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Polarized radiance fields under a dynamic ocean surface: a three-dimensional radiative transfer solution Yu You,1,* Peng-Wang Zhai,2 George W. Kattawar,1 and Ping Yang3 1

Department of Physics, Texas A&M University, College Station, Texas, 77840, USA

2

NASA Postdoctoral Program Fellow, NASA Langley Research Center, Hampton, Virginia, 23681, USA 3

Department of Atmospheric Sciences, Texas A&M University, College Station, Texas, 77843, USA *Corresponding author: [email protected] Received 25 February 2009; revised 23 April 2009; accepted 30 April 2009; posted 5 May 2009 (Doc. ID 107952); published 1 June 2009

The hybrid matrix operator, Monte Carlo (HMOMC) method previously reported [Appl. Opt. 47, 1063–1071 (2008)] is improved by neglecting higher-order terms in the coupling of the matrix operators and by introducing a dual grid scheme. The computational efficiency for solving the vector radiative transfer equation in a full 3D coupled atmosphere–surface–ocean system is substantially improved, and, thus, large-scale simulations of the radiance distribution become feasible. The improved method is applied to the computation of the polarized radiance field under realistic surface waves simulated by the power spectral density method. To the authors’ best knowledge, this is the first time that the polarized radiance field under a dynamic ocean surface and the underwater image of an object above such an ocean surface have been reported. © 2009 Optical Society of America OCIS codes: 010.5620, 290.4210, 290.7050, 010.1290, 010.4450.

1. Introduction

Radiative transfer theory [1,2] has found applications in a variety of scientific disciplines. In most realistic cases, numerical solutions are the only option for solving the radiative transfer equation (RTE), as analytical solutions lack flexibility on both source configurations and boundary conditions. Because of limited computational power, the RTE or, if polarization is involved, the vector RTE (VRTE) is traditionally solved in an idealized plane-parallel configuration [3–12], where the medium is assumed to have varying optical properties in one dimension and to be homogeneous in the other two dimensions. Thanks to increasing computational power, threedimensional (3D) solutions to the RTE and the VRTE

0003-6935/09/163019-11$15.00/0 © 2009 Optical Society of America

[13–24] become possible and give more precise predictions of the natural systems under study. Among numerous research topics involving radiative transfer, solutions to the RTE in a coupled atmosphere–ocean (CAO) system [9–12,24] are of great interest. It is quite challenging to consider a continuously varying ocean surface in a realistic CAO system because an appropriate treatment of the dynamic surface waves is essential. To show the importance of surface wave coupling in the RTE solution of a CAO system, it is necessary to consider a coupled atmosphere–surface–ocean (CASO) system. In the traditional plane-parallel treatment [9] and an earlier 3D treatment [24] of the surface waves, a statistical wave model developed by Cox and Munk [25] was used. The solutions based on this model correspond to temporal- and spatial-averaged radiation fields but are not able to capture the temporal and spatial dependence of the radiation fields when a dynamic surface wave is present. 1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS

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Recently, Hedley [26] reported a 3D radiative transfer model for solving the scalar RTE for a coupled surface–ocean system based on the geometric optical method. This model accounts for both temporal and spatial dependence, but it does not include a coupled atmosphere layer. Instead, the sky downwelling radiances are used to specify the boundary conditions, which do not include the interactions between the atmosphere and ocean layers. The solution from this scalar model does not provide any information about the polarization state of the radiance fields. To effectively solve the VRTE in a CASO system with a dynamic surface and relatively slowly varying atmosphere and ocean layers, Zhai et al. [27] reported a hybrid matrix operator, Monte Carlo (HMOMC) method involving two processes. The CASO system is partitioned into three layers, namely, the atmosphere layer, the surface boundary layer, and the ocean layer. The impulse response functions for the slow-varying atmosphere and ocean layers are computed from a 3D Monte Carlo model [28], and the impulse response function for a fastvarying surface layer is determined by the Fresnel formulas. Furthermore, the three impulse response relations are coupled by a 3D generalization of the 1D matrix operator method [5,6,29,30]. The impulse response functions for the atmosphere and ocean, computed from the Monte Carlo method in the first process, can be reused in the second process for all time instances with different wave profiles, mitigating the computational burden inherent to the direct Monte Carlo method. Therefore, the hybrid method is much more efficient in comparison with the direct 3D Monte Carlo method [21,24]. The main idea of the HMOMC method for a CASO system is based on the fact that the atmosphere and ocean layers vary much more slowly in the time domain than does the surface layer. This method can be further improved considering that the atmosphere and ocean layers also vary much more slowly in the space domain. In the HMOMC method reported in [27], all three layers are discretized with the same spatial resolution, and the coupling of layers is straightforward. However, the wavelengths of real ocean surface waves can be of the order of centimeters and millimeters, while the optical properties of the atmosphere and ocean are usually homogeneous within the extent of meters or tens of meters in the horizontal dimensions. Therefore, the spatial resolution required for the atmosphere and ocean layers is much lower than that for the surface layer, implying that the previous discretization scheme (hereafter referred to as the “single grid” scheme) requires computational effort to discretize the atmosphere and ocean layers into unnecessary grids. In this study, we introduce a dual grid scheme, which discretizes the atmosphere and ocean layers and the surface layer with different spatial resolutions. The atmosphere and ocean layers can be specified with lower spatial resolution, and the surface layer 3020

