Underwater passive manoeuvring target tracking with isogradient sound speed profile

IET Radar, Sonar & NavigationVolume 16, Issue 9 p. 1415-1433 ORIGINAL RESEARCHOpen Access Underwater passive manoeuvring target tracking with isogradient sound speed profile Jun Su, Jun Su orcid.org/0000-0001-9143-7467 School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, Shaanxi, ChinaSearch for more papers by this authorYaan Li, Corresponding Author Yaan Li [email protected] School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, Shaanxi, China Correspondence Yaan Li, School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China. Email: [email protected]Search for more papers by this authorWasiq Ali, Wasiq Ali School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, Shaanxi, ChinaSearch for more papers by this author Jun Su, Jun Su orcid.org/0000-0001-9143-7467 School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, Shaanxi, ChinaSearch for more papers by this authorYaan Li, Corresponding Author Yaan Li [email protected] School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, Shaanxi, China Correspondence Yaan Li, School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China. Email: [email protected]Search for more papers by this authorWasiq Ali, Wasiq Ali School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an, Shaanxi, ChinaSearch for more papers by this author First published: 10 May 2022 https://doi.org/10.1049/rsn2.12269AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Compared with the traditional tracking problem, the underwater 3-D passive manoeuvring target tracking faces the following challenging problems: (1) Since the signal propagation delay is varying and unknown, the relationship between the two adjacent target states is hard to describe; (2) limited by the underwater environment, it is more difficult to establish a target manoeuvring motion model; (3) the propagation speed of the underwater acoustic signal varies with the depth, and the propagation path will be curved instead of straight. Various parameters including signal propagation delay, target pitch angle and receiving frequency measured by the observer cannot be calculated as usual. In this paper, we improved a 3-D constant turn (3D-CT) model to match the underwater target motion. For the problems (1) and (3), this paper proposes a novel interacting multiple model underwater Gauss-Helmert unscented Kalman filter (IMM-UGHUKF) with isogradient sound velocity profile, which analytically describes the relationship between the state to be estimated and the passive measurement. This method augments the unknown signal emission time into the state with implicit equations, and transits the state by the Gauss–Helmert model. Simulation tests have been carried out, and the results show that IMM-UGHUKF can significantly reduce the estimation error compared with other approaches. 1 INTRODUCTION In recent years, underwater passive target tracking has become a frontier research field for its wide application and vigorous development in marine safety, underwater engineering operations, fishery fishing and other military and civilian uses [1-4]. The basic issue of passive target tracking is to estimate the target kinematic state [5]. The motion parameters of the target, such as location, speed and turn rate will be outputted by filters (such as unscented Kalman filter and particle filter) with the measurement data corrupted by noise [6-8]. Passive target tracking can generally be divided into angle-only target tracking and Doppler-angle target tracking according to the measurement category. The angle-only target tracking problem with a single observer will be affected by the system observability [9, 10]. The use of frequency measurement can extend the dimension of the measurement to improve the observability performance, which allows Doppler-angle tracking to be more reliable than