Inverse problems arise when the quantity of interest cannot be measured directly. Instead, another quantity y is measured that is related to the unknown parameters x by a forward model F. The forward model maps from parameter space X to measurement space Y. For most problems, X = Rn and Y = Rm. The forward problem is to calculate simulated measurement data y from known parameters x by evaluating the potentially nonlinear forward model, y = F(x). Estimating realistic parameters that explain the given measurements means solving the inverse problem.

The problem of finding parameters that best explain noisy measurements yδ in a least squares sense is equivalent to solving x*=arg⁡min⁡x∈X12‖F(x)-yδ‖22. Often the inverse problem is ill posed. This means that the minimizer x*, if one exists, might be nonunique, and particularly that the inversion is unstable. “Unstable” denotes that small changes in the measurements result in large changes in estimated parameters. In the real world, measurements are never exact. A common assumption is that the noisy measurements yδ can be split up into exact data, y ∈ Y, and noise, δ ∈ Y, such that yδ = y + δ. The exact data are defined by the true parameters x+ via the underlying forward model, i.e., y = F(x+). In this definition, the noise includes errors from the measurements, the forward model, and numerical approximations. Solving Eq. () means fitting the parameters to the exact data and the noise. In ill-posed problems, the retrieved parameters are very sensitive to data, so the true solution x+ is typically far away from the least squares solution x*. Thus, the inverse mapping using Eq. () is not suitable for ill-posed problems (see the Supplement).

The inversion using Eq. () is unstable for ill-posed problems. Tikhonov regularization, by contrast, stabilizes the inversion by adding a convex penalty function ϕ: X → R see, e.g.,: x*=arg⁡min⁡x∈X12‖F(x)-yδ‖22+αϕ(x). Classical Tikhonov regularization uses ϕ(x) = 12‖x-xa‖22 with xa = 0. The regularization parameter α, with α > 0, weights the data fitting and the penalty term. A greater value forces the solution to stay close to the a priori solution xa, while a small value results in a better model–data fit. A number of methods are available to automatically choose a balancing regularization parameter see, e.g.,. We use Morozov's discrepancy principle (see Eq.  in Sect. ), which requires knowledge of the noise level ‖δ‖. A suitable regularization parameter prevents overfitting of the estimated parameters x* to the noisy data via the forward model.

If more detailed noise characteristics are known, these can be introduced by adaptation of the data fitting term. In case of Gaussian noise, penalized, weighted least squares x*=arg⁡min⁡x∈X12‖LδF(x)-yδ‖22+αϕ(x) with a noise covariance R and R-1 = LδtLδ best incorporate this information. The covariance is especially useful for weighting measurements with different uncertainties. For a suitable penalty function, this approach translates to Bayesian inverse modeling (see Sect. ).

We solve the inverse problem using Tikhonov regularization, Eq. (). Information about the measurement noise is introduced in the data fitting term of the Tikhonov functional, and prior information about the unknown parameters is formulated in the penalty function ϕ. In the absence of any prior information, the classical Tikhonov approach uses a 2-norm penalty with zero a priori, ϕ(x) = 12‖x‖22. Among all possible solutions, it chooses the solution that is closest to the origin, i.e., (0, …, 0)t ∈ X, but still reproduces the data. Proximity to the origin means that the solution is simple. In particular, it prevents the oscillating behavior of the parameters as encountered when the inversion is unstable e.g.,chap. 4. If an a priori estimate xa of the parameters is available, it can be included by setting ϕ(x) = 12‖x-xa‖22.

Sometimes it can be useful to penalize the components of the parameter vector x differently. This results in a weighted 2-norm penalty: ϕ(x) = 12‖Lax‖22. In a Bayesian inversion setup Q = 1α(LatLa)-1 gives the covariance of a multivariate Gaussian a priori distribution. Diagonal elements in La weight the parameters, while off-diagonal entries correlate parameters, resulting in smoother estimates in case of positive correlation and vice versa.

