## Proposed EDOVE Scheme
### Estimation of \(T_{h_{i}}\)  
Let, node \(S\) sends or forwards the data packet and node \(i\) within its \(t_{r}^{S}\) receives that data packet. Then, the node \(i\) first compares its depth to the depth of \(S\). If \(i\) has the depth larger than that of \(S\), \(z_{i}>z_{S}\), then it simply discards the data packet. On the other hand, if node \(i\)'s depth is less than the depth \(S\) and has one or more than one neighbors with smaller depth (i.e., \(z_{i}<z_{S}\) and \(\mid\kappa_{i}^{NF}\mid\geq 1\)), then it computes the holding time \(T_{h_{i}}\). However, if node \(i\) has no neighbor at the depth lower than \(i\), then it also discards the data packet. The holding time of node \(i\) should be very small compared to its neighboring nodes if it has the following characteristics:  
* larger residual energy than its neighbors. * larger depth difference than sender. * larger depth difference to its minimum depth neighbor. * many neighbors with large variance in their depth differences. EDOVE considers all the above mentioned characteristics to calculate the holding time of \(i\in\kappa_{S}\) as follows:  
\[T_{h_{i}}=\frac{2\times t_{r}^{S}}{v_{Sound}}(H) \tag{3}\]  
where,  
\[H=e_{i}\times\left(1-\frac{d_{i}}{t_{r}^{S}}\right)+\left(1-\frac{max(d_{j}^{ NF})}{t_{r}^{i}}\right)\times\delta_{i}, \tag{4}\]  
\[j=1,\ldots,\mid\kappa_{i}^{NF}\mid,\]  
\[v_{Sound}\approx 1500\,\mathrm{m/s}.\]  
Figure 4: EDOVE network parameters to compute holding time. Equation (4) consists of two parts. First part prioritizes a node from the 1-hop neighbors of the data sender that has smaller depth with larger depth difference \(d_{i}\) relative to the _Sender_ node and large normalized residual energy, \(e_{i}\), which is explained later in this section. The main objective of prioritizing the smaller depth potential forwarder with larger residual energy is to achieve energy efficiency, energy balancing, and disseminating data further in the network. In result, it extends the network lifetime and minimizes the data broadcast overhead. Next part of Equation (4) considers the depth difference between the potential forwarder \(i\) relative to its neighbors, \(max(d_{j}^{NF})\), and depth variance among its neighbors \(\delta_{i}\). Any data receiving node that has the large depth difference to its lowest depth neighbor and has large depth variance becomes the potential forwarder. This part of the expression intends to avoid the packet collision and prevents the void region when it further forwards the packet. The factor, \(e_{i}\), in Equation (4) is the normalized residual energy of \(i\) computed using the residual energy of all the neighbors of \(i\), \(\kappa_{i}\). \[e_{i}=\frac{E_{max}-E_{i}}{E_{max}-E_{min}} \tag{5}\]  
where,  
\[E_{i} =\text{Residual Energy of node }i,\] \[E_{max} \
eq E_{min},\] \[E_{min} =min\left(E_{k}|\forall k\in\kappa_{r}\right),\] \[E_{max} =max\left(E_{k}|\forall k\in\kappa_{r}\right),\text{and}\] \[H=e_{i}\times\left(1-\frac{d_{i}}{t_{r}^{S}}\right)+\left(1-\frac{max(d_{j}^{ NF})}{t_{r}^{i}}\right)\times\delta_{i}, \tag{4}\]0  
In case, \(E_{max}=E_{min}\), then \(e_{i}=1.0\). The value of \(e_{i}\) will be smaller for the node \(i\) with large residual energy and that node will be prioritized to forward the data packet. Similarly, the second factor, \(\delta_{i}\), in Equation (4) is the normalized variance of the depth differences of the next forwarding nodes of \(i\), \(\delta_{i}\). \[H=e_{i}\times\left(1-\frac{d_{i}}{t_{r}^{S}}\right)+\left(1-\frac{max(d_{j}^{ NF})}{t_{r}^{i}}\right)\times\delta_{i}, \tag{4}\]1  
where,  
\[H=e_{i}\times\left(1-\frac{d_{i}}{t_{r}^{S}}\right)+\left(1-\frac{max(d_{j}^{ NF})}{t_{r}^{i}}\right)\times\delta_{i}, \tag{4}\]2  
The value returned by Equation (6) will be smaller for the receiver \(i\) that has the large normalized depth difference variance, and that node will be prioritized to act as the forwarder node.