Monte Carlo radiative transfer simulation for the nearocean-surface high-resolution downwelling irradiance
statistics
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Citation
Xu, Zao, and Dick K. P. Yue. “Monte Carlo Radiative Transfer
Simulation for the Near-Ocean-Surface High-Resolution
Downwelling Irradiance Statistics.” Opt. Eng 53, no. 5 (February
5, 2014): 051408. © 2014 SPIE.
As Published
http://dx.doi.org/10.1117/1.OE.53.5.051408
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Society of Photo-Optical Instrumentation Engineers (SPIE)
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Final published version
Accessed
Thu May 26 09:15:47 EDT 2016
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http://hdl.handle.net/1721.1/88497
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Detailed Terms
Monte Carlo radiative transfer
simulation for the near-ocean-surface
high-resolution downwelling
irradiance statistics
Zao Xu
Dick K. P. Yue
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Optical Engineering 53(5), 051408 (May 2014)
Monte Carlo radiative transfer simulation for
the near-ocean-surface high-resolution downwelling
irradiance statistics
Zao Xu* and Dick K. P. Yue
Massachusetts Institute of Technology, Department of Mechanical Engineering, Cambridge, Massachusetts 02139
Abstract. We present a numerical study of the near-surface underwater solar light statistics using the state-ofthe-art Monte Carlo radiative transfer (RT) simulations in the coupled atmosphere-ocean system. Advanced
variance-reduction techniques and full program parallelization are utilized so that the model is able to simulate
the light field fluctuations with high spatial [Oð10−3 mmÞ] and temporal [Oð10−3 sÞ] resolutions. In particular, we
utilize the high-order correction technique for the beam-surface intersection points in the model to account for
the shadowing effect of steep ocean surfaces, and therefore, the model is able to well predict the refraction and
reflection of light for large solar zenith incidences. The Monte Carlo RT model is carefully validated by data-tomodel comparisons using the Radiance in a Dynamic Ocean (RaDyO) experimental data. Based on the
model, we are particularly interested in the probability density function (PDF) and coefficient of variation
(CV) of the highly fluctuating downwelling irradiance. The effects of physical factors, such as the water turbidity
of the ocean, solar incidence, and the detector size, are investigated. The results show that increased turbidity
and detector size reduce the variability of the downwelling irradiance; the shadowing effect for large solar zenith
incidence strongly enhances the variability of the irradiance at shallow depths. © 2014 Society of Photo-Optical
Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.53.5.051408]
Keywords: ocean optics; radiative transfer; irradiance statistics; rough surface scattering.
Paper 131433SS received Sep. 17, 2013; revised manuscript received Dec. 20, 2013; accepted for publication Jan. 7, 2014; published online Feb. 5, 2014.
1 Introduction
The near-ocean-surface underwater irradiance fluctuations
are mainly induced by the refraction of solar incident
light at the random ocean surface and further diffused by
the multiple scattering of ocean particles. Understanding
the physics behind the temporal and spatial statistical characteristics of underwater irradiance, including probability
distribution and power spectrum density, have drawn potential interests in many areas, such as near-surface radiometric
sensing,1–3 phytoplankton photosynthetic physiology,4–6 and
underwater imaging.7 The difficulty of understanding the
statistical characteristics results from the complexity of
both dynamic ocean surface waves and light radiation transfer in the turbid ocean. Since the 1970s, extensive experimental studies have been conducted,8–14 mainly focusing
on the effects of ambient conditions, such as inherent optical
properties (IOPs), depths, and sky radiance diffuseness, on
the statistics of the underwater downwelling irradiance.
Recently, under the Radiance in a Dynamic Ocean (RaDyO)
project, researchers presented a preliminary experimental
effort on the various forms of probability distributions of
the downwelling irradiance.15,16 On the other hand, theoretical investigations of the probability distribution of the
downwelling irradiance have been made through analytical
derivations.17 However, theoretical study does not consider
certain key factors affecting probability distribution of the
irradiance such as the shadowing effect at the steep surface
for a large solar zenith incidence. Numerical simulations
of radiative transfer equation (RTE) have been the most
important approaches to understand the underwater light
fields.18 Among them, the Monte Carlo method has been
known as the most popular approach to simulate light RT
in the ocean for decades,19–21 but most of the previous
work does not focus on the light field fluctuations induced
by the deterministic surface wave fields. Only recently,
numerical investigations about the near-surface light field
fluctuations have been made utilizing Monte Carlo RT simulations.22–25 However, those studies have not provided
a systematic investigation of the probability distribution of
the light fields as the function of various factors, e.g., the
shadowing effect and the detector size.
In this paper, we present the work of developing the stateof-the-art Monte Carlo RT models [two-dimensional (2-D)
and three-dimensional (3-D)] to simulate the high-resolution
fluctuations of the near-surface downwelling irradiance. The
spatial resolution of the detector can reach to Oð10−1 mmÞ
and temporal resolution up to Oð10−1 sÞ. In particular, the
shadowing effect of the surface is also considered in the
numerical model for the case of large solar zenith angle
incidence. The paper is presented in the following manner:
Sec. 2 describes briefly the key steps of the Monte Carlo RT
model including light-surface interactions; Sec. 3 presents
a careful validation of the numerical model using comparisons with the field data; Sec. 4 demonstrates the results of
key factors influencing the statistical characteristics of irradiance using the numerical simulations, including turbidity
of the ocean, detector size, and shadowing effect of the
surface.
*Address all correspondence to: Zao Xu, E-mial: xuzao@mit.edu
0091-3286/2014/$25.00 © 2014 SPIE
Optical Engineering
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2 Method
directions before and after scattering. For each scattering
event after the photon passing through the air-sea interface,
the modified phase function can be written as
2.1 Monte Carlo RT Simulation
The propagation of solar radiation at the upper-ocean column
is well described by the RTE in the water/air column and raytracing at the air-sea interface (where the wavelength of light
is much shorter than the characteristic length of surface
waves). One of the most efficient ways to simulate these
two processes is the Monte Carlo method. Since the RTE
is an integro-differential equation, it can be solved by first
solving every order of Neumann series of an integral equation and then summing up solutions of all orders which are
solved with Monte Carlo integration. More importantly, each
order of the Neumann series can be physically interpreted as
the scattering and absorption events of a photon, and the
summation of solutions of all orders can be performed by
tracing the path of photons during scattering and absorption
events.
Therefore, the basic steps of Monte Carlo RT simulations
can be simply concluded in the following manner. First,
launch a photon with unit energy from the source in the incidence direction and determine the traveling path length l
using the formula l ¼ −ðln r1 Þ∕c, where r1 is the random
number between 0 and 1 and c is the beam attenuation coefficient. After traveling a distance l, we generate another random number r2 ; if r2 is greater than the single scattering
albedo ω0 which in general indicates the turbidity of ocean,
we assume that the photon is deceased and then start launching a new photon; otherwise, we assume the photon is
changing the direction due to the scattering event. The
new direction is determined by the scattering phase function.
After each scattering event, we repeat the process of photon
traveling until the photon is deceased. If the photon hits the
air-sea interface, it splits into two parts: reflection and refraction, whose energies can be obtained by multiplying the
Fresnel reflection and refraction coefficients, respectively.
The advantage of Monte Carlo method to simulate the RT
process is that it has a very clear physical interpretation
and it is relatively simple to program.
In order to obtain high-resolution spatial/temporal distributions of the underwater radiation fields, several approaches
are utilized to increase the efficiency of the Monte Carlo RT
simulations including the variance-reduction techniques and
the parallelization of the program.
The variance-reduction techniques we use in the study are
the photon path length sampling and the detector directional
importance sampling.26 For photon path length sampling, it
is assumed that the bottom of the deep ocean is a highly
absorbing boundary. Therefore, it is computationally wasted
if a lot of photons collide with the bottom. Using the biased
density function, we can determine the photon path l as
l ¼ lb −
ln½eclb þ r1 ð1 − eclb Þ
;
c
(2)
where ϵ ∈ ½0; 1 is a free parameter to be optimized, μs0 is the
cosine of the angle between the direction to the detector and
the photon direction after scattering. The modification of the
phase function results in a weight factor for the photon
wi ¼ pðμs Þ∕p 0 ðμs Þ:
(3)
With the help of biased sampling of the phase function, a
large amount of the scattered photon tends to travel toward
the detector so that we are able to achieve the faster convergence during the Monte Carlo RT simulation.
Another way to improve the convergence is to use parallel
computing. The parallelization of the Monte Carlo program
is straightforward. We utilize message passing interface to
realize the parallelization of the program. Each individual
photon is launched and traced at the different processors
and the results from every processor are collected at the
end of the program. Using the paralleled program, we are
able to launch up to Oð1011 Þ photons and achieve convergence for extremely high spatial resolutions (detector size
R ∼ 10−3 mm) for the large spatial simulation domain
[Oð102 Þ m].
Among many quantities that we simulate to obtain to
describe the solar underwater light fields, the downwelling
irradiance I d is the most commonly used and is collected
by a plane detector. It is defined by
Z 2π Z π∕2
Lðx; θ; ϕÞj cos θj sin θdθϕ;
(4)
I d ðxÞ ¼
0
0
where the radiance Lðx; θ; ϕÞ is the function of both position
x and direction ðθ; ϕÞ; here, θ and ϕ are the polar angle and
azimuthal angle, respectively.
However, for both field measurements and Monte Carlo
simulations, due to finite size of the detector and the spatial
fluctuation of I d induced by wavy surface, the downwelling
irradiance is the spatially averaged quantity which strongly
depends on the size of the detector R. Therefore, we define
the spatially averaged downwelling irradiance Ed ðxÞ for the
later studies. For the 2-D case, the horizonal coordinate is x
and Ed ðxÞ is defined as
Z
1 xþR∕2
I ðx 0 Þdx 0 :
(5)
Ed ðxÞ ¼
R x−R∕2 d
2.2 Ocean Surface Reconstruction
(1)
where lb is the distance from the scattering event to the bottom along the direction of travel.
For the detector directional importance sampling, it is
mainly used for the simulation of temporal light field fluctuations where only a small number of detectors are placed.
Assume the original scattering phase function is pðμs Þ,
where μs ¼ cosðθs Þ and θs is the angle between photon
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p 0 ðμs Þ ¼ ð1 − ϵÞpðμs Þ þ ϵp½μs0 ;
Ocean surface slope determines the photon refraction and
reflection features when it passes through the air-sea interfaces. However, for realistic free ocean surface with steep
surface waves, the slope information alone is not accurate
enough to predict the focusing of light by the surface.
Therefore, we choose to utilize the deterministic surface
wave elevations in the numerical model. Under the linear
wave assumption, the wave elevations are reconstructed
through linear superposition of many wave modes based
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on the empirical wave spectrum, e.g., the ECKV (Elfouhaily,
Chapron, Vandemark, and Katsaros) spectrum.27 Assume the
wavenumber spectrum of the 3-D waves are Sðkx ; ky Þ, where
ðkx ; ky Þ are the 2-D wavenumbers. To numerically reconstruct the ocean surfaces, we need to discretize the 2-D
wavenumber space so that kxn ¼ nΔkx and kym ¼ mΔky
ð−∞ < n < ∞; −∞ < m < ∞Þ. Then the amplitude of the
wave component with a wavenumber of ðkxn ; kym Þ can be
obtained as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(6)
Aðkxn ; kym Þ ¼ 2Sðkxn ; kym ÞΔkx Δky :
The reconstructed surface elevation based on the linear
wave approximation is expressed as
∞
∞
X
X
ηðx; yÞ ¼
Aðkxn ; kym Þ cosðkx x þ ky y þ ψÞ;
(7)
n¼−∞ m¼−∞
where ψ is the phase of ocean surface waves and is assumed
to be uniformly distributed between 0 and 2π. Under
such linear reconstruction, the surface wave slopes have
a Gaussian distribution based on the central limit theory.
A more realistic wave can be obtained using the highorder spectrum method to consider the nonlinear effect of
free surface ocean waves.28
Figure 1 shows an example of spatial patterns of Ed under
rough ocean surfaces for a normal solar incidence (where
the solar zenith angle is small). In this figure, the beam
−2
Ed(W m
6
(a)
0
Z (m)
−1
nm )
4
5
2
10
−10
0
−5
0
X (m)
5
10
6
Z (m)
(b)
4
0.5
2
1
1
Y (m)
0
1.2
1.4
X (m)
1.6
1.8
2
(c)
0.1
0.8
0.2
0.6
0.3
0.4
0.4
0.2
0.5
0
0.1
0.2
X (m)
0.3
0.4
2.3 High-Order Correction to Photon-Surface
Intersection
For very large solar zenith angles (solar light coming from
horizon), the shadowing of the steep surface waves must be
considered. To take into account the shadowing effect of
ocean surface on the refracted light, we utilize the faceted
surface to represent the real ocean surfaces. We assume
the minimum length of the ocean surface waves is down
to O(1) mm; the size of the facet has been determined
based on the smoothed surface. The smoothed surface is
obtained by applying a low-pass filter for the ocean spectrum. Under the faceted surface treatment, both shadowing
and multiple refraction effects of the steep rough surface can
be considered when photons pass through.
However, approximating the smooth surface as the discrete facets results in errors of locating the exact positions
of intersection of the photon beams and the ocean surfaces.
It also affects the computational efficiency at the same time,
especially for a large solar zenith incidence. As Fig. 2 shows,
for a 2-D ocean surface, a photon beam which is defined by
the equation z ¼ αx þ z0 hits the n’th facet of ocean surface
ηðxÞ with the intersection point located at x ¼ xi . However,
the position of the true intersection point of the photon beam
and the ocean surface is xt . The error is Δx ¼ xt − xi . One
solution is to make the facet as small as possible, but this will
greatly increase the computational effort and damage the
efficiency of program. To achieve both efficiency and accuracy for Monte Carlo RT simulations at the air-sea interface,
we apply the technique of high-order correction to intersection points to find the Δx so that for even sparsely discretized
ocean surface we are able to achieve relatively high accuracy
for the beam-surface intersection points. To obtain Δx, we
expand the ocean surface elevation ηðxt Þ into a Taylor series
at xi for the n’th facet intersection
0.5
Fig. 1 (a) Vertical spatial distribution of E d under rough ocean
surface waves obtained by the Monte Carlo RT simulation.
(b) Detailed patterns of near-surface E d . (c) A horizontal spatial
distribution of the E d under a wavy surface obtained by the threedimensional Monte Carlo RT simulation.
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attenuation coefficient is chosen to be as small as 0.02 m−1
so that strong focusing effect by the surface waves can be
clearly demonstrated. Figure 1(a) presents a typical vertical
patterns of Ed at the upper ocean where a large amount of
high-frequency capillary waves exists, which are obtained
from the 2-D Monte Carlo RT simulation. Figure 1(b) illustrates a detailed view of the pattern near the surface. It is
shown that due to the focusing of radiation by the extremely
small waves, the patterns of Ed are quite chaotic and must be
quantified using statistical approaches. Figure 1(c) presents
a horizontal view of Ed obtained by the 3-D Monte Carlo
RT simulation under a wavy surface with relatively small
amount of capillary waves. The patterns of Ed due to the
focusing of light by relatively large irregular waves can
be clearly observed.