A large number of methods are available to solve optimization problems of the type see, e.g.,chap. 5 x*=arg⁡min⁡x∈X12‖LδF(x)-yδ‖22+α2‖Lax-xa‖22. The minimizer x*, if it exists (see Sect. ), can also be interpreted as a maximum a posteriori solution to a Gaussian prior with Gaussian noise in a Bayesian inversion framework seechap. 3.

Localized structures like point sources or edges in the true solution x+ are smoothed out by regularization with the 2-norm and thus disappear in the estimate x*.

The sparsity constraint has become very popular for regularization of inverse problems over the last decade. The 1-norm is used to constrain parameters instead of taking the 2-norm as a penalty function. This results in the optimization problem x*=arg⁡min⁡x∈X12‖LδF(x)-yδ‖22+α‖x‖1. The constraint produces solutions with only a few nonzero components, which are called sparse solutions. Replacing the 2-norm by the 1-norm adds a greater penalty on small components in the solution and favors the inclusion of larger components within the solution. The effect is that the 1-norm constraint selects a sufficient number of components to explain the data while setting the others to zero. In contrast, the 2-norm penalty uses all the components to reproduce the data and avoids large components. Figure  illustrates why parameters that are not sufficiently constrained by the data are set to zero when using the sparsity constraint. Due to this property, the method in Eq. () is called sparse reconstruction.

Flux fields with sporadic local hot spots are a prime example of sparse signals and sparse reconstruction is well suited to identifying such sparse but nonsmooth signals. However, if the true solution x+ is not sparse, reconstruction by Eq. () will still produce a sparse approximation of the solution. Of course, the same is true for other penalty functions promoting certain properties of the estimate. Whether or not a signal is sparse is a matter of the representation system used. The solution might be non-sparse in the natural parameter space but have a sparse representation when transformed into a different space. To make use of the sparsity constraint, we present a representation system that allows for sparse representation for all possible solutions.

The field of signal and image processing offers a variety of transforms designed for sparse representation of oscillations, localized signals, edges, and the like. Options include regular basis transforms, Fourier transforms, wavelets, shearlets, and curvelets see, e.g.,. However, it is not straightforward to find a sparsifying transform for a given application, and often a basis with its unique representation of the state is too restrictive. We consider a more flexible representation system called a dictionary, described in detail in the next section.

A dictionary is a collection of N elementary functions dk ∈ X, called atoms, that can be combined linearly to represent the parameters, so x=∑k=1Nckdk. These atoms can be thought of as building blocks of the signal. By the choice of the atoms, it can be ensured that there is at least one representation for each state x ∈ X. Dictionaries are typically redundant representation systems, meaning that N > dim(X). This quality means that there are infinitely many representations c for the same parameter vector x. We expect that at least one representation in a suitable dictionary is sparse, which means that the true signal x+ is a linear combination of only a few atoms. The sparse reconstruction approach, Eq. (), can be used to select these atoms.

Consider an example for X = R3 with the dictionary D=d1,d2,d3,d4=12100120100001. Each column of D is an atom of norm one (‖dk‖2 = 1). The vector x = (1, 1, 1)t can be represented in the dictionary by coefficients c ∈ R4 in infinitely many different ways as the dictionary is redundant. Some possible representations include 0111,-2221,2001,1212121,… The first representation is a somewhat natural choice as it represents each dimension with a different atom. The third representation has minimal 1-norm and the fourth minimal 2-norm. All other choices have more complicated structures. The third representation is also the sparsest possible representation. This example illustrates not only that the sparsest solution often coincides with the minimum 1-norm solution, but also how a redundant representation system is able to sparsify the signal with fewer nonzero entries than the vector it represents.

We assume that the estimated state in the AIM problem can be sparsely represented in a given dictionary D, leading to the optimization problem c*=arg⁡min⁡c∈C12LδFDc+xa-yδ22+αc1,x*=Dc*, where xa is an a priori estimate of the state. Again, the assumption when solving Eq. () is that the difference between true solution and a priori, x+ - xa, can be approximated by a linear combination of a small number of dictionary atoms dk. In contrast to the sparse reconstruction approach, Eq. (), this assumption does not require that the flux field is sparse. A suitable dictionary provides a sparse approximation to many signals that are non-sparse in the standard representation system see, e.g.,. The approach is thus particularly well suited to sparse problems, but it can also adeptly estimate non-sparse signals. We refer to this approach as sparse dictionary reconstruction.

