=Paper=
{{Paper
|id=Vol-2485/paper45
|storemode=property
|title=The Model of Reflective Surface based on The Scattering Layer with Diffuse Substrate and Randomly Rough Fresnel Boundary
|pdfUrl=https://ceur-ws.org/Vol-2485/paper45.pdf
|volume=Vol-2485
|authors=Vladimir Budak,Anton Grimailo
}}
==The Model of Reflective Surface based on The Scattering Layer with Diffuse Substrate and Randomly Rough Fresnel Boundary==
https://ceur-ws.org/Vol-2485/paper45.pdf
The Model of Reflective Surface Based on the Scattering Layer
with Diffuse Substrate and Randomly Rough Fresnel Boundary
V.P. Budak1, A.V. Grimailo1
BudakVP@gmail.com|GrimailoAV@gmail.com
1
National Research University “Moscow Power Engineering Institute”, Moscow, Russia.
In this article, we describe the mathematical model of the reflective surface as a scattering layer with the diffuse substrate and
randomly rough Fresnel boundary. This model opens the way for a physically correct description of the light reflection processes with
polarization account and hence enables engineers and designers to obtain much more precise results in their work. The algorithm of
Fresnel boundary modeling based on the method of mathematical expectations reduces calculation time by constructing the randomly
rough surface only at the ray trajectory nodes instead of constructing realizations of a random field. As a part of the complete reflective
surface model, the algorithm made it able for us to model the effect of the average lens emergence.
Keywords: mathematical model, reflection, refraction, polarization, reflective surface, light scattering.
1. Introduction
Nowadays it is a tradition for light engineering that the light
polarization state is not considered when modeling light
distribution. This neglection is acceptable when we deal with
diffusely reflecting surfaces and a small number of re-reflections.
On the contrary, we must account the influence of light
polarization when considering surfaces with a significant
specular part. The very first reflection changes the state of light
polarization and this fact affects all the following processes of
light distribution.
To date, a series of proceedings devoted to the light
polarization account has been published [3, 6, 7]. Basing on use Fig 1. Representation of the reflective surface
of ray tracing and local estimations of the Monte-Carlo Method in the mathematical model.
they show that accounting of the light polarization state leads to
quite significant changes not only in the qualitative results but in Generally, scattering media are characterized by matrix
the quantitative results as well.
scatter coefficient , matrix absorption coefficient and matrix
However, the mathematical model of multiple reflections
with polarization account used for estimating the influence of extinction coefficient . Neglection of dichroism,
polarization showed just the first approximation for the birefringence, and similar effects of the same kind, which are
quantitative results. Therefore, the following step of the model inherent only for several materials, enables us to transit to the
development is to create and use the physically correct model of scalar analogs of the matrix coefficients .
the reflective surface. For modeling the reflective surface with the assumptions
We must consider that the light is always reflected from both taken above, we need to solve the boundary value problem for
of the faces of the material surface and the material volume. The the vector radiative transfer equation. Let us consider the plane-
light penetrates the near-surface layers of the material where the parallel system of the scattering layer with a diffuse substrate.
light scattering by the material particles occurs. Then, a certain The layer is irradiated at the angle by the plane
fraction of the initial luminous flux re-enters the surrounding monodirectional source with random polarization state. Then one
space. At this point, the role of polarization account takes an can write the problem as
exceedingly significant part as it influences all the processes of
the light scattering.
L( , l ) L( , l ) 4 R( l l N l )
Thus, the authors decided to develop the model of reflective ˆ ˆ ˆ ˆ ˆ ˆ
surface, which would account the effects described above.
Further, the physically correct model will enable us to obtain x ( , ˆl, ˆl )R( N
ˆ ˆl ˆl ˆl )L( , ˆl )dˆl ,
more precise results of light distribution modeling. (1)
L(0, 0, ) L( ˆl ˆl ),
2. Mathematical model of the reflective surface
L( 0 , 0, ) [E / 0 0 0] ,
T
When the light penetrates the near-surface layers, the
processes occurring have the same nature as the radiative transfer
in turbid media. Additionally, we must account that a real where l̂ and ˆl are the unit vectors of the scattered and incident
material surface is always uneven owing to the most varied ray directions respectively;
causes (corpuscular structure of the material, surface treatment N̂ is the normal vector;
defects, etc.) and never reflects the light according to only L is radiance (Stokes vector);
specular or only diffuse law but there take place both of them at cos ;
the same time. z1
Thus, we decided to represent the reflective surface as a ( z )dz is the optical track thickness in the section
scattering layer with a diffusely reflecting plane at the bottom z0
and randomly rough Fresnel boundary above (Fig. 1).