Fig. 2 Higher-order correction of the intersection point of the light
beam and a two-dimensional ocean surface.
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CV ¼
(8)
where η 0 ðxÞ, η 0 0 ðxÞ, and η 0 0 0 ðxÞ are the first, second and third
derivatives of ηðxÞ, respectively. Recall that equation of the
photon beam is
ηðxt Þ ¼ αΔx þ ηðxi Þ:
(9)
If we keep up to the second-order term of Δx and ignore
the higher orders, we can solve Δx as
Δx ¼ 2
α − η 0 ðxi Þ
:
η 0 0 ðxi Þ
(10)
Readers can keep higher orders to achieve higher accuracy. Since we already know the information of the surface,
it is easy to find all orders of derivative of ηðxÞ. Therefore, by
choosing sparsely discretized ocean surfaces and applying
the high-order corrections to beam-surface intersection
points, we are able to enhance both efficiency and accuracy
of simulations at the same time.
Figure 3 shows an example of the vertical patterns of
Ed (in logarithm) for the case of large zenith incidence
(θsun ¼ 85 deg) both when the shadowing effect is ignored
[Fig. 3(a)] and the shadowing effect is considered [Fig. 3(b)].
It is clear that the shadow effects have great influence on the
pattern and statistics of the irradiance fields, especially at
the near-surface depths.
3 Model Validation
In order to quantitatively investigate the near-surface statistics of Ed , it is necessary to carefully validate the model
using model-to-data comparisons. For the specific problem
of fluctuating Ed , we compare the simulated probability
density function (PDF) of the normalized downwelling irradiance Ed ∕hEd i and coefficient of variation (CV) of Ed
obtained from the RaDyO Santa Barbara channel experiments.15,22 Here, h·i means statistical average. The CV of
Ed is defined as
z (m)
0
1
−4
10
−6
10
Field data
Fitting
0
10
z (m)
λ = 532 nm
6
Measurements
8
−8
10
λ = 670 nm
4
2
4
10
Wavenumber (rad m−1)
10
10
0
MC simulations
0.2
0.4
0.6
CV of E
0.8
d
(c)
Measurements
Probability density
MC simulations
−2
5
10
15
20
(b) 0
1
−1
10
Z = 0.87 m
−2
10
Z = 2.83 m
−3
10
0
1
Z =1.70 m
−1
2
−4
10
0
1
5
10
15
2
3
4
5
6
Ed /⟨Ed ⟩
−2
20
x (m)
Fig. 3 Vertical spatial distributions of E d (in logarithm) under a rough
free ocean surface obtained by the Monte Carlo RT simulations for a
large solar zenith incidence (θsun ¼ 85 deg) (a) without considering
the shadowing effect and (b) with considering the shadowing effect.
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(b) λ = 670 nm
2
0
x (m)
3
0
0
(a)
10
−1
2
3
0
(11)
−2
10
1
1
:
The ocean surface waves used for comparisons are based
on the omnidirectional wave number slope spectra obtained
from the RaDyO field measurements in the same day of Ed
measurements,29 when the wind speed increased only
slightly from 4.6 to 6.5 m∕s and the wave number ranges
from 7 to 1500 rad m−1 . To well represent the surface
waves, the horizontal grid for surface is small (0.76 mm).
The corresponding mean square slope of the surface is
around 0.03 rad2 , which agrees with the experiments.29
Figure 4(a) shows the slope spectra used in the comparison,
where the circle represents the experimental data. Based on
the experimental data, we fit the spectra and reconstruct the
ocean surface wave elevations. To erase the effects of the
periodical conditions imposed on the Monte Carlo model,
we choose the 2-D ocean wave with the horizontal domain
as wide as 50 m, and the interested water depth up to 10 m.
