## 3 Acoustic Models
### Delay Computation Model
#### 3.3.2 Monterey-Miami Parabolic Equation  
Monterey-Miami Parabolic Equation (MMPE) is also an accurate model that computes channel losses in acoustic channel. It is formulated by using the main principle of wave equation. It is highly complex model, however more accurate than the Thorp model. It also shows the impacts of variations in the depth information of the sender and receiver nodes on the signal quality. Moreover, it considers the Euclidean distance between the communicating nodes and the frequency of the transmitted signal.  
The basic formula of MMPE model is given below [30]:  
\[PL(t)=m(f,s,d_{A},d_{B})+w(t)+e() \tag{23}\]  
where:  
\(PL(t)\): propagation loss occurs during transmission from node A to node B.  
\(m()\): propagation loss with random and periodic components; obtained from regression of MMPE data.  
\(f\): frequency of transmitted acoustic signal in kHz.  
\(s\): Euclidean distance between node A and node B in meters.  
\(w(t)\): periodic function to approximate signal loss due to wave movement.  
\(d_{A}\): sender's depth in meters.  
\(d_{B}\): receiver's depth in meters.  
\(e()\): signal loss due to random noise error.  
The first term of the Equation (23) calculates the propagation loss caused by the random and periodic components. It also performs nonlinear regression on the data to obtain _A(n)_ coefficients. By using the resulting data, \(m(f,s,d_{A},d_{B})\) function computes the propagation loss [31] as follows:  
\[m(f,s,d_{A},d_{B})=log\left(\left|\frac{\left(\frac{s}{0.9}\right) ^{A0}\!d_{A}{}^{A9}s^{A7}\!\left(\left(d_{A}-d_{B}\right)^{2}\right)^{A10}}{ \left(s*d_{B}\right)^{10.45}}\right|\right)\right.\] \[\left.+\left(f^{2}\left(\frac{A1}{1+f^{2}}+\frac{41}{4100+f^{2}} +0.002\right)+0.003\right)*\right.\] \[\left.\left(\frac{s}{914}\right)+A6*d_{B}+A8*s\right. \tag{24}\]  
In the second part of the Equation (23), MMPE model estimates the losses caused due to wave motion. For this purpose, it considers the sinusoidal movement of the water particles around the acoustic signal. The _w(t)_[30] function considers scale factor function, wavelength and wave effects function to predict the signal loss caused by the wave movement. It can be mathematically written as:  
\[w(t)=h(l_{w},t,d_{B},h_{w},T_{w})*E(t,T_{w}) \tag{25}\]  
where:  
_h(s)_: scale factor function.  
\(l_{w}\): ocean wavelength in meters.  
\(h_{w}\): wave height in meters.  
\(d_{B}\): receiver's depth in meters.  
\(T_{w}\): wave period in seconds.  
\(E()\): function of wave effects in nodes.  
As, we assume the continuous node's movement in aqueous environment, so _h(s)_ depicts the effect on wavelength of transmitted signal due to receiver node's movement. It also scales the movement of signal with the distance. We compute scale factor by using the following formula:  
\[h(T_{w},l_{w},t,h_{w},d_{B})=\left(\frac{\left(h_{w}\left(1-( \frac{2d_{B}}{l_{w}})\right)\right)}{0.5}\right)*\] \[\left|sin\left(\frac{2\pi(mod(T_{w}))}{T_{w}}\right)\right| \tag{26}\]  
Third term of Equation (23), _e(_ ) describes the random background noise. In order to estimate the noise in dense conditions, the random noise function used by _e(_ ) follows a Gaussian distribution. The random noise function is based on the proportion of the distance between communicating nodes and the source transmitter range.  
\[e()=20\left(\frac{s}{s_{max}}\right)R_{N} \tag{27}\]  
where:\(e()\): random noise function.  
\(s_{max}\): maximum transmission range in meters.  
\(R_{N}\): random number from a Gaussian distribution centered at 0 with variance 1.