*4.1.1 Simulation results*

As expected, due to Nyquist sampling theorem, the maximum frequency that can be approximated in the frequency domain will be related to the time step by:

*<sup>f</sup> max* <sup>¼</sup> <sup>1</sup>

In this section, we will provide some simulation results for several emitter

In the second part, we will study the effect of different phase functions in Harbor waters with a perfectly collimated beam. We believe that this is an interesting approach since in these waters', due to the higher scattering coefficient *b*, scattering will be a much larger factor and the amount of scattering that occurs at a given angle is an important factor for the received optical power and the delay spread. We will also provide numerical results for the CIR and an estimative for the 3-dB underwater channel bandwidth. The simulation parameters for Sections 4.1

**Parameter Value** Simulated beam Gaussian beam Beam divergence at 1*=e*<sup>2</sup> 0, 10, 20 and 30 mrad

Beam width 1 mm Receiver diameter 0.1 m Receiver FOV 180° Photons per simulation <sup>100</sup> � <sup>10</sup><sup>6</sup> Phase function Fournier-Forand

**Parameter Value** Simulated beam Gaussian beam Beam divergence at 1*=e*<sup>2</sup> 0 Beam width 1 mm Receiver diameter 0.1 m Receiver FOV 180<sup>o</sup> Photons per simulation <sup>100</sup> � 106

Phase functions Petzold average, FF, SS and HG g ¼ 0*:*924

In the first part, we will simulate the received optical power for clear and coastal waters for different beam divergence values and compare it with the popular Beer-Lambert law. It can be seen in [14] that in these water conditions, the delay spread is so small that they can be considered as a nondispersive medium even for high data rate communications, and for this reason the channel impulse response (CIR)

parameters, underwater conditions, and different phase functions.

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will be studied only for conditions where the medium is dispersive.

and 4.2 are listed in **Tables 3** and **4**, respectively.

**4. Simulation**

**Table 3.**

**Table 4.**

**118**

*Simulation parameters for Section 4.1.*

*Simulation parameters for Section 4.2.*

2*tstep*

*:* (74)

In **Figure 12**, we compare the received power for clear ocean waters with varying beam divergence parameters of 0, 10, 20, and 30 mrad for an emitted beam with a FWHM width of 1 mm. For the phase function, we used the FF, since, as seen in Section 3.2.4, it provides a good fit for Petzold's phase function. However, as mentioned in the "Introduction," when discussing simulation, the impact of different phase functions, especially those which are a good approximation for the Petzold's average phase function, is small for the clear and coastal waters due to the low contribution of scattered photons on the final received power.

Our simulation matches very well the Beer-Lambert law for the case of an emitted Gaussian beam with no divergence, since one of the main assumptions of the law is that the emitted beam is perfectly collimated, and as the beam divergence increases the power loss becomes more accentuated.

We note that for a beam with no divergence, after 10 m, the normalized received optical power would be 2*:*<sup>2</sup> � <sup>10</sup>�<sup>1</sup> for a receiver aperture of 10 cm. Since the beam spread after a distance *z* can be calculated by:

$$w(z) = 2z \times \tan\left(\frac{\phi}{2}\right),\tag{75}$$

**Figure 12.** *Received power for clear ocean waters with varying beam divergence parameter.*

we may expect a beam spread of 20 cm after 10 m for a beam divergence of 10 mrad, potentially reducing by more than half the received optical power (due to the limited aperture of 10 cm). In fact, we can verify from **Figure 12** that the received power for a 10 mrad divergence is 8*:*<sup>1</sup> � <sup>10</sup>�<sup>2</sup> , �40% of the power when the beam was perfectly collimated. The power loss due to the beam spread can be further confirmed by analyzing the position of the received photons along the receiver axis of both configurations, the result being shown in **Figure 13**. From this figure, we can see that for the case of a beam with a divergence of 10 mrad, the photons are spread all over the receiver; while, in the case of a collimated beam, the photons are concentrated along the initial FWHM width of 1 mm.

