Probability distribution functions (PDFs) of model inputs that affect the
transport and dispersion of a trace gas released from a coastal California
nuclear power plant are quantified using ensemble simulations, machine-learning algorithms, and Bayesian inversion. The PDFs are constrained by
observations of tracer concentrations and account for uncertainty in
meteorology, transport, diffusion, and emissions. Meteorological uncertainty
is calculated using an ensemble of simulations of the Weather Research and
Forecasting (WRF) model that samples five categories of model inputs
(initialization time, boundary layer physics, land surface model, nudging
options, and reanalysis data). The WRF output is used to drive tens of
thousands of FLEXPART dispersion simulations that sample a uniform
distribution of six emissions inputs. Machine-learning algorithms are trained
on the ensemble data and used to quantify the sources of ensemble
variability and to infer, via inverse modeling, the values of the 11 model
inputs most consistent with tracer measurements. We find a substantial
ensemble spread in tracer concentrations (factors of 10 to 10

Although the probability of a nuclear power plant accident is low, the risks
associated with accidental releases of radioactive materials from nuclear
power plants are expected to remain elevated worldwide through the coming
decades

An overview of our ensemble weather and dispersion system for inverse modeling and uncertainty applications. The system uses WRF to calculate wind fields and FLEXPART to transport materials in the atmosphere. Information about atmospheric releases is inferred by minimizing differences between plume predictions and field observations through Monte Carlo sampling loops (red dashed arrows). The loops sample different reanalysis datasets and model configurations in WRF and source term inputs in FLEXPART.

A variety of atmospheric models have been developed for simulating the
transport and dispersion of releases from nuclear power plants, starting from
the accidents at Three Mile Island in 1979 and Chernobyl in 1986

Atmospheric models used for nuclear power plant applications also use
emissions modules to estimate the release rates of radionuclides based on
specific reactor conditions

Inverse modeling can provide a safe way to infer information about
radioactive emissions from nuclear power plants and can also help estimate
uncertainty in the meteorological fields used to transport the radioactive
materials. Emissions and winds are constrained in an inverse method by
minimizing differences between dispersion model predictions and observations
of materials transported and deposited downwind from the source location

Maps showing the geographical location of the Diablo Canyon nuclear power plant on the coast of central California, near San Luis Obispo and Pismo Beach.

Starting from the left-hand side of the diagram in Fig.

The WRF-FLEXPART ensemble provides a set of plume predictions that are
compared with field measurements (right-hand side of Fig.

Measurements from a tracer release experiment conducted
in September 1986 at the Diablo Canyon nuclear power plant

To our knowledge, this work represents the first joint inversion capability
applied to FLEXPART dispersion simulations that provides probability
distribution functions of categorical inputs in WRF and continuous inputs in
FLEXPART and that has been verified with tracer release data. This capability can
be useful for other applications beyond releases from nuclear power plants,
including compliance monitoring of the nuclear test ban treaty

The non-hydrostatic, fully compressible Weather Research and Forecasting
(WRF) atmospheric model

High-resolution winds are needed to simulate the dispersion for the Diablo
Canyon tracer release test problem (see Fig.

Panel

Using this nesting capability, five WRF model domains are used to downscale
and generate high-resolution meteorological fields over the Diablo Canyon
region. The WRF domain configuration and geographic coverage are shown in
Fig.

By downscaling, a horizontal grid spacing of 300

Several features also make WRF ideal for creating an ensemble of plausible
atmospheric conditions for uncertainty assessments. Ensemble modeling
approaches have been shown to be effective at quantifying physically
plausible states of the atmosphere in a probabilistic manner

Categorical random variables for WRF ensemble.

Table

Several variations are included in the weather ensemble to account for
uncertainty related to model initialization and meteorological reanalysis
inputs. WRF simulations were started at either 15 or 9 h before the
beginning of the tracer release (i.e., at 00:00 or 06:00 UTC on
4 September 1986) to investigate the sensitivity of model solutions to
initialization start time and model spin-up duration. All of the WRF
simulations ended at 13 h after the end of the tracer release (i.e., at
12:00 UTC on 5 September 1986). The three reanalysis variations included are
the North American Regional Reanalysis (NARR) data

The FDDA weighting of meteorological data fields during the WRF ensemble
simulations was varied to account for uncertainty associated with the
assimilation of gridded reanalysis fields and irregularly spaced weather
observations. Weather simulations were performed with WRF FDDA options for
analysis and observational nudging options either turned off, using default
weighting factors as suggested by WRF guidance, or with a high option with the
weighting factors 1 order of magnitude higher than the default values.
Additionally, FDDA analysis nudging was used only on the two outer
course-resolution model domains, while FDDA observational nudging was used
only on the two innermost model domains (see WRF domains in
Fig.

