Numerical analysis of light reflection and transmission in poly-disperse sea fog

Chi Zhang; Jianqi Zhang; Xin Wu; Xin Wu; Melin Huang; Melin Huang; Melin Huang

doi:10.1364/OE.400002

1. Introduction

Sea fog occurs in the lower layer of atmosphere over the sea or coastal areas [1], which is one type of specific atmospheric aerosols, existing frequently in maritime backgrounds. Detections of maritime targets may thus fail due to sea fogs. In general, opto-electronic images captured by a detector in a foggy sky over a sea can be very different from those captured in a clear sky because radiation propagation through a sea fog area experiences multiple scatterings and its trajectory would be disturbed by particles in the sea fog. Therefore, the presence of sea fogs would affect the detection of both reflectance and transmittance of radiation propagation through targets (such as ships) and backgrounds (such as a sea), which reflects on the detections of maritime targets. Thus, studies on the characteristics of radiation transfer through sea fogs have significances on the detections of maritime targets, scene simulation, and remote sensing.

Methods such as approximations to the radiative transfer equation [e.g., 2–7] and Monte Carlo (MC) simulation [e.g., 8–16] are two main approaches to investigate the multiple scattering effects of radiation (such as visible light, infrared light, microwave, etc.) propagation in scattering media (such as rain, cloud, soot aerosols, etc.). A few examples of applying these approaches on the studies of multiple scattering effects are given as follows. The effects of mixing states on the multiple scatterings of soot aerosols were presented in [2–4,10]. The multiple scattering effects of radiation propagation in a complex media (such as a coupled atmosphere-ocean system) were inspected in [5–9,12]. The behaviors of polarization and depolarization of lights in scattering media were probed in [13–16] as well. Furthermore, the influences of multiple scatterings on the optical signal transmission though clouds have been probed based on the hypothesis of independent scatterings among all scattered particles [17–22]. On the contrary, a dependent scattering induced by turbulent fluctuations due to both temperature and humidity of the air in the cloud was instead considered in the study on the multiple scattering effects of the microwave radiation through the cloudy atmosphere [23–24].

Researches mentioned above have further been applied to examine the multiple scattering effects when radiation transfers through the foggy atmosphere. For instance, approximation to the radiative transfer equation has been employed to investigate the effects of multiple scatterings caused by radiation fogs and advection fogs [25–26], MC simulation has been applied to the studies on the multiple scattering effects of light through rain or fog media [27–30] and to the studies on the optical properties of polarized light scattering in fog environments [31–33].

In summary, generally previous studies focused on the influence of multiple scattering on the reflection or transmission of radiation propagation through media (e.g., [10,11,27–29]), but few paid attention to the bidirectional scattering properties of sea fog. Although most of earlier studies concentrated on poly-disperse case to imitate the reality (e.g., [16,27–29,32]), some used mono-disperse cases (e.g., [14,30,31,34]). Furthermore, previous studies (e.g., [27,28,35,36]) employed one value for each attenuation, asymmetry factor, absorption coefficient, and scattering coefficient by averaging over particle size distribution in the analysis of radiation transfer through media of interest, which may result in lack of representativeness of reality. Note that mono-disperse stands for particles of a uniform radius comprised in a medium of interest (e.g., rain, fog), while poly-disperse represents particles of various radii in a medium. Apart from above, previous studies usually dealt with the situations of radiation propagation incident from air, through a medium of the same refractive index as air, then exiting back to air again. In other words, the refractive index is the same all the way along the radiation transfer path.

Due to effectiveness, flexibility, and easy implementation, MC simulation has become a common approach for simulating radiation propagation through atmosphere aerosols. We thus adopt MC simulation for our study by considering diverse attenuations, asymmetry factors, absorption coefficients, and scattering coefficients of radiation when the radiation passes through a poly-disperse sea fog medium. This is in an attempt toward more reality. Here briefly describe our MC simulation processes as follows. Firstly, a poly-disperse sea fog system is created using MC simulation. Next, calculations on the optical parameters of sea fog particles are performed based on Mie theory. This is followed by simulation of interactions (such as multiple scatterings, absorption, reflection, transmission, etc.) between radiation and sea fog particles. Previous studies used one value for each of attenuation ($\bar{{A}}$), asymmetry factor ($\bar{{g}}$), absorption coefficient (${{\bar{\mu }\textrm{}}_{{abs}}}$), and scattering coefficient (${{\bar{\mu }\textrm{}}_{{sca}}}$) of radiation in a medium of interest by averaging over the particle size distribution. Our MC simulation instead employs a variety of values for each of ${A}$, ${g}$, ${\mathrm{\mu }_{{abs}}}$, and ${\mathrm{\mu }_{{sca}}}$ of radiation for the reason that the optical properties of radiation would alter when radiation interacts with sea fog particles of various radius sizes. Furthermore, previous studies considered media of the same refractive index through which the radiation traveled. One particular feature implemented in our MC simulation is considering media of two different refractive indices, where the air above the sea fog layer and the sea fog layer itself are taken as possessing the same refractive dex while the sea beneath the sea fog layer holds a bigger refractive index. Upon such a circumstance, radiation would experience both refraction and reflection simultaneously when crossing the boundary between the sea fog layer and the sea beneath. All of our MC simulation processes are detailed in Section 2.

In Section 3, three different MC methods were employed to simulate the propagation of the photons through the sea fog layer. The techniques (termed as previous method and our method) were applied in these three MC methods for the calculations of reflectance and transmittance. The influence of thicknesses and the water contents of the poly-disperse sea fog layer on both reflectance and transmittance of the radiation are presented in Section 3 and Section 4, respectively. Comparisons of our MC results with those using previous method are also presented therein. Conclusions are made in Section 5. Finally, future work is briefed in Section 6.

2. Theoretical basis

MC simulation, a method based on statistics, has been used in many researches in recent years to simulate the radiation transport process through media for radiative transfer problems (e.g., [8–16]). We here adopt MC simulation to simulate a layered system of poly-disperse sea fogs (see Section 2.1) by considering the effects of the multiple scatterings of radiation by sea fog particles of various radii in the layer system. Furthermore, the interaction between radiation and sea fog particles are outlines in Section 2.2. This is followed by the description of our MC simulation processes in Section 2.3, which includes tracing the interactions of radiation with sea fog particles. In addition, comparisons between our MC simulation method and previous method are given at the end of this section.

In the following, a layered system of poly-disperse sea fogs is termed as poly-disperse sea fog layer or simply sea fog layer. We here make two additional notes. The first note is that we have combined those previous techniques used in MC methods as one term “previous method”. To improve the MC methods, the techniques we proposed in this work are termed as “our method”. These techniques are described in Section 2.1–2.3. The second note is that, as those studies in [27–33], the hypothesis of independent scattering of light by single water droplets in the sea fog is also adopted in our work.

2.1 Poly-disperse sea fog system

Sea fogs, which are advection fogs in general, can be characterized by several microphysics parameters such as particle size distributions, particle radii, and liquid water content (LWC) of the sea fogs. For the maritime and coastal atmosphere aerosols, the presence of small sea-salt particles in the sea fog layer [37] due to the evaporation of the sea water droplets is possible. Such a situation can make the radiation transfer in the sea fog medium complicated if the salinity is high enough. However, according to Ref. [38], the difference in refractive indices between standard sea water at room temperature for salinity of 3.5% and pure water is very small in the wavelength range of $\textrm{1} - \textrm{6}$ μm. Therefore, the sea fog model in our work is approximated by only considering pure water. Note that the refractive indices of the sea fog particles in our MC simulation are based on the value published by [39]. Also by taking sea fogs as advection fogs in our study, each water particle is regarded as a sphere. Due to the complex physical process of formation and dissipation, in general, the particle size distribution ${n(r)}$ (${\textrm{m}^{\textrm{ - 3}}}$μm^-1) of sea fogs varies with time and region. To compare our method with the previous method, we have kept the ${n(r)}$ fixed in our work. The widely-used ${n(r)}$ that is based on the well-known Gamma distribution [28,40] is thus chosen for our study:

(1)$$n(r) = 3.73 \times {10^5}{W^{ - 0.804}}{r^2}\exp ( - 0.2392{W^{ - 0.301}}r),$$

where r(μm) is the particle radius size and $W\textrm{(g/}{\textrm{m}^\textrm{3}}\textrm{)}$ is the LWC.