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can be specified with higher spatial resolution. We will show that this scheme requires much less computational time without loss of accuracy. The development of the HMOMC method and the current study are part of the Radiance in a Dynamic Ocean (RaDyO) program [31,32], which is devoted to the construction of a radiance-based surface boundary layer model and the validation of this model with ongoing field observations of relevant quantities such as downwelling sky light radiance above the ocean surface, surface wave slopes, and polarized radiance under the surface [31]. The current study is a continued effort toward the goal of this program. In this paper, we introduce the dual grid scheme in the HMOMC method to reduce the computational time. Furthermore, we compare the temporal- and spatial-averaged radiation field simulated from the HMOMC method with its Cox–Munk counterpart and show the temporal evolution of simulated radiation fields when a dynamic ocean surface is involved. 2.

Dual Grid Scheme

As discussed in detail in [27], the CASO system is partitioned into three layers: atmosphere, ocean surface, and ocean. Levels 0—3 are used to label consecutively the top to bottom of the atmosphere layer and the top to bottom of the ocean layer. A degenerate surface boundary layer lies between levels 1 and 2. A finite computational domain in the horizontal dimension is discretized into grids and infinitely extended by using a periodic boundary condition. The impulse response function is a function of the impulse position ri , impulse direction ni , response position rr , and response direction nr , which can be compactly written as a multidimensional matrix FðρðrÞ ; ρðiÞ Þ, where ρ ¼ ðr; nÞ denotes a combination of the position and direction vectors. Each subset of this matrix is a 4 × 4 Mueller matrix. For example, in the atmosphere layer, T01 ðρðrÞ ; ρðiÞ Þ represents the transmission response function at level 1 when an impulse is incident on level 0, and R10 ðρðrÞ ; ρðiÞ Þ represents the response function at level 1 in conjunction with the reflections between levels 1 and 0 when the impulse is from below level 1. Response functions in the surface and ocean layers are denoted in a similar way. When the underwater radiation field is of interest, the detector response function D2d ðρðrÞ ; ρðiÞ Þ at an optical depth d below the ocean surface is also required when an impulse is incident on level 2. All response functions in the atmosphere and ocean layers are calculated from the 3D Monte Carlo method, while those in the surface layer are determined by the Fresnel formulas. According to the matrix operator theory [5,6,29,30], the detector response function D0d ðρðrÞ ; ρðiÞ Þ at level d when the radiation is incident on level 0, the top of the atmosphere, can be written, for example, as follows (Eq. (1) in [27]):

D0d ¼ D2d · T12 · T01 þ D2d · T12 · R10 · R12 · T01 þ D2d

D0d ¼ D2d · T12 · T01;eff þ D2d · R21 · R23 · T12 · T01;eff ; ð5Þ

· R21 · R23 · T12 · T01 þ D2d · R21 · R23 · T12 · R10 · R12 · T01 þ D2d · T12 · R10 · R12 · R10 · R12 · T01 þ D2d · R21 · R23 · R21 · R23 · T12 · T01 þ D2d · T12 · R10 · T21 · R23 · T12 · T01 þ …;

ð1Þ

where the arguments ρðrÞ and ρðiÞ are omitted for simplicity. Numerical computation based on Eq. (1) will involve a huge number of multiplications of multidimensional matrices and is computationally costly, rendering large-scale simulations of a dynamic CASO system impractical. Fortunately, one does not need all of these terms to achieve satisfactory accuracy in the specific CASO system under study. In a real CASO system, the oceanic scattering phase function is always highly anisotropic with most radiance scattered into the forward directions and very little radiance into the backward directions. The most extensively used Petzold phase function [33] has a backscatter factor (the probability a photon will be scattered at a scattering angle larger than 90°) of B ¼ 0:0183, implying that the reflected response function in the ocean R23 is very small. Therefore, we can neglect all terms with R223 or higher orders. Equation (1) can be rearranged as D0d ¼ D2d · ðI þ R21 · R23 Þ · T12 · ðI þ R10 · R12 þ ðR10 · R12 Þ2 þ R10 · T21 · R23 · T12 Þ · T01 ;

