for each pixel of the plume. The resulting spectrum is then multiplied with the radiance spectrum from the L1 data to obtain the simulated plume.

We used 48 background samples from different locations in North America, the Middle East, and northern of Africa. Those three areas are places where methane plumes are frequently detected with PRISMA and EnMAP and will therefore allow us to recreate real conditions. The different locations from which the background samples were taken are summarized in Table .

These areas allow us to have homogeneous and heterogeneous backgrounds in our dataset. We compare three of these background samples in Fig. , where methane enhancement maps are computed using the matched filter technique . The enhancement maps show mainly noise but also retrieval artifacts associated with elements of the original scene, such as a road for the Uzbekistan image. By comparing the distributions in ΔXCH₄ between the three scenes, we can observe three Gaussian distributions with a mean close to zero but with very different standard deviations. This is caused by the different types of surface between the scenes. In particular, a very heterogeneous scene can lead to a noisy enhancement map. Source rate estimation will be harder on those backgrounds as the signal-to-noise ratio between the plume and the background will be lower. This means that we will have fewer spatial features of the plume available in those cases. Including different background distributions in the dataset will help the network to be robust to different noise levels.

Training a deep learning model requires a large amount of data. One of the main constraints in source rate estimation from satellite imagery is the lack of ground truth, which prevents us from using a dataset of real images. Therefore, we used our real plume images for testing purposes only. To train the model, we built a dataset of simulated plumes. To complete this dataset, we used the plume dataset generated by as a testing only dataset.

We created a dataset of simulated methane plumes with large eddy simulation (LES). The LES procedure allows one to simulate realistic plumes exposed to wind turbulence. We used the MicroHH model , which has already been used for methane plume simulations . We simulated at a spatial resolution of 30 m in a 6×6 km² domain. We used 61 different wind speeds between 0.5 and 6.5 m s⁻¹, and for each wind speed we conducted four simulations with different temperature profiles, resulting in 244 different simulations. Each simulation lasted 3 h, with the first hour being used as spin-up. In the remaining 2 h, we sampled one plume every 2 min. Our dataset thus contains 14 640 methane plumes, and we used 12 444 of them for training and 2196 for validation. Splitting the dataset before data augmentation ensured that the network would not see a plume with exactly the same shape during training as during testing. The dataset was randomly split from the combined 14 640 plumes and not from the 244 simulations. This ensures that for each wind speed we have plumes simulated with different temperature profiles in the training set. During the simulation process, all plumes were generated with the same constant emission source rate.

As mentioned above, to verify that our model can generalize to a diversity of plumes, we also tested it on the simulations performed in . The dataset of , originally designed for Sentinel-2, was generated with WRF-LES . It contains 1200 methane plumes simulated at various wind speeds ranging between 1.5 and 5 m s⁻¹ and at a 25 m horizontal and 15 m vertical resolution over a 9×9×2.4 km³ domain. The simulations were obtained from five different wind speeds and sampled with a 30 s time gap. Before testing, the plumes were resampled at a resolution of 30 m with cubic spline interpolation. We will refer to this dataset as S2Test.

To study the performance of our model with a wide range of source rates, we performed data augmentation by randomly scaling all the plumes in the dataset. The plumes in the training set were generated from the result of the LES by randomly scaling each plume 10 times to obtain source rates between 50 and 33 000 kg h⁻¹. It is very difficult to detect emissions at a source rate of 50 kg h⁻¹ with satellites such as PRISMA or EnMAP .

However, neural networks usually suffer from a threshold effect associated with the training range that prevents them from predicting values outside said range. This leads to a bias in the predictions near the limits of the training range. Therefore, we propose to train the network on emission rates as low as possible. If we only considered source rates starting at 1000 kg h⁻¹, it would not be possible to know if a plume for which we estimate 1000 kg h⁻¹ is not actually at a lower source rate. Training from 50 kg h⁻¹ up ensures that the plumes that can actually be detected will not suffer from the threshold effect, with the detection threshold for EnMAP being between 100 and 500 kg h⁻¹ depending on the background .

The plumes in the testing set were scaled between 100 and 25 000 kg h⁻¹. Due to the threshold effect described above, the range of source rates for the testing set needs to be smaller than the range of source rates for the training set. Otherwise, the network would underestimate the source rate close to 33 000 kg h⁻¹ and overestimate the source rate close to 50 kg h⁻¹. This would introduce a bias in the evaluation, which is avoided by evaluating on a, more realistic, narrower range.

To generate our dataset, we used a time gap of 120 s between two plumes from the simulation, while in the dataset of the time gap is only 30 s. Datasets can even be found with shorter time gaps, such as the one used by , which has a time gap of only 10 s. Increasing the time gap between simulated plumes in a dataset reduces their correlation and therefore increases the diversity of the dataset. If we consider plumes taken with a small time gap (less than 30 s), we can observe the same turbulence patterns; thus, they can hardly be considered different samples in the dataset. We can observe this redundancy in Fig. , where we show the same plume at different time steps and for different wind speeds. We can easily notice that after 10 s and for any wind speed the plume is almost identical to the initial image, whether it is in terms of shape or concentration. After 30 s, the shape is still quite similar, but there are some changes in the distribution of the concentration. This observation is mostly true around the source of the plume. In 30 s, the new concentration distribution has not yet spread to the tail of the plume. After 60 s, the changes in the distribution of the concentration have increased, and we start to see some noticeable changes at the beginning of the plume tail. This is visible for the plumes at 1 and 3 m s⁻¹. After 120 s, most of the plumes are globally different from their original image. However, we still see residuals from the turbulence that were occurring in the initial image. For example, for the plume at 2 m s⁻¹, even if the concentration distribution is different; the overall shape of the plume after 120 s looks similar to the one in the initial image.

Thus, using small time gaps leads to a low plume variety in the dataset. This can lead to severe network overfit. In addition, the training set and testing set will be strongly correlated; thus, overfitting will be more difficult to notice. To show the overfitting effect caused by small time gaps, we generated a second dataset following the methodology of : we performed one simulation per wind speed and used a time gap of 10 s. We will refer to the dataset with a large time gap as MicroL (for MicroHH-Large) and to the dataset with a short time gap as MicroS (for MicroHH-Short). The dataset of referred to as S2Test will be used only for testing. The parameters for each dataset are summarized in Table . Note that MicroL does not result from subsampling MicroS. Otherwise, we would not be able to show the overfitting when working with MicroS, as performing well on MicroS would also lead to performing well on MicroL. Therefore, there is no plume in common between MicroL and MicroS, and the plumes come from different simulations. The MicroL dataset is publicly available for download at 10.5281/zenodo.15618044 .

In hyperspectral imaging, any object in a scene can be assigned a spectral signature. In the case of methane, this spectral signature is the absorption spectrum of the gas. To determine whether an observation contains an excess of methane, it is therefore natural to look for a deviation in the observation spectrum in the direction of the methane spectral signature. The amplitude of the observed deviation then provides a measure of the gas concentration. This idea sums up how the matched filter retrieval works for methane concentration. It is used on many hyperspectral instruments such as AVIRIS and PRISMA .

A standard hypothesis for hyperspectral images is that background pixels follow a Gaussian multivariate distribution . With this assumption, the maximum likelihood estimator of the methane mixing ratio is given by the matched filter .

Let us denote by K_CH4 the diagonal matrix whose diagonal components are the methane absorption coefficient values, and let μ and Σ be, respectively, the mean vector and the background covariance matrix. We define the target vector by 1t=-KCH4⋅μ.

With this notation, the excess methane concentration α corresponding to an observed pixel x is given by the matched filter formula: 2α(x)=tTΣ-1(x-μ)tTΣ-1t.

Parameters μ and Σ are computed with their empirical unbiased estimates. They are calculated across-track, which means that we compute a different set of parameters for each detector element in the sensor. This applies for both PRISMA and EnMAP images.

