Gas-Leak Localization Using Distributed Ultrasonic Sensors

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Keywords: ultrasonic source localization, MEMS microphone, maximum likelihood ... first application of microphone arrays to speech and speaker recognition in ... combining optical and acoustic sensors for detecting and localizing gas leaks.
Gas-Leak Localization Using Distributed Ultrasonic Sensors Javid Huseynova,b , Shankar Baligab , Michael Dillencourta , Lubomir Bica , Nader Bagherzadeha a Bren

School of Information and Computer Science, University of California, Irvine, CA 92697 b General Monitors Transnational, 26776 Simpatica Circle, Lake Forest, CA 92630 ABSTRACT

We propose an ultrasonic gas leak localization system based on a distributed network of sensors. The system deploys highly sensitive miniature Micro-Electro-Mechanical Systems (MEMS) microphones and uses a suite of energy-decay (ED) and time-delay of arrival (TDOA) algorithms for localizing a source of a gas leak. Statistical tools such as the maximum likelihood (ML) and the least squares (LS) estimators are used for approximating the source location when closed-form solutions fail in the presence of ambient background nuisance and inherent electronic noise. The proposed localization algorithms were implemented and tested using a Java-based simulation platform connected to four or more distributed MEMS microphones observing a broadband nitrogen leak from an orifice. The performance of centralized and decentralized algorithms under ED and TDOA schemes is analyzed and compared in terms of communication overhead and accuracy in presence of additive white Gaussian noise (AWGN). Keywords: ultrasonic source localization, MEMS microphone, maximum likelihood estimator, distributed algorithms, ultrasonic gas detection, sensor networks

1. INTRODUCTION The reliable detection and localization of gas leaks is essential for ensuring safety and minimizing property damage. In industry, a primary approach for detecting gas leaks is based on the physical phenomenon of absorption of infrared energy by combustible gases such as methane, propane or ethane. Another approach is to measure the current generated by electrochemical cells or catalytic sensors to determine the amount of toxic or combustible gas present. The third, and relatively new, method is detecting the acoustic wave emitted during a gas leak. As opposed to the other detection methods described, acoustic gas detection cannot, as yet, measure the amount or type of gas from an observed signal. However, and again unlike the other detection methods, the acoustic method can be developed to locate the source of a leak. ULTRASONIC SOURCE

LIILII

((

MICROPHONES

Figure 1. Distributed Source Localization

Use of ultrasonic frequencies, with the audio frequencies filtered out by an electrical bandpass filter, enables the measurement and localization of high-pressure gas leaks in an industrial environment while avoiding the disturbance and nuisance at audio frequencies caused by reverberation, natural and man-made background phenomena.1 To the best of our knowledge, the distributed localization of broadband gas-leak sources using MEMS microphones in the ultrasonic frequency range has not been attempted before. Primary author contact: [email protected], +1 949 581 4464 Smart Sensor Phenomena, Technology, Networks, and Systems 2009, edited by Norbert G. Meyendorf, Kara J. Peters, Wolfgang Ecke, Proc. of SPIE Vol. 7293, 72930Z · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.812058 Proc. of SPIE Vol. 7293 72930Z-1

The localization of an ultrasonic source is accomplished by using differences in ultrasonic signals received at spatially distributed microphones. There are two parameters that characterize these differences:2 • Interaural Level Difference (ILD), or the differential Energy Decay (ED) • Interaural Time Difference (ITD), or the differential Time Delay of Arrival (TDOA) The ILD, a difference in sound pressure level (SPL) measured at two different receivers (ears, microphones), is helpful for humans to localize a sound source. It is based on a simple physical property that a sound wave has an amplitude which decays at a rate inversely proportional to the square of the distance from the source to measurement location.3 The ITD-based methods rely on phase differences in acoustic signals arriving at neighboring microphone sensors at the speed of sound.4 The localization of utrasonic gas leak sources is an inherently distributed target detection problem (Fig. 1). To localize a gas leak source in any signal frequency spectrum, individual sensors must first detect the presence of a source from their local data. Then, a collaborative signal processing at the network level involves routing information through the network and fusing the data from different sensors to generate an estimate of the source location. Therefore, solutions proposed for ultrasonic localization are naturally applicable in the context of wireless sensor networks (WSN). Due to low-power nodes in WSN, efficient in-network data processing is a key factor for enabling WSN to extract useful information and to ensure efficiency and accuracy of detection.5–7 In terms of efficiency, radio communication is an energy consuming task and is identified in many deployments as the primary factor for sensor node’s battery exhaustion8 because emitting or receiving a packet is far more energy consuming than local computations. Thus, the reduction of the amount of data transmissions and decentralized in-network computation are the recognized design priorities for WSN algorithms.9 In this work, we propose and implement a suite of ED- and TDOA-based algorithms for localizing a broadband ultrasonic gas-leak source using distributed MEMS microphones. This research problem is novel from three aspects: 1) the localization of gas leaks using ultrasonic broadband signals as an input; 2) the deployment of MEMS microphones for a distributed ultrasonic localization, and 3) the application of Newton’s iterative method as a decentralized implementation of ML acoustic localization.