The detector size R used for the comparison is consistent
with the experiment (R ¼ 2 mm). Three light wavelengths
are chosen to represent three different IOPs. They are
λ ¼ 443, 532, and 670 nm. The beam absorption coefficients
a for the three wavelengths are 0.126, 0.089, and
0.4787 m−1 , respectively; the beam scattering coefficients
b for the three wavelengths are 0.67, 0.610, and
0.4478 m−1 , respectively. The scattering phase function is
chosen as the Petzold phase function. The solar incidence
angle is chosen to be θsun ¼ 30 deg according to the condition of measurements. Considering the clear sky condition,
we assume the diffuseness of the sky radiance is 0 and the
10
(a) 0
1∕2
Depth (m)
Δx3 0 0 0
η ðxi Þ þ OðΔx4 Þ;
6
hE2d i
−1
hEd i2
k
þ
Δx2 0 0
η ðxi Þ
2
S (m)
ηðxt Þ ¼ ηðxi Þ þ Δxη 0 ðxi Þ þ
Fig. 4 Comparisons of statistical quantities obtained by measurements and Monte Carlo simulations. (a) Measured ocean surface
slope spectral density utilized in the simulation. (b) Coefficient of
variation (CV) as a function of depth for three light wavelengths.
(c) Probability density function (PDF) at three depths for light wavelength λ ¼ 532 nm.
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Xu and Yue: Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling. . .
4.1 Effects of Ocean Turbidity and Detector Size
As a demonstration of the Monte Carlo RT model for the
study of the statistical properties, we systematically investigate two important factors that potentially affect the statistics
of near-surface downwelling irradiance Ed : detector size R
and turbidity or the single scattering albedo ω0 . In this
study, ω0 ranges from 0 to 0.9 and R is chosen to be 10,
1, 0.1, and 0.01 mm. As described, we use ECKV spectra
as the free ocean surface spectra. The minimum and maximum wavenumbers are chosen as 0.628 and 6283 rad m−1 ,
respectively. In the simulation, we assume that the solar
zenith angle is zero (normal incidence on the ocean) and
the sky radiance is approximated as the black sky in order
to maximize the focusing effects of the surfaces; the horizontal domain is 50 m and the vertical domain is 10 m which
corresponds to five optical depths (c ¼ 0.5 m−1 ). The phase
function used is Henyey–Greenstein phase function with
asymmetry factor g ¼ 0.924. The probability distribution
of the normalized downwelling irradiance Ed ∕hEd i is calculated based on its spatial distribution for different depths
under various detector sizes. Similar to previous case, for
each condition, all results are ensemble averaged using 10
surface wave realizations.
Figures 5(a) and 5(b) demonstrate the PDFs of Ed ∕hEd i
for R and ω0 , respectively. We can see that the shape of
PDF is strongly dependent on both R and ω0 . For the
case of small detector size and clean water, the probability
distribution is highly asymmetric and has a long tail for
high values of Ed ∕hEd i, e.g., when the single scattering
albedo ω0 ¼ 0.1, the highest value of Ed ∕hEd i can reach
to 10, with the corresponding probability density down to
Oð10−1 Þ. With increasing turbidity and decreasing detector
size, the probability of Ed ∕hEd i for large values (tail of PDF)
diminishes. For extremely large R and ω0 , the PDF of
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R =10 mm
R =1 mm
R = 0.1 mm
R = 0.01 mm
Probability density (log)
0
−1
−2
−3
−4
−5
(a)
−6
0
1
2
3
4
5
6
7
Probability density (log)
1
0
ω 0 = 0.9
−1
ω 0 = 0.5
ω 0 =0.1
−2
−3
−4
−5
−6
0
(b)
2
4
6
8
10
Ed /⟨Ed ⟩
Fig. 5 The PDFs of E d ∕hE d i for (a) R ¼ 10, 1, 0.1, and 0.01, mm and
(b) with ω0 ¼ 0.1, 0.5, and 0.9.
