On the Link Modeling of Static Wireless Sensor Networks in Ocean Environments

Transcription

1 On the Link Modeing of Static Wireess Sensor Networks in Ocean Environments Aireza Shahanaghi, Yaing Yang, and R. Michae Buehrer Department of Eectrica and Computer Engineering, Virginia Tech Emai: Abstract Despite of the advantages that ocean surface Wireess Sensor Networks WSN have over traditiona ocean monitoring methods, ocean WSN research suffers from ack of an accurate mode that describes the stabiity of wireess inks among sensor nodes. In this paper, we are going to investigate ocean surface waves effects on the Line-of-Sight LoS ink between sensors in a homogeneous WSN. Specificay, we wi derive the bocking probabiity of LoS inks between a transmitter and receiver pair due to wave movements, and anayze how environmenta effects ike wind speed affects it. Simuation resuts aong with oceanographic measurements vaidates our anayses, making our mode appicabe in design and panning of WSN in ocean environment. Index Terms Wireess Sensor Network, marine communications, Line-Of-Sight ink stabiity, maximum wave-height, ink stabiity I. INTRODUCTION Comparing to conventiona marine research vesses, ocean surface wireess sensor networks WSN have many advantages for ocean monitoring studies, incuding dramatic improvement in access to rea-time data for ong periods of time, arger geographica coverage of ocean areas, higher resoution monitoring of marine environments, more rapid processing of coected data and instantaneous transmission of data to monitoring centers in the shore [1]. Consisting of sensor nodes with simpe structures, ocean WSN can decrease the cost by at east one order of magnitude comparing to marine research vesse approach []. Despite of the advantages of ocean WSN, ocean WSN research suffers from ack of an accurate mode that describes the wireess inks among sensor nodes [3] [5]. Whie some existing iteratures have proposed accurate modes of the radio-wave propagation and refection over ocean surface [6], these modes assume either one or both transmitter and receiver antennas are much higher than the ocean surface eve, e.g., a vesse and an onshore base station. However, the heights of antennas of most sensors in ocean WSN are short. When the weather is rough, the height of ocean waves can exceed the antenna height. Since eectromagnetic waves cannot penetrate sat water, occurrence of waves higher than antenna height bocks the Line-of-Sight LoS ink between the transmit sensor and the receiver, causing disruption to wireess communications. Unfortunatey, existing iteratures have not taken into account of this kind of disruptions to wireess inks. Without proper ink-stabiity understanding, network protocos designed for ocean WSN are not robust to disruptions in bad weather, which resuts in decreasing network reiabiity [4]. To fi in this void, in this paper, we are going to present a mode that captures the effects of wave-height on the ine-ofsight LoS ink between the transmitter and receiver. Then, using the mode, we wi obtain the bocking probabiity of LoS ink and verify that with simuation resuts. Our work is the first that provides a reaistic and tractabe mode of ink stabiity under different weather conditions in ocean WSN. This resut can provide important guideines for network protoco designs in ocean WSN. Buiding an accurate mode to describe ocean waves impact on LoS wireess ink is a non-trivia task. The very few research papers in the area of ocean WSN that does not ignore the probabiity of ink bocking due to wave-height use an over-simpified ocean mode [7]. They assume ocean surface fuctuates according to a singe sine wave and ignore environmenta parameters ike wind speed which have an important effects on ocean surface dispacement. Rea ocean waves are much more compicated than a sine wave, making these papers resuts unreaistic. Whie statistica modes that accuratey describe the ocean surface behavior are avaiabe in oceanographic research, a of these modes focus ony on tempora behavior of ocean waves. Such tempora behavior modes are usefu in traditiona ocean and coasta engineering, which ony needs to consider the tempora characteristics of ocean surface at a specific site. However, in ocean WSN, the quaity of a wireess ink depends on the states of the entire ocean surface area between the transmitter and receiver, which can span many kiometers. Describing the LoS wireess inks for the ocean WSN, thus, requires a tractabe mode of ocean surface in spatia domain, which is acking in existing iteratures. To further compicate the situation, existing oceanographic works focus on deriving statistics that are usefu in ocean engineering fieds. Many statistics that are significant for communication systems cannot be found in oceanographic studies. For exampe, in order to understand statistica simiarities between different parts of a wireess ink, spatia autocorreation function behavior shoud be determined; however, oceanographic works mosty do not focus on auto-correation function and efforts in specific cases ike [8] [1] resuts in intractabe integras and compicated functions. In this paper, we address the chaenges and buid a tractabe and accurate mode for LoS ink in ocean WSN through

2 the foowing steps. First, we obtain the average wave-ength and derive a simpe mode for describing the spatia autocorreation behavior based on existing oceanographic modes. Using the mode of auto-correation, we define the coherence distance and divide the waves between transmitter and receiver into semi-independent profies using the coherence distance. Then, we derive the Probabiity Density Function PDF of oca maximum wave-height of each profie. Next, using the derived oca maximum PDF and concept of semi-independent wave profies, we obtain the PDF of goba maximum waveheight occurring between transmitter and receiver. Finay using the PDF of goba maximum wave-height, we obtain the probabiity of LoS bocking and vaidate that with simuation resuts. The rest of this paper is organized as foows. In Section II we wi have a brief review about ocean wave modeing and spectra that are widey accepted in oceanographic studies and used in ocean and coasta engineering. In Section III, we wi discuss spatia autocorreation function, and approximate its enveope with respect to the average waveength. Using the approximated enveope of spatia autocorreation, we wi obtain the PDF of oca and goba maximum wave-height in Section IV and use that for cacuating the bocking probabiity of LoS ink between a transmitter and receiver pair in a homogeneous WSN. Finay, in Section V, we wi evauate our mathematica anaysis regarding LoS ink bocking and vaidate them by simuation resuts. II. OCEAN WAVE MODELING In this section, we are going to have a brief review about inear wave theory, which is the foundation for our anaysis of wireess ink stabiity. Linear wave theory is an ocean wave mode that has been widey used in ocean and coasta engineering and has been shown to perfecty match with the spectra description of rea ocean waves in deep water, where the wave ampitude is sma comparing to the waveength and water depth. [11], [1] Specificay, inear wave theory modes the ocean surface dispacement as a inear summation of arge number of harmonic waves, each of which is propagated independenty without any effect on the others. Denoting x, t as the ocean surface dispacement in Cartesian coordination of x = x, y and time t, inear wave theory caims x, t = κ aκ cos κ x ωt + θκ, 1 where κ = κ 1, κ is caed wave-vector. Constants aκ and θκ are independent random variabes corresponding to ampitudes and phases of harmonic waves respectivey. ω is the anguar frequency, which reates to its corresponding wave-vector by dispersion equation ω = gκ tanhκd, where D represents the depth of ocean water, g 9.87m/s is the standard acceeration of gravity, and κ = κ 1 + κ is the Frobenius norm of κ. For the case of deep water waves, which is the focus of this paper, we have κd 1 and the dispersion equation can be simpified to ω = gκ. Theoretica anaysis using centra imit theorem as we as statistica anaysis of data sets obtained from wave-buoys and remote sensing measurements have a reveaed that x, t is a wide sense stationary WSS Gaussian random process of mean [13]. For WSS processes, the tota power P can be obtained by integration of power spectra density. Appying change of variabes and dispersion equation on this integration for ocean waves, the tota power is given by [11], [1] as : P = ω= ϑ= π Eω, ϑdωdϑ. 3 Eω, ϑ is caed the ocean waves directiona spectrum which represents the distribution of ocean wave energy in frequency ω and direction ϑ. Oceanographic investigations [11], [14] have reveaed that Eω, ϑ can be factorized as Eω, ϑ = SωΩϑ such that Ωϑdϑ = 1. Ωϑ is π caed directiona distribution and Sω is caed ocean wave spectrum. Setting the Cartesian coordinate system such that the x-axis is aong the wind direction, Ωϑ is an even function defined over π ϑ < π with a maximum at ϑ =. Sω is a singe-sided function defined on ω <. The exact forms of Ωϑ and Sω differ among different ocean wave modes. The expression of Sω is reated to wind speed, and Ωϑ is determined by wind direction. For detaied information on different directiona distributions and wave spectra, readers can refer to [14] and [15]. III. SPATIAL AUTOCORRELATION FUNCTION In Section II, we had a brief review on the state-of-art mode of ocean waves and spectra. As we discussed, ocean surface dispacement is a WSS zero mean Gaussian process. In genera, WSS Gaussian processes can be competey characterized by their mean and inear autocorreation function. In this section, we are going to investigate the autocorreation function in space domain, which then wi be used to anayze the LoS ink stabiity of ocean WSN in section IV. Providing an exact cosed form expression for the spatia autocorreation function of ocean wave is not mathematicay possibe. To cope with this chaenge, we wi use an approximation approach in this section. In essence, there is no need to have the exact expression of spatia autocorreation function for anayzing the LoS ink stabiity. In our foow-up anaysis of ink stabiity, we wi derive the enveope of the autocorreation function and use it as the approximation of autocorreation function. In Section III-A, we wi first identify the expression of the enveope of normaized autocorreation function in space domain based on oceanographic measurements in time domain [11], [16]. In Section III-B, we wi further show how to derive average waveength, which is the key parameter in this enveope expression.