The previous section formulates the AIM problem as an optimization problem using Tikhonov functionals. In the following paragraphs, we focus on efficient methods to solve problems (Eqs. –).

Henceforth, we only consider linear forward models F(x) := Ax. Nonlinear forward models require additional properties for the existence of a minimizer and might have local minima. They are typically addressed by solving a sequence of linearized problems. Theory for nonlinear inverse problems is still an active field of study. summarize the basic results.

For linear forward models the classical Tikhonov functional, Eq. (), is strictly convex, and thus a unique global minimizer exists. The optimization problem can be solved by exerting the necessary conditions of first order, i.e., setting its derivative equal to zero. This setup leads to the linear equation AtLδtLδA+αLatLax=AtLδtLδyδ+αLatLaxa. The solution to this equation is the minimizer of problem 5. A variety of methods exist to solve this linear equation. Our choice is a conjugate gradient method. Note the similarity of Eq. () to Bayesian inversion when LδtLδ = R-1 and αLatLa = Q-1.

For the sparse reconstruction problem, Eq. (), the functional to minimize is only convex and no longer differentiable everywhere. For this kind of problem, subgradient methods can be applied to find a minimizer. A fundamental contribution was the Iterative Shrinkage Thresholding Algorithm (ISTA) , which is a simple iterative scheme consisting of a gradient and a shrinkage step: xk+1=Sαβxk-βAtLδtLδAxk-yδ,Sλ(x)i:=max⁡xi-λ,0signxi. The step size β must be chosen such that 0 < β < 2/‖LδA‖2. The gradient step adds a non-sparse update to the current iterate. Subsequently, the shrinkage operator Sλ: X → X shrinks the updated parameters componentwise by λ towards zero. This step ensures that only dominant components can increase to non-zero values. The algorithm converges rather slowly, but faster algorithms have been developed . We use the Fast Iterative Shrinkage Thresholding Algorithm (FISTA) .

The sparse dictionary reconstruction problem (Eq. ) translates into the sparse reconstruction problem (Eq. ) by defining Ã: = AD and yδ̃: = yδ - Axa. This reformulation allows the use of the methods stated.

Some problems require a bound on the parameter space: x ∈ M ⊂ X. Examples are nonnegative physical quantities like atmospheric mixing ratios. We only consider a nonnegativity constraint here, but the approach works for general closed convex subsets. When enforcing positivity, we use iterative methods to solve the optimization problems (Eqs. –) and couple the update scheme T with a projection step PM onto the set of permitted parameters M: xk+1=PMTxk. For a positivity constraint, the projection is straightforward when the iteration is carried out in the state space X by setting all negative parameters to zero. However, it can be complicated to translate these constraints to the corresponding space when sparsity is assumed in a different representation system (e.g., a dictionary). For our sparse dictionary reconstruction, we calculate the corresponding parameters xk to the current iterate ck before projecting: xk = Dck. Subsequently, the projection can be performed in parameter space, xk+ = PM(xk). Finally, we have to translate xk+ back into dictionary space. Note that there are infinitely many representations for the same state.

The iterative scheme of the sparse dictionary reconstruction creates a sparse representation ck. Hence, one should also choose the sparsest representation for the projected state xk+, which requires solving for ck+1=arg⁡min⁡c∈C12‖Dc-xk+‖22+α‖c‖1. The problem can be solved using the iterative shrinkage algorithm (see Eq. ). We can speed up the convergence with a good initial value, which is given by the current iterate ck. Moreover, the iteration does not need to run until convergence is reached, as the outcome will change in the next update step. Still, solving Eq. () for each iteration of the update scheme is a costly operation.

