Sniper Localization Using Acoustic Sensors

Allison Doren

Anne Kitzmiller

Allie Lockhart

Under the Direction of Dr. Arye NehoraiDecember 11, 2013

[6]

Outline

Background

Muzzle Blast Model

Sniper Localization Maximum Likelihood Cramér-Rao Bound Mean Square Error

Results

Detection

Conclusions

Background

Existing Work: “Shooter Localization in Wireless Microphone Networks,” comparing muzzle blast and shock wave

models and using Cramér-Rao lower bound analysis[1]

“Analysis of Sniper Localization for Mobile, Asynchronous Sensors”, relying on time difference of arrival measurements, and providing a Cramér-Rao bound for the models[2]

“ShotSpotter” uses acoustic sensors to detect outside gunshot incidents in the D.C. area[5]

Applications: Military Operations: can be worn by soldiers or placed in vehicles Civilian Environments: can detect gunfire to alert local authorities

Example of a sensor network[2]

= sensor= shooter

Types of Models1. Shockwave Model (SW)

Exploits the shockwave of a gun shot, which comes about as a result of the supersonic bullets

2. Muzzle Blast Model (MB) Exploits the “bang” of a gun shot

3. Combined Model (Shockwave and Muzzle Blast)

The shockwave from the supersonic bullet reaches the microphone before the muzzle blast [1]

Muzzle Blast Model: First Step

Time of Arrival (TOA), for the ith sensor and the mth measurement:

Define Parameters: N = total number of sensors (N = 6) iter = number of iterations (iter = 100) m = total number of measurements (m = 500) i = ith sensor (i = 1, 2, …, N) c = speed of sound (330 m/s) = time origin of the muzzle blast (normal distribution) = distance from the ith sensor at to the sniper position

at

Muzzle Blast Model: Second Step

Muzzle Blast Time Difference of Arrival (TDOA): Uses sensor 1 as a reference, for time synchronization purposes = time origin of muzzle blast for ith sensor , as defined below, where and are assumed to be independent, , and

, for i = 2, 3, …, N

𝜃 e

Muzzle Blast Model: Second Step

Maximum Likelihood Estimation, using the conditional probability distribution p:

Maximum Likelihood (ML) and Least Squares (LS) equivalent in this simulation, because using deterministic ML method, where is the unknown parameter

Therefore, maximizing for the ML method was equivalent to minimizing the error for the LS method.

Cramér-Rao Bound

The Cramér-Rao Bound (CRB) is a lower bound on the variance of an unbiased estimator

We use a Multivariate Normal Distribution, because TDOA vector has a length equal to N-1

Cramér-Rao Bound

CRB for Multivariate Case The Fisher Information Matrix (FIM) for N-variate multivariate normal

distribution 𝜇ሺ𝜃ሻ= [𝜇1ሺ𝜃ሻ,𝜇2ሺ𝜃ሻ,…,𝜇𝑁ሺ𝜃ሻ]𝑇

𝐿𝑒𝑡 ∑ሺ𝜃ሻ 𝑏𝑒 𝑡ℎ𝑒 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑇ℎ𝑒 𝑡𝑦𝑝𝑖𝑐𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝐽𝑚,𝑛,𝑜𝑓 𝑡ℎ 𝐹𝐼𝑀 𝑓𝑜𝑟 𝑋 ~ 𝑁൫𝜇ሺ𝜃ሻ,∑ሺ𝜃ሻ൯𝑖𝑠:

𝐽𝑚,𝑛 = 𝜕𝜇𝑇𝜕𝜃𝑚 ∑−1 𝜕𝜇𝜕𝜃𝑛 + 12𝑡𝑟൬∑−1 𝜕∑𝜃𝑚 ∑−1 𝜕∑𝜃𝑛൰

𝐹𝑜𝑟 𝑡ℎ𝑒 𝑠𝑝𝑒𝑐𝑖𝑎𝑙 𝑐𝑎𝑠𝑒 𝑤ℎ𝑒𝑟𝑒 ∑ሺ𝜃ሻ= ∑,𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝐽𝑚,𝑛 = 𝜕𝜇𝑇𝜕𝜃𝑚 ∑−1 𝜕𝜇𝜕𝜃𝑛

Cramér-Rao Bound

In our case,

Cramér-Rao Bound

Fisher Information Matrix

For T independent measurements,

Compare MSE with CRB

N = number of sensors

iter = number of iterations

= our parameter

= the estimate of our parameter

Also find the MSE of our sniper position (x, y)

