Energy-Efficient Deterministic Approach for Coverage Hole Detection in Wireless Underground Sensor Network: Mathematical Model and Simulation

Abstract

Wireless underground sensor networks (WUSNs) are being used in agricultural applications, in border patrol, and in the monitoring of remote areas. Coverage holes in WUSNs are an issue which needs to be dealt with. Coverage holes may occur due to random deployment of nodes as well as the failure of nodes with time. In this paper, a mathematical approach for hole detection using Delaunay geometry is proposed which divides the network region into Delaunay triangles and applies the laws of triangles to identify coverage holes. WUSNs comprise static nodes, and replacing underground nodes is a complex task. A simplistic algorithm for detecting coverage holes in static WSNs/WUSNs is proposed. The algorithm was simulated in the region of interest for the initially randomly deployed network and after energy depletion of the nodes with time. The performance of the algorithm was evaluated by varying the number of nodes and the sensing radius of the nodes. Our scheme is advantageous over other schemes in the following aspects: (1) it builds a mathematical model and polynomial-time algorithm for detecting holes, and (2) the scheme does not work on centralized computation and therefore provides better scalability, (3) is energy-efficient, and (4) provides a cost-effective solution to detect holes with great accuracy and a low detection time. The algorithm takes less than 0.1 milliseconds to detect holes in a 100 m × 100 m-size network with 100 sensor nodes having a sensing radius of 8 m. The detection time shows only a linear change with an increase in the number of nodes in the network, which makes the algorithm applicable for every network size from small to large.

1. Introduction

Wireless underground sensor networks (WUSNs) [1,2] are sensor networks in which sensor nodes are buried under the surface. They are an extension of wireless sensor networks and are a potential field with enormous applications. WUSNs [3] have a large number of applications in the areas of irrigation monitoring, garden automation systems, sports ground maintenance, border patrol, and toxic substance and environment monitoring. WUSNs are becoming increasingly popular because of their contribution to accessing and monitoring toxic substances and the environment in remote areas. As compared to earlier wired underground networks, wireless networks provide more reliability and better concealment and are easy to deploy. However, understanding and modeling the radio propagation [4] among sensor nodes deployed underground are very critical for the operation of WUSNs. Network designers can estimate the most accurate communication radius and network capacity of WUSNs using an underground radio propagation model. Moreover, the characteristics of the underground communication medium (i.e., soil) differ from those of terrestrial networks, necessitating the use of an underground radio signal propagation model to regulate and validate the impacts of soil properties. Reliable antennas and the link quality [5,6,7,8] of deployed nodes play a key role for a range of communication protocols in WSNs. Data collection is easy with wireless underground sensor networks and provides real-time data. Besides the potential usage of WUSNs, there are many issues which need to be dealt with. The hole problem [9] is one of the major issues which WUSNs face.

A hole [10] is a region in the network where communication with other nodes is disconnected. The hole problem may occur due to a number of reasons. Firstly, holes can be created due to random deployment of nodes. When sensor nodes are deployed randomly in the network, during the placement of these nodes, a few areas may remain uncovered/disconnected. Secondly, sensor nodes have limited energy, and after continuous data collection and dissipation for a long time, they become exhausted and stop responding, leading to the failure of the nodes. The failure of nodes in a hefty figure generates holes in the network. Thirdly, there may be an oddity in the network due to a disaster. Nodes can fail due to anomalies in the network and hence can hamper the operation of the network. For example, a volcano in a region may damage sensor nodes buried underground. Damage to nodes can create a void in the region and generate holes. Holes can be divided into different categories. A region that is supposed to be fully covered may have some uncovered areas in it due to random deployment, an obstruction, or node failure; these uncovered regions are called coverage holes [11]. In the same way, there may be regions where nodes are not able to communicate, known as routing holes [12]. Other than these two categories, sometimes there may be some adversaries in the network which may create oddities. There may be malicious nodes that may jam communication to create jamming holes or that can seize the network by denial-of-service attacks, namely, sink/black/wormholes, and deteriorate the functionality of the network. Network performance is affected by holes as they act as barriers to communication. The purpose of hole identification [13] is to detect attacked, damaged, or inaccessible voids or nodes in the network. Unidentified holes lead to greater energy consumption and congestion, which ultimately increases the hole size. Identification of holes saves the network from additional energy exhaustion and congestion and therefore enhances the network life [14].

Thus far, we have discussed different types of holes; out of these, coverage holes are salient ones as they play a key role in maintaining quality of service [13]. Identification of coverage holes helps in finding voids/uncovered regions in the network. In uncovered regions, more nodes are set to be deployed to guarantee reliability by patching the holes. Hole detection assures the quality of service by preventing the loss of data. Therefore, this paper focused on the identification of coverage holes in wireless underground sensor networks.

We propose a Delaunay-based deterministic mathematical model which covers all required cases to detect coverage holes. The region of interest is divided into empty triangles (triangles with no node inside) using Delaunay triangulation. Then, using the proposed mathematical model, holes are detected in the network. Major contributions of this work are as follows:

1

A polynomial-time algorithm is proposed which can effectively detect all the holes presented in the network. Most of the existing solutions are centralized and work for hybrid networks. Such solutions are not scalable and therefore not suitable for WUSNs. WUSNs consist of static nodes and require solutions that possess low complexity and are scalable. Our algorithm is very well suited for WUSNSs.

2

Our algorithm requires only the location/coordinate information of the nodes.

3

Our algorithm is energy-efficient. The proposed coverage hole detection algorithm was run for the network lifetime, where all nodes communicate using the LEACH protocol. The algorithm avoids unnecessary computation and extra communication overheads which reduce the network lifetime. The residual energy after every round was noted. Based on the residual energy in the network, a new node deployment can be pre-planned. Considering the energy factor provides a comprehensive and wide-ranging viewpoint. The algorithm performs efficiently during the complete lifetime of the network. Extensive simulations were conducted to validate the performance of our proposed algorithm. The algorithm was tested by varying the number and sensing radius of the nodes. It works very well in spite of the nodes’ distribution and density. The algorithm has a short run time with the lowest cost and provides an accurate solution. The results show that the method is reliable, cost-effective, and energy-efficient.

The rest of this paper is structured as follows. Section 2 presents details of the work already conducted to identify coverage holes in WUSNs. The proposed mathematical model and related assumptions are presented in Section 3. In Section 4, the coverage hole identification algorithm is detailed. The simulation results are presented in Section 5. In Section 6, the proposed approach is discussed, and key contributions are highlighted. Section 7 concludes the paper.

2. Related Work

A significant amount of research has been conducted in the area with a preview of challenges faced in the deployment of wireless underground sensor networks. Hole detection in WUSNs is a promising area of research, although few works have been conducted in this domain.

Coverage hole identification techniques for wireless sensor networks can be categorized into three types: range-based, connectivity-based, and location-based techniques. In range-based/statistical techniques [15,16], nodes need to know the relative distance between neighbor nodes. In connectivity-based techniques [17,18,19,20], topology/connectivity information between nodes is required. The primary requirement of location-based/geometric techniques [5,21,22] is that all the sensor nodes need to know their location.

As per our acquaintance, there is only one model proposed by Kaushik [5] for coverage hole detection in wireless underground sensor networks. Therefore, to go into detail, studies conducted on coverage hole detection in terrestrial wireless sensor networks were also explored.

Kaushik [9] proposed location-based algorithms to find the boundary nodes and for the detection of holes in WUSNs. The author used the Voronoi diagram to find boundary nodes. After identifying the boundary nodes, the author used these boundary nodes to identify holes. With the help of the hole detection algorithm, approximate polygons using boundary nodes were created. The concept of the ‘flattening increment’ was used to approximate the hole edge point to the actual edge point, and the ’discretizing point’ was used to approximate the sensing edge.

In terms of range-based approaches for coverage hole detection in wireless sensor networks, Silva et al. [15] proposed a blind swarm of robots to determine coverage and holes. As no information about coordinates and connectivity was used by the authors, the algorithm only provides information about coverage, not about the minimum number of nodes required to fully cover the network or placement of nodes to maximize the coverage. Relative homology and simplical complexes [23] were used to determine holes in the network. The algorithm is based on centralized computation.

Bejerano [24] proposed two distributed coordinate-free k-coverage solutions (range-based) for hole detection, namely, a two-anchor (TA) scheme and a cyclic segment sequence (CSS) algorithm, for coverage verification and coverage hole detection. In the TA scheme, a local polar coordinate system is created by every internal node in the network by selecting two of its neighbor nodes as reference points, named anchors. Then, the location of other neighbor nodes is calculated with respect to this local polar coordinate system to verify the coverage. When all the neighbor nodes are determined, intersection points of the sensing border are calculated by each node to check the local coverage in its vicinity. The two-anchor solution works only if the transmission/communication range of the sensor node is greater than or equal to four times the sensing range. The cyclic segment sequence algorithm can be used to verify the coverage and detect holes even when the sensor nodes have a transmission range less than four times the sensing range (for a transmission range greater than or equal to two times the sensing range). It is a localized and efficient algorithm for detecting coverage holes without using local polar coordinates and works for 1-coverage. CSS may be susceptible to inaccurate distance measures and generate false positives.

In terms of connectivity-based approaches for coverage hole detection in wireless sensor networks, Khan et al. [20] proposed an approach to identifying the boundaries of holes as well as the boundaries of the sensor network. The basis of this approach is to check the connectivity [25] of the x-hop neighbor nodes adjoining every node. Here, the value of x can be set static and predetermined or dynamic and can vary as per the application requirement depending on a number of criteria such as the network density, degree of connectivity, and accuracy parameters. Though the algorithm has a low communication overhead in comparison to other similar techniques, deciding the value of x requires a thorough study of the network.

Li et al. [26] proposed a connectivity-based triangular mesh distributed hole recovery algorithm that detects and recovers coverage holes with only connectivity information. To find the accurate boundary of any hole, redundant nodes around the hole are activated using the algorithm. The algorithm is also said to be an enhanced 3MeSH algorithm. In the algorithm, sensor nodes are divided into two categories: active nodes and redundant nodes. A 3MeSH ring is formed in the graph of the active nodes to find the hole, if it exists. Nodes adjacent to hole boundary nodes are self-activated. The network reaches saturation when all redundant nodes become active and cannot perform any further hole recovery.

