Abstract

We present an analysis of a number of different approximations for the diffuse reflectance (spherical and plane albedo) of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from full absorption to full scattering and from fully forward to fully backward scattering) at collimated, diffuse, and combined illumination conditions were considered. The approximations were analyzed from the point of view of their physical limitations and compared to the numerical radiative transfer solutions, whenever it was possible. The main factors impacting the spherical and plane albedo were revealed for the known and unknown scattering phase functions. The main criterion for inclusion of the models in analysis was the possibility of practical use, i.e., approximations were well parameterized and only included easily measured or estimated parameters. We give a detailed analysis of errors for different models. An algorithm for recalculation of results under combined (direct and diffuse) illumination also has been developed.

© 2013 Optical Society of America

1. Introduction

An estimation of the reflected and transmitted light in turbid medium as a function of layer thickness, illumination, and observation conditions, along with the inherent optical properties (IOPs) of the medium, is a main step to a solution of many different problems. Examples of such problems are an estimation of the amounts of various substances dissolved and suspended in natural waters, noninvasive determination of the optical absorption and scattering properties of human tissue, evaluation of particle size distribution, and refractive indices of pigments and other particles in paint and varnish, calibration of optical instruments, etc. However, in many cases direct measurements of IOPs are difficult; for example, in cases of medical diagnostics or ocean color remote sensing. In such cases, measurements of apparent optical properties such as reflectance and transmittance remain the main sources of information about the target under investigation. Therefore the use of indirect methods and algorithms for the conversion of measured reflectance and transmittance into IOPs is still an utmost issue.

Though a large number of such methods have been developed over the last century in the radiative transfer theory and such applicative fields as astrophysics, ocean and atmospheric optics, biomedical optics, chemistry-technological optics etc., the selection of simple and accurate methods remains a nontrivial task for many investigators. One of the reasons for this consists in using very different systems of approaches and nomenclatures in different scientific fields. Our purpose was to express these different approaches and nomenclatures to the common language. In the current work, we focus on an analysis of the accuracy for different optical models and their combinations, limiting ourselves to consideration of reflected light in plane-parallel homogeneous unbounded layers. Additionally, we considered only models for collimate (at incidence angles smaller than 45°) or diffuse incoming light and diffuse collected light. Such conditions are fulfilled in the most real situations, especially if we consider propagation and reflection of light in the media with the refractive indices different from this of the air. The majority of the models have been selected from numerous literature sources, while several other models have been developed for the first time.

We consider this study as a continuation of the previous studies attempting to compare different numerical and analytical radiative transfer reflectance approximations [113] with the aim of seeking simple but reliable solutions. We put forward a task to yield a comprehensive review of the radiative transfer approximations for diffuse reflectance, especially in their analytical form, which is convenient for fast routine calculations. This purpose responds to the utmost requirements of many various fields of knowledge.

2. Main Definitions

We will consider diffuse reflectance (plane and spherical albedo) in plane-parallel homogeneous unbounded layers illuminated by an external light source. Thus we neglect the refractive index mismatch between the plane-parallel layer and the surrounding medium. However, readers interested in how to account accurately for this mismatch are invited to use algorithms described by numerous authors beginning from Saunderson [14] and his followers [1517]. Both types of illumination (direct and diffuse) with the diffusely collected reflected light will be considered. Finally, we show how to deal with the combined (direct and diffuse) illumination.

A. Plane Albedo

The plane albedo Rp is defined as the ratio of radiation reflected diffusively from the layer to the incoming direct radiation. This optical property is often called by other names such as “directional-hemispherical reflectance,” “hemispherical albedo,” “hemispherical reflectance,” and “diffuse reflectance of the surface, illuminated by the direct rays.” We use the term “plane albedo,” which is more acceptable in publications on radiative transfer. The plane albedo for an infinite layer is defined as [9,1820]

Rp(μi)=1π02π01R(μi,μv,φ)μvdμvdφ,
where μi is the cosine of the incidence angle θi in the medium, μv is the cosine of the viewing angle θv in the medium, and φ is the azimuthal angle between the incident and scattered beam directions. The reflectance factor R(μi,μv,φ) is defined [20,21] as the ratio of the intensity of light reflected from a given layer to the intensity of light reflected from the Lambertian absolutely white surface. For the practical aims, it is suitable to use an azimuthally averaged reflectance factor R¯(μi,μv) [11] as follows:
Rp(μi)=201R¯(μi,μv)μvdμv,R¯(μi,μv)=12π02πR(μi,μv,φ)dφ.
Five physical limitations may be imposed on Rp(μi) [2224]: (1) Rp(μi)=0 at the single-scattering albedo ω0=b/(a+b)=b/c=0; (2) Rp(μi)=1 at ω0=1; (3) 0<Rp(μi)<1 for 0<ω0<1; (4) Rp(μi,ω0)/ω00; and (5) Rp(μi,B)/B0.

Here a, b, and c are coefficients of absorption, scattering, and attenuation, respectively. Table 1 contains these and several other IOPs used in this study. Note also that although the Rp(μi)1, R(μi,μv,φ) may be >1 at ω0 close to 1 [23,24].

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Table 1. Compendium of IOPs Used in the Work and Their Mathematical Definitions

B. Spherical Albedo

The spherical albedo r is another type of reflectance at which incoming and reflected light are diffused. In the literature, other names of this physical quantity like “global albedo,” “diffuse-diffuse reflectance,” and “spherical reflectance” may be met. The spherical albedo for an infinite layer is defined as [9,18,20,25]

r=201Rp(μi)μidμi=40101R¯(μi,μv)μiμvdμidμ.
From Eq. (3) it follows that r does not depend on the angular conditions of illumination or observation; thus it may be considered as an IOP of the medium. Five physical limitations similar to this were imposed on the Rp(μi) and may also be applied for r: (1) r=0 at the ω0=0; (2) r=1 at ω0=1; (3) 0<r<1 for 0<ω0<1; (4) r(ω0)/ω00; and (5) r(μi,B)/B0.

3. Calculation Methods and Numerical Results

A. Scattering Phase Functions

Three different scattering phase functions p(θ) (Fig. 1) have been used in our study for modeling reflective and transmitted properties of the layer:

 figure: Fig. 1.

Fig. 1. Scattering phase functions p(θ) used for modeling. The main selected p(θ) is shown by solid lines, while corresponding Henyey–Greenstein p(θ) (with the same B values) is shown by dash lines.

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We used two optical properties for characterization of p(θ); namely, scattering asymmetry parameter g and backscattering probability B as follows: (1) g=0.0019, B=0.4986 (the p(θ) with such parameters corresponding to the case of the balance between the forward and back scattering); (2) g=0.5033, B=0.1559 (process of forward scattering is prevailing); (3) g=0.9583, B=0.0087 (process of backscattering is almost negligible compared to forward scattering). The p(θ) have been calculated using exact Mie theory for spherical particles distributed in the medium according to the gamma particle size distribution [9,24] for different values of the effective radius reff and particles (relative to medium) of refractive indices m=niχ (at the wavelength of 550 nm) as specified in Table 2. Only models satisfying the above-mentioned physical limitations were included in the final tables. For comparison, we plot also Henyey–Greenstein p(θ) [26] computed for the same values of B as selected p(θ), but for g values ensuring a maximal closness to the selected p(θ).

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Table 2. Parameters Used for the p(θ) Generation

Accuracy of the approximated models was evaluated by computing the mean absolute percentage error (MAPE) and the normalized (to the standard deviation s) root-mean-square error (NRMSE):

MAPE(%)=100%i=1n|(x˜ixi)/x˜i|n,
NRMSE(%)=100%i=1n(x˜ixi)2n1x¯,
where x˜i and xi are the analytical (approximated) and numerical (accepted as a reference) values of the optical property under investigation, respectively; x¯ is the averaged value for all xi values derived for a given phase function p(θ). The MAPE yields an averaged absolute error, while the NRMSE indicates whether the prediction is better than a simple mean prediction. An NRMSE=0 indicates predictions are perfect, and an NRMSE=1 indicates that the prediction is no more accurate than taking the mean of numerical results for given modeling parameters.

B. Plane Albedo Modeling

Seven various approximations for Rp derived by a number of authors and three approximations derived currently for the first time were examined for their accuracy (Table 3, Figs. 25). We used for computations three different p(θ) described above. The values of ω0 for modeling were taken from the range of 0.1,0.2…0.9, 0.95, 0.99, 0.999, 0.9999, while values of μi were taken almost evenly (51 values) within the range from 0.70 to 1. However, in a final analysis we used only three different ranges for the ω0; namely, 0.1ω00.6, 0.6ω00.9, and 0.9ω01, and only one value of μi=cos30.5°=0.862. A reason for the last choice is a rather flat dependence Rp(μi), excluding the range of θi>180°θr where θr is the rainbow scattering angle. The reasonable range for the θr values [2729] is from 114° to 124° at the selected values of refractive indices n (Table 2). Therefore, θi=30.5° is approximately the middle of the zone free from the rainbow effect. This θi value corresponds to 42.8° for the angle of incidence from the air to water (with refractive index n=1.34) and 49.6° for the incidence angle from the air to glass (n=1.5), which are typical values for many optical applications. Such choice of parameters allows obtaining some conclusions for the very different scattering media.

 figure: Fig. 2.

Fig. 2. Plane albedo as a function of transport single-scattering albedo ωtr. (a) Computations performed by the numerical (IIM). (b)–(d) Selected analytical methods at incidence angle θi=30.5° shown for three different phase functions with g=0.00 [(a), (b)]; 0.50 [(a), (c)]; and 0.96 [(a), (d)].

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 figure: Fig. 3.

Fig. 3. Errors of selected plane albedo approximations compared to the IIM-derived values at θi=30.5° for different phase functions versus ωtr.

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 figure: Fig. 4.

Fig. 4. Plane albedo as a function of Gordon’s parameter G computed by numerical and selected analytical methods at θi=30.5° for different phase functions.

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 figure: Fig. 5.

Fig. 5. Same as Fig. 3, but errors versus G.

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Table 3. Accuracy of Selected Models for the Plane Albedo Rp(θi) at θi=30.5°a

All plane albedo approximation models have been compared with the results derived by using the invariant imbedding method (IIM) described in detail by Mishchenko et al. (1999) [30] and Sokoletsky et al. (2009) [12]. This method has been verified previously [12] for two very distinct scattering phase functions (the Rayleigh and Fournier–Forand–Mobley phase functions with g=0.00 and g=0.94, respectively) by comparison of the plane albedo IIM computations with two other numerical methods (namely, different variants of the discrete ordinate method). Divergence of the results was within 1.8% for any combination of geometrical and optical parameters.

Below we give a derivation of new approximations used for the plane albedo modeling.

1. Replacement Method

An idea of this method is a replacement of computation for the plane albedo Rp(μi) by the computation for the spherical albedo r. Taking into account that dependences of Rp(μi) are strictly monotonic, and the “effective” angles θi,ef=arccosμef at which Rp(μef)=r lie generally in the range from 48° to 61° [10], a procedure of Rp(μi) calculation may be presented as

Rp(μi)=rRp(μi)/Rp(1)Rp(μef)/Rp(1),
where an effective angle θef is a function of the similarity parameter as follows:
θef=48+14.12s22.77s2+19.24s3(deg),
while the Rp(μi)/Rp(1) ratio approximated (with the relative errors generally smaller than 10%) by the following expression:
Rp(μi)Rp(1)=exp{[3.599ln(1s)0.550ln2(1s)+0.0416ln(1g)×ln(1s)](1μi)2}
at any values of g and μi>0.7. An expression for calculation of r used in the form developed by Hulst [18,31]:
r=(10.139s)(1s)1+1.170s.
The numerator of the fraction in Eq. (6) is the plane albedo at the given illumination angle normalized by the plane albedo at the vertical illumination, and the denominator of the fraction is the plane albedo at the effective angle normalized by the plane albedo at the vertical illumination. Calculations carried for five very different scattering phase functions [10] show that the relative errors of Eq. (7) are generally smaller than 2%.

2. Hapke and van de Hulst (HH) Method

This method was derived from the Hapke [20] expression obtained for the azimuthally averaged reflectance factor in the case of the isotropic scattering:

R¯(μi,μv)=0.25ω0H(μi)H(μv)μi+μv,
where the Ambartsumian–Chandrasekhar function H(μ) is expressed as
H(μ)=1+2μ1+2μ1ω0.
To generalize Eqs. (10) and (11) for media with arbitrary phase functions, we apply Hulst’s [18,31] similarity rule using the replacement of the single-scattering albedo ω0 by the transport (reduced) single-scattering albedo ωtr and then by the similarity parameter s as follows:
ω0=ba+bωtr=btra+btr=1s2.
Then
R¯(μi,μv)=0.25(1s2)H(μi)H(μv)μi+μv
and
H(μ)=1+2μ1+2μs.
Finally, an expression for the Rp(μi) has been derived by substitution of Eqs. (13) and (14) into Eq. (2) and taking the definite integral:
Rp(μi)=0.5(1s2)(1+2μi)s(1+2μis)×{1+(1s)ln(1+2s)2s(2μis1)+μis(2μi1)[lnμiln(1+μi)]2μis1}.

