**3. Doppler Measurement Association**

Based on the LM procedure [31], we develop the DDA method in a joint model for LM tracking for IPDA. To solve the LM tracking problem, we need to estimate the joint posterior density of individual target state **x***<sup>k</sup> <sup>p</sup>* conditioned on measurement sequences up to the *p*-th scan **Z***<sup>p</sup>* as follows

$$p(\mathbf{x}\_{p\prime}^k \chi\_p^k | \mathbf{Z}^p), k = 1, \ldots, K. \tag{7}$$

In Equation (7), χ*<sup>k</sup> <sup>p</sup>* denotes the existence of the *k*-th target at the *p*-th scan. *K* describes the total number of tracks. We can express the joint posterior density of the track state as follows:

$$p\left(\mathbf{x}\_{p^\prime}, \chi\_p | \mathbf{Z}^l\right) = p\left(\chi\_p | \mathbf{Z}^p\right) p\left(\mathbf{x}\_{p^\prime} | \chi\_{p^\prime} \mathbf{Z}^l\right). \tag{8}$$

When *l* = *p*, the above formula can be considered as an estimation model. And *l* = *p* − 1, the above formula is a prediction model. Based on the Bayesian theory, *p* , **x***p*, χ*<sup>p</sup>* - - -**Z***l* can be calculated recursively. At time *p*, given prior density *P* % <sup>χ</sup>*p*−<sup>1</sup> - - - -**Z***p*−<sup>1</sup> & and prior density of the target state *p* % **<sup>x</sup>***p*−1, - - - <sup>χ</sup>*p*−1,**Z***p*−<sup>1</sup> & , the iterative procedure can be expressed by the following stages.

(1) Calculate the predicted prior density *P* , χ*p* - - -**Z***p*−<sup>1</sup> and the predicted prior density of the target state *p* , **x***p* - - χ*p*,**Z***p*−<sup>1</sup> - .

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An illustrative diagram for an iteration process of the above four stages is presented in Figure 1.

**Figure 1.** An illustrative diagram for an iteration process of synthetic tracking.

Step 1: Prediction

Based on the Gaussian hypothesis, every target measurement can be considered as a single Gaussian density function, which is expressed as:

$$p\left(\mathbf{x}\_{p}|\boldsymbol{\xi}\_{p}(\boldsymbol{c}),\boldsymbol{\chi}\_{p},\mathbf{Z}^{p-1}\right) \sim \mathcal{N}\left(\mathbf{x}\_{p};\mathbf{\hat{x}}\_{p|p-1}\left(\boldsymbol{\xi}\_{p}(\boldsymbol{c})\right),\boldsymbol{P}\_{p|p-1}\left(\boldsymbol{\xi}\_{p}(\boldsymbol{c})\right)\right) \tag{9}$$

where ξ*p*(*c*) denotes event that the *c*-th out of *Cp* track components is true at the *k*-th sonar scan. **x**ˆ*p*|*p*−<sup>1</sup> , ξ*p*(*c*) and *Pp*|*p*−<sup>1</sup> , ξ*p*(*c*) describes the mean and covariance of predictive prior density of the target track state, respectively.

Step 2: Measurement Selection

The **<sup>Z</sup>***<sup>p</sup>* <sup>=</sup> **z***p*,1, **z***p*,2, ... , **z***p*,*mp* measurement selection procedure is implemented at a component level. During the *p*-th scan, the *c*-th of *Cp* components chooses its measurements using a range measurement confirmation gate, which is concentrated at the expected range measurement ˆ**y***<sup>c</sup> p* , ξ*p*(*c*) as follows

$$\left(\mathbf{y}\_{p,i}^{c} - \mathbf{y}\_{p}^{c}(\boldsymbol{\xi}\_{p}(\boldsymbol{c}))\right)' \left[\mathbf{S}\_{p}^{c}(\boldsymbol{\xi}\_{p}(\boldsymbol{c}))\right]^{-1} \left(\mathbf{y}\_{p,i}^{c} - \mathbf{y}\_{p}^{c}(\boldsymbol{\xi}\_{p}(\boldsymbol{c}))\right) \leq \boldsymbol{\gamma},\tag{10}$$

