A DNS Capability For Obtaining Underwater
Light Field And Retrieving Upper Ocean
Conditions Via In-Water Light Measurements
by
Zao Xu
Submitted to the Department of Mechanical Engineering
T
in partial fulfillment of the requirements for the
Mechanical Engineer's degree
ARCHIVES
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JUL 29?2011
at the
L RARI ES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2011
@ Massachusetts Institute of Technology 2011. All rights reserved.
Author ..........
Department of Mechanical Engineering
May 6, 2011
Certified by ................
Dick K.P. Yue
Philip J. Solondz Professor of Engineering
Thesis Supervisor
Accepted by ....................
1INSTTUTF
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Har. .
David E. Hardt
Chairman, Committee of Graduate Student
A DNS Capability For Obtaining Underwater Light Field
And Retrieving Upper Ocean Conditions Via In-Water Light
Measurements
by
Zao Xu
Submitted to the Department of Mechanical Engineering
on May 6, 2011, in partial fulfillment of the
requirements for the
Mechanical Engineer's degree
Abstract
Predicting the ocean surface conditions (surface elevation, temperature, wind speed,
etc.) becomes more and more important for both real life and military applications.
This thesis presents a direct numerical simulation (DNS) capability of solving
complicated natural light field patterns in the ocean-atmosphere system. The DNS is
applied by means of Monte Carlo method to solve radiative transfer for both unpolarized and polarized natural light radiation, especially strongly affected with dynamic
air-sea boundary conditions and inhomogeneous ocean turbulence. In the thesis, radiative transfer theory and Monte Carlo method are introduced. The realization and
rigorous code validation are given.
In order to apply this software to engineering, applications of radiative transfer
theory in ocean-atmosphere system is briefly introduced. To achieve most of the
engineering of retrieving ocean surface properties, systematical investigations of how
dynamic air-sea boundaries affect the underwater radiance and polarization are taken
and discussed.
To predict the upper-level ocean conditions based on radiometric underwater measurements, a scheme of inversion algorithm of reconstructing inherent optical properties based on a underwater radiance and irradiance radiometric measurements are
described. The key step of the inversion is an analytical solution of Green's function
of RTE under the approximation of single scattering. The preliminary trial of the
inversion are being taken.
Thesis Supervisor: Dick K.P. Yue
Title: Philip J. Solondz Professor of Engineering
Acknowledgments
This thesis and related work would not have been possibly finished without the support of many people. Among whom, I firstly would like to sincerely thank my thesis
advisor Professor Dick K.P. Yue for his long-lasting advising and support. Even when
I was deeply entangled by uncertainty and confusion of future, he gave me the opportunity to keep chasing my goals of life and advised me with passion and kindness.
I have been learning from him of being professional as a researcher. His rigorousness
in science and upstanding personality will set a model for me for my entire life.
I would also like to thank Dr. Yuming Liu who introduced me to the group. During
the past years, he sent me supports by sharing with me his expertise in research work
and giving me encouragement all the time.
I would deeply appreciate my parents and my girlfriend Lin Yang for their unconditional support along the way. For years, whenever I failed, they always sent their
comfort and encouragement at the earliest time and never gave up on me. Their love
and companionship are always my motivation of life.
Finally, I would thank my colleagues, professor Lian Shen, Dr. Yu Guang, Dr.
Hongmei Yan, Grgur Tokic, Wenting Xiao, Meng Shen, etc. Their help and discussion
helped me overcome difficulties in my research work.
6
Contents
1
Introduction
2
Radiative Transfer Theory for Ocean-Atmosphere System
21
2.1
Light radiation in the ocean-atmosphere system . . . . . . . . . . . .
21
2.2
Scattering and Absorption of Light . . . . . . . . . . . . . . . . . . .
24
2.3
Optical Properties of Water . . . . . . . . . . . . . . . . . . . . . . .
33
2.4
Principle of Radiative Transfer in Natural Water . . . . . . . . . . . .
37
51
3 Monte Carlo Radiative Transfer Simulation
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
3.2
Photon propagation in homogeneous medium
. . . . . . . . . . . . .
53
3.3
Scattering process . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
3.4
Radiance and irradiance detecting . . . . . . . . . . . . . . . . . . . .
64
3.5
Surface interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.6
Backward Monte Carlo simulations . . . . . . . . . . . . . . . . . . .
68
3.7
Polarized Light Radiative Transfer Equation . . . . . . . . . . . . . .
70
4 Program Realization and Validations
85
4.1
Program introduction . . . . . . . . . . . . . . . . . . . . . I . . . . .
85
4.2
Radiative transfer simulation . . . . . . . . . . . . . . . . . . . . . . .
86
4.2.1
Geometry and brief descriptions . . . . . . . . . . . . . . . . .
86
4.2.2
Methods and techniques . . . . . . . . . . . . . . . . . . . . .
88
4.2.3
Truncation approximations . . . . . . . . . . . . . . . . . . . .
88
4.3
Polarized MCRT: variables and subroutines
. . . . . . . . . . .
4.3.1
V ariables . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2
Key Subroutines
. . . . . . . . . . . . . . . . . . . . . .
.
.
.
90
4.4
Parallelization of program:Message Passing Interface (MPI) . . .
95
4.5
Validation of the Monte Carlo Radiative Transfer Code . . . . . . . .
95
4.5,1
Scattering properties in the ocean . . . . . . . . . . . . . . . .
95
4.5.2
Radiance profile in the diffusion region . . . . . . . . . . . . .
95
4.5.3
Radiance profile just below the surface . . . . . . . . . . . . .
98
4.5.4
Refraction and reflection of light at the surface . . . . . .
.
.
99
4.5.5
Polarization distribution in diffusion region . . . . . . . .
.
. .
101
. . .
106
4.6
Underwater Stokes vector compared with field measurements
.
5 Investigation of Ocean Surface Wave Effects on Underwater Light
Field-Forward Problem
107
5.1
Generating air-sea boundary: ocean surface waves . . . . . . . . . . .
107
5.2
Generating linear wave elevations from wave spectrums . . . . . . . .
112
5.3
Gravity-Capillary wave surface elevation generation . . . . . . .
. .
114
5.4
Underwater radiance and polarization affected by ocean surface v aves
116
5.4.1
Irradiance patterns affected by 2D surface waves . . . . . . . .
116
. . . . . . .
118
.
5.5
Radiance distribution induced by Airy and Stokes waves
5.6
Irradiance patterns induced by IHT FST and Shear Flow FST
. . .
126
5.7
Effects of roughness of ocean surface on underwater polarization . . .
133
6 Radiative transfer in ocean turbulent flow
145
6.1
Radiative transfer with varying refractive index . . . . . .
. . . 145
6.2
Monte Carlo solution and numerical ray-tracing technique
. . . 146
6.3
Empirical IOPs models . . . . . . . . . . . . . . . . . . . .
. . . 148
6.3.1
A. Model of Absorption Coefficient a(T, S, Cc, A)
. . . 148
6.3.2
B. Model of Scattering Coefficient b(T, S, Cc, A)
. . . 150
6.3.3
C. Model of Refractive Index n(T, S, A) . . . . . . .
. . . 151
Simulation results . . . . . . . . . . . . . . . . . . . . . . .
. . . 151
6.4
6.4.1
Clear ocean:uniform chlorophyll concentration . . . . . . . . .
152
7 Inversion of IOPs and applications of RT prediction in engineering159
Radiometric measurement: Radiance and polarization
7.2
Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.2.1
7.3
8
. . . . . . . . 159
7.1
Inverse problem in ocean optics . . . . . . . . . . . . . . . . . 165
Inverse algorithm of reconstructing IOPs . . . . . . . . . . . . . . . . 177
7.3.1
Single scattering approximation . . . . . . . . . . . . . . . . .
178
7.3.2
Differentiation techniques
. . . . . . . . . . . . . . . . . . . .
180
7.4
Application of RT Prediction in Engineering . . . . . . . . . . . . . .
184
7.5
Remote sensing of cloud properties via radiative transfer
. . . . . . .
184
7.6
Application of retrieving ocean conditions to fishery . . . . . . . . . .
186
Conclusion
187
10
List of Figures
2-1
Schematic design for measuring unpolarized spectral radiance
. . . .
23
2-2
Elliptical polarization . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2-3
Geometry used to define IOP's . . . . . . . . . . . . . . . . . . . . . .
34
2-4
Depletion of the radiance in passing through an extinction medium
.
38
2-5
(1)True emissiion; (2)Multiple scattering; (3)Single Scattering
. . . .
42
2-6
Understanding of integral form of the RTE . . . . . . . . . . . . . . .
45
. . . . . . . . . . . . . . . . . . . . . . . . . . .
48
. . . . . . . . . . . . .
49
2-7 Interaction principle
2-8
Reflectance and transmittance of level surface
3-1
Polar scattering angle versus random number q for isotropic scattering
(g = 0) and Henyey-Greenstein scattering with anisotropy factors of
g = 0.5 and 0.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
61
Cosine of polar scattering angle versus random number q for Petzolds
San Diego Harbor scattering phase function. . . . . . . . . . . . . . .
62
3-3
Generating Backward Monte Carlo functions for various quantities . .
70
3-4
Meridian planes geometry
. . . . . . . . . . . . . . . . . . . . . . . .
73
3-5
Initially the electrical field E is defined with respect to theimeridian
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3-6
The electrical field is rotated so that Ell is parallel to BOA . . . . . .
78
3-7
After a scattering event the Stokes vector is defined respect to the
plane C O A
plane B O A
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3-8
The electric field is now rotated so that Ell is in the meridian plane COB 80
4-1
Flow chart of 3D MCRT simulation . . . . . . . . . . . . . . . . . . .
87
4-2
Coordinate definition . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4-3 Trunction approximation . . . . . . . . . . . . . . . . . . . . . . . . .
89
4-4 An example of the rough water surface discretized to facets by Preisendorfer and M obley (1986)
. . . . . . . . . . . . . . . . . . . . . . . . . .
91
4-5
Radiance profile v.s. Phase function . . . . . . . . . . . . . . . . . . .
96
4-6
Radiance distribution: Invariant Imbedding method vs Monte Carlo
m ethod
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4-7 Irradiance relative to surface (a) T. Adams et al (b) Our simulation
98
4-8 Irradiance relative to surface (a) T. Adams et al (b) Our simulation
99
4-9
Irradiance relative to surface (a) Zaneveld et al (b) Our simulation .
100
4-10 Irradiance for 3D sinusoidal surface wave . . . . . . . . . . . . . . . .
101
4-11 Vector radiance just below the surface for 0, = 0 . . . . . . . . . . . .
102
4-12 Degree of polarization just below the surface for 0, = 0 . . . . . . . ..
103
4-13 Degree of polarization (# = 0)just below the surface for 0, = 0'
103
. .
4-14 Degree of polarization (# = 90')just below the surface for 0, = 600
104
.
4-15 Degree of polarization (# = 180')just below the surface for 0, = 60
.
4-16 Irradiance relative to surface (a) Zaneveld et al (b) Our simulation.
104
105
4-17 Direct comparisons of underwater polarization field under a dynamic
wavy ocean surface between (A) RaDyO field measurement (Voss 2010)
and (B) Monte Carlo RT prediction.
5-1
Wave spectra of a fully developed sea for different wind speeds according to Moskowitz (1964)
5-2
. . . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . . 109
Significant wave-height and period at the peak of the spectrum of a
fully developed sea calculated from PM spectrum . . . . . . . . . . . 110
5-3
Wave spectra of a developing sea for different fetches according to
Hasselmann et al (1973)
. . . . . . . . . . . . . . . . . . . . . . . . .111
5-4
Downwelling irradiance patterns induced by Airy waves with KA=0.19 116
5-5
Downwelling irradiance patterns induced by Stokes waves with KA=0.19117
5-6
Downwelling irradiance patterns induced by Airy waves with KA=0.38 117
5-7
Downwelling irradiance patterns induced by Stokes waves with KA=0.38118
5-8
Downwelling irradiance patterns induced by Narrow band irregular
waves with KA, = 0.2 in normal incidence case . . . . . . . . . . . . 119
5-9
Downwelling irradiance patterns induced by Broad band irregular waves
with KAp = 0.2 in normal incidence case
. . . . . . . . . . . . . . . 119
5-10 Downwelling irradiance patterns induced by Broad band irregular waves
with KpA, = 0.2 with incidence angle of 20 . . . . . . . . . . . . . . .
120
5-11 Radiance distributions under a Airy vs (fully nonlinear) Stokes wave
as a function of position and depth . . . . . . . . . . . . . . . . . . .
120
5-12 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under crest . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121
5-13 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under trough . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
5-14 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
. . . . . . . . . . . . . .
122
5-15 Spatial spectrum of Downward Irradiance at Z = 1 . . . . . . . . . .
123
Z = 1 under 900 phase.. . . . . . . . . .
5-16 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 5 under crest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5-17 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 5 under trough . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5-18 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 5 under 900 phase . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5-19 Spatial spectrum of Downward Irradiance at Z = 5 . . . . . . . . . .
125
5-20 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under 90' phase with KA = 0.1
. . . . . . . . . . . . . . . . .
125
5-21 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under 90' phase with KA = 0.25 . . . . . . . . . . . . . . . . .
125
5-22 Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under 90' phase with KA = 0.4
. . . . . . . . . . . . . . . . .
126
5-23 Downwelling irradiance patterns under isotropic homogeneous turbulence (IHT) with Froude number = 0.8 . . . . . . . . . . . . . . . . .
127
5-24 Downwelling irradiance patterns under isotropic homogeneous turbulence (IHT) with Froude number = 0.2 . . . . . . . . . . . . . . . . .
128
5-25 Downwelling irradiance patterns under shear flow free surface turbulence with Froude number = 2.25 . . . . . . . . . . . . . . . . . . . .
129
5-26 Downwelling irradiance patterns under shear flow free surface turbulence with Froude number = 1 . . . . . . . . . . . . . . . . . . . . . .
130
5-27 Downwelling irradiance patterns under ship wake with Froude number
= 0 .9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 1
5-28 Downwelling irradiance patterns under ship wake with Froude number
= 0 .6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
5-29 Downwelling sky polarizations just above the ocean surface and underwater polarizations at the optical depth of 0.1 below the surface
predicted by Monte Carlo simulation under different solar zeniths and
ocean surface conditions. (a) and (b) are degree polarization P and
e-vector x respectively when solar zenith is 0,
= 100; similarly, (c)
and (d) are degree polarization P and e-vector X respectively in the
case that solar zenith is 0,, = 700; (e), (f) and (g) are degree of polarization, e-vector and ellipticity respectively at the solar zenith equal
to 10' under the flat surface assumption; (h), (i) and (j) take the same
sequence but with solar zenith of 700 for flat ocean surface; similarly,
(k), (1) and (m) are polarization with 0,, = 100 under roughened
ocean surface with wind speed U10 = 8m/s; (n), (o) and (p) are same
as (k), (1) and (m) except in the case of 0,,, = 70 . . . . . . . . . ..
135
5-30 Maximum degree of polarization P,,,a within Snell's window affected
by different surface roughness represented by wind speed U10 in the
case of low solar zenith O, = 100 (a) At the optical depth
T
= -0.1
(b) At the optical depth T = -1 . . . . . . . . . . . . . . . . . . . . .
137
5-31 Maximum degree of polarization Pma, within Snell's window affected
by different surface roughness represented by wind speed UiO in the
case of low solar zenith Os,
= 700 (a) At the optical depth -r = -0.1
(b) At the optical depth r = -1 . . . . . . . . . . . . . . . . . . . . .
138
5-32 Maximum degree of polarization Pmax within Snell's window varies
with solar zenith angles for several surface conditions at the optical
depth T = -0.1
wo
= 0.8
(a) Single scattering albedo of ocean wcn -= 0.2 (b)
138
. . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .
5-33 Distribution of azimuthally averaged ellipticity tan 3. . . . . . . . . . 140
5-34 Dependence of maximum ellipticity tan
.max
on solar zenith angle 0,,,
under different ocean surface conditions (wind speed varies from Om/s
to 20m /s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
140
5-35 Effects on ocean surface roughness on maximum ellipticity tan Omax
under conditions of different single scattering albedo of ocean: wo
=
0.2,0.5 and 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-36 ........
6-1
...
...........................
.......
141
142
The mean value and standard deviation of IOPs of ocean turbulence
at various depths (a) Mean value of absorption coefficient < a > (b)
Mean value of scattering coefficient < b > (c) Mean value of refractive
index < n > (a) Standard deviation of absorption coefficient Ua (e)
Standard deviation of scattering coefficient o-b (f) Standard deviation
of refractive index o-, . . . . . . . . . . . . . . . . . . . . ..
6-2
. . . . . 153
The light field pattern below the calm ocean surface in the presence of
ocean turbulence(a) Downwelling irradiance Ed just below the ocean
surface(b) Upwelling irradiance E, just below the ocean surface(c)
Downwelling irradiance Ed at z = -20m(d) Upwelling irradiance Ed
at z = -20m
6-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
154
Mean values of downwelling irradiance and upwelling irradiance at different ocean depth. (a) lg < Ed(z) > (b) lg < E,(z) > . . . . . . . . .
155
6-4
Standard deviations of downwelling irradiance and upwelling irradiance
at different ocean depth. (a)
6-5
6
Ed
(b)
CE. . . . . . . . . . . . . .
. .-
156
The light field pattern below the progressive ocean surface waves under progressive ocean waves with kpa, = 0.1 in the presence of ocean
turbulence(a) Downwelling irradiance Ed just below the ocean surface(b)
Upwelling irradiance E, just below the ocean surface(c) Downwelling
irradiance Ed at z = -20m(d) Upwelling irradiance Ed at z = -20m
6-6
157
Mean values of downwelling irradiance and upwelling irradiance at different ocean depth under progressive ocean waves with kpa, = 0.1. (a)
lg < Ed(z) > (b) lg < E,(z) > . . . . . . . . . . . . . . . . . . . . . . 157
6-7
Standard deviations of downwelling irradiance and upwelling irradiance
at different ocean depth under progressive ocean waves with kpap = 0.1.
(a)
UEd
(b) JE
.
. . . . . . . . . . .
.
.
.. . . . . . . . .
.
. . . . .158
.
7-1
Block diagram of radiometer (Voss, 1991) . . . . . . . . . . . . . . . . 160
7-2
Fisheye optic description (D. Antoine et al, 2009) . . . . . . . . . . . 160
7-3
Radiometer scheme description (D. Antoine et al) . . . . . . . . . . . 161
7-4
The first prototype of the LOV-CIMEL radiance camera (D. Antoine
et a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7-5
Au radiance field measurement taken by K. Voss 1988 with Radiometer) 162
7-6
Measured polarization radiance, degree of polarization and e-vector
orientation taken by Sabbah et al (2006) . . . . . . . . . . . . . . . . 164
7-7
Single scattering light propagation geometry . . . . . . . . . . . . . . 179
7-8
Geometry of two-dimensional radiance measurement setup . . . . . . 181
7-9
Example of using single scattering approximation and radiance measurement to reconstruct ocean turbulent structure (a) Original turbulent flow structure of attenuation coefficient (b) Reconstructed attenuation coefficient with 50 sources (c)Reconstructed attenuation coefficient with 100 sources (d) Reconstructed attenuation coefficient with
150 sources .......
................................
183
Chapter 1
Introduction
Ocean optics research covers a wide range of subjects. Among all the subjects, the
problem of specifying the radiation field in the ocean water is both the most fundamental and interesting one. As an investigating method of radiation distribution,
the theory of Radiative Transfer has long been explored since 1905 when Arthur
Schuster formulated in a problem in Radiative Transfer in an attempt to explain the
appearance of absorption and emission lines in stellar spectra and 1906 when Karl
Schwarzschild introduced the concept of radiative equilibrium in stellar atmospheres.
Huge volume of work has been done explaining the Radiative Transfer process of
light in the atmosphere of the Earth. Due to the close relationship of atmosphere
and ocean, Radiative Transfer theory is currently thought as the most reliable and
promising tool to predict the radiation field in an atmosphere-ocean system, especially
under the ocean surface. In this thesis, we manifest a direct numerical simulation capability to understand the radiative transfer in ocean and atmosphere system and the
influence of the air-sea dynamic boundaries.
The polarization of underwater light has drawn the attention of oceanographers
for decades because it contains rich information of ocean and atmosphere constituents,
solar condition, and dynamic ocean surface wave-fields. Some properties of the polarization have been found as a navigation tool used by many marine animals. Measurements of polarization can also help oceanographers to retrieve information of inherent
optical properties (IOP's) and the concentrations of water and air particulate.
While having realized the potential applications of underwater polarization, for a
long period researchers are faced with the practical difficulties of making systematic
observations with advanced instruments due to the complications and high costs of
experimental setups.
Watermann
[3] and Ivanoff and Timofeeva
[2] performed
pioneering measurements to understand of the polarized light field distribution in
water and its dependence on solar direction, depth and light wavelength. In recent
decades, Voss and co-workers (see for example Voss and Fry
[5]) used 'fish-eye'
polarimeters to measure the Stokes vector components of sky and the underwater
polarization patterns with high angular resolutions. Sabbah and Shashar [6] [16]
extended the experimental work to hyperspectral region and started a preliminary
investigation of the fluctuation of polarization fields.
Besides experimental work, considerable amount of theoretical work has been
done to understand the polarized light propagation in atmosphere and oceans. Chandrasekhar
[1] applied the Stokes vector, which represents the polarization state of
light fields, to classic radiative transfer equation and derived the exact solution of the
vector radiative transfer equation for a plane-parallel atmosphere with Rayleigh scattering. Kattawar and Adams [9] [13] [12] introduced a Monte Carlo (MC) method
in solving the Stokes vector radiative transfer equation (VRTE) and provided expressions of transmission and reflection Muller matrices for the dielectric interface. Fast
development in computing power makes the MC method the most prevalent tool in
solving the polarized light fields in atmosphere-ocean system.
Previous modeling works on underwater polarized light fields were mainly focused
on the relations between polarization properties and water turbidity, solar incidence,
and light wavelength. Very few considered the effect of dynamic ocean surfaces due to
complication of describing and understanding a realistic phase-resolved ocean surface
wave field. With the assumption of a flat ocean surface, Horvath and Varju
[4]
studied theoretically the polarization patterns affected by the air-sea interface. In
the work by Kattawar and Adams [18], the Cox-Munk [15] ocean surface model was
used, which gives an empirical slope distribution. These studied are important and
the first step towards the statistical description of underwater polarization properties.
For instantaneous patterns and higher order statistics of polarization, the dynamics
of more realistic wave field should be incorporated to the study.
The studies on fluctuations of the in-water irradiance and radiance at the upper
ocean have been taken both theoretically and experimentally for half century. Both
analytical (Snyder,1970) and numerical models (Kattawar, 2010) are based on the
assumption that irradiance and radiance fluctuation at the upper ocean are mainly
caused by the focusing of sunlight at the dynamic air-water interface and light scattering and absorption within ocean body. Scattering of light performs as low-band
filter and smooths out the high frequencies of light fields. Standard deviation of downwelling irradiance decreases when scattering effects is considered. For those methods
and models, the ocean is thought as a homogeneous isotropic thick scattering medium
with uniform scattering and absorption coefficients. However, most oceans are turbulent, which causes fluctuations of passive scalars, such as temperature, salinity and
chlorophyll concentrations and therefore lead to variations of inherent optical properties, mainly absorption and scattering coefficients and refractive indices. The inhomogeneity in inherent optical properties have strong effects on light radiative transfer
mechanism in ocean bodies and change the statistics feature of in-water irradiance
and radiance originally induced by surface wave focusing effects. Experiments also
showed correlations between light fields and temperature and salinity distributions at
the same ocean depth.
Modeling light propagation in turbulent medium is an old topic and has been well
studied in micro scale by solving Maxwell's equations or scalar Hehholtz equation
directly. However, in large scale turbulent environment such as ocean turbulence
flow, the most commonly used method to describe and predict the light propagation
is the radiative transfer equation (RTE) with inhomogeneous scattering and absorption coefficients.. Many methods have been introduced to solve the inhomogeneous
RTE, such as spectral method and Monte Carlo method. Conventional Monte Carlo
method is regarded as the most straight-forward means to solve RTE, but it is only
accurate under the assumption of the first-order perturbation. On the other side, as
an important optical property, refractive index changes with temperature and salin-
ity, and therefore affects the light beam translation. Unfortunately, the refractive
index does not appear in the conventional RTE. Previous treatment of the fluctuation of refractive index and its effects on radiative transfer properties was to assume
the medium is an incoherent addition of two scatterers, particle and turbulence. The
turbulence phase function is obtained by solving the Helmholtz equation. However,
Monte Carlo simulation wit this strategy can not give the phase-resolved light fields
determined by ocean turbulence. In recent years, several authors reported the modified radiative transfer equation by taking into account the light bending effects when
light beam passes through materials with continuously changed refractive index. Our
work follows the derivation of the modified RTE with varying refractive index and
use it to solve light radiation transfer in ocean turbulent flow.
Chapter 2
Radiative Transfer Theory for
Ocean-Atmosphere System
2.1
Light radiation in the ocean-atmosphere system
The wave-particle duality tells us, in one hand, light is reasonably envisioned as
consisting of numerous localized packets of electromagnetic energy, called photons,
which move through empty space with speed c = 2.998 x 108 m s-'. Each of these
"particles" of light also carries certain momentum and angular momentum. On the
other hand, we also envision light as an electromagnetic wave with a wavelength A
and a frequency v based on the relation v = c/A.
The photon viewpoint of light is well suited to our development of radiative transfer theory. The energy q of a photon is related to its frequency V and corresponding
wavelength A by
hc
q = hv = h(2.1.1)
A
where h = 6.626 x 10 3 4 J s is Planck's constant.
