It is generally assumed that the SRS matrix M is correct and the true source term minimizes error of y=Mx. However, M is prone to errors due to a number of approximations in the formulation of the atmospheric transport model and use of uncertain weather analysis data as input to the atmospheric transport model. Therefore, one should rather consider a hypothetical model, 2y=M+ΔMx, where M is the available estimate of the sensitivity matrix from a numerical model, and the term ΔM is the deviation of the estimate from the true generating matrix, Mtrue=(M+ΔM). Exact estimation of ΔM is not possible due to a lack of data; however, many existing regularization techniques can be interpreted as various simplified parameterizations of ΔM.

The L2 norm

The analysis can be generalized to a quadratic norm with arbitrary kernel R; however, we will discuss its simpler version for clarity.

In , the source term inversion problem is formulated as in Eq. () with choices 6Ξ=B+ϵDTD, where matrix B is the selected or estimated precision matrix, the matrix D a discrete representation of the second derivative with diagonal elements equal to -2 and equal to 1 on the first sub-diagonals, and the scalar ϵ is the parameter weighting the smoothness of the solution x.

Minimization of Eq. () does not guarantee the non-negativity of the estimated source term x. To solve this issue, an iterative procedure is adopted whereby minimization of Eq. () is done repetitively with reduced diagonal elements of B related to the negative parts of the solution, thus tightening the solution to the prior source term, which is assumed to be non-negative. The diagonal elements of B related to the positive parts of the solution can, on the other hand, be enlarged up to a selected constant. This can be iterated until the absolute value of the sum of negative source term elements is lower than 0.01 % of the sum of positive source term elements. Formally, Bj,j in the ith iteration is calculated as 7Bj,j(i)=0.5Bj,j(i-1)for xj(i-1)<0,min⁡{1.5Bj,j(i-1),errx}for xj(i-1)≥0. We observed very low sensitivity to the choice of the recommended values of 0.5 and 1.5 in Eq. (). In most cases, varying these values does not lead to any serious differences in the resulting estimate. However, the selection of parameters xa, errx, and ϵ is crucial and will be discussed in Sect. .

The method is summarized as Algorithm 1 and will be denoted as the optimization method in this study. The maximum number of iterations is set to 50, which was enough for convergence in all our experiments. To solve the minimization problem (), we use the CVX toolbox for MATLAB.

In , the problem was addressed using a Bayesian approach. The difference from the optimization approach is twofold. First, it has a different approximation of the covariance matrix Ξ: 8Ξ=LΥLT, where matrix L models smoothness and matrix Υ models closeness to the prior source term xa. Matrix Υ=diagυ1,…,υn is a diagonal matrix with positive entries, while matrix L is a lower bi-diagonal matrix: 9L=1000l11000⋱⋱000ln-11. Second, the Bayesian approach allows us to estimate the hyper-parameters Υ and L from the data.

Specifically, it formulates a hierarchical probabilistic model: 10p(y|x,ω)=NMx,ω-1Ip11pω=Gϑ0,ρ0,12px|L,Υ=tNxa,LΥLT-1,[0,+∞],13pυj=Gα0,β0,j=1,…,n,14plj|ψj=N-1,ψj-1,j=1,…,n15pψj=Gζ0,η0,j=1,…,n. Here, N(μ,Σ) denotes a multivariate normal distribution with a given mean vector and covariance matrix, tN(μ,Σ,[a,b]) denotes a multivariate normal distribution truncated to given support [a,b] (for details, see Appendix in ), and G(α,β) denotes a gamma distribution with given scalar parameters. Prior constants α0 and β0 are selected similarly to ϑ0 and ρ0 as 10-10, yielding a noninformative prior, and prior constants ζ0 and η0 are selected as 10-2 to favor a smooth solution (equivalent to lj prior value -1); see the discussion in . To consider the prior vector xa is novel in the LS-APC model.

To estimate the parameters of the prior model (), (), and ()–(), we will use two inference methods, variational Bayes' approximation and Gibbs sampling.

All mentioned methods are sensitive to a certain extent to the selection of their parameters. Here, we will identify these tuning parameters and discuss their settings in the following experiments. Moreover, we will discuss the selection of the prior source term.

The optimization approach is summarized in Algorithm 1 wherein two key tuning parameters are needed: parameter errx, which affects the closeness of a solution to the prior source term through the matrix B, and parameter ϵ, which affects the smoothness of a solution. In the following experiments, we select the parameter ϵ by experience, while it can be seen that the solution is similar for a relatively wide range of values (a few orders of magnitude). The parameter errx seems to be crucial for the optimization method and sensitivity to the choice of this parameter will be studied, while errx will be referred to as the tuning parameter. Note that heuristic techniques such as the L-curve method cannot be used here because of the modification of the matrix B within the algorithm. This will be demonstrated in Sect.  (Fig. ). The LS-APC-VB method, summarized in Algorithm 2, also needs the selection of initial errx; however, relatively low sensitivity to this choice was reported . The LS-APC-G method, summarized in Algorithm 3, is also initialized using errx, while its sensitivity to this choice is negligible due to the Gibbs sampling mechanism.

To select the prior source term seems to be an even more difficult problem, especially in cases of releases with limited available information. Therefore, we will investigate various errors in the prior source term, which can be considered thanks to controlled experiments in which the true source term is available. We consider the time shift of the prior source term in contrast with the true source term, different scales, and a blurred version of the true source term. These errors can be examined alone or combined, which will probably be more realistic.

While the tuning parameters selected in the previous section are often selected manually, statistical methods for their selection are also available. One of the most popular is cross-validation , which we will investigate in the context of source term determination. The method is really simple: all available data are split into training and testing datasets ytrain, Mtrain, ytest, and Mtest. The training dataset is then used for source term estimation, while the test dataset is used for computation of the norm of the residue of the estimated source term, ||ytest-Mtestx||2. Such an estimate is known to be almost unbiased but with large variance. Therefore, the procedure is repeated several times and the tuning parameters are selected based on statistical evaluation of the results. In this experiment, we repeat the random selection of 80 % of the measurements as the training set and using the remaining 20 % as the test set. For each tuning parameter errx, this is repeated 100 times in order to reach statistical significance of the selected tuning parameter.

We observe that for all choices of the optimization method, the results exhibit two notable modes of solution: the data mode for tuning parameters with minimum impact on the loss function and the prior mode for tuning parameter values that cause the prior to dominate the loss function. This is notable for the results in the range of errx in Fig. . For errx=10-10 the data term dominates the loss function, and all methods converge to a similar answer (note that the data mode is different for different smoothing parameters in the optimization method).

For errx=105, the loss function is dominated by the prior and all estimates converge to xa. Although there are typically only two visibly stable modes of all the solutions (the data and prior mode), we also observe a third mode in the optimization solution, best seen, e.g., in Fig.  in the second row and the fourth column or in the third row and the fifth column, where the error significantly drops. These “sweet spots” are the desired locations that we hope to find by tuning of the hyper-parameters. While they are obvious when we know the ground truth, the challenge is to find them without this knowledge.

An attempt to find the optimal tuning via the L-curve method (i.e., dependence between the norm of the solution and the norm of the residuum between measurement and reconstruction) is displayed and demonstrated in two cases: ETEX ERA-40 with xa 2.0 scaled (Fig. , left) and ETEX ERA-Interim with xa shifted, scaled, and blurred (Fig. , right) for the optimization method with ϵ=10-5. In these cases (and all others), L-curve shapes were not reached at all and thus an optimum regularization parameter cannot be chosen from these plots. The red crosses denote the value corresponding to minima of MAEs. One can see that the sweet spots are on the transition between the data mode and the prior mode of solutions with no specific feature in these measures. More detailed analysis is presented in the next section.

The LS-APC-VB method also exhibits modes of solution; however, the transition between the data mode and the prior source term mode seems to be rather fast. Notably, no such transitions are observed in the case of the LS-APC-G method. This is caused by the fact that the Gibbs sampling is not sensitive to the selection of the initial state, as discussed in Sect. . With the exception of xa as a constant activity (Fig. , eighth column), the LS-APC-VB method performs better than the optimization method when approaching the data mode of a solution. The LS-APC-G method suffers from overestimating the source term in time steps when the true source term is zero and not enough evidence is available in the data. This can clearly be seen in Figs.  and where estimates from the LS-APC-G method are displayed using green lines; see especially the time steps between 15 and 45 h. This is closely related to the low sensitivities in SRS matrices between the 15th and 45th columns; see Fig.  for an illustration.

