Plume Tracking in Sensor Networks

1. Plume Tracking in Sensor Networks Glenn Nofsinger PhD Thesis Defense August 22, 2006

2. Outline • Motivation and Problem Statement • Other Work • Theoretical Background • 2-Step Algorithm • Experiments • Results and Conclusions

3. Motivation • Current monitoring lacks information sharing and high sampling density • Method needed for estimating highly unpredictable events: chemical, biological, radioactive agents • Many current sensors for such agents are binary

4. Problem Statement (1) • Gedanken-experiment: city with fixed, binary sensors of harmful agent • At an unexpected time a series of sensors activated, cause of release unknown • Where was the release? • How many release sources? • How are observations correlated?

5. Problem Statement (1) t=o

6. Problem Statement (1) t=1

7. Problem Statement (1) t=2

8. Problem Statement (1) t=3

9. Problem Statement (1) t=4 • What is the best estimate of the true source locations given these observations?

10. Problem Statement (1) t=1 • True initial state: two source locations • Thesis work estimates this truth state

11. Problem Statement (2) • This problem is hard! • Having an unknown number of sources and only binary detections at a large number of nodes is a new type of problem

12. Problem Statement (3) • Problem Summary: • Use a sensor network capable of only binary detection to estimate source locations • Evaluate performance of this estimation • As a function of wind • As a function of sensor density

13. Other work Mobile scout robots Swarm robots II I Model Complexity III IV Static sensor networks with high density cheap fixed sensors Traditional environmental techniques with high resolution sensors, low sensor density (Our approach) Mobility

14. Graphical Conventions: Theoretical Background (1) • Source • Sensor • Sensor with detection • Track • A collection of sensors with detections believed to originate from the same event • Each track has different color +

15. Plot Conventions: Theoretical Background (2) • Agent concentration for some area, A • Likelihood map given sensor observations

16. Theoretical Background (3) • Fick’s Law for diffusion and linear wind • First order approximation to process • Standard Gaussian solution Advection-diffusion Model

17. Theoretical Background (4) Plume Model • Solution of differential equations for advection-diffusion lead to a superposition of Gaussians • Peclet number measures relative strengths of diffusion to wind. A typical Peclet number is 10. This ratio determines plume width in our model

18. Theoretical Background (5) • Assume a spatially uniform wind over the matrix A • Concentration state matrix A is designed to simulate an area of size 25mi x 25mi • Decorrelation length scale in wind data indicates the distances over which spatially uniform assumption holds • Typical values are on the order of 50-200 miles, therefore to first order we can assume spatially uniform wind Wind Model

19. Classic Analytical Approaches Theoretical Background (6) • Unique response per source location • Relative differences of Tmax unique • 3 sensors for 2D location • Can solve for (X0,Y0)

20. Analytical Approach Theoretical Background (7) • Can solve differential equations for advection-diffusion • Solution of the source (X0,Y0) based on measurements of C(t) • Method breaks down: • No continuous time series available • Very noisy, possibly binary data

23. C(x,y,t) A B t Theoretical Background (8) Sensors Radii of location for rising and falling edges of agent detection – one for each edge is possible in binary sensor Analytical approach no longer useful, need statistical methods. Leads to Bayesian formulation A B Typical Sensor Response Curve

24. Bayesian Estimation Theoretical Background (9) • Goal to obtain good estimate of target state Xt based on measurement history Zt • p(x) – a priori probability distribution function of state x (plume concentration) – assumed uniform • p(z|x) – the likelihood function of z given x • p(x|z) – the a posteriori distribution of x given measurement z, also called the current belief

25. Relationship between a posteriori distribution, a priori distribution, and the likelihood function Bayesian Formulation Theoretical Background (10) • Our state estimate • True state • Want our state estimate to be as close to true state as possible • Given observation set, what is:

26. MMSE –minimum-mean-squared error. It is the mean posterior density. Equal weight to obs. MAP- maximum a posteriori, maximizes the posterior distribution ML- maximum likelihood, considers information in measurement only Estimators Theoretical Background (11)

27. Estimator Example: Source Localization Theoretical Background (12) • Each sensor measurement produces independent likelihood function • Cone shaped likelihood function • Localization based on sequential Bayesian estimation • Measurements combined, assuming independence of likelihood

28. Uniform State Estimation Theoretical Background (13)

29. MHT Theoretical Background (14) • The previous uniform estimator can be improved with advanced data association (DA) techniques such as multiple hypothesis tracking (MHT) • By maintaining multiple “tracks” observations partitioned into subsets which correspond to unique “targets” – in this case unique plume sources