2. BACKGROUND AND RELATED RESEARCH Acoustic localization strategies can be traced back to the earliest radar and sonar localization systems.10 With the first application of microphone arrays to speech and speaker recognition in 1960s, beamforming techniques became widely popular for computing the direction of arrival (DOA) of sound.11–13 Yet the distributed applications of acoustic localization in an industrial setup were not addressed until the development of low-cost microphones and WSN.14 In recent years, low-cost MEMS microphones have been introduced to the consumer market. Such microphones surpass traditional Electret Condenser Microphones (ECM) in frequency response: many can be operated up to 100 kHz in the utrasonic range, thereby opening up novel applications beyond the consumer audio market. MEMS microphones can also operate in a wide range of temperatures and consume less than a milliwatt of power in operation, making them ideal for use in battery-powered wireless sensors. Figure 2 shows the ultrasonic performance of such a microphone out to 70 kHz, compared with an expensive 14 inch ultrasonic precision condenser pressure microphone (Type 40 BD) from G.R.A.S. Sound & Vibration.15 The experiments in this paper were conducted using the omni-directional MSM2RM-S3035 MEMS microphone from Memstech Berhad.16 A number of ED- and TDOA-based algorithms have been developed for localizing sound sources in open spaces and reverberant room environments using distributed microphones. Yonak et al17 proposed a system combining optical and acoustic sensors for detecting and localizing gas leaks. A decentralized incremental gradient algorithm for ED-based localization with known source energy was proposed by Rabbat and Nowak.18 This algorithm relies on a priori knowledge of acoustic energy measured at the source to estimate its location ρˆ by incrementally solving the error ML function at each sensor using steepest

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-40 1/4 GRAS microphone, 40BD MEMS microphone

et 50 C

>

60 a

70 80 90 0

10

30

20

40

50

60

70

Frequency. kliz

Figure 2. Frequency response of MEMS and ECM microphones to a broadband ultrasonic source

gradient descent. The estimation begins with an arbitrary initial location and the circulation continues till the location estimation converges to a limit point or the itineracy exceeds a predefined cycle number. A drawback of this algorithm rests with the assumption of known source energy, which is often not the case in reality. Li and Hu19 proposed a centralized localization method using energy ratios which mathematically eliminate unknown source energy from the computation. Wang and Yang20 proposed a decentralized version of this energy-ratio based algorithm using the ML gradient descent method. In practical applications, acoustic energy is expressed in terms of a sound pressure level (SPL) in decibels (dB). Hence, we define the ED-based localization method in terms of SPL measurements and distance. Instead of using energy ratios,20 resulting in extra communication between pairs of sensors, we guess the value of an unknown source SPL based on a reference sensor SPL measurement and use it in the ML error function minimized at the remaining sensors. In addition, we make use of Newton’s method which converges faster than gradient descent, thus further reducing the amount of communication and computation overhead.

3. ED-BASED LOCALIZATION The signal received at a microphone observing an ultrasonic source at time t is modeled as: x(t) = h ∗ s(t) + w(t),

(1)

where s(t) is a broadband source signal at time t, h is a channel impulse response, and w(t) is the additive noise.21 The AWGN is a primary factor impacting the performance of the localization scheme. Using ultrasonic frequencies, we eliminate audible noise with a frequency bandpass filter in hardware, significantly lowering the impact of AWGN on the SPL measurements. Li and Hu22 formulated the energy-decay model for an audible sound signal as: Es + wi , (2) Ei = gi β ρ − ri  where Ei is the energy measured at sensor i, Es is the energy at the source, gi is electronic gain of sensor i, ρ is the target location, ri is the location of sensor i, wi is the noise estimate, which can be approximated by a Gaussian random variable, and β is the power of energy decay over distance. The β parameter is dependent on environmental conditions such as temperature, humidity, wind, equipment, walls, and other obstructions. In our work, we assume that the ultrasonic source is omni-directional and the ultrasonic waves propagate in free space, so that β is assumed to be constant at 2.