The magnitudes of the linear
momentum p and angular momentum 1 of a photon are give by
p=
(2.1.2)
and
1=-
h
21r
(2.1.3)
respectively.
By puttipg radiant energy detectors in watertight assemblies and by appropriately
channeling the direction of the photons arriving at the detector, we can measure the
flow of radiant energy as a function of direction at any location within a water body.
By adding appropriate filters to the instrument, we can also measure the wavelength
dependence and state of polarization of the light field. From such measurements we
can develop precise descriptions of radiative transfer in natural waters. Thus, here,
we introduce some basic radiometric quantities.
Solid Angle is the angle subtended at the center of a sphere by an area on its surface
numerically equal to the square of the radius
Q
(2.1.4)
=(
where a is the area of the sphere surface and r is radius of the sphere. Unit of
a solid angle is steradian (sr).
A differential solid angle can expressed as
dQ -
do
- sin(O)dOd$
(2.1.5)
Radiance (or Intensity) is defined as radiant energy in a given direction per unit
time per unit wavelength (or frequency) range per unit solid angle per unit area
perpendicular to the given direction:
I;
t;
)(J
S-1
m-2 sr-1 nm')
(2.1.6)
Where i is the position of detector; (is the direction perpendicular to the detecting area. AQ is the radiant energy which has the unit of J; AA and AQ mean
the definition is taken in a very small detecting area and solid angle respec-
Detector
Figure 2-1: Schematic design for measuring unpolarized spectral radiance
tively. The radiance is also called spectral radiance or monochromatic radiance
which indicates the measurement is limited in a small range of wavelength. The
definition can be understood by figure 1.
Radiance, as function of position and direction, gives a complete description
of the electromagnetic field. If radiance does not depend on the direction, the
electromagnetic field is said to be isotropic. If it does not depend on position,
the field is called to be homogeneous.
Irradiance (or Flux) is defined as radiant energy in a given direction per unit
time per unit wavelength (or frequency) range per unit area perpendicular to
the given direction:
(W m- 2 nm- 1 )
F(; t; A) = At A
AAA A
(2.1.7)
The relation between Irradiance and radiance can be expressed by
F
Ilcos() dQ
=
23
(2.1.8)
As the radiance, we call irradiance with such definition as spectral or monochromatic. There are several kinds of spectral irradiance which are important in
measurement.
Spectral downward irradianceis related to radiance by
Fd(z'; t; A) =
I I(X; t; ; A)|Icos(0)|1 dQ~f
(2.1.9)
where Ed points to beams with downward direction so the integration is taken
from upper hemisphere.
Spectral upper irradianceis related to radiance by
I(; t; ;A)|cos(0)| dQ(f)
Fd(; t;A) =
(2.1.10)
Spectral downward scalar irradiance is related to radiance by
Fod( ;t; A) =
J
I(i; t;;A)dQ(f)
(2.1.11)
Spectral vector irradianceis related to radiance by
od('; t;
2.2
A)
=
I(F; t; ; A)(dQ()
(2.1.12)
Scattering and Absorption of Light
Radiometric quantities are important because later calculation and measurements
are taken mostly based on those quantities. However, to fully understand the radiative transfer process in the Atmosphere-ocean system, it is necessary to get back
to the electromagnetic wave viewpoint of light in order to investigate two important
interactions between light and ocean water: Scattering and Absorption.
Before digging into the scattering and absorption process, we first present an
important properties based on wave viewpoint: Polarization.
Let us consider a plane electromagnetic wave, propagating in the direction specified by the vector es. We obtain a transverse electric vector E
E = E1
1
+ E 2 62
(2.2.1)
where complex oscillating functions E1 and E 2 represent components of the electric
vector in the plane perpendicular to the propagation direction along vector 6
,2.
It
follows for unit vectors 21,22,e 3 that
e1 x
(2.2.2)
e2 = e3
Now we define a important quantity I which is called the Stokes vector. It has 4
elements in terms of the components Ei and E2 .
I
=
E1 E* + E 2E2*
Q = E1 E*
- E 2E2*
(2.2.3)
(2.2.4)
U= E1 E* + E*E2
(2.2.5)
V = i(EIE2*- E*E2 )
(2.2.6)
The values of I, Q, U, V completely characterize the arbitrarily light beam in terms
of the intensity, degree of polarization and characteristics of the polarization ellipse
(the ellipticity, the azimuth, and the direction of rotation).
The components of the Stokes vector can be written in terms of the amplitudes
ai, a 2 and phases
U1 , 0 2
of a simple electromagnetic wave as well:
I + =2a
a 2(2.2.7)
2
Q=a
2
-
(2.2.8)
a22
U = 2aia 2 cos(Oi
- 92)
(2.2.9)
Figure 2-2: Elliptical polarization
V = 2aia2 sin(o-
-
(2.2.10)
0-2 )
where we use the following representation of complex amplitudes E1 and E 2 :
E1 = aei(kz-Wt+) , E2 = a2e-i(kz-t+o2)
(2.2.11)
Here k = 27r/A, z is the propagating distance along e, w = kc is the circular frequency.
Figure 2 shows the characteristics of the polarization ellipse using the Stokes vector
I = a2 , Q = a 2 cos 20 cos 2T, U = a 2 cos 20 sin 24, V = a 2 sin 2 3
(2.2.12)
where a2 = a + a2. Equation (2.24) provide the geometrical interpretation of the
Stokes vector I. Thus, the angle of the polarization plane(or the azimuth) IF and the
ellipticity angle # can be found from the following equations:
U
1
WI=- arctan , --
2
Q'
1
2
arctan
V
Q2+U2
(2.2.13)
The angle
#
can also be calculated from the following formula
3=
1
-
2
arcsin
V
2
2
VQ +U
+y
2
(2.2.14)
(
The degree of polarization DP of a light beam is defined as
fQ 2 + u 2 +
DP =
v 2)
(2.2.15)
The degree of linear polarization LP of a light beam is defined by neglecting U and
V as
LP =
" Completely unpolarized light:]
Q=
(2.2.16)
U = V =0
" Fully polarized light:1 2 = Q2 + U2 + V 2
" Linear polarized light:V = 0
" Circular polarized light:IVl = 1
Let us first start from Maxwell's Wave Equations which governs the interaction
of electromagnetic waves with a particle in source free space.
V x V x $(r)
- k2 n()
5(r) = 0
2
(2.2.17)
where E is the electric vector, k is the wave number equal to 27r/A and n is the
refractive index which depends on position vector r.
The boundary conditions require that components of the electric vector E and
magnetic vector H = (i/k)nabla x E, which are continuous at the boundary of a
particle. The integral form of the solution of equation (2.29) is
5 = EO(f) +
$- r =(n2(
k 2 eqir
47r
IV(y
-)
$,(r)
1)eik(-')$( ') dai'r(..9
(2.2.18)
(2.2.19)
where V is the volume of a particle, Eo is the incident electric field and E is the
scattered electric field. The radiance can be expressed by the electric field as
I
=
1E2
AQ 47r
(2.2.20)
The expression of scattering field explains that scattering happens because of variation
of refractive index n(i')in space, while absorption happens due to the imaginary part
of square of refractive index n 2 (i ). If n 2 (?) = 1, scattered field will disappear, which
means the light will keep traveling the free space.
Assuming a beam of radiation passes through a slab, it interacts with the particles
through either absorption or scattering and a reduced amount of energy emerges at
the opposite side. The beam of radiation is said to have suffered extinction. Thinking
about the process happening within a very small length ds, it is found experimentally
that the degree of weakening depends linearly upon both the incident intensity and
the amount of optically active matter along the beam direction (proportional to the
length ds). This is called differential form of The Extinction Law:
dl
- plds
(2.2.21)
where p is the density of the material. The quantity , introduced in this manner
defines the mass extinction coeffjcient. If there is only scattering happening and the
cross-section of slab is do-, the energy is scattered from it at the rate
t'pIds x cos 8dudo-dQ
(2.2.22)
in all directions. Since the mass of the element is
dm = p cos Eduds
(2.2.23)
kldmdvdQ
(2.2.24)
we can also write from(2.34)
Now we introduce a phase function p(cos 8) such that
kIp(cos 8)
dQ'
47r
dmdvdQ
(2.2.25)
gives the rate at which energy is being scattered into an element of solid angle dQ'
and in a direction inclined at an angle 8 to the direction of incident radiant beam on
an element of mass din. According to (2.21), the total loss through the slab due to
scattering in all directions is
kldmdvdQ
p(cos 8) dQ'
47r
(2.2.26)
When both scattering and absorption are present, we have the following expression
p(cos 8) dQ'
47r
(2.2.27)
where r < 1 is referred as albedo for single scattering. It is evident from the definition
that w represents the fraction of the light lost from an incident beam due to scattering,
while (1-
) represents the remaining fraction which has been transformed into other
forms of energy (or of radiation of other wavelengths).
The simplest example of a phase function is
p(cos 8) = costant = w
(2.2.28)
In this case the radiation scattered by the element is isotropic. Otherwise, it is called
nonisotropic scattering.
As is mentioned before, the scattering electromagnetic field can be precisely predicted by solving Maxwell's Equations with boundary condition. However, in radiative transfer theory, it is much more interesting to calculate the scattering phase
function from various existing scattering theory of electromagnetic field. Here, we
present the S-Matrix method introduced by Van de Hulst (1967).
Here, we are interested in the scattering process Considering the transverse inci-
dent wave and scattered wave have the following forms respectively:
Ei =
En 1' + Ei 2e 2
(2.2.29)
5s =
Es 1 -+ E2( 2
(2.2.30)
it easily follows that:
Esi
Es2 J
eik(z-r)
S2 S3
ikr
S4 S1
E
J kE J
(2.2.31)
Thus, we can see that the process of light scattering by a single particle is completely
defined if the elements Sij of the amplitude scattering matrix (S-matrix) are known.
The next step is to transform the S-matrix into a 4-component matrix which links
the Stokes vector of the incident light Ii with that of the scattered light I,.
Is = PhI
(2.2.32)
Here, P is called Scattering Phase Matrix or Mueller Matrix. To simplify the calculation, the scattering are carried out under the same reference plane.
According to Bohren and Huffman, 1983, when the plane of reference is the scattering plane, the Mueller matrix is presented by
Pi I P12 P13
P21 P22 P23
P31 P32 P33
P41 P42 P43
(2.2.33)
where
P11 = 1/2(|S 11 2 + |S 1 2 |12
2
P12 = 1/2(|Sn|1
_
S12|2
S2 1 + S22|2)
+ S21 12
P13 = Re(SIIS*2 +
S22
P 14 = Im(Sn S1*2
S22 S 1)
P21 = 1/2(|S 11
2
-
P22 = 1/2(|Sn| 2 + |S 2212
P23 =
S 22
-
S 21 12)
2)
S 1)
S 22 |2 +| S 12
-
-
-
2
12 2
2112)
Re(SuS*2 - S 22 S 1 )
P24 = Im(Sn1S1*2 + S 22S3 1)
P31 = Re(SuS21 + S22S1*2)
(2.2.34)
P32 = Re(SulSi1 - S22 S1*2 )
P33 = Re(S*1 S 22 + S 12 S2*1)
P34 = Im(SIS
2
+ S21
P41 = Im(S* 1 S21 + S*2S
*2)
22 )
P42 = Im(S*1S 2 1 - S1*2S22)
P43 = Im(S 22S*1 - 312321)
P44 = Re(S 22S *1 + S1232*1)
The above calculation is affected by the following factors:
" The definition of the Stokes vector
" Selection of reference plane
" Approximation of scattering theory
" Shape of the particle
Rayleigh scattering and Mie scattering are the most two commonly used models
in the radiative transfer calculation. They have different the conditiois.
Rayleigh scattering
Light scattering and absorption characteristics of particles that are small compared with the wavelength of the incident radiation outside and inside the a particle
can studied using Rayleigh approximation (Rayleigh, 1871). It is assumed within the
framework of the approximation that:
x < 1, x~nl < 1
where x = ka is the size parameter, k = 27r/A, a is the size of a particle, n is the
complex refractive index of a particle.
For such a small particle the electric field within it is constant. Thus, a particle
can be replaced by a single oscillating dipole with the polarizability tensor 6 and
a simple theory of the dipole scattering can be applied for determination of light
scattering and absorption features. The tensor has the diagonalized form with three
components (1, a 2 anda3 .
For randomly oriented particles it follows that
A + Bcos2 E) B(cos 2 e _ 1)
P
B(cos 2 8
3
3A+B
-
2
1) B(cos 8 + 1)
0
2B cos E 0
0
0
where P is the Mueller matrix,
(12-
e is the scattering angle.
10A) cos 8
(2.2.35)
Then the Scattering Phase
Function is
p(cos8) = P1 1(cose) =
3(A + Bcos 2 g)
3AB
(2.2.36)
3 A +B
and
6- M
5
For spheres: A = B
2+3M
Re(a*a2 + a*a 3 + aa
lo
'
'5
3
12 +|a2|2 + |a3|2
)
(2.2.37)
1, then we get the final form of Mueller matrix and Scattering
Phase Function as
1 + cos 2 E cos 2 e
3
3A+B
cos 2 E
-
1 0
0
1 cos 2 E + 1 0
0
-
0
0
2 cos
0
0
0
E
(2.2.38)
0
2 cos
e
and
3
p(cos E) = Pu(cos 8) = - (1 + cos 2 8)
4
(2.2.39)
The Mueller matrix depends on the internal structure of particle, their refractive
index and shape. However, it does not depend on the size of particles.
Mie scattering
For particles not small compared with the wavelength, the interaction properties
must be found by solving a complicated boundary-value problem for the electric and
magnetic fields. Therefore, Mie scattering model is built under the assumption that
scattering happens with a spherical particle with a radius r > A.
The Phase Function for Mie scattering is given by
1
(2.2.40)
p(8) = 2-(IS1|21 +S22)
where Si and S2 are the scattering amplitudes,
SZ
nn(n=
0-
2n-+1
2n+1
(an7rn + bn), S2 =ar+bir)
1)n1(n+1)
(2.2.41)
and where
7rn(8) =
pit(e)
sin e
, 7rn(8) =
dPn(8
dsin8
(2.2.42)
P'(8k) is the associated Legendre polynomial.
2.3
Optical Properties of Water
After knowing how to quantitatively describe light radiation field and its traveling
features, it remains to focus on the optical properties of the medium through which
light propagates, natural water.
the large-scale optical properties of water are conveniently divided into two mutually exclusive classes: inherent and apparent. Inherent optical properties (IOP's) are
those properties that depend only upon the medium , and therefore are independent
of the ambient light field within the medium. The two fundamental IOP's are the
AV
I
vaI
Qi
IQt
Ar
Figure 2-3: Geometry used to define IOP's
absorption coefficient and the volume scattering function. Other IOP's include the
index of refraction, the beam attenuation coefficient and single-scattering albedo.
As figure 3 shows, consider a small volume AV of water, of thickness Ar, illuminated by a narrow collimated beam of monochromatic light of spectral radiant power
Qj.
Some part
Q,
is absorbed within the volume of water. Some part
out of the beam at an angle
e, and remaining power Qt
Qs
is scattered
is transmitted through the
volume with no change in direction. Then by conservation of energy,
Qi = Qa + Qs + Qt
(2.3.1)
Thus, the spectral absorption coefficient a is defined as
a
lim
4
Ar-+O Q4Ar
(M-1)
(2.3.2)
and the spectral scattering coefficient b is
Qr (m- 1)
Ar-+0 QjAr
b A lim
34
(2.3.3)
The spectral extinction coefficient c is
c = a + b (m'1)
(2.3.4)
Now take into account the angular distribution of the scattered power. As we
have known,
E
is called scattering angle. It's value lies in the interval 0 <
< 7r.
The volume scattering function in such point of view is defined by
When integrating
0(8)
E
/3(E)
over all directions gives the total scattered power per
lim lim
Q
Ar+OE2+OQjAr/XQ
(m sr-1)
(2.3.5)
unit incident irradiance and unit volume of water which is the spectral scattering
coefficient:
b=
#(8) dQ
(2.3.6)
Compared with volume scattering function, the scatteringphase function p(8) is
defined by
p(8) EO(E)
b
(2.3.7)
From this definition, we can see that p(8) only gives a angular distribution of the
scattered photons, but scattering coefficient b concerns the strength of scattering.
Volume scattering function 0(0) combined the two together.
In the former part, the single scattering albedo zu was introduced, while here, it
is defined in another way.
b
IU = c
(2.3.8)
In waters where the beam extinction is due primarily to scattering, w is near one.
In waters where the beam extinction is due primarily to absorptiori,w is near zero.
The single scattering albedo is the probability that a photon will be scattered in any
given interaction, hence w is also known as the probability of photon survival.
Apparent optical properties (AOP's) are those properties that depend both on
the medium (IOP's) and on the geometric structure of the ambient light field, and
that display enough regular features and stability to be useful descriptors of the water
body. Commonly used AOP's are the irradiance reflectance, the average cosines, and
the various diffuse attenuation coefficients.
Spectral downwelling average cosine
fs
Fd(Z; )
f
I (z; 0, #;- A) cos 0 dQ
d I(z; 0, #; A) dQ
Fd (z; A)
Fod(z; A)
(2.3.9)
Compared with the former definitions of the irradiance, we are just interested
in it in a certain depth x 3 = z and in a steady state.
Spectral upwelling average cosine
pt(z; A) =
,
-
F(z;A)
= Fu (z;A)
(2.3.10)
Fd (z; A) - F(z; A)
(2.3.11)
I(z; 0,#0; A) cos 0 dQ
fI(z; 0, #;1A)
dQ
Fu (z; A)
Spectral average cosine
p(z; A)
f
I(z;0, #; A) cos 0 dQ
f
J(z; 0, #;A) dQ
Fo(z; A)
The definition of ft gives a distribution of radiance around a point. For example,
if the radiance is collimated in one direction (0o, 0), then Ad = cos 0. Typical
values of the average cosines for natural waters illuminated by the sun and sky
are Ad ~~3/4 and fLu ~ 3/8. Also, in natural, the sunlit waters, ft is always
positive.
Spectral irradiance reflectance
R (z; A) = Fu (z; A)
Fd(z; A)
R(z; A) is often evaluated in the water just below the surface.
(2.3.12)
Spectral diffuse attenuation coefficients
Kd(z; A)
d In F(z; A)
dz
1 dF(a; A)
F(z; A) dz
_
(2.3.13)
this expression is derived from the extinction law which points that for the
various radiances and irradiances all decrease approximately exponentially with
depth, at least when far enough below the surface. This law can expressed by
the euqatioin
F(z; A) = F(0; A)exp[-
f
K(z'; A)dz']
(2.3.14)
Other diffusion attenuation coefficients, e.g. Kd, K., KOdaandKos,,, are defined in
terms of corresponding Fd, F, FodandFs.
The distinction between beam attenuation coefficient(IOP's) and diffuse attenuation coefficient(AOP's) is important. the beam attenuation coefficient c(A) is
about the radiant power lost from a single, narrow, collimated beam of photons.
The downwelling diffuse attenuation coefficient Kd(z; A) is about the decrease
with depth of the ambient downwelling irradiance Ed(z; A), which comprises
photons heading in all downward directions.
An ideal AOP changes only slightly with external environmental changes, but
changes enough from one water body to the next. Unlike IOP's, AOP's cannot be
measured on water samples because they depend on the ambient radiance distribution
found in the water body itself.
2.4
Principle of Radiative Transfer in Natural Water
Based on the knowledge of how light is scattered by a single particle and the optical
properties of water, we will use them to figure out how a phone passes air-water
surface and interacts with a group of concentrated particles. The particles in natural
water have size distributions ranging from water molecules of about 0.1 nm to small
I
I+dI
I I
I I
I \
I |
li|ds
I I
Figure 2-4: Depletion of the radiance in passing through an extinction medium
organic molecules of size 100nm, to bacteria of about 1 pm, to phytoplankton of about
100 gm. So the natural water could be thought as clouds of particles with different
particle size and concentrations.
The interaction of radiation with particles clouds-Radiative Process in the water
can be described by the successive scattering, absorption and emission with particle
clouds. The equation governing those interactions of radiance with water bodies is
called radiative transfer equation(RTE). We will show how all kinds of interactions
contribute and are described in the framework of RTE.
Recalling the simplified case describe in section 2, a pencil of monochromatic
radiation traversing a medium will be weakened by its interaction with matter. If the
radiance I becomes I + dl after passing through a thickness ds in the direction of its
propagation, according to (2.33)
dI = -pIds
(2.4.1)
where p is the density of the material, and K denotes the mass extinction coefficient(in
units of area per mass) for radiation of certain wavelength.
We know the mass
extinction coefficient is the sum of the mass absorption and scattering coefficient
very similar to the spectral extinction coefficient c. The relation between r, and c is
c = np. Thus, the reduction in the intensity is due to absorption and scattering by
the material.
On the other hand, the radiance of given direction may be strengthened by emission from the material and diffusive energy from other areas around this small we are
interested. The diffusion here is called multiply scattering from all other direction
in the original propagation direction. Those radiation energy acquired from other
sources is given by dI1
dI, = jpds
(2.4.2)
where source function coefficient j has the same physical meaning as the mass extinction coefficient. Therefore, we obtain change of radiance after the thickness ds
dI = -pIds
+ jpds
(2.4.3)
Moreover, it is convenient to define the source path function J such that
J= jp (W m-3 sr- 1)
(2.4.4)
Finally, we get the fundamental radiative transfer function without any coordinate
system imposed:
dI- =
-cI + J
ds
(2.4.5)
Before that, we recall the n 2 law which means that when light travels through a
boundary of two medias with different refractive index ni and n 2 , then the radiances
within two medias satisfy
t(
2= -(2.4.6)
where t(( 1 . h) is the transmission coefficient which is determined by the direction of
incident radiance t and normal unit vector of the boundary h.
When a beam of photons traveling an infinitesimal distance across the boundary
from one media to another. We can take t ~1 without loss of rationality, since we
can always see this case as the normal incidence to the boundary. Thus, we have
12
=
1
(2.4.7)
Applying the assumption to RTE, we can just replace all the I in the previous
fundamental RTE with I/n2 in order to fit the radiative transfer problem in inhomogeneous media.
Then we take 3D case, time-dependence and wavelength-dependence into account.
The radiance now is written as I(; t; ; A)/n 2 and I/n
2
for short.
The different operator in RTE should be expressed in terms of the usual substantive derivative by
D
D=s
-Ds
_
a-
(2.4.8)
-vaot
Where v is the traveling speed in the direction of ( in the media.
So the general RTE is expressed as
vOt
q1q+
n2
-V
)
n2
=-c(
n2
+J
(2.4.9)
The above equation is the most general form of the radiance transfer equation for
unpolarized radiance. It governs the time-dependent, three dimensional behavior of
the radiance. It is also valid for inhomogeneous and anisotropic media, if we take the
inherent optical properties to be functions of position and direction respectively, e.g.
=n(; t; ; A) and c = c(; t; ; A).
For many ocean radiative transfer applications, it is physically appropriate to
consider that the water body in localized portions is plane-parallel such that variations
in radiance and optical properties of water are permitted only in the vertical direction.
In this case, there is only one spatial variable,z, remains in RTE. So we can change
the partial derivative to an ordinary derivative.
Assuming our interest here is in
time-independent radiative transfer in horizontally homogeneous water bodies with a
constant index of refraction. We get RET for plane-parallel waters
dI
p = -cI + J
(2.4.10)
dz
where y
=
cos 0 To solve the RTE, we need to know the form of a very important
term, source path function, J. There are quite a few forms of the sources. In different
medias, we have different source functions.
The true emission source means the
creation of radiance by conversion of nonradiant energy into light. Such process in
water can be found, e.g. bioluminescence or an underwater artificial light source. We
use a source path function JS to model it:
Js~ (; A)= So (Y; A)ps(j
(2.4.11)
Here,So, gives the spectral radiant power emitted at 7 and A;and pS give the directional distribution of the emitted light.
Now we discuss the source function which is contributed from the scattering of
radiances of other directions. We know that in random media, such as atmosphere
and water, where the particles scatter light independently of one another, their contribution then add together in one direction. Consider the illumination from the Sun.
Assume that the particles are well separated, so that each is subjected to direct solar
radiation. Only a small portion of the direct radiation incident on the particle will be
scattered again and then gives rise to diffusion radiation. Thus, if the diffuse radiation arriving from all directions is negligible compared with the direct radiation from
the Sun, the medium is said to be optically thin and diffuse radiation is unimportant,
as the case (3) shown in figure 5. At this time, we call this kind of source appearing
in the RTE as single scattering or First-Orderscattering. Usually, the light from the
Sun is refracted at the air-water surface. So, the form of source path function is like:
Jss
b
47r
t( o - h;
n)p(z;
Pon
on -
A)e-c(z0-z)/ft0n
(2.4.12)
0z
1
2
3
Figure 2-5: (1)True emissiion; (2)Multiple scattering; (3)Single Scattering
where b and c are scattering and extinction coefficients respectively; po = cos 0 and
/on = cos 0on; p(z; o
-+
(; A) is the scattering phase function. The last exponential
term e-c(z'O-)/POn means that the direct light radiation is attenuated exponentially
with distance c(zo - z)/po.
When the particles in the medium are close enough that a largo portion of the
direct radiation incident on the particles will be scattered several times before totally
attenuated. Thus it will give rise to diffuse radiation. Then, the diffuse radiation
itself is an important additional source. In this case, this kind of radiation source is
called multiple scattering,and the medium is seen to be optically thick. The form of
multiple scattering source path function is written as
JMS
=
b JJ(z;
'; A)p(z-
A) d(( ))
(2.4.13)
Compared with atmosphere, the natural water can be thought as optically think
medium, since water molecule are close to each other. Therefore, in the later parts,
we only deal with multiple scattering. For convenience, we write multiple scattering
source path function as J instead of JMS. The first-order scattering source path
function Jss will not appear in the RTE. The refracted light radiation from the Sun
will not be used as a boundary condition when solving the RTE.
