**2. The radiative transfer equation**

One important law of geometrical radiometry is the n-squared law for radiance. To derive it, we consider two mediums separated by a transparent surface; in that case, Snell's law states that the angles of an incident ray from medium 1 to medium 2 are related by:

$$n\_1 \sin \left(\theta\_1\right) = n\_2 \sin \left(\theta\_2\right), \tag{1}$$

As seen in [3], this result is known as the fundamental theorem of radiometry and it states that the radiance divided by the refraction index squared is constant along any path; however, because all real substances will cause some absorption and scattering on the incident photons, the validity of theorem is restricted for paths in

*T j* ð Þ *; <sup>j</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*<sup>2</sup>

where *T j* ð Þ *; j* þ 1 is the Fresnel transmittance of the interface between the media with refractive indexes *nj* and *nj*þ1. From this, we can see that the radiance along a path will change due to variations in the real index of refraction along that same path. The index of refraction in a water body will change from point to point by random molecular motions, by organics or inorganic particulate matter and by

<sup>Δ</sup>*<sup>r</sup>* <sup>¼</sup> *D L=n*<sup>2</sup> ð Þ

being *D=Dr* the total rate of change along the path. The total rate of change can be expressed in terms of the advective derivative, or the substantive derivative, as

*s n*2 1 *,* (10)

! to point *x*

*Dr ,* (11)

*Dt ,* (12)

� �*,* (13)

! �∇*,* (14)

<sup>þ</sup> ^<sup>ξ</sup> � <sup>∇</sup>*,* (15)

! *=v:*

! at time *t* for

! <sup>þ</sup><sup>Δ</sup> *<sup>x</sup>*

! and

If we consider that a ray of radiance *L*<sup>1</sup> starts in a medium with index of refraction *n*<sup>1</sup> and goes through successive refractions and ends up with radiance *LS* in a medium of refractive index *ns*, we can come up with a formulation for succes-

sive crossings of a ray from one medium to another:

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

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

turbulent fluctuations in temperature and salinity.

! � � �

taking the limit as Δ*r* ¼ Δ *x*

referred by Mobley:

wavelength λ:

The operator *D=Dt* is:

RTE for unpolarized radiance:

**101**

Considering a beam of radiance traveling from point *x*

lim Δ*r*!0

where *v* is the speed of light in the medium at position *x*

v *x* !; *t*; λ

and the gradient operator ∇ is defined in the usual way. From these developments, we can rewrite Eq. (11) as:

> *D Dr* <sup>¼</sup> <sup>1</sup> *v D Dt* <sup>¼</sup> <sup>1</sup> *v ∂ ∂t*

valid for photons traveling with speed *<sup>v</sup>* in the direction ^<sup>ξ</sup> <sup>¼</sup>*<sup>v</sup>*

Writing the expression for the change in *L=n*<sup>2</sup> along a path that combines all the physical phenomena causing that change, we obtain the most general form of the

*D Dt* <sup>¼</sup> *<sup>∂</sup> ∂t* þ *v*

<sup>Δ</sup> *<sup>L</sup>=n*<sup>2</sup> ð Þ

*D Dr* <sup>¼</sup> <sup>1</sup> *v D*

� � <sup>¼</sup> <sup>c</sup>*=*<sup>n</sup> *<sup>x</sup>*

!; *t*; λ

� � � ! 0:

*LS L*1 <sup>¼</sup> <sup>Y</sup>*s*�<sup>1</sup> *j*¼1

vacuum.

where *n* is the index of refraction of the medium. Considering that a ray is simply a narrow beam of photons traveling in almost the same direction, and if we consider two narrow rays, incident and refracted, that will have solid angles given by:

$$
\Delta\Omega\_1 = \sin\theta\_1 \Delta\theta\_1 \Delta\phi,\tag{2}
$$

$$
\Delta\Omega\_2 = \sin\theta\_2 \Delta\theta\_2 \Delta\phi,\tag{3}
$$

and as seen in [3], squaring each side of Eq. (1) and multiplying by the Azimuthal spread, Δ*ϕ*, we obtain:

$$n\_1^2 \cos \theta\_1 \Omega\_1 = n\_2^2 \cos \theta\_2 \Omega\_2,\tag{4}$$

which is known as the Straubel's invariant. Consider now the radiances of the two rays, incident and refracted, defined as:

$$L\_1 = \frac{\Delta P\_1}{\Delta A\_1 \Delta \Omega\_2},\tag{5}$$

$$L\_2 = \frac{\Delta P\_2}{\Delta A\_2 \Delta \Omega\_2},\tag{6}$$

where Δ*P*1*,* <sup>2</sup> is the spectral radiant power and Δ*A*1*,* <sup>2</sup> is the cross section area of incident and refracted rays. The ratio between the refracted and incident rays is called the Fresnel transmittance:

$$\frac{\Delta P\_1}{\Delta P\_2} = T,\tag{7}$$

from which we can obtain:

$$\frac{L\_2}{n\_2^2} = T \frac{L\_1}{n\_1^2},\tag{8}$$

which is the n-squared law for radiance. For the case of *T* ¼ 1, which is the case of normal incidence on an air-water surface, Eq. (8) becomes:

$$\frac{L\_2}{n\_2^2} = \frac{L\_1}{n\_1^2}.\tag{9}$$

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

integrodifferential equation involving different independent variables, an exact analytical solution is hard to find. In view of this, solving the RTE numerically is a preferred approach, being the most popular one the Monte Carlo simulation.

One important law of geometrical radiometry is the n-squared law for radiance. To derive it, we consider two mediums separated by a transparent surface; in that case, Snell's law states that the angles of an incident ray from medium 1 to medium 2

where *n* is the index of refraction of the medium. Considering that a ray is simply a narrow beam of photons traveling in almost the same direction, and if we consider two narrow rays, incident and refracted, that will have solid angles

and as seen in [3], squaring each side of Eq. (1) and multiplying by the Azi-

which is known as the Straubel's invariant. Consider now the radiances of the

*<sup>L</sup>*<sup>1</sup> <sup>¼</sup> <sup>Δ</sup>*P*<sup>1</sup> Δ*A*1ΔΩ<sup>2</sup>

*<sup>L</sup>*<sup>2</sup> <sup>¼</sup> <sup>Δ</sup>*P*<sup>2</sup> Δ*A*2ΔΩ<sup>2</sup>

> Δ*P*<sup>1</sup> Δ*P*<sup>2</sup>

*L*2 *n*2 2

> *L*2 *n*2 2 ¼ *L*1 *n*2 1

of normal incidence on an air-water surface, Eq. (8) becomes:

where Δ*P*1*,* <sup>2</sup> is the spectral radiant power and Δ*A*1*,* <sup>2</sup> is the cross section area of incident and refracted rays. The ratio between the refracted and incident rays is

> <sup>¼</sup> *<sup>T</sup> <sup>L</sup>*<sup>1</sup> *n*2 1

which is the n-squared law for radiance. For the case of *T* ¼ 1, which is the case

<sup>1</sup> cos *<sup>θ</sup>*1Ω<sup>1</sup> <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>

*n*2

two rays, incident and refracted, defined as:

*n*<sup>1</sup> sin ð Þ¼ *θ*<sup>1</sup> *n*<sup>2</sup> sin ð Þ *θ*<sup>2</sup> *,* (1)

ΔΩ<sup>1</sup> ¼ sin *θ*1Δ*θ*1Δ*ϕ,* (2) ΔΩ<sup>2</sup> ¼ sin *θ*2Δ*θ*2Δ*ϕ,* (3)

<sup>2</sup> cos *θ*2Ω2*,* (4)

*,* (5)

*,* (6)

¼ *T,* (7)

*,* (8)

*:* (9)

**2. The radiative transfer equation**

*Wireless Mesh Networks - Security, Architectures and Protocols*

are related by:

given by:

muthal spread, Δ*ϕ*, we obtain:

called the Fresnel transmittance:

from which we can obtain:

**100**

As seen in [3], this result is known as the fundamental theorem of radiometry and it states that the radiance divided by the refraction index squared is constant along any path; however, because all real substances will cause some absorption and scattering on the incident photons, the validity of theorem is restricted for paths in vacuum.

