# An Improved Forwarding of Diverse Events with Mobile Sinks in Underwater Wireless Sensor Networks
## 4 Problem Statement and Motivation
### Forwarding Functions Analysis
Anyhow if there comes a lower bound on \(E_{r}\) defined as \(min(E_{r}\geq 1)\) and \(D_{i}\) defined as \(min(D_{i}\geq 1)\) it causes a significant change on the upper bound of all the respective forwarding function \(W_{i}\) as in Table 2. Since forwarding function of AMCTD are independent of depth threshold so maximum values are same regardless of \(d_{th}\). Dimensional analysis of each forwarding function describes that \(W_{1}\) and \(W_{3}\) are dimensionally equivalent (=Newton) while \(W_{3}\) is dimensionally reciprocal of \(W_{1}\) and \(W_{3}\). Forwarding function in iAMCTD is defined differently for different depth regions. \(SQI\) based forwarding function is defined in Equation (15a). \[SQI=\frac{LSNR\times E_{r}}{l_{i}}\] (15a) In this equation \[l_{i}\] is depth difference between two successive nodes. \[LSNR\] is Localization free Signal to Noise Ratio defined in [41 ] and given by the following equation. \[LSNR=\frac{P_{t}}{A(l,f)\times N(f)} \tag{15b}\]  
where \(P_{t}\) is transmitted power and \((A()\times N())\) is the attenuation noise product factor, which is product of path loss and ambient noise. \(A(l,f)\) is the sum of absorption coefficient (Abc) and scattering coefficient (Sc) whose minimum and maximum values for different depth thresholds are given in Table 1.  
\(SQI\) based forwarding decisions are made if depth of the node is less than \(D_{1}\) which is top shallow region of underwater environment. This region involves more shipping activities. We have considered the statistical fact that variance is the measure of randomness in a set of observations. Careful analysis \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Protocol**} & \multirow{2}{*}{**Forwarding Function**} & \multirow{2}{*}{**Minimum Value**} & \multicolumn{4}{c}{**Maximum Values**} \\ \cline{3-6}  & & & \(d_{th}=20\) & \(d_{th}=40\) & \(d_{th}=60\) \\ \hline \multirow{4}{*}{AMCTD} & \(W_{1}\) & 0.0020 & 70 & 70 & 70 \\ \cline{2-6}  & \(W_{2}\) & 0.0143 & 500 & 500 & 500 \\ \cline{2-6}  & \(W_{3}\) & 0.0020 & 70 & 70 & 70 \\ \hline \multirow{4}{*}{iAMCTD} & \(LSNR\) & \(4.52\times 10^{6}\) & \(6.07\times 10^{7}\) & \(2.05\times 10^{7}\) & \(1.06\times 10^{7}\) \\ \cline{2-6}  & \(SQI\) & 0 & \(3.03\times 10^{5}\) & \(1.33\times 10^{5}\) & \(1.01\times 10^{5}\) \\ \cline{2-6}  & \(ECF\) & 0 & 70 & 70 & 70 \\ \cline{2-6}  & \(DDF\) & 0 & \(3.03\times 10^{4}\) & \(1.33\times 10^{4}\) & \(1.01\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Forwarding functions ranges. of \(SQI\) reveals that the range of output from this function is in the order of \(10^{5}\) as shown in Table 1. Such values must not be used directly as forwarding function as holding time calculated from these values would be very much incomparable, having high value of variance. So these values are needed to be made comparable before being used as forwarding function and holding time. This is explained as follows:  
If \(f=30\) kHz and depth difference \(l\) has range in the interval [\(d_{th}=60,R=100\)] absorption coefficient has range [0.5969 dB, 0.9949 dB]. A(l,f) eventually comes out with in the range of [533.2, 1245.4]. For \(f=30\) kHz noise \(N(f)\) comes out to be \(1.12\times 10^{-5}\) using this, Attenuation Noise product range is calculated and finally \(LSNR\) is in range of [\(4.52\times 10^{6},\ 1.06\times 10^{7}\)] making \(SQI\) in range of [\(0,\ 1.01\times 10^{5}\)]. This analysis is for \(d_{th}=60\) m but when \(d_{th}\) is modified to 40 and 20 value of forwarding function for different nodes would be with larger values of variance. Direct use of values in this range is impractical and is needed to made comparable with other forwarding functions. Hence, in our solution linearized version of \(SQI\) is proposed to be used as forwarding function. \(ECF\) based forwarding function is utilized when depth of node is between \(D_{1}=150\) and \(D_{2}=350\). \(ECF\) is defined as:  
\[ECF=\frac{p_{v}\times E_{r}}{D_{i}} \tag{15c}\]  
where \(D_{i}\) can be between \(D_{1}\) and \(D_{2}\).