30. Theoretical Background (15) MHT • MHT handles the combinatorial growth of possible track assignments via accurate pruning • Once tracks are built in the plume problem, assume 1 target per track, therefore focusing the custom estimation on one exclusive source observations Tracks

31. 2-Step Algorithm (1) • 2-Step track-estimate algorithm • Step 1 is track building • Step 2 is state estimation of tracks • Custom Estimator based on tracks, ignoring observations not associated to a track • Able to work in two scenarios: • Sources distant, distributed sensor groups • Overlapping tracks, mixing sensor groups

32. End of Background • (20 minutes)

33. 2-Step Algorithm (2) Input: Sensor “Hits” (x,y,t) Step 1: Track estimation Output: N Tracks M(Track1) Step 2: State Estimation For each track M(Track2) … M(TrackN)

34. Step 1: Track Formation 1.1 Track Initialization –All new observations potentially create tracks. The terminal node on track is designated leader node 2-Step Algorithm (3)

35. 2-Step Algorithm (4) • Step 1: • Track Formation • 1.2 Data Association – All sensors with new observations calculate a likelihood function based on wind history. Function evaluated at all leader nodes

36. 2-Step Algorithm (5) • Step 1: • Track Formation • 1.3 Track extension – observations that were associated in step 1.2 become the new leader nodes.

37. 2-Step Algorithm (6) • Step 1: • Track Formation • 1.4 Track termination – The track is terminated once simulation ends or no new associations within cutoff parameter. Track outputs sent to Step 2

38. Track A Leader nodes New Observation Track B Likelihood Function 2-Step Algorithm (7) Detail of likelihood function for track association

39. 2-Step Algorithm (8) • Step 2: • State Estimation • Each track sequence produces an individual likelihood map • In this case only 4 sensor observations used to form belief map

40. 2-Step Algorithm (9) • Step 2: • State Estimation • Each track sequence produces an individual likelihood map. • Only subset of observations applied to belief

41. 2-Step Algorithm (10) Track Assisted State Estimation Gradual update of estimated source position, as sensor data is aggregated along the path ABCD.

42. Track Assisted State Estimation 2-Step Algorithm (11) Final estimated likelihood map after integration of ABCD, and renormalization for easier viewing. Final update of estimated source position, as sensor data is aggregated along the path ABCD.

43. End of 2-Step Algorithm • (40 minutes)

44. Experiments (1) • Experimental Setup • Originally intended on collecting data from a field of physical sensors, however this hardware component distracted from analytical purpose of thesis • Forward data generated based on real wind data, numerical approximation to diffusion, on a grid size m=n=250 • All code implemented in LabVIEW graphical programming language, allows for easy future hardware integration

45. Initialize, setup scenarios And control batch runs Main loop – heavy computation Result Outputs, statistical calculations Experiments (2) • LabVIEW simulation design 2-Step Alg.

46. Experiments (3) DiffuseNumerical implementation Fick’s law for diffusion implemented numerically using standard 2D centered difference scheme Concentration of Agent assumed=0 at boundaries, agent “floats off screen” Same code used for forward diffusion and backward belief state propagation

47. Large Batch Study #1: Wind Study Experiments (4) • Likelihood as a function of wind direction standard deviation • As wind variability increases tracks become critical and perform dramatically better, operating in regions of high wind shift • A dataset containing 40,000 samples of real wind data are used to generate samples of length 200 spanning 5 degrees to 90 degrees

48. Experiments (5) • Wind Data Example, heavy processing needed Data Imported from web http://www.ndbc.noaa.gov/ YYYY MM DD hh mm DIR SPD GDR GSP GTIME 2004 12 31 23 00 116 7.5 999 99.0 9999 2004 12 31 23 10 115 6.7 999 99.0 9999 2004 12 31 23 20 134 7.2 999 99.0 9999 2004 12 31 23 30 136 8.2 999 99.0 9999

49. Large Batch Study #2: Sensor Density Study Experiments (6) • Increase number of sensors from N=50, 100, 150, 200, 250, 300 for a 250x250 grid. Random addition of new sensors to existing set. • Source fixed • Same wind series for each trial • Compared performance of belief maps generated by sensor network using tracks Vs. No Tracks

50. Results and Conclusions • Maximum likelihood, ML(M), in each belief map compared to likelihood value at true source M(i,j) Likelihood performance metrics Belief M(i,j) Source A(i,j) ML(M)