Source localization and tracking in a dispersive medium using wireless sensor network
© Hakim and Jayaweera; licensee Springer. 2013
Received: 12 July 2012
Accepted: 18 May 2013
Published: 18 September 2013
In this paper, we address the issue of collaborative information processing for diffusive source localization and tracking using wireless sensor networks capable of sensing in dispersive medium/environment. We first determine the space-time concentration distribution of the dispersion from physical modeling and mathematical formulations of an underwater oil spill scenario, considering the effect of laminar water velocity as an external force. For static diffusive source localization, we propose two parametric estimation techniques based on maximum-likelihood (ML) and best linear unbiased estimator for the special case of our physical dispersion model. We prove the consistency and asymptotic normality of the obtained ML solution when the number of sensor nodes and samples approach infinity, and derive the Cramér-Rao lower bound on its performance. We also propose a particle filter-based target tracking scheme for moving diffusive source and derive the posterior Cramér-Rao lower bound for the moving source state estimates as a theoretical performance bound. The performance of the proposed schemes are shown through numerical simulations and compared with the derived theoretical bounds.
The release of liquid petroleum hydrocarbon into the ocean or coastal water due to human activity has attracted tremendous attention because of its environmental, biological, and economical impact. Recent BP oil disaster in the Gulf of Mexico is a perfect example of how spill stemmed from a sea-floor oil gusher can severely damage the marine and wildlife habitats as well as the Gulf’s fishing and tourism industries. Research in modeling and predicting the extent of such oil spill can assist in planning and emergency decision-making, thereby reducing the threats and hazardous effects on the environment as well as the economic cost. Considering the fact that this is a diffusive source estimation and tracking problem, such research can in general be applicable in many other similar contexts such as homeland security, environmental and industrial monitoring, pollution control, servers, and data center temperature monitoring as well [1–8]. For example, the spread of chemical and biological agents as homeland security problems are discussed in [5, 9–11].
Recent advances in sensor technology, such as smart/intelligent nodes with cognitive abilities, on-board sensors, and wireless networking capabilities have triggered the use of wireless sensor networks (WSNs) in monitoring various physical phenomena [12–14]. Though sensor nodes are capable of a limited amount of local processing and wireless communication, when a large number of sensors communicate and share information among themselves, they can measure a desired phenomenon-of-interest in great detail. Also with the developments of unmanned autonomous vehicles, WSNs are gaining popularity due to their potential to be useful for a wide range of applications including environmental monitoring, intrusion detection, and various military and civilian applications [12, 15, 16]. Due to advanced microelectromechanical systems, many of the state-of-the-art sensors are now more accurate, robust against noise, and energy efficient [17, 18]. These new cutting-edge sensors can withstand severe unfavorable conditions in hazardous areas where human deployment is impossible. All these useful and exciting features in recently developed sensors make them suitable candidates for the set of applications involving monitoring of diffusion phenomena that we are interested in.
Source or target localization using distributed sensor arrays is an area of active research interest for a considerable period of time [19, 20]. In the past, detection and localization problems of diffusive sources in WSN have been a topic of interest, specially in the case of chemical/biological threat detection. Interesting research in this context can be found in [3, 4, 9, 10], where biochemical concentration distribution in space and time for different types of diffusive sources, diffusion models, and/or sensor networks is estimated. For instance, remotely localizing a gas or odor source using mobile robot was proposed in  by fitting the gas distribution model to the gas sensor response at the sensor locations. However, the mobile sensor dynamics model therein was obtained empirically, which does not allow for dynamic environment and moving diffusive source. In , a maximum-likelihood (ML) estimator was developed for localizing vapor-emitting sources, and its asymptotic normality of the obtained ML estimator was proved when the signal-to-noise-ratio (SNR) approaches infinity. Many other estimation techniques have also been used in diffusive source parameters estimation literature [9, 10, 21–23]. In particular, Bayesian estimation has been applied in [9, 21] in a sequential manner, which is not suitable in many practical scenarios where faster estimation and immediate actions based on the estimation are top priorities. A real-time maximum-likelihood estimation method was proposed in  for estimating diffusive source parameters, where consistency and asymptotic efficiency of the obtained estimator were proved when the density of sensors becomes infinite. In , the problem of impulsive diffusive source localization was solved assuming the spatial sensor measurements at any sensor location as a scaled and shifted version of a common prototype function, leading to solving a set of linear equations. However, the physical diffusion models used in [23, 24] are oversimplified with the diffusive sources assumed to be impulsive or instantaneous in nature.
