HT TOC
Proceedings of IMECE’03
2003 ASME International Mechanical Engineering Congress
Washington,
D.C., November
15–21, 2003
Proceedings
of IMECE'03
2003 ASME International Mechanical Engineering Congress & Exposition
Washington, DC, November 16-21, 2003
IMECE2003-41932
IMECE2003-41932
REVERSE MONTE CARLO METHOD FOR TRANSIENT RADIATIVE TRANSFER
IN PARTICIPATING MEDIA
Xiaodong Lu and Pei-feng Hsu
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
150 West University Blvd., Melbourne, FL
ABSTRACT
The Monte Carlo (MC) method has been
widely used to solve radiative transfer problems
due to its flexibility and simplicity in simulating
the energy transport process in arbitrary geometries
with complex boundary conditions. However, the
major drawback of the conventional (or forward)
Monte Carlo method is the long computational
time for converged solution. Reverse or backward
Monte Carlo (RMC) is considered as an alternative
approach when solutions are only needed at certain
locations and time. The reverse algorithm is similar
to the conventional method, except that the energy
bundle (photons ensemble) is tracked in a timereversal manner. Its migration is recorded from the
detector into the participating medium, rather than
from the source to the detector as in the
conventional MC. There is no need to keep track
of the bundles that do not reach a particular
detector. Thus, RMC method takes up much less
computation time than the conventional MC
method. On the other hand, RMC will generate
less information about the transport process as only
the information at the specified locations, e.g.,
detectors, is obtained. In the situation where
detailed information of radiative transport across
the media is needed the RMC may not be
appropriate. RMC algorithm is most suitable for
diagnostic applications where inverse analysis is
required, e.g., optical imaging and remote sensing.
In this study, the development of a reverse Monte
Carlo method for transient radiative transfer is
presented. The results of non-emitting, absorbing,
and anisotropically scattering media subjected to
an ultra short light pulse irradiation are compared
with the forward Monte Carlo and discrete
ordinates methods results.
INTRODUCTION
The recent research on the propagation of
ultra-short light pulse inside the absorbing and
scattering media has lead to some interesting
applications in the area of material properties
diagnostic, optical imaging, remote sensing, etc.
The time scales of such processes are usually in the
order of 10-12 to 10-15 seconds. In the case of
remote sensing using short light pulse, the pulse
width is in the order of 10-9 seconds. The
corresponding spatial and temporal variations of
radiation intensity in these processes are
comparable. Therefore, the consideration of the
transient term in the radiation transport equation is
necessary. The simulation of transient radiation
process is more complex than that in the steady
state due to the hyperbolic wave equation coupled
with the in-scattering integral term.
Several
numerical strategies have been developed, which
include discrete ordinate method (Sakami, et al.
2000; 2002), finite volume method (Chai, 2003;
Lu, et al. 2003), integral equation models (Tan and
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Copyright © 2003 by ASME
becomes unnecessarily inefficient. The RMC is
based on the reciprocity principle in radiative
transfer theory (Case, 1957).
The reverse
algorithm is similar to the conventional method,
except that the energy bundle (photons) is tracked
in a time-reversal manner. Its migration is recorded
from the detector into the participating medium,
rather than from the source to the detector as in the
conventional MC. There is no need to keep track
of the bundles that do not reach a particular
detector. Thus, RMC method takes up much less
computation time than the conventional MC
method. On the other hand, one should always
note that RMC will generate far less information
than the corresponding MC method.
Collins, et al. (1972) reported the earliest work
on RMC relevant to radiation transfer calculations.
Adam and Kattawar (1978) adopted the same
concept of Collins, et al. in their study of spherical
shell atmospheric radiation. They also provided
justifications of RMC, which will be expanded in
this study. In a series of rocket plume base heating
calculations by Nelson (1992), RMC was also
found to be a practical tool to include various
effects, e.g., spectral, scattering. To provide a
more rigorous theoretical foundation to RMC,
Walters and Buckius (1992) utilized the reciprocity
relations developed by Case (1957). Recently,
Modest (2003) used their results and applied to
singular light source problems. The only RMC
treatments for transient radiation processes
reported so far, to the authors' knowledge, are by
Wu and Wu (2000) and Andersson-Engels, et al.
(2000).
However, the details of the RMC
algorithm were not given in either paper and the
RMC results of the latter work were inconsistent.
This study will present a detailed and validated
RMC method to simulate transient radiation
process, particularly for light pulse propagation
within scattering media, with validated solutions.
The utilization of a Beowulf cluster for simulation
demonstrates the RMC method has the potential for
real time inverse analysis. Although only onedimensional geometry is considered in this study,
the algorithm can be readily extended into multidimensional geometry, which is to be treated in a
follow-up study.
Hsu, 2001; Wu, 1999; Wu and Wu, 2000), and
Monte Carlo method (Brewster and Yamada, 1995;
Hsu, 2001a).
The Monte Carlo method is a numerical
technique of solving various sciences and
engineering problems by the simulation of random
variables. Monte Carlo is one of the most versatile
and widely used numerical methods, very suitable
for solving multidimensional problems, especially
when deterministic solutions are difficulty to
obtain. Monte Carlo simulations have in the past,
and continue to, consume a significant fraction of
high performance computing time. With the
advancement of the low cost Beowulf cluster
(Sterling, et al. 1995) and inherently high parallel
efficiency of Monte Carlo method (Siegel and
Howell, 2002; Sawetprawichkul, et al. 2002), its
usage will only grow over time.
The theoretical foundation of the method had
been known long before digital computers were
available. Of course, the use of the Monte Carlo
method became practical only with the advent of
computers and high-quality pseudorandom number
generators. In the early days of the digital
computer, significant amount of computational
work was carried out with Monte Carlo simulations
of the neutron transport. Radiation heat transfer is
one of the disciplines that take great advantage of
the Monte Carlo method (Howell, 1968).
However, the Monte Carlo method can be
computationally very time consuming due to its
error bound being limited by the number of
sampling in the form of O(N-0.5), in which N is the
sampling number. That is why much of the effort
in Monte Carlo development has been in
construction of variance reduction methods that
speed up the computation. The other approach,
which has shown promising results, is the
utilization of deterministic number sequences
instead of random numbers for integration
(O’Brien, 1992; Hsu, 2001)– this belongs to the socalled quasi-Monte Carlo (QMC) method (Holton,
1960; Niederreiter, 1978). In theory, the error
bound of QMC is proportional to O(N-1(logN)D),
where D is the dimension of integration or the
number of deterministic number sequence. In
practice, depending on the quality of the number
sequences and the nature of the problem, the error
bound is typically between O(N-0.5) to O(N-1) (Hsu,
2001b; Bratley, et al. 1992).
In the case of detecting radiation signals at
some selected positions and/or given time intervals,
the reverse Monte Carlo (RMC) method becomes
very advantageous. Since computational results in
the whole spatial and temporal domains are not
always necessary, then either MC or QMC
TRANSIENT RADIATION TRANSPORT
Consider a one-dimensional slab containing an
absorbing and scattering medium, the radiative
transfer equation (RTE) in a given direction ŝ is
(Ozisik, 1973):
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∂I (rˆ, sˆ, t ) ∂I (rˆ, sˆ, t )
+
= −κ (rˆ ) I (rˆ, sˆ, t )
∂s
c∂t
σ (rˆ )
+
I (rˆ, sˆ ′, t ) Φ (sˆ ′, sˆ ) dΩ′
4π ∫Ω′= 4π
(
∂I d z * , sˆ, t *
c∂t *
(1)
=
where κ and σ are the extinction coefficient and the
scattering coefficient, respectively, c the speed of
light in the medium, ŝ the light propagation
direction, and Φ ( sˆ′, sˆ) the scattering phase
function.