3. Hapke (Hapke 2) Method

This method uses the R¯(μi,μv) in the form derived by Hapke [20]:

R¯(μi,μv)=0.25ω0[p(θ)+H(μi)H(μv)1]μi+μv
and H(μ) in the form derived by Hapke [36] for anisotropic scattering with any g:
H(μ)={1μω0[r0+(0.5μr0)ln(1+1/μ)]}1,r0=11ω01+1ω0.
Substitution of Eqs. (16) and (17) into Eq. (2) leads to expression for the Rp(μi):
Rp(μi)=0.5ω0{[p(θ)1][1+μilnμiln(1+μi)]+H(μi)01H(μv)μvμi+μvdμv}.
An integral in Eq. (18) cannot be taken analytically; instead, we estimated it by the standard numerical method using a composite of Simpson’s rule with 250 subintervals.

4. Plane Albedo Modeling Results

Numerical results demonstrate a great variability of different approximations for the plane albedo (Table 3 and Fig. 2), an accuracy of which mostly depends on the selected model and values of μi, g, and ω0. As shown for calculations carried out by different models, the best result demonstrates the HKS 2 model at ωtr>0.1; however, at smaller values of ωtr, the best result was obtained by the GKS model (Fig. 3). The HKS 2 model has been first derived by Kokhanovsky and Sokoletsky [10] as a modification of Hapke [20] solution. This modification may be reduced to two subsequent steps: (1) the replacement ω0 by the ωtr according to Hulst’s [18,31] similarity rule and (2) using the additional multiplicative factor Φ(ζi) to improve accuracy of the model. The GKS model for the plane albedo was derived by Kokhanovsky and Sokoletsky [10] from the Gordon [33] and Golubitsky et al. [34] quasi-single-scattering approximation obtained initially as a solution for the reflectance factor.

Another finding is that the Rp(μi) is rather a function of the Gordon’s parameter G (Fig. 4) than of the transport albedo ωtr [Fig. 2(a)]. This fact is confirmed by the numerical and approximated computations. Especially good results were obtained by applying the Gordon, Haltrin, and GKS models. Note that in a practice if, for example, the ωtr and ω0 are known, then it is easy to find an asymmetry parameter g (see Table 1) and then to estimate a backscattering contribution B (for a given scattering phase function) and G. To show better the impact of G on Rp(μi), we used an alternative absciss axis with G values (Figs. 4 and 5) for demonstrating the absolute values of Rp(μi,G) and relative errors for several models. Note that since the GKS and HKS 2 models give values of Rp(μi) almost independent on selected scattering phase functions, we have shown here the values of Rp(μi) obtained only for one phase function; namely, with g=0.50 (Fig. 4).

Overall, among all considered approximations, the best results show the GKS approximation at G<0.05 and HKS 2 approximation at all other values of G (see Fig. 5). Combined application of these two models gives a relative error δ<16%.

5. Plane Albedo Modeling Under Lack of p(θ) Information

In real practice, a scattering phase function is often inaccessible. This is also means inaccessibility of information about the g, ωtr, and s parameters. However, it is possible to estimate g (and, hence, ωtr and s) from measured B=bb/b values. Below we suggest an algorithm for the p(θ) and g estimation and test for the impact of this inaccuracy in g on the accuracy of estimated Rp(μi).

For current modeling, we selected the widely applied Henyey–Greenstein p(θ):

p(θ)=1g2(12gcosθ+g2)1.5.
The relationship between g and B is easily determined by integrating within the equation for B (see Table 1) leading to [37]
B=1g2g(1+g1+g21).
Inverting this equation and fitting it by the 5° polynomial leads to the following result:
g=14.440B+12.11B223.87B3+23.52B49.317B5,
with the NRMSE=1.2% and R2=0.99998 over the whole possible ranges of parameters B and g: 0B1 and 1g1.

The values of the expansion coefficients xj needed for computation Rp(μi) by the GKS model (see Table 3) may be now easily obtained from the expansion of p(θ) into the Legendre polynomial series [37]:

xj=(2j+1)gj.
The selection of Henyey–Greenstein p(θ) may be explained by its convinience for computations along with its closeness to the initial p(θ) at given values of B (see Fig. 2). The errors for the Rp(μi) computed for the same models that were used for plotting Fig. 6 at μi=0.862 and three values of B taken from the Table 2 are shown in Fig. 6 as a function of parameter G. As it follows from the computations, the inexact values of p(θ) and g lead typically to small decreases in the Rp(μi) values (up to 12% and 10% for GKS and HKS 2 models, respectively) comparative to the Rp(μi) values computed at exact (known) values of p(θ). However, an accuracy of the models under consideration is still high. The HKS 2 model is obviously better than the other models at any p(θ) and G0.05, while at G<0.05 the best results yield a GKS model. Combined application of these two models (or only the HKS 2 model) gives a relative error δ<15% (Fig. 6) that is even better than an error obtained for the case of known p(θ) and g.

 figure: Fig. 6.

Fig. 6. Errors of selected plane albedo approximations compared to the IIM-derived values at θi=30.5° for different phase functions versus G in situations when the scattering phase function is unknown.

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C. Spherical Albedo Modeling

Table 4 and Fig. 7 show various radiative transfer approximations for the spherical albedo r derived by a number of authors. The separate values of ω0, their ranges, and scattering phase functions were selected the same as were used for the Rp(μi) computations. Again, all models were compared with the numerical calculations (IIM). The results show that the transport albedo ωtr (or, alternatively, Hulst’s similarity parameter s) better related to r than Gordon’s parameter G. A logical explanation of this fact may be found on p. 211 by van de Hulst [18]. Hulst also gave numerical confirmation of this fact for the wide ranges of ω0 (from 0.2 to 0.99) and g (from 0 to 7/8). Our work confirms this fact again.

 figure: Fig. 7.

Fig. 7. Spherical albedo as a function of ωtr computed by numerical and selected analytical methods for different phase functions.

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Table 4. Accuracy of Selected Models for the Spherical Albedo ra

Numerical results contained in Table 4 and shown in Figs. 7 and 8 demonstrate an excellent accuracy for Hulst’s and six other testing approximations. Five of these models (Hulst, GKM+MR, Flock, HKS 1, and GSK) were considered earlier [7,9,10,43] and also demonstrated encouraging results. Two new models were analyzed now for a first time and also show reasonable results; namely, the model by Gemert and Star (“GS”) and the “GKM-new” models. Both models are similar (as well as several other models listed in Table 4) and present actual attempts to express the parameters of absorption (K) and scattering (S) of the two-flux GKM theory via well-established IOPs: a, btr, and ωtr.

 figure: Fig. 8.

Fig. 8. Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus ωtr.

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The “GKM-new” model was developed by following [7,41,42,60] findings. More specifically, for this model we have solved a system of the equations

r=1+KSKS(KS+2),r=ln(1+ξ)ξln(1ξ)+ξ,ωtr=2ξln[(1+ξ)/(1ξ)],
with the boundary conditions
η=a/K=1atωtr=0,η=0.5atωtr=1,andχ=btr/S=4/3atωtr=1.
The first equation in Eq. (23) is a classical GKM, while two others were derived from the Chandrasekhar–Klier equations with Hulst’s replacement: ω0ωtr (CKS model) to the better accounting of a scattering anisotropy. The boundary conditions [Eq. (24)] were derived by Yudovsky and Pilon [60], and they seem as reasonable and close to conditions derived by other investigators (see Table 4). The solution has been found in the form of strictly decreasing η(ωtr) and χ(ωtr) polynomial values that maximally satisfy Eqs. (23) and (24). A full solution presented in the Table 4 yields r values almost coinciding with the values following from the CKS model and the Yudovsky and Pilon (2009) model as well; however, their solutions failed at values ωtr<0.06 (or s>0.97). Contrarily, the GKM-new model has a solution at any values of ωtr or s. Overall, among all considered approximations, the GSK approximation demonstrated the best results at ωtr<0.36 (or s>0.80), and the Hulst approximation is the best at the other values of ωtr or s (see Fig. 8). Combined application of two these models yields a relative error δ<3% at any values of parameters. However, taking into account the relative complexity of the GSK model, application of only Hulst’s model (with δ<7%) also is reasonable. It is worth mentioning that the GSK approximation for the spherical albedo was derived by the direct integration (Eq. 3) from the GKS model for the plane albedo (Table 3) by Sokoletsky and Kokhanovsky [43], while Hulst’s model has been developed by van de Hulst [18].

1. Spherical Albedo Modeling Under Lack of p(θ) Information

This is the case similar to that for the plane albedo (Subsection 3.B.5). Again, we replaced the initial p(θ) by the Henyey–Greenstein p(θ) [Eq. (19)] and the g values by their calculated values [Eq. (21)] in the cases when knowledge of p(θ) and/or g was necessary for model computations. Results show that, similarly to the Rp(μi) calculations, inaccurate g values lead typically to a small decrease in the r values (up to 11%) comparative to the r values computed at known (exact) values of g (Fig. 9). As before, Hulst’s model demonstrates superior results in most cases with the relative errors <7%. Thus we recommend to use this spherical albedo model in the case, if the precise values of p(θ) is unknown.

 figure: Fig. 9.

Fig. 9. Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus G in situations when the scattering phase function is unknown. The values of backscattering ratio B are (a) 0.4986, (b) 0.1559, and (c) 0.0087.

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D. Diffuse Reflectance Under Combined Illumination

Let us consider now a case of combined (direct and diffuse) illumination with the diffuse irradiance Ii,dif contribution dE into the total (Ii) irradiance. Then the reflectance may be easily determined as follows:

Rc(μi)=IrIi=Ii,dir(μi)Rp(μi)+Ii,difrIi=(1dE)Rp(μi)+dEr.

4. Conclusions

A large number of different approximate analytic models for plane and spherical albedo (specifically, 10 and 28 for Rp and r, respectively) of unbounded plane-parallel turbid layers were considered to reveal the most theoretically grounded and accurate models. For this aim, all models were checked for their correspondence with the physical limitations and/or compared when it was possible with the accurate numerical results. As concluded elsewhere [7] and confirmed again, among spherical albedo approximations, the Hulst and the GKM+MR models are the most accurate ones with the errors NRMSE <6.8% and 7.5%, respectively, under any optical conditions (see Table 4). The current study revealed a number of other approximations, which may also yield accurate solutions in definite situations. Such an impressive result may be explained by the fact that the r is safely governed by only one parameter, namely, s (or ωtr) (Fig. 7).

A much more difficult situation is with plane albedo modeling. This property is governed by three optical parameters: μi, g (or B), and ω0, though this also may be expressed by only two parameters: μi and ωtr (Fig. 2) or μi and G (Fig. 4). The best result here is demonstrated in the HKS 2 and Gordon’s model with the NRMSE <28% and 34%, respectively (Table 3). However, using a replacement method [Eqs. (6) to (9)] at which a calculation of the plane albedo Rp(μi) replaced by the calculation of the spherical albedo r also seems to be a perspective. The replacement method yields good results only at strong scattering (i.e., ω0 close to 1) in its current form. Probably, the more complicate (and accurate) approximation for Rp(μi)/Rp(1) and θi,eff may help in improvement of the current solution.

The study also considers briefly the issue of the layers illuminated by combined (collimated and diffuse) light (Subsection 3.D).

The obtained results may be useful for the solution to many problems relating to the light reflected from turbid media—from very clear skies and oceanic waters to extremely turbid inland waters, biological tissues, and paint and varnishes. Better knowledge of the relationships between the measured coefficients of reflectance and the modeled IOPs (beam scattering, absorption, asymmetry parameters, etc.) will allow a more accurate solution to such problems as remote monitoring of water environments and developing multifunctional laser systems for noninvasive diagnostics.

The authors would like to acknowledge the valuable comments of Dr. Dmitrii A. Rogatkin (Moscow Regional Research and Clinical Institute “MONIKI”) on earlier versions of the manuscript. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7-PEOPLE-2009-IAPP) under grant agreement number no. 251531 (MEDI-LASE project). During the final stage of the manuscript preparation, it was also supported by the State Key Laboratory of Estuarine and Coastal Research (SKLEC) grant 2012KYYW02 and by the 111 project (B08022).

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8. IOCCG Report 5, “Remote sensing of inherent optical properties: fundamentals, tests of algorithms, and applications”, Z. P. Lee, ed. (Naval Research Laboratory, Stennis Space Center, 2006).