Although research has been done in tracking and/or estimating time-varying parameter estimation in general [25–28], to the best of our knowledge, very few attempts have been made in time-varying diffusive source parameter estimation. Some of these methods cannot be applied directly into our time-varying parameter estimation model since, e.g., for a moving source, the concentration at the current time is affected by all past values of source position. Therefore, time-cumulation effects on the concentrations (i.e., observations) must be taken into account to estimate time-varying parameters. Among previous works, a parametric moving path model for a diffusive moving source was discussed in , where the moving source path was approximated using finite number basis functions. Tracking performance in this case depends on the smoothness of the source trajectory, prior information about the moving source trajectory, and choosing a suitable finite set of basis functions. In , a novel recursive algorithm was proposed to track the intensity of a diffusive point source, but the source location was considered as an unknown static value.
The aforementioned limitations may be overcome by developing or exploiting state-of-the-art Bayesian-based location tracking methods suitable for handling highly nonlinear diffusion processes. In the Bayesian approach, the key is to construct the posterior probability density function (PDF) of the underlying state vector based on all available information. For linear and Gaussian state dynamics and observation models, the optimal minimum mean squared error (MMSE) solution is tractable and is given by the well-known Kalman filter . However, for most of the real-world scenarios, dynamic state estimation problems are nonlinear and non-Gaussian, and obtaining optimal closed-form solution is not tractable under the Bayesian approach. In these cases, suboptimal approaches such as extended Kalman filter and Gaussian-sum filter  are used with certain approximations. These suboptimal algorithms become inefficient for highly nonlinear and non-Gaussian systems. In these cases, numerical techniques based on sequential Monte Carlo methods are used to achieve better performance for highly nonlinear systems. To that end, the idea of particle filtering was introduced in  as an effective method of representing PDF in terms of a set of random sampling.
In this paper, our main objectives are to efficiently estimate and track diffusive source location using a wireless network of chemical sensor capable of sensing in diffusive environment. To cater to the objectives, we formulate and derive a physical model for the space-time substance dispersion mechanism of an underwater diffusive source. The modeling and the proposed solution methods can also be extended to other important diffusion phenomena involving biochemical contaminant materials as well. We propose and implement ML and best linear unbiased estimator (BLUE)-based parameter estimation techniques for a static diffusive source which is continuously emitting substance . In the previous literature, such as in , the asymptotic normality of the obtained ML estimator was proved when the SNR approaches infinity. We prove both the consistency and asymptotical normality of our obtained ML-based solution when the number of sensor nodes and time samples go to infinity, thus allowing for the option of tweaking these two parameters. We derive the Cramér-Rao lower bound (CRLB) as a theoretical performance bound for a special case of our obtained physical dispersion model. We also propose a particle filter (PF)-based target tracking method for moving diffusive source. To the best of our knowledge, moving diffusive source tracking using particle filtering approach has not been attempted before. The posterior Cramér-Rao lower bound (PCRLB) for the moving source state estimates is also derived as a theoretical performance bound .
The remainder of this paper is organized as follows: Sections 2 and 3 discuss, respectively, modeling of an underwater oil spill scenario and measurement model for static diffusive source localization using sensor network. The proposed statistical methods for static diffusive source localization and corresponding theoretical performance bound are discussed in Section 4. Section 5 presents the proposed particle filter-based method for moving diffusive source tracking with theoretical performance bound analysis in detail. Section 6 shows the validity and effectiveness of our proposed methods for diffusive source localization and tracking through numerical simulations. Finally, Section 7 concludes the paper by summarizing our results.
2 Physical model for dispersion
For parametric estimation case, it is to be noted that from the concentration measurements taken by the sensors, we can first estimate the source parameters of interest and then predict its cloud evolution in space and time by inserting the estimated parameters into (1).
Derivation to (9) is given in Appendix 1. For the sake of simplicity from here on, we denote the diffusivity constant κ w = κ.
2.1 Moving diffusive source
where r 0(t) = [x 0(t),y 0(t),z 0(t)] T represents the source moving path. The advantage of solving the physical diffusion model corresponding to a moving diffusive source using (10) is that the initial, boundary, and other necessary conditions can be taken into account to solve for the stationary case in the first step before extending it to the moving source case.
3 Measurement and system models for static diffusive source localization
where y j,k = y(r j ,t k ), e j,k = e(r j ,t k ), c j,k (θ) = c(r j ,t k ), is the unknown source and medium parameter vector that we are interested to estimate, and b is the bias or clutter term representing the sensor’s response to foreign substances that may be present in a diffusive field of interest. The bias term is assumed to be space and time-invariant such that the foreign substances interfering with the actual measurements are in steady state. If we want to localize a static diffusive source, then only [x 0,y 0,z 0] are the parameters of interest. It is to be noted that some of the parameters, such as the diffusivity constant κ, bias term b, and noise variance σ 2, can be measured at the calibration stage, thereby reducing the cost of computation during the detection/estimation phase.
We assume that the sensor nodes are in sleep mode until they are activated by some central control (i.e., FC) due to a possible release of a substance from a diffusive source. The activated sensor nodes take measurements of the substance’s concentration at time instants t k s and then return to sleep mode. For N number of nodes in a WSN and with each node taking T number of time samples of the substance concentrations at their respective locations, let be the measurement vector received at the FC.
4 Static diffusive source localization
where erfc(.) is the complementary error function.
4.1 Maximum-likelihood-based source localization
Since the system of equations in (14) is nonlinear, there is no closed-form solution to it. We can obtain an ML estimation of the source location using any suitable nonlinear optimization technique. In this case, (14) is solved using simplex search algorithm .
where (16) was obtained using the independence assumption of observations in space and in time.
A sequence of estimators to an unknown parameter vector θ is said to be consistent if the sequence converges in probability to θ, i.e., , where n is the sample size . It is desirable to have a consistent MLE as consistency ensures that for large data sets, the MLE will converge to the true parameter. The obtained MLE to our source localization problem is consistent when the number of sensor nodes in any non-negligible open subset of and time samples go to infinity.
If the number of sensors N increases in such a way that for any open subset having positive area, the number of sensors N and/or the number of time samples T tend to infinity, the obtained ML estimator is consistent.
Proof. See Appendix 2.
Once consistency for the obtained MLE is established, the next important thing is to check the asymptotic normality. An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter θ approaches a normal distribution with standard deviation shrinking in proportion to as the sample size n grows, i.e., , where I θ and I are the Fisher information and identity matrices, respectively . It ensures that the estimator not only converges to the unknown parameter, but it converges fast enough at a rate . We address this issue with the following theorem on asymptotic normality. □
Proof. See Appendix 3. □
4.2 Best linear unbiased estimator-based source localization
The advantages of using the BLUE for static diffusive source localization are that there are no constraints on the PDF and also knowing only the mean and covariance of the measurements is enough. However, observations have to be linear for the BLUE algorithm. In this subsection, we assume that the distributed sensing nodes are capable of estimating their respective distances from the source using BLUE.
with estimator variance .
To solve for the source location from (20), simplex search algorithm  has been used.
5 Moving diffusive source tracking
5.1 State dynamics model
which models the acceleration terms in the spatial directions, and is the variance of the process noise.
5.2 Observation model
where ε j,k is the received noise which is assumed to be Gaussian with mean 0, variance and e j,k = ε j,k + ν j,k and . We denote y j,1:k as the measurement vector from j th node up to the time t k , and as the collection of all measurements at the FC from N-distributed sensor nodes.
5.3 Target tracking using particle filters
In Bayesian belief update, to estimate state vector s k at time instant k, we need to construct posterior distribution p(s k |y c,1:k ) with initial PDF p(s 0). The Bayesian belief update is done in two stages: prediction and update.
Since the observation model is highly nonlinear, the analytical solution for the optimal estimator is not tractable in our case. Hence, we use sequential Monte Carlo method to approximate the posterior PDF (27) with particle filters .
Let us denote to be the random measure that characterizes the posterior PDF p(s k |y c,1:k ), where P is the number of particles. Then , where δ(.) is the Dirac delta function. The state vector estimate at time t k can be obtained as , and the covariance matrix U k|k of the estimate is . The predicted state and the corresponding covariance matrix U k+1|k can be obtained from the state dynamics in (21) as and U k+1|k = F U k|k F T + Q.
5.4 Posterior Cramer-Rao lower bound analysis
Let I(S k ) ∈ R 4k × 4k be the information matrix derived from the joint distribution p k . We wish to solve for the information submatrix for estimating s k , denoted by I k . The following theorem gives a two-step recipe for computing I k .
and with ∇ being the Laplacian operator.
Proof. See Appendix 4.
Note that the information submatrix computation in (28) involves computation of the inverse of a matrix of size 4k × 4k. This is because of the output y j,k+1 at the j th node at (k + 1)th time instant being a function of all the previous states S k+1. □
6 Simulation results
In the following, we show the performances of our proposed models and schemes through numerical simulations.
6.1 Simulations for the physical model in Section 2
6.2 Static diffusive source localization
Here, we show the simulation results in estimating the location of a static diffusive source using the proposed MLE and BLUE-based methods from the concentration observations taken by the sensing nodes. For the sake of simplicity, we consider a 2D diffusive field volume of Λ = [-50,50] × [-50,50]m2. We assume that the sensors are placed in a uniform 2D grid such that the distance between adjacent sensors along the same ordinate is approximately 14.3 m. Parameters used for simulations are number of nodes N = 64, r 0 = [0,0] T , μ = 1,000 kg/s, b = 10-4 kg/m2, t I = 0 s, and κ = 25 m2/s. The observation noise is assumed to have Gaussian distribution with mean 0 and variance σ 2 = 1 × 10-4 kg/m2. The total number of random realizations used for simulations is 100. The measurements are taken at every 0.5 s time-step starting from 0.5 s ending at 30 s. In the case of BLUE estimator, the received noise variance at the fusion center is assumed to be m2.