3 A. Spatia Autocorreation Enveope At a given time t, the spatia autocorreation function of ocean surface is defined as: R δ x, δ y, t = E{x + δ x, y + δ y, tx, y, t}, 4 where E{.} represents the statistica expected vaue. R δ x, δ y, t describes the joint statistica behavior of two ocean surface positions whose distance is defined by vector δ x, δ y. In the remainder of this paper, for simpicity of notation, we wi use R δ x, δ y for representing the spatia autocorreation for a given fixed t. According to Wiener-Khinchin theorem [17], the autocorreation function of a WSS random process can be obtain by inverse Fourier transform of its power spectra density. Combining Wiener-Khinchin theorem and the concept of directiona spectrum, it can be shown that [8], [1] R δ x, δ y for deep water waves can be obtained by semi-inverse cosine Fourier transform of Eω, ϑ such that : R δ x, δ y = Eω, ϑ ω= ϑ= π ω cos g [cosϑδ x + sinϑδ y ] dωdϑ. 5 Since we are going to use the spatia autocorreation expression for describing the ink between a transmitter and receiver, we change Cartesian distance vector of δ x, δ y to the poar coordination, φ to faciitate the expression of transmitter-receiver distance. The x-axis is directed aong the wind direction downwind and y-axis is perpendicuar to wind direction crosswind. Without oss of generaity, we assume the origin of coordinate system is set to the transmitter ocation. Therefore, denotes the distance between transmitter and receiver, and φ is the ange from the positive side of x- axis such that δ x = cosφ and δ y = sinφ. Substituting Eω, ϑ = SωΩϑ, 5 is transformed to the foowing poar representation: R, φ = ω= SωΩϑ ω cos cosϑ φ dωdϑ. 6 g ϑ= π In the remainder of this paper, we assume that sensors are not moving horizontay such that the poar ange φ is fixed. Thus, we can treat φ in 6 as a constant and use R instead of R, φ to show that the spatia autocorreation is ony a function of. In genera, integras of 6 can not be simpified to a cosedform tractabe expression. Some existing efforts [8], [1] in the iterature tried to simpify these integras for specific cases of Sω and Ωϑ. But, their resuts are not generay appicabe for most ocean wave modes. In addition, these efforts mosty resuted in some series and specia functions which are too compicated for further anaysis, making them not usefu for deriving ink stabiity metrics. To dea with this chaenge, we note that for ink stabiity anaysis, it is not essentia to have the exact mathematica integration resut from 6. Thus, in this work, we are going to approximate the enveope of spatia autocorreation function, which is sufficient for anayzing the ink behavior. This approximation method can greaty simpify the issue of ink stabiity anaysis and can aso potentiay faciitate simiar future works. Our simuation resuts in Section V have shown that this approximation method produces very accurate resuts. In order to approximate the autocorreation function in space domain, we take a coser ook at the genera stochastic process of ocean surface dispacement expression in 1. Consider 1 in poar coordination, which yieds, φ, t = aκ 1, κ κ 1,κ cos [κ 1 cosφ + κ sinφ] ωt + θκ 1, κ, 7 where x = cosφ and y = sinφ. For a constant φ, by fixing one of or t in 7,, φ, t wi turn into a one dimensiona WSS zero-mean Gaussian process in time or space domain. Without oss of generaity, we can set and t to in 7 which yieds =, φ, t = aκ 1, κ cos ωt + θκ 1, κ 8 and, φ, t = = κ 1,κ κ 1,κ aκ 1, κ cos [κ 1 cosφ + κ sinφ] + θκ 1, κ, 9 as ocean surface dispacement in time and space domain. As can be seen, both of 8 and 9 are generated by summation of infinite number of independent harmonic waves each of which has random ampitude and phase corresponding to the wave-vector. This simiarity brings the tempora and spatia autocorreation about to have the same behavior. Oceanographic measurements in time domain revea that the enveope of normaized tempora autocorreation decays exponentiay with respect to the average period of ocean waves [11], [16] in the form of R tempora τ exp τ/t, where τ is the time difference, and T is the average period of ocean waves. Based on the stochastic simiarities between processes of ocean surface dispacement in time and space domains shown in 8 and 9, we can mode the enveope of normaized spatia auto-correation by an exponentia function of average spatia period of ocean waves as we. Fixing the poar ange in 6, we have R exp /λ, 1 where λ is the average wave ength i.e. the average spatia wave period, which we wi discuss how to compute it in the next subsection.