We have already know that in the elastic scattering process, there is no wavelength
shift and all the discuss above are based on the elastic scattering, so only direction
of radiance changes in the expression of scattering phase function p(z;
' -+
(; A).
However, inelastic scattering is also called transpectral scattering, which means
the wavelength A of scattered light is different from the wavelength A' of the incident
light. Thus, the wavelength transition should be considered in the forms of so called
spectral inelastic scattering phase function p(z; j'
source path function JI.
-+
(; A' -+ A) and corresponding
b (z; j'; A' -+
J
A)
-
(2.4.14)
I(z; '; A')p(z; j' -+ (; A' -+ A) dQ((())dA'
47r
where b'(z; j'; A' -+ A) is inelastic scattering coefficient.
The inelastic scattering exists in the natural water, but takes only small portion
of all the interactions. Therefore, in the later discussion, we will put more energy in
elastic scattering.
Standard form of the RTE is written under the assumption of optically thick and
plane-parallel medium and elastic scattering. We get
-cI(z; ; A) + b
y d(z;-;
dz
4r
f2
; A)p(z;'
(z;
-
A) dQ((()) +±;Js(z;
A)
(2.4.15)
Another form of standard RTE is to use the optical depth:d( = cdz and recall the
single-scattering albedo z = b/c, then gives
dI (z;
-
;A) A) +
d(
47r
A
I(z; '; A)p(z;'
-
A) dQ((c)) +
J
c((; A)
A)
(2.4.16)
The integrodifferential equation is the form of the RTE which is the basis of our
future work. The form of RTE yields an important observation: Any two water bodies
having the same single-scattering albedo, phase function, extinction coefficient and
source path function will have the same radiance distribution.
To get additional insight into radiative transfer theory, it is necessary to obtain a
integral form of the RTE.
First, we introduce Beer' Law based on the existing RTE. For the idealized case
of source-free(Js = 0), non-scattering(w
=
dI
y_
d(
=
0) medium, the RTE reduces to
-I
(2.4.17)
I(;
&)
Geoinettic
Optical
depth %
0
depth
1=0
IL
Figure 2-6: Understanding of integral form of the RTE
This equation is easily integrated and gives solution like
)=I(0;
1((
) exp
mu
(2.4.18)
Here, 1(0; ) is the known boundary condition at a certain depth which a assume
C = 0. Equation shows that if there are no addition sources and scattering radiance
adding to the original direction, the radiance has the value of product of the boundary
radiance and an exponential factor that depends on the optical depth ( and traveling
direction y = cos 0. This result is known as the Beer's Law.
Then we defined quantity 1 by
dl =- d= - cdz
(2.4.19)
p
p
where 1 is called optical path length in the direction of p. Note that dl is always
positive. Then it will be convenient to write 1 as
1=
1
(
-
Ia
I
\pio
fz
(2.4.20)
c(z')dz'
and Beer's Law becomes
I(1)
=
I(0)e-'
(2.4.21)
which shows clearly that radiance decrease along optical path.
///
Let us return to
the monochromatic RTE and repeat the integration process that led to Beer' law.
Multiplying both sides of Eq. (4.10) and rearranging gives
d(
I[exp(-)=
pt
-exp
p
-)J
(2.4.22)
p
Integrating both sides and express them in terms of the geometric depth z, we get
I(z;
=(0;
1) ()exp -
c(z")dz" +
J(z'; )exp
c(z")dz"1 dz'
(2.4.23)
The above equations is known as integral form of the RTE. As figure 6 shows,
suppose an radiance 1(0; () starts from the boundary with depth z = 0. The radiance
I(z;() is observed at the depth z. Equation (4.23) shows that I(z;
of two parts. The first part is just the boundary radiance 1(0;)
)
is composed
attenuated by the
factor exp(-i). The second part consists of the radiance generated at the each depth
z' along the optical path from 0 to ', and then attenuated by a factor exp[-(l - 1')].
The first part is inherent radiance related to boundary condition, and the second part
is apparent radiance related to the ambient contributions.
Radiative transfer theory is a linear theory of the interaction of light with matter
on a phenomenological level. Therefore, there are at least two limitations of radiant
transfer equation in the application of predicting radiance distribution.
One limitation is that the theory be concerned only with electromagnetic radiation of low irradiance and low photon energies. Since all our derivations of equations
are based on linearity of radiance transportation within the medium. In low energy
conditions, the response of interactions between light radiation and matter is proportional to the magnitude of electromagnetic field. This linear response is inherited
from linearity of Maxwell's equations. However, at very high irradiance, e.g. generated with lasers, the materials may exhibit responses that are proportional to the
square of the electric field magnitude. This phenomena, are the subject matter of
nonlinear optics. In this situation, the theory of radiative transfer does not work
properly any more.
Another limitation is that radiative transfer theory is a macroscopic level approximation. Its development is based on the measurement of variables and physical
phenomenon. For example, in theory, we suppose that the scattering particles were
far away so that the scattering lights of two particles are not tangled by each other.
If two particles touching each other, and the electromagnetic wave scattered by one
particle and then immediate encounters neighboring particle. At this time, even if we
know the optical properties of an isolated particle, we can not expect to obtain the
IOP's of a dense collection of such particles by means of radiative transfer equation.
Our aim is to find the radiance distribution below the air-water surface. The tools
we are using is the RTE. As an integrodifferential equation, the boundary condition
which here means the radiance distribution just below the air-water surface is a must
for the solution.
The mathematical model of surface of the ocean is very complicated. Since our
initial light is incident from the Sun or other sources, the calculation of radiance just
below the air-water surface need the model of capillary gravity waves. Besides, for
simplification, we suppose the wavelengths of free surface waves are much larger than
the wavelength of the light source such that the diffraction be properly neglected and
geometric optics can do well in our calculation. Therefore, in our discussions, the ray
tracing techniques will be applied in numerical calculation.
Water
Figure 2-7: Interaction principle
Like radiative transfer theory itself, the interaction principle is built based on
linearity of radiance and physical phenomena. We know when light travels through
the boundary of the two medium with different refractive index, according to the
Snell's Law, some of the light beam will be refracted through the boundary and the
others will be reflected in a same angle to the normal as the incident beam.
In our discussion, we are more interested in the distribution of radiance in the
water body, Thus, considering the part of light which refracted from the downwelling
light sources in the air and the part of light which is reflected by the upwelling light
sources is nore important, as figure 7 shows. Assuming the transmission coefficient of
light passing downward from air to water is written as t(a, w;
' -
coefficient of light passing upward from water to air is r(w, a;("
(); the reflection
-+
(), then, we
have the following equation about the total radiance I(w;() within the water in the
direction of
in terms of the radiance from air I(a; ') and radiance from the deep
Water
Water
Air to Water
Water to Air
Figure 2-8: Reflectance and transmittance of level surface
water I(w;(").
I(w;
j
1) I(a; ')t(a, w; j'
-
() dQ((') +
I(w; $') r(w, a; i';
-
() dQ((")
(2.4.24)
Now the important thing is to find the transmittance and reflectance between the
air and water.
For the simplest case, we think of the air-water surface as a flat horizontal plane,
the level surface.
Since the derivations of the reflectance and transmittance is based on the electromagnetic theory, here, we just present the transmittance and reflectance directly.
As figure 8 shows, there are two cases of incident: air-to-water and water-toair. But,when using same variables for the incident angle O, refraction angle Ot and
reflection angle Or. we get the same expressions of the reflectance and transmittance
in the two cases.
The reflectance is calculated from the equation:
1 ~sin(6O
-
-2 1 tan(6 -
[ sin(Oi + Ot)
2
2)
2 tan(Oi + O6)
And the tmnsmittance is calculated from the equation:
t -
1
2 sin(0j) cos()
iO
2 _sin(0i ) cos(Gi ) + sin(0t) cos(0t)
12+
1 [2 sin(0t) cos(i)- 2
2
sin(Gi + Oe )
_
(2.4.26)
Chapter 3
Monte Carlo Radiative Transfer
Simulation
3.1
Introduction
Monte Carlo simulations in this context involve the tracing of the fates of millions of
virtual photons or photon packets according to statistical probabilities. The source of
the photons being simulated can be either the sun or an artificial light. Each simulated
photon path is randomly distinct from the others as determined by the probabilities of
absorption and scattering in the water and air and by the probabilities of transmission,
reflection, or absorption at air-liquid and liquid-solid interfaces. Example ray-tracing
diagrams of Monte Carlo simulations are shown in Figure 1 for the open ocean water
column and for a point-source integrating cavity absorption meter (PSICAM). In the
ocean case, photons enter at the sea surface, scatter within the water, reflect internally
at the sea surface or seafloor, and eventually either escape into the atmosphere or
are absorbed by the water or seafloor. In the PSICAM case, photons are initiated at
the center of the cavity, internally scattered by the water, internally reflected by the
cavity wall, and eventually absorbed by the water or cavity wall. The light intensity
at any region in these modeled systems is determined by counting the number of
photon paths that intersect that region.
The Monte Carlo approach provides the most general and most flexible technique
for numerically solving the equations of radiative transfer. Because Monte Carlo techniques are computationally expensive, they have historically been avoided whenever
alternative solution techniques are available. However, as computer speeds continue
to dramatically improve, the Monte Carlo approach is becoming more practical and
popular for all types of radiative transfer calculations. Furthermore, it is no longer imperative that Monte Carlo codes be complicated with tricks designed to save computer
memory and to avoid expensive commands (i.e., logical statements and trigonometric
functions). This makes Monte Carlo techniques accessible now to anyone with basic
programming skills. Monte Carlo techniques have been used by various researchers
in the ocean optics community over the last several decades , and introductions to
the use of Monte Carlo techniques in ocean optics are provided in Refs. 21 and 22.
References on the use of Monte Carlo methods for the study of light propagation in
the atmosphere and in biological tissue are also relevant. Most of the cited references
discuss very specific applications and fail to provide sufficient detail to reproduce
their Monte Carlo simulations; however, they at least provide insight into general
Monte Carlo methodology. In this report we compile much of the practical information provided in the references and add lessons learned from our own research. We
introduce some basic ocean optics and Monte Carlo concepts in Section 2, describe
how to move a photon from one interaction point to another in Section 3, and show
how to compute scattering angles in Sections 4. A description of how to simulate
photon sources and detectors is provided in Sections 5 and 6. The interactions of
photons with surfaces, such as the sea surface and seafloor are discussed in Section 7.
Backward Monte Carlo techniques are introduced in Section 8. A short summary of
the most frequently used Monte Carlo equations is provided in Section 9. In Section
10 we provide many examples of Monte Carlo codes relevant to Ocean Optics and
evaluate their performance.
3.2
Photon propagation in homogeneous medium
Because it is computationally wasteful to trace the full paths of photons that never
reach a spatial region of interest, it is frequently prudent to sample from a biased
cumulative distribution function. Biasing the distribution function makes it possible
to trace more photons that are likely to find their way to areas of interest and fewer
photons that are unlikely to, without changing the final computed result. This makes
it possible to converge more quickly to a result of specified accuracy. If the probability
function is p(x), but we prefer to trace photons according to the biased probability
density function pb(x) so that more photons reach regions of interest, we can use pb(x)
as long as we multiply the photon weight by the ratio of the correct distribution to
the biased one
w
w(
p(x)
)
(3.2.1)
We use a similar idea to treat the absorption of photons within the water column
and at surfaces.
Rather than computing the probability of absorption and then
terminating the photon path if the photon is absorbed, we trace a photon to an
event (scattering or reflecting) and then reduce its weight based by the statistical
probability that absorption would have taken place. We then continue to trace the
photon path as if it had not been absorbed. An alternative way to think of this is that
we initially launch a packet of many photons traveling in the same direction and we
reduce the number of photons in the packet at each event by the number of photons
that would have been absorbed by the event. Specifically, we multiply the photon
weight by the value of the single-scattering albedo wo each time it is scattered and by
the value of the surface albedo each time it is reflected by a surface. Only when the
weight of a photon is reduced to below a certain specified level do we stop tracing the
photon path.
From the definition of optical distance 1, the probability density function for the
attenuation of light with respect to optical distance traveled is
p(l) = e',l ;> 0
(3.2.2)
The cumulative distribution function is
e- dl' = 1 - e-l
P(l) =
(3.2.3)
To determine 1 for Monte Carlo simulations, let P(l) = q and get
1 = -ln(1 - q) = -Inq,0 < 1 < 1
(3.2.4)
In homogenous waters, the geometric pathlength s (in meters) can be computed
1
lnq
C
c
(3.2.5)
where c is the attenuation coefficient (m( - 1)). In nonhomogenous waters, the
relationship between s and 1 is given by
I=
c(s')ds'
(3.2.6)
and the value of s must be computed by starting at zero and increasing its value
until the total value of I computed with Eq. (3.5) equals that given by Eq. (3.3). In
a layered system, the integral in Eq. (3.5) becomes a summation.
Equation (3.3) gives the probability density function for the distance an unimpeded photon travels before being scattered or absorbed. However, if there is a highly
absorbing boundary present, it is computationally wasteful to track a high number
photons that collide with this boundary. We can instead sample from a biased density
function that only considers photons that are scattered before reaching the boundary
()
exp(-
1 - exp
(3.2.7)
-lb
where lb is the optical distance to the boundary along the direction of travel. To
compensate for the biased probability function, we multiply the weight of the photon
packet by [1 - exp -b],
which is the fraction of the photons that do not reach the
boundary. The cumulative distribution function is
P(l) =
I
I
p(l')dl =
C-l-elb-
e 1± b
1
C 2--
e'b
e'b +
(3.2.8)
Note that P(lb) = 1. Solving P(l) = q, we obtain
1 = lb - ln[exp lb + (1 - exp lb)q]
The direction cosines pt,
(3.2.9)
py, and yi are the x, y, andzcomponents of the unit
vector that points in the direction of photon travel. By definition, they satisfy
[L + p
+ I
1
=
(3.2.10)
Given the direction cosines and the photon path length s, we can advance the
photon from its previous position (xo, yo, zo) to its next location with
x = xo + p/s
(3.2.11)
Y =YO + pyS
(3.2.12)
z = zo + pzS
(3.2.13)
For a direction defined by polar angle 0 and azimuthal angle
0
0 < r 0 < q K 27r
#,
(3.2.14)
the corresponding direction cosines are
#
(3.2.15)
pv = sin 0sin#
(3.2.16)
px = sin 0 cos
pz =
cos 0
(3.2.17)
BecauseO <0 <q r, sin Ois always positive and can be determined from pz with
(3.2.18)
sin 0 = V1 -PIZ
Frequently we know fizz and
#,
from which we can find
To compute 0 and
#
(3.2.19)
#
=
l-p
Cos
Ay=
l-
sin#
yx
(3.2.20)
from the direction cosines we can use
(3.2.21)
0 = cos y
=
A
cos
, yf
-p
z1
=27r - cos
y
1, y
=
(3.2.22)
;> 0
1, yV <
0
(3.2.23)
1- p
If p
3.3
= 1, then yx = yy = 0, making
#
set randomly.
Scattering process
The scattering phase function
#(P, @) is
the probability density function that gives
the probability that a photon, when scattered, will scatter at polar angle T and
azimuthal angle D away from the incident direction [21]. From Eq. (2.1), the integral
of
#(T,
D) over all directions is unity,
7r
27T
#(x,
D) sin Pdqd = 1
(3.3.1)
The sin term in this equation arises from the definition of the spherical coordinate
system. For seawater and for air, the scattered azimuthal angle <Dwith respect to the
incident direction is uniformly distributed over [0, 27r]. Therefore p(4 ') and p(<b) are
independent of one another and
P(P,
<b)
= p(WF)p(<b)
(3.3.2)
In order to satisfy Eq. (2.1)
(3.3.3)
027r
the probability density function for <1must be
1
p(<b)
(3.3.4)
27r
From Eq. (2.2)
P(<b) =j
10
* 1
27r
db = <d/27r
(3.3.5)
and from Eq. (2.3)
<b = 27rq
(3. 3.6)
Alternatively, one can use two random numbers q1andq2 to determine <b with
W1 = 1 - 2qi, W 2 = 1 - 2q2
d=
±W2
W22
cosIb = W1 ld, sin
Because for seawater and air
# does
= W 2/d
(3. '3.7)
(3.3.8)
(3.3.9)
not depend on <b, it can be simplified to
# (T)
27r
sin fdT = 1
(3.3.10)
From Eq. (2.1) and above equation,
(3.3.11)
p(I) = 27r#(4f) sin 'If
And then
P('I) = 27r
J
3(') sin Td
=q
(3.3.12)
Alternatively, if we express the scattering phase function in terms of the cosine of the
scattering angle,
ps = cos IF,dpt, = - sin TJd'J
(3.3.13)
then, Eq.(4.5)-(4.7) become
#(pu)dys
(3.3.14)
p(p-s) = 27#(p1)
(3.3.15)
-27rf
Ps)=
p(,p)dyu
=
2c
$( pG)dy = q
(3.3.16)
For a given phase function,one must solve for T or pu in terms of q.
For all specific cases we will consider the azimuthal scattering angle is determined
with
<D =
For isotropic scattering,
#
27q
(3.3.17)
is constant by definition. In order to satisfy above equa-
tions,
= --
(3.3.18)
From Eqs. (4.6) and (4.9)
p(qI)
= 1/2 sin T
(3.3.19)
1/2
(3.3.20)
sin xdT = -(1 - cos'I) = q
(3.3.21)
p(p,) =
and
Thus the polar scattering angle is determined with
ps = 1 - q
(3.3.22)
Rayleigh, Raman, and Pure-water scattering phase function are of the form
#
where
f
cX1 + f p 2
(3.3.23)
is a scalar constant. Specifically, scattering for pure water obeys
/3 oc
1 + 0.835p/2
(3.3.24)
and Raman scattering at 490 nm follows
3 cx 1 + 0.55pt2
(3.3.25)
For a given value of f (e.g., 0.835 for pure water) we can normalize the probability
density function such that its integral from p = -1 to 1 is unity,
3 1 + f p2
2 3+f
(3.3.26)
Integrating, we obtain the cumulative distribution function,
Pp) =-
f3
-
6 +2f
-P -1
+_
f
2
6 +2f
(3.3.27)
To find a value for pin Monte Carlo simulations, we set P(p) = q and solve for q.
Therefore, the value of pis the real root of
P(-
3
P3_-
p+-q=0
6 +2f
6 +2f
(3.3.28)
2
Above equation can be solved either analytically (which has a complicated form)
or numerically.
The Henyey-Greenstein phase function is defined in terms of the asymmetry factor
g,
p(g
8
)=
-- 1 < g <1
47 (1 +
,
g2 -2g
1
p(g, p)=
where g
1 -g
(3.3.29)
cos qf,)/2
2
3 2 -I-1
< g < 1
,
47 (1 + g - 2gp,) /
2
(3.3.30)
0 reduces to isotropic scattering and g close to 1 describes very forward
scattering. Then we can get
(1 - g2 ) sin T3
1
2 (1 + g 2 - 2g cos xji,)3/2
1
1 -g 2
2 (1 + g2 - 2g cos p,) 3 /2
p(ys) =(3.3.32)
Then from Eq. (4.10), solving for p,
9
pS =I [1+g2 _( 1+g --2gR
2g
p=1
- 2,g = 0
2
g]g 0
(3.3.33)
(3.3.34)
Note that if we substitute 1 - q for q. We obtain an alternative form
===
[1 + gq2
_ (
2g
1 -92
)2], g/0(..5
1 - g + 2g
The value of p, s is plotted versus random number q for several different values of g
in Figure 3
10.8
0 .6
-
----------------
-- - - -
-- - -
0.4 ------- 0.4 ---
--0
-0.2
-0.2---
-
- - -
-- - --
- -I-
-------------------
g=--0.5
-
-----
-----
-=O
0.2--
----
-----
.4--.
i-
-
-
-0.
-
g=0.9I
--
-
-1 f
0
Feiesrm
0.2
0.4
nts o h satrnphefuc
0.6
q
0.8
ihv
1
enhaei
aua
Figure 3-1: Polar scattering angle versus random number q for isotropic scattering
(g = 0) and Henyey-Greenstein scattering with anisotropy factors of g = 0.5 and 0.9
Few measurements of the scattering phase function have been miade in natural
waters. The most well known are those made by Petzold ; other early measurements
are summarized by Jerlov . One way to use such measurements in Monte Carlo
codes is to tabulate the cumulative distribution function as a function of q and use
interpolation to compute T for each q.
Kirk computed the cumulative distribution function for the Petzold measurement
made in San Diego Harbor. This is included here in Figure 4. Similarly, the Petzold
average particle phase function can be integrated to obtain its CDF.
To avoid having to interpolate tables, Haltrin suggested using an empirical fits to
these functions. He provides mathematical expressions for many different measured
scattering phase functions, including those obtained by Petzold, Kopelevich, and
--- ------
---
0.5
- - - -- - - -- - - - -- - -0.5 -- - - - if-1
0.6
0.4
0.2
0
0.8
1
q
Figure 3-2: Cosine of polar scattering angle versus random number q for Petzolds San
Diego Harbor scattering phase function.
others.
Given an initial direction defined by 6 and
#
and the scattering angles T and <D
with respect to the inital direction, we need to determine the new direction cosines,
',Iy and p'.
If we let a represent the unit vector in the direction of the initial
photon direction,
a
=
(3.3.36)
[p, py, yz]
then the new direction unit vector is given by
a = sin T cos <Daj x a + sin T sin Da I + cos Ta
(3.3.37)
Where a 1 is an unit vector perpendicular to A. Because a1 is not unique, there
is not a unique formula for updating the direction cosines. If we choose a 1 to lie in
the x-y plane, then
ap= [-y,yo 0]/1 - p
a1 x i:=
+i[px yz,Iy
pz, -(1
-
p2t)]/1 -p
(3.3.38)
(3.3.39)
If the phton is very close to the z-axis (e.g. |jz > 0.99999) then it is preferable
to update the direction cosines with
/X = sin Qcosl '
(3.3.40)
= sin T sin @
p=
(3.3.41)
PZ cos<
(3.3.42)
Putting this all together, the new direction cosines can be detertined from the
initial photon direction, the cosine of the polar scattering angle p,(p, = cos T) and
the azimuthal scattering angle <b with
pxptz/
1
Zy
-
1-j p
[Lx!
21
yI zI p-
cos
V1 -- p
F-
0
p sil 4)
, 2 <1
PS
(3.3.43)
-
p'V1f
-2Ul sin @
p =signpz
p'z
pU cos1
,p
l
(3.3.44)
p
In plane prarllel problems where Iz is the only direction cosine whtat we need to
keep track of, we can update pz simply with
/I
= pzp-s T-
1 -LZpVS
2cos<
(3.3.45)
As a consistency check, any transformation should satisfy
ps = [pX, pL, P.] -[p', py, 41
(3.3.46)
and
(pz)
+ (pY)2 + (p's)2
=1
(3.3.47)
3.4
Radiance and irradiance detecting
Generally we want to measure either the light that is absorbed by the water or the
light field that remains. The former is easier to implement in Monte Carlo codes;
each time a photon is absorbed we note its location and increase the counter for the
location by the absorbed photon's weight. To measure the light field, on the other
hand, we need to use logical statements to determine if each photon path crosses the
area of the sensor in an appropriate direction.
In three-dimension problems, we generally treat Monte Carlo photons as power [W.
The downwelling and upwelling irradiance measurements are computed with
1
0
(3.4.1)
E,(z) o A Eiwi, (PA) < 0
(3.4.2)
Ed(Z)
oc
AN
EiAN
i (pz)i
1
AN
Where A is the area of the detector and N is the number of photons traced. For
radiance detectors we collect the photon weights that reach the detector within the
appropriate solid angle and divide the result by the sensor area and solid angle Q.
By definition, dQ = sin OdO.
For a sensor with a conical field of view (FOV) with
half angle Of, the solid angle of its FOV is
sin OdO = 27r(1 - cos Of)
Q = 27rj
(3.4.3)
For exanple, the upwelling radiance is
La(z)
=
2,(
1
-
EwIi) NA' (pz)i
where pf is the cosine of the half-angle FOV (pf
pf
=
(3.4.4)
cos f). The radiance and
irradiance at non-vertical orientations are computed in the same manner except that
the logical check for admitting photons changes to a comparison between p2t and the
FOV about the normal to the detector.
3.5
Surface interactions
When photons hit an air-water interface, a fraction of them will be reflected and the
rest will be transmitted. We therefore need to compute the fraction that is reflected
and the directions of the reflected and transmitted packets. The angle of incidence
with respect to the normal to the surface h is
= cCos- 1 |pz - h|
(3.5.1)
The reflected angle Or (with respect to the normal to the surface) is the same as
the incident angle,
Or = Qi
(3.5.2)
And the transmitted angle O6is
nt
I
6, = sin~-(sin Ot)
ni
(3.5.3)
where ni and nt are the indices of refraction for the incident side (water) and for
the transmitted side (air) of the interface.
For the special case of a horizontal surface,
0= cos- 1 /z
= sin-
(3.5.4)
1 - p')
(
ni
(3.5.5)
The fractional reflectance for unpolarized light is given by
p(63, 6t)
=
-{
20 sn711(0,_+
p(Oi)
)2
)22 2}, 6i /
+ tan(O+Ot)
(
= (il__ni2
n 1 + n2
65
2
,6
= 0
(3.5.6)
(3.5.7)
If we wish to model polarization effects, we can use
n' cosO - n cos Ot 2
ni cosO +nt cosO(5
0
t
an
1 cos O,-ni cos
t)2
o
p"(6i, 6t) = (,,csO
ut cos 6%+ ni cos Ot
(3.5.9)
(3.5.10)
P±IP)
2
p(
(39
To attract both photons that are reflected and those that are transmitted, we
must determine each time a photon reaches that surface whether it is reflected or
transmitted. We can do this simply by drawing a random number q and letting the
p(6j,O) and transmit if and only if q > p(Ot,
photon reflect if and only if q
).