Three PBL models and three LSM schemes were used to construct the WRF
ensemble to account for uncertainty associated with turbulent mixing and
surface momentum, moisture, and thermodynamic fluxes. The PBL models included
the Yonsei University (YSU) scheme

The FLEXPART Lagrangian dispersion particle model

We used FLEXPART-WRF version 3.1

In addition to the wind field variations generated by the WRF ensemble, the
inverse modeling system in Fig.

Continuous random variables for FLEXPART tracer ensemble.

The location, timing, and magnitude of the Diablo Canyon release are inferred
by sampling the six emissions inputs shown in Table

The FLEXPART ensemble contains 40 000 dispersion simulations that were run
and analyzed for the Diablo Canyon release. These ensemble simulations were
generated by randomly sampling both the WRF ensemble and the FLEXPART
emissions variables. Random samples were drawn using a Latin hypercube design

Machine learning is used to train statistical regression functions to approximate the input–output relationships in the WRF-FLEXPART ensemble. Once trained, the machine-learning functions can be evaluated very efficiently at new input values and used for uncertainty propagation, parameter estimation, Bayesian inference, and other types of statistical analysis. These functions are used for two primary purposes in our work. They are used to identify and rank the effects of input features in WRF and FLEXPART on the tracer responses (i.e., a form of sensitivity analysis) and to determine the values of the inputs that yield responses that are similar to tracer observations (i.e., optimization and inverse modeling). These applications are described in more detail below.

The WRF-FLEXPART ensemble is mathematically expressed as

We use a method called gradient boosting (GB) that fits statistical
regressions to Eq. (

As noted, a GB model is a sum of decision trees of the form

Bayesian inversion method for constraining the WRF and FLEXPART inputs. Samples are drawn from the uniform prior distribution on the left and then evaluated in WRF-FLEXPART and compared to measurements. Gradient-boosting regression trees are fit to model–measurement differences and used to infer the posterior distributions of the inputs.

Although other statistical regression methods could be used, GB offers two
clear advantages for fitting the WRF-FLEXPART ensemble. First, as shown in
Eq. (

The goal of the inverse modeling is to determine the values of the inputs to
WRF and FLEXPART that provide output concentrations that best match the
tracer measurements. The inversion uses an extension of our approximate
Bayesian computation algorithm described in

The inverse method applies Bayes' rule,

The remaining term in Eq. (

Following our previous work

To account for the two metrics in the likelihood function, we use an
expression of the form

Before computing the mse and corr in Eq. (

The Bayesian inversion is performed using GB regressions, instead of actual
model simulations, to predict

To verify the Bayesian inversion scheme, we performed a series of “synthetic
data” tests using model-generated inputs and outputs. These tests are
important because inverse problems often have multiple solutions and may be
poorly constrained (i.e., ill-posed and ill-conditioned).
Section

Field measurements from the Diablo Canyon nuclear power plant tracer release
experiment

PG&E conducted eight tracer release tests between 31 August and
17 September 1986. Although the large-scale wind patterns for the eight tests
showed relatively similar onshore flow from the northwest, the third tracer
test on 4 September experienced a strong sea breeze that presents a challenge
for dispersion modeling. We therefore use the third tracer release for our
inversion testing. Starting at 08:00 Pacific Daylight Time (PDT) on
4 September (15:00 UTC), 146

One-hour average

Thirty-minute average plumes of

The measurement network was designed to monitor the expected tracer transport
paths near terrain gaps, the entrances and exits to the inland valley, and
the coastal boundary. An arc of 24 sampling sites was positioned very close
to the nuclear power plant release point, at a radial distance of
840

Figure

Before presenting results from the large Latin hypercube ensemble, we first
show dispersion results using the actual tracer release parameters with the
162 wind fields from the WRF ensemble. Figure

The upper portion of the figure shows the plumes using NARR and ECMWF 3 h after the release. At this stage of the simulations, there is a large
spatial difference between the plumes. The dispersion using NARR is directed
eastward, is spatially more confined, and does not extend downwind of Pismo
Beach, as compared to the southeast directed ECMWF plume. The ECMWF plume
covers a much wider region, though most of the extended area is over the
ocean. Because there are not many measurement sensors over the ocean, we
expect there to be smaller differences between NARR and ECMWF in the
inversion algorithm than the plumes in upper part of Fig.