3.1 Localization using ED-based MLE The ML estimator (MLE) helps to approximates the location of an ultrasonic source in presence of AWGN, given the observed sensor SPLs. MLE is a popular method for highly non-linear estimation due to its assymptotic

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efficiency in the limit T → ∞, where T denotes the observation interval.23 In order to apply the MLE, certain assumptions need to be made about the statistical behavior of the measurements, namely, that the measurement error is modeled as Gaussian.21 In the ED-based localization, the MLE seeks to find a source position ρ(u, v) which maximizes the likelihood of observing SPL values already measured at each sensor: ρˆML = arg max Pr(L0 , L1 , . . . , Ln |ρ) ρ

(3)

Using the SPL definition and assuming β = 2, the sound energy degradation defined in (2) can be expressed as: Li = LS − 20 log

di + i , dS

(4)

where dS is a unit distance from the source, di is a distance from the source to the microphone i, LS and Li are the SPLs at the source and node i respectively, and i is the additive dB noise, expressed as a logarithm of wi from equation (2). In order to solve the problem with unknown source SPL LS , we assume L0 , the SPL measured by one of the sensors, to be a reference for LS . Knowing L0 measured by a sensor located at r0 (x0 , y0 ) and an initial estimate of the source location ρ(u, v), the LS at a unit distance, dS = 1, from the source is calculated:  LS = L0 + 20 log10 (x0 − u)2 + (y0 − v)2 (5) This value of LS is then used either to maximize the likelihood of SPL values Li observed by the remaining sensors i = 1, . . . , n, or, equivalently, to minimize the square of error function i between the assumed reading of local SPL given LS and the actual received Li as follows: 2

fi (ρ) = 2i = [Li − LS + 20 log10 dρi ] ρˆML = arg min ρ

where dρi =

n 

fi (ρ)

(6) (7)

i=1

 (xi − u)2 + (yi − v)2 is the distance between sensor i and current source location estimate ρ(u, v).

Under ideal circumstances (i = 0), the SPL measured by a microphone must be exactly the amount of source SPL reduced by a logarithm of the square of distance from the source to that microphone. But in reality, there is always AWGN present in measurements from background nuisance and processing in electronics. So the numerical approximation methods such as the steepest gradient descent (ML Gradient Descent) and the Newton’s iterative method (ML Newton) need to be used for the convergence of MLE.

Figure 3. ED ML convergence at 0 dB with Exhaustive Search, Gradient Descent and Newton’s Method

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Figure 3 shows the ML error surface generated by an exhaustive search of (13) on a 500 × 500 coordinate square as well as the convergence of ML Gradient Descent and ML Newton methods on that surface. This result is for a sample case with n = 5 sensors and an unknown source positioned at S(200, 200) with no additive noise. The global minimum on this error surface represents the source position, so the ML Gradient Descent and ML Newton solutions described in the following subsections seek to find this minimum point in more efficient ways than exhaustive search. 3.1.1 ML Gradient Descent Steepest gradient descent is an algorithm for finding the nearest local minimum of a function, assuming the gradient of the function can be computed. If the real-valued ML error function f (ρ) is defined and differentiable in a neighborhood of a point ρ(u, v), then f (ρ) decreases fastest from point ρ in the direction of the negative gradient of f at point ρ, which is −∇f(ρ). Redefining this in iterative terms, where ρk and ρk+1 are iterative points on the gradient line: ρk+1 = ρk − μk ∇f(ρk ) (8) where μk > 0 is a step size or value of the gradient at each step. After several iterations of this algorithm on the function surface, with f (ρ0 ) ≥ f (ρ1 ) ≥ . . . ≥ f (ρn ) the sequence of ρk converges to the desired local minimum of the ML function. The value of step size μk can change at every iteration k of the algorithm. While the direction of gradient descent can be definitively computed as the first derivative of the ML function, the amount of offset in the direction of the gradient, or the step size, requires additional pre-processing. Various heuristics exist for estimating the step size, and in our work, we make use of binary search for determining the right step size μk . 3.1.2 ML Newton Guessing an initial value and readjusting the step size in ML Gradient Descent requires the application of time-consuming search heuristics. Also, depending on error function, gradient descent can take many iterations to converge to the solution, yielding it to be inefficient for decentralized implementations. Alternatively, ML Newton method can be applied to iteratively approximate the source location using a tangent line of the gradient. Depending on error function, ML Newton can be unstable, but it does converge to a solution much faster than ML Gradient Descent, making it attractive for a decentralized localization. Given the source location at ρk (u, v) and the error gradient ∇f(ρk ), Newton’s iterative method can be formulated as: (9) −f  (ρk )(ρk+1 − ρk ) = ∇f(ρk ) Computing the first derivative of the gradient, or the second derivative of the error function, essentially provides the step size and direction at each iteration of the algorithm. Since the gradient ∇f(ρ) is two dimensional, its first derivative would be the Jacobian J(ρ) defined as: ⎤ ⎡ 2 2 J(ρ) = ⎣

∂ f ∂u2

∂ f ∂u∂v

∂2 f ∂u∂v

∂2f ∂v 2



(10)

Our implementations of centralized and decentralized ML Newton methods are presented below. In centralized case, each step of the algorithm is computed and the source position is updated at a central processor based on SPL data collected from all sensor nodes. In decentralized version, each step of the ML Newton method is taken on a distinct sensor using only its local measured SPL and the values of source SPL LS and ρ(u, v) received from the neighbor.

3.2 Localization using ED-based LSE Another way of eliminating the unknown source energy was proposed by Li and Hu.19 Approximating the additive noise by its mean value μi , ratio φij of energies received by sensors i and j can be expressed as follows:  φij =

Ei − μi Ej − μj

− β1 =

ρ − ri ρ − rj

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(11)

Algorithm 1 ED-Based ML Newton Centralized Require: Li , ri (xi , yi ) for i = 0, . . . , n; ρ(u, v) set k = 0; i = 1 while (k < max) and (∇fi (ρ) > tolerance) do update LS for i : 1 → n do ∇f(ρ) = add ∇fi (ρ) J(ρ) = add fi (ρ) end for ρ = ρ− Gaussian Eliminate J(ρ), ∇f(ρ) k =k+1 end while Algorithm 2 ED-Based ML Newton Decentralized Require: Li , ri (xi , yi ) for i = 0, . . . , n; ρ(u, v) set k = 0; i = 1 while (k < max) and (∇fi (ρ) > tolerance) do update LS compute ∇fi (ρ) compute Ji (ρ) = fi (ρ) ρ = ρ− Gaussian Eliminate Ji (ρ), ∇fi (ρ) k =k+1 i = (k mod n) + 1 end while When 0 < φij = 1, the source must reside on a circle described by 2 ρ − cij 2 = Rij

(12)

where cij is the center and Rij is the radius of the circle, so that: cij =

ri − φ2ij rj φij ri − rj  , Rij = 2 1 − φij 1 − φ2ij

(13)

In absence of noise, the source can be located using the closed-form intersection of all generated circles. But since AWGN is present, the source location ρ(u, v) can be computed as a point closest to the surface of all circles using the LS estimator (LSE). Given circles k = 1, · · · , n − 1, where n is the number of sensors, the LS criterion seeks to minimize the following cost function: J(ρ) =

n−1  k

(ρ − Rk )2 =

n−1 

 (u − ak )2 + (v − bk )2 − 2 (u − ak )2 + (v − bk )2 Rk + Rk2 ,

(14)

k=1

where ρ(u, v) is the source location, (ak , bk ) and Rk are the center and the radius of circle k, respectively. The LS solution is obtained by making the sum of the partial derivatives of equation (14) equal to 0 and solving in closed form.

n−1  Rk ) =0 (15) (u − ak )(1 −  (u − ak )2 + (v − bk )2 k

n−1  Rk ) =0 (16) (v − bk )(1 −  (u − ak )2 + (v − bk )2 k Thus, as also shown in Figure 4, the ED-based LS algorithm seeks to compute the source location ρ(u, v) which is closest to the surface of all circles by minimizing the sum of cumulative distances from the estimated location to the centers of circles.

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\

p(u,v)

R1\

\ \

\I

'- _/

Figure 4. Source position with regard to circles

4. TDOA-BASED LOCALIZATION The TDOA-based localization makes use of the relative time delay, or a phase shift, between signals from the same observed source arriving at two distinct microphones. The TDOA approach is fundamental in microphone array processing,14 and the estimation of TDOAs is the first step in source localization. Similar to ED-based localization, estimating source position from a set of measured TDOA data represents a nonlinear inverse problem which can be tackled using numerical approximation.

Figure 5. Sensors A, B observing source S.

As shown in Figure 5, the TDOA localization is described by a hyperbola, where sensors A and B are the foci, and the source S is located anywhere on the hyperbolic surface defined by the difference ΔdAB = dAS − dBS . The ratio of this distance (or range) difference with a constant speed of sound c represents the TDOA, or a phase shift, τAB = ΔdAB /c. Ideally, in the absence of AWGN, the source location can be found by simply intersecting a set of hyperbolas. However, in presence of an additive noise, there is a need for applying non-linear estimation techniques such as the MLE or the least-squares estimator (LSE).

4.1 Localization using TDOA-based MLE In the TDOA-based localization, MLE is applied to approximate the location of the source using the computed delay times as an input. Speaking in geometric terms, if the computed relative TDOA hyperbolas do not intersect at a single point due to additive noise, the MLE helps to approximate such intersection. The application of MLE to the TDOA estimation is usually based on three fundamental assumptions: constant delay, stationary processes, and long observation interval T >> τc , |d|/c, where τc is the correlation time and d/c is the differential time delay.24 Given a set of time delay estimates τ = (τ10 , τ20 , · · · , τn0 )T and an assumed source location ρ(u, v), the maximum likelihood estimator is defined as follows: ρˆML = arg max (ρ) = arg max P r(τ10 , τ20 , · · · , τn0 |ρ). ρ

ρ

Assuming that the additive noise is zero-mean and jointly Gaussian distributed, the joint probability density function (PDF) of τ conditioned on ρ is given as: f (τ |ρ) =

exp{− 21 [τ − τm (ρ)]T Σ−1 }  (2π)N det(Σ)

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The iterative error in the MLE can be modeled as: =

N 2  [τ0i − τˆ0i (ρ)] i=1

2 σ0i

(17)

Figure 6. TDOA ML convergence at 0 dB with Exhaustive Search, Gradient Descent and Newton’s Method

An exhaustive search solution of the ML error function 17 can be found by sweeping through the entire space of possible source positions to search for one, where  is closest to 0. Figure 6 shows the error surface generated by the Exhaustive Search and the convergence plots of Gradient Descent and Newton’s Method. 4.1.1 ML Gradient Descent Gradient descent method applied to ML error function seeks to find the point where the gradient (or the first derivative of the ML error function) is equal to 0. It is applied iteratively as follows: ρn+1 = ρn −

1 μ ∇ 2

For a given set of two sensors A and B, observing a source at location ρ(u, v):   (xA − u)2 + (yA − v)2 − (xB − u)2 + (yB − v)2 , τˆAB (ρ) = c

(18)

(19)

where c is a speed of sound, assumed to be a constant. To obtain the gradient ∇ for steepest descent formulation, we take a partial derivative of the error function  with respect to u and v coordinates of the source. If we assume one of the microphones located at position r0 (x0 , y0 ) as a reference, using TDOAs between other i = 1 . . . N microphones and this reference microphone 0, we can obtain the following iterative solution for ρ(u, v) using the steepest gradient descent method: un+1 = un +

N  i=1

2cμ

  τ0i − τ0i (ρ) x0 − un xi − un − 2 σ0i dρ0 dρi

  τ0i − τ0i (ρ) y0 − vn yi − vn 2cμ − vn+1 = vn + 2 σ0i dρ0 dρi i=1   = (x0 − un )2 + (y0 − vn )2 and dρi = (xi − un )2 + (yi − vn )2 . N 

where dρ0

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(20)

(21)

4.1.2 ML Newton Similar to the ED-based localization, Newton’s method can also be in TDOA approach to localize the source using the tangent line of the gradient. As shown for ED-based methods, given the estimate ρn (u, v) of a source location at iteration n and the error gradient ∇ = f (ρn ), the iterative ML Newton solution is defined as: −f  (ρn )(ρn+1 − ρn ) = f (ρn )

(22)

Hence computing the first derivative of the gradient, or the second derivative of the error function, essentially provides us with a step size. Since the gradient is two dimensional, it can be redefined in terms of partial ∂ ∂ derivatives as f (ρn ) = { ∂u ; ∂v }. And the first derivative of the gradient function f (ρn ) is the Jacobian: ⎡ 2 ⎤

2 J =⎣

∂  ∂u2

∂  ∂u∂v

∂2 ∂u∂v

∂2 ∂v 2

⎦=

∂f1 ∂u

∂f1 ∂v

∂f2 ∂u

∂f2 ∂v

(23)

0i (ρ) using the gradient values For a given set of two microphones r0 (x0 , y0 ) and ri (xi , yi ), TDOA error ei = τ0i −τ 2 σ0i of the steepest descent solution for TDOA-based method as defined in previous subsection, we obtain: 

 2  N  u − x0 ∂f1 dρi − (u − xi )2 u − xi dρ0 − (u − x0 )2 = 2 c ei − − + (24) ∂u d3ρ0 d3ρi dρ0 dρi i=1

N

 ∂f2 = 2c ∂v i=1



∂f1 ∂f2 = ∂v ∂u

ei

=

+

 2  v − y0 dρi − (v − yi )2 v − yi dρ0 − (v − y0 )2 − − + d3ρ0 d3ρi dρ0 dρi 

N 

(u − x0 )(v − y0 ) (u − xi )(v − yi ) 2 c ei − d3ρ0 d3ρi i=1   N  u − x0 v − y0 u − xi v − yi 2c − − dρ0 dρi dρ0 dρi i=1

(25)

 +

(26)

  where dρ0 = (x0 − u)2 + (y0 − v)2 and dρi = (xi − u)2 + (yi − v)2 are the distances between the source ρ and microphones 0 and i, respectively. ⎡ 2 ⎤

∂ ∂  ∂2  un+1 − un 2 ∂u∂v ∂u ⎣ ∂u ⎦ = (27) ∂ ∂2 ∂2  v − v n+1 n 2 ∂v ∂u∂v

∂v

Algorithm 3 TDOA-based ML Newton Centralized Require: τ0i (i = 1, . . . , n) for i = 0, . . . , n; ρ(u, v) set k = 0; i = 1 while (k < max) and (∇fi (ρ) > tolerance) do for i : 1 → n do ∇f(ρ) = add ∇fi (ρ) J(ρ) = add fi (ρ) end for ρ = ρ− Gaussian Eliminate J(ρ), ∇f(ρ) k =k+1 end while

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4.2 Localization using TDOA-based LSE The TDOA-based ultrasonic source localization model was initially proposed in Walworth and Mahajan.25 The LS solution seeks to minimize misfit in the measured data but it may be complicated by multiple TDOAs being very close to the same value. This can be caused by microphone placement geometry, measurement uncertainty, additive noise or a combination of all three factors. Node positions are treated as known parameters when solving the inversion process using LS, however, error in node‘position measurements may introduce significant error to the solution. Given a set of n microphones with known locations ri (xi , yi ), 0 ≤ i ≤ n and an unknown source located at ρ(u, v), we assume that microphone 0 is a reference. The TDOA between any other microphone i and reference microphone is then defined as τ0i = Ti − T0 , where T0 and Ti are the absolute arrival times of the wave at reference and microphone i, respectively. Assuming a constant speed of sound as c, distance from assumed source location ρ(u, v) to reference microphone 0 as d, we can formulate a set of equations: d2 = (x0 − u)2 + (y0 − v)2 , (d + cτ01 )2 = (x1 − u)2 + (y1 − v)2 , .. .

(28)

(d + cτ0n )2 = (xn − u)2 + (yn − v)2 Expanding these equations we obtain: ⎡ 2x0 − 2x1 2y0 − 2y1 ⎢ 2x0 − 2x2 2y0 − 2y2 ⎢ ⎢ .. ⎣ . 2x0 − 2xn

2y1 − 2yn

⎤ ⎡ 2 2 −2cτ01 c τ01 + x20 + y02 − x21 − y12 ⎡ ⎤ 2 2 2 2 2 2 u ⎥ ⎢ −2cτ02 ⎥ ⎢ c τ02 + x0 + y0 − x2 − y2 ⎥×⎣ v ⎦=⎢ .. ⎦ ⎣ . d 2 −2cτ0n + x20 + y02 − x2n − yn2 c2 τ0n

⎤ ⎥ ⎥ ⎥ ⎦

(29)

The set of equations (29) of the form Xb = y can be solved in closed form using Gaussian elimination when n = 3, that is if there are four microphones. When more than four microphones are present, the system is overdetermined and can be solved using LS fit using the matrix transpose X T X b = X T y.

5. SIMULATION AND EXPERIMENTAL RESULTS A Java-based simulation program was developed for evaluating and comparing the proposed distributed algorithms under varying levels of AWGN, sensor and source placement topology, and the number of sensors. MEMS microphones in the simulation were randomly positioned within the field of size 500 × 500 sq. ft.

5.1 ED vs. TDOA comparison The source positions estimated in 1000 runs of the algorithm on a sample setup with a gas leak inside the convex hull of sensors at S(200, 200) and outside the convex hull at S(143, 54) with AWGN levels of 1 and 5 dB are presented in Figures 9 and 10 for ED-based methods and Figures 11 and 12 for TD-based methods. The starting guess for the source location in all presented cases is S(100, 50). On all figures, circles for 50th, 90th and 95th percentile location estimates in 1000 repetitive cases are presented in black, red and blue respectively. The experimental results show that generally all methods perform better if the source is located inside the convex hull of sensors. In ML solutions, as the source is moved away from the centroid of sensors toward the edge of the convex hull and closer to reference sensor, the performance of centralized ML Newton method degrades less gracefully than centralized ML Gradient Descent or decentralized ML Newton. If the source is moved away from centroid of sensors in the direction further from reference sensor, all methods degrade in performance. Both ML Gradient Descent and ML Newton produce more accurate results with time-delay case than with energy-decay. ML Newton converges to the solution faster than ML Gradient Descent, but is also unstable with increasing AWGN. In LS approach, for either TDOA or ED-based solution, placement of the source outside the convex hull results in drastic performance degradation. When the source is placed inside the convex hull, TDOA LS method

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outperforms ED LS method in accuracy with increasing AWGN, though evens out with the same accuracy at AWGN of 7 dB. The number of steps taken by each numerical method to localize the source to within the gradient tolerance of 0.25 are presented in Table 1 for ED-based localization and Table 2 for TDOA-based localization with two source placement options. Table 1. Convergence of ED-based ML algorithms at ∇f (ρ) ≤ 0.25

ML Method

S(200,200)

S(143,54)

13 4 8

2 2 7

Gradient Descent Newton Centralized Newton Decentralized

Although the ML Gradient Descent method converges better in general, in cases where the source is located inside the convex hull of sensors, ML Newton method can achieve the same result 3 times faster. Clearly decentralized ML Newton converges slower than centralized ML Newton, but it should be kept in mind that each step of centralized ML Newton requires SPL data from all sensors, while decentralized method uses only local SPL value at each sensor without extra communication. As shown by Rabbat and Nowak,18 such decentralization also offers significant savings in terms of complexity and power consumption. Table 2. Convergence of TDOA-based ML algorithms at ∇f (ρ) ≤ 0.25

ML Method

S(200,200)

S(143,54)

16 10

7 5

Gradient Descent Newton Centralized

5.2 Testing with a nitrogen leak source

\.p.