[ z0 , z1 ];
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
practical cases, the shape of a randomly rough surface is
is the single scatter albedo; described by a random function of coordinates (and sometimes
of time). Therefore, scattering on a real surface should be
x ( , ˆl , ˆl ) is the scatter matrix; considered as a statistical task, which consists in finding the
R(ˆl ˆl N ˆ ˆl ) is the matrix of the reference plane probabilistic characteristics of a scattered field from known
statistical characteristics of the random surface. The methods of
rotation from ˆl ˆl to N
ˆ ˆl.
solving such a problem are the same regardless of the physical
nature of the roughness [1].
3. Randomly rough Fresnel boundary Thus, researchers of the radiative transfer processes in the
One of the most important components of the reflective ocean-atmosphere system face a similar task when describing the
surface model described above is the construction of the effect of the perturbed sea surface on the radiation field.
randomly rough Fresnel boundary. To show the importance of There are two ways for solving this problem [4]. In the first
taking into account properties of a randomly rough surface one one, realizations of a random field are constructed according
can give the cases of observing ocean currents, underwater to the randomization principle, then one simulates random
mountain ranges, and shoals by people from outer space. trajectories l̂ and on their basis calculate the random estimates
For the first time, the deep-sea bottom topography from of the sought-for functionals. The main difficulty of employing
space was observed by American astronaut Gordon Cooper from this approach is the necessity to find the intersection points of the
the Gemini 5 spacecraft in August 1965. The first of the Soviet ray and surface at each trajectory node. In the general case, the
cosmonauts were A. G. Nikolaev and V. I. Sevastyanov from the determination of the intersection coordinates costs much
Soyuz-9 spacecraft in June 1970. At the same time, they first computational time.
drew attention to the fact that the sea waves, ripples on its surface The second approach is preferable, therefore. It is based on
are not an obstacle when observing the topography of the seabed the method of mathematical expectations (Fig. 2). Here, to
from space. construct an N-component random trajectory we need to have the
Then there was a series of other known observations through realizations of the random surface only at N points, calculated in
the rough surface of the ocean: a certain way [4]. At the points of rays reaching a random
1. August 1974. G. V. Sarafanov and L. S. Demin observed the interface, the selection is made of random realizations of normals
bottom relief at depth of hundreds of meters from the to the surface.
Soyuz-15 spacecraft. They succeeded to see the bottom of
the Mozambique Gulf that separates the island of
Madagascar from the African continent. The cosmonauts saw
a bottom, covered with shafts that stretch along the strait. The
structure of the strait bottom resembles the structure of that
of a small river, but the dimensions are many times larger
than in the river.
2. June 1975. From the board of the Salyut-4 orbital station,
P. I. Klimuk and V. I. Sevastyanov observed the bottom of
the seas and oceans. When flying over the Atlantic Ocean
from Newfoundland to the Canary Islands, they clearly saw Fig 2. To the second approach
ocean currents. Along the European coast of the of the randomly rough Fresnel boundary modeling.
Mediterranean Sea with an emerald strip of subtropical
greenery, they saw under the water a continuation of the 4. Algorithm of Fresnel boundary modeling
continent relief. Continuation of the relief was also visible on
the eastern coast of South America - three terraces extending In order to make clear the way one can use the second
deep into the Atlantic Ocean. It was visible how far the approach from the latter section, let us consider [4] an arbitrary
Amazon River carried its muddy waters into the ocean, how trajectory of n nodes
they were carried away by deep currents under a layer of
clean water. {(r0 , ˆl0 , Q0 ), (r1 , ˆl1 , Q1 ), , (rn , ˆl n , Qn )}, (2)
3. June 1978. Underwater relief of the Pacific Ocean bottom in
the region of the Solomon Islands at depths of up to 400
meters was observed by V. V. Kovalenok and where ri is the i-th ray collision point on the surface or in the
A. S. Ivanchenkov. During the flight, the cosmonauts first medium;
made an attempt to derive the laws of the most favorable ˆl is a unit vector of the ray direction after the i-th
i
conditions for observing underwater formations. These
collision;
observations were carried out from an orbit close to the solar
one at a small height of the Sun above the horizon. Qi is the vector weight after i-th collision (its
Experience shows that the best conditions for observation are components correspond to those of the Stokes vector).