angle-only tracking [11-13]. The research in this manuscript is focussed on the manoeuvring targets in an underwater 3-D environment. Compared with the traditional passive target tracking, it will face more challenging problems: The underwater acoustic signal propagation delay, the underwater 3-D target motion model, and the acoustic signal propagation path will be curved with the varying sound speed. Under ideal conditions, it is considered that the measurement data is transmitted to the sensor without any time delay, and the current measurement can directly correspond to the current state [14]. However, this assumption is unreasonable in a practical underwater environment [15, 16]. Signals such as light and electromagnetic waves decay quickly in water, and sound is the only signal can effectively reach to long distances. However, the underwater sound propagation speed is approximately equal to 1480 m/s, which is far lower than the previous two. The negligence of the signal time delay is equivalent to the target holding its position during the signal propagation until the observer receives the measurement data. This is obviously inconsistent with the facts which lead to the significantly increasing of the error between the filtering estimation and the target true state. In recent years, various algorithms have been proposed to handle passive target tracking with latency measurements. In Ref. [17, 18], the online parameter estimation method uses a linear search method to calculate the local optimal solution of the signal transmission time interval before filtering. This method may be computationally expensive. In Ref. [19, 20] directly calculates the signal propagation delay through implicit equations. These methods apply particle filter, however, each particle generates a different time delay, which will cause extra errors when calculating the estimate mean. In Ref. [21, 22], the dimension of the state is expanded by the signal emission time variable. The Gauss–Helmert model (GHM) is applied to handle the previous state and the current state occurred in the implicit state equation set simultaneously. The interacting multiple model (IMM) is a manoeuvring target tracking algorithm most commonly used in engineering [23-25]. It estimates the target motion state by several models in parallel, and the final estimate is the combined output of all model filters. For the 3-D manoeuvring target tracking, the first step of the IMM method is to select a model set that matches to the target motion. The 3D constant speed coordinated turn (3D-CSCT) model applied in Ref. [26, 27] which has a restriction that the target must always move in the same plane during the period of CT motion. Once the above conditions are not met, the filtering accuracy will be greatly degraded. In Ref. [28], the angular velocities in the x, y, and z axis directions are simultaneously treated as unknown variables of the state. This method equips high tracking accuracy performance; however, it needs more measurement data, including bearing angle, elevation angle, receiving frequency, the angular varying rate of the bearing angle and pitch angle. The 3D-CT model suggested by Ref. [29] divides the target's turn rate into two parts, the horizontal plane and the vertical plane turn rate. Compared with the method in Ref. [26], this method does not require the angle varying rate as the measurement data. The advantage over 3D-CSCT is that the target will not be limited to move in the same plane during the CT motion period. Under the influence of temperature, salinity, pressure and other factors, the underwater sound speed cannot be kept constant, and its propagation trajectory is not a straight line. This phenomenon will have a continuous and strong impact on the entire underwater target tracking process. The measurement of angle and frequency will be affected by the curved sound ray. The signal propagation delay is not simply proportional to the distance between the two points. Ref. [30-32] apply an isogradient sound speed profile (ISSP) model to describe the underwater varying sound propagation velocity. This model only relies on depth, which is suitable for deep sea areas where salinity and temperature are almost constant. In order to simultaneously estimate the unknown signal propagation delay and the horizontal and vertical turn rates, this paper incorporates them into the state as variables to be predicted, and establishes a discrete implicit constraint equation system to connect two adjacent states based on GHM. The time delay constraint is an important equation that reflects the relationship between the target state and the delayed measurement data. Literature [21] is mainly aimed at target tracking with acoustic signal delay in the air and on the water surface, while the discussions in this paper are mainly faced to the underwater environment, which involves the isogradient sound speed profile model and the underwater manoeuvre target motion model in three-dimensional space. Although the authors in Ref. [31] also studied the underwater signal propagation delay and the curved sound ray, it is mainly focussed on tracking the constant velocity motion target with double stationary observers, which solves the problem of the time offset between two observers caused by signal propagation delay. Different from Ref. [31], this paper is dedicated to tracking a manoeuvring target by a single observer. For the scene of 3-D underwater manoeuvring target tracking, we introduced the 3D-CT model in Ref. [29] to describe the target turn motion, and propose the modified 3D-CT model combined with the signal propagation delay problem. The effectiveness of the motion model is verified based on the isogradient sound speed profile model. In order to improve the tracking performance of the single observer, this paper further studies the isogradient sound velocity profile model raised in Ref. [30] and investigates the analytical relationship between frequency measurement and the target state with the curved underwater acoustic ray. We developed the IMM-UGHUKF method under the above conditions to recursively estimate the state of the manoeuvring target with delayed measurement. This method uses the Gauss–Newton numerical algorithm to find the local optimal solution of the implicit constraint equations, and proves that the Jacobian matrix is always non-singular in the underwater environment. The remaining parts are organised as follows. The second part addresses the system model, signal propagation delay model and target motion model. The third section discusses the isogradient sound speed profile model and related measurement equations. The novel IMM-UGHUKF algorithm is derived in Section 4. The simulation results and conclusions are in Section 5 and Section 6, respectively. 2 PROBLEM DESCRIPTION The unknown time delay of signal propagation For underwater tracking systems, the propagation speed of acoustic signals is much lower than that in the air, such as optical signals and electrical signals. This phenomenon will cause a certain delay in the propagation of underwater acoustic signals. Figure 1 illustrates the relationship between the signal propagation delay and other time parameters in the tracking process. t k e ${t}_{k}^{e}$ denotes the instant that the signal is emitted from the target. t k o ${t}_{k}^{o}$ represents the instant when the observer obtains the signal, and t k p ${t}_{k}^{p}$ is the elapsed time of signal propagation. FIGURE 1Open in figure viewerPowerPoint Illustration of signal propagation delay The relationship between these three time parameters can be described as t k e = t k o − t k p , ${t}_{k}^{e}={t}_{k}^{o}-{t}_{k}^{p},$ (1)where the index k represents a certain moment in the discrete system. For two adjacent moments, the time interval is defined as T k e = t k e − t k − 1 e , ${T}_{k}^{e}={t}_{k}^{e}-{t}_{k-1}^{e},$ (2) T = t k o − t k − 1 o , $T={t}_{k}^{o}-{t}_{k-1}^{o},$ (3) δ k = T k e − T , ${\delta }_{k}={T}_{k}^{e}-T,$ (4) t k p = d t k e / c p , ${t}_{k}^{p}\,=d\left({t}_{k}^{e}\right)/{c}_{p},$ (5)where T k e ${T}_{k}^{e}$ denotes the time interval of two adjacent signals originated from the target, T represents the observer sampling interval which is a fixed and known parameter, and δk is a varying offset of the above two intervals. d t k e $d\left({t}_{k}^{e}\right)$ denotes the distance from the target to the observer, cp is the underwater sound propagation speed. In Figure 1, the distance varies with time, which leads to the signal propagation delay cannot be kept as constant, and this phenomenon further generates an unknown offset between the interval T k e ${T}_{k}^{e}$ and the observer sampling interval T. System models For the traditional target tracking problem, the state transition equation of the system and measurement equation satisfy the GMM, in which the time interval between adjacent states is constant. The state transition equation and measurement equation for GMM are given by x t k = F t k t k − 1 x t k − 1 + Γ t k t k − 1 w t k − 1 , $\mathbf{x}\left({t}_{k}\right)=\mathbf{F}\left({t}_{k}\left\vert {t}_{k-1}\right.\right)\mathbf{x}\left({t}_{k-1}\right)+{\Gamma}\left(\,{t}_{k}\left\vert {t}_{k-1}\right.\right)\mathbf{w}\left({t}_{k-1}\right),$ (6) z t k = θ o t k ϕ o t k f o t k ′ = h x t k + v t k , $\mathbf{z}\left({t}_{k}\right)={\left[{\theta }_{o}\left({t}_{k}\right)\,\,{\phi }_{o}\left({t}_{k}\right)\,\,{f}_{o}\left({t}_{k}\right)\right]}^{\prime }=h\left[\mathbf{x}\left({t}_{k}\right)\right]+\mathbf{v}\left({t}_{k}\right),$ (7)where x(tk) denotes the target motion state at the instant tk. z(tk) is the measurement vector. In this paper, it includes the horizontal bearing angle measurement θo(tk), the vertical pitch angle measurement ϕo(tk), and the frequency measurement fo(tk). The specific form for these measurements will be derived in section 3. F(tk|tk−1) and Γ(tk|tk−1) are the state transition matrix and system noise transition matrix, respectively. w(tk−1) represents the zero-mean Gaussian white noise for the process noise. Similarly, v(tk) is the measurement noise. h(⋅) denotes the nonlinear measurement function. Note that the time tk of the state equation and the measurement equation in the GMM are the same, and satisfy with following equation t k − t k − 1 = T . ${t}_{k}-{t}_{k-1}=T.$ (8) However, the above model does not match the tracking scenario of signal propagation delay. In equation (2), the state time interval T k e ${T}_{k}^{e}$ is an unknown varying parameter due to the adjacent instants t k e ${t}_{k}^{e}$ and t k − 1 e ${t}_{k-1}^{e}$ are also unknown parameters related to the target state. In order to accurately describe the state transition equation affected by the propagation delay, t k e ${t}_{k}^{e}$ is treated as a variable in the state vector to be estimated from the beginning. Then the state transition matrix F t k e t k − 1 e $\mathbf{F}\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right)$ and noise transition matrix Γ t k e t k − 1 e ${\Gamma}\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right)$ will contain two adjacent state vectors x t k e $\mathbf{x}\left({t}_{k}^{e}\right)$ and x t k − 1 e $\mathbf{x}\left({t}_{k-1}^{e}\right)$ simultaneously; such an equation is called implicit equation and usually has no analytical solution. According to relationship of equations (1) to (6), the state transition equation based on GHM is given by s x t k e , x t k − 1 e + Γ t k e t k − 1 e w t k − 1 e = 0 , $s\left[\mathbf{x}\left({t}_{k}^{e}\right),\mathbf{x}\left({t}_{k-1}^{e}\right)\right]+\mathbf{\Gamma }\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right)\mathbf{w}\left({t}_{k-1}^{e}\right)=\mathbf{0},$ (9)where 0 denotes the nx × nx zero vector, and nx is the dimension of the target state. Since the sampling interval T of the measurement equation is determined by the observer itself and remains constant, it still satisfies the GMM. To distinguish from the time parameter t k e ${t}_{k}^{e}$ in Equation (9), the measurement equation is rewritten as z t k o = h x t k e + v t k o . $\mathbf{z}\left({t}_{k}^{o}\right)=h\left[\mathbf{x}\left({t}_{k}^{e}\right)\right]+\mathbf{v}\left({t}_{k}^{o}\right).