Number of wireless sensors needed to detect a wildfire

Page 1 Number of wir eless senso rs needed to detect a w ild fire Pablo I. Fierens Instituto Tecnológico de Buenos Aires (ITBA) Physics and Mathematics Department Av. Madero 399, Buenos Aires, (C110 6ACD) Argenti na pfierens@itba.edu.ar Abstract The lack of extensi ve research in the application of inexpensive wireless sensor nodes for the earl y detection of wildfires m otivated us to i nvestigate the cost of su ch a network. As a first s tep, in thi s paper we present s everal results which relate th e time to d etection and the burned area to the num ber of senso r nodes in the region which is protected. W e prove that the probabilit y di stribution of t he burned area at the moment of d etection is approximately exponential, given that some hypotheses hold: the positi ons of the sensor nodes are independent random variables uniforml y distribut ed and the number of sensor nod es is large. This conclusion depends neithe r on th e n umber of i gnition poi nts nor on the propagation model of the fire. Page 2 1 Introduction The well-established li terature on wireless sensor networks cont inuousl y mentions prevention and e arly detection of wildfires as a t ypical application of the fi eld. Howeve r, t o the best of our knowledge, on ly a few proposals o f inexpensive sensor networks have been actually made in the liter ature (e.g., R odriguez et al. (2000); Yu et al. (2005 )) and onl y o ne has been implemented and t ried to a small scale (Chen et al. (2003); Doolin et al. (2004); Glaser (2004); Doolin an d Sitara (2005)). The lack o f a ctual exp eriences on the implementation of wirel ess sensor networks for the detection of wild fires an d t he current problemati c of forest fires in Argenti na lead a gr oup of res earchers at ITBA t o become involved in a mi d-term project for t he developm ent of such a network. As p art of the p roject, fo restry companies were consult ed about their needs and experiences on fire d etection. Comp anies in the re gion use mainl y two alt ernative wa ys of fire detection (priv ate comm unication). On on e hand, the mos t extend ed practice is t he visual inspection of large areas (with a coverage radius of up to 20 km) from high towers and the dail y walk o f personnel t hrough pre-established p aths during the fire-se ason. This type o f system is v ery c heap because its main co st is represented b y th e low wa ges o f the few people involved in the direct obse rvation. On the ot her hand, a few companies h ave also implemented the o bservation through cameras in t he visual and infrared ra nges. However, this class of system is usuall y co nsidered too ex pensive because the r elativel y high initial cost of installati on of the infrared cameras. Under this situation, se veral companies were i nterested in the idea of a wireless s ensor network for the detection o f wildfires, but the y were also concerned on th e cost of the system pe r unit of area. This probl em can be d ecomposed mainl y into tw o p arts: a) the cost Page 3 of each individual node; b) the number of nodes which are need ed per uni t of area (i. e., the number of nodes per s quared m eter). In this paper we investigate part b), while part a) will be presented elsewhere. 1.1 Variables under analysis We shall work with simple two dimensional models and we shall deal only with propagation of surface fi res. M oreover, w e shall keep the model of the wireless sensors as simple as possible, that i s, we sh all assume that all sensors allow for th e detection of the fire as soon as the fire reaches their loc ation. Finally, we shall not concern ourselves with the difficulties of th e communication amo ng t he sensors, whi ch ma y be impaired b y the activity of the fire itself (Heron and Mph ale (2004), Mphale et al. (2007 )). Under this setting, there are man y variables related to the number of wireless sensors th at can b e studied. We choo se to of them, the time to detection ( T d ) and th e area already burnt at the time of the detection ( A d ). Both variables can be related, in t urn, t o the resources needed for contention of th e fire after it has been detected. Since the number of wireless sensors m ay var y according to t he ex tension of the re gion that must be protected, we sh all use the characteristic distance between sensors ( D ) as reference independent of the actual area. Although the definition of the characteristic di stance between s ensors may va ry sli ghtl y from one s etting to another, in all cases it subsumes under a single value how widely spaced the sensors are. The rest of the pap er i s structured as follows. In Section 2, we present so me simp le results for the case wher e the nodes ar e distrib uted in a regular p attern a cross th e ar ea of interest. In S ection 3, we anal yze the expected time to detection and the burned area before Page 4 detection when the s ensor nodes are randoml y distributed in the protected area. Se ction 4 summarizes the main conclusi ons of the paper and m entions some ideas for future work. 2 Regularly distributed sensors In this se ction, we an alyze the case where the senso rs are lo cated in a re gular grid i n such a way that the distance bet ween an y pair of them in the same row or the sa me colum n is D , the characteristic distance in this setting. As a further sim plification, we shall assume that the region t o b e protected is a rectan gle whose sides are integer mult iples of D . I t is eas y to extend the work in thi s se ction to more gener al regions b y p artitioning them into small rectangular pieces. 