Bayesian source reconstruction is a powerful tool for determining atmospheric releases. It can be used, amongst other applications, to identify a point source releasing radioactive particles into the atmosphere. This is relevant for applications such as emergency response in case of a nuclear accident or Comprehensive Nuclear-Test-Ban treaty verification. The method involves solving an inverse problem using environmental radioactivity observations and atmospheric transport models. The Bayesian approach has the advantage of providing an uncertainty quantification on the inferred source parameters. However, it requires the specification of the inference input errors, such as the observation error and model error. The latter is particularly hard to provide as there is no straightforward way to determine the atmospheric transport and dispersion model error. Here, the importance of model error is illustrated for Bayesian source reconstruction using a recent and unique case where radionuclides were detected on several continents. A numerical weather prediction ensemble is used to create an ensemble of atmospheric transport and dispersion simulations, and a method is proposed to determine the model error.

Nuclear facilities release a certain amount of anthropogenic radioactive particulates or gases into the atmosphere, which are transported and dispersed by the wind. These releases can either be routine or accidental.
Several countries run a network of stations to monitor airborne levels of environmental radioactivity

On the international scale, the radionuclide component of the International Monitoring System will consist of 80 stations measuring radioactive particulates (of which at least 40 will be equipped with radioactive noble gas detectors). This network is being set up to verify compliance with the Comprehensive Nuclear-Test-Ban Treaty once it enters into force. In the past, anomalous radionuclide detections were made that are likely linked to a nuclear explosion

If an anomalous

Anomalous radionuclide detections are detections of anthropogenic radionuclides originating from upwind nuclear facilities, where the detected concentration of (a) specific radionuclide(s) and/or the combination of several detected radionuclides are anomalous with respect to the station's detection history and/or with respect to what can be expected from these upwind nuclear facilities operating under normal conditions.

detection occurred (either from a nuclear accident or a clandestine nuclear weapon test), methods are needed that relate the detection with its source or potential sources if the source is unknown. One of these methods is atmospheric transport and dispersion modelling. An atmospheric transport and dispersion model typically simulates the transport, dispersion, dry and wet deposition, and radioactive decay of radionuclides released in the atmosphere. These processes establish a linear relationship between the concentrations at receptors and the release amount at the source. One can calculate such source–receptor relationshipsA significant event of interest will often be accompanied by multiple detections taken at multiple stations.
Statistical methods can then be employed to combine the information from all these detections (and possibly non-detections – observations where the activity concentration is below a minimum detectable concentration) in a meaningful way in order to infer relevant information on the source. In cases with an unknown source, the objective is often to find the source location, release time and release amount. In case of a known source, the source location and perhaps also the release times are known. In that case, the release amount and release height can be inferred to refine a previous release estimate obtained through other ways (for instance, an estimation could be made based on accident scenarios and the known or estimated inventory of a reactor).
The process of inferring information on the source based on observations is called inverse modelling.
Several methods exist, ranging from simply calculating correlations between observations and source–receptor sensitivities to locate the source

Of these methods, the Bayesian inference has the advantage of readily providing an uncertainty quantification on the outcome. However, the quality of the inference and the uncertainty quantification depends on the quality of input uncertainties. Typically, one specifies the observation error and model error. Here, the model error relates to errors in the atmospheric transport and dispersion model. These errors are very hard to readily quantify, mainly because of the underlying numerical weather prediction data that are used to calculate the transport and dispersion

A well-established method to quantify uncertainties in numerical weather predictions is the ensemble method

In Sect.

In autumn 2017, several national and international monitoring networks reported the detection of

Here, we revisit the modelling data used in

Locations of the five stations from which

A total of 12

List of 12 observations with their corresponding sampling times and location.

