Four-flux model and a Monte Carlo code: comparisons between two simple, complementary tools for multiple scattering calculations B. Maheu, J. P. Briton, and Gérard Gouesbet
the description of multiple light scattering. Recently, we built a Monte Carlo code, called MUSCAT (for multiple scattering), turned toward multiple scattering computations. Both these works are aimed at developing efficient and easily utilizable tools for computations of multiple light scattering. Here we report the first comparisons between the four-flux model and the code MUSCAT showing good agreement and outlining the applicability fields of both models. We also mention other validations for both tools which strengthen our confidence in these simple approaches of the complex multiple scattering phenomena.
The four-flux model1,2 handles multiple scattering inside a plane-parallel slab with, possibly, a background surface and optical interfaces between the slab and the surrounding medium (see figures in Refs. 1 and 2). The monochromatic radiation field at location z is modeled as constituted of two collimated beams and of two semi-isotropic diffuse radiations propagating in opposite directions. The homogeneous and isotropic scattering medium is characterized by its width Z, its absorption and scattering coefficients k and s, its forward scattering ratio ζ, and the average crossing parameter ε (definitions in Ref. 1). The scattering process is assumed to be quasielastic (no emission nor quantum transitions inside the slab) without any dependent scattering. Polarization is not taken into account. The four-flux model provides formulas for the various reflectances Rcc, Rcd, Rdd and transmittances τcc, τcd, Tdd of the slab (cc for collimated-collimated; cd for collimated-diffuse; dd for diffuse-diffuse). The main interest of the four-flux theory comes from the simplicity of the formulas. In a recent comparison with rigorous multiple scattering theories,3 Niklasson concluded that the four-flux model provides sufficient accuracy in most situations when spectrophotometric integrating sphere measurements are made. In this Letter, we bring further comparisons of the four-flux theory with Monte Carlo computations for the case of scattering slabs without optical interfaces and without background surface. With these assumptions the general four-flux formulas of Ref. 1 simplify to those of Ref. 2 [Sec. VI. A, Eqs. (47)-(52)].
MUSCAT is a Monte Carlo program which has been de-
signed to handle multiple scattering under the following assumptions: (i) homogeneous and isotropic scattering medium, (ii) quasielastic scattering process without any account for polarization nor for dependent scattering (arbitrary phase function), (iii) monochromatic source, (iv) arbitrary geometrical characteristics and locations for the source, the medium, and the detectors.
In the case of multiple scattering, the Monte Carlo method essentially consists of following each photon from its emission by the source up to its absorption inside or escape from the scattering medium. The process has to be repeated as many times as imposed by the accuracy required for the results. Random numbers are used to determine the successive events of a photon travel (absorption/scattering/escaping). MUSCAT was first compared with exact theoretical results of van de Hulst4 obtained by the doubling method. From the nearly perfect agreement we inferred the validity of the code MUSCAT and began other comparisons with the four-flux model which we present hereafter.
The first comparison between MUSCAT and four-flux holds on diffuse reflectances of opaque atmospheres. In the limit of infinite optical thickness, the scattering slab only reflects (and absorbs) the incident radiation. The reflectances R'∞ and R∞ for, respectively, collimated and diffuse illumination are Rcd and Rdd [Eqs. (51) and (52), Ref. 2] in the limit Z → ∞. Isotropic scattering is assumed (ζ = 0.5). Reflectances computed by both methods are shown vs scat-terer diameter d in Fig. 1 (note the monotonic relation: diameters can be deduced from reflectance measurements). The single scattering albedo
was obtained using Lorenz-Mie computations5 with a scat-terer complex refractive index equal to 2-5i. For computing R'∞, we also assumed ε = 2. Owing to other applications, the wavelength is high (λ = 337 μm), but the relevant numerical values are those of the size parameter α = πd/λ. We note very good agreement between MUSCAT and the four-flux results, the discrepancies always being under 2% of the asymptotic reflectance value (collimated illumination) or under 5% (diffuse illumination).
Fig. 2. Reflectance of a slab vs optical thickness (collimated illumination); comparisons between MUSCAT (*) and four-flux with = 1 (dashed line) or ε = 2 (continuous line). The albedo (a), the asymmetry parameter (g), and the forward scattering ratio (ζ) values are
given beside the curves.
Fig. 1. Reflectances of an opaque slab vs scatter diameter and size parameter (isotropic scattering; refractive index 2-5i; λ = 337 μm); comparisons between MUSCAT (*) and four-flux for diffuse (dashed
line) or collimated illumination (continuous line). Fig. 3. Same as Fig. 2 but for transmittance instead of reflectance.
For other situations, like scattering slabs with finite optical thicknesses b [b = (k + s)Z), we have selected a few examples of comparisons between MUSCAT and the four-flux theory in Figs. 2-4.
Figure 2 (respectively, 3 and 4) shows the collimated-diffuse reflectance (respectively, collimated-diffuse trans-mittance/diffuse-diffuse reflectance) as a function of the optical thickness for four sets of scattering parameters (a;ζ). For each set, we give the results of MUSCAT (stars) and two curves computed from the four-flux theory with ε = 1 (dashed line) and ε = 2 (continuous line). MUSCAT computations have been carried out assuming Henyey-Greenstein phase functions with asymmetry parameter g.
This comparison between MUSCAT and four-flux exhibits some interesting features:
(i) Both qualitative and quantitative agreement for all cases, with discrepancies ranging from 0% up to 15% provided that the optimized ε value has been chosen.
(ii) Validity of the ε ≃ 2 assumption in most cases. From other computations which have not been shown here, we observed that this assumption almost always holds except for highly peaked forward ( ζ ≃ 1) or backward (ζ ≃ 0) scattering. For these latter cases, the assumption ε ≃ 1 better accounts for the actual scattering process.
(iii) For the diffuse-diffuse transmittance (Fig. 4), if one decreases the albedo for a given forward scattering ratio ζ, one must simultaneously use lower ε values to preserve the agreement between MUSCAT and four-flux.
(ii) and (iii) have recently been pointed out by Niklasson3
for the collimated-diffuse transmittance. As a conclusion, let us emphasize the main interest of the
above comparisons: the close agreement between the four-flux model and MUSCAT validates both approaches. Concerning the four-flux model it is a further confirmation which strengthens previous validations. For MUSCAT it means that further applications will be reached in the near future.
Fig. 4. Same as Fig. 2 but for diffuse instead of collimated illumination.
From the user's point of view, the four-flux model is a very simple and efficient model for multiple scattering as far as integral quantities, like total reflectance or transmittance, are measured. This statement holds for collimated as well as for diffuse illumination, for isotropic as well as for anisotropic scattering, and for tenuous as well as for dense clouds. In all these situations, the measurements can be interpreted with a simple pocket calculator in the limit of ~10% accuracy. Keeping the user's point of view, soon we expect to publish an improved version of MUSCAT allowing easy and fast computations with PCs. MUSCAT is currently being compared with experiments using narrow beam sources and we aim at producing a versatile tool for multiple scattering computations with arbitrary geometry and sources. Such a code will handle more realistic problems than the four-flux model does and these two complementary tools will cover a broad range of multiple scattering problems. Among a lot of concrete applications let us mention a single example we are faced with in our research work as well as in everyday life in Normandy: the determination of the visibility on foggy roads as a function of the illumination conditions, the fog composition, and the target nature.
Part of this work was done under a contract between the Institut National des Sciences Appliquées de Rouen and the Laboratoire Central des Ponts et Chaussees (L.C.P.C.). We acknowledge J. M. Caussignac (L.C.P.C.) and J. Menard (Laboratoire Regional CETE de Rouen) for helpful collaboration in the framework of the above contract.
References 1. B. Maheu, J. N. Letoulouzan, G. Gouesbet, "Four-Flux Models to
Solve the Scattering Transfer Equation in Terms of Lorenz-Mie Parameters," Appl. Opt. 23, 3353 (1984).
2. B. Maheu and G. Gouesbet, "Four-Flux Models to Solve the Scattering Transfer Equation: Special Cases," Appl. Opt. 25, 1122 (1986).
3. G. A. Niklasson, "Comparisons Between Four Flux Theory and Multiple Scattering Theory," Appl. Opt. 26, 4034 (1987).
4. H. C. van de Hulst, Multiple Light Scattering. Tables, Formulas and Applications Vols. 1 and 2 (Academic, New York, 1980).
5. G. Gouesbet, G. Gréhan, and B. Maheu, "Single Scattering Characteristics of Volume Elements in Coal Clouds," Appl. Opt. 22, 2038 (1983).
