# Underwater Electromagnetic Sensor Networks, Part II: Localization and Network Simulations
## 2 Parametrization of the Communication Channel  
We consider a U-WSN consisting of \(N\) nodes, and we refer to the set of nodes as \(\mathcal{N}=\{1,\dots,N\}\). We performed real measurements in the underwater medium to characterize the channel, where the specific procedure is described in Part I of this paper, presented in [9]. Figure 1 shows the specific signal strengths measured between receiver and transmitter with loop antennas of an 80 cm radius at the seabed. We only show the lowest frequency range (from 10 to 150 kHz), since it offers the lowest attenuation. Figure 2 illustrates the temporal variation and reception noise at 40 kHz along almost two hours. Note also that since the attenuation increases in higher frequencies, the overall bandwidth must be narrow. For the previous reasons, the communication frequencies should be established at a low frequency (\(\sim\)10 kHz) and present a narrow bandwidth (\(\leq\)3 kHz). In Part I of this paper (see [9]), we considered a sophisticated simulator of EM fields to validate the measurement procedure. For the purpose of this paper, we use the EM field simulator to extrapolate the attenuation levels at distances greater than those considered during the measurement campaign. The simulation results appear in Table 1, which require a high computation time. In order to simulate a complete U-WSN with multiple nodes transmitting multiple packets each, we need to express the channel model in a manner that can be simulated with low computational cost. In this section,  
Figure 1: Measurements of the underwater wireless channel: frequency response at multiple distances. Figure 2: Measurements of the underwater wireless channel: temporal variation at 40 kHz. we propose to approximate the EM model with a parametric path loss model, which basically extends the log-distance path loss model by including a linear term. Let \(L(d)\) denote the path loss (dB) at \(d\) meters from the transmitter. Let \(d_{0}\) denote the reference distance (in meters), and let \(X\) be a random variable that considers random variations in the channel. Then, the proposed model with parameter \(\eta_{1}\), \(\eta_{2}\) has the following form:  
\[L(d)=L(d_{0})+\eta_{1}\frac{d}{d_{0}}+\eta_{2}\cdot 10\cdot\log\left(\frac{d}{d _{0}}\right)+X \tag{1}\]  
We remark that in general, the path loss model should be frequency dependent, but we have assumed the simpler narrow band case. As we will see below, this parametric form provides an efficient characterization of the underwater wireless channel for our network requirements. In order to find parameters \(L(d_{0})\), \(\eta_{1}\) and \(\eta_{2}\) for the model, we have considered carrier frequency \(10\,\mathrm{kHz}\) and \(d_{0}=3\). Let \(\mathbf{x}=[L(d_{0}),\eta_{1},\eta_{2}]^{\top}\) be the vector of unknown parameters. Now, for the parametric model, we introduce the matrix \(\mathbf{A}=[\mathbf{1},\mathbf{d}/d_{0},10\log(\mathbf{d}/d_{0})]\) of size \(14\times 3\), where \(\mathbf{1}\) is a vector of ones. Thus, we have a system of equations, \(\boldsymbol{\ell}=\mathbf{A}\mathbf{x}\), which, using the mean square error (MSE) criterion, can be solved as follows: \(\mathbf{x}=(\mathbf{A}^{\top}\mathbf{A})^{-1}\mathbf{A}^{\top}\boldsymbol{\ell}\), obtaining path loss terms \(\eta_{1}=7.2525\), \(\eta_{2}=3.6678\) and offset \(L(d_{0})=63.0633\,\mathrm{dB}\). The MSE results in \(0.3225\,\mathrm{dB}\). Figure 3 shows the real measurements at \(10\,\mathrm{kHz}\) (black dots) as presented in [9], the simulated EM channel (solid blue) also presented in [9] and the proposed parametric model (dashed red) described by Equation (1). Table 1 presents the measuring distances and the corresponding losses. It is apparent that the parametric model (1) fits the measurements accurately in the given distances. The additive noise \(X\) can be also estimated from the trace of measurements displayed in Figure 2. Figure 4 shows the histogram for such a trace together with a normal distribution fit.