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Effective scattering phase functions for the
multiple scattering regime
Jacek Piskozub1 and David McKee
2,*
1Institute of Oceanology, Polish Academy of Sciences, ul. Powstancow Warszawy 55, 81-712 Sopot, Poland 2Physics Department, University of Strathclyde, 107 Rottenrow, Glasgow, G4 0NG, Scotland
Abstract: The propagation of light through turbid media is of fundamental
interest in a number of areas of optical science including atmospheric and
oceanographic science, astrophysics and medicine amongst many others.
The angular distribution of photons after a single scattering event is
determined by the scattering phase function of the material the light is
passing through. However, in many instances photons experience multiple
scattering events and there is currently no equivalent function to describe
the resulting angular distribution of photons. Here we present simple
analytic formulas that describe the angular distribution of photons after
multiple scattering events, based only on knowledge of the single scattering
albedo and the single scattering phase function.
© 2011 Optical Society of America
OCIS codes: (290.4210) Multiple scattering; (290.7050) Turbid media; (010.0010) Atmospheric
and oceanic optics.
References and links
1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981). 2. H. C. van de Hulst, Multiple Light Scattering, Vol. 1 (Academic Press, 1980).
3. H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic Press, 1980).
4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge Univ. Press, 2006).
5. A. A. Kokhanovsky, Cloud Optics (Springer, 2006).
6. M. Sydor, “Statistical treatment of remote sensing reflectance from coastal ocean water: proportionality of reflectance from multiple scattering to source function b/a,” J. Coast. Res. 235, 1183–1192 (2007).
7. N. Pfeiffer, and G. H. Chapman, “Successive order, multiple scattering of two-term Henyey-Greenstein phase
functions,” Opt. Express 16(18), 13637–13642 (2008). 8. P. Flatau, J. Piskozub, and J. R. V. Zaneveld, “Asymptotic light field in the presence of a bubble-layer,” Opt.
Express 5(5), 120–124 (1999).
9. Z. Otremba, and J. Piskozub, “Modelling the bidirectional reflectance distribution function (BRDF) of seawater polluted by an oil film,” Opt. Express 12(8), 1671–1676 (2004).
10. J. Piskozub, A. R. Weeks, J. N. Schwarz, and I. S. Robinson, “Self-shading of upwelling irradiance for an
instrument with sensors on a sidearm,” Appl. Opt. 39(12), 1872–1878 (2000). 11. D. McKee, J. Piskozub, and I. Brown, “Scattering error corrections for in situ absorption and attenuation
measurements,” Opt. Express 16(24), 19480–19492 (2008).
12. G. Marsaglia, The Diehard Battery of Tests of Randomness (1995), http://www.stat.fsu.edu/pub/diehard/. 13. L. C. Henyey, and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
14. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters. (Academic, 1994).
15. T. J. Petzold, T. J. SIO Ref. 72–78, Scripps Institute of Oceanography (U. California, 1972). 16. G. R. Fournier, and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
17. J. L. Forand, and G. R. Fournier, “Particle distributions and index of refraction estimation for Canadian waters,”
Proc. SPIE 3761, 34–44 (1999). 18. I. Turcu, and M. Kirillin, “Quasi-ballistic light scattering – analytical models versus Monte Carlo simulations,” J.
Phys.: Conf. Ser. 182, 012035 (2009).
1. Introduction
Many natural media, including the oceans, clouds, nebulae, skin and other living tissues,
exhibit strong effects of multiple scattering. Optical text books often discuss single scattering
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4786
theory in depth, but give only a cursory introduction to multiple scattering [1]. There are
specialist books on multiple scattering [2–5] that treat the subject in much greater detail, but
there remains a need for simple analytical relationships to describe the phenomenon. We
present a number of new formulations relating multiple scattering signals to well-understood
single scattering parameters that go some way to satisfying this demand.
The scattering phase function describes the angular distribution of scattered
photons in the case of single scattering. However, in many cases photons undergo more than
one scattering event and there is a need to develop effective phase functions ms that
describe the angular distribution of photons after multiple scattering events [6]. The
asymmetry parameter for the single scattering phase function is given by
1
0
2 sin cosg d
(1)
Pfeiffer and Chapman [7] showed that the Henyey-Greenstein, HG, phase function which is
parameterised on the asymmetry parameter could be easily modelled in the multiple scattering
case as the phase function for nth
order scattering has gn = g1n. They also showed that the
resulting nth
order phase function takes the form of a HG phase function.
Fig. 1. Schematic diagram of scattering angles used in Eq. (2) for calculating multiple
scattering trajectories.
Multiple scattering is a 3D process that cannot, in the general case, be reduced to 2D by
assuming azimuthal symmetry. After two scattering events at θ1 and θ2, a photon will
propagate at an angle θn to the original incident direction, where
1 2 1 2cos cos cos sin sin cosn (2)
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4787
and ψ is the azimuthal angle of scattering for the second scattering event relative to the first.
After two scattering events the photon will have a direction of propagation between θ1 - θ2 and
θ1 + θ2 (Fig. 1). Here we assume that the medium is infinite, isotropic and homogeneous.
The purpose of this paper is to determine if the results of Pfeiffer and Chapman [7] are
valid for the general case of phase functions that are not defined on the asymmetry parameter
and to develop a robust method for determining an effective scattering phase function for the
case of multiple scattering. In the first instance Monte Carlo simulations are used to provide
fully parameterised light fields that can be used to test the above hypotheses and to establish
general relationships that are applicable across the range of sciences where multiple scattering
is a feature. Here we define effective scattering phase functions in the same sense as Pfeiffer
and Chapman [7] as describing the angular distribution of scattered photons for nth
order
scattering, as determined from Monte Carlo simulations. Furthermore, we define an effective
phase function for multiple scattering which describes the angular distribution of photons after
all orders of scattering have been simulated (limited only by the number of photons used in
the Monte Carlo simulations). We believe this name represents a logical extension of the
classical, single scattering phase function.
2. Methods
The forward directed Monte Carlo code has been extensively tested previously and used in
both studies of marine environment [8,9] and for correction of measurement errors of ocean
optics instrumentation [10,11]. Absorption and scattering parameters as well as the scattering
phase functions were treated as statistical properties, which makes it possible to decide the
fate of each virtual photon using random numbers. A pseudo-random number generator of
tested uniformity [12] and a period of 2127
(KISS by George Marsaglia) was used. In the setup
used for this study, the modelled light source was a parallel light beam placed in a virtually
unbounded space (no virtual photons reached the 3D model walls which were several
thousands of optical depths from the light source). The beam is assumed to be vertical for
simplicity. For each run, one billion (109) virtual photons were used. The zenith angle of the
virtual photon and the scattering order were recorded for each scattering event. This made
possible the estimation of phase functions for each scattering order as well as the effective
phase function (the latter using all scattering events).
The Henyey-Greenstein [13] scattering phase function is calculated from
2
3/22
1 1
4 1 2 cos
g
g g
(3)
where g is the mean cosine of the scattering phase function. Mobley [14] showed that g =
0.924 gave a best-fit to the „particle phase function‟ derived from Petzold‟s [15]
measurements of the scattering phase function of seawater. Fournier and Forand [16,17] gave
an alternative analytic expression for the scattering phase function based upon a population of
Mie scatterers with a hyperbolic (Junge) particle size distribution (slope μ) and a real
refractive index, nr,
2 2
1 41 1 1 1
4 1 u
(4)
where
2
2
3, , 2sin 2
2 3 1r
uu
n
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4788
By assuming a linear relationship between n and μ, and integrating Eq. (4), Mobley et al [18]
were able to obtain a formulation of the Fournier-Forand (FF) scattering phase function using
only the particle backscattering ratio, Bp = bbp / bp, as input. This formulation has been used
widely in ocean optics and is adopted here. For the purpose of this paper the key difference
between HG and FF scattering phase functions is that the former are defined on the
asymmetry parameter, g, while the latter are not. This difference facilitates testing the general
applicability of the Pfeiffer and Chapman [7] result for any arbitrary scattering phase function.
