## 2 Materials and Methods
### A-VTTCES-KF Algorithm
#### 2.1.2 System Model for A-VTTCES-KF
\right]\] (10) where \(\otimes\) stands for the Kronecker product operator; \(\mathbf{I}_{x}\) denotes the identity matrix with the dimension of \(x\). The transition matrix can be transferred from the dynamic matrix which is attained as [66] \[\mathbf{\Phi}=e^{\mathbf{F}\cdot\Delta t}\approx\mathbf{I}_{12}+\mathbf{F} \Delta t+\frac{1}{2}\mathbf{F}^{2}\Delta t^{2}\] (11) The approximation at the right of (11) is used for the practical case in this work. Since the transition matrix does not change with time in this system, it is denoted as \(\mathbf{\Phi}\) after the subscript is omitted. - 1 pseudo
- range error}\\ \vdots\\ \text{Channel
- n${}_{1}$ pseudo
- range error}\\ \text{Channel
- 1 ${}_{2}$ Doppler error}\\ \vdots\\ \text{Channel
- n${}_{2}$ Doppler error}\end{array}\right]\] (12) where \(\tilde{\rho}_{k}^{(n_{1})}\) and \(\tilde{f}_{k}^{(n_{2})}\) denote the pseudo-range and the Doppler measurements at the \(k\)th epoch of the \(n\)th tracking channel, respectively. Then, the design matrix can be attained as follow \[\mathbf{H}_{k}=\left[ \begin{array}{cccc}\mathbf{D_{1}}&\mathbf{0_{3\times 3 }}&\mathbf{0_{3\times 3}}&\mathbf{I_{3}}&\mathbf{0_{3\times 3}}&\mathbf{0_{3 \times 3}}\\ \mathbf{0_{3\times 3}}&\mathbf{D_{2}}&\mathbf{0_{3\times 3}}&\mathbf{0_{3 \times 3}}&\mathbf{I_{3}}&\mathbf{0_{3\times 3}}\end{array} \right]\] (13) where \(\mathbf{D_{1}}\) is a DCM in terms of the pseudo-range measurements of which the dimension is \(n_{1}\times 3\), and \(n_{1}\) denotes the number of available measurements; the counterpart \(\mathbf{D_{2}}\) is a DCM for the Doppler measurement with the dimension of \(n_{2}\times 3\) where \(n_{2}\) is the available Doppler measurement number. * axis acceleration random process white noise}\\ \text{y
- axis acceleration random process white noise}\\ \text{z
- axis acceleration random process white noise}\end{array}\right]\] (15) All these system process noise terms are assumed to be independent with each other. Again, given \[ \begin{cases}S_{f}=\mathbf{E}\left[w_{f}w_{f}^{T}\right],\;S_{g}=\mathbf{E} \left[w_{g}w_{g}^{T}\right],\;S_{t}=\mathbf{E}\left[w_{t}w_{t}^{T}\right],\\ S_{ax}=\mathbf{E}\left[w_{ax}w_{ax}^{T}\right],\;S_{ax}=\mathbf{E}\left[w_{ay}w_ {ay}^{T}\right],\;S_{ax}=\mathbf{E}\left[w_{az}w_{az}^{T}\right]\end{cases} \] (16) and, in this work, we assume that \(S_{a}=S_{ax}=S_{ay}=S_{ay}\), where \(S_{f}\), \(S_{g}\), \(S_{t}\), and \(S_{a}\) are the spectral densities related to the clock bias error, clock drift error, VT time-correlated error, and acceleration random process, respectively. Therefore, the general process noise matrix is provided by \[\mathbf{Q}=\left[ \begin{array}{cccc}\mathbf{0_{3\times 3}}&&&\\ &\mathbf{0_{3\times 3}}&&&\\ &&\mathbf{S_{a}}&\\ &&&\mathbf{S_{c}}\end{array} \right]\] (17) with \[\mathbf{Q}=\left[ \begin{array}{cccc}\mathbf{0_{3\times 3}}&&&\\ &\mathbf{0_{3\times 3}}&&&\\ &&&\mathbf{S_{a}}&\\ &&&\mathbf{S_{c}}\end{array} \right]\] (18) with \[\mathbf{Q}=\left[\begin{array}{cccc}\mathbf{0_{3\times 3}}&&&\\ &\mathbf{0_{3\times 3}}&&&\\ &&&\mathbf{0_{3\times 3
\[ \begin{cases}\mathbf{S}_{a}=S_{a}\times\mathbf{I}_{3}\\ \mathbf{S}_{c}=\text{diag}\left(\left[\begin{array}{cc}S_{f}&S_{g}&S_{t}\\ \end{array}\right]\right)\end{cases}  \tag{18}\]  
where \(\text{diag}\left(\cdot\right)\) denotes the diagonal matrix operator; \(\mathbf{Q}\) is the diagonal matrix. The discrete form of the process noise matrix, \(\mathbf{Q}_{k}\), is well-known to be calculated based on the linear system theory, so, the related computation process will be not be described in this paper any more. It is worth noting that the PSD of \(S_{a}\) associated with the white noise of the dynamic or acceleration driving process is set to \(2\times 2\)\(\left(m/s^{2}\right)^{2}\) according to the empirical experience; \(S_{f}\) and \(S_{g}\) can be computed by (A15); based on (A18) and (A21), \(S_{t}\) can be attained as \(2\beta\sigma_{c}^{2}\), where \(\sigma_{c}^{2}\) can be estimated through (A12).