A three-dimensional radiative transfer model for shallow water

Transcription

A three-dimensional radiative transfer model for shallow water
A three-dimensional radiative transfer
model for shallow water environments
John Hedley
School of Biosciences, University of Exeter, EX4 4PS, UK
[email protected]
Abstract:
A geometric optical model for three-dimensional radiative
transfer capable of handling arbitrary arrangements of surfaces within
anisotropic scattering media is described. The model operates by discretizing surfaces and volumes into patches and voxels and establishing
the radiative transfer relationship between every pair of elements. In a
plane-parallel configuration results for directional radiance agree closely
with the numerical integration invariant imbedded method. Model accuracy
for two examples incorporating surface water waves and complex benthic
structures were assessed by conservation of energy, errors were less than
1%. Potential applications in remote sensing or photobiological studies of
structurally complex benthos in shallow water environments are illustrated.
© 2008 Optical Society of America
OCIS codes: (010.5620) Radiative transfer; (080.1753) Computation methods; (280.0280) Remote sensing
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1.
Introduction
Many shallow water ecosystems such as coral reefs or kelp forests exhibit complex underwater
three-dimensional structures with vertical scales comparable to the height of the water column
itself. The penetration of light into these ecosystems plays a significant role in influencing their
productivity and biological composition [1] and the backscatter of this same light provides
the primary means for monitoring these systems by optical remote sensing [2]. In addition to
benthic complexity, the propagation of water surface waves causes fluctuations in depth and
water surface slope which may have radiative transfer consequences at time scales relevant to
photobiology and remote sensing: wave focusing [3] and spatial sunglint patterns [4] are just
two phenomenological examples. Many current important research questions relate to these
structurally complex aquatic ecosystems and so there is an imperative to develop the capability
for accurate modeling of the interaction light with three-dimensional structures underwater.
Some of these active research areas include the adaptive role of coral morphology [5, 6, 7];
primary productivity in macroalgal and seagrass canopies [8]; advancing and understanding
the limitations of remote sensing of coral reefs [9].
A variety of models for aquatic radiative transfer appropriate for radiometrically planeparallel environments with time-averaged horizontal homogeneous boundary conditions have
been developed [10, 11, 12, 2], but Monte-Carlo methods [13] remain the only general method
for handling arbitrary three-dimensional aquatic radiative transfer. However, Monte-Carlo
methods face computational efficiency limitations and thus far their application has been restricted to plane-parallel systems or examples of simple inhomogeneities in benthic reflectance,
substrate slope or one-dimensional water surface patterns [14, 15, 16, 17]. Application-specific
geometrical optics models [18] have also been devised to address specific questions but are by
definition limited in scope.
In this paper I present a radiative transfer model based on the Inherent Optical Properties
(IOPs) of source-free waters which is capable of accommodating arbitrary surfaces and participating volumetric media within a three-dimensional domain. The algorithm applies a spatial and
volumetric discretization, initializes the system with incident solar energy, establishes the radiant power transfer between every pair of elements and converges to a solution by Gauss-Seidel
iteration [19] (Fig. 1). The model has design similarities to radiosity or global illumination
methods, which had early applications in modeling the physics of thermal transfer in furnaces
[20], then were subsequently developed for realistic computer rendering of three-dimensional
scenes [21, 22]. Radiosity methods have also been applied to photobiology and remote sensing of terrestrial vegetation canopies [23, 24]. Here the method is extended to allow surface
structures to be embedded in volumetric scattering media and incorporate boundary surfaces
between compartments of differing refractive index. The model contains two important features: 1) the capability for arbitrary anisotropic volumetric scattering functions, thus allowing
the use of Petzold’s, Fournier-Forand or any form of phase function [25, 26] and 2) an analytical
derivation of the radiative transfer relationships involving volumetric elements that significantly
improves the accuracy compared to the original radiosity Zonal Method [27]. Other aspects of
the model are identical to, or bear strong similarities to, existing methods from both hydrological [2] and atmospheric optics [28].
The next section establishes the model formulation in general terms, followed by the method
of spatial and angular discretization (Sec. 3) and the solution algorithm (Sec. 4). Section 5.1
demonstrates that in a plane-parallel configuration, model outputs for directional radiance are
virtually identical to the invariant imbedded method (as used in the commercial software HydroLight [29]). When three-dimensional structures are introduced (Secs. 5.2 and 5.3) one method
for assessing accuracy is energy conservation. For the coral-like structures assessed here losses
or gains are only a few percent and can be further constrained at computational expense.
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Fig. 1. Overview of the 3D model. Surfaces are spatially discretized into patches while
regions of space containing scattering media are discretized into cubic voxels (only a subset
of the voxels are illustrated).
2.