4 oca maximums eevation λ goba maximum Fig. 1: A simpe iustration of surface dispacement in space domain B. Average Waveength To define the average waveength of ocean waves, consider the ocean surface dispacement of 9 in space domain when φ is constant. With a sight abuse of notation, we use for demonstrating ocean surface dispacement in this case. Fig. 1 depicts a simpe iustration of fuctuating around zero eve since it is a zero-mean stochastic process. As can be seen, there exists two kinds of zero-crossing points in the figure; upward zero-crossing point after which the surface eevation is above the zero mean eve, and downward zerocrossing after which the surface eevation goes beneath the zero eve. Using the concept of zero-crossing, waveength λ can be defined as the distance between two successive upward downward zero-crossing points. In order to obtain the average waveength, we are going to derive the average number of zero-crossings first. Let a be the set of a random ampitudes of harmonic waves, and θ be the set of a random phases in 1. can be interpreted as a random curve which is a function of stochastic sets a and θ, and a deterministic variabe, i.e., F a, θ;. Let be the derivative of with respect to, i.e., = F a, θ; /. We define and as sampes of and in a certain respectivey. is a WSS zeromean Gaussian process which necessitates to be a WSS zero-mean Gaussian process. Moreover, and wi be jointy WSS Gaussian processes [17]. Therefore, and are zero-mean jointy Gaussian random variabes described by bivariate norma distribution function of f, ; [17]. From [18], it can be shown that the average number of zerocrossings N z of a random curve over a given interva 1, can be obtained as N z = = 1 = f, ; d d. 11 To compute the above integration, note that and are jointy Gaussian random variabes. Hence, f, 1 ; can be jointy described by their first order mean and second order variance and correation coefficient statistics. As it has been mentioned before, and are both zero mean. Next, we wi show how to compute their second order statistics. Define the rth moment of wave spectrum Sω as mr := ω r Sωdω. 1 Variance of σ can be obtained by substituting = in 6 which yieds σ = ω= ϑ= π SωΩϑdωdϑ = m. 13 In order to obtain the variance of, first note that for WSS stochastic processes, the autocorreation function of, which is denoted as R, φ, can be computed as foows [17]: R, φ = R, φ. 14 Considering 6 and 1, variance of σ can be obtained by substituting = in 14, which yieds σ = ω= = m4 g ϑ= π π ω 4 g Sω cos ϑ φωϑdωdϑ cos ϑ φωϑdϑ. 15 In order for computing the correation coefficient of and, the distribution of random sets a and θ shoud be known. One of the typica assumptions usuay made for WSS stochastic processes of form 9 is that each harmonic has a phase uniformy distributed over [ π, π, which is vaid for ocean waves [19]. This assumption is a sufficient condition for and to be uncorreated and consequenty independent since both of these random variabes are Gaussian. Therefore, f, ; wi be equivaent to the mutipication of two norma PDFs corresponding to and, i.e., f, ; 1 = exp πσ σ σ + σ, 16 where σ and σ are given in 13 and 15 respectivey. As can be seen in 16, f, ; is not a function of. By Using 16 in 11, after some manipuations see Appendix I average number of zero-crossings over interva 1, wi be derived as m4 N z = 1 πg π cos ϑ φωϑdϑ m. 17 On average, haf of the N z zero-crossings is dedicated to upwards and haf to downwards over interva 1,. Thus, the average waveength can be obtained as λ = 1 N z / = πg m m4 π cos ϑ φωϑdϑ, 18 which can be interpreted as the average distance between two successive upward downward zero-crossings. Substituting 18 in 1, we can approximate the enveope of normaized autocorreation function in space domain. In Section V i.e. the evauation section, we wi compare the

5 spatia autocorreation function numericay cacuated from 6 and the exponentia enveope approximation expressed by 1 and 18. We wi see the exponentia approximation with respect to the average wave-ength can propery describe the behavior of autocorreation enveope. IV. MAXIMUM WAVE-HEIGHT BLOCKING In this section, we are going to investigate bocking probabiity of the ink between a transmitter and a receiver ocated distance L away from each other. We wi use the same poar coordinate setup we defined in Section III. Hence, transmitter is ocated at the origin and poar coordination of receiver wi be L, φ, where φ is the ange from the positive side of x- axis. Consider a homogeneous ocean WSN, where antenna heights of the transmitter and receiver are the same. According to [4], in such a network, an LoS ink is bocked if there exists at east one ocean wave between the transmitter and receiver pair whose height exceeds a certain threshod H th. In other words, a ink is bocked if the maximum wave-height between transmitter and receiver exceeds the threshod H th, where the threshod H th is determined by the transmitter and receiver s antenna height. Hence, the bocking probabiity of the LoS ink is equivaent to the probabiity that the maximum waveheight is arger than H th. Deriving the above probabiity is not easy since obtaining the goba maximum of a curve is not a simpe probem. It is even more compicated if the curve is random. To find the goba maximum of a curve, one shoud first determine its oca maxima and then compute the goba maximum as the maximum of oca maxima. Using this idea, we wi first obtain the PDF of oca maximum and then derive the PDF of goba maximum based on the PDF of oca maxima. Specificay, we wi divide the distance between transmitter and receiver into semi-independent wave profies, each of which has their own oca maximum. Then, assuming identica and independent distribution i.i.d. for each profie s maximum, the goba maximum PDF can be derived. Existence of one oca maximum in each wave profie and these oca maxima are independent are two of the approximations we made in order to compute the goba maximum PDF. These approximations are necessary since finding the exact PDF of maximum of random variabes is generay not soube. In Section V, we wi compare the statistics of approximated maximum wave-height with simuation resuts, which wi show this approximation does not introduce significant errors. In the remainder of this section, we are going to first drive PDF of the oca maximum height of ocean surface dispacement in Section IV-A and then derive the goba maximum PDF in Section IV-B. A. Loca Maximum of Wave-height In this section, we are going to derive PDF of the oca maximum wave-height in ocean. Loca maxima of ocean surface dispacement occur in the points at which the first derivative of curve is zero whie the second derivative is negative. As can be seen in Fig. 1, oca maximum is not necessariy a positive vaue. Consider the spatia surface dispacement expression in poar coordinate system in 9. Fixing the poar ange φ, in Section III, we defined and as the ocean surface random curve and its derivative with respect to distance, respectivey. and denoted spatia sampes of and at a certain. Simiary, now et us define as the second order derivative of with respect to, and as the sampe of at a certain. Since is a WSS zero-mean Gaussian process, wi be a WSS zeromean Gaussian process and moreover,, and wi be jointy WSS Gaussian processes [17]. Therefore,, and are jointy zero-mean Gaussian random variabes that can be described by a mutivariate norma distribution function f,, [17]. In [], it is shown that the PDF of oca maxima f max m for the random curve can be obtained as f max m = f m,, d. 19 f,, d d Since,, and are jointy zero-mean Gaussian random variabes, f,, has the genera form of f,, 1 = exp 1 π 3 Σ 1 T Σ 1, where = [,, ]T, and Σ = E{ T } is the covariance matrix. T denotes transpose and. denotes the absoute determinant. Σ is a 3 3 Hermitian matrix with eements corresponding to variances and covariances. In order to obtain Σ s eements, we shoud obtain auto/cross-correation functions first. In III-B, we obtained autocorreation functions of and in 6 and 14 respectivey, which eaded to find variances σ in 13 and σ in 15. As it has been mentioned, having i.i.d. set of harmonics phases θ uniformy distributed over [ π, π is the sufficient condition for and to be uncorreated. We wi go through the same procedure for finding statistics of as we. Based on probabiity theory about WSS stochastic process [17], the autocorreation function of for a constant φ, denoted as R, φ, can be computed as: R, φ = 4 4 R, φ. 1 Considering 6 and 1, variance of denoted as σ can be obtain by substituting = in 1 which yieds σ = ω= = m8 g 4 ϑ= π π ω 8 g 4 Sω cos4 ϑ φωϑdωdϑ cos 4 ϑ φωϑdϑ. Simiar to what we had for and, i.i.d. uniform phase assumption in harmonic waves makes and uncorreated, which impies that and have zero correation. However, and are correated. It can be shown that for

6 two jointy WSS stochastic processes and, the crosscorreation function R, φ can be obtained as [17] R, φ = R, φ, where φ is a constant phase. Covariance of and wi be obtain by choosing = in which wi be equa to σ. Having a second order statistics of,, and, covariance matrix Σ wi be given as σ σ Σ = σ σ σ. 3 By substituting 3 in, after some manipuations see Appendix II.A, 19 wi give the PDF of oca maximum as f max m = exp 1 m πm m + 1 m m exp 1 m m 1 m Φ, < m < 4 m where = 1 σ 4 /σ σ, and Φ. denotes the Cumuative Distribution Function CDF of the standard norma distribution, i.e., Φx = 1 x π exp α / dα. Using Cauchy-Schwarz inequaity [17], it can be shown < 1. CDF of the oca maximum is derived as Fmax m = m f max αdα, which after some manipuations See Appendix II.B yieds m F max m = Φ m 1 exp m 1 Φ m. 5 m m In the next section, we wi use the obtained PDF and CDF of oca maximum of ocean dispacement curve for deriving the PDF of goba maximum wave-height between transmitter and receiver. Then, we can use the goba maximum PDF for cacuating the ink bocking probabiity. B. Goba Maximum of Wave-height In this section, we are going to derive the PDF of goba maximum