If, on the other hand, we are not interested in the photons that are reflected off the
water, we can treat all photons as if they are transmitted and multiply the photon
weight by [1
-
p(O6,
Ot)].
In either case, the total optical path length traveled by the
photon from one scattering point to the next should equal the value of 1 computed
with Eq.
l
-ln(q),
1
0 <q
(3.5.11)
b. On the water side
The angle of incidence with respect to the normal to the surface t is
63 = cos~-
|pI
.
hi
(3.5.12)
The reflected angle 6 , (with respect to the normal to the surface) is the same as
the incidence angle,
Or = Oj
(3.5.13)
and the transmitted angle Ot is
Ot = sin-(
(3.5.14)
- sin j)
ni
where ni and nt are the indices of refraction fro the incident side(water) and for the
transmitted side (air) of the interface. For the special case of a horizontal seasurface,
O6= cos- 1 p,
(3.5.15)
2
(3.5.16)
Ot = sin-(
1p4-)
-
ni
Because ni > n (i.e. traveling from water to air), there is a critical incident angle
O, above which there is 100% reflection,
(3.5.17)
n-i
The fraction reflected for unpolarized light is given by
1 sin( -Ot 2]2
2
2 sin(Oi±+O)
p(ni, nt2)
=
t an(i - Ot)2]2
,A6 < Oc, OB/ 0
tan(Oj + t)
(3.5.18)
(
(3.5.19)
- n2 )2,
ni +n2
, = 0
p(OA,1 ) = 1,Oi > Oc
(3.5.20)
Reflectance for polarized light is provided before. If the surface is horizontal, we
can check if the photon is beyond the critical angle by comparing pz to pc = cos(Oc)
pc =
I
(3.5.21)
If we wish to trace both photons that are reflected and those that are transmitted,
then we must determine each time a photon reaches the surface whether or not it is
internally reflected. If it arrives at an angle greater than the critical angle then it
reflects. If not, we can simply draw a random number q and let the photon internally
reflect if and only if q
p(Oj, O). If, on the other hand, we are not interested in the
photons that transmitted, we can treat all photons as if they are internally reflected
and multiply the photon weight by p(Oj, Ot). In either case, the total optical path
length traveled by the photon from one scattering point to the next should equal the
value of 1computed with Eq.
1= -ln(q),O < q
3.6
1
(3.5.22)
Backward Monte Carlo simulations
When modeling three-dimension ocean-atmosphere problems, it is often necessary to
use Backward Monte Carlo simulations. The Backward Monte Carlo (BMC) approach
is most useful for an extended source (e.g. sky radiance on to the sea surface) and a
point (or small) detector, which is what commonly exists in oceanography. Only BMC
lets us simulate a point-sized detector. Backward Monte Carlo simulation generally
refers to ray tracing of photons in the reverse direction; that is, to trace photons from
the detector to the source rather than from the source to the detector. Backward MC
is useful because it is wasteful to follow simulated photons in the forward direction
when very few what are incident on the sea surface are actually collected by the
simulated detector. Backward techniques allow us to only follow photons that are
pertinent to the final outcome of the simulation. In addition, in the forward direction
it is impossible to know how large an area of the sea surface should receive incident
photons because that area depends on the local factors of sky conditions, water IOP's,
and the location and orientation of the simulated instrument.
Here we will only discuss backward Monte Carlo simulations in teh context of
problems consisting of a body of water that is illuminated at the sea surface. To
implement a BMC simulation, we initiate photons at eh position of the detector. The
photon's initial direction is determined by sampling from a cumulative distribution
function (CDF) designed to reflect the radiance response of the detector. Photons
that reach the air-side of the air-water interface are weighted by the probability that
a photon would be incident on the seasurface in the exact opposite direction for the
given illumination conditions.
For example, consider an upward-facing irradiance detector that measures Ed.
Because an irradiance detector has cosine response to the radiance, i.e.
1
Ed
=
pzL(j)d p
(3.6.1)
the probability density function for emitting photons from an irradiance sensor in
a BMC simulation is proportional to pz and the CDF is proportional to p'
1z
P(pz)
P(ps) = 27r
1s
(3.6.2)
7r
p(1p)d p = 2
p = = -p
=q
(3.6.3)
We therefore generate initial photon directions for a BMC simulation with
pz = -q
<b= 2,7rq
(3.6.4)
(3.6.5)
Shown in following Table are example functions for generating the initial directions
for a variety of ideal sensors. As always in this report, yz = 1 is taken to be the
downwelling direction.
For non-ideal sensors, a CDF can be constructed using laboratory measurements.
forward-problem
detector type
backward MC source
function
Ed
:~
EOd
/1
Eon
P =q
Eo
p: =1-2q
q
_ 1q
cosine response
sensor
=
__
Figure 3-3: Generating Backward Monte Carlo functions for various quantities
3.7
Polarized Light Radiative Transfer Equation
In the former part, we have mentioned that the Stokes Vector contains the full information of the light radiation. Therefore, in order to find the full information of
the distribution of light in the ocean water, it is necessary to develop the radiative
transfer theory of Stokes vector. The equation governs this is called Vector Radiative
Transfer Equation (VRTE).
Fortunately, we don't need to introduce more information about the VRTE. The
form of VRTE is just replace the radiance I in the RTE with the Stokes vector I.
Recall the equation (2.44), at this time, the scattering phase function becomes Phase
Matrix P. Thus, the Vector Radiative Transfer Equation for a plane-parallel system
is written as
dz
yd = -I+
dz
4=r _1, fo
Np
, #)I dp'd#'
(3.7.1)
where I is the Stokes vector; (p = cos 0, #) is the incident direction; (p' = cos 0', #')
is the multiple scattering source; P is the Phase Matrix; = is the single scattering
albedo.
I
|E|||2 +|E 1 |2
Q
|E||| 2 - |E1 |2
U
EIE
1
*+El*
El
(3.7.2)
i(EIIEi * -Eli * E 1 )
V
P(p', d'; p, #) = R(<b)M(O)R(F)
(3.7.3)
ai b1 b3 b5
M(8sca)
cl
a
2 b4 b6
c
a24be(3.7.4)
=
c3 c4 a3 b2
(
c5 c6 c2 a4
and R(J) is the rotation matrix corresponding to the rotation f Stokes vector by
a certain angle T.
R(T) =
0
0
0
cos(2 T)
sin(2T)
- sin(24f)
cos(2xF)
0 0
(3.7.5)
0
Proper treatment of the Stokes vector requires very careful consideration of reference frames. Here, we consider all the scattering under the frame called Meridian
plane which is defined by the z-axis and propagation direction of a beam. Both incident and scatted beam are treated based on meridian plane, as Figure shows. The
photon directions, before and after scattering, are represented as points A and B
respectively on the unit sphere. Each photon direction is uniquely described by two
angles 0 and
#.
The first angle,O, is the angle between the initial photon direction and
Z-axis. The angle
# is the
angle between the meridian plane and the X-Z plane. The
photon direction is also specified by a unit vector I1 whose elements are specified by
the direction cosines [ux, uy, Uz]. The directions before and after scattering are called
respectively I1 and
12.
The unit vector I1 and the Z-axis determine a plane COA.
The COA plane is the meridian plane. the incident field can be decomposed into two
orthogonal components Eli and Ei, that describe the vibration of the electrical field
parallel and perpendicular to the meridian plane. When a scattering event occurs a
new meridian plane is created by the new direction of propagation I2 and the Z-axis,
the Stokes vector must be transformed so that the polarization is properly represented
with respect to the new meridian plane.
The first step is to launch one of several million photons in a media. Depending
on the source geometry and distribution, the strategy of sampling is the same as
that of unpolarized light. The polarized light source includes some additional steps.
The reference frame of the field must be defined and the Stokes vector describing the
incident light source polarization must be declared. For illumination perpendicular
to the X-Y plane, the initial photon direction is defined by the following directioin
cosines [z, uI uz] = [0, 0, 1].
The polarization of the source field is relative to the meridain plane. The meridian
plane begins with
#=
0, and the reference frame is initial equal to the x-z plane. The
initial Stokes vector is relative to this meridian plane. Then the status of polarization
at launch is defined by the Stokes vector I = [IQUV] specified relative to the X-Z
plane. For example, if the user selects a Stokes vector I
The move step in polarized light Monte Carlo programs is executed as in an
unpolarized Monte Carlo program. The photon is moved to a propagation distance s
that is calculated based on the random number q generated in the interval [01].
In q
c
(3.7.6)
where c is the absorption coefficient. The mean free path between every scattering
and absorption event is 1/c.
The trajectory of the photon is characterized by the direction cosines [ux,
Uy, Uz].
The photon position is updated to a new position [x', y', z'] with the following equations:
'
x
Figure 3-4: Meridian planes geometry
X =
x
s,y'
Y+Uvsz = Z+Uzs
(3.7.7)
As in unpolarized Monte Carlo code the absorption of light by a dye or an absorbing material present in the scattering media, e.g. ocean, is tracked by giving a weight
after every absorption step according to the single scattering albedo 0o. The photon
propagates through an absorbing media; after n steps the weight of the photon is
equal to Wo. When the weight of the photon reaches a threshold level the photon
is terminated and is considered completely absorbed. If the photon exits the media
with a certain weight W the corresponding Stokes vector is multiplied by W to keep
into account the photon attenuation.
I
I -W
Q
U
V
-
ut
- O-
Q.
U-W
(3.7.8)
V -W
We determine the scattering angle through which it is scattered by sampling from
the phase function. If several types of scatterers are found, the fraction of the total
scattering that is due to each type is compared with a random number to determine
which phase function is used. Ocean the type of scattering is determined, we can then
sample the appropriate phase function to determine the scattering angle and hence
the new packet direction.
There are several ways to sample the phase function to obtain the scattering angles
E8s
and I(rotation angle from the initial meridian plane into the scattering plane).
We can select 8,,, from the phase function from unpolarized light and I randomly
between 0 and 27r. This introduces a bias into the sampling for which we must correct
by multiplying the scattered result by the phase function evaluated at the scattering
angle 1,,. To reduce the bias, another way for sampling the phase function is to
consider the radiance after a single scattering. For the most general polarization of
light scattered by the most general scatterer, the first element of Stokes vector If
after scattering and Stokes vector 1 o
if = a1 o + bi(Qo cos 24I + Uo sin 24) + b3(-Qo sin 2T + Uo cos 2T) + b5 V
(3.7.9)
The radiance shows a bivariate dependence on the angles 8,c and T. This result
is the phase function p(e8
, iT)
from which the scattering angles will be selected. The
bivariate nature of the phase function illustrates the effect of polarization on the scattering of polarized or partially polarized light. This is exactly why polarization must
be included to obtain the full, correct solution to the problem of radiative transfer. It
is important to note that the phase function for scattering initially unpolarized light
(Qo = Uo = Vo = 0) has the form (divided by Io for normalization)
p( 8,) = 27ra 1(8,.)
(3.7.10)
Then the light has some polarization, and the phase function has the form of above
equation.
It is necessary to understand how to sample correctly a bivariate probability density function p(X, Y) when X and Y are functionally independent variables. The
conditional probability p(XIY) can be expressed as
p(X|Y) = p(X, Y)/p(Y)
(3.7.11)
where
p(Y) =
f
p(X,Y)dX; a < X K b
(3.7.12)
Then unconditional probability for 8,c, is
p(E8 sca) = j
p(Osca, 'T)d4
= 27[a1(8sca) + b5(esca)Vo/I]
(3.7.13)
(3.7.14)
which was normalized by our dividing by 10. Rearranging some terms, we obtain
the conditional probability density function for 'Ifgiven e
8,a
s
sea)
a1 Io + (b1Qo + b3 Uo) cos 24' + (biUo - b3 Qo) sin 2' - b5 V
2-7r(a 1 + b5 Vo/Jo)
(3.7.15)
To sample above equation, we use rejection technique which was developed by Von
Neumann and it relatively straightforward. Consider p(x) defined on the interval [a, b]
and constrained such that 0 < p(x)
1. Select a random number x(a < x < b) and
find p(x) corresponding to it. Now choose another random number and compare it
with p(x), we reject it and select another random number and try again.
To apply the rejection method to conditional phase function, we must calculate
the maximum value of above equation over the interval (0, 27) and divide by it.
Differentiating it with respect to xF and setting the result to zero, we obtain tan(241) =
biUo - b3Qo/biQo + b3 Uo. Solving this equation for cos(24') and substituting back
into equation, we can get
a1 Io + [b((Ql + U 2) + b2(Q2 + U2)] 1/ 2 + b5 V(
030(3.7.16)
pmnax(IF|8sca) = aIo+[,
27r(a 1 + b5 Vo/Io)
Dividing the previous equation by Pmax ensures that on the T interval (0, 27r), the
maximum value of the density function does not exceed unity. We can then sample
this function using a point rejection technique. It is important to note that this
bivariate sampling technique can not be used in the case of a backward Monte Carlo
simulation because the initial state of the photon is not known. For spherical particles
the Mueller matrix assumes a block diagonal form with the following relations between
teh matrix elements: ai = a2 , a3 = a4 , bi = ci, b 2 = -c 2 . and the rest of the elements
are equal to zero. As a result, the conditional phase function becomes
p(Ie8sm) = Io + i(Qo cos 21 + Uo sin 2T)
a1
(3.7.17)
For Rayleigh scattering,
ai =
1 + cos 2 esca
2
,,bbi =
cos 2 esca _ I
2
(3.7.18)
Now to use rejection method for bivariate distribution, p(|e 8 ,m) can be writen
as p(8,,
it) and three random numbers are generated. ar, P,, !,.
The angle a, is
uniformly distributed between 0 and 7r and /, is uniformly distributed between 0 and
27r. If P, <; p(ar,,
) then both ar,,
are accepted as the new angles.
After the new scattering angles (,2,
xI) are determined, the scattering step will
include three operations. These operations are graphically described in the following
figures.
The E field is originally defined with respect to a meridian plane COA. The field
can be decomposed into its parallel and perpendicular components Ell and E1 . Once
the scattering angle E
8 , and azimuthal angle IFhave been generated with the rejection method, the Stokes vector is manipulated three times, the three manipulation
will be described in the next three paragraphs.
a. Rotation of the reference frame into the scattering plane
First the Stokes vector is rotated so that its reference plane is BOA. This is the
scattering plane. This rotation is necessary because the scattering matrix, that defines
the in-elastic interaction of a photon with a sphere, is specified with respect to frame
of reference of the scattering plane. The new Stokes vector I1 defined relative to the
X7
Figure 3-5: Initially the electrical field E is defined with respect to the meridian plane
COA
Figure 3-6: The electrical field is rotated so that Ell is parallel to BOA
scattering plane (BOA) is found by multiplying by a rotational matrix R(I).
This action corresponds to a counterclockwise rotation by an angle XFabout the
direction of propagation.
b. Scattering of the photon at an angle 8ca in the scattering plane
the Stokes vector at this point is referred to the scattering plane ACB. We can
then use the scattering matrix, or Mueller matrix M(E8 c), that is also related to this
plane. The scattering matrix M(8,c,) determines the polarization properties of the
scattering sphere. The status of polarization of the field after a scattering event at an
angle 8 a, and the relative Stokes vector can be obtained by the multiplication of the
incident Stokes vector I to the scattering matrix M(ec"). The angle Ec, depends
on the phase function and is obtained as described above.
The direction cosines are updated to take into account the new direction. The
new direction cosines will be called (u). The new trajectory [ux,
, uZ] is calculated
based on the angle esc and T and on the current trajectory [u, uY, uZ].
x/
Figure 3-7: After a scattering event the Stokes vector is defined respect to the plane
BOA
if |Ipz ~ 1
ax =
sin Osacos 'I
(3.7.19)
dy = sin Gca sin IF
(3.7.20)
A~z
esca I I
(3.7.21)
sin esc [uxuy cos T - UY sin T] + ux cos e8 ca
(3.7.22)
uz
=
COS
for all other cases:
a2
=
=
=z
1
Q1-p
1-
2 sin
E
8,ca[uuz cos T -
ux sin T] + uy cos Esca
1l- pl sin EOca cos T[uyuz cos4'IF-u sin
c. Return the reference frame to a new meridian plane
+u coseca
(3.7.23)
(3.7.24)
ZI
a
A
X/
Figure 3-8: The electric field is now rotated so that Ell is in the meridian plane COB
The Stokes vector is multiplied by the rotational matrix R(-7Y) so that it is referenced to the meridian plane COB. The angle -y was calculated by Hovenier.
cos 1y
-Uz +
Ud
=
tV(1
-
cos
2
cos Esca
8,a)(1
-
(3.7.25)
dz2)
where the sign is taken when ir < q, < 27r and the minus sign is taken when
0 < T < r. In summary the Stokes vector (I)f after scattering is obtained from
Stokes vector before the scattering (I)o by
If = R(--y)M(E8
ca)R(P)Io
(3.7.26)
where the Mueller matrix with Rayleigh scattering form is
U2 +1
M(8)
=
2
I2-1
2
-1
2
,U + 1
2
0
0
0
0
(3.7.27)
where y = cos 0sca
The reflection Mueller matrix Raw and transmission Mueller matrix Taw for light
going from air to water have derived from first principles and are as follows:
0
a -o
Raw=
Taw
a -
a+ n7 0
0
0
'ye
0
0
0
0
(3.7.28)
YRe )
0
a'+r'
a'-
a' -
a' +
0
0
0
'yD
0
0
0
I'
y'
(3.7.29)
e
7yRe
where
1I tan(i - Ot)
2 tan(Oi + O1)
a
YRe
1I sin(Gi =
tan(i
(3.7.30)
t) 2
(3.7.31)
± O~
[
~ tan(Oi
-
2
-
Ot) sin(Oj
-
Ot)
+ 6t) sin(Oi + Ot)
(3.7.32)
and
,
1[
2sintcos
]2
2 sin(Oi + t) cos(Oi - 6
O)
,
0
1 2sinOtcos .
= -[
22
n
sin(Gi + Ot) 1]2
(3.7.33)
(3.7.34)
f
sin 2
YRe
4 sin 2 6t cos 2 0,
(0, + 0t) cOS(0i - 0t)
(3.7.35)
where 0, and 6, refer to the incident and transmitted angles and are related by
Snell's law; namely sin O6= n sin 0, where n is the refractive index of the water relative
to air. The element R11
=
a + q, which is the effective reflectivity.
In going from medium into air, as long as 0 < Ocrit where
c,it
is the critical angle
(crit = arcsin(1/n) where n is the refractive index of water, we can still use above
a, a', j, 7', 7, 'y'for Ra and Twa.
However, for the region where 0 > Orit which is the region where total internal
reflection takes place, 0, becomes complex, Twa becomes the null matrix, and the
following equations must be used to compute Rewa.
+ 77 a- 77 0
0
a+ 7 0
0
0
0
YRe
7Im
0
0
YIm
YRe
a'+9'
0
0
a' -r'
0
0
Raw
Taw =
0
(3.7.36)
(3.7.37)
-Yim
Y~mm
The
0
7Re
1 1/n 2 cos(04) - i sin 2 6, - 1/n 2 2
2 1/n 2 cos(06) - i Vsin 2 0, - 1/n2
1 cos(06) - i /sin 2 0,
2 cos(06) - i fsin 2 0
82
-
1/n 2 2
1/n2
(3.7.38)
(3.7.39)
7m
=
2
Im{[ 1/n 2 cos(j)
cos(O) - iV/sin -
1/n2cos(2)
--
1/n 2]
iv/sin2 0, _ 1/n2
cos(62) - i Vsin2 ]-
I/n2
2
cos(64) - i v/sin 0 -
1/n2
cos(6) - i V/sin 2O6
1/n2
(3.7.40)
and
YRe
=
2
2
Re{ [1/n 22 cos(,) - i /sin 2 iO- 1/n 2]
1/n cos(0,) - i Vsin 02 - 1/n
2
-
cos(62) - i V/sin 0i - i/rn2
(3.7.41)
where now 62 > Oit. the corresponding Brewster angle in going from water into
air is defined by 0%a = arctan(1/n).
84
Chapter 4
Program Realization and
Validations
4.1
Program introduction
MCARaTS is an scientific software package to simulate the three-dimensional radiative transfer (3DRT) in the coupled atmosphere-ocean system using the Monte Carlo
methods. The codes can be applied to simulations of natural light radiation above
and below the dynamic ocean surface. Several state-of-the-art schemes are incorporated to make the code flexible and fast. A major purpose to use the software is to
simulate accurate radiance and irradiance as virtual observation data and compared
with field measurement of radiative quantities for the Radyo project. Features of the
radiative transfer code can be summarized as follows:
What's good: Easy to use, fast, and parallelized.
Basic algorithm: Forward-propagating Monte Carlo radiative transfer algorithm
Radiative transfer solvers: Fully-3-D Monte Carlo method
Techniques:
Spline fit the ocean surface
The local estimation method
Variance reduction techniques Biased sampling of photon path length
Flexible truncation of forward peak of phase function
Numerical diffusion
Parallelization using MPI
Input:
Three-dimensionally inhomogeneous ocean and atmosphere IOPs
Dynamic ocean surface wave elevations
Output:
Upward/downward/direct irradiances at any point
Detector-size related radiance at any point
Plain averaged radiances and irradiance at arbitrary layer interfaces
Requireients Linux/UNIX-like operating system Fortran 77 compiler compatible
with GCC (GNU's compiler collection)
MPI is needed for parallelization.
4.2
Radiative transfer simulation
Used algorithms are described by Curtis D. Mobley. The model uses Monte Carlo
methods for simulating photon trajectories and samples fluxes, heating rates, radiances, and other radiometric quantities. The local estimation method is used for
radiance averaged over some specific area or over specific angular region. The maximum cross section method is used for acceleration of photon tracing in inhomogeneous
media. Other methods useful for variance reduction are also used and documented
in the scientific paper. The flowing figure is an algorithm flow chart.
4.2.1
Geometry and brief descriptions
The position of photon is defined in the 3D Cartesian coordinate system. The direction of photon transport is defined by zenith angle (theta) and azimuth angle (phi)
from the X-axis, as in the following figure, The incident and emergent directions for
pixel radiances can be specified by the user. The source zenith and azimuth angles
are defined as theta0 and phiO, respectively. Similarly, emergent zenith and azimuth
angles are defined as thetal and phil, respectively. The x-, y- and z-coordinate is
Simulation flow chart
Figure 4-1: Flow chart of 3D MCRT simulation
defined in
-xmax, O-ymax and zmin-zmax, respectively. The horizontal boundary
condition is cyclic. The vertical boundaries z=zmin and zmax correspond to bottom
and top, respectively.
The surface is modeled as dynamic, BRDF model. Scattering media is decided by
the input ocean and atmosphere IOPs.
z
Direction of photon transfer
y
Figure 4-2: Coordinate definition
4.2.2
Methods and techniques
Various methods and techniques are used in the code, for better performance. Some
of them are described in Mobley's book: Light in the water.
"
Spline fit the ocean
surface " The local estimation method " Variance reduction techniques
Biased sampling of photon path length
Flexible truncation of forward peak of phase function
Numerical diffusion " Parallelization using MPI
4.2.3
Truncation approximations
The method of truncation of forward peak of phase function is incorporated. The
truncation fraction increases with the order of scattering.
The user specifies the
number of truncation regimes , diffusivity thresholds, and maximum truncation factor
Direct Truncation Approximation
Me Scattering
&-Isotropic Approximation
Figure 4-3: Trunction approximation
4.3
4.3.1
Polarized MCRT: variables and subroutines
Variables
NBGX: Number of air-water boundary grid at the x-coordinate
NBGY: Number of air-water boundary grid at the y-coordinate
NREX: Number of receiver grid at the x-direction
NREY: Number of receiver grid at the y-direction
NTETA: Number of receiver polar angle grid
NPHI: Number of receiver azimuthal angle grid
NPHOTON: Number of photon used in the Monte Carlo simulation
NQUEUE: Maximum number of queues used to store split photons in one launch
ZSURF: Z value of sea level
ZMAX : Maximum of z value used in simulation
ZMIN: Minimum of z value used in simulation
XMIN: Minimum of x value used in simulation
XMAX: Maximum of x value used in simulation
YMIN: Minimum of y value used in simulation
YMAX: Maximum of y value used in simulation
4.3.2
Key Subroutines
BOUND()
The function of this subroutine is to input a two-dimensional air-water boundary data
from hard drive and calculate the normal unit vector at every point, then store them
into a NBGX x NBGY x 3 matrix NORMSURF.
INTERFACE()
This subroutine is called when a photon comes across the ocean surface boundaries.
The purpose of it is to calculate the directions of the reflected and refracted photons
based on Snell's law. The refraction and reflection of light must be categorized to
three types:
1) From air to water
2) From water to air with incidence angle smaller than critical angle
3) From water to air with incidence angle larger than critical angle and totally internal
reflection happens
Besides directions of reflected and refracted photons, the energy of those photons
are also calculated according to Snell transmission and reflection coefficients.
For polarization codes, the interface transmission and reflection involve two coordinate rotations and one matrix multiplication.
TRANSFERO
This is the key subroutine of programs. It controls the moving and scattering of
photons. In an atmosphere-ocean system, before each photon moves, it has to be
determined that if the photon will hit the air-sea boundaries based on it's original
direction and moving distance.
