I Introduction
Many military communication scenarios involve the deployment of heterogeneous teams of mobile assets that execute joint tasks in challenging environments, such as forests, where randomized obstacles are located. Often these tasks are autonomous and the assets may be robotic. Potential application scenarios range from exploring and monitoring sensitive areas to search and rescue [16, 24, 9, 3, 23, 20]. In such scenarios, different mobile assets in the same team need to coordinate with each other and perform tasks jointly. Information sharing is an essential piece to the coordination of the team members [4] and can be realized via data streaming from one asset to another.
Computeraided localization and mapping [19, 2] relying on cameras and laser devices have been developed to realize highquality environment modeling in forests. Camera or laser enabled modeling and monitoring techniques usually require the mobile asset to continuously see the objects (e.g., its teammates) being monitored. Cuttingedge communication technologies use short wavelengths, such as in millimeterwave (mmwave) or even terahertz (THz) bands, to enhance the transmission speed, but the penetration capability is limited and usually requires a line of sight (LoS) to close a link. Motivated by these examples and others like it, it is clear that maintaining a line of sight (LoS) is preferred between these kinds of mobile assets.
Various existing works related to environments with obstacles focus on planning for the best routes for mobile assets or the optimal locations for stationary assets to avoid the connections between certain pairs of assets being blocked by the obstacles [7]. Such methods rely on perfect knowledge of the locations of the obstacles. In reality, this type of knowledge may be expensive to acquire, especially for largescale areas.
Locations of obstacles randomly distributed in a cluttered 2D environment can be seen as generated via a Poisson point process (PPP). Such a model is usually referred to as a Poisson Forest. Extensive work has been done related to navigating mobile assets in such forests [13, 14, 17]. Probabilities of having LoS of a minimum length in such a 2D environment are also computed [13] and related to the density of the PPP.
Because a given team may deploy heterogeneous mobile assets located at different altitudes (e.g. both ground assets and air assets), it is essential to consider the 3D nature of the environment. Not only the thickness and the locations of the obstacles need to be considered, but also the heights of the obstacles are important. References [11, 12, 10] address the height distribution of obstacles in an urban environment with semirandomly distributed obstacles. References [1, 6] model urban areas with obstacles of random heights using the Manhattan Point Line Process to address the gridlike patterns of the obstacles’ locations. This series of work points out that only obstacles taller than a critical height can block the LoS between a ground asset and an air asset. The critical height is a locationdependent function that depends on the ground and the air assets’ locations as well as the height that the air asset is flying at. Any obstacle that is taller than the critical height evaluated at that obstacle’s location will block the LoS.
In this paper, we study the 3D LoS between heterogeneous assets in a Poisson forest. We calculate the critical height, i.e., the height above which an obstacle is able to block the LoS, at any given location between a pair of assets. The distribution of the obstacles above the critical height is then modeled as an inhomogeneous PPP. By finding the void probability of the inhomogeneous PPP, we are able to determine the probability of obtaining LoS. Examples and expressions that are in closed form (up to known standard functions) are provided for two obstruction height distributions: uniform and truncated Gaussian. The examples are validated through simulation. Additionally, we establish throughput as a metric that balances communication distance with the impact of LoS blockage. Throughput is used to determine the range at which it is better to relay communications through the air asset, and, when the air asset is deployed, its optimal height.
The paper is organized as follows. Sec. II formulates the problem. Sec. III analyzes the probability of obtaining LoS between a pair of assets. Sec. IV discusses the throughput of the communication among assets. Sec. V provides numerical results derived from our methods as well as simulation validations. Sec. VI concludes the paper.
Ii Problem formulation
Consider a pair of ground assets, each equipped with a communication device (e.g., an antenna, a camera, etc.) at a height of , deployed in a planar task space (e.g., a forest) with stochastically distributed obstacles (e.g., trees) of a nontrivial thickness. The locations of the obstacles are generated by a twodimensional Poisson Point Process with a fixed density . Let be the number of obstacles in a task space of area and be the intensity of the PPP representing the expected number of obstacles per unit area. From the basic properties of a PPP
Such a task space is referred to as a Poisson forest.