$ (10) The-form of the CV model had been derived in Ref. [31]. The following part of this section will discuss underwater Modified 3-Dimensional coordinated turn model. For 3-D manoeuvring target tracking, it is generally assumed that the target follows a constant turn-rate motion in the same plane, as the literature in Ref. [26, 27]. In other words, there is only one turn-rate parameter in this 3-D model. However, this assumption is difficult to hold in underwater environment. In this paper, a modified 3D-CT model is applied, where the turn rate of the depth direction and the horizontal direction are separate, and there is no restriction that the target must turn in the same plane. The state vector x t k e $\mathbf{x}\left({t}_{k}^{e}\right)$ for the CT model is denotes as x c t t k e = x y z x ̇ y ̇ z ̇ f t k e w h w d ′ , ${\mathbf{x}}_{ct}\left({t}_{k}^{e}\right)={\left[x\,y\,z\,\dot{x}\,\dot{y}\,\dot{z}\,f\,{t}_{k}^{e}\,{w}_{h}\,{w}_{d}\right]}^{\prime },$ (11)where x t k e , y t k e , z t k e , x ̇ t k e , y ̇ t k e $x\left({t}_{k}^{e}\right),y\left({t}_{k}^{e}\right),z\left({t}_{k}^{e}\right),\dot{x}\left({t}_{k}^{e}\right),\dot{y}\left({t}_{k}^{e}\right)$ and z ̇ t k e $\,\dot{z}\left({t}_{k}^{e}\right)$ are the motion parameters of the target at the instant t k e ${t}_{k}^{e}$ , the first three variables denote the position of the target on the corresponding coordinate axis, and the last three variables are the target velocity components on the corresponding coordinate axis. f t t k e ${f}_{t}\left({t}_{k}^{e}\right)$ is the radiation frequency of the target, and t k e ${t}_{k}^{e}$ is the time when the target emits signal. w h t k e ${w}_{h}\left({t}_{k}^{e}\right)$ and w d t k e ${w}_{d}\left({t}_{k}^{e}\right)$ represent the angular velocity in the horizontal direction and the depth direction, respectively. Note that the time t k e ${t}_{k}^{e}$ of variables are omitted for simplicity. In the following part, sometimes the time t k e ${t}_{k}^{e}$ of the vector or the variable will be similarly omitted; this paper will not describe it repeatedly unless necessary. s c t ( ⋅ ) = s 1 ( ⋅ ) s 2 ( ⋅ ) s 3 ( ⋅ ) s 4 ( ⋅ ) s 5 ( ⋅ ) s 6 ( ⋅ ) s 7 ( ⋅ ) s 8 ( ⋅ ) s 9 ( ⋅ ) s 10 ( ⋅ ) ′ , ${s}_{ct}(\cdot )={\left[{s}_{1}(\cdot )\,{s}_{2}(\cdot )\,{s}_{3}(\cdot )\,{s}_{4}(\cdot )\,{s}_{5}(\cdot )\,{s}_{6}(\cdot )\,{s}_{7}(\cdot )\,{s}_{8}(\cdot ){s}_{9}(\cdot ){s}_{10}(\cdot )\right]}^{\prime },$ (12)where s 1 = x t k e − x t k − 1 e − V t k − 1 e ∫ t k − 1 e t k e cos ϕ v − w d τ sin θ v − w h τ d τ = x t k e − x t k − 1 e − x ̇ t k − 1 e 2 sin w 1 T k e w 1 + sin w 2 T k e w 2 − y ̇ t k − 1 e 2 1 − cos w 1 T k e w 1 + 1 − cos w 2 T k e w 2 − z ̇ t k − 1 e 2 1 − cos w 1 T k e w 1 − 1 − cos w 2 T k e w 2 sin θ v − z ̇ t k − 1 e 2 sin w 1 T k e w 1 − sin w 2 T k e w 2 cos θ v , $\begin{array}{rl}\hfill {s}_{1}& =x\left({t}_{k}^{e}\right)\,-x\left({t}_{k-1}^{e}\right)-V\left({t}_{k-1}^{e}\right){\int }_{{t}_{k-1}^{e}}^{{t}_{k}^{e}}\mathrm{cos}\left({\phi }_{v}-{w}_{d}\tau \right)\mathrm{sin}\left({\theta }_{v}-{w}_{h}\tau \right)d\tau \hfill \\ \hfill & =x\left({t}_{k}^{e}\right)\,-x\left({t}_{k-1}^{e}\right)-\left[\frac{\dot{x}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{\mathrm{sin}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}+\frac{\mathrm{sin}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]-\frac{\dot{y}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{1-\mathrm{cos}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}+\frac{1-\mathrm{cos}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]\right.