Time-dependent polarized radiative transfer in an atmosphere-ocean system exposed to external illumination

Cun-Hai Wang; Yan-Yan Feng; Xun Ben; Kai Yue; Xin-Xin Zhang

doi:10.1364/OE.27.00A981

1. Introduction

Ignoring the polarization effect of radiation usually leads to significant errors in the field of atmosphere radiative transfer [1–3]. Polarized radiative transfer in the atmosphere-ocean system has attracted a great deal of attention since 70% of our Earth’s surface is covered by ocean. Kattawar et al. [4,5] were among the first researchers who studied the polarized radiative transfer in the atmosphere-ocean system. After that, He et al. [6,7] developed a vector radiative transfer model for the atmosphere-ocean system with a rough surface. Based on the discrete ordinate method, Sommersten et al. [8] and Cohen et al. [9] studied the polarized radiative transfer in a coupled system with different refractive indices. To make the polarized radiative transfer model easy-to-use and computationally efficient, Zhai et al. [10–13] developed an SOS (successive order of scattering) based vector radiative transfer model and studied the polarized radiative transfer in the coupled atmosphere-ocean system with specific scenarios.

The governing equation of polarized radiative transfer process in an emitting, absorbing, and scattering medium is referred to as the vector radiative transfer equation (VRTE). It is an integral-differential equation for the Stokes vector components and is usually difficult to solve analytically; thus many numerical algorithms [14–19] have been proposed for polarized radiative transfer problems. Among various algorithms, the ray-tracing based Monte Carlo [20–22] is popular due to its easy implementation when dealing with the polarized radiative transfer processes such as reflection and scattering events. As a statistical algorithm based on tracing a large number of radiation bundles, the shortcoming of Monte Carlo lies in its lack of computational efficiency. To overcome this shortcoming, Yi et al. [23] and Wang et al. [24,25] introduced a time shift and superposition principle to the traditional Monte Carlo method and developed a modified Monte Carlo algorithm for solving the scalar radiative transfer problems. Recently, Ramon et al. [26] described a GPU-based Monte Carlo technique to improve the computational performance and modeled the atmosphere-ocean polarized radiative transfer problem.

In recent years, due to the availability of short-pulsed illumination and advanced detectors, time-dependent radiation signals [27–29] have been extended to more and more practical applications. However, most of the current work on polarized radiative transfer in the atmosphere-ocean system ignores the effect of time and only solves the steady-state radiation signals. Up to now, only a few investigations [30–35] on time-dependent polarized radiative transfer have been reported but none of them is about the atmosphere-ocean system involving interface reflection and vertically inhomogeneous properties.

The objective of this paper is to fill the gap of time-dependent polarized radiative transfer in the atmosphere-ocean system exposed to external illumination. By introducing a time shift and superposition principle to improve the computational efficiency of the traditional Monte Carlo (TMC) method, we developed a modified Monte Carlo (MMC) which fully considers the effects of time and polarization in the atmosphere-ocean radiative transfer. This paper is organized as follows. In Section 2, the ray tracing theory of polarized radiation by taking advantage of time shift and superposition principle is introduced. In Section 3, the accuracies of our MMC are established and the time-resolved Stokes components for the atmosphere-ocean system are presented. Finally in Section 4, the conclusions are summarized.

2. Theory of MMC for time-dependent polarized radiative transfer

In this section, the ray tracing technique for a polarized radiation bundle is presented. The polarization information of a radiation bundle is described by the Stokes vector

I = {(I, Q, U, V)}^{T},

where the superscript “T” denotes the matrix transpose, the Stokes vector component I is the radiation intensity, Q is the linear polarization aligned parallel or perpendicular to the z-axis, U is the linear polarization aligned ± 45° to the z-axis and V is the circular polarization.

To analyze the effect of time, the top surface of the atmosphere is considered to be exposed to illumination with a duration of t_p and incident direction of Ω⁰ = (θ⁰, φ⁰). In the implementation of the traditional Monte Carlo algorithm, a number of radiation bundles are assumed to emit uniformly within this duration, and the initial time of a bundle is given by

t_{0} = R_{t} t_{p},

where R_t is a uniform pseudo-random number within [0, 1].

When the radiation bundle propagates in the atmosphere-ocean system, the location and polarization state of a bundle varying with time are recorded. For a bundle at the current position and time of (x, y, z, t), a free propagation distance L is calculated by

L = - \frac{\ln (1 - R_{L})}{β},

where β is the extinction coefficient of the current position and R_L is a uniform pseudo-random number.

