A Strategy for Creating Probabilistic Radiation Maps in Areas Based on Sparse Data

1. A Strategy for Creating Probabilistic Radiation Maps in Areas Based on Sparse Data Robin McDougall, Ed Waller and Scott Nokleby Faculties of Engineering & Applied Science and Energy Systems & Nuclear Science

2. 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”

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

4. Method - Overview 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create Map) • 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) 5. Data Visualization (Create Map) Overhead Isobars Augmented Reality • 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 scenario Choice of method for calibration used should consider: Order of model (Linear? Non-Linear?) Variability, Uncertainty, Sparsity of Data Type of map being generated Regardless 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 • We propose using a five-stage procedure for generating radiation maps based on sparse or incomplete radiation sensor data

5. An Example…. 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

6. Example – Radiation Model 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create Map) • Select Radiation Model • Choose a radiation model based on the environment • Parameters In • Source Position(s) • Source Intensities • Data Out • Sensor Readings for Sample Locs • Radiation Intensity for grid locations Model

7. Example – Radiation Model 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create Map) • 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.

8. Example – Radiation Model 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create Map) • 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)

9. Example – Element Details 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create Map) • 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)

10. Example – Element Details 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create Map) • 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 • Calibrate Radiation Model • Two additional design considerations • Which likelihood function to use • How to explore candidate solutions • 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 Calibrate Radiation Model Likelihood function LF = m - # of points n - # of Samples • 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 Compare with Obs. Compare again

11. 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.

12. Bayes Theorem Bayes Theorem 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: May 30, 2008

13. The Posterior Distribution • Contains everything we want to know about the distributions of the parameters • Get 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

14. How to Sample Posteriors • Conventional 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 samples • We propose using MCMC techniques based on the Gibbs and Metropolis-Hastings sampling algorithms

15. MCMC Configuration • 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 15

16. MCMC Results • For this study: • Recall the system has two sources of Radiation Resulting Chains from the MCMC Study - 6 Parameters / 6 Chains

17. Example – Element Details 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create 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

18. Example – Element Details 2. Gather Radiation Data 3. Calibrate Radiation Model 1. Select Radiation Model 4. Generate Data for Map 5. Data Visualization (Create 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

19. 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.

20. Acknowledgements: • University Network of Excellence in Nuclear Engineering • Natural Sciences and Engineering Research Council Thank you for your time! Questions?