When taking into account the multiple reflection and refraction, the way to deal
with the air-sea boundaries is to discretize it into a series of continuous triangular
facets, as following figure shows:
2
Figure 4-4: An example of the rough water surface discretized to facets by Preisendorfer and Mobley (1986)
For each move of a photon, if it hits one facet, it is divided to two parts: refracted
and reflected photons. The reflected photon is stored into a queue and treated as a
new photon at a later time of one launch. The queue contains all the information of
photons, position, direction, scattered times and energy or weight.
Then a periodic conditions process will be called. If the biased sampling of moving
distance is applied, no periodic condition process is needed. In this case, when first
sampled distance is beyond the boundaries of the computation regions, it has to be
calculated the distance of a photon to its intersecting point with the boundaries.
However, no biased sampling is used here, the periodic condition must be used. The
part of photon traveling outside of computation regions will travel back from the
other side of regions. This process is repeated until it reaches the end point in the
region.
Then at the end of a moving process, the program will generate a random number
limited to [0, 1] to decide if the scattering happens or not. If the random number is
bigger than the single scattering albedo, we think the photon stops and dies. If the
random number is smaller than the single scattering albedo, a subroutine "scatter"
is called to calculate the direction of the scattered photon.
Before every scattering, we have to calculate the total energy distribution of the
region by adding the energy of currently passing photons into every receiver. This is
process is done by calling subroutine ENERGY(;
ENERGY()
This subroutine is to calculate the energy contribution of a single photon to the
receivers. We first preselect the region of interest and put receivers in it. When a
photon passes by, if it hits the receiver, we will count it and add its energy to current
ones.
CUBIC0
One important step in subroutine ENERGY( is to determine if the photon has passed
the receiver. In our Monte Carlo simulation, we consider the region of computation
is a cubic. Therefore, for every receiver, we treat it as a small cubic too.
The subroutine CUBIC() does the work of determine if a line segment has a
intersection with a small cubic. If so, the photon energy is counted and on the hand
is ignored.
SURFACE()
This subroutine determine if a photon hits the boundary. Because the air-sea boundaries is composed of complicated waves. There are two ways to deal with it.
One way is the most approximated one. Due to the relatively small elevations
of ocean surface in contrast to the scale of computation region, the boundary will
always be treated as flat to determine if it hits the boundary, e.g.,the intersection
point of photon traveling line segment with surface z = 0. The intersection point,
say, (Xo, yo, 0), then the surface elevation is q(xo, yo, 0) and the slopes are qx(xo, yo, 0)
and iy.(x
0 ,y,
0). In addition to not consider multiple reflection, the error for this
treatment is the position of intersection.
One improvement is to expand the surface to a 2 "d order expression. The procedure of this analysis starts by obtaining the linear solution of intersection points.
Assuming (xo, yo) is the linear solution which is the intersection point of plane z = Zo
and the photon path. The corresponding surface elevation is q(Xo, yo). The real intersection is (X0 + Ax, Yo + Ay) and its corresponding elevation is r(Xo + Ax, yo+ Ay).
1
q(Xo+Ax, yo+Ay)
=
Tj(Xo, yo)+AXr+AyIY+
1 [(AX)2+2AxAyqv+(Ay)
qyy]+...
(4.3.1)
Where 77, r7y, i22, r/2y, (vy are known.
To find Ax, Ay, two more equations must be introduced. The projections of slopes
of the line into x and y directions are m and n. Therefore,
I(-o
+ Ax, yo + Ay) - Zo
(4.3.2)
m
Ax_=
(Xo + Ax, yo + Ay) - Zo
Ay=
Ay
(4.3.3)
n
Now by solving equations above we can get Ax and Ay and further obtain the
exact solutions of intersection points (xo + Ax, Yo + Ay).
The first order solution are
A
Ay =
-
m -
- mr
- m/n y
(4.3.4)
- qo Zo
n m - r/2 - m/nr/,
(4.3.5)
The higher order solutions can be acquired iteratively. For example, the second
order iteration equation can be written as
\
-
2
2Z
0
-
/o
- Ax(0 m + m/nY + ?r/)]
+ 2m/nr/7 + m2/
2
(4.3.6)
(46
Most accurate way to deal with boundaries is to discretize surface to triangular
facets (3D) or line segment (2D) as described above.
TABLETETAO
This subroutine is to pre-calculate the scattering phase function and store them into
table. This table contains two corresponding variable: Random number q and cumulative distribution function.
INTEGRAL()
This subroutine is to calculate the cumulative distribution function from integration
of phase function. Three kinds of phase function are presented. One is Rayleigh scattering phase function. One is Henyey-Greenstein scattering phase function. Another
is Petzold scattering phase function.
SCATTER()
This subroutine is to input a random number which is limited to [0, 1]. Then look up
the table by calling the subroutine tableteta and obtain a corresponding polar angle.
MAIN PROGRAM()
The main program mainly consists of four parts. The first part is define the sky
radiance.
Three different definitions of sky radiance are applied in the program:
Black sky radiance, sky radiance defined by thickness of atmosphere, empirical sky
radiance model.
The second part is the radiative transfer part. This part determines when where
a photon finish transport and dies.
The third part is the MPI part. The calculated data will be communicated with
each other and finally transport to the
0 "h.
At this nodes, all information will be
integrated and processed.
The fourth part is the export part. The interested data is written to data files for
further post-processing.
4.4
Parallelization of program:Message Passing Interface (MPI)
4.5
Validation of the Monte Carlo Radiative Transfer Code
Before any study of ocean surface effects on the underwater light field distributions,
it is necessary to validate the Monte Carlo Radiative Transfer Code at. the first step.
As described in the previous parts, our code is able to simulate both polarized and
unpolarized radiative transfer problem in Atmosphere-Ocean system, especially in
the case of complicated gravity and capillary ocean waves. Usually, this includes two
basic processes: light interaction with surface and light diffusion at a deeper region
in both ocean and atmosphere. Therefore, it is important to separate the validation
into several categories in order to test the every feature of radiative transfer. The
following features will be validated one by one.
" Scattering properties in the ocean
" Refraction and reflection of light at the surface
" Apparent Optical properties of the ocean
" Polarization distribution in diffusion region
4.5.1
Scattering properties in the ocean
4.5.2
Radiance profile in the diffusion region
As pointed out by Kattawar', when scattering albedo w is small, the shape of the
radiance curve depends on the shape of the phase function in the diffusion region.
Therefore, in our simulation, we choose w as small as 0.09 and scattering phase
function as Henyey-Greenstein phase function with g equal to 0.9.
1
1 -g 2
19(4.5.1)
2 1+g2 2gcosE(
p(8)) = 1
In the simulation, parameters are chosen as:
i) Absorption coefficient of the ocean: a = 0.02(m-1)
ii) Scattering coefficient of the ocean : b = 0.002(m- 1)
iii) Measurement depth:Zo = -150m
iv) Number of photons: 5 x 106
Figure 1 shows the agreement of radiance profile with scattering phase function
with a small w.
10
102
radiance
lSimulated
of
10
Scattering phas function
10
10
~
000
000
0L
10%
10
10
00000
00000
-150
-100
-50
0
50
100
150
Polar angle [i
Figure 4-5: Radiance profile v.s. Phase function
We then compare the radiance distribution of Mont Carlo RT code with that of
classical invariant imbedding method. The
In the simulation, the conditions are:
1) Flat ocean surface
2) Black sky, with solar zenith 0, = 600 and solar azimuthal angle
3) Depth independent IOP's for light of A = 550um
4) Petzold phase function
#,,=
0'
5) Single scattering albedo wo = 0.8
6) Monte Carlo RT: number of photonM = 107 and 2562 horizontalgrid
102
.
:
.
.
.
0
Monte Carlo, Optical depth=1
a Monte Carlo, Optical depth=5
100
Invariant Imbedding, Optical depth=1
-
Invariant Imbedding, Optical depth=5
-
10
10-2
10
150
100
50
0
Polar angle [o]
50
100
150
Figure 4-6: Radiance distribution: Invariant Imbedding method vs Monte Carlo
method
To further check our code, we justify it with experiment data of radiance distribution of Fig. 5 from James T. Adams' paper 2 in deeper ocean. The conditions in
the simulation are described as:
1) Sky model:SEMI-EMPIRICAL SKY RADIANCE MODEL
This model was described by Harrison and Coombes(1998) 3 . It is also applied by
the software HYDROLIGHT. It is presented by the following equations:
N(0, #)
=
CNo(6, #) + (1 - C)Ne(0, #)
No(0, #) = 0.45 + 0.120* + 0.43 cos 0 + 0.72e-1.880
Nc(0, #)
=
Where N(0,
[1.63 + 53.7e -5.49 0 +2.04 cos 2 V cos 0*] [1
#)
-
(4.5.2)
(4.5.3)
e.19see] [1 - e 0 .53 secO*] (4.5.4)
is the sky radiance; 0* is the solar zenith angle; (0, #) is the sky
radiance direction;
Q is the scattering angle
between sky and sun directions; and C is
the prevailing opaque cloud cover, clear skies (C = 0.0) and overcast skies (C = 1.0).
In the simulation, for clear skies, we choose C = 0.1 and O*= 250.
2) Internal Optical Properties
i) Absorption coefficient of the ocean: a = 0.0472(m- 1)
ii) Scattering coefficient of the ocean : b = 0.1106(m- 1 )
iii) Phase function: Petzold function
3) Measurement depth: Zo = -50m
4) Number of photons:5 x 106
o
o Experiment
--- Simulation
10
0
-150
-100
50
-50
100
150
Polar angle
Figure 4-7: Irradiance relative to surface (a) T. Adams et al (b) Our simulation
Figure 2 shows the agreement of experiment data with our simulation results.
4.5.3
Radiance profile just below the surface
This is to validate the scattering feature of code just below the ocean surface. Likewise, we use the same experiment source from Fig. 4 of James T. Adams' paper. The
conditions are as following:
1) Same sky model with C = 0.1, but solar angle is 35.
2) Internal Optical Properties
i) Absorption coefficient of the ocean: a = 0.0277(m- 1)
ii) Scattering coefficient of the ocean : b = 0.0458(m- 1 )
iii) Phase function: Petzold function
3) Measurement depth: Zo = -25m
4) Number of photons:5 x 106
at"
I
U
-1
~
9;
10-
9****
~'.
-2.
10
-1
W
-600
60
120
180
Gobs
Polar angle
Figure 4-8: Irradiance relative to surface (a) T. Adams et al (b) Our simulation
Figure 3 shows the above comparison.
4.5.4
Refraction and reflection of light at the surface
The light refraction and reflection at the ocean surface are predicted under approximation of geometric optics. In this validation step, ray tracing schemes without and
with energy loss are both used in order to clearly demonstrate the effects of surfaces.
We will start from the simplest sinusoid wave form.
2D sinusoidal surface wave
We compare the refraction pattern of our code with that of Prof. Zaneveld 4 (2000).
The surface pattern is expressed by equationr/(x) = 1.2 sin(0.157rx). In the simulation,
the absorption and scattering coefficients are considered to be infinitesimal so that
absorption and scattering can be ignored.
rt~n4rK
N60tr
WOVAQ~e
4
Iradience reatie to surface
4
3.5
3
2.5
2
1.5
0.5
0
Figure 4-9: Irradiance relative to surface (a) Zaneveld et al (b) Our simulation
100
xy
Figure 4-10: Irradiance for 3D sinusoidal surface wave
3D sinusoidal surface wave
The 3D irradiance pattern is shown in figure 5. Like the 2D case, we choose simple
sinusoidal wave to check the focusing effects of ocean surface wave on light beams.
4.5.5
Polarization distribution in diffusion region
We compare vector radiance and the degree of polarization of our Monte Carlo polarized RT code with Multicomponent Approximation method. The simulation conditions are:
1) Atmosphere-ocean system
2) Atmosphere optical thickness r = 0.15 and refractive index na = 1
3) Ocean optical thickness tau, = 1 and refractive index n=1.338
4) Flat air-water interface
5) Detector put at the depth just below the interface
6) Henyey-Greenstein phase function with g = 0.75 for both layers
7) Mueller matrix used is Rayleight scattering
8) Solar zenith 0,
0 or60
0.
0.30.2
0.1
0
50
100
150
Figure 4-11: Vector radiance just below the surface for 0,
00
Here we compare the degree of polarization in the diffusion region. Similarly, we
applied the experimental data from T. Adams's paper. We choose the figure 9 which
has the following conditions:
1) Same sky model with C = 0.1, but solar angle is 25 .
2) Internal Optical Properties
i) Absorption coefficient of the ocean: a = 0.0472(m- 1 )
ii) Scattering coefficient of the ocean : b = 0.1106(m- 1 )
iii) Phase function: Petzold function
3) Initial polarization: Polarized light with degree equal 1.0
3) Measurement depth: Zo = -50m
4) Number of photons:5 x 106
102
0.
0.3
0.2
0.1
0
50
100
150
Figure 4-12: Degree of polarization just below the surface for 0,
Figure 4-13: Degree of polarization
=
103
=
0'
0*)just below the surface for 0, = 0*
0. 0.4-
0.3
0.20.1
50
0
100
0
150
Figure 4-14: Degree of polarization (# = 90')just below the surface for 0,
=
600
1
-- MCA
0.8
0.60.4
0.2
15
050
100
150
F0
Figure 4-45: Degree of polarization (#b
=
180 0)just below the surface for 6,
104
=
60~
0.60
0.50-
Im**
it
w
UN
I
0.40-
S
x
0
4&
0.30
x
N
N.
E
"
N
0.20 -
N
NC
0.10
Nh
a.
0.00
x
Xe
C
K
-1 0
-120
-60
6b
0
Iio
Oobs
0.6 -
0
C, 03 -
0.0.2 Ch
0
0
0
-160
-100
-60
0
Polar angle {*
60
100
160
Figure 4-16: Irradiance relative to surface (a) Zaneveld et al (b) Our simulation
105
4.6
Underwater Stokes vector compared with field
measurements
The field measurements of underwater polarization patterns were taken by Ken Voss in
Hawaii. We take their results as an reference to validate our polarization code. Instead
of using temporal or spatial averaged Stokes vectors, we choose the measurement at
one shot taken by one receiver. Due to lack of surface information, the comparison
can only be made qualitatively.
The figure below shows the comparison of Stokes vector elements I,
Q, U,
degree
of polarization P and e-vector X.
Polarized radiance
Normalized
Q
Normalid t!
Degmi of Foltriln
E-vector
31k
Figure 4-17: Direct comparisons of underwater polarization field under a dynamic
wavy ocean surface between (A) RaDyO field measurement (Voss 2010) and (B)
Monte Carlo RT prediction.
We can see that even without the detailed knowledge of surface condition, we can
still see that key features of polarized radiance, normalized
Q, normalized
U, degree
of polarization P and e-vector x shows the agreement between field measurements
and simulated data.
106
Chapter 5
Investigation of Ocean Surface
Wave Effects on Underwater Light
Field-Forward Problem
5.1
Generating air-sea boundary: ocean surface
waves
In order to study the effects of air-sea boundaries to underwater light field distribution and fluctuation, it is important to reconstruct the deterministic ocean surface
elevations based on empirical surface wave spectrum models.
Since ocean waves have been measured and analyzed for a long time based on
statistics of data, the frequency spectrum can be obtained. Therefore, the frequency
spectrum is mostly used to describe the state of an ocean wave field and is applied
to reconstruct the one realization of ocean wave elevations.
In 1958 based on the dimensional analysis Phillips suggested that when gravitational restoring forces dominate surface wave hydrodynamics the displacement
frequency spectral density for a sea in equilibrium with the wind should have the
functional dependence
107
SU = rag2/W
((5.1.1)
where a is a dimensionless constant. This formula is expected to be valid at frequencies well above the dominant frequency wo and well below frequencies influenced
by surface tension restoring forces.
Phillips' spectral model has been used and improved by oceanographers for the
past decades. In 1964, Pierson and Moskowitz assumed that if the wind blew steadily
for a long time over a large area, the waves would come into equilibrium with the
wind. This is the concept of a fully developed sea. A long time here is about tenthousand dominant wave periods, and a large area is about five-thousand dominant
wave-length.
To obtain a spectrum of a fully developed sea, they used measurements of waves
made by accelerometers on British weather ships in the North Atlantic. First, they
selected wave data for times when a wind had blown steadily for long times over large
areas of the North Atlantic. Then they calculated the wave spectra for various wind
speeds, and they found that the spectra were of the form.
S(w) = '9 exp[-O("0)4]
(5.1.2)
where w = 27rf, f is the wave frequency in Hertz, a = 8.1 x 10-3, b = 0.74, wo = g/U19.5
and U19 .5 is the wind speed at a height of 19.5m above the sea surface, the height of
the anemometers on the weather ships used by Pierson and Moskowitz. For most air
flow over the sea the atmospheric boundary layer has nearly neutral stability, and
U19.5 ~: 1.026U10
(5.1.3)
assuming a drug coefficient of 1.3 x 10- 3 . The dominant frequency of PM spectrum
is calculated by solving d = 0 and obtain
=
0.877g/U
108
19 .5
(5.1.4)
20-
6
1WmS
0-
0
0.10
0,05
0.20
0,5
0 30
025
Frqency aHz)
Figure 5-1: Wave spectra of a fully developed sea for different wind speeds according
to Moskowitz (1964)
The speed of waves at the peak is
c, - 9- 1.14U 19.5
WP
1.17U1o
(5.1.5)
the significant wave-height is calculated from the integral of S(w) to obtain:
<,2 >= j
H1/3 = 4 <
712 >1/2,
S(w)dw = 2.74 x 10- 3 (U19.5 ) 4
(5.1.6)
the significant wave-height calculated from the Pierson-
Moskowitz spectrum is
H1/3 = 0.21
(U19.5 )2
9
(U10 ) 2
~ 0.22
9
(5.1.7)
A improvement to the Pierson-Moskowitz spectral density evolved from analysis
by Hasselmann et al (1973) using data from Joint North Sea Wave Project- commonly
referred as the JONSWAP spectrum. In order to accommodate observed departures
109
to
Wied SpEed U
otws)
Figure 5-2: Significant wave-height and period at the peak of the spectrum of a fully
developed sea calculated from PM spectrum
from the basic Pierson-Moskowitz spectral density attributed to a sea that is not
in equilibrium they allowed Phillips' constant and the peak frequency to be fetch
dependent. Specifically, they define a = a(f*) and wo = wo(f*) where f* = lg/U
is the dimensionless fetch and added a spectral-shaping multiplier. the JONSWAP
spectral density replaces the PM spectrum with
S(w)
= 0,9
W5
exp[-5 (P)4]r
4 w
2
r = exp[-
]
(5.1.8)
(5.1.9)
Wave data collected during the JONSWAP experiment were used to determine
the values for parameters:
a = 0 .0 7 6 ( U120 )0.22
Fg
w = 22( g
)1/3
U1OF
110
(5.1.10)
(5.1.11)
(5.1.12)
7 = 3.3
o-= 0.07,
<;p
=0.09 W > WP
(5.1.13)
(5.1.14)
Frequency (HW)
Figure 5-3: Wave spectra of a developing sea for different fetches according to Hasselmann et al (1973)
Recently, BAttjes, Zitman, and Holthuijsen (1987) reexamined the JONSWAP
data base by comparing the JONSWAP spectral density with the Toba spectral density modified for a low-frequency cutoff and peak shaping,
S(W) =
exprogu*
W4
5(P)4]fr
4w
(5.1.15)
where u, is the wind-friction velocity. It is common in the literature to reference
to dimensionless constant o- as Toba's constant.
Donelan, Hamilton, and Hui implement Toba's spectral form by substituting w4 wo
for w in the JONSWAP spectral density and rewriting it as
S(w) = 7rag2 exp-5 (P)4]
4 w
WWho
(5.1.16)
For gravity waves, the frequency spectral density transforms into the wavenumber
spectral density for deep water via dispersion relation w = v/gk
S(k) Skk)
= 2S(w) dw
=(5.1.17)
k
dk
5.2
Generating linear wave elevations from wave
spectrums
A linear wave elevation realization can be generated from a given wave spectrum.
A wave is usually given as a frequency spectrum instead of wavenumber spectrum.
Therefore, we will use frequency spectrum of JONSWAP as an example.
Two-dimension case:
For a two-dimensional wave if a JONSWAP spectrum, we recall the linear dispersion relation for low frequency and deep water,
w2 =gk
(5.2.1)
and remember the fact that the integral of the frequency spectrum equals the
integral of the wavenumber spectrum
S(w)dw = S(k)dk
(5.2.2)
Then we have the relation to convert frequency spectrum to wavenumber spectrum
as
S(k) = g S(w)
2w
112
(5.2.3)
To generate the surface eleveation, we discretize the wave number space and the
amplitude of the nth wave component can be obtain as
A(kn) = V2S(kn)Ak, n = 1, ... , N
(5.2.4)
The corresponding phase angle, a(kn), is obtained from a uniform random distri-
bution in [0, 27] for linear waves. Then the surface wave elevation 77(x) is the discrete
Inverse Fourier transform of A(k,)eio(kn)
q(x) =
-OA(kn)eia(kn)cixkL Ak
(5.2.5)
Three-dimensional case:
For three-dimensional waves, we have to include spreading function for directional
wave field. The directional spectrum can be written as the frequency spectrum times
a spreading function. In a general case, the spreading function can be frequency
dependent. However, based on field observations and measurements, in most cases
the spreading function can be approximated as frequency-independerit and having a
form of cosine power. For example, if the wave energy spreads within an angle range
of [-8/2,8/2], the spreading function can have a form of:
D(O)
=
2
y
cos2(,o), 18
8
(5.2.6)
2
D(O) = 0, 101 > -)
2
(5.2.7)
We use a cos 20 angular distribution as an example. The spectruth in frequency
domain can be written as
S(w)
=
ag2
5
W
7r
r
2
2
,
5
<0 < exp[---( 2 )4]-y'- cos 0-2
'
2
4 w
(5.2.8)
We need to convert the spectrum from frequency-angle domain to wavenumber
domain.
from the relation between the frequency-angle and the two-dimensional
113
wavenumber,
(kx, kv)
=
9
(cos0, sin 0)
(5.2.9)
we have
dkxdky
=
9kx
OW_ Oky
O9ky
OW_
Cos 0
dwdO
0
--
2w
g
g
9
sin
0
dwd0 =
2w 3
dwd0
(5.2.10)
Cos 0]
Then, according to S(w, 0)dwd0 = S(kx, kv)dkxdky, we have
2
S(kX, kY) = gS(w, 0)
(5.2.11)
Then we discretize the two-dimensional wavenumber space, the amplitude of the
wave component with a wavenumber of (k2i, kyj) can be obtained as
A(kxi, kyj) =
2S(kxj, kg)AkxAk,
(5.2.12)
The corresponding phase is a(k , kyj) which is obtained from a uniform random
distribution in [0, 27r]. Then the surface elevation 7(x, y) is the 2D discrete inverse
Fourier transform of A(kxi, kvj)ei(xkxiYkY).
(x,
5.3
y)=
En
r 0 A(kn,
kyn)ei(knkym) ei(xkxn + ykym)ZAkxAky
(5.2.13)
Gravity-Capillary wave surface elevation generation
The most accepted capillary wave model in ocean optics society is Cox-Munk slope distribution model, which can be described by the probability density function P(q, 77y),
114
P~1,1)
P2(,)
71y2=
1
27ra
a2
2
-2
exp[- 21 ( 77
+ U-2 )]
or
(5.3.1)
where ?7 and qy are the slopes in the upwind and crosswind directions, respectively.
And U and o-2 are the associate variance given by
3.16 x 10- 3 U
(5.3.2)
a = 1.92 x 10-3U
(5.3.3)
=
where U is the wind speed in meters per second measured at a height of 12.5m
above sea level.
Unfortunately, the Cox-Munk model only gives the slope statistics which can not
be used to generate the continuous surface elevations. Hence, a alternative model
developed by Elfouhaily et al (1997, called ECKV) is accepted. Mobley in order to
prove the agreement of Cox-Munk statistics and ECKV wave elevation statistics. It
tested the mean square slope (mss) of sea surface
115
Underwater radiance and polarization affected
5.4
by ocean surface waves
5.4.1
Irradiance patterns affected by 2D surface waves
Here is the downwelling irradiance patterns under the 2D Airy waves and Stokes
waves. The steepness varies from KA = 0.19 to KA = 0.38. The simulation conditions are:
e
NX=1024, NZ=1024
" Absorption and scattering are ignored
" Normal incidence and black sky
" The x and z scale is normalized by the peak wavelength of surface waves.
Xmial=-5, Xmax=5, Zmin=-5, Zmax=5
aedwnos oole d regular surace wave
4
os
1
;~
j
q
v.
Q0
0
1
a
Figure 5-4: Downwelling irradiance patterns induced by Airy waves with KA=0.19
The following it the irradiance patterns under irregular waves. We compared the
irregular waves with narrow band and broadband. The simulation conditions and
parameters are:
116
10
1
2
3
4
5
a
Figure 5-5: Downwelling irradiance patterns induced by Stokes waves with KA=0.19
of regular suefae wav
kiadiarme pWiEJlB
1
2
3
4
5
5
7
a
Figure 5-6: Downwelling irradiance patterns induced by Airy waves with KA=0.38
117
Irradiance profile di regular sunae vmve
1
j
4
Figure 5-7: Downwelling irradiance patterns induced by Stokes waves with KA=0.38
" NX=1024, NZ=1024
" a = 0,016m-1, b = 0.016mn 1
* Petzold phase function
" JONSWAP spectrum. kp = 0.11m- 1 ,a
=
0.0081,
=
3.3
" Normal incidence and black sky
" The x and z scale is normalized by the peak wavelength of surface waves.
