Number of wireless sensors needed to detect a wildfire

Transcription

1 Number of wireless sensors neee to etect a wilfire Pablo I. Fierens Instituto Tecnológico e Buenos Aires (ITBA) Physics an Mathematics Department Av. Maero 399, Buenos Aires, (C1106ACD) Argentina pfierens@itba.eu.ar Abstract The lack of extensive research in the application of inexpensive wireless sensor noes for the early etection of wilfires motivate us to investigate the cost of such a network. As a first step, in this paper we present several results which relate the time to etection an the burne area to the number of sensor noes in the region which is protecte. We prove that the probability istribution of the burne area at the moment of etection is approximately exponential, given that some hypotheses hol: the positions of the sensor noes are inepenent ranom variables uniformly istribute an the number of sensor noes is large. This conclusion epens neither on the number of ignition points nor on the propagation moel of the fire. Page 1

2 1 Introuction The well-establishe literature on wireless sensor networks continuously mentions prevention an early etection of wilfires as a typical application of the fiel. However, to the best of our knowlege, only a few proposals of inexpensive sensor networks have been actually mae in the literature (e.g., Roriguez et al. (000); Yu et al. (005)) an only one has been implemente an trie to a small scale (Chen et al. (003); Doolin et al. (004); Glaser (004); Doolin an Sitara (005)). The lack of actual experiences on the implementation of wireless sensor networks for the etection of wilfires an the current problematic of forest fires in Argentina lea a group of researchers at ITBA to become involve in a mi-term project for the evelopment of such a network. As part of the project, forestry companies were consulte about their nees an experiences on fire etection. Companies in the region use mainly two alternative ways of fire etection (private communication). On one han, the most extene practice is the visual inspection of large areas (with a coverage raius of up to 0 km) from high towers an the aily walk of personnel through pre-establishe paths uring the fire-season. This type of system is very cheap because its main cost is represente by the low wages of the few people involve in the irect observation. On the other han, a few companies have also implemente the observation through cameras in the visual an infrare ranges. However, this class of system is usually consiere too expensive because the relatively high initial cost of installation of the infrare cameras. Uner this situation, several companies were intereste in the iea of a wireless sensor network for the etection of wilfires, but they were also concerne on the cost of the system per unit of area. This problem can be ecompose mainly into two parts: a) the cost Page

3 of each iniviual noe; b) the number of noes which are neee per unit of area (i.e., the number of noes per square meter). In this paper we investigate part b), while part a) will be presente elsewhere. 1.1 Variables uner analysis We shall work with simple two imensional moels an we shall eal only with propagation of surface fires. Moreover, we shall keep the moel of the wireless sensors as simple as possible, that is, we shall assume that all sensors allow for the etection of the fire as soon as the fire reaches their location. Finally, we shall not concern ourselves with the ifficulties of the communication among the sensors, which may be impaire by the activity of the fire itself (Heron an Mphale (004), Mphale et al. (007)). Uner this setting, there are many variables relate to the number of wireless sensors that can be stuie. We choose to of them, the time to etection (T ) an the area alreay burnt at the time of the etection (A ). Both variables can be relate, in turn, to the resources neee for contention of the fire after it has been etecte. Since the number of wireless sensors may vary accoring to the extension of the region that must be protecte, we shall use the characteristic istance between sensors (D) as reference inepenent of the actual area. Although the efinition of the characteristic istance between sensors may vary slightly from one setting to another, in all cases it subsumes uner a single value how wiely space the sensors are. The rest of the paper is structure as follows. In Section, we present some simple results for the case where the noes are istribute in a regular pattern across the area of interest. In Section 3, we analyze the expecte time to etection an the burne area before Page 3

4 etection when the sensor noes are ranomly istribute in the protecte area. Section 4 summarizes the main conclusions of the paper an mentions some ieas for future work. Regularly istribute sensors In this section, we analyze the case where the sensors are locate in a regular gri in such a way that the istance between any pair of them in the same row or the same column is D, the characteristic istance in this setting. As a further simplification, we shall assume that the region to be protecte is a rectangle whose sies are integer multiples of D. It is easy to exten the work in this section to more general regions by partitioning them into small rectangular pieces..1 Circular propagation at constant rate We assume that surface fires propagate at a constant rate of sprea (R) in all irections an that the probability of ignition is uniformly istribute insie the protecte area. Both the uniform istribution of the probability of ignition an the fact that the noes are locate in a regular lattice enable us to reuce the stuy of the etection of a fire to a much smaller area corresponing to a square elimite by four sensor noes, one on each vertex. Furthermore, this square can be split up into four smaller squares, as shown in Figure 1. Given the assumption that a fire propagates in all irections at the same spee, if a fire originates in Region i (i =1,, 3, 4 see Figure 1), it will be etecte by sensor i first. Therefore, we can further limit our analysis to only one of the four regions an the corresponing noe, say, Region 1 an sensor noe 1. In other wors, P ( T x) = P[ T x the fire originate in Region 1]. Page 4

