UAV-Assisted Maritime Legitimate Surveillance: Joint Trajectory Design and Power Allocation

This paper investigates a novel maritime wireless surveillance scenario, where a legitimate monitor vessel moves around to eavesdrop the suspicious communication from a suspicious unmanned aerial vehicle (UAV) to a suspicious vessel with the help of a cooperative UAV. Specifically, the cooperative UAV can adjust its jamming power and trajectory to exactly control the transmission rate of the suspicious link, thus improving the monitor vessel's surveillance performance. Furthermore, since the cooperative UAV cannot land or replenish energy on the sea surface, its jamming power allocation on the ocean should be carefully designed by the energy thresholds. Under such setup, we formulate a sum eavesdropping rate maximization problem, which jointly optimizes the jamming power and three-dimensional (3D) trajectory of the cooperative UAV, as well as the two-dimensional (2D) trajectory of the monitor vessel. To address this non-convex optimization problem, we decompose the design problem into three subproblems and propose an iterative algorithm to find its suboptimal solution. Numerical results show that the proposed jamming-assisted 3D joint design can significantly improve the eavesdropping rate and save the jamming power compared to the benchmark schemes.

the monitor can simultaneously eavesdrop the suspicious links and purposely send jamming signals.However in practice, the self interference cancellation is challenging to achieve as perfect.Such jamming-assisted proactive eavesdropping was then extended to a half-duplex parallel fading channels in [3].In [4], the multi-antenna techniques were applied at the monitor to achieve more efficient jamming.Moreover, in [5], the authors proposed a spoofing relay-assisted proactive eavesdropping strategy to overhear the suspicious link.
In contrast to the aforementioned works, e.g., [1], [2], [3], [4], [5], mainly focus on the ground surveillance scenarios, the unmanned aerial vehicles (UAVs)-assisted legitimate surveillance has recently drawn significant attentions due to the agile maneuverability [6], [7], [8].In [9], the authors first investigated a wireless surveillance in a UAV-aided suspicious communication network.In [10], the authors considered a UAV-assisted proactive eavesdropping in a suspicious multi-relay system.This UAV-assisted proactive eavesdropping was then extended to multiple suspicious links in [11], where the sum eavesdropping rate was maximized by jointly designing the jamming power and UAV's position.In [12], the authors studied a wireless surveillance of UAV communication system, where a legitimate UAV tried to eavesdrop the transmission of suspicious UAVs.Furthermore, in [13], the authors considered a multi-UAV-enabled wireless surveillance system, where there were three UAVs collaboratively eavesdropped on suspicious wireless transmitters on the ground.
Recently, UAVs have been extended to maritime communication scenarios [14], [15], [16], which makes maritime safety more vulnerable to threats, since these maritime UAVs may be misused by the illegal authorities to commit crimes.However, the works in [1], [2], [3], [4], [5], [9], [10], [11], [12], [13] only considered the terrestrial cases, which may not be applicable to UAV-assisted maritime surveillance scenarios.This is mainly due to the fact that the vessels on the ocean, unlike the users in the terrestrial case with random distributions and static patterns, often have unique characteristics in terms of both distribution and mobility.Furthermore, since the UAV cannot land or replenish energy on the sea surface, its power allocation on the ocean should be carefully designed by the energy thresholds.Motivated by the aforementioned facts, in this paper we study a new maritime surveillance system, where a legitimate monitor vessel moves around to eavesdrop the suspicious communication from a suspicious UAV to a suspicious vessel with the help of an energy-constrained cooperative UAV.In particular, the mobility of both maritime vessels and UAVs are taken into account.Moreover, we adopt a practical UAV-vessel channel model that includes both large-scale and small-scale fading components.Our objective is to maximize the sum eavesdropping rate over all time slots by jointly optimizing the cooperative UAV three-dimensional (3D) trajectory and jamming power, and the monitor vessel two-dimensional (2D) trajectory under the mobility, security, and power-limited constraints.The main contributions are summarized as follows: 1) In the existing literature, this paper is the first attempt to investigate the proactive eavesdropping in the UAV-assisted maritime surveillance scenarios.Specifically, we consider a more practical UAV-vessel channel model and formulate an optimization problem by taking into account the mobility of both vessels and UAVs.2) In each iteration, we derive closed-form expressions for the jamming power, UAV and vessel trajectories.Different from the terrestrial proactive eavesdropping scheme, the proposed 0018-9545 © 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.maritime proactive jamming scheme cannot only covertly disturb the suspicious receivers with the channel qualities, but also effectively save the jamming power by the energy thresholds.3) For the energy-constrained UAV-assisted maritime surveillance scenario, the proposed jamming-assisted 3D joint design can significantly improve the eavesdropping rate and save the jamming power compared to the benchmark schemes in the literature.Notations: Non-boldface and boldface letters denote scalars and vectors, respectively.| • |, || • || and E(•) denote the absolute value, Euclidean norm, and statistical expectation, respectively.The distribution of a circular symmetric complex Gaussian vector with mean vector x and covariance matrix Σ is denoted by CN (x, Σ).