Ed ∕hEd i becomes more symmetric and close to Gaussian
with a unit mean value.
The effects of the R and ω0 on the PDF of the Ed ∕hEd i
can also be better shown by looking at the CV of Ed (Fig. 6).
Figures 6(a) and 6(b) demonstrate CVs as functions of ω0
and R, respectively. Three different depths are chosen in
order to understand the depth dependence on these two
effects. It can be seen that CV of Ed generally decreases
with increasing turbidity (ω0 ) and detector size (R). The
decreasing rates of CV of Ed for increasing ω0 are strongly
dependent on depths. At the shallower depth, CV of Ed
almost linearly drops with ω0 ; while at deeper locations,
1
CV
4 Results and Discussion
1
cZ = 5
cZ = 3.5
cZ =2
0.5
(a)
0
0
0.2
0.4
ω0
0.6
0.8
1
0.6
cZ = 5
cZ = 3.5
cZ = 2
0.5
0.4
CV
sky is black for simplicity. For each simulation setup, 10 realizations of surface wave fields are utilized to generate the
downwelling irradiance fields. Therefore, all interested statistical quantities, e.g., PDF of Ed ∕hEd i and CV, are ensemble
averaged. For each Monte Carlo RT simulation, 3.2 × 1010
photons are launched to achieve full convergence for all
depths.
Figure 4(b) shows the CV of Ed as a function of depth z
for both experimental data and simulation results at three different light wavelengths. It can be seen that the numerical
simulations are able to make very precise predictions of
CV for all depths (up to 10 m) and for all three wavelengths
compared to experiments. Figure 4(c) shows the probability
distribution (in logarithm) at three depths z ¼ 0.87, 1.70, and
2.83 m for the light wavelength λ ¼ 532 nm. We can see that
the model predictions agree with the experimental data very
well, especially for the greater depths (z ¼ 1.70 and 2.83 m).
There are some discrepancies between model predictions and
measurements at the shallower depth z ¼ 0.87 m for the
large value of Ed ∕hEd i (>4). This is likely due to the fact
that the rough ocean surfaces are treated as linear free surface
waves. However, during field experiments all surface waves
are fully developed and nonlinear. In general, the nonlinear
waves have stronger focusing of light and therefore lead to
higher values of Ed at a shallower depth.
0.3
0.2
(b)
0.1 −2
10
−1
10
0
R (mm)
1
10
10
Fig. 6 (a) The CV of normalized E d as a function of ω0 at different
optical depths cZ ¼ 2, 3.5, and 5 and (b) CV as a function of R at
above three depths.
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Xu and Yue: Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling. . .
CV quickly drops and reaches the asymptotic value as ω0 is
increased. For increasing R, the dropping rate of CV is relatively larger at the shallower depth and smaller at the
deeper depth.
4.2 Shadowing Effect
Probability density
To understand the shadowing effect on the underwater
downwelling irradiance fluctuations, we compare the PDF
and CV of Ed with and without considering the shadowing
effect in the Monte Carlo simulations (Fig. 7). In this study,
similar to above cases, the ECKV ocean spectra are applied
and the detector size is R ¼ 2 mm. The solar zenith angle is
θsun ¼ 85 deg to represent the solar light from the horizon
and demonstrate the difference of the two cases. Figures 7(a)
and 7(b) show the comparisons for the PDFs between considering shadowing effect and not considering shadowing
effect at z ¼ 1 m and z ¼ 4 m, respectively. We can see
that at the shallower depth, due to the shadowing effect,
the PDF with consideration of shadowing effect is strongly
modified by the block of incident light by surface waves. The
PDF neglecting shadowing effect is quite like those under
normal incidence or small solar incidence, while PDF
with shadowing effect taken into account becomes irregular.
For larger depths, however, it can be observed that the PDFs
for both cases do not deviate from each other very much.