5 Notice that T is simply the istance from the ignition point to the sensor noe ivie by the constant rate of sprea R. Since the istribution of the ignition point is uniform insie Region 1, we have [ T x the fire originate in Region 1]= P [ Distance from sensor 1 to the ignition point Rx the fire originate in Region ] = = P 1 Area of the surface whose points are at a istance Rx of sensor 1 = Area of Region 1 After some work, the latter expression can be foun to be π Rx P( T x) si Rx D =, 4 D (1) π 1 ( ) tan Rx 1 Rx Rx P T + x = 1 si D D, 4 Rx () D D D ( T x) = 1 si Rx D. P (3) Further algebraic work leas to D π P T = , (4) R 4 + log [ ] ( 1+ ) D D E T = 0.386, (5) 6 R R E 1 D 6 [ ] ( ) ( + log( 1+ ) D D T =, var T = 0.003, (6) 6 R 36 R R π E[ A ] = D 0.5 D. (7) 6. Page 5

6 Some comments are ue. Equation (4) points out that more than 80% of fires will be etecte in less than D/R an, hence, their area will be smaller than πd /4. Equation (5) says that, as expecte, the mean etection time is proportional to the ratio D/R. Finally, while Equations (1)-(3) imply that no fire will have an area greater than πd / at the moment of etection, Equation (7) says that the expecte value of such area is approximately D /. In particular, note that the expecte value of the area of the fire at the moment of etection oes not epen on the rate of sprea of the fire, but it only epens on the characteristic istance D. 3 Ranomly istribute sensors In this section, we analyze the case where the sensors are ranomly istribute in the region to be protecte. In this case, the characteristic istance D is the mean istance between sensors compute as A D =, (14) N where A is the total area of the protecte region an N is the total number of sensors. We shall first show that, when the number of sensors is large, the burne area at the moment of etection A has a simple probability istribution which is inepenent of the propagation moel. We shall then show simple approximations for the statistics of the time to etection for simple propagation moels. 3.1 Probability istribution of the burne area The probability that the burne area A is greater than a given number x is equal to the probability that the fire has not foun any sensor insie the burne region. Since sensor Page 6

7 noes are ranomly istribute across the protecte region, an assuming that the position of each sensor is inepenent of that of the others, we have P ( A > x) = 1, where A is the total area of the protecte region an N is the total number of sensor noes. Using Equation (14), we can write Note that, as N, P P x A ( A > x) = 1 = 1. x D N x ND N N N x D N ( A x) = exp. N x D > 1 N Formally, we have the following Theorem 1. Assume the locations of sensor noes are inepenent an ientically istribute ranom variables with uniform istribution across the protecte region. Furthermore, assume that the position of the sensor noes an the ignition points (there may be more than one) are inepenent ranom variables. Let N be the number of sensor noes an A(N) the surface area of the protecte region, which varies with N in such a way that A(N) = D N, where D is a positive constant. Then, as the number N of sensor noes goes to infinity, the ranom variable A corresponing to the burne area at the moment of etection converges in istribution to an exponential ranom variable with parameter λ=1/d. Page 7

8 Probably, the most salient feature of Theorem 1 is that it oes not epen on the propagation law of the fire an it epens neither on the number nor on the istribution of the ignition points. In other wors, Theorem 1 states that, if the protecte area is large an the number of sensor noes is also large, then P x D ( A > x) exp, E[ A ] D, var( A ). D 3. Probability istribution of the time to etection Theorem 1 leas to the following simple Corollary 1. Assume that there is a eterministic law F such that, at each time instant t, F(t) represents the total burne area at time t. If the hypotheses of Theorem 1 are satisfie, then, as the number of sensor noes N goes to infinity, P F( t) exp D ( T t). > N In the following paragraphs, we consier two simple examples of application of Corollary Circular propagation at constant rate Note that, as the surface area of the protecte region increases, we may ignore the cases where the fire evelops near the borers of the region. Then, it is easy to see that the function F in Corollary 1 for the case of circular propagation at constant rate of sprea R is given by F ( t) = π ( Rt). Page 8

