## A novel approach for simulating light interaction with particulate materials: application to the modeling of sand spectral properties

Optics Express, Vol. 15, Issue 15, pp. 9755-9777 (2007)

http://dx.doi.org/10.1364/OE.15.009755

Acrobat PDF (3051 KB)

### Abstract

In this paper, we present a new spectral light transport model for sand. The model employs a novel approach to simulate light interaction with particulate materials which yields both the spectral and spatial (bidirectional reflectance distribution function, or BRDF) responses of sand. Furthermore, the parameters specifying the model are based on the physical and mineralogical properties of sand. The model is evaluated quantitatively, through comparisons with measured data. Good spectral reconstructions were achieved for the reflectances of several real sand samples. The model was also evaluated qualitatively, and compares well with descriptions found in the literature. Its potential applications include, but are not limited to, applied optics, remote sensing and image synthesis.

© 2007 Optical Society of America

## 1. Introduction

3. Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, “Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf,” Applied Optics **35**, 6585–6598 (1996). [CrossRef] [PubMed]

*s*pectral

*li*ght transport model for

*s*and, hereafter referred to as SPLITS. We evaluate the model using virtual spectrophotometric [4

4. G. Baranoski, J. Rokne, and G. Xu, “Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering,” The Visual Computer **17**, 506–518 (2001). [CrossRef]

*characterization data*, as input.

6. S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, “Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model,” Remote Sensing of Environment **56**, 194–202 (1996). [CrossRef]

7. R. Shuchman and D. Rea, “Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data,” Remote Sensing of Environment **11**, 295–310 (1981). [CrossRef]

8. R. Morris and D. Golden, “Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars,” Icarus **134**, 1–10 (1998). [CrossRef]

9. R. Singer, “Spectral Evidence for the Mineralogy of High-Albedo Soils and Dust on Mars,” Journal of Geophysical Research **87**, 10,159–10,168 (1982). [CrossRef]

## 2. Properties of Sand

*et*

*al*. [10

10. F. Pettijohn, P. Potter, and R. Siever, *Sand and Sandstone*, 2nd ed. (Springer-Verlag, New York, NY, 1987). [CrossRef]

*soil*. Soil is composed of particles of weathered rock and sometimes organic matter immersed in a medium of air and water (the

*pore space*) [12].

*soil separates*, according to their size. The relative masses of each of the soil separates are then compared to determine the

*texture*of a soil sample.

*sand*,

*silt*, and

*clay*, as delineated in Table 1. Particles larger than 2mm are classified as rock fragments and are not considered to be part of the soil. A sand textured soil contains at least 85% sand-sized particles.

*sand*is used to describe both a soil

*separate*and a soil

*texture*may be a source of confusion. In the remainder of this paper, the term

*sand*is used to refer to a soil texture unless otherwise stated (for instance, by referring to

*sand-sized*particles).

### 2.1. Factors Affecting Light Transport

14. K. Coulson and D. Reynolds, “The Spectral Reflectance of Natural Surfaces,” Journal of Applied Meteorology **10**, 1285–1295 (1971). [CrossRef]

15. J. Norman, J. Welles, and E. Walter, “Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils,” IEEE Transactions on Geoscience and Remote Sensing **GE-23**, 659–667 (1985). [CrossRef]

16. K. Coulson, G. Bouricius, and E. Gray, “Optical Reflection Properties of Natural Surfaces,” Journal of Geophysical Research **70**, 4601–4611 (1965). [CrossRef]

*retro-reflection*[16

16. K. Coulson, G. Bouricius, and E. Gray, “Optical Reflection Properties of Natural Surfaces,” Journal of Geophysical Research **70**, 4601–4611 (1965). [CrossRef]

16. K. Coulson, G. Bouricius, and E. Gray, “Optical Reflection Properties of Natural Surfaces,” Journal of Geophysical Research **70**, 4601–4611 (1965). [CrossRef]

17. M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, “Reflectance Properties of Soils,” Advances in Agronomy **38**, 1–43 (1985). [CrossRef]

10. F. Pettijohn, P. Potter, and R. Siever, *Sand and Sandstone*, 2nd ed. (Springer-Verlag, New York, NY, 1987). [CrossRef]

*parent material*is the rock that is the source of the mineral part of the sand. This is typically a silicate mineral [11] such as quartz, gypsum or calcite, with quartz being the most common [11, 18

18. D. Leu, “Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship,” Remote Sensing of Environment **6**, 169–182 (1977). [CrossRef]

*yellow ochre*[20] or

*limonite*[21], is one of the most common minerals found in soils [20]. It colours soils yellow to brown [22

22. J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, “Quantitative Relationships Between Soil Color and Hematite Content,” Soil Science **136**, 354–358 (1983). [CrossRef]

*red ochre*[20], imparts a red colour to soils and may mask the colour of goethite except when in small quantities [22

22. J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, “Quantitative Relationships Between Soil Color and Hematite Content,” Soil Science **136**, 354–358 (1983). [CrossRef]

23. R. Cornell and U. Schwertmann, *The Iron Oxides*, 2nd ed. (Wiley-VCH GmbH & Co. KGaA, Weinheim, Germany, 2003). [CrossRef]

*21*]. They are also found, typically within a kaolinite or illite matrix, as coatings, approximately 1–5

*µ*m thick, that form on the grains during aeolian (

*i*.

*e*., by wind) transport [24

24. H. Wopfner and C. Twindale, “Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia,” Geologische Rundschau **77**, 815–834 (1988). [CrossRef]

#### 2.1.2. Water

26. S. Twomey, C. Bohren, and J. Mergenthaler, “Reflectance and Albedo Differences Between Wet and Dry Surfaces,” Applied Optics **25**, 57–84 (1986). [CrossRef]

27. M. Kühl and B. Jørgensen, “The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments,” Limnology and Oceanography **39**, 1368–1398 (1994). [CrossRef]

#### 2.1.3. Grain Size and Shape

28. R. Vincent and G. Hunt, “Infrared Reflectance from Mat Surfaces,” Applied Optics **7**, 53–59 (1968). [CrossRef] [PubMed]

17. M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, “Reflectance Properties of Soils,” Advances in Agronomy **38**, 1–43 (1985). [CrossRef]

18. D. Leu, “Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship,” Remote Sensing of Environment **6**, 169–182 (1977). [CrossRef]

17. M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, “Reflectance Properties of Soils,” Advances in Agronomy **38**, 1–43 (1985). [CrossRef]

*sphericity and roundness*. Sphericity refers to the general shape of a particle by expressing its similarity to that of a sphere [29

29. H. Wadell, “Volume, Shape, and Roundness of Rock Particles,” Journal of Geology **40**, 443–451 (1932). [CrossRef]

*projection*sphericity measure, meaning its definition is based on the projection of the particle onto a plane. The

*Riley sphericity*, Ψ, of a particle is given by

*D*is the diameter of the largest inscribed circle and

_{i}*D*is the diameter of the smallest circumscribed circle, as shown in Fig. 1.

_{c}*roundness*can loosely be described as a measure of detail in the features on the grain surface [29

29. H. Wadell, “Volume, Shape, and Roundness of Rock Particles,” Journal of Geology **40**, 443–451 (1932). [CrossRef]

#### 2.1.4. Additional Factors

## 3. Related Work

32. H. Zhang and K. Voss, “Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models,” Applied Optics **44**, 597–610 (2005). [CrossRef] [PubMed]

*et*

*al*. [33

33. Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, “Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance,” Journal of Quantitative Spectroscopy and Radiative Transfer **100**, 457–469 (2006). [CrossRef]

*et al*. [34, 35

35. M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, “Multiple Scattering by Random Particulate Media: Exact 3D Results,” Optics Express **15**, 2822–2836 (2007). [CrossRef] [PubMed]

### 3.1. General Models

36. B. Hapke, “Bidirectional Reflectance Spectroscopy. 1. Theory,” Journal of Geophysical Research **86**, 3039–3054 (1981). [CrossRef]

37. B. Hapke and E. Wells, “Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations,” Journal of Geophysical Research **86**, 3055–3054 (1981). [CrossRef]

38. A. Emslie and J. Aronson, “Spectral Reflectance of Particulate Materials. 1: Theory,” Applied Optics **12**, 2563–2572 (1973). [CrossRef] [PubMed]

*asperities*on the surface of the particle. However, their model was only applicable to the far infrared [39

39. W. Egan and T. Hilgeman, “Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering,” Applied Optics **17**, 245–252 (1978). [CrossRef] [PubMed]

39. W. Egan and T. Hilgeman, “Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering,” Applied Optics **17**, 245–252 (1978). [CrossRef] [PubMed]

41. L. Wolff, “Diffuse-Reflectance Model for Smooth Dielectric Surfaces,” Journal of the Optical Society of America A (Optics, Image Science, and Vision) **11**, 2956–2968 (1994). [CrossRef]

*V*-shaped cavities. Oren and Nayar compared the output from their model to a sand sample. Their model, however, only simulates reflectance in the spatial domain. Mishchenko

*et al*. [42

42. M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces,” Journal of Quantitative Spectroscopy and Radiative Transfer **63**, 409–432 (1999). [CrossRef]

43. D. Stankevich and Y. Shkuratov, “Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure,” Journal of Quantitative Spectroscopy and Radiative Transfer **87**, 289–296 (2004). [CrossRef]

44. J. Peltoniemi, “Spectropolarised Ray-Tracing Simulations in Densely Packed Particulate Medium,” Journal of Quantitative Spectroscopy and Radiative Transfer (2007). In Press, accepted, URL http://dx.doi.org/10.1016/j.jqsrt.2007.05.009. [CrossRef]

### 3.2. Simulation of Specific Effects

#### 3.2.1. Moisture

*et al*. [45] use an extension of the Henyey-Greenstein phase function [46

46. L. Henyey and J. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophysical Journal **93**, 70–83 (1941). [CrossRef]

47. Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, “A Modeling Study of Backscattering from Soil Surfaces,” IEEE Transactions on Geoscience and Remote Sensing **34**, 264–271 (1996). [CrossRef]

*et al*. [42

42. M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces,” Journal of Quantitative Spectroscopy and Radiative Transfer **63**, 409–432 (1999). [CrossRef]

48. D. Lobell and G. Asner, “Moisture Effects on Soil Reflectance,” Soil Science Society of America Journal **66**, 722–727 (2002). [CrossRef]

*et al*. [49

49. D. Neema, A. Shah, and A. Patel, “A Statistical Optical Model for Light Reflection and Penetration Through Sand,” International Journal of Remote Sensing **8**, 1209–1217 (1987). [CrossRef]

#### 3.2.2. Iron Oxides

50. V. Barron and L. Montealegre, “Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory,” American Journal of Science **286**, 792–802 (1986). [CrossRef]

*et al*. [22

22. J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, “Quantitative Relationships Between Soil Color and Hematite Content,” Soil Science **136**, 354–358 (1983). [CrossRef]

52. D. Nickerson, “History of the Munsell Color System and its Scientific Application,” Journal of the Optical Society of America **30**, 575–645 (1940). [CrossRef]

*et al*. call the

*redness rating*. Okin and Painter [53

53. G. Okin and T. Painter, “Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces,” Remote Sensing of Environment **89**, 272–280 (2004). [CrossRef]

## 4. The SPLITS Model

*measurement of appearance*[55]. Furthermore, the SPLITS model is dependent on the physical and mineralogical characterization data for sand. The appearance of sand depends on many such parameters, and, to the best of our knowledge, there is no single source describing all of the requied characterization data for several sand samples. Therefore, data was gathered from several sources. With any research effort, this data gathering process represents a cruicial step. As pointed out by several researchers, “good science requires both theory and data — one is of little use without the other” [56

56. G. Ward, “Measuring and Modeling Anisotropic Reflection,” Computer Graphics **26**, 262–272 (1992). [CrossRef]

### 4.1. Concept

*e.g*., wavelength dependent refractive indices) of the sand medium (Fig. 2). The wavelength of light, a wave (physical) optics parameter, is included in its formulation by associating a wavelength value with each ray. It is assumed that each ray carries the same amount of radiant power, and the energies associated with different wavelengths are decoupled (

*i.e*., phenomena such as fluorescence are not addressed). Although the geometric approach adopted in the model design does not favour the direct computation of wave optics phenomena, such as interference

^{1}, it is more intuitive and allows an adequate description of the large scale light behaviours, such as reflection and refraction, in environments characterized by incoherent radiation fields, which represent the main targets of our simulations. It is also assumed that the relevant distances are much larger than the wavelength of the light. This assumption holds over the visible region of the spectrum for sand particles, as they are larger than 0.05mmin diameter (Section 2).

59. T. Nousiainen, K. Muinonen, and P. Räisänen, “Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation,” Journal of Geophysical Research **108**, AAC 12–1–17 (2003). [CrossRef]

### 4.2. Construction

#### 4.2.1. Extended Boundaries

*extended boundaries*(Fig. 2). The distance,

*D*, between them is set high enough so that light penetration to that depth is negligible [49

49. D. Neema, A. Shah, and A. Patel, “A Statistical Optical Model for Light Reflection and Penetration Through Sand,” International Journal of Remote Sensing **8**, 1209–1217 (1987). [CrossRef]

#### 4.2.3. Particle Geometry

*Left*).

65. M. Vepraskas and D. Cassel, “Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties,” Soil Science Society of America Journal **51**, 1108–1112 (1987). [CrossRef]

*h*, that is proportional to the particle size, with

*h′*=

*h*/2

*c*. The coating thickness is assumed to be small relative to the size of the particle, so that light travelling within the coating does not stray far from the point of entry. Hence, the coating may be approximated locally as a flat slab.

*Right*). The orientations of the facets are distributed such that the dot product between the microfacet normal,

**n**′, and the interface normal,

**n**, is given by

**X**is normally distributed with zero mean and standard deviation

*R*<

**n**′·

**n**≤1 for 95% of the facets [67]. Additionally, the microfacet normals are constrained so that

**n**′·

**n**>0. The particle roundness,

*R*, is therefore used to control roughness. When

*R*=1, the interface reduces to a smooth surface, as one would expect based on the concept of roundness [29

29. H. Wadell, “Volume, Shape, and Roundness of Rock Particles,” Journal of Geology **40**, 443–451 (1932). [CrossRef]

#### 4.2.4. Particle Composition

*dielectric constant*, the square of the complex refractive index, of a mixture to those of its constituents. This theory originally addressed the optical properties of media containing minute metal spheres and it was developed to predict the colours of metal glasses and metallic films [69

69. J. Maxwell Garnett, “Colours in Metal Glasses and in Metallic Films,” Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character **203**, 385–420 (1904). [CrossRef]

53. G. Okin and T. Painter, “Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces,” Remote Sensing of Environment **89**, 272–280 (2004). [CrossRef]

*ε*, of a mixture is given by

_{avg}*ε*is the dielectric constant of the matrix,

_{m}*ε*is the dielectric constant of the inclusions, and

*ν*is the volume fraction of the inclusions [68].

_{i}#### 4.2.5. Particle Types

*pure*(quartz (pq), hematite (ph), goethite (pg), and magnetite (pm)),

*mixed*(hematite with quartz (mh) and goethite with quartz (mg)), and

*coated*(hematite coated quartz (ch) and goethite coated quartz (cg)). The hematite and goethite in the coatings are present in the form of inclusions in a kaolinite matrix (Section 2.1.1).

*ν*(1-

_{j}*P*), occupied by particles of type

*j*, as

*j*varies over each of the eight particle types (pq, ph, pg, pm, mh, mg, ch, cg). For the mixed particles, we must also determine the volume concentration,

*ν*, of each of the constituent minerals

_{j,ħ}*ħ*within the particle, required to apply Eq. (4). For the coated particles, we must determine the volume concentration of the inclusions within the coating, which may be computed from the overall volume concentrations,

*ν*, of each of the mineral constituents of the particle. We begin by determining the mass fraction,

_{j,ħ}*µj*, of each of the various types of particles (Table 3), and the mass concentration,

*ϑ*, of each mineral,

_{j,ħ}*ħ*, within particles of each type,

*j*(Table 4).

*ϑ*,

_{h}*ϑ*, and

_{g}*ϑ*. In addition, we define some additional quantities. We define the iron oxide concentration as

_{m}*ϑ*. The remainder of the mineral matter is quartz (within particle cores) and kaolinite (within coatings).

_{hg}*µ′*,

_{p}*µ′*, and

_{m}*µ′*, partition the particles by mass into the pure, mixed, and coated particles respectively, with

_{c}*µ′*+

_{p}*µ′*+

_{m}*µ′*=1. These parameters are further constrained by the concentrations of the various mineral constituents, since, for example, a particle consisting of a quartz core coated by a mixture of hematite and kaolinite has an upper bound on hematite concentration within that particle.

_{c}*µ*, of each of the various categories,

_{j}*j*, of particles. These are indicated in Table 3, with two unknowns,

*β*and

_{q}*β*. Since we are given the total mass concentration of hematite,

_{hg}*ϑ*, and the mass concentration of hematite in each of the particle types (provided in Table 4), we have the relationship

_{h}*r*+

_{hg}β_{hg}*ϑ*+

_{hg}r_{hg}µ′_{m}*ϑ*=ϑ

_{hg}r_{hg}µ′_{c}*.*

_{h}*=*

_{hg}r_{hg}*ϑ*. Substituting and solving for

_{h}*β*yields

_{hg}*µ′*=1-

_{p}*µ′*-

_{m}*µ′*, we get

_{c}*β*. Since top row in Table 3 must sum to

_{q}*µ′*, we have

_{p}*ϑ′*and

_{k}*ϑ′*, in Table 4. These quantities correspond to mass concentrations of kaolinite in the hematite coated quartz and the goethite coated quartz particles, respectively. Recall that the hematite coated quartz particles consist of a pure quartz core coated in a mixture of hematite and kaolinite. Hence, to determine

_{k}*ϑ′*, corresponding to the mass concentration of kaolinite within the particle (Table 4), we need to know the volume of the coating as a fraction of the total volume of the particle,

_{k}*ν*. Solving for

_{coat}*ϑ′*[66] then yields

_{k}*A*, where

_{S}h*A*is the surface area of the particle, since we are assuming that the

_{S}*h*≪

*s*. Therefore,

*V*is the volume of the particle. Dividing the numerator and denominator by

*V*yields

*AV*(

*s*,Ψ)=

*A*(

_{S}*s*,Ψ)/

*V*(

*s*,Ψ) is the

*surface area to volume*ratio of the particle. Noting that

*i.e., AV*(1,

*ν*, and thus

_{coat}*ϑ′*and

_{k}*ϑ′*, vary with sphericity.

_{k}*j*, the mass concentration

*ϑ*of each mineral

_{j,ħ}*ħ*(Table 4), we may compute the density of a particle of type

*j*,

*γ*(

*i.e*., the mean density over all particles), knowing the mass fractions

*µ*of each particle type

_{j}*j*(Table 3). This is given by

_{ch},

*/(ν*

_{h}_{ch},

*+ν*

_{k}_{ch},

*). The volume fraction of goethite within the coating of the goethite coated particles is determined similarly.*

_{h}#### 4.2.6. Shape and Size Distribution

65. M. Vepraskas and D. Cassel, “Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties,” Soil Science Society of America Journal **51**, 1108–1112 (1987). [CrossRef]

*et al*. [70

70. M. Shirazi, L. Boersma, and J. Hart, “A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale,” Soil Science Society of America Journal **52**, 181–190 (1988). [CrossRef]

*d*

*, and its standard deviation*

_{g}*σ*, which are functions of soil texture (as defined in Section 2). It is important to note that, since the distribution is specified in a

_{g}*piecewise*manner, there will be a different

*d*and

_{g}*σ*for each soil separate. Defining

_{g}*a*=log

_{g}*d*and

_{g}*b*=logσ

_{g}*, the mass fraction of the particles with sizes ranging from*

_{g}*s*1 to

*s*2 is then given by

*(*

_{m}*s*1,

*s*2) may also be interpreted as a volume fraction. This approximation is justified by the fact that the silt and sand-sized particles are dominated by quartz [11]. Shirazi

*et al*. [70

70. M. Shirazi, L. Boersma, and J. Hart, “A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale,” Soil Science Society of America Journal **52**, 181–190 (1988). [CrossRef]

*d*and σ

_{g}*from texture.*

_{g}### 4.3. Light Transport Simulation

^{2}according to the following algorithm. After computing the Fresnel coefficient at the boundary, a random number uniformly distributed on (0,1) is computed. If the Fresnel coefficient is greater than the random number, then a reflected ray is generated applying the law of reflection. Otherwise, a refracted ray is generated according to Snell’s Law.

#### 4.3.1. Extended Boundaries

#### 4.3.2. Pore Space

*S*, defined in Section 4.2.2, is used to partition the pore space into air and water. Water is selected with probability

*S*to represent the pore space. Air is selected with probability 1-

*S*. This selection is made when an incident ray approaches, as well as when the ray reaches the outer interface of a particle from the inside.

#### 4.3.3. Generating a Sand Particle

*on the fly*. The light interaction with this particle is then simulated, and the particle is then discarded. In a conventional ray tracing approach, particles would be stored explicitly. Random variables, such as the distance to the next particle, the shape, size, and composition of the particle that is intercepted, and the point on the surface of the particle that is intercepted, arise implicitly as a consequence of the ray tracing simulation. Conversely, in the SPLITS model, we compute the probability distribution functions for these random variables and sample them explicitly. In addition, we ensure that the particle lies completely between the extended boundaries and that the particle does not intersect with the previous ray.

*category j*of particles and then selecting the category for which that distance is minimal. The distance,

*d*, to the next particle of type

_{j}*j*is an exponential random variable with a mean of 1/

*K*, where

_{j}*K*is the

_{j}*geometric attenuation coefficient*for particles of type

*j*. This distance is given by

*K*reduces [66] to

_{j}_{g}(

*s*), is given by Eq. (19),

*AV*(1,Ψ) is given by Eq. (14), and Φ′(

*x̄*,σ

^{2})(

*x*) is the probability density function for the normal distribution with mean

*x̄*and variance σ

^{2}[67].

*s*, is then chosen according to the probability density function

*C*

_{1}and

*C*

_{2}are the constants that ensure that

*f*and

_{s}*f*Ψ (respectively) integrate to one over their domain [66]. Note that Eq. (25) does not match the distribution provided by Shirazi et al. [70

70. M. Shirazi, L. Boersma, and J. Hart, “A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale,” Soil Science Society of America Journal **52**, 181–190 (1988). [CrossRef]

*mass*, provided by Shirazi

*et al*. [70

**52**, 181–190 (1988). [CrossRef]

*volume*as we are interpreting it), and the distribution of particle sizes struck by rays travelling randomly through the medium [66].

*h*=

*h′s*, where

*h′*is the relative coating thickness specified in Table 2. Additionally, the particle roundness,

*R*is given by a normal random variable with mean

*R̄*and standard deviation σ

*given in Table 2.*

_{R}*z*=

*cF*

^{-1}(2ξ

_{1}-1), where ξ

_{1}is a uniform random number on (0,1),

*F*(

*u*) is the fraction of the surface area of the spheroid above the plane

*z*=

*u*, given by

*F*

^{-1}is difficult to compute analytically,

*F*may be inverted using numerical techniques. The other two coordinates are then given by

*x*=

*ar*cos

*θ*and

*y*=

*ar*sin

*θ*, with

*θ*=2

*πξ*

_{2}(ξ

_{2}being another canonical random variable) and

*v*into the particle’s local coordinate space to get

*v′*, and compute the normal

**n**at the chosen point. We then use the point (

*x*,

*y*,

*z*) if

**n**·

*v′*<0 and (-

*x*,-

*y*,-

*z*) otherwise.

**q**, that is

*d*units along the ray lies outside the extended boundaries, the ray will instead interact with the boundary as described in Section 4.3.1. If the randomly generated particle intersects with either boundary, the particle is rejected. To account for the opposition effect [72

72. B. Hapke, “Bidirectional Reflectance Spectroscopy. 4. The Extinction Coefficient and the Opposition Effect,” Icarus **67**, 264–280 (1986). [CrossRef]

*i.e*., the increase in reflectance toward the source of illumination due to shadow hiding), the particle is also rejected if it intersects with the last leg of the path. If the particle is rejected, the above process of generating a distance and particle intersection point is repeated.

#### 4.3.4. Light Propagation Within a Sand Particle

*A*. The

*projected*area with respect to the incident ray,

*v*, is therefore

*A*|

**n**′·

**v**|. Hence, the probability that an incident ray

**v**strikes a microfacet with normal

**n**′ should be scaled by |

**n**′·

**v**|. This is accomplished using the rejection method [67]. The ray is then reflected in

**n**′ with a probability once again determined using the Fresnel equations

^{3}[58], considering the media on either side of the interface. Otherwise, the ray is refracted according to Snell’s Law for absorbing media [73]. If the ray was to be reflected and

**n**·

**r**<0 (where

**r**is the reflected vector), or if the ray was to be refracted and

**n**·

**t**>0, then multiple scattering is approximated using a cosine lobe centered around

**n**(respectively -

**n**) [41

41. L. Wolff, “Diffuse-Reflectance Model for Smooth Dielectric Surfaces,” Journal of the Optical Society of America A (Optics, Image Science, and Vision) **11**, 2956–2968 (1994). [CrossRef]

*k*, of the coating or of the core, absorption within the coating or within the core is then simulated according to Lambert’s Law [54]. The light is transmitted a distance

*d*with probability

*T*=exp(-

*αd*), where

*α*is the absorption coefficient, given by

*d*=

*h*/|

**n**·

**t**|, where

**t**is the ray transmitted through the coating. For the core,

*d*is determined from the ray-particle intersection.

### 4.4. Extensibility

## 5. Evaluation

4. G. Baranoski, J. Rokne, and G. Xu, “Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering,” The Visual Computer **17**, 506–518 (2001). [CrossRef]

### 5.1. Comparisons with Measured Data

_{j}9823), a magnetite rich beach sand from central Peru (TEC #10039240), and a sample from a dike outcrop in San Bernardino county, California (TEC #19au9815). In our experiments, we attempted to simulate the actual measurement conditions as accurately as possible according to the measurement setup outline provided by Rinker

*et al*. [75].

#### 5.1.1. Results of Comparisons

6. S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, “Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model,” Remote Sensing of Environment **56**, 194–202 (1996). [CrossRef]

*S*, was also varied from

*S*=0 to

*S*=1 within each image to show the darkening effect simulated by the model.

### 5.2. Qualitative Characteristics

**38**, 1–43 (1985). [CrossRef]

*i*.

*e*.,

*r*is increased), as confirmed in the literature [76

_{hg}76. T. Cudahy and E. Ramanaidou, “Measurement of the Hematite:Goethite Ratio Using Field Visible and Near-Infrared Reflectance Spectrometry in Channel Iron Deposits, Western Australia,” Australian Journal of Earth Sciences **44**, 411–420 (1997). [CrossRef]

**136**, 354–358 (1983). [CrossRef]

**38**, 1–43 (1985). [CrossRef]

28. R. Vincent and G. Hunt, “Infrared Reflectance from Mat Surfaces,” Applied Optics **7**, 53–59 (1968). [CrossRef] [PubMed]

*S*=0 and

*S*=1 for the four TEC sand samples. The darkening effect, reported in the literature [17

**38**, 1–43 (1985). [CrossRef]

48. D. Lobell and G. Asner, “Moisture Effects on Soil Reflectance,” Soil Science Society of America Journal **66**, 722–727 (2002). [CrossRef]

## 6. Conclusion

*along with*the characterization data for those surfaces. Despite the efforts of numerous researchers [36

36. B. Hapke, “Bidirectional Reflectance Spectroscopy. 1. Theory,” Journal of Geophysical Research **86**, 3039–3054 (1981). [CrossRef]

39. W. Egan and T. Hilgeman, “Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering,” Applied Optics **17**, 245–252 (1978). [CrossRef] [PubMed]

77. J. Cierniewski, “A Model for Soil Surface Roughness Influence on the Spectral Response of Bare Soils in the Visible and Near-Infrared Range,” Remote Sensing of Environment **23**, 92–115 (1987). [CrossRef]

*vice versa*. It is therefore difficult to verify quantitatively any models for these surfaces without making assumptions about the properties of the material in question. However, one can still validate the model qualitatively by demonstrating that modifying input parameters have the expected effect, and quantitatively by showing that it is possible to obtain matches between reflectances from real samples and from the model using plausible input parameters, as we have done.

## Acknowledgements

## Footnotes

1 | It had been demonstrated that the Monte Carlo simulation of the optical path of light propagating through n scattering events in an inhomogeneous medium is equivalent to the calculation of the n
^{th} order ladder diagram, and the Monte Carlo approach can be extended to the computation of time correlation functions and coherent backscattering [5757. V. Kuz’min and I. Meglinski, “Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering,” Optics & Spectroscopy |

2 | Recall that the extended boundary represents the interface between the pore space (air or water) and the ambient medium. It does not represent the rough sand surface. This arises from the random positioning of the simulated particles. |

3 | Note that the Fresnel equations are applied to the microfacets, not the overall particle surface. Since the microfacets themselves are planar surfaces, the Fresnel equations may be applied. |

## References and links

1. | S. Prahl, “Light Transport in Tissue,” Ph.D. thesis, University of Texas at Austin (1988). |

2. | S. Prahl, M. Keijzer, S. Jacques, and A. Welch, “A Monte Carlo Model of Light Propagation in Tissue,” SPIE Institute Series |

3. | Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, “Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf,” Applied Optics |

4. | G. Baranoski, J. Rokne, and G. Xu, “Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering,” The Visual Computer |

5. | A. Krishnaswamy, G. Baranoski, and J. G. Rokne, “Improving the Reliability/Cost Ratio of goniophotometric measurement,” |

6. | S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, “Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model,” Remote Sensing of Environment |

7. | R. Shuchman and D. Rea, “Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data,” Remote Sensing of Environment |

8. | R. Morris and D. Golden, “Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars,” Icarus |

9. | R. Singer, “Spectral Evidence for the Mineralogy of High-Albedo Soils and Dust on Mars,” Journal of Geophysical Research |

10. | F. Pettijohn, P. Potter, and R. Siever, |

11. | N. Brady, |

12. | J. Gerrard, |

13. |
Soil Science Division Staff, |

14. | K. Coulson and D. Reynolds, “The Spectral Reflectance of Natural Surfaces,” Journal of Applied Meteorology |

15. | J. Norman, J. Welles, and E. Walter, “Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils,” IEEE Transactions on Geoscience and Remote Sensing |

16. | K. Coulson, G. Bouricius, and E. Gray, “Optical Reflection Properties of Natural Surfaces,” Journal of Geophysical Research |

17. | M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, “Reflectance Properties of Soils,” Advances in Agronomy |

18. | D. Leu, “Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship,” Remote Sensing of Environment |

19. | G. Hunt and J. Salisbury, “Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals,” Modern Geology |

20. | P. Farrant, |

21. | A. Mottana, R. Crespi, and G. Liborio, |

22. | J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, “Quantitative Relationships Between Soil Color and Hematite Content,” Soil Science |

23. | R. Cornell and U. Schwertmann, |

24. | H. Wopfner and C. Twindale, “Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia,” Geologische Rundschau |

25. | G. Hunt, J. Salisbury, and C. Lenhoff, “Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides,” Modern Geology |

26. | S. Twomey, C. Bohren, and J. Mergenthaler, “Reflectance and Albedo Differences Between Wet and Dry Surfaces,” Applied Optics |

27. | M. Kühl and B. Jørgensen, “The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments,” Limnology and Oceanography |

28. | R. Vincent and G. Hunt, “Infrared Reflectance from Mat Surfaces,” Applied Optics |

29. | H. Wadell, “Volume, Shape, and Roundness of Rock Particles,” Journal of Geology |

30. | N. Riley, “Projection Sphericity,” Journal of Sedimentary Petrology |

31. | W. Krumbein, “Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles,” Journal of Sedimentary Petrology |

32. | H. Zhang and K. Voss, “Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models,” Applied Optics |

33. | Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, “Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance,” Journal of Quantitative Spectroscopy and Radiative Transfer |

34. | M. Mishchenko, L. Travis, and A. Lacis, |

35. | M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, “Multiple Scattering by Random Particulate Media: Exact 3D Results,” Optics Express |

36. | B. Hapke, “Bidirectional Reflectance Spectroscopy. 1. Theory,” Journal of Geophysical Research |

37. | B. Hapke and E. Wells, “Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations,” Journal of Geophysical Research |

38. | A. Emslie and J. Aronson, “Spectral Reflectance of Particulate Materials. 1: Theory,” Applied Optics |

39. | W. Egan and T. Hilgeman, “Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering,” Applied Optics |

40. | M. Oren and S. Nayar, “Generalization of Lambert’s Reflectance Model,” in |

41. | L. Wolff, “Diffuse-Reflectance Model for Smooth Dielectric Surfaces,” Journal of the Optical Society of America A (Optics, Image Science, and Vision) |

42. | M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces,” Journal of Quantitative Spectroscopy and Radiative Transfer |

43. | D. Stankevich and Y. Shkuratov, “Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure,” Journal of Quantitative Spectroscopy and Radiative Transfer |

44. | J. Peltoniemi, “Spectropolarised Ray-Tracing Simulations in Densely Packed Particulate Medium,” Journal of Quantitative Spectroscopy and Radiative Transfer (2007). In Press, accepted, URL http://dx.doi.org/10.1016/j.jqsrt.2007.05.009. [CrossRef] |

45. | H. Jensen, J. Legakis, and J. Dorsey, “Rendering ofWet Materials,” |

46. | L. Henyey and J. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophysical Journal |

47. | Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, “A Modeling Study of Backscattering from Soil Surfaces,” IEEE Transactions on Geoscience and Remote Sensing |

48. | D. Lobell and G. Asner, “Moisture Effects on Soil Reflectance,” Soil Science Society of America Journal |

49. | D. Neema, A. Shah, and A. Patel, “A Statistical Optical Model for Light Reflection and Penetration Through Sand,” International Journal of Remote Sensing |

50. | V. Barron and L. Montealegre, “Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory,” American Journal of Science |

51. | P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche (An Article on Optics of Paint Layers),” Zeitschrift für Technische Physik |

52. | D. Nickerson, “History of the Munsell Color System and its Scientific Application,” Journal of the Optical Society of America |

53. | G. Okin and T. Painter, “Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces,” Remote Sensing of Environment |

54. | F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, |

55. | R. Hunter and R. Harold, |

56. | G. Ward, “Measuring and Modeling Anisotropic Reflection,” Computer Graphics |

57. | V. Kuz’min and I. Meglinski, “Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering,” Optics & Spectroscopy |

58. | F. Pedrotti and L. Pedrotti, |

59. | T. Nousiainen, K. Muinonen, and P. Räisänen, “Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation,” Journal of Geophysical Research |

60. | G. Hale and M. Querry, “Optical Constants of Water in the 200-nm to 200- |

61. | W. Tropf, M. Thomas, and T. Harris, “Properties of Crystals and Glasses,” in |

62. | I. Sokolik and O. Toon, “Incorporation of Mineralogical Composition into Models of the Radiative Properties of Mineral Aerosol from UV to IR Wavelengths,” Journal of Geophysical Research |

63. | W. Egan and T. Hilgeman, |

64. | A. Schlegel, S. Alvarado, and P. Wachter, “Optical Properties of Magnetite (Fe |

65. | M. Vepraskas and D. Cassel, “Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties,” Soil Science Society of America Journal |

66. | B. Kimmel, “SPLITS: A Spectral Light Transport Model for Sand,” Master’s thesis, School of Computer Science, University of Waterloo (2005). |

67. | S. Ross, |

68. | C. Bohren and D. Huffman, |

69. | J. Maxwell Garnett, “Colours in Metal Glasses and in Metallic Films,” Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character |

70. | M. Shirazi, L. Boersma, and J. Hart, “A Unifying Quantitative Analysis of Soil Texture: Improvement of Precision and Extention of Scale,” Soil Science Society of America Journal |

71. | J. Snyder, |

72. | B. Hapke, “Bidirectional Reflectance Spectroscopy. 4. The Extinction Coefficient and the Opposition Effect,” Icarus |

73. | M. Born and E. Wolf, |

74. | C. Gribble and A. Hall, |

75. | J. Rinker, C. Breed, J. McCauley, and P. Corl, “Remote Sensing Field Guide — Desert,” Tech. rep. , U.S. Army Topographic Engineering Center, Fort Belvoir, VA (1991). |

76. | T. Cudahy and E. Ramanaidou, “Measurement of the Hematite:Goethite Ratio Using Field Visible and Near-Infrared Reflectance Spectrometry in Channel Iron Deposits, Western Australia,” Australian Journal of Earth Sciences |

77. | J. Cierniewski, “A Model for Soil Surface Roughness Influence on the Spectral Response of Bare Soils in the Visible and Near-Infrared Range,” Remote Sensing of Environment |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(280.0280) Remote sensing and sensors : Remote sensing and sensors

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Scattering

**History**

Original Manuscript: May 24, 2007

Revised Manuscript: July 2, 2007

Manuscript Accepted: July 13, 2007

Published: July 20, 2007

**Citation**

Bradley W. Kimmel and Gladimir V. G. Baranoski, "A novel approach for simulating light interaction with particulate materials:
application to the modeling of sand spectral properties," Opt. Express **15**, 9755-9777 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9755

Sort: Year | Journal | Reset

### References

- S. Prahl, "Light Transport in Tissue," Ph.D. thesis, University of Texas at Austin (1988).
- S. Prahl, M. Keijzer, S. Jacques, and A. Welch, "A Monte Carlo Model of Light Propagation in Tissue," SPIE Institute Series 5, 102-111 (1989).
- Y. Govaerts, S. Jacquemoud, M. Verstraete, and S. Ustin, "Three-Dimensional Radiation Transfer Modeling in a Dycotyledon Leaf," Applied Optics 35, 6585-6598 (1996). [CrossRef] [PubMed]
- G. Baranoski, J. Rokne, and G. Xu, "Virtual Spectrophotometric Measurements for Biologically and Physically Based Rendering," The Visual Computer 17, 506-518 (2001). [CrossRef]
- A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools 9, 31-51 (2004).
- S. Jacquemoud, S. Ustin, J. Verdebout, G. Schmuck, G. Andreoli, and B. Hosgood, "Estimating Leaf Biochemistry Using PROSPECT Leaf Optical Properties Model," Remote Sensing of Environment 56, 194-202 (1996). [CrossRef]
- R. Shuchman and D. Rea, "Determination of Beach Sand Parameters Using Remotely Sensed Aircraft Reflectance Data," Remote Sensing of Environment 11, 295-310 (1981). [CrossRef]
- R. Morris and D. Golden, "Goldenrod Pigments and the Occurrence of Hematite and Possibly Goethite in the Olympus-Amazonis Region of Mars," Icarus 134, 1-10 (1998). [CrossRef]
- R. Singer, "Spectral Evidence for the Mineralogy of High-Albedo Soils and Dust on Mars," Journal of Geophysical Research 87, 10,159-10,168 (1982). [CrossRef]
- F. Pettijohn, P. Potter, and R. Siever, Sand and Sandstone, 2nd ed. (Springer-Verlag, New York, NY, 1987). [CrossRef]
- N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).
- J. Gerrard, Fundamentals of Soils (Routledge, New York, NY, 2000).
- Soil Science Division Staff, Soil Survey Manual (Soil Conservation Service, 1993). United States Department of Agriculture Handbook 18.
- K. Coulson and D. Reynolds, "The Spectral Reflectance of Natural Surfaces," Journal of Applied Meteorology 10, 1285-1295 (1971). [CrossRef]
- J. Norman, J. Welles, and E. Walter, "Contrasts Among Bidirectional Reflectance of Leaves, Canopies, and Soils," IEEE Transactions on Geoscience and Remote Sensing GE-23, 659-667 (1985). [CrossRef]
- K. Coulson, G. Bouricius, and E. Gray, "Optical Reflection Properties of Natural Surfaces," Journal of Geophysical Research 70, 4601-4611 (1965). [CrossRef]
- M. Baumgardner, L. Silva, L. Biehl, and E. Stoner, "Reflectance Properties of Soils," Advances in Agronomy 38, 1-43 (1985). [CrossRef]
- D. Leu, "Visible and Near-Infrared Reflectance of Beach Sands: A Study on the Spectral Reflectance/Grain Size Relationship," Remote Sensing of Environment 6, 169-182 (1977). [CrossRef]
- G. Hunt and J. Salisbury, "Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals," Modern Geology 1, 283-300 (1970).
- P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).
- A. Mottana, R. Crespi, and G. Liborio, Simon and Schuster’s Guide to Rocks and Minerals (Simon and Schuster, Inc., New York, NY, 1978).
- J. Torrent, U. Schwertmann, H. Fechter, and F. Alferez, "Quantitative Relationships Between Soil Color and Hematite Content," Soil Science 136, 354-358 (1983). [CrossRef]
- R. Cornell and U. Schwertmann, The Iron Oxides, 2nd ed. (Wiley-VCH GmbH & Co. KGaA, Weinheim, Germany, 2003). [CrossRef]
- H. Wopfner and C. Twindale, "Formation and Age of Desert Dunes in the Lake Eyre Depocentres in Central Australia," Geologische Rundschau 77, 815-834 (1988). [CrossRef]
- G. Hunt, J. Salisbury, and C. Lenhoff, "Visible and Near-Infrared Spectra of Minerals and Rocks: III. Oxides and Hydroxides," Modern Geology 2, 195-205 (1971).
- S. Twomey, C. Bohren, and J. Mergenthaler, "Reflectance and Albedo Differences Between Wet and Dry Surfaces," Applied Optics 25, 57-84 (1986). [CrossRef]
- M. Kühl and B. Jørgensen, "The Light Field of Microbenthic Communities: Radiance Distribution and Microscale Optics of Sandy Coastal Sediments," Limnology and Oceanography 39, 1368-1398 (1994). [CrossRef]
- R. Vincent and G. Hunt, "Infrared Reflectance from Mat Surfaces," Applied Optics 7, 53-59 (1968). [CrossRef] [PubMed]
- H. Wadell, "Volume, Shape, and Roundness of Rock Particles," Journal of Geology 40, 443-451 (1932). [CrossRef]
- N. Riley, "Projection Sphericity," Journal of Sedimentary Petrology 11, 94-95 (1941).
- W. Krumbein, "Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles," Journal of Sedimentary Petrology 11, 64-72 (1941).
- H. Zhang and K. Voss, "Comparisons of Bidirectional Reflectance Distribution Function Measurements on Prepared Particulate Surfaces and Radiative-Transfer Models," Applied Optics 44, 597-610 (2005). [CrossRef] [PubMed]
- Y. Xie, P. Yang, B.-C. Gao, G. Kattawar, and M. Mishchenko, "Effect of Ice Crystal Shape and Effective Size on Snow Bidirectional Reflectance," Journal of Quantitative Spectroscopy and Radiative Transfer 100, 457-469 (2006). [CrossRef]
