Plume Tracking in Sensor Networks

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Plume Tracking in Sensor Networks

• Glenn Nofsinger
• PhD Thesis Defense
• August 22, 2006

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Outline

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

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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

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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?

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Problem Statement (1)
to
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Problem Statement (1)
t1
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Problem Statement (1)
t2
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Problem Statement (1)
t3
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Problem Statement (1)
t4

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

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Problem Statement (1)
t1

• True initial state two source locations
• Thesis work estimates this truth state

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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

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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

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Other work
Mobile scout robots
Swarm robots
I
II
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
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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

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Plot Conventions
Theoretical Background (2)

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

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Theoretical Background (3)
Advection-diffusion Model

• Ficks Law for diffusion and linear wind
• First order approximation to process
• Standard Gaussian solution

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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

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Theoretical Background (5)
Wind 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

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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)

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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

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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
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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(zx) the likelihood function of z given x
• p(xz) the a posteriori distribution of x given
  measurement z, also called the current belief

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Bayesian Formulation
Theoretical Background (10)

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

• Our state estimate
• True state

• Want our state estimate to be as close to true
  state as possible
• Given observation set,
  what is

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Estimators
Theoretical Background (11)

• 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

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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

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Uniform State Estimation
Theoretical Background (13)
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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

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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
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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

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End of Background

• (20 minutes)

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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)
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2-Step Algorithm (3)

• Step 1
• Track Formation
• 1.1 Track Initialization All new observations
  potentially create tracks. The terminal node on
  track is designated leader node

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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

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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.

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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

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2-Step Algorithm (7)
Detail of likelihood function for track
association
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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

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2-Step Algorithm (9)

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

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

• (40 minutes)

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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 mn250
• All code implemented in LabVIEW graphical
  programming language, allows for easy future
  hardware integration

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Experiments (2)

• LabVIEW simulation design

2-Step Alg.
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Experiments (3)

• DiffuseNumerical implementation
• Ficks law for diffusion implemented numerically
  using standard 2D centered difference scheme
• Concentration of Agent assumed0 at boundaries,
  agent floats off screen
• Same code used for forward diffusion and backward
  belief state propagation

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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

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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
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Large Batch Study 2 Sensor Density Study
Experiments (6)

• Increase number of sensors from N50, 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

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Results and Conclusions
Likelihood performance metrics

• Maximum likelihood, ML(M), in each belief map
  compared to likelihood value at true source M(i,j)

Belief M(i,j)
Source A(i,j)
ML(M)
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Results and Conclusions

• 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

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Typical M for same data
Results and Conclusions (2)
2-Step predictor
ZT
Observation set
MMSE predictor
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Results and Conclusions KEY RESULT

• Wind Experimental Results Summary

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Results and Conclusions KEY RESULT

• Sensor Node Density Results Summary

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Density Conclusion
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

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Results and Conclusions

• 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
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ZONE 1

ZONE 2
ZONE 3
Worst performance Zone low wind Speed, with
Frequent shifts
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low
high
Wind direction Std. deviation
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Results and Conclusions

• 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.

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Future Plans

• 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).

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Questions?
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Sensor Density Study N50 Sensors
Results and Conclusions (3)
P(M)1 E-4
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N100
Results and Conclusions (4)
P(M)1 E-4
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N200
Results and Conclusions (5)
P(M)1.3 E-4
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N300
Results and Conclusions (6)
P(M)1E-4
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N400 Sensors
Results and Conclusions (7)
P(M)9E-5
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Results - Belief Map From Uniform Predictor
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Results - Belief Map, Track 1
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Results - Belief Map Track2
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Future Plans

• 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).

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Questions?
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My papers

• SPIE 2004
• MILCOM 2005
• SPIE 2006

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Backup Slides
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Source Separation Problem
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
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Belief Map Without Tracks
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Inverse Belief Map of Sensor Network
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
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Belief No Tracks
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Belief No Tracks
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Track Formation

• 3 sources
• MN250
• 300 sensors
• Constant Wind

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Example Likelihood (Belief) Map
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
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Known release event A
Forward Probability, P(BA)
No Wind
Variable Wind
Constant Wind
Inverse Probability, P(AB)
Known detection event B
Constant Wind
No Wind
Variable Wind
Bayes Rule