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 adequately compared to simulated concentrations to limit the weight of modelling errors due to the meteorology used to analyse the images. For such comparisons, the usual pixel-wise

All the metrics are evaluated using first a catalogue of analytical plumes and then more realistic plumes simulated with a mesoscale Eulerian atmospheric transport model, with an emphasis on the sensitivity of the metrics to position error and the concentration values within the plumes. As expected, the metrics with the upstream correction are found to be less sensitive to position error 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. Hence the contribution of the latter to our new metrics is reduced, thus limiting misinterpretation. Furthermore, the new metrics also avoid the double penalty issue.

Near-real-time monitoring of atmospheric gaseous compounds at the scale of power plants, cities, regions, and
countries would allow decision-makers to track the effectiveness of emission reduction policies in the context of climate change mitigation

Here we focus on the use of such images to update the emissions sources on a smaller timescale. This can be done using an inverse method relying on comparisons between the images and the predictions of a CTM. A better match between the observed concentration fields and the simulated one will result from a more accurate source. However, the CTM prediction is bounded by the meteorological conditions used. It takes as inputs temperature, pressure, winds, cloud cover fields, etc. Usually, these atmospheric fields are provided by predictions previously obtained with mesoscale numerical weather prediction models

A better account of position error for observation versus simulation comparison of coherent features is a subject of active research

The objective of this paper is to develop a simple and efficient metric for urban-scale plume images which can level down the difference due to the meteorology while fitting into an inverse framework

In this section, we start by introducing the notation in Sect.

In the present article, we focus on two-dimensional images of the enhancement of the total column of CO

It is also possible to obtain a continuous representation of the image using interpolation (e.g. bilinear). In this case, the image is represented by a two-dimensional field

For each metric definition, we will use either the discrete or the continuous representation of the images, but this will be explicitly mentioned.

Our Gaussian puff model is a simplified model of a concentration field (e.g. concentration at a given altitude or total column concentration in specific conditions). It has the advantage of yielding analytical expressions for the Wasserstein metrics (see Sect.

Example of pixel-wise comparison.

In the Gaussian puff model, we assume that

To compare two concentration fields, one can see to what extent the fields overlap. This provides a pixel-wise (i.e. local) assessment of the discrepancies. The

To identify the origin of the discrepancies,

For this specific value of

Comparison between the distances for the example in Fig.

In the following sections, we further extend the classification of

Flow chart of error splitting.

We propose to address the double penalty issue while still relying on the

The new distance is defined in a way that involves finding the rotation and translation that minimise

Finally, with the optimal transformation

In this section, we introduce the Wasserstein distance, the distance of the optimal transport, as a non-local alternative to the pixel-wise

The optimal transport theory was first introduced in the 17th century by Monge in his famous memoir

In this section, we follow the Kantorovich approach, which means that we will use the discrete representation (see Sect.

The set of couplings

Two interesting properties of the Wasserstein distance can be highlighted. First, this metric is defined for normalised vectors only. This means in our case that the difference in total mass between two images is entirely ignored. Alternative solutions have been proposed to take into account this difference, e.g. the one proposed by

Comparison between the optimal transport interpolation (top panels) and the liner interpolation (bottom panels). In both cases, we interpolate between two puffs using a pseudo time ranging from

Second, following

To compute the Wasserstein distance, we have to determine the optimal coupling matrix

It is possible to show that minimising Eq. (

The

Sinkhorn's algorithm provides a simple and quick solution to the optimal transport problem. However, this formulation raises technical issues. The first is that for small values of

The convergence speed is measured here by the number of iterations.

To accelerate the convergence, we use a high value ofAnother numerical issue appears when

Let us introduce

Combining the log-Sinkhorn algorithm while decreasing

Following the derivation of Sect.

More specifically, we assume that we have two continuous concentration fields

By construction,

Following the approach of Sect.

Finally, an issue with both

In this section, we evaluate and compare the metrics with a database of images built using a set of Gaussian puffs. The database is introduced in Sect.

The database consists of

Both components of

Characteristics of the Gaussian puff database.

The characteristics of the database are shown in Fig.

For our Gaussian puff database, there are four different ways to compute the Wasserstein distance:

The analytical formula Eq. (

The analytical formula Eq. (

The network simplex algorithm

The log-Sinkhorn iterations can be applied using Appendix

Comparison of the different ways to compute the Wasserstein distance over the Gaussian puff database. Relative errors in percent between

The fractional bias over all pairs is no more than 5 % when we compare

In this subsection, we compare the behaviour of the metrics with respect to three error categories: the translation error, the orientation error, and the shape error. Note that the behaviour with respect to the scale error cannot be compared since the

For each pair of images in the database, we define

Correlations between the distances

As expected, the Wasserstein distance

For each pair of images in the database, we define

For each pair of images in the database, we define

Both

To go deeper in our analysis, we now compare the metrics using realistic plumes. This section follows the same organisation as Sect.

We use a simulation database of hourly 3D fields of CO

Experimental setup. The simulation grid from

We ensure that the daily evolution of the hourly emission rate from the source is the same for all plumes. Hence, for a given hour of the day, the difference between two simulated plumes from two different days is due to the meteorology. We build a database that regroups per pair simulated plumes at a given hour but from different days (e.g. day 1 hour 10 versus day 3 hour 10). To get a realistic two-dimensional concentration field, we compute the vertical mean of the concentration weighted by the width of the vertical levels. We ignore the first 2 h of the simulation, to ensure that a plume appears in the image. This leaves 2093 pairs of distinct plumes. The images are cropped to 100

We have applied all three methods, and the differences are shown in Fig.