ð2Þ

where I is an identity matrix with corresponding dimensionality. Furthermore, a response function in the surface layer, e.g., R12 , can be expanded as a ð0Þ first-order term R12 corresponding to a flat surface boundary case plus higher-order corrections induced ðwaveÞ by the surface waves R12 , ð0Þ

ðwaveÞ

R12 ¼ R12 þ R12

:

ð3Þ

Equation (3) applies to the response functions T12 and T21 as well. For such response functions in the second line of Eq. (2), they are higher-order corrections to the identity matrix I. Therefore, we can simply keep the flat surface terms and omit surface wave correction terms. The second line can then be approximated by an effective transmitted response function T01;eff , ð0Þ

ð0Þ

which includes the major part of the underwater radiance with much less computational effort in the matrix operator coupling. To achieve a good resolution, it is required that the size of each discrete grid δl be much smaller than the smallest wavelength λ constituting the ocean waves, e.g., δl ∼ λ=10. The HMOMC method reported in [27] uses a single grid discretization scheme by assuming that the numbers of grids are the same, N, in all three layers. Figure 1(a) illustrates the single grid scheme with all three layers discretized into 5 × 5 grids (N ¼ 25). In this scheme, a matrix operator coupling involves impulse response functions from all N impulse grids for each of the N response grids, meaning that the computational time is proportional to N 2 . Therefore, although this scheme gives good results with reasonable computational effort when one specifies a moderate N, it could be prohibitive for a large N. However, the wavelengths of capillary waves and gravity waves in a real ocean are of the order of centimeters or even millimeters. To resolve these waves, small grids must be specified, increasing the likelihood that the number of grids is very large. On the other hand, the large-scale optical properties of the atmosphere and ocean layers are homogeneous in the horizontal dimension. Therefore, these layers can be discretized into fewer and larger grids without losing details in the atmospheric or oceanic optical properties. The size of the atmospheric and oceanic grids can be determined by requiring that the corresponding optical properties not vary much across a grid. Typically, a grid size of the order of meters is sufficient for open ocean waters. It can be as small as desired, say, for studying highly dynamic coastal waters. Based on these specific features of a CASO system, we introduce the dual grid scheme by discretizing the atmosphere and ocean layers and the surface boundary layer in different spatial resolutions. A schematic figure of the dual grid scheme is shown in Fig. 1(b), where the atmosphere and ocean layers are discretized into 5 × 5 medium grids, while the surface layer is discretized into 20 × 20 surface grids. In this scheme, the small-scale structure in the surface waves can be

ð0Þ

T01;eff ¼ ðI þ R10 · R12 þ ðR10 · R12 Þ2 þ R10 · T21 ð0Þ

· R23 · T12 Þ · T01 ;

ð4Þ

which involves no surface wave corrections. Therefore, it can be conveniently computed from the 3D Monte Carlo method for a CAO system with a flat ocean surface. Equation (2) now becomes

Fig. 1. Schematic configurations of the discretizations in the horizontal dimension for the (a) single grid and (b) dual grid schemes. Starting at the top, the three planes consecutively represent the atmosphere, surface, and ocean layers. 1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS

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perfectly represented in terms of the fine surface grids, while no computational resources are wasted in discretizing the atmosphere and ocean layers with an unnecessarily fine resolution. To make the dual grid scheme more efficient, we first separate D2d in Eq. (5) into two terms: ð0Þ

D2d ¼ D2d þ Ddiff 2d ;

ð6Þ

ð0Þ

where D2d ∼ e−τ corresponds to the exponential extinction of the direct impulse function, with τ the optical thickness from the impulse position to the response position, while Ddiff 2d is the diffuse part due to multiple scattering in the ocean, given by the Monte Carlo method. Substitution of D2d into Eq. (5) with Eq. (6) gives ð0Þ

ð0Þ

D0d ¼ D2d · T12 · T01;eff þ Ddiff 2d · T12 · T01;eff þ D2d · R21 · R23 · T12 · T01;eff þ Ddiff 2d · R21 · R23 · T12 · T01;eff : ð7Þ In the single grid scheme, each matrix multiplication in Eq. (7) is a regular matrix multiplication. For ð0Þ example, the components of the second term D2d · T12 · T01 that correspond to the lth impulse location ðiÞ ðrÞ rl and the mth response location rm can be explicitly written as N X

ðrÞ

ðiÞ

ðrÞ

ðiÞ

ðrÞ

ðiÞ

Ddiff 2d ðrm ; ro Þ · T12 ðro ; rp Þ · T01 ðrp ; rl Þ;

ð8Þ

o;p¼1

where p and o are location indices at levels 1 and 2, respectively, and N ¼ 25 if the configuration shown in Fig. 1(a) is considered. For simplicity we omitted the angular arguments. In the dual grid scheme, the transmitted response function through the ocean surface T12 and the reflected response function at the ocean surface R21 have more matrix elements in the dimensions of the impulse and response locations. Therefore, extra effort is needed to correctly couple the layers discretized into different grid sets, i.e., to rewrite the matrix multiplications in Eq. (7). It is clear that each surface grid falls into a larger medium grid, while a medium grid corresponds to multiple surface grids. For the Fig. 1(b) configuration, there are 25 medium grids and 400 surface grids, and each medium grid corresponds to 16 surface grids. It is helpful to label a surface grid by a combination of two indices as r0½o;o1  , where o is the index of the corresponding medium grid and o1 labels the surface grids that correspond to the same medium grid. For example, in the Fig. 1(b) configuration, T12 can be 0ðrÞ 0ðiÞ explicitly written as T12 ðr½o;o1  ; r½p;p1  Þ, where indices o and p run from 1 to 25, and o1 and p1 run from 1 to 16. Equation (8) should now be rewritten as 3022

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N  X ðrÞ ðiÞ Ddiff 2d ðrm ; ro Þ o;p¼1 N0 X

·

 0ðrÞ 0ðiÞ ðrÞ ðiÞ ½T12 ðr½o;o1  ; r½p;p1  Þ · T01 ðrp ; rl Þ ;

ð9Þ

o1 ;p1 ¼1

where N ¼ 25 and N 0 ¼ 16 if the Fig. 1(b) configuration is considered. The same modification applies to the multiplications by T12 and R21 in the third term of Eq. (8). The physical implication of Eq. (9) is as follows. When computing the transmission of radiance through the ocean surface, for all surface grids corresponding to the same medium grid, the impulse funcðrÞ ðiÞ tion T01 ðrp ; rl Þ is assumed to be the same. This is reasonable, as the optical properties of the atmosphere and ocean are assumed not to vary much across a medium grid. This impulse function is first operated on by the transmitted response function 0ðrÞ 0ðiÞ T12 ðr½o;o0  ; r½p;p0  Þ, which may vary a lot within the medium grid and lead to significant difference in the angular distribution of the transmitted radiation field. 0ðrÞ 0ðiÞ The transmitted radiation field T12 ðr½o;o0  ; r½p;p0  Þ · ðrÞ

ðiÞ

T01 ðrp ; rl Þ is then averaged over a medium grid and operated by the same in-the-ocean transmitted ðrÞ ðiÞ response function D2d ðrm ; ro Þ, which, again, does not vary much across the medium grid. Moreover, now one can see the advantage of writing D0d as Eq. (7) instead of Eq. (5). Unlike Ddiff 2d , the direct part ð0Þ D2d is not obtained from Monte Carlo calculations. Therefore, one does not have to average the response functions over the medium grid for the first and third terms in Eq. (7). Consequently, the surface wave information in the direct part is completely preserved, while that in the diffuse part is partly smoothed out due to the average over the medium grid. By calculating the direct and diffuse parts separately, the surface waves are resolved to the best extent with a reasonable computational effort. Unlike increasing the number of medium grids N, which implies tremendously increased computational time in both the Monte Carlo calculation and the matrix operator coupling, increasing the number of surface grids N × N 0 with N fixed does not require any extra time in the Monte Carlo calculation and introduces very little extra time in the matrix operator coupling, as there are only N × N 0 multiplications of T12 and R21 rather than ðN× N 0 Þ2 . In this way, the more homogeneous medium part and the more inhomogeneous surface part are properly coupled, leading to a more precise radiation field than the single grid scheme gives with the same number of medium grids at the expense of little extra computational effort. To test the error introduced by neglecting the higher-order terms in Eq. (5), we calculated the radiance field in a CAO system using both models. In this

study, the dimensions of the computational domain are 10:5 m < x < 10:5 m, 10:5 m < y < 10:5 m, and 0 m < z < 20 m. The ocean has a physical depth of 10 m and an extinction coefficient of 1 m−1. The atmosphere has a physical depth of 10 m and an extinction coefficient of 0:025 m−1. The corresponding optical depths for the ocean and atmosphere are 10 and 0.25, respectively. The light scattering in the atmosphere is determined by the Rayleigh phase matrix. The light scattering in the ocean is determined by the Henyey–Greenstein phase function [34]: 1 − g2 ; ð1 − 2gμ þ g2 Þ3=2

Pðμ; gÞ ¼

ð10Þ

where μ ¼ cosðθÞ, θ is the scattering angle, and g is the asymmetry factor defined by g¼

1 4π

Z PðθÞ cosðθÞdΩ: 4π

ð11Þ

The asymmetry factor used in this study is g ¼ 0:95, which is consistent with average water phase function measurements [35]. All other phase matrix elements for the ocean are determined by letting the ~ ij ¼ elements of the reduced phase matrix (P Pij =P11 ) be equal to those of the reduced Rayleigh phase matrix, which is consistent with field measurements reported by Voss and Fry [36]. The singlescattering albedo is 0.5 for the atmosphere, the ocean, and the Lambertian ocean bottom. The refractive indices of the atmosphere and the ocean are 1.0 and 1.338, respectively. In the matrix operator computation, the downwelling responses have 12 polar angles θi ¼ 92:5°; 100°; 110°; 120°; 130°; 135°; 140°; 150°; 160°; 170°; 175°; 180°, and the azimuthal angle ϕi ranges in 15° increments from 0° to 360°. Figure 2 shows the comparison of the underwater downwelling diffuse radiance and reduced Stokes vector component Q=I computed from the single grid and the dual grid models. For comparison, we also include results computed from the direct Monte Carlo method [28]. The sun is at the zenith, i.e., θs ¼ 0°, the ocean surface is assumed to be flat, and the optical thickness between the surface and the detector is τdet ¼ 1. The dual grid calculation used 9 × 1 medium and wave grids, and the single grid calculation also used 9 × 1 grids. A good agreement between results from the single and dual grid schemes and those from the direct Monte Carlo method is observed. Figure 2(a) further shows that the diffuse radiance given by the dual grid model is exactly the same as that given by the single grid model at polar angles larger than 130°, and is slightly lower at polar angles smaller than 130°, because the terms involving R223 and higher orders are neglected in the dual grid scheme. Figure 2(b) implies that, considering the degree of polarization, neglecting the higher-order terms introduces no noticeable error and gives perfect results for the dual

Fig. 2. Distributions of underwater downwelling (a) radiance and (b) ratio of Stokes parameters Q=I for a CAO system with a flat ocean surface computed from both single grid and dual grid models. The radiance is normalized to unit incident solar irradiance. Results from a direct Monte Carlo computation are included for comparison. The optical thickness between the surface and the detector is τdet ¼ 1. See text for specifics of the scattering system.

grid model. In both plots, the jump around 135° indicates the sharp boundary of the Snell cone, the angular region that directly refracted sky light can reach. The radiance observed outside the Snell cone comes solely from multiple scattering in the ocean. Figure 3 shows the radiance and Q=I component in the same CASO system except that a 1D cosine wave is present, i.e., zðxÞ ¼ 0:33 cosð2π3x=Lx Þ, where Lx ¼ 21 m is the length of the computational domain in the x direction. The detector is located at x ¼ 2:33 m, y ¼ 0 m, and τdet ¼ 1 below the ocean surface. The shape of the cosine wave and the location of the detector in the x direction are illustrated in Fig. 3(a). Results from three calculations are shown: a single grid calculation with 81 × 1 grids (circles), used as a benchmark, a dual grid calculation with 9 × 1 medium grids and 81 × 1 wave grids (squares), and a single grid calculation with 9 × 1 grids (crosses). All three calculations give similar results, except that the 9 × 1 single grid results fail to capture the blurred boundary of the Snell cone at polar angles 120° and 130°, as the number of grids is insufficient to describe the cosine waves. The dual grid calculation, on the other hand, gives much closer approximations. The discrepancy between the dual grid and the 81 × 1 single grid results are due mainly to the average of the response functions over medium grids. This comparison shows that a significant improvement in accuracy is obtained by using the dual grid scheme over the single grid scheme with the same number of

Fig. 3. Same as Fig. 2 except that a 1D cosine surface wave is present. The detector is at x ¼ 2:33 m, y ¼ 0 m and the same level below the ocean surface. The scattering azimuthal angle is ϕ ¼ 0°. 1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS

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medium grids. For an Apple Mac Pro running on 3 GHz Dual-Core Intel Xeon processors, the computational time was 13 s for the dual grid calculation and 106 s for the 9 × 1 single grid calculation when a single processor was used. As a reference, it took 2.6 h for the 81 × 1 single grid calculation. The dual grid calculation is much faster than the single grid calculation with the same number of medium grids because the terms involving R223 and higher orders are neglected and the effective response function T 01;eff is used in the matrix operator coupling. At the same time, it gives an accuracy similar to that of the single grid scheme, making large-scale simulations of a dynamic ocean practical. 3. Power Spectral Density Wave Model

With the dual grid scheme, it becomes feasible to simulate the CASO system with a dynamic ocean surface much faster than with the previous single grid scheme. This study is limited to linear gravity waves and swells generated by a power spectral density (PSD) approach [37]. This approach expresses the temporal- and spatial-dependent wave heights Zðr; tÞ at grid points r ¼ ðx; yÞ as a superposition of plane waves Zðr; tÞ ¼

X ZðkÞeiðk·rþωðkÞtþΦk Þ ;

ð12Þ

k

where k ¼ ðkx ; ky Þ is a discrete 2D wave vector whose components kx and ky are multiples of 2π=Lx and 2π=Ly , respectively. Here Lx and Ly are the sizes of the surface patch along the x and y directions, respectively. The dispersion relation for gravity waves and swells in deep water is approximately given by ωðkÞ ¼

pffiffiffiffiffiffiffiffiffi G · k;

ð13Þ

where G is the acceleration of gravity. The phase factor Φk characterizes the random nature of surface waves and satisfies a Gaussian distribution with zero mean value and unit standard deviation. It is given by the inverse error function Φk ¼

pffiffiffi 2Erf −1 ðζÞ where ζ are random numbers uniformly distributed between −1 and 1. For gravity waves, the Fourier amplitude ZðkÞ of the wave with wavenumber k is given by Zg ðkÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sz;g ðkÞcos2 Θ;

ð14Þ

where the 1D semiempirical Pierson–Neumann wave height PSD is a function of the wavenumber k and the wind speed v, given by Sz;g ðkÞ ¼

0:00506 −2G=kv2 e ; πk4:5

ð15Þ

and cos2 Θ is a directional weighting function that converts the 1D PSD to two dimensions. Figure 4 shows simulated gravity wave patches driven by wind with speeds v ¼ 5 m=s and 10 m=s, with the height of the waves exaggerated. Wave heights at 255 × 255 points are generated within a simulation domain of an area 21 m × 21 m. For swells, the Fourier amplitude is Zs ðkÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sz;s ðkÞ;

ð16Þ

where the PSD satisfies a 2D Gaussian distribution, Sz;s ðkÞ ¼

A2 −½ðk−kp Þ·ðk−kp Þ=2σ2 k e 2πσ 2k

ð17Þ

with A the swell amplitude, kp the wave vector of the swell peak in Fourier space, and σ k the swell peak standard deviation sffiffiffiffiffi kp : σ k ¼ 0:8494 × FWHM × G

ð18Þ

Here the FWHM characterizes the width of the swell peak in the frequency domain. Using the HMOMC method, we simulated the spatial-averaged radiance distribution under the wave field shown in Fig. 4 in the CASO system for

Fig. 4. (Color online) Model generated gravity waves for wind speeds (a) v ¼ 5 m=s and (b) v ¼ 10 m=s at time t ¼ 0. The simulation domain shown here is 21 m × 21 m. The height distribution is calculated for 255 × 255 regularly distributed points. 3024

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Fig. 5. Normalized downward radiance distributions just below the ocean surface (τdet ¼ 0:001) when wind driven surface waves shown in Fig. 4 are present. The atmosphere and ocean are the same as that for Fig. 2, and the sun is at the zenith θs ¼ 0°. Symbols are averaged HMOMC results using the gravity wave fields; lines are Monte Carlo results using the Cox–Munk statistical wave model. Results corresponding to a flat ocean surface are also shown for comparison.

Fig. 2. The results are compared with their counterparts computed from the Monte Carlo method [28] using the Cox–Munk statistical wave model. The detectors are located just below the ocean surface τdet ¼ 0:001. In the HMOMC calculation, we used the same set of impulse and response angles as in the previous calculation and specified 15 × 15 medium grids and 255 × 255 surface grids. The HMOMC gives radiance distributions at all 225 detectors, each of which is at the center of a medium grid. These spatialdependence results are then averaged to give statistical results comparable with Cox–Munk results. Figure 5 shows the comparison of HMOMC results using the gravity wave fields (symbols) and Monte Carlo results using the Cox–Munk statistical model (lines). The sun is at the zenith, i.e., θs ¼ 0°. Surface waves corresponding to wind speeds v ¼ 5 m=s and 10 m=s are considered, and a flat surface case is included for comparison. The angular distributions of

the statistical radiance field computed from the two models have the same dependence on wind speed. There are some discrepancies outside of the Snell cone, as the radiance there is so small that an average over more spatial positions is needed to achieve a good statistical agreement. This comparison reveals that the gravity wave fields generated by the PSD method give the same radiance field as the Cox– Munk model as far as the statistical characteristics are concerned. The Cox–Munk model will be inadequate when the temporal and spatial dependences of the radiance field are of interest, while the PSD wave model, incorporated with the HMOMC method, can provide all possible temporal- and spatial-dependent information. 4.

Simulation Results

We used the dual grid HMOMC model to calculate time series of the radiation field under surface waves generated by the PSD method. To make the surface waves more realistic, we modulate the gravity waves by swells when the wind speed is v ¼ 10 m=s, as shown in Fig. 6(a) (Media 1). The wavelength of the swell peak is 6:7 m, the amplitude of the swells is 0:15 m, and the FWHM is 0.5. We generated a time series of surface waves within the time duration from t ¼ 0 s to t ¼ 8 s, with a frame rate of 5 frames per second. This time series of model simulation serves as an excellent representation of the time evolution of surface waves in a real ocean. Figure 6(b) illustrates the locations of 255 underwater detectors (circles) at the same horizontal level. The solid circles indicate the locations of nine detectors, for which the radiation fields will be shown. We simulated the angular distribution of the polarized downwelling radiance field as this horizontal level moves from just below the ocean surface τdet ¼ 0:001 to halfway within the ocean layer τdet ¼ 5, and we study how the surface wave effects vary accordingly. Figure 7 (Media 2) shows the time evolution of the angular distribution of downwelling radiance just below the ocean surface with τdet ¼ 0:001, when the

Fig. 6. (Color online) (a) Snapshot from a time series (Media 1) of gravity waves modulated by swells, modeled by the PSD approach; (b) locations of underwater detectors (circles) and nine of them for which the radiation fields will be shown (solid circles). 1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS

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Fig. 7. (Color online) Snapshot from a time series (Media 2) of the angular distributions of the normalized downwelling radiance in unitsofW sr−1 m−2 atninedetectorsjustbelowtheoceansurfacewith τdet ¼ 0:001, when the surface waves shown in Fig. 6(a) (Media 1) is present. The locations of the nine detectors in the horizontal dimensions are as illustrated in Fig. 6(b).

PSD generated wave fields shown in Fig. 4 are present. The atmosphere and ocean are the same as those for Fig. 2. Here the downwelling hemisphere is represented by a round disk, where the center of the disk corresponds to the downwelling radiance with a polar angle of 180°, and the outer edge corresponds to a polar angle of 90°. The azimuthal angle is mapped such that the rightmost point, the top, the leftmost point, and the bottom of the disk correspond to ϕ ¼ 0°; 90°; 180°; 270°, respectively. The results from the array of nine detectors are arranged such that the pattern shown in the upper left-hand corner

of Fig. 7 is measured by the leftmost detector shown in Fig. 6(b). A single snapshot at t ¼ 0 s shows that the boundary of the Snell cone measured by each detector is blurred by the surface waves with a unique shape determined by the local wave slope. From the animation showing the time evolution of the radiation fields measured by the array of detectors, one can even perceive the movement of the surface waves from right to left based on the correlation between the movements of the radiation fields at different detectors. In addition to the radiance, in Fig. 8 (Media 3 and 4) we show the angular distributions of downwelling Stokes vector components Q=I and U=I observed by the same array of detectors in the same CASO system. Figure 8(a) (Media 3) shows that the Q=I distribution has strong dependence on the shape of the local wave slope. As seen from the snapshot, the Q=I patterns at different locations are almost the same within the Snell cone but vary a lot outside of the Snell cone. The corresponding time evolution shows more variation both inside and outside of the Snell cone. Specifically, the boundary of the Snell cone moves around in the same fashion as seen in the movement of the radiation field, and the Q=I patterns outside of the Snell cone circle around. Figure 8(b) (Media 4) shows that the U=I component is even more sensitive to the local wave slope across the full 2π downwelling hemisphere. However, there is no obvious boundary of the Snell cone. The movement of the U=I pattern looks more related to the local wave slope field. We next move the array of detectors to halfway within the ocean layer at τdet ¼ 5 in the same CASO system. Figure 9 (Media 5) shows the simulated time evolution of the downwelling radiation field. A snapshot at t ¼ 0 s suggests that the surface wave effects become very small as far as the radiance is concerned, because the multiple scattering plays an

Fig. 8. (Color online) Snapshots from time series of the angular distributions of downwelling Stokes vector components (a) Q=I (Media 3) and (b) U=I (Media 4) measured by the same array of detectors as for Fig. 7. 3026

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Fig. 9. (Color online) (Media 5) Same as Fig. 7 except that the detectors are halfway within the ocean layer at τdet ¼ 5.

essential role at this level and removes most of the surface wave effects. This model simulation agrees with previous results [27] where a one-dimensional cosine surface wave was considered. A visual examination of the time evolution reveals that one can tell the movement of the surface wave patterns from the movement of the radiance distributions. This demonstrates the importance of the capability to model the temporal- and spatial-dependence of the wave fields. Figure 10 (Media 6 and 7) shows the angular distributions of downwelling Stokes vector components Q=I and U=I measured by the same array of detectors as for Fig. 9. At this level, a snapshot of the distribution of the Q=I component also loses most of its dependence on the local wave slope, but the variation in the time series still shows sufficient information

about the local wave slopes. On the other hand, a snapshot of the U=I distribution is strongly spatial-dependent, and the time series clearly indicates how the surface wave moves. This is because the U component involves the phase difference between the parallel and perpendicular components of the electric field, which is more sensitive to the surface waves. Finally, we study the surface wave effects on an underwater image of an object above the ocean surface. We place a round disk centered at x ¼ 0 m and y ¼ 0 m just above the ocean surface and focus on the distribution of the downwelling radiance measured by the detector right at the center of the computational domain, which is placed at the same level as it was for Figs. 9 and 10, τdet ¼ 5. The corresponding physical depth is zdet ¼ 5 m. Shown in Fig. 11(a) is the implementation of the disk, where the response function corresponding to a medium grid is turned off when the distance from the center of this grid to the center of the computational domain is less than the radius of the disk, r ¼ 3:5 m. These grids are represented by closed circles, while the grids not covered by the disk are represented by open circles. Here the inner circle indicates the edge of the disk, and the outer circle indicates the foot print of the Snell cone at the ocean surface when the detector is at zdet ¼ 5 m (the radius of the footprint is then 5 × tanðarcsin 1=1:338Þ ¼ 5:62). Figure 11(b) shows the simulated distribution of downwelling radiance observed by this detector for a flat surface case when the atmosphere and ocean are the same as that for Fig. 7. At this level, multiple scattering becomes a major part of the downwelling radiance, substantially blurring the boundary of the Snell cone. Under the disk there is a dark region as the direct sunlight and single scattered light are blocked by the disk. The boundary of the dark region is more octagonal than circular, since the medium grids covered by

Fig. 10. (Color online) (Media 6 and 7) Same as Fig. 8 except that the detectors are halfway within the ocean layer at τdet ¼ 5. 1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS

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Fig. 11. (Color online) (a) Illustration of the implementation of a disk with radius r ¼ 3:5 m placed just above the ocean surface in the CASO system. (b) The simulated angular distribution of the normalized downwelling radiance in units of W sr−1 m−2 observed by a detector at the center of the computational domain at τdet ¼ 2 below the surface. The mapping from the downwelling hemisphere to a round disk is the same as that for Fig. 7.

the round disk actually form an octagon instead of a circle. Figure 12 (Media 8) shows the time evolution of the downwelling radiance observed by the same detector in the same CASO system as for Fig. 11, except that the wave fields shown in Fig. 6 (Media 1) are present. The snapshot itself does not show much sensitivity on the surface waves, other than a little asymmetry that one can observe at the boundary of the disk image. However, the time evolution of the radiation distribution reveals that this image moves around appreciably in accordance with the movement of the local wave slopes, even if the detector is halfway within the ocean layer where the radiance distribution without the disk shows little sensitivity to the surface wave slopes. 5. Discussion and Conclusions

The HMOMC method reported in a previous paper for solving the vector radiative transfer equation

(VRTE) in a three-dimensional coupled atmosphere– surface–ocean (CASO) system has been improved by neglecting higher-order terms in the coupling of the matrix operators and introducing the dual grid scheme. The computational time is substantially reduced while the simulated results are the same as those from the previous algorithm. With this improvement, it is feasible to simulate temporal- and spatial-dependent distributions of polarized radiation fields in a dynamic CASO system. The dual grid HMOMC method is used to simulate temporal evolution of the downwelling polarized radiation field observed by detectors in a CASO system with surface wave slope fields generated by the power spectral density (PSD) method. Our model simulation shows that inclusion of polarization gives us higher sensitivity to the surface wave slopes deep under the ocean surface. We also simulated the underwater image of an object located just above the ocean surface, which shows great sensitivity to the surface waves if a time series of the radiance distribution is available. With the effects of a dynamic surface boundary layer well understood, potential applications of this study include detection and imaging of objects across dynamic ocean surfaces from the spatio-temporal variations of radiance and polarization. In the current dual grid scheme, we used an effective transmitted response function T01;eff , where some higher-order corrections due to surface waves were neglected. This effective function can be directly measured with all surface wave corrections included [31]. The model simulations will be more accurate once a measured T01;eff is incorporated. This research was partially supported by the Office of Naval Research under contract N00014-06-10069. Ping Yang acknowledges support from the National Science Foundation (ATM-0239605) and the NASA Radiation Sciences Program (NNX08AF68G). Peng-Wang Zhai is currently a NASA Postdoctoral Program fellow at the NASA Langley Research Center administered by Oak Ridge Associated Universities through a contract with NASA. References

Fig. 12. (Color online) Same as Fig. 11(b), but shows a snapshot from a time series (Media 8) of downwelling radiance distributions when the wave field shown in Fig. 6 (Media 1) is present. 3028

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