II. SYSTEM MODEL AND PROBLEM FORMULATION
A UAV-assisted maritime legitimate surveillance system is considered as shown in Fig. 1, where a monitor vessel B tries to overhear the suspicious communication from a suspicious UAV S to a suspicious vessel D with the help of a cooperative UAV U .In practice, the vessel D and UAV S on the ocean often follow a set of predefined sea lanes for safety [14], [15], [16].Specifically, since the cooperative UAV U cannot land or replenish energy on the sea surface, its jamming power allocation on the ocean should be carefully designed among all time slots.We consider a particular UAV's flight time T , which is discretized into N time slots with equal duration d t = T /N, where N {1, 2, . . ., N} denotes the set of slots.Furthermore, a 3D Cartesian coordinate system is considered, where the coordinates of the vessel B and UAV U can be denoted as ) represent the location of the suspicious vessel D and UAV S, respectively.Under the above setting, the mobility constraints of the vessel B and UAV U can be formulated as where ) represent the horizontal coordinate in time slot n, respectively.The symbols z min and z max denote the minimum and maximum flight altitudes of UAV U , respectively.The symbols V B , V U,h and V U,v represent the maximum horizontal and vertical speed of the vessel B and UAV U , respectively.Moreover, to ensure the secrecy of eavesdropping activities, we define the following security distance constraint: where In addition, we consider a more practical UAV-vessel channel model [14], [15], [16], which can be defined by where g ij and h ij [n] represent the large-scale and small-scale fading coefficients, respectively.The symbol φ ij ∈ CN (0, 1), and β 0 and K[n] denote the reference power gain and Rician factor, respectively.In general, the sea lanes of the vessels can be obtained from the Automatic Identification System (AIS) and thus the coefficients g ij and K[n] are available [14], [15], [16].Since the UAV U is powered by an energy-limited onboard battery and cannot replenish energy on the sea surface, the jamming power P U [n] needs to meet the following constraints: where E and P U max denote the total and the maximum jamming power, respectively.Based on the above setting, at time slot n ∈ N , the achievable average rate of the eavesdropping link and the suspicious link can be derived respectively as where P S [n] denotes the transmit power of the UAV S. The symbols N 0 denotes the noise power and λ represents the interference cancellation factor.Therefore, as defined in [9], [10], the effective eavesdropping rate can be expressed as Our objective is to maximize the sum eavesdropping rate over all time slots by jointly optimizing the jamming power and 3D trajectory of the UAV U , and the 2D trajectory of the vessel B, subject to the mobility, security, and power-limited constraints.This optimization problem is formulated as 1), ( 2), ( 4). ( Since the problem defined in (7) contains highly nonlinear objective functions (both numerator and denominator of R B [n] and R D [n] depend on the optimization variables {q B [n], q U [n], z U [n]}); it is challenging to obtain the global optimal solution.In the following section, we propose an iterative algorithm to circumvent this non-convexity issue.
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In the following, we propose an iterative algorithm to find an approximated optimal solution.