Xmin=-5, Xmax=5, Zmin=-5, Zmax=5
5.5
Radiance distribution induced by Airy and Stokes
waves
Here we compared the radiance distribution induced by Airy waves and Stokes waves
by putting receivers at different locations under the the waves. The setup can be
described by following pictures:
The simulation conditions and parameters are:
118
iradiWmos
proli of random knoar
waves
14
Iso'0
Figure 5-8: Downwelling irradiance patterns induced by Narrow band irregular waves
with KpA, = 0.2 in normal incidence case
Irradiance prile Ofrandom hlmeer
Vae
2X
10
x
2
4
3
4
Figure 5-9: Downwelling irradiance patterns induced by Broad band irregular waves
with KA, = 0.2 in normal incidence case
119
Irradiance profide of random linear wae
-0.36
-1.4
1
2
XI)
Figure 5-10: Downwelling irradiance patterns induced by Broad band irregular waves
with KA, = 0.2 with incidence angle of 200
Figure 5-11; Radiance distributions under a Airy vs (fully nonlinear) Stokes wave as
a function of position and depth
120
*
Black sky and normal incidence
" Coastal water: a = 0.179m-1, b = 0.219m-1
" Petzold phase function
" Wave parameters: KA = 0.4, wavelength A = 1m; 10 waves.
" Photon number: M = 10'; 2562 horizontal grid; 10 polar angle resolution
Here is the radiance distribution at optical depth Z = 1 for locations are crest,
trough and 900 phase.
2
10
-e-Airy wave, C=1
-Stokes wave,C=1
0
10
10-2
1 25
-20
-15
-10
5
0
-5
Viewing angle U1
10
15
20
25
Figure 5-12: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under crest
Here is the radiance distribution at optical depth Z = 5 for locations are crest,
trough and 90* phase.
We can see that the nonlinearity of ocean surface waves have strong effects at the
phase of 900. Therefore, we consider the point of the phase of 90' for different KAs.
121
102
--- Airy waveC=1
-- Stokes waveC=1
10
10
-25
-20
-15
-10
-5
0
Viewing angle [j
5
10
15
20
25
Figure 5-13: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under trough
102
wave,C=1
Stokes wave,"=1
+Airy
0
10
10-1
10-
-25
-20
-15
-10
-5
0
5
Viewing angle [I
10
15
20
25
Figure 5-14: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under 90' phase
122
E-
-Airy wave, 1
-Stokes wave,C=1
80.8
0.
I::n
0.4
0.2
5
"O
10
15
20
25
30
35
40
Spatial frequency k[m ]
Figure 5-15: Spatial spectrum of Downward Irradiance at Z = 1
102
--- Airy wave, C=5
Stokes wave. C=5
100
10-2
10-
-25
-20
-15
-10
0
-5
5
Vewing angle ()
10
15
20
25
Figure 5-16: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 5 under crest
123
10 2
--- Airy wave,'=5
-- Stokes wave.C=5
0
10
102
-25
-20
-15
-10
-5
0
5
Viewing angle ["
10
15
20
Figure 5-17: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 5 under trough
2
-- Airy wave,(=5
-*-Stokes wave,;'=5
10
10-2
-25
-20
-15
-10
-5
0
5
Viewing angle []
10
15
20
25
Figure 5-18: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 5 under 90' phase
124
S0.16 -E
-
-Airy wave,C=5
-Stokes wave,-=5
0.14
0.12
0.1
0.08
E
0.06,
0.04
0.020-
0
10
5
15
30
25
20
35
40
Spatial frequency k[M']
Figure 5-19: Spatial spectrum of Downward Irradiance at Z = 5
Airy wave.4=1
Stokes wave. q1
10"
10-2
1074
-20
-15
-10
-6
0
VIvewang angle
5
[
10
is
20
I
Figure 5-20: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
1 under 900 phase with KA = 0.1
Z
102
---
Airy wave.4=i
Stokes waveq,1
104
10-
-15
-10
-5
0
5
Vewmng angle { ]
10
15
20
Figure 5-21: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under 90' phase with KA = 0.25
125
102
-Airy
wave.g=n1
Stokes wave.C=
10,
10
-25
-20
-15
-10
-5
Vs'
0
ng
5
10
1
15
20
25
]nQ1,
Figure 5-22: Effect of Wave Nonlinearity on Underwater Radiance Distribution at
Z = 1 under 900 phase with KA = 0.4
5.6
Irradiance patterns induced by IHT FST and
Shear Flow FST
Here we studied the downwelling irradiance patterns under complicated free surface
turbulence(FST) at different depth. The free surface turbulence we considered are
isotropic homogeneous turbulence(IHT) and shear flow FST.
The simulations condition for downwelling irradiance induced by IHT FST are:
" Black sky and normal incidence
" Coastal water: a = 0.179m-1, b = 0.219m 1
" Petzold phase function
" Surface waves: Isotropic homogeneous turbulence (IHT), X=Y=6.28m; Froude
number= 0.8 and Froude number = 0.2
" Photon number: M
=
10'; 2562 horizontal grid;
With Froude number
=
0.8. The irradiance patterns are:
With Froude number
=
0.2. The irradiance patterns are:
The simulations condition for downwelling irradiance induced by shear flow FST
are:
* Black sky and normal incidence
126
2
Surface
x
V
Z=5
Figure 5-23: Downwelling irradiance patterns under isotropic homogeneous turbulence (IHT) with Froude number = 0.8
127
z
Surface
V
Z=5
Figure 5-24: Downwelling irradiance patterns under isotropic homogeneous turbulence (IHT) with Froude number = 0.2
128
"
Coastal water: a = 0.179m- 1 , b = 0.219m-1
" Petzold phase function
" Surface waves: Shear flow free surface turbulence, X=Y=6.28m; Froude number=
2.25 and Froude number = 1
" Photon number: M
With Froude number
10'; 1962 horizontal grid;
=
=
2.25. The irradiance patterns are:
Figure 5-25: Downwelling irradiance patterns under shear flow free surface turbulence
with Froude number = 2.25
With Froude number = 1.. The irradiance patterns are:
We also investigated the downwelling irradiance patterned induced by ship wake.
* Black sky and normal incidence
129
Figure 5-26: Downwelling irradiance patterns under shear flow free surface turbulence
with Froude number = 1
130
*
Coastal water: a = 0.179m- 1 , b = 0.219m- 1
" Petzold phase function
" Ship model: Cylinder with radius of 0.98m, X = 12.56m, Y = 6.28m
" Weber number = 100, Re = 1000, Density ratio = 1000, Viscosity ratio 100,
Froude number= 0.9 and Froude number = 0.6
" Photon number: M
With Froude number
=
107; 1282 horizontal grid;
0.9. The irradiance patterns are:
Surface
x
Y
Figure 5-27: Downwelling irradiance patterns under ship wake with Froude number
=0.9
With Froude number = 1.. The irradiance patterns are:
131
Surface
x
Figure 5-28: Downwelling irradiance patterns under ship wake with Froude number
= 0.6
132
5.7
Effects of roughness of ocean surface on underwater polarization
The roughness of ocean surface is represented mainly by the standard deviation of
surface slopes in x and y directions as ox and o- respectively. At the same time,
we assume both of the surface slopes (r/x, ry) are of Gaussian distribution with mean
value of zero. A frequently used empirical formula was given by Cox and Munk,13
who claimed that for a wind driven ocean surface the Gaussian distributed slopes of
surface have following relation with wind speed U1o
or = 0.000 + 3.16 x 10- 3 U10 t 0.004
(5.7.1)
og = 0.003 + 1.92 x 10-3U 1 0 ± 0.002
In the simulation, we choose atmosphere optical depth as
Tatm
= 1 to mimic the
a cloudy sky condition and ocean optical depth as Toc = 10 to represent the deep
water assumption. The aerosol is assumed to be atmosphere molecule only therefore
we assume the scattering phase function take the Rayleigh phase function.
p(E8
ca) =
3
167r
(1 + cos 2 19ca)
(5.7.2)
In the ocean, we use the Henyey-Greenstein phase function
P(sca) =
47r(1 +g
2
Where g is the anisotropy factor and g
g 2 cos ,ca) 3/ 2
- 2g
8
=
(5.7.3)
0.924 for ocean water. We focus on
the air-sea boundary effects on polarizations patterns therefore only one wavelength
(A = 514nm) is considered. For such wavelength in clear ocean water, we choose its
absorption coefficient a0c, and scattering coefficient boc, measured by Petzold in 1972
which is a0 c, = 0.114m/s, boa = 0.037m/s, co,
= 0.151m/s. The single scattering
albedo of clear ocean won = 0.247. We assume conservative scattering for atmosphere
so that Watm = 0.99.
Wavy surfaces causes the major fluctuation of underwater polarization pattern at
133
the upper ocean. Marine animals could possibly integrate Sun's positions over time
to navigate. In our simulation,in stead of integration over time, we use Stokes vector
parameters integrated over space to represent the polarization patterns according
to the ergodic property of ocean surface waves. The four spatially averaged Stokes
parameters are expressed as I(Q, r), Q(Q, -r), U(, r) and /(Q,
T).
Based on above definition, we defined the plane-parallel degree of polarization P
as
( )Q2(Q, r) + U 2 (Q, T) +
P(0,r)
V 2 (Q, 7()
-(5.7.4)
I(Q, T)
Another important quantity e-vector or orientation of polarization is defined as
1
X=
2
tan-
U
(5.7.5)
Q
The ellipticity of polarization is also important to understand the pattern of underwater polarization just below the ocean surface. The ellipticity is defined as
tan #
Both neasurements'
2
=
tan
1
-
2
1
tan-I
V_
_
g_2 + U2
(5.7.6)
and theories3 1 5 indicate that the downwelling underwater
light includes two parts, one within Snell's window 0 < 480 and the other outside
the Snell's window. The two parts react differently with environmental parameters.
The polarization pattern within Snell's window is mainly determined by the sky polarization pattern just above the surface and ocean optical properties for a flat ocean
surface. This was well investigated theoretically by Horvath and Sabbah. Light outside the Snell's window comes from direct scattering of that inside Snell's window and
that reflected back to ocean from upwelling light by ocean surface. The polarization
here is mainly governed by the phase matrix and the single scattering albedo woca.
To qualitatively demonstrate above facts, in figure 2, we show a picture of sky polarization just above the surface and underwater polarization at a very shallow optical
depth r = -0.1
in the cases of small solar zenith 0,,,
134
= 100 and large solar zenith
= 700 and in the cases of flat surface U10 = Om/s and wind toughen surface
U10 = 8m/s. For both cases, the principle plane lies on the upwind direction so that
the solar azimuthal angles is
4=
180'.
(c)
0sun = 704
0sun = 104
P
P
1800
k
180*
1800
(a)
(b)A
P
N
1800
1800
tan3
X
P
tan3
(e(
tan-
P
0
-0.1
0.1
-90*
900
Figure 5-29: Downwelling sky polarizations just above the ocean surface and underwater polarizations at the optical depth of 0.1 below the surface predicted by
Monte Carlo simulation under different solar zeniths and ocean surface conditions.
(a) and (b) are degree polarization P and e-vector x respectively when solar zenith is
0,,, = 100; similarly, (c) and (d) are degree polarization P and e-vector x respectively
in the case that solar zenith is 0,,, = 70'; (e), (f) and (g) are degree of polarization,
e-vector and ellipticity respectively at the solar zenith equal to 100 under the flat
surface assumption; (h), (i) and (j) take the same sequence but with solar zenith of
70' for flat ocean surface; similarly, (k), (1) and (m) are polarization with Osun = 10'
under roughened ocean surface with wind speed U10 = 8m/s; (n), (o) and (p) are
same as (k), (1) and (m) except in the case of 6,u = 70'
The three important quantities, degree of polarization P, e-vector x and ellipticity
tan3 are presented in figure 2. We know the downwelling sky light, after scattering
by atmosphere molecules and aerosols, is linearly polarized. It has the largest degree
of polarization 90' to solar zenith based on Rayleigh model. For 0n
one maximum degree of polarization appears at the point of 0 = 80', #
=
=
10*,
#un
0 which is in
the downwelling direction and is refracted to the ocean. The other is at 0 = 100*,
135
= 0'
4
=
1800 which doesn't directly contributed to refracted light and thus is considered here.
Similarly, for solar zenith equal to 700, the maximum degree polarization is located
at the point 0 = 20',
#=
0'. In this case, we can clearly see from the e-vector pattern
of sky polarization that Babinet neutral point located at around 0 = 500 near the
principle plane. Since all polarized sky light is linearly polarization. The ellipticity
of sky polarization equals to zero.
The underwater polarization pattern under a flat ocean just below the surface
(Fig 2 (e)-(j)) demonstrates strong dependence on the sky polarization. The degree
of polarization and e-vector have similar patterns with sky polarization but they
are limited to the Snell's window. Inside the Snell's window, there is a point with
maximum degree of polarization. For 0,,, = 10' and 0O, = 70' the maximum degree
of polarization occurs at around 0 = 470 and 0 = 15' respectively. This can be derived
from maximum locations of sky polarization of these two cases based on Fresnel law.
The solar zeniths seen from underwater are 7.5' and 44.6'. Thus, due to scattering of
water molecules and hydrosol, another maximum degree of polarization exist outside
900 away from underwater solar zenith. In above two
the Snell's window at 0
cases, only that with 0,,,
=
10' has this kind of maximum point outside Snell's law
in downwelling directions which is located at 0 = 82.50,
#
= 00.
There is a small
portion of elliptical polarization just below the ocean surface. Ellipticity is caused by
total internal reflection of those upwelling light from deep ocean. For a calm ocean,
the elliptical polarization occurs exactly outside the Snell's window. This is largely
affected by the shape of ocean single phase function. In the case of the smaller solar
zenith, elliptical polarization is evenly distributed around all azimuths with relatively
small degree (less than 0.1). For higher solar zenith angle, more ellipticity takes place
at the anti-solar meridian with much higher value than small solar zenith case.
To quantify the effects of roughness of ocean surface with coupling effects of turbidity of ocean and solar incidence, we investigated the underwater polarization pattern
with different single scattering albedo off ocean as wc, = 0.2,0.5 and 0.8. The maximum degree of polarization within the Snell's window is examined at the optical
depth of T= -0.1 and
T=
-1.
136
0.75
0.65
&o
0.7-
oen
=0.2
o
co =0.5
(o
0.6
0.2
ocn
on=0.5
eoi =0.8
=0.8
0.65
4"05
0.55-
0.6 -
0.5
0.45
0.5
0.45.
050
Wind Speed U10 (mis)
15
04
05
20
10
Wind Speed
1U(nis)
15
20
Figure 5-30: Maximum degree of polarization Pmax within Snell's window affected
by different surface roughness represented by wind speed U10 in the case of low solar
zenith 0, = 10' (a) At the optical depth T = -0.1 (b) At the optical depth T = -1
The above figure demonstrates the effects of surface roughness of ocean on underwater maximum degree of polarization within Snell's window with smaller solar zenith
angle
0
,
= 100 which represents noon conditions. As expected, the maximum degree
of polarization drops with increasing wind speed almost linearly for both r = -0.1
and
= -1. The Not very obvious fact is that the maximum degree of polarization
behave differently for different water turbidity. We can see that at the depth just
below the surface, say,
T
-0.1, for calm water, the ocean turbidity doesn't influ-
ence much on Pmax. However, with ascending surface roughness, the ocean surface
effects couple with scattering process, which cause different dropping rate of Pmax.
Unlike maximum degree of polarization outside Snell's window which shows strong
decaying with depth under larger single scattering albedo, here the maximum degree of polarization Pmax within Snell's window has different trend. We see with any
rough surface, the bigger wc, is, the larger Pma is. At deeper depth tau = -1, Pmax
demonstrate the same trend with almost same decreasing rate for any water turbidity.
Above figure represents quite similar physics with figure 3 but under different solar
zenith angle Ou = 700. We can see that the ocean surface roughness have different
affects on Pmax within Snell's window even both of them have same trends.
We see that solar zenith plays an very important role on underwater degree of
137
0.8
0.75Ocn
-F)
E
0.2
o
-0.5
OCn
0.2
*oen=0.5
0.
So
=0.8
oen
0.65
=0.8
0.4-
0.6
0.50.55
0.4-
0.45-
0
Wind Speed U10 (m/s)
20
15
0
5
10
Wind Speed U10 (m/s)
15
20
Figure 5-31: Maximum degree of polarization Pmax within Snell's window affected
by different surface roughness represented by wind speed U10 in the case of low solar
(b) At the optical depth
zenith 0,,, = 70' (a) At the optical depth T = -0.1
0.9
-U
-U 0 m/s
0.8
=0 m/s
0.8 -&U1=8 m/s
-9U 0= 12 m/s
12m/s
0.7 -- U10 16 mns
= -1
-(U10=4 ma/s
U =4 m/s
&U1=8 m/s
U
T
----
0.7
A-U10
-
0.6
0.5
0.5
0.-
0
10
20
30
40
50
Solar Zenith 8.
60
(')
70
80
0.40
90
10
20
50
30
40
Solar Zenith
60
(')
70
80
90
Figure 5-32: Maximum degree of polarization Pmax within Snell's window varies with
solar zenith angles for several surface conditions at the optical depth T = -0.1 (a)
Single scattering albedo of ocean w,, = 0.2 (b) w,, = 0.8
138
polarization. Its influence on degree of polarization is also coupled with roughness
of ocean surface. Figure 5 shows the how maximum degree of polarization within
Snell's window Pmax reacts with various solar zenith for different wind speeds. We
can see that under the calm water condition and at the very shallow depth (r =
-0.1),
Pmax increases with solar zenith from nadir (00) to horizon (900). It is noted
that water turbidity doesn't have influence on maximum degree of polarization in
this case. However, under the conditions of wind roughened surface, effects of solar
zenith, surface roughness and water turbidity are coupled. Pmax is not monotonically
increased with solar zenith. Pmax reaches two peaks at the solar zeniths with around
0,,, = 100 and 0 ,,, = 70'. Rougher surface leads to smaller degree of polarization.
More turbid ocean causes slower drop rate of degree of polarization within the Snell's
window.
Experiments has shown that the part of the downwelling light just below the airsea interface is elliptically polarized. Those elliptically polarized light is induced by
the totally internal reflection where a phase change happens for the two perpendicular
components of the light. Ellipticity represents the ratio of the amplitude of minor
electric vector component to that of major electric vector component. Among the
four components of Stokes vector, V determines the main characteristics of ellipticity.
Though ellipticity is a function of azimuthal angle, we take an average of ellipticity
over azimuthal angle from 00 to 1800 and from 180' to 3600 in order to understand
its dependence of polar angle. Figure 6 is the ellipticity at the very shallow depth
T
= -0.1 with wind driven surface U10 = 8m/s. In this case, the single scattering
of ocean is chosen as wc, = 0.8 so that more upwelling light backscattered to the
surface.
As expected, ellipticity only have values outside the Snell's wind. The maximum
value of ellipticity exists at around 0 = 70'. It is noted that this position of maximum
ellipticity doesn't change with solar zenith. However, the maximum value of ellipticity
increase with solar zenith from 00 to 700. The following figure will demonstrate how
maximum value of ellipticity depend on solar zenith in different cases of ocean surface
roughness.
139
0.15r
:0
-0.05
"90
60
30
0
Polar Angle
0 (0)
30
60
90
Figure 5-33: Distribution of azimuthally averaged ellipticity tan#.
0.1
0.08-0.06-- U10=0 mn/s
.0.04 -
'-
- - U 10=4 mn/s
+ U10=8 m/s
/
+
U10=12 m/s
-E0- U10=16 m/s
-A U10=20 m/s
0
20
60
40
Solar Zenith Osun (0)
80
Figure 5-34: Dependence of maximum ellipticity tan /3max on solar zenith angle 6
under different ocean surface conditions (wind speed varies from Om/s to 20m/s)
140
Figure 7 gives 6 curves of maximum ellipticity vs solar zenith, each of which
represents one roughness from complete calm surface Uio = Om/s to highly roughen
surface U10 = 20m/s. The single scattering albedo of this case is wa, = 0.5. We can
see that the maximum value of ellipticity becomes the biggest when solar zenith is
between 60' 700. When solar zenith gets larger from 70* to horizon the maximum
of ellipticity drop to almost the same value with that for the Sun is at the zenith.
For different surface roughness, we see that maximum ellipticity slightly drops with
increasing wind speed. This variation of solar zenith effects on ellipticity is caused
by the feature of reflection muller matrix of the surface and the effects of solar zenith
angle on upwelling radiance distribution at the range of 0
=
48' 900 just below the
ocean surface.
0.12(o
=0.8
ocn
0.11. 0.1
0.09m 0.5
0.08
ocn
S0.07
0.8.8
0.06oocn =0.2
0.05
0
5
10
15
Wind Speed U 10 (m/s)
20
Figure 5-35: Effects on ocean surface roughness on maximum ellipticity tan~m
under conditions of different single scattering albedo of ocean: w,,, = 0.2,0.5 and
0.8.
In figure 7, we can see that ellipticity monotonically decreases with surface roughness. The roughness effect on ellipticity behave differently in different turbidities.
It is easy to understand that ellipticity is bigger in more turbid water due to more
upwelling radiance backscattered from the deeper ocean, which leads to more totally
internal reflections. The rates of decreasing ellipticity are also different for different
141
single scattering albedos of water. Larger water turbidity have bigger effects when
the roughness of ocean surface is increasing. At relatively clean water,say wo,
= 0.2,
we can see that the ocean surface roughness almost doesn't change the ellipticity.
70.5
-
--
30
-
70 -
en = 0.2
i69.5e- 25-3n
cn = 0.8
69
68.
68on
=0.2
=0.5
67.57-5 67 6.0
0-
E3
oen
5
15 -
=0.8
10
15
Wind Speed U 10 (m/s)
20
0
5
10
15
Wind Speed U 10 (m/s)
20
Figure 5-36:
Other than magnitude of ellipticity, there are another two features characterizing the distribution of ellipticity: the location of maximum value of ellipticity and
the full width at the half maximum value(FWHM) of ellipticity profile along polar
angle.
Figuire 9 shows that they also strongly affected by the ocean surface and
ocean turbidity. Figure 9 (a) show the how the location of maximum ellipticity
changes with wind speed under three ocean turbidity conditions. For relatively calm
surface(Uio < 5m/s), the maximum ellipticity shifts to larger polar angle with same
shifting rate in three different water turbidity cases. However, when continuing to
increase the surface roughness, the shifting rate varies with ocean turbidities. (a)
shows for larger water turbidity, say wo,
= 0.8, the maximum value of ellipticity
take places at bigger and bigger polar angle. On the other hand, for smaller water
turbidity(w,, = 0.2), the location of maximum ellipticity shifts to closer to critical
angle.
The FWHM of ellipticity gives a quantitative range of elliptical polarization distribution. FWHM strongly depends on the ocean surface roughness. In the case
of flat surface, FWHM of ellipticity has the largest value. Ellipticity spreads more
evenly over at the range of polar angle 0 >
0
eitica,.
When the ocean surface gets
rougher, more ellipticity accumulates to the area of maximum ellipticity and FWHM
142
becomes smaller and smaller. However, the three different water turbidity leads to
different feathers. For high ocean turbidity, FWHM tends to increase at very large
wind speeds.
143
144
Chapter 6
Radiative transfer in ocean
turbulent flow
6.1
Radiative transfer with varying refractive index
Consider a time-independent inhomogeneous radiative transfer equation (RTE) with
varying refractive index which was derived by several authors with slightly different
forms. The general form of the inhomogeneous RTE can be expresse(d as:
Vn(r)I = -c(r)I
Q -VrI + F . Vol - 2
+ b(r)
n(r)
O')I(')d7'
'p(,
+Q
(6.1.1)
where I is the radiance which is the function of location r and direction Q. n(r)
, c(r) and b(r) are refractive index, absorption coefficient and scattering coefficient
respectively.
Q
is the source term. All of them are variables of location r. We
assume the volume scattering phase function p(Q, Q') remains the same in turbulent
environment.
F has different forms under different approximations. Here we take Tualle' and
145
Shendelva's results which is
F
6.2
Vn(r)
n(r)
(6.1.2)
n(r)
Monte Carlo solution and numerical ray-tracing
technique
Among many ways to solve RTE, Monte Carlo method is the most physically straightforward one. To solve the above RTE with varying refractive index, we have to first
write it as an integral equation with Green's function theory. Change the material
derivative as a path derivative, we can obtain the following from from equation (1),
LI =
f
(6.2.1)
where L is the operator which has the form of
L =
as
+ F - Vo + [c(s) - 2
ds
(6.2.2)
]
here s is the arc length of photon propagations.
f = b(s)
j
p(,
Q')I(7')dO' + Q(Q, s)
(6.2.3)
Consider the Green's function G = G(Q, s; Qo, so)) which satisfies
(6.2.4)
LG = 6(s - so)6(O - Go)
The solution of above equation (2.6) can be expressed as a
G(Q, s; Qo, so)) = 6(Qo - W(Q, s, so))e-
"o")s"e
2
in
(6.2.5)
where W(Q, s, so) is the direction of light beam after traveling by a distance s -so.
Then the solution of RTE can be written as
146
I(A, s) =
+
j
s
dQoG(Q, s; Qo, so)b(so)
dso j
doG(Q, s; Qo, so)Q(Qo, so)
is dsoe
+
f
dso j
oc(s" )ds" e2n
so
dsoe~ fo c(s")ds"e 2 n nO)
d'p(Q,
d'p(W(7, s, so),
SO)
so')J(',
')I(Q, so)
Q(W( , s, so), so)
(6.2.6)
Where ws0 is the single scattering albedo.