Model theoretical background
2.1. Optical properties of volumetric media
In the absence of inelastic scattering the parameters necessary and sufficient to calculate radiative transfer through a volumetric media are the spectral absorption coefficient a(λ , p) and spectral volume scattering function (VSF), β (λ , p, ψ ) where λ is the wavelength, p = p x , py , pz is a point in space, and ψ is the scattering angle, 0 ≤ ψ ≤ π [2]. The co-efficients of scattering
b(λ , p) and attenuation c(λ , p) are derived from the absorption co-efficient and the VSF [2]
b(λ , p) = 2π
π
0
β (λ , p, ψ ) sin ψ dψ
c(λ , p) = a(λ , p) + b(λ , p)
(1)
(2)
In some literature c is referred to as the extinction co-efficient and denoted k [20].
Scattering of the radiance distribution at a point p in space, L(λ , p, v) by the VSF gives rise
to the path radiance at that point,
L∗ (λ , p, vout ) = L∗E (λ , p, vout ) +
Ξ
L(λ , p, vin )β (λ , p, cos−1 [vin · vout ]) dΩ(vin )
(3)
where dΩ(vin ) is the solid angle associated with v in over which the integral is performed and
L∗E allows for radiant emission from the medium.
The radiance distribution at a point L(λ , p, v) is the sum of the path attenuated radiance
received from point p S on the first surface encountered in the direction looking back along v,
plus any radiance received from volumetric scattering or emission in the intervening path.
L(λ , p, v) = K(λ , pS , p)LEX−G (λ , pS , v) +
pS
p
K(λ , p , p)L∗ (λ , p , v) dx(p )
(4)
where LEX−G (λ , pS , v) is the radiance exitant from the surface at point p S in the direction vector
v, expressed in the global co-ordinate system. The function K(λ , p 1 , p2 ) is the transmittance of
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radiance from point p 1 to p2 in the medium,
p
2
K(λ , p1 , p2 ) = exp
−c(λ , p ) dx(p )
p
d1
x
= exp
−c(λ , p1 + (p1 − p2 )) dx
d
0
(5)
where d = |p1 − p2 | is the distance between p1 and p2 . If the attenuation coefficient is spatially
homogeneous then Eq. (5) reduces to K(λ , p 1 , p2 ) = exp[−c(λ )d].
Note that there is no need to consider the path along the vector negative v beyond the first
intersecting surface even if that surface is transparent, since the effect of radiance sources on
the far side of the surface is expressed as radiance exitant the surface itself. A key part of the
algorithm described below is the efficient detection of the first surface encountered looking
back along a given incident vector from a point in space.
2.2. Optical properties of surfaces
Surfaces in the model may take arbitrary shapes in space subject to the assumption that at some
small scale they may be approximated as locally flat. At this locally flat scale the optical properties of general two-sided reflecting and transmitting surface, such as the air-water interface, can
be described by two directional reflection functions, r 1|2 (λ , p, uin , uout ), for sides 1 and 2, and
two directional transmission functions t 12|21 (λ , p, uin , uout ) for transmission in both directions
[2]. Here uin and uout represent incident and exitant vectors in the surface’s local co-ordinate
system which is defined with the z-axis normal to the surface side 1 and the x-axis and y-axis
lying in the tangential plane. Premultiplying a vector by the 3 × 3 matrix M G→L (p) performs
the rotation from global co-ordinates to local surface co-ordinates at point p,
⎞
⎛
Xx (p) Xy (p) Xz (p)
MG→L (p) = ⎝ Yx (p) Yy (p) Yz (p) ⎠
(6)
nx (p) ny (p) nz (p)
Where n(p) = nx (p), ny (p), nz (p) is the unit normal on side 1 at point p. Thus defined, the local surface tangential x-axis and y-axis vectors X x (p), Xy (p), Xz (p) and Yx (p),Yy (p),Yz (p)
have rotational freedom about n(p). For surface types for which the optical properties are rotationally dependent, e.g. a wind-blown air-water interface [2], the orientation of x and y must be
appropriately aligned across the surface. The inverse rotation, from local to global co-ordinates
can similarly be derived and is denoted M L→G (p).
Since the four functions r 1 , r2 , t12 and t21 jointly cover all hemispherical domain combinations in uin and uout it is convenient to combine them into a single surface transfer function with
spherical domains in u in and uout ,
⎧
r (λ , p, uin , uout ), if uin ∈ Ξz− and uout ∈ Ξz+
⎪
⎪
⎨ 1
r2 (λ , p, uin , uout ), if uin ∈ Ξz+ and uout ∈ Ξz−
s(λ , p, uin , uout ) =
(7)
t
⎪
12 (λ , p, uin , uout ), if uin ∈ Ξz− and uout ∈ Ξz−
⎪
⎩
t21 (λ , p, uin , uout ), if uin ∈ Ξz+ and uout ∈ Ξz+
The surface transfer function establishes the relationship between radiance exitant from
a surface in its local co-ordinate system L EX−L (λ , p, uout ) and the incident radiance,
LIN−L (λ , p, uin ),
LEX−L (λ , p, uout ) = LE (λ , p, uout ) +
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Ξ
LIN−L (λ , p, uin ) s(λ , p, uin , uout ) dΩ(uin )
(8)
Received 17 Oct 2008; revised 8 Dec 2008; accepted 9 Dec 2008; published 17 Dec 2008
22 December 2008 / Vol. 16, No. 26 / OPTICS EXPRESS 21891
This expression also allows for emissive surfaces when L E (λ , p, uout ) = 0 and treats both sides
of the surface simultaneously, so u out ∈ Ξ. The radiance incident on a surface is calculated in
an identical manner to the radiance at a point in volumetric space (Eq. (4)), but note that Eq.
(4) is defined in the global co-ordinate system while L IN−L (λ , p, uin ) is in the local surface
co-ordinate system, so rotation of the incident vector by M L→G (p) is required,
LIN−L (λ , p, uin ) = L[λ , p, ML→G (p)uin ]
(9)
The inverse rotation from global to local co-ordinates using M G→L (p) is similarly required
to establish LEX−G as used in Eq. (4) from L EX−L (λ , p, uout ). Hence, the co-ordinate system
rotations defined by M G→L (p) and its inverse map the local surface radiance transfer functions
onto a non-flat surface. The functions r 1 , r2 , t12 and t21 with the p-dependence removed are
identical in form to the air-water interface radiance transfer functions presented by Mobley [2].
For an opaque boundary surface, such as a the substrate in a shallow water application, only
r1 is non-zero and,
r1 (λ , p, uin , uout ) = R(λ , p, uin , uout ) cos θ
(10)
where R(λ , p, uin , uout ) is the bi-directional reflectance distribution function (BRDF) [30] of
the surface at point p, and cos θ is the z-component of u in . For many applications it may be
sufficient to consider the BRDF as constant for different surface types, so models can be built
with a small set of BRDFs which in themselves are not p-dependent. Further, if it is appropriate
to consider surfaces as locally Lambertian both R and the surface exitant radiance L EX−L lose
their directional dependence. In this case significant computational storage savings can be made
and r1 (λ , p, uin , uout ) = [RD (λ , p) cos θ ]/π , where RD (λ , p) is the diffuse reflectance at point p
on the surface [2].
Fig. 2. Directional discretization schemes. (a) HydroLight standard spherical partition, (b)
the cubic 8-partition and (c) as a spherical projection, (d) the corresponding hemicube
8-partition for surface directional functions and (e) its unfolded visualization as used in
Fig. 4.
3.
Discretization of model
3.1. Directional discretization
Like the plane-parallel invariant imbedded solution method [2] the model presented here calculates directional radiance as the mean over segments of finite solid angle defined by a discretization of the surface of a sphere (Fig. 2). However, instead of a system based on angles
of constant zenith and azimuth [2] here the sphere surface is partitioned into cells (or ‘quads’
[2]) numbered q = 1, 2, 3, . . . 6n 2 by a spherical projection of a cubic grid with n-cells on a side,
referred to as a cubic n-partition (Fig. 2). This partitioning scheme is identical to the hemicube
and fullcube methods developed for computer graphics radiosity methods [21, 22] and is a
requirement of the algorithm. The cubic 8-partition contains 384 cells and offers comparable
directional resolution to the standard 482-cell spherical discretization used in the commercial
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invariant imbedded implementation HydroLight [29]. One advantage of the cubic partition over
the constant zenith-azimuth angle partition is that directional resolution can be increased, i.e.
n = 16, 24, 32, while retaining a relatively uniform sampling of the sphere surface. Ensuring n
is even allows trivial conversion between functions defined on the sphere and functions of surfaces that only require a hemispherical or ’hemicubic’ n-partition, such as BRDFs. For a given
n-partition the solid angle Ω(q) of each cell can be pre-calculated by numerical integration.
The cubic n-partition of the sphere enables the cell-averaged tabulation of functions of one
vector direction, such as incident solar radiance or the radiance distribution at point, and of
functions of two directions, such as the VSF. A mapping from an outward orientated vector
direction u = ux , uy , uz to a cell index q can be implemented highly efficiently, requiring just
a single division as the most expensive computational operation,
⎧
for ux < uy and ux < uz
⎪
⎪ 02+ n
f xuy + h + fxuz + h,
⎪
⎪
n
+
n
f
u
+
h
+
f
u
+
h,
for
ux > uy and ux > uz
⎪
x
y
x
z
⎪
⎨ 2
2n + n
fy ux + h + fyuz + h, for uy < ux and uy < uz
q(u) = 1 +
(11)
3n2 + n
fy ux + h + fyuz + h, for uy > ux and uy > uz
⎪
⎪
⎪
⎪
4n2 + n
fz ux + h + fzuy + h, for uz < ux and uz < uy
⎪
⎪
⎩ 2
5n + n
fz ux + h + fzuy + h, for uz > ux and uz > uy
where h = n/2 and f x = h/|ux |, fy = h/|uy |, fz = h/|uz |. Each of the six options in Eq. (11)
corresponds to one face of the projected cube. The situation in which a direction vector lies exactly on an edge between cube faces can be detected easily by the equality of the u values and
is handled by consistently assigning such vectors to one of the adjacent faces. The computational efficiency of converting a vector direction to cell index is important in the implementation
because it occurs a huge number of times in the solution process.
Using Eq. (11), directionally dependent quantities expressed in terms of vectors can be tabulated into q-dependent quantities, e.g. L(λ , p, v in ) in Eq. (3) becomes L(λ , p, q in ). In particular
the VSF, β (λ , p, ψ ), is tabulated as, β (λ , p, qin , qout ) by numerical integration [2]. If the VSF or
scattering phase function are considered spatially invariant, by assuming Petzold’s [25] phase
function, for example, then considerable computational savings can be made by pre-calculation
of the tabulated VSF [2]. The current implementation of the model works directly with the
tabulated directional values of radiance and does not utilize representation by orthogonal basis
functions such as Spherical Harmonics [28, 31] or the Discrete Fourier Transform [2] to efficiently exploit the rotational invariance of the VSF [2, 28]. However the 48-fold symmetry of
the cubic partition substantially reduces the required VSF data structure size.
3.2. Surface discretization
The scheme used to represent surfaces as a mesh of polygonal patches requires that 1) the
radiance distribution at a patch center is representative of the whole surface of the patch, and 2)
surfaces appear perfectly sealed when rendered from any view direction by a computer graphics
algorithm. The model operates by establishing the radiance incident onto a patch from the
perspective of a single view-point at the patch center, and then translating that into an uniform
exitant radiance over the surface of the patch using Eq. (7). Requirement 1 implies that patches
must be sufficiently small, especially on curved surfaces or on the boundaries of shadows. A
consequence of the second requirement is that patches must be exactly flat to prevent rendering
errors where cracks could allow leakage of radiance through opaque surfaces. This can be
ensured by using only triangular patches on curved surfaces, restricting the use of quadrilaterals
to flat surfaces. Depending on how the scene is constructed a polygon intersection algorithm
may also be required as a pre-processing step to subdivide any intersecting pairs of patches
along the intersection line. Overall, the requirements for surface meshing are identical to those
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for many computer graphics applications and are discussed in that literature [32].
Once the surface mesh has been established all positionally dependent surface properties can
be tabulated over the finite number i = 1, 2, 3, . . . n P of patches. E.g. with directional discretization applied as well, LEX−L (λ , p, uout ) in Eq. (8) becomes L EX−L (λ , i, qout ). The spatial and
directional properties of all surfaces are thus finitely described ready for algorithmic manipulation.
3.3. Volumetric discretization
Volumetric discretization is initiated by subdividing the model domain into cubic voxels (Fig.
1) with each voxel view-point at the voxel center. However for voxels the translation of incident
radiance into scattered radiance is not as straightforward as the direct application of the path
radiance equation at the voxel center (Eq. (3)). This approach is taken in Zonal Method for
computer graphics [27] and while producing visually acceptable solutions it ignores processes
which occur over the path length through the voxel itself, potentially leading to substantial
violations of conservation of energy.
Due to the voxels cubic shape (Fig. 3), the incident radiance at a voxel center has a qdependent voxel path length, x(q) (note that both the orientation of the voxel and its directional
discretization are global-axes aligned). The distance x(q) can be numerically pre-calculated for
all voxels of given size as the mean over the solid angle of the cell q. Rather than using the
radiance distribution at the voxel center, L(λ , i, q in ), to estimate the scattered radiant energy
from the whole voxel, a more accurate estimation is obtained by integrating the source function
(Eq. (3)) along the entire path length d = x(q). This method takes into account both the incident
radiance attenuation and increase due to in-scattering along the path in the voxel. If emission is
zero the form of the source function (Eq. (3)) based on these path integrals is,
∗
L (λ , i, qout ) =
6n2
∑
qin =1
β (λ , i, qin , qout ) [LEXT (λ , i, qin ) + LSELF (λ , i, qin )] Ω(qin )
where
LEXT (λ , i, qin ) =
and
LSELF (λ , i, qin ) =
d x
0
0
d/2
−d/2
L (λ , i, qin ) exp[−c(λ , i)x] dx
L∗ (λ , i, qin ) exp[−c(λ , i)t] dt exp[−c(λ , i)x] dx
(12)
(13)
(14)
Here L (λ , i, qin ) in Eq. (13) is the radiance distribution at the voxel center from external
sources, i.e. excluding scattered radiance from the voxel itself. This is the quantity applied
directly in Eq. (12) by the Zonal Method [27]. Eq. (13) extrapolates the radiance at the center point forward and backward to the voxel perimeter and integrates over the path length.
Internal in-scattering to the incident radiance direction is handled by Eq. (14) which simultaneously integrates and attenuates the current voxel path radiance along the path through the
voxel. Since the voxel-averaged VSF, β (λ , i, q in , qout ), is assumed constant within the voxel it
has been removed from both path integrals (Eqs. (13) and (14)) and incorporated into Eq. (12).
The integrals can then be analytically derived as,
c(λ , i)x(qin )
−c(λ , i)x(qin )
L (λ , i, qin )
LEXT (λ , i, qin ) =
exp
− exp
(15)
c(λ , i)x(qin )
2
2
LSELF (λ , i, qin ) =
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L∗ (λ , i, qin )
(c(λ , i)x(qin ) + exp[c(λ , i)x(qin )] − 1)
c(λ , i)2 x(qin )
(16)
Received 17 Oct 2008; revised 8 Dec 2008; accepted 9 Dec 2008; published 17 Dec 2008
22 December 2008 / Vol. 16, No. 26 / OPTICS EXPRESS 21894
For a given beam attenuation, c(λ , i), and voxel size, x(q in ), Eqs. (15) and (16) reduce to simple
directional q-dependent multiplicative factor for L (λ , i, qin ) and L∗ (λ , i, qin ). In many model
applications the range of voxel sizes and beam attenuation coefficient values will be very restricted, with, for example, spatially uniform attenuation and a single uniform voxel size, so
these multiplicative factors can be pre-calculated and tabulated for q with little computational
overhead. Note that while Eqs. (12), (15) and (16) are circular for L ∗ , the solution method is
iterative, so L∗ is updated from the current value on each iteration.
Fig. 3. Voxelization of volumetric radiative transfer. (a) Scattered energy resulting from a
beam of radiance passing through the voxel through solid angular cell qin and the viewpoint p is found by integrating radiance along the in-voxel path length d. The resulting
directional path radiances L∗ (qout ) are assumed uniform throughout the voxel volume, and
the radiance leaving the voxel is found by evaluating the integral of Eq. (4) along the exitant
direction path length. (b) When surfaces and voxels intersect the path lengths are truncated
and the view-point is shifted to the center of the remaining voxel segment.
3.4. Voxel and surface intersections
The above discussion applies to voxels in open space filled only with volumetric scattering
media. A specific modification is required to accommodate situations in which a voxel has
surface elements within it:
1. Where a surface element intersects into a voxel it is determined if the voxel is completely
bisected by the surface and if scattering is non-zero on both sides. If so, two notional
voxels are superposed in space, one for each side of the surface.
2. If voxels are subdivided by a surface the view-point of the voxel is shifted to most closely
represent the center point of the remaining voxel segment on the appropriate side of the
surface.
3. The path length x(q) for each directional cell is re-calculated based on the path length
through the view-point to the first intersecting surface or voxel perimeter in both directions (Fig. 3).
The above procedure maintains good theoretical accuracy only when voxels are wholly subdivided by surfaces of relatively low curvature compared to the voxel size. Where a large voxel
contains a complex surface errors are introduced due to parts of the voxel itself being no longer
visible to the view-point. However in practice, these introduced errors often do not seriously
confound overall accuracy for two reasons: 1) the error only pertains to the volumetrically
scattered component of the light in small interstices of complex structures, for the intended
applications in waters of relatively high clarity this energetic component is extremely small;
and 2) the error can be reduced to an arbitrary level anyway by recursively subdividing voxels.
The model implementation has the capability to adaptively refine both the surface and volume
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22 December 2008 / Vol. 16, No. 26 / OPTICS EXPRESS 21895
discretizations in areas of high radiance gradient by subdividing polygons and voxels between
solution iterations [33]. However, to simplify the discussion this feature is disabled for the
results presented in this paper.
3.5. Domain boundaries
In the model implementation the boundaries in the x, y, and z directions may be independently
specified as either periodic, so that the model structures are repeated for a set number of times,
or a fixed incoming boundary radiance distribution may be specified. For shallow water applications the most appropriate configuration is that the domain repeats horizontally (x and y
directions) but the upper z boundary is initialized with a sky downwelling radiance model. Note
that this scheme requires that opposite sides of the model domain match exactly, so for example
the water surface must be periodic at the width of the model domain.
4.
Algorithm operation
The key process in the algorithm is establishing the incident radiance distribution at the viewpoint of an element by rendering the entire model domain as seen through a hemispherical or
spherical field of view (Fig. 4). This is achieved by projecting the current elements onto each
face of high resolution hemicubic or cubic partition (e.g. n = 128), using standard computer
graphics techniques, and then down-sampling to the directional resolution of the elements scattering function. The approach is conceptually identical to that of to the hemicube radiosity
method [21]. Figure 4 illustrates the steps taken in establishing the incident radiance distribution for a surface patch and translating it into the patch exitant radiance distribution. The overall
sequence of events for model setup and solution is as follows:
1. The input surface vector meshes are pre-processed (§3.2)
2. The volumetric subdivision into voxels is established and pre-processed (§3.3, §3.4)
3. The upper z-boundary incident radiance is initialized with a sky radiance model (§3.5)
4. All elements have their exitant radiance distributions set to zero.
5. For each element in the model, exitant radiance is updated based on the current incident
radiance distribution at the element view-point (Fig. 4). The change in exitant radiant
energy for this iteration is summed over all elements (§3.2, §3.3).
6. Iteration continues from Step 5 until either the total change in exitant energy for the last
iteration falls below a predefined fraction of the total energy (called the solution stability
threshold) or the maximum permitted number of iterations has occurred.