The height of any single obstacle in the Poisson forest is represented by a nonnegative random variable
. The distribution ofmay vary. We denote the cumulative distribution function (cdf) of
as . Evaluating at a given height gives the probability that a given obstacle has a height that is less than or equal to .The analysis in this paper does not require that assumes any particular distribution. In fact, the height distribution of trees in a forest varies [15, 5, 18]. Bellshaped distributions [5]
and positiveskewed distributions
[15]can both be found. To provide specific realistic cases, without any loss of generality, we consider a singlesided truncated Gaussian distribution and a uniform distribution as examples to illustrate our methods. Both examples are constructed such that the heights are nonnegative.
The truncated Gaussian uses a Gaussian random variable with mean
as its parent distrution, and is truncated to the range . The cdf of this variable is(1) 
for , and zero elsewhere, where is the Qfunction.
For the uniform distribution, the cdf is
(2) 
where is the maximum height of the obstruction.
Now consider the onedimensional space between the two ground assets. Let the locations of the two ground assets be and on this 1D coordinate system. Any obstacle with a height above located along the interval would potentially block the unobstructed view of one ground asset on the other. If there is no such obstacle, we consider there is a line of sight (LoS) between the two ground assets.
In the abovementioned Poisson forest, the expected number of obstacles located exactly along the onedimensional space should be zero since the Lebesgue Measure of a straight line in a twodimensional space is always zero. However, when the obstacles’ thicknesses are nontrivial, obstacles located in a finite area with a nontrivial width around the line may also block the LoS between the two ground assets. Therefore we consider the distribution of potential obstacles along a straight line in this Poisson forest to be characterized by a 1D Poisson Process with a fixed density capturing the expected number of obstacles located along a straight line of unit length. is determined by , where is the average thickness of obstacles.
Now consider the case that an air asset at an altitude or height of is available to aid in the communication, for instance, by receiving the signal transmitted to it by the first ground asset and relaying it to the second ground asset. In this paper, we limit the horizontal location of the air asset to be along the straight line that connects the two ground assets. If we define to be the horizontal location of the air asset, then .
In the following sections, we calculate the probabilities of obtaining a LoS in groundground, groundair, and groundairground (i.e., airaided) connections, and calculate the throughput in all these scenarios. We furthermore find the throughput for these cases. Our calculations provide conditions under which air assets should be deployed to aid the connections between ground assets and provide insight into the optimal height of the air asset when one is deployed.
Iii Probability of Obtaining LoS
In this section, we provide methods for calculating the probabilities of obtaining a LoS (i.e., the LoS probability), between a pair of ground assets as well as between one ground asset and one air asset.
Iiia Critical height
In the Poisson forest introduced in the previous section, only obstacles that are above a certain height will block the view between two assets. If the two assets are both on the ground and have their communication devices at the same height , only obstacles with heights greater than may block the LoS. In this case, we say is the critical height of the obstacles.
When considering the LoS between a ground asset of height and an air asset of height , the critical height is determined by the straight line connecting the communication devices of the ground asset and the air asset which is located at , as shown in Fig. 1. Given both assets are fixed, the critical height is a function of the location along the horizontal coordinate. As can be seen in Fig. 1, the critical height is low close to the ground asset, where even short obstructions can block the LoS, but is high further away from the ground asset, where only the tallest obstructions can block the LoS. The criticalheight function is
(3) 
Notice that increases linearly with increasing from to .
IiiB Inhomogeneous Poisson Point Process
As introduced in Sec. II, the location of obstacles are generated by a homogeneous PPP with density . Consider the case that the critical height is a constant, such as with the direct groundground link. Since not all the obstacles are tall enough to block the LoS between assets, we must retain only those obstacles whose heights are higher than . For any random obstacle, the probability that it has a height greater than can be calculated by
Therefore, the distribution of obstacles with heights greater than in this Poisson forest can be modeled as a Poisson process with its density defined as
(4) 
When is locationdependent as in (3), this model becomes an inhomogeneous Poisson process with a locationdependent density defined as
(5) 
IiiC LoS between a pair of ground assets
As discussed in Sec. IIIA, only obstacles with heights above will block the direct view between a pair of ground assets. Therefore we set the critical height . Since is constant, the blockages are described by the homogeneous PPP given by (4). Therefore, the probability of obtaining a LoS between a pair of ground assets that are located at and is found from the void probability of the homogeneous PPP of density over an interval of length . This probability is
(6) 
IiiD LoS between a ground asset and an air asset
The distribution of obstacles along the straight line from the ground asset () to the air asset () can be modeled as a Poisson process with a density of . The critical height, above which an obstacle may block the LoS between the two assets, varies depending on the location of the obstacle. Therefore, the distribution of the obstacles that are tall enough to disrupt the unobstructed view fits an inhomogeneous Poisson process with a locationdependent density defined as in (5).