\hfill \\ \hfill & \left.-\,\frac{\dot{z}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{1-\mathrm{cos}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}-\frac{1-\mathrm{cos}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]\mathrm{sin}\,{\theta }_{v}-\frac{\dot{z}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{\mathrm{sin}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}-\frac{\mathrm{sin}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]\mathrm{cos}\,{\theta }_{v}\right],\hfill \end{array}$ (13)similarly s 2 = y t k e − y t k − 1 e − x ̇ t k − 1 e 2 1 − cos w 1 T k e w 1 + 1 − cos w 2 T k e w 2 + y ̇ t k − 1 e 2 sin w 1 T k e w 1 + sin w 2 T k e w 2 + z ̇ t k − 1 e 2 sin w 1 T k e w 1 − sin w 2 T k e w 2 sin θ v − z ̇ t k − 1 e 2 1 − cos w 1 T k e w 1 − 1 − cos w 2 T k e w 2 cos θ v , $\begin{array}{rl}\hfill {s}_{2}& =y\left({t}_{k}^{e}\right)\,-y\left({t}_{k-1}^{e}\right)-\left[\frac{\dot{x}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{1-\mathrm{cos}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}+\frac{1-\mathrm{cos}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]+\frac{\dot{y}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{\mathrm{sin}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}+\frac{\mathrm{sin}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]\right.\hfill \\ \hfill & \,\,\,\,\,\,\left.+\,\frac{\dot{z}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{\mathrm{sin}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}-\frac{\mathrm{sin}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]\mathrm{sin}\,{\theta }_{v}-\frac{\dot{z}\left({t}_{k-1}^{e}\right)}{2}\left[\frac{1-\mathrm{cos}\left({w}_{1}{T}_{k}^{e}\right)}{{w}_{1}}-\frac{1-\mathrm{cos}\left({w}_{2}{T}_{k}^{e}\right)}{{w}_{2}}\right]\mathrm{cos}\,{\theta }_{v}\right],\hfill \end{array}$ (14) s 3 = z t k e − z t k − 1 e − z ̇ t k − 1 e sin w d T k e w d − V x y t k − 1 e 1 − cos w d T k e w d , ${s}_{3}=z\left({t}_{k}^{e}\right)\,-z\left({t}_{k-1}^{e}\right)-\left[\dot{z}\left({t}_{k-1}^{e}\right)\frac{\mathrm{sin}\left({w}_{d}{T}_{k}^{e}\right)}{{w}_{d}}-{V}_{xy}\left({t}_{k-1}^{e}\right)\frac{1-\mathrm{cos}\left({w}_{d}{T}_{k}^{e}\right)}{{w}_{d}}\right],$ (15) s 4 = x ̇ t k e − x ̇ t k − 1 e cos w d T k e cos w h T k e − y ̇ t k − 1 e cos w d T k e sin w h T k e + z ̇ t k − 1 e sin w d T k e cos w h T k e sin θ v − z ̇ t k − 1 e sin w d T k e sin w h T k e cos θ v , $\begin{array}{r}\hfill {s}_{4}=\dot{x}\left({t}_{k}^{e}\right)-\left[\dot{x}\left({t}_{k-1}^{e}\right)\mathrm{cos}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{cos}\left({w}_{h}{T}_{k}^{e}\right)-\dot{y}\left({t}_{k-1}^{e}\right)\mathrm{cos}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{sin}\left({w}_{h}{T}_{k}^{e}\right)\right.\\ \,\,\,\,\,\,\left.+\,\dot{z}\left({t}_{k-1}^{e}\right)\mathrm{sin}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{cos}\left({w}_{h}{T}_{k}^{e}\right)\mathrm{sin}\,{\theta }_{v}-\dot{z}\left({t}_{k-1}^{e}\right)\mathrm{sin}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{sin}\left({w}_{h}{T}_{k}^{e}\right)\mathrm{cos}\,{\theta }_{v}\right],\hfill \end{array}$ (16) s 5 = y ̇ t k e − x ̇ t k − 1 e cos w d T k e sin w h T k e + y ̇ t k − 1 e cos w d T k e cos w h T k e + z ̇ t k − 1 e sin w d T k e cos w h T k e cos θ v + z ̇ t k − 1 e sin w d T k e sin w h T k e sin θ v , $\begin{array}{rl}\hfill {s}_{5}& =\dot{y}\left({t}_{k}^{e}\right)-\left[\dot{x}\left({t}_{k-1}^{e}\right)\mathrm{cos}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{sin}\left({w}_{h}{T}_{k}^{e}\right)+\dot{y}\left({t}_{k-1}^{e}\right)\mathrm{cos}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{cos}\left({w}_{h}{T}_{k}^{e}\right)\right.\hfill \\ \hfill & \,\,\,\,\,\,\left.