Analytical source term optimization for radioactive releases with approximate knowledge of nuclide ratios
Radek Hofman, University of Vienna, Vienna, Austria, [email protected]
Petra Seibert, University of Vienna, Vienna, Austria; University of Natural Resources and Life Sciences (BOKU), [email protected]
Ivan Kovalets, Ukrainian Center of Environmental and Water Projects, Kiev, Ukraine, [email protected]
Spyros Andronopoulos, National Center for Scientific Research Demokritos, Athens, Greece, [email protected]
1. Introduction
2. Illustrative example — contd.
We are concerned with source term retrieval in the case of an accident
in a nuclear power with off-site consequences. The goal is to optimize
atmospheric dispersion model inputs using inverse modeling of gamma
dose rate (GDR) measurements (instantaneous or time-integrated).
These are the most abundant type of measurements provided by various radiation monitoring networks across Europe and available continuously in near-real time. Usually, a source term of an accidental release
comprises of a mixture of nuclides. Unfortunately, gamma dose rate
measurements do not provide a direct information on the source term
composition; however, physical properties of respective nuclides (deposition properties, decay half-life) can yield some insight. In the method
presented, we assume that nuclide ratios are known at least approximately, e.g. from nuclide specific observations or reactor inventory and
assumptions on the accident type. The source term can be in multiple
phases, each being characterized by constant nuclide ratios.
The method is an extension of a well-established source term inversion
approach ([2]) based on minimisation of the following cost function:
J1(x) = (y
Mx)>R 1(y
Mx) + (x
xa)>B 1(x
xa).
(1)
The first quadratic form on r.h.s of (1) measures departures between
observations y 2 Rm and model predictions Mx (source-receptor
sensitivities ⇥ source vector) weighted by inverted observation error
covariance matrix R
I The second is a regularization term and ensures convexity of the
cost function and unambiguity of the solution (regularization with
respect to a first-guess xa and its error covariance matrix B)
I For simplicity, R and B are often assumed to be diagonal.
I Analytical minimization of the cost function leads to a liner system of
equations (LSE). This means that no special software packages are
needed — just a LSE solver, e.g. from LAPACK.
I Possible negative parts of the solution are iteratively removed by the
means of first guess error variance reduction
I
Features of the modified method are:
I It can be used for inversion of a source term with multiple nuclides
using bulk GDR measurements
I Nuclide ratios enter the problem in the form of additional linear
equations, where the deviations from prescribed ratios are weighted
by factors; the corresponding error variance allows us to control how
strongly we want to impose the prescribed ratios
I This introduces some freedom into the problem and nuclide release
rates can accommodate to both measurements (and their errors) and
prescribed ratios.
The method is demonstrated on simulated releases. Meteorological
fields and topology of receptors are inspired by ETEX tracer experiment.
Both source-receptor sensitivities (SRS) and measurements are generated by different runs of Lagrangian puff dispersion model DIPCOT.
2. Illustrative example
Similarly, we augment the observation error covariance matrix:
2
3
2
6 y1 .
..
6
6
R̃ 1 = 6
6
6
4
2
I and
M̃x)>R̃ 1(ỹ
xa)>B 1(x
2
2
Cs
10x2)2],
2
Cs = E[(x1
(2)
Figure 1: Nuclide ratios strictly forced (R augmented by sigmas ⇡ 1 ⇥ 1010), perfect
knowledge of nuclide ratios. We obtains a perfect estimate.
Figure 5: Estimated source term assuming four phases (columns of M aggregated).
I
I
Important properties of the test case are listed below:
I m=12744 10-minute GDR measurements form 59 stations, we
assume estimation time step 1h
I Vertically, the release is uniformly distributed between 0 and 50 m
using 10 point sources
6. x = 0
I Model error assumed as y = 0.1y + 1.0 ⇥ 10
a
What follow are results for different settings.
Figure 2: Nuclide ratios strictly forced, nuclide ratios perturbed by factor of 10. Estimate is biased because we strictly force wrong nuclide ratios.
Figure 3: Nuclide ratios loosely forced, nuclide ratios perturbed by factor of 10. Wrong
nuclide ratios are compensated by influence of GDR data.
Figure 6: Four phases: vertical profiles for selected nuclides.
3.2 Experiment D
2.1 Experiment A
We assume a prior knowledge of the homogeneity of the release
(both in time and vertical)
I It can be fully described by 3 numbers (dim x = 3) representing
releases of 3 nuclides (augmented cost function is identical to that in
the previous Section)
m⇥3 was created from M by aggregating corresponding
IM 2 R
1
columns
I
I
Nuclide
133
Xe
I
Cs
131
137
I
TABLE 1: True and estimated release rates.
Ground truth
Estimates of release rates (Bq/s)
Cumulative
Release
A1: Linear
A2: Min. of
A3: Min. of
rel. (Bq)
rates (Bq/s)
regression
cost (1)
cost (2)
1.00 ⇥ 1018
2.32 ⇥ 1013
5.24 ⇥ 1011 1.80 ⇥ 1013
2.28 ⇥ 1013
1.00 ⇥ 1017
2.32 ⇥ 1012
8.53 ⇥ 1012
4.44 ⇥ 1012
2.28 ⇥ 1012
5.00 ⇥ 1017
1.16 ⇥ 1013
6.67 ⇥ 1012
1.03 ⇥ 1013
1.14 ⇥ 1013
Linear regression without any assumptions on positivity of x provided
unrealistic negative emissions
Methods A2 an A3 gave more reasonable solutions, where A3 was
very close to true values due to additional information on nuclide
2
ratios, which were strictly required by lower values of I2 and Cs
⇡ (1 ⇥ 1013)2 = 1 ⇥ 1026
Each release hour now as a separate ”phase” (but vertically
aggregated, dim x = 36)
I y, M augmented by 24 rows and R by 24 rows and columns (12 for
I-131 and 12 for Cs-137 ratios).
I
Higher emissions were estimated correctly.
Big differences in vertical levels, cannot be described by a the same
number in B and R for all levels (needs additional prior information)
2x3)2].
2.2 Experiment B
xa).
3.1 Experiment C — contd.
2
Cs determine how large variances we allow for er-
2
I = E[(x1
10x2 = 0
2x3 = 0.
M̃x) + (x
I
7
7
7
7.
7
7
5
Contrary to the original algorithm minimizing (1), the new one requires a
first guess of nuclide ratios and their variances, which is essentially not
different of what is required by the numerical algorithm of [1]. These inputs can be given by some nuclide specific measurements, prior source
terms, or user specified.
I
These conditions deterministically relate release rates of 137Cs and 131I
to 133Xe. For N nuclides we can define up to N-1 conditions. An alternative cost function including nuclide ratios reads
J(x) = (ỹ
2
ym
Coefficients
rors of residuals in ratios conditions:
Let’s illustrate the method on a simple case (a 12-hour homogeneous
release of three nuclides, see Table 1). Let release rates of 133Xe, 131I
and 137Cs be x1, x2 and x3. Nuclide ratios criteria are:
x1 = 10x2 , x1
x1 = 2x3 , x1
2.2 Experiment B — contd.
Here, ỹ and M̃x are augmented versions of y and M:
2
3 2
3
y1
M11 M12 M13 2 3
6 . 7 6 .
.
. 7 x1
6
7 6
7
7 6
74 5
ỹ M̃x = 6
6 ym 7 6 M1m M2m M3m 7 x2 .
4 0 5 4 1
5 x3
10
0
1
2
I
I
Inversion of full SRS matrix, each time slot is a separate ”phase”
dim x is 21 nuclides ⇥ 10 levels ⇥ 17 time steps = 3570
R augmented with (1 ⇥ 1010)2 for all steps ) no phase assumptions
Figure 4: Nuclide ratios loosely forced (R augmented by sigmas ⇡ 1 ⇥ 1015). Nuclide
ratios perturbed by factors 100 for I and 0.01 for Cs. Even here we obtain a useful
estimate.
3 More realistic example
Scenario with 21 nuclides and two release phases 0.5 and 4 h
(source term Muehleberg-1 from flexRISK,
http://flexrisk.boku.ac.at/)
12 Bq/s
I Release rates from 0 to 1 ⇥ 10
I 10 vertical levels with Gaussian profile (0–100m, mode at 50m)
I SRS fields with time step 0.5 hours and 2-hour no-release periods
before and after the release =) 4 phases with 17 time steps in total
I
3.1 Experiment C
Bulk estimate for whole phases — releases of nuclides in phases
aggregated (columns of M)
I dim x = 21 nuclides ⇥ 10 vertical levels ⇥ 4 phases = 840
11 for 2nd phase, 1.0 ⇥ 108 otherwise.
I std. of xa is 1.0 ⇥ 10
10 for 2nd phase, 1 ⇥ 108 otherwise.
I R augmented with sigmas 1 ⇥ 10
I
Research was supported by FP7 project PREPARE — Innovative integrative tools and platforms to be prepared for radiological emergencies and post-accident response in Europe. Poster created with beamerposter.
Figure 7: Evaluation of estimated source term with no assumption on phases.
Timing of releases is correct (periods with no releases were more or less preserved).
4 Discussion
Properties of the the method were examined on a synthetic release with
21 nuclides, 4 phases and 10 vertical levels using GDR data. It is not
possible to go through all the possible configurations but we tried to
show that under some moderate assumptions we get useful estimates.
References
Olivier Saunier, Anne Mathieu, Damien Didier, Marilyne Tombette, Denis Quélo, Victor Winiarek, and Marc Bocquet.
An inverse modeling method to assess the source term of the fukushima nuclear power plant accident using gamma
dose rate observations.
Atmospheric Chemistry and Physics, 13(22):11403–11421, 2013.
A Stohl, P Seibert, G Wotawa, D Arnold, JF Burkhart, S Eckhardt, C Tapia, A Vargas, and TJ Yasunari.
Xenon-133 and caesium-137 releases into the atmosphere from the fukushima dai-ichi nuclear power plant:
determination of the source term, atmospheric dispersion, and deposition.
Atmospheric Chemistry and Physics, 12(5):2313–2343, 2012.
WWW: http://www.univie.ac.at/theoret-met/prepare/