Coverage Improvement of Wireless Sensor Networks via Spatial Profile Information
11 Coverage Improvement of Wireless SensorNetworks via Spatial Profile Information
Kaushlendra Pandey, Abhishek GuptaIndian Institute of Technology Kanpur, Kanpur (India)Email: { kpandey,gkrabhi } @iitk.ac.in Abstract
This paper considers a wireless sensor network deployed to sense an environment variable witha known spatial statistical profile. We propose to use the additional information of the spatial profileto improve the sensing range of sensors while allowing some tolerance in their sensing accuracy. Weshow that the use of this information improves the sensing performance of the total WSN. For this, wefirst derive analytical expressions for various performance metrics to measure the improvement in thesensing performance of WSN. We then discuss the sensing gains quantitatively using numerical results.
I. I
NTRODUCTION
Modern wireless sensor networks (WSNs) consist of a large number of inexpensive wirelesslyconnected sensor nodes, each enabled with a limited sensing capability. In some applications ofWSN, such as monitoring of a macro-environmental variable (MEV), we need to deploy the sen-sors with considerable density and monitor the area for an extended period. Macro-environmentalvariables generally have little variation over the space. For example, e.g. environmental humidity,temperature of the earth surface observe very little change between two locations that are onlymeters away. Since MEVs have a spatial correlation, there must be a tremendous amount ofredundancy in the sensed data when sensors are relatively dense. Due to slow spatial variationof these variables, one can estimate their value at one location from their value at another locationwith high accuracy. However, we would need knowledge of the spatial profile of these variablesfor the estimation. The estimation would also introduce some errors, and hence, an error in thesensing accuracy needs to be tolerated while using the spatial correlation information.
This work is supported by the Science and Engineering Research Board (DST, India) under the grant SRG/2019/001459. a r X i v : . [ c s . I T ] A p r In the past literature, the performance of WSNs has been studied both numerically via sim-ulations and analytically using tools from stochastic geometry [1]. In [2] authors presented amodel for the k -coverage for Boolean-Poisson model for the intrusion detection. A survey oncoverage control algorithms, along with the relation of coverage and connectivity, is available in[3]. A comprehensive survey on directional and barrier coverage is available in [4]. Comparativecoverage analysis for WSNs when sensors locations are according to different point processesis performed in [5]. The researchers have also studied various ways to increase the coverageregion, including the densification of sensors, optimal deployment, and increasing the sensorrange. For example, in [6] focuses on the early detection of a forest fire. In [7] authors showedthat sensing coverage performance can be improved by using data fusion techniques where eachsensor sends its measurements about the sensed signal to the cluster head, which makes thedetection decision based on the received measurements. One way to increase the coverage ofWSN without deploying more sensors is to avoid redundancy in the sensed data. In WSNs that aredeployed to sense a target MEV, the spatial variation profile of the MEV may help us increasecoverage by estimating the value of MEV in the uncovered regions. The spatial variation ofMEVs has been studied in the past works. For example, in [8], authors present an approximaterelation between the temperature of the soil and its depth. Similarly, the numerical study ofvariation of forest temperature over space and time was performed in [9]. It is interesting tostudy if the spatial profile information can be used to achieve better coverage of a region withoutincreasing the sensor density by avoiding redundancy in the sensed data. There are only a fewpast works that focus on the spatial correlation of variables. In [10], clusters of closely locatedsensors are formed as they have similarity in the sensed data. For cluster formation, the sinknode observes the reading of source nodes for a time-period, and based on the observation, itcreates the clusters. In [11], the authors used spatio-temporal correlation among the sensed datafor dual prediction and data compression where sensors predict data based on past observations.However, a comprehensive analytical framework for the sensing performance of a WSN deployedto sense an environmental variable with known spatial profile information, is not still availablewhich is the main focus of this paper.In this paper, we consider a WSN deployed to sense an environment variable with a knownspatial statistical profile. We propose to use the additional information of the spatial profileto estimate the value of the environmental variable in the uncovered regions from the value atcovered regions. This can improve the sensing range of sensors while introducing some estimation errors in the sensing accuracy which need to be tolerated. We first derive analytical expressionsfor various performance metrics to measure the improvement in the sensing performance ofWSN. Using these analytical expressions and quantitative results, we show that the use of thisinformation improves the WSN’s sensing performance. Notation: B ( y , a ) denotes a - d ball of radius a centered at location y . X i denotes the locationof i -th sensor. Let R S be the sensing range of each sensor. Hence, S ≡ B (o , R S ) denotes thesensing region of the sensor located at the origin o . Let C ≡ B (o , r ) be the region of interest.Let | A | denote the Lebesgue measure of the set A . The Minkowski sum of any two set A ⊕ B isdefined as { a + b : a ∈ A , b ∈ B } . The Minikowski difference of the two set A (cid:9) B = ( A c ⊕ B ) c .II. S YSTEM MODEL
In this paper, we consider a WSN deployed in R to sense an environmental variable Θ thatvaries spatially. Examples include the temperature in a forest, soil moisture in an agriculturalfield, humidity in a city. The locations of sensors can be modeled as homogeneous Poisson pointprocess (PPP) Ψ = { X i } with intensity λ [1]. We assume that each sensor X i has a circularsensing region S i = B ( X i , R S ) around it. The region sensed by WSN is Boolean-Poisson model ξ given as ξ = (cid:91) X i ∈ Ψ X i + S , (1)where S ≡ B (o , R S ) denotes the sensing region of each sensor around itself. A. Profiling of spatial variation of the environmental variable
Let the value of the environmental variable at a location x be denoted by Θ( x ) . For real-worldcases it can be assumed that the spatial variation of Θ is bounded which means that it can varyonly by a finite value in a finite distance. One example can be found in [12] where the spatialprofile of the soil moisture S with depth y is given as S ( y ) = A ( y ) + S (0)[1 + B ( y ) ] + S c . Here, S ( y ) is the soil moisture at the depth y , S (0) the soil moisture at or near the surfacelayer at depth , and A , B and S c are some constants. If S ( x ) , y and x are known, S can beestimated. This assumption results in variable having a spatial profile, and the spatial correlation at thevariable’s value at two points. We assume the knowledge of this spatial profile. In particular, weassume the following spatial profile that for any two points x and y , the variation in the valueof Θ | Θ( x ) − Θ( y ) | ≤ f ( (cid:107) x − y (cid:107) , w ) , (2)where f ( (cid:107) x − y (cid:107) , w ) is a tolerance function and w is the spatial variation rate of Θ . Hence, theuncertainity in the value of Θ( y ) conditioned on the knowledge of Θ( x ) is Uncert (Θ( y ) | Θ( x )) = f ( (cid:107) x − y (cid:107) , w ) . B. Use of the environmental variable’s spatial profile
Due to this additional information about the correlation, the variable’s value at locationsthat are not covered in the sensing range of any sensor, can be guessed/estimated within sometolerance as shown in Lemma 1.
Remark 1. f is an increasing function with respect to the first argument which can be seen asfollows. If y is close to x , Θ( y ) is equal to Θ( x ) ( i.e. f (0 , w ) = 0 ), In other words, Θ( y ) canbe predicted exactly. As we move y away from x , the correlation between Θ( x ) and Θ( y ) willreduce and the certainty in the value of Θ decreases. Lemma 1.
If the value of Θ at x ( i.e. Θ( x ) ) is known, then the set of points where uncertaintyin Θ is within τ tolerance, is given as P ( x , τ ) = { y : | Θ( y ) − Θ( x ) | < τ } = B ( x , R ( τ, w )) , where R ( τ, w ) is given as R ( τ, w ) = f − ( τ, w ) . (3) Here, inverse of f is with respect to the first argument.Proof. To ensure that Θ( y ) does not vary more than τ from Θ( x ) , f ( (cid:107) x − y (cid:107) ) must satisfy f ( (cid:107) x − y (cid:107) , w ) ≤ τ = ⇒ (cid:107) x − y (cid:107) ≤ f − ( τ, w ) . Lemma 2.
If the value of Θ at all points in a set A is known, the set of points where Θ can bepredicted within τ tolerance, is given as P ( A , τ ) = { y : | Θ( y ) − Θ( x ) | < τ, for at least a point x ∈ A } = A ⊕ B (o , R ( τ, w )) . Proof.
Owing to increasing nature of f , it is best to use the closest point in A to estimate thevalue of Θ( y ) at a location y . Hence, P ( A , τ ) = ∪ x ∈ A P ( x , τ ) = A ⊕ B (0 , R ( τ, w )) . C. τ − tolerance sensed region From (1), ξ is the region where value of Θ is exactly known. From Lemma 2, the region inwhich Θ can be sensed within τ tolerance is given as ˜ ξ = ξ ⊕ B (o , R ( τ, w )) = (cid:91) X i ∈ Ψ X i + ˜ S , (4)where ˜ S = B (o , R S ) ⊕ B (o , R ( τ, w )) = B (o , R S + R ( τ, w )) . We term ˜ ξ as τ − tolerance sensedregion and ˜ S i = X i + ˜ S as τ − tolerance sensing zone of the sensor at X i . Note that ˜ ξ is also aBoolean Poisson model. Note that ξ which is the exact sensed area, can be obtained from ˜ ξ bysubstituting τ = 0 . Hence, ξ ( i.e. the 0-tolerance sensed area) may be seen as a special caseof ˜ ξ .We now analyze the sensing and covering performance of WSN. We give the following twodefinitions for their use in next sections. Definition 1.
A point z ∈ R is said to be ˜ ξ − covered if z falls in the τ − tolerance sensing zoneof at least one sensor i.e. z ∈ ˜ ξ . Definition 2.
A point z ∈ R is said to be ˜ ξ − covered by exactly k sensor if z falls in the τ − tolerance sensing zones of exactly k sensors. III. S
ENSING P ERFORMANCE A NALYSIS
In this section, we analyze the sensing performance of the WSN. Let C be a set denoting aregion of interest. A. m − sensed area fraction We will first derive the τ − tolerance at-most- m -sensed area fraction ( ν m ( τ ) ) which is definedas average fraction of C falling under the τ − tolerance sensing region of at-most m sensors i.e. ν m ( τ ) = 1 |C| E (cid:34) m (cid:88) k =1 (cid:90) C γ k ( z )d z (cid:35) , (5)where γ k ( z ) is defined as γ k ( z ) = (cid:16) z is ˜ ξ − covered by exactly k sensors (cid:17) . Lemma 3.
The probability that a point y ∈ R is ˜ ξ − covered by exact k sensors is equal to P [ γ k ( z )] = exp (cid:0) − λπR ( τ ) (cid:1) ( − λπR ( τ )) k k ! , where R S ( τ ) = R S + R ( τ, w ) .Proof. z will fall in τ − tolerance sensing zone of k sensors if there are exactly k sensors in the B ( y , R S ( τ )) . Since sensors follow PPP, we get the desired result.Applying Lemma 3 in (5), we get the following theorem. Theorem 1. τ − tolerance at-most- m -sensed area fraction ν m ( τ ) of the WSN is given as ν m ( τ ) = m (cid:88) k =1 e − λπR ( τ ) ( λπR ( τ )) k k ! . Note that the above result is independent of the set C owing to the stationarity of the WSN[13]. B. Sensed area fraction
We now derive the τ − tolerance sensed area fraction, also termed τ − SAF, ( ν ( τ ) ) which isdefined as the fraction of a set C that can be sensed by ξ within τ − tolerance. Mathematically,it is equal to ν ( τ ) = E (cid:104) | ˜ ξ ∩ C| (cid:105) |C| and it can be expressed in terms of ν m ( τ ) as ν ( τ ) = lim m →∞ ν m ( τ ) . (6) Corollary 1.
The τ − SAF ν ( τ ) is given as ν ( τ ) = (1 − exp (cid:0) − λπR ( τ ) (cid:1) ) . Proof.
The result can be obtained Theorem 1 and (6).
Corollary 2. ν o ( τ ) = 1 − ν ( τ ) = exp ( − λπR ( τ )) , represents the average vacant fraction. The gain due to the use of additional correlation information can be expressed in terms ofthe coverage improvement factor (CIF) which is defined as the relative improvement in ν whenallowing tolerance in sensing with the use of spatial correlation information. Mathematically, η ( τ ) = ν ( τ ) ν (0) , and is given as η ( τ ) = 1 − exp( − λπR ( τ ))1 − exp( − λπR ) . Now, we focus on how well a network of sensors can cover a region of interest C . Let C ≡ B (o , r ) . C. m − intersection probability τ -tolerance m -intersection probability ( µ m ( τ ) ) is defined as the probability that C has non-empty intersection with the τ − tolerance sensing zone of exactly m sensors. The τ − tolerancesensing zone of a sensor intersects with C if and only if the location of sensor falls in theMinkowski sum of C and ˜ S [14]. Hence µ m ( τ ) is equal to the probability that there are exactly m sensors in C ⊕ ˜ S i.e. µ m ( τ ) = P (cid:104) Ψ(˜ S ⊕ C ) = m (cid:105) . Now, C = B (o , r ) . Noting that ˜ S ⊕ C = B (o , R S ( τ )) ⊕ B (o , r ) = B (o , R S ( τ ) + r ) , we get µ m ( τ ) = P [Ψ( B (o , R S ( τ ) + r )) = m ] . Now, since Ψ is a PPP, we get the following theorem. Theorem 2.
The τ -tolerance m -intersection probability µ m ( τ ) for C = B (o , r ) is µ m ( τ ) = e − λπ ( R S ( τ )+ r ) ( λπ ( R S ( τ ) + r ) ) m m ! . (7) Remark 2.
The density λ opt that maximizes the τ -tolerance m -intersection probability is equalto λ opt = mπ ( R S ( τ ) + r ) . and the maximum value of µ m ( τ ) is µ m ( τ ) max = me − m /m ! . (8)There are applications where sensors cooperatively decide the value of the environmentalvariable and there may a minimum limit on number of sensors require to build a consensus.The m -sensed area fraction and m -intersection probability are useful metrics for these cases.Maximizing the metrics µ m ( τ ) or β m ( τ ) for a particular value of m may help optimizing theperformance of network to build the optimal consensus among sensors. D. Intersection probability
We now derive the τ -tolerance intersection probability ( µ ( τ ) ) which is defined as the proba-bility that C has non-empty intersection with the τ -tolerance sensing zone of at least one sensor i.e. µ ( τ ) = P (cid:104) Ψ(˜ S ⊕ C ) ≥ (cid:105) = ∞ (cid:88) m =1 µ m ( τ ) . Corollary 3.
The τ -tolerance intersection probability for C = B (0 , r ) is µ ( τ ) = 1 − e − πλ ( r + R S ( τ )) . (9) E. m-cover probability
We now derive the τ -tolerance m -cover probability β m ( τ ) which is defined as the probabilitythat C lies entirely inside the τ -tolerance sensing zone of exactly m sensors i.e. β m ( τ ) = P (cid:34)(cid:32) (cid:88) X i ∈ Ψ (cid:16) C ⊂ X i + ˜ S (cid:17)(cid:33) = m (cid:35) . (10)Now, the location of the sensors that can fully cover C are the one inside the Minkowski difference ˜ S (cid:9) C . Hence, β m ( τ ) is equal to the probability that there are exactly m sensors in ˜ S (cid:9) C i.e. β m ( τ ) = P (cid:104) Ψ(˜ S (cid:9) C ) = m (cid:105) . Now, C = B (o , r ) . Note that ˜ S (cid:9) C = B (o , R S ( τ )) (cid:9) B (o , r ) . This is equal to B (o , R S ( τ ) − r ) if R S ( τ ) > r , otherwise is equal to the null set φ . Substituting the value, we get µ m ( τ ) = P [Ψ( B (o , R S ( τ ) − r )) = m ] . Now, since Ψ is a PPP, we get the following theorem. Theorem 3.
The τ -tolerance m -cover probability β m ( τ ) is β m ( τ ) = ( R S ( τ ) > r ) exp (cid:0) − λπ ( R S ( τ ) − r ) (cid:1) × ( λπ ( R S ( τ ) − r ) ) m m ! . (11) Note that a sensor can only cover B (o , r ) if R S ( τ ) > r .F. Cover probability We now derive the τ -tolerance cover probability ( β ( τ ) ) which is defined as the probabilitythat C lies entirely inside the τ -tolerance sensing zone of at least one sensor i.e. β ( τ ) = P (cid:104) Ψ(˜ S (cid:9) C ) ≥ (cid:105) = ∞ (cid:88) m =1 β m ( τ ) . Corollary 4.
