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Numerical Heat Transfer, Part B, 62: 203 222, 2012 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7790 print=1521-0626 online DOI: 10.1080/10407790.2012.709163 PHASE-FUNCTION NORMALIZATION IN THE 3-D DISCRETE-ORDINATES SOLUTION OF RADIATIVE TRANSFER PART I: CONSERVATION OF SCATTERED ENERGY AND ASYMMETRY FACTOR Brian Hunter and Zhixiong Guo Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey, USA The conditions for which conversation of scattered energy and phase-function asymmetry factor after discrete-ordinates methods (DOM) directional discretization for 3-D radiative transfer in anisotropic scattering media breaks down are examined. Directional discretization in anisotropic scattering media is found to alter the scattering asymmetry factor a second-type of false scattering. Phase-function normalization which conserves scattered energy alone cannot correct this problem, and conservation of the asymmetry factor is simultaneously required. A normalization technique developed by the authors, which was successfully tested in 2-D asymmetric cylindrical-coordinate radiative transfer analysis, is intensively examined and validated with benchmark problems in 3-D Cartesian coordinates. In Part I of this study, the degree of anisotropy for which normalization is necessary to conserve these inherent quantities is presented for various phase-function approximations and discrete quadrature sets. INTRODUCTION Accurate yet efficient modeling of radiative heat transfer has become increasingly demanded [1 12]. Processes requiring precise and complete solutions to the equation of radiative transfer (ERT) range from traditional high-temperature combustion modeling and material thermal processing [3 7] to ultrafast interactions of light with biological tissue for therapeutic and imaging applications [8 12]. Due to the integro-differential nature, the ERT is extremely difficult to solve analytically. As a result, various numerical methods have been fashioned to provide accurate solutions to the ERT. One of the most commonly implemented numerical methods for determining radiative transfer via solution of the ERT is the S N discrete-ordinates method (DOM). Fiveland [13] determined steady-state radiative transfer in 2-D rectangular enclosures using the DOM. Further works by Fiveland [14] and Truelove [15] extended the method to determine radiative transfer in 3-D enclosures containing Received 20 May 2012; accepted 21 June 2012. Address correspondence to Zhixiong Guo, Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, USA. E-mail: guo@ jove.rutgers.edu 203

204 B. HUNTER AND Z. GUO NOMENCLATURE normalization coefficients c speed of light in medium g asymmetry factor I radiative intensity, (W=m 2 sr) M total number of discrete directions N number of terms in Legendre approximation r position vector ^s unit direction vector w discrete direction weight h radiation direction polar angle, deg H scattering angle, deg m, g, n direction cosines r a absorption coefficient, m 1 scattering coefficient, m 1 A l0 l r s / radiation direction azimuthal angle, deg U scattering phase function ~U normalized scattering phase function x scattering albedo [¼r s =(r a þ r s )] Subscripts b blackbody HG henyey-greenstein L legendre LA linear-anisotropic w boundary wall Superscripts B ballistic component l, l 0 discrete direction index l 0 l from direction l 0 into direction l absorbing, emitting, and scattering media. In order to accurately determine ultrafast radiative transfer in participating media due to the propagation of laser light, Guo and Kumar pioneered the solution of the transient ERT using the DOM, applying it to both 2- [16] and 3-D [17] geometries containing participating media. Guo and co-authors further expanded the use of the DOM for determining ultrafast radiative transfer in 2-D axisymmetric cylindrical coordinates [10], as well as using Duhamel s superposition theorem to accurately represent various pulses [7] or pulse train irradiation [18]. In most practical participating media, radiation scattering is anisotropic. This is especially true for turbid media in nature, such as biological tissues, where scattering can be highly anisotropic. When scattering is anisotropic, the directional discretization of the ERT using numerical methods, such as the DOM, can result in a lack of conservation of scattered energy in the system. In order to maintain scattered-energy conservation, phase-function normalization techniques were implemented [19, 20]. However, work by Boulet et al. [21] indicated that while these techniques do indeed accurately conserve scattered energy, they greatly distort both the phase-function shape and the overall phase-function asymmetry factor in highly anisotropic scattering media, which results in greatly distorted radiative predictions compared to benchmark Monte Carlo simulations. In order to correct this issue, Hunter and Guo [22 24] developed a new technique for phase-function normalization, which was devised to conserve both scattered energy and the overall phase-function asymmetry factor after directional discretization. This technique was applied to the DOM [22, 24] and the finite-volume method (FVM) [23] in a 2-D, axisymmetric cylindrical medium. However, it is important to investigate the impact of the new technique in 3-D computations, as many processes cