Estimation of the temporal profile of an atmospheric release, also
called the source term, is an important problem in environmental sciences.
The problem can be formalized as a linear inverse problem wherein the
unknown source term is optimized to minimize the difference between
the measurements and the corresponding model predictions. The problem
is typically ill-posed due to low sensor coverage of a release and
due to uncertainties, e.g., in measurements or atmospheric transport
modeling; hence, all state-of-the-art methods are based on some form
of regularization of the problem using additional information. We
consider two kinds of additional information: the prior source term,
also known as the first guess, and regularization parameters for the shape
of the source term. While the first guess is based on information
independent of the measurements, such as the physics of the potential
release or previous estimations, the regularization parameters are
often selected by the designers of the optimization procedure. In
this paper, we provide a sensitivity study of two inverse methodologies
on the choice of the prior source term and regularization parameters
of the methods. The sensitivity is studied in two cases: data from
the European Tracer Experiment (ETEX) using FLEXPART v8.1 and the
caesium-134 and caesium-137 dataset from the Chernobyl accident using
FLEXPART v10.3.
Introduction
The source term describes the spatiotemporal distribution of an atmospheric
release, and it is of great interest in the case of an accidental
atmospheric release. The aim of inverse modeling is to reconstruct
the source term by maximization of agreement between the ambient measurements
and prediction of an atmospheric transport model in a so-called top-down
approach . Since information provided by the
measurements is often insufficient in both spatial and temporal domains
, additional information and regularization
of the problem are crucial for a reasonable estimation of the source
term . Otherwise, the top-down determination
of the source term can produce artifacts, often resulting in some
completely implausible values of the source term. One common
regularization is the knowledge of the prior source term, also known
as the first guess, considered within the optimization procedure .
However, this knowledge could dominate the resulting estimate and
even outweigh the information present in the measured data. The aim
of this study is to discuss drawbacks that may arise in setting
the prior source term and to study the sensitivity of inversion methods
to the choice of the prior source term. We utilize the ETEX (European
Tracer Experiment) and Chernobyl datasets for demonstration.
We assume that the measurements can be explained by a linear model
using the concept of the source–receptor sensitivity (SRS) matrix
calculated from an atmospheric transport model (e.g., ).
The problem can be approached by a constrained optimization with selected
penalization term on the source term
and further with an additional smoothness constraint
used for both spatial and temporal
profile smoothing. The optimization terms are typically weighted by
covariance matrices whose forms and estimation have been studied in
the literature. Diagonal covariance matrices have been considered by
and its entries estimated using the maximum likelihood method. Since the
estimation of full covariance matrices tends to diverge ,
approaches using a fixed common autocorrelation timescale parameter
for non-diagonal entries has been introduced
for atmospheric gas inversion. Uncertainties can be also reduced with
the use of ensemble techniques (see, e.g., ,
and references therein) when such an ensemble is available in the
form of several meteorological input datasets and/or variations in
atmospheric model parameters. Even with only one SRS matrix, the problem
can be formulated as a probabilistic hierarchical model with unknown
parameters estimated together with the source term with constraints
such as positivity, sparsity, and smoothness .
The drawback of these methods is the necessity of selection and tuning
of various model parameters, with the selection of the prior source
term and its uncertainty being the most important.
There are various assumptions on the level of knowledge of the prior
source term used in the inversion procedure. Assumption of the zero
prior source term is common
in the literature with a preference for a zero solution on elements whereby no
sufficient information from data is available. This assumption is
typically formalized as the Tikhonov or
LASSO regularizations or their variants.
Soft assumptions in the form of the scale of the prior source term ,
bounds on emissions , or even knowledge
of total released amount as discussed, e.g., in
can be considered. One can also assume the ratios between species
in multispecies source term scenarios .
However, the majority of inversion methods explicitly assume knowledge of
the prior source term .
This is more or less justified by appropriate construction of the
prior source term based, for example, on a detailed analysis of an
inventory and accident , on previous estimates
when available , or on measured or observed
data . While in well-documented cases
this approach could be well-justified, in cases with very limited
available information or even complete absence of information on the
source term, such as the iodine occurrence over Europe in 2017
and the unexpected detection of ruthenium in Europe in 2017 ,
the use of strong prior source term assumptions could lead to prior-dominated results with limited validity. Although the choice of the
prior source term is crucial, few studies have discussed the choice
of the prior source term in detail and provided sensitivity studies
on this selection as in for the
temporal profile of sulfur dioxide emissions and in
for the spatial distribution of greenhouse gas emissions.
The aim of this paper is to explore the sensitivity of linear inversion
methods to the prior source term selection coupled with tuning of
the covariance matrix representing modeling error. We considered the
optimization method proposed by , as well as its
probabilistic counterpart formulated as the hierarchical Bayesian
model, extended here by a nonzero prior source term with
variational Bayes' inference and with Monte Carlo
inference using a Gibbs sampler .
Two real cases will be examined: the ETEX and
Chernobyl datasets. ETEX
provides ideal data for a prior source term sensitivity
study since the emission profile is exactly known. We propose various
modifications of the known prior source term and study their influence
on the results of the selected inversion methods. The Chernobyl dataset, on the other hand, provides a very demanding case in which only consensus
on the release is available and the source term is more speculative.
Inverse modeling using the prior source term
We are concerned with linear models of atmospheric dispersion using
an SRS matrix ,
which has been used in inverse modeling .
Here, the atmospheric transport model calculates the linear relationship
between potential sources and atmospheric concentrations. The source–receptor
sensitivity is calculated as mij=ci/xj, where xj
is the assumed release from the release site at time j and ci
is the calculated concentration at a receptor ci at the respective
time period. The measurement yi at a given time and location can
be explained as a sum of contributions from all elements of the source
term weighted by mij. In matrix notation,
y=Mx+e,
where y∈ℜp is a vector
aggregating measurements
from all locations and times (in arbitrary order), and x∈ℜn
is a vector of all possible releases from a given time period and all
possible source–receptor sensitivities form the SRS matrix M∈ℜp×n.
The residual model, e∈ℜp, is a sum of model and
measurement errors. While the model looks trivial, its use in practical
applications poses significant challenges. The key reason is that
all elements of the model are subject to uncertainty
and the problem is ill-posed.
In the rest of this section, we will discuss an influence of the modeling
error and show how existing methods approach compensation of such
error. We will analyze in detail two methods for the source term estimation:
(i) the optimization model proposed by
with a prior source term already considered and (ii) a Bayesian model
extended here by prior source term information
and solved using both the variational Bayes' method and the Gibbs sampling
method.
Influence of atmospheric model error
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,
y=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.
of the residuum between measurement and reconstruction for Eq. ()
would become
||y-Mx-ΔMx||22=||y-Mx||22-2yTΔMx+xTΦx3Φ=MTΔM+ΔMTM+ΔMTΔM.
The ideal optimization problem (right-hand side of Eq. )
can be decomposed into the norm of residues of the estimated model
||y-Mx||22, with both
linear (i.e., -2yTΔMx)
and quadratic terms in x (i.e., xTΦx).
Both of the additional terms contribute to incorrect estimation of
x when ΔM is significant.
An common attempt to minimize the influence of the linear term is
to define the prior source term xa and subtract Mxa
from both sides of Eq. (). This yields a new decomposition
derived in Appendix (with substitutions
x‾=x-xa and y‾=y-Mxa)
as
||y-Mx-ΔMx||22=||y‾-Mx‾||22-2y‾TΔM-ΔMxaTM-ΔMxaTΔMx‾+x‾TΦx‾,
where Φ is the same term as in Eq. ().
In the ideal situation, we would like to optimize the left-hand side
of Eqs. () and (). However,
due to unavailability of ΔM the linear term is
typically ignored (assumed to be negligible) and the quadratic term
is approximated using a parametric form of Φ≈Ξ. The
optimization criterion is then
J=||y‾-Mx‾||22+x‾TΞx‾.
The estimation error caused by approximation () can
be influenced by two choices of the user: (i) first guess xa
and (ii) regularization matrix Ξ. The purpose of choosing xa
is minimization of the linear term in Eq. (). The
choice of the parametric form of Ξ corresponds to choosing a model
of the SRS matrix error ΔM, since Ξ is an approximation
of Φ, which is determined by ΔM.
In the following sections, we will discuss methods that estimate
Ξ from the data using parameterization of Ξ by tri-diagonal
matrices with a limited number of parameters. Specifically, we will
investigate if the choice of xa has an impact on better
estimation of Ξ.
Optimization approach
In , the source term inversion problem
is formulated as in Eq. () with choices
Ξ=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
Bj,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.
Bayesian approach
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:
Ξ=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:
L=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.
Variational Bayes' solution
The variational Bayes' solution seeks the posterior
in the specific form of conditional independence such as
p(x,υ,l,ψ1,…,n-1,ω|y)≈p(x|y)p(υ|y)p(l|y)p(ψ1,…,n-1|y)p(ω|y).
The best possible approximation minimizes Kullback–Leibler
divergence between the estimated solution
and hypothetical true posterior. This minimization uniquely determines
the form of the posterior distribution:
17p̃(x|y)=tNxμx,Σx,18p̃(υj|y)=Gυjαj,βj,∀j=1,…,n,19p̃(lj|y)=Nljμlj,Σlj,∀j=1,…,n-1,20p̃(ψj|y)=Gψjζj,ηj,∀j=1,…,n-1,21p̃(ω|y)=Gωϑ,ρ,
for which the shaping parameters μx,Σx,αj,βj,μlj,Σlj,ζj,ηj,ϑ, and ρ
are derived in Appendix . The shaping
parameters are functions of standard moments of posterior distribution,
which are denoted here as x^ and indicate the expected
value with respect to the distribution on the variable in the argument.
The standard moments together with shaping parameters form a set of
implicit equations solved iteratively; see Algorithm 2.
Note that only convergence to a local optimum is guaranteed; hence,
a good initialization and iteration strategy are beneficial (see Algorithm 2 and the discussion in ). The
algorithm is denoted as the LS-APC-VB algorithm.
Gibbs sampling solution
The Gibbs sampler is a Markov chain Monte Carlo method for obtaining sequences
of samples from distributions for which direct sampling is difficult
or intractable . It is a special case
of the Metropolis–Hastings algorithm with the proposal distribution derived
directly from the model . Given
a joint probability density p(x,υ,l,ψ1,…,n-1,ω,y),
a full conditional distribution needs to be derived for each variable
or a block of variables; i.e., for x, distribution p(x|υ,l,ψ1,…,n-1,ω,y)
has to be found. These full conditionals then serve as proposal generators
and have the same form as Eqs. ()–().
We use the original Gibbs sampler from .
Having samples from the last iteration, or a random initialization
for the first iteration, the algorithm sweeps through all variables
and draws samples from their respective full conditional distributions.
It can be shown that samples generated in such a manner form a Markov
chain whose stationary distribution, the distribution to which the
chain converges, is the original joint probability density. Since
the convergence of the algorithm can be very slow, it is common practice
to discard the first few obtained samples. This is known as a burn-in
period. The advantage of this algorithm is its indifference to the
initial state from which sampling starts.
Tuning parameters and prior source term
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.
Tuning by cross-validation
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.
Sensitivity study on the ETEX dataset
The European Tracer Experiment (ETEX) is one of a few large controlled
tracer experiments (see https://rem.jrc.ec.europa.eu/etex/, last access: 26 April 2020). We use
data from the first release in which a total amount of 340 kg of nearly
inert perfluoromethylcyclohexane (PMCH) was released at a constant
rate for nearly 12 h at Monterfil in Brittany, France, on 23 October
1994 . Atmospheric concentrations of PMCH were
monitored at 168 measurement stations across Europe with a sampling
interval of 3 h and a total number of measurements of 3102. The ETEX
dataset has been used as a validation scenario for inverse modeling
(see, e.g., ).
The great benefit of this dataset is that the estimated source terms
can be directly compared with the true release given in Fig.
(first row) using dashed red lines.
The uppermost row of panels shows eight different
prior source terms xa (black lines) used for the ETEX
source term estimation. The true ETEX source term is repeated in every
panel (red dashed line). The middle and the lowermost rows display
the mean absolute error between estimated and true source terms for the ETEX
ERA-40 and ETEX ERA-Interim datasets, respectively.
To calculate the SRS matrices, we used the Lagrangian particle dispersion
model FLEXPART version
8.1. We assume that the release period occurred during 120 h period;
thus, 120 forward calculations of a 1 h hypothetical unit release
were performed and SRS coefficients were calculated from simulated
concentrations corresponding to the 3102 measurements. As a result,
we obtained the SRS matrix M∈R3102×120.
FLEXPART is driven by meteorological input data from the European
Center for Medium-Range Weather Forecasts (ECMWF) from which different
datasets are available. We used two: (i) data from the 40-year reanalysis
(ERA-40) and (ii) data from the continuously updated ERA-Interim reanalysis.
The computed matrices for ETEX are given in Appendix together with their associated singular
values to demonstrate conditioning of the problem.
The tested method will be compared in terms of the mean absolute error
(MAE) between the estimated and the true source term for different
tuning parameters errx. We select two representative
values of the smoothing parameter ϵ for the optimization
method. Specifically, we selected ϵ=10-5 and ϵ=10-4,
while higher values lead to overly smooth and lower values to non-smooth
solutions. We tested eight different prior source terms xa;
see Fig. , top row, black lines. These included the following: xa
equal to (i) the true source term; (ii) zeros for all elements; (iii)
true source term right-shifted by five time steps; (iv) true source term
scaled by a factor of 2.0; (v) true source term blurred using a convolution
kernel of five time steps, left-shifted by five time steps, and
scaled by a factor of 1.3; (vi) true source term substantially right-shifted;
(vii) true source term scaled by a factor of 10; and (viii) source term
with constant activity. The resulting MAEs for all tested methods
and for all eight prior source terms are displayed in Fig.
for ETEX with the ERA-40 dataset in the second column and for ETEX with
ERA-Interim in the third row. The figures in the second and third
rows are accompanied by the MAE between the true source term and the
prior source term used, displayed with dashed red lines.
L-curve-type plots using the optimization algorithm
with ϵ=1e-5 from ETEX ERA-40 with xa 2.0
scaled (a) and from ETEX ERA-Interim with xa shifted,
scaled, and blurred (b). The red crosses denote “sweet spots”.
Results
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.
Desired optima of the estimated source term
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 .
The uppermost panel shows mean absolute errors
between estimated and true source terms for the ETEX ERA-40 dataset with
xa 2.0 scaled for all methods. Certain settings of
the tuning parameter are highlighted and labeled with (a), (b), (c),
and (d). Estimated source terms for these tuning parameter choices
are given in the panels below. The lowermost panel displays the estimated
source terms from the LS-APC-VB and LS-APC-G algorithms for comparison.
Same as Fig. for the ETEX ERA-Interim
dataset with xa shifted, scaled, and blurred.
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.
The top left panel (a) shows the sensitivity of
MAE to the tuning parameter for the ETEX ERA-40 dataset with xa
2.0 scaled. This is a repetition from Fig. . The chosen
optimal setting based on CV is shown with a thick red vertical line.
The top right panel (b) shows the error residuals of the CV experiments
as a function of the tuning parameter. Residuals are shown as box-and-whisker
plots, where the boxes extend between the 25th and 75th percentiles (whiskers
between 2.7 sigmas) and medians are marked with red lines, while mean
values are displayed using a solid magenta line. The lowermost panel (c)
shows the source term obtained with the tuning parameter setting chosen
via CV.
Tuning by cross-validation
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).
Same as Fig. for the ETEX
ERA-Interim dataset with xa shifted, scaled, and blurred.
Same as Fig. for the ETEX
ERA-40 dataset with xa equal to the true source term.
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.
Box-and-whisker plots of the MAE averaged
over all explored prior source terms, with the tuning parameter settings
determined by CV for the optimization method (left) and the LS-APC-VB
method (middle), as well as for the LS-APC-VB method using a default errx
setting of 100.
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.
Sensitivity study on the Chernobyl dataset
We demonstrate the sensitivity of the tuning methods to estimation
of the source term for the Chernobyl accident, which became, together
with the Fukushima accident, a widely discussed case in the inverse modeling
literature. Specifically, we study caesium-134 (Cs-134) and caesium-137
(Cs-137) releases from the Chernobyl nuclear power plant (ChNPP).
For this purpose, we use a recently published dataset
consisting of 4891 observations of Cs-134 and 12 281 observations of
Cs-137. We use the same experimental setup as in ,
which will now briefly be described.
Atmospheric modeling
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 .
Prior source term and measurement uncertainties
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.
Results
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.
Estimated total released activities for both
meteorological reanalyses (ERA-40 and ERA-Interim) and both nuclides
(Cs-134 and Cs-137) using all tested methods; see the label bar on the right
for a line description.
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.
Cross-validation for Chernobyl Cs-134 (top
panels) and Cs-137 (bottom panels) source terms using FLEXPART driven
with ERA-40 meteorological reanalyses. Optima in the sense of cross-validation
are denoted using red circles with total estimated releases reported
in the legends.
Tuning by cross-validation
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.
Conclusions
Methods for the determination of the source term of an atmospheric release
have to cope with inaccurate prediction models often represented by
the source–receptor sensitivity (SRS) matrix. Relying solely on the
SRS matrix using a best estimate of weather and dispersion parameters
may lead to highly inaccurate results. We have shown that various
regularization terms introduced by different inversion methods are
essentially coarse approximations of the error of the SRS matrix, and
thus we can evaluate their suitability using methods of statistical
model validation. We have performed sensitivity tests of inverse modeling
methods to the selection of the prior source term (first guess) and
other tuning parameters for two selected inversion methods: the optimization
method and the LS-APC method .
These were preformed on datasets from the ETEX controlled release and the Chernobyl releases
of caesium-134 and caesium-137.
We have observed that the results have two strong modes of solution:
the data mode for minimal influence of the prior on the loss and the
prior mode for the loss function with significant influence of the prior.
The prior mode is naturally significantly influenced by the choice
of the prior source term. However, the dominant impact on the resulting
estimate has the choice of the regularization. In the case of the
ETEX dataset, good estimates were obtained for every choice of the
prior source term; however, the regularization has to be carefully
tuned. For some choices of the prior source term, the error of the
estimated source term was exceptionally low for good selection of
the tuning parameters. After analyzing these minima, we conjecture
that they are caused by coincidence. These minima are visible only
in comparison with the ground truth; they have no visible impact on
the common validation metrics such as the L-curve or cross-validation
and thus cannot be objectively identified.
We have tested the suitability of the cross-validation approach for selection
of the tuning parameters for both methods. In the case of the
ETEX release, we have observed that this approach tends to select
modes closer to the data mode than the prior mode of solution. However,
this is not the case of the Chernobyl Cs-134 release for which cross-validation
selects solutions close to the prior-dominated mode. This may be caused
by the fact that the prior source term used here fits the measurements well,
and only small corrections by the inversion are needed.