The Monte Carlo solution can be performed based on equation (2.8). Unlike the
homogeneous cases, the photon propagation in the inhomogeneous medium is treated
with biased sampling techniques.
p
= e f-3(ocs)s+
e
e
o
"
(6.2.7)
Each propagating are length for photon can be determined by the first part of
right side of equation (2.9)
l
nR
c(so)
(6.2.8)
where R is the random number between [0, 1]. Accordingly, a weight function
must be multiplied to the energy of photon by a factor ef. A"("")d"
The integral in the weight function is solved discretely as the numerical ray tracing
method id employed. Then numerical ray tracing method treats photon migration in
inhomogeneous medium with variable refractive index by solving ray equation
d[ n(r)
ds
= Vn(r)
ds
147
(6.2.9)
Empirical IOPs models
6.3
The direct impact of ocean turbulence flow on light propagation is due to the fluctuation of inherent optical properties (IOPs) of the ocean caused by the turbulence
fluctuations of passive scalars, such as temperature, salinity and chlorophyll concentrations, The relations between IOPs and temperature, salinity and chlorophyll
concentration has been intensively investigated and various empirical models have
been built. In this study, we used empirical models of absorption coefficient, scattering coefficient and refractive index to perform the inhomogeneous radiative transfer
simulation. It is assumed that the shape of scattering phase function is irrelevant
to turbulence passive scalars. Here we apply Henyey-Greenstein phase function with
anisotropic factor g = 0.924 for all depths of the ocean.
A. Model of Absorption Coefficient a(T, S, Cc, A)
6.3.1
The model of the absorption coefficient a(T, S, Cc, A) is applied based on that of
V. Haltrin('1998) and W. Pegau (1997).
Haltrin gave the following one-parameter
equation:
. 2 + ay Cf -kf A+ alCe-kA
a(A) = a,(A) + a' (A)(Cc/CD) 060
(6.3.1)
where a,(A) is the pure water absorption coefficient in m-', A is the wavelength
of light in unm, a8 is the specific absorption coefficient of chlorophyll in m'.
Cc is the total concentration of chlorophyll in mg/m
3
and C is the constant with
value CX = 1mg/m 3 . ao and a' are the specific absorption coefficient of fulvic acid
and the specific absorption coefficient of humic acid respectively with values afo
35.959m 2/mrig, ao = 18.828m 2 /mg. The two constants kf = 0.0189nm 1 and k=
0.01105nmn.
mg/m
3
Cf and Ch are concentrations of fulvic and humic acids respectively in
respectively. The relations between Cc, Cf and Ch are given as
148
Cf = 1.74098- Cc - exp(0.12327. Cc)
Ch = 0.19334- Cc - exp(0.12343. Cc)
(6.3.2)
According to Pegau, we can write the pure water absorption coefficient aw as the
function of temperature T and salinity S.
aw(A) = aw(T, S, A) = a,(T,, 0, A) +
T(T - T,) + JsS
(6.3.3)
where T is the temperature in C, T, is the reference temperature. S is the salinity
in per thousand. Values for WT and Is at different wavelength were given by table 2
and table 4 of (Pegau, 1997).
aw(T, 0, A) and a' are given by Bricaud (1995) with following relation:
agc(A) = A(A\) C
(6.3.4)
(A
The value of A and B is listed in following table.
Table 6.1: Coefficints A(A) and B(A) used in Eq. (2.14)
A(nm)
400
450
500
550
a,(T,0, A)A(A)
B(A)
0.00663
0.00922
0.0204
0.0565
0.282
0.359
0.052
0.092
0.0263
0.0371
0.023
0.008
Put all equation above together, we have the following relation
a(T, S,Ce, A) = aw(Tr,0, A) +IT(T
- Tr) + 4 sS
+ ac(A)(C/Ce) 0 6. 0 2 + aoCf e-k/A + a Che-k"A
(6.3.5)
Here we treat the total chlorophyll concentration as one passive scalar which is
not dependent on temperature and salinity.
149
6.3.2
B. Model of Scattering Coefficient b(T, S, Cc, A)
The model of scattering coefficient b(T, S, Cc, A) we used by are presented by Haltrin
and Kattawar (1991) and X. Zhang (2009).
According to Haltrin, the scattering
coefficient cap be written as
b(A) = bw(A) + bo(A)Cs + b? (A)C,
(6.3.6)
where b,(A) is the scattering coefficient of pure water in m- 1 . b (A) and b?(A) are
the specific scattering coefficients for small and large particular matter respectively
with unit g/m 3 . The spectral dependencies of these two coefficients are given by
Kopelevich.
4001.
bo(A) = (1.1513m 2/g)( A
b'(A) = (0.3411m 2 /
(6.3.7)
400 0.3
C, and C1 are concentrations of small and large particulate matter respectively in
g/m. These two parameters can be determined with
C, = 0.01739 C- exp(0.11631 Cc)
C, = 0.76284 Cc exp(0.03092
c,)
(6.3.8)
According to Morel(1974), the scattering coefficient of pure water can be obtained
by
bA(A)
Here
#(90;
16.06()4.32w (90; Ao)
(6.3.9)
AO) is the volume scattering function at 90 degree. Zhang (2009) gave
the expression of 0(90; AO) by equations below.
150
0(90; A0 ) =
7
(
2A _
ap
)2 kT
+ (f
2
0S
2
p
f(6)
-a Ina/S
(6.3.10)
Therefore, the scattering coefficient b(T, S, Cc, A) can be obtained with
b(A) = 16.06(
6.3.3
)4-#1 (90; Ao) + b (A)C8 + b0(A)C
(6.3.11)
C. Model of Refractive Index n(T, S, A)
The model of refractive index n(T, S, A) applied in the study is based on the expression
of McNeil(1977) who used the A&H data to obtain an empirical equation for the
refractive index of sea water as a function of wavelength, temperature and pressure.
At the upper ocean level(< 100m) the influence of pressure on refractive index is very
small, therefore, the empirical equation is manifested as
n(T, S, A) = 1.3247-2.5 x 10- 6T 2+S(2x 10-4-8 x 10- T)+
33 00
A2
A4
(6.3.12)
(..2
where n is the refractive index, S is the salinity in parts per thousand, T is the
temperature in 'C and A is the wavelength in nm.
6.4
Simulation results
To understand how ocean turbulence the features of underwater radiance and irradiance distribution, we investigated two types of waters. One is very clear ocean with
uniform chlorophyll concentration. The other is the turbid ocean with fluctuating
chlorophyll concentration.
Since interaction of ocean turbulence flow and dynamic ocean surface is an important factor of statistic properties of passive scalars, we investigate both conditions
of calm ocean and progressive ocean surface waves are considered in the study.
151
For underwater irradiance with ocean surface waves, previous studies show under
linear assumption of Snell's law, the spectrum of downwelling irradiance is a linear
combination of that of the surface elevation, that of the surface slopes and that of the
surface curvature. Also, downwelling irradiance is governed by Poisson distribution
with mean values as a function of surface slope distribution and surface curvature
distribution.
Due to the perturbation of ocean turbulence to the light propagation, the spectrum and probability density function of downwelling irradiance are influenced by the
spatially varying refractive index, absorption coefficient and scattering coefficient.
We examine the radiance angular distribution and irradiance statistics for both
cases.
6.4.1
Clear ocean:uniform chlorophyll concentration
In order to characterize the influence of temperature and salinity on IOPs and further
affect the radiative transfer properties, it is assumed that the ocean water is clear so
that from depth of around 100m to the surface, chlorophyll concentration Cc is considered as uniform everywhere with value of Cc = O.1mgm- 3 . Here we show the case
of IOPs' vertical distribution under the condition of a dynamic surface wave field with
steepness of ka = 0.1. For different temperature difference (or TKE dissipation rate
c), the absorption coefficient, scattering coefficient and refractive index have following
vertical profiles of mean value and standard deviation of normalized variables defined
as,
r
((
-
(a)
(a))2)
(6.4.1)
We can see that variations of inherent optical properties take place mostly at the
depth from Om to 40m. Below the depth of -40m, the inherent optical properties
become uniform because of the standard deviations decrease to one order lower than
those of shallower depth. The absorption coefficient a decreases along the vertical
direction by around 1% to 15% corresponding to temperature change AT = 5' and
152
0
-20
-40
-60
-60
SC
0
-80
--
-80
AT-20 *C
--
120023 0.024 0.025 0.026 0.027 0.028 0.029
340
.032
0034
T-20*C
0.0348 0.0348
003
T348
1,3485
<A>(m)
1.349 13495
<n>
1.35
1.3505
6-40
(a)
(b)
(c)
-80AT-10
-20-
-20
20
+-40T2
-80
-so iAT-10r
-0
0.00
001
0015
002
AT-20*
0025
2
0
003
.10,
(d)
(e)
Figure 6-1: The mean value and standard deviation of IOPs of ocean turbulence at
various depths (a) Mean value of absorption coefficient < a > (b) Mean value of
scattering coefficient < b > (c) Mean value of refractive index < n > (a) Standard
deviation of absorption coefficient -a (e) Standard deviation of scattering coefficient
Ub (f) Standard deviation of refractive index o
AT = 200 respectively. Horizontal fluctuation of a varies from 0.5% to 2.5%. The
scattering coefficient b decrease by from round 0.1% to 1% and varies horizontally
from 0.1% to 0.4%. Refractive index increase by 0.15% and 0.075% respectively. It
horizontally fluctuates by around 0.01% and 0.02% at the depth of -20m.
It is noted
that the absorption coefficient has the largest variations caused by the temperature
change.
Calm ocean surface
In the simulation, the sky is assumed clear and the solar incidence is normal to the
ocean surface. We use the empirical model of atmosphere which has a total optical
depth of ratm = 0.15, and volume scattering function with the form of Rayleigh
scattering. The single scattering albedo watm is chosen to be 1. For a calm ocean
with turbulence flow below the surface, we got the typical field pattern of downwelling
irradiance and upwelling irradiance as figure 2 as an example.
153
-100
1-05
-50
1.01
0025
-50
0.024
1.005
1000
0
-100
0.995
50
0.023
0
0
0.022
50
0.99
-100
-50
0
X(m)
50
0.021
-100
100
-50
0
X(m)
100
50
0.01
(a)
(b)
007
-0.017
-100
-100 *.6
100
0.67
-50
10.66
100850100
-0.0165
--
-50
0.016
0.65
-0.0155
0
0
4*
0.64
50
100
-100
-50
0
X(m)
50
-0.015
0.63
50
-0.0145
0.62
10.61
100
;0 0135
0.014
-100
100
-50
0
X(M)
50
100
(d)
(c)
Figure 6-2: The light field pattern below the calm ocean surface in the presence
of ocean turbulence(a) Downwelling irradiance Ed just below the ocean surface(b)
Upwelling irradiance E, just below the ocean surface(c) Downwelling irradiance Ed
at z = -20m(d) Upwelling irradiance Ed at z = -20m
154
It is noted from figure 2 (a) that the downwelling irradiance pattern just below
the surface is affected mildly due to the small distance of light propagation in the
turbulence. At the deeper depth as shown in figure 2 (c) we can see the clear pattern
of downwelling irradiance. It can been seen that pattern of upwelling irradiance has
resemblance of the downwelling irradiance.
In Figure 3, the vertical profile of the mean value and the standard deviation
of downwelling and upwelling irradiance under the calm ocean surface are given.
Similarly, the standard deviation of irradiance is defined as
O
((Ed - (Ed)) 2 )
(Ed)
0
0
-20-
-20
-40 --
(6.4.2)
40
-60 -
-60
-1
-0.8
-1000
-0.6
-04
log(E d)
-0.2
- -AT=5*
-*-- AT=10*
--- AT=20*
-80
-+AT=1 0*
-e-AT=200
-80
0
100
-3
(a)
- 5
-2.5
lgE
-2
-1.5
(b)
Figure 6-3: Mean values of downwelling irradiance and upwelling irradiance at different ocean depth. (a) lg < Ed(z) > (b) lg < Es(z) >
It is shown that larger AT leads to faster attenuation of downwelling irradiance
along the ocean depth. This can be understood with the help of figure 1 which shows
larger attenuation coefficient for bigger AT. For upwelling irradiance, the attenuation
rate at three different AT are the same at the shallower depth.
Figure 4 gives the standard deviation of downwelling irradiance and upwelling
irradiance at different ocean depth. The standard deviations of the normalized downwelling irradiance are on the order of 10-3. Bigger temperature change leads to higher
standard deviation. It reaches the maximum value at the depth of around -40 meters.
155
-40-
'j-40-
-60
-60
-80-
-
-1 00_
2
4
6
AT=0*
-80
AT=1
10
8
x10
-1006
120
0
3
4
0.01
0.02
0.03
0.04
0.05
Eu
(a)
(b)
Figure 6-4: Standard deviations of downwelling irradiance and upwelling irradiance
at different ocean depth. (a) oEd (b) CEu
Under dynamic ocean surface
Under the dynamic ocean surface, the turbulence play different role in affecting underwater light field statistics.
The downwelling and upwelling irradiance has the following patterns under a
typical progressive waves as figure 5.
Figure 3 gives the mean value of downwelling irradiance and upwelling irradiance
at different ocean depth.
Figure 4 gives the standard deviation of downwelling irradiance and upwelling
irradiance at different ocean depth.
156
1.3
0.025
1.2
0.024
0.023
1.1
5(
0.022
1
10(
0.021
X(m)
(b)
0.019
0.018
0.017
0.8
0.016
0.015
0.014
-100
xen)
-50
Xe)
50
100
Figure 6-5: The light field pattern below the progressive ocean surface waves under
progressive ocean waves with kpa, = 0.1 in the presence of ocean turbulence(a) Downwelling irradiance Ed just below the ocean surface(b) Upwelling iriadiance E" just
below the ocean surface(c) Downwelling irradiance Ed at z = -20m(d) Upwelling
irradiance Ed at z = -20m
-0.6
log(E )
-0.4
-2.5
log(E.)
Figure 6-6: Mean values of downwelling irradiance and upwelling irradiance at different ocean depth under progressive ocean waves with kpa, = 0.1. (a) Ig < Ed(z) > (b)
lg < E,(z) >
157
0
0.01
0.02
0.03
~Ed
0.04
0.05
~Eu
-
(b)
(a)
Figure 6-7: Standard deviations of downwelling irradiance and upwelling irradiance
at different ocean depth under progressive ocean waves with ka, = 0.1. (a) oEd (b)
UE,
158
Chapter 7
Inversion of IOPs and applications
of RT prediction in engineering
7.1
Radiometric measurement: Radiance and polarization
There are various types of devices for capturing the solar radiant energy. Here we
will mainly introduce the devices to measure radiance and polarization.
Radiometer: measuring radiance
The device to measure radiance is called Radiometer. The current radiometer system
usually consists of Fish eye optics, CCD array cameras and position correction system.
Take the example of the radiance distribution camera built by Voss (1991), the figure
shows a block diagram the radiometer instrument.
The central feature of this camera is CCD arrays in the imaging plane. These
cameras are 480x542 arrays and are cooled to approximately -30 degrees C. the cooling
reduces the dark noise significantly thus increasing the allowable integration times.
Camera images are digitized by a 16 bit frame grabber thus significantly increasing
the intrascene dynamic range.
One of the fisheye optics built by D. Antoine et al (2009) is shown in the following
figure.
159
kmlaba
Collector
whwow
ftwe
Sensor
nWawce
Collector
Plasde Dame
WIdow
Figure 7-1: Block diagram of radiometer (Voss, 1991)
7
o
i
Figure 7-2: Fisheye optic description (D. Antoine et al, 2009)
160
The light collected by the fisheye lens is transmitted trough spectral and neutral
density filters before being imaged on the surface of the array. The spectral filters
which allow 4 different spectral wavelengths to be investigated. The 4 neutral density
filters allow the overall scene intensity to be varied as the radiance field increases
or decreases due to depth or changes in the incident light field. The filters may be
changed remotely at the surface.
Inside the underwater housing, a single board computer controls the camera and
frame grabber. Hard drive is also enclosed in the underwater housing. Since the
computer controlling the camera and storage device are contained in the housing,
data communication to the surface is greatly reduced. Only subsmapled low resolution
images are needed at the surface to check data quality and set exposure times. Data
communication between the surface and the underwater unit is performed using serial
9600 baud transmission using standard software. This allows the surface computer
to display the complete operations of the underwater computer without additional
complicated software.
Another radiometer system built by D. Antonie et al is shown below:
Fish eye optic
W
UMME 1
logos
T
Bandpass filters
(on a filter wheel)
CMOS matrix
Auxiliary sensors
L11AuxI Icommands Data transfer &
[ =OM_ 200m depth
t-or
200m electrical/optical cable
container
Figure 7-3: Radiometer scheme description (D. Antoine et al)
The spectral filter allows a small incidence angle.
6 wavelength light can be
measured with dynamic range from 406nm to 628nm. One the iumage plane, two
kinds of sensors are chosen and have a good performance on dynamic, sensitivity and
161
Figure 7-4: The first prototype of the LOV-CIMEL radiance camera (D. Antoine et
al)
no blooming: CCD which has best sensitivity and CMOS which has less sensitive and
more linear and no blooming. The commercial CMOS used is monochrome, 12 bit
digitization and has hight definition format(1920x1080) with pixel size of 5pm.
The Auxiliary sensors include compass, depth sensor, tilt sensor and internal
temperature and humidity sensors. The data transfer uses optical connection with
transfer rate of 15frames/s. Here is the picture of first prototype of the Radiometer.
The angle resolution can be better than 10 An in-water radiance field taken by K.
Voss 1988 is shown below.
Figure 7-5: An radiance field measurement taken by K. Voss 1988 with Radiometer)
Polarization radiance measurement
162
The polarization radiance measurement is usually conducted by using a three
channel polarimeter. Take Shashar et al as an example, a fiber optic with an acceptance angle restrictor of 50 was connected to each channel, and on each restrictor a
polarizer was mounted. the polarizers were adjusted to three different orientations:
00, 450, 90' from the horizontal. On top of each polarizer, an polarization-neutral,
colored filter was mounted to flatten the natural spectrum. The fibers, along with
their polarizers and colored filters, were inserted into a submersible housing that was
fixed on a rotating apparatus, attached to a stable vertical pole. When recording
radiance at different viewing zenith angles, we encountered a wide range of intensity
levels. Thus to attain a high signal-to-noise ratio in the system, then measuring in
the spectral range of 350 - 650nm, the spectraophotometer was set to integration
times of 50ms(at view zenith angles of 700 and 900). Owing to an inherent delay of
the spectrophotometer between successive integrations the actual sampling frequency
was 3.2 and 1.7Hz.
Prior to each experimental seeion and deply, fibers were cross calibrated by examining an evenly illuminated white defusing fabric. Polarization analysis can be done
as following:
the phase 0 is given
= (1)
2 arctan(
2
L9o - Lo)
)
(7.1.1)
Then if L 90 < Lo, 0 = 0 + 90', otherwise 0 = 0 + 900.
Since the phase 0 represents the e-vector shift from the vertical to obtain the
absolute e-vector orientation a, the following condition was applied:
If (0 > 900), a = 0 - 90 , otherwise a = 0 + 900.
The total radiance is given by
L = Lo + L9 0
while the degree of polarization is given by
163
(7.1.2)
fi%=64V
Os -31'
(d))
eto
Prepl.rato
sky
~
~
nntto
ielh
etge.
'c
10
cacuacd4
injkkSk
PercentwSy
po
iat
iand
4.0ena
i-0
-,-oma
degree
Figure 7-6: Measured polarization radiance,
L AOte of polarization and e-vector oriC%Lo +1A1
entation taken by Sabbah et al (2006)
y'(Lo
-
Ligo)2 + (2L 4 6
-
L9 o --
(7.1.2
The e-vector orientation scale ranges between 0 and 180, with 0 and 1800 corresponding to horizontal polarization, and 90 corresponding to a vertical e-vector.
The degree of polarization ranges between [0, 1].
An demonstration of measured degree of polarization and e-vector taken by Sabbah et al (2006).
7.2
Inverse problem
In 1971, the first clinical machine for detection of head tumors based on X-ray Computed Tomgraphy (CT) was built at the Atkinson Morley's Hospital in Great Britain.
The announcement of this machine by G.H. Hounsfield at the 1972 British Institute
of Radiology annual conference, as been considered the greatest achievement in radi164
ology since the discovery of X-rays. In 1979 G.H. Hounsfield shared with A. Cormack
the Nobel Prize for Physiology and Medicine. The CT computer-generated image
provided the first example of images obtained by solving a mathematical problem
which belong to the class of so-called inverse and ill-posed problems. The topics considered as inversion methods are applied to passive and active atmospheric sounding,
ionsospheric sounding, particle scattering, electromagnetic scattering and seimology,
etc.
As for definition of inverse problem, The mathematician frequently quoted statement of J.B. Keller: "We call two problems inverses of one another if formulation of
each involves all or part of the solution of the other. Often, for historical reasons, one
of the two prolems has been studied extensively for some time, while the other has
never been studied and is not so well understood. In such cases, the former is called
the direct problems, while the latter is the inverse problem". For physicist, however,
the situation is quite different because the two problems are not on the same level:
one of them, and precisely that called the direct problem, is considered to be more
fundamental than the other and, for this reason, is also much more investigated. In
other words, the historical reasons mentioned by Keller are basically physical reasons.
7.2.1
Inverse problem in ocean optics
After introducing a general inverse problem and general method to deal with it, we
get back to the inverse problem for ocean optics. The first problem we come across is
also the uniqueness of the solution. For example, a body of water with a particular
set of IOP's and certain boundary conditions has an underwater radiance distribution
L 1. If the boundary conditions now change, perhaps because the wind ruffles the sea
surface or the sun moves, there will be a different radiance distribution L 2 within
the water, even though the IOP's remain unchanged. It is necessary to know if we
can get obtain the same set of IOP's from the two different light field. Can we
distinguish between L 1 # L 2 because of a change in boundary conditions, as opposed
to L1 f L 2 because of a change in IOP's? If the same set of IOP's cas yield different
radiance distributions.
Therefore, question comes up that if two different sets of
165
IOP's and boundary conditions can lead to the same radiance distribution. Another
problem often encountered with inverse solutions is the stability of the solution, or its
sensitivity to errors in the measured radiometric variables. In forward problems we
usually find that a small error in the IOP's leads to a correspondingly small errors in
the computed radiance. With inverse problems lead to large errors, or even unphysical
results, in the retrieved IOP's. Such extreme sensitivity of the inversion scheme to
small errors in the input data often renders inversion algorithms useless, even though
they appear in principle to be quite elegant and satisfactory.
From a practical standpoint, if we have to measure the entire radiance distribution
throughout the water body, we could measure the IOP's themselves.
An inverse
method is useful only when is saves us time, money,or effort. What would be of real
value is a recovery of the IOP's from measured irradiance and radiance.
Categories of inverse problem in ocean optics
There are many kinds of inverse problem in ocean optics. For example, there medium
characterization problems. It's objective is to acquire information about the IOP's of
the medium, which in our case is the water body with all of its constituents. There
are also in-water object characterization problems, for which the goal is to detect
or obtain information about an object imbedded within the medium, for example a
submerged submarine. Inverse probmes may use optical measurements made in situ.
Also, remote sensing uses measurements made outside the medium, typically from a
satellite or aircraft.
Another type of inverse problem seeks to determine the proerties of individual
particles fron light scattered by single particles. In such problems, the inversion
algorithms usually assume that the detected light has been singly scattered. Even
these hight constrained problems can be very difficult.
The solutions to inverse problems fall into two categories: explicit and implicit.
Explicit solutions are formulas that give the desired IOP's as function of measured
radiometric quantities. The simplest example is Gershun's law when solved for the
absorption in terms of the irradiance. Such solutions are few. Implicit solutions
166
are obtained by solving a sequence of forward problems. At first, we solve forward
problems to predict the the radiance for each of many different sets of IOP's. Each
predicted radiance is compared with the measured radiance. the IOP's associated
with the redicted radiance that most closely matches the measured radiance are then
taken to be the solution of the inverse problem.
Inversion of RTE
From integral equation relating the asymptotic radiance distribution L" to IOP's Wo
and phase function p(p', #', y, #).
(1 where ,
Io)Lo(p)
= o /27r
7r
L(')p(p', #', y, #)dy'd#'
(7.2.1)
is the rate of decay with optical depth of the asymptotic radiance
distribution and L,
is asymptotic radiance distribution. We assume L.
and
,
can be determined by measurements at large optical depth, and then solving above
equation for W-o and phase function p(p', #', y,
#).
However, use of this equation
requires us to be in homogeneous water, so that an asymptotic radiance distribution
exists. In addition, we must make our measurements within the asyniptotic regime.
These two requirements rules out the use of this equation near the surface or for
inhomogeneous waters, which are precisely the situations of greatest interest in ocean
optics.
Let's assume that we have measured the radiance distribution L(z, y,
out a water body. We then azimuthally average L(z, y,
L(z, p) =
1
[2W
27r 0
#)
L(z, y, #)d#
#) through-
to get
(7.2.2)
Zaneveld(1974) expanded L(z, p) and the volume scattering function p as series
of Legendre polynomials:
L(z, p) = E"'Aa(z)Pa(cos
167
p)
(7.2.3)
p(T) = E'OBP(cosT)
(7.2.4)
Substituting these forms into the azimuthally averaged, source free RTE led to a
set of different equations for the expansion coefficients A, and B,. These equations
eventually led to explicit formulas for the beam attenuation coefficient c,
r =0
Cm[f
n-+oo
-dzPn(cos 0) cos
0sinOdO
f" L(z, 0)Pn(cos0)sin OdO
(7.2.5)
and for the phase function
p(T9) =
2n-71
_ n + IP
4
A0
cr
Pn(Cos
fJ dL(zp)
&
ifco [C + fo
O)Cos 6 sinOdO
]
fJo L(z,0)Pn(cos 0) sinOdO
(7.2.6)
Above equations gives us a very important result: the radiance and its depth
derivative together uniquely determine the IOP's c and p. We can thus lay to rest
our philosophical concerns about the uniqueness of IOP's associated with a particular
light field. However, having L(z, 0,
#)
at only one depth is not sufficient to uniquely
specify the IOP's at that depth. dL/dz should be estimated from measurements at
two closely spaced depths.