In terms of location-based approaches for coverage hole detection in wireless sensor networks, Wang et al. [21] designed two distributed protocols and three algorithms for handling sensor nodes’ movement. Voronoi diagrams were used to detect coverage holes. The first protocol is a basic protocol in which sensor nodes are moved iteratively for optimizing coverage. After each iteration, coverage holes are detected using the Voronoi diagram, and sensor nodes are moved by using one of the following algorithms: the VEC (VECtor-based), the VOR (VORonoi-based), or the minmax algorithm. In the VEC algorithm, sensor nodes are moved away from denser areas. In the VOR algorithm, nodes are moved toward holes. Nodes move toward another node that is farthest in the Voronoi diagram. VOR is a greedy algorithm and may result in the further creation of new holes. In the minmax algorithm, sensor nodes are also moved towards the hole, but the movement is kept such that the node in the Voronoi diagram for which the farthest movement is minimized is chosen. The second protocol is a virtual protocol in which sensor nodes are moved only virtually after each iteration; no actual physical movement is carried out. In the virtual protocol, the energy consumption is lower as the actual movements are deferred. The authors used a distributed approach to make it applicable for large networks. The simulation results showed that the minmax approach performed better than the other two. This protocol is only applicable for sensor nodes with mobility.

Zhang et al. [22] proposed two new computational geometric techniques, namely, localized Voronoi polygons (LVPs) and neighbor embracing polygons (NEPs), and using these techniques, two algorithms were designed. The first algorithm is localized and needs both distance and directional information. It uses localized Voronoi polygons (LVPs) and tentative localized Voronoi polygons (TLVPs) to find boundary nodes. A hole is detected only when any node which was being used in the LVP or TLVP construction dies. The second algorithm is an NEP-based algorithm which only needs direction information but cannot identify all holes. Both the algorithms can be combined for more efficient and cost-effective detection.

After reviewing various studies (as detailed above and also shown in Table 1) and analyzing scenarios of wireless underground sensor networks, we made the following observations:

Table 1. Literature review of various coverage hole detection algorithms and their limitations.

Type of Technique	Paper	Model/Concept Used	Limitations/Constraints
Range-Based Techniques	[15]	Blind swarm-based algorithm	Centralized computation. Minimum node requirement for optimal coverage.
	[24]	Two-anchor (TA) and cyclic segment sequence (CSS) algorithms	TA algorithm requires that the transmission range must be more than or equal to four times the sensing range. CSS may be susceptible to inaccurate distance measures and generate false positives.
Connectivity-Based Techniques	[19]	Cech complex, Rips complex, and connectivity-based distributed algorithm	In many scenarios, the algorithm is not sufficient in discovering triangular holes.
	[20]	Hop-based approach for hole and network boundary detection	A set of x-hop neighbors is discerned by each node in the network. Deciding the value of x needs a thorough study of the network.
	[26]	Triangular mesh distributed hole recovery algorithm	Redundant nodes need to be placed in the network. The selection of an optimal number of nodes for hole recovery is NP-hard. The algorithm can identify large holes but is unable to detect small bores.
	[27]	Hole detection algorithm with cooperative neighbors	The algorithm will not work with non-uniformly distributed nodes. While checking the hole boundary, a threshold from the average degree of neighbor nodes is taken into account; an incorrect value of the threshold may bring an extra communication overhead or false information, or may lead to the loss of boundary nodes.
	[28]	Hole detection based on homology	Centralized approach, computationally complex, and may not always detect holes.
Location-Based Techniques	[9]	Voronoi diagram—with the concept of the flattening increment	The value of the flattening increment and approximation is not clear.
	[21]	Voronoi diagram, two algorithms—VOR and VEC	Performance depends upon the number of mobile nodes in the network, which is not suitable for large-sized holes.
	[22]	Localized Voronoi polygon (LVP) and neighbor embracing polygon (NEP)	LVP needs both direction and distance. NEP cannot identify all holes.
	[29]	Convex hull-based algorithms	One algorithm possesses an extra communication overhead to maintain the state information; the second algorithm requires routing details.
	[30]	Scan-based movement-assisted sensor deployment method (SMART)	The number of scans leads to an extra communication overhead. The scan process may not work correctly if consecutive empty clusters increase in the network.
	[31]	Empty i-Cone-based hole detection algorithm	The algorithm requires 2-hop connectivity information along with the location information. Node synchronization is needed.
	[32]	Virtual force-based HEAL (hole detection and healing) algorithm using the concept of the Gabriel graph	Unable to detect holes present on the network boundary.

─

Uniform and dense deployment of sensor nodes with a statistical distribution is required in range-based techniques which is not always feasible.

─

In connectivity-based techniques, topology/connectivity information between nodes is needed which adds an extra communication overhead in the network and thus reduces the lifetime of the network. This will increase the deployment frequency of underground nodes and increase the cost and complexity.

─

In many techniques, sensor networks are hybrid, i.e., a network containing both static and mobile nodes. However, in WUSNs, all the sensor nodes are static, so the algorithm needs to be designed with this factor taken into consideration.

Therefore, after reviewing various studies, we provide a geometry/location-based coverage hole detection technique for randomly deployed nodes in wireless underground sensor networks.

3. Mathematical Model and Methodology

In this section, we first discuss the property of Delaunay triangulation, which is adapted in our model. This is followed by the description of our network parameters, mathematical formulation, and methodology used to identify the coverage holes in the network.

The target region is divided into Delaunay triangles with each sensor node as a point in the geometry. Delaunay triangulation (DT) and Voronoi diagrams (VDs) [33,34] are two imperative and fundamental data structures of computational geometry that share the property of duality and play a key role in solving many problems. The property of duality states that the Delaunay triangulation of any set of points P corresponds to the dual graph of those set of points in P. Circumcenters of DT correspond to the vertices of VD. Delaunay triangulation for a given set of points P is a triangulation DT(P) with the property that no point in P lies inside the circumcircle (a circle passing through the vertices) of any triangle in the triangulation. In other words, any two points connected by an edge in DT exist iff a circle passing through these points exists that does not contain any other point inside. The minimum angle amongst all achievable triangulations is maximized in DT. We took a target region of 100 m × 100 m (shown in Figure 1) with 100 sensor nodes randomly deployed in the network.