wave-height between transmitter and receiver, and use that for cacuating the probabiity of a ink to be bocked. In a random curve, goba maximum can be obtained by finding the maximum of a oca maxima. In genera, the oca maximum heights may not be independent, and finding the maximum among dependent random variabes is generay mathematicay insoube. To resove this chaenge, we wi divide the waves between the transmitter and receiver into semi-independent profies. Assuming each profie has ony one oca maximum whose distribution is described by 4 and 5, we can obtain the approximated PDF of goba maximum wave-height between a pair of transmitter and receiver. As it has been mentioned before, is a WSS zeromean Gaussian process each of its sampes on is a zeromean Gaussian random variabe. For this case, zero correation is equivaent to independence of Gaussian sampes. Considering the spatia autocorreation approximation of 1, zero correation wi not happen uness tends to infinity. Therefore, simiar to approaches usuay used in wireess communication context [1], we define coherence distance c as the distance within which waves behave statisticay simiar. In other word, c corresponds to the distance inside which normaized correation is cose to 1. Therefore, those ocean waves whose distance from each other is farther than c are weaky correated and we consider them semi-independent. In this paper, we choose the concept of 3-dB decay for defining the coherence distance and in Section V, we wi show goba maximum statistics can be very accuratey described using this definition. Specificay using the exponentia approximation of normaized autocorreation enveope in 1, 3-dB decay occurs when the enveope reaches to / of its maximum. Note that og 1 / = 3dB. Therefore, the coherence distance can be obtained as c = og e λ.3466λ, 6 where og e. is natura ogarithm and λ is the average waveength obtained from 18. Using the concept of coherence distance, we divide waves between the transmitter and receiver into N = L/ c semiindependent profies, assuming ony one oca maximum is ocated in each profie and these oca maxima of the semiindependent profies are independent of each other. Let Ξ max denotes the goba maximum within the distance L. For N independent random variabes corresponding to oca maxima, probabiity theory [17] says that PDF of Ξ max can be computed as: f max m = N f max m F max m N 1, 7 where f max m and F max m are obtained in 4 and 5 respectivey. For goba maximum CDF we have F max m = Pr{Ξ max m } = F max m N. 8 From now on, when we use maximum wave-height, we refer to goba maximum of wave-height uness we mention oca maximum expicity. As we discussed before, a ink is bocked if the maximum wave-height exceeds a certain threshod H th. Using 8, we have P b = Pr{Ξ max > H th } = 1 F max H th, where P b denotes probabiity of bocking. V. EVALUATION In this section, we are going to evauate our mathematica anaysis resuts regarding LoS ink bocking caused by ocean

7 normaized spatia auto-correation waves. First, we demonstrate our simuation setup in Subsection V-A. Then we compare the exponentia approximation of autocorreation enveope we obtained in 1 with exact numerica resuts of 6 in Subsection V-B. Subsection V-C is dedicated for vaidating the approximations we made for obtaining the maximum wave-height and investigating the effect of wind speed in our anaytica resuts. Finay in Subsection V-D we investigate the bocking probabiity of LoS ink for different wind speeds. A. Simuation Setup In our simuations, we chose the genera mode of wave spectrum proposed by oceanographers which has the form of [11], [15] Sω = Aω p exp Bω q, 9 where A, B, p and q are constants that may differ between different wave modes. In our simuation evauation, we used Neumann spectrum, which is a we-known wave spectra widey used for theoretica anaysis of wind generated waves. This wave spectrum sets A = 3.5m /s 5 π, B = g /Uw, p = 6, and q = in 9 []. U w denotes the surface wind speed in ocean. For Sω of form 9, it can be shown that the rth moment can be expressed as [11] mr = ABr p+1/q p r 1 Γ q q, 3 where Γ. is the gamma function defined as Γx = α x 1 e α dα. For directiona distribution, we chose the genera cosinepower mode widey used for anaytica purposes which has the form of [14] Ωϑ = Γs + 1 πγs + 1 coss ϑ where s is a parameter reated to the anguar frequency which is out of the context of this paper to discuss about. One can find more detais about this mode in [14] and references therein. We set s = in our simuations. As it was discussed in Section IV, we used a poar coordinate system whose x-axis is set to the downwind and y-axis is crosswind. In this system, transmitter is ocated at the origin and receiver has the coordination of L, φ. We set φ equa to π/4 in our simuations. Our simuation is done in MATLAB wherein we created a script fie to generate instantiations of ocean waves using the poar mode of 9. B. Accuracy of the Enveop Approximation of Autocorreation Function In our first set of