In the fall of 2011, iodine-131 (

In the fall of 2011,

Although some ambient concentration measurements are available for
this case, they are quite sparse, poorly resolved in time (typically
sums over 7 days), and cover many orders of magnitude. This makes
an analysis of the impact of the event based on measurement data alone
very difficult. For example, if no measurements are available in the
area of the largest impact, the severity of the event may be grossly
underestimated. Given accurate release information, atmospheric transport
models can simulate the radioiodine dispersion and give a more comprehensive
view of the situation than the measurements alone. For instance, simulations
with atmospheric transport models were used previously to study the
distribution of radioactive material after the Chernobyl (e.g.,

To our best knowledge, the exact source term in the case of the Hungary
iodine release in 2011 is unknown and only approximate and vague information
is available

The range of possible regularization techniques starts with positivity
constraint of the source, simple Tikhonov penalty (see e.g.,

An application of the inverse modeling problem is the source location
problem. If the release site is unknown, the inverse modeling is performed
for many potential release sites and their likelihood of being the
correct site is compared. The simplest scenarios assume a constant
release rate

Measurements of

List of the sampling sites
from which

Typically, the inverse modeling problem is recast as an optimization
problem such as the weighted linear or nonlinear least squares (e.g.,

Recently, a Bayesian inverse method called least-squares method with
adaptive prior covariance (LS-APC) was proposed

In this paper, we use the LS-APC method for inversion for the case
of the iodine release in Hungary in 2011. Moreover, we derive the
variational Bayesian model selection for the LS-APC model. Using this
methodology, we can compare the reliability of each SRS matrix from
the selected spatial domain at a reasonable computational cost. The
same methodology can be used to quantify uncertainty in the evaluation
of the SRS matrix. Specifically, if several possible variants of the
SRS matrix computation are available, the Bayesian model selection
can evaluate their posterior probability, providing an objective guideline
for selection of the most likely dispersion model or weather data.
In this study, we evaluate the probability of the SRS matrices obtained
using backward runs of the dispersion models FLEXPART

Iodine can exist in the atmosphere both as a gas and in the aerosol
phase. Measurements of particulate phase

Atmospheric aerosol sampling was performed using various types of
high-volume samplers with flow rates ranging from 150 to 900 m

After the sampling completion and decay of short-lived radon decay
products, the filters are measured without additional chemical preparation
in laboratories equipped with a high-resolution gamma ray spectrometer.
Since

We follow the concept of linear modeling of the atmospheric dispersion
using a SRS matrix (e.g., see

An estimate of the unknown vector

Before reviewing the full probabilistic model, we would like to illustrate
its relation to the conventional cost optimization. Consider the quadratic
norm of the residues of Eq. (

After reviewing the selected Bayesian inverse method, we will derive a lower bound on its marginal likelihood which will be used for selection of the most suitable model structure. Specifically, we will use this tool to select from multiple SRS matrices arising from different settings of the dispersion model. Multiple SRS matrices may arise, for example when multiple atmospheric transport models are available, when varying model parameters, when multiple meteorological input data are available, or when SRS matrices are computed for each potential release site. The marginal likelihood measure is able to select the most suitable model, with natural penalization for complex models due to the principle of marginalization. Thus, the influence of the estimated tuning parameters (hyper-parameters of the prior) is minimized.

The probabilistic model of

The prior distribution of the source term

The original LS-APC model (Eqs.

The LS-APC model is a hierarchical Bayesian model designed to estimate
its hyper-parameters from the data. For a given model (SRS matrix)

Under the VB approximation

Approximation of the marginal likelihood (Eq.

Source location via marginal log-likelihood
where the observed data are explained by a release from a grid cell
using the LS-APC-VB algorithm for all four tested combinations of
dispersion model and meteorological data: Flexpart-GFS-0.5

An alternative approximation of the posterior (Eq.

The SRS matrices in this work were computed using backward runs of
two alternative models, namely HYSPLIT

As a result, the domain was discretized into

Radioiodine can be present in the atmosphere as molecular

Estimated source terms at locations
selected by the marginal likelihood method, shown in Fig.

FLEXPART (FLEXible PARTicle dispersion model) is a scientific model
used worldwide by many research groups and also operationally, e.g.,
at the Comprehensive Nuclear-Test-Ban Treaty Organization for routine atmospheric backtracking

Simulations in FLEXPART can be carried out on two different output
grids in a single run. The so-called mother grid is usually a global
grid with coarser resolution, whereas the nested grid is a smaller
subdomain with higher horizontal resolution (vertical resolution must
be the same for both grids). Our domain of interest was a nested output
grid with horizontal resolution

The HYSPLIT (HYbrid Single-Particle Lagrangian Integrated Trajectory)
model is a model widely used to simulate atmospheric transport and
dispersion on various levels of complexity. Its applications range
from simple estimation of forward and backward trajectories of air
parcels, to advanced modeling of transport, dispersion and deposition
of air masses on large domains. HYSPLIT adopts a hybrid approach combining
the Lagrangian (moving frame of reference for diffusion and advection)
and Eulerian (fixed model grid for calculation of air concentration)
model methodologies. In this study we applied HYSPLIT model version
4

The model was forced with GFS analyses with horizontal resolution
of

Scatter plots of the measurements

Estimated source terms at locations selected by
the marginal likelihood method, shown in Fig.

In this section, we apply the Bayesian inverse modeling method introduced
in Sect.

Maps of daily concentrations of

The LS-APC-VB inversion method, described in Sect.