3. Results
Monte Carlo simulations were performed for HG scattering phase functions with a range of
asymmetry values (0.5 < g1 < 0.99). In every case the angular distribution of nth
order
scattering was well described by a HG function with gn = g1n, closely reproducing the results
of Pfeiffer and Chapman [7] (Fig. 2). To test the general validity of this result, we repeated the
simulations with FF scattering phase functions that are used to describe scattering in marine
environments and are not specifically parameterised on the asymmetry parameter [16,17].
Figure 3 shows that gn = g1n is true for all orders of scattering up to and including 50th order
scattering (the result is only limited by the number of photons used in the simulation). We
note that this result is consistent with taking the mean of both sides of Eq. (2) over all values
of ψ.
Scattering angle (deg)
0 20 40 60 80 100 120 140 160 180 200
Scatt
ering p
hase f
unction (
sr-1
)
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
1e+3
1e+4
1e+5
n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10
Scattering angle (deg)
0 20 40 60 80 100 120 140 160 180 200
n=1
n=10
n=20
n=30
n=40
n=50
Scatt
ering p
hase f
unction (
sr-1
)
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10
n=1
n=10
n=20
n=30
n=40
n=50
Henyey-Greenstein g1=0.9
Fournier-Forand bbp/bp=0.018 Fournier-Forand bbp/bp=0.018
Henyey-Greenstein g1=0.9
(a) (b)
(c) (d)
Fig. 2. nth order scattering phase functions for (a) and (b) HG (g1 = 0.9) and (c) and (d) FF (bbp / bp = 0.018) phase functions up to 50th order scattering obtained from Monte Carlo simulations
(lines). Symbols in (a) and (b) are calculated using Eq. (3) with gn = g1n from Pfeiffer and
Chapman [7]. FF phase functions are not defined on the asymmetry parameter, so it is not possible to use the Pfeiffer and Chapman relationship in this manner for these functions.
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4789
Order of scattering, n0 10 20 30 40 50
gn
0.0001
0.001
0.01
0.1
1
10
bbp
/bp=0.001 g
1=0.9941
bbp
/bp=0.018 g
1=0.9304
bbp
/bp=0.100 g
1=0.7182
log(gn) = log(0.9941).n
log(gn) = log(0.9303).n
log(gn) = log(0.7177).n
gn = g
1
n log(g
n) = log(g
1).n
Fig. 3. Asymmetry parameters for nth order scattering phase functions, gn, plotted against scattering order for 3 different FF scattering phase functions. Best-fit regression lines confirm
that gn = g1n holds for FF phase functions.
Single scattering albedo,
0.0 0.2 0.4 0.6 0.8 1.0
Eff
ective a
ssym
etr
y p
ara
me
ter,
gm
s
0.0
0.2
0.4
0.6
0.8
1.0
1.2
bbp
/bp=0.001 g
1=0.9941
bbp
/bp=0.018 g
1=0.9304
bbp
/bp=0.100 g
1=0.7182
Fig. 4. Asymmetry parameters for effective multiple scattering phase functions for three FF single scattering phase functions calculated from Monte Carlo simulations (symbols) and Eq.
(5) (lines) over a wide range of single scattering albedos.
The single scattering albedo, ω, is defined as the ratio of scattering, b, to attenuation, c,
where c = a + b and a is the absorption coefficient. From Monte Carlo simulations we were
able to resolve the final angular distribution of photons that had experienced multiple
scattering and hence could determine an effective scattering phase function ms . Figure 4
shows that this effective scattering phase function has an asymmetry parameter gms, that is
related to the single scattering asymmetry parameter g1 through
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4790
11
1
1
1
1
ni
i
ims n
i
i
gg
gg
(5)
Equation (5) holds for 1 < g1 < 1 and 0 ω 1.
The probability of single scattering into an angular bin centred on θ1 is given by
1 1 1sin . The phase function for nth
order scattering can be calculated from the single
scattering phase function by iterating through each order of scattering n, using
1 2
1 2
1 1 1 1 1 2 2 2
, ,
1 1 2 2
, ,
sin sin
4sin sin
n
n n
(6)
with θn determined for each combination of θ1, θ2, ψ using Eq. (2). The validity of this
parameterisation is demonstrated in Fig. 5a where nth
order HG phase functions are accurately
reproduced up to order 50 using Eq. (6) and using the result of Pfeiffer and Chapman [7] for
HG functions. Figure 5b shows that Eq. (6) accurately reproduces nth
order FF phase functions
obtained from Monte Carlo simulations up to at least 50th order.