9. A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006). [CrossRef]  

10. A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006). [CrossRef]  

11. A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka–Munk theory,” J. Phys. D 40, 2210–2216 (2007). [CrossRef]  

12. L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009). [CrossRef]  

13. L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012). [CrossRef]  

14. J. L. Saunderson, “Calculation of the color of pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942). [CrossRef]  

15. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996). [CrossRef]  

16. C. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999). [CrossRef]  

17. A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011). [CrossRef]  

18. H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).

19. T. Nakajima and M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates method,” Appl. Opt. 31, 7669–7683 (1992). [CrossRef]  

20. B. Hapke, Theory of Reflectance and Emittance Spectroscopy (University Press, 1993).

21. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

22. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975). [CrossRef]  

23. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).

24. A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

25. J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974). [CrossRef]  

26. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941). [CrossRef]  

27. H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977). [CrossRef]  

28. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

29. A. A. Kokhanovsky, “The determination of the effective radius of drops in water clouds from polarization measurements,” Phys. Chem. Earth B 25, 471–474 (2000). [CrossRef]  

30. M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999). [CrossRef]  

31. H. C. van de Hulst, Multiple Light Scattering (Academic, 1980), Vol. 2.

32. O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

33. H. G. Gordon, “Simple calculation of the diffuse reflectance of ocean,” Appl. Opt. 12, 2803–2804 (1973). [CrossRef]  

34. B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

35. H. G. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989). [CrossRef]  

36. B. Hapke, “Bidirectional reflectance spectroscopy. 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002). [CrossRef]  

37. V. I. Haltrin, “One-parameter two-term Henyey–Greenstein phase function for light scattering in seawater,” Appl. Opt. 41, 1022–1028 (2002). [CrossRef]  

38. M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift 31, 753–763 (1930).

39. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Zeitschrift für Technische Physik. 12, 593–601 (1931).

40. P. Kubelka, “New contributions to the optics of intensely light-scattering material. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948). [CrossRef]  

41. S. Chandrasekhar, Radiative Transfer (Dover, 1960).

42. K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1972). [CrossRef]  

43. L. G. Sokoletsky and A. A. Kokhanovsky, “Reflective characteristics of natural waters: The accuracy of selected approximations,” in Proceedings of the D. S. Rozhdestvensky Third International Conference: Current Problems in Optics of Natural Waters, I. Levin and G. Gilbert, eds. (Saint-Petersburg, Russia, 2005), pp. 56–63.

44. G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).

45. L. W. Richards, “The calculation of the optical performance of paint films,” J. Paint Technol. 42, 276–286 (1970).

46. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971). [CrossRef]  

47. W. E. Meador and W. R. Weaver, “Diffusion approximation for large absorption in radiative transfer,” Appl. Opt. 18, 1204–1208 (1979). [CrossRef]  

48. R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987). [CrossRef]  

49. M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

50. V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988). [CrossRef]  

51. S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989). [CrossRef]  

52. J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors I: coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992). [CrossRef]  

53. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992). [CrossRef]  

54. J. Wu, F. Partovi, M. S. Field, and R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993). [CrossRef]  

55. G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998). [CrossRef]  

56. S. L. Jacques, “Diffuse reflectance from a semi-infinite medium,” 1999, http://omlc.ogi.edu/news/may99/rd/index.html.

57. G. Zonios and A. Dimou, “Modeling diffuse reflectance from semi-infinite turbid media: application to the study of skin optical properties,” Opt. Express 14, 8661–8674 (2006). [CrossRef]  

58. G. L. Rogers, “Multiple path analysis of reflectance from turbid media,” J. Opt. Soc. Am. A 25, 2879–2883 (2008). [CrossRef]  

59. S. N. Thennadil, “Relationship between the Kubelka–Munk scattering and radiative transfer coefficients,” J. Opt. Soc. Am. A 25, 1480–1485 (2008). [CrossRef]  

60. D. Yudovsky and L. Pilon, “Simple and accurate expressions for diffuse reflectance of semi-infinite and two-layer absorbing and scattering media,” Appl. Opt. 48, 6670–6683 (2009). [CrossRef]  

