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This paper presents the development and application of a deep learning-based method for inverting CO2 atmospheric plumes from power plants using satellite imagery of the CO2 total column mixing ratios (XCO2). We present an end-to-end CNN approach, processing the satellite XCO2 images to derive estimates of the power plant emissions, that is resilient to missing data in the images due to clouds or to the partial view of the plume due to the limited extent of the satellite swath. The CNN is trained and validated exclusively on CO2 simulations from 8 power plants in Germany in 2015. The evaluation on this synthetic dataset shows an excellent CNN performance with relative errors close to 20 %, which is only significantly affected by substantial cloud cover. The method is then applied to 39 images of the XCO2 plumes from 9 power plants, acquired by the Orbiting Carbon Observatory-3 Snapshot Area Maps (OCO3-SAMs), and the predictions are compared to average annual reported emissions. The results are very promising, showing a relative difference of the predictions to reported emissions only slightly higher than the relative error diagnosed from the experiments with synthetic images. Furthermore, the analysis of the area of the images in which the CNN-based inversion extract the information for the quantification of the emissions, based on integrated gradient techniques, demonstrates that the CNN effectively identifies the location of the plumes in the OCO-3 SAM images. This study demonstrates the feasibility of applying neural networks that have been trained on synthetic datasets for the inversion of atmospheric plumes in real satellite imagery of XCO2, and provides the tools for future applications. [ABSTRACT FROM AUTHOR]

We propose denoising diffusion models for data-driven representation learning of dynamical systems. In this type of generative deep learning, a neural network is trained to denoise and reverse a diffusion process, where Gaussian noise is added to states from the attractor of a dynamical system. Iteratively applied, the neural network can then map samples from isotropic Gaussian noise to the state distribution. We showcase the potential of such neural networks in proof-of-concept experiments with the Lorenz 1963 system. Trained for state generation, the neural network can produce samples that are almost indistinguishable from those on the attractor. The model has thereby learned an internal representation of the system, applicable for different tasks other than state generation. As a first task, we fine-tune the pre-trained neural network for surrogate modelling by retraining its last layer and keeping the remaining network as a fixed feature extractor. In these low-dimensional settings, such fine-tuned models perform similarly to deep neural networks trained from scratch. As a second task, we apply the pre-trained model to generate an ensemble out of a deterministic run. Diffusing the run, and then iteratively applying the neural network, conditions the state generation, which allows us to sample from the attractor in the run's neighbouring region. To control the resulting ensemble spread and Gaussianity, we tune the diffusion time and, thus, the sampled portion of the attractor. While easier to tune, this proposed ensemble sampler can outperform tuned static covariances in ensemble optimal interpolation. Therefore, these two applications show that denoising diffusion models are a promising way towards representation learning for dynamical systems. [ABSTRACT FROM AUTHOR]

Because optimal transport (OT) acts as displacement interpolation in physical space rather than as interpolation in value space, it can avoid double-penalty errors generated by mislocations of geophysical fields. As such, it provides a very attractive metric for non-negative, sharp field comparison – the Wasserstein distance – which could further be used in data assimilation (DA) for the geosciences. However, the algorithmic and numerical implementations of such a distance are not straightforward. Moreover, its theoretical formulation within typical DA problems faces conceptual challenges, resulting in scarce contributions on the topic in the literature. We formulate the problem in a way that offers a unified view with respect to both classical DA and OT. The resulting OTDA framework accounts for both the classical source of prior errors, background and observation, and a Wasserstein barycentre in between states which are pre-images of the background state and observation vector. We show that the hybrid OTDA analysis can be decomposed as a simpler OTDA problem involving a single Wasserstein distance, followed by a Wasserstein barycentre problem that ignores the prior errors and can be seen as a McCann interpolant. We also propose a less enlightening but straightforward solution to the full OTDA problem, which includes the derivation of its analysis error covariance matrix. Thanks to these theoretical developments, we are able to extend the classical 3D-Var/BLUE (best linear unbiased estimator) paradigm at the core of most classical DA schemes. The resulting formalism is very flexible and can account for sparse, noisy observations and non-Gaussian error statistics. It is illustrated by simple one- and two-dimensional examples that show the richness of the new types of analysis offered by this unification. [ABSTRACT FROM AUTHOR]