(a)
z =1 m
0
10
−2
10
−4
10
0
0.5
1
1.5
2
2.5
3
3.5
4
E /⟨E ⟩
d
d
(b)
Probability density
z =4 m
0
10
−2
10
−4
10
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
E /⟨E ⟩
d
d
0
Depth (m)
1
2
Without shadow effect
3
With shadow effect
4
5
0
(c)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CV of Ed
Fig. 7 Comparisons of statistics of E d obtained without and with consideration of shadowing effect for large solar zenith angle θsun ¼
85 deg. (a) PDF obtained at z ¼ 1 m, (b) PDF obtained at
z ¼ 4 m, and (c) CV of E d as the function of z.
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Both of them become Gaussian-like with unit mean value.
This is due to the fact that at greater depths, the multiple scattering of ocean particles becomes a dominant effect on the Ed
and it smooths out its fluctuations. Figure 7(c) presents a
more straightforward picture of the effect of surface shadowing on variability of Ed and its dependence on depth z. For
the depth z < ∼3 m, the CV of Ed with the shadow effect is
dramatically larger than that without considering shadow
effect due to the low-intensity areas right behind the steep
surface waves. For depths z < ∼3 m, the two cases become
almost the same when light is fully diffused.
Recent study shows that under small solar zenith incidences for clear oceans and at very shallow depth, the probability distribution of Ed ∕hEd i is close to gamma distribution,
while for turbid oceans or at great depths, the probability distribution is more like lognormal. However, Fig. 7 indicates
that, under large solar zenith incidences, the gamma distribution fails to describe the PDF of Ed ∕hEd i at the shallow
depth, but lognormal is likely to work for greater depths.
It is also noted that detector and ocean’s turbidity play
a similar role in influencing the variability of Ed as spatial
filters. It is observed that current theoretical models of PDF
usually work better for greater R and ω0 . For extremely small
R and ω0 , the PDF usually becomes very complicated and
should be treated numerically.
5 Conclusion
In conclusion, we develop a state-of-the-art Monte Carlo RT
simulation capability to study solar underwater light variability, with particular interest in PDF and CV of the downwelling irradiance Ed caused by the focusing effects of the rough
ocean surface waves. In addition to various advanced variance-reduction techniques and full program parallelizations,
the numerical model is able to handle reflection/refraction of
light for the complicated surface and solar incidence conditions, such as the shadowing of steep surface waves for large
solar zenith angles. We particularly apply the high-order
correction of the beam-surface intersection points to obtain
relatively high accuracy for the interaction of light and airsea interfaces and increase the computational efficiency at
the same time.
To study the statistics of high fluctuated downwelling
irradiance near the surface, the model is capable of detecting
light field with high spatial [Oð10−3 mmÞ] and temporal
[Oð10−3 sÞ] resolutions. We carefully validate the models
using data-to-model comparisons. The comparisons between
simulated results and field data obtained during the RaDyO
experiments for multiple light wavelength and at various
depths indicate very good agreements.
Based on the model, we take a systematic investigation of
PDF and CV under the influence of two important physical
factors, water turbidity of the ocean and the detector size.
Effects of shadowing of the surface waves on the PDF
and CV are also studied. We show that increased turbidity
and detector size reduce the variability of the downwelling
irradiance; the shadow effects for large solar zenith strongly
enhance the variability of irradiance at shallow depths. It
seems that for large solar incidence the shadowing effects
play a more important role than other factors on the statistics
of the near-surface light field statistics. Therefore, the
systematic understanding of this effect indicates possibility
of directly obtaining the topology of the air-sea interface.
051408-6
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Xu and Yue: Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling. . .
On the whole, this numerical capability of Monte Carlo RT
simulation provides a great potential for future studies of
both forward and inverse problems about surface effects
on underwater light fields fluctuations.
Acknowledgments
This study was supported by the Office of Naval Research
through the RaDyO project.
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