The geometry of a collimated pulsed
irradiation on the top surface of the slab medium is
shown in Fig. 1.
I
o
*
*
(
*
d
*
)
*
(
*
*
*
*
∫ Φ( sˆ′, sˆ) I d z , sˆ, t dΩ' + S z , sˆ, t
)
(3)
4π
(
)
S z* , sˆ,t* =
ω
4π
∫4π Φ(sˆ′, sˆ) I c (z
*
)
, sˆ, t * dΩ'
(4)
where Ic(z*, ŝ , t*) is obtained by solving the RTE
governing the attenuation of the collimated beam.
In the case of a collimated beam in the z-direction
the solution is
(
)
I c z * , sˆ, t * = I o e − z* [ H (t * − z * )
z
− H (t * − t *p − z * )]δ ( sˆ − sˆo )
i
(5)
with t *p = κ c t p and tp is the pulse width, δ the
Dirac delta distribution, and H the Heaviside step
function. Equation (4) becomes
Figure 1. Geometry of the collimated irradiation.
(
)
ω
I o e − z* [ H ( t * − z * )
4π
− H (t * − t *p − z * )]Φ (sˆo , sˆ )
S z * , sˆ, t * =
I( ŝo , t) is the collimated irradiation intensity
in the direction ŝo . Io is the intensity leaving the
wall toward the medium and Ii the intensity coming
from the medium toward the wall. It is convenient
to use a concept similar to Olfe's (1970) modified
differential approximation by splitting the intensity
into two components: one contributed by incident
radiation and the other by medium emission and
scattering. The former can be solved exactly with
the boundary condition containing singular
emission source and the latter by various numerical
schemes. The approach has successfully treated
various radiative transport problems, e.g., Liou and
Wu (1997) and Ramankutty and Crosbie (1997;
1998). With the time-dependent, incident
collimated irradiation, e.g., a light pulse, one can
separate the intensity into two parts
I ( x , sˆ, t ) = I c ( x, sˆ, t ) + I d ( x, sˆ, t )
d
where t* = κct and z* = κz. In the above equation,
the out-scattered radiation source formed from the
collimated irradiation contributes to the source
term and is described by
ŝo
I
ω
4π
) + µ ∂I (z , sˆ, t ) + I (z , sˆ, t )
∂z
(6)
Φ ( sˆ ′, sˆ) = 1 + aµµ' where µ is the direction cosine
of ŝ . If the medium is isotropically scattering,
then a = 0. The collimated source function is then
equivalent to the isotropic blackbody emission
term. However, it is a time-dependent "emission"
term.
REVERSE MONTE CARLO METHOD
Figure 2 demonstrates the geometry of the
intensity received at the detector from a collimated
source. For simplicity, consider the collimated
irradiation to be a steady state source. This
generality can be easily extended to transient
sources. The collimated source irradiates normally
on a semi-infinite, absorbing, and scattering
medium. The detector is located at origin, O. The
source is attenuated in the medium and, in
particular, the source at a cylindrical coordinate
position (r, z) shown in the figure is considered.
The problem can be treated as axisymmetric with
respect to the r = 0 axis.
(2)
In this equation, Ic is the attenuated intensity of
the collimated beam, and Id is the diffuse intensity,
which is the in-scattered radiation into direction ŝ .
Substitution of Eq. (2) into Eq. (1) and normalizing
the terms give the dimensionless RTE,
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Copyright © 2003 by ASME
Converting the (r, φ, z) coordinate into spherical
coordinate of (ρ, φ, θ) with r = ρsinθ and z = ρcosθ
and the Jacobian of the transformation, Eq. (10)
becomes
I (r, φ)
o
Detector
O
G (0 ) =
θ
z'
ρ
Fig. 2 Geometry of integrated intensity received at
the detector.
Integrating the steady state RTE within the
isotropic scattering and non-emitting medium, the
intensity at any position z, including z = 0, can be
found as
I (z, sˆ ) = I (0, sˆ )e −κz
+
κ
4π
∞
∫ ∫ I (z ' , sˆ )e
−κ ( z − z ' )
dΩ s dz '
(7)
0 4π
If only singly scattered photons are considered, that
is, the collimated intensity being attenuated at
position z' and then scattered to the detector, then
the I(z', ŝ ) inside the integral will be the attenuated
collimated intensity, i.e., Io(r, φ)e-κz'. The intensity
at the detector, I(z = 0, ŝ ), doesn't include the
direct contribution from the collimated source in
direction ŝo , i.e., the first term on the right hand
side of Eq. (7). The integrated intensity over the
incoming (lower) hemisphere at the detector can
then be expressed as
G (0 ) =
κ
4π
∞
∫ ∫ I o (r,φ )e
−κz ′ −κρ
e
dΩ d dz '
ρ2
0 0
−κρ cosθ −κρ
e
sin θ d sin θ dφ dρ (11)
0
That is, the intensity at r̂1 and direction - ŝ1 due to
the point source at r̂2 in the direction ŝ2 equals to
the intensity at r̂2 and direction - ŝ2 due to the
point source at r̂1 in the direction ŝ1 . Several
other reciprocity relations can be obtained from
this. In the sense of Monte Carlo simulation, the
bundles (photons) that finally reached the detector
can be backtracked in both temporal and spatial
manner.
As the bundle travels from the detector
position back into the medium, the diffuse intensity
caused by the collimated source (Eq. 6) must be
integrated along the path length if encountered by
the bundle:
(8)
It should be pointed out that the dΩs in Eq. (7)
is the solid angle from all in-scattered directions
and dΩd in Eq. (8) is the solid angle viewed from
the detector to position z'. The G(0) represents the
signal received at the detector. From Fig. 2,
dr (rdφ )cosθ
∫ ∫ ∫ I o (r,φ )e
I (rˆ1 ,− sˆ1 ; rˆ2 , sˆ2 ) = I (rˆ2 ,− sˆ2 ; rˆ1 , sˆ1 )
0 2π
dΩ d =
∞ 2π π / 2
Equation (10) is the equivalent of Monte Carlo
integration and Eq. (11) is the counter part of
reverse Monte Carlo integration. Two problems
are immediately observed in Eq. (10): First, a
truncation of r has to be determined, because if r is
too large then sampling in some regions may not
contribute much to the detector signal. Second, the
ρ2 in the denominator causes large statistical
oscillation in the results for small ρ. By
considering the same problem with the viewpoint
from the detector in Eq. (11), both of these
problems are eliminated. It should be noted that
the determination of radiative flux at the detector
could be similarly obtained, although the sampling
of the bundle direction will be different due to
different cumulative distribution function.
Case (1957) derived a fundamental identity
that was based on the RTE. From that identity, the
reciprocity principle used in this study is simply a
special case. Implicitly assumed in the principle is
Φ ( sˆ′, sˆ) = Φ ( − sˆ,− sˆ′) , i.e., the scattering phase
function has time reflection symmetry. The most
basic reciprocity principle states that:
r
z
κ
4π
(9)
(
)
l*
(
) (
)
I d z , sˆ, t * = ∫ S z * , sˆ, t * exp − κl *' dl *′
Substitute into Eq. (8),
κ ∞ 2π ∞ I o (r,φ )e −κz′ e −κρ
G (0) =
cosθ r dr dφ dz (10)
4π ∫0 ∫0 ∫0
ρ2
*
(12)
0
RMC Algorithm
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2) ( z p ′ − ct p ) > z1: the first point that the
bundle and the pulse meet is between z1
and z2, then
µc (t1 − t p ) + z1
(15)
zL =
1+ µ
In this case, since the bundle and pulse
will never meet again, go to step 1 and
start a new bundle from the detector;
4. At the end of path length, calculate the
absorption decay and determine whether the bundle
is absorbed or scattered from the albedo. If
scattered, pick a new direction based on phase
function and return to step 2. If absorbed, go to
next step;
5. Go to step 1 for a new bundle and complete
the tally of all N bundles.