After a free traveling process, the bundle reaches a new state of

{\begin{cases} x' = x + L \sin θ \cos φ \\ y' = y + L \sin θ \sin φ \\ z' = z + L \cos θ \\ t^{'} = t + \frac{n L}{c_{0}} \end{cases},

where θ and φ denote the polar and azimuth angle of the propagation direction, respectively, n is the refractive index of the current medium (atmosphere or water), and c₀ is the light speed in a vacuum.

For the Monte Carlo simulation of the polarized radiative transfer problem, the key is to determine the polarization state after the radiation bundle is scattered by the atmosphere or water. As shown in Fig. 1, Ωⁱ denotes the incident direction and Ω^s denotes the scattered direction. The Stokes vector of the scattered bundle can be obtained by

I (Ω^{s}) = Z (Ω^{i} \to Ω^{s}) I (Ω^{i}),

where Z is the scattering phase matrix and is usually obtained by rotating the single scattering Mueller matrix P as [36–38]

Z (Ω^{i} \to Ω^{s}) = ℜ (π - ψ^{s}) P (Ω^{i}, Ω^{s}) ℜ (- ψ^{i}),

where ψⁱ and ψ^s denotes the rotation angles as shown in Fig. 1, and the rotation matrix

ℜ (ψ)

is defined as [39]

Fig. 1 The incident direction, scattering direction, and rotation angle in the scattering event.

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ℜ (ψ) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos 2 ψ & \sin 2 ψ & 0 \\ 0 & - \sin 2 ψ & \cos 2 ψ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

The scattering matrix P for the Rayleigh scattering atmosphere-ocean considered in this paper is written as [40–42]

P (Θ) = \frac{3}{4} (\begin{matrix} \cos^{2} Θ + 1 & - \sin^{2} Θ & 0 & 0 \\ - \sin^{2} Θ & \cos^{2} Θ + 1 & 0 & 0 \\ 0 & 0 & 2 \cos Θ & 0 \\ 0 & 0 & 0 & 2 \cos Θ \end{matrix}),

where Θ denotes the scattering angle between the incident and scattering directions, as shown in Fig. 1.

The first component of the Stokes vector I denotes the radiation intensity and the other components describe the polarization state. To ensure energy conservation, the Stokes vector is normalized as

I^{*} = \frac{I}{I} = {(1, \frac{Q}{I}, \frac{U}{I}, \frac{V}{I})}^{T},

then the ray-tracing technique can be extended to the polarized radiation bundle by applying the normalization before each propagating event (scattering, reflection, and transmission).

The interface of the atmosphere and ocean is considered to be specular and the reflection matrix is calculated by [43]

R (θ) = \frac{1}{2} (\begin{matrix} r_{h}^{2} + r_{v}^{2} & r_{h}^{2} - r_{v}^{2} & 0 & 0 \\ r_{h}^{2} - r_{v}^{2} & r_{h}^{2} + r_{v}^{2} & 0 & 0 \\ 0 & 0 & 2 Re (r_{v} r_{h}^{*}) & 2 Im (r_{h} r_{v}^{*}) \\ 0 & 0 & 2 Im (r_{v} r_{h}^{*}) & 2 Re (r_{v} r_{h}^{*}) \end{matrix}),

where the superscript ‘*’ denotes conjugate transpose, Re (•) and Im (•) indicate the real and imaginary part, respectively. The vertical and horizontal reflection coefficients r_v and r_h are given by

r_{v} (θ) = \frac{\cos θ - \sqrt{n^{2} - \sin {}^{2}θ}}{\cos θ + \sqrt{n^{2} - \sin {}^{2}θ}},

r_{h} (θ) = \frac{n^{2} \cos θ - \sqrt{n^{2} - \sin {}^{2}θ}}{n^{2} \cos θ + \sqrt{n^{2} - \sin {}^{2}θ}},

where n denotes the relative refractive index of the water to the atmosphere. With the above definitions, the transmission matrix is calculated by

T (θ) = \frac{n}{2} \frac{\cos ϑ}{\cos θ} (\begin{matrix} t_{h}^{2} + t_{v}^{2} & t_{h}^{2} - t_{v}^{2} & 0 & 0 \\ t_{h}^{2} - t_{v}^{2} & t_{h}^{2} + t_{v}^{2} & 0 & 0 \\ 0 & 0 & 2 Re (t_{v} t_{h}^{*}) & 2 Im (t_{h} t_{v}^{*}) \\ 0 & 0 & 2 Im (t_{v} t_{h}^{*}) & 2 Re (t_{v} t_{h}^{*}) \end{matrix}),

where ϑ = arcsin (sinθ/n) denotes the refractive angle, and the transmission coefficients t_v and t_h are given by

t_{v} (θ) = \frac{2 \cos θ}{\cos θ + \sqrt{n^{2} - \sin^{2} θ}},

t_{h} (θ) = \frac{2 n \cos θ}{n^{2} \cos θ + \sqrt{n^{2} - \sin^{2} θ}} .

Monte Carlo simulation is based on statistics. In order to obtain the time-resolved Stokes vector distributions in spatial and angular space, time averaging of the bundles within a finite time period is necessary. If the simulated bundles within the duration of the illumination is N, among which N_t bundles reach a location P within the solid angle of dΩ centered at Ω = (ϑ, φ) and a time period of Δt centered at iΔt, then the time-domain vector radiation distribution (TDVRD) is given by

M_{i, Ω, P} = \frac{\sum_{j = 1}^{N_{t}} I_{j}^{*} / Δ t}{N / t_{p}},

where

I_{j}^{*}

is the normalized Stokes vector of the jth bundle. The TDVRD represents the intensity and polarization state of the statistical radiation bundles. Thus the time-resolved Stokes vector for time t, direction Ω, and location P of the one-dimensional atmosphere-ocean can be written as

I_{i, Ω, P} = \frac{I_{0} | \cos θ_{0} |}{| \cos θ | d Ω} \cdot M_{i, Ω, P},

where I₀ is the intensity of the incident illumination and θ₀ is the incident angle.

In the traditional Monte Carlo (TMC) algorithm for time-dependent radiative transfer, the simulated bundles are assumed to be uniformly emitted within the duration of the external illumination. If one wants to obtain smooth and satisfactory results using this treatment, the simulated bundles within each finite time period (used for time average) need to be compared with those used for a steady problem. Thus the bundle number for the time-dependent problem is many more times (the number of time steps) than that for a steady problem and the simulation time is usually too long to be acceptable, which prohibits the practical application of the TMC for time-dependent polarized radiative transfer problems. To improve the computational efficiency of the Monte Carlo technique, we develop a modified Monte Carlo (MMC) algorithm combined with time shift and superposition (TSS) principle and then extend it for solving the time-dependent polarized radiative transfer in the atmosphere-ocean system exposed to an external illumination source.

In this work, the illumination duration t_p is divided into M time steps of Δt = t_p/M. By applying the TSS described in [23–25], only the radiation bundles emitted within the first time step Δt need to be traced, thus the number of simulation bundles is decreased by M times and this significantly shortens the computational time. By setting t_p in Eq. (16) as Δt, the corresponding Stokes vector at a time of t = iΔt by the first time step is written as

I_{i, Ω, P}^{Δ} = \frac{I_{0} | \cos θ_{0} |}{| \cos θ | d Ω} \cdot M_{i, Ω, P}^{Δ} = \frac{I_{0} | \cos θ_{0} |}{| \cos θ | d Ω} \cdot \frac{\sum_{j = 1}^{N_{t}} I_{j}^{*}}{N^{Δ}},

where N^Δ = N/M denotes the number of emitted bundles within the first time step Δt. Then the illumination induced Stokes vector at time t = mΔt can be written as

I_{m, Ω, P}^{} = \frac{1}{\min (m, M)} \sum_{k = 1}^{\min (m, M)} I_{_{(m - k + 1), Ω, P}}^{Δ} .

Here, it is noted that if m < M, the computational time mΔt is within the illumination duration t_p = MΔt, only the previous time periods (k from 1 to m) contribute to the Stokes vector at the current time. Otherwise, if m > M, all the time periods of the illumination (k from 1 to M) contribute to the Stokes vector at current time. Therefore, the symbol k in Eq. (19), which denotes the index of shifted and superposed time step, is from 1 to min(m, M).

3. Results and discussions

In this section, the MMC is applied to solve the time-dependent polarized radiative transfer in a general atmosphere-ocean system exposed to illumination on the top of the atmosphere. The computational performance of the MMC is verified and time-resolved Stokes distributions are presented. In the following results, a widely-used dimensionless time defined as t* = βc₀t is applied for time-resolved polarized results. All the following simulations are taken on a laptop with the configuration of an Intel Core (TM) i7-6700 processor with 3.40 GHz CPU and 8 GB RAM.

3.1 Validations of the MMC for polarized radiative transfer

To verify the accuracy of the MMC for polarized radiative transfer, the case of one-dimensional atmosphere-ocean system studied by Sommersten et al. [8] is considered in this paper. As shown in Fig. 2, the optical thickness of the upper atmosphere and the lower water is 0.15 and 1.0, respectively. The interface between the atmosphere and water is specular and the relative refractive index of water to the atmosphere is taken as n = 1.338. Both the atmosphere and water are Rayleigh scattering with an extinction coefficient of β = 1.0 and a scattering albedo of ω = 1.0. The bottom of the water is completely absorbing and the top of the atmosphere is exposed to downward illumination with a unity density I₀ = 1.0 W/m², a polarization state of I₀ = (I₀, Q₀, U₀, V₀)^T = (1.0, 0, 0.5, 0.5)^T, and an incident direction of Ω⁰ = (120°,0°).

Fig. 2 Schematic of the atmosphere-ocean system exposed to an external illumination.

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For the time-dependent polarized radiative transfer in the atmosphere-ocean system exposed to external illumination, if the duration of the illumination is long enough, the radiation energy within the system will reach a steady state. The MMC solutions after a sufficient number of time steps are compared with the steady solutions for the validations since no data of time-resolved solutions for the atmosphere-ocean system are readily available in the literature.

The computational accuracy and efficiency of the MMC are first compared with those of the TMC. By dividing the illumination duration into M = 1000 time steps and using N^Δ = 5 × 10⁹ simulation bundles (which is proved to be enough for stable results after several tests with different bundle numbers), the MMC is applied to solve the above-described polarized radiative transfer. With a viewing azimuthal angle of φ_V = 90°, the zenith distribution of Stokes vector components just below the atmosphere-ocean interface (location A in Fig. 2) at a dimensionless time of t* = 20 (a steady state has been reached) obtained by our MMC are compared with those by Sommersten et al. [8] using a discrete ordinate method (DOM), as shown in Fig. 3. The TMC results using the same bundles are also plotted in Fig. 3 for comparison. The computational time is 4h 39 min for our MMC and 5h 18 min for TMC, respectively. Here it is announced that the TMC calculation with N = 5 × 10¹¹ bundles needs more than 390 hours and the results fluctuations are still obvious. It is seen from Fig. 3 that with the same simulated bundles, the results obtained by the MMC are smooth and agree well with the DOM results, while the TMC results have strong fluctuations due to the statistical noise. The comparisons in Fig. 3 indicate that the MMC is accurate for solving polarized radiative transfer problems in the atmosphere-ocean system and has obvious superiority over the TMC on numerical results.

Fig. 3 Stokes vector components just below the interface obtained by different algorithms.

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For different viewing azimuth angles φ_V = 0°, 90°, and 180°, the zenith distributions of Stokes vector component at the location A obtained by our MMC and the DOM are compared in Fig. 4. Here, the negative direction cosine denotes downward radiation, while the positive direction cosine denotes upward radiation. When the radiation propagates from the water to the atmosphere-ocean interface, the direction cosine of the critical angle is $μ_{c} = \sqrt{1 - {(1 / n)}^{2}} = 0.66$ , which means any radiation with a direction cosine of μ = [0, 0.66] incident to location A will be totally reflected back to the water with a direction cosine of μ = [-0.66, 0]. Thus all of the Stokes vector components shown in Fig. 4 have step changes at the direction of μ = −0.66, and this demonstrates our MMC has accurately simulated the polarized propagation at the refractive interface. It is also seen from Fig. 4 that for different viewing azimuth angle φ_V = 0°, 90°, and 180°, the zenith distributions of Stokes vector component I have similar intensity changing trends, while the other three components Q, U, and V have an obvious difference in both intensity and changing trends, which indicates that the polarized radiation signals Q, U, and V are easier to distinguish compared with the radiation intensity. The good comparisons between the MMC results with the DOM results [8] verify the accuracy of our MMC code and the following time-dependent Stokes vector components obtained by the MMC are believed to be credible and accurate.

Fig. 4 Comparisons of Stokes vector components obtained by MMC at a long time against the DOM results.