The matched filter is the optimal detector for an additive target in a Gaussian background. This assumption on the background is not necessarily true in methane plume detection. Several variations of the matched filter are designed to improve the quantification provided. Here, we use the MAG1C method proposed by for retrieving methane concentration. The MAG1C method introduces two improvements to the matched filter formulation. The first is a spatial L1 regularization to take into account the fact that most observations are not part of a plume. The second is the estimation of a different albedo coefficient for each pixel. The latter is defined by 3r(x)=xTμμTμ. This albedo coefficient is used to scale the target spectrum. Thus, the target spectrum used in the matched filter for pixel x becomes r(x)t instead of t.

The MAG1C method can be applied to images from both PRISMA and EnMAP. The main difference when applying it to those instruments is the target spectrum. To calculate the target spectrum, we need to compute a unit methane absorption spectrum through radiative transfer simulation . This requires us to compute convolutions with the spectral response function of the instrument, which is not the same for PRISMA and EnMAP. However, we obtain similar enhancement maps when applying MAG1C to images from both satellites.

The classical method for estimating point source methane emissions is integrated mass enhancement (IME) . This method is already widely used for EnMAP and PRISMA images . The source rate Q is calculated as 4Q=Ueff⋅IME⋅3600L, where IME is the total mass of excess methane (in kg) contained in the plume, L is the plume length (in m), and Ueff (in m s⁻¹) is the effective wind speed. The factor 3600 results from the conversion from kg s⁻¹ to kg h⁻¹. The effective wind speed Ueff is usually estimated from the wind speed at 10 m altitude U10. The relationship between Ueff and U10 is obtained by fitting a regression model on simulated data with large eddy simulations . Several expressions exist for Ueff with linear or logarithmic models . A model suited for source rate estimation with PRISMA or EnMAP is 5Ueff=0.34⋅U10+0.44.

In Eq. (), the IME is obtained from the estimated CH₄ concentration in the plume. The length of the plume is usually calculated by taking the square root of the plume area . This allows one to deal with the fact that the length of the plume is not always properly defined. Indeed, because of turbulence and wind variations, the plume does not necessarily follow a straight path. However, this implies using a plume mask to compute L. Therefore, the quality of the estimation of Q will depend on the quality of the mask. A good mask (one that provides a good estimate of Q) is difficult to obtain. The varying plume shapes and the amount of noise in the images make it difficult to distinguish the contours of the plumes. This means that two different human operators can label the same plume in very different ways. This can affect not only the quality of the estimation of Q, but also the reproducibility of the method.

To obtain U10, a standard practice is to calculate it with an external measurement coming from a dataset of wind data at a global scale such as GEOS-FP or the ECMWF-ERA5 dataset . However, these datasets provide wind data with a low spatial resolution (around 25×25 km² for GEOS-FP and 30×30 km² for ERA5) and a low temporal sampling (hourly data). Hence, these datasets are not ideal as wind data sources to characterize CH₄ emissions: the temporal gap between the emission and the wind data point can be up to 30 min and most plumes studied with PRISMA or EnMAP will not exceed 5 or 6 km.

To compare our method with the IME, we considered two cases. In the first case, we estimate the source rate by using the effective wind given by Eq. () obtained by with LES. In the second case, we fit our own effective wind model using the MicroL dataset. We obtain the following equation for Ueff: 6Ueff=0.17⋅U10+0.49.

To fit our effective wind model, we use a different masking procedure than . We first apply a threshold to the plume image, corresponding to the quantile at 85 %. Then, we apply a 4×4 mean filtering and a thresholding to the resulting image. To remove remaining potential false positives, we finally apply a 3×3 mean filtering followed by another thresholding. Once the set of plume masks is calculated, the wind model is obtained by performing a linear regression. Using different masking procedures explains why our model is different from the one used by . We will refer to our version of the IME as IME-MicroL.