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4791
Scattering angle (deg)
0 20 40 60 80 100 120 140 160 180 200
Scatt
erin
g p
hase f
unctio
n (
sr-1
)
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
1e+3
1e+4
1e+5
n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10
Scattering angle (deg)
0 20 40 60 80 100 120 140 160 180 200
n=1
n=10
n=20
n=30
n=40
n=50
Scatt
erin
g p
hase f
unctio
n (
sr-1
)
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
n=1
n=2
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n=10
n=1
n=10
n=20
n=30
n=40
n=50
Henyey-Greenstein g1=0.924 Henyey-Greenstein g1=0.924
Fournier-Forand bbp/bp=0.018 Fournier-Forand bbp/bp=0.018
Fig. 5. nth order scattering phase functions calculated with 0.1° increments in θ using Eq. (6) (symbols - only every 2° shown for clarity) and from Monte Carlo simulations (solid lines) for
a HG single scattering phase function with g1 = 0.924 (a and b), and for a FF single scattering
phase function with bbp / bp = 0.018 (c and d).
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4792
scattering angle (deg)
0 30 60 90 120 150 180
eff
ective
mu
ltip
le s
ca
tte
rin
g p
ha
se
fu
nctio
n (
sr-1
)
1e-3
1e-2
1e-1
1e+0
1e+1
1e+2
1e+3
1e+4
=0.3
=0.7
=0.95
n=1
Fournier-Forand bbp/bp=0.018
Fig. 6. Effective multiple scattering phase functions for a FF single scattering phase function (bbp / bp = 0.018) from Monte Carlo simulations (lines) and calculated using Eq. (7) (symbols)
with nth order phase functions derived from Monte Carlo simulations. Increasing single
scattering albedo results in more scattering at wide angles for a given single scattering phase function. The single scattering phase function (solid line, no symbols, n = 1) is shown for
reference.
The effective scattering phase function for multiple scattering can be calculated from nth
order phase functions using Eq. (7)
2 1
1 2 3
2 11
n
n
ms n
(7)
Figure 6 shows effective multiple scattering phase functions obtained from Monte Carlo
simulations for a single FF scattering phase function and a range of scattering albedo values,
match effective scattering phase functions calculated using Eq. (7) and nth
order phase
functions obtained from Monte Carlo simulations.
4. Conclusions
Multiple scattering is a phenomenon that affects the propagation of light through turbid media.
Examples are easily found in marine and atmospheric optics, biomedical optics, astrophysics
and plasma physics amongst others. Derivation of approximate analytical solutions for
multiple scattering is an active area of research, with one potential benefit (under limited
circumstances) being the ability to vastly reduce computation time in comparison to standard
Monte Carlo approaches [18]. Equations (5), (6), and (7) are, to the best of our knowledge,
new developments that provide excellent fits to simulated data and are consistent with rational
analysis of the multiple scattering problem. The results presented in this paper are of general
relevance and provide a simple framework in which the relationship between single scattering
events and multiple scattering effects can be easily parameterised and understood. We note
that the asymmetry parameter (also known as the mean cosine) is a particularly useful
descriptor of the scattering phase function and further effort to parameterise popular phase
functions (such as Fournier-Forand) in terms of g would be well placed. It is anticipated that
these results will facilitate new insights into the effect of multiple scattering on a variety of
applications such as generation of remote sensing signals, impact on optical measurements
and interpretation of scattering signals from turbid media.
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4793
Acknowledgments
The authors would like to thank the Royal Society of Edinburgh International Exchange
Programme for funding a number of visits between laboratories. This work was also supported
by the Natural Environment Research Council through the award of an Advanced Fellowship
to D. M. (NE/E013678/1) and by statutory task I.3.3 of Institute of Oceanology PAS.
#141685 - $15.00 USD Received 26 Jan 2011; accepted 16 Feb 2011; published 25 Feb 2011(C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4794