simuation, we set wind speed U w equa to 5m/s. The purpose of this simuation set is to vaidate the enveope of the spatia autocorreation function of ocean wave that we have derived in 1. Fig. compares the exponentia enveope approximation of autocorreation function in 1, which is the red ine, with the actua vaue of the autocorreation functions under different spatia distance. Two ines of the autocorreation functions are simuation theory enveope approximation distance[m] Fig. : Normaized spatia autocorreation function and its enveope approximation Fig. 1. Your TEXtified figure shown. The green ine I. isintroduction obtained from simuated ocean wave instantiations and the back ink is numericay computed from 6. It can be seen that this enveope approximation is not tight for short distances. However, it provides a conservative and easiy computabe estimation of the coherence distance c. Using this simpe expression, we can obtained the number of semi-independent wave profies and consequenty maximum II. CONCLUSION wave-height PDF and CDF, which our foowing simuations show to match accuratey with the simuation resuts. figureadjuster is a MATLAB program provided in order to modify a.fig fie in terms of font weights, font sizes, font names,and other objects of a reguar figure. After a changes, you can TEXtify your picture and find whether it fufis your expectations in the pdf format or not. Just watch your modified figure. If it is not acceptabe, just run figureadjuster again and enjoy : C. Accuracy of the maximum wave-height approximation Best Wishes Shahan As it has been mentioned in Section IV, our ink bocking probabiity is estimated based on an approximated estimation of maximum wave-height Aireza distribution. Shahanaghi For vaidating the PDF Schoo and CDF of Eectrica of maximumand wave-height Computerand Engineering, having a sense University about its behavior when of Tehran, other Tehran, parameters Iran, change, 14174in our second set of simuations, Emai:a.shahanaghi@ut.ac.ir we investigated the expected vaue of maximum wave-heightnowroz in two different 1396 scenarios. The first scenario examines the average vaue of maximum wave-height when the distance between transmitter and receiver changes. Fig. 3 depicts the average vaue of maximum wave-height when L varies from 35m to.7km whie U w = 3, 4, 5 and 6 m/s. As can be seen, the theoretica approximated resuts match perfecty with the acuta maximum wave-height obtained in ocean wave simuations. It can be observed that the average maximum wave-height gets arger as the distance between transmitter and receiver increases. This is because the farther receiver is ocated from the transmitter, the more probabe occurrence of arger waves in between. Another important observation is that by increasing the distance between transmitter and receiver, average maximum wave-height tends to a saturated vaue since the energy of waves is constant. In the second scenario, we set the distance between transmitter and receiver to L = 4m and swept the wind speed from U w = 1m/s to U w = m/s and obtained average maximum wave-height accordingy. Fig. 4 compares the theoretica resuts with simuation, which match very we. As can be seen, increasing the wind speed is equivaent to have more power in ocean waves, which resuts in arger average

8 average max wave height[m] average max wave-height[m] soid-ine: theoretica approximation markers: simuation resuts U w = 6 m/s U w = 5 m/s U w = 4 m/s U w = 3 m/s distance[m] Fig. 3: Average maximum wave-height for different distances between transmitter and receiver P b theoretica approximation U w = 3m/s simuation U w = 3m/s theoretica approximation U w = 4m/s simuation U w = 4m/s theoretica approximation U w = 5m/s simuation U w = 5m/s theoretica approximation U w = 6m/s simuation U w = 6m/s H th [m] Fig. 5: Link bock probabiity in different wind speeds simuation theoretica approximation wind speed[m/s] Fig. 4: Average maximum wave-height for different wind speeds maximum wave-heights. In order to vaidate our simuation and theoretica approximation with reaity, we compared our resuts with significant wave-height 1 measurements in [, Fig. 8.4-]. In these measurements, it has been shown that by increasing the wind speed from U w = m/s to U w = m/s, significant wave-height starts from m and exponentiay grows to 1.5m. Therefore, the vaues of average maximum wave-height and its semi-exponentia behavior we obtained in Fig. 4 grows in the same scae trend as rea-word measurement and can be appied for rea ocean waves. D. Accuracy of ink bocking probabiity After vaidating the PDF of maximum wave-height, for the fina step, we are going to evauate our mode of the LoS ink bocking probabiity. Sweeping H th, Fig. 5 depicts pots of P b for U w = 3, 4, 5, and 6 m/s. It can be seen that simuation resuts have fairy accurate matching with our theoretica approximation of maximum wave-height bocking. As it was shown in Fig. 4, by increasing the wind speed, average maximum wave-height wi grow. Thus, for faster wind speeds, it is more probabe to have higher waves, which resuts in higher probabiity of bocking and shifting P b to the right. 1 In oceanography, significant wave-height is defined as the mean of the highest one-third of waves [] VI. CONCLUSION Link stabiity is an important network design consideration for ocean surface WSN. In this paper, we present a thorough anaysis on the ink stabiity for ocean surface communications based on reaistic ocean surface mode and derive a highy accurate cosed-form expression for the LoS ink s bocking probabiity. The accuracy of the expression is vaidated through extensive simuation under different ocean weather conditions. Our future work wi focus on how to everage this ink stabiity mode in the panning and design of ocean WSN. APPENDIX I AVERAGE NUMBER OF ZERO-CROSSINGS In this section, we are going to derive the average number of zero-crossings in 17. As it has been mentioned in Section III-B, the joint PDF of f, ; in 16 is not a function of. Hence, substituting f, ; in 11 we have N z = 1 = πσ σ = 1 πσ σ exp σ d exp σ d = 1 σ, σ which yieds 17 by substituting σ and σ with 13 and 15 respectivey. APPENDIX II PDF AND CDF OF LOCAL MAXIMUM OF OCEAN SURFACE A. PDF DISPLACEMENT In this subsection, we are going to derive the PDF of oca maximum of wave-height in 4. For the first step, we shoud obtain the expression corresponding to the mutivariate norma

9 distribution of. Substituting the covariance matrix of 3 in we have f,, 1 = π 3/ σ exp σ + σ + σ where = σ σ 31 σ 4 Having f.,,, we are be abe to use 19 for obtaining f max m. Substituting 31 in 19, the numerator becomes f m,, d [ σ πσ = πm exp + 1 m and denumerator becomes m exp Φ 1 m m m 1 m 1 ] m m f,, d d = π σ σ Finay, by substituting 3 and 33 in 19, fmax m wi be given in form of 4. B. CDF After deriving the PDF of oca maximum in 4, CDF can be obtained as F max m = = m m m + f max αdα πm exp 1 α m α 1 m exp 1 Φ In 34, I 1 wi be computed as Φ dα I 1 α 1 m α m dα I. 34 m m. For cacuating I, we can use the integration by part rue. Specificay, by choosing du = Φ 1 α, we have m I = 1 exp + 1 m α m exp α m dα and v = m m exp α 1 Φ m exp m m 1 1 π m α 1 m dα. 35 After some manipuations, integra of 35 wi be simpified as 1 Φ. Hence, the CDF of 5 wi be given m m by F max m = I 1 + I. REFERENCES [1] G. Xu, W. Shen, and X. Wang, Appications of wireess sensor networks in marine environment monitoring: A survey, Sensors, vo. 14, no. 9, pp , 14. [] J. Tateson, C. Roadknight, A. Gonzaez, T. Khan, S. Fitz, I. Henning, N. Boyd, C. Vincent, and I. Marsha, Rea word issues in depoying a wireess sensor network for oceanography, REALWSN 5, 5. [3] M. Jiang, Z. Guo, F. Hong, Y. Ma, and H. Luo, Oceansense: A practica wireess sensor network on the surface of the sea, in Pervasive Computing and Communications, 9. PerCom 9. IEEE Internationa Conference on. IEEE, 9, pp [4] D. C. Steere, A. Baptista, D. McNamee, C. Pu, and J. Wapoe, Research chaenges in environmenta observation and forecasting systems, in Proceedings of the 6th annua internationa conference on Mobie computing and networking. ACM,, pp [5] S. M. Genn, T. D. Dickey, B. Parker, and W. Boicourt, Long-term rea-time coasta ocean observation networks, Oceanography, vo. 13, no. 1, pp. 4 34,. [6] L. Yee Hui, F. Dong, and Y. S. Meng, Near sea-surface mobie radiowave propagation at 5 ghz: measurements and modeing, Radioengineering, vo. 3, no. 3, p. 85, 14. [7] T. Yapiciogu and S. Oktug, Effect of wave height to connectivity and coverage in sea surface wireess sensor networks, in Computer and Information Sciences, 9. ISCIS 9. 4th Internationa Symposium on. IEEE, 9, pp [8] J. G. de Boer, On the correation functions in time and space of windgenerated ocean waves, SACLANT ASW RESEARCH CENTRE LA SPEZIA ITALY, Tech. Rep., [9] G. Latta and J. Baiie, On the autocorreation functions of wind generated ocean waves, Zeitschrift für angewandte Mathematik und Physik ZAMP, vo. 19, no. 4, pp , [1] C. Bourier, J. Saiard, and G. Berginc, Intrinsic infrared radiation of the sea surface, Journa of eectromagnetic waves and appications, vo. 14, no. 4, pp ,. [11] S. R. Masse, Ocean surface waves: their physics and prediction. Word scientific, 13, vo. 36. [1] L. H. Hothuijsen, Waves in oceanic and coasta waters. Cambridge university press, 1. [13] J. Tessendorf et a., Simuating ocean water, Simuating nature: reaistic and interactive techniques. SIGGRAPH, vo. 1, no., p. 5, 1. [14] S. F. Barstow, J.-R. Bidot, S. Caires, M. A. Donean, W. M. Drennan, H. Dupuis, H. C. Graber, J. J. Green, O. Gronie, C. Guérin et a., Measuring and anaysing the directiona spectrum of ocean waves, 5. [15] S. K. Chakrabarti, Hydrodynamics of offshore structures. WIT press, [16] P. Rudnick, Correograms for pacific ocean waves, UNIVERSITY OF CALIFORNIA MARINE PHYSICAL LABORATORY San Diego United States, Tech. Rep., [17] A. Papouis and S. U. Piai, Probabiity, random variabes, and stochastic processes. Tata McGraw-Hi Education,.

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