Here, we will discuss the behavior of the methods around the regions of the tuning parameter with minimum MAE (sweet spots) observed in the case of the optimization method. Note that no such regions are observed in the case of the LS-APC-VB and LS-APC-G methods. The temporal profiles of the estimated source term at different penalization coefficients selected around two different sweet spots are displayed in Figs.  and .

Figure  displays results for the ETEX ERA-40 dataset with the prior source term selected as the 2.0 times scaled true source term. The top graph is a copy of sensitivity to tuning in terms of MAE from Fig.  (second row, fourth column), and labels (a), (b), (c), and (d) indicate selected values of tuning parameters for which the resulting estimated source terms are shown in Fig. . The four estimates illustrate the transition from the data mode of solution (a) to the prior mode of solution (d). The data mode underestimates the true release, while the prior mode overestimates it. As displayed in Fig. b and c, the slow transition between these two modes allows us to approach the true source term closely, since the chosen prior term is only a scaled version of the true release and the sweet spot lies exactly between the two modes. Both the LS-APC-VB and the LS-APC-G methods diverge from the “good” solution since they consider it to be very unlikely with respect to the observed data. Since no heuristics such as the L-curve can identify this tuning as providing good results (see Fig. , left), we argue that choosing the optimal setting of the tuning parameter is not possible without knowledge of the true source term and the occurrence of the sweet spot is only a coincidence.

Figure  displays results for the ETEX ERA-Interim dataset with the prior source term shifted, scaled, and blurred in the same way as in the third row and fifth column of Fig. . Here, the transition is not so sharp as in the previous case since the true source term does not lie exactly on the transition between the data mode (panel a) and the prior mode (panel d). The data mode (a) also contains nonzero elements, mainly in the first half of the source term. The transition can be seen in Fig. b and c where the nonzero activity at the beginning of the data mode is eliminated by using prior source term information, while the nonzero elements are relatively close to the true release (b). In (c), the zero activity in the first half remains due to the prior source term; however, the estimated activity within the true release period moves toward the assumed prior source term. In (d), the estimation is already very close to the chosen prior source term. Once again, the improvement appears to be coincidental rather than systematic.

We note that the two discussed sweet spots are selected as representative cases and other observed sweet spots (see, e.g., Fig. , the second or eighth column) are very similar in nature. By analyzing the sweet spots, we conclude that they represent a transition from the data mode to the prior mode of solution. In some cases, the transition is very close to the true release (see, e.g., Fig. ), while in some cases, no point on the transition path approaches the true solution (see, e.g., Fig. ), and the data or prior mode is the closest.

Since the LS-APC-VB and LS-APC-G methods provide rather stable estimates of the source term, we will investigate the use of cross-validation (CV) in optimization-based approaches. The results of CV for the optimization method with ϵ=10-5 for selected combinations are displayed in Figs. , , and in the top right panels: (i) ERA-40 with xa 2.0 scaled; (ii) ERA-Interim with xa shifted, blurred, and scaled; and (iii) ERA-40 with xa equal to the true source term. The results are displayed using box plots where medians are displayed using red lines inside boxes, while the boxes cover the 25th and 75th quantiles. The mean values of the residuals for each tuning parameter are displayed using magenta lines. The value of the tuning parameter that minimizes the CV error is also visualized in the top left panels using solid vertical red lines inside the graphs of MAE sensitivity from Fig. . Bottom panels of figures display the estimated source terms using the tested methods for the tuning parameter selected by cross-validation together with the true source term (dashed red line) and the prior source term used (full black line).

The results demonstrate significant differences between the prior mode and the data mode of the solution, which can be seen in all cross-validation box plots. This is also the case for xa, which is not displayed here. Notably, the minima of cross-validation are not reached in the positions of the sweet spots, indicating that the observed MAE minima are coincidental. In all tested cases, the minima of cross-validation are reached closer to the data mode than to the prior mode. This is demonstrated for the extreme case of xa equal to the true source term in Fig. . Even for this case, the minimum of cross-validation is associated with the data mode rather than the prior mode.

To evaluate the overall results, we compute the mean MAE over all estimated source terms using the optimization method with ϵ=10-5 and the tuning parameter errx selected using cross-validation (CV) for each prior source term xa. This result is given in Fig.  using a box-and-whisker plot. The same box-and-whisker plots are also computed for the LS-APC-VB method with the same scheme of selection of tuning parameters errx using the cross-validation method (denoted CV in Fig. ) and for the LS-APC-VB algorithm with the tuning parameter errx set to 100 as recommended in (denoted default in Fig. ). These results suggest that the LS-APC-VB method with fixed start (but weighted to data using selection of ω(0)) slightly outperforms other methods in terms of the mean MAE for ETEX data with various assumed prior source terms without the necessity of exhaustive tuning.