Examples of radiations are such as visible light, infrared light, microwave, radio wave, electromagnetic wave, etc. Therefore, when passing through a sea fog medium, radiations would interact with sea fog particles, experiencing free-path propagation, absorption, and scatterings. On account of differences in absorption and scattering coefficients when radiations interact with sea fog particles of various radii, the radii of the sea fog particles in our MC simulation are then divided into eight regions. The number of eight regions was randomly chosen merely for computational time though it is believed that the more number of regions would result in more accurate calculation. The eight regions are $\textrm{0}\textrm{.01}\;\mathrm{\mu}\textrm{m} - \textrm{1}\; \mathrm{\mu}\textrm{m}$, $\textrm{1}\; \mathrm{\mu}\textrm{m} - \textrm{2}\; \mathrm{\mu}\textrm{m}$, $\textrm{2}\; \mathrm{\mu}\textrm{m} - \;\textrm{5}\; \mathrm{\mu}\textrm{m}$, $\textrm{5}\; \mathrm{\mu}\textrm{m} - \textrm{10}\; \mathrm{\mu}\textrm{m}$, $\textrm{10}\; \mathrm{\mu}\textrm{m} - \textrm{15}\; \mathrm{\mu}\textrm{m}$, $\textrm{15}\; \mathrm{\mu}\textrm{m} - \textrm{18}\; \mathrm{\mu}\textrm{m}$, $\textrm{18}\; \mathrm{\mu}\textrm{m} - \textrm{30}\; \mathrm{\mu}\textrm{m}$, and $\textrm{30}\; \mathrm{\mu}\textrm{m} - \textrm{60}\; \mathrm{\mu}\textrm{m}$. The i-th radius region of the sea fog particles is denoted as ${{r}_{i}}$ hereafter. As such, the attenuation, asymmetry factor, absorption coefficient, scattering coefficient, and the absorbed probability $\textrm{(}{\alpha}\textrm{)}$ of the radiation would accordingly vary with the sea fog particle sizes, which reflects the poly-disperse feature in our considered sea fog system. Let us denote the attenuation, asymmetry factor, absorption coefficient, scattering coefficient, and absorption probability for the radiation interacting with sea fog particles in the i-th region as ${{A}_{i}}$, ${{g}_{i}}$, ${{\mu }_{{abs,i}}}$, ${{\mu }_{{sca,i}}}$, and ${{\alpha}_{i}}$, respectively. The relation among ${\; }{{A}_{i}}$,$\; {\mathrm{\mu }_{{abs,i}}}$, and ${{\mu }_{{sca,i}}}$ are given by

(2)$${A_i}(\lambda ) = {\mu _{sca,i}}(\lambda ) + {\mu _{abs,i}}(\lambda ),$$

(3)$${\mu _{sca,i}}(\lambda ) = \int_{{r_{\min ,i}}}^{{r_{\max ,i}}} {\pi {r^2}} {q_{sca}}(r,\lambda )n(r)dr,$$

(4)$${\mu _{abs,i}}(\lambda ) = \int_{{r_{\min ,i}}}^{{r_{\max ,i}}} {\pi {r^2}} {q_{abs}}(r,\lambda )n(r)dr.$$

Besides, ${{\alpha}_{i}}$ and ${{g}_{i}}$ are defined as follows

(5)$${\alpha _i}(\lambda ) = \frac{{{\mu _{abs,i}}(\lambda )}}{{{A_i}(\lambda )}},$$

(6)$${g_i}(\lambda ) = \frac{{\int_{{r_{\min ,i}}}^{{r_{\max ,i}}} {{r^2}{q_{sca}}(r,\lambda )g(r,\lambda )n(r)dr} }}{{\int_{{r_{\min ,i}}}^{{r_{\max ,i}}} {{r^2}{q_{sca}}(r,\lambda )n(r)dr} }},$$

where ${{r}_{{\textrm{min},i}}}$ and ${{r}_{{\textrm{max},i}}}$ respectively stand for the minimum and maximum radii of the sea fog particles in the i-th region; ${{q}_{{sca}}}({{r},\; {\lambda}} )$, ${{q}_{{abs}}}({{r},\; {\lambda}} )$ and ${g(r},\; {\lambda}{)}$ respectively represent scattering efficiency, absorption efficiency and asymmetry factor for radiation interacting with sea fog particles of radius ${r}$, which are obtained using the Mie theory [41,42]. The attenuation of radiation propagation through a poly-disperse sea fog medium is thus given by ${{A}^{{tot}}}{ = }\mathop \sum \limits_{{i = 1}}^{m} {{A}_{i}}{,}$ where ${m}$ represents the total regions of the sea fog particle radii. Once ${{A}_{i}}$, ${{g}_{i}}$, ${\mathrm{\mu }_{{abs,i}}}$, ${\mathrm{\mu }_{{sca,i}}}$, and ${{\alpha}_{i}}$ are obtained, the reflectance (${R}$), transmittance (${T}$), bidirectional distributions of reflectance (${{R}^{{bd}}}$) and transmittance (${{T}^{{bd}}}$) of the radiation can then be computed by tracing the interactions of radiation with the sea fog particles of various sizes in the poly-disperse medium. These are presented in Section 2.2.

Let us take a look at previous studies (e.g., [27,28,35,36]). Although many previous studies employed a poly-disperse system (such as sea foam, rain, fogs) in their work, values of $\bar{{A}}$, $\bar{{g}}$, ${{\bar{\mu }\textrm{}}_{{abs}}}$, ${{\bar{\mu }\textrm{}}_{{sca}}}$, and $\bar{\alpha}$ by averaging over all particle sizes were used, where

(7)$$\overline A (\lambda ) = {\overline \mu _{abs}}(\lambda ) + {\overline \mu _{sca}}(\lambda ),$$

(8)$${\overline \mu _{sca}}(\lambda ) = \int_{{r_{\min }}}^{{r_{\max }}} {\pi {r^2}} {q_{sca}}(r,\lambda )n(r)dr,$$

(9)$${\overline \mu _{abs}}(\lambda ) = \int_{{r_{\min }}}^{{r_{\max }}} {\pi {r^2}} {q_{abs}}(r,\lambda )n(r)dr,$$

(10)$$\overline \alpha (\lambda ) = \frac{{{{\overline \mu }_{abs}}(\lambda )}}{{\overline A (\lambda )}},$$

(11)$$\overline g (\lambda ) = \frac{{\int_{{r_{\min }}}^{{r_{\max }}} {{r^2}{q_{sca}}(r,\lambda )g(r,\lambda )n(r)dr} }}{{\int_{{r_{\min }}}^{{r_{\max }}} {{r^2}{q_{sca}}(r,\lambda )n(r)dr} }},$$

with ${{r}_{\textrm{min}}}$ and ${{r}_{\textrm{max}}}$ respectively symbolizing the minimum and maximum radii of the sea fog particles.

2.1.1 Demonstration of ${\alpha}$ versus ${r}$, g versus ${r}$, and ${{A}^{{tot}}}$ versus ${W}$

With ${W = 0.2{\textrm {g}}/}{{\textrm m}^{3}}$, Figs. 1(a)−(b) shows ${\alpha}$ and ${g}$ respectively versus ${r}$ in eight regions for ${{\lambda}_{{in}}}{\; = \; 3,\; 4,\; 5\; }\mathrm{\mu}\rm{m}$. The average values, $\bar{\alpha}$ and $\bar{{g}}$, for each ${{\lambda}_{{in}}}$ using previous methods are also displayed for comparison. They clearly show that both ${\alpha}$ and ${g}$ vary with ${r}$. Furthermore, given ${{\lambda}_{{in}}}{\; = \; 3,\; 4,\; 5\; }\mathrm{\mu}\rm{m}$, Fig. 1(c) displays ${{A}^{{tot}}}$ versus ${W}$ after the radiation passes through the sea fog layer. The results in Fig. 1 are calculated from Mie scattering formulas. Since the calculations of ${R}$, ${T}$, ${{R}^{{bd}}}$, and ${{T}^{{bd}}}$ are related to ${\alpha}$, ${g}$, and ${{A}^{{tot}}}$, the findings we summarize below are useful for explaining the results presented in Section 3.

1) In Figs. 1(a)−(b), as presented in Eqs. (2)−(6) and Eqs. (7)−(11), all ${A}$, ${g}$, and ${\alpha}$ depend on ${{q}_{{sca}}}{(r},\; {{\lambda}_{{in}}}{)}$and/or ${{q}_{{abs}}}{(r},\; {{\lambda}_{{in}}}{)}$, and thus both ${r}$ and ${{\lambda}_{{in}}}$ are involved in the calculation of ${{A}_{i}}$, ${{g}_{i}}$, and ${{\alpha}_{i}}$ (our methods) and $\bar{{A}}$, $\bar{{g}}$, and $\bar{\alpha}$ (previous methods). The values of $\bar{{A}}$, $\bar{{g}}$, and $\bar{\alpha}$ are obtained by averaging over all ${r}$’s and hence each of them acquires a unique value for each given ${{\lambda}_{{in}}}$.
2) In Figs. 1(a)−(b), given ${{r}_{i}}$, there apparently show discrepancies in ${A}$, ${g}$, and ${\alpha}$ when using our methods and previous methods. However, the discrepancies for ${{\lambda}_{{in}}} = {3}$ μm is much smaller than those for ${{\lambda}_{{in}}} = {4},\; 5$ μm when ${{r}_{i}} \ge {5}$ μm. These discrepancies subsequently affect the calculation in the reflectance and transmittance of the photons.
3) Again in Figs. 1(a)−(b), given ${{r}_{i}}$, ${\alpha}$ and g generally signifies the largest, medium, and the least for ${{\lambda}_{{in}}} = {3}$ μm, ${{\lambda}_{{in}}} = {5}$ μm, and ${{\lambda}_{{in}}} = {4}$ μm accordingly. That is, ${\alpha}$ and g for ${{\lambda}_{{in}}} = {3}$ μm are always higher than those for ${{\lambda}_{{in}}} = {4},\; 5$ μm.
4) In Fig. 1(c), the higher ${W}$, the more ${{A}^{{tot}}}$. This is expected because higher ${W}$ implies more sea fog particles especially these large particles and hence photons have higher collision possibility and experience more ${{A}^{{tot}}}$.

Fig. 1. Demonstration for (a) ${\alpha}$ versus ${r}$, (b) ${g}$ versus ${r}$, and (c) ${{A}^{{tot}}}$ versus ${W}$. For (a) and (b), ${W} = \textrm{0}\textrm{.2}\; \textrm{g/}{\textrm{m}^\textrm{3}}$ is employed. Note that solid curves are for using our methods and dashed curves for using previous methods.

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2.2 Interactions between radiation and sea fog particles

Before we process our MC simulation, a few words about the sea fog layer and how radiation interacts with sea fog particles are given as follows. Firstly, the sea fog layer can usually extend to a vast area in the horizontal direction. Even so, the ${n(r)}$ is assumed to be the same everywhere along the horizontal direction. For easily tracing the scattering radiation in our MC calculations, the whole sea fog layer is divided into several components along the horizontal directions (see Fig. 2), though those partitioned components do not physically exist in reality. In our MC simulation, the upper boundary ${(z} = {d)}$ and the lower boundary ${(z} = {0)}$ of each component are the top surface and the bottom surface of the sea fog layer, respectively. All components have equal volume and equal water content, and each component is a basic unit in our MC simulation. Hereafter, the term “upper boundary (interface)” stands for the interface between the sea fog layer and the air above, and the term “lower boundary (interface)” represents the interface between the sea fog layer and the sea beneath.

Fig. 2. The sea fog layer is divided into many components, all of which have equal volume and equal water content.

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Next, when a radiation beam propagates through the sea fog layer, photons are launched at the upper boundary at a certain zenith angle (${{\theta}_{{in}}}$) and an azimuth angle (${\varphi _{{in}}}$). Six possible occurrences would happen to the photons as depicted in the following. The first occurrence is that the photon gets reflected back to the sky at the upper boundary of the sea fog layer without going further into the sea fog layer (see Fig. 3(a)). The second case is that the photon travels through the sea fog layer, succeeded by a few scatterings with sea fog particles, and finally ending up with exiting at the upper boundary (see Fig. 3(b)). The third course is similar to the second course except that the photon exits at the lower boundary followed by entering the sea with some refraction (see Fig. 3(c)). The fourth event is that the photon is absorbed by a sea fog particle without further adventure (see Fig. 3(d)). The fifth happening is that the photon goes through the sea fog layer without collisions with any sea fog particle and directly enter the sea at the lower boundary with some refraction (see Fig. 3(e)). The last possibility is that the photons would enter the neighboring component if the photons leave from one lateral of a component (see Fig. 3(f)). This photon subsequently follows one or more of above five courses until it is reflected from the upper boundary, transmitted through the sea fog layer at the lower boundary, or absorbed by a sea fog particle. Usually such photons only undergo a couple of components. Among the six occurrences, the first and the second occurrences correspond to the reflection of photons, the third and the fifth occurrence to the transmission of photons, and the fourth occurrence to the absorption of photons.

Fig. 3. Six possible occurrences when photons travel through the sea fog layer.

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Two notes need to be explained. Firstly, the refractive indices of the air and the sea fog layer are the same because both media are taken as air. As a result, the first occurrence would not happen in our simulation study. Secondly, the refractive index of the sea water is greater than that of the sea fog layer. Therefore, when photons reach the lower boundary, part of the photons are reflected back to the sea fog layer, and part of the photons are transmitted into the sea with some refraction. This situation is also considered in our MC simulation.

2.3 Monte Carlo simulation

The procedure of our MC simulation consists of following steps:

1) Each photon is launched at the upper boundary with an assigned direction (${{\theta}_{{in}}}$, ${\varphi _{{in}}}$).
2) Each photon is given a collision length (${s}$) and is propagated according to this s. See details in Section 2.3.1.
3) For each movement of the photon, check whether the photon reaches or is closest to the lower boundary. If so, a decision is made for the photon: to be reflected back to the sea fog layer or to be transmitted into the sea with some refraction. For the former case, see details in Section 2.3.1 for the photon’s further movement in the sea fog layer. For the latter case, no further movement takes place for the photon but counting the transmission of the photon once.
4) Followed by step 3), the decision of reflection or transmission for the photon is given by a random number based on Fresnel’s law and Snell’s law. See details in Section 2.3.2.
5) Also check whether the photon leaves from one lateral of a component. If so, the photon would enter the neighboring component and keep moving without absorption.
6) During the photon’s movement, the probability for the photon colliding with sea fog particles is calculated and the ${{r}_{i}}$ radius region is determined as well. See details in Section 2.3.3.
7) Followed by step 6), once the photon has a collision with sea fog particles, there is a chance that the photon would be absorbed by a sea fog particle. The absorption probability is calculated according to the ${{r}_{i}}$ radius region given in step 6). See details in Section 2.3.4.
8) Again followed by steps 6) and 7), if the photon is not absorbed, the photon may be scattered by a sea fog particle. The direction of the scattered photon is determined according to the ${{r}_{i}}$ radius region given in step 6). See details in Section 2.3.5.
9) Up to here, if the photon still survives within the sea fog layer, steps 2) − 8) are repeated. For each repetition, a new ${s}$ is assigned to the photon for further movements. The repetition is performed until the photon is absorbed or exits the sea fog layer.
10) The steps 1) – 9) are repeated until all photons are launched.
11) Once all photons are launched, the physical quantities (e.g., ${R}$, ${T}$, ${{R}^{{bd}}}$, and ${{T}^{{bd}}}$) are then computed. See details in Section 2.2.6.

In our MC simulation, multiple scatterings of photons with sea fog particles are presented. In addition, Mie scattering theory is applied to calculate ${{q}_{{sca}}}$, ${{q}_{{abs}}}$, ${g}$, ${A}$, and ${\alpha}$ of photons when photons pass through the sea fog layer. Below present the details of some steps in our MC simulation. At the end of this section, a comparison between our method and previous methods is given.

2.3.1 Collision length and remaining distance

At the entrance in which a photon enters the sea fog layer, a collision length, ${s}$, is generated using a random number ${{\xi}_{1}}$ between 0 and 1 [27,36]:

(12)$$s ={-} \frac{1}{{{A^{tot}}}}\ln {\xi _1}.$$

This ${s}$ represents the distance between any two collisions. In other words, the photon is propagated with ${s}$ for its each movement. Note that ${{A}^{{tot}}}$ is used and thus ${s}$ does not involve any particle radius region. Besides, the unit of ${s}$ is km in our work since the unit of ${{A}^{{tot}}}$ is km^-1 (see Fig. 1).

When the photon reaches or is closest to the lower boundary, the remaining distance (${s_{rem}}$) for the photon is calculated as ${{s}_{{rem}}} = {s} - {l}$, where ${l}$ is the distance from the photon’s current location to the point at which it hits the lower interface. Two situations would happen to ${s_{rem}}$. Firstly, if ${{s}_{{rem}}} \ne 0$, the photon is moved to the lower interface. At the interface, if the photon is reflected back to the sea fog layer, its new ${s}$ becomes ${{s}_{{rem}}}$. If the photon is transmitted into the sea with some refraction, the transmission is counted once with no further movement for the photon. Secondly, if ${{s}_{{rem}}} = {0}$, which means that the photon has reached the lower interface. Again, reflection or transmission needs to be determined for the photon. If the photon is reflected back to the sea fog layer, then the photon receives a new ${s}$ that may not be the same as the original ${s}$, and the photon keeps moving with this new ${s}$. If the photon is transmitted into the sea with some refraction, again transmission is counted once without further movement for the photon. All of these situations are summarized in Fig. 4.