The projection step for the dictionary is difficult because the dictionary D is not invertible. We suggest the following heuristic approach: we select a subset of atoms from the dictionary that form a basis. The projection update is then only calculated for these components. This procedure might damage the sparsity of the current iterate. However, sparsity will be created by the next shrinkage step, if the update by the projection was not too large. It is important to note that this idea is heuristic, meaning that the algorithm may not converge against a minimizer of problem  restricted to nonnegative parameters x in some cases.

The methods presented in this paper are formulated as Tikhonov regularizations. The inverse modeling community may be more familiar with the statistical formulation, namely Bayesian inverse modeling. In the following section, we briefly describe how both formulations overlap.

In a Bayesian inverse modeling setup, noise and unknown parameters are assumed to be realizations of known probability distributions. Given these distributions and the forward model, Bayes' theorem is used to infer the a posteriori distribution. The maximizer of the posterior probability density function, called the maximum a posteriori solution, is often presented as a best estimate. Further evaluation of the posterior distribution also yields uncertainty bounds for the estimate.

We previously explained that covariance matrices for the noise or prior translate into weighting matrices for the norms in the Tikhonov formulation (see Sects.  and ). For proper weighting matrices, Tikhonov regularization with 2-norm penalty as in Eq. () is equivalent to a Gaussian prior and a Gaussian noise model. By contrast, Tikhonov regularization with 1-norm penalty from Eq. () translates into a Laplacian prior and Gaussian noise. The probability density functions for Gaussian and Laplacian distributions are shown in Fig. . The Tikhonov approach only aims at the calculation of a best estimate, which compares to the maximum a posteriori solution in the Bayesian approach. Uncertainties can be assessed by additional calculations, which we present in Sect. .

Inversions with non-Gaussian priors, like the Laplacian in Eqs. () and (), rarely have an analytical solution simplifying the calculation of the posterior distribution. The posterior distribution can also be approximated by samples created by Markov chain Monte Carlo methods e.g.,chap. 2. In those cases the computational cost for Bayesian inversion methods can become intractable. Tikhonov methods calculate a best estimate without further information about the underlying distribution. This property makes them more suitable for computationally demanding nonlinear or large-scale problems if an uncertainty analysis is not required.

To judge the quality of an estimate, it is necessary to know the uncertainty associated with each estimated parameter. For Bayesian methods, these uncertainties and the best estimate are deduced from samples of the posterior distribution if no analytical expressions exist. For the Tikhonov methods used in this work, uncertainty estimates are an extra calculation performed after the retrieval of a best estimate. In this section, we present an uncertainty analysis for Tikhonov methods based on chap. 3.

We call the true parameters x+ and the a priori xa. Let R denote a general inversion method, R: Y → X, and F a general forward model, F: X → Y. Then, the best estimate is given by x*=xa+Ryδ-Fxa=xa+RFx+-Fxa+δ. We linearize the forward model and reconstruction method to determine the first-order terms x*=xa+∂R∂y∂F∂xx+-xa+∂R∂yδ. For linear forward models we have ∂F∂x = A, but the reconstruction methods remain nonlinear. Thus the analysis depends on the point of linearization. The total error can be differentiated between smoothing and (total) measurement error: x*-x+=∂R∂y∂F∂x-Ix+-xa︸smoothingerror+∂R∂yδ︸measurementerror. The measurement error describes how noise on the measurement data propagates to errors in the estimated parameters. Recall that the definition of noise used here includes errors in the measurement, the forward model, and numerical approximations. Reconstruction methods try to suppress the effect of noise on the parameters by stabilizing the unstable inversion (e.g., via a penalty term or a Bayesian prior). This modification introduces the smoothing error. For ill-posed problems, a smaller smoothing error results in a greater measurement error and vice versa. The smallest total error is expected when both terms are approximately balanced see, e.g.,chap. 5.

The exact total, smoothing, and measurement error can be calculated from the true solution, x+, the estimate under noisy data, x*, and the estimate to noiseless data using the same regularization parameter as in the noisy case. The errors are given by x*-x+︸totalerror=Rα*yδ-Fxa-Rα*y-Fxa︸measurementerror+xa+Rα*y-Fxa-x+︸smoothingerror. Note that this equation does not require a sensitivity matrix.

In real data problems, the error terms in Eqs. () and () are impossible to calculate as they require knowledge of the true solution x+ as well as the noise δ. Instead, it is common to estimate reliable uncertainties that bound the actual errors.

To find such bounds for the smoothing error, Bayesian methods make additional assumptions about the true solution by applying so-called a priori knowledge. Comparable source conditions also exist for Tikhonov methods e.g.,, but such assumptions are often hard to guarantee.

Without applying a priori knowledge, the best one can do is to analyze the sensitivity matrix S: = ∂R∂y∂F∂x, sometimes called the averaging kernel matrix. For linear forward models A it can be approximated by S:,k:=Ryδ-Axa+AΔxk-Ryδ-Axa‖Δxk‖. The kth column of this matrix expresses how the estimate reacts on a perturbation in the kth parameter of the true fluxes, Δxk. Its structure gives an insight into the smoothing error. For an ideal sensitivity matrix equal to the identity, the smoothing error vanishes (see Eq. ). Thus, the closer the sensitivity matrix is to the identity, the smaller the smoothing error can be expected to be. For nonlinear reconstruction methods R, interpretation of the sensitivity matrix is difficult. The information is only local and cannot predict reactions of the estimate for perturbations different from Δxk via Eq. (). Even the normalization in Eq. () might be misleading, if the amplitude of the perturbation, ‖Δxk‖, is not provided. However, sensitivities are still meaningful for assessing the accuracy of the estimate. We describe further details on how we analyze the sensitivity matrix in Sect. . The numerical computation requires one to solve a reconstruction problem per parameter, which can be done in parallel but might still be infeasible for large-scale problems.

Even in real data problems, one often has access to the noise characteristics. Uncertainty bounds for the measurement error can be approximated via resampling of the noise and recalculation of the estimate under this noise for a sufficient number of samples. The distribution of estimates to different realizations of the noise yields the uncertainties commonly expressed by standard deviations. If the noise characteristics are unknown, resampling can be achieved by bootstrapping of the residual of the estimate see. This numerical approach is computationally demanding, but can be run in parallel.

We estimate emissions for the North American mainland (25–55∘ N and 145–51∘ W) on a 1∘ by 1∘ grid (land grid cells only). We use a combination of synthetic in situ aircraft and tall tower measurements that were available during May to September 2008 from operations by the NOAA Earth Systems Research Laboratory , the US Department of Energy , and the START08 aircraft campaign . Footprints for these measurements, which define the forward model, are calculated using the WRF-STILT model (see Sect. ).

Synthetic methane emissions are generated from the Emission Database for Global Atmospheric Research (EDGAR). We project anthropogenic methane emissions from the EDGAR v3.2 FT2000 inventory onto our model grid. These emissions are constant in time during our observation period. As discussed in , newer versions of the EDGAR inventory are available, but we use this version for reasons of comparison to the previous study. Moreover, we work with simulated data only, so there is no strict need to use the most recent inventory version. Rather the solution should comprise typical features, which we assume holds for this version as well.

The simulated noisy measurements are calculated by applying the linear WRF-STILT forward model to the EDGAR fluxes and adding Gaussian noise of realistic magnitude. The noise vector is sampled from the multivariate Gaussian distribution with a diagonal covariance matrix. estimate the values of this matrix using real observations (Fig. ) and restricted maximum likelihood.

The fluxes are temporally constant in the inversion setup here, so each of the 1469 land grid cells has only one unknown emission parameter (x ∈ R1469). On the other hand, there are 4600 total measurements, so yδ ∈ R4600. Despite the fact that there are more measurements than unknowns, the inverse problem is still ill posed, as many measurements yield similar pieces of information. We estimate the surface fluxes x only by knowledge of the forward model A defined by the footprints, the noisy measurements yδ, and the noise characteristics determined by the noise covariance matrix R. In this scenario R does not have any off-diagonal entries; thus, it is easy to calculate Lδ from LδtLδ = R-1. In general cases we recommend using the Cholesky factorization. For all problems, we use zero a priori, so xa = 0. More advanced a priori models should be considered in real data scenarios.