9 Then, for a large area covere with a large number of sensors, we can make the following approximation: πr t exp D ( T > t). P So we can estimate the expecte value of the time to etection as E πr t D = R 0 0 D [ T ] P( T > t) t exp t, = where we have omitte the etails of the calculation. In a similar fashion, we get E D 4 π D D [ T ] var( T ) πr, Figures an 3 show the agreement of the previous equations an the results of simulations (10 5 Monte Carlo runs) with D=1 m, R=1 m/s, N varying from 10 to an one ranom ignition point. 4π R R 3.. Elliptical propagation at constant rate In this section, we consier the case of a surface fire that propagates with an elliptical shape an at a constant rate of sprea. In this case, the function F is given by (see Finney (1998)) F HB 4LB ( t) = π R t, where R is the rate of sprea, HB is the Hea-to-Back ratio an LB is the Length-to-Breath ratio. In a similar fashion as before, we can compute Page 9

10 LB D [ T ], E[ T ] 4LB D E, 1 R 1 1 πr + HB 1 + HB var 4 π 4LB D 4LB D ( T ) π 1 R 1 R HB HB 4 Conclusions The lack of extensive research in the application of inexpensive wireless sensor noes for the early etection of wilfires motivate us to investigate the cost of such a network. As a first step, in this paper we present several results which relate the time to etection an the burne area to the number of sensor noes in the region which is protecte. Our main result is Theorem 1 which states that the probability istribution of the burne area is approximately exponential, given that some hypotheses hol: the positions of the sensor noes are inepenent ranom variables uniformly istribute an the number of sensor noes is large. It is important to remark that this conclusion epens neither on the propagation moel of the fire nor on the number an istribution of ignition points. Our next step in the investigation of the cost of a network of sensor noes for the etection of wilfires is to actually buil prototypes of such noes an to test their behavior uner controlle fires. Acknowlegements This work was partially supporte by anonymous contributors through the project Prevention an early etection of forest fires by means of sensor networks which is being evelope at the Instituto Tecnológico e Buenos Aires (ITBA). Page 10

11 References Chen MM, Majii C, Doolin DM, Glaser SD, Sitar N (003) Design an construction of a wilfire instrumentation system using networke sensors. Presente in Network Embee Systems Technology (NEST) Retreat 003. (Oaklan, California) Doolin DM, Glaser SD, Sitar N (004) Software Architecture for GPS-enable Wilfire Sensorboar. Presente in TinyOS Technology Exchange, February 6, 004. (University of California: Berkeley, California) Doolin DM, Sitara N (005) Wireless sensors for wilfire monitoring. In Proceeings of SPIE Symposium on Smart Structures & Materials, SPIE 5765, Finney MA (1998) FARSITE: Fire Area Simulator-Moel evelopment an evaluation. USDA Forest Service, Rocky Mountain Research Station Research Paper RMRS-RP-4. (Ogen, UT) Glaser SD (004) Some real-worl applications of wireless sensor noes. In Proceeings of SPIE Symposium on Smart Structures & Materials, SPIE 5391, Heron ML, Mphale K (004) Raio Wave Attenuation in Bushfires, Tropical Cyclones an Other Severe Atmospheric Conitions. Final Report on EMA Project 60/001. (School of Mathematical an Physical Sciences, James Cook University: Australia) Page 11

12 Mphale K, Heron M, Verma T (007) Effect of Wilfire-Inuce Thermal Bubble on Raio Communication. Progress In Electromagnetics Research 68, Roriguez N, Bistue G, E. Hernanez, Egurrola D (000) GSM front-en to forest fire etection. In Proceeings of the IEEE International Symposium on Technology an Society 000: University as a Brige from Technology to Society, Yu L, Wang N, Meng X (005) Real-time Forest Fire Detection with Wireless Sensor Networks. In Proceeings of the International Conference on Wireless Communications, Networking an Mobile Computing 005,, Page 1

13 Figure 1 Page 13

14 Figure Page 14

15 Figure 3 Page 15