Fig. 4. Two situations happened to ${{s}_{{rem}}}$ when a photon reaches the lower interface.

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2.3.2 Reflection or transmission at lower interface

When a photon reaches the lower interface, a random number is generated to decide whether the photon is reflected or is transmitted. Due to the influence of wind, the sea surface usually is a flat surface, but a collection of randomly titled and locally flat sea-wave facets instead [43]. In our MC simulation, the Cox-Munk wave model [44] was employed for the slope distribution of the sea-wave facets, where the wind speed is set as 5 m/s when calculating the variance of the sea-wave facet slopes. Only single-reflection is considered on account of the error caused by ignoring multiple reflections is less than ${1}\%$ for an incident zenith angle is less than ${7}{{0}^{\circ }}$ [36]. Note that even due to the influence of wind upon the sea and the sea fog, the particle size distribution in the sea fog layer would not change and thus would not have impacts on our studies.

At the lower interface, the reflection and the transmission of the photon comply with Fresnel’s law [45,46]:

(13)$$\rho ({\beta _i},{\beta _t}) = \left\{ {\begin{array}{cc} {\frac{1}{2}\left[ {\frac{{{{\sin }^2}({\beta_i} - {\beta_t})}}{{{{\sin }^2}({\beta_i} + {\beta_t})}} + \frac{{{{\tan }^2}({\beta_i} - {\beta_t})}}{{{{\tan }^2}({\beta_i} + {\beta_t})}}} \right],}&{{\beta_i} \ne 0}\\ {{{\left( {\frac{{{n_{air}} - {n_{sw}}}}{{{n_{air}} + {n_{sw}}}}} \right)}^2},}&{{\beta_i} = 0} \end{array}} \right.,$$

The notations are explained as follows. The symbols ${{n}_{{sw}}}$ and ${{n}_{{air}}}$ are the refractive indices of the sea water and the air, respectively. Note that following refractive indices are used in our MC simulation: ${{n}_{{air}}} = {1}$ for the air and ${{n}_{{sw}}} = {1}{.338}$ for the sea water. The notations ${{\beta}_{i}}$ and ${{\beta}_{t}}$ respectively stand for the incident angle and the transmitted angle with respect to the sea-wave facet. Computation of ${{\beta}_{i}}$ can be found in [6], and ${{\beta}_{t}}$ can be acquired from Snell’s law: ${\beta}_{t}\; = \; {\arcsin(}{n}_{{air}}{\sin}{{\beta}_{i}}/{n}_{{sw}})$.

A random number ${{\xi}_{2}}{\; }$ between 0 and 1 is generated. If ${{\xi}_{2}}{\; } \le {\; }{\rho}{(}{{\beta}_{i}}{},\; {{\beta}_{t}}{)}\cdot {{S}_{h}}$, the photon is reflected back to the sea fog layer; otherwise the photon is transmitted into the sea water with a certain refraction. The factor ${{S}_{h}}$ is the shadowing factor, which can be found in Ref. [36].

2.3.3 Collision probability and regional radii

Two actions need mention here: i) the action of collision pertaining which sea fog particle is picked for interacting with a photon; ii) the action of being scattered (relating to ${{q}_{{sca}}}{(r)}$) or being absorbed (relating to ${{q}_{{abs}}}{(r)}$) for the photon. Physically, action ii) always occurs after action i), and thus these two actions would never take place simultaneously. In other words, information of optical parameters ${{q}_{{sca}}}{(r)}$ and ${{q}_{{abs}}}{(r)}$ would not involve in the equation of picking-up (i.e., collision) probability when photons collide with sea fog particles. Inspired by the thought of the collision probability presented in [13] and the thought that the collision probability increases as the radii of sea fog particles increase, the collision probability for photons colliding with sea fog particles in the ${{r}_{i}}$ radius region in our case is thus defined by

(14)$${P_i} = \frac{{\int_{{r_{\min ,i}}}^{{r_{\max ,i}}} {rn(r)dr} }}{{\sum\nolimits_{k = 1}^m {\int_{{r_{\min ,k}}}^{{r_{\max ,k}}} {rn(r)dr} } }},$$

The accumulative collision probability for photons colliding with sea fog particles from the first radius region up to the i-th radius region is hence expressed as [47]

(15)$${P_{C,i}} = \sum\nolimits_{k = 1}^i {{P_i}} .$$

A random number ${{\xi}_\textrm{3}}$ between 0 to 1 is generated. If ${{\xi}_\textrm{3}}$ falls between ${{P}_{{C,i - 1}}}$ and ${{P}_{{C,i}}}$ with ${i}\; > 1$, the photon would collide with sea fog particles in the ${{r}_{i}}$ radius region. For contrast, collision probability = 1 is used in previous methods for all ${{r}_{i}}$’s of the sea fog particles with which the photons collide.

2.3.4 Absorption probability

When a photon collides with a sea fog particle in the ${{r}_{i}}$ radius region, the photon has a chance to be absorbed. If is not absorbed, there is a possibility that it is scattered in a specific direction [7]. To decide whether the photon is absorbed or scattered, a random number ${{\xi}_\textrm{4}}$ between 0 and 1 is generated. If ${{\xi}_\textrm{4}} \le {{\alpha}_{i}}$, the photon is absorbed and the photon is no longer traced; otherwise the photon is scattered (see Section 2.3.5).

In our MC simulation, due to the collision of a photon with sea fog particles of various radii, the absorption probability would vary with the ${{r}_{i}}$ radius region and is hence given in Eq. (5). Whenever a photon experiences collisions with sea fog particles, all of ${{\mu }_{{abs,i}}}$, ${{A}_{i}}$, and ${{\alpha}_{i}}$ would be updated according to which radius region of the sea fog particles with which the photon collides. For contrast, the previous studies used an average value, $\bar{{\alpha}}$, for all sea fog particles, see Eq. (10).

2.3.5 Scattering direction

Followed by previous section, if the photon is not absorbed, there is a chance that it would be scattered in a specific direction. The scattering angle $\theta $ is determined by the approximation to the Mie scattering phase function with the Henyey-Greenstein model, which is given by [48]:

(16)$$\theta = \left\{ {\begin{array}{cc} {\arccos (2{\xi_5} - 1),}&{{g_i} = 0}\\ {\arccos \left( {\frac{1}{{2{g_i}}}\left( {1 + g_i^2 - \frac{{1 - g_i^2}}{{1 - {g_i} + 2{g_i}{\xi_5}}}} \right)} \right),}&{{g_i} \ne 0} \end{array}} \right.,$$

where ${{\xi}_\textrm{5}}$ is a random number between 0 and 1. Note that the azimuth angle ${\varphi }$ is uniformly distributed between 0 and $\textrm{2}\mathrm{\pi }$. The scattering direction is updated after each scattering occurrence. Again for contrast, rather than using ${{g}_{i}}$, the asymmetry factor used in previous studies was a value $\bar{{g}}$, by averaging over all sea fog particle sizes, see Eq. (11).

2.3.6 Computation of reflectance and transmittance

Photons are detected by detectors at the upper boundary (referred to photons’ reflection) or the lower boundary (referred to photons’ transmission) though infrared radiation may decay quickly in the sea or water. In our work, we assume that the detectors are designed to detect un-polarized radiation. The detected ${R}$ and ${T}$ of the photons can respectively be calculated by [49,50]

(17)$$R = \frac{{\sum\nolimits_{{\theta _{out}} = 0}^{{\pi / 2}} {\sum\nolimits_{{\varphi _{out}} = 0}^{2\pi } {{N_{ref}}({\theta _{out}},{\varphi _{out}})} } }}{{{N_{in}}}},$$

(18)$$T = \frac{{\sum\nolimits_{{\theta _{out}} = {\pi / 2}}^\pi {\sum\nolimits_{{\varphi _{out}} = 0}^{2\pi } {{N_{tra}}({\theta _{out}},{\varphi _{out}})} } }}{{{N_{in}}}},$$

where ${{N}_{{in}}}$, ${{N}_{{ref}}}$, and ${{N}_{{tra}}}$ represent the number of incident photons, the number of detected reflected photons, and the number of detected transmitted photons, respectively.

Similarly, the bidirectional distribution of ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$ of the photons are respectively expressed by [49,50]

(19)$${R^{bd}} \equiv R({\theta _{in}},{\varphi _{in}};{\theta _{out}},{\varphi _{out}}) = \frac{1}{{{N_{in}}}} \cdot \frac{{{N_{ref}}({\theta _{out}},{\varphi _{out}})}}{{\sin {\theta _{out}}d{\theta _{out}}d{\varphi _{out}}}},$$

(20)$${T^{bd}} \equiv T({\theta _{in}},{\varphi _{in}};{\theta _{out}},{\varphi _{out}}) = \frac{1}{{{N_{in}}}} \cdot \frac{{{N_{tra}}({\theta _{out}},{\varphi _{out}})}}{{\sin {\theta _{out}}d{\theta _{out}}d{\varphi _{out}}}},$$

where ${0}\; \le {\; }{{\theta}_{{out}}}{\; < \; }\mathrm{\pi}{/2}$ and ${0}\; \le {\; }{\varphi _{{out}}}{\; } \le {\; 2}\mathrm{\pi}$ are for ${{R}^{{bd}}}$; $\mathrm{\pi}{/2}\; < \; {{\theta}_{{out}}}{\; } \le \mathrm{\pi}$ and ${0}\; \le {\; }{\varphi _{{out}}}{\; } \le {\; 2}\mathrm{\pi}$ are for ${{T}^{{bd}}}$. The second terms in the right sides of Eqs. (19) and (20) respectively represent the number of reflected photons and the number of transmitted photons per unit sold angle in the outgoing directions.

2.3.7 Comparison of our method and previous method

Throughout the descriptions for our MC simulation and previous studies, Table 1 summarizes the comparison of our MC method and previous method.

Table 1. Comparison of our MC method and previous method

View Table

3. Comparison of reflection and transmission with different MC methods

Two kinds of absorption need consideration. The first type of absorption by the atmospheric gases can be simulated through several well-known atmospheric radiative transfer models such as MODTRAN [51]. The second type of absorption by the sea fog particles has been considered in our study using three different MC methods, which are presented in this section. These three MC methods are employed with the previous method and our method for the calculations of reflectance and transmittance. These two types of absorption are not completely independent from each other. To demonstrate the performance of our proposed method, like the studies done in [14,21,27,31], we only focused on the interaction of the radiation with the sea fog particles by neglecting the interaction of the radiation with the atmospheric gases. At the end of this section, our studies show that our method has an advantage over the previous method. Once the demonstration is done, when coming to the reality in the consideration of both types of absorption, we can employ our method in the second type of absorption calculation, while MODTRAN (or other atmospheric radiative transfer models) can be used for the first type of absorption calculation.

In the investigation of how different methods affect the calculations of reflection and transmission, we here apply three different MC methods: H-G MC method, Mie MC method, and transport MC method. The H-G MC method stands for a MC method using the optical parameters from the Mie theory [41,42] and Henyey-Greenstein scattering phase function [48] in a single collision occurrence. The Mie MC method is another MC method adopting the Mie scattering phase function and the optical parameters as used in the H-G MC method. The transport MC method represents a MC method employing the transport optical parameters and the transport approximation of the scattering phase function, where the latter consists of a sum of the isotropic component and a term describing the peak of forward scattering [52,53]. In this study, the results of ${R}$ and ${T}$ acquired from the three MC methods along with the previous method are referred as “previous H-G MC”, previous Mie MC”, and previous transport MC”, respectively. A single value for each optical parameter is adopted for all considered radii of the sea fog particles in the previous method. Besides, the results of ${R}$ and ${T}$ obtained from the three MC methods together with our method are named as “new H-G MC”, “new Mie MC”, and “new transport MC”, respectively. In contrary to the previous method, the considered radii of the sea fog particles were copped into eight radius regions in our method and each radius region possessed its own value for each optical parameter. Hence, for the new H-G MC and the new Mie MC, the optical parameters in the i-th radius region are defined in Eqs. (2)−(6).

The key parameters needed in the new and previous MC methods are the optical parameters and the scattering direction. Pertaining to the optical parameters, Eqs. (2)−(5) were employed in the new H-G MC and the new Mie MC, while Eqs. (7)−(10) were adopted in the previous H-G MC and the previous Mie MC. For the new transport MC, the transport optical parameters in the i-th radius region are instead defined by:

(21)$$\begin{array}{ccc} {\alpha _i^{tr} = \frac{{{\mu _{abs,i}}}}{{A_i^{tr}}},}&{A_i^{tr} = \mu _{sca,i}^{tr} + {\mu _{abs,i}},}&{\mu _{sca,i}^{tr}} \end{array} = {\mu _{sca,i}}(1 - {g_i}),$$

where ${\alpha}$, ${A}$, $\mathrm{\mu }$, and ${g}$ have the same physical meanings as those defined in Eqs. (2)−(6), but the superscript tr indicates transport. Note that ${\mathrm{\mu }_{{sca,i}}}$, ${\mathrm{\mu }_{{abs,i}}}$, and ${{g}_{i}}$ have been defined in Eqs. (3), (4), and (6), respectively. For the previous transport MC, those parameters with subscript i (i.e., the i-th radius region) in Eq. (21) are replaced by the values averaged over all radii, which are similar to those defined in Eqs. (7)−(11).

Regarding the scattering direction generation, the azimuth angle $\varphi \; $is uniformly distributed between 0 and ${2}\mathrm{\pi}$ for all MC methods in this work. For the scattering angle $\theta $, Eq. (16) was used for the new H-G MC if ${{g}_{i}}$ was applied and used for the previous H-G MC if ${{g}_{i}}$ is replaced by $\bar{{g}}$. The ${\theta}$ in the Mie MC can be obtained from the scattering phase function ${\Phi}^{Mie}$ using the Mie theory [41,42]. For the previous Mie MC, the scattering phase function integrated over all radii of the sea fog particles is given by

(22)$${\overline \Phi ^{Mie}}(\theta ,\lambda ) = \frac{{\int_{{r_{\min }}}^{{r_{\max }}} {{r^2}{q_{sca}}(r,\lambda )p(r,\lambda ,\theta )n(r)dr} }}{{\int_{{r_{_{\min }}}}^{{r_{\max }}} {{r^2}{q_{sca}}(r,\lambda )n(r)dr} }}.$$

Similarly, for the new Mie MC, the scattering phase function integrated over the radii in the i-th radius region is given by

(23)$$\Phi _i^{Mie}(\theta ,\lambda ) = \frac{{\int_{{r_{\min ,i}}}^{{r_{\max ,i}}} {{r^2}{q_{sca}}(r,\lambda )p(r,\lambda ,\theta )n(r)dr} }}{{\int_{{r_{_{\min ,i}}}}^{{r_{\max ,i}}} {{r^2}{q_{sca}}(r,\lambda )n(r)dr} }}.$$

Note ${p(r},\; {\lambda}{,}\; \theta )$ is the Mie scattering phase function of the sea fog particle radius ${r}$, the wavelength of the incident radiation ${\lambda}$, and the ${\theta}$. The ${\theta}$ can then be determined through [21]:

(24)$${\eta _j} = \frac{{\int_{ - 1}^{{\mu ^j}} {{\Phi ^{Mie}}(\mu )d\mu } }}{{\int_{ - 1}^1 {{\Phi ^{Mie}}(\mu )d\mu } }},$$

where ${0}\; \le {\; }{{\eta}_{j}}{\; } \le {\; 1}$, $\mathrm{\mu }{\; = \; \cos(\theta })$, and ${j}$ runs from 1 to 1800 (denoted as ${N}$) in our simulations. The ${\Phi}^{Mie}$ in Eq. (24) can be the scattering phase function in Eq. (22) or Eq. (23). Following describes how to obtain ${\theta}$ in the Mie scattering calculation. Firstly, the value, $- {1}\; \le {\; }\mathrm{\mu }{\; } \le {\; 1}$, is divided into 1800 values with equidistance. Given ${r}$ and ${\lambda}$, there is no analytical expression of ${\Phi}^{Mie}$ in the Mie theory. As such, an analytical expression of ${{\eta}_{j}}$ is not available, either. Up to this stage, a look-up table for ${{\eta}_{j}}$, can be created for the determination of $\theta $. Secondly, a random number ${{\xi}_\mathrm{\mu }}$ is then generated between 0 and 1. Based on the pre-calculated look-up table, one can find out ${{\xi}_\mathrm{\mu }}$ falling in which corresponding ${j}$. Once index j is found, the sampled ${\mathrm{\mu }^{j}}$ can be determined, and ${\theta}{\; = \; \arccos(}{\mathrm{\mu }^{j}}{)}$ is hence determined. Searching through such a look-up table during each scattering occurrence can be time consuming, depending on the size of the look-up table (i.e., ${N}$.) This is why the calculation with the Mie MC is generally time consuming.