We start by comparing Tikhonov regularization with the classical 2-norm penalty x*=arg⁡min⁡x∈Rn12‖LδAx-yδ‖22+α*2‖x‖22(L2) and Tikhonov regularization with sparsity constraint x*=arg⁡min⁡x∈Rn12‖LδAx-yδ‖22+α*‖x‖1.(L1) The optimal regularization parameter α* is approximated by Morozov's discrepancy principle for each problem; we start with a value α0 that is certainly too large, so the corresponding minimizer is the a priori xa, here xa = 0. Then, the regularization parameter is reduced iteratively by αk = qαk-1 with 0 < q < 1 and the corresponding minimizer xαk is calculated. For each minimizer we check whether ‖LδAxαk-yδ‖2<τδ‾, where δ‾ = ‖δ‖2 is the expected noise level and τ>1. If it holds, an appropriate regularization parameter α*: = αk is found and the corresponding minimizer xα* is our best estimate x*. In Eq. (), we have δ‾ = m, m = 4600 as the noise is normalized by Lδ.

For a fixed α-value, we use a conjugate gradient method to solve the problem L2 via Eq. (). We include a speed-up by , which detects in early iterations whether or not Morozov's criterion, Eq. (), can be reached for the given parameter. This speed-up allows one to continue with the subsequent smaller regularization parameter α before convergence is reached.

We use FISTA, which is an accelerated version of ISTA (see Sect. ), to determine the sparse reconstruction solution, Eq. (L1), to a fixed α-value. It uses a weighted combination of the last two iterates to calculate an update instead of using the last iteration only.

Figure  shows the methane emission estimates by Tikhonov methods L2 and L1. Small sinks appear in both estimates because both methods are not restricted to positive emissions. It is worth noting that the measurements include some negative values, due to the fact that they are enhancements above a background level and are perturbed by noise. Even without negative observations, the estimated emissions may include negative values, if not explicitly enforced. An overestimation in one grid cell and an underestimation in another might still be consistent with the data because the data are not sufficient to fully constrain all flux parameters in an ill-posed inverse problem.

The estimates differ (Fig. ) but explain the data up to the noise. Most inverse problems are underconstrained by the data. As a result, the penalty term has a large effect on the final estimate and explains many of the differences between the L1 and L2 estimates. As expected, L2 produces an emission field that is smooth. Large sources are avoided to minimize the 2-norm. In contrast, L1 produces emissions that are larger in magnitude but more concentrated to a few pixels. The resulting estimated emission field is sparse.

Often, the sparse emission field better estimates large sources such as those from major cities (see, e.g., Salt Lake City emissions at 111∘ W, 40∘ N). Such large isolated pixel emissions are smeared to regional emissions by L2. However, L1 misplaces some of these large emitters (see, e.g., the San Francisco Bay area). This particular misplacement is present in all methods and is caused by a combination of small footprint information and the measurement noise. The source is placed to the correct grid cell for other realizations of the noise. Also, L1 estimates regional emissions such as those in Kansas or Arkansas falsely as large pixel emitters and neglects many small sources. The estimate given by L2 much better reconstructs these regional emissions. A grid cell by grid cell comparison favors the L2 estimate. L2 also produces a better estimate for the total emissions (see Table ).

The histogram of the EDGAR fluxes in Fig.  supports the use of the sparsity constraint. We assign a random sign to each flux as fluxes are nonnegative, in contrast to the a priori models of L2 and L1. The resulting empirical distribution agrees much better with a Laplacian than with a Gaussian prior. Recall that L1 corresponds to a Laplacian and L2 corresponds to a Gaussian prior.

Based on these preliminary results we conclude that the estimate using L2 is closer to the true EDGAR emissions than L1, but the estimate is not satisfying for the reconstruction of large emitters due to the smoothing effect. Also, this methane emissions case study is not suitable for sparse reconstruction in the standard representation system.