If we consider that a ray of radiance *L*<sup>1</sup> starts in a medium with index of refraction *n*<sup>1</sup> and goes through successive refractions and ends up with radiance *LS* in a medium of refractive index *ns*, we can come up with a formulation for successive crossings of a ray from one medium to another:

$$\frac{L\_S}{L\_1} = \prod\_{j=1}^{s-1} T(j, j+1) \frac{n\_s^2}{n\_1^2},\tag{10}$$

where *T j* ð Þ *; j* þ 1 is the Fresnel transmittance of the interface between the media with refractive indexes *nj* and *nj*þ1. From this, we can see that the radiance along a path will change due to variations in the real index of refraction along that same path. The index of refraction in a water body will change from point to point by random molecular motions, by organics or inorganic particulate matter and by turbulent fluctuations in temperature and salinity.

Considering a beam of radiance traveling from point *x* ! to point *x* ! <sup>þ</sup><sup>Δ</sup> *<sup>x</sup>* ! and taking the limit as Δ*r* ¼ Δ *x* ! � � � � � � ! 0:

$$\lim\_{\Delta r \to 0} \frac{\Delta(L/n^2)}{\Delta r} = \frac{D(L/n^2)}{Dr},\tag{11}$$

being *D=Dr* the total rate of change along the path. The total rate of change can be expressed in terms of the advective derivative, or the substantive derivative, as referred by Mobley:

$$\frac{D}{Dr} = \frac{1}{v} \frac{D}{Dt},\tag{12}$$

where *v* is the speed of light in the medium at position *x* ! at time *t* for wavelength λ:

$$\mathbf{v}\left(\overrightarrow{\mathbf{x}};t;\lambda\right) = \mathbf{c}/\mathbf{n}\left(\overrightarrow{\mathbf{x}};t;\lambda\right),\tag{13}$$

The operator *D=Dt* is:

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + \overrightarrow{v} \cdot \nabla,\tag{14}$$

and the gradient operator ∇ is defined in the usual way. From these developments, we can rewrite Eq. (11) as:

$$\frac{D}{Dr} = \frac{1}{v} \frac{D}{Dt} = \frac{1}{v} \frac{\partial}{\partial t} + \hat{\xi} \cdot \nabla,\tag{15}$$

valid for photons traveling with speed *<sup>v</sup>* in the direction ^<sup>ξ</sup> <sup>¼</sup>*<sup>v</sup>* ! *=v:*

Writing the expression for the change in *L=n*<sup>2</sup> along a path that combines all the physical phenomena causing that change, we obtain the most general form of the RTE for unpolarized radiance:

$$\frac{1}{v}\frac{\partial}{\partial t}\left(\frac{L}{n^2}\right) + \hat{\xi} \cdot \nabla \left(\frac{L}{n^2}\right) = -c \left(\frac{L}{n^2}\right) + L^E + L^I + L^S. \tag{16}$$

hand, if the excited molecule emits a photon of lower energy, i.e., a photon with a longer wavelength, the incident photon goes through a process called inelastic scattering. As seen in [3], the absorption coefficient, that will be introduced later in this chapter, accounts for both the conversion of radiant energy into heat and the loss of power at wavelength λ by inelastic scattering to other wavelengths. Mobley [3] refers to this as true absorption. In view of this, in the remaining of this chapter, we will treat the scattering as elastic, the loss of energy due inelastic scattering being

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

To derive a mathematical coefficient for scattering and absorption, we assume

*PI*ð Þ¼ λ *PA*ð Þþ λ *PS*ð Þþ λ*; θ PT*ð Þλ *:* (21)

*:* (22)

*:* (23)

*c*ð Þ¼ λ *a*ð Þþ λ *b*ð Þλ *:* (24)

*<sup>c</sup>*ð Þ*<sup>λ</sup> :* (25)

A volume of water Δ*V* with thickness Δ*r* is illuminated by a light beam with wavelength λ*:* Some part of the incident power, *PI*ð Þλ , will be absorbed within the volume of water, *PA*ð Þλ ; some part, PSð Þ λ*; θ* , is scattered out of the beam at an angle *θ* and the remaining light power, *PT*ð Þλ , is transmitted. Then by conservation of energy:

Using Eq. (22), we can define the ratio between the absorbed and incident power as the absorptance and the ratio between scattered power and incident power as scatterance. As already stated, when solving the RTE the IOPs usually employed are the absorption and the scattering coefficients which are the absorptance and scatterance per unit distance of the medium. Taking the limit of these terms as the

> *a*ð Þ¼ λ lim *Δr*!0

> *b*ð Þ¼ λ lim *Δr*!0

The beam attenuation coefficient present in the RTE (Eqs. (16) and (20)) is

Another useful IOP showing in the RTE is the single scattering albedo, ω0, that is defined as the ratio between the scattering and the beam attenuation coefficient:

*<sup>ω</sup>*<sup>0</sup> <sup>¼</sup> *<sup>b</sup>*ð Þ*<sup>λ</sup>*

In waters where the beam attenuation is due primarily to scattering, ω<sup>0</sup> is near one and when the beam attenuation is due primarily to absorption, ω<sup>0</sup> is near zero.

*PA*ð Þ*λ PI*ð Þ*λ Δr*

*PS*ð Þ*λ PI*ð Þ*λ Δr*

water thickness Δ*r* becomes infinitesimally small, we have:

included in the absorption coefficient.

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

the model shown in **Figure 1**.

written as:

**Figure 1.**

**103**

*Geometry of IOPs for a volume of water ΔV.*

In Eq. (17), *c* is the beam attenuation coefficient, *LE, LI ,* and *L<sup>S</sup>* are the path functions for elastic scattering, inelastic scattering, and spontaneous emission, respectively. The path function for spontaneous emission is also referred in [3] as the source path function and can be considered as the contribution to total radiance from a light source and in the case of UOWC systems a laser diode or a light emitting diode (LED).

For UOWC systems, we are interested in the time-independent RTE in horizontally homogenous water bodies with a constant index of refraction. Because of this, the factor *<sup>n</sup>*�<sup>2</sup> will divide both sides of Eq. (15). Hence, with *<sup>∂</sup>L=∂<sup>t</sup>* <sup>¼</sup> 0 and using the notation adopted by Mobley where *x*<sup>3</sup> ¼ *z* and ξ<sup>3</sup> ¼ cosθ*,* we obtain:

$$
\hat{\xi} \cdot \nabla L = \mathfrak{k}\_3 \frac{\partial L}{\partial \mathfrak{x}\_3} = \cos \theta \frac{dL(\theta, \phi)}{dz}. \tag{17}
$$

It is also usual [3] to combine *LI* and *L<sup>S</sup>* as an effective source function *<sup>S</sup>* <sup>¼</sup> *LI* <sup>þ</sup> *<sup>L</sup><sup>S</sup>* . The RTE then becomes:

$$\cos\theta \frac{d\mathcal{L}(\theta,\phi)}{dz} = -\mathcal{c}\mathcal{L}(\theta,\phi) + \mathcal{L}^{E}(\theta,\phi) + \mathcal{S}(\theta,\phi),\tag{18}$$

and the path function for elastic scattering, *L<sup>E</sup>*:

$$L^E(\theta, \phi) = \int\_{4\pi} L(\theta', \phi') \mathfrak{f}(\theta', \phi' \to \theta, \phi) d\Omega(\theta', \phi'), \tag{19}$$

Combining all these terms in Eq. (18), we can come up with the standard form of the RTE. As a simple statement, the IOPs and boundary conditions will go into the RTE and the final radiance will come out as a result. The standard form of the RTE, as shown in [4], is given by:

$$\cos\theta \frac{d\mathcal{L}(\theta,\phi)}{dz} = -\mathcal{L}(\theta,\phi) + \alpha\_0 \left[ \int\_{4\pi} \tilde{\mathfrak{J}}(\theta',\phi' \to \theta,\phi) \mathcal{L}(\theta',\phi') d\mathfrak{A}(\theta',\phi') + \mathcal{S}(\theta,\phi), \tag{20}$$

where *L*ð Þ *θ; ϕ* represents the radiance in the direction ð Þ *θ; ϕ* , ω<sup>0</sup> is the single scattering albedo, c is the beam attenuation coefficient, and e*β θ*ð Þ *; ϕ* is the scattering phase function. Each one of these terms will be explained in more detail in the following sections.

### **2.1 Inherent optical properties**

During the interaction of a photon with a water molecule, one of two things may happen: the photon may be absorbed, leaving the water molecule in a state with higher internal energy, or the photon may undergo scattering. Such scattering occurs when there is a change in the direction of the propagation of the photon or in the photon energy—or in both.

When scattering happens, if the molecule that interacted with the photon returns to its original internal energy state by emitting a photon of the same energy as the absorbed one, the scattering process is called elastic scattering; on the other

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

1 *v ∂ ∂t*

emitting diode (LED).

*<sup>S</sup>* <sup>¼</sup> *LI* <sup>þ</sup> *<sup>L</sup><sup>S</sup>*

cosθ

*L n*2 � �

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<sup>þ</sup> ^<sup>ξ</sup> � <sup>∇</sup>

In Eq. (17), *c* is the beam attenuation coefficient, *LE, LI*

notation adopted by Mobley where *x*<sup>3</sup> ¼ *z* and ξ<sup>3</sup> ¼ cosθ*,* we obtain:

^<sup>ξ</sup> � <sup>∇</sup>*<sup>L</sup>* <sup>¼</sup> <sup>ξ</sup><sup>3</sup>

. The RTE then becomes:

*LE*ð Þ¼ *<sup>θ</sup>; <sup>ϕ</sup>*

*cdz* ¼ �Lð Þþ *<sup>θ</sup>; <sup>ϕ</sup>* <sup>ω</sup><sup>0</sup>

RTE, as shown in [4], is given by:

**2.1 Inherent optical properties**

the photon energy—or in both.

*dL*ð Þ *θ; ϕ*

following sections.

**102**

*dL*ð Þ *θ; ϕ*

and the path function for elastic scattering, *L<sup>E</sup>*:

ð 4π *L θ*<sup>0</sup> *; ϕ*<sup>0</sup> ð Þβ *θ*<sup>0</sup>

> ð 4π eβ *θ*<sup>0</sup>

cos *θ*

*L n*2 � �

¼ �*c*

For UOWC systems, we are interested in the time-independent RTE in horizontally homogenous water bodies with a constant index of refraction. Because of this, the factor *<sup>n</sup>*�<sup>2</sup> will divide both sides of Eq. (15). Hence, with *<sup>∂</sup>L=∂<sup>t</sup>* <sup>¼</sup> 0 and using the

¼ cos*θ*

Combining all these terms in Eq. (18), we can come up with the standard form of the RTE. As a simple statement, the IOPs and boundary conditions will go into the RTE and the final radiance will come out as a result. The standard form of the

where *L*ð Þ *θ; ϕ* represents the radiance in the direction ð Þ *θ; ϕ* , ω<sup>0</sup> is the single scattering albedo, c is the beam attenuation coefficient, and e*β θ*ð Þ *; ϕ* is the scattering phase function. Each one of these terms will be explained in more detail in the

During the interaction of a photon with a water molecule, one of two things may happen: the photon may be absorbed, leaving the water molecule in a state with higher internal energy, or the photon may undergo scattering. Such scattering occurs when there is a change in the direction of the propagation of the photon or in

When scattering happens, if the molecule that interacted with the photon returns to its original internal energy state by emitting a photon of the same energy as the absorbed one, the scattering process is called elastic scattering; on the other

*; ϕ* ð Þ <sup>0</sup> ! *θ; ϕ L θ*<sup>0</sup>

*dL*ð Þ *θ; ϕ*

*dz* ¼ �*cL*ð Þþ *<sup>θ</sup>; <sup>ϕ</sup> LE*ð Þþ *<sup>θ</sup>; <sup>ϕ</sup> <sup>S</sup>*ð Þ *<sup>θ</sup>; <sup>ϕ</sup> ,* (18)

*; ϕ*<sup>0</sup> ð Þ*d*Ω *θ*<sup>0</sup>

*; ϕ* ð Þ <sup>0</sup> ! *θ; ϕ d*Ω *θ*<sup>0</sup>

functions for elastic scattering, inelastic scattering, and spontaneous emission, respectively. The path function for spontaneous emission is also referred in [3] as the source path function and can be considered as the contribution to total radiance from a light source and in the case of UOWC systems a laser diode or a light

> *∂L ∂x*<sup>3</sup>

It is also usual [3] to combine *LI* and *L<sup>S</sup>* as an effective source function

*L n*2 � �

<sup>þ</sup> *<sup>L</sup><sup>E</sup>* <sup>þ</sup> *<sup>L</sup><sup>I</sup>* <sup>þ</sup> *LS*

*:* (16)

*,* and *L<sup>S</sup>* are the path

*dz :* (17)

*; ϕ*<sup>0</sup> ð Þ*,* (19)

*; ; ϕ*<sup>0</sup> ð Þþ *S*ð Þ *θ; ϕ ,*

(20)

hand, if the excited molecule emits a photon of lower energy, i.e., a photon with a longer wavelength, the incident photon goes through a process called inelastic scattering. As seen in [3], the absorption coefficient, that will be introduced later in this chapter, accounts for both the conversion of radiant energy into heat and the loss of power at wavelength λ by inelastic scattering to other wavelengths. Mobley [3] refers to this as true absorption. In view of this, in the remaining of this chapter, we will treat the scattering as elastic, the loss of energy due inelastic scattering being included in the absorption coefficient.

To derive a mathematical coefficient for scattering and absorption, we assume the model shown in **Figure 1**.

A volume of water Δ*V* with thickness Δ*r* is illuminated by a light beam with wavelength λ*:* Some part of the incident power, *PI*ð Þλ , will be absorbed within the volume of water, *PA*ð Þλ ; some part, PSð Þ λ*; θ* , is scattered out of the beam at an angle *θ* and the remaining light power, *PT*ð Þλ , is transmitted. Then by conservation of energy:

$$P\_I(\lambda) = P\_A(\lambda) + P\_S(\lambda, \theta) + P\_T(\lambda). \tag{21}$$

Using Eq. (22), we can define the ratio between the absorbed and incident power as the absorptance and the ratio between scattered power and incident power as scatterance. As already stated, when solving the RTE the IOPs usually employed are the absorption and the scattering coefficients which are the absorptance and scatterance per unit distance of the medium. Taking the limit of these terms as the water thickness Δ*r* becomes infinitesimally small, we have:

$$\mathfrak{a}(\lambda) = \lim\_{\Delta r \to 0} \frac{P\_{\mathbb{A}}(\lambda)}{P\_{\mathbb{I}}(\lambda)\Delta r}. \tag{22}$$

$$b(\lambda) = \lim\_{\Delta r \to 0} \frac{P\_{\mathbb{S}}(\lambda)}{P\_{\mathbb{I}}(\lambda)\Delta r}. \tag{23}$$

The beam attenuation coefficient present in the RTE (Eqs. (16) and (20)) is written as:

$$
\sigma(\lambda) = a(\lambda) + b(\lambda). \tag{24}
$$

Another useful IOP showing in the RTE is the single scattering albedo, ω0, that is defined as the ratio between the scattering and the beam attenuation coefficient:

$$a\rho\_0 = \frac{b(\lambda)}{c(\lambda)}.\tag{25}$$

In waters where the beam attenuation is due primarily to scattering, ω<sup>0</sup> is near one and when the beam attenuation is due primarily to absorption, ω<sup>0</sup> is near zero.

**Figure 1.** *Geometry of IOPs for a volume of water ΔV.*

In [3], it is possible to find the absorption, scattering, and the attenuation coefficients for several water types and wavelengths. The usual parameters considered for UOWC systems are the ones presented in **Table 1**.

As mentioned before, the ratio between the scattered power and incident power is referred as scatterance, that is the fraction of incident power scattered out of the beam through an angle *θ* into a solid angle *ΔΩ*. Defining the angular scatterance per unit distance and unit solid angle, βð Þ *θ;* λ , as:

$$\beta(\theta,\lambda) = \lim\_{\Delta r \to 0} \lim\_{\Delta \Omega \to 0} \frac{P\_S(\theta,\lambda)}{P\_I(\lambda)\Delta r \Delta \Omega}.\tag{26}$$

are the most precise to this date, to the best of the author's knowledge, and widely cited in the literature. The measurements were carried in the Bahamas, San Pedro in California, and San Diego harbor also in California, corresponding to clear, coastal,

**3. Solving the radiative transfer equation: Monte Carlo simulation**

sciences, economics, engineering, and pure mathematics.

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

tion and scattering in the underwater channel.

the microscopic Beer-Lambert law method (mBLL).

approach was favored by several research papers [11–16].

tional cost of Monte Carlo simulations.

RTE, step by step, will be shown.

the absorbed photons are wasted.

**105**

The Monte Carlo name was coined by Nicholas Metropolis [7]. The modern version of the method was invented in the late 1940s and found use early use in the studies of neutron transport for the design of nuclear weapons [8, 9]. More recently, Monte Carlo techniques evolved and are now used to solve problems in physical

Monte Carlo methods may be used to solve any problem that have a probabilistic interpretation, meaning that if the probability of occurrence of each separate event in a sequence of events is known, one can determine the probability that the entire sequence of events will occur. By tracing the fate of millions of photons according to statistical probabilities, a solution for the RTE is built. Each simulated photon path is randomly distinct from the others, as determined by the probabilities of absorp-

The Monte Carlo simulations of photon propagation offer a flexible yet rigorous approach toward solving numerically the RTE. In the last decades, the exponential improvement in computer speeds attenuated the problem of the high computa-

In the following subsections, a detailed description for building a solution for the

As seen in [10], there are four main Monte Carlo methods for photon migration in turbid media, that is, four different ways to build photon's trajectories. The four methods are referred as the albedo-weight method (AW), the albedo-rejection method (AR), the absorption-scattering path length rejection method (ASPR), and

The AW method considers a source emitting packets of many photons and after each interaction a fraction of the photons in the packet are absorbed, i.e., lost. In this type of simulation, the packet is emitted with an initial weight of 1 and at each interaction the current weight is multiplied by the albedo to account for the loss of photons by absorption and the packet continues propagating in the scattered direction with a reduced weight. This method has in its favor that there are no wasted computations because all photon packets eventually reach the detector. This

In this chapter, we present a simulation using the AR method; the main advantage of this method is that it mimics what happens in nature by tracking individual photons emitted by some source. Naturally for this case, the computational time of

It is seen in the investigation conducted by Sassaroli that the total probability of

detecting a photon is equivalent in the four methods and they have statistical equivalence, with some differences regarding the convergence time. When comparing the AW to the AR method, as one may expect, the AR will require, in most scenarios, more launched photons for convergence. However, it is also noted that in the AR method, the photon is detected or lost similarly to what happens in a timeresolved experiment in which photons are detected using the time-correlated single photon counting technique and the noise on the simulated response reproduces the Poisson statistics that typically characterize the noise in experiments. Therefore, as we already stated, the AR method reproduces a more physical simulation of

and harbor waters, respectively.

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

In [3], we can see that the spectral power scattered into the given solid angle *ΔΩ* is just as the spectral radiant intensity, *IS*ð Þ *θ;* λ *,* scattered into the angle *θ* times the solid angle:

$$P\_S(\theta,\lambda) = I\_S(\theta,\lambda)\Delta\Omega,\tag{27}$$

if the incident power falls on an area Δ*A*, then the corresponding irradiance *Ei*ð Þ¼ λ *PI*ð Þλ *=*Δ*A*. Recalling that Δ*V* is the volume of water that is illuminated by the incident beam:

$$\mathfrak{P}(\theta,\lambda) = \lim\_{\Delta V \to 0} \frac{I\_{\mathbb{S}}(\theta,\lambda)}{E\_i(\lambda)\Delta V}. \tag{28}$$

As said in [5], the form of Eq. (28) suggests the name volume scattering function (VSF), that is, the scattered intensity per unit incident irradiance per unit volume of water. Integrating βð Þ *θ;* λ over all directions gives the total scattered power per unit of incident irradiance and unit volume of water, or as defined in Eq. (23), the scattering coefficient:

$$\mathbf{b}(\lambda) = 2\pi \int\_0^\pi \beta(\theta, \lambda) \sin \theta \mathrm{d}\theta. \tag{29}$$

Normalizing Eq. (26) with the scattering coefficient, we obtain the scattering phase function (SPF):

$$
\widetilde{\mathfrak{B}}(\theta,\lambda) = \frac{\mathfrak{B}(\lambda,\theta)}{b(\lambda)}.\tag{30}
$$

The scattering phase function can be interpreted as a probability density function (PDF) for scattering from an incident direction *θ*<sup>0</sup> *; ϕ*<sup>0</sup> ð Þ to a final direction ð Þ *θ; ϕ* [5]. In general, the SPF must be solved numerically, or using tabulated data like the ones given by Petzold [6]. The scattering measurements made by Petzold


**Table 1.** *Measured parameters for different water types.*

In [3], it is possible to find the absorption, scattering, and the attenuation coefficients for several water types and wavelengths. The usual parameters consid-

As mentioned before, the ratio between the scattered power and incident power is referred as scatterance, that is the fraction of incident power scattered out of the beam through an angle *θ* into a solid angle *ΔΩ*. Defining the angular scatterance per

> lim *ΔΩ*!0

In [3], we can see that the spectral power scattered into the given solid angle *ΔΩ* is just as the spectral radiant intensity, *IS*ð Þ *θ;* λ *,* scattered into the angle *θ* times the

if the incident power falls on an area Δ*A*, then the corresponding irradiance *Ei*ð Þ¼ λ *PI*ð Þλ *=*Δ*A*. Recalling that Δ*V* is the volume of water that is illuminated by

Δ*V*!0

ðπ 0

Normalizing Eq. (26) with the scattering coefficient, we obtain the scattering

<sup>e</sup>βð Þ¼ *<sup>θ</sup>;* <sup>λ</sup> β λð Þ *; <sup>θ</sup>*

The scattering phase function can be interpreted as a probability density func-

ð Þ *θ; ϕ* [5]. In general, the SPF must be solved numerically, or using tabulated data like the ones given by Petzold [6]. The scattering measurements made by Petzold

**Water type** *<sup>c</sup> <sup>λ</sup>*<sup>Þ</sup> *<sup>m</sup>*�**<sup>1</sup>** <sup>ð</sup> *<sup>ω</sup>***<sup>0</sup>** Clear waters 0*:*15 0.25 Coastal waters 0*:*4 0.55 Harbor I waters 1*:*1 0.83 Harbor II waters 2*:*19 0.83

As said in [5], the form of Eq. (28) suggests the name volume scattering function (VSF), that is, the scattered intensity per unit incident irradiance per unit volume of water. Integrating βð Þ *θ;* λ over all directions gives the total scattered power per unit of incident irradiance and unit volume of water, or as defined in Eq. (23), the

*IS*ð Þ *θ;* λ

βð Þ¼ *θ;* λ lim

bð Þ¼ λ 2π

tion (PDF) for scattering from an incident direction *θ*<sup>0</sup>

*PS*ð Þ *θ; λ*

*PS*ð Þ¼ *θ;* λ *IS*ð Þ *θ;* λ *ΔΩ,* (27)

*PI*ð Þ<sup>λ</sup> *<sup>Δ</sup>rΔ<sup>Ω</sup> :* (26)

*Ei*ð Þ<sup>λ</sup> <sup>Δ</sup>*<sup>V</sup> :* (28)

βð Þ *θ;* λ sin*θ*d*θ:* (29)

*<sup>b</sup>*ð Þ<sup>λ</sup> *:* (30)

*; ϕ*<sup>0</sup> ð Þ to a final direction

ered for UOWC systems are the ones presented in **Table 1**.

*Wireless Mesh Networks - Security, Architectures and Protocols*

*β θ*ð Þ¼ *;* λ lim

*Δr*!0

unit distance and unit solid angle, βð Þ *θ;* λ , as:

solid angle:

the incident beam:

scattering coefficient:

phase function (SPF):

**Table 1.**

**104**

*Measured parameters for different water types.*

are the most precise to this date, to the best of the author's knowledge, and widely cited in the literature. The measurements were carried in the Bahamas, San Pedro in California, and San Diego harbor also in California, corresponding to clear, coastal, and harbor waters, respectively.
