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
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 [1–13] 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  and his followers [15–17]. 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 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 [9,18–20]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  as follows: 22–24]: (1) at the single-scattering albedo ; (2) at ; (3) for ; (4) ; and (5) .
Here , , and 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 , may be at close to 1 [23,24].
B. Spherical Albedo
The spherical albedo 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]3) it follows that 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 and may also be applied for : (1) at the ; (2) at ; (3) for ; (4) ; and (5) .
3. Calculation Methods and Numerical Results
A. Scattering Phase Functions
Three different scattering phase functions (Fig. 1) have been used in our study for modeling reflective and transmitted properties of the layer:
We used two optical properties for characterization of ; namely, scattering asymmetry parameter and backscattering probability as follows: (1) , (the with such parameters corresponding to the case of the balance between the forward and back scattering); (2) , (process of forward scattering is prevailing); (3) , (process of backscattering is almost negligible compared to forward scattering). The 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 and particles (relative to medium) of refractive indices (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  computed for the same values of as selected , but for values ensuring a maximal closness to the selected .
Accuracy of the approximated models was evaluated by computing the mean absolute percentage error (MAPE) and the normalized (to the standard deviation ) root-mean-square error (NRMSE):
B. Plane Albedo Modeling
Seven various approximations for derived by a number of authors and three approximations derived currently for the first time were examined for their accuracy (Table 3, Figs. 2–5). We used for computations three different described above. The values of 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 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 ; namely, , , and , and only one value of . A reason for the last choice is a rather flat dependence , excluding the range of where is the rainbow scattering angle. The reasonable range for the values [27–29] is from 114° to 124° at the selected values of refractive indices (Table 2). Therefore, is approximately the middle of the zone free from the rainbow effect. This value corresponds to 42.8° for the angle of incidence from the air to water (with refractive index ) and 49.6° for the incidence angle from the air to glass (), which are typical values for many optical applications. Such choice of parameters allows obtaining some conclusions for the very different scattering media.
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)  and Sokoletsky et al. (2009) . This method has been verified previously  for two very distinct scattering phase functions (the Rayleigh and Fournier–Forand–Mobley phase functions with and , 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 by the computation for the spherical albedo . Taking into account that dependences of are strictly monotonic, and the “effective” angles at which lie generally in the range from 48° to 61° , a procedure of calculation may be presented as18,31]: 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  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  expression obtained for the azimuthally averaged reflectance factor in the case of the isotropic scattering: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 by the transport (reduced) single-scattering albedo and then by the similarity parameter as follows: 13) and (14) into Eq. (2) and taking the definite integral:
3. Hapke (Hapke 2) Method
This method uses the in the form derived by Hapke :36] for anisotropic scattering with any : 16) and (17) into Eq. (2) leads to expression for the : 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 , , and . As shown for calculations carried out by different models, the best result demonstrates the HKS 2 model at ; however, at smaller values of , the best result was obtained by the GKS model (Fig. 3). The HKS 2 model has been first derived by Kokhanovsky and Sokoletsky  as a modification of Hapke  solution. This modification may be reduced to two subsequent steps: (1) the replacement by the according to Hulst’s [18,31] similarity rule and (2) using the additional multiplicative factor to improve accuracy of the model. The GKS model for the plane albedo was derived by Kokhanovsky and Sokoletsky  from the Gordon  and Golubitsky et al.  quasi-single-scattering approximation obtained initially as a solution for the reflectance factor.
Another finding is that the is rather a function of the Gordon’s parameter (Fig. 4) than of the transport albedo [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 and are known, then it is easy to find an asymmetry parameter (see Table 1) and then to estimate a backscattering contribution (for a given scattering phase function) and . To show better the impact of on , we used an alternative absciss axis with values (Figs. 4 and 5) for demonstrating the absolute values of and relative errors for several models. Note that since the GKS and HKS 2 models give values of almost independent on selected scattering phase functions, we have shown here the values of obtained only for one phase function; namely, with (Fig. 4).
Overall, among all considered approximations, the best results show the GKS approximation at and HKS 2 approximation at all other values of (see Fig. 5). Combined application of these two models gives a relative error .