2.1 Circular propagation at constant rate We assume that surface fires propa gate at a constant rate of spread ( R ) i n all directions and that the probabilit y of igni tion i s uniforml y distri buted ins ide the protected area. Both t he uniform distributio n of the probabilit y of ignition and t he fact that the nod es are lo cated in a regular latti ce enabl e us to reduce the study of the detection of a fire to a much smaller area corresponding to a s quare delimited b y four sensor nodes, on e on each v ertex. Furthermore, this square can be split up into four smaller s quares, as shown in Figure 1 . Gi ven th e assumption that a fire propagates in all directions at t he same speed, if a fire originates in Region i ( i =1, 2, 3, 4 – see Figure 1), it wil l be d etected b y sensor i first. T herefore, we can further l imit our analysis to only on e of th e four regions and the corresponding node, say, Region 1 and sensor nod e 1. In other words, ( ) [ ] . 1 Region in originated fire the x T P x T P d d ≤ = ≤ Page 5 Notice that T d is simpl y the distance from the ignition point to the sensor node divid ed b y the constant rate of spread R . Since the di stribution of the ignition point is uniform inside Region 1, we have [ ] = ≤ 1 Region in originated fire the x T P d [ ] = ≤ = 1 Region in originated fire the point ignition the to 1 sensor from Distance Rx P . 1 Region of Area 1 sensor of distance a at are points whose surface the of Area Rx ≤ = After some work, the latt er expression can be foun d to be ( ) , 2 si 2 4 D Rx D Rx x T P d ≤         ⋅ π = ≤ (1) ( ) , 2 D 2 D si 1 2 2 1 2 tan 4 2 2 2 1 ≤ ≤ −         +         ⋅           −         − π = ≤ − Rx D Rx D Rx D Rx x T P d (2) ( ) . 2 si 1 D Rx x T P d ≥ = ≤ (3) Further algebraic work le ads to , 215 . 0 4 1 2 ≈ π − =       ≥ R D T P d (4) [ ] ( ) , 3826 . 0 6 2 1 log 2 R D R D T E d ⋅ ≈ + + = (5) [ ] ( ) ( ) ( ) , 0203 . 0 36 2 1 log 2 6 var , 6 1 2 2 2 2 2       ⋅ ≈       + + − =       = R D R D T R D T E d d (6) [ ] . 52 . 0 6 2 2 D D A E d ⋅ ≈ π = (7) Page 6 Some comments ar e due. Equation ( 4) points o ut that more than 80% of fires wil l be detected in less than D /2 R and , hence, t heir area will be small er than π D 2 /4. Equ ation (5) says that, as expected, t he mean detect ion ti me is proportional to the ratio D/R. Finally, while Equati ons (1)-(3) impl y that no fir e will have an area grea ter t han π D 2 /2 at the moment of detection, Equation (7) says th at the expected value of such a rea is approximatel y D 2 /2. In particular, n ote that the expected value of t he area of the fire at the moment of detection does not depend on the rate of spread of the fire, but it only depends on the characteristi c distance D . 3 Rando mly distrib uted sensors In this section, w e anal yze the case whe re the sens ors are random ly distribut ed in the re gion to be protected. I n this c ase, the c haracteristi c distance D is the mean distance between sensors computed as , N A D = (14) where A is the total a rea of the prot ected region and N is the total number of sensors. We shall first s how that, when the number of senso rs is l arge, the bu rned area at the moment of detection A d has a simple prob ability dist ribution w hich is i ndependent of the propagation mo del . We shall then show sim ple approx imations for the statist ics of t he t ime to detection for simp le propagation models. 3.1 Probability distribution of the burned area The probabili ty that the burned ar ea A d is g reater than a given number x is equal to the probabilit y that the fir e “has not found” an y sen sor inside the bu rned region . Since sensor Page 7 nodes ar e r andomly dis tributed across th e prot ected re gion, and assumin g that the position of each sensor is ind ependent of that of t he others, we have ( ) , 1 N d A x x A P       − = > where A is the total area of the prot ected region and N is the tot al num ber of sensor nodes. Using Equation (14 ), we can write ( ) . 