## 7 Calculation of Cramer Rao Lower Bound for the Proposed Algorithm  
The proposed \(E^{2}\)_JSL_ use least square estimation, which is an unbiased estimator, which means that the Cramer-Rao lower bound can be derived from the LS estimator [6]. To calculate the CRLB for \(E^{2}\)_JSL_ method we provide the performance reference as,  
Suppose \(V~{}=~{}[\eta_{A},\,T_{A},\,\eta_{B},\,T_{B}]\) is a vector of measurement with mean \(\mu(\phi)\) and covariance C. Also \(\phi\) is defined as the measurement of timestamps of ordinary node location \(x,\ y\) and clock parameters \(\alpha,\ \beta\) represented as,  
\[\mu(\phi)~{}=~{}[\alpha\eta_{1,1}~{}+~{}\beta,\,\ldots~{}\ldots~{}\ldots~{} \ldots~{}\ldots~{}\alpha\eta_{1,n}~{}+~{}\beta]^{T} \tag{44}\]  
or Equation (33) can be written as,  
\[\mu(\phi)~{}=~{}\left[\frac{T_{2,1}^{1}-\beta}{\alpha}-t_{1},\,\ldots~{} \ldots~{}\ldots~{}\frac{T_{2,n}^{n}-\beta}{\alpha}-t_{n}\right] \tag{45}\]  
where \(C=\sigma^{2}\) denotes the F \(\times\) F identity matrix. The Fisher matrix can be represented as,  
\[J(\phi)~{}=~{}\left[\frac{\partial\mu}{\partial\varnothing}\right]_{i}^{T}C^{- 1}\left[\frac{\partial\mu}{\partial\varnothing}\right]_{j}^{T},i,j~{}=~{}1,2,3,4,5 \tag{46}\]  
where \(i,j\) stands for \((i,j)\) element of the matrix, \([.]_{i}\) stands for \(i\)th element of the vector.  
\[\left[\frac{\partial\mu}{\partial\varnothing}\right]_{i}~{}=~{}\left\{\begin{array}[ ]{ll}\frac{\partial\mu}{\partial x}&if~{}i~{}=~{}1\\ \frac{\partial\mu}{\partial x}&if~{}i~{}=~{}2\\ \frac{\partial\mu}{\partial z}&if~{}i~{}=~{}3\\ \frac{\partial\mu}{\partial z}&if~{}i~{}=~{}4\\ \frac{\partial\mu}{\partial\beta}&if~{}i~{}=~{}5\end{array}\right.\]  
Similarly,  
\[\frac{\partial C}{\partial\varnothing}~{}=~{}\left\{\begin{array}{ll}\text{ diag}\sigma^{2},\,\ldots~{}\ldots\,\sigma^{2},&if~{}i~{}=~{}4\\ 0&otherwise\end{array}\right.\]  
The partial derivative \(\left[\frac{\partial\mu}{\partial\varnothing}\right]^{T}~{}=~{}\left[\frac{x -x^{\ast}}{c\gamma_{1}}-\frac{y-y^{\ast}}{c\gamma_{1}}-\frac{T_{2,\,1}-\beta}{ \alpha^{2}}-\frac{1}{\alpha},\,\frac{x-x^{\ast}}{c\gamma_{2}}-\frac{y-y^{ \ast}}{c\gamma_{2}}-\frac{T_{2,\,2}-\beta}{\alpha^{2}}-\frac{1}{\alpha}\, \ldots~{}\ldots\right]\triangleq K\) where \(\gamma_{i}~{}=~{}\sqrt{\left(x-x^{\ast}\right)^{2}~{}+~{}\left(y-y^{\ast} \right)^{2}~{}+~{}\left(z-z^{\ast}\right)^{2}}\)  
\[\left[J^{-1}\right]_{i,j}~{}=~{}\sigma^{2}\left(K^{T}K\right)^{-1}\]  
co-variance \(C=\sigma^{2}I\), therefore,  
For the simulation of time synchronization error, initial skew and clock offset are considered as \(\Delta_{thresh}=15\) and the initial time slot was set to be \(T_{slottur}=5\) s. Initial sound speed at the surface is b m = 1420 m/s with environment constant a and average sound speed is taken as c = 1500 m/s. To prove the validity of the proposed algorithm we have made a comparison with some previously developed techniques such as STSL, JSL, and GNJSL since all of them are joint localization and time synchronization techniques. In the simulation setting, we have used 20 number of messages, where \(t_{all}\) is taken to be 106 s, 40 bytes \(l_{p}\) is taken for both JSL and STSL as they employ a two-way message exchange and 25 \(l_{p}\) is set for both GN-JSL [19] and \(E^{2}SL\) as both of them employ one-way messaging approach. The performance of the proposed algorithm is evaluated on the basis of synchronization accuracy, localization accuracy, and energy efficiency by calculating some parameters such as propagation delay measurement noise, the number of messages, and the number of reference nodes that directly affect these algorithms. We conduct Monte Carlo (MC) simulations, and all simulation results are averaged of \(N_{mc}=2000\) independent runs.