Figure 1. WUSN with a size of 100 m × 100 m with 100 randomly deployed sensor nodes. The network plot was created using MATLAB software.

Delaunay triangles are drawn using the sensor nodes as points in the network (shown in Figure 2). As can be seen from Figure 1, we placed a few nodes on the boundaries and corners of the network so that the Delaunay triangles can cover the full region. The complete network region was divided into a mesh of Delaunay triangles. Using the laws and properties of triangles, a mathematicalmodel was built. Holes in the network are detected using models applied to Delaunay triangles in the region of interest (ROI). A few assumptions were made, as follows:

Figure 2. Delaunay triangulation of sensor nodes deployed in the target region. The Delaunay triangles of the deployed sensor nodes were generated using MATLAB software.

3.1. Assumptions

1

Each node knows its location.

2

The communication radius (Rc) is larger than or equal to two times the sensing radius (Rs).

(Note that the sensing radius of any sensor node shows its coverage capacity, whereas the transmission/communication radius shows connectivity. A node can sense any object in its sensing range. Two sensor nodes can communicate with each other if they are within communication range of each other and are connected, called neighbor nodes.)

3

The sensing area of every sensor node is a circular disc with a sensing radius of Rs.

3.2. Mathematical Formulations

There are different types of Delaunay triangles in the network; we categorized them on the basis of their angles/circumcenter, as acute, right-angle, and obtuse triangles. In an obtuse triangle, one angle is more than 90°, whereas in an acute triangle, no angle is more than 90°. A triangle with one angle of 90° is a right-angle triangle. As it can be seen in Figure 3, a circumcircle is a circle drawn around the triangle passing through all the points of the triangle; it is drawn using the center at perpendicular bisectors of the edges of the triangle, and the distance from the circumcenter to any vertex is the radius of the circle.

Figure 3. Circumcircle around a triangle using ⊥ bisectors: (a) acute triangle; (b) obtuse triangle; (c) right-angle triangle.

CASE 1:

There is no hole in the triangular region if the sensing radius of nodes (Rs) is larger than or equal to the radius of the circumcircle (R). As shown in Figure 4, the full area of the triangular region is covered by sensing areas of three vertex nodes of the triangle when the sensing radius is more than the radius of the circumcircle of the triangle.

Figure 4. There is no coverage hole when Rs > R: (a) acute triangle; (b) obtuse triangle; (c) right-angle triangle.

In acute and right-angle triangles, the circumcenter is inside/on the edge of the triangle, so the maximum distance which is needed to cover is the circumcircle radius (the circumcenter being the farthest point from every sensor node). Therefore, if the sensor radius is more than the circumcircle radius, then every point in the triangular region will be covered.

In an obtuse triangle, the circumcenter is outside the triangle and farthest from every sensor node. If the circumcenter is covered by the sensor nodes’ sensing radius, then every point inside the triangle will be covered as every point inside the triangle is at a shorter distance than the circumcenter.

CASE 2:

If the sensing radius of nodes (Rs) is less than the radius of the circumcircle (R) drawn around the triangle, then there may or may not be holes present in the network. To manifest that, we first need to check if the triangle is obtuse or not.

A triangle is obtuse if it follows Equation (1):

ArΔAOB + ArΔBOC + ArΔAOC > Ar ΔABC

(1)

where Ar represents the area of the triangle. To compute the area of △ABC, we apply a formula to ‘calculate the area of the triangle for coordinate geometry’ evinced in Equation (2). As shown in Figure 1, each point has its coordinates on the x- and y-axis. Therefore, taking sensor nodes A(${\mathrm{x}}_{\mathrm{A}}$,${ \mathrm{y}}_{\mathrm{A}}$), B(${\mathrm{x}}_{\mathrm{B}}$, ${\mathrm{y}}_{\mathrm{B}}$), and C(${\mathrm{x}}_{\mathrm{C}}$, ${\mathrm{y}}_{\mathrm{C}}$), and circumcenter O(${\mathrm{x}}_{\mathrm{O}}$, ${\mathrm{y}}_{\mathrm{O}}$),

$$\mathrm{Ar}\mathsf{\Delta }\mathrm{ABC}=\left|\frac{{\mathrm{x}}_{\mathrm{A}}\left({\mathrm{y}}_{\mathrm{B}}-{\mathrm{y}}_{\mathrm{C}}\right)+{\mathrm{x}}_{\mathrm{B}}\left({\mathrm{y}}_{\mathrm{C}}-{\mathrm{y}}_{\mathrm{A}}\right)+{\mathrm{x}}_{\mathrm{C}}\left({\mathrm{y}}_{\mathrm{A}}-{\mathrm{y}}_{\mathrm{B}}\right)}{2}\right|$$

(2)

Similarly, we calculate the area of △AOB, △BOC, and △AOC and compare the sum of the three with the area of △ABC. Now, there are two cases: in the first case, the sum is equal, and in the second, some of the areas of the three triangles are more than the area of △ABC (shown in Figure 5).

Figure 5. Checking if a triangle is obtuse or not, with three cases: (a–c).

CASE 2a.

If the sum of areas of the triangles made with the circumcenter as one point (ArΔAOB + ArΔBOC + ArΔAOC) is equal to the area of ΔABC and the sensing radius of the sensor nodes (Rs) is less than the circumcircle radius (R), then there must be a hole (shown in Figure 6). As the sensing radius is less than the circumcircle radius, there must be some area around the circumcenter that remains uncovered. In acute and right-angle triangles, this area falls within the triangular region generated by sensor nodes and is called a coverage hole.

Figure 6. There is a coverage hole when Rs < R: (a) acute triangle; (b) right-angle triangle.

CASE 2b.

If the sum of areas of the triangles made with the circumcenter as one point (ArΔAOB + ArΔBOC + ArΔAOC) is more than the area of ΔABC, then ΔABC is an obtuse triangle. If Rs < R, then, the region around O is not covered. In an obtuse triangle, the circumcenter lies outside the triangle; therefore, it may be that the uncovered area lies outside the triangular region or inside the target area to be covered.

As shown in Figure 7, even if Rs < R, there may be no hole (Figure 7a), or a hole may be present (Figure 7b). To find out if there is a coverage hole or not, we can perform some mathematical calculations with the help of geometry.

Figure 7. Estimation of a coverage hole in an obtuse triangle when Rs < R: (a) no hole; (b) hole.

To estimate coverage holes in an obtuse triangle, ⊥ bisectors play a foremost role. It is a well-known fact that in obtuse triangles, the circumcenter is always outside the triangle and is opposite to the largest angle (shown in Figure 8). Perpendicular bisectors dissect the largest side at two points (in Figure 8 and Figure 9, E and F), and there will be no hole if these points are covered by any of the sensor nodes (Figure 9a,b).

Figure 8. Obtuse triangle with ⊥ bisectors GE and HF.

Figure 9. Coverage hole estimation in an obtuse triangle: (a) no hole when E and F are covered; (b) hole when E and F are not covered.

To see if points E and F are covered or not, we need to find the distances of BE, BF, AE, and CF. If these distances are smaller than or equal to the sensing radius of the sensor nodes, these points are covered; otherwise, they are not. We have three sensor nodes as three vertices, A(${\mathrm{x}}_{\mathrm{A}}$, ${\mathrm{y}}_{\mathrm{A}}$), B(${\mathrm{x}}_{\mathrm{B}}$, ${\mathrm{y}}_{\mathrm{B}}$), and C(${\mathrm{x}}_{\mathrm{C}}$, ${\mathrm{y}}_{\mathrm{C}}$), of △ABC (shown in Figure 10 for representation).

Figure 10. Obtuse triangle with crucial points and sides marked.

We need to find ⊥ bisectors of sides other than the side made by the largest angle, i.e., the largest side. Therefore, to find two smaller sides, we need to calculate the length of each side:

$$\mathrm{AB}=\sqrt{\left(\left({\mathrm{x}}_{\mathrm{B}}{}^2-{\mathrm{x}}_{\mathrm{A}}{}^2\right)+\left({\mathrm{y}}_{\mathrm{B}}{}^2-{\mathrm{y}}_{\mathrm{A}}{}^2\right)\right)}$$

(3)

Using Equation (3), we can find all three sides, AB, BC, and AC, and take two smaller sides (here AB and BC) for further calculation. GE and HF are ⊥ bisectors of sides AB and BC, respectively (shown in Figure 10).

∵ GE is ⊥ AB, ∴ AG = GB

∵ GE is ⊥ AB, ∴ ∠AGE = ∠BGE = 90°

∵ AG = GB and ∠AGE = ∠BGE= 90°

∴ using Pythagoras′ rule, AE=BE and ◺AGE= ◺BGE

Similarly, BF = FC

Using the law of cosine,

$${\mathsf{\theta }}_1=\frac{{\left(\mathrm{AB}\right)}^2+{\left(\mathrm{AC}\right)}^2-{\left(\mathrm{BC}\right)}^2}{2·\mathrm{AB}·\mathrm{AC}}$$

(4)

$${\mathsf{\theta }}_2=\frac{{\left(\mathrm{BC}\right)}^2+{\left(\mathrm{AC}\right)}^2-{\left(\mathrm{AB}\right)}^2}{2·\mathrm{BC}·\mathrm{AC}}$$

(5)

Now, as

$$\begin{gathered} {\tan \mathsf{\theta }}_1=\raisebox{1ex}{$\mathrm{GE}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{AG}$}\right. \\ \Rightarrow \mathrm{GE}= \mathrm{AG}\times {\tan \mathsf{\theta }}_1 \\ \Rightarrow \mathrm{GE}= \mathrm{AB}/2\times {\tan \mathsf{\theta }}_1 \end{gathered}$$

(6)

we already have the value of AB (using Equation (3)) and θ₁ (using Equation (4)), which we can use to compute GE.

Similarly,

$$\mathrm{HF}= \mathrm{BC}/2\times {\tan \mathsf{\theta }}_2$$

(7)

Now, using Pythagoras’ rule for ◺AGE,

$$\mathrm{AE}=\sqrt{{\left(\mathrm{AB}/2\right)}^2+{\left(\mathrm{GE}\right)}^2}$$

(8)

Similarly,

$$\mathrm{CF}=\sqrt{{\left(\mathrm{BC}/2\right)}^2+{\left(\mathrm{HF}\right)}^2}$$

(9)

As proved earlier, AE = BE and BF = FC, so now we have the values of AE, BE, BF, and CF. If these values are less than the sensing radius Rs, then there is a coverage hole in the target region. Otherwise, there is no hole.

The proposed mathematical model is very robust. The approach is very simple and energy-efficient. The mathematical formulations were built using the inherent properties of triangles, the rule of Pythagoras, the law of cosine, and triangle geometry. Use of the model is not only limited to WUSNs; it can also be very effectively applied to all static WSNs.

4. Coverage Hole Detection Algorithm

We used these mathematical models presented in the previous section to devise an algorithm for coverage hole detection. We randomly placed sensor nodes under the surface of the earth. The coverage hole detection algorithm (Algorithm 1) was run to find holes at the beginning and in every subsequent iteration in the case of node failure. Algorithm 2 was used to facilitate nodal communication, wherein the LEACH protocol was deployed for said purpose. We evaluated the performance of our algorithm by measuring parameters such as the number of holes, detection time, and residual energy. The hole detection time was observed with an increase in the number of nodes, as well as during the lifetime of the network, which proves the algorithm is scalable and suitable for networks with sizes varying from small to large. Residual energy after every round was observed, which proves that our techniques are energy-efficient.

Algorithm 1: Coverage Hole Detection for WUSNs.

Input: Location coordinates of sensor nodes deployed under the surface of earth Node (xi, yi), i = 1, 2, …, n. Sensing radius Rs Target Area Steps: Generate Delaunay Triangulation in the target area using locations of sensor nodes. (To cover the full area using Delaunay triangles, we have placed a few nodes at boundaries and corners of the network) Find out coordinates of circumcenter (CCxj, CCyj) and circumradius (Rj) for circumcenters of all the Delaunay triangles (Let the total number of Delaunay triangles generated be m). for each Delaunay triangle, DT do if circumradius (Rj) ≤ Rs then There is no Hole; (shown in CASE 1 of Section 3) else Find out if Triangle is an acute/right angle or obtuse using CASE 2 of the mathematical model in Section 3. if Triangle is acute/right angle then There is a Hole; (mathematically proven in CASE 2a of Section 3) else Find out the largest side of an obtuse triangle Find out points where perpendicular bisectors of the other two sides dissect (Let E and F). if E & F are covered by any of the three sensor nodes of DT then There is no Hole; (mathematically proven in CASE 2b of Section 3) else There is a Hole; (mathematically proven in CASE 2b of Section 3) end if end if end if end for Output: Marking of Holes in Delaunay triangles H = [H1, H2,…Hm], if Hi is NULL, then there is no Hole else There is a Hole and Hi contains coordinates of the circumcenter of DTi.

In Algorithm 2, holes are calculated whenever the energy level of any node falls to zero. We used the LEACH protocol for communication in the network. Any other protocol can also be used to lower energy consumption, if possible. The LEACH [35] protocol uses a clustering mechanism to improve the life of the network. Nodes in the network are divided into different clusters. It is a hierarchical protocol in which nodes in one cluster share data with the cluster head, where the cluster head aggregates the received data and transmits them to the sink. Cluster heads are decided by the use of a stochastic algorithm with a random probability distribution and are updated after each round to maintain the energy balance in the network. Nodes join the cluster with the closest cluster head. Once a node is chosen as a cluster head, it cannot become a cluster head again for P rounds, where P is the percentage of cluster heads. Subsequently, each node will have a probability of 1/P of becoming the cluster head. We implemented Algorithm 2 to assess the performance of our coverage hole detection algorithm with an increasing number of node failures in the network.

Algorithm 2: Estimation of Coverage Holes in the Lifetime of a WUSN.