To further illustrate this, we show, in **Figure 14**, the geometrical losses corresponding to beam divergence. Given that for a perfectly collimated beam in clear ocean waters, power loss is mainly due to absorption and the additional power loss solely due to the spread of the Gaussian beam.

For the case of coastal waters with *<sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*4*m*�1, we can see from **Figure 15** that the Beer law is still a good approximation for a collimated beam, but the law starts to underestimate the received optical power due to the increasing contribution of scattered photons to the final received power.

As was the case for clear waters, the received power drops considerably with an increase in the beam divergence parameter.

Here, the contribution of scattered photons to the final received power increases: after 20 m, 25% of the received power came from scattered photons, and after 30 m, this contribution is elevated to almost 30%. This is further illustrated in **Figure 16**, where the scattering histograms for *z* ¼ 20 and 30 *m* are depicted, and in **Figure 17**, where we compare the received beam for the same distances in clear and coastal waters. For coastal waters, the effect of scattering is already noticeable at the distance of *z* ¼ 10 *m* and becomes more accentuated at *z* ¼ 20 *m*.

**Figure 14.**

**Figure 15.**

**121**

*Beam spread loss in clear waters for beam divergence parameters of 10, 20, and 30 mrad.*

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*DOI: http://dx.doi.org/10.5772/intechopen.85961*

*Received power for coastal waters with varying beam divergence parameter.*

**Table 5** gives the received normalized power for clear ocean waters for various distances and beam divergence parameters. In the following table (**Table 6**), the same results are given for coastal waters.

### **4.2 Harbor waters**

In this section, we compare the performance of different phase functions for Harbor I and Harbor II waters. First, we interpolated values for the Petzold average

**Figure 13.** *Beam spread in clear waters for a receiver located at* z *= 10 m. 0 rad (left) and 10 mrad (right).* *Monte Carlo Radiative Transfer Modeling of Underwater Channel DOI: http://dx.doi.org/10.5772/intechopen.85961*

we may expect a beam spread of 20 cm after 10 m for a beam divergence of 10 mrad, potentially reducing by more than half the received optical power (due to the limited aperture of 10 cm). In fact, we can verify from **Figure 12** that the

the beam was perfectly collimated. The power loss due to the beam spread can be further confirmed by analyzing the position of the received photons along the receiver axis of both configurations, the result being shown in **Figure 13**. From this figure, we can see that for the case of a beam with a divergence of 10 mrad, the photons are spread all over the receiver; while, in the case of a collimated beam, the

To further illustrate this, we show, in **Figure 14**, the geometrical losses corresponding to beam divergence. Given that for a perfectly collimated beam in clear ocean waters, power loss is mainly due to absorption and the additional power

For the case of coastal waters with *<sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*4*m*�1, we can see from **Figure 15** that the Beer law is still a good approximation for a collimated beam, but the law starts to underestimate the received optical power due to the increasing contribution of

As was the case for clear waters, the received power drops considerably with an

Here, the contribution of scattered photons to the final received power increases: after 20 m, 25% of the received power came from scattered photons, and after 30 m, this contribution is elevated to almost 30%. This is further illustrated in **Figure 16**, where the scattering histograms for *z* ¼ 20 and 30 *m* are depicted, and in **Figure 17**, where we compare the received beam for the same distances in clear and coastal waters. For coastal waters, the effect of scattering is already noticeable at the

**Table 5** gives the received normalized power for clear ocean waters for various distances and beam divergence parameters. In the following table (**Table 6**), the

In this section, we compare the performance of different phase functions for Harbor I and Harbor II waters. First, we interpolated values for the Petzold average

*Beam spread in clear waters for a receiver located at* z *= 10 m. 0 rad (left) and 10 mrad (right).*

, �40% of the power when

received power for a 10 mrad divergence is 8*:*<sup>1</sup> � <sup>10</sup>�<sup>2</sup>

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loss solely due to the spread of the Gaussian beam.

scattered photons to the final received power.

increase in the beam divergence parameter.

same results are given for coastal waters.

