Acoustic localization with multi-layer isogradient sound speed profile using TDOA and FDOA*

Dongzhou ZHAN,Sitian WANG,Shougui CAI,Huarong ZHENG,Wen XU

1College of Information Science and Electronic Engineering,Zhejiang University,Hangzhou 310027,China

2Key Laboratory of Ocean Observation-Imaging Testbed of Zhejiang Province,Ocean College,Zhejiang University,Zhoushan 316021,China

1 Introduction

The emergence of underwater acoustic sensor networks(UASNs)has greatly improved the effi-ciency of marine environmental monitoring(Kong et al.,2005).In most UASN applications,precise node localization is fundamental and important.The node location information is the key to accomplishing other underwater tasks(Pompili and Akyildiz,2009;Fan et al.,2011).In location-dependent data acquisition tasks,the data is useful only when the location information is correct(Ferguson et al.,2005;Li et al.,2018).Because the speed of sound underwater varies with water depth,salinity,and temperature,acoustic localization methods based on the assumption of straight-line sound propagation have inherent biases.More accurate localization algorithms with realistic sound speed profiles(SSPs)need to be developed.

Various methods have been proposed to solve the underwater localization problem.Basically,these methods can be divided into two major classes,range-free and range-based methods(Erol-Kantarci et al.,2011;Han et al.,2012).Range-free methods localize the target node according to the connectivity and topology of UASN without the range and bearing information(Chen et al.,2018).The accuracy of the range-free localization methods is not high and is limited by the number of sensor nodes.Range-based localization methods include time of arrival(TOA)(Luo et al.,2018),time difference of arrival(TDOA)(Liang et al.,2013),angle of arrival(AOA)(Huang and Zheng,2018),frequency difference of arrival(FDOA)(Zhang et al.,2018),and received signal strength indicator(RSSI)(Sun et al.,2019).These methods first estimate the ranges or angles between the target node and the sensor nodes.Trilateration,triangulation,or multilateration methods are then used to localize the target node.Beaudeau et al.(2015)proposed a new RSSIbased multi-target tracking approach,and demonstrated the effectiveness of the approach in tracking a relatively large number of targets.However,the RSSI method performs terribly for long-range localization in an underwater environment.Huang and Zheng(2018)proposed a new multi-hop localization algorithm with AOA,which outperforms the conventional algorithms,even if the AOA measurement error is large.

However,accurate time synchronization is required between the sensors and the target,remaining a challenging task in UASNs.Although the TOA-based methods are simple to implement,a time synchronization problem exists.For unsynchronized UASNs,TDOA and FDOA are used commonly,especially for localization of moving sources(Ho et al.,2007;Tan et al.,2011).

All the range-based methods that use the straight-line propagation model ignore the soundray bending phenomenon.To mitigate the influence of the varying sound speed on localization,different sound models have been proposed.The effective sound velocity(ESV)model(Vincent and Hu,1997)was proposed to compensate for the error caused by the bending of sound ray.However,ESV works well only in the deep-sea environment.Ameer and Jacob(2010)proposed that when the target depth is known,the TOA between the target node and each sensor node can be transformed into a constant range surfaces using an SSP.The position of the target is then derived as the point whose sum of the squared distances from all these surfaces is minimum.Although this method has a high localization accuracy,the main drawback is the significant computational complexity.In the case of the isogradient SSP,Ramezani et al.(2013)used geometric relationships to establish path equations among the nodes.The computational complexity is acceptable,because it is analytic.However,this method cannot be applied to more complex SSP scenarios.

In most applications of underwater localization using the TDOA and FDOA methods,the localization problem is nonlinear(Ho and Xu,2004;Jiang et al.,2020).Therefore,various nonlinear methods have been proposed to localize the target node.Some of them are implemented iteratively based on maximum likelihood methods(Vankayalapati et al.,2014),such as the Gauss-Newton algorithm(GNA)(Doˇgan?ay and Hashemi-Sakhtsari,2005).Ho et al.(2007)introduced closed-form solutions that require low computation capability.Linearization methods can be implemented in localization.Ho and Xu(2004)employed several weighted least-square minimizations to transform the nonlinear problem into a linear one.Furthermore,closed-form solutions of the nonlinear measurement equations were constructed step by step in Jia et al.(2019).However,considering the multi-layer isogradient SSP,the scenario we discuss in this paper is relatively complicated,and cannot be solved using linear methods or easily transformed into a linear one.