6.3 Moving diffusive source tracking
In this subsection, we analyze the performance of our proposed moving diffusive source tracking scheme. We use the same sensor network setup as described in Section 6.2. The initial source state vector is assumed to be Gaussian with mean μ = [0,0,0,0] T and covariance matrix Σ0 = diag([ 0.01,0.01,0.01,0.01] T ). The intensity of the state process noise is . The sampling time is assumed to be T s = 0.5 s, and the total number of random realizations used for simulations is 50. The tracking is performed for 30 s and the number of particles in the PF is N p = 1,000. The rest of the parameters is same as in Section 6.2. The performance measure is taken as the root mean squared error (RMSE) of the moving source position estimate given by . The RMSE is compared with the square root of the PCRLB components of the position error, .
In this paper, we obtained spatio-temporal distribution of the substance concentration by solving physical diffusion model for an underwater oil spill scenario which considers laminar water velocity as an external force. The obtained mathematical model was found to be capable of modeling satisfactorily the underlying physical diffusion phenomenon. We proposed two parametric estimation methods based on MLE and BLUE for determining static diffusive source location using wireless sensor network. We also obtained the CRLB as theoretical performance bound for source localization. It was observed that though the MLE performs better than the BLUE-based diffusive source localization method, the latter shows satisfactory performance trend for large number of sensing nodes and time samples. We also proposed a particle filter-based target tracking method for moving diffusive source-emitting substance continuously into the dispersive medium. PCRLB corresponding to the moving diffusive source tracking was obtained as a theoretical performance measure and was compared with the simulation results. The effect of sampling time on the moving source tracking was also investigated. The performance of the proposed estimation and tracking methods are shown to be excellent using numerical simulations. In future research, we plan to combine our obtained analytical results with non-model-based numerical techniques to make them applicable for more realistic and complex scenarios.
Derivation of spatio-temporal concentration in (9)
Similarly, we can also verify the expressions for c 2(r,t) and c 4(r,t). Therefore, the spatio-temporal concentration distribution c(r,t) given in (9) is valid. ■
Proof of Theorem 1
If d N,T = N 3 T 3 > 0 for N ≥ 1,T ≥ 1, then we can claim that exists.
Choosing d N,T = N T > 0 for N ≥ 1,T ≥ 1, we have . Similarly, for y 0 and z 0, we can also claim that the MLE to the diffusive source localization problem is consistent when the number of sensor nodes and time samples go to infinity. ■
Proof of Theorem 2
To prove the asymptotic normality of the MLE, we define Φ j,k (y j,k |θ) = logp(y j,k |θ), , and . Below, we verify the necessary conditions mentioned in  for our obtained MLE to be asymptotically normal.
From practical point of view, there is no loss in generality in assuming that , where is an open subset of Λ. Also because the obtained MLE to source localization is consistent, it is also consistent even when . Thus conditions N 1 and N 2 are satisfied.
From the notations defined above, since and exist for u,v = 1,2,3, it can be easily verified that both and exist almost surely. Therefore N 3 is satisfied.
Since θ ∈ Λ and is a continuous mapping of θ, we can claim that is indeed uniformly continuous on θ in j and k. Also, because is a continuous function of y j,k with y j,k being Lebesgue measurable function, is also a measurable function of y j,k and condition N 4 is satisfied. To satisfy N 5, it is easy to verify that for all j,k and u. Since and p(y j,k |θ) is continuous and Lebesgue measurable in y j,k , is valid for all j,k,u, and v, and thus N 6 is satisfied. From Appendix 2, it can be claimed that and exist and are bounded for all u,v. Hence, using the Cauchy-Schwarz inequality, all the leading principle minors of (in Theorem 2) can be shown to be positive. Thus, we can claim that is also positive-definite and therefore N 7 is satisfied. Because , we have , ∀j,k,u, where K 1 is some real positive finite number and N 8 is satisfied.
To prove condition N 9 since is a uniformly continuous function of θ (shown in condition N 4), for any ε > 0, there exists one δ > 0 such that . Therefore, for all ||θ - θ 0|| ≤ ε since and are continuous functions of and v, we have , where and are some finite real numbers and B j,k,u,v (e j,k ) is a random variable. Since and , hence , where K 2 is a finite real number.
Therefore, the obtained MLE of the diffusive source location is asymptotically normal when the number of sensor nodes and time samples go to infinity. ■