Comparison of the different ways to compute the Wasserstein distance over the realistic database. Relative errors in percent between

In this subsection, we compare the behaviour of the metrics with respect to the same three error categories as in Sect.

Correlations between the distances

While the correlation between

In this case, the correlations between the metrics and

By construction,

This second study with realistic cases shows that the behaviour of each metric slightly differs from what has been seen in the Gaussian case. Nevertheless, the results confirm that both

As stated in the introduction, the goal of this article is to develop and test metrics that can discriminate errors stemming from imperfect meteorology from other sources of discrepancies. Therefore, following the approach used in the previous sections, we define here four indicators that we consider representative of the difference in meteorological conditions between the two images. We then examine the correlation between these indicators, the previous indicators (

To simplify the analysis, we define four scalar indicators that characterise the meteorological conditions. These indicators focus on the direction and the norm of the wind as experienced by the pollutant during its transportation. For each image, we proceed as follows.

We first average the wind components (three-dimensional fields) in the vertical direction between the surface and the planetary boundary layer (PBL) height. Indeed, the realistic database has been simulated with summer conditions, and hence the plumes are assumed to be dispersed within the PBL. This results in two-dimensional fields for each time snapshot.

We compute the norm and the direction of the averaged winds. This results in two two-dimensional fields for each time snapshot.

We average the norm and the direction over the

We finally compute the time average and time standard deviation of the averaged norm and direction between midnight (the time at which the emissions started) and the time of the image. This results in four scalars:

Using the realistic database, we compute the correlation between

Correlations between

One can notice that

To conclude our study, we now compare the different metrics to the meteorological indicators. The results are reported in Table

Correlations between

According to the correlations shown in Table

The lack of correlation to our meteorological indicators for

In this article, we discussed the use of new metrics for comparing passive gas plumes, practically CO

At first, we emphasised how critical the double penalty issue related to pixel-wise comparison is. The traditional

These four metrics were compared on a specifically designed Gaussian puff database and evaluated according to their correlations with respect to translation error, orientation error, and shape error. The numerical experiments showed that the resolution of the images tends to impact the optimal transport problem. As expected, the two metrics designed to be freed from position errors are not correlated to translation and orientation errors. The

Then we discussed the link between a position error and a variation within the mesoscale meteorology using the same realistic database. Designing relevant scalar indicators related to meteorological variance, we evaluated how specific changes in meteorological conditions lead to an increase in the distance between the plumes. We have seen that the meteorological changes can be correlated to position errors as well as amplitude errors between plumes. This means that removing the position error from the metrics will not make the comparison insensitive to a meteorological change. However, some metrics were found to be more sensitive to specific changes in meteorological conditions. For instance, while the Wasserstein metric is sensitive to changes in the main direction or intensity of the winds, the Hellinger metric is more sensitive to changes in the spread of the wind direction both in time and space. This provides guidelines to enlighten the choice of a metric for a given meteorological situation. By composing this with these new metrics freed from position error and additional scaling terms, we get more manageable metrics that will level down in the weight of modelling error due to the meteorology used in the comparison.

These metrics are used to quantify the proximity of a couple of plumes and could hence be used in an inverse framework, in particular for processing XCO

For an operational purpose, optimising on non-local metrics is much more difficult than on pixel-wise metrics because it requires the computation of non-trivial gradients. The three non-local metrics that we proposed are parameterised. These parameters usually balance a trade-off between computational efficiency and accuracy. In the case of the pixel-wise distance with an upstream correction, the choice of the optimal isometry to apply depends on these parameters. Even though this study could be done with a personal computer, further computation optimisation developments are needed for operational use. Here we are only considering passive tracers, but an extension of the study should be using these metrics for reactive pollutants. However, it requires quantifying the relative impact of chemistry on the shape, scale, and position of the plume.

The key idea here is that meteorology is fixed and bounds our model predictions. We choose to develop metrics that aim to remove the weight of such bound within the comparison to the observation. We could instead consider that meteorology is not fixed and can be seen as additional degrees of freedom to estimate. Thus the Wasserstein metric is interesting because it penalises the position error linearly, but it remains numerically costly compared to pixel-wise metrics. Yet, we have seen that approximating the plume by Gaussian puffs yields a cheap estimate of the true Wasserstein distance. To ease the computation, we suggest using an approximation of the Wasserstein distance, assuming Gaussian puff-like plumes or separable into a Gaussian mixture as in

Log-Sinkhorn algorithm with decreasing

Cost matrix

To minimise Eq. (

Let us define the cost function

Let us first consider the case

If this is not the case, we just have to change

Let us now consider the non-isotropic case:

All the data required to get the presented results are available in the Zenodo repository (

Writing process: mainly PJV and EP with inputs from all co-authors. System and experiment design: PJV, JDLB, AF, MB, and YR. Implementation: PJV. Support in development and use of data: PJV and EP. Analysis: mainly PJV, JDLB, AF, and MB, with feedback from YR, EP and GB.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This study has been funded by the national research project ANR-ARGONAUT no. ANR-19-CE01-0007 (PollutAnts and gReenhouse Gases emissiOns moNitoring from spAce at high ResolUTion). Joffrey Dumont Le Brazidec is supported by the European Union's Horizon 2020 research and innovation programme under grant agreement no. 958927 (Prototype system for a Copernicus CO2service). All the figures were drawn using CVD-friendly colour maps. This was made possible using a Python wrapper around Fabio Crameri's perceptually uniform colour maps

This research has been supported by the national research project ANR-ARGONAUT (grant no. ANR-19-CE01-0007, PollutAnt and gReenhouse Gases emissiOns moNitoring from spAce at high ResolUTion).

This paper was edited by Lok Lamsal and reviewed by two anonymous referees.

^{®}in Machine Learning, 11, 355–607,