A novel generation of sea-ice models with elasto-brittle rheologies, such as neXtSIM, can represent sea-ice processes with an unprecedented accuracy at the mesoscale for resolutions of around 10 km. As these models are computationally expensive, we introduce supervised deep learning techniques for surrogate modeling of the sea-ice thickness from neXtSIM simulations. We adapt a convolutional U-Net architecture to an Arctic-wide setup by taking the land–sea mask with partial convolutions into account. Trained to emulate the sea-ice thickness at a lead time of 12 h, the neural network can be iteratively applied to predictions for up to 1 year. The improvements of the surrogate model over a persistence forecast persist from 12 h to roughly 1 year, with improvements of up to 50 % in the forecast error. Moreover, the predictability gain for the sea-ice thickness measured against the daily climatology extends to over 6 months. By using atmospheric forcings as additional input, the surrogate model can represent advective and thermodynamical processes which influence the sea-ice thickness and the growth and melting therein. While iterating, the surrogate model experiences diffusive processes which result in a loss of fine-scale structures. However, this smoothing increases the coherence of large-scale features and thereby the stability of the model. Therefore, based on these results, we see huge potential for surrogate modeling of state-of-the-art sea-ice models with neural networks. [ABSTRACT FROM AUTHOR]

The quantification of emissions of greenhouse gases and air pollutants through the inversion of plumes in satellite images remains a complex problem that current methods can only assess with significant uncertainties. The anticipated launch of the CO2M (Copernicus Anthropogenic Carbon Dioxide Monitoring) satellite constellation in 2026 is expected to provide high-resolution images of CO2 (carbon dioxide) column-averaged mole fractions (XCO2), opening up new possibilities. However, the inversion of future CO2 plumes from CO2M will encounter various obstacles. A challenge is the low CO2 plume signal-to-noise ratio due to the variability in the background and instrumental errors in satellite measurements. Moreover, uncertainties in the transport and dispersion processes further complicate the inversion task. To address these challenges, deep learning techniques, such as neural networks, offer promising solutions for retrieving emissions from plumes in XCO2 images. Deep learning models can be trained to identify emissions from plume dynamics simulated using a transport model. It then becomes possible to extract relevant information from new plumes and predict their emissions. In this paper, we develop a strategy employing convolutional neural networks (CNNs) to estimate the emission fluxes from a plume in a pseudo- XCO2 image. Our dataset used to train and test such methods includes pseudo-images based on simulations of hourly XCO2 , NO2 (nitrogen dioxide), and wind fields near various power plants in eastern Germany, tracing plumes from anthropogenic and biogenic sources. CNN models are trained to predict emissions from three power plants that exhibit diverse characteristics. The power plants used to assess the deep learning model's performance are not used to train the model. We find that the CNN model outperforms state-of-the-art plume inversion approaches, achieving highly accurate results with an absolute error about half of that of the cross-sectional flux method and an absolute relative error of ∼ 20 % when only the XCO2 and wind fields are used as inputs. Furthermore, we show that our estimations are only slightly affected by the absence of NO2 fields or a detection mechanism as additional information. Finally, interpretability techniques applied to our models confirm that the CNN automatically learns to identify the XCO2 plume and to assess emissions from the plume concentrations. These promising results suggest a high potential of CNNs in estimating local CO2 emissions from satellite images. [ABSTRACT FROM AUTHOR]

Because optimal transport acts as displacement interpolation in physical space rather than as interpolation in value space, it can potentially avoid double penalty errors. As such it provides a very attractive metric for non-negative physical fields comparison – the Wasserstein distance – which could further be used in data assimilation for the geosciences. The algorithmic and numerical implementations of such distance are however not straightforward. Moreover, its theoretical formulation within typical data assimilation problems face conceptual challenges, resulting in scarce contributions on the topic in the literature. We formulate the problem in a way that offers a unified view on both classical data assimilation and optimal transport. The resulting OTDA framework accounts for both the classical source of prior errors, background and observation, together with a Wasserstein barycentre in between states that stand for these background and observation. We show that the hybrid OTDA analysis can be decomposed as a simpler OTDA problem involving a single Wasserstein distance, followed by a Wasserstein barycentre problem which ignores the prior errors and can be seen as a McCann interpolant. We also propose a less enlightening but straightforward solution to the full OTDA problem, which includes the derivation of its analysis error covariance matrix. Thanks to these theoretical developments, we are able to extend the classical 3D-Var/BLUE paradigm at the core of most classical data assimilation schemes. The resulting formalism is very flexible and can account for sparse, noisy observations and non-Gaussian error statistics. It is illustrated by simple one– and two–dimensional examples that show the richness of the new types of analysis offered by this unification. [ABSTRACT FROM AUTHOR]

We propose denoising diffusion models for data-driven representation learning of dynamical systems. In this type of generative deep learning, a neural network is trained to denoise and reverse a diffusion process, where Gaussian noise is added to states from the attractor of a dynamical system. Iteratively applied, the neural network can then map samples from isotropic Gaussian noise to the state distribution. We showcase the potential of such neural networks in experiments with the Lorenz 63 system. Trained for state generation, the neural network can produce samples, almost indistinguishable from those on the attractor. The model has thereby learned an internal representation of the system, applicable on different tasks than state generation. As a first task, we fine-tune the pre-trained neural network for surrogate modelling by retraining its last layer and keeping the remaining network as a fixed feature extractor. In these low-dimensional settings, such fine-tuned models perform similarly to deep neural networks trained from scratch. As a second task, we apply the pre-trained model to generate an ensemble out of a deterministic run. Diffusing the run, and then iteratively applying the neural network, conditions the state generation, which allows us to sample from the attractor in the run's neighboring region. To control the resulting ensemble spread and Gaussianity, we tune the diffusion time and, thus, the sampled portion of the attractor. While easier to tune, this proposed ensemble sampler can outperform tuned static covariances in ensemble optimal interpolation. Therefore, these two applications show that denoising diffusion models are a promising way towards representation learning for dynamical systems. [ABSTRACT FROM AUTHOR]

A novel generation of sea-ice models with Elasto-Brittle rheologies, such as neXtSIM, can represent sea-ice processes with an unprecedented accuracy at the mesoscale, for resolutions of around 10 km. As these models are computationally expensive, we introduce supervised deep learning techniques for surrogate modeling of the sea-ice thickness from neXtSIM simulations. We adapt a convolutional UNet architecture to an Arctic-wide setup by taking the land-sea mask with partial convolutions into account. Trained to emulate the sea-ice thickness on a lead time of 12 hours, the neural network can be iteratively applied to predictions up to a year. The improvements of the surrogate model over a persistence forecast prevail from 12 hours to roughly a year, with improvements of up to 50 % in the forecast error. The predictability of the sea-ice thickness measured against a daily climatology additionally lays by around 8 months. By using atmospheric forcings as additional input, the surrogate model can represent advective and thermodynamical processes, which influence the sea-ice thickness and the growth and melting therein. While iterating, the surrogate model experiences diffusive processes, which result into a loss of fine-scale structures. However, this smoothing increases the coherence of large-scale features and hereby the stability of the model. Therefore, based on these results, we see a huge potential for surrogate modelling of state-of-art sea-ice models with neural networks. [ABSTRACT FROM AUTHOR]