Convergence is guaranteed provided some absorption occurs, however there is no implicit
constraint that the final solution will observe conservation of energy. In fact, small errors in
the radiative transfer calculations can escalate to substantial discrepancies between the energy
absorbed and input, e.g. a 10% loss in the radiant transfer between elements will increase the
rate of convergence but result in nearly 70% energy loss over 10 iterations. In a shallow water
application the radiant energy transferred upward through the water surface should be equal
the energy transferred downward through the surface minus the energy absorbed by patches
and voxels. Any discrepancy in the final energy balance is therefore a useful metric for overall
model accuracy.
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Fig. 4. Processing chain for one surface patch from the model example run of Sec. 5.2. Sequential rendering processes are shown on an unfolded hemicubic 128-partition, an identical process occurs for voxels but the rendering occurs on a cubic partition data structure.
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5.
Model performance and applications
5.1. Plane-parallel performance
The outputs of the 3D model in a plane-parallel configuration were compared to the invariant
imbedded numerical integration method [2] for radiative transfer through non-stratified IOPs
with a total optical depth of 5 and single scattering albedos (ω 0 ) of 0 to 1 in 0.1 steps (an optical
depth of 5 corresponds to approximately 10 m depth at 500 nm for the IOP data set used in the
next section). Whereas the full 3D model is constrained by design to use cubic directional
partitions, in a plane-parallel configuration the standard HydroLight spherical partition can
be used (Fig. 2) and the horizontal periodic repeat can be effectively infinite. This enabled
the accuracy and stability of the iterative solution of the volumetrically discretized radiative
transfer equations (Eqs. (12), (13), (14)) to be assessed without the complicating factors of
differing directional discretizations and finite horizontal periodic repeat.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
10-2
10-3
10-4
-90-60-30 0 30 60 90
nadir angle, θ, degrees
(a)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
10-2
10-3
10-4
3Dpp
II
3Dpp
II
-1
-1
10
-1
1.0
0.07
87°
87°
60°
60°
-2
10-1
ω0 =
1.0
3Dpp
II
Lb, Wm sr nm
100
10
0.08
ω0 =
0
3Dpp
II
Lb(60°), Wm-2sr-1nm-1
Lb, Wm-2sr-1nm-1
10
0.06
0.05
1 2
5 10
30 100
iterations
(b)
0
50 100
200
no. of layers
(c)
Fig. 5. Modeled substrate-incident radiance distribution in the plane of the sun, for optical
depth 5.0, substrate diffuse reflectance RD = 0.7 and single scattering albedos, 0 ≤ ω0 ≤ 1,
with Petzold’s phase function. Sun zenith angle is 30◦ in a HydroLight idealized sky model
(C = 0, Rdif = 0.3, irradiance = 1.0 for all bands) [29]. The water surface is perfectly flat. (a)
Comparison between invariant imbedded method (II) and 3D model algorithm (3Dpp) with
10 layers and 20 iterations. (b) 3D model convergence rate to invariant imbedded solution
for substrate incident radiance at 60◦ from nadir. (c) Effect of vertical spatial discretization
(no. of layers) on modeled substrate-incident radiance at 60◦ and 87◦ from nadir, runs of
100 iterations.
The results are shown in Fig. 5, and the caption gives further details on the model input
parameters. In Fig. 5(a), two points of disagreement between the invariant imbedded method
and the 3D model can be identified. Firstly, the 3D model underestimated radiance levels resulting from a single scattering albedo of 1.0 over the majority of substrate incident angles.
Figure 5(b) shows that this is not a fundamental inaccuracy but simply due to insufficient solution iterations, after 100 iterations the solution converges to the invariant-imbedded solution
even for ω 0 equal to unity. Secondly, the invariant imbedded solution predicted a small increase
in radiance incident on the substrate in directions close to horizontal across the substrate, i.e.
near −90◦ and 90◦ . This phenomena, due to multiple scattering between the highly reflective
substrate (RD = 0.7) and the bottom of the water column, was less evident in the 3D model
because the bottom 0.5 optical depths from the substrate were averaged into a single layer radiance distribution. Discretizing the water column into progressively thinner layers reduces this
discrepancy to an arbitrarily small amount (Fig. 5(c)). Twenty uniform layers were sufficient to
get close agreement with the invariant imbedded solution for substrate incident radiance at 60 ◦
from nadir, whereas radiance at 87 ◦ required 200 layers.
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The results demonstrate that the volumetrically discretized treatment of radiative transfer
(Eqs. (12), (13), (14)) is fundamentally accurate, since even the slightest inaccuracy would
cause the result to drift away from the invariant imbedded solution as the number of iterations
or layers increased. In addition, degradation in accuracy for lower numbers of layers and iterations is small and acceptable for many applications. A substrate reflectance of 0.7 is very
high compared to most naturally occurring substrate materials but nevertheless even with only
10 layers and 20 iterations the overall error introduced by the discrepancy at substrate-incident
angles greater than 60 ◦ magnitude is extremely small, corresponding to approximately 0.2% of
the total downwelling irradiance on the substrate for ω 0 = 0.9, or 0.008% for ω 0 = 0.3.
5.2. Three-dimensional structures - example model setup
This section presents two example model setups incorporating three-dimensional substrate
structures and a macro-scale water surface wave structure, the intention is to illustrate the model
potential for remote sensing and photobiology applications in shallow water coral reefs. While
in an actual application it would be desirable to parametrize the three-dimensional structures
from empirical data, here the water surface is modeled as a superposition of three Gerstner
waves [36] of mean amplitude 0.025 m and wavelengths from 1 m to 1.4 m. Two differing substrate structures are used, being simplified representations of dome shaped corals (e.g. Porites
sp.) and branching corals (e.g. Acropora sp.) [37] standing approx. 0.5 m high on a sand substrate at 1 m depth (Fig. 6). The dome structures were formed by superposing two spheres using
a patch intersection algorithm, giving a scene total of 3733 patches. The branches were composed of tubular segments with the patch intersection algorithm disabled, giving 3904 patches.
The two model setups also differ with respect to downwelling sky radiance distributions which
were modeled [34] from field-collected total and diffuse sky spectral irradiances collected under
clear sky conditions at Glover’s Reef, Belize, with sun zenith angles of 15 ◦ and 45◦ . The two
model setups are identical in every other respect and will henceforth be referred to as ‘domes
high sun’ and ‘branches low sun’.
In the model setups the water surface was considered locally smooth, a Fresnel reflection
and refraction function [2] was directionally discretized over a cubic 12-partition (864 cells)
and mapped onto the triangulated Gerstner wave surface according to the local co-ordinate
system (Eq. (9)). Lambertian-assumed coral and sand spectral reflectances, R D (λ ), were from
a previously published dataset [38], to facilitate comparison the same coral reflectance was
mapped onto both the domes and branching structures. Water IOPs were from a dataset collected at Glover’s Reef lagoon with a WETLabs AC-S and ECOBB3 IOP profiling package
[40], giving volumetrically homogeneous a(λ ), c(λ ) and the VSF, β (λ , ψ ), was modeled using a wavelength-dependent Fournier-Forand phase function [26, 39]. All spectral data were
resampled to 17 wavelength bands of 20 nm width from 400 nm to 740 nm. Over these bands
the range of single scattering albedos (ω 0 ) was 0.13 to 0.82, and optical depths to the sand
substrate ranged from 0.3 to 2.1. Both setups had 1200 voxels arranged in a 10 × 10 × 12 grid.
The 3D model implementation allows a variety of virtual sensors to be placed anywhere
within the model domain, such as cameras or spectroradiometers with a defined field of view,
or orthographic projection area integrators for simulating a single above-water remote sensing
pixel. After the model solution is obtained the virtual sensors estimate collected light by a
modification of the same rendering method used in the solution algorithm (Fig. 4). In the two
example setups, two remote sensing pixel sensors of 1 m × 1 m were placed above and below
the water surface, and 72 camera sensors were placed in a 360 ◦ circle to provide views of the
benthic structures from all directions (Fig. 6). Each camera sensor collected a spectral radiance
image of 600 × 600 pixels. A PNG or JPEG image requires a linear scale of 0-255 in the red,
green and blue, conversion was performed by weighting the spectral data by the human eye
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(a)
(b)
(c)
(d)
Fig. 6. Input vector meshes (a) and (c), and example camera sensor outputs (b) and (d) for
the domes high sun and branches low sun model setups. On the left, red pyramids show a
sample of the virtual camera FOV locations and blue rectangles are remote sensing pixel
sensors above and below the water surface. Animations over the 24 time step solutions
show effects of water movement (Media 1), (Media 2), and the 72 circularly placed cameras give a 360◦ rotational view of the 3D light field for a single time point (Media 3),
(Media 4). Solution algorithm hemicubic and cubic resolutions (Fig. 4) were 96 and 48
respectively. Solution stability threshold was 0.001 with runs limited to 5 iterations. For
clarity volumetric scattering is directionally interpolated in these visualizations.
tristimulus functions [2] and log transforming the result multiplied by a user-defined parameter
akin to camera exposure time. For each setup, 24 separate model solution runs were performed
with the wave surface structure adjusted each time by propagating the Gerstner waves forward
by 1/24 of the slowest wave time period. This enabled the variation of water surface effects,
such as sun-glint patterns, to be estimated over a cyclic repeat of the water surface movement.
5.3. Three-dimensional structures - results
Model outputs from the virtual camera sensors display a high level of visual realism, Figs. 6(b)
and 6(d). Note that these images and movies are directly calculated model outputs with no postprocessing except for the choice of overall camera sensitivity. The capability of the model to
capture various radiometric processes is apparent, such as: shading on structure sides and the
surrounding substrate; total internal reflection on the underside of the water surface; directional
volumetric scattering of light, especially when viewing towards the sun in the branches low
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sun setup; multiple scattering effects, where the water column horizontally scattered radiance
becomes colored by the coral substrate; and wave focusing effects. An orthographic projection
of a 1 m2 above-water view simulates a remote sensing pixel (Figs. 7(a) and 7(b)). The significant contribution of sunglint to variance in above-water reflectance and its dependence on sun
elevation is evident, but in addition the water surface also spatially distorts the substrate, and
the branching structures are relatively darker due to within-structure shading.
(a)
(b)
(c)
Fig. 7. Upwelling radiance, Lu integrated over 1 m2 above the water surface by the virtual
remote sensing pixel sensor for (a) domes high sun (Media 5) and (b) branches low sun
(Media 6). The third panel (c) shows minimum and maximum percentage energy loss over
the 24 runs for the domes and branches setups, ‘isec’ is one run of the branches setup using
a patch intersection algorithm.
Energy conservation over all 24 solution runs for the domes high sun setup was within ± 1%
for almost all wavelengths (Fig. 7(c)). For the branches low sun solutions there was a net energy
gain of 2% to 4% corresponding in spectral shape to the water attenuation. This error was due
to poor construction of the input vector mesh. Since a patch intersection algorithm was not
applied parts of individual patches were embedded behind surfaces, so the coral surface area
and hence total absorbed light were overestimated. When a single solution run of the branches
setup was performed with a pre-processing patch intersection algorithm, energy conservation
improved to within 0.5% for almost all wavelengths (Fig. 7(c)). This highlights the importance
of having a well constructed vector mesh for input when assessing conservation of energy.
Figure 8 illustrates the potential use of the model for investigating three-dimensional
processes in the vertical transmission of light through the water surface and benthic canopies.
The periodic distortion of the sun image in the associated movies is the basis of the wavefocusing phenomena visible in Figs. 6(b) and 6(d). However, note that the model has some
limitations with respect to high spatial and directional resolution phenomena such as wave
focusing, since a finite directional resolution translates to reduced spatial resolution as the distance between elements increases. Consequently, with increased depth the narrow wave focusing peaks become spatially smeared out, although the overall spatially-averaged light energy
remains accurate.
For either setup average solution time was 2.5 hours on a standard Linux based workstation.
The basic time complexity of the algorithm is ∼ O(n 2 ) but in practice performance is better
due to various optimizations, e.g. exploiting the fact that patches on flat surfaces cannot see
other patches on the same surface. The current implementation is not yet fully optimized, a
GPU (Graphics Processing Unit) based implementation is feasible [41]. The current rapid rate
of GPU development implies that scenes of the complexity presented here could be solvable in
under minute on a standard workstation within the next five years.
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(a)
(b)
(c)
(d)
Fig. 8. Hemispherical ‘fish-eye’ projections of downwelling radiance, Ld , from (a) the sky,
(b) directly under the water surface (Media 7), (c) on the substrate under the branching
structures without water column scattering included in the visualization (Media 8) and (d)
with scattering (Media 9).
6.
Conclusion
A geometrical optics algorithm for solving three-dimensional radiative transfer for arbitrary
surfaces in anisotropic scattering media has been presented. The 3D model is based on a spatial and directional discretization of radiative transfer theory. In a plane-parallel configuration
the method produces essentially identical results to the invariant imbedded numerical integration technique, provided that sufficient volumetric resolution and solution iterations are used.
Though general in its design, the model was primarily designed for applications in natural
shallow water environments with complex benthic structures. For media with single scattering
albedo less than 1.0 and benthic reflectances less than 70% solution convergence is sufficiently
rapid to render many problems of interest computationally tractable. Model validation for complex structures is challenging, but internal conservation of energy implies that overall error can
be constrained by careful model setups, e.g. within 1% for the example benthic structures presented here. Potential applications are numerous. Forward-modeling of structural effects and
spatial patterns in water leaving radiance can be used to inform inversion methods for shallow water remote sensing, and since the model simultaneously provides light absorption by
the benthic canopy there are clear applications in remote sensing of benthic photobiology. With
suitably chosen input radiance distributions the model can construct a BRDF function for either
the total water column or below-water structures, which can then be used as input to a planeparallel atmospheric or water column model. Modeling of instrument shading by boats, buoys
or cages is also possible and could inform instrument design or deployment practices. The main
limitation in the model is capturing phenomena at small spatial scales that also rely on a fine
directional resolution, such as wave focusing in deep water, since greater amounts of computer
memory are required than most current workstations provide. Run times required for obtaining
a solution are a less serious issue, a GPU-based implementation of the model is feasible and
could reduce run times for problems of moderate complexity to a few minutes.
Acknowledgments
The author thanks Eugenio Méndez and an anonymous reviewer for comments on the manuscript, Stuart Phinn and Chris Roelfsema for field data on sky irradiances, and Alan Lim and
Ellsworth LeDrew for validation of the invariant imbedded solution code against HydroLight.
Reflectances and IOPs were collected using equipment held by the NERC Field Spectroscopy
Facility. This work was funded by three NERC grants, NER/Z/S/2001/01029, NE/C513626/1,
NE/E015654/1 (held in conjunction with Peter Mumby, Mark Cutler and Paul Blackwell) and
also by the World Bank / GEF funded Coral Reef Targeted Research Program.
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