An interesting question is whether it is beneficial to use a nonzero
prior source term at all. Considering ETEX, for which the
true release is known, one can see that the estimates in data modes
are often even better than the considered prior source terms. On the
other hand, when the prior source term used is close to the true release,
which is probably the case for the Chernobyl Cs-134 release, its use
seems beneficial. Also, the prior source term could be valuable in
cases when the release is not fully seen by the measurement network
and thus the measurements do not provide a good constraint for the
source term estimation. However, determining the reliability of the
prior source term is difficult and even impossible in real-world scenarios,
and the prior source term would probably be shifted, scaled, and/or
blurred. We recommend tackling this task using the cross-validation approach,
providing a reasonable although computationally expensive tool for
determination at least between a prior-dominated mode or a data-dominated
mode of solution. A more sophisticated approach is to design a different
regularization of the error term ΔM exploiting, e.g., sensitivities
to local changes in concentrations around the measuring sites. The
information about sensitivity is already available from an atmospheric
transport model but it is not fully exploited with current source
term determination methods.
Derivation of residuum between measurement and reconstruction
From Eq. (), y=M+ΔMx
can be rewritten using the subtraction of Mxa and
ΔMxa from both sides, yielding
y-Mxa-ΔMxa=Mx-xa+ΔMx-xa,
which can be read as
y‾=Mx‾+ΔMx‾+ΔMxa
for commonly used substitutions y‾=y-Mxa
and x‾=x-xa. This means
that the minimization of Eq. () is equivalent to
the minimization of the former Eq. (). Thus,
minx||y-Mx-ΔMx||22⟺minx||y‾-Mx‾-ΔMx‾-ΔMxa||22==minxy‾Ty‾-2y‾TMx‾+x‾TMTMx‾︸||y‾-Mx‾||22-2y‾TΔM-ΔMxaTM-ΔMxaTΔMx‾︸linear in x‾+x‾TMTΔM+ΔMTM+ΔMTΔM︸Φx‾,
where terms independent of x‾ are omitted.
SRS matrices used for the ETEX and Chernobyl experiments
SRS matrices for ETEX are displayed in Fig.
for illustration. The SRS matrix computed using ERA-40 reanalyses
is in the left column, while the SRS matrix computed using ERA-Interim
is in the right column. The matrices are associated with their singular
values displayed in the bottom row. These illustrate properties of
the matrices and, importantly, their ill conditionality.
ETEX SRS matrices computed using FLEXPART driven
by meteorological input data from the European Center for Medium-Range
Weather Forecasts (ECMWF). (a, c) Data from the 40-year reanalysis
(ERA-40) and (b, d) data from the continuously updated ERA-Interim reanalysis.
The matrices are associated with their singular values (bottom row).
Code and data availability
All data used for the present publication can be freely downloaded from https://rem.jrc.ec.europa.eu/etex/ (last access: 26 April 2020, ) and from the Supplement of . The FLEXPART model versions 8.1 and 10.3 are open-source and freely available from their developers at https://www.flexpart.eu/ (last access: 26 April 2020, ). Reference MATLAB implementations of algorithms can be downloaded from http://www.utia.cas.cz/linear_inversion_methods/ (last access: 26 April 2020, ).
Author contributions
OT designed and performed the experiments and wrote the paper. LU performed Gibbs sampling experiments and wrote parts of the paper. VŠ designed and supervised the study and wrote parts of the paper. NE prepared the Chernobyl dataset and commented on the paper. AS commented on the paper and wrote parts of the final version.
Competing interests
The authors declare that they have no conflict of interest.
Financial support
This research has been supported by the Czech Science Foundation (grant no. GA20-27939S).
Review statement
This paper was edited by Slimane Bekki and reviewed by two anonymous referees.
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