The quantification of emissions of greenhouse gases and air pollutants through the inversion of plumes in satellite images remains a complex problem that current methods can only assess with significant uncertainties. The anticipated launch of the CO2M satellite constellation in 2026 is expected to provide high-resolution images of CO2 column-averaged mole fractions (XCO2), opening up new possibilities. However, the inversion of future CO2 plumes from CO2M will encounter various obstacles. A challenge is the CO2 plume low signal-to-noise ratio, due to the variability of the background and instrumental errors in satellite measurements. Moreover, uncertainties in the transport and dispersion processes further complicate the inversion task. To address these challenges, deep learning techniques, such as neural networks, offer promising solutions for retrieving emissions from plumes in XCO2 images. Deep learning models can be trained to identify emissions from plume dynamics simulated using a transport model. It then becomes possible to extract relevant information from new plumes and predict their emissions. In this paper, we employ convolutional neural networks (CNN) to estimate the emission fluxes from a plume in a pseudo XCO2 image. Our dataset used to train and test such methods includes pseudo images based on simulations of hourly XCO2, NO2 and wind fields near various power plants in Eastern Germany, tracing plumes from anthropogenic and biogenic sources. CNN models are trained to predict emissions from three power plants that exhibit diverse characteristics. The power plants used to assess the deep learning model's performance are not used to train the model. We find that the CNN model outperforms state of the art plume inversion approaches, achieving highly accurate results with an absolute error about half of that of the cross-sectional flux method. Furthermore, we show that our estimations are only slightly affected by the absence of NO2 fields or a detection mechanism as additional information. Finally, interpretability techniques applied to our models confirm that the CNN automatically learns to identify the XCO2 plume and to assess emissions from the plume concentrations. These promising results suggest a high potential of CNNs in estimating local CO2 emissions from satellite images. [ABSTRACT FROM AUTHOR]

We introduce a proof of concept to parametrise the unresolved subgrid scale of sea-ice dynamics with deep learning techniques. Instead of parametrising single processes, a single neural network is trained to correct all model variables at the same time. This data-driven approach is applied to a regional sea-ice model that accounts exclusively for dynamical processes with a Maxwell elasto-brittle rheology. Driven by an external wind forcing in a 40km×200km domain, the model generates examples of sharp transitions between unfractured and fully fractured sea ice. To correct such examples, we propose a convolutional U-Net architecture which extracts features at multiple scales. We test this approach in twin experiments: the neural network learns to correct forecasts from low-resolution simulations towards high-resolution simulations for a lead time of about 10 min. At this lead time, our approach reduces the forecast errors by more than 75% , averaged over all model variables. As the most important predictors, we identify the dynamics of the model variables. Furthermore, the neural network extracts localised and directional-dependent features, which point towards the shortcomings of the low-resolution simulations. Applied to correct the forecasts every 10 min, the neural network is run together with the sea-ice model. This improves the short-term forecasts up to an hour. These results consequently show that neural networks can correct model errors from the subgrid scale for sea-ice dynamics. We therefore see this study as an important first step towards hybrid modelling to forecast sea-ice dynamics on an hourly to daily timescale. [ABSTRACT FROM AUTHOR]

We introduce a scalable approach to parametrise the unresolved subgrid-scale of sea-ice dynamics with deep learning techniques. We apply this data-driven approach to a regional sea-ice model that accounts exclusively for dynamical processes with a Maxwell-Elasto-Brittle rheology. Our channel-like model setup is driven by a wave-like wind forcing, which generates examples of sharp transitions between unfractured and fully-fractured sea ice. Using a convolutional U-Net architecture, the parametrising neural network extracts multiscale and anisotropic features and, thus, includes important inductive biases needed for sea-ice dynamics. The neural network is trained to correct all nine model variables at the same time. With the initial and forecast state as input into the neural network, we cast the subgrid-scale parametrisation as model error correction, needed to correct unresolved model dynamics. We test the here-proposed approach in twin experiments, where forecasts of a low-resolution forecast model are corrected towards high-resolution truth states for a forecast lead time of about 10 min. At this lead time, our approach reduces the forecast errors by more than 75 %, averaged over all model variables. The neural network learns hereby a physically-explainable input-to-output relation. Furthermore, cycling the subgrid-scale parametrisation together with the geophysical model improves the short-term forecast up to one hour. We consequently show that neural networks can parametrise the subgrid-scale for sea-ice dynamics. We therefore see this study as an important first step towards hybrid modelling to forecast sea-ice dynamics on an hourly to daily timescale. [ABSTRACT FROM AUTHOR]