To estimate the emission source rate from the methane concentration retrieval image, we used a deep neural network. The use of a neural network enables the estimation of the source rate without relying on an external data source for the wind speed. It also removes the variability associated with the manual labeling of the plume that is needed when using methods such as integrated mass enhancement . The different steps of the source rate estimation pipeline are summarized by the diagram presented in Fig. .

We start by studying the influence of the network architecture and of the plume orientation. To do so, we compare different networks for which we select an architecture between EfficientNetV2-B0 and ConvNeXt-Tiny and a plume orientation between random rotations and alignment with the x axis as described in the previous section. This leads to four networks: EffNet+rotation, EffNet+align, CNext+rotation, and CNext+align. These four networks are trained with MSE loss, on the dataset MicroL.

In Table , we compare the results of EffNet+rotation, EffNet+align, CNext+rotation, and CNext+align in terms of RMSE and MAPE. Overall, the methods with plume alignment outperform the other networks both in RMSE and in MAPE. We can also observe that, for a fixed pre-processing, the networks based on ConvNeXt seem to perform better than those based on EfficientNet. Whereas it is clear that CNext+rotation outperforms EffNet+rotation, CNext+align outperforms EffNet+align only in MAPE. The networks with plume alignment have a very close RMSE with a gap of only 16 kg h⁻¹, which is not statistically significant. However, a gap of 0.7 in MAPE shows a real difference in performance. Indeed, the low source rates have very little impact on the RMSE but can have a high impact on the MAPE. A lower value in MAPE but not in RMSE therefore means that the estimation is improved for low source rates.

We can observe the evolution of the MAPE with respect to the source rate in Fig. . We see that the four tested networks have very similar performance levels for high source rates, from 4000 kg h⁻¹ upwards. Above 4000 kg h⁻¹, the MAPE hardly decreases at all, remaining around 10 % for all networks. Indeed, at high source rates, the methane concentration looks like the ground-truth image (the output of LES), meaning that the noise is negligible with respect to the plume concentration. Therefore, for the network, there is no difference (in terms of additional information) between an image with a plume at 10 000 kg h⁻¹ or at 20 000 kg h⁻¹. Thus, the networks differ only in their performance at medium- and low-source rates, with the gaps between them narrowing as the source rate increases. In particular, we can see that CNext+align is outperforming EffNet+align for low source rates. Between 100 and 200 kg h⁻¹, the MAPE of CNext+align is at least half the MAPE of any other network. However, even if we dismiss the case of source rates below 500 kg h⁻¹, CNext+align still outperforms the other methods. Since MAPE is a better representation of the networks performance over the entire range of source rates, from now on we will focus only on the ConvNeXt-Tiny architecture.

As we saw in Fig. , the differences in performance between the methods lie in the 0–4000 kg h⁻¹ range. However, when training with MSE loss, this is the range that has the least weight in the loss. To improve performance in the 0–4000 kg h⁻¹ range, we train the network directly with MAPE loss, which gives more weight to low source rates than MSE loss. This is also possible because performance in the 4000–25 000 kg h⁻¹ range is stable for all the networks used, so we can expect the same result when changing the loss. In Fig. , we observe the influence of the loss. We compared a network, CNext+RMSE, trained with the RMSE loss on aligned plumes with CNext+MAPE, the same network but trained with the MAPE loss.

As expected, CNext+MAPE outperforms CNext+RMSE in terms of MAPE. We can see a significant improvement in the 0–10 000 kg h⁻¹ range. Note that changing the loss affects not only plumes with small source rates but also those with higher rates. Moreover, it also outperforms CNext+RMSE in terms of RMSE with a lower RMSE in the 0–10 000 kg h⁻¹ range. Beyond 10 000 kg h⁻¹, the two networks have similar performance.

The network CNext+MAPE trained with aligned plume on MicroL is the best version of our different networks and we name it MetFluxNet.

