To localize and quantify greenhouse gas emissions from cities, gas concentrations are typically measured at a small number of sites and then linked to emission fluxes using atmospheric transport models.
Solving this inverse problem is challenging because the system of equations often has no unique solution and the solution can be sensitive to noise.
A common top–down approach for solving this problem is Bayesian inversion with the assumption of a multivariate Gaussian distribution as the prior emission field.
However, such an assumption has drawbacks when the assumed spatial emissions are incorrect or not Gaussian distributed.
In our work, we investigate sparse reconstruction (SR), an alternative reconstruction method that can achieve reasonable estimations without using a prior emission field by making the assumption that the emission field is sparse.
We show that this assumption is generally true for the cities we investigated and that the use of the discrete wavelet transform helps to make the urban emission field even more sparse.
To evaluate the performance of SR, we created concentration data by applying an atmospheric forward transport model to CO

Understanding anthropogenic greenhouse gas (GHG) emissions is important for scientists and decision makers fighting climate change.
Based on a growing amount of atmospheric observations, studies estimating emission fields of GHG sources and sinks from these observations have been performed on local

determining the difference between the real emissions and the emissions captured by inventories

determining differences between the real and bottom–up estimated emissions for individual emitters

finding emitters which are not captured by inventories (unknown emitters).

An alternative to overcoming these issues are sparse reconstruction (SR) methods

SR for the recovery of GHGs has been proposed by several recent studies.

In this paper, we apply SR to assessing urban GHG emissions. Current GHG emission monitoring systems in cities, such as

As urban emissions, we are using anthropogenic emission inventories from multiple European cities. To overcome the sparsity constraints of SR for non-sparse emissions, we use a wavelet transformation.

We are the first to apply SR to the estimation of urban GHG emissions.
The findings of our work are the following:

Urban emissions are mostly sparse and a third-level wavelet transform performs well in sparsifying urban emissions further.

SR needs fewer measurements than Gaussian prior methods to achieve a similar performance if the emissions are sparse enough.

SR performs well in localizing and quantifying large emitters, leading to the application of finding unknown emitters not captured by emission inventories.

The paper is structured as follows.
Section

This section gives the problem statement of atmospheric inverse problems (Sect.

An inverse problem is a problem in which input parameters should be determined from the observation of a process.
For the problem in this paper, those input parameters

A typical approach in atmospheric sciences to solve inverse problems is Bayesian inversion.
In such a setup, the unknown emissions are assumed to follow a known probability distribution. This probability distribution is referred to as a priori.
Measurements are used to update the a priori, which results in a new probability distribution referred to as a posteriori.
From this posteriori distribution a parameter estimation can be made using a maximum likelihood (ML) detector on the a posteriori.
Since the ML detector acts on the a posteriori, this is commonly referred to as Maximum a posteriori (MAP) detector.
Let us call the probability distribution of the a priori

To show the relation between Bayesian inversion and regularization methods, we show how a Bayesian inversion problem, using Gaussian priors, can be converted to a regularization problem.
Assume that

In this paper, we investigate sparse reconstruction (SR) methods.
To achieve SR, the regularization term has to be changed so that sparse solutions are preferred over non-sparse solutions.
In statistics, such a regularization function is the Lasso regularization function, presented by

Compressed sensing (CS) is a theory which provides sufficient conditions to guarantee best possible reconstruction using the Lasso regularizer, therefore, preventing false discoveries.
The conditions of CS apply to the forward model

CS states that an

In real-world scenarios, coefficients are rarely sparse but often compressible.
This means that the coefficients can be well approximated by sparse coefficients.
A more detailed explanation for compressible coefficients is found in Sect.

Sparsity is one of the key elements for SR.
However, emissions are not always sparse.
In order to make a non-sparse emission field sparse, a transformation into a different domain can be used.
Such transformations include the Fourier transform, wavelet transform, transformations tailored to specific data sets, e.g., by SVD truncation (see

There are several ways to quantify the sparsity of an emission field.
One possibility is to measure the error, using any

Visualization of the compressibility of emission fields by plotting

An example is depicted in Fig.

In this paper, a DWT is used to sparsify emissions, using Haar wavelets.
Throughout the paper, we use the third-level wavelet transform and refer to its matrix as

The emission map

There have been many algorithms proposed for the task of CS and SR.
However, our goal in this paper is not to compare these algorithms, instead, we want to demonstrate the applicability of SR in general for urban GHG emission assessments.
We, therefore, only solve the initial SR problem, given in Eq. (

For the noisy case, we solve a modified form of Eq. (

We compare our results to regularized least squares, which we refer to as least squares (LSs) hereafter. The equation of the LS is given by

Table of symbols and measures used in the paper.

Table

The mean relative error is used if the relative error of emission grid points independent of their contribution to the total amount of emissions is of interest. This metric is useful to evaluate how well a reconstruction method estimates emitters of a certain kind.

Sparsity of the reported emission fields in different European cities. In the wavelet domain, in all of the cases a better or at least as good a sparse approximation of the emission map exists.

In the following, we determine the sparsity and compressibility of real-world emissions.
To do so, we study the CO

In the following, we define the estimation problem (Sect.

In the following, we assume that the influence of the background GHG concentration on the measurements is fully known.
Subtracting this influence from the measurements yields the enhancement produced by the GHG emissions in the domain of interest.
Therefore, the background can be ignored, and the model is simplified to a pure transport emission system.
Let the sensitivity of the GHG concentration measurements

Visualization of a sensing matrix, where the values in the matrix are color encoded.
The entry

Figure

From a physical point of view, these removed emission grid cells are not situated upwind of the measurement stations and, therefore, cannot be well reconstructed using the measurements. For the wavelet domain, we perform this step on the wavelet transformed sensing matrix. Therefore, wavelet coefficients which are weakly sensed and below the threshold are removed.

Map of exemplary measurement locations for Munich using seven measurement stations, which represents a ratio

In our study, we use both artificially created emission fields and real emission fields

The measurements

Visualization of the

Reconstruction results for Munich in the noiseless case using LS, SR, and SR in the wavelet domain are depicted in Fig.

In the following, we identify properties of SR and compare them to the LS using the European city emission inventories.
The properties we identify are interconnection between SR and CS (Sect.

Additionally, in Appendix

We examine the effect of wind coverage on the effectiveness of the sensing matrix for SR.
The term wind coverage is used to measure the range of wind directions during the measurement period.
A wind coverage of 0

Estimation errors for SR in the spatial and wavelet domain and LS reconstruction of

The wind coverage is changed in an interval from 96 to 360

This major improvement in SR for higher wind coverage can be explained using the incoherence property from CS.
The coherence parameter, given in Eq. (

Even though for both domains the coherence for a wind coverage of 360

In the following, we compare reconstruction results of SR in the wavelet and spatial domains.
Accordingly,

Comparison of the LS (yellow) and SR in the spatial (blue) and wavelet (red) domains in terms of relative sparsity

Figure

With fewer measurements, the error in the wavelet domain already plateaus for

Our result demonstrates that the wavelet domain can help to improve SR in those instances for which the spatial domain is not well suited. However, changing the domain also alters the conditions of CS; therefore, conclusions made for CS in the spatial domain cannot be directly transferred to the wavelet domain.

Qualitative and quantitative measure of how well SR and LS estimate the highest emissions for

In the following, we evaluate how well SR can identify unknown emitters. Assuming a good prior estimation of the emission field, where the relative error for the real emissions for each emitter is approximately constant and small, unknown emitters can be assumed to make a huge contribution to the difference between the prior expected measurements and real measurements. Therefore, we evaluate how well the highest emitters are reconstructed to assess the performance of finding unknown emitters.

To do this, we consider the emission inventory data for Munich and Paris. While Munich's emissions inventory is highly compressible, the emission inventory of Paris is not. For the setup, seven measurement stations are used in Munich and 39 in Paris.
This results in

We employ a qualitative and a quantitative measure, both of which are depicted in Fig.

The qualitative measure (see upper panels in Fig.

The quantitative measure (see lower panels in Fig.

The results indicate that SR in the spatial domain works particularly well for the highest emitters (largest 0.4 % in our examples), while SR in the wavelet domain performs well for a broader range of high emitters. The LS performance is much less sensitive to the emission strength of individual strong emitters, which is not beneficial for estimating large unknown emitters.

These results for the same scenario but with fewer measurement stations (

Previous scenarios have presented comparisons between SR and LS using

Relative error for SR, SR with DWT, and LS at different degrees of undersampling, for

Figure

For Paris, where the emissions have a low compressibility, SR in the spatial domain performs worst for

These trends are also confirmed for the emissions of other cities (see Supplement). These results demonstrate that when using SR instead of the LS, a higher undersampling with fewer errors is possible, especially for highly compressible emissions (as in Munich).

To make the results applicable to real-world scenarios, noise also has to be taken into consideration.
There are different types of noise, including measurement noise, transport error, and representation error.
In the following, we assume measurement errors with an SNR typical of column measurements.

Reconstruction errors of SR in the spatial and wavelet domain as well as LS for Munich when varying the SNR and

As emission fields, we use the inventory data for Munich.
Compared to the noiseless case, we increase the number of measurement stations to

Reconstruction errors use SR in the spatial and wavelet domain as well as the LS for Munich with an SNR of 20 dB, where

Here, we show the relative error of the estimated emission fields for the noisy case at different spatial resolutions of the emissions, both for SR in the spatial domain and wavelet domain and for the LS (see Fig.

As a result, they do not provide accurate localization of the emitters.
To overcome this, more measurements are needed for SR in the wavelet domain and LS.
To determine the number of measurements required, we show the relative errors for the highest resolution (1 km

Qualitative and quantitative measure of how well SR and LS estimate the highest emissions with a noise of 20 dB for Munich. The qualitative plot shows how many percent of the highest emissions in the inventory are also contained in the same highest amount of emission of the reconstruction. The quantitative plot shows the mean relative errors for different emission strengths.

