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Plume Tracking in Sensor Networks

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Plume Tracking in Sensor Networks

Glenn Nofsinger

PhD Thesis Defense

August 22, 2006

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

- 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

- 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?

t=o

t=1

t=2

t=3

t=4

- What is the best estimate of the true source locations given these observations?

t=1

- True initial state: two source locations
- Thesis work estimates this truth state

- 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

- 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

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

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

+

Theoretical Background (2)

- Agent concentration for some area, A
- Likelihood map given sensor observations

- Fick’s Law for diffusion and linear wind
- First order approximation to process
- Standard Gaussian solution

Advection-diffusion Model

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

- 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

Theoretical Background (6)

- Unique response per source location
- Relative differences of Tmax unique
- 3 sensors for 2D location
- Can solve for (X0,Y0)

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

C(x,y,t)

A

B

t

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

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

Relationship between a posteriori distribution, a priori distribution, and the likelihood function

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:

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

Theoretical Background (11)

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

Theoretical Background (13)

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

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

- 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

- (20 minutes)

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)

Step 1:

Track Formation

1.1 Track Initialization –All new observations potentially create tracks. The terminal node on track is designated leader node

- 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

- Step 1:
- Track Formation
- 1.3 Track extension – observations that were associated in step 1.2 become the new leader nodes.

- 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

Track A

Leader nodes

New Observation

Track B

Likelihood Function

Detail of likelihood function for track association

- Step 2:
- State Estimation
- Each track sequence produces an individual likelihood map
- In this case only 4 sensor observations used to form belief map

- Step 2:
- State Estimation
- Each track sequence produces an individual likelihood map.
- Only subset of observations applied to belief

2-Step Algorithm (10)

Track Assisted State Estimation

Gradual update of estimated source position, as sensor data is aggregated along the path ABCD.

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.

- (40 minutes)

- 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

Initialize, setup scenarios

And control batch runs

Main loop –

heavy computation

Result Outputs,

statistical calculations

- LabVIEW simulation design

2-Step Alg.

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

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

- 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

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

- 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)

- Performance Metric Definition, For a Single Source:
- [M(i,j) / ML(M) ] = P(M), the performance of M
- For P(M)=1, sensor occurs at the position (i,j) within M of maximum likelihood. 1 is considered a perfect score, while 0 is considered the lowest score
- This is the metric used in wind study and sensor node density study

Results and Conclusions (2)

2-Step predictor

ZT

Observation set

MMSE predictor

- Wind Experimental Results Summary

- Sensor Node Density Results Summary

Results and Conclusions

- Identical network with tracks can achieve sharper maps with lower densities of sensors
- Major advantage of using tracks is the ability to establish number of unique sources
- Theoretical information content of a sensor network grows as log(N), therefore diminishing returns as N gets large. Both estimators approach this limit but at different rates

- Summary of Wind performance zones

Best performance

zone

Mean wind Speed scaled into 4 groups

Standard deviation of direction divided into 4 groups

This produced 16 total wind categories

high

Intermediate performance

3

4

1

2

Mean wind speed

5

…

ZONE 1

…

ZONE 2

ZONE 3

Worst performance

Zone: low wind

Speed, with

Frequent shifts

16

low

high

Wind direction Std. deviation

- The 2-Step tracking based algorithm allows provides enhanced performance compared to uniform estimator
- Sensor density – on average the tracker based maps received a likelihood metric better by a factor of 2
- High Wind variability – in conditions of high wind direction variability, the tracking based estimator performs much better than uniform estimator. Maintaining tracks and therefore estimates up to 30 degrees Std. deviation higher.

- Application of sensor network physical process tracking to extreme remote environments
- The computationally intensive data association portion of the 2-Step algorithm method could be exported to existing MHT/PQS infrastructures and improved (pruning, track maintenance, hypothesis management).

Results and Conclusions (3)

P(M)=1 E-4

Results and Conclusions (4)

P(M)=1 E-4

Results and Conclusions (5)

P(M)=1.3 E-4

Results and Conclusions (6)

P(M)=1E-4

Results and Conclusions (7)

P(M)=9E-5

- Application of sensor network physical process tracking to extreme remote environments
- The computationally intensive data association portion of the 2-Step algorithm method could be exported to existing MHT/PQS infrastructures and improved (pruning, track maintenance, hypothesis management).

- SPIE 2004
- MILCOM 2005
- SPIE 2006

To what extent can

we differentiate two sources

as a function of sensor

density?

In this example, two sources in constant wind can superimpose to create a 3rd peak

The goal of this sensor network is to correctly identify exactly 2 sources, not 3

We want to construct a belief map after each trial, and look at

the value of the cell where the actual source was released.

Once we introduce tracking, we get sharper regions with higher

Values per cell. This allows us to compare the predicted map

with ground truth on any selected trial.

Forward simulation

Likelihood Map M

3 sources

M=N=250

300 sensors

Constant Wind

The inverse scale here is E-5, which is likelihood that

The source was released from that particular cell.

Typical values for a single cell are between 10E-3 and 10E-5

Known release event : A

Forward Probability, P(B|A)

No Wind

Constant Wind

Variable Wind

Inverse Probability, P(A|B)

Known detection event : B

No Wind

Constant Wind

Variable Wind

Bayes Rule: