DETECTION AND LOCALIZATION OF A MOVING ... - CiteSeerX

DETECTION AND LOCALIZATION OF A MOVING BIOCHEMICAL SOURCE IN A SEMI-INFINITE MEDIUM Tong Zhao

Arye Nehorai

Electrical and Computer Engineering Departmant University of Illinois at Chicago 851 South Morgan Street, Chicago, IL, 60607 ABSTRACT We develop methods for detecting and localizing a moving biochemical source in a semi-infinite medium (e.g., in air above the ground) using sensor array measurements. Potential applications include security, environmental monitoring, pollution control, and explosives detection. We derive models for the source substance concentration diffusion under different boundaries and environment conditions. A maximum likelihood estimation algorithm to localize the moving source is presented and its performance is analyzed by the Cram´er-Rao bound. We derive two detectors (generalized likelihood ratio test as well as a mean-difference detector) and determine their performances in terms of the probabilities of false alarm and detection. Numerical examples illustrate the source diffusion model and the performances of the proposed methods. 1. INTRODUCTION A current prominent security problem is the prospect of an adversarial biochemical attack [1], [2]. An early and reliable detection of such an attack is critical for minimizing its potential damage. Research in this area involves the development of new sensors, physical models, and information analysis [3]-[10]. In our earlier work we have developed diffusion models and techniques for detecting and localizing a stationary biochemical source using sensor array measurements [3]-[6]. In this paper, we consider these problems when the source is moving. An important security application of this problem is the detection of a biochemical attack, for example from a crop-duster spreading toxins in aerosol. Other potential applications include the detection of explosives mounted on a vehicle or monitoring car pollution. We first derive physical models for the spatial and temporal concentration distribution of the diffused biochemical substance from a moving source in a semi-infinite medium (e.g., in air above the ground). We make the assumptions This work was supported by the National Science Foundation Grants CCR-0105334 and CCR-0330342 .

that the distances between the source and sensors are much larger than the source dimensions and model the boundary as a flat surface. Such assumptions enable analytical solutions, which are good approximations to many practical situations, and can also be generalized to more complex environments using, for example, numerical methods. Under these assumptions we develop the physical (diffusion) models for permeable and impermeable boundaries, and include the effects of wind and gravity. We then transform the physical models to parametric statistical measurement models and develop the biochemical signal estimation and detection methods. This paper is organized as follows: In Section 2 we derive the physical models of the released substance distribution and present numerical examples. In Section 3 we consider the signal processing methods, namely the statistical measurement models, the source estimation and detection. Numerical examples are used to demonstrate the performance of the proposed methods. Conclusions are given in Section 4. 2. PHYSICAL MODELS In this section we derive the physical models for the concentration distribution of a substance emitted by a moving biochemical source. We consider the environment of a semi-infinite medium, i.e. z > 0, in a Cartesian coordinate system (x, y, z) such that the x-y plane lies in the boundary surface. We first develop a model for a stationary source in a semi-infinite medium under different types of boundary conditions (permeable and impermeable boundaries) and environmental effects (wind and gravity). Then, we extend the results to a moving source by considering the time-cumulative effect of the substance diffusion. 2.1. Stationary Source in a Semi-infinite Medium To derive the distribution model of a stationary source in a semi-infinite medium, we first ignore the environmental

effects of wind and gravity; then we generalize it to more realistic models. 2.1.1. Permeable Boundary Ignoring Wind and Gravity Let c(r, t) denote the substance concentration diffusion at a position r = (x, y, z) and time t. For a source-free volume and space-invariant diffusivity κ, the concentration of a diffusing substance follows the well-known diffusion equation  2  ∂c ∂ c ∂2c ∂2c = κ∇2 c = κ + + , (2.1) ∂t ∂x2 ∂y 2 ∂z 2 where we omitted the dependence on r and t to simplify the presentation, see [11]. Carslaw and Jaeger [12] have proved that for certain types of initial and boundary conditions, the solution of (2.1), which is a three spatial-variable differential equation, is the product of the solutions of the three single spatial-variable problems in a rectangular parallelepiped space [12]. Also note that in a semi-infinite medium boundary conditions exist only at the z dimension. Hence, for the remainder of this section we will consider only the diffusion equation in the z dimension ∂c(z, t) ∂ 2 c(z, t) =κ . ∂t ∂z 2

(2.2)

In practice, the released substance cannot be totally reflected by the boundary surface. Some substance will transfer across the surface of the medium, whereas another part may stop at the surface, i.e., there is always a loss of the diffusing substance. Therefore, permeable boundary conditions are more appropriate in practical situations. A reasonable assumption for the permeable boundary conditions is that the rate of loss of the diffusing substance is proportional to the actual concentration in the surface at any time [11], i.e., the boundary condition at the surface is −κ

∂c = −αc ∂z

at z = 0,

(2.3)

where α is a constant of proportionality. Under the boundary condition (2.3) we now calculate the corresponding Green function of the concentration distribution. Assume there is an impulse point source of unit mass stationary at z = z0 , i.e. the initial condition is c(z, t) = δ(z−z0 )δ(t−tI ) where tI is the initial source emission time.

Then, the solution of (2.2) is given by (2.4) at the bottom of this page, where “erfc” is the error-function complement (see [13] for details.) The first term on the r.h.s. of (2.4) is the solution for an impermeable boundary, which can also be interpreted as a superposition of contributions from the actual source and a mirror-image source of equal magnitude at z = −z0 ; the second term represents the loss effect of the diffusing substance. 2.1.2. Permeable Boundary with Wind and Gravity Consider the effects of wind and gravity on the diffusion of vapors in a semi-infinite medium with a permeable boundary. Gravity will cause an initial downward acceleration of the released substance. However, due to the air viscosity it will soon reach a limit speed vg in the z direction. Let vw denote the wind speed in the z direction. In the presence of wind and gravity, the diffusion equation (2.2) takes the form [14] ∂2c ∂c ∂c =κ 2 −v , (2.5) ∂t ∂z ∂z where v = vw − vg assuming the substance is heavier than air (otherwise, v = vw + vg ). Under the influence of external forces of wind and gravity, the flux density is given by f (z, t) = −κ∇c + cv [15]; therefore, the permeable boundary condition (2.3) becomes −κ

∂c + cv = −αc ∂z

at z = 0.

(2.6)

Solving the diffusion equation (2.5) under the boundary condition (2.6) we obtain the solution (2.7) as shown at the bottom of the next page (see [13] for details.) 2.2. Moving Source in a Semi-infinite Medium In this section we derive the diffusion models for the concentration distribution of a substance continuously released from a moving biochemical source in a semi-infinite medium, by extending the above solutions for a stationary source. Note that the solution at time t should depend on the past source positions, i.e., the concentration is affected by all past values of the source positions and release rates. Therefore, to obtain the substance diffusion model for a moving source, we need to consider the time-cumulation effects on the concentrations.

     (z − z0 )2 (z + z0 )2 exp − c(z, t) = p + exp − 4κ(t − tI ) 4κ(t − tI ) 2 πκ(t − tI ) ( )   2 α α(z + z0 ) + α (t − tI ) z + z0 αp p + κ(t − tI ) − exp erfc κ κ 2 κ(t − tI ) κ 1

(2.4)

Fig. 1. Concentrations on the ground as a function of time with different source heights based on Example 1. The dash lines represent the concentrations under the impermeable boundary; the solid lines are for the permeable boundary.

Fig. 2. Concentrations as a function of time at the positions: (a) x = 1000 m; (b) x = 2000 m; (c) x = 4000 m, based on Example 2.

Suppose we have a moving source continuously releasing a substance at a mass rate µ(t). Let cGreen (r, t|r 0 (t)) denote the Green function of the stationary source case where we replaced the source position r 0 = (x0 , y0 , z0 ) by r 0 (t) = (x0 (t), y0 (t), z0 (t)). Then, according to the above discussion, the concentration of a moving source in a semi-infinite medium is obtained as a convolution between the substance release rate and the above Green function Z t c(r, t) = µ(τ )cGreen (r, t − τ |r 0 (τ )) dτ. (2.8)

significant impact on the concentration distribution. However, the above solution still holds as a reasonable approximation in many cases, with only a larger diffusivity which is on the order of 1-100 m2 /s and even more [3]. The above method to derive the physical model for moving biochemical sources in two steps (stationary and moving source) can be applied also to other different boundary conditions and environments. In this way, the boundary conditions and environmental effects are analyzed only in the first step of the stationary source case, which simplifies solving the diffusion equation for the moving source.

tI

Using the 3D solution under the impermeable boundary condition as an example and suppose, for simplicity, that the substance releasing rate is constant, i.e. µ(t) = µ, we obtain the concentration distribution    Z t |r − r 0 (τ )|2 1 exp − c(r, t) = µ 3/2 4κ(t − τ ) tI 8[πκ(t − τ )]   2 |r − r 1 (τ )| + exp − dτ, (2.9) 4κ(t − τ ) where r 1 (t) = (x0 (t), y0 (t), −z0 (t)). We note that in most problems, air turbulence, due to thermal effects, people’s movements, wind, etc., can have

2.3. Numerical Examples of Concentration Distributions We present numerical examples to illustrate the above results and examine how the source movement affects the current distribution. Numerical Example 1: We compare the concentration distributions of a stationary source in a semi-infinite medium between permeable and impermeable boundaries. To simplify the computations, we locate the sensor at x = y = z = 0, i.e. on the ground, and the source is vertically above the sensor. We use a temporal impulse point source with unit release mass. We vary the source height to be 50, 100, 200 or 400 m above the ground. The parameters κ, α, and tI are

       (z − z0 )2 (z + z0 )2 v v 2 (t − tI ) exp − c(z, t) = p + exp − exp (z − z0 ) − 4κ(t − tI ) 4κ(t − tI ) 2κ 4κ 2 πκ(t − tI ) " #   2 v+α 2v + α α (2καv + α )(t − tI ) z + z0 + (v + α)(t − tI ) p − exp z+ z0 + erfc 2κ 2κ 2κ 4κ 2 κ(t − tI ) 1

(2.7)

taken to be 100 m2 /s, 1m/s, and 0 s, respectively. Figure 1 shows the concentration at z = 0 as a function of time, from t = 0 to 300 s. We find that the concentrations for the impermeable boundaries are always higher than those for the permeable boundaries, but as the height of the source increases, the concentrations under the permeable boundary assumption approach those for the impermeable boundaries. This is expected because when the source is sufficiently high above the ground, the concentration near the ground is very small, which causes the effects of the boundary permeability to be negligible (the effects of the boundary permeability are proportional to the concentration near the ground.) Hence, we can ignore the effects of the boundary permeability when the source is expected to be sufficiently high to simplify computing the concentration. Numerical Example 2: We investigate the effect of the source movement on the concentration distribution. Assume the source and sensors at y = 0 and the boundary is impermeable. The source is moving along a straight line which parallels the x axis at a height h = 100 m and constant speed v = 60 m/s along the x direction. The initial position of the moving source is at the point (-100, 100) m. There are three sensors at the points (1000, 0), (2000, 0), and (4000, 0) m. The other parameters µ, κ, and tI are the same as in the first example. The concentrations at the three sensors as a function of time are shown in Figure 2. As expected, the concentration distribution at each sensor reaches its peak at different time instant, related to its distance from the moving source. In other words, these instants are related to the time when the moving source is vertically above each sensor. 3. SOURCE DETECTION AND PARAMETER ESTIMATION In this section, we develop statistical signal processing techniques for detecting the moving biochemical source and estimating the unknown source and medium parameters by employing temporal measurements from an array of spatially distributed biochemical sensors. 3.1. Statistical Measurement Model To model the measurements, we suppose a spatially distributed array of m biochemical sensors located at known positions {r i , 1 ≤ i ≤ m} and each sensor takes measurements at times {tj , 1 ≤ j ≤ p}, where p is the number of time samples. Therefore, a measurement model for a sensor at a point r i and time tj is [3] e(r i , tj ) ∼ N (0, σ 2 ), (3.10) where b denotes a bias term invariant in space and time, and e(r i , tj ) is the measurement noise.

y(r i , tj ) = c(r i , tj )+b+e(r i , tj ),

Assume that the source’s substance releasing rate µ is time-invariant, and let β denote the unknown parameter vector of its moving path. We can partition the unknown parameter vector as [θ T , xT , σ 2 ]T , where θ = [β T , κ, α, tI ]T represents the nonlinear parameters (the relationships between y and these parameters are nonliner), and x = [µ, b]T represents the linear parameters. With this notation we can lump the measurements in the vector form y = A(θ)x + e

(3.11)

where y is an (mp)-dimensional vector whose (m(j − 1) + i)th component is y(r i , tj ) and similarly for e. Also A(θ) = [a(θ), 1], where 1 = [1, 1, . . . , 1]T is an (mp)-dimensional vector whose entries are 1, and a(θ) is a source-to-sensor transfer vector function of dimension mp, whose (m(j − 1) + i)th component is the concentration at the location r i and time tj arising from a unit release rate from a moving source. Using the physical model (2.9) as an example, this element is    Z tj |r i − r 0 (τ )|2 1 exp − 3/2 4κ(tj − τ ) tI 8[πκ(tj − τ )]   2 |r i − r 1 (τ )| dτ. (3.12) + exp − 4κ(tj − τ ) Modeling the moving source path by samples at each time snapshot, the resulting number of unknown parameters to estimate, hence also the computational cost, would be very high, and the results would be inaccurate. Furthermore, as the observation time increases the complexity and computational cost would increase as well. Therefore, we propose to use a parametric moving path model. Such a model allows for a smooth estimate of the source trajectory and reduces the number of parameters to be estimated, thus improving the location accuracy. In this model, we approximate the moving path as a linear combination of a set of basis functions: r 0 (t) = W φ(t), (3.13) where φ(t) is an l × 1 vector whose entries are temporal basis functions which are known or depend only on a limited number of unknown parameters, and W is an unknown coefficient matrix of dimension 3 × l. In this case, we need to estimate only 3l parameters, and the number of the unknown parameters will not increase with the observation time. The temporal basis functions should be chosen based on prior information on the trajectory. Their number should be chosen as a trade-off between desired accuracy and computational complexity. An example of a simple choice of these functions is a polynomial with φ(t) = [1, (t − tI ), . . . , (t − tI )l−1 ]T .

(3.14)

In practice, the source can be mounted on a plane or a car, hence its trajectory will usually be smooth. Therefore, we

can use a small number of basis functions and unknown parameters to have a good approximation of the path. After parameterizing the trajectory, the resulting form of the measurement model is the same as (3.11), except that the path parameter vector becomes β = vec(W ). Example: Consider a source moving along a straight line that parallels the land surface at an unknown height h and constant velocity v. We create a Cartesian coordinate system (x, y, z), such that the x-y plane lies in the land surface. Let ϕ denote the angle between the source moving direction and the x axis. Then, we obtain the source path model as r 0 (t) = [x0 (t), y0 (t), z0 (t)]T = [xI + v(t − tI ) cos ϕ, yI + v(t − tI ) sin ϕ, h]T    xI v cos ϕ  1   yI v sin ϕ = . (3.15) t − tI h 0 Here, r I = [xI , yI , h]T is the source position at the initial emission time tI . The unknown nonlinear parameter vector is θ = [β T , κ, α, tI ]T , where β = [xI , yI , h, v, ϕ]T is the parameter vector of the source trajectory. Hence here     xI v cos ϕ 1   and φ(t) = . W = yI v sin ϕ t − tI h 0 Note that in practice some of the source parameters may be known. For example, for an airborne source some of its path parameters can be measured by radar.

FIM can be derived similarly to [17], resulting in   2 ∂ ln p(y; ζ) FIM(ζ) = −E ∂ζ∂ζ T   A(θ)T A(θ) µA(θ)T D(θ) 0 1  0 = 2  µD(θ)T A(θ) µ2 D(θ)T D(θ) σ 0 0 mp/2σ 4 where ζ = [θ T , xT , σ 2 ]T denotes the unknown parameter vector; D(θ) = ∂a(θ)/∂θ, an (mp × n) matrix; and n = dim(θ). By using the standard result on the inverse of a partitioned matrix, we obtain the CRBs for θ, x and σ 2 as σ2 T [D (θ)PA⊥ (θ)D(θ)]−1 µ2 ⊥ CRB(x) = σ 2 [AT (θ)PD (θ)A(θ)]−1 4 2σ CRB(σ 2 ) = . mp CRB(θ) =

(3.17)

3.3. Source Detection The problem of detecting the source can be defined as a binary hypothesis test. Under the hypothesis H0 , the source is absent and only the bias term and noise are present. Under the hypothesis H1 , the source is also present. Mathematically, this binary hypothesis test for the present case can be represented as H0 : y = b1 + e H1 : y = A(θ)x + e = a(θ)µ + b1 + e. (3.18) 3.3.1. GLR Detector

3.2. Parameter Estimation To estimate the source and medium parameters we use the maximum-likelihood (ML) estimator. The ML estimates of θ, µ, and σ 2 are b = argmax{y T PA (θ)y} θ b b −1 AT (θ)y b b = [AT (θ)A( x θ)] b σ b2 = (mp)−1 y T PA⊥ (θ)y

(3.16)

where PA (θ) is a projection matrix on the column space of A(θ), and PA⊥ (θ) is its complementary projection matrix (see [3].) Since the MLE is asymptotically efficient, i.e. its variance attains the Cram´er-Rao bound (CRB) asymptotically, we analyze the performance of the proposed estimator by calculating the CRB, which is a lower bound on the variance of any unbiased estimators. To find the CRB, first we derive the Fisher information matrix (FIM). The FIM can be viewed as a measure of intrinsic accuracy of a distribution [16] and its inverse is the CRB. For the present problem, the

We consider the generalized likelihood ratio test (GLRT), which replaces the unknown parameters by their maximum likelihood estimates (MLEs) in the likelihood ratio test (LRT) [19]. Although there is no optimality associated with the GLRT, in practice it is widely used and known to perform well. Asymptotically, it can be shown that the GLRT is uniformly most powerful (UMP) among all tests that are invariant [18]. In our problem, the generalized likelihood ratio is (see [3] for details) #mp/2 " y T y − (mp)−1 (y T 1)2 GLR(y) = . (3.19) b y T y − y T PA (θ)y We make the decision by comparing this ratio with a threshold γ chosen to achieve a desired constant false-alarm probability. To compute γ to achieve this goal would require knowledge of the probability distribution of GLR(y) under H0 . This distribution cannot be computed in a closed form, due to the nonlinear dependence of A(θ) on θ. However, it can be estimated by Monte Carlo simulations. In some cases the nonlinear parameter vector θ can be known from

some prior information or can be measured by other ways. Under these situations, the exact threshold for the GLRT can be computed. If θ is known, the measurement model (3.11) becomes a classical linear model. Then we can rewrite the GLR(y) in (3.19) as (omitting the dependence on θ for convenience) ˜Ty ˜ y GLR(y) = T ˜ y ˜−y ˜ T Pa y y ˜ (θ)˜ 

model is not available. It also enables analytical determination of the threshold without Monte Carlo simulations. In the mean-difference detector, we do not assume any structure of the physical process, which makes it more robust with respect to the model assumptions. Here, the measurement model is y = ξ + b1 + e,

mp/2 ,

(3.20)

where 1 11T )y = P1⊥ y, mp 1 ˜ = (I − a 11T )a = P1⊥ a. mp ˜ = (I − y

GLR(y) =

˜ Pa y y ˜ (θ)˜ . ˜ [I − Pa y y /(mp − 2) ˜ (θ)]˜ T

(3.21)

Referring to [19], we find that under H0 , GLR(y) has a central F distribution with 1 numerator degree of freedom and mp − 2 denominator degrees of freedom, denoted by F1,mp−2 ; under H1 , GLR(y) has a noncentral F distribution with 1 numerator degree of freedom and mp − 2 denominator degrees of freedom, with noncentrality parame0 ˜ Ta ˜ /σ 2 , denoted by F1,mp−2 ter λ = µ2 a (λ). Therefore, the probabilities of false alarm and detection are given by PFA = = PD = =

P {GLR(y) > γ; H0 } QF1,mp−2 (γ) P {GLR(y) > γ; H1 } 0 QF1,mp−2 (λ) (γ)

(3.22) (3.23)

where we define the function QF1,mp−2 (x) as the probability that an F1,mp−2 random variable exceeds x and similarly 0 for QF1,mp−2 (λ) (x). From (3.22) we can calculate the exact threshold as γ = Q−1 F1,mp−2 (PFA ) and equation (3.23) represents the performance of this detector.

H0 : y = b1 + e H1 : y = ξ + b1 + e.

For the above GLRT detector to have good performance, the assumed physical model in Section 2.9 has to be reliable and the unknown parameters should be estimated accurately. Also, because of the existence of unknown nonlinear parameters, usually we cannot compute the GLRT threshold precisely and analyze the performance analytically. Therefore, by extending the ideas in [5] we now propose a meandifference detector, which makes less assumptions on the model than the GLRT, and hence is useful when a reliable

(3.25)

The basic idea of this detector is that we first calculate the difference between the means of the measurements obtained before and during the detection phase, and then compare this mean-difference with a threshold γ to determine whether a biochemical moving source is present or not. Denote the signal-free N × 1 measurement vector as y 0 . Hence, we create such a statistic y¯ − y¯0 1 · p (3.26) T (y, y 0 ) = p c2 1/mp + 1/N σ Pmp where y¯ = (1/mp) i=1 yi is the mean of the measurePN ments in the detection phase, y¯0 = (1/N ) i=1 y0i is the mean of the measurements before the detection phase which c2 = 1 PN (y0i − is also an estimate of the bias term b, and σ i=1 N −1 y¯0 )2 is an estimate of the noise variance σ 2 . Rewrite the above statistic as follows T (y, y 0 ) = q