In this study,we first propose an analytical multi-layer isogradient SSP model for the sound ray path between nodes.We then propose a method to calculate the gradient of the frequency shift of the arrival signals.On this basis,we propose an FDOAbased node localization algorithm,and a location and velocity joint estimation algorithm based on TDOA and FDOA.The proposed algorithm tracks the sound ray effectively.Results from the simulations prove the effectiveness of the proposed algorithms.More accurate and reliable node localization can be attained,compared with the straight-line propagation method.

2 Multi-layer isogradient sound speed profile model and sound ray tracking

In this section,the mathematical multi-layer isogradient SSP model is described in detail.

As shown in Fig.1,the green line represents the true SSP,and the red line represents the multi-layer isogradient SSP.The multi-layer model segments the true SSP into multiple layers with a piecewise linear function of water depth.Each layer has a constant gradient.There are a total ofPsegmented lines to approximate the true SSP(Fig.1).Then,the sound speedcp(z)in thepthsound ray layer can be expressed as

Fig.1 A true sound speed profile(SSP)and a multilayer isogradient SSP(References to color refer to the online version of this figure)

wherezis the water depth ranging fromzp-1tozp,apis the gradient,andbpis the horizontal-axis intercept.Bothapandbpare determined by the true SSP.As the number of layers becomes larger,the difference between the true SSP and the multilayer SSP decreases.

According to Snell’s law(Shirley,1951)and the geometric relationship of the sound ray,the starting pointSp=(xSp,ySp,zSp)Tand the ending pointEp=(xEp,yEp,zEp)Tin a single layer are related as(Cai,2019)wheredpis the horizontal distance between the starting and ending points of the sound ray,Lpis a constant determined by the characteristics of SSP,XpandYpare the defined auxiliary variables,αpis the angle between the actual acoustic ray path and the straight-line path,βpis the angle between the straight-line path and the horizontal direction,andθSpandθEpare the glancing angles at the starting and ending points of the sound ray,respectively.The anglesαp,βp,θSp,andθEpare illustrated directly in Fig.2.The description of linetype is the same as in Fig.1.The traveling time fromSptoEpis denoted astp.

Fig.2 Glancing angles of θEp and θSp+1(References to color refer to the online version of this figure)

Regardless of the reflection of the seabed and sea surface(Fig.2),the glancing angles at the ending point of thepthlayer and the starting point of the(p+1)thlayer are related as

Using Eqs.(3)-(11)and the properties of trigonometric functions,we can obtain

Eq.(12)is a high-order polynomial ofXp(ordp).The calculated value ofXpis substituted into Eqs.(5)-(9)to find theθSpandθEpof each layer,thereby realizing sound ray tracking.Therefore,with the multi-layer model,the sound ray tracking is equivalent to determining the roots of thep-1 polynomials.

3 Moving target node localization

This section introduces two algorithms in detail,i.e.,the FDOA localization algorithm and the joint estimation algorithm based on the TDOA(Cai,2019),and FDOA for localization and velocity estimation of a moving target node.To fulfill either of the two algorithms,we need to know the gradient of certain spatial positions first.

3.1 Frequency gradient calculation

Assume that the velocities of the target and sensor nodes are not zero,and define the target position asx=[x,y,z]T,the target velocity at positionxasvx=[vx,vy,vz]T,the sensor position asa=[xa,ya,za]T,and the sensor velocity asva=[vax,vay,vaz]T.With the multi-layer isogradient SSP model,the analytical expression of the frequency of arrival(FOA)from the transmitter to the receiver is

wheref1is the shifted signal frequency related to the movement of the transmitter andfDis the Doppler frequency shift produced by the movement of the receiver(Bogushevich,1999):

wheref0is the carrier frequency,csandcrare the speed of sound at the depths of the target and sensor nodes,respectively,andv⊥sandv⊥rare the radial velocities of the target and sensor nodes,respectively.Moreover,v⊥sandv⊥rare given by

DenoteAT＇as the line that connects the target’s projectionT＇and sensorA.φis the angle betweenAT＇and thex-axis,andvs=[vsx,vsy,vsz]Tandvr=[vrx,vry,vrz]Tare the velocity of the transmitter and receiver nodes,respectively(Fig.3).