References

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  1. M. King and Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
    [Crossref]
  2. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
    [Crossref]
  3. P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
    [Crossref]
  4. V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
    [Crossref]
  5. E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
    [Crossref]
  6. L. Sokoletsky, “A comparative analysis of selected radiative transfer approaches for aquatic environments,” Appl. Opt. 44, 136–148 (2005).
    [Crossref]
  7. L. Sokoletsky, “A comparative analysis of simple radiative transfer approaches for aquatic environments,” in Proceedings of 2004 ENVISAT & ERS Symposium, H. Lacoste, ed. (Noordwijk, 2005), pp. 1–7.
  8. , “Remote sensing of inherent optical properties: fundamentals, tests of algorithms, and applications”, Z. P. Lee, ed. (Naval Research Laboratory, Stennis Space Center, 2006).
  9. A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
    [Crossref]
  10. A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
    [Crossref]
  11. A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka–Munk theory,” J. Phys. D 40, 2210–2216 (2007).
    [Crossref]
  12. L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
    [Crossref]
  13. L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
    [Crossref]
  14. J. L. Saunderson, “Calculation of the color of pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
    [Crossref]
  15. A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
    [Crossref]
  16. C. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999).
    [Crossref]
  17. A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
    [Crossref]
  18. H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).
  19. T. Nakajima and M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates method,” Appl. Opt. 31, 7669–7683 (1992).
    [Crossref]
  20. B. Hapke, Theory of Reflectance and Emittance Spectroscopy (University Press, 1993).
  21. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).
  22. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
    [Crossref]
  23. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).
  24. A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).
  25. J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
    [Crossref]
  26. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [Crossref]
  27. H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
    [Crossref]
  28. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  29. A. A. Kokhanovsky, “The determination of the effective radius of drops in water clouds from polarization measurements,” Phys. Chem. Earth B 25, 471–474 (2000).
    [Crossref]
  30. M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
    [Crossref]
  31. H. C. van de Hulst, Multiple Light Scattering (Academic, 1980), Vol. 2.
  32. O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).
  33. H. G. Gordon, “Simple calculation of the diffuse reflectance of ocean,” Appl. Opt. 12, 2803–2804 (1973).
    [Crossref]
  34. B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).
  35. H. G. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [Crossref]
  36. B. Hapke, “Bidirectional reflectance spectroscopy. 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
    [Crossref]
  37. V. I. Haltrin, “One-parameter two-term Henyey–Greenstein phase function for light scattering in seawater,” Appl. Opt. 41, 1022–1028 (2002).
    [Crossref]
  38. M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift 31, 753–763 (1930).
  39. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Zeitschrift für Technische Physik. 12, 593–601 (1931).
  40. P. Kubelka, “New contributions to the optics of intensely light-scattering material. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [Crossref]
  41. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  42. K. Klier, “Absorption and scattering in plane parallel turbid media,” J. Opt. Soc. Am. 62, 882–885 (1972).
    [Crossref]
  43. L. G. Sokoletsky and A. A. Kokhanovsky, “Reflective characteristics of natural waters: The accuracy of selected approximations,” in Proceedings of the D. S. Rozhdestvensky Third International Conference: Current Problems in Optics of Natural Waters, I. Levin and G. Gilbert, eds. (Saint-Petersburg, Russia, 2005), pp. 56–63.
  44. G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).
  45. L. W. Richards, “The calculation of the optical performance of paint films,” J. Paint Technol. 42, 276–286 (1970).
  46. P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
    [Crossref]
  47. W. E. Meador and W. R. Weaver, “Diffusion approximation for large absorption in radiative transfer,” Appl. Opt. 18, 1204–1208 (1979).
    [Crossref]
  48. R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [Crossref]
  49. M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).
  50. V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988).
    [Crossref]
  51. S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
    [Crossref]
  52. J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors I: coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
    [Crossref]
  53. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [Crossref]
  54. J. Wu, F. Partovi, M. S. Field, and R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
    [Crossref]
  55. G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
    [Crossref]
  56. S. L. Jacques, “Diffuse reflectance from a semi-infinite medium,” 1999, http://omlc.ogi.edu/news/may99/rd/index.html .
  57. G. Zonios and A. Dimou, “Modeling diffuse reflectance from semi-infinite turbid media: application to the study of skin optical properties,” Opt. Express 14, 8661–8674 (2006).
    [Crossref]
  58. G. L. Rogers, “Multiple path analysis of reflectance from turbid media,” J. Opt. Soc. Am. A 25, 2879–2883 (2008).
    [Crossref]
  59. S. N. Thennadil, “Relationship between the Kubelka–Munk scattering and radiative transfer coefficients,” J. Opt. Soc. Am. A 25, 1480–1485 (2008).
    [Crossref]
  60. D. Yudovsky and L. Pilon, “Simple and accurate expressions for diffuse reflectance of semi-infinite and two-layer absorbing and scattering media,” Appl. Opt. 48, 6670–6683 (2009).
    [Crossref]

2012 (1)

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[Crossref]

2011 (1)

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[Crossref]

2009 (2)

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

D. Yudovsky and L. Pilon, “Simple and accurate expressions for diffuse reflectance of semi-infinite and two-layer absorbing and scattering media,” Appl. Opt. 48, 6670–6683 (2009).
[Crossref]

2008 (2)

2007 (1)

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka–Munk theory,” J. Phys. D 40, 2210–2216 (2007).
[Crossref]

2006 (3)

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[Crossref]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[Crossref]

G. Zonios and A. Dimou, “Modeling diffuse reflectance from semi-infinite turbid media: application to the study of skin optical properties,” Opt. Express 14, 8661–8674 (2006).
[Crossref]

2005 (1)

2003 (1)

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[Crossref]

2002 (2)

B. Hapke, “Bidirectional reflectance spectroscopy. 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[Crossref]

V. I. Haltrin, “One-parameter two-term Henyey–Greenstein phase function for light scattering in seawater,” Appl. Opt. 41, 1022–1028 (2002).
[Crossref]

2000 (1)

A. A. Kokhanovsky, “The determination of the effective radius of drops in water clouds from polarization measurements,” Phys. Chem. Earth B 25, 471–474 (2000).
[Crossref]

1999 (2)

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[Crossref]

C. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999).
[Crossref]

1998 (2)

1997 (1)

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[Crossref]

1996 (1)

1993 (2)

1992 (3)

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors I: coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[Crossref]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[Crossref]

T. Nakajima and M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates method,” Appl. Opt. 31, 7669–7683 (1992).
[Crossref]

1989 (2)

H. G. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[Crossref]

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[Crossref]

1988 (1)

1987 (2)

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[Crossref]

M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

1986 (1)

M. King and Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[Crossref]

1979 (1)

1977 (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[Crossref]

1975 (1)

1974 (3)

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[Crossref]

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).

1973 (1)

1972 (1)

1971 (2)

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

1970 (1)

L. W. Richards, “The calculation of the optical performance of paint films,” J. Paint Technol. 42, 276–286 (1970).

1962 (1)

G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).

1948 (1)

1942 (1)

1941 (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

1931 (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Zeitschrift für Technische Physik. 12, 593–601 (1931).

1930 (1)

M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift 31, 753–763 (1930).

Bass, L. P.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

Bonner, R. F.

Brown, O. B.

Budak, V. P.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

Bushmakova, O. V.

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Campos Velho, H. F.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[Crossref]

Chalhoub, E. S.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Cornet, J. F.

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors I: coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[Crossref]

Cuppo, F. L. S.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[Crossref]

Dimou, A.

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[Crossref]

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[Crossref]

Dubertret, G.

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors I: coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[Crossref]

Dussap, C. G.

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors I: coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[Crossref]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[Crossref]

Field, M. S.

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[Crossref]

Garcia, R. D. M.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[Crossref]

García-Valenzuela, A.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[Crossref]

Gemert, M. J. C.

M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

Gentili, B.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Golubitsky, B. M.

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

Gordon, H. G.

H. G. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[Crossref]

H. G. Gordon, “Simple calculation of the diffuse reflectance of ocean,” Appl. Opt. 12, 2803–2804 (1973).
[Crossref]

Gordon, H. R.

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Gurevich, M.

M. Gurevich, “Über eine rationelle klassifikation der lichtenstreuenden medien,” Physikalische Zeitschrift 31, 753–763 (1930).

Haltrin, V. I.

Hapke, B.

B. Hapke, “Bidirectional reflectance spectroscopy. 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[Crossref]

B. Hapke, Theory of Reflectance and Emittance Spectroscopy (University Press, 1993).

Harshvardhan,

M. King and Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[Crossref]

Havlin, S.

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Jacobs, M. M.

Jin, Z.

Katsev, I. L.

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Kattawar, G. W.

King, M.