σ

√

y¯−¯ y0 1/mp+1/N

c2 (N −1)σ /(N σ2

− 1)

Z(y, y 0 )

=p

S(y 0 )/(N − 1)

,

(3.27) where Z(y, y 0 ) =

3.3.2. Mean-difference (MD) Detector

(3.24)

where we use ξ to replace the substance concentration c = a(θ)µ in (3.11) and we do not make any assumption on the structure of ξ. We also assume a scenario in which we can obtain a set of measurements before the detection phase, i.e. in the absence of the signal of interest (in the practice, this is possible), which are independent of the measurements in the detection phase. In this case, we will make the decision between the following two hypotheses:

Applying the monotonic transformation (mp − 2)(x2/mp − 1), we redefine the GLR as T

e ∼ Nmp (0, σ 2 I)

y¯ − y¯0 p σ 1/mp + 1/N

and S(y 0 ) =

c2 (N − 1)σ . 2 σ

It is easy to show that the numerator Z(y, y 0 ) has the probability distribution functions (PDF’s)  N (0, 1) under H0 Z(y, y 0 ) ∼ (3.28) ¯ 1) under H1 N (ξ, Pmp where ξ¯ = (1/mp) i=1 ξi , and S(y 0 ) in the denominator has the PDF’s  2 χN −1 under H0 S(y 0 ) ∼ (3.29) χ2N −1 under H1

Fig. 3. Receiver operating characteristics for the (a) GLR detector; (b) MD detector, for the example in Section 3.3.3.

Fig. 4. Probability of detection vs. the number of total samples: (a) GLR detector; (b) MD detector, for the example in Section 3.3.3.

where χ2N −1 denotes a Chi-Square distribution with N − 1 degrees of freedom, and Z(y, y 0 ) and S(y 0 ) are independent with each other. Hence, we find that under H0 , the statistic T (y, y 0 ) has a central t distribution with N − 1 degrees of freedom, denoted by TN −1 ; under H1 , T (y, y 0 ) has a noncentral t distribution with N − 1 degrees of freedom ¯ denoted by T 0 (λ), and noncentrality parameter λ = ξ/σ, N −1 see [16]. Therefore, the probabilities of false alarm and detection are given by

of the number of total samples (the product of the number of sensors m and of time samples p), where the noise standard deviation is σ = 10−6 Kg/m3 . Here, the thresholds are chosen to yield a probability of false alarm PFA = 5%. As expected, the performance of the GLR detector is always better than the MD detector, since in GLR we fully utilize the physical model information. However, the performance of the MD becomes close to that of GLR when the number of total samples is large.

PFA = = PD = =

P {T (y, y 0 ) > γ; H0 } QTN −1 (γ) P {T (y, y 0 ) > γ; H1 } QTN0 −1 (λ) (γ).

(3.30) (3.31)

From (3.30) we can calculate the exact threshold as γ = Q−1 TN −1 (PFA ) and equation (3.31) represents the performance of this detector. 3.3.3. Numerical Examples of Detection Performance We present numerical examples to compare the performances of the GLR detector when the nonlinear parameters are known with the proposed mean-difference detector. We consider the same source moving scenario in the example of Section 2.3, and install the sensors along the x axis starting at the origin and with an interval of 100 m. The measurements are taken every 10 seconds at each sensor. The other nonlinear parameters κ and tI are taken to be 120 m2 /s and 0 s, respectively. In Figure 3, we compare the receiver operating characteristics (ROC) of the GLR and MD detectors using 25 sensors and each sensor taking 100 measurements, where the noise standard deviation is σ = 1.5 × 10−6 Kg/m3 . In Figure 4, we show the PD of the two detectors as a function

4. CONCLUSION We addressed the problems of detection and localization of a moving biochemical source in a semi-infinite medium (e.g. in air) using the measurements of a sensor array. Such a problem appears, for instance, in applications of security and environmental monitoring. We derived the physical models by solving the diffusion equation under different environment assumptions. We included the boundary permeability in the model and took into account the effects of wind and gravity. The maximum likelihood estimation method was employed to localize the moving source and determine its substance release rate. We used the Cram´er-Rao bound as the performance measure. We also presented two source detection methods, the generalized likelihood ratio test and mean-difference detector, and analyzed their performances using the probabilities of false alarm and of detection. Numerical examples were used to illustrate the results. In future research, we will consider the temperature effects on the substance distribution and deal with more complex environments by solving the diffusion equation using numerical methods; we will also combine the analytical results with empirical formulas to make them applicable to more realistic scenarios. In the signal processings part we

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