Fig.3 Decomposition of speed vs

When the target is the transmitter,the partial derivative of FOA to the target position can be obtained from Eq.(13)as

where the partial derivatives are

wherevh=vxcosφ+vysinφ,andvah=vaxcosφ+vaysinφ(a refers to the sensor node related parameters).Similarly,the partial derivatives offwith respect to the target’sy-andz-axis coordinates can be obtained.

The partial derivative of FOA to the target velocity can also be obtained from Eq.(13)as

Similarly,the partial derivatives offwith respect to the target’s velocity components of theyandz-axis can be obtained.

3.2 Frequency difference of the arrival localization algorithm

Consider a three-dimensional UASN withNsensor nodes.The positions and velocities of all the sensor nodes are known based on the global positioning system(GPS).Meanwhile,the velocity of the target node is assumed to be known for the FDOA localization algorithm for four sensor nodes.Recall that the position of the target node isx;FOAs of the positioning signals received by the sensor node can then be modeled as

where the actual signal arrival frequencyg(·)=[g1(·),g2(·),...,gN(·)]Tis a function of the target positionx=[x,y,z]Trepresented by Eq.(13),ff=[ff1,ff2,...,ffN]Tis the measured FOAs between the target and the individual sensors,andN-dimensional column vectornis the measurement noise,obeying the Gaussian distribution with a mean value of zero and a variance of.The corresponding measurement noises of individual sensor nodes are independent of each other.

Taking the first sensor node as the reference node,and subtracting the arrival frequencies of the reference node from the arrival frequencies of the otherN-1 nodes,the FDOA measurement model is

whereΔf=[Δf21,Δf31,...,ΔfN1]T,g＇(·)=[g21＇(·),g31＇(·),...,gN1＇(·)]T,ζ=[ζ21,ζ31,...,ζN1]T,and

The covariance matrix of the noise vector can be obtained from Eq.(31)as

where diag(σ2f2,σ2f3,...,σ2fN)stands for a diagonal matrix,whose diagonal values areσ2f2,σ2f3,...,σ2fN.

Because it is assumed that the measurement noise is Gaussian,the maximum likelihood estimate of target positionxcan be derived as

Obviously,the solution represented by Eq.(33)is nonlinear,and GNA(Cai,2019)can be used to find the solution.The GNA method uses Taylor series approximation to minimize Eq.(33),through multiple iterations.Thekthiteration can be expressed as

whereJf=?g＇(x(k)),rf=g＇(x(k))-Δf,and

According to Eq.(18)and the solution methods proposed by Cai(2019),it can be obtained that(i=2,3,...,N).The pseudocode of this algorithm is listed in Algorithm 1.

Note that the target velocity is assumed to be known in the scenario of four sensor nodes for the FDOA localization algorithm.This assumption is actually not necessary if more(at least seven)sensor nodes are available,because both position and velocity information can be estimated from FDOAs.In other words,the considered four-sensor-node scenario cannot provide velocity information.Instead of using more sensor nodes,we propose a joint FDOA and TDOA method to estimate both the position and velocity information in the following subsections.

3.3 Joint estimation using both TDOA and FDOA

When the TDOA and FDOA measurements are combined to estimate the position and velocity of the moving target,the estimation can be achieved by increasing the dimension of the matrices mentioned above.The variable to be estimated isΦ=[x,y,z,vx,vy,vz]T,and GNA is again used to solve the problem.First,the TOAs of the positioning signals received by the sensor node can be modeled as

where the actual signal arrival timeh(·)=[h1(·),h2(·),...,hN(·)]Tis a function ofΦ,t=[t1,t2,...,tN]Tis the measured TOAs between the target and the individual sensors,andN-dimensional column vectorntis the measurement noise,obeying the Gaussian distribution with a mean value of zero and a variance of.The corresponding measurement noises of individual sensor nodes are independent of each other.Each item ofh(·)can be described in detail as

wherePis the number of the total layers through which the sound ray travels.

Taking the first sensor node as the reference node,and subtracting the arrival times of the reference node from otherN-1 nodes,the TDOA measurement model is

whereΔt=[Δt21,Δt31,...,ΔtN1]T,h＇(·)=[h21＇(·),h31＇(·),...,hN1＇(·)]T,ρ=[ρ21,ρ31,...,ρN1]T,and

The covariance matrix of the noise vector can be obtained from Eq.(43)as

Following the same procedure as mentioned in the FDOA localization algorithm,the maximum likelihood estimate ofΦcan be derived as and thekthiteration of the GNA method can be expressed as

where