In all four cases, the source location mechanism of the LS-APC-VB
method works very well and the maxima of the variational lower bound

We would like to point out that the Bayesian model selection allows us
to compare the likelihood of models for any set of matrices

The Gibbs sampling is computationally too expensive to run it for the full set of potential source locations. However, we ran it for a very small neighborhood around the best location identified with the LS-APC-VB method. The results closely correspond to those of the VB approximation, with occasional changes between the best and second best location. The differences in log-likelihood between models are smaller than in the case of the VB method. The main difference from VB is that the GS approach selects the most likely release to be that of the best location for the Hysplit-GFS-1.0 model.

Sensitivity study of the source location
using measurements without the Budapest station. Marginal log-likelihood
that the observed data are explained by a release from a grid cell
using the LS-APC-VB algorithm for each tested combination of dispersion
model and meteorological data: Flexpart-GFS-0.5

With selected location of the release, we proceed to estimate the
release profile using both approximations, the VB method and the GS method.
Source term estimates for the most likely locations obtained by the
VB approximation for each dispersion model are given in Fig.

The same data were processed using the LS-APC-GS method, which provides
results in the form of samples from the posterior distribution of
the source term. The best values of the marginal likelihood for this
approximation was obtained for model Hysplit-GFS-1.0. The posterior
distributions of the source term for each of the tested models is
displayed in Fig.

With respect to the time variation of the release, all source terms
estimated by the VB method have an emission activity peak around the
reported maximum activity period from 12 to 14 October, confirming
this aspect of the official report. The main difference between the
VB and GS approximations of the source term estimation is that the
results of the VB approach are concentrated around a selected mode
of the posterior distribution, while the GS approach considers all
possible modes. Therefore, the GS results are a collection of many
possible profiles. The posterior distribution in the period of
12 to 14 October is not so narrow but contains a smooth bump. This
is due to low informativeness of the data at temporal resolution,
since the sampling period of the measurements is 7 days for the majority
of the data. The estimates provided by the GS method also provide
higher values of the total release amount than the VB method. We conjecture
that this is due to the property of the VB approximation to yield
a zero source term when the measurements are insensitive to its choice.
The posterior distribution of the Gibbs sampler has also maximum at
zero, but the median is positive. See

The officially reported total release activity was 342 GBq with a maximum
release intensity between 12 and 14 October of 108 GBq and a total release
period from 8 September until 16 November

First, one has to consider uncertainty due to the long sampling period of the
measurements, mostly 7 days. This may lead to a large uncertainty in
estimated source terms since the inversion method tries to capture a source
term with resolution of one day from such a time-insensitive measurement.
The second source of uncertainty is the relatively coarse discretization of the
studied domain and the proximity of the IoI facility and the measuring
station Budapest (approximately 10 km). Since concentration gradients cannot
be resolved within one grid cell, the inversion may try to compensate this by
overestimation of the source term to fit the Budapest measurements. The third
source of uncertainty of the source term is the selected atmospheric
dispersion models (and their parameterizations). For example, both
atmospheric transport models may simulate too short a lifetime of particulate
iodine. This, as for many other models, was found for Cs-137 attached to
particles after the Fukushima Dai-ichi accident

Given all these uncertainties and also the fact that in our study
different atmospheric transport models driven with different meteorological
reanalyses provide different source terms, one should be cautious
in comparing the total estimated release with the reported release
amount. An agreement of the total amount of released

This high sensitivity is at least partly related to the small number of
available

Estimated source terms at locations selected by the marginal
likelihood method, shown in Fig.

Estimated source terms at locations selected by the marginal
likelihood method, shown in Fig.

Using the estimated source location and source term, we can perform a forward
run of the model and study the simulated consequences of the accidental
release. For this purpose, we identify the most probable location of the
release from all cases evaluated by the LS-APC-VB method
(Fig.

The computed concentrations of

The cumulated gamma dose for the whole 3-month period is computed for the
most probable source terms computed using LS-APC-VB and LS-APC-GS methods.
The cumulated gamma dose for the LS-APC-VB estimate is displayed in
Fig.

Since the distance between the measuring site in Budapest (denoted by the
letter A in Fig.

To test the sensitivity of the results to the values from the Budapest
station, we run the source location excluding those measurements. The results
are given in Fig.

Source term estimates done without using Budapest data for the most likely
locations for each dispersion model are given in
Fig.

Similar results are obtained using the GS method, Fig.

Low concentrations of iodine

The performance of the Bayesian methodology was also tested when using less informative data. For this, we removed the most informative measurements from the nearest measurement station. Even in this case, the algorithm was able to locate the source with high accuracy but with significantly higher uncertainty, and the source strength was particularly uncertain. The main reason for this large uncertainty was that all available measurement data (except for those taken at the one close-by station) were collected to the north of the release location. Therefore, releases could not be detected by this network during periods with northerly winds. This demonstrates the importance of the spatial distribution of measurement stations.

Since not all of the laboratories agreed with publication of the used measurement data, the data are available upon request to the corresponding author (for academic purposes).

Truncated normal distribution, denoted as

The moments of truncated normal distribution are

The authors declare that they have no conflict of interest.

This research is supported by EEA/Norwegian Financial Mechanism under project MSMT-28477/2014 Source-Term Determination of Radionuclide Releases by Inverse Atmospheric Dispersion Modelling (STRADI). The authors would like to thank all the laboratories who provided monitoring data. Edited by: Ronald Cohen Reviewed by: two anonymous referees