For a nonhomogeneous medium, step 2 needs
to be modified to consider the variation of radiation
properties in the path length.
1. Assume N number of bundles arrive at the
detector.
Start the first bundle and pick a
backtracking direction and a t1, which is the bundle
arrival time at the detector. The direction can be
uniformly distributed to minimize the fluctuation in
the result.
2. Pick a path length (lκ) based on κ, the
extinction coefficient;
3. At the end of the path length, check the
local time t2 (= t1 – lκ /c) > 0 and see if the pulse
will travel to the local position at t2. If so, calculate
the integration of the collimated source.
Since the pulse has a certain temporal width tp
that spans over a finite spatial distance ctp, it is
necessary to check whether the bundle will meet
the pulse, which is at position zp = ct2, at the new
position z2. There are three possible cases:
(a) ( z p − ct p ) ≤ z 2 ≤ z p - The bundle
lands inside the pulse. The upper limit of
the source integration (Eq. 12) will be zU =
z2.
As for the lower limit of the
integration zL, there are two possibilities:
1) ( z p′ − ct p ) ≤ z1 ≤ z p′ : The first point
that the bundle and pulse meet is also
inside the pulse, then zL = z1 . Note that
z p′ = ct1 is the position of the pulse’s
leading edge at time t1.
2) ( z p ′ − ct p ) > z1: the first point that the
bundle and the pulse meet is between z1
and z2, then
µc (t1 − t p ) + z1
(13)
zL =
1+ µ
where µ is the direction cosine of the path
from z1 to z2. The derivation of Eq. (13)
can be easily obtained with the aid of Fig.
3. The concept is similar to the domain of
influence described in Tan and Hsu
(2001).
Go to step 4;
(b) z2 < (zp - ctp) - the pulse hasn't met the
bundle yet, go to step 4;
(c) z2 > zp - the pulse meets and move
across the backtracking bundle. Their last
meeting point will be the upper limit of
the source term, zU, which must be
between z1 and z2,
µct1 + z1
(14)
zU =
1+ µ
As for the low limit of the source
integration n term zL, there are also two
possibilities:
1) ( z p ′ − ct p ) ≤ z1 ≤ z p ′ : the first point
that the bundle and pulse meet is inside
the pulse, then zL = z1.
RESULTS AND DISCUSSION
The parallel computer architecture used in this
study is simply a collection of computers with
commodity processors and parts. The coding is
based on the Single Program Multiple Data model
and use the Message Passing Interface (MPI)
library (Lu and Hsu, 2003). The system consists of
one root or head node and many slave or compute
nodes. The communication among nodes, in this
case, is through a private, channel-bonded Fast
Ethernet. A 48-node IBM PC based Linux cluster
was used in this study. This system can be
extended to 96 processors with dual processors in
each node. Currently, except the head node, only
one processor is installed in each of the compute
nodes. Each compute node is an IBM x330 xSeries Pentium III 866 MHz processor with 512
MB SDRAM. The total system memory space is
24 GB.
Detailed hardware and software
configuration information is given in the website at
http://olin.fit.edu/beowulf/.
As a first test of the RMC algorithm used in
this study, it is used to solve a typical medium of τ
= 10 and large scattering albedo 0.998 subjected to
a collimated pulse irradiation (Hsu, 2000). The
incident pulse width tp* = 1 is modeled by two
consecutive Heaviside unit step functions at the
boundary z = 0 and the normalized collimated
intensity is equal to 1. The solution improves as
the standard deviation is reduced from 5%, 2%, to
0.5% (Fig. 4). The 5% curve shows moderate
statistical oscillations and 0.5% curve is very
smooth. The temporal transmittance curve is
consistent with those reported by our earlier work.