The τ -tolerance cover probability for C = B (0 , r ) is β ( τ ) = ( R S ( τ ) > r ) (1 − e − πλ ( R S ( τ ) − r ) ) . (12)IV. N UMERICAL R ESULTS
In this section, we will numerically evaluate the presented analysis to find the impact ofdifferent system parameters on the performance of WSN. For numerical simulation purposes,we consider the following tolerance function f ( · , · ) : f ( (cid:107) x − y (cid:107) , w ) = Ae w (cid:107) x − y (cid:107) if x (cid:54) = y , if x = y (13)where w > , is the spatial variation rate of Θ . This form of f is inspired by the relation of soiltemperature at the surface and at a given depth presented in [8]. For this function, τ -tolerancesensing radius R S ( τ, w ) is given as R S ( τ, w ) = ln( τ /A ) /w if τ > A if ≤ τ ≤ A . (14) Note that lower value of w corresponds to slow variation of Θ and hence large region can besensed within a certain tolerance. On the other hand, higher value of w corresponds to fastvariation of Θ resulting in a high uncertainty in sensed data beyond the sensing range of R S .The case w = ∞ corresponds to complete uncertainty beyond R S and hence, R S ( τ, ∞ ) = 0 resulting in no additional coverage. Impact of sensor density on τ -SAF: Fig. 1 shows the variation of τ -SAF with the sensordensity λ . The maximum value of τ -SAF is 1 (corresponding to the case when C ⊂ ˜ ξ ). It canbe observed that increasing λ increases both ν ( τ ) and ν (0) , however, the relative gain decreaseswith λ beyond a certain density. When sensing tolerance is allowed, for moderate value of λ ,the use of spatial profile information can give us significant gain (up to 76% for w = . in Fig.1) in the average sensed area. If we fix the target SAF at a certain value (for example, γ = 0 . ),the required density to achieve this target SAF can be reduced (82/km to 8/km for w = . in Fig. 1) owing to the reduction in data redundancy. If spatial variation is slower, a less sensordensity would be required due to increased redundancy. Impact of the sensor density on τ -tolerance at-most- m -sensed area ν m ( τ ) and vacancy ν o ( τ ) : Fig. 2 shows the variation ν m ( τ ) with the sensor density λ . It can be observed that ν m ( τ ) increases with m (consistent with (5)). As we increase λ , ν m ( τ ) first increases. However, thereis an optimal density beyond which, ν m ( τ ) starts decreasing. This is because an increase inthe density beyond a certain point is not helpful, due to increased redundancy in sensing data.However, with spatial correlation this redundancy is reached beforehand which can be seenwhen comparing the optimal density value for the curves ν ( τ ) and ν (0) . Fig. 2 also showsthe variation of average vacancy ν o ( τ ) with λ . It can be seen that the vacancy decreases with λ . However, when spatial information is used, vacancy is reduced. Effect of the sensor density on m -intersection probability: Fig. 3 shows the variation of m -intersection probability with λ . It can be observed that there exists an optimal value λ opt ofdensity for a particular value of m . Below λ opt , µ m ( τ ) is less as there are not enough sensorsto intersect C . When density is higher than λ opt , there are more than required sensors whichcreates redundancy and therefore, µ m ( τ ) is also less for this case. Sensors can decide which m can be sufficient for the coverage of C . In the case when there are more number of sensors thanrequired, some sensors may choose to turn-off themselves. It can be observed that the maximumvalue of µ m ( τ ) does not depend on the radius of C . The value of r only shifts the value of λ opt .Increasing r or tolerance τ decreases the value of λ opt that can also be seen from Remark 2. -4 Fig. 1. τ -tolerance sensed area fraction for WSN with R S = 80 . Here A = 1 and allowed tolerance τ = 10 . Allowing tolerancecan reduce the required density to achieve the same target SAF, which can help reducing cost and improve the lifetime of WSN. -5 Fig. 2. τ − tolerance at-most- m -sensed area fraction and vacancy ν o ( τ ) for a WSN with τ = 5 , w = . , R S = 150 , A = 1 . ν o ( τ ) reduces considerably with spatial correlation information. This is due to the fact that data redundancy is reached earlier when spatial correlation is usedor when sensors with larger sensing range are used.