Input: Locations of sensor nodes deployed under surface of earth Node (xi, yi), I = 1, 2, …, n, and Sink. Sensing radius Rs Target Area Energy (E) of sensor nodes Probability P of a node to become cluster head Transmission Energy ETX, Receiving Energy ERX, Data Aggregation Energy, Amplifying Energies (Efs & Emp depending upon the type of amplifier used, where Efs > Emp) Number of rounds (Rmax) Steps: Computation of threshold distance (d0), when distance among transmitting and receiving nodes is more than d0, then Efs is the amplifying energy otherwise Emp. $${\mathrm{d}}_0=\sqrt{\frac{\mathrm{Efs}}{\mathrm{Emp}}}$$ for r = 1 to Rmax if (mod(r, round(1/P)) == 0) then for i = 1 to n Reset all cluster heads //flag is reset after every P round after that all nodes can participate in becoming cluster head end for end if for each node in the network if there is any dead node then                  //which was not dead in the previous round Update dead nodes Use Algorithm 1 to detect Holes in the Network Record time taken to detect a hole in rth round Update number of Holes in the N/w //If there is any hole that was not present in the last round end if if the energy level of any node reaches 0 then Set Node to dead. end if Compute the remaining energy in the network. end for for each active node in the network Generate a random number—rand1 $\mathrm{if} \mathrm{rand}1\le \left(\frac{\mathrm{P}}{\left(1-\mathrm{P} \times \mathrm{mod}\left(\mathrm{r}, \mathrm{round}\left(1/\mathrm{P}\right)\right)\right)}\right)$ AND Sensor Node was not a cluster head in previous P rounds then Node is set to Cluster Head//Cluster heads selection using random number probability-based fun. end if end for for each non-cluster active node in the network Calculate the distance of node to Sink (distanceSink) and every other cluster node (distancec), to choose closest node (with minimum distance- distancemin) for transmitting data $$\begin{gathered} distanc{e}_C=\sqrt{{\left(C_x-Nod{e}_x\right)}^2+{\left(C_y-Nod{e}_y\right)}^2} \\ distanc{e}_{Sink}=\sqrt{{\left(Sin{k}_x-Nod{e}_x\right)}^2+{\left(Sin{k}_y-Nod{e}_y\right)}^2} \\ distanc{e}_{min}=\mathit{min}\left(distanc{e}_C,distanc{e}_{Sink}\right) \end{gathered}$$   if (distancemin > d0) then                            //Node Energy consumption in transmitting data $${\mathrm{Energy}}_{\mathrm{Dissip}\_\mathrm{Node}}=\mathrm{ETX}\times 4000+ \mathrm{Emp}\times 4000\times {\mathit{distance}}_{\mathit{min}}{}^4$$ else $${\mathrm{Energy}}_{\mathrm{Dissip}\_\mathrm{Node}}=\mathrm{ETX}\times 4000+ \mathrm{Efs}\times 4000\times {\mathit{distance}}_{\mathit{min}}{}^2$$ end if E = E $-{ \mathrm{Energy}}_{\mathrm{Dissip}\_\mathrm{Node}}$                                      //Remaining energy of node is updated if the Cluster node is chosen to transmit Data Packet then ${\mathrm{Energy}}_{\mathrm{Dissip}\_\mathrm{C}}=\left(\mathrm{ERX}+\mathrm{EDA}\right)\times 4000$   //Cluster node Energy consumption in receiving data E = E $-{ \mathrm{Energy}}_{\mathrm{Dissip}\_\mathrm{C}}$                                                   //Remaining energy of cluster node is updated end if end for for each cluster node in the network $\mathrm{distance}=\sqrt{{\left({\mathrm{C}}_{\mathrm{x}}-{\mathrm{Sin}\mathrm{k}}_{\mathrm{x}}\right)}^2+{\left({\mathrm{C}}_{\mathrm{y}}-{\mathrm{Sin}\mathrm{k}}_{\mathrm{y}}\right)}^2}$      //Calculate distance of node to Sink node if (distance > d0) then               //Cluster Node Energy consumption in transmitting data to Sink  $${\mathrm{Energy}}_{\mathrm{Dissipation}}=\left(\mathrm{ETX}+\mathrm{EDA}\right)\times 4000+ \mathrm{Emp}\times 4000\times {\mathrm{distance}}^4$$ else $${\mathrm{Energy}}_{\mathrm{Dissipation}}=\left(\mathrm{ETX}+\mathrm{EDA}\right)\times 4000+ \mathrm{Efs}\times 4000\times {\mathrm{distance}}^2$$ end if E = E $-{ \mathrm{Energy}}_{\mathrm{Dissipation}}$                              //Remaining energy of cluster node is updated end for end for Output: ─ Marking of Holes in each round ─ Energy Dissipation in every round ─ Time taken to detect holes in every round

Holes can be healed by placing new sensor nodes at the location of the circumcenters of Delaunay triangles with a hole.

5. Simulation Results

To simulate our algorithms and evaluate the performance, we randomly deployed 100 sensor nodes under the earth’s surface in a 100 m × 100 m area (as shown in Figure 1). During the sensor deployment, we intentionally deployed a few nodes on the boundary of the network so that the full target area is covered by Delaunay triangles in the network (as shown in Figure 2). The sensing radius (Rs) of the sensor nodes was set to 8. A complete list of the parameters is shown in Table 2.

Table 2. Network parameters for simulation.

Parameter	Value
Target Area	100 m × 100 m
Sensor Nodes (n)	100
Sensing Radius (Rs)	8 m
Initial Energy of Nodes (E)	0.1 J (for 90% nodes) 0.2 J (for 10% nodes)
Cluster Head Probability (P)	0.1
Packet Size	4000 bits
Transceiver Energy (ETX, ERX)	50 × 10−9 J/bit (i.e., 50 nJ/bit)
Data Aggregation Energy (EDA)	5 × 10−9 J/bit (i.e., 5 nJ/bit)
Amplification Energy (Emp), When d > d0	0.0013 × 10−12 J/bit/m4 (i.e., 0.0013 pJ/bit/m4)
Amplification Energy (Efs), When d ≤ d0	10 × 10−12 J/bit/m2 (10 pJ/bit/m2)
No. of Rounds (Rmax)	500
No. of Sink Nodes	1 (center of the network)

We used MATLAB for the implementation of our algorithms. Using Algorithm 1, we identified holes in the network after deployment. As shown in Figure 11, sensing nodes with a sensing radius of 8 m were deployed in the network; our algorithm marks holes in the network with very good accuracy.

Figure 11. Hole detection in a randomly deployed network. Red circles show hole centers marked by the algorithm.

After the identification of holes in the beginning, we ran Algorithm 2 for 500 rounds and recorded the performance of our coverage hole detection algorithm. Figure 12a,b represent two intermediate hole detection results that show an increasing number of holes due to the failure of nodes with time. Here, in our simulation, we set 10% of the total nodes as advanced nodes having energy (Ea) two times that of the energy (E) of other nodes. The motivation behind capturing this scenario is to view the change in the network with time, viewing the failure of nodes and increasing holes.

Figure 12. Increasing holes in the network with time (due to node failure). Failing nodes and increasing holes are shown from (a) to (b) to capture real-time scenarios.

In Figure 13, we can see that up to the first 175 rounds, there was no new hole. The nodes communicated with each other using Algorithm 2, and no node failure was observed until 175 rounds. The number of holes exponentially increased in the next 100 rounds as the sensor nodes in the network started failing due to the loss of energy. Increasing the number of holes in the network clearly shows that node failure will increase with an increase in the hole size due to the higher energy consumption of other active nodes and will ultimately lead to more failures. To further analyze this, we checked how many times our coverage hole detection algorithm had to run in the network lifetime. As mentioned in Algorithm 2, the hole detection algorithm is run whenever there is a new dead node in the network.

Figure 13. Number of rounds vs. holes in the network. The figure depicts no change in the holes for the first 175 rounds and a huge increase in the number of holes when nodes start failing, and again, very few change as communication decreases due to the failing of nodes in the range of each other.

Figure 14 shows that the hole detection algorithm was run around 95 times during the entire lifetime of the network, i.e., there was a new dead node found; this shows the detection time taken by the algorithm in every run. When we compare Figure 13 and Figure 14, it can be seen that the detection time was longer when there were more active nodes in the network. After a time when there was a sudden rise in the number of dead nodes/holes (as shown in Figure 13), the detection time reduced. The algorithm detected holes in the network with a size of 100 m × 100 m, with 100 sensor nodes having a sensing radius of 8 m, in less than 0.1 millisecond, and this was further reduced when the number of nodes reduced in the network.

Figure 14. Number of detections vs. detection time. The detection time is less than 0.1 millisecond and decreases with the decrease in the number of nodes (nodes failing with time) in the network.

Figure 15 shows the total residual energy of the network after every round. It can be seen that up to around 250 rounds, there was a linear and fast consumption of energy, as more nodes were communicating. After around 250 rounds, the number of dead nodes/holes increased (shown in Figure 13), so the communication reduced, as there were a few nodes in the communication range of each other and the sink.

Figure 15. Number of rounds vs. residual energy. Residual energy is measured after every fifty rounds.

We changed the sensing radius of the sensor nodes to 2 m and varied the number of nodes in the network in a range of 50–500 with an increase of 40 nodes after each iteration. After randomly deploying nodes in the 100 m × 100 m area, we ran the hole detection algorithm to observe the change in the hole detection time with an increase in the number of nodes in the network.

Figure 16 shows that the hole detection time increased linearly with an increase in the number of nodes; this shows that our algorithm’s performance is not affected by the increase in the network size and proves its suitability for every network size from small to large.

Figure 16. Number of nodes vs. detection time. The hole detection time shows a linear increase with an increase in the number of nodes, proving the scalability of the algorithm.

6. Discussion

Coverage holes are one of the most imperative challenges in wireless underground sensor networks, which can very badly affect the performance and quality of a network. Therefore, hole detection algorithms with a low cost and low complexity play a very important and crucial role in high-performance networks. In this paper, we proposed a Delaunay-based model for hole detection in wireless underground sensor networks. The simulation results show that our technique is advantageous over other schemes in many aspects.

A very simplistic and easy-to-implement mathematical model for static sensor nodes with a small capacity was built to detect coverage holes. The target region is divided into triangles using Delaunay triangulation. The model was developed using the properties of DT and geometry. All the cases were included in the model and were very well formulated using computational geometry. The results show that our algorithm can identify all holes present in the network.

We evaluated the consumption of energy in the network. We used the LEACH protocol for communication and recorded the residual energy after each round. The energy consumption in the network showed a linear depletion with the number of rounds. The energy consumption and the number of holes showed a comparative similarity when analyzed (Figure 13 and Figure 15). When the overall energy of the network fell below a threshold, a sudden increase in the number of holes could be seen due to the failure of the nodes. Residual energy can be helpful in pre-planning the deployment of additional nodes. The results show that the technique is energy-efficient, with less communication overhead. The algorithm performs very well during the lifetime of the network.

We analyzed the average hole detection time by varying the density of nodes in the network. Different numbers of nodes ranging from 50 to 500 having a sensing radius of 2 m were deployed in the region of interest. The hole detection time observed during all the experiments is presented in Figure 16, in Section 5. It can be seen that there was almost a linear rise in the hole detection time with increasing network density. A linear rise in time is very reasonable as, when the number of nodes in the network increases, the communication and computation time also increase. The lack of an exponential rise shows that our algorithm does not add any additional overhead with increasing network size. Therefore, the algorithm is scalable and can fit well with an increasing network size. The repeatability of the results was verified by generating one set of data points and running the hole detection algorithm multiple times; the algorithm is deterministic.

7. Conclusions

In this paper, we proposed a Delaunay-based coverage hole detection algorithm for randomly deployed sensor nodes in wireless underground sensor networks (WUSNs). We built rigorous mathematical models to realize the algorithm. The proposed algorithm detected all holes present in the network. The algorithm is easy to implement and possesses no hurdles in scalability. To analyze the performance of the coverage hole detection algorithm during the lifetime of the network, we implemented the algorithm using LEACH as a communication protocol. Using both algorithms, we simulated our network and found that holes in the network were captured with very good accuracy during the whole network lifetime. We calculated the residual energy after each round; the algorithm is very energy-efficient. The residual energy after every round provides a baseline for low-energy nodes and can be very useful in planning the deployment of new nodes in the network. The proposed algorithm took less than 0.1 milliseconds to detect holes in a 100 m × 100 m network with 100 sensor nodes having a sensing radius of 8 m. The hole detection time increased linearly with an increase in the number of nodes, which proves that the algorithm is scalable and suitable for small as well as large networks. Our scheme is cost-effective and shows fine performance over time when nodes start failing during communication. We developed an algorithm for nodes deployed in the same plane considering x and y coordinates in a 2D environment. The model can be extended to the 3D environment by considering depth as the third dimension. Future work can be conducted in this direction.