We have used the source–receptor sensitivities associated with the 12 observations from

The unknown source is described by the following five source parameters:

the longitude of the source (

the latitude of the source (

the accumulated release (

the release start time (

the release end time (

Uninformative bounded uniform priors are used for the source parameters. The prior is designed to allow for all plausible scenarios given the sparse measurement network and under the assumption the detected radionuclides are from the same release.
For the current study, the source longitude is assumed to be between 20 and 80

Definition of a true non-detection, a miss, a false alarm, and a true detection based on the Currie critical level

Likelihood function for two detections:

The posterior distribution was sampled using the general-purpose Markov chain Monte Carlo algorithm MT-DREAM

The detected activity concentration

In this section, only the unperturbed member is used. The model uncertainty thus needs to be specified manually, and the impact of different choices on the posterior is discussed here.
We focus on the inferred source location and not the release period or release amount. This is because the release location is of primary interest in the context of treaty verification. Once a location is found (for instance, based on the location of known nuclear facilities within the posterior source region or based on a seismic signal associated with a nuclear explosion), a new inference could be performed fixing the release location as was done in

In this section, the effect of model error on the posterior is illustrated by varying the parameter

Source location probability maps obtained from the Bayesian source reconstruction. The model uncertainty parameters of the inverse gamma distribution were fixed a priori, and different values for

In this and the next subsection, an alternative model error structure will be discussed involving multipliers or scale factors. Besides being an alternative model error, multipliers could also be used to take into account errors that were not fully captured by the model (such as errors due to local atmospheric features not resolved by the model, measurement errors due to sample inhomogeneity)

Here, multipliers (

The multipliers are additional parameters that need to be estimated during the Bayesian inference.
For the sampling of the parameter, we work with

We apply the multipliers using three different values for the model uncertainty parameter

The results in this section lead to the following question: which of the source location maps shown in Fig.

In this subsection, it is assessed whether the atmospheric transport model error structure can be obtained from the ensemble of source–receptor sensitivities (SRSs). Note that the ensemble is set up to deal with errors arising from the meteorological input data only. While this type of error likely adds the largest contribution to the total model error, other sources of model error are not included.

As our ensemble contains 51 members (one unperturbed member and 50 perturbed members), there are 51 SRS values available for each spatio-temporal grid box and each observation.
In order to obtain the error structure, the data of all spatial grid boxes are aggregated into an uncertainty distribution. This does not necessarily destroy the spatial error correlations in the numerical weather prediction data, since the SRS are the result of an integrated trajectory through the atmosphere associated to a specific observation.
The following procedure is applied in order to find the error structure.

For each SRS file (associated with a certain observation) and for each spatio-temporal grid box, the ensemble median SRS is calculated; each of the 51 SRS values is scaled by its ensemble median.

A Lagrangian particle model can only track a finite number of particles due to computational constraints, and this causes stochastic uncertainty when there are very few particles passing through a geotemporal grid box. However, the SRS variations between ensemble members should represent meteorological uncertainty and should not be impacted by stochastic uncertainty. Therefore, a threshold

Since the SRS files are output every 3 h, the maximum value for the SRS is 10 800 s.

ofThe natural logarithm is applied to all SRSs since these span many orders of magnitude. If any ensemble member has an SRS equal to 0 for a specific grid box, all its 51 SRSs are omitted from the analysis.

The remaining data points are used to make an uncertainty distribution (as in Fig.

Probability density function showing how the atmospheric transport model ensemble members are distributed around the ensemble median (solid black line). Also shown are two fits, one using Eq. (

In this subsection, the SRS ensemble is used to determine the parameters

The following cases are considered to obtain the uncertainty parameters.

Case 1: the parameters are fixed by a priori chosen values; for the fit using Eq. (

Case 2: the parameters are fitted once for all release time intervals and all observations (data are aggregated for all release times and all observations).

Case 3: the parameters are fitted for each observation (data are aggregated for all release time intervals).

Case 4: the parameters are fitted for each observation and each release time interval.

The uncertainty parameters are obtained using the procedures outlined in cases 1 to 4. Following this, for each case the overlap in density is calculated by comparing

Box plots of the overlap between the ensemble densities and the fitted densities using uncertainty parameters obtained in four ways (cases 1 to 4; see Sect.

In this subsection, it is assessed how the fitted uncertainty parameters vary among different observations and different release time intervals.
The motivation for this is as follows: first (somewhat trivially), we can expect the model uncertainty to increase as a function of simulation time. Second, uncertainties are expected to be observation-dependent, since observations are made on different times and at different distances from the source; uncertainties on the trajectories between the receptor and the source will also be affected by the atmospheric conditions along the trajectory, which are expected to be observation-specific.
The interplay of the three uncertainty parameters

The fitted standard deviation for each observation (averaged over all release time intervals) is shown in Fig.