Time series of the distribution of

Seven hours after the release, as shown in the lower part of the figure, the plumes using NARR and ECMWF begin to resemble each other. Both are directed to the southeast, and both have about the same spatial extent. The higher concentration area of the plume using ECMWF is a little more dispersed near the release location (see red contour), but otherwise the differences between the two reanalysis cases are minor.

To see the variability associated with the full WRF ensemble with the actual
tracer release, Fig.

Starting with the distribution at Site 325, which is closest to the release
point, the simulated

Correlation and mean squared error between the simulations and
measurements of

Time series of the prior probability distribution of

Figure

To further examine the WRF variations, we compute the observational metrics
mse and corr described in Eq. (

To determine the primary causes of variation in Fig.

The results presented in this section and in the rest of the paper are
based only on the 40 000-member Latin hypercube ensemble. We exclude the 162
cases using the known release parameters that were analyzed previously in
Sect.

Distributions of the time series of

Comparing Figs.

Time series of the

The

Figure

Overall, the feature scores suggest that the prior uncertainty in the source term inputs are more critical than the prior uncertainty in the meteorological inputs for this particular tracer release experiment. Although the WRF inputs are not the dominant source of variability, the combined effects of the sources of meteorological uncertainty still cannot be neglected. It is also important to note that, for tracer release simulations conducted under different meteorological conditions, at different scales, or that consider additional sources of uncertainty or observational constraints (i.e., posterior uncertainty), the input sensitivities will likely differ from those estimated here. Moreover, the contribution of meteorological uncertainty is expected to be larger for forecast problems that are not constrained by reanalysis data.

In addition to being useful for understanding the drivers of variance in the prior probability distribution, the feature scores are also useful for interpreting the results of the Bayesian inversion in the following sections. Inputs with relatively high feature scores are often easier to constrain with observations. On this basis, therefore, we expect the posterior probability distributions for the FLEXPART longitude, latitude, and source amount inputs to be relatively narrower than the other FLEXPART terms because they have the highest feature scores.

Diablo Canyon

Figure

Figure

Correlation and mean squared error between the simulations and
measurements of

Inversion estimates versus actual source parameters.

The points in the figure are used to estimate the terms in the likelihood
function in Eq. (

Before performing an inversion with the Diablo Canyon tracer data, we first
apply the algorithm to “synthetic” data with known inputs and outputs as
a verification test. Synthetic data are generated by adding noise to the
output concentrations from a randomly selected ensemble member. The posterior
probability distribution of parameter values is computed using the previously
described methods (i.e., fitting GB regressions to the mse and
corr and estimating the covariance matrix as in Sect.

Figure

For this particular test, the synthetic source was located about
850

The marginal posterior distribution of WRF and FLEXPART parameters
for the synthetic data inversion. Diagonal components show univariate
continuous distributions for FLEXPART (top left) and univariate categorical
distributions for WRF (bottom right). Off-diagonal components show bivariate
distributions for the pair of parameters in the corresponding row and column.
Probability density is normalized, with red colors denoting regions of high
probability in the bivariate distributions. Known input values are shown by
the black lines and circles in the diagonal and off-diagonal components,
respectively. The FLEXPART parameter values have been scaled to [0,1] using
the inversion ranges in Table

Table

These results, along with other synthetic data tests that are not shown, provide confidence that the inversion algorithm appears to be functioning adequately. The algorithm returns values for the WRF and FLEXPART inputs that are close to the actual values for most of the parameters. For the release height, we are also satisfied with the non-informative values provided by the algorithm because we expected a relatively flat posterior distribution. In future work, we will broaden the range of release heights to test the algorithm for elevated and surface releases.

The marginal posterior distribution of WRF and FLEXPART parameters
for the Diablo Canyon tracer data inversion. Actual source term input values
used in the tracer release experiment are denoted by the black lines and
circles. Refer to the caption in Fig.

Time series of the posterior probability distribution of

For the inversion using the

As is the case with the synthetic inversion tests, the actual values for the
location, start, duration, and amount of the

Referring to Fig.

The posterior distribution also shows a strong covariance relationship
between the release latitude and longitude (see bivariate distribution in the
upper left of Fig.

The WRF configurations in the posterior distribution are displayed on the
right-hand side of Fig.