Variation of m -cover probability with the size of C : Fig. 4 shows the variation of m -coverprobability with the radius r of set C . It can be observed that it is not possible to cover C when r > R S with 0 tolerance. However, with use of spatial information, it is possible to cover setswith higher radius (200-300 in this example) with some tolerance.V. C ONCLUSION
In this paper, we consider a WSN deployed to sense a target environment variable. We showedthat the sensing performance of a WSN can be improved using the information on spatial profileof the target environment variable while allowing some error tolerance in the sensing accuracy. -5 Fig. 3. m -intersection probability µ m ( τ ) of a WSN with w = . , R S = 150 for C = B (0 , r ) . The maximum value of µ m ( τ ) does not depend on the size of C . Fig. 4. m -cover probability of a WSN with w = . , R S = 150 for set C = B (0 , r ) . With use of spatial information, it ispossible to cover sets with higher radius. We also saw that the required density to achieve a certain sensing performance can be reducedwhen the information on spatial profile of the target environment variable is used. This canhelp us reduce the cost of network, both- capital (by deploying less sensors) and operating (bykeeping some sensors off to save their life-time) costs.R
EFERENCES [1] J. G. Andrews, A. K. Gupta, and H. S. Dhillon, “A primer on cellular network analysis using stochastic geometry,” arXivpreprint arXiv:1604.03183 , 2016.[2] S. K. Iyer, D. Manjunath, and D. Yogeshwaran, “Limit laws for k-coverage of paths by a markov–poisson–boolean model,”
Stochastic Models , vol. 24, no. 4, pp. 558–582, 2008.[3] A. More and V. Raisinghani, “A survey on energy efficient coverage protocols in wireless sensor networks,”
Journal ofKing Saud University-Computer and Information Sciences , vol. 29, no. 4, pp. 428–448, 2017. [4] F. Wu, Y. Gui, Z. Wang, X. Gao, and G. Chen, “A survey on barrier coverage with sensors,” Frontiers of Computer Science ,vol. 10, no. 6, pp. 968–984, 2016.[5] K. Pandey and A. Gupta, “On the coverage performance of boolean-poisson cluster models for wireless sensor networks,” in Proc. WCNC May 2020 . [Online]. Available: https://arxiv.org/abs/2001.11920.pdf[6] K. K. Pandey and A. K. Gupta, “On detection of critical events in a finite forest using randomly deployed wirelesssensors,” in Proc. SpasWin June 2019 . [Online]. Available: https://arxiv.org/pdf/1904.09543.pdf[7] R. Tan, G. Xing, B. Liu, J. Wang, and X. Jia, “Exploiting data fusion to improve the coverage of wireless sensor networks,”
IEEE/ACM Trans. Netw. , vol. 20, no. 2, pp. 450–462, 2012.[8] S. Kaushik, G. Sharma, and B. Mokhashi, “Preliminary studies on subsoil temperatures at jodhpur,”
Defence ScienceJournal , vol. 15, no. 1, pp. 30–35, 1965.[9] H. Kawanishi, “Numerical analysis of forest temperature. I. Diurnal variations,”
Ecological modelling , vol. 33, no. 2-4,pp. 315–327, 1986.[10] C. Liu, K. Wu, and J. Pei, “An energy-efficient data collection framework for wireless sensor networks by exploitingspatiotemporal correlation,”
IEEE Trans. Parallel Distrib. Syst. , vol. 18, no. 7, pp. 1010–1023, 2007.[11] A. Jarwan, A. Sabbah, and M. Ibnkahla, “Data transmission reduction schemes in wsns for efficient iot systems,”
IEEE J.Sel. Areas Commun. , vol. 37, no. 6, pp. 1307–1324, 2019.[12] B. Biswas and S. Dasgupta, “Estimation ofsoil moisture at deeper depth from surface layer data,”
Mausam , vol. 30, pp.511–516, 1979.[13] S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke,
Stochastic geometry and its applications . John Wiley & Sons, 2013.[14] M. Haenggi,