# Node Depth Adjustment Based Target Tracking in UWSNs Using Improved Harmony Search
## 4 Depth Adjustment Based Target Tracking in UWSNs
### Relationship between FIM and Node Depth  
FIM is widely utilized in estimation and fusion issues, and it is the inverse matrix of PCRLB. Both of them stand for information contained in measurements [32; 33; 34]. During the tracking processing, the larger the FIM is, the more useful information we can get from measurements, and the better tracking performance we will achieve.  
Therefore, to improve tracking accuracy via adjusting nodes' depth, FIM can be utilized as a metric to determine the optimal depth. The FIM \(J_{k+1}\) has the following definition:  
\[\mathrm{E}[\mathbf{\hat{x}}_{k+1}-\mathbf{\boldsymbol{x}}_{k+1}][\mathbf{ \hat{x}}_{k+1}-\mathbf{\boldsymbol{x}}_{k+1}]^{\mathrm{T}}\geq\mathbf{J}_{k+1 }^{-1}. \tag{9}\]  
FIM \(\mathbf{J}_{k+1}\) is a the \(4\times 4\) matrix with elements  
\[\mathbf{J}_{k+1}(i,j)=\mathrm{E}_{p\left(\mathbf{M}_{k+1,\boldsymbol{x}}_{k+1 }\right)}\left[-\frac{\partial^{2}\mathrm{log}\;p(\mathbf{\boldsymbol{Z}}_{k+1 }|\mathbf{\boldsymbol{x}}_{k+1})}{\partial\mathbf{\boldsymbol{x}}_{k+1}(i) \partial\mathbf{\boldsymbol{x}}_{k+1}(j)}\right], \tag{10}\]  
where \(\mathbf{J}_{k+1}(i,j)\) denotes the \(i\)th row and \(j_{th}\) column element of \(\mathbf{J}_{k+1}\), \(\mathrm{E}_{p\left(\mathbf{M}_{k+1,\boldsymbol{x}}_{k+1}\right)}\) denotes the expectation respect to \(p\left(\mathbf{M}_{k+1,\boldsymbol{x}},\mathbf{\boldsymbol{x}}_{k+1}\right)\), and \(\mathbf{\boldsymbol{x}}_{k+1}(i)\) denotes the \(i\)th element of vector \(\mathbf{\boldsymbol{x}}_{k+1}\).  
Since the measurements in vector \(\mathbf{\boldsymbol{Z}}_{k+1}\) is Gaussian with nonzero mean, FIM \(\mathbf{J}_{k+1}\) can be calculated as follows [35]:  
\[\mathbf{J}_{k+1}(i,j)=\left[\frac{\partial\mathbf{\boldsymbol{H}}_{k+1}( \mathbf{\boldsymbol{x}}_{k+1})}{\partial\mathbf{\boldsymbol{x}}_{k+1}(i)} \right]^{\mathrm{T}}\mathbf{R}^{-1}\left[\frac{\partial\mathbf{\boldsymbol{H }}_{k+1}(\mathbf{\boldsymbol{x}}_{k+1})}{\partial\mathbf{\boldsymbol{x}}_{k+1 }(j)}\right], \tag{11}\]  
where \(\mathbf{\boldsymbol{H}}_{k+1}(\mathbf{\boldsymbol{x}}_{k+1})\) is the true distance vector of \(\left[h_{k+1}^{1},\cdots,h_{k+1}^{N_{s}}\right]^{\mathrm{T}}\) in Equation (5). \(\mathbf{\boldsymbol{R}}\) is the measurement noise matrix. Consequently, we can get  
\[\begin{split}\mathbf{J}_{k+1}=&\sum_{s=1}^{N_{t}} \begin{bmatrix}\frac{(x^{s}-\mathbf{x}_{k+1}^{1})^{2}}{h_{k+1}^{s}R}& \frac{(x^{s}-\mathbf{x}_{k+1}^{1})(y^{s}-\mathbf{x}_{k+1}^{1})}{h_{k+1}^{s}R }\\ \frac{(y^{s}-\mathbf{x}_{k+1}^{1})(x^{s}-\mathbf{x}_{k+1}^{1})}{h_{k+1}^{s}R}& \frac{(y^{s}-\mathbf{x}_{k+1}^{1})^{2}}{h_{k+1}^{s}R}\end{bmatrix}\\ =&\sum_{s=1}^{N_{t}}\{\frac{1}{[(\mathbf{x}_{k+1}^{1}-x^{s})^{2}+( \mathbf{x}_{k+1}^{3}-y^{s})^{2}+(h-z^{s})^{2}]R}\begin{bmatrix}(x^{s}-\mathbf{ x}_{k+1}^{1})^{2}&(x^{s}-\mathbf{x}_{k+1}^{1})(y^{s}-\mathbf{x}_{k+1}^{3})\\ (y^{s}-\mathbf{x}_{k+1}^{3})(x^{s}-\mathbf{x}_{k+1}^{1})&(y^{s}-\mathbf{x}_{k+ 1}^{3})^{2}\end{bmatrix}\},\end{split} \tag{12}\]  
where \(h_{k+1}^{s}\) is the true distance from sensor \(s_{th}\) to the target's position, and it is calculated via Equation (5). It can be seen that the node's depth \(z^{s}\) is related to the value of \(h_{k+1}^{s}\), and the term \(h-z^{s}\) represents relative depth between node and target, which has much effect on the value of FIM. Since FIM is always used to present estimation accuracy, the value of FIM can be utilized as a metric to determine nodes' depth. It should be noted that nodes will move before estimating target's position at each time according to predicted FIM. To calculate FIM, the predicted target's position \(\mathbf{\boldsymbol{x}}_{k+1|k}\) will be considered in Equation (12) as target state.