The Lagrangian particle dispersion model FLEXPART version 10.3 was used to simulate the transport of radiocesium and to calculate the SRS matrices. FLEXPART was driven by two meteorological reanalyses so that these can be compared. First, ERA-Interim , a European Center for Medium-Range Weather Forecast (ECMWF) reanalysis, was used with a temporal resolution of 3 h and spatial resolution of circa 80 km on 60 vertical levels from the surface up to 0.1 hPa. Second, the ERA-40 ECMWF reanalysis was used at a 125 km spatial resolution. The emissions from the ChNPP are discretized into six 0.5 km high vertical layers from 0 to 3 km. The assumed temporal resolution is 3 h from 00:00 UTC on 26 April to 21:00 UTC on 5 May 1986, for which the forward runs of FLEXPART are done. This period is selected according to the consensus that the activity declined by approximately 6 orders of magnitude after 5 May . This discretized the temporal–spatial domain to 480 assumed releases (80 temporal elements times 6 vertical layers) for each nuclide. For each spatiotemporal element, concentration and deposition sensitivities are computed using 300 000 particles. Following , the aerosol tracers Cs-134 and Cs-137 are subject to wet and dry deposition depending on different particle sizes with aerodynamic mean diameters of 0.4, 1.2, 1.8, and 5.0 µm. The distribution of mass is assumed as 15 %, 30 %, 40 %, and 15 % for 0.4, 1.2, 1.8, and 5.0 µm particle sizes, respectively, following measurements of and computation results of .

The exact temporal profile of the Chernobyl release is uncertain, although certain features are commonly accepted . The first 3 d correspond to higher release with an initial explosion and release of part of the fuel. The next 4 d correspond to lower releases, and the last 3 d correspond to higher releases again due to fires and core meltdown. After these 10 d, the released activity dropped by several orders of magnitude .

We follow the setup of and consider six previously estimated Chernobyl source terms of Cs-134 and Cs-137 for which the criteria of consideration were their sufficient temporal resolution and emission height information. The source terms are taken from (two cases with the same amount of release but slightly different release heights), , , , and . See for detailed descriptions and profiles. The prior source term in our experiment is computed as their average emissions at a given time and height. In sum, the total prior releases of Cs-134 and Cs-137 are 54 and 74 PBq, respectively.

The uncertainties associated with measurements are relatively high since both concentration and deposition measurements are used from the dataset . As was pointed out by and , deposition measurements may be biased by an unknown mass of radiocesium already deposited over Europe from, e.g., nuclear weapons tests. This mass has, however, been reported and already subtracted from the dataset. Still, similarly to , we consider relative measurement errors of 30 % for concentration measurements and 60 % for deposition measurements, while the absolute measurement errors are handled in the same way as in . Here, the measurement vector and SRS matrix are preconditioned (scaled) using the matrix R, which is a diagonal matrix with σabs2+σrel∘y2-1/2 on its diagonal, where σabs is assumed absolute error, σrel is assumed relative error, and ∘ denotes the Hadamard product (element-wise multiplication). The scaling is then yscaled=Ry and Mscaled=RM.

In this case, direct comparison of the estimates with the true emission profile is not possible since this remains unknown. Therefore, we will provide results of the tested methods as the sensitivity of the total estimated release activity to tuning parameters in the same way as in Sect. . Note that the total release activity is a sum of releases from all six vertical layers and all four aerosol size fractions. Due to this composition of the problem, the selection of the smoothness parameter ϵ in the case of the optimization approach is relatively difficult since specific selection may fit better for one vertical layer than for another. We will provide results for two settings of this parameter, ϵ=10-2 and ϵ=10-4, leading to two different behaviors of the optimization method.

The resulting estimates of the total released activity are displayed in Fig.  where the total of the prior source term used xa is displayed with a dashed red line (same for all tested settings of the tuning parameter errx). The estimated total release activity with the use of the prior source term xa is displayed using full lines with colors given in the legend in Fig. , while estimations without the use of this prior source term, i.e., with xa=0, are given using dashed lines and respective colors.