+\,\dot{z}\left({t}_{k-1}^{e}\right)\mathrm{sin}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{cos}\left({w}_{h}{T}_{k}^{e}\right)\mathrm{cos}\,{\theta }_{v}+\dot{z}\left({t}_{k-1}^{e}\right)\mathrm{sin}\left({w}_{d}{T}_{k}^{e}\right)\mathrm{sin}\left({w}_{h}{T}_{k}^{e}\right)\mathrm{sin}\,{\theta }_{v}\right],\hfill \end{array}$ (17) s 6 = z ̇ t k e − z ̇ t k − 1 e cos w d T k e − V x y t k − 1 e sin w d T k e , ${s}_{6}=\dot{z}\left({t}_{k}^{e}\right)\,-\left[\dot{z}\left({t}_{k-1}^{e}\right)\mathrm{cos}\left({w}_{d}{T}_{k}^{e}\right)-{V}_{xy}\left({t}_{k-1}^{e}\right)\mathrm{sin}\left({w}_{d}{T}_{k}^{e}\right)\right],$ (18) s 7 = f t t k e − f t t k − 1 e , ${s}_{7}={f}_{t}\left({t}_{k}^{e}\right)-{f}_{t}\left({t}_{k-1}^{e}\right),$ (19) s 8 = t k e + t k p − t k o , ${s}_{8}={t}_{k}^{e}+{t}_{k}^{p}-{t}_{k}^{o},$ (20) s 9 = w h t k e − w h t k − 1 e , ${s}_{9}={w}_{h}\left({t}_{k}^{e}\right)-{w}_{h}\left({t}_{k-1}^{e}\right),$ (21) s 10 = w d t k e − w d t k − 1 e . ${s}_{10}={w}_{d}\left({t}_{k}^{e}\right)-{w}_{d}\left({t}_{k-1}^{e}\right).$ (22) Figure 2 describes the relationships of the target motion direction and the angles in the Cartesian coordinate. V denotes the target velocity in the 3D space; θv and ϕv are, respectively, the course angle in the horizontal plane and pitch angle in the vertical direction. Let w1 = wh − wd, w2 = wh + wd, denote Vxy as the target horizontal velocity, and the implicit state equations for the modified 3D-CT model are given by (see (12) to (22)) FIGURE 2Open in figure viewerPowerPoint Target motion direction in 3-dimensional space Note that the time t k − 1 e ${t}_{k-1}^{e}$ of the angular velocities wh, wd, w1, w2 in equations s1 to s6 are omitted for simplicity. The noise transition matrix Γ c t t k e t k − 1 e ${\mathbf{\Gamma }}_{ct}\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right)$ and the noise covariance qct are given by Γ c t t k e t k − 1 e = T k e 2 / 2 ∗ I 3 0 3 , 4 T k e ∗ I 3 0 3 , 4 0 4 , 3 I 4 , ${\mathbf{\Gamma }}_{ct}\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right)=\left[\begin{array}{cc}\hfill {\left({T}_{k}^{e}\right)}^{2}/2\ast {\mathbf{I}}_{3}\hfill & \hfill {\mathbf{0}}_{3,4}\hfill \\ \hfill {T}_{k}^{e}\ast {\mathbf{I}}_{3}\hfill & \hfill {\mathbf{0}}_{3,4}\hfill \\ \hfill {\mathbf{0}}_{4,3}\hfill & \hfill {\mathbf{I}}_{4}\hfill \end{array}\right],$ (23) q c t = diag σ x ̈ 2 σ y ̈ 2 σ z ̈ 2 σ f 2 σ t e 2 σ w h 2 σ w d 2 , ${\mathbf{q}}_{ct}=\mathrm{diag}\left({\sigma }_{\ddot{x}}^{2}\,{\sigma }_{\ddot{y}}^{2}\,{\sigma }_{\ddot{z}}^{2}\,{\sigma }_{f}^{2}\,{\sigma }_{te}^{2}\,{\sigma }_{wh}^{2}\,{\sigma }_{wd}^{2}\right),$ (24)where 0n,m denotes a zero matrix with n rows and m columns. Moreover, In is the n × n identity matrix. σ x ̈ 2 ${\sigma }_{\ddot{x}}^{2}$ , σ y ̈ 2 ${\sigma }_{\ddot{y}}^{2}$ and σ z ̈ 2 ${\sigma }_{\ddot{z}}^{2}$ are the process noise variances, which are modelling the disturbance during the state transition. Similarly, σ f 2 ${\sigma }_{f}^{2}$ and σ t e 2 ${\sigma }_{te}^{2}$ are the system process noise variances of the target radiation frequency and signal emission time respectively. σ w h 2 ${\sigma }_{wh}^{2}$ and σ w h 2 ${\sigma }_{wh}^{2}$ are the noise variances to describe the uncertainty of the target manoeuvring. Correspondingly, the system process noise matrix is written as Q c t t k e t k − 1 e = Γ c t t k e t k − 1 e q c t Γ c t t k e t k − 1 e ′ . ${\mathbf{Q}}_{ct}\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right)={\mathbf{\Gamma }}_{ct}\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right){\mathbf{q}}_{ct}{\left[{\mathbf{\Gamma }}_{ct}\left(\,{t}_{k}^{e}\left\vert {t}_{k-1}^{e}\right.\right)\right]}^{\prime }.$ (25) 3 ISOGRADIENT SOUND SPEED PROFILE AND MODIFIED MEASUREMENT EQUATIONS For the target tracking problem in the ground or in the air, it can be approximately considered that the signal propagates in the homogeneous medium. However,