1 1 2 2 N N d N D x ND x x A P           − =       − = > Note that, as N →∞, ( ) . exp 1 2 2       −   →            − = > ∞ → D x N D x x A P N N d Formall y, we have the followi ng Theorem 1. Assume the lo cations of sensor nodes are independent and identically distributed random variables with unifor m distribution across the p rotected region. Furthermore, assume th at the pos ition of the s ensor nodes and the ignition points (ther e may be more than one) are independent random variables. Let N be the number of sens or nodes and A(N) the s urface area of the protected regio n, which varies with N in such a w ay that A (N) = D 2 N, w here D is a positi ve constant. Then, as the number N of sensor nodes goes to infinity, th e ran dom variable A d cor responding t o the burned area at the moment of detection converges in distribution to an exponential random variable with parameter λ=1/D 2 . Page 8 Probably, the mos t salient feature of Theorem 1 is th at it does not depend on t he propagation law of the f ire and it d epends n either o n the nu mber nor on the distributi on of the igniti on points. I n ot her words, Th eorem 1 st ates that, if t he protected area is l arge and the number of senso r nodes is als o large, then ( ) [ ] ( ) . var , E , exp 2 2 2 D A D A D x x A P d d d ≈ ≈       − ≈ > 3.2 Probability distribution of the time to detection Theorem 1 leads to the followi ng simple Corollary 1. Assume that there is a determini stic law F such t hat, at each time inst ant t, F(t) represents the total burned area at time t. If t he hypotheses of Th eorem 1 are satisfied, then, as the number of sensor nod es N goes to infinity, ( ) . ) ( exp 2       −   →  > ∞ → D t F t T P N d In the followin g paragraphs, we consider two sim ple examples of application of Corollar y 1. 3.2.1 Circular pr opagation at consta nt rate Note that, as the surface area of the p rotected region i ncreases, we ma y ignore the cases where the fire dev elops near the borde rs of t he re gion. Then, it is easy to see that the function F in Corol lary 1 for t he case of circular propa gation at const ant rate of spread R i s given by ( ) ( ) . 2 Rt t F ⋅ π = Page 9 Then, for a l arge area covered with a lar ge numb er of sensors, we can make the followi ng approxim ation: ( ) . exp 2 2 2         π − ≈ > D t R t T P d So we can estimat e the ex pected value of th e time to d etection as [ ] ( ) , 2 exp 0 2 2 2 0 R D t d D t R t d t T P T E d d =         π − ≈ > = ∫ ∫ ∞ ∞ where we have om itted the det ails of the calcul ation. In a similar fashion, we get [ ] ( ) . 068 . 0 4 4 var , 2 2 2 2 2 2 2 R D R D T R D T E d d ≈ ⋅       π π − ≈ π ≈ Figures 2 and 3 show the agreement of th e previo us equati ons and the resu lts of simulations (10 5 Monte Carlo run s) with D= 1 m, R= 1 m /s, N var ying from 10 t o 10000 and one random ignition poin t. 3.2.2 Elliptical pro pagation at consta nt rate In this section, we consider the case of a s urface fi re that propagates wi th an elliptical shape and at a constant rate of spread. In this case, the fu nction F is given by (see Finney (1998)) ( ) , 4 1 1 2 2 2 t LB HB R t F       + ⋅ π = where R i s the rate of spread, HB is the Head-to-Back ratio and LB is the Length-to-Breadth ratio. In a similar fashio n as be fore, we can com pute Page 10 [ ] [ ] , 1 1 4 , 2 1 1 2 2 2 2 2 R D HB LB T E R D HB LB T E d d π       + ≈ + ≈ ( ) . 1 1 4 068 . 0 1 1 4 4 4 var 2 2 2 2 2 2 R D HB LB R D HB LB T d ⋅       + ≈ ⋅       + ⋅       π π − ≈ 4 Conclusio ns The lack of ex tensive research in the appli cation of inex pensive wireless sensor nodes for the earl y detection of wil dfires motivated us to investigate the cost of su ch a network. As a first step, i n t his paper w e present sever al results which relate th e t ime to d etection and the burned area to the numb er of sensor nodes in the region which is protected. Our main r esult is Theorem 1 which states t hat the pr obabilit y di stribution of t he burned area i s approximatel y exponential, g iven that some hypotheses h old: the positions of the sensor nodes are independent random variables uniformly distribu ted a nd the number of senso r nodes is large. It is im portant to remark that this conclusion depends neit her on t he propagation model of the fire nor on the numbe r and distribut ion of ignition point s. Our next step in the inve stigation of the cost of a network of s ensor nodes for th e detection of wildfires is to actually build prototypes of s uch nodes and to test t heir behavior u nder controlled fires. Acknow ledge ments This work was partially supported by anonymous contributors through the project “ Prevention and early de tection of forest fires b y means of senso r networ ks ” which is b eing developed at the Instituto Tecnológico de Buenos Aires (ITBA). Page 11 References Chen MM, M ajidi C, Doolin DM, Glaser SD, Sit ar N (2003) Design and construction of a wildfire instrumentation s y stem using networked sensors. Present ed in ‘Network Embedded S ystems Technolog y (NEST) Retreat 2003.’ (Oakland, C alifornia) Doolin DM, Glaser SD, S itar N (2004) Softw are Archi tecture fo r GPS -enabled W ildfire Sensorboard. 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