The simplest way to estimate the uncertainty on the source rate estimate provided by the neural network is to compute the empirical standard deviation of the estimation. To compute it for a given prediction, we consider a sample of the true source rate distribution corresponding to this prediction, and we compute the standard deviation of this distribution with its usual non biased empirical estimate. The sample of the true source rate distribution is obtained from the MicroL testing set. Under the assumption that the source rate distribution corresponding to a prediction made by the network locally follows a Gaussian distribution, we can then obtain a confidence interval on the prediction.

Under the same assumption, another way to obtain a confidence interval is to train the network for a probabilistic regression . For a given plume P, let us denote by Q its emission source rate. Then, the prediction made by the network for P follows a Gaussian distribution N(Q^,σ), where Q^ is an estimator of Q. When using a probabilistic regression, we want to estimate both Q^, which will be the predicted source rate, and σ, which will be the standard deviation of the estimation. This standard deviation yields confidence intervals.

Predicting the standard deviation requires a small change in the network architecture. The previous networks used a fully connected layer with one unit as the last layer to perform the source rate estimation. To output both the predicted source rate and the standard deviation, we add in parallel of this last layer a fully connected layer with one unit set to the power of two. Squaring the layer ensures that the output is positive. Therefore, we consider that the output of this second layer will be the variance of the distribution, that is, σ2.

To ensure that σ is an estimate of the standard deviation, we use the negative log likelihood (NLL) as loss. Indeed, if (Q^,σ) minimizes the NLL, then (Q^,σ) is the maximum likelihood estimator for the parameters of the output distribution of the network. The NLL is defined by 7NLL(Q,Q^,σ)=12log⁡2πσ2+Q^-Q2σ2.

We obtain a similar performance for the predicted source rate when comparing CNext+MAPE with CNext+NLL. The CNext+MAPE has a RMSE of 1388 kg h⁻¹ and a MAPE of 8.3 %, whereas CNext+NLL has a RMSE of 1369 kg h⁻¹ and a MAPE of 8.3 %.

In Fig. , we compare the empirical standard deviation computed with the CNext+MAPE output, denoted by σemp, with the network estimated standard deviation computed with CNext+NLL, denoted by σNLL. Note that the plot depends on the predicted source rate and not on the true source rate, because we look at the distribution of the network output. We can notice that σNLL and σemp have the same behavior. The proximity between the values of σemp and σNLL is due to the fact that the empirical standard deviation is the maximum likelihood estimator of the standard deviation for a Gaussian distribution. Since σNLL is an approximation of the maximum likelihood estimator of the standard deviation, we deduce that σNLL should be close to σemp. From now on, we name ProbMetFluxNet the CNext+NLL network.

The standard deviations computed for our networks in Fig.  are way lower than those provided by the IME for the same source rates . This is mainly due to the fact that our neural networks do not rely on external data. With IME, a large part of the uncertainties comes from the wind speed. It is common to assume that there is around 50 % of error on the wind speed . Hence, removing the wind speed from the estimation pipeline allows us to significantly reduce the standard deviation on the estimation and therefore the size of the associated confidence interval.

All the presented networks estimate the emission source rate using all the information in the image, including background data. However, the background can contain false positives, typically pixels not belonging to the plume that could be considered plume pixels because of their high retrieved concentration. When estimating the source rate, we first want to remove those false positives before giving the image to the network. Removing a part of the background pixels will change the overall distribution of the background. In particular, the resulting distribution will be different from the ones the network has seen in training. This might lead to errors in the source rate estimation.

In Fig. , we observe two images of the same simulated plume. The source rate corresponding to this plume is 2192 kg h⁻¹. In the left image, we have the original methane retrieval image. In the right image, we removed the top and bottom edges, which contained only background pixels. Even if these images do not include false positives, this example shows how a change in the background far from the plume can impact the source estimation.

We can see that although the two plumes are identical, we have two significantly different source rate predictions with an increase of almost 10 % in the predicted source rate when removing a part of the background pixels. Moreover, this increase in the predicted source rate widens the gap between the prediction and the ground truth.