Let us turn to the transport MC. The transport scattering phase function is given by [53]

(25)$${\Phi ^{tr}}(\mu ) = (1 - g) + 2g\delta (1 - \mu ),$$

where $\mathrm{\delta }$ is the Dirac function. Similar to above, $\bar{{g}}$ is used in the previous transport MC, while ${{g}_{i}}$ is employed in the new transport MC. Again, the ${\theta}$ can be acquired through [21]:

(26)$${\xi _\mu } = \frac{{\int_{ - 1}^{{\mu ^j}} {{\Phi ^{tr}}(\mu )d\mu } }}{{\int_{ - 1}^1 {{\Phi ^{tr}}(\mu )d\mu } }} = \frac{{F(g,\mu )}}{C},$$

where the result in the numerator ${F(g},\; \mathrm{\mu }\textrm{)}$ is a function of g and $\mu $, and the result in the denominator C is simply a number. Given a random number ${{\xi}_\mathrm{\mu }}$, since C and g are known, $\mathrm{\mu }$ can then be easily acquired, and ${\theta}{\; = \; \arccos(\mu )}$ can be determined much easier than the Mie MC.

The comparisons presented here apply these six MC methods to the calculations of ${R}$ and ${T}$ respectively as a function of ${d}$, where the situations of ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$, ${{\theta}_{{in}}}{\; = \; 3}{\textrm{0}^\textrm{o}}$, ${W}\; = \; 0\textrm{.2}\; \textrm{g/}{\textrm{m}^\textrm{3}}$ and a total of $\textrm{1}{\textrm{0}^\textrm{8}}$photons are used. With these MC methods, ${R}$ as a function of ${d}$, ${T}$ as a function of ${d}$, and the computational time as a function of d are displayed in Fig. 5. Besides, relative errors in ${R}$ and ${T}$ as a function of ${d}$, respectively, are shown in Fig. 6. The relative error $\Delta $ is defined by

(27)$$\Delta = \left|{\frac{{{t_0} - t}}{t}} \right|,$$

where ${t}$ represents ${R}$ or ${T}$ results calculated from new Mie MC method, while ${{t}_\textrm{0}}$ can be those results obtained from the previous H-G MC, previous Mie MC, previous transport MC, new H-G MC or new transport MC. In Figs. 5 and 6, note that solid curves are referred to the results obtained from new MC methods, while the dashed curves are referred to the results acquired from previous MC methods. Below discuss our findings from these studies.

Fig. 5. Leftmost graph:$\; {R}$ versus ${d}$; middle graph: ${T}$ versus ${d}$; rightmost graph: computational time (in second) versus d. Note that three MC methods (H-G, Mie, and Transport) are employed for the case using λ_in = 4 µm, θ_in = $\textrm{3}{\textrm{0}^{\circ }}$, and ${W}\; = \; 0\textrm{.2}\; \textrm{g/}{\textrm{m}^\textrm{3}}$. Solid curves are results using our method and dashed curves are results using previous method.

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Fig. 6. Relative errors ($\Delta $) as functions of ${d}$ for ${R}$ calculation (left) and ${T}$ calculation (right). The relative errors are calculated with respective to the ${R}$ or ${T}$ results obtained from the “new Mie MC” (i.e., $t$). Therein the solid curves represent the relative errors for ${{t}_\textrm{0}}$ using “new H-G MC” (in red) or using “new transport MC” (in blue), and the dashed curves stand for ${{t}_\textrm{0}}$ using “previous H-G MC” (in red), “previous Mie MC” (in green), or “previous transport MC” (in blue).

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As displayed in Fig. 5, the trends of ${R}$ or ${T}$ results obtained from the three new MC methods are close to one another. The closeness results among the three new MC methods also occur in the relative errors in Fig. 6, where the two solid curves fall in relatively small areas. However, the results obtained from the three previous MC methods appear to be very different from one another, which can be seen in the dashed curves in Figs. 5 and 6. For why such outcomes happen is explained below.

For any given MC method (e.g., H-G MC method) used in this study, the only difference between the new MC method and the previous MC method is at the techniques pertaining to whether the radii of the sea fog particles are chopped into several regions and accordingly the optical parameters in each radius region hold their own particular values. The three new MC methods adopt our method, and the three previous MC methods employ the previous method. The details about our method and the previous method have been described in Section 2. In theory, regardless of which approach (i.e., H-G, Mie, or transport) is applied, one would expect to attain very close outcomes among all MC methods of different approaches. Positively, it demonstrates that our method is able to bring the outcomes very close to one another among H-G MC, Mie MC, and transport MC methods, while the previous method is unable. This is because the optical parameters in different radius region possess their own properties. Additionally, the more pieces the radii are chopped into, the more accurate calculations and the closer to the reality. This shows manifestation that our method has an advantage over the previous method.

Next, let us take a look at the computational time for different MC methods. Measured with a desktop computer Intel Core CPU I5-8250U @ 1.6 GHz, Fig. 5(c) presents the computational time as functions of ${d}$ using the six MC methods. In spite of which MC method is employed, the computational time increases as ${d}$ increases. Furthermore, our method by chopping the radii of the sea fog particles into eight radius regions indeed consumes more computational time if comparing the same MC methods (e.g., comparing the new H-G MC with the previous H-G MC). Notwithstanding, the transport MC methods consume the least time, while the Mie MC methods consume much more time than the other two MC methods, especially when ${d}$ gets thicker. This is because the Mie theory does not provide analytical expressions for the scattering phase function, resulting in time consuming during choosing the scattering direction for a single collision occurrence. Besides, the Mie scattering phase function relates to the computation of the Bessel function, which is also time consuming. If we look at the case at ${d}\; = \; 400\; \textrm{m}$ and compare the computational time with that obtained by the new Mie MC, we find that the computational time obtained by the new H-G MC is reduced by ∼81%, while that obtained by the new transport MC is even in reduction of 1 to 2 orders in magnitude. In addition, the computational time obtained by the previous Mie MC is reduced by ∼11% for ${d}\; = \; 400\; \textrm{m}$ if it is compared with that obtained by the new Mie MC.

As shown in Fig. 5(c), in spite of the reliance on different MC methods for R and ${T}$ calculations, the transport MC method indeed advantages the computational time [52,53]. Because of its less computational time within an acceptable accuracy, the transport MC method usually can be employed to solve many engineering issues, such as the radiation transfer problems when the radiation travels through a medium of large optical thickness with a predominate forward scattering [21,23,52,53]. The H-G MC method also performs well in terms of computational time though it is not so fast as the transport MC method. Nevertheless, we chose the new H-G MC method and the previous H-G method for other situation studies, which are presented in Sections 4.

4. Further experiments

As mentioned previously, the sea fog layer and the air above are assumed to possess the same refractive index in our MC simulation. In this case, the reflection of the photons at the upper boundary without going into the sea fog layer (see Fig. 3(a)) would not happen and thus is not considered in our MC simulation. Furthermore, our simulations only focus on the effects of the absorption and scattering due to Mie scattering. The absorption and scattering due to the atmospheric gas molecules are neglected in these simulations.

Absorption and scattering are the two major physical mechanisms that the photons would experience when they collide with the sea fog particles of various radii in the sea fog layer. Hence, the impacts of the absorption and scattering mechanisms on the reflectance and transmittance of the photons are investigated in our studies. However, the absorption and scattering of the radiation by the sea fog particles are influenced by several factors, such as ${W}$ and sea fog thickness (${d}$). The influence of ${d}$ on ${R}$ and ${T}$ of the radiation has been presented in Section 3. The influence of ${W}$ on ${R}$ and ${T}$ of the radiation is also inspected using our method and previous method (or the previous H-G MC and the new H-G MC) in the MC simulation. Apart from that, the influence of ${W}$ on ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$ of the radiation is examined in this section as well. This is the focus of this section.

4.1 Influence of ${W}$ on ${R}$ and ${T}$

Figures 7(a) – (f) display the ${R}$ and ${T}$ of the photons versus ${W}$, where three different wavelengths (${{\lambda}_{{in}}}{\; = \; 3}$, $\textrm{4}$ and $\textrm{5}\; \mathrm{\mu}\textrm{m}$) together with three different incident zenith angles (${{\theta}_{{in}}}{\; = \; }{\textrm{0}^\textrm{o}}$, $\textrm{3}{\textrm{0}^\textrm{o}}$ and $\textrm{6}{\textrm{0}^\textrm{o}}$), ${\varphi _{{in}}}{\; = \; 18}{\textrm{0}^\textrm{o}}$, and ${d}\; = \; 200\; {\textrm{m}}$ are taken into account in our simulation study. Some learnings are discussed below.

