## 2 Acoustic Propagation Models
### Monterrey Miami Parabolic Equation (MMPE)  
The Monterey-Miami Parabolic Equation model [11] is used to predict underwater acoustic propagation using a parabolic equation which is closer to the Helmholtz equation (wave equation) [12]; this equation is based on Fourier analysis. The sound pressure is calculated in small incremental changes in range and depth, forming a grid. It incorporates randomness and wave motion to the approximation, using a dynamic propagation loss calculation. The authors show that small changes in depth and node distances can drive to big differences in the path loss as a result of the ocean wave's motion impact on acoustic propagation (more details in [3]). The propagation loss formula based on the MMPE model is the following one:  
\[PL(t)=m(f,s,d_{A},d_{B})+\ w(t)+\ e() \tag{2}\]  
where:  
* _PL(t)_: propagation loss while transmitting from node A to node B.
* _m( )_: propagation loss without random and periodic components; obtained from regression of MMPE data.
* _f_: frequency of transmitted acoustic signals (in kHz).
* _d_: sender's depth (in meters).
* _d_: receiver's depth (in meters).
* _s_: Euclidean distance between nodes A and B (in meters).
* _w(t)_: periodic function to approximate signal loss due to wave movement.
* _e()_: signal loss due to random noise or error.  
The _m()_ function represents the propagation loss provided by the MMPE model. According to the logarithmic nature of the data, a nonlinear regression is the best option to provide an approach to the model based on the coefficients, _An_, supplied by the preliminary model. The proposed expression to calculate this function is the following one:  
\[\begin{split} m(f,s,d_{A},d_{B})=\log\left(\left|\frac{\left( \frac{s}{0.914}\right)^{A0}(d_{A})^{A9}s^{A7}((d_{A}-d_{b})^{2})^{A10}}{(s*\ d_{B})^{10\ A5}}\right|\right)+\\ \left(f^{2}\left(\frac{A1}{1+f^{2}}+\frac{40}{4100+f^{2}}+0.00275 \right)+0.003\right)*\left(\frac{s}{914}\right)+A6*d_{B}+A8*s\end{split} \tag{3}\]  
The _w()_ function approximates the signal loss due to the wave movement. It considers the movement of a particle that will oscillate around its location in a sinusoidal way [13]. That movement is represented as circular oscillations that reduce their radius as the depth of the particle increases. The length of that radius is dependent of the energy of the wave and is related to its height value. The common waves have hundreds of meters of wavelength and have an effect up to 50 m of depth.  
For the calculation of the effects of the wave we will consider:  
\[w(t)=h(l_{w},d_{B},t,h_{w},T_{w})\,E(t,T_{w}) \tag{4}\]  
where:  
* _h0_: scale factor function.
* _lw_: ocean wave length (meters).
* _dB_: depth of the receiver node (meters).
* _hw_: wave height (meters).
* _Tw_: wave period (seconds).
* _E0_: function of wave effects in nodes.  
This function contains the elements that resemble the node movement, first by calculating the scale factor _h( )_ and then the effect of the wave in a particular phase of the wave motion. The calculation of the scale factor is as follows:  
\[h\big{(}l_{w},d_{B},t,h_{w},T_{w}\big{)}=\frac{\left(h_{w}\left(1-\big{(}\frac{2 d_{B}}{l_{w}}\big{)}\right)\right)}{0.5}\ast\left|sin\left(\frac{2\pi(mod\ T_{w})}{T_{w}}\right)\right| \tag{5}\]  
The _e( )_ function represents a random term to explain background noise. As the number of sound sources is large and undetermined, this random noise follows a Gaussian distribution and is modeled to have a maximum of 20 dB at the furthest distance. This function is calculated by the following equation:  
\[e(\ )=20\left(\frac{s}{s_{max}}\right)R_{N} \tag{6}\]  
where:  
* _e(_): random noise function
* _s_: distance between the sender and receiver (in meters).
* _smax_: maximum distance (transmission range in meters)
* _RN_: random number from a Gaussian distribution centered in 0 and with variance 1.