Unfortunately, the two equations are of little use for actually retrievals of c and
p from measured radiances. The reason is that an expansion of natural water phase
functions in Legendre polynomials requires hundreds of terms. We must have the
same number of terms for inversion equations. Computing the high-order coefficients
requires that L and dL/dz be measured with extreme accuracy and directional resolution in order to evaluate the integrals. In practice, we can at best measure L to an
accuracy of a few percent on a 0-grid with a resolution of a few degrees. This is by
no means sufficient for a retrieval of p. Similarly, we note that c is obtained from the
limit of an infinite sequence of terms like those involved with p.
Wells(1983) performed a decomposition of L and p similar to Zaneveld. He defined
moments of the radiance distribution as
168
L.(z) = 27r
j
L(z, O)Pn(cos 0) sin OdO
(7.2.7)
where L(z, 0) is the azimuthally averaged radiance and n = 0, 1, 2,... Likewise,
the moments Sn of the volume scattering function p are defined by
Sn = 27r
j
p(I)Pn(cos T) sin xIdxJ
(7.2.8)
A related set of coefficients Dn is defined by
D
= c - Sn
(7.2.9)
where c is the beam attenuation coefficient. So = b and S, = 0; hence, Do = a
and D, = c. The Dn are IOP's, and knowing them is equivalent to knowing a, c, p.
Physically, the Dn can be loosely interpreted as decay rates with depth for the Ln.
Wells showed that Dn are determined from the moments of radiance through
na
D
-i+
=_
Dn
(n +
1) dLn+1
( + I)Lnz
(7.2.10)
(2n±+1)La
where L- 1 = 0.
Wells realized that the radiance moments L, can be measured directly by an
instrument whose angular response is proportional to Pn (cos 0) and independent of 0.
Such an instrument has been constructed in 1992. The prototype instrument uses then
specially shaped mirrors to measure Lo, ..., L 9 , from which, Do,
... ,
Ds are computed
by above equation. The needed derivatives of Ln are estimated from measurements of
Ln at closely spaced depths. The instrument thus gives the first nine terms of the Dn
sequence. An independent measurement of c, which is easily made, gives D.. One
can then estimate the remaining Dn, n > 9 , by interpolation based on Do,
...
, Ds and
D,. The phase function is then recovered by
2n±+1
)= E
47r
(c - Dn)Pn(cosT)
(7.2.11)
The initial sea trial of the this instrument showed a good recoe-vefy of the total
169
absorption coefficient a = Do when compared to a values determined by filter pad
techniques. It should be noted the instrument gives an in-situ, real-time determination of the absorption. The recovered p(P) looked realistic except at T > 1600. The
unrealistic behavior of p for large angle was probably due to inaccurate estimation of
the D, for large n values.
Inversion based on the irradiance quartet
After seeing that inverting the RTE given measured radiances is in principle possible,
we not investigate what information can be recovered from measured irradiance. Here
we assume that downwelling and upwelling plane and scalar irradiance are measured
as functions of depth z at the wavelength of interest.
These quantities from the
irradiance quartet
[Ed(z), Eu(z), Eod(z), Eo(z)]
(7.2.12)
where Ed is downwelling irradiance; F is upwelling irradiance; Eod is scalar downwelling irradiance; E0 u is t scalar upwelling irradiance. Because these irradiances contain less information than the radiance, we expect to recover less information than
can in principle be obtained by inverting the RTE. But since the governing two-flow
equations have a simpler mathematical form than the integro-differential RTE, they
may yield more easily to inversion. We first consider formal inversions of the two-flow
and related equations.
From Gershun's law, we can solve that in a source-free water the absorption coefficient a
a(z)
=
)
d
1
Eod (Z) + Eu (z) dz
[E(z) - Eu(z)]
(7.2.13)
This is tle simplest possible inversion of the two flow equations. Errors in the
measured irradiance, or the presence of internal sources, will result in errors in the
recovered absorption coefficient.
Gershun's law gives us one equation in one unknown, and a measured irradiance
170
quartet provides sufficient information to effect the inversion. The needed derivative
can be estimated from measurements at different depths.
Ed
dz
Eu
= -(a
Eod
+ bdu) + budEu
Ed
Eo
= -(a Ed + bad) + bduEu
(7.2.14)
(7.2.15)
There three unknown a, bad, bdu in two equations. In order to proceed, we can
conjure up a third independent equation relating the unknows and the irradiance
quartet, so that we have three equations for three unknowns. Or second, we can add
an additional measurement to the irradiance quartet, so that we have left only two
unknowns.
Preisendorfer and Mobley (1984) chose the first path. The different directional
structures of the upwelling and downwelling radiance distributions cause bud to differ
from bdu. Because these directional structures are parameterized in terms of the mean
cosine jii and I'd. They assumed that
(7.2.16)
bundg = bdugTd = b
Above equation is exact for isotropic scattering, where bb = b. It was hoped
that this would be a useful approximation for anisotropic phase function, and bb
was termed a "mean backscattering coefficient". Note that bb is an AOP, which was
presumed to be roughly equal to the IOP bb. This equation provides the needed third
equation. Then reduced two-flow equation to two equations in the two unknowns a
and bb:
dEd
dz
1
(a + 6b) -Ed
dEu
*E =--(a +
dz
pd
1
5 ) -Eu
pau
+
1
bb_-Eu
(7.2.17)
yTu
1
+ 6b_ E
(7.2.18)
pTu
the mean cosines are computable from their definitions if the irradiance quartet is
171
measured.However, the fundamental assumption leading to above equations is incorrect. Later it was shown that for oceanic water bdffu is typically two to four times
bduAd.
ru
bbd/)
bb
_
bb/ta
bb
(7.2.19)
The proper view of the Preisendorfer-Mobley inversion algorithm is that is recovers two numbers which can be by evaluated from the measured plane irradiances.
However, it is not possible to relate the AOP bb to the IOP bb in any simple way.
Therefore, the two-flow equations can not be inverted without making an a priori
assumption, which usually is cast as an assumption about the diffuse backscatter
coefficients.
McCormick and Rinaldi (1989) developed an algorithm that recovers the similarity
parameter
s = [1 + (1
-
g)]-
2
(7.2.20)
from measurements of the irradiance quartet at two depths. Here g is the mean
cosine of the scattering phase function. The significance of s is that quantities such as
the reflectance of a water body are nearly the same for water bodies having different
a, bandg values, but having the same value of s. Similarity parameters are discussed
in general by Van de Hulst (1980). Let E(z) = Ed(z) - Eu(z) be the net irradiance
and let
AE 2(z1, z2) = E 2(zi) - E 2 (z2 )
(7.2.21)
z EO(z1, z2) = E (zi) - E (z2 )
(7.2.22)
where zi, z2 are two depths. Then s is the solution of
2
S2 - 23[1 - s( ai1 - a2 s )]2 LXE (zi, z 2 ) = 00
a3 - a4s
172
AE (zi, z 2 )
(7.2.23)
where sisintheinterval[0, 1].
Here ai, ..., a 4 are constants that depend on the
scattering phase function, but only weakly. For the Petzold turbid-harbor phase
function, we have
ai = 0.30633, a2 = -0.32405, a3 = 6.2114, a 4 = 4.5767
(7.2.24)
Therefore, all the quantities except s are known if the shape of the phase function
is assumed and the irradiance quartet is measured at two depths. For most natural
waters, the a' values given above should be acceptably close to the actual a's .For
the Petzold phase function, g = 0.93. Numerical tests of the McCormick-Rinaldi
algorithm show that s is recovered to within 1%of its true values for depths greater
than 14m when Wo < 0.84 , if there are negligible errors in the measured irradiances.
The recovered s becomes less accurate near the sea surface, if o > 0.84, and if there
are random errors in the measured irradiances.
Inversions based on the plane irradiance
Though can be measured theoretically, Eod and E0 , are difficult to measure. On the
other hand, researchers are more interested in measuring AOP's like Kd or R values,
which can be obtained from plane irradiance. Firstly recall that the various relations
between Kd, R, a, b and bb. For example,
K 10 =
a [1 + (0.473tsw - 0.218) ]1/2
pSW
(7.2.25)
b
where pw is the cosine of the sun's zenith angle after refraction by a level water
surface. The value of K 10 must be determined by measurements of Ed just above and
just below the z1o depth.
If coupled with a measurement of beam attenuation c and observation that b/a =
c/a - 1, gives a quadratic equation for a:
(1.218 - 0.473isw)a
2
+ [(0.473tsw - 0.218)c]a - Kp
173
=
0
(7.2.26)
The positive root for a then yields b - c - a. The backscatter coefficient then can
be determined from
b =
where R(0) = E,(z = O)/Ed(z
aR(0)
(7.2.27)
R
0.975 - 0.629p,(.
0).
Gordon (1991) used used numerical simulations for wide range of oceanic conditions to develop explicit algorithms for retrieving a, b and bb from measurements of
the plane irradiance Ed and E,, and of the beam attenuation coefficient c. Let K and
R denote the respective values of Kd(z) and R(z) measured just below the sea surface.
The measured value of K in first normalized in the manner described by deviding
K by a certian donwwelling distribution function Do from field measurements. The
normalized quantity K/Do corresponds to the value of K that would be measured
with the sun at the zenith, with a flat water surface, and with a black sky.
Let bf
bf/b be the probability of forward scattering for the scattering phase
function p; bf is the forward scattering coefficient. Gordon first shows that the quantity 1 - Wobf can b expanded as
1-
=
ERik,(
nob-f
K
)fl
cDo
(7.2.28)
where the expansion coefficients are
ki = 0.89670, k 2 = 0.20271, k3
=
-0.13506
(7.2.29)
bf/c to about 1% accuracy. Gordon next shows
This expansion recovers LLobf
that
b
K/DO
1
-
r[R(1)]"
(7.2.30)
where the expansion coefficients are
r1 = 2.8264, r 2
=
-3.8947, r 3 = 36.232
174
(7.2.31)
Here R(1) = ROD, = 1 is the irradiance reflectance that would be measured in the
zenith-sun, level-surface, no atmosphere case. The value of R(1) must be determined
by extrapolation from value of R(Do) made for two or more D, values. For example,
R can be measured at several different solar zenith angle 0, during the course of a
day; each 0, value gives a different D, value. The relation between R(Do) is linear for
a given scattering phase function. The estimated values of R(1) and the measured
K and Do values yield bb from above equation This estimate of bb is available even if
c is not measured. If c is also measured, then a and b can be determined as follows.
The value of bb obtained gives us
b
K/Do
W0 6obb(..2
K/(cDo)
So that 0obb is known. Since bf + b+= 1, we can determine
Lo
from
wo = Jobf + 0obb
(7.2.33)
after obtaining Wobf. We get b from b = cJo, and a from a = c - b.
Gordon estimates that his algorithm will recover a and b to better than 1% accuray, and bb to better than 10%. It should be noted that this algorithm requires no
assumptions about the shape of the scattering phase function. The only irradiance
measurements needed are Ed and E, just below the surface, and Ed just above the
surface. Gordon also shows how the phase function can be estimated over a limited
range of scattering angles, typically 60' < T < 150'. However, the i-ecovery of the
phase function is not as satisfactory as is the recovery of a, b and bb.
As one of the important inversion problem example, the oceanic remote sensing
measures the ocean surface wave conditions, IOP's or constituents of natural waters
from aircraft or satellite. The remote sensing using electromagnetic signals is commonly performed from the near UV to various radar bands, whose wavelengths range
from 1 cm to 1 m. This remote sensing can be active or passive. Active remote
sensing means that a signal of known characteristics is sent from the sensor platforman aircraft or satellite - to the ocean, and the return signal is then detected after
175
a short time delay determined by the distance from the platform to the ocean and
by the speed of light. An example of active remote sensing at visible wavelengths is
the use of laser induced fluorescence to detect chlorophyll, yellow matter, or pollutants. In laser fluorosensing, a pulse of UV light is sent to the ocean surface, and
the spectral character and strength about the location, type and concentration of
fluorescing substances in the water body. In passive remote sensing we simply observe the electromagnetic radiation that is naturally emitted or reflected by the water
body. One example at visible wavelengths would be the night time detection of bioluminescence from aircraft. Another example is the detection of sunlight that has
been backscattered by the water.
As an example, the color of a water body can be computed if a spectral radiometric
quantity is nmeasured over the visible wavelengths. In practice this is seldom done.
Spectral signals usually are measured at only a few selected wavelengths, and other
forms of information than the color itself can be obtained from such data. The term
ocean color is therefore loosely used to mean radiometric data at two or more visible
wavelengths, from which useful information about water bodies can be extracted.
Sunlight, whose spectral properties are known, enters a natural water body. The
spectral character of the sunlight is then altered, depending on the absorption and
scattering properties of the water body, which of course depend on the types and
concentrations of the various constituents of the particular water body. Part of the
altered sunlight eventually makes its way back out of the water, and can be detected
from an aircraft or satellite. If we know how different substances spectrally alter
sunlight, for example by wavelength dependent absorption or by fluorescence, then
we can hope to deduce from the altered sunlight what substances must have been
present in the water, and in what concentrations.
The water-leaving radiance, which is commonly denoted by L" is the upwelling
radiance measured in the air just above the water surface.
L ,(0, #, A) = L-(z = a, 0, #, A)
176
(7.2.34)
where 7r/2 < 0 < 7r in the coordinate system. A sensor looking straight down
at the water surface sees the zenith radiance heading straight upward. We could
computed the color associated with L,(A) for case 1 waters. From the computed
results, we can see the color shifts from blue to gree as the chlorophyll concentration
the chlorophyll concentration C increases. By plotting the color on CIE chromaticity
diagram for different values of C, we could determine the path followed across the
diagram as C increases. If we then measured L, (A) from a water body with an
unknown C values, and determine the value of C that most closely corresponds to
the color of the water body in question. We then could estimated the chlorophyll
concentration of the water body using only the remotely sensed spectral radiance.
7.3
Inverse algorithm of reconstructing IOPs
The light propagation in a turbulent flow can be well described by the radiative transfer equation (RTE) with spatially variable attenuation coefficient a(r) and scattering
coefficient c(r).
Q -VI + c(r)I = b(r)
Jp(Q, Q')I(r, Q')dQ'
(7.3.1)
The boundary condition of RTE here is described as
I (r, Q) = linc (r, Q), r E (9V
(7.3.2)
where &V is the boundary and Inc is the incident specific intensity at the boundary
towards the inward direction. Together with boundary condition, the RTE can be
written as an integral form
I(r, Q) = Io(ri, QI)G(r,Q; ri, Q1)
+
J G(r, Q; r',Q')b(r')p(Q', Q"I(r',Q")d r'd Q'd Q"
3
177
2
2
(7.3.3)
where Io is the incident light source at the point r1 in the direction Q1
Io(ri, Q1) = 6(r - ri)6(Q - Q 1 )
(7.3.4)
and G is the Green's function for the ballistic RTE whose scattering effects are
ignored which is
Q . VI + c(r)I = 0
(7.3.5)
The ballistic Green's function has the following form
G(r, Q; r', Q')
=
x 6 r
1
- jr-r'I
rf 2x
[X9J
- r - r/
c(r' +
(
r r)dl
lir -r
|
-O)
|r - r |)
(7.3.6)
The integral equation can be written as a born series with convergence requirement
satisfied.
I(r, Q) = I(0) (r, Q) + I) (r, Q) + 1(2) (r, Q) ...
(7.3.7)
Where each term of the series is given by
jG(r, Q; r', D')b(r')p(Q',
Qf")I(n-1)
(r', Q/")d3r'd 2Q'd2Q"1
(7.3.8)
The exact solution of this integral equations needs recursive iterations and it is
usually treated numerically.
7.3.1
Single scattering approximation
Instead of using the flux measurement to retrieve information of the inherent optical
properties of turbulence, the radiance data reveal more information of the internal
178
Figure 7-7: Single scattering light propagation geometry
structure of turbulence. Schotland and Markel were the first to treat the problem
with the single scattering approximation. This approximation assume higher order
Born series I(')(r, Q) where n > 2.
Under this approximation, the radiance I,(ri, Q1 ; r 2 , Q2 ) at the point r 2 in the
direction Q2 can be expressed in the following equation
Is (x1, 1; x 2 ,0 2) = iince(7r - 01
x rps
7
- 02)6(1#1 - 021 - 7r)
'
r21 sin 1sin(
exp
2
fo
L- a(r 1 + 1I)dl
-L2
x exp I- JOa(R21 +
1Q2)dl
(7.3.9)
Where 8(x) is the step function. R 21 can be understood with above figure. the
following relations can be obtained as some math manipulation.
R21 = r1 + L1Q1 = r2 - L2 Q 2
si(
179
sin 02
0 1
+02)
(7.3.10)
(7.3.11)
L2 = T2 1 s.
sin 01
in 02
smn(61
(7.3.12)
+ 02)
From the figure, we can find the physical interpretation of the single scattering
radiative transport of light beams. The light photon travels in the direction Q1 and
attenuated at the exponential rate. At every position where it pass through the
beam is broken into second ray for all directions. The scattered ray directly enters
the detector without interference of scattered and only decays exponentially.
From the single scattering solution, we can write the following path integral of
attenuation coefficient c(r)
#(ri, Q1; r2,
Q2)
= J(jL)c(r(l))dl
(7.3.13)
The path integral is performed in the path Li and L 2 consecutively. Based on this
integral, an inversion scheme can be employed by discretize the attenuation coefficient
on a rectangular grid. Then the equation can be written as
AmnCn =
#nt
(7.3.14)
where Anzn is the element of the transfer matrix A. The matrix form of it is
Az = #
(7.3.15)
where i2has the element cn. Above equation can be easily solved with optimization
method since the element of A can be analytical acquired form single scattering
approximation theory. For example, use a regularized pseduoinverse.
7.3.2
Differentiation techniques
The above technique introduced by Schotland and Markel solves the equation with
classical numerical optimization method. Therefore, iteration must be done and it
requires an efficient forward solution of three-dimensional radiative transport equation
with high accuracy. This becomes computationally expensive. Noticing that the form
180
0.
rx
xx
L2
X2
Single or multiple detectors
2
Figure 7-8: Geometry of two-dimensional radiance measurement setup
of path integral of attenuation coefficient, it is possible to apply spatial and angular
differentiations in reconstruction process under proper assumptions.
We assume the perturbation of attenuation coefficient of turbulence is weak. Priori
information of the background attenuation coefficient is known. The two-dimensional
geometry of measurement setup is shown above.
The idea of using the single scattering based on light beam translation in turbulence is to obtain the optical property information by differentiating measurement
angularly or spatially. Under this geometry, we have the expressions of the following
parameters,
01 = arctan(
h
)
(7.3.16)
)
(7.3.17)
h
02
=
arctan(
r2l = vh
2
I2 - XiI
+ (X2
181
-X)2
(7.3.18)
L
1= h - (x 2 - xi) tan 0
-
L2 =
/(1I2)
(7.3.19)
(7.3.20)
X
cos 0
= |7r/2 - 01
(7.3.21)
Then the measurement can be made on the parameter f(x 1 , x 2 ; 0) with expression
as
fx;(x,
fI
)=
r21 sin 01 sin 02
-I
bp(|7r/2 - 01)
c(l)dl +
=
By taking the derivative of
af
ao
F(xi,x 2 ;0)
+
JL
f
c(l)dl
(7.3.22)
over angular variable 0, we have the following form.
(R
ao c(R21)
2
j
+
aL2c(r
90 2)
[(xi + 1 cos 0)xc + (-L1 - i sin 6)1 dl
(7.3.23)
Under weak perturbation assumption, the third term of right hand side of equation
can be neglected due to that it is the second order term. Based on this, the light
source is the shined sequentially normal to the turbulence flow from left to right and
recorded by the detector at the bottom of turbulence. We differentiate F spatially
between two light sources. We have
(X"),
0)F(n) X2;
F(n-1)(X2
) , X 2 ; 0) =
aa(R")Ax1 1ao
a(R1-1 )Axi
(7. 3.24
(7.3.24)
182
Define GC") = FC) - F(4-) we have following recursive relation under weak
perturbation approximation where the third term is neglected.
G(")
09
c(090 -
_
_
_
c(R 1
)
(7.3.25)
To demonstrate the method, we show a simple reconstruction of turbulent attenuation coefficient.
Figure 7-9: Example of using single scattering approximation and radiance measurement to reconstruct ocean turbulent structure (a) Original turbulent flow structure
of attenuation coefficient (b) Reconstructed attenuation coefficient with 50 sources
(c)Reconstructed attenuation coefficient with 100 sources (d)Reconstkticted attenuation coefficient with 150 sources
The forward problem is executed with inhomogeneous Monte Carlo method. We
can see that from figure 7.9 that the number of sources determine the resolution
of reconstructed images. It is understood that the angular resolution of radiance
measurement is another important factor of resolution of images.
183
7.4
Application of RT Prediction in Engineering
Radiative transfer theory has many applications in astro-physics, particle physics,
meteorology and oceanography, laser fabrication techniques and bio-imaging technology. The neutron and electron scattering,light diffusion in cloud and ocean, photon
migration in bio-tissues, laser ablation can all be described by the radiative transfer
equation. In this thesis, we briefly introduce the practical applications of radiative
transfer theory in atmosphere science and oceanography.
7.5
Remote sensing of cloud properties via radiative transfer
Clouds play a very important role in the climate system. A cloud is composed of
millions of little droplets of water or ice crystals, when temperature is very low, suspended in the air. Clouds could form when water vapor becomes liquid, then humid
air is cooled the water vapor condenses onto tiny particles. The major influence of
clouds on climate is done through the interaction of sunlight and cloud particles. Sunlight consists of visible light, ultra-violet and infra-rad radiation. The atmosphere,
the ocean and most of the clouds reflect part of the sunlight which reaches the Earth
back to space. Around 70% of energy from the Sun reaches the surface of the Earth
and most of this is absorbed by the surface. Just as our skin warms up when the Sun
shines on it, the Earth warms up and emits infra-rad radiation and this heats the air
above the ground. This infra-red radiation has a different energy to the sunlight absorbed by the surface of Earth. If all the infrared radiation emitted from the Earth's
surface escaped directly back into space without being trapped, our planets temperature would be -18 o C. The averaged temperature of our planet keeps warm because
of the Clouds, the so called Greenhouse effects. The presence of clouds and greenhouse gases such as water vapor and carbon dioxide keep our average temperature
much higher.Clouds cover about 50% of the sky and these clouds absorb radiation
emitted from the surface of the Earth. The clouds then re-emit a portion of the en184
ergy into outer space and a portion back toward the surface. It is this portion which
warms our planet and is why clouds have the capability for reducing temperature
differences between the day and the night. During the day the ground is warmed
up by the sunshine. The fewer clouds there are in the sky, the more the surface of
the Earth is heated by the Sun. If there are no clouds during the night, most of the
infra-red radiation emitted by the Earth goes back into space and the night is cold.
If the sky is cloudy, part of the infra-red radiation from the Earth is trapped by the
clouds. Some of this radiation is then reflected back to the Earth's surface and the
temperature of the air above the ground is warmer that it would have been if the
night had been cloudless. High thin clouds like cirrus clouds contribute to heating,
whereas low thick stratocumulus clouds tend to cool our planet.
Therefore, in order to successfully forecast the weather or to understand the climate change of the planet, it is critical to acquire the vertical and horizontal properties
of the clouds.
Using satellite remote sensing techniques to retrieve clouds information has been
studied and widely used for decades. Many satellite remote sensing application involves the three-dimensional radiative transfer. Most frequently, the remote sensing
is used to retrieve the optical properties of clouds such as optical depth (or attenuation coefficient). The three-dimensional RT even was applied to acquire the rainfall
rates from passive microwave radiances by Weinman and Davies (1978). More other
type of applications of radiative transfer in remote sensing have been made about
retrieving temperature and humidity of atmosphere or land features.
Because of the strong scattering and absorption of solar radiation by clouds particles and the fact the clouds are highly heterogeneous in optical properties, the radiation detected by spaceborne or airborne sensor complicated. Accurate model of radiation model is required to ensure precise inversion algorithm. The three-dimensional
radiative transfer in atmosphere provides such forward model. The most popular
atmosphere radiative transfer model is MODTRAN. MODTRAN is based on the solution of radiative transfer equation with discrete ordinate method. Now MODTRAN
is widely used in many areas.
185
7.6
Application of retrieving ocean conditions to
fishery
Both commercial and naval need of knowing ocean surface conditions are increasing.
The surface waves conditions are critical to fishery fleet. The surface roughness is
directly associated with wind conditions above the ocean. To measure the surface
elevation and slope in large areas, the reflectance of the ocean surface as an function
of frequency can be done by satellite remote sensing. Both experiments and theories
have proved the dependence of surface reflectance and ocean wave features, such as
slope distribution and its statistic characteristics.
The measurement can provide weather information over a wide area at a given
time. This could assit fisherman to plan their fishing operations. For those in higher
latitudes, ice and icebergs are major hazards.
186
Chapter 8
Conclusion
This thesis thorough go through the radiative transfer theory in the use of coupled
ocean and atmosphere system. The Monte Carlo codes of several different radiative
transfer equations are developed to solve the light scattering and absorption within
the turbid ocean body. The main focuses of the thesis are on the influences of dynamic ocean surface and turbulent upper-ocean level flow on underwater light field
distribution and fluctuation. Systematic studies have been done based on the DNS
capability of radiative transfer predictions. With the understandings of those effects
on light fields, an inversion scheme is proposed to reconstruct the turbulent inherent optical properties of upper-level oceans based on the measurement of underwater
radiance.
Both DNS capability and inversion algorithm provide various potential applications to retrieve ocean surface information with long-term observation and real time
predictions.
187
188
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