A Strategy for Creating Probabilistic Radiation Maps in

Areas Based on Sparse Data Robin McDougall, Ed Waller and Scott

NoklebyFaculties of Engineering & Applied

Science and Energy Systems & Nuclear Science

1

Overview• Motivation

• What is a radiation map?• Potential problems in their generation

• Overview of the Proposed Strategy• Core components• Their functional relationship

• Illustrative Case Study • Present a preliminary simulation study • “Sanity-check”

2

Motivation

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• Radiation Maps characterize a radiation field in an easily understood format• Can help minimize total exposure by identifying areas with relatively high radiation intensities

• Number of options for generation

Workflow Optimization

• For these instances it becomes necessary to use modeling based techniques to predict the radiation intensities in areas where a sensor cannot be positioned.

• We propose using a five-stage procedure for generating radiation maps based on sparse or incomplete radiation sensor data

5. Data Visualization (Create Map)

Overhead Isobars Augmented Reality

1. Select an Appropriate Radiation Model for the environment

• Representative of the physical environment• Two (at least!) concerns:

• Radiation Intensity Field• Radiation Sensor

• Can support multiple radiation modeling tools:• Simplistic; Inverse square law • More Complex – MCNP, Microsheild

• Parameterized appropriately• [Radiation Intensities] = f (Source Locations, Source Intensities)

2. Gather Radiation Data

• Use an suitable scheme• Static Sensor Net• Human Operators• Mobile Robots

• Take Readings and Methodically Transfer to Physical Layout Maps• Record multiple readings for each location (Sensor Variability)

3. Calibrate Radiation Model

Use data collected in (2) to calibrate the “Generic” radiation model to the specific instantaneous radiation exposure scenarioChoice of method for calibration used should consider:

Order of model (Linear? Non-Linear?)Variability, Uncertainty, Sparsity of Data

Type of map being generatedRegardless of technique, want to infer intensity and location(s) of sources most likely to cause the data observed

4. Generate Data for Map

• Use the “Calibrated” model from (3) to calculate the “predicted” values at an interval sufficient enough to characterize the radiation intensity field in the area

Method - Overview

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

An Example….

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Let’s Consider a 20 x 20 area:

- Exposed to two radiation sources. One placed at P1(2,5) with an intensity of (I=650) and another at P2(16,6) and intensity of (I=350).

- Sensor data taken from two edges

Objective: Generate Radiation Map for the entire region using Radiation modeling

Example – Radiation Model

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

1. Select Radiation Model

• Choose a radiation model based on the environment

Model

Parameters In

• Source Position(s)• Source Intensities

Data Out

• Sensor Readings for Sample Locs

• Radiation Intensity for grid locations

Example – Radiation Model

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

1. Select Radiation Model

• Two elements – want to model radiation intensity and sensor readings

• In the preliminary study we used the simplistic inverse square modeling method

• Radiation intensity at a point “P” was found by summing the contribution from the two sources.

• Each Contribution was equal to the Intensity at the source divided by the square of the distance.

Example – Radiation Model

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

1. Select Radiation Model

• The sensor was modeled by sampling the Poisson distribution with using the intensity “P” from the radiation model as the mean.

• Radiation at Sensor “S” = Pois( Ps)

Example – Element Details

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

2. Gather Radiation Data

• In the “real-world” this would be where the sensor readings are acquired.• Sensor Net• Operator• Robot

• In this simulated study, the radiation model from (1) was used to synthesize (simulate) radiation readings for each point (5 for each of the 13 pts)

3. Calibrate Radiation Model

• Two additional design considerations

• Which likelihood function to use

• How to explore candidate solutions

3. Calibrate Radiation Model

Likelihood function

LF =

m - # of pointsn - # of Samples

3. Calibrate Radiation Model

• Many quantitative techniques to perform this task, choice depends on system being studied

• Unique characteristics of this scenario:• Potential for very non-linear radiation models• Radiation sensing is a discrete probabilistic process

• More samples taken the more likely mean -> actual• Sparse available data imparts uncertainty as well

• Desirable to capture and characterize these in our estimates for the parameters which describe the source(s)

• We propose using Bayesian Inference Techniques to sample the LF

3. Calibrate Radiation Model

• Use an optimization based routine to infer where the sources are using the model from (1) and the data from (2) by maximizing the Likelihood function

• Iterative process

3. Calibrate Radiation Model

• Candidate source locations and intensities are proposed

• The values for their modeled sensor readings (1) are compared with the observed data (2)

• Updated iteratively until some termination criteria is reached

Example – Element Details

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

Compare with Obs.Compare again

Bayesian Inference Background Instead of point estimates for locations and intensities, we

want to find distributions of the parameters as implied by a likelihood function

Also want to incorporate prior knowledge. Simplest case is upper/lower bounds on parameters, but could be distributions as well

Idea is to use Bayes Theorem to find the Joint Posterior distribution, then draw samples from this distribution

Once we have the collection of samples, we can get any other kind of statistical information we need:

– Marginal densities of parameters– Correlations– Means, modes, etc.

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Bayes Theorem

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Bayes Theorem

)(p

)|( xL

)|( xp

Prior distribution: prior knowledge regarding the distribution of the parameters

Likelihood function: the probability of observing a set of model outputs (x) given a set of parameters (theta)

Posterior distribution: the distribution of the parameters taking into account the likelihood and prior information.

Bayes Theorem relates these quantities as follows:

)()|()|( pxLxp

The Posterior DistributionContains everything we want to know about the

distributions of the parametersGet information about the parameter distributions by

sampling from the posterior, the performing various analyses of the collections of samples Marginals, correlations, etc

Usually impossible to sample from directly Multivariate, no analytic form, usually involves running the

simulation

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How to Sample PosteriorsConventional techniques don’t work

E.g., rejection sampling: posterior is close to zero in most places, so almost all samples will be rejected

Use Markov Chain Monte Carlo techniques provide a way to sample from the posterior One sample is used to generate the next sample in a “smart”

way... this produces “chains” of samplesWe propose using MCMC techniques based on the Gibbs

and Metropolis-Hastings sampling algorithms

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MCMC Configuration

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For this study:

• Use uniform priors to constrain locations within the area

• Use uniform prior for intensities constraining it a range which certainly contains the “real” value

MCMC ResultsFor this study:

• Recall the system has two sources of Radiation

Resulting Chains from the MCMC Study- 6 Parameters / 6 Chains

Example – Element Details

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

4. Generate data for map

• Use a Forward Monte Carlo (FMC)• For each FMC iteration, randomly

select a value from the posterior distributions for each parameter (3)

• Run the model (1) with this candidate-set and record predicted intensities for a sufficient number of points

Example – Element Details

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1. Select Radiation

Model

2. Gather Radiation

Data

5. Data Visualization (Create Map)

3. Calibrate Radiation

Model

4. Generate Data for

Map

5. Visualization / Map Generation

• Process FMC results appropriately and generate map

• For example:• Take 90th percentile radiation

intensities for each grid intersection (400 pts)

• Plot intensity isobars

Final Thoughts….In preliminary study presented here the

model used in the Map Generation Tool was “perfect” – the same model was used to synthesize the data in the study

The platform and procedure themselves are generic enough that extension to more sophisticated radiation models should be straight-forward.

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Acknowledgements:

University Network of Excellence in Nuclear Engineering

Natural Sciences and Engineering Research Council

Thank you for your time! Questions?