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3.2 Time dependent Stokes vector components for the atmosphere-ocean system exposed to external illumination

In order to analyze time-resolved Stokes vector component distributions, we consider the atmosphere-ocean system described in Section 3.1 and assume continuous illumination incident on the atmosphere top at time t = 0. All the following Stokes vector components are plotted for the viewing azimuth angle of φ_V = 90°.

The zenith distributions of Stokes vector components for the location just below the interface (location A in Fig. 2) at different dimensionless time moments t* = 0.5, 1.0, 2.0, 4.0, 8.0, 12.0, 20.0, and 30.0 are plotted in Fig. 5. It is seen that for an early time such as t* = 0.5, the radiation intensity, i.e., the Stokes component I has only one peak near μ = −0.66. This is due to the fact that any radiation transmitted from the atmosphere layer has a direction cosine smaller than −0.66. For an early time, most of the radiation at location A accumulates from the bundles transmitted from the atmosphere layer and thus the radiation intensity for the direction μ > −0.66 remains quite weak, as shown in Fig. 5(a). As the simulation proceeds, more and more bundles reach the water layer and the multiple scattering leads to more bundles propagating in the directions of μ > −0. 66, and the Stokes component I for μ > −0.66 increases. Finally, another peak arises at the direction of μ = 0. It is also seen that the I distribution in the region μ = [-0.66, 0] is symmetric with that in the region μ = [0, 0.66]. The reason is that all the radiation bundles with a direction cosine of μ = [0, 0.66] incident at location A from the water will be totally reflected back with a direction cosine of μ = [-0.66, 0].

Fig. 5 Zenith distributions of Stokes vector components at different time moments for the location just below the atmosphere-ocean interface.

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The zenith distributions of Stokes component Q at different time moments are plotted in Fig. 5(b). Two valleys can be found at the direction of μ = −0.8 and μ = 0, respectively. For an early time such as t* = 0.5, 1.0, 2.0, and 4.0, only a very few radiation bundles propagate to location A, and thus the valley at the direction of μ = 0 is very low. With the simulation proceeds, more and more radiation bundles reach location A, when the time goes to t* = 8.0, the valley value accumulated from the many multiple-scattered bundles becomes bigger than zero and the Q values for all directions are positive. The distribution of the U component of any time plotted in Fig. 5(c) starts from a negative value and increases quickly until a step drop at μ = −0.66. The U distribution also has an obvious peak at μ = 0 for all time except for t* = 0.5, as shown in Fig. 5(c). Different from I, Q, and U, the Stokes component V plotted in Fig. 5(d) remains at a relatively restricted value for the region of μ = [-1.0, −0.8] and then decreases quickly to a sharp valley at μ = −0.66. For a longer time before the steady state, the V distribution reaches a higher peak value near μ = −0.5 but decreases faster after the peak, and thus reaches a lower value at μ = 1.0, as shown in Fig. 5(d).

The Stokes vector component contours at location A varying with time and direction are presented in Fig. 6. A similar analysis for the selected distribution lines in Fig. 5 can be applied to the results in Fig. 6 for Stokes vector varying with zenith angle at any time (or Stokes vector varying with time for any zenith angle). It is seen that the contour of I is found to clearly change before t* = 20 and then keeps nearly unchanged for the later time, and this indicates the I component reaches the steady state at the time of t* = 20. This steady-state time is found to be t* = 15 for the Q component, and t* = 10 for the U and V components, which means the time-resolved Stokes vector component Q (U or V) reaches a steady state at a time moment earlier than that for the Stokes vector component I. This phenomenon indicates that all the Stokes components (other than only one of the components) must be confirmed to be steady for the steady-state time estimate for the polarized radiative transfer problems with continuous illumination.

Fig. 6 Stokes vector contour varying with time and direction for the location just below the atmosphere-ocean interface.