A. Optimization of the Vessel B Trajectory q B [n]
For given UAV U trajectory {q U [n], z U [n]} and transmit power P U [n], problem ( 9) is simplified as For the given R D [n], it is obvious that if we want the objective function in (10) to increase, the number of R[n] = 1 slots increases, which needs more slots satisfy R B [n] ≥ R D [n].Thus, we rewrite problem (10) into the following form: Problem ( 11) is challenging to be directly solved due to the smallscale fading h ij [n] constraints in the numerator and denominator of R B [n].Thus, we first derive the lower bound of R B [n] as follows: where Γ max represents the upper bound of small-scale fading between UAV U and vessel B, and R lb B [n] denotes the lower bound of R B [n].Then, by introducing a slack variable r n [n] = we have the first-order Taylor series expansions of as follows: where d SB fea [n] is the feasible solution at the (l − 1)th iteration.As a result, based on ( 12) and ( 13), problem (11) is approximately transformed into the following form: Note that problem ( 14) is a convex optimization problem, which can be optimally solved by the interior-point method [14], [15].

B. Optimization of the UAV U Trajectory {q U [n], z U [n]}
For given q B [n] and P U [n], problem ( 9) is recast as Based on 5(a) and 5(b), to guarantee the UAV U should be close to the suspicious vessel D with the given q B [n] and P U [n], which implies that the achievable rate R D [n] will decrease.Therefore, problem ( 15) is rewritten as follows: Similarly, we can derive the upper bound of R D [n] as follows: where Γ min represents the lower bound of small-scale fading between UAV U and vessel D, and R up D [n] denotes the upper bound of R D [n].Then, by substituting ( 17) into (16), problem (16) reduces to an equivalent form as It is obvious that the problem in (18) can be solved by standard convex optimization technique, such as CVX [17].

C. Optimization of the UAV U Transmit Power P U [n]
With the fixed variables q B [n] and {q U Algorithm 1: The Proposed Iterative Algorithm.
Input: (x 0 B , y 0 B ), (x 0 U , y 0 U , z 0 U ), P 0 U and E; 1: Set l max = 50, l = 0, γ = 10 −5 , R l 0 = 0, and R l f = 100; 2: while R l f > γ and l < l max do 3: Let l = l + 1 4: Calculate (x l B , y l B ) of ( 14) for given (x l−1 U , y l−1 U , z l−1 U ) and P l−1 U ; 5: Calculate (x l U , y l U , z l U ) of (18) based on (x l B , y l B ) and P l−1 U ; 6: Calculate P l U of (20) under given (x l B , y l B ) and To solve problem (19), we provide the following important theorem.Theorem 1: The optimal transmit power P U [n] of problem ( 19) can be derived as follows: where the UAV U should send the jamming signal with the power is strictly monotonically decreasing with respect to P U [n], and it can be observed that when , the optimal transmit power P U [n] can be derived as in (20).Next, if E is limited, the UAV U cannot send the optimal jamming power at every time slot n, which leads to R B [n] < R D [n] in some time slots.Thus, in this case, the jamming signal will not be sent for saving power to obtain more effective eavesdropping rate, i.e., we have P U [n] = 0.