when the height of the Sun above the horizon is 30°—60°; A set of the random surface point corresponds to the nodes
direction from the Sun 90°—130°; viewing angles do not exceed of the trajectory:
30°—40° from the direction of the nadir and, of course, outside
the glare zone. ˆ (r ) ),
{( (r,1 ), N ˆ (r ) )},
, ( (r,n ), N (3)
,1 ,n
In the cases described above, the so-called statistical lens
effect appeared due to a randomly rough Fresnel surface at the
ocean-atmosphere boundary. This effect allowed cosmonauts where r ( x, y) are horizontal coordinates so that r (r , z);
and astronauts to observe the bottom of the seas and oceans from (r ) is the random surface roughness function;
outer space at great depths.
ˆ (r ) is the outer normal to the surface at the point
N
The problem of the randomly rough Fresnel boundary
modeling is unavoidably encountered in the solution of a large r (r , (r )).
number of physical problems in various fields. In the majority of
At the first step, one samples a random value of deviation 1
when 1
from the interval (hm , hm ) according to the probability density
based on the normal distribution. Then one evaluates the distance
t1
from the point r0 to the plane z (r ) in the direction ˆl 0 . P(ri 1 , ri ) exp K (t )dt ,
0
The coordinates of the first trajectory node are evaluated by using
the formula
t1 [ zi h i 1 ][( ˆli 2 , k ) 2 1], k [0 0 1],
r1 r0 ˆl0 . (4)
when 1
Thus, the point r1 is the first point of the ray intersection with
the random surface, provided that there have been no
intersections before. We allow for this condition by multiplying
e
the vector weight Q0 by the probability P(r0 , r1 ) for the ray P(ri 1 , ri )
e 2
3
r1 r0 ˆl 0 to have no intersections with the surface on the track
between the points r0 and r1 . 1 z x (r ,i 1 ) z y (r ,i 1 )
2 2
i 1 1 [( ˆli 2 , k ) 2 1] ,
The last statement requires some clarification. In general e e 2
case, instead of P(r0 , r1 ), one needs to use another probability
P(r0 , r1 | ζ1 ), the expression for which has the following form: x
1
e
t 2 /2
( x) dt.
2
P(ri 1 , r | ζi 1 , , ζ1 ) (ri 1 , r, ζ)dP(ζ), (5)
A random realization of the normal vector N̂1 is sampled
where ζ (, x , y ) : [hm , hm ], x , y ( , ); ˆ : (N
ˆ , ˆl ) 0, N
ˆ } by the
from the set of the unit vectors {N 0
, x , y are the random functions possessing normal
following way. Using normal distribution, one model zx and z y
one-dimensional distributions with the parameters
with the distribution parameters (0, x ) and (0, y )
(0, ), (0, x ) and (0, y ) respectively;
respectively.
1, r r 0 ,
(r, r ) The quantities zx and z y are substituted into the following
0 r r 0 ;
formula and one evaluates the components of the vector N̂
0 is the minimal distance from the point r to the
medium boundary in the direction ˆl. ˆ (r ) k e (r )
N , (8)
The standard deviations x and y are not independent and 1 | e (r ) |2
connected with the standard deviation of the random quantity
by the following expression e(r ) [ z x z y 0].
x y | K(0) |, (6) Having obtained the values of N̂1 and ˆl 0 , we are able to gain
[5] the Fresnel reflection factor R ( ˆl0 , N
ˆ ):
1
where K K ( r ) is the correlation function of deviations for
isotropic undulation. In case of anisotropic undulations, we have (| A | B)2 ( A2 B 2 C 2 )
R(ˆl, N
ˆ) , (9)
to set functions K ,x and K, y . In applied calculations, functions (| A | B)2 (| A | B C )2
of the form are often used as a correlation function:
A (ˆl, N
ˆ ), B 1/ 2 1 A2 , C 1 A2 ,
r 2
K ( r ) exp , (7)
1 2 (1 A2 ) A,
where (and ) are the parameters determining the force and 1 / n, (ˆl, N
ˆ ) 0,
shape of undulation.