Mean Square Error

Signal-to-Noise Ratio (SNR)

Compare signal power to noise power

Signal Power: , where is as defined previously

Noise Power:

Results

Iterations, iter = 100

Number of measurements (shots), m = 500

Number of sensors, N = 6

= 0:0.04:0.36, standard deviation of noise

-100 -80 -60 -40 -20 0 20 40 60 80 100-20

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or(a) Sensor network and shooter position (b) Localization error of position

Placement of sensors in Matlab model and localization error

Variance = 0.01

Minimum values of error at (0,0), our true sniper location

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Comparison of localization performance on various six sensor geometries

Sensor Network Geometry

Shooter surrounded by sensors is ideal, but not practical

Line of sensors does not provide sufficient information

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Comparison of localization performance on various random sensor geometries

Sensor Network Geometry

Increased number of sensors increases accuracy, but not realistic to have this many sensors in close range

5 10 15 20 250

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SNR

MS

E (

met

ers)

SNR vs MSEMSE of sniper position (x, y) vs. SNR

As the signal-to-noise ratio increases, error decreases

Thus as noise increases, error increases

MSE of position vs. SNR

MSE of vs. SNR, with CRB

5 10 15 20 250

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SNR vs MSE with CRB

MSE

CRB

r

MSE converges to the CRB as SNR increases

MSE of , the TDOA, vs. SNR with CRB

Detection - general

The Neyman-Pearson Lemma [7] uses a likelihood-ratio test to choose a critical region that maximizes the power of a hypothesis test

=, false alarm

If are independent and identically distributed random samples of , and the following hypothesis test is given

.

It follows that the critical region is

where k is calculated from

Detection of a shot

For this simulation,

, where

, where .

If

then the critical region is of the form

Detection of a shot is rejected if and a is calculated from where . Then,

and

Therefore,

will be rejected if , and will be accepted if

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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Alpha (False Positive Rate)

Pow

er (

Tru

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ROC Curve

ROC Curve

ROC Curve generated from detection applied in the scalar case (2 sensors)

Power, PD =

As increases, the critical region also increases, and thus power increases.

PD

𝛼

Conclusions

We used the Maximum Likelihood Method, Cramér-Rao Bound, and Mean Square Error in the Muzzle Blast Model to analyze our simulated shooter data, with different values of variance (noise) As predicted, MSE increases as noise increases MSE converges to the CRB as SNR increases

We studied the concept of detection and applied it to the scalar case of detecting a sniper with two sensors

We would have liked to compare our results to actual data obtained from sensors

Further Research Adding walls or other obstacles to sensor model Using different types of sensors, ie. optical, infrared Explore shockwave or combined MB-SW model Compare results to real data

References

1. D. Lindgren, O. Wilsson, F. Gustafsson, and H. Habberstad, “Shooter localization in wireless sensor networks,” Information Fusion, 2009, FUSION ’09, 12th International Conference on, pp. 404-411, 2009.

2. G. T. Whipps, L. M. Kaplan, and R. Damarla, “Analysis of sniper localization for mobile, asynchronous sensors,” Signal Processing, Sensor Fusion, and Target Recognition XVIII, vol. 7336, 2009.

3. P. Bestagini, M. Compagnoni, F. Antonacci, A. Sarti, and S. Tubaro, “TDOA-based acoustic source localization in the space-range reference frame,” Multidimensional Systems and Signal Processing, Vol. March, 2013.

4. Stephen, Tan Kok Sin. (2006). Source localization using wireless sensor networks (Master’s thesis). Naval Postgraduate School, 2006. Web. Sept 2013.

5. Berkowitz, Bonnie, Emily Chow, Dan Keating and James Smallwood. “Shots heard around the District.” The Washington Post 2 Nov. 2013. Investigations Web. Nov. 2013.

6. Photograph of Sniper. Photograph. n.d. Shooter Localization Mobile App Pinpoints Enemy Snipers. Vanderbilt School of Engineering. Web. 11 Nov 2013.

7. Hogg, Robert V., and Allen T. Craig. Introduction to Mathematical Statistics. New York: Macmillan, 1978. 90-98. Print.

Thank You!

Thank you to Keyong Han, the PhD student who has been guiding us throughout this project.

Thank you to Dr. Arye Nehorai for all of his help in overseeing our work and our progress.

Questions?