The probability of having no obstacle blocking the LoS between a ground asset at and an air asset at is found from the void probability of the inhomogeneous PPP, which is found as follows
(7) 
Finding a solution to (7) depends on the difficulty of performing the integration, which depends on the nature of . In some cases, such as the two examples provided below, closedform solutions can be found, at least up to being expressed in terms of wellknown expressions. Alternatively, when closedform solutions cannot be readily obtained, (7) can be solved numerically, for instance, by using numerical integration.
IiiE for truncated Gaussian distribution
For the truncated Gaussian distribution, the density of the inhomogenous PPP is given by (5), where is given by (1) and is given in (3). The resulting density is
where
(8) 
is the cdf of the standard normal distribution.
IiiF Analytical solution of for uniform distribution
For the uniform distribution, the density of the inhomogenous PPP is given by (5), where is given by (2) and is again given in (3). Evaluating we obtain
(11) 
where and is the critical distance at which any obstacle located at a distance cannot block the LoS since . This distance is given by
since is negative it is inconsequential, since the integral of (7) does not cover negative . The density of the inhomogeneous PPP is found by substituting (11) into (5) resulting in
(12) 
From (7), requires that be integrated from 0 to . If , then
(13) 
When , for and thus the integral from to is zero. It follows then that the integral will take the same form as in (13) but the upper limit can be tightened to since the integral beyond that point is zero.
Rather than expressing the integral separately for the two cases of and , we can express them as the following single expression
defining as the solution of the previous integral we have
Then, using we get that the probability of obtaining LoS for groundair communication at is given by
Iv Calculating the Throughput
While the LoS probability is useful for predicting the existence of a LoS between two assets, it does not characterize the quality of the link, which is also affected by the transmission distance of each link. For instance, if one wants to determine the height of an air asset that maximizes only the endtoend LoS probability with a ground asset, the solution would be to place the air asset at an infinite height so that the critical heights at any location between the two assets are infinite (i.e., no obstacle is able to block the unobstructed view between them). Such solution is neither practical nor efficient in reality due to the significant signal loss caused by the infinite transmission distance.
An appropriate metric that captures the loss of signal power at distance is the expected throughput, which we here define to be the maximum achievable data rate when accounting for the possibility of blockage. This definition makes sense considering a mixed team of ground and air assets navigating through the forest and maintaining communication. The probability of LoS between any pair of assets yields an expected communication time, which contributes to an expected throughput that can be achieved. For a single hop, the expected throughput is
(14) 
where is the capacity of the link, and the multiplication by accounts for the expectation being with respect to LoS. Here, we set as the Shannon Capacity, which is the maximum achievable rate of an unblocked link
(15) 
where
is the signal bandwidth, and the signaltonoise ratio, when expressed in dB, is
(16) 
where is the pathloss exponent, is a reference distance typically set to meter, and is the SNR when the receiver is placed at distance assuming freespace propagation up to that distance. The value of can be measured, or it can be calculated from the transmit power, carrier frequency, bandwidth, receiver’s noise figure, and antenna gains.