The preliminary results in the previous section show that the classical 2-norm regularization estimates a flux field that is too smooth. Large pixel sources such as those from cities are smoothed out. Sparse reconstruction improves the estimate of these large sources. However, we also observe that regional sources are likely incorrectly represented as point sources and that the total emissions are underestimated. The EDGAR solution is neither smooth nor naturally sparse because we expect methane emissions of differing sizes in most grid cells. We seek to find a representation system that is able to sparsely represent such emission fields.

We employ a dictionary to achieve this goal. We therefore need to select atoms such that the dictionary can sparsely approximate all methane emission patterns. Efficient dictionaries can be created using learning algorithms, but a set of training data is required see, e.g.,. We could extract training data from the EDGAR inventory for other regions, learn a dictionary, and use it for the US setup, but the results could be too optimistic as this will not be an option for real data scenarios. Our approach is to identify typical source shapes and include these shape functions as atoms in our dictionary. The sparse reconstruction approach will then select those atoms that explain the data in the sparsest way.

The 1∘ by 1∘ model grid is too coarse to identify individual sources. Many typical methane sources such as cities, landfills and waste, industrial facilities, and mining do not extend beyond one grid cell. To represent these grid cell emissions efficiently, we include the pixel basis in our dictionary. Metropolitan areas, livestock areas, and oil and gas fields might extend over several pixels though. Thus, we also add circular peak shape functions (see Fig. ). We could add more functions with bigger and more complicated shapes. That approach not only requires more computational time, but also leads to more redundancy. It is our intention to create some redundancy, but only to the degree that it helps to sparsify the representation of our possible emission fields. From numerical experiments, we find that including bigger shape functions does not add value to the reconstruction.

Another option to sparsify the representation is to use atoms that cover a large portion of the domain. A background is best represented by a constant function. With the same argument we could add regional background functions. We in fact find that a division into regions as shown in Fig.  would improve the estimate, but the placement of those regions is partly inspired by looking at the true EDGAR fluxes.

present a geostatistical inversion as an extension of L2. This approach uses a model of the mean in place of a traditional prior emissions estimate. It spatially correlates regions based on geostatistical information such as population density or agricultural use. This setup results in shape functions, which we could include in our dictionary as well. The difference between both approaches is that our coefficients receive penalization to select the atoms during the inversion, whereas the shape functions are preselected and unconstrained in the geostatistical inversion. Weighting the penalty on the coefficients individually would translate from one approach to the other.

For our experiments, we decide not to include atoms that are constructed from EDGAR or geostatistical data. We use a pixel basis, a basis with peaks that extend into the direct neighbors (see Fig. ) and a background function for the entire domain. Thus, we have D ∈ Rn×N with N ≈ 2n + 1. All atoms are normalized in the 2-norm. Generally speaking, our dictionary holds functions to represent processes at different spatial scales.

To estimate the flux parameters x with sparse dictionary reconstruction, we solve c*=arg⁡min⁡c∈RN12‖LδADc+xa-yδ‖22+α*‖c‖1,x*=Dc*.(L1DIC) As before, we use a zero a priori, xa = 0, the discrepancy principle, Eq. (), to determine the optimal regularization parameter α* and FISTA to solve problem L1 DIC for a given α. A pseudocode is included in the Supplement.

For further analysis, we add a positivity constraint on the flux parameters, x ∈ R+n, which we denote by the suffix POS. We use the projection approach described in Sect.  for all methods to enforce positivity. Projecting the iterates of a conjugate gradient method may lead to poor performance, as the special structure of the search directions is lost. Thus, we also use FISTA with a shrinkage operator for the 2-norm penalty when solving L2 POS. Note that the projection step for L1 DIC POS is more demanding as it involves the transition from parameter to dictionary space (see Eq. ). Instead, we apply the suggested heuristic nonnegativity update (see Sect. ) using the pixel basis as an invertible submatrix to correct for negative fluxes. Further details are included in the Supplement.

By enforcing positive parameters, three different constraints determine the final estimate: positivity, data, and minimal norm. Often, these constraints may pull the estimate in different directions. The final estimate depends on the balance between them. Using a projection, positivity is always enforced. The emissions will also explain the given data up to the noise as long as Morozov's discrepancy principle is fulfilled. The most flexible constraint in this setup is thus the smoothness or sparsity assumption defined through the penalty term because it is the most uncertain of all constraints.