Firstly, because of point 4) given in Section 2.1.1, the enhancement in ${{A}^{{tot}}}$ hinders the photons to pass through the sea fog layer. For each ${{\lambda}_{{in}}}{\; }$, the overall variation trends of ${R}$ and ${T}$ decrease as ${W}$ rises except that peaks around ${W}\; = \; 0\textrm{.06}\; \textrm{g/}{\textrm{m}^\textrm{3}}$ appear in ${R}$ curves.

Fig. 7. ${R}$ and ${T}$ of photons as functions of ${W}$ for three wavelengths (${{\lambda}_{{in}}}{\; = \; 3}$, $\textrm{4}$ and $\textrm{5}\; \mathrm{\mu}\textrm{m}$) and three incident angles (${{\theta}_{{in}}}{\; = \; }{\textrm{0}^\textrm{o}}$, $\textrm{3}{\textrm{0}^\textrm{o}}$ and $\textrm{6}{\textrm{0}^\textrm{o}}$). Note that the solid curves are for using our method and dashed curves for using previous method.

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Secondly, peaks appear around ${W}\; = \; 0\textrm{.06}\; \textrm{g/}{\textrm{m}^\textrm{3}}$ in the ${R}$ curves when ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$ at ${{\theta}_{{in}}}{\; = \; }{\textrm{0}^\textrm{o}},\; \textrm{3}{\textrm{0}^\textrm{o}},\; \textrm{6}{\textrm{0}^\textrm{o}}$ and when ${{\lambda}_{{in}}}{\; = \; 5\; }\mathrm{\mu}\textrm{m}$ at ${{\theta}_{{in}}}{\; = \; }{\textrm{0}^\textrm{o}}\textrm{,}\; 3\textrm{0}^\textrm{o}$. The reason is explained as follows. Three quantities mainly affecting ${R}$ calculation are g, ${\alpha}$, and ${{A}^{{tot}}}$. We have investigated which quantity would cause the peaks in ${R}$ around ${W}\; = \; 0\textrm{.06}\; \textrm{g/}{\textrm{m}^\textrm{3}}$. It was found that none of these three quantities would individually bring about the peaks. Since ${R} = \textrm{1} - {{\alpha}^{{tot}}} - {T}$, we then looked into the total effect of ${{\alpha}^{{tot}}} + {T}$ on ${R}$ as ${W}$ varies, which is displayed in Fig. 8 using ${{\theta}_{{in}}}{\; = \; }{\textrm{0}^\textrm{o}}$ and ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$ as inspection. One can see that the valley in the curve of ${{\alpha}^{{tot}}} + {T}$ is around ${W}\; = \; 0\textrm{.06}\; \textrm{g/}{\textrm{m}^\textrm{3}}$, which corresponds to the peak in ${R}$ at the same location of ${W}$. Thirdly, the results reveal that both ${R}$ and ${T}$ receive the largest, medium, and lowest significances corresponding to ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$, ${{\lambda}_{{in}}}{\; = \; 5\; }\mathrm{\mu}\textrm{m}$ and ${{\lambda}_{{in}}}{\; = \; 3\; }\mathrm{\mu}\textrm{m}$, accordingly. The reason can be explained using point 3) given in Section 2.1.1.

Fig. 8. Curves of ${{\alpha}^{{tot}}}$, ${T}$, ${{\alpha}^{{tot}}} + {T}$, and ${R} = \textrm{1} - {{\alpha}^{{tot}}} - {T}$ of the photons as functions of ${W}$ using our method (solid curves) and previous method (dashed curves), where ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$ and ${{\theta}_{{in}}}{\; = \; }{\textrm{0}^\textrm{o}}$ are applied.

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Fourthly, again, when ${{\lambda}_{{in}}}$ goes higher, it was found that the discrepancies in ${R}$ and ${T}$ using our method and previous method become bigger. Such a phenomenon can be explained using points 1) and 2) of Section 2.1.1.

4.2 Influence of ${W}$ on ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$

Next, we studied the 3D distributions of ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$ respectively versus ${{\theta}_{{out}}}$ and ${\varphi _{{out}}}$ of the photons for three water contents (${W}\; = \; 0\textrm{.06}$, 0.2 and $\textrm{0}\textrm{.4g}/{\textrm{m}^\textrm{3}}$), which are displayed in Fig. 9. Only transmission with multiple scatterings was considered in the calculation of ${{T}^{{bd}}}$ in our study. Here the following conditions are employed in this simulation study: ${{\theta}_{{in}}}{\; = \; 3}{\textrm{0}^\textrm{o}}$, ${\varphi _{{in}}}{\; = \; 18}{\textrm{0}^\textrm{o}}$, ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$, and ${d}\; = \; 200\; {\textrm{m}}$. Note that the 3D distributions are plotted using our method only, whereas distributions of ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$ as a function of ${{\theta}_{{out}}}$ are plotted in Fig. 10 by averaging all ${\varphi _{{out}}}$’s for comparison using our method and previous method. Some learnings from this study are discussed as follows.

Firstly, it was found that the peak ${\theta}_{{out}}^{{max}}\; \sim \; {{\theta}_{{in}}} + {\theta}_{{shift}}^{R}$ for the ${{R}^{{bd}}}$ distribution and the peak (${\theta}_{{out}}^{{max}} - \textrm{9}{\textrm{0}^\textrm{o}}\textrm{)}\; \sim \; (9{\textrm{0}^\textrm{o}} - {{\theta}_{{in}}} + {\theta}_{{shift}}^{T}\textrm{)}$ for the ${{T}^{{bd}}}$ distribution. This is because of ${W}$’s influence in this simulation study. The quantities ${\theta}_{{shift}}^{R}$ and ${\theta}_{{shift}}^{T}$ are introduced to describe the shift of ${\theta}_{{out}}^{{max}}$ with respect to ${{\theta}_{{in}}}$ in the ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$ distributions, respectively.

Fig. 9. Three-dimensional distributions of ${{R}^{{bd}}}$ (upper graphs) and ${{T}^{{bd}}}$(lower graphs) of photons as functions of ${{\theta}_{{out}}}$ and ${\varphi _{{out}}}$ for three different values of ${W}$’s, where ${{\theta}_{{in}}}{\; = \; 3}{\textrm{0}^\textrm{o}}$, ${\varphi _{{in}}}{\; = \; 18}{\textrm{0}^\textrm{o}}$, ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$, and ${d}\; = \; 200\; {\textrm{m}}$.

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Fig. 10. Distributions of ${{R}^{{bd}}}$ (left graph) and ${{T}^{{bd}}}$ (right graph) of photons as a function of ${{\theta}_{{out}}}$ for three different values of ${W}$’s, where ${{\theta}_{{in}}}{\; = \; 3}{\textrm{0}^\textrm{o}}$, ${\varphi _{{in}}}{\; = \; 18}{\textrm{0}^\textrm{o}}$, ${{\lambda}_{{in}}}{\; = \; 4\; }\mathrm{\mu}\textrm{m}$, and ${d}\; = \; 200\; {\textrm{m}}$. Note that the solid curves are for using our method and dashed curves for using previous method.

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Given ${\lambda _{in}}$ in this case, the quantities, ${\theta}_{{shift}}^{R}$ and ${\theta}_{{shift}}^{T}$, depend on the amount of ${W}$ in the sea fog layer. When ${W}$ is near zero, the photons would experience very few scattering occurrences. As a result, the photons pass through the sea fog layer with very few collisions with sea fog particles until the photons reach the lower interface. At the lower interface, if the photons get reflected back, most of the reflection angle, ${\theta}_{{out}}^{{max}}$, would be close to ${\theta _{in}}$, and the photons keep travelling up by this ${\theta}_{{out}}^{{max}}$ with very few scatterings until they go out the upper interface. If the photons get transmitted to the sea, most of the transmission angle, ${\theta}_{{out}}^{{max}} - \textrm{9}{\textrm{0}^\textrm{o}}$, would be close to $\textrm{9}{\textrm{0}^\textrm{o}} - {{\theta}_{{in}}}$. In this case, both ${\theta}_{{shift}}^{R}$ and ${\theta}_{{shift}}^{T}$ are near zero. See the illustration displayed in the left graph of Fig. 11.

Fig. 11. Illustrations for the reflections angles and transmission angles of the photons when the photons propagate through a layer of sparse sea fog particles (left graph) and through a layer of dense sea fog particles (right graph).