D. Overall Algorithm
In this subsection, based on the results presented in the previous three subsections, we develop an iterative algorithm, which is summarized in Algorithm 1.In each iteration, the problems defined in ( 14), ( 18) and ( 19) are alternately solved by using the existing standard convex optimization techniques, and thus a suboptimal solution of problem (9) can be obtained by the proposed iterative algorithm in Algorithm 1; the relevant details can be found in [15].Furthermore, the whole complexity of the proposed algorithm in Algorithm 1 is O[N ite (N ) 3.5 ], where N ite and N denote the number of required iterations and time slots, respectively.
Fig. 2 depicts the trajectories of the UAVs and vessels onto the 3D plane during different time durations T , respectively.The symbols and represent the initial and final positions of the UAVs and vessels, respectively.From the simulation results illustrated in Fig. 2, when T = 20 s, both vessel B (dashed curves) and UAV U (solid curves) almost directly move (fly) to the final position.However, when T increases, the vessel B and UAV U first move quickly to the vessel D, then they follow the vessel D as long as possible.The reason is that the vessel B needs to follow the mobile vessel D to eavesdrop more information while UAV U moves closer to the vessel D to get the best position for jamming accordingly.
Fig. 3 illustrates the achievable eavesdropping rate versus different vessel speed V B and time slots N , respectively.As shown in in Fig. 3, the achieved eavesdropping rate first increases and then decreases as the time slots N increases for all considered values of V B .This is because the vessel B first comes closer to the vessel D, and then moves to the final location as the time slots N increases.Moreover, from Fig. 3, when the maximum vessel speed V B is large, i.e., V B = 15 m/s and V B = 18 m/s, the achieved maximum eavesdropping rate remains the same.This is because when V B is large enough, the vessel B cannot fully use the maximum speed to approach the vessel D due to the fact that there is a minimum safe distance constraint.Fig. 4 presents the jamming power of the UAV U versus different E thresholds and flight time N .As expected, the jamming power of the UAV U first decreases and then increases as the flight time N increases, regardless of the total jamming power E value.This is because when N is small or large, the UAV U is far from the vessel D, and thus a larger jamming power P U needs to be allocated to satisfy R In addition, it is worth noting that when E is not large enough, i.e., E = 200 mW and E = 100 mW, the UAV U reduces the jamming power P U to zero for a certain period of time.This is because when E is limited, the UAV U cannot send the enough jamming power to satisfy R B [n] ≥ R D [n] at every time slot n.Thus, for those time slots that far from the vessel D, the UAV U will not send jamming signal for saving power to obtain a more effective eavesdropping rate.Fig. 5 compares the performance of our proposed jamming-assisted 3D joint design (denoted as Proposed scheme) with other three benchmark schemes with N = 30, namely: 1) The 2D joint design scheme (denoted as 2D scheme), i.e., the UAV U flight altitude z U [n] = 70 m; 2) The time-slot sequence allocation scheme (denoted as TS scheme), i.e., the UAV U jamming power allocation in time slot sequence; 3) The fixed power design scheme (FP scheme), i.e., the UAV U jamming power P U [n] = E N mW, ∀n ∈ N .As shown in Fig. 5, when the total jamming power E is small, both TS and FP schemes cannot achieve the effective eavesdropping rate.This is because when E is limited, the UAV U still sends the jamming signal with the optimal power or the fixed power in some time slots that far from the vessel D, which leads to R B [n] < R D [n] on all time slots.Moreover, it is also observed that the Proposed scheme only requires less jamming power to achieve the same or better performance compared to the benchmark schemes.The reason is that the proposed algorithm can jointly optimize the jamming power and 3D trajectory of the UAV U , and thereby can save the jamming power based on the channel qualities.

V. CONCLUSION
This paper has investigated the legitimate surveillance in a UAVassisted maritime communication scenario.In the considered scheme, with the help of a cooperative UAV transmitting jamming noise, the jamming power and 3D trajectory of the cooperative UAV, as well as the 2D trajectory of the monitor vessel have been jointly optimized to maximize the sum eavesdropping rate under the mobility, security, and power-limited constraints.Simulation results have confirmed that the proposed jamming-assisted 3D joint design significantly improved the eavesdropping rate and saved the jamming power compared to the considered reference schemes.
D}, indicates the distance from i to j at slot n.The symbols d B min and d U min are the minimum safe distance.

Fig. 4 .
Fig. 4. Jamming power of the UAV U with different E thresholds.

Fig. 5 .
Fig. 5. Achieved effective eavesdropping rates of different algorithms versus E.