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The zenith distributions of Stokes vector components for the location just above the interface (location B in Fig. 2) at different dimensionless time moments are plotted in Fig. 7. The refractive index of the atmosphere is smaller than that of the water, and total reflection won’t occur for any radiation propagating from the atmosphere to the interface, thus the step change appeared in the Stokes vector components at location A cannot found for the Stokes vector components at location B. All the zenith distributions of the Stokes vector component I plotted in Fig. 7(a) have one peak, but the direction cosine of the peak is bigger for a later time, which is different from the situation in Fig. 5(a) where the direction cosine of the peak remains the same. Figure 7(b) plots the zenith distributions of the Q component at location B. For an early time such as t* = 0.5, two peaks and a platform can be found. With time increasing, these two peaks move closer and finally, only one peak can be observed. The zenith distributions of U component at location B are plotted in Fig. 7(c) and they have similar changing trends with the I component plotted in Fig. 7(a). Figure 7(d) plots the zenith distributions of V component at location B. For the direction cosine region of μ = [-1.0, 0], the V distributions for all the chosen dimensionless time except t* = 0.5 share quite similar values and changing trends. In contrast, for the direction cosine region of μ = [0, 1.0], the V distributions before the steady state have different value stages, as shown in Fig. 7(d).

Fig. 7 Zenith distributions of Stokes vector components at different time moments for the location just above the atmosphere-ocean interface.

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The Stokes vector component contours at location B varying with time and direction are presented in Fig. 8. A striped area denoting the high value can be found in the contours of I, Q, and U components, as shown in Figs. 8(a)~8(c). It is also seen that the V value for μ = [-1.0, 0] reaches a plateau [denoted by the red area in Fig. 8(d)] faster than any other components, this means the V components responses fastest to the incident illumination compared with other components. The dimensionless time for Stokes vector component contours at location B to reach a steady is t* = 10 for the I/Q component and t* = 8 for U/V component, which is earlier than those for the Stokes vector component contours at location A. By comparing the steady-state time for locations A and B, we can get a conclusion that the reflection of the atmosphere-ocean interface delays the steady-state time for the Stokes vector components at location A in the ocean.

Fig. 8 Stokes vector contour varying with time and direction for the location just above the atmosphere-ocean interface.

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The zenith distribution of Stokes vector components at the top of the atmosphere (location C in Fig. 2) at different dimensionless time moments are plotted in Fig. 9. As the top surface is non-reflecting, all the radiation bundles propagating to the top surface will escape with a positive direction cosine, the Stokes vector component at location C remains zero for the direction cosine region of μ = [-1.0, 0]. For the I, Q and U components plotted in Figs. 9(a)~9(c), respectively, the start value at μ = 0 is bigger and the zenith distribution line is higher for a later time moment. All the V components start from the same value of zero. For a later time, it is seen form Fig. 9(d) that the zenith distribution line of V component decreases faster and reaches a lower value at μ = 1.0.

Fig. 9 Zenith distributions of Stokes vector components at different time moments for the location at the top of the atmosphere.

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The Stokes vector component contours at location C varying with time and direction are presented in Fig. 10. The I, Q and U components are found to be non-negative while the V component is found to be non-positive for the entire contour. It is seen from Figs. 10(a) and 10(b) that even after t* = 20 which is bigger than the steady time for Stokes vector component contours at locations A and B, a slight change can be found for the I and Q contours. No significant variation can be found after t* = 10 for the contours of U and V components, as shown in Figs. 10(c) and 10(d). Compared with those for location A or B, the steady-state time for location C is longer, which is due to the influence of the reflected radiation by the atmosphere-ocean interface.

Fig. 10 Stokes vector contour varying with time and direction for the location at the top of the atmosphere.

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4. Conclusions

A modified Monte Carlo algorithm combined with the time shift and superposition principle is extended for solving the time-dependent polarized radiative transfer problem in an atmosphere-ocean system exposed to external illumination. It is found that the MMC can significantly improve the resolution of traditional Monte Carlo by using the same simulation bundles. The time-resolved Stokes vector components are calculated and presented for the first time for an atmosphere-ocean system with a refractive interface. The simulation results confirm that the Stokes vector component distributions varying with zenith angle at early time can be significantly different from the steady-state one. The time to reach the steady-state is found to be various for different components and for the locations just below and above the atmosphere-water interface.

Funding

China Postdoctoral Science Foundation (2018M641196), National Natural Science Foundation of China (51890891), and Fundamental Research Funds for the Central Universities (FRF-TP-18-072A1, FRF-BD-18-015A).

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Time-dependent polarized radiative transfer in an atmosphere-ocean system exposed to external illumination

Abstract

1. Introduction

2. Theory of MMC for time-dependent polarized radiative transfer

3. Results and discussions

3.1 Validations of the MMC for polarized radiative transfer

3.2 Time dependent Stokes vector components for the atmosphere-ocean system exposed to external illumination

4. Conclusions

Funding

References

Cited By

Figures (10)

Equations (19)

Optics Express