For a direct groundground transmission, the expected throughput is computed from (14) with in (16) set to . For a twohop groundairground transmission, it depends on the LoS probabilities of both hops. Let and be the LoS probabilities of the groundtoair and airtoground links, respectively, and similarly define and as the two capacities, the expected throughput for the groundairground communication is
(17) 
where the multiplication by accounts for the timedivision duplexing (TDD) operation at the air asset (i.e., the air asset spends half its time receiving from the first ground asset and half its time transmitting to the second ground asset). Alternatively, frequencydivision duplexing (FDD) can be used, but in that case, the perlink capacities should be scaled accordingly since only half of the band could be used for each hop. Each capacity is found from (15) with the distance in (16) set as the Euclidean distance between the air antenna and the corresponding ground antenna, where each distance is the hypotenuse of a right triangle formed with one leg being the horizontal distance, either for the groundair link or for the airground link, and the other leg being the difference in antenna heights, . Notice that the groundairground throughput is determined by the minimum capacity of the two hops, which motivates us to consider deploying the aiding air asset always above the midpoint of the two ground assets.
V SimulationValidated Numerical Results
In this section, we present numerical results generated by our methods. The key part of these methods is the calculation of the probability of obtaining a LoS between different types of assets. We consider a Poisson forest where the locations of the obstacles along a straight line are generated via a Poisson point process with . The distributions of the obstacle height were chosen to be a truncated Gaussian distribution with cdf defined as in (1) with m and m and a uniform distribution with cdf defined as in (2) with m. The choice of the parameters is consistent with the parameters chosen in [6].
To validate the numerical results, we performed Monte Carlo simulations involving the repeated drawing of Poisson forests. Let be the interval of a simulated Poisson forest, where for a groundground link or for a groundair link. Drawing a Poisson forest involves first determining the random number of obstructions in the interval, which is done by drawing
from a Poisson distribution of mean
. Next, each of the obstructions is placed uniformly over the interval. Then, each obstruction’s height is determined by randomly drawing its from the corresponding distribution. Once the forest was constructed, it was determined whether or not the LoS path was blocked by checking to see if any of the obstructions were above the critical height at that location. This process was repeated for 500 thousand trials for each data point reported.Fig. 2 shows the probability of obtaining LoS between a ground asset and an air asset, , with the horizontal distance . In this figure, only the truncated Gaussian distribution is considered, as results for the uniform distribution are similar. The air asset flies at different fixed heights of , , and meters. The ground asset has a communication device fixed at the height of meters.
The results in Fig. 2 show that decreases as increases. The reason is straightforward. A longer distance between the two assets simply allows a greater probability of having obstacles in between. Meanwhile, increases as increases. This is because the critical height will increase with a greater (as in (3)). A greater critical height rejects more obstacles from potentially blocking the LoS between the two assets. Therefore flying the air asset at a higher altitude generally increases the probability of obtaining LoS.
Increasing the height of the communication devices carried by the ground asset will improve the probability of obtaining LoS as well. Generally, increasing the height of the ground assets’ communication devices is a more expensive and less efficient way to enhance the probability of LoS as compared with increasing the height of the air asset. For practical purposes, a big increase of is not preferable, but a small increase may result in an acceptable increase of the for shorter distances.
We then compare the endtoend LoS probability of direct groundground communication with airaided groundairground communication. When making this comparison, the air asset is always deployed above the midpoint of the two ground assets; i.e., . In this scenario, the probability to obtain LoS from the air asset to both ground assets synchronously is the square of . We assume that the air asset is flying at a height of m, while all ground assets have their communication devices fixed at a height of m. Both the truncated Gaussian and the uniform height distributions are considered. Fig. 3 shows the results of this comparison. For direct groundground communication, the probability of obtaining LoS decreases much faster as a function of distance than in the case of airaided groundairground communication. For the truncated Gaussian distribution, when = m and = m, (1) suggests that most of the obstacles will be taller than m. Thus, almost all obstacles can block the unobstructed view between a pair of ground assets, severely decreasing the probability of obtaining the LoS. For the uniform distribution, according to (2), there is a probability greater than that the heights of the obstacles are taller than . This causes a fast decrease in the for the groundground communication, which is similar to what is observed for the truncated Gaussian distribution.
