# American Institute of Mathematical Sciences

September  2019, 9(3): 361-382. doi: 10.3934/naco.2019024

## Optimum sensor placement for localization of a hazardous source under log normal shadowing

 1 GE Global Research Center, 1 Research Circle, Niskayuna, NY, 12308, USA 2 Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242 USA 3 Shandong Computer Science Center, Shandong Provincial Key Laboratory of Computer Networks, China 4 Department of Electrical and Computer Engineering, University of California, Davis, CA 95616, USA

* Corresponding author: dasgupta@engineering.uiowa.edu

Received  June 2018 Revised  April 2019 Published  May 2019

We consider the problem of optimum sensor placement for localizing a hazardous source located inside an $N$-dimensional hypersphere centered at the origin with a known radius $r_1$. All one knows about the probability density function (pdf) of the source location is that it is spherically symmetric, i.e. it is a function only of the distance from the center. The sensors must be placed at a safe distance of at least $r_2>r_1$ from the center, to avoid damage. Localization must be effected from the strength of a signal emanating from the source, as received by a set of sensors that do not lie on an $(N-1) -$ dimensional hyperplane. Under the assumption that this signal strength experiences log normal shadowing, we characterize non-coplanar sensor positions that optimize three distinguished parameters associated with the underlying Fisher Information Matrix (FIM): maximizing its smallest eigenvalue, maximizing its determinant, and minimizing the trace of its inverse. We show that all three have the same set of optimizing solutions and involve placing the sensors on the surface of the hypersphere of radius $r_2.$ As spherical symmetry of the pdf precludes uniqueness we provide certain canonical optimizing solutions where the $i$-th sensor position $x_i = Q^{i-1}x_1$, with $Q$ an orthogonal matrix. We provide necessary and sufficient conditions on $Q$ and $x_1$ for $x_i$ to be non-coplanar and optimizing. In addition, we provide a geometrical interpretation of these solutions. We observe the $N$-dimensional solutions for $N>3$ have implications for optimal design of sensing matrices in certain compressed sensing problems.

