# Misalignment-Resilient Propagation Model for Underwater Optical Wireless Links +
## 2 Underwater Photon Propagation Model
### Monte Carlo Algorithm  
In this sense, two different probability density functions provide the photon path length between events and the scattering angle after a given collision. In developing this statistical algorithm, some assumptions were made: (1) the underwater medium was homogeneous (i.e., the optical medium had a uniform composition throughout it, and the scattering and absorption effects did not have a spatial dependency); (2) only elastic scattering existed (i.e., when the scattering occurred, there were no shifts in wavelength and consequently no change in photon energy); (3) the photon generator was perfectly random; (4) no triple backscattering occurred (i.e., a scattered photon was not backscattered to its initial position); and (5) the simulation boundaries were perfectly absorbing (i.e., no photon propagated backward after passing the sensor plane). The first two assumptions were a result of the RTE equation used, which expresses the conservation of energy as the beam propagates underwater and accounts for the losses in the received beam due to scattering and absorption. The other three assumptions were associated with (1) the limitations of numerical methods, which are never truly random, and (2) the actual implementation of the numerical model itself, which sets some limits on the scattering phenomena, neglecting third-order scattering and less likely events. The assumptions were shown to be valid since it was shown previously by [PERSON] that its results were in good agreement with the experiment [4]. There are several Monte Carlo approaches to solving the RTE, and their difference lies in what happens to the photon at each event (absorption, scattering or both) and how to quantify the absorption. Four equivalent methods capable of solving this problem are the albedo weight (AW) method, albedo rejection (AR) method, microscopic Beer-Lambert law (mBLL) method, and absorption-scattering path length rejection (ASPL) method [16]. In the first method, a weight factor given by the albedo coefficient is assigned at the interaction point to the traveling photon propagated through the free path length along the initial direction of propagation in order to propagate to the next one. The weight accounts for the loss of photons by absorption, and after the interaction, it continues propagating in the scattered direction with a reduced weight. In the AR method, the path length of the photon is affected by the extraction of a random number uniformly distributed between 0 and 1. Thus, whether this number is lower or higher than the albedo coefficient, the photon is propagated again, or its trajectory is terminated, respectively. In this case, the weight applied to the photon trajectory is one. In the mBLL method, the detected photons will favor the same trajectories independent of the absorption coefficient distribution, and the weight factor assigned to those trajectories is calculated through the exponential of the absorption coefficient integral along the whole trajectory. The latter consists of extracting two random numbers uniformly distributed between 0 and 1, which allows us to obtain the scattering and absorption coefficients. With these, the individual path lengths are obtained and compared for the two situations. If the path length associated with the scattering event is less than or equal to the path length associated with the absorption event, then the photon is propagated, and if it is greater, then the photon is terminated. Note that for the AW, AR and ASPL methods, the actual trajectories chosen by the detected photons depend on the distributions of both the scattering and absorption coefficients [16]. In this work, the AW method was adopted since it exhibits superior convergence performance when compared with the AR and ASPL methods [16]. As referred to earlier, in this approach, each photon has a given amplitude starting at one, and each time it is absorbed, its amplitude is multiplied by a weight given by the water albedo coefficient.