Next, we analyze the performance of finding unknown emitters in the noisy case.
For this purpose, we use the same setup as in Sect.

In the following, case studies on emission fields of all cities considered in Sect.

Reconstruction performances for SR in the spatial and wavelet domain as well as for LS by measuring the relative error (1 km

Reconstruction performances for SR in the spatial and wavelet domain as well as for LS by using

To compare the performance, we measure relative errors, relative smoothed errors (for a spatial resolution of 5 km

These results support our previous findings that SR performs well at high resolutions for cities with highly compressible emissions.

Next, we consider the noisy case with an SNR of 20.0 dB and

For the cities that are slightly compressible, SR in the spatial domain performs the worst, while SR in the wavelet domain and the LS perform similarly.

For the highly compressible cities, SR in the spatial domain produces the best results for the 1 km

In this paper, we introduced sparse reconstruction (SR) as a novel method for the inversion of urban GHG emissions and further provided key examples to identify the advantages of SR to inversely model emissions.

SR can be easily integrated into existing top–down frameworks for estimating emission. We examined the applicability of this method by evaluating the sparseness of emissions from several European cities. Our results indicate that the emissions from most of these cities are sparse and that SR is applicable. We also showed that a wavelet transform increases the sparsity of urban emissions, making SR applicable to cities with less sparse emissions. SR is known to have reconstruction guarantees if conditions of compressed sensing (CS) are satisfied. We tested different CS conditions for various wind fields and showed that wind fields with better CS conditions do significantly increase the performance of SR.

Compared to state-of-the-art inversion methods using Gaussian priors, our method requires fewer measurements and provides better localization and quantification of unknown emitters.

SR works best if the underlying representation of the emissions is sparse. In this paper, we employed the wavelet transform to increase the sparsity; however, other transformations, such as a curvelet transform or more general dictionary representations, might be even more suited for specific spatial domains. Finding such transformations which also work well with CS conditions is challenging, and future studies should be devoted to them.

Table

Longitude and latitude boundaries used to create the CO

The sensing matrices we use consist of vectorized footprints generated by the Gaussian plume model with artificial wind data. The Gaussian plume model provides a good enough approximation for Lagrangian particle dispersion models for the purpose of this paper. The footprints are generated for a high-resolution grid using a dynamic Gaussian plume with varying wind speed, direction, and diffusion. The footprints are then scaled to the domain size of the city. Because of the scaling, the wind speed and diffusion changes for each city. Nevertheless, we think that this approach makes the reconstruction performances for different cities more comparable than using different footprints for every city.

For Sect.

Figure

Comparison of a STILT footprint to a Gaussian plume footprint, where different wind fields have been used.

Consider

Reconstruction errors for sparse cities with

Visualization of falsely discovered (dark blue), undiscovered (yellow), and discovered (cyan) emissions in

These results demonstrate the power of SR for sparse emission fields and illustrate the link between SR and CS.
As shown in Sect.

Next, we consider the case with

Therefore, both methods produce the same relative error, which allows us to compare the spatial distribution of the error on an equal basis.
This is the case for

which emitters are found correctly (discovered)?

which emitters are not found (undiscovered)?

which emitters are discovered, even though there is no emitter there (falsely discovered)?

These results demonstrates the advantage of SR for the localization of emitters. This property is especially useful to find unknown emitters, since the spatial certainty of emitters for SR is much higher.

In Sect.

Qualitative and quantitative measure of how well SR and LS estimate the highest emitters for Munich

The results are depicted in Fig.

Uncertainty quantification for sparse reconstruction is distinctly different compared to uncertainty quantification for LS reconstruction. First, while for the LS fit with Gaussian noise a closed-form solution exists, there does not exist any closed-form solution for sparse reconstruction.

In the following we use the uncertainty quantification approach from

Uncertainty of the emission estimates for Munich with

In the following, we show that the changes in relative error for LS in Sect.

Sensitivity map of LS in Munich for a wind coverage of

The code for this paper is written in Matlab 2021a and available on Zenodo:

The supplement related to this article is available online at:

JC and FD conceived the study. BZ and MS performed the data analysis supervised by JC and FD. BZ, JC, and FD wrote the paper. BZ and JC revised the paper according to the reviewer comments. MS contributed to the content of the paper. JC acquired the funding and guided the project.

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.

We would like to thank Alihan Kaplan from the chair of Theoretical Information Technology for his comments and useful discussions. We would also like to thank Stephen Starck for helping us to improve the language of the paper. We are thankful to the editor and the reviewers for their valuable comments and suggestions, which helped to improve the paper. Benjamin Zanger, Jia Chen, and Florian Dietrich were supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (grant nos. CH 1792/2-1 and INST 95/1544).

This research has been supported by the Deutsche Forschungsgemeinschaft (grant nos. CH 1792/2-1 and INST 95/1544).This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Jinkyu Hong and reviewed by Scot Miller and two anonymous referees.