n, ˆ ˆ ) 0,
( l, N
Calculation of probabilities (5) is necessary on each step of
modeling trajectory. Since the formula (5) is extremely
complicated for direct calculation and practically inapplicable, where n is the refractive index of the material with respect to air.
the formulae obtained in [1] are often used when solving such We consider R( ˆl0 , N
ˆ ) as a probability for the ray l̂ having
1 0
problems provided that P(ri 1 , r1 | ζi 1 , , ζ1 ) P(ri 1 , r). These
collided with the facet of normal N̂1 to undergo the mirror
formulae can be applied in two extreme cases:
reflection, and 1 R ( ˆl0 , N
ˆ ) as a probability for the ray to
1
undergo refraction.
In this way, the coefficient R( ˆl0 , N
ˆ ) is used to choose the
1
[5] Marchuk, G.I., et al. The Monte Carlo Methods in
Atmospheric Optics. Springer-Verlag, Berlin, 1980.
type of interaction with the surface from two possible outcomes: [6] Mojzik, M., Skrivan, T., Wilkie, A., Krivanek, J. Bi-
reflection and refraction. Having made the choice, one defines Directional Polarised Light Transport. Eurographics Symposium
the vector l̂1 according to the formulae on Rendering 2016.
[7] Wolff, L. B., Kurlander, D. J. Ray tracing with polarization
parameter // IEEE Computer Graphics and Applications. 1990.
ˆl0 2( ˆl0 , N
ˆ )N
ˆ , for reflection,
ˆl
(10) V. 10, No. 6, P. 44-55.
1
ˆ ˆ
l0 N,
for refraction.
After that, one can make the transformation of the vector
weight according to the following expression
Q1 R( ˆl0 ˆl1 N
ˆ ˆl )(r , ˆl , ˆl ) R( N
1 1 1 1 0
ˆ ˆl ˆl ˆl )Q , (11)
0 0 0 1 0
ˆ ˆl ˆl ˆl ) is the matrix of the reference plane
where R( N 0 0 0 1
ˆ ˆl to ˆl ˆl ;
rotation from N 0 0 0 1
(r1 , ˆl1 , ˆl0 ) is the Mueller matrix for reflection or
refraction depending on the choice based on the
coefficient R( ˆl0 , N ˆ ).
1
The next steps of the algorithm have the same logic with the
exception that we need to account light scattering in the material
medium and thus, solve a rather complicated problem of
estimating probabilities and evaluating ray weights.
5. Conclusion
At the current stage, the realization of the algorithm
described above enabled us to model the effect of the average
lens emergence. Further, we are going to use the model of the
randomly rough Fresnel boundary as a part of the reflective
surface model (Section 2).
Basing on the reflective surface representation as a scattering
layer with the diffuse substrate and randomly rough Fresnel
boundary above, we will be able to construct a complete model
of reflection with the account of scattering in the material
volume.
The main role in solving the problem (1) should be given to
the analytical methods [2], as they are much faster than numerical
those. This approach will pave the way for us to integrate the
model into existing methods, used in computer graphics (e. g. ray
tracing, photon maps, local estimations, etc.). It will enable
engineers and designers to account polarization when solving
practical tasks and thus obtain much more precise results of light
distribution calculation and visualization.
Nevertheless, analytical methods always imply the use of
certain assumptions, the effect of which on the result is currently
possible to estimate only by using the Monte-Carlo Methods. In
the future, it is interesting to compare the two variants of the
mathematical model and, possibly, combine them, taking into
account the advantages of each of the variants.
6. References
[1] Bass, F.G., Fuks, I.M. Wave Scattering from Statistically
Rough Surfaces. Pergamon, 1979.
[2] Budak, V.P., Basov, A.Y. Modeling of a scattering slab with
diffuse bottom and top reflecting by the Snell law. In Proceedings
of GraphiCon 2018, p. 399-401.
[3] Budak, V.P., Grimailo, A.V. The influence of the light
polarization account on the result of multiple reflections
calculation. In Proceedings of GraphiCon 2018, p. 409-410.
[4] Kargin, B.A., Rakimgulov, K.B. A weighting Monte-Carlo
method for modelling the optical radiation field in the ocean-
atmosphere system. Russ. J. Numer. Anal. Math. Modelling,
Vol.7, No.3, pp. 221-240 (1992).