5. Plane Albedo Modeling Under Lack of Information
In real practice, a scattering phase function is often inaccessible. This is also means inaccessibility of information about the , , and parameters. However, it is possible to estimate (and, hence, and ) from measured values. Below we suggest an algorithm for the and estimation and test for the impact of this inaccuracy in on the accuracy of estimated .
For current modeling, we selected the widely applied Henyey–Greenstein :1) leading to  2). The errors for the computed for the same models that were used for plotting Fig. 6 at and three values of taken from the Table 2 are shown in Fig. 6 as a function of parameter . As it follows from the computations, the inexact values of and lead typically to small decreases in the values (up to 12% and 10% for GKS and HKS 2 models, respectively) comparative to the values computed at exact (known) values of . However, an accuracy of the models under consideration is still high. The HKS 2 model is obviously better than the other models at any and , while at the best results yield a GKS model. Combined application of these two models (or only the HKS 2 model) gives a relative error (Fig. 6) that is even better than an error obtained for the case of known and .
C. Spherical Albedo Modeling
Table 4 and Fig. 7 show various radiative transfer approximations for the spherical albedo derived by a number of authors. The separate values of , their ranges, and scattering phase functions were selected the same as were used for the computations. Again, all models were compared with the numerical calculations (IIM). The results show that the transport albedo (or, alternatively, Hulst’s similarity parameter ) better related to than Gordon’s parameter . A logical explanation of this fact may be found on p. 211 by van de Hulst . Hulst also gave numerical confirmation of this fact for the wide ranges of (from 0.2 to 0.99) and (from 0 to 7/8). Our work confirms this fact again.
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, , 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 () and scattering () of the two-flux GKM theory via well-established IOPs: , , and .23) is a classical GKM, while two others were derived from the Chandrasekhar–Klier equations with Hulst’s replacement: (CKS model) to the better accounting of a scattering anisotropy. The boundary conditions [Eq. (24)] were derived by Yudovsky and Pilon , 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 and polynomial values that maximally satisfy Eqs. (23) and (24). A full solution presented in the Table 4 yields 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 (or ). Contrarily, the GKM-new model has a solution at any values of or . Overall, among all considered approximations, the GSK approximation demonstrated the best results at (or ), and the Hulst approximation is the best at the other values of or (see Fig. 8). Combined application of two these models yields a relative error at any values of parameters. However, taking into account the relative complexity of the GSK model, application of only Hulst’s model (with ) 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 , while Hulst’s model has been developed by van de Hulst .
1. Spherical Albedo Modeling Under Lack of Information
This is the case similar to that for the plane albedo (Subsection 3.B.5). Again, we replaced the initial by the Henyey–Greenstein [Eq. (19)] and the values by their calculated values [Eq. (21)] in the cases when knowledge of and/or was necessary for model computations. Results show that, similarly to the calculations, inaccurate values lead typically to a small decrease in the values (up to 11%) comparative to the values computed at known (exact) values of (Fig. 9). As before, Hulst’s model demonstrates superior results in most cases with the relative errors . Thus we recommend to use this spherical albedo model in the case, if the precise values of is unknown.
D. Diffuse Reflectance Under Combined Illumination
Let us consider now a case of combined (direct and diffuse) illumination with the diffuse irradiance contribution into the total () irradiance. Then the reflectance may be easily determined as follows:
A large number of different approximate analytic models for plane and spherical albedo (specifically, 10 and 28 for and , 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  and confirmed again, among spherical albedo approximations, the Hulst and the models are the most accurate ones with the errors NRMSE 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 is safely governed by only one parameter, namely, (or ) (Fig. 7).
A much more difficult situation is with plane albedo modeling. This property is governed by three optical parameters: , (or ), and , though this also may be expressed by only two parameters: and (Fig. 2) or and (Fig. 4). The best result here is demonstrated in the HKS 2 and Gordon’s model with the NRMSE and 34%, respectively (Table 3). However, using a replacement method [Eqs. (6) to (9)] at which a calculation of the plane albedo replaced by the calculation of the spherical albedo also seems to be a perspective. The replacement method yields good results only at strong scattering (i.e., close to 1) in its current form. Probably, the more complicate (and accurate) approximation for and 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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