\O

e

- - -N

U

MICROPHONE

-

SOURCE

Figure 7. Test setup with nitrogen leak

We have also tested the decentralized ED-based ML Newton method in a 20 ft × 20 ft room with 4 MEMS microphone-mounted sensors arranged in a square as shown in Figure 7. Nitrogen gas was released from an orifice at 150 psi at four different locations inside the convex hull of the sensors. Raw signal data was sampled at 200 kHz from the four sensors for periods ranging from 5 to 10 seconds using a National Instruments’ (NI) 16-bit data acquisition card on a PC. The NI LabView program on the PC used this data to simultaneously compute the SPL readings at each microphone. The SPL data observed by the four microphones was fed as an input

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into the Java-based program for localization. The application of decentralized ML Newton method resulted in a successful localization of the nitrogen leak source with an offset of less than 1 ft.

6. CONCLUSION AND FUTURE RESEARCH We have developed and analyzed a suite of algorithms for localization of ultrasonic sources of gas leak using distributed MEMS microphones. The algorithms make use of the multiplicity of ultrasonic waves received at distributed microphones observing a single source and estimate the source location using relative time delays (TDOA) or energy decays (ED). Apart from the algorithmic design, the application of MEMS microphones in the context of a distributed localization of ultrasonic sources is novel on its own. Figure 8 presents the classification of analyzed algorithms. The proposed localization schemes define ML

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error functions based on either the differences in SPL values or in arrival times of the ultrasonic signal from a single source at various sensor nodes. For the ED-based approach, we define the LS and ML models using SPL variables measured in dB instead of an abstract definition of energy. Due to presence of AWGN in the input data and background nuisance, we employed numerical approximation methods such as the steepest gradient descent and Newton’s method for converging to the ultrasonic ML solution. We also implemented LS-based methods for both ED-based and TDOA-based approaches, and drew comparisons in presence of the varying levels of AWGN. The correctness of all implemented methods was verified using exhaustive search on the ML and LS error functions. In addition to centralized implementation, a decentralized implementation of the most efficient ED-based ML Newton method was developed. Experimental results indicate the advantage of using decentralized ML Newton method for fast localization of sources located inside the convex hull of microphones. Although less precise than ML Gradient solution, decentralized ML Newton offers significant savings in power consumption due to lesser number of convergence steps, hence - a lower communication overhead and better scalability, making this solution more attractive for a sensor network implementation.

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Although the ED-based methods perform poorer in accuracy with increasing AWGN, they do not require a high-precision time synchronization like the TDOA-based methods do. The ED-based methods also involve less communication overhead as a single sensor can produce a data observation (SPL), as opposed to TDOA-based cases where a pair of sensors is needed to compute the time delay. These advantages make the ED-based methods more attractive for implementation in distributed sensor networks. There are several directions for the future research on the subject of ultrasonic gas-leak localization using distributed sensor networks. Development of standardized network middleware for distributed ultrasonic localization would assist in the development of large and scalable clusters for detection of multiple gas leak sources at an industrial facility. Of particular interest in this case is the TinyOS operating system26, 27 for wireless sensor networks, which already provides some necessary utilities and interfaces. Since the TinyOS was implemented over the Texas Instruments’ low-power MSP430 16-bit RISC microcontrollers,28 further research into hardware integration of MEMS microphones with this platform would be of interest as well. In terms of distributed algorithms, the focus of future work could be on the development and performance analysis of the TDOA-based decentralized ML Newton as well as the high-precision clock mechanism for computing time delays. TDOA methods which are not as efficient as ED-based methods can still offer a viable alternative in powered sensor networks. The application of microphone arrays21, 29 and beamforming techniques,30 in conjunction with the ultrasonic gas-leak localization using sensor networks is also an interesting future direction to explore.

REFERENCES [1] Huseynov, J., Baliga, S., Dillencourt, M., Bic, L., and Bagherzadeh, N., “Energy-based localization of ultrasonic gas-leak sources using distributed MEMS microphones and a maximum-likelihood estimator,” in [the 29th IEEE International Conference on Distributed Computing Systems (ICDCS)], (June 2009 (submitted)). [2] Van De Par, S., Schimmel, O., Kohlrausch, A., and Breebaart, J., “Source segregation based on temporal envelope structure and binaural cues,” in [Hearing From Sensory Processing to Perception ], 143–153, Springer Berlin Heidelberg (2007). [3] Guentchev, K. and Weng, J., “Learning-based three dimensional sound localization using a compact noncoplanar array of microphones,” in [1998 AAAI Symposium of Intelligent Environments], (1998). [4] Aarabi, P., “Robust multi-source sound localization using temporal power fusion,” in [Proceedings of Sensor Fusion: Architectures, Algorithms, and Applications V (Aerosense’01), Orlando, Florida], (April 2001). [5] Intanagonwiwat, C., Govindan, R., and Estrin, D., “Directed diffusion: A scalable and robust communication paradigm for sensor networks,” in [Proceedings of the ACM/IEEE International Conference on Mobile Computing and Networking], 56–67 (2000). [6] Pattem, S. and Zhao, F., “The impact of spatial correlation on routing with compression in wireless sensor networks,” in [Proceedings of the third international symposium on Information processing in sensor networks,], 28–35 (2004). [7] Kumar, S., Zhao, F., and Shepherd, D., “Collaborative signal and information processing in microsensor networks,” IEEE Signal Processing Magazine 19, 13–14 (2002). [8] Akyildiz, I., Su, W., Sankarasubramaniam, Y., and Cayirci, E., “Wireless sensor networks: a survey,” Computer Networks 39, 393–422 (2002). [9] Ilyas, M., Mahgoub, I., and Kelly, L., [Handbook of Sensor Networks: Compact Wireless and Wired Sensing Systems], ch. Dynamic Source Routing in Ad Hoc Wireless Networks, CRC Press (2004). [10] Bian, X., Abowd, G. D., and Rehg, J. M., “Using sound source localization in a home environment,” Lecture Notes in Computer Science 3468, 19–36 (2005). [11] Widrow, B., Mantey, P., Griffiths, L., and Goode, B., “Adaptive antenna systems,” in [Proceedings of the IEEE], 55, 2143–2159 (December 1967). [12] Frost, O. L., “An algorithm for linearly constrained adaptive array processing,” in [Proceedings of the IEEE], 60, 926–935 (August 1972). [13] Johnson, D. H. and Dudgeon, D. E., [Array Signal processing: Concepts and Techniques], Prentice Hall (1993).

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[14] Chen, J. C., Yip, L., Elson, J., Wang, H., Maniezzo, D., Hudson, R. E., Yao, K., and Estrin, D., “Coherent acoustic array processing and localization on wireless sensor networks,” in [Proceedings of the IEEE], 91, 1154–1162 (August 2003). [15] GRAS, “Product data sheets and specifications.” http://www.gras.dk (2008). [16] MEMSTech, “Product data sheets and specifications.” http://www.memstech.com (2008). [17] Yonak, S. and Bowling, D., “Multiple microphone photoacoustic leak detection and localization system and method,” (2001). U.S. Patent No. 6,227,036. [18] Rabbat, M. and Nowak, R., “Decentralized source localization and tracking,” in [Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’04)], 3, 921–924 (2004). [19] Li, D. and Hu, Y. H., “Energy-based collaborative source localization using acoustic microsensor array,” EURASIP Journal of Applied Signal Processing , 321–337 (2003). [20] Wang, S. and Yang, J., “Decentralized acoustic source localization with unknown source energy in a wireless sensor network,” Measurement Science and Technology 18, 3768–3776 (2007). [21] Huang, Y., Benesty, J., and Chen, J., [Acoustic MIMO Signal Processing], Springer Berlin Heidelberg (2006). [22] Li, D. and Hu, Y. H., “Least square solutions of energy based acoustic source localization problems,” in [Proceedings of the International Conference on Parallel Processing (ICPP)], (August 2004). [23] Bickel, P. J. and Doksum, K. A., [Mathematical Statistics: Basic Ideas and Selected Topics], Holden Day, San Francisco (1977). [24] Champagne, B., Eizenman, M., and Pasupathy, S., “Exact maximum likelihood time delay estimation on short observation intervals,” IEEE Transactions on Signal Processing 39, 285–298 (June 1991). [25] Walworth, M. and Mahajan, A., “An ultrasonic 3d position estimation system using the differences in the time-of-flights from the transmitter to various receivers,” in [The 8th International Conference on Advanced Robotics (ICAR’97)], (1997). [26] Levis, P. and Culler, D., “Mat´e: A Tiny Virtual Machine for Sensor Networks,” in [International Conference on Architectural Support for Programming Languages and Operating Systems], (2002). [27] UC Berkeley, “TinyOS - an open-source operating system designed for wireless embedded sensor networks.” http://www.tinyos.net (2004). [28] Texas Instruments, “MSP430 low-power 16-bit risc microcontroller.” http://www.ti.com (2008). [29] Wax, M. and Kailath, T., “Optimum localization of multiple sources by passive arrays,” IEEE Transactions on Acoustics, Speech and Signal Processing 31, 1210–1218 (1983). [30] Ward, D. B. and Elko, G. W., “Mixed nearfield-farfield beamforming: a new technique for speech acquisition in reverberant environments,” in [the IEEE ASSP Workshop on Applied Signal Processing Audio Acoustics ], (1997).

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