M. King and Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[Crossref]

King, M. D.

Klier, K.

Kokhanovsky, A. A.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

A. A. Kokhanovsky, “Physical interpretation and accuracy of the Kubelka–Munk theory,” J. Phys. D 40, 2210–2216 (2007).
[Crossref]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[Crossref]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[Crossref]

A. A. Kokhanovsky, “The determination of the effective radius of drops in water clouds from polarization measurements,” Phys. Chem. Earth B 25, 471–474 (2000).
[Crossref]

A. A. Kokhanovsky, Light Scattering Media Optics (Springer-Praxis, 2004).

L. G. Sokoletsky and A. A. Kokhanovsky, “Reflective characteristics of natural waters: The accuracy of selected approximations,” in Proceedings of the D. S. Rozhdestvensky Third International Conference: Current Problems in Optics of Natural Waters, I. Levin and G. Gilbert, eds. (Saint-Petersburg, Russia, 2005), pp. 56–63.

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light-scattering material. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
[Crossref]

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Zeitschrift für Technische Physik. 12, 593–601 (1931).

Kuznetsov, V. S.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

Levin, I. M.

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Lunetta, R. S.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[Crossref]

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

Marcus, R. T.

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[Crossref]

Masoliver, J.

G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
[Crossref]

Meador, W. E.

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[Crossref]

Mobley, C. D.

Morel, A.

Mudgett, P. S.

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche,” Zeitschrift für Technische Physik. 12, 593–601 (1931).

Nakajima, T.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Nikolaeva, O. V.

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

Nossal, R.

Nussenzveig, H. M.

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[Crossref]

Olivares, J. A.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[Crossref]

Paerl, H. W.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[Crossref]

Partovi, F.

Patterson, M. S.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[Crossref]

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[Crossref]

Pierce, P. E.

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[Crossref]

Pilon, L.

Porrà, J. M.

G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
[Crossref]

Rava, R. P.

Reinersman, P.

Richards, L. W.

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

L. W. Richards, “The calculation of the optical performance of paint films,” J. Paint Technol. 42, 276–286 (1970).

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical considerations and nomenclature for reflectance” (National Bureau of Standards, U.S. Department of Commerce, 1977).

Rogers, G. L.

Rozenberg, G. V.

G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).

Saunderson, J. L.

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, 1975).

Sokoletsky, L.

L. Sokoletsky, “A comparative analysis of selected radiative transfer approaches for aquatic environments,” Appl. Opt. 44, 136–148 (2005).
[Crossref]

L. Sokoletsky, “A comparative analysis of simple radiative transfer approaches for aquatic environments,” in Proceedings of 2004 ENVISAT & ERS Symposium, H. Lacoste, ed. (Noordwijk, 2005), pp. 1–7.

Sokoletsky, L. G.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[Crossref]

L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S. Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters,” J. Quant. Spectrosc. Radiat. Transfer 110, 1132–1146 (2009).
[Crossref]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[Crossref]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[Crossref]

L. G. Sokoletsky and A. A. Kokhanovsky, “Reflective characteristics of natural waters: The accuracy of selected approximations,” in Proceedings of the D. S. Rozhdestvensky Third International Conference: Current Problems in Optics of Natural Waters, I. Levin and G. Gilbert, eds. (Saint-Petersburg, Russia, 2005), pp. 56–63.

Stamnes, K.

Star, W. M.

M. J. C. Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

Stavn, R. H.

Tantashev, M. V.

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

Thennadil, S. N.

van de Hulst, H. C.

H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

H. C. van de Hulst, Multiple Light Scattering (Academic, 1980), Vol. 2.

Vihena, M. T.

E. S. Chalhoub, H. F. Campos Velho, R. D. M. Garcia, and M. T. Vihena, “A comparison of radiances generated by selected methods of solving the radiative-transfer equation,” Transp. Theory Stat. Phys. 32, 473–503 (2003).
[Crossref]

Weaver, W. R.

Weiss, G. H.

G. H. Weiss, J. M. Porrà, and J. Masoliver, “The continuous-time random walk description of photon motion in an isotropic medium,” Opt. Commun. 146, 268–276 (1998).
[Crossref]

R. F. Bonner, R. Nossal, S. Havlin, and G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[Crossref]

Wetz, M. S.

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[Crossref]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[Crossref]

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[Crossref]

Wu, J.

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[Crossref]

Yanovitskij, E. G.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[Crossref]

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[Crossref]

Yudovsky, D.

Zakharova, N. T.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[Crossref]

Zege, E. P.

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Zonios, G.

Appl. Opt. (14)

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[Crossref]

V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
[Crossref]

L. Sokoletsky, “A comparative analysis of selected radiative transfer approaches for aquatic environments,” Appl. Opt. 44, 136–148 (2005).
[Crossref]

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
[Crossref]

C. D. Mobley, “Estimation of the remote-sensing reflectance from above-surface measurements,” Appl. Opt. 38, 7442–7455 (1999).
[Crossref]

T. Nakajima and M. D. King, “Asymptotic theory for optically thick layers: application to the discrete ordinates method,” Appl. Opt. 31, 7669–7683 (1992).
[Crossref]

H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[Crossref]

H. G. Gordon, “Simple calculation of the diffuse reflectance of ocean,” Appl. Opt. 12, 2803–2804 (1973).
[Crossref]

V. I. Haltrin, “One-parameter two-term Henyey–Greenstein phase function for light scattering in seawater,” Appl. Opt. 41, 1022–1028 (2002).
[Crossref]

P. S. Mudgett and L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[Crossref]

W. E. Meador and W. R. Weaver, “Diffusion approximation for large absorption in radiative transfer,” Appl. Opt. 18, 1204–1208 (1979).
[Crossref]

V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988).
[Crossref]

J. Wu, F. Partovi, M. S. Field, and R. P. Rava, “Diffuse reflectance from turbid media: an analytical model of photon migration,” Appl. Opt. 32, 1115–1121 (1993).
[Crossref]

D. Yudovsky and L. Pilon, “Simple and accurate expressions for diffuse reflectance of semi-infinite and two-layer absorbing and scattering media,” Appl. Opt. 48, 6670–6683 (2009).
[Crossref]

Astron. Astrophys. (1)

H. C. van de Hulst, “The spherical albedo of a planet covered with a homogeneous cloud layer,” Astron. Astrophys. 35, 209–214 (1974).