where γ denotes a fixed threshold, **y***<sup>c</sup> p*,*i* , *i* = 1, ... , *mp* describes the *i*-th confirmed measurement and **S***c p* , ξ*p*(*c*) -2 describes the innovation covariance of the *c*-th of *Cp* components. In the gating process, only the range measurements can be utilized to produce a tracking gate for choosing a set of confirmed observations.

Step 3: Predictive Measurement PDF

*p* , **x***p*, - - χ*p*,**Z***p*−<sup>1</sup> - is assumed to be a sum of *Cp* mutually exclusive components *C* \$*p c*=1 *p* , ξ*p*(*c*) - - χ*p*,*Zp*−<sup>1</sup> *p* , **x***p* - - ξ*p*(*c*), χ*p*,**Z***p*−<sup>1</sup> - . Consequently, the predicted measurement PDF Λ*p*,*<sup>i</sup>* under the assumption of each measurement's **z***p*,*<sup>i</sup>* can be expressed as

$$\begin{array}{ll} \Lambda\_{p,i} & \triangleq p\Big(\mathbf{x}\_{p,i'} \Big| \chi\_{p'} \mathbf{Z}^{p-1}\Big) \\ & \overset{\circlearrowright}{\mathbf{C}\_p} \\ & = \sum\_{c=1} p\Big(\xi\_p(c) \big| \chi\_{p'} \mathbf{Z}^{p-1}\Big) p\Big(\mathbf{y}\_{p,i'}^c \mathbf{y}\_{p,i}^d \big| \xi\_p(c) \ll\_{\times \times p'} \mathbf{Z}^{p-1}\Big) \end{array} . \tag{11}$$

In Equation (11), the term *p* % **y***c p*,*i* , *yd p*,*i* - - - ξ*p*(*c*), χ*p*,**Z***p*−<sup>1</sup> & ≈ *p* % **y***c p*,*i* - - - ξ*p*(*c*), χ*p*,**Z***p*−<sup>1</sup> & *p* % *yd p*,*i* - - - ξ*p*(*c*), χ*p*,**Z***p*−<sup>1</sup> & , which is the measurement likelihood function conditioned on the event ξ*p*(*c*). The joint likelihood can be expressed as a product of target range likelihood and target velocity likelihood.

Step 4: The Probability of Target Existence Update

From the Reference [30], the data association factor can be written as:

$$\delta\_{\mathcal{P}} = P\_d P\_{\mathcal{S}} \left( 1 - \sum\_{i=1}^{m\_p} \frac{\Lambda\_{p,i}}{\rho\_{p,i}} \right) \tag{12}$$

where *Pd* and *Pg* denote the probability of detection and the probability of the tracking gate, respectively. ρ*p*,*<sup>i</sup>* denotes the clutter density of a measurement in Equation (6). Then, we can write the probability of target existence as follows:

$$P\{\chi\_p \middle| \mathbf{Z}^p\} = \frac{\left(1 - \delta\_p\right) P\{\chi\_p \middle| \mathbf{Z}^{p-1}\}}{1 - \delta\_p P\{\chi\_p \middle| \mathbf{Z}^{p-1}\}} \tag{13}$$

and the data association probabilities can be expressed as:

$$\beta\_{p,i} \triangleq \frac{1}{1 - \delta\_p} \begin{cases} 1 - P\_d P\_{\mathcal{S}'} & i = 0 \\ P\_d P\_{\mathcal{S}} \frac{\Lambda\_{p,i}}{\rho\_{p,i}} & i > 0 \end{cases} \tag{14}$$

It needs to be emphasized that the contribution of target Doppler information is embodied in the factor of data association Equation (12), the probability of target existence Equation (13), and the probabilities of data association Equation (14).