**4.2 Harbor waters**

**Figure 13.**

**120**

photons are concentrated along the initial FWHM width of 1 mm.

distance of *z* ¼ 10 *m* and becomes more accentuated at *z* ¼ 20 *m*.

**Figure 14.** *Beam spread loss in clear waters for beam divergence parameters of 10, 20, and 30 mrad.*

**Figure 15.** *Received power for coastal waters with varying beam divergence parameter.*

### **Figure 16.**

The SS phase function, seen in Section 3.2.4, predicts scattering for all forward

1 ð Þ *<sup>n</sup>* � <sup>1</sup> <sup>2</sup>

!*<sup>P</sup>*<sup>2</sup>

**Distance**

**Distance**

*ϕ* **(mrad) 10 m 30 m 50 m 70 m** <sup>2</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>1</sup> <sup>1</sup>*:*<sup>09</sup> � <sup>10</sup>�<sup>2</sup> <sup>5</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>6</sup> � <sup>10</sup>�<sup>5</sup> <sup>8</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>2</sup> <sup>8</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>4</sup> <sup>1</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>5</sup> <sup>5</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>7</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>4</sup> <sup>4</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>7</sup> <sup>1</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>2</sup> <sup>1</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>6</sup> <sup>7</sup> � <sup>10</sup>�<sup>8</sup>

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

*DOI: http://dx.doi.org/10.5772/intechopen.85961*

*ϕ* **(mrad) 10 m 20 m 30 m 40 m** <sup>2</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>2</sup> <sup>4</sup>*:*<sup>4</sup> � <sup>10</sup>�<sup>4</sup> <sup>8</sup>*:*<sup>6</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>7</sup> <sup>1</sup>*:*<sup>13</sup> � <sup>10</sup>�<sup>2</sup> <sup>1</sup>*:*<sup>15</sup> � <sup>10</sup>�<sup>4</sup> <sup>1</sup>*:*<sup>4</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>6</sup> � <sup>10</sup>�<sup>8</sup> <sup>4</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>3</sup> <sup>4</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>5</sup> <sup>7</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>7</sup> <sup>1</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>3</sup> <sup>2</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>5</sup> <sup>4</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>7</sup> <sup>1</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>8</sup>

*Pm* <sup>¼</sup> *am*ð Þ <sup>μ</sup> � <sup>3</sup> <sup>2</sup> <sup>þ</sup> *bm*ð Þþ <sup>μ</sup> � <sup>3</sup> *cm,* (77)

*Bp* ¼ *P*<sup>1</sup>

with the values for *am*, *bm*, and *cm* being the ones extracted from [25]. Using the values of *n* ¼ 1*:*16 and μ ¼ 3*:*4319, the backscattering ratio

, similarly to what was done in [14].

of Harbor II waters and how it impacts the received beam just after 4 m.

*Bp* ¼ 0*:*0185 is obtained, considering the scattering to be uniformly distributed over

For the received optical power for Harbor I and Harbor II waters, we can see in **Figures 18** and **19** that the results obtained when using the FF and the SS phase functions are closely matched to the results obtained using the interpolated Petzold average PF, meanwhile the one term HG severely underestimates the received optical power. The main reason for this, as we have seen before, is that the HG is unable to reproduce the high amount of scattering that occur at small angles, as is the case in the Petzold average PF and matched by the other analyzed phase functions. In **Figure 20**, the received beams are compared for Harbor I and Harbor II waters for *z* ¼ 4 and 6 *m*. This figure clearly shows the higher scattering coefficient

, we first calculated the backscattering

*,* (76)

angles. For the angles between 90 and 180°

ratio, as given in [25]:

*Received power for coastal waters.*

where

**Table 5.**

**Table 6.**

*Received power for clear ocean waters.*

all angles above 90°

**123**

*4.2.1 Simulation results*

phase function and considered, as in Section 3.2.5, the benchmark for all other phase functions. The amount of unique scattering angles generated by this interpolation was enough for each photon to have a unique scattering angle.


*Monte Carlo Radiative Transfer Modeling of Underwater Channel DOI: http://dx.doi.org/10.5772/intechopen.85961*

### **Table 5.**

*Received power for clear ocean waters.*


### **Table 6.**

*Received power for coastal waters.*

The SS phase function, seen in Section 3.2.4, predicts scattering for all forward angles. For the angles between 90 and 180° , we first calculated the backscattering ratio, as given in [25]:

$$B\_p = P\_1 \left(\frac{1}{(n-1)^2}\right)^{p\_1},\tag{76}$$

where

$$P\_m = a\_m(\mu - \mathfrak{Z})^2 + b\_m(\mu - \mathfrak{Z}) + c\_m. \tag{77}$$

with the values for *am*, *bm*, and *cm* being the ones extracted from [25].

Using the values of *n* ¼ 1*:*16 and μ ¼ 3*:*4319, the backscattering ratio *Bp* ¼ 0*:*0185 is obtained, considering the scattering to be uniformly distributed over all angles above 90° , similarly to what was done in [14].

### *4.2.1 Simulation results*

For the received optical power for Harbor I and Harbor II waters, we can see in **Figures 18** and **19** that the results obtained when using the FF and the SS phase functions are closely matched to the results obtained using the interpolated Petzold average PF, meanwhile the one term HG severely underestimates the received optical power. The main reason for this, as we have seen before, is that the HG is unable to reproduce the high amount of scattering that occur at small angles, as is the case in the Petzold average PF and matched by the other analyzed phase functions.

In **Figure 20**, the received beams are compared for Harbor I and Harbor II waters for *z* ¼ 4 and 6 *m*. This figure clearly shows the higher scattering coefficient of Harbor II waters and how it impacts the received beam just after 4 m.

phase function and considered, as in Section 3.2.5, the benchmark for all other phase functions. The amount of unique scattering angles generated by this interpolation

was enough for each photon to have a unique scattering angle.

**Figure 16.**

**Figure 17.**

**122**

*Received beams in clear and coastal waters.*

*Scattering histogram for coastal waters with a perfectly collimated beam.*

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**Figure 18.** *Received power for Harbor I waters for selected phase functions.*

When looking at the CIR for Harbor I waters with *z* = 15 m, plotted in **Figure 21**, we can see that the results for the Petzold, FF, and the SS phase functions show similar behavior on the delay profile, while the CIR for the HG phase function

The same analysis is now applied to Harbor II waters. The impulse response is shown in **Figure 23**. The interpretation is that the HG phase function exhibits a response where the power is much more distributed over time when comparing to what is seen in the other phase functions where most of the power is located at the instant *tdirect* ¼ 44 *ns*. The effect of the delay spread is even more noticeable when we analyze the transfer function. In **Figure 24**, this analysis is done for *z* = 10 and 12 m, where we see that in these conditions, due to higher number of scattering events that each photon undergoes, the bandwidth is lower than what we found at Harbor I waters, even when considering smaller propagation distances. In this case, the phase

**Figure 22** plots the transfer function of the channel, from which the 3-dB bandwidth can be extracted, for Harbor I waters corresponding to *z* = 15 and 20 m using the method described in Section 3.3.2. It is possible to verify from these results that the behavior of the SS phase function closely matches the one for the Petzold PF for both distances and the HG underestimates the bandwidth by a large margin in both scenarios. The FF phase function on the other hand, overestimates the bandwidth for the case *z* = 15 m. The 3-dB bandwidth is limited to �300 and

shows a considerable number of delayed photons.