Astrophys. J. (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Biotechnol. Bioeng. (1)

J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors I: coupling between light transfer and growth kinetics,” Biotechnol. Bioeng. 40, 817–825 (1992).
[Crossref]

Color Res. Appl. (3)

P. E. Pierce and R. T. Marcus, “Radiative transfer theory solid color-matching calculations,” Color Res. Appl. 22, 72–87 (1997).
[Crossref]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 1: spherical albedo,” Color Res. Appl. 31, 491–497 (2006).
[Crossref]

A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light from semi-infinite absorbing turbid media. Part 2: plane albedo and reflection function,” Color Res. Appl. 31, 498–509 (2006).
[Crossref]

Doklady Acad. Sci. BSSR (1)

O. V. Bushmakova, E. P. Zege, and I. L. Katsev, “On asymptotic equations for brightness coefficients of optically thick light scattering layers,” Doklady Acad. Sci. BSSR 15, 309–311 (1971).

Doklady Acad. Sci. USSR (1)

G. V. Rozenberg, “Light characteristics of thick layers of a weakly absorbing scattering medium,” Doklady Acad. Sci. USSR 145, 775–777 (1962).

Icarus (2)

B. Hapke, “Bidirectional reflectance spectroscopy. 5. The coherent backscatter opposition effect and anisotropic scattering,” Icarus 157, 523–534 (2002).
[Crossref]

J. M. Dlugach and E. G. Yanovitskij, “The optical properties of Venus and the Jovian planets. II. Methods and results of calculations of the intensity of radiation diffusely reflected from semi-infinite homogeneous atmospheres,” Icarus 22, 66–81 (1974).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[Crossref]

Isr. J. Plant Sci. (1)

L. G. Sokoletsky, R. S. Lunetta, M. S. Wetz, and H. W. Paerl, “Assessment of the water quality components in turbid estuarine waters based on radiative transfer approximations,” Isr. J. Plant Sci. 60, 209–229 (2012).
[Crossref]

Izv. Atmos. Ocean. Phys. (1)

B. M. Golubitsky, I. M. Levin, and M. V. Tantashev, “Brightness coefficient of a semi-infinite layer of sea water,” Izv. Atmos. Ocean. Phys. 10, 1235–1238 (1974).

J. Atmos. Sci. (1)

M. King and Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
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Figures (9)

Fig. 1.
Fig. 1. Scattering phase functions p(θ) used for modeling. The main selected p(θ) is shown by solid lines, while corresponding Henyey–Greenstein p(θ) (with the same B values) is shown by dash lines.
Fig. 2.
Fig. 2. Plane albedo as a function of transport single-scattering albedo ωtr. (a) Computations performed by the numerical (IIM). (b)–(d) Selected analytical methods at incidence angle θi=30.5° shown for three different phase functions with g=0.00 [(a), (b)]; 0.50 [(a), (c)]; and 0.96 [(a), (d)].
Fig. 3.
Fig. 3. Errors of selected plane albedo approximations compared to the IIM-derived values at θi=30.5° for different phase functions versus ωtr.
Fig. 4.
Fig. 4. Plane albedo as a function of Gordon’s parameter G computed by numerical and selected analytical methods at θi=30.5° for different phase functions.
Fig. 5.
Fig. 5. Same as Fig. 3, but errors versus G.
Fig. 6.
Fig. 6. Errors of selected plane albedo approximations compared to the IIM-derived values at θi=30.5° for different phase functions versus G in situations when the scattering phase function is unknown.
Fig. 7.
Fig. 7. Spherical albedo as a function of ωtr computed by numerical and selected analytical methods for different phase functions.
Fig. 8.
Fig. 8. Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus ωtr.
Fig. 9.
Fig. 9. Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus G in situations when the scattering phase function is unknown. The values of backscattering ratio B are (a) 0.4986, (b) 0.1559, and (c) 0.0087.

Tables (4)

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Table 1. Compendium of IOPs Used in the Work and Their Mathematical Definitions

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Table 2. Parameters Used for the p(θ) Generation

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Table 3. Accuracy of Selected Models for the Plane Albedo Rp(θi) at θi=30.5°a

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Table 4. Accuracy of Selected Models for the Spherical Albedo ra

Equations (25)

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Rp(μi)=1π02π01R(μi,μv,φ)μvdμvdφ,
Rp(μi)=201R¯(μi,μv)μvdμv,R¯(μi,μv)=12π02πR(μi,μv,φ)dφ.
r=201Rp(μi)μidμi=40101R¯(μi,μv)μiμvdμidμ.
MAPE(%)=100%i=1n|(x˜ixi)/x˜i|n,
NRMSE(%)=100%i=1n(x˜ixi)2n1x¯,
Rp(μi)=rRp(μi)/Rp(1)Rp(μef)/Rp(1),
θef=48+14.12s22.77s2+19.24s3(deg),
Rp(μi)Rp(1)=exp{[3.599ln(1s)0.550ln2(1s)+0.0416ln(1g)×ln(1s)](1μi)2}
r=(10.139s)(1s)1+1.170s.
R¯(μi,μv)=0.25ω0H(μi)H(μv)μi+μv,
H(μ)=1+2μ1+2μ1ω0.
ω0=ba+bωtr=btra+btr=1s2.
R¯(μi,μv)=0.25(1s2)H(μi)H(μv)μi+μv
H(μ)=1+2μ1+2μs.
Rp(μi)=0.5(1s2)(1+2μi)s(1+2μis)×{1+(1s)ln(1+2s)2s(2μis1)+μis(2μi1)[lnμiln(1+μi)]2μis1}.
R¯(μi,μv)=0.25ω0[p(θ)+H(μi)H(μv)1]μi+μv
H(μ)={1μω0[r0+(0.5μr0)ln(1+1/μ)]}1,r0=11ω01+1ω0.
Rp(μi)=0.5ω0{[p(θ)1][1+μilnμiln(1+μi)]+H(μi)01H(μv)μvμi+μvdμv}.
p(θ)=1g2(12gcosθ+g2)1.5.
B=1g2g(1+g1+g21).
g=14.440B+12.11B223.87B3+23.52B49.317B5,
xj=(2j+1)gj.
r=1+KSKS(KS+2),r=ln(1+ξ)ξln(1ξ)+ξ,ωtr=2ξln[(1+ξ)/(1ξ)],
η=a/K=1atωtr=0,η=0.5atωtr=1,andχ=btr/S=4/3atωtr=1.
Rc(μi)=IrIi=Ii,dir(μi)Rp(μi)+Ii,difrIi=(1dE)Rp(μi)+dEr.

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