To reduce the impact of the background distribution, we trained a version of our network while removing different parts of the background. This aims at reproducing the background distribution we would obtain when removing false positives in real plume images. To create those sparse images to train the network, we draw random bounding boxes that include the entire plume, and we remove the pixels outside of it. This avoids mistakenly removing plume pixels. The random bounding boxes are used during training to be robust to different bounding box sizes and therefore different levels of sparsity in the background distribution. The idea behind the use of bounding boxes is that it is easier to draw a bounding box than a fine mask of the plume. This comes with an important drawback: if a false positive is close to the plume, it might not be possible to remove it with a rectangular bounding box. However, this limit also exists for fine plume masks when the plume intersects a false positive.

In Fig. , the right-hand image corresponds to the bounding box applied to the left-hand plume. We name MicroL-sparse the MicroL dataset with the partial background removal. In the same way, we name MetFluxNet-sparse the version of MetFluxNet trained on MicroL-sparse.

In Table , we compare MetFluxNet and MetFluxNet-sparse on MicroL and MicroL-sparse. As could be expected, MetFluxNet has the best performance on MicroL and MetFluxNet-sparse obtains the best performance on MicroL-sparse. In particular, the performance of MetFluxNet on MicroL is similar to the performance of MetFluxNet-sparse on MicroL-sparse. Hence, removing background pixels when there is no false positive to remove does not improve the results, but it does not decrease them either (the gap between a RMSE of 1388 and 1374 kg h⁻¹ is not statistically significant). However, the results of MetFluxNet-sparse are much better on MicroL than the results of MetFluxNet on MicroL-sparse. This is because MetFluxNet-sparse is trained on images with various degrees of sparsity; therefore, it generalizes better when there is no added sparsity in the images. However, MetFluxNet has the advantage of being able to be used without any manual intervention on the background.

Since MetFluxNet-sparse requires additional pre-processing to create bounding boxes and because those bounding boxes do not necessarily allow us to remove all the false positives, our main focus in the following sections will be on the standard MetFluxNet method.

Another way to look at the influence of the background is to compare the network performance on several different backgrounds. In Table , we compare the MetFluxNet results in three locations: North America, the Middle East and northern Africa. We obtain very similar results for the three locations, in terms of both RMSE and MAPE. We note that the RMSE and MAPE are slightly higher for North America than for the other two areas. This might be due to the more desertic background we can have in the Middle East and northern Africa, which usually are less noisy. Moreover, the heterogeneous backgrounds we can find in North America make the estimation more difficult . The increase in RMSE between North America and the other locations is about 35 kg h⁻¹, which represents only a 2.5 % increase compared to the results in the Middle East and northern Africa.

To validate predictions of our networks, we want to test it on images of real plumes. However, without ground truth, which is generally not available, it is difficult to measure the quality of our prediction. Therefore, we will work with methane plumes observed after the controlled methane releases carried out by . In , researchers conducted single-blind controlled methane release experiments to evaluate the performance of satellite-based methane detection and quantification methods. They released methane plumes in Arizona between October and November 2021 and October and November 2022. These releases occurred during overpasses of several satellites with methane detection capabilities, including PRISMA and EnMAP. In 2021, three methane plumes were released during PRISMA overpasses. In 2022, one methane plume was released during EnMAP overpasses and three plumes were released during PRISMA overpasses. The plume released on 27 October 2022, during a PRISMA overpass, is not visible in our methane enhancement retrieval image, as the order of magnitude of retrieved concentrations is lower than the background noise level. Hence, we will test our networks on the six remaining plumes for which we have a ground truth.

In Fig. , we can observe four of the plumes detected by PRISMA or EnMAP. The plumes have been rotated to be aligned with the x axis to comply with the alignment pre-processing required for the different versions of MetFluxNet. The red bounding boxes are used for the sparse versions of the networks, and the non-sparse networks used the whole image. In Fig. d, no pixels needed to be removed; therefore, the bounding box includes the whole image.

In Table , we compare the predictions made by MetFluxNet, ProbMetFluxNet, and their sparse versions with state-of-the-art methods. Those predictions are provided with 95 % confidence intervals. The confidence interval is empirically computed for MetFluxNet and MetFluxNet-sparse. For ProbMetFluxNet and ProbMetFluxNet-sparse, it is computed with the standard deviation estimated by the network. We reproduced the results of S2MetNet by training a version of the network on MicroL; the corresponding confidence interval is computed empirically. The work of does not introduce any new methods but gathers the results of different research teams. Therefore, for each plume, we selected the best result obtained among all different teams. To select the best result for a given plume, we considered all the proposed 95 % confidence intervals that contain the true source rate and selected the one for which the prediction is closest to the true flux rate. For the four plumes considered here, the best results have been produced with the integrated mass enhancement method . The 95 % confidence intervals were calculated by , and the corresponding data and code are publicly available.

For the MetFluxNet network, the ground truth is within the 95 % confidence interval for the six plumes. In particular, MetFluxNet makes the best prediction in three cases out of six with predictions very close to the exact value. This shows that false positives, such as the ones we can see in Fig. b and c, do not prevent a good source rate estimation. A possible explanation is the plume alignment: as the position of the plume is fixed in the image, pixels far from it should have a small weight in the final source rate computation. We show in Fig.  that variations in the background could lead to significant prediction changes. However, when applying bounding boxes, we modify the value of a high number of pixels, but the brightest false positives visible in Fig.  represent only a few dozen pixels.

It should be noted that even if MetFluxNet can correctly estimate the source rate in the presence of a false positive in the image, it might not be the case if the plume intersects the false positive. Indeed, here the false positive is outside the plume and does not match the plume alignment criterion. However, if the false positive intersects the plume, it might be interpreted as a local methane concentration maximum which are commonly observed due to eddies. In this case, it can lead to an overestimation of the source rate, as the false positive would increase the apparent total mass of methane.

Overall, the results of MetFluxNet are closer to ground truth than those presented in for the PRISMA and EnMAP plumes. Even though IME gives the best result for two of the plumes, the ground truth is within the confidence interval produced by MetFluxNet for those cases. Moreover, most of our confidence intervals are also smaller than those of . This shows that MetFluxNet works not only on simulations, but also for real plumes.

To show the improvement brought about by MetFluxNet, we compare it with popular methods to estimate the source rate of point source methane emissions detected with satellite imagery such as IME and S2MetNet . S2MetNet is a deep learning model based on the EfficientNetV2-L architecture which is then fine-tuned on a simulated dataset generated with LES. Here, we reproduce a version of S2MetNet on MicroL to compare it with MetFluxNet. The methods described here are tested on the datasets MicroL and S2Test.

The results of the above methods are presented in Table . The standard IME version presented here is computed with the effective wind model of and the corresponding masking procedure. The IME-MicroL version is computed with our effective wind model and our masking procedure. IME and IME-MicroL give similar results on S2Test but not on MicroL. This is due to the effective wind model of IME-MicroL being fitted on a dataset closer to the testing set of MicroL than IME. With the same methodology for IME and an effective model fitted on the training set of MicroL, the standard IME could obtain better results.

We can also observe that both versions of the IME are widely outperformed by deep learning methods. When comparing the deep learning methods, MetFluxNet has a lower RMSE and MAPE than S2MetNet on both datasets. On MicroL, the RMSE of MetFluxNet is about 150 kg h⁻¹ lower, and the MAPE is more than 1 % lower. On S2Test, the RMSE of MetFluxNet and S2MetNet are very close to each other, but in terms of MAPE the gap is the same as for MicroL. This means that MetFluxNet significantly outperforms S2MetNet for the low source rates. Moreover, MetFluxNet relies on a much lighter model than S2MetNet. The ConvNeXt-Tiny architecture has only 28.6 million parameters, whereas EfficientNetV2L has 119 million parameters. Hence, MetFluxNet is easier to train than S2MetNet and also performs better.

When comparing the results of MetFluxNet on MicroL and S2Test, we see that MetFluxNet performances are worse on S2Test than on MicroL. The RMSE is about 850 kg h⁻¹ higher, and the MAPE is 4.4 % higher. This can be explained by the fact that the dataset of comes from a different simulation setup and is therefore farther from the training set than the data from our simulations. This difference in RMSE and MAPE does not mean that MetFluxNet cannot generalize to different plumes. As we saw in the previous section, it accurately estimated the source rates for the real plumes we tested. Moreover, our method performs better on S2Test than S2MetNet or the IME; this makes MetFluxNet a method well suited for real applications.

The main reason why MetFluxNet outperforms S2MetNet is the plume alignment, which simplifies the source rate estimation problem. S2MetNet is trained to estimate the source rate of a plume in any direction. This can be useful because it can be difficult to perfectly align plumes. In Fig. , we compare the RMSE and MAPE of MetFluxNet and S2MetNet with respect to the angle between the shape of the plume and the x axis. We consider angles between -45 and 45° as this range represents a reasonable error range for the plume alignment.

We can observe that MetFluxNet outperforms S2MetNet for all angles tested. In particular, MetFluxNet gives the best results even when there is a large angle between the plume and the x axis. This is due to the fact that the training set naturally contains examples with a non-zero angle between the plume and the x axis, as explained in Sect. . Moreover, it is still easier for MetFluxNet to learn to predict in a range of 140° (which is the range of angles in MicroL) than for S2MetNet to learn to predict for any possible angle. Hence, even without perfect alignment, it is preferable to restrain the plume to a cone rather than allowing unrestricted rotations; the restriction to a cone being easier to satisfy than an exact plume alignment.

We also notice that the MetFluxNet estimation error is rather stable in the range -45 to 45°. The order of magnitude of the MAPE variation is 0.5 %. Thus, the prediction quality is the same with an angle of 0 and 45°. There was no overfit of the network around 0°. Therefore, MetFluxNet is robust to a misalignment of the plumes.

To show that training with MicroS necessarily leads to overfitting, we compare a network trained on MicroS to a network trained on MicroL. We name MicroSnet and MicroLnet the networks trained on MicroS and MicroL, respectively. Both networks are trained without any plume mask, on aligned plumes, and with MSE loss.

In Fig. , we can compare the results of MicroLnet and MicroSnet to MicroL, MicroS, and S2Test. The RMSE and MAPE values are also summarized in Table . MicroLnet gives results of the same order of magnitude as MicroL and MicroS. MicroSnet outperforms MicroLnet on MicroS, which was to be expected, but has a higher RMSE and MAPE than MicroLnet on MicroL. In particular, the RMSE of MicroSnet almost triples between MicroS and MicroL.

MicroSnet performs well on MicroS because the training set and the testing set are too similar. As we saw in Fig. , with a 10 s time step, the testing set contains plumes that are practically identical to those found in the training set. Even if MicroSnet is trained on more samples than MicroSnet (according to Table ), it generalizes poorly on MicroL because the training samples are too similar. On the other hand, MicroLnet has RMSE and MAPE of the same order of magnitude on MicroL and MicroS, which shows that the network did not overfit. On S2Test, MicroSnet slightly outperforms MicroLnet, but the performance of both networks is way lower than those trained on their respective testing set.

Thus, MicroSnet clearly overfits the MicroS dataset. It performs very well on the testing set of MicroS, but the performance for this dataset does not correctly represent the ability to quantify source rate under real conditions. Even if the RMSE and MAPE of MicroSnet are of the same order of magnitude as those of MicroLnet when tested on MicroL and S2Test, it is necessary to have a dataset additional to MicroS, to be able to properly evaluate the results of MicroSnet. Therefore, we can simply work with MicroL, as working with MicroS would require using another dataset anyway.