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The quantities of ${g}$, ${\alpha}$, and collisions occurrences are affected by ${W}$. Thus, when the ${W}$ gets higher, the photons would have more probability to be absorbed or experience more scatterings with sea fog particles. In this case, if the photons survive, most of the reflection angles and transmission angles at the lower interface may depart from ${{\theta}_{{in}}}$ and $\textrm{9}{\textrm{0}^\textrm{o}} - {{\theta}_{{in}}}$, respectively. That is, the absolute values $\textrm{|}{\theta}_{{shift}}^{R}\textrm{|}$ and $\textrm{|}{\theta}_{{shift}}^{T}\textrm{|}$ may increase as ${W}$ increases. See the illustration depicted in the right graph of Fig. 11. Therefore, by looking at the peaks in the 2D distributions, it was found that ${\theta}_{{shift}}^{R}{\; = \; }{\textrm{8}^\textrm{o}}\textrm{},\; - \textrm{2}{\textrm{2}^\textrm{o}}\textrm{},\; - \textrm{3}{\textrm{0}^\textrm{o}}$ and ${\theta}_{{shift}}^{T}{\; = \; }{\textrm{9}^\textrm{o}}\textrm{,}\; 3{\textrm{0}^\textrm{o}}\textrm{,}\; 3{\textrm{0}^\textrm{o}}$ corresponding to ${W}\; = \; 0\textrm{.06},\; 0\textrm{.2},\; 0\textrm{.4}\; \textrm{g/}{\textrm{m}^\textrm{3}}$, respectively.

In addition, the high ${W}$ would increase the photons’ scattering occurrences with sea fog particles, leading to more variety of reflection angles and transmission angles. This indicates that the higher ${W}$ usually brings wider distributions of ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$ though ${{R}^{{bd}}}$ barely exposes this feature (see Fig. 10).

Lastly, as presented previously, Fig. 10 exhibits discrepancies in ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$ when using our method and previous method, which can be explained by points 1) and 2) of Section 2.1.1.

5. Conclusions

Our proposed method of investigating the reflection and transmission characteristics of radiation transfer through a poly-disperse sea fog system are presented in this paper. First of all, a poly-disperse sea fog system was taken as a poly-disperse system comprising sea fog particles of various radii. As such, the radii of the sea fog particles were divided into eight radius regions. After that, by using Mie scattering theory, the optical parameters (such absorption probability (${\alpha}$), asymmetry factor (${g}$), attenuation (${A}$)) and collision probability of the photon colliding with sea fog particles in each radius region (${{r}_{i}}$) were calculated based on the sea fog particle size distribution. Subsequently, an improved Monte Carlo (MC) method was employed to solve the radiation transfer in the poly-disperse sea fog layer.

Furthermore, it is worthwhile to mention that different refractive indices at the lower interface between the sea fog layer and the sea were considered in our study such that both reflection and transmission would occur simultaneously when the photons reached the lower interface. For all the calculations of reflectance and transmittance, the results obtained using our method were compared with those obtained using the previous method which employed one single value for each ${\alpha}$, ${g}$, and ${A}$ by averaging over all sea fog particle sizes.

The first consideration of calculations is for comparison among three different MC methods: Mie MC method (a MC method directly allied with the optical parameters from the Mie theory and Mie scattering phase function), H-G MC method (a MC method utilizing the optical parameters from the Mie theory and Henyey-Greenstein scattering phase function), and transport MC method (a MC method adopting the transport optical parameters together with the transport approximation of the scattering phase function). The results of reflectance and transmittance calculations using the three MC methods along with our method are very close to one another. However, significant discrepancies appear in the results if the three MC methods along with the previous methods are applied. The results demonstrate that our method is able to bring the outcomes very close to one another among H-G MC, Mie MC, and transport MC methods, while the previous method is unable. Besides, it was found that the transport MC method advantages the computational time over the other two MC methods. In spite of the best computational performance for the transport MC method, the H-G MC method also performance well in terms of computational time. We thus adopted the H-G MC method together with our method and the previous method for some more studies.

Followed by the H-G MC method, we investigated the influences of the water contents (${W}$) of the sea fog layer on the reflectance (${R}$), transmittance (${T}$), and on the bidirectional distributions of reflectance (${{R}^{{bd}}}$) and transmittance (${{T}^{{bd}}}$). Calculations using our MC method and previous MC method are displayed for comparison.

Our studies show that ${W}$ and ${d}$ have significant impacts on the results of ${R}$, ${T}$, ${{R}^{{bd}}}$, and ${{T}^{{bd}}}$. On the other hand, the calculations of ${R}$, ${T}$, ${{R}^{{bd}}}$, and ${{T}^{{bd}}}$ are related to the ${A}$, scattering, and ${\alpha}$ when the radiation propagates through the sea fog layer, where the scattering directions are determined by ${g}$. However, ${\alpha}$, ${g}$, and ${A}$ involve ${{r}_{i}}$ and ${{\lambda}_{{in}}}$. In our MC simulation, ${\alpha}$, ${g}$, and ${A}$ are computed according to each radius region ${{r}_{i}}$ of the sea fog particles with which the photons collide. On the contrary, the previous method uses one single value for each$\; {\alpha}$, ${g}$, and ${A}$ by averaging over all sea fog particle sizes. As a result, discrepancies are seen in ${R}$, ${T}$ ${{R}^{{bd}}}$, and ${{T}^{{bd}}}$ when using our method and using previous method. Although the radii of the sea fog particles were divided into only eight radius regions in our MC simulation, it is expected that the more divided radius regions, the more accurate ${\alpha}$, ${g}$, and ${A}$ and thus the more accurate ${R}$, ${T}$, ${{R}^{{bd}}}$, and ${{T}^{{bd}}}$. Therefore, in consideration of accuracy, the calculation of$\; {\alpha}$, ${g}$, and ${A}$ as a function of ${{r}_{i}}$ is needed.

The proposed MC simulation in this paper is in an attempt to fulfill the quantitative analysis of radiation transfer in more complex media. Beside the sea fog layer, our method presented in this paper can be further applied to investigate the optical characteristics (such as reflection, absorption, and transmission, etc.) of the radiation propagation through poly-disperse media such as cloud, rain and other aerosols, guide the object detections in maritime scenes during foggy weather, and provide support for remote sensing in oceanic environment.

6. Future work

In the future, to propagate the radiation through the sea fog layer, we will incorporate MODTRAN (or other atmospheric radiative transfer models) with a MC method using our method. The former model is to handles the interaction of radiation with the atmospheric gases, and the latter method is to deal with the interaction of radiation with sea fog particles. The total effects of ${R}$ and ${T}$ from these two types of interactions can be compared with in-situ measurements if the measurements are available.

Funding

Natural Science Basic Research Plan in Shaanxi Province of China (2019JQ-235); National Science Fund for Distinguished Young Scholars (61904138,61705179); Fundamental Research Funds for the Central Universities (20101186281); Higher Education Discipline Innovation Project (111 Project (B17035)).

Disclosures

The authors declare no conflicts of interest.

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Technique No.	Our method	Previous method
1	Consider reflection and refraction at two media of different refractive indices	Usually only a uniform medium is considered
2	Collision probability varies with the radii of sea fog particles with which the photons collide	Collision probability = 1 is used for all $r_{i}$ ’s of the sea fog particles with which the photons collide
3	$α_{i}$ varies with the radii of sea fog particles with which the photons collide	One uniform $\bar{α}$ is used, which does not depend on the radii of sea fog particles
4	$g_{i}$ used for determining photons’ scattering directions varies with the radii of sea fog particles with which the photons collide	$\bar{g}$ used for determining photons’ scattering directions is a uniform value

Numerical analysis of light reflection and transmission in poly-disperse sea fog

Abstract

1. Introduction

2. Theoretical basis

2.1 Poly-disperse sea fog system

2.1.1 Demonstration of ${\alpha}$ versus ${r}$, g versus ${r}$, and ${{A}^{{tot}}}$ versus ${W}$

2.2 Interactions between radiation and sea fog particles

2.3 Monte Carlo simulation

2.3.1 Collision length and remaining distance

2.3.2 Reflection or transmission at lower interface

2.3.3 Collision probability and regional radii

2.3.4 Absorption probability

2.3.5 Scattering direction

2.3.6 Computation of reflectance and transmittance

2.3.7 Comparison of our method and previous method

3. Comparison of reflection and transmission with different MC methods

4. Further experiments

4.1 Influence of ${W}$ on ${R}$ and ${T}$

4.2 Influence of ${W}$ on ${{R}^{{bd}}}$ and ${{T}^{{bd}}}$

5. Conclusions

6. Future work

Funding

Disclosures

References

Cited By

Figures (11)

Tables (1)

Equations (27)

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