We carry out two types of analysis to measure the quality of the estimates. First, we perform an uncertainty analysis based on knowledge about the noise characteristics but without knowledge about the true fluxes, as would be the case for many real data scenarios. As discussed in Sect. , we assess smoothing and measurement error separately by analyzing the sensitivity matrix and by resampling of the noise, respectively. In a second analysis, we make use of the true EDGAR solution and calculate the exact total, smoothing, and measurement errors.

The smoothing error describes the error that results from stabilizing the inversion. It can only be estimated if additional assumptions about the true fluxes are made. Without such assumptions, the best one can do is to analyze the sensitivity matrix (see Eq. ). As mentioned before, an ideal sensitivity matrix is equal to the identity. We address two measures: the column sum and the diagonal. The column sum of the sensitivity matrix should be close to one. Otherwise, the method overestimates (> 1) or underestimates (< 1) in that region. The diagonal of the sensitivity matrix shows the sensitivity of the parameter that is perturbed. Values close to one indicate high confidence in the reconstruction. Smaller values are either a consequence of smoothing or of not being sensitive at all. The latter is captured by looking at the column sum as well.

The measurement error shows the influence of the noise on the estimated parameters. We estimate uncertainty bounds for the measurement error based on 1000 samples of noise as described in Sect. . We express the uncertainties by 2 standard deviations of the empirical distribution of estimates corresponding to these samples. The exact total, smoothing, and measurement error are calculated using Eq. ().

We compare our approaches to state-of-the-art methods studied in . The scope of that article is to analyze different formulations to enforce positive parameters. The methods are the following.

Standard inversion: this is a geostatistical approach following . It does not include a positivity constraint and is taken as a benchmark method in .

Figure  shows the estimated emissions using L2 POS, L1 POS, and L1 DIC POS. Our first observation is that there are no sinks as positive fluxes are enforced by all methods. The results for L2 POS and L1 POS are close to the ones obtained by L2 and L1 (see Fig.  and Sect. ), setting negative fluxes to zero. However, projecting the final estimate (see Sect. ) should be avoided as the mismatch between modeled and measured data increases and thus does not make full use of the information in the data. A projection step in every iteration allows the algorithm to correct for this problem in the next update. We measure significant improvement excluding sinks, especially for L2 (see Table ).

In contrast to L1 POS, the solution of L1 DIC POS does not look sparse, as sparsity is enforced on the coefficients of the dictionary. The background function in the dictionary is selected to represent a base level of small emissions (not visible in the color map). It improves the inversion’s ability to accurately estimate total US emissions. Regionally, other atoms are added and subtracted from this background level. The smooth character of the estimate in many regions is a result of the broader dictionary functions (see Fig. ). The method adds pixel sources where local hot spots are assumed.

L1 DIC POS shows significant improvement in the estimate of localized sources against L2 POS (e.g., when looking at West Coast emissions or Salt Lake City), but a slight setback against L1 POS. Sometimes, these emission peaks might be misplaced (e.g., the San Francisco Bay area). Regions of significant emissions like in the Midwest are often reasonably well reconstructed, but the method still tends to spatially concentrate these sources. This localization property is a consequence of the sparsity constraint because the flux field is represented by as few atoms as possible.

We observe that the locations of significant sources agree much better with both sparsity methods L1 POS and L1 DIC POS than with the classical L2 approach. This result can be explained by the fact that the sparse schemes look for the dominant sources. Even if the magnitude is not captured exactly, the method L1 might be used in applications to identify the center of source locations.

As described in Sects.  and , we assess smoothing and measurement error separately. In a first analysis, we ignore the known, true emission field and analyze the sensitivity matrix and the measurement uncertainties as if we faced a real data problem. In the error analysis, we study the exact total, smoothing, and measurement errors.

In this section, we evaluate the estimates of our Tikhonov-based methods by comparing them to the estimates of the methods studied in (see Sect. ).