The fitted standard deviation for each release time interval (averaged over all observations) is shown in Fig.

The standard deviation obtained after fitting a Gaussian distribution to the SRS ensemble.

In this subsection, the Bayesian source reconstruction is applied using fitted observation-specific parameters

For comparison, the average probability of the six panels in Fig.

Source location probability map obtained from the Bayesian source reconstruction.

In this section, it is assessed whether additional information can be acquired by considering each ensemble member as an independent scenario, thus performing the Bayesian source reconstruction for each ensemble member separately.
No fitting of uncertainty parameters is applied here, and thus these need to be set a priori. The experiment is performed twice, once using

The Bayesian inference is repeated using the SRS of each ensemble member, so that 51 different posteriors are obtained. These posteriors then need to be aggregated in some way. As before, we focus on the source location probability. While several aggregation methods are possible, here the grid-box-wise mean and maximum probability is taken (normalization is required to ensure that the probabilities sum up to 1). Equal weights were assigned to each ensemble member, since our ensemble is constructed to yield equally likely scenarios or ensemble members. Note that in the case of multi-model ensembles, the latter might not be true, and thus a weighting should be applied based on the skill of each model.

Figure

It seems that overall a similar picture is obtained when running the Bayesian inference for each ensemble member separately compared to the procedure explained in Sect.

Source location probability maps for

Finally, we perform a brief assessment on whether or not each ensemble member is adding new information to the ensemble mean source location probability.
For each perturbed member

Figure

Overlap in source location probability as a proxy for how much new information each ensemble member is adding to the ensemble mean (see text).

Model error has a huge impact on the posterior obtained through Bayesian source reconstruction, a conclusion in agreement with other studies

Both the non-uniformity of the broadening and the shift in the source location probability imply that one cannot simply predict beforehand what the result will be when the Bayesian source reconstruction is repeated with different model error.

In the absence of a way to determine the model error, one could perform multiple Bayesian source reconstructions using different model error formulations as shown in Fig.

Multipliers can be used to represent model error

We found that the ensemble members of source–receptor sensitivities are distributed around their ensemble median (Fig.

The ensemble showed that model error varies among different observations (up to a factor of 2 in the standard deviation when fitting a Gaussian distribution). Therefore, it is expected that having available model error information that is observation-specific can improve the quality of the Bayesian source reconstruction. The model error is also shown to increase when going further backward in time (for this specific case, there was an increase of 30 % during a 3 d period in the standard deviation when fitting a Gaussian distribution).

The source location probability using the fitted model error obtained from the ensemble (Fig.

A scenario-based approach (where each ensemble member is used as input for the Bayesian source reconstruction, instead of using the ensemble to fit the uncertainty parameters) gives results that are more robust against the choice of the uncertainty parameters but are more costly compared to directly fitting the uncertainty parameters. This is because the ensemble introduces model uncertainty that may predominate against the uncertainty prescribed by arbitrarily choosing the uncertainty parameter. No new information is obtained for the source location probability (in other words, one does not lose information when using the ensemble only to fit the uncertainty parameters and to calculate the ensemble median for use in the Bayesian inference). The scenario-based approach might be best in case of a small multi-model ensemble, since the fitting of uncertainty parameters might be difficult due to the difference in skill of each ensemble member.

In a future study, we will apply the different approaches and methods presented in this paper to situations in which the source characteristics are known unambiguously. This will help to better evaluate the different approaches proposed in this paper.

The Flexpart model that was used to generate the SRS data is open source and is available for download

The supplement related to this article is available online at:

All authors contributed to the conceptualization of the study. PDM conducted the simulations and performed the analysis. IH and KU supervised the research. All authors contributed to the manuscript. IH took care of the project administration.

The authors declare that they have no conflict of interest.

The authors would like to thank the reviewers for their constructive comments.

This research has been supported by the Defense Research and Development Canada's Canadian Safety and Security Program (project no. CSSP-2018-TI-2393).

This paper was edited by Slimane Bekki and reviewed by Patrick Armand and two anonymous referees.