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Copyright © 2003 by ASME
time interval is needed for time-gated
measurements. This particular advantage of RMC
greatly reduces the amount of computation and
may have the potential of being used in the real
time signal processing and inverse analysis for
optical image re-construction.
Many existing numerical schemes to simulate
transient radiative transport simply fail to resolve
the wave front correctly, e.g., Mitra, et al. (1997),
leading to incorrect propagation speed. A part of
the rigorous tests of transient radiation transfer
numerical schemes, including Monte Carlo
algorithms, is to demonstrate the capability of
wave front resolution. Figure 6 depicts the
excellent agreement between RMC and DOM and
the sharp collimated pulse front in this regard. The
medium has unity optical thickness and isotropic
scattering with albedo being equal to 0.9. The
DOM solution was validated with our prior integral
equation and Monte Carlo solutions.
Figure 7 shows the variation of the bundle
number used in each time instant and for three
different phase functions. The result is based on
the 2% standard deviation setting. A similar trend
is found in other standard deviation settings. First,
the bundle number decreases as the transmittance
signal reaches its peak. Then, the bundle number
increases as the transmittance signals gradually
levels off. Secondly, for t* > 30, i.e., after the pulse
passes the medium, the bundle number used is
higher for forward scattering medium and lower for
backward scattering medium. In the backward
scattering medium, the bundles tend to stay inside
the medium longer, and the chance to encounter
pulse is thus higher. This leads to quicker
convergence with fewer number of bundles used.
The behavior of bundle movement is similarly
described in our earlier work on a discrete
ordinates method (Sakami, et al. 2000).
A series of tests of the proposed RMC
algorithm shows that it can provide accurate
simulations with significant reduction in
computational time when compared to the
traditional MC method. The RMC will be used to
compare pulsed laser measurements of threedimensional scattering media in a follow-up
publication.
In transient problems, one of the forward
Monte Carlo methods can carry out the time
stepping of the simulation of all the N bundles
simultaneously at a given time step (Hsu, 2001a).
While this approach consumes very large amount
of memory, it can achieve good parallel efficiency.
A different approach is to trace the bundle one at a
time and record the time history accordingly
(Sawetprawichkul, et al. 2002). This second
To further verify the reverse Monte Carlo
algorithm,
the
simulations
of
forward
anisotropically, isotropically, and backward
anisotropically scattering media are compared with
the discrete ordinates method with which accurate
transient solutions were obtained (Sakami, et al.
2000). Comparison with a forward Monte Carlo
method (Sawetprawichkul, et al. 2000) is also
made on the same plot (Fig. 5). A 5% RMC
solution takes about 12 seconds using MPI parallel
library on 10 nodes of the afore-mentioned
Beowulf cluster. It takes about two hours by
Monte Carlo method using a serial code on a 633
MHz 21164A Alpha processor workstation (Hsu,
2001a). The run time is shorter if the albedo is
smaller.
Figure 5 shows good agreement between the
MC and RMC solutions and somewhat small
difference with the discrete ordinates solutions in
three cases of linear anisotropic scattering media.
The temporal transmittance curves of three
different cases are shown. The number of energy
bundles used in the simulation at different time
step varies from 105 to 2.105 by setting the standard
deviation of the solution to be 5% and it takes
about 12 seconds to finish. It takes 47 seconds by
setting standard deviation at 2% with the bundle
number varying from 105 to 1.6.106.
In
comparison the maximum number of bundles used
by Monte Carlo is 2.107. It is noted that the results
at long time by Monte Carlo method with this
bundle number still show large degree of
oscillation (Hsu, 2001a). This is caused by the
insufficient number of energy bundles used in the
MC simulation. It is obvious that if one sets the
standard deviation to be 0.5%, the solution become
much smoother and the corresponding bundle
numbers will vary from 3.2.106 to 1.28.107. The
total computational time is 11 minutes with 0.5%
standard deviation setting. It should be pointed out
that the computational time saving over the MC
method is not significant as this is one-dimensional
slab geometry. We expect the saving will be much
more in the two- and three-dimensional geometries
with detectors.