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

*DOI: http://dx.doi.org/10.5772/intechopen.85961*

90 MHz, for 15 and 20 m, respectively.

*Received beams in Harbor I and Harbor II waters.*

**Figure 20.**

**125**

**Figure 19.** *Receiver power for Harbor II waters for selected phase functions.*

*Monte Carlo Radiative Transfer Modeling of Underwater Channel DOI: http://dx.doi.org/10.5772/intechopen.85961*

**Figure 20.** *Received beams in Harbor I and Harbor II waters.*

When looking at the CIR for Harbor I waters with *z* = 15 m, plotted in **Figure 21**, we can see that the results for the Petzold, FF, and the SS phase functions show similar behavior on the delay profile, while the CIR for the HG phase function shows a considerable number of delayed photons.

**Figure 22** plots the transfer function of the channel, from which the 3-dB bandwidth can be extracted, for Harbor I waters corresponding to *z* = 15 and 20 m using the method described in Section 3.3.2. It is possible to verify from these results that the behavior of the SS phase function closely matches the one for the Petzold PF for both distances and the HG underestimates the bandwidth by a large margin in both scenarios. The FF phase function on the other hand, overestimates the bandwidth for the case *z* = 15 m. The 3-dB bandwidth is limited to �300 and 90 MHz, for 15 and 20 m, respectively.

The same analysis is now applied to Harbor II waters. The impulse response is shown in **Figure 23**. The interpretation is that the HG phase function exhibits a response where the power is much more distributed over time when comparing to what is seen in the other phase functions where most of the power is located at the instant *tdirect* ¼ 44 *ns*.

The effect of the delay spread is even more noticeable when we analyze the transfer function. In **Figure 24**, this analysis is done for *z* = 10 and 12 m, where we see that in these conditions, due to higher number of scattering events that each photon undergoes, the bandwidth is lower than what we found at Harbor I waters, even when considering smaller propagation distances. In this case, the phase

**Figure 18.**

**Figure 19.**

**124**

*Received power for Harbor I waters for selected phase functions.*

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*Receiver power for Harbor II waters for selected phase functions.*

**Figure 21.** *Channel impulse response for Harbor I waters with* z *= 15 m.*

**Figure 23.**

**Figure 24.**

**127**

*Channel impulse response for Harbor II waters with* z *= 10 m.*

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*Transfer function for Harbor II waters.* z *= 10 and 12 m.*

**Figure 22.** *Transfer function for Harbor I waters. z = 15 and 20 m.*

*Monte Carlo Radiative Transfer Modeling of Underwater Channel DOI: http://dx.doi.org/10.5772/intechopen.85961*

**Figure 23.** *Channel impulse response for Harbor II waters with* z *= 10 m.*

**Figure 24.** *Transfer function for Harbor II waters.* z *= 10 and 12 m.*

**Figure 21.**

**Figure 22.**

**126**

*Channel impulse response for Harbor I waters with* z *= 15 m.*

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*Transfer function for Harbor I waters. z = 15 and 20 m.*


### **Table 7.**

*Results for received power, rms delay, and 3-dB BW for Harbor I and Harbor II waters.*

functions FF and SS give similar results to the Petzold average PF, the bandwidth being limited to �150 and 100 MHz, for distances of 10 and 12 m, respectively.

In the following table, **Table 7**, we summarize the most relevant numerical results for Harbor I and Harbor II waters as we did for the case of clear and coastal waters in Section 4.1. For brevity, we only show the results obtained using the Petzold average phase function; however, as noted in the results presented previously, one may expect similar numerical results when using the FF or the SS phase function as they predict scattering angles similar to those predicted by Petzold.

### **4.3 Validation of simulation results**

For further validation of the simulation model, we compared the values obtained in our simulation with the ones obtained by Sahu in [14]. For this, we used the same parameters used in his simulations, a Gaussian beam profile with FWHM beam width of 2 mm, a beam divergence parameter of 1.5 mrad, and a receiver aperture of 10 cm.

different propagation distances and the comparison against his intercept points for these distances are depicted in **Figure 27**. The CIR in that figure is normalized by the straight-line time of propagation (*tdirect*) for a better visualization of the delay spread. The black line in the figure is the 20 dB intercept point as given in [14], and we can see that again the values obtained in the simulation are quite close to the

In this chapter, a simple yet powerful tool for modeling the propagation of photons in an underwater channel is presented, which provides more accurate results than the conventional Beer-Lambert law. The algorithm presented here highlights the fundamental processes of absorption and scattering, hopefully helping the reader to better understand the physics of photon propagation in an underwater environment. In the first part of the simulation, we showed how important the collimation of a

beam is and how the beam spread can cause losses up to 13 dB even after only 10 m. It was also possible to verify that in clear waters, a good signal to noise ratio is

In the second part of the simulation, the impact of the water turbidity on the underwater link was assessed, where it was concluded that increasing turbidity limits not only the received optical power, but also the maximum bandwidth of the channel. An adequate choice of phase function is important as simpler phase functions, such as the Henyey-Greenstein, may wrongly predict the numerical results in these conditions. For the first time results for the received power, channel impulse response and 3-dB bandwidth are obtained from Monte Carlo RTE simulations for different phase functions. The SS and FF are compared with the Petzold average PF and results are shown to

achievable for several tens of meters in both clear and coastal waters.

ones obtained by Sahu, which validate the model proposed here.

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

*DOI: http://dx.doi.org/10.5772/intechopen.85961*

*Comparison between this simulation model and [14].*

**5. Conclusion**

**129**

**Figure 26.**

The comparison for clear and coastal waters between our simulation and [14] is shown in **Figure 25**, and we can see that our simulation closely matches the results obtained by Sahu.

For Harbor waters, Sahu only considered Harbor II waters with *<sup>c</sup>* <sup>¼</sup> <sup>2</sup>*:*<sup>19</sup> *<sup>m</sup>*�1, hence we could not compare the values we obtained for Harbor I waters with

*<sup>c</sup>* <sup>¼</sup> <sup>1</sup>*:*1*m*�1. The results are illustrated in **Figure 26**, where a good agreement is seen. Furthermore, Sahu defined a delay spread quantity which is the time period over which the CIR falls to �20 dB below its peak. He provided numerical values for

**Figure 25.** *Comparison between this simulation model and [14].*

*Monte Carlo Radiative Transfer Modeling of Underwater Channel DOI: http://dx.doi.org/10.5772/intechopen.85961*

**Figure 26.** *Comparison between this simulation model and [14].*

different propagation distances and the comparison against his intercept points for these distances are depicted in **Figure 27**. The CIR in that figure is normalized by the straight-line time of propagation (*tdirect*) for a better visualization of the delay spread. The black line in the figure is the 20 dB intercept point as given in [14], and we can see that again the values obtained in the simulation are quite close to the ones obtained by Sahu, which validate the model proposed here.

### **5. Conclusion**

functions FF and SS give similar results to the Petzold average PF, the bandwidth being limited to �150 and 100 MHz, for distances of 10 and 12 m, respectively. In the following table, **Table 7**, we summarize the most relevant numerical results for Harbor I and Harbor II waters as we did for the case of clear and coastal waters in Section 4.1. For brevity, we only show the results obtained using the Petzold average phase function; however, as noted in the results presented previously, one may expect similar numerical results when using the FF or the SS phase function as they predict scattering angles similar to those predicted by Petzold.

*Results for received power, rms delay, and 3-dB BW for Harbor I and Harbor II waters.*

HI *<sup>z</sup>* = 15 m <sup>2</sup>*:*<sup>6</sup> � <sup>10</sup>�<sup>6</sup> 2.93 <sup>310</sup> HI *<sup>z</sup>* = 20 m <sup>1</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>7</sup> 6.18 <sup>80</sup> HII *<sup>z</sup>* =8m <sup>5</sup>*:*<sup>4</sup> � <sup>10</sup>�<sup>6</sup> 2.06 <sup>440</sup> HII *<sup>z</sup>* = 10 m <sup>7</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>7</sup> 3.51 <sup>185</sup> HII *<sup>z</sup>* = 12 m <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>7</sup> 4.78 <sup>110</sup> HII *<sup>z</sup>* = 14 m <sup>2</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>8</sup> 7.87 <sup>53</sup>

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**Received power** *τRMS* **(ns) 3-dB BW (MHz)**

For further validation of the simulation model, we compared the values obtained in our simulation with the ones obtained by Sahu in [14]. For this, we used the same parameters used in his simulations, a Gaussian beam profile with FWHM beam width of 2 mm, a beam divergence parameter of 1.5 mrad, and a receiver aperture of 10 cm. The comparison for clear and coastal waters between our simulation and [14] is shown in **Figure 25**, and we can see that our simulation closely matches the results

For Harbor waters, Sahu only considered Harbor II waters with *<sup>c</sup>* <sup>¼</sup> <sup>2</sup>*:*<sup>19</sup> *<sup>m</sup>*�1, hence we could not compare the values we obtained for Harbor I waters with *<sup>c</sup>* <sup>¼</sup> <sup>1</sup>*:*1*m*�1. The results are illustrated in **Figure 26**, where a good agreement is seen. Furthermore, Sahu defined a delay spread quantity which is the time period over

which the CIR falls to �20 dB below its peak. He provided numerical values for

**4.3 Validation of simulation results**

obtained by Sahu.

**Figure 25.**

**128**

*Comparison between this simulation model and [14].*

**Table 7.**

In this chapter, a simple yet powerful tool for modeling the propagation of photons in an underwater channel is presented, which provides more accurate results than the conventional Beer-Lambert law. The algorithm presented here highlights the fundamental processes of absorption and scattering, hopefully helping the reader to better understand the physics of photon propagation in an underwater environment.

In the first part of the simulation, we showed how important the collimation of a beam is and how the beam spread can cause losses up to 13 dB even after only 10 m. It was also possible to verify that in clear waters, a good signal to noise ratio is achievable for several tens of meters in both clear and coastal waters.

In the second part of the simulation, the impact of the water turbidity on the underwater link was assessed, where it was concluded that increasing turbidity limits not only the received optical power, but also the maximum bandwidth of the channel. An adequate choice of phase function is important as simpler phase functions, such as the Henyey-Greenstein, may wrongly predict the numerical results in these conditions.

For the first time results for the received power, channel impulse response and 3-dB bandwidth are obtained from Monte Carlo RTE simulations for different phase functions. The SS and FF are compared with the Petzold average PF and results are shown to

**Figure 27.** *CIR for various distances in Harbor II waters, where the delay spread is highlighted.*

agree well. Moreover, for validation purposes, our simulation results are further compared with those obtained by Sahu [14] and good agreement is shown to exist.

**Author details**

Portugal

**131**

Rafael M.G. Kraemer1,2, Luís M. Pessoa2 and Henrique M. Salgado1,2\*

2 INESC TEC—INESC Technology and Science (Formerly INESC Porto), Porto,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

1 Faculty of Engineering, University of Porto, Portugal

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

*DOI: http://dx.doi.org/10.5772/intechopen.85961*

provided the original work is properly cited.

\*Address all correspondence to: henrique.salgado@inesctec.pt

We believe that the approach presented here provides a general and flexible technique for numerically solving the radiative transfer equation, accessible to any reader with basic programming skills. Besides the conditions simulated in this chapter, the simulation program may be used to simulate innumerous other conditions, like misalignment between receiver and transmitter, smaller aperture sizes, limited FOVs, and other light sources other than Gaussian beams.

### **Acknowledgements**

This work is financed by the ERDF—European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation —COMPETE 2020 Programme, and by National Funds through the FCT— Portuguese Foundation for Science and Technology, I.P., within project POCI-01-0145-FEDER-031971.

*Monte Carlo Radiative Transfer Modeling of Underwater Channel DOI: http://dx.doi.org/10.5772/intechopen.85961*