Citation: Hema K. Achanta, Soura Dasgupta, Raghuraman Mudumbai, Weiyu Xu, Zhi Ding. Optimum sensor placement for localization of a hazardous source under log normal shadowing. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 361-382. doi: 10.3934/naco.2019024
##### References:
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Dasgupta and B. D. O. Anderson, Adaptive source localization by mobile agents, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 2045–2050. Google Scholar [12] S. H. Dandach, B. Fidan, S. Dasgupta and B. D. O. Anderson, A continuous time linear adaptive source localization algorithm, robust to persistent drift, Systems and Control Letters, 58 (2009), 7-16.  doi: 10.1016/j.sysconle.2008.07.008.  Google Scholar [13] S. Dasgupta, S. R. Ibeawauchi and Z. Ding, Optimum sensor placement for source monitoring under log-normal shadowing, Proceedings of IFAC Workshop on System Identification, 42 (2009), 1710-1714.   Google Scholar [14] S. Dasgupta, S. C. Ibeawuchi and Z. Ding, Optimum sensor placement for localization under log-normal shadowing, Proceedings of the International Symposium on Communications and Information Technologies (ISCIT), (2010), 204–208. Google Scholar [15] S. Dasgupta, S. R. C. Ibeawuchi and Z. Ding, Optimum sensor placement for source monitoring under log-normal shadowing in three dimensions, Proceedings of the 9th International Conference on Communications and Information Technologies, (2009), 376–381. Google Scholar [16] B. Fidan, S. Dasgupta and B. D. O. Anderson, Guaranteeing practical convergence in algorithms for sensor and source localization, IEEE Transactions on Signal Processing, 56 (2008), 4458-4469.  doi: 10.1109/TSP.2008.924138.  Google Scholar [17] G. H. Forman and J. Zahorjan, The challenges of mobile computing, IEEE Computer, 27 (1994), 38-47.   Google Scholar [18] J. T. Isaacs, D. J. Klein and J. P. Hespanha, Optimal sensor placement for time difference of arrival localization, Proceedings of the 48th IEEE Conference on Decision and Control, (2009), 7878–7884. Google Scholar [19] D. B. Jourdan and N. Roy, Optimal sensor placement for agent localization, ACM Transactions on Sensor Networks, 4 (2008), 13: 1–13: 40 Google Scholar [20] B. Karp and H. T Kung, GPSR: greedy perimeter stateless routing for wireless networks, Proceedings of the 6th Annual International Conference on Mobile Computing and Networking, (2000), 243–254. Google Scholar [21] P. Lancaster and M. Tismenetsky, The Theory of Matrices,, Computer Science and Applied Mathematics, Academic Press, 1985.  Google Scholar [22] A. W. Marshall, I. Olkin and B. Arnold, Inequalities: Theory of Majorization and Its Applications, , Springer Series in Statistics, Springer, 2010. doi: 10.1007/978-0-387-68276-1.  Google Scholar [23] S. Martinez and F. Bullo, Optimal sensor placement and motion coordination for target tracking, Automatica, 42 (2006), 661-668.  doi: 10.1016/j.automatica.2005.12.018.  Google Scholar [24] W. Meng, L. Xie and W. Xiao, Optimality analysis of sensor-source geometries in Heterogeneous sensor networks, IEEE Transactions on Wireless Communications, 12 (2013), 1958-1967.   Google Scholar [25] J. Neering, M. Bordier and N. Maizi, Optimal passive source localization, Proceedings of International Conference on Sensor Technologies and Applications (SensorComm), (2007), 295–300. Google Scholar [26] T. O'Donovan, J. O'Donoghue, C. Sreenan, D. Sammon, P. O'Reilly and K. A. O'Connor, A context aware wireless body area network (ban), Proceedings of the 3rd International Conference on Pervasive Computing Technologies for Healthcare, (2009), 1–8. Google Scholar [27] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, R. L. Moses and N. S. Correal, Locating the nodes: cooperative localization in wireless sensor networks, IEEE Signal Processing Magazine, 22 (2005), 54-69.  doi: 10.1109/MSP.2005.1458287.  Google Scholar [28] F. Pukelsheim, Optimal Design of Experiments, , Wiley, 1993.  Google Scholar [29] M. Rabbat and R. Nowak, Distributed optimization in sensor networks, Proceedings of International Symposium on Information Processing in Sensor Networks (IPSN), (2004), 20–27. Google Scholar [30] M. G. Rabbat and R. D. Nowak, Decentralized source localization and tracking [wireless sensor networks], Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 3 (2004). Google Scholar [31] L. Ran, S. Helal and S. Moore, Drishti: An integrated indoor/outdoor blind navigation system and service, Proceedings of the Second IEEE International Conference on Pervasive Computing and Communications (PerCom), (2004), 23–30. Google Scholar [32] T. S. Rappaport, Wireless Communications: Principles and Practice, , Prentice Hall communications engineering and emerging technologies series, Prentice Hall PTR, 2002. Google Scholar [33] A. H. Sayed, A. Tarighat and and N. Khajehnouri, Network-based wireless location: challenges faced in developing techniques for accurate wireless location information, IEEE Signal Processing Magazine, 22 (2005), 24-40.  doi: 10.1109/MSP.2005.1458275.  Google Scholar [34] L. L. Scharf and L.vT. McWhorter, Geometry of the Cramer-Rao bound}, Signal Processing, 31 (1993), 301-311.   Google Scholar [35] O. Tekdas and V. Isler, Sensor placement for triangulation-based localization, IEEE Transactions on Automation Science and Engineering, 7 (2010), 681-685.   Google Scholar [36] H. L. Van Trees, Detection, Estimation, and Modulation Theory: Detection, Estimation, and Linear Modulation Theory, Wiley, 2001. Google Scholar [37] Y. Wang and W. Xiong, Anchor-based three-dimensional localization using range measurements, Proceedings of International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM), (2012), 1–5. Google Scholar [38] S. Wang, B. R. Jackson and R. J. Inkol, Impact of emitter-sensor geometry on accuracy of received signal strength based geolocation, Proceedings of IEEE Conference on Vehicular Technology, (2011), 1–5. Google Scholar [39] M. Weiser, Some computer science issues in ubiquitous computing, Communications of the ACM - Special Issue on Computer Augmented Environments: Back to the Real World, 36 (1993), 75–84. Google Scholar [40] E. Xu, Z. Ding and S. Dasgupta, Reduced complexity semidefinite relaxation algorithms for source localization based on time difference of arrival, IEEE Transactions on Mobile Computing, (2011), 1276-1282. doi: 10.1109/TMC.2010.263.  Google Scholar [41] E. Xu, Z. Ding and S. Dasgupta, Robust and low complexity source localization in wireless sensor networks using time difference of arrival measurement, 2010 IEEE Wireless Communication and Networking Conference, (2010), 1-5. Google Scholar [42] E. Xu, Z. Ding and S. Dasgupta, Source localization in wireless sensor networks from signal time-of-arrival measurements, IEEE Transactions on Signal Processing, 59 (2009), 2887-2897.  doi: 10.1109/TSP.2011.2116012.  Google Scholar

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##### References:
 [1] J. S. Abel, Optimal sensor placement for passive source localization, Proceedings of International Conference on Acoustics, Speech, and Signal Processing(ICASSP), 5 (1990), 2927-2930.   Google Scholar [2] H. K. Achanta, S. Dasgupta and Z. Ding, Optimum sensor placement for localization in three dimensional under log normal shadowing, Proceedings of the International Congress on Image and Signal Processing (CISP), (2012), 1898–1901. Google Scholar [3] H. Achanta, W. Xu and S. Dasgupta, Matrix design for optimal sensing, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (2013), 4021–4025. Google Scholar [4] I. F. Akyildiz and M. C. Vuran, Wireless Sensor Networks, Advanced Texts in Communications and Networking, John Wiley & Sons, 2010. Google Scholar [5] A. N. Bishop, B. Fidan, B. D. O. Anderson, K. Dogancay and P. N. Pathirana, Optimality analysis of sensor-target localization geometries, Automatica, 46 (2010), 479-492. doi: 10.1016/j.automatica.2009.12.003.  Google Scholar [6] R. Bodor, A. Drenner, P. Schrater and N. Papanikolopoulos, Optimal camera placement for automated surveillance tasks, Journal of Intelligent and Robotic Systems, 50 (2007), 257-295.   Google Scholar [7] G. Calafiore, F. Dabbene and R. Tempo, Radial and uniform distributions in vector and matrix spaces for probabilistic robustness, Topics in Control and its Applications, Springer London, (1999), 17–31.  Google Scholar [8] C. D. Cordeiro and D.P. Agrawal, Ad Hoc and Sensor Networks: Theory and Applications, , World Scientific, 2011. Google Scholar [9] C. T. Chen, Linear System Theory and Design, 4th Edition, 2013. Google Scholar [10] J. Chaffee and J. Abel, GDOP and the Cramer-Rao bound, Proceedings of IEEE Symposium on Position Location and Navigation, (1994), 663–668. Google Scholar [11] S. H. Dandach, B. Fidan, S. Dasgupta and B. D. O. Anderson, Adaptive source localization by mobile agents, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 2045–2050. Google Scholar [12] S. H. Dandach, B. Fidan, S. Dasgupta and B. D. O. Anderson, A continuous time linear adaptive source localization algorithm, robust to persistent drift, Systems and Control Letters, 58 (2009), 7-16.  doi: 10.1016/j.sysconle.2008.07.008.  Google Scholar [13] S. Dasgupta, S. R. Ibeawauchi and Z. Ding, Optimum sensor placement for source monitoring under log-normal shadowing, Proceedings of IFAC Workshop on System Identification, 42 (2009), 1710-1714.   Google Scholar [14] S. Dasgupta, S. C. Ibeawuchi and Z. Ding, Optimum sensor placement for localization under log-normal shadowing, Proceedings of the International Symposium on Communications and Information Technologies (ISCIT), (2010), 204–208. Google Scholar [15] S. Dasgupta, S. R. C. Ibeawuchi and Z. Ding, Optimum sensor placement for source monitoring under log-normal shadowing in three dimensions, Proceedings of the 9th International Conference on Communications and Information Technologies, (2009), 376–381. Google Scholar [16] B. Fidan, S. Dasgupta and B. D. O. Anderson, Guaranteeing practical convergence in algorithms for sensor and source localization, IEEE Transactions on Signal Processing, 56 (2008), 4458-4469.  doi: 10.1109/TSP.2008.924138.  Google Scholar [17] G. H. Forman and J. Zahorjan, The challenges of mobile computing, IEEE Computer, 27 (1994), 38-47.   Google Scholar [18] J. T. Isaacs, D. J. Klein and J. P. Hespanha, Optimal sensor placement for time difference of arrival localization, Proceedings of the 48th IEEE Conference on Decision and Control, (2009), 7878–7884. Google Scholar [19] D. B. Jourdan and N. Roy, Optimal sensor placement for agent localization, ACM Transactions on Sensor Networks, 4 (2008), 13: 1–13: 40 Google Scholar [20] B. Karp and H. T Kung, GPSR: greedy perimeter stateless routing for wireless networks, Proceedings of the 6th Annual International Conference on Mobile Computing and Networking, (2000), 243–254. Google Scholar [21] P. Lancaster and M. Tismenetsky, The Theory of Matrices,, Computer Science and Applied Mathematics, Academic Press, 1985.  Google Scholar [22] A. W. Marshall, I. Olkin and B. Arnold, Inequalities: Theory of Majorization and Its Applications, , Springer Series in Statistics, Springer, 2010. doi: 10.1007/978-0-387-68276-1.  Google Scholar [23] S. Martinez and F. Bullo, Optimal sensor placement and motion coordination for target tracking, Automatica, 42 (2006), 661-668.  doi: 10.1016/j.automatica.2005.12.018.  