However, even with onedimensional geometry the result of 0.5% standard
deviation is better than the MC solution at
comparable computational time.
Another advantage of the reverse Monte Carlo
method is that it can provide the solution at a
specific time or over a time interval. It is unlike
other solution methods or MC that the complete
time history from t = 0 to any time of interest has
to be computed. The information over specific
6
Copyright © 2003 by ASME
approach doesn't perform time marching for all
bundles simultaneously, but rather it "assembles"
the temporal information of all bundles at the end.
In this way, the memory requirement is not an
issue, but it can't provide the variance of the
simulations during the calculation. It should be
pointed out that the coarse grain parallelism is
retained in all of the above MC and RMC
algorithms. As far as the requirements of reducing
memory usage and obtaining solution variance
during the calculation, the reverse Monte Carlo
algorithm is more suitable.
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Finite Volume Method and Integral Equation
Model,," Proceedings, the ASME Summer
Heat Transfer Conf., Las Vegas, NV, 2003b.
Mitra, K., M.-S. Lai, and S. Kumar, "Transient
Radiation Transport in Participating Media
within a Rectangular Enclosure," AIAA J.
Thermophysics and Heat Transfer, Vol. 11,
No. 3, pp. 409-414, 1997.
Modest, M. F., "Backward Monte Carlo
Simulations in Radiative Heat Transfer,"
CONCLUSIONS
The reverse Monte Carlo simulations are
conducted for transient radiative transfer process
within the absorbing, scattering, and non-emitting
participating media.
Detailed algorithm is
developed and discussed.
The solutions are
validated with the existing solutions obtained by
Monte Carlo and discrete ordinates methods. The
method was shown to be a very effective way to
reduce the long computational time and memory
requirements in the Monte Carlo method when
only certain spatial and temporal domain
information is needed. This method provides a
computationally efficient scheme for remote
sensing and optical imaging inverse analysis. It is
possible to apply the method in real time
applications.
ACKNOWLEDGEMENTS
This work is partially supported by Sandia
National Laboratories with contract No. AW-9963.
The parallel system is provided by a grant from
National Science Foundation MRI Program grant
No. EIA-0079710 and Dr. Rita V. Rodriguez is the
program director.
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&
Pulse trailing edge
Pulse leading edge
(z - z )/(-µ)
1
L
µ
0
z ,t
2
1
L
ct
2
+z
z ,t
z
2
1
z -z
1
ct
L
1
c(t - t ) - z
1
p
L
c(t - t )
1
p
Fig. 3 The relation of backtracking bundle path
length and pulse movement. The path length of (zL
- z1)/-µ equals the pulse travel distance at the same
time interval. The equality leads to the
determination of zL.
8
Copyright © 2003 by ASME
-2
10
Transmittance
Transmittance
10
-3
10
Standard deviation
5%
2%
0.5%
0
10
-1
10
-2
10
-3
10
-4
DOM
RMC
τ = 1, ω = 0.9, isotropic
-4
10
0
20
40
60
80
100
0
2
4
t*
6
10
-2
a = 0.9
a=0
a = -0.9
Bundle Number
a=0
-3
10
7
a = 0.9
a = -0.9
Transmittance
10
8
Fig. 6 Wave front resolution of the RMC method.
Fig. 4 The convergence of solutions with different
variance. The medium has an isotropic scattering
phase function, scattering albedo = 0.998, and
optical thickness = 10.
10
t*
τ = 10, ω = 0.998
10
6
10
5
DOM
10
RMC,std dev 0.5%
MC
-4
0
20
40
t*
60
80
100
0
20
40
t*
60
80
100
Fig. 7 Bundle number used in three difference
scattering phase functions. The converged
solutions have 2% standard deviation.
Fig. 5 Comparison of RMC, MC, and Discrete
Ordinates Method solutions for isotropic scattering
slab with a collimated pulse irradiation.
9
Copyright © 2003 by ASME