As shown in the figure, the maximum likelihood configuration consists of the 06:00 UTC initialization time, the ECMWF reanalysis fields, the YSU PBL scheme, the RUC land surface model, and no data assimilation nudging. Some of these configuration settings may, at first, seem surprising. For example, the NARR reanalysis fields have a higher spatial resolution than the ECMWF fields and therefore may be expected to perform better. Likewise, the option to run without data assimilation seems to outperform the options with assimilation. Referring to the figure for these cases, the posterior distribution still has significant probability density for both the NARR and low nudging options. Compared to the posterior distribution in the synthetic data inversion, the WRF inputs are not as strongly constrained using the tracer data, especially the inputs for the land surface model and nudging. Only two of the WRF inputs have settings with negligible probability: the earlier initialization time and CFSR reanalysis data. The alternate PBL schemes also have relatively low probability. We therefore conclude that the winds generated using the 06:00 UTC initialization time, the NARR or ECMWF reanalysis fields, and the YSU PBL scheme will optimize our likelihood metrics, and that there is not a preferred land surface model or nudging option.

The ensemble time series in Fig.

Figure

Other than at site 325, where a large spread remains, the 5–95 % range covers about 1 order of magnitude, or a reduction of 2 orders of magnitude. Even with the reduced range, most of the measurements still fall within the light blue area. We do not expect all of the measurements to lie in the 5–95 % range because the likelihood metrics consider the aggregation of all of the sites and times. In order to achieve an overall higher likelihood, individual measurement points may be farther away from the median in the posterior distribution than they were in the prior.

In addition to the reduction in the variance, Fig.

We have developed an ensemble-based Bayesian inverse modeling system that can determine information about an atmospheric release from a nuclear power plant using measurements collected a relatively safe distance downwind from the plant. The system uses an ensemble of WRF simulations to capture uncertainty in meteorological fields and an ensemble of FLEXPART dispersion simulations to vary factors related to emissions. Machine-learning algorithms are trained on the input–output relationships in the meteorological and dispersion ensemble, resulting in statistical surrogate models that mimic the behavior of the actual WRF and FLEXPART models, but that can be evaluated very rapidly at millions of new input value combinations.

Using our system, we can determine the input factors that are most important for understanding and reducing uncertainty in the ensemble (i.e., sensitivity analysis) and can estimate the values of the model inputs that provide likely matches between model output and field measurements (i.e., inverse modeling). Bayes' rule is used for the inversion, which provides probability distribution functions of model inputs and outputs constrained by observations and that serve as a quantitative assessment of model performance. The inversion is designed to estimate the location, timing, and amount of material released to the atmosphere, and to determine the best categories of settings for running a meteorological model. The inversion system should be able to handle, without difficulty, additional factors related to the transport and dispersion of potential materials released during a nuclear power plant accident (e.g., wet and dry deposition of soluble radioactive products).

Our ensemble system is tested against a tracer release experiment conducted
near the Diablo Canyon nuclear power plant located in the rugged terrain of
coastal California

By calculating the mean squared error and correlation between the tracer
measurements and the surrogate model predictions, the Bayesian inversion
algorithm produces a posterior distribution of model inputs and outputs for
the tracer release experiment. Even though the source term parameters are
initially unknown in the inversion (i.e., we used a non-informative prior),
the most likely posterior values of the FLEXPART inputs closely estimate the
actual values used in the tracer release experiment, which demonstrates
a successful inversion. Table

It is important to keep in mind that the ensemble and inversion methods can be applied to problems other than nuclear power plant releases. While the location of a nuclear power plant release is generally restricted to reactor buildings or other nearby facilities, the inversion algorithm can also determine an arbitrary release location from within a large area (e.g., hundreds of square kilometers) if suitable observations are available.

The number of measurements can also affect the quality of the inversion. The
Diablo Canyon field experiment had a large number of sensors to measure

The Diablo Canyon ensemble dataset used for machine
learning and Bayesian inversion is available for public download through
anonymous FTP at the Lawrence Livermore National Laboratory Green Data Oasis,

The authors declare that they have no conflict of interest.

This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and was funded by Laboratory Directed Research and Development at LLNL under project tracking code PLS-14ERD006. The paper is released under UCRL number LLNL-JRNL-710162. The authors thank PG&E for access to the Diablo Canyon measurement data, and Livermore Computing for providing computational resources through an institutional allocation for the Monte Carlo simulations. The authors also thank Devin Francom and Bruno Sansó from UC Santa Cruz and Vera Bulaevskaya from LLNL for invaluable discussions about the statistical analysis. Edited by: Manvendra K. Dubey Reviewed by: two anonymous referees