- M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, Cambridge, 2006).
- M. Mishchenko, L. Liu, D. Mackowski, B. Cairns, and G. Videen, "Multiple Scattering by Random Particulate Media: Exact 3D Results," Optics Express 15, 2822-2836 (2007). [CrossRef] [PubMed]
- B. Hapke, "Bidirectional Reflectance Spectroscopy. 1. Theory," Journal of Geophysical Research 86, 3039-3054 (1981). [CrossRef]
- B. Hapke and E. Wells, "Bidirectional Reflectance Spectroscopy. 2. Experiments and Observations," Journal of Geophysical Research 86, 3055-3054 (1981). [CrossRef]
- A. Emslie and J. Aronson, "Spectral Reflectance of Particulate Materials. 1: Theory," Applied Optics 12, 2563-2572 (1973). [CrossRef] [PubMed]
- W. Egan and T. Hilgeman, "Spectral Reflectance of Particulate Materials: A Monte Carlo Model Including Asperity Scattering," Applied Optics 17, 245-252 (1978). [CrossRef] [PubMed]
- M. Oren and S. Nayar, "Generalization of Lambert’s Reflectance Model," in Computer Graphics Proceedings, Annual Conference Series, pp. 239-246 (1994).
- L. Wolff, "Diffuse-Reflectance Model for Smooth Dielectric Surfaces," Journal of the Optical Society of America A (Optics, Image Science, and Vision) 11, 2956-2968 (1994). [CrossRef]
- M. Mishchenko, J. Dlugach, E. Yanovitskij, and N. Zakharova, "Bidirectional reflectance of flat, optically thick particulate layers: An efficient radiative transfer solution and applications to snow and soil surfaces," Journal of Quantitative Spectroscopy and Radiative Transfer 63, 409-432 (1999). [CrossRef]
- D. Stankevich and Y. Shkuratov, "Monte Carlo Ray-Tracing Simulation of Light Scattering in Particulate Media with Optically Contrast Structure," Journal of Quantitative Spectroscopy and Radiative Transfer 87, 289-296 (2004). [CrossRef]
- J. Peltoniemi, "Spectropolarised Ray-Tracing Simulations in Densely Packed Particulate Medium," Journal of Quantitative Spectroscopy and Radiative Transfer (2007). In Press, accepted, URL http://dx.doi.org/10.1016/j.jqsrt.2007.05.009. [CrossRef]
- H. Jensen, J. Legakis, and J. Dorsey, "Rendering ofWet Materials," in Proceedings of the Eurographics Workshop on Rendering, pp. 273-282 (1999).
- L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophysical Journal 93, 70-83 (1941). [CrossRef]
- Z. Li, A. Fung, S. Tjuatja, D. Gibbs, C. Betty, and J. Irons, "A Modeling Study of Backscattering from Soil Surfaces," IEEE Transactions on Geoscience and Remote Sensing 34, 264-271 (1996). [CrossRef]
- D. Lobell and G. Asner, "Moisture Effects on Soil Reflectance," Soil Science Society of America Journal 66, 722-727 (2002). [CrossRef]
- D. Neema, A. Shah, and A. Patel, "A Statistical Optical Model for Light Reflection and Penetration Through Sand," International Journal of Remote Sensing 8, 1209-1217 (1987). [CrossRef]
- V. Barron and L. Montealegre, "Iron Oxides and Color of Triassic Sediments: Application of the Kubelka-Munk Theory," American Journal of Science 286, 792-802 (1986). [CrossRef]
- P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche (An Article on Optics of Paint Layers)," Zeitschrift fur Technische Physik 12, 593-601 (1931).
- D. Nickerson, "History of the Munsell Color System and its Scientific Application," Journal of the Optical Society of America 30, 575-645 (1940). [CrossRef]
- G. Okin and T. Painter, "Effect of Grain Size on Remotely Sensed Spectral Reflectance of Sandy Desert Surfaces," Remote Sensing of Environment 89, 272-280 (2004). [CrossRef]
- F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometrical Considerations and Nomenclature for Reflectance (National Bureau of Standards, United States Department of Commerce, 1977).
- R. Hunter and R. Harold, The Measurement of Appearance, 2nd ed. (JohnWiley and Sons, New York, NY, 1987).
- G. Ward, "Measuring and Modeling Anisotropic Reflection," Computer Graphics 26, 262-272 (1992). [CrossRef]
- V. Kuz’min and I. Meglinski, "Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering," Optics & Spectroscopy 97, 100-106 (2004). [CrossRef]
- F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).
- T. Nousiainen, K. Muinonen, and P. Räisänen, "Scattering of Light by Large Saharan Dust Particles in a Modified Ray Optics Approximation," Journal of Geophysical Research 108, AAC 12-1-17 (2003). [CrossRef]
- G. Hale and M. Querry, "Optical Constants of Water in the 200-nm to 200- m Wavelength Region," Applied Optics 12, 555-563 (1973). [CrossRef] [PubMed]
- W. Tropf, M. Thomas, and T. Harris, "Properties of Crystals and Glasses," in Handbook of Optics, M. Bass, E. Van Stryland, D. Williams, and W.Wolfe, eds., vol. 2, 2nd ed., chap. 33 (McGraw-Hill, 1995).
- I. Sokolik and O. Toon, "Incorporation of Mineralogical Composition into Models of the Radiative Properties of Mineral Aerosol from UV to IR Wavelengths," Journal of Geophysical Research 104, 9423-9444 (1999). [CrossRef]
- W. Egan and T. Hilgeman, Optical Properties of Inhomogeneous Materials: Applications to Geology, Astronomy, Chemistry, and Engineering (Academic Press, New York, NY, 1979). [PubMed]
- A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979). [CrossRef]
- M. Vepraskas and D. Cassel, "Sphericity and Roundness of Sand in Coastal Plain Soils and Relationships with Soil Physical Properties," Soil Science Society of America Journal 51, 1108-1112 (1987). [CrossRef]
- B. Kimmel, "SPLITS: A Spectral Light Transport Model for Sand," Master’s thesis, School of Computer Science, University of Waterloo (2005).
- S. Ross, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).
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