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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 15 — Jul. 23, 2007
  • pp: 9755–9777
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A novel approach for simulating light interaction with particulate materials: application to the modeling of sand spectral properties

BradleyW. Kimmel and Gladimir V.G. Baranoski  »View Author Affiliations


Optics Express, Vol. 15, Issue 15, pp. 9755-9777 (2007)
http://dx.doi.org/10.1364/OE.15.009755


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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

Monte Carlo simulations of light transport in natural materials are usually performed under the assumption that such materials may be represented using layers and that the directional changes of travelling rays can be modeled using precomputed phase functions. In this paper, we provide the mathematical framework for an alternative approach in which these directional changes are computed on the fly, without explicitly storing the material constituents. This approach strikes a balance between the use of precomputed phase functions [1

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

, 2

2. 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).

], which do not relate to any particular material, and a conventional ray tracing approach, which, among other difficulties, requires much storage space if one wants to geometrically represent the material constituents [3

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]

]. Although in the context of this paper we focus on its application to the modeling of sand optical properties, this approach can potentially be applied to other materials.

In this paper, we present a spectral light transport model for sand, 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]

] and virtual goniophotometric [5

5. A. Krishnaswamy, G. Baranoski, and J. G. Rokne, “Improving the Reliability/Cost Ratio of goniophotometric measurement,” 9, 31–51 (2004).

] methods, and comparing its results to measured data. SPLITS represents, to the best of our knowledge, the first attempt to simulate the spectral reflection properties of sand using its physical and mineralogical properties, henceforth referred to as characterization data, as input.

There are several reasons for modeling sand in terms of its characterization data rather than in terms of arbitrary parameters not directly relating to sand. Hyperspectral remote sensing with satellite or aircraft based equipment is used to investigate properties of land surfaces without having to physically survey the area [6

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

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]

]. For other planetary bodies, such as Mars, a field survey may not be possible. In this case, spectral signatures may be the only clue available for determining surface characteristics [8

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

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]

]. For these scientific applications, this relationship between model parameters and the physical properties is necessary so that sand characterization data may be derived from predictive simulations provided by the model. Finally, for the model to be evaluated objectively, the parameters must relate to the material that one is trying to simulate.

2. Properties of Sand

In this section, the relevant background information on sand is provided. We begin by defining precisely what sand is. The pertinent factors affecting light transport with sand are then described. The following, however, is not a complete treatise on sand. Interested readers are referred to works by Pettijohn et al. [10

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

], Brady [11

11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

], or Gerrard [12

12. J. Gerrard, Fundamentals of Soils (Routledge, New York, NY, 2000).

].

Sand is a particular type of 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

12. J. Gerrard, Fundamentals of Soils (Routledge, New York, NY, 2000).

].

Soils are classified according to the size distribution of the mineral particles [11

11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

]. This is accomplished first by assigning individual particles to classes, called 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.

Various agencies have differing definitions for soil separates and textural classes [11

11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

]. In this work, we use the system developed by the United States Department of Agriculture (USDA) [13

13. Soil Science Division Staff, Soil Survey Manual (Soil Conservation Service, 1993). United States Department of Agriculture Handbook 18.

]. The USDA defines three soil separates, called 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.

That the term 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).

Table 1:. Soil separates (particle size classes) defined by the United States Department of Agriculture [13].

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2.1. Factors Affecting Light Transport

The reflectance of sand generally increases with wavelength [14

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

]. Additionally, sand reflection is non-Lambertian [15

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

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

]. In particular, sand exhibits 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]

], which is reflection in the direction toward the source of illumination. Sand also reflects light in the forward direction, peaking at an angle to the normal greater than that of the direction of specular 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]

].

The most important factors contributing to the reflectance of soils in the visible range are its mineral composition (and iron oxides in particular), moisture, and particle size and shape [17

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

].

2.1.1. Mineral Composition

As sand is composed primarily of weathered rock [10

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

], the optical properties of that rock may influence light transport within sand. The parent material is the rock that is the source of the mineral part of the sand. This is typically a silicate mineral [11

11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

] such as quartz, gypsum or calcite, with quartz being the most common [11

11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

, 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]

]. While these minerals are colourless in pure form, trace amounts of contaminants may substantially affect their colour [19

19. G. Hunt and J. Salisbury, “Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals,” Modern Geology 1, 283–300 (1970).

].

Iron oxide gives sand its distinctive hues. Indeed, the two most significant minerals that determine soil colour, hematite and goethite [20

20. P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).

], are iron oxides. Goethite, also known as yellow ochre [20

20. P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).

] or limonite [21

21. A. Mottana, R. Crespi, and G. Liborio, Simon and Schuster’s Guide to Rocks and Minerals (Simon and Schuster, Inc., New York, NY, 1978).

], is one of the most common minerals found in soils [20

20. P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).

]. 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]

]. Hematite, or red ochre [20

20. P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).

], 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]

]. Hematite is usually found in tropical regions and is rare in temperate or cool climates [23

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

]. Iron oxides may be present as contaminants in the parent material [21]. They are also found, typically within a kaolinite or illite matrix, as coatings, approximately 1–5µm 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]

].

Additionally, magnetite and ilmenite are often found in beach and river sands [25

25. 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).

]. These minerals are spectrally similar. They are opaque and are black in colour [25

25. 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).

].

2.1.2. Water

The presence of water darkens sand. The principal reason for this is that the reduced contrast between the refractive index of the pore space and that of the mineral particles (typically quartz) induces stronger forward scattering at particle interfaces [26

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

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

Grain size affects reflectance by influencing the number of scattering interfaces per unit distance through the medium [28

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

]. Smaller particles, and thus a higher density of scattering interfaces, result in higher reflectances [17

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

, 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]

].

Grain shape may also affect scattering properties [17

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

]. The shape of the grain is described by two quantities: 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]

]. In this paper, we use the sphericity measure proposed by Riley [30

30. N. Riley, “Projection Sphericity,” Journal of Sedimentary Petrology 11, 94–95 (1941).

], which is a 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

Ψ=DiDc,
(1)

where Di is the diameter of the largest inscribed circle and Dc is the diameter of the smallest circumscribed circle, as shown in Fig. 1.

Fig. 1. Projection of a sand grain onto a plane. The circles indicate measurements used in the calculation of the Riley sphericity [30].

In contrast, 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]

]. A higher roundness value indicates a smoother surface. Roundness is determined by comparing the particle to images on a roundness chart [31

31. W. Krumbein, “Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles,” Journal of Sedimentary Petrology 11, 64–72 (1941).

].

2.1.4. Additional Factors

There are several other factors that may effect the reflectance of soils, such as organic matter content, surface roughness due to aggregation of soil particles, or human activities like tillage or transportation. These factors, however, either show themselves primarily outside the visible region of the spectrum, are less applicable to sands than to finer soils, or affect reflectance indirectly.

3. Related Work

In this section, we provide an overview of general purpose models which simulate optical characteristics relating to the appearance of sand. We then outline methods available for simulating some of the factors affecting light transport in sand described in Section 2.1. The reader interested in more information about general particulate scattering models based on radiative transfer theory is referred to the recent works by Zhang and Voss [32

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]

], Xie 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]

], and Mishchenko et al. [34

34. M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, Cambridge, 2006).

, 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

Hapke [36

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

] proposed an analytic radiative transfer model for particulate surfaces, which was then used to simulate reflectances from planetary surfaces [37

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

]. Emslie and Aronson [38

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

] modeled particulate surfaces, treating large particles separately from those smaller than the wavelength of light. Large particles were modeled using spheres of the same volume as actual particles and absorption was simulated at small scale 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]

]. Egan and Hilgeman [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]

] subsequently rectified this by simulating scattering, in addition to absorption, at these asperities. Oren and Nayar [40

40. M. Oren and S. Nayar, “Generalization of Lambert’s Reflectance Model,” in Computer Graphics Proceedings, Annual Conference Series, pp. 239–246 (1994).

] generalized the widely used Lambertian model [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]

] by representing a surface using a collection of symmetrical 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]

] proposed a radiative transfer technique for computing the bidirectional reflectance of a semi-infinite random media composed of nonabsorbing or weakly absorbing particles. Stankevich and Shkuratov [43

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]

] simulate light scattering in optically inhomogeneous particulate media using a geometric optics approach which combines ray tracing techniques and Monte Carlo methods. Recently, Peltoniemi [44

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]

] used a similar framework to simulate light scattering in densely packed particulate media and account for polarization. It is worth noting that the preceding models are intended to simulate a wide variety of materials. As such, their parameters do not specifically relate to sand.

3.2. Simulation of Specific Effects

3.2.1. Moisture

Various methods exist to achieve the darkening effect caused by water. Jensen et al. [45

45. H. Jensen, J. Legakis, and J. Dorsey, “Rendering ofWet Materials,” in Proceedings of the Eurographics Workshop on Rendering, pp. 273–282 (1999).

] 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]

], and adjust the degree of forward scattering to achieve varying levels of wetness. We remark that the Henyey-Greenstein phase function is an empirical function with a single parameter which bears no relation to the sand characterization data [47

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]

]. It is also worth noting that it has been demonstrated by 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]

] that its application in the simulation of soil scattering properties results in large errors, which can exceed a factor of twenty at backscattering geometries and a factor of three near forward scattering geometries. Lobell and Asner [48

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

] use an exponential function to empirically model reflectance in terms of moisture content. Neema 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]

] analytically model the sand medium using layers of reflective spheres coated by a thin film of water, the thickness of which is a function of moisture content.

3.2.2. Iron Oxides

Barron and Montealegre [50

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]

] apply Kubelka-Munk theory [51

51. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche (An Article on Optics of Paint Layers),” Zeitschrift für Technische Physik 12, 593–601 (1931).

] to model the influence of iron oxides on soil colour. Torrent 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]

] relate hematite and goethite concentrations to a function of the Munsell colour [52

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]

] of the soil, which Torrent 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]

] use spherical, hematite coated quartz particles to study the effect of grain size on the reflectance of desert sands.

4. The SPLITS Model

The SPLITS model is a comprehensive light transport model for sand. That is, both the spectral (colour) and spatial (bidirectional reflectance distribution function, or BRDF [54

54. 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).

]) responses are simulated. These two components together constitute the measurement of appearance [55

55. R. Hunter and R. Harold, The Measurement of Appearance, 2nd ed. (JohnWiley and Sons, New York, NY, 1987).

]. 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]

].

The remainder of this section is organized as follows. We begin with a general overview of the model in Section 4.1. The construction of the model is described in Section 4.2. In Section 4.3, the light transport simulation is described. We conclude with an outline of possible extensions supported by the model in Section 4.4.

Fig. 2. Modeled sand medium.

4.1. Concept

The SPLITS model consists of sand particles that are randomly distributed within the half space below a horizontal plane representing the sand surface. The pore space, defined in Section 2, consists of a mixture of air and water. This is depicted in Fig. 2. The particles are composed of the most important mineral constituents of sands: quartz, hematite, goethite and magnetite. In addition, quartz particles may be contaminated by hematite or goethite, or coated by hematite or goethite in a kaolinite matrix (Section 2.1.1).

Light propagation in the model is described in terms of ray (geometric) optics. Its implementation is based on an algorithmic process combining ray tracing techniques and standard Monte Carlo methods, which are applied to generate a scattered ray given the incident ray and the properties (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 interference1, 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).

Ultimately, an exact simulation of a particulate material would involve computing the solution of Maxwell’s equations [58

58. F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).

]. However, this is not feasible given the current state of computer technology and the complexity of the media involved. Any model must therefore be approximate: either in that the medium being simulated is an ideal one, or in that the light transport simulation is approximate. For instance, it could be argued that an approach based on Mie theory may yield better results. However, simulating Mie theory may add undue complexity and, due to the large size and irregular shape of sand particles, may not result in an improvement over a ray optics approximation [59

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]

].

Table 2. A summary of the data required in the simulations, along with typical ranges and values used.

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4.2. Construction

We begin by describing the representation of the medium in which the particles are immersed. Next, details are provided for the geometry and composition of individual particles. We then describe how those particles are distributed by size, shape, and composition. Table 2 lists the data used by the model along with typical ranges and the values used.

4.2.1. Extended Boundaries

As previously stated, the sand medium is bounded above by a horizontal plane representing the surface. The medium is additionally bounded below by another horizontal plane to prevent runaway paths from occurring during the light transport simulation. These two planes are called the 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]

].

4.2.2. Pore Space

Fig. 3. An example of the light propagation through a modeled sand particle. Left: The particle consists of a core and an optional coating. The small box around the ray path at the coating interface corresponds to the layer interface diagram to the right. Right: A closer look at the modeled interface between the core, coating, and/or the surrounding medium.

4.2.3. Particle Geometry

Individual particles consist of a core and an optional coating, as depicted in Fig. 3 (Left).

Ψ=ac.
(2)

One might suggest that because sand particles are irregularly shaped, one should use more complex particle shapes. To justify the added model complexity, it would be necessary to have the data describing the particle shapes at that level of detail. To the best of our knowledge, such data is not available in the literature. Since roundness and Riley sphericity data is readily available [65

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]

], we decided to use prolate spheroids. However, SPLITS is not limited to the use of prolate spheroids. The use of arbitrary particle distributions with SPLITS is described by Kimmel [66

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

]. Optionally, the core may be coated by a layer of uniform thickness, h, that is proportional to the particle size, with h′=h/2c. 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.

The interfaces between the core, coating, and surrounding medium are modeled using randomly oriented microfacets of equal area to simulate a rough surface, as shown in Fig. 3 (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

n·n=1X,
(3)
Fig. 4. Prolate spheroid.

where X is normally distributed with zero mean and standard deviation

σ=1R2.

This standard deviation was chosen so that R<n′·n≤1 for 95% of the facets [67

67. S. Ross, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).

]. 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

The minerals that are used in the model are quartz, hematite, goethite, and magnetite. Additionally, kaolinite is used as the coating matrix (Section 2.1.1). These minerals may occur in pure form or as a mixture.

To represent a mixed material, we use an equation from Maxwell Garnett theory [68

68. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, NY, 1983).

] that relates the 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]

]. Recently, it was applied to assess the effect of grain size on the reflectance of sandy materials [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]

]. According to Maxwell Garnett theory, the dielectric constant, εavg, of a mixture is given by

εavg=εm(1+3νi(εεmε+2εm)1νi(εεmε+2εm)),
(4)

where εm is the dielectric constant of the matrix, ε is the dielectric constant of the inclusions, and νi is the volume fraction of the inclusions [68

68. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, NY, 1983).

].

4.2.5. Particle Types

The sand particles are divided among the following eight 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).

Ultimately, we must determine the fraction of volume, νj(1-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, νj,ħ, of each of the constituent minerals 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, νj,ħ, of each of the mineral constituents of the particle. We begin by determining the mass fraction, µj, of each of the various types of particles (Table 3), and the mass concentration, ϑj,ħ, of each mineral, ħ, within particles of each type, j (Table 4).

The overall mass concentrations of hematite, goethite, and magnetite in the simulated sand is controlled, respectively, by ϑh, ϑg, and ϑm. In addition, we define some additional quantities. We define the iron oxide concentration as

ϑFe=ϑh+ϑg+ϑm.
(5)

Because of the particular importance of hematite and goethite (Section 2.1.1), we also define the concentration of these two minerals,

ϑhg=ϑh+ϑg,
(6)

and a ratio describing the relative proportions of hematite and goethite,

rhg=ϑhϑh+ϑg.
(7)

The concentrations of hematite and goethite may be given equivalently by the above two quantities. For simplicity, the concentration of hematite (respectively goethite) in the mixed and coated particles containing hematite (respectively goethite) is fixed at ϑhg. The remainder of the mineral matter is quartz (within particle cores) and kaolinite (within coatings).

Three parameters, µ′p, µ′m, and µ′c, partition the particles by mass into the pure, mixed, and coated particles respectively, with µ′p+µ′m+µ′c=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.

From the above parameters, we may now derive the mass fraction, µj, of each of the various categories, j, of particles. These are indicated in Table 3, with two unknowns, βq and βhg. Since we are given the total mass concentration of hematite, ϑh, and the mass concentration of hematite in each of the particle types (provided in Table 4), we have the relationship

rhgβhg+ϑhgrhgµ′m+ϑhgrhgµ′ch.

From Equations (6) and (7), we see that ϑhgrhg=ϑh. Substituting and solving for βhg yields βhg=ϑh(1μm'μc')rhg.

Noting that µ′p=1-µ′m-µ′c, we get

βhg=ϑhμprhg=ϑhgμp.
(8)

The reader may wish to verify that we arrive at the same result if we relate the concentration of goethite in each particle types to the total goethite concentration. Now we may solve for βq. Since top row in Table 3 must sum to µ′p, we have

βq=μpϑmβhg.
(9)

Table 3. Mass fractions, µj, of each of the different classes, j, of particles. The quantities βhg and βq are unknowns, given by Eqs. (8) and (9) respectively.

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Table 4. Mass concentration, ϑj,ℓ, of the constituent minerals, , in each of the different classes j of particles. The quantities ϑ′k and ϑ″k are unknowns given by Eqs. (10) and (11) respectively.

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Next, we must determine the two unknowns, ϑ′k and ϑ′k, 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 ϑ′k, 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, νcoat. Solving for ϑ′k [66

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

] then yields

ϑk=γkγh(γhνcoat(1ϑhg)γq(1νcoat)ϑhgγq(1νcoat)+γkνcoat).
(10)

Similarly,

ϑk=γkγg(γgνcoat(1ϑhg)γq(1νcoat)ϑhgγq(1νcoat)+γkνcoat).
(11)

The volume of the coating may be estimated as ASh, where AS is the surface area of the particle, since we are assuming that the hs. Therefore,

νcoatAS(s,Ψ)hV(s,Ψ)+AS(s,Ψ)h,

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

νcoatAV(s,Ψ)h1+AV(s,Ψ)h,
(12)

where AV (s,Ψ)=AS(s,Ψ)/V(s,Ψ) is the surface area to volume ratio of the particle. Noting that

AV(s,Ψ)=AS(s,Ψ)V(s,Ψ)=s2AS(1,Ψ)s3V(1,Ψ)=s1AV(1,Ψ),

and that, by definition, h=h′s the particle size,s, may be eliminated from Eq. (12), yielding

νcoatAV(1,Ψ)h1+AV(1,Ψ)h.
(13)

The surface area to volume ratio of a prolate spheroid with s=2c=1 is given by [66

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

]

AV(1,Ψ)=3(1+sin11Ψ4Ψ21Ψ4).
(14)

For simplicity, we use the mean sphericity, Ψ¯ , in the above (i.e., AV (1,Ψ¯ )) rather than having νcoat, and thus ϑ′k and ϑ′k, vary with sphericity.

Now that we know, for each particle type j, the mass concentration ϑj,ħ of each mineral ħ (Table 4), we may compute the density of a particle of type j,

γj=(ϑj,γ)1.
(15)

Similarly, we may compute the particle density, γ (i.e., the mean density over all particles), knowing the mass fractions µj of each particle type j (Table 3). This is given by

γ=(jμjγj)1.
(16)

Knowing the mass fractions and densities of the components of a mixture allows us to compute the volume fractions. Thus, we may now compute the volume concentrations of all the minerals within a particle, as well as the volume fractions of each particle type. The former is given by

νj,=γjγϑj,,
(17)

and the latter by

νj=γγjμj.
(18)

For hematite coated particles, the volume fraction of hematite within the coating, required to compute the complex refractive index of the mixture of hematite and kaolinite, is νch,h/(νch,kch,h). The volume fraction of goethite within the coating of the goethite coated particles is determined similarly.

4.2.6. Shape and Size Distribution

The size and shape of the particles are randomly distributed and are independent of one another. That is, the conditional size distribution for any two shapes is the same.

The sphericity is normally distributed, with the mean and the standard deviation derived from data provided by Vepraskas and Cassel [65

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]

], and presented in Table 2. The sphericity is also constrained to fall within a range derived from the same data.

The particle size is distributed according to a piecewise (one piece for each soil separate) log-normal distribution as suggested 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]

]. That is, logs is normally distributed. This distribution is characterized by two parameters: the geometric mean particle diameter, d g, and its standard deviation σg, which are functions of soil texture (as defined in Section 2). It is important to note that, since the distribution is specified in a piecewise manner, there will be a different dg and σg for each soil separate. Defining

fg(s)=1bg2πexp((logsag)22bg2),
(19)

where ag=logdg and bg=logσg, the mass fraction of the particles with sizes ranging from s1 to s2 is then given by

𝕱m(s1,s2)=s1s2s1fg(s)ds.
(20)

Assuming that particle density does not vary significantly with size, 𝕱m(s1, s2) 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

11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

]. 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]

] provide a table for determining dg and σg from texture.

4.3. Light Transport Simulation

When a ray is incident at the outer extended boundary of the sand medium, it is either reflected or refracted, with the probability of reflection given by the Fresnel equations [58

58. F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).

], 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.

Once a ray penetrates the outer boundary, it has entered the medium. The ray is then repeatedly scattered by the sand particles until it escapes the medium or is absorbed. Rather than explicitly represent and store each particle, however, they are generated stochastically as needed and subsequently discarded. The light transport simulation process is depicted in Fig. 5.

4.3.1. Extended Boundaries

A ray reaching the surface extended boundary from the inside, as from the outside, is reflected with a probability given by the Fresnel equations [58

58. F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).

] as described above. If internally reflected, the ray continues traversing the medium. Otherwise, the ray escapes the medium and the scattered ray is returned. When a ray reaches the lower extended boundary, it is absorbed.

4.3.2. Pore Space

Each time a ray reaches an interface to the pore space, the medium representing the pore space is selected randomly. The degree of saturation, 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

Fig. 5. Flowchart outlining the light transport algorithm used by the SPLITS model. The dashed line indicates the main loop. The term inner boundary refers to the inside of either extended boundary, whereas the outer boundary is the outside of the surface extended boundary.

dj=1Kjlogξ,
(21)

where ξ is uniformly distributed on (0,1) [67

67. S. Ross, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).

].

For prolate spheroids with normally distributed sphericity and log-normally distributed size, Kj reduces [66

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

] to

Kj=K1K2,
(22)

where

K1=νjsminsmaxs2fg(s)dssminsmaxs1fg(s)ds,
(23)

and

K2=ΨminΨmaxAV(1,Ψ)Φ(Ψ¯,σΨ2)(Ψ)dΨ4ΨminΨmaxΦ(Ψ¯,σΨ2)(Ψ)dΨ.
(24)

The particle size distribution, fg(s), is given by Eq. (19), AV (1,Ψ) is given by Eq. (14), and Φ′(2)(x) is the probability density function for the normal distribution with mean and variance σ2 [67

67. S. Ross, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).

].

The particle size, s, is then chosen according to the probability density function

fs(s)=1C1fg(s)s2,
(25)

and the sphericity, Ψ, is chosen according to

fΨ(Ψ)=1C2AV(1,Ψ)Φ(Ψ¯,σΨ2)(Ψ),
(26)

where C 1 and C 2 are the constants that ensure that fs and fΨ (respectively) integrate to one over their domain [66

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

]. 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]

] shown in the integrand of Eq. (20). This follows from the distinction between the particle size distribution by mass, 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]

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

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

].

Now that we have the particle size, s, the coating thickness is then given by 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 and standard deviation σR given in Table 2.

The point on the surface of the particle which is struck is chosen uniformly on the side of the particle surface facing the ray origin. To select a point uniformly on a prolate spheroid (Fig. 4), we set z=cF -1(2ξ1-1), where ξ1 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(u)=12(1+eu1e2u2+sin1eue1e2+sin1e),
(27)

and e=1Ψ4 is the eccentricity of spheroid [71

71. J. Snyder, Map Projections: A Working Manual (United States Government Printing Office, Washington, 1987). U.S. Geological Survey Professional Paper 1395.

]. Since F -1 is difficult to compute analytically, F may be inverted using numerical techniques. The other two coordinates are then given by x=ar cosθ and y=ar sinθ, with θ=2πξ 22 being another canonical random variable) and r=1z2c2. To restrict the generated point to the side of the particle facing the ray origin, we transform the ray direction 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.

Before we simulate light interaction with this particle, we first check the validity of the particle. If the point, 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

At each interface, the microfacet normal is selected according to Eq. (3). As mentioned in Section 4.2.3, the microfacets are of equal area, 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

67. S. Ross, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).

]. The ray is then reflected in n′ with a probability once again determined using the Fresnel equations3 [58

58. F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).

], considering the media on either side of the interface. Otherwise, the ray is refracted according to Snell’s Law for absorbing media [73

73. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, Oxford, England, 1980).

]. 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]

].

Given the extinction index, 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

54. 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).

]. The light is transmitted a distance d with probability T=exp(-αd), where α is the absorption coefficient, given by

α=4π(η)k

[58

58. F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).

]. For the coating, 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

The framework for SPLITS supports several possible extensions. The model may be extended to include other minerals, given their densities and spectral complex refractive indices. The model can also be extended to include other particle shape and size distributions [66

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

]. Additionally, multiple coatings may be applied to the particles.

While polarization was not simulated, it can be incorporated into the SPLITS model by tracing the Stokes vector instead of the intensity. It is worth mentioning that during the model development an effect of polarization was examined, namely birefringence, since it may affect the accuracy/cost ratio of the simulations. In the case of quartz, for example, the birefringence is low [74

74. C. Gribble and A. Hall, Optical Mineralogy: Principles and Practice (Chapman & Hall, New York, NY, 1993).

], and therefore only the ordinary ray refractive index was used. A full description of the data and optics equations used to implement the SPLITS model is provided by Kimmel [66

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

].

5. Evaluation

To evaluate the SPLITS model objectively, it is necessary to perform comparisons with measured data directly. Thus, comparisons were made between measured data [75

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).

] and results obtained from the model using virtual goniophotometric [5

5. A. Krishnaswamy, G. Baranoski, and J. G. Rokne, “Improving the Reliability/Cost Ratio of goniophotometric measurement,” 9, 31–51 (2004).

] and 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]

] methods.

Table 5. The parameters used in the SPLITS model for each of the four sand samples, and the root-mean-square error (RMSE) between the actual and simulated reflectances.

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One might suggest comparing SPLITS to existing models that have been used to represent sand. There are several problems with this approach. First, to the best of our knowledge, SPLITS is the first model proposed that provides both spectral and spatial (BRDF) responses for sand. Many existing models generate only the spatial response and use the spectral data as input. Furthermore, SPLITS uses sand characterization data as input. Candidates for comparison in the literature are generally designed to handle a wide variety of materials, and therefore use parameters not relating to sand. There is, therefore, no basis for comparison. Finally, even if two models yield similar responses, both models may be wrong. It is therefore best to compare directly with measured data.

These comparisons are made in two manners. First, we directly compare the output from the model to measured curves. Additionally, we demonstrate the effects caused by varying individual parameters to show that the resulting changes are in agreement with what is reported in the literature. We also compare the spatial response from the model to qualitative descriptions found in the literature. Additionally, images are presented in this section to illustrate the colours corresponding to the spectral responses obtained from the model.

5.1. Comparisons with Measured Data

Because the TEC database [75

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).

] contains a wide variety of sand samples and since this data covers the entire visible region of the spectrum, this data was chosen for comparison with the SPLITS model. Due to the lack of characterization data, however, precise quantitative comparisons are not feasible. Instead, the parameters were selected from within acceptable ranges (Table 2) as reported in the literature to attempt to obtain a good match between the measured and modeled curves. This allows us to demonstrate qualitative agreement with measured data, as well as to show that reflectance curves of actual sands are reproducible using the SPLITS model. Four samples were selected from the TEC database [75

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).

]: two dune sands, one from Australia (TEC #10019201), and one from Saudi Arabia (TEC #13j9823), 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

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).

].

5.1.1. Results of Comparisons

Figure 6 shows comparisons between the reflectance curves of these samples and the corresponding curves simulated using SPLITS. The reflectance is provided in the form of directional-hemispherical reflectance [54

54. 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).

] with an incident angle of zero degrees. The corresponding parameters, along with root-mean-square (RMS) errors, are provided in Table 5. The low magnitude of the RMS errors, below 0.03, indicate a good spectral reconstruction even for remote sensing applications demanding high accuracy [6

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]

]. Computer generated images for each of the sand samples are depicted in Fig. 7. The sand grain pattern was extracted from a photograph and the spectral responses were provided by the SPLITS model. The degree of saturation, S, was also varied from S=0 to S=1 within each image to show the darkening effect simulated by the model.

Fig. 6. Comparisons between real and simulated sand. The solid line indicates the reflectance of the sand sample. The dashed line indicates the modeled reflectance. The parameters used and corresponding RMS errors are provided in Table 5. Top Left: TEC #10019201, Top Right: TEC #10039240, Bottom Left: TEC #13j9823, Bottom Right: TEC #19au9815.
Fig. 7. Computer generated images showing variation of sand colour with moisture as predicted by the model. The degree of saturation varies from S=0 at the top of each image to S=1 at the bottom. From left to right, the samples are TEC #10019201, TEC #10039240, TEC #13j9823, and TEC #19au9815.
Fig. 8. Qualitative behaviour of simulated sand. The solid line in each plot indicates the spectral response from the SPLITS model with ϑhghg=0.01, rhgh/(ϑhg)=0.00, ϑm=0.00, ζ1=0.00, ζ2=0.10, ζ3=0.90, and µ′p=µ′m=µ′c=1/3. Top Left: Varying ϑhg, Top Right: Varying rhg, Bottom Left: Varying ϑm, Bottom Right: Varying the texture (ζ 123). See Table 2 for symbol definitions.

5.2. Qualitative Characteristics

To show that the SPLITS model qualitatively behaves like sand, we have also conducted simulations varying individual parameters. The variation in the spectral responses resulting from changes to these parameters are shown in Fig. 8. As expected [17

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

], increasing the iron oxide concentration lowers the reflectance. Note also the shift in reflectance toward the red end of the spectrum when goethite is replaced with hematite (i.e., rhg is increased), as confirmed in the literature [76

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]

, 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]

]. Also, as the particle size was decreased, the reflectance predicted by the model increased, in agreement with the literature [17

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

, 28

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

]. The resulting variation in colour is depicted in Fig. 9. Additionally, the degree of saturation varied between S=0 and S=1 for the four TEC sand samples. The darkening effect, reported in the literature [17

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

, 48

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

], is reflected in the SPLITS model. Simulated reflectance curves for two of these samples are given in Fig. 10.

A scattering simulation using SPLITS was also conducted to show the spatial distribution predicted by the model. This is shown in Fig. 11. While the predicted BRDF is diffuse, there is forward scattering and retro-reflection. This is also in agreement with the literature (Section 2.1).

Fig. 9. Computer generated images showing variation of sand colour as various parameters are changed. The image on the left (base image) corresponds to the solid lines in Fig. 8. The remaining images correspond to the spectral responses from the SPLITS model with the same parameters as in the base image except, from left to right, ϑhghg=0.05, rhgh/(ϑhg)=0.90, ϑm=0.30, ζ3=1.0. See Table 2 for symbol definitions.
Fig. 10. Spectral reflectance curves predicted by the SPLITS model for two of the TEC sand samples, varying the water content, expressed as the degree of saturation, S. Left: TEC #13j9823, Right: TEC #19au9815.

6. Conclusion

We have presented a new model that simulates spectral light transport for sand. The model applies a novel technique for simulating light interaction with particulate materials by generating these particles on the fly, rather than storing them explicitly. Although, in the context of this paper, we have applied this technique to the simulation of light transport with sand, the technique may potentially be applied to other heterogenous materials (organic and inorganic). Both the spectral and spatial responses (the measurement of appearance [55

55. R. Hunter and R. Harold, The Measurement of Appearance, 2nd ed. (JohnWiley and Sons, New York, NY, 1987).

]) are represented using SPLITS. The model is controlled by parameters that relate to physical and mineralogical properties of sand, and the effects of these parameters on the model show good quantitative and qualitative agreement with observations reported in the literature.

While the results show good quantitative and qualitative agreement with measured data, they also indicate that there is still room for further improvement which is likely to depend on data availability. In fact, this paper also serves to point out the lack of and need for spectral BRDF measurements of natural surfaces 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

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

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]

], much work still needs to be done in this area. In most cases where the spectral reflectance or BRDF data is available, the corresponding characterization data is not, and 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.

Fig. 11. BRDF of simulated sand at 600nm. Three dimensional plots of the BRDF are shown on the top. Profiles along the principal plane are shown on the bottom (the dashed line indicates the incident direction). Left: Normal incidence, Middle: For light incident 30° from the normal, Right: For light incident 60° from the normal.

Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful comments. The work presented in this paper was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC grant 238337) and the Canada Foundation for Innovation (CFI grant 33418).

Footnotes

1It 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 [57

57. V. Kuz’min and I. Meglinski, “Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering,” Optics & Spectroscopy 97, 100–106 (2004). [CrossRef]

].
2Recall 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.
3Note 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 5, 102–111 (1989).

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]

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.

A. Krishnaswamy, G. Baranoski, and J. G. Rokne, “Improving the Reliability/Cost Ratio of goniophotometric measurement,” 9, 31–51 (2004).

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]

10.

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

11.

N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).

12.

J. Gerrard, Fundamentals of Soils (Routledge, New York, NY, 2000).

13.

Soil Science Division Staff, Soil Survey Manual (Soil Conservation Service, 1993). United States Department of Agriculture Handbook 18.

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]

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]

19.

G. Hunt and J. Salisbury, “Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals,” Modern Geology 1, 283–300 (1970).

20.

P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).

21.

A. Mottana, R. Crespi, and G. Liborio, Simon and Schuster’s Guide to Rocks and Minerals (Simon and Schuster, Inc., New York, NY, 1978).

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]

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]

25.

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).

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]

28.

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

29.

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

30.

N. Riley, “Projection Sphericity,” Journal of Sedimentary Petrology 11, 94–95 (1941).

31.

W. Krumbein, “Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles,” Journal of Sedimentary Petrology 11, 64–72 (1941).

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]

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]

34.

M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, Cambridge, 2006).

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]

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]

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]

40.

M. Oren and S. Nayar, “Generalization of Lambert’s Reflectance Model,” in Computer Graphics Proceedings, Annual Conference Series, pp. 239–246 (1994).

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]

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]

45.

H. Jensen, J. Legakis, and J. Dorsey, “Rendering ofWet Materials,” in Proceedings of the Eurographics Workshop on Rendering, pp. 273–282 (1999).

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]

48.

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

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]

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]

51.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche (An Article on Optics of Paint Layers),” Zeitschrift für Technische Physik 12, 593–601 (1931).

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]

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]

54.

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).

55.

R. Hunter and R. Harold, The Measurement of Appearance, 2nd ed. (JohnWiley and Sons, New York, NY, 1987).

56.

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

57.

V. Kuz’min and I. Meglinski, “Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering,” Optics & Spectroscopy 97, 100–106 (2004). [CrossRef]

58.

F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).

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]

60.

G. Hale and M. Querry, “Optical Constants of Water in the 200-nm to 200-µm Wavelength Region,” Applied Optics 12, 555–563 (1973). [CrossRef] [PubMed]

61.

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).

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 104, 9423–9444 (1999). [CrossRef]

63.

W. Egan and T. Hilgeman, Optical Properties of Inhomogeneous Materials: Applications to Geology, Astronomy, Chemistry, and Engineering (Academic Press, New York, NY, 1979). [PubMed]

64.

A. Schlegel, S. Alvarado, and P. Wachter, “Optical Properties of Magnetite (Fe3O4),” Journal of Physics C: Solid State Physics 12, 1157–1164 (1979). [CrossRef]

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]

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, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).

68.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, NY, 1983).

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]

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]

71.

J. Snyder, Map Projections: A Working Manual (United States Government Printing Office, Washington, 1987). U.S. Geological Survey Professional Paper 1395.

72.

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

73.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, Oxford, England, 1980).

74.

C. Gribble and A. Hall, Optical Mineralogy: Principles and Practice (Chapman & Hall, New York, NY, 1993).

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 44, 411–420 (1997). [CrossRef]

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]

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


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References

  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 5, 102-111 (1989).
  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]
  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. A. Krishnaswamy, G. Baranoski, and J. G. Rokne, "Improving the Reliability/Cost Ratio of goniophotometric measurement," Journal of Graphics Tools  9, 31-51 (2004).
  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]
  10. F. Pettijohn, P. Potter, and R. Siever, Sand and Sandstone, 2nd ed. (Springer-Verlag, New York, NY, 1987). [CrossRef]
  11. N. Brady, The Nature and Properties of Soils, 8th ed. (Macmillan Publishing Co., Inc., New York, NY, 1974).
  12. J. Gerrard, Fundamentals of Soils (Routledge, New York, NY, 2000).
  13. Soil Science Division Staff, Soil Survey Manual (Soil Conservation Service, 1993). United States Department of Agriculture Handbook 18.
  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]
  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]
  19. G. Hunt and J. Salisbury, "Visible and Near-Infrared Spectra of Minerals and Rocks: I. Silicate Minerals," Modern Geology 1, 283-300 (1970).
  20. P. Farrant, Color in Nature: A Visual and Scientific Exploration (Blandford Press, 1999).
  21. A. Mottana, R. Crespi, and G. Liborio, Simon and Schuster’s Guide to Rocks and Minerals (Simon and Schuster, Inc., New York, NY, 1978).
  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]
  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]
  25. 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).
  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]
  28. R. Vincent and G. Hunt, "Infrared Reflectance from Mat Surfaces," Applied Optics 7, 53-59 (1968). [CrossRef] [PubMed]
  29. H. Wadell, "Volume, Shape, and Roundness of Rock Particles," Journal of Geology 40, 443-451 (1932). [CrossRef]
  30. N. Riley, "Projection Sphericity," Journal of Sedimentary Petrology 11, 94-95 (1941).
  31. W. Krumbein, "Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles," Journal of Sedimentary Petrology 11, 64-72 (1941).
  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]
  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]
  34. M. Mishchenko, L. Travis, and A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University Press, Cambridge, 2006).
  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]
  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]
  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]
  40. M. Oren and S. Nayar, "Generalization of Lambert’s Reflectance Model," in Computer Graphics Proceedings, Annual Conference Series, pp. 239-246 (1994).
  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]
  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]
  45. H. Jensen, J. Legakis, and J. Dorsey, "Rendering ofWet Materials," in Proceedings of the Eurographics Workshop on Rendering, pp. 273-282 (1999).
  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]
  48. D. Lobell and G. Asner, "Moisture Effects on Soil Reflectance," Soil Science Society of America Journal 66, 722-727 (2002). [CrossRef]
  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]
  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]
  51. 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).
  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]
  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]
  54. 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).
  55. R. Hunter and R. Harold, The Measurement of Appearance, 2nd ed. (JohnWiley and Sons, New York, NY, 1987).
  56. G. Ward, "Measuring and Modeling Anisotropic Reflection," Computer Graphics 26, 262-272 (1992). [CrossRef]
  57. V. Kuz’min and I. Meglinski, "Numerical Simulation of Coherent Effects under Conditions of Multiple Scattering," Optics & Spectroscopy 97, 100-106 (2004). [CrossRef]
  58. F. Pedrotti and L. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, Upper Saddle River, NJ, 1993).
  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]
  60. 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]
  61. 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).
  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 104, 9423-9444 (1999). [CrossRef]
  63. W. Egan and T. Hilgeman, Optical Properties of Inhomogeneous Materials: Applications to Geology, Astronomy, Chemistry, and Engineering (Academic Press, New York, NY, 1979). [PubMed]
  64. A. Schlegel, S. Alvarado, and P. Wachter, "Optical Properties of Magnetite (Fe3O4)," Journal of Physics C: Solid State Physics 12, 1157-1164 (1979). [CrossRef]
  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]
  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, A First Course in Probability, 5th ed. (Prentice Hall, Upper Saddle River, NJ, 1998).
  68. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, NY, 1983).
  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]
  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]
  71. J. Snyder, Map Projections: A Working Manual (United States Government Printing Office, Washington, 1987). U.S. Geological Survey Professional Paper 1395.
  72. B. Hapke, "Bidirectional Reflectance Spectroscopy. 4. The Extinction Coefficient and the Opposition Effect," Icarus 67, 264-280 (1986). [CrossRef]
  73. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, Oxford, England, 1980).
  74. C. Gribble and A. Hall, Optical Mineralogy: Principles and Practice (Chapman & Hall, New York, NY, 1993).
  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 44, 411-420 (1997). [CrossRef]
  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]

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