Google Scholar [24] W. Meng, L. Xie and W. Xiao, Optimality analysis of sensor-source geometries in Heterogeneous sensor networks, IEEE Transactions on Wireless Communications, 12 (2013), 1958-1967.   Google Scholar [25] J. Neering, M. Bordier and N. Maizi, Optimal passive source localization, Proceedings of International Conference on Sensor Technologies and Applications (SensorComm), (2007), 295–300. Google Scholar [26] T. O'Donovan, J. O'Donoghue, C. Sreenan, D. Sammon, P. O'Reilly and K. A. O'Connor, A context aware wireless body area network (ban), Proceedings of the 3rd International Conference on Pervasive Computing Technologies for Healthcare, (2009), 1–8. Google Scholar [27] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, R. L. Moses and N. S. Correal, Locating the nodes: cooperative localization in wireless sensor networks, IEEE Signal Processing Magazine, 22 (2005), 54-69.  doi: 10.1109/MSP.2005.1458287.  Google Scholar [28] F. Pukelsheim, Optimal Design of Experiments, , Wiley, 1993.  Google Scholar [29] M. Rabbat and R. Nowak, Distributed optimization in sensor networks, Proceedings of International Symposium on Information Processing in Sensor Networks (IPSN), (2004), 20–27. Google Scholar [30] M. G. Rabbat and R. D. Nowak, Decentralized source localization and tracking [wireless sensor networks], Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 3 (2004). Google Scholar [31] L. Ran, S. Helal and S. Moore, Drishti: An integrated indoor/outdoor blind navigation system and service, Proceedings of the Second IEEE International Conference on Pervasive Computing and Communications (PerCom), (2004), 23–30. Google Scholar [32] T. S. Rappaport, Wireless Communications: Principles and Practice, , Prentice Hall communications engineering and emerging technologies series, Prentice Hall PTR, 2002. Google Scholar [33] A. H. Sayed, A. Tarighat and and N. Khajehnouri, Network-based wireless location: challenges faced in developing techniques for accurate wireless location information, IEEE Signal Processing Magazine, 22 (2005), 24-40.  doi: 10.1109/MSP.2005.1458275.  Google Scholar [34] L. L. Scharf and L.vT. McWhorter, Geometry of the Cramer-Rao bound}, Signal Processing, 31 (1993), 301-311.   Google Scholar [35] O. Tekdas and V. Isler, Sensor placement for triangulation-based localization, IEEE Transactions on Automation Science and Engineering, 7 (2010), 681-685.   Google Scholar [36] H. L. Van Trees, Detection, Estimation, and Modulation Theory: Detection, Estimation, and Linear Modulation Theory, Wiley, 2001. Google Scholar [37] Y. Wang and W. Xiong, Anchor-based three-dimensional localization using range measurements, Proceedings of International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM), (2012), 1–5. Google Scholar [38] S. Wang, B. R. Jackson and R. J. Inkol, Impact of emitter-sensor geometry on accuracy of received signal strength based geolocation, Proceedings of IEEE Conference on Vehicular Technology, (2011), 1–5. Google Scholar [39] M. Weiser, Some computer science issues in ubiquitous computing, Communications of the ACM - Special Issue on Computer Augmented Environments: Back to the Real World, 36 (1993), 75–84. Google Scholar [40] E. Xu, Z. Ding and S. Dasgupta, Reduced complexity semidefinite relaxation algorithms for source localization based on time difference of arrival, IEEE Transactions on Mobile Computing, (2011), 1276-1282. doi: 10.1109/TMC.2010.263.  Google Scholar [41] E. Xu, Z. Ding and S. Dasgupta, Robust and low complexity source localization in wireless sensor networks using time difference of arrival measurement, 2010 IEEE Wireless Communication and Networking Conference, (2010), 1-5. Google Scholar [42] E. Xu, Z. Ding and S. Dasgupta, Source localization in wireless sensor networks from signal time-of-arrival measurements, IEEE Transactions on Signal Processing, 59 (2009), 2887-2897.  doi: 10.1109/TSP.2011.2116012.  Google Scholar
(a) Illustration of optimum sensor placement in two dimensions using four sensors. (b) Illustration of optimum sensor placement in three dimensions using six sensors and sphere of radius of 2 (i.e.r2 = 2)
Plot of determinant of FIM Versus Number of sensors in the network
Plot of Minimum Eigenvalue of FIM Versus Number of sensors in the network
Plot of $10log_{10}$(Average Normalized Mean square error in the source location) Versus Signal to Noise Ratio (dB). Red dotted line represents the performance of the random placement. Blue line represents the performance of the proposed optimum sensor placement
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