When m, (i.e. m), the probability of obtaining LoS between ground assets using direct groundground communication is approximately 0.1 for both the truncated Gaussian and the uniform distributions. However, when an air asset is used, the probability that it obtains a LoS with both ground assets is approximately and times greater than the probability of the two ground assets obtaining LoS over a direct link considering the truncated Gaussian and uniform distributions, respectively. Fig. 3 shows that the choice of distribution does not have a significant impact on the probability of obtaining the LoS between ground assets using the direct link, since the communication devices of the ground assets are fixed at a relatively low height and therefore the LoS would be easily blocked by most obstacles. On the other hand, when an air asset is used, the height distribution has a bigger impact on the LoS probability since the differences of the distributions become more pronounced.
In addition, we computed the throughput performance for the same scenarios previously discussed. The additional parameters required to compute the throughput are a reference SNR of dB at a reference distance of meter, a pathloss coefficient of , and a bandwidth of MHz. This pathloss coefficient corresponds to the one reported in [22] for the measured LoS pathloss at GHz. The reference SNR is computed for a transmit power of dBm, a receiver noise figure of dB, and antenna gains of dBi for both the transmit and receive antennas, which are the gains reported for a compact 6element array operating at GHz in [21]. We consider the same obstacle models as before, with and height distributions that are either a truncated Gaussian (with and ) or a uniform (with ). The ground asset’s antenna height is set to m.
Fig. 4 shows throughput as a function of the height of the air asset, , for several different distances between ground assets, . The air asset is located at the midpoint between the two ground assets, i.e. , and this figure shows results for just the truncated Gaussian height distribution (results for the uniform distribution are similar). As expected, the throughput is higher when the ground assets are closer to each other. However, for each curve, a peak value can be observed. Lowering the altitude of the air asset below this peak makes it prone to blocking, but raising it above the peak value causes a loss in signal power which translates to a loss of capacity. The peak value balances the assets’ capability of obtaining LoS and the signal power, which is a key tradeoff as both contribute to the throughput. For equal to m, m, and m, the peak values are Mbps, Mbps, and Mbps, respectively, and these peaks occur at of m, m, and m, respectively.
Fig. 5 shows throughput as a function of the horizontal distance between the ground assets. The figure shows results for both truncated Gaussian and uniform height distributions and for both direct groundground communication and relayed groundairground communication. For groundairground communication, the throughput is optimized at each distance by maximizing its value over the height of the air asset . For direct groundground communication, no such optimization is possible. The plot shows that, for sufficiently far distances, the throughput of the groundairground communication is higher than that of the direct groundground communication. However, for shorter distances, groundground communication has a higher throughput. When the height distribution is a truncated Gaussian, this crossover occurs at a distance of m, where the throughput for both direct groundground and relayed groundairground communications is Mbps. The reason that direct groundground communications performs better at ranges closer than this crossover distance is primarily due to the need for the air asset to duplex the signal received from the first ground asset and transmitted to the second ground asset. The direct link does not need to duplex. However, at longer distances, maintaining a direct link between the two ground assets suffers from a lower probability of obtaining a LoS and a weaker signal power due to the long single transmission path.
Vi Conclusions and future work
In this paper, we studied issues related to deploying an air asset in a Poisson forest to aid the connection between a pair of ground assets. The key contribution is a framework for calculating the LoS probability and the throughput. This framework depends on carefully considering the locationdependent critical height, which is the minimum height required for an obstruction at that location to block the LoS. Because the critical height is distance dependent, the distribution of obstacles that are above the critical height is an inhomogeneous Poisson point process even if the location of the obstructions themselves is a homogeneous PPP. Closedform results are provided for two height distributions: truncated Gaussian and uniform. Simulation results validate the theoretical expressions.
The theory enables the solution to two particular problems. First, it allows the determination of a crossover distance, below which it is better for the ground assets to communicate directly, and above which airassistance is desirable. Second, it allows for the determination of the optimal height of the air asset when one is used. The key to solving these problems is to use throughput as the performance metric, as throughput can properly balance the tradeoff wherein higher altitudes increase the LoS probability but reduce signal power.
This work is a gateway to optimally deploying a heterogeneous team of mobile air assets and ground assets in a dense forest. Future works include planning for the optimal deployment of the air assets given a known or unknown layout of the ground assets, and planning the optimal trajectories for the joint team to navigate through the task space while staying connected and coordinating on task delivery.
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