Under the Copernicus programme, an operational CO2 monitoring system (CO2MVS) is being developed and will exploit data from future satellites monitoring the amount of CO2 within the atmosphere. Methods for estimating CO2 emissions from significant local emitters (hotspots, i.e. cities or power plants) can greatly benefit from the availability of such satellite images, displaying atmospheric plumes of CO2. Indeed, local emissions are strongly correlated to the size, shape and concentrations distribution of the corresponding plume, the visible consequence of the emission. The estimation of emissions from a given source can therefore directly benefit from the detection of its associated plumes in the satellite image. In this study, we address the problem of plume segmentation, i.e. the problem of finding all pixels in an image that constitute a city or power plant plume. This represents a significant challenge, as the signal from CO2 plumes induced by emissions from cities or power plants is inherently difficult to detect since it rarely exceeds values of a few ppm and is perturbed by variable regional CO2 background signals and observation errors. To address this key issue, we investigate the potential of deep learning methods and in particular convolutional neural networks to learn to distinguish plume-specific spatial features from background or instrument features. Specifically, a U-net algorithm, an image-to-image convolutional neural network, with a state-of-the-art encoder, is used to transform an XCO2 field into an image representing the positions of the targeted plume. Our models are trained on hourly 1 km simulated XCO2 fields in the regions of Paris, Berlin and several German power plants. Each field represents the plume of the hotspot, the background consisting of the signal of anthropogenic and biogenic CO2 surface fluxes near or far from the targeted source and the simulated satellite observation errors. The performance of the deep learning method is thereafter evaluated and compared with a plume segmentation technique based on thresholding in two contexts: the first where the model is trained and tested on data from the same region, and the second where the model is trained and tested in two different regions. In both contexts, our method outperforms the usual segmentation technique based on thresholding and demonstrates its ability to generalise in various cases: city plumes, power plant plumes, and areas with multiple plumes. Although less accurate than in the first context, the ability of the algorithm to extrapolate on new geographical data is conclusive, paving the way to a promising universal segmentation model, trained on a well-chosen sample of power plants and cities, and able to detect the majority of the plumes from all of them. Finally, the highly accurate results for segmentation suggest a significant potential of convolutional neural networks for estimating local emissions from spaceborne imagery. [ABSTRACT FROM AUTHOR]

In the next few years, numerous satellites with high-resolution instruments dedicated to the imaging of atmospheric gaseous compounds will be launched, to finely monitor emissions of greenhouse gases and pollutants. Processing the resulting images of plumes from cities and industrial plants to infer the emissions of these sources can be challenging. In particular traditional atmospheric inversion techniques, relying on objective comparisons to simulations with atmospheric chemistry transport models may poorly fit the observed plume due to modelling errors rather than due to uncertainties in the emissions. The present article discusses how these images can be properly compared to simulated concentrations to limit the weight modelling errors due to the meteorology used to analyse the images. For such comparisons, the usual pixel-wise L2 norm may not be a good option, because it is subject to the double penalty issue inherent to its local definition. This issue is characterised by a mutation of any position shift into significant amplitude discrepancies. To circumvent this issue, we propose to either provide an upstream correction of the position misfit between the observed and simulated plumes in the usual L2 norm or use a non-local metric based on the optimal transport theory, such as the Wasserstein distance. All the metrics are evaluated using first a catalogue of analytical plumes and then more realistic plumes simulated with mesoscale Eulerian atmospheric transport model, with an emphasis on the sensitivity of the metrics to position mismatch and the concentration values within the plumes. As expected, the metrics with the upstream correction are found to be less sensitive to position errors in both analytical and realistic conditions. Furthermore, in realistic cases, we evaluate the weight of changes in the norm and the direction of the four-dimensional wind fields in our metric values. This comparison highlights the link between differences in the synoptic-scale winds direction and position error. It is found that discrepancies between two plume images due to wind direction errors in the meteorological conditions are less penalised by our new metrics with the upstream correction than without, thus avoiding the double penalty issue. [ABSTRACT FROM AUTHOR]