Similarly to the ETEX results, the results in Fig.  suggest the occurrence of two main modes of solution, the data mode and the prior mode, with a smooth transition between them in the case of the LS-APC-VB and optimization methods. The LS-APC-G method (evaluated only at four points denoted by green squares due to high computational costs) has, again, low sensitivity to the initialization of the tuning parameter. However, the results of the LS-APC-G method are close to the data mode of the remaining method, or higher than those. Contrary to the previous results, the LS-APC-VB algorithm does not provide a stable solution and suffers from the need to select the tuning parameter. This signifies that the problem is ill-conditioned even with the proposed regularization term; thus, VB converges to various local minima. The optimization method with both settings of the smoothness parameter also has two modes of solution. In the prior mode of solution (higher values of the tuning parameter), both settings approach the same total release activity for both nonzero (full lines) and zero (dashed lines) prior source terms. The prior mode is dominated by the prior source term used for an arbitrary smoothness parameter ϵ. The difference can be seen in the data mode whereby about one-third higher total released activity was estimated for smoothness parameter ϵ=10-4 than for smoothness parameter ϵ=10-2 on the same level of tuning parameter errx. This is caused by the penalization of high peaks of activity in the case of ϵ=10-2. Thus, in the data mode of solution, the smoothness parameter is much more important than the prior source term used, which plays almost no role here.

Notice that the estimated mass is higher in the data mode than in the prior mode. This means that the model constrained by the measurement data alone would support a higher total release amount than the a priori source term. The true source term is not known; however, it is likely that the data mode overestimates the true total release. This can happen when the SRS matrix is biased. For instance, removal of radiocesium that is too rapid would lead to estimated air concentrations with the correct source term that are too low, and the inversion would compensate for the bias by increasing the posterior source term (notice, though, that deposition values would in this case be overestimated at least close to the source, leading to the contrary effect for the deposition data). Regardless, this effect shows that in the data mode, the resulting source term is heavily influenced by possible biases in the transport model.

The same cross-validation scheme as in the case of ETEX (Sect. ) was used here for the Chernobyl datasets. The train–test split was once again 80 %–20 %, and the CV was performed 50 times for each tuning parameter errx. The cross-validation errors are displayed in Fig.  using box plots and associated mean values of the residue errors ||ytest-Mtestx||2. Here, the results are given for the datasets of Cs-134 (top row) and Cs-137 (bottom row) with FLEXPART driven with ERA-40 meteorological fields. We will investigate CV for the tuning of parameters for the optimization and the LS-APC-VB method. The results are presented for two xa configurations in Fig. : LS-APC-VB with xa, LS-APC-VB with xa=0, the optimization method with xa and with smoothness parameter ϵ=10-2, and the optimization method with xa=0 and with smoothness parameter ϵ=10-2. For these, box plots are displayed together with mean residuals using the same types of lines as in Fig. . Moreover, minimal mean residuals are identified and denoted using red circles in Fig.  for each graph, and their associated total activities are displayed in the legend of each graph.

In the case of Cs-134 (top row), the cross-validation was able to determine optimal values of tuning parameters in the case of all tested methods. The total estimated releases associated with these tuning parameters are 87.1 PBq (LS-APC-VB with xa), 56 PBq (LS-APC-VB with xa=0), 69.7 PBq (the optimization method with xa), and 43 PBq (the optimization method with xa=0), which are in accordance with the mean of previously reported total activity of 54 PBq used as a prior. Note that the prior-dominated modes have lower residuals than the data-dominated modes in all cases. This suggests that the prior source term used and applied to the FLEXPART/ERA-40 simulation matches the measurements well. On the other hand, this is not the case for Cs-137 for which the prior-dominated modes have, with the exception of LS-APC-VB with xa=0, significantly higher residuals than the data-dominated modes. This may be caused by two factors. First, the prior source term is less adequate for interpretation of measurements of Cs-137 than those of Cs-134. Second, all methods assume a quadratic loss function, which may be less appropriate for this dataset and could cause overestimation of the source term with the tuning parameter selected using cross-validation in comparison with the previously reported 74 PBq used as a prior. We note that similar results were also observed with the ERA-Interim dataset.

The results suggest that a well-selected prior source term can bind the solution to acceptable values and prevent the occurrence of extreme outliers. On the other hand, we observed that the regularization terms commonly used to compensate for errors of the SRS matrices are not able to compensate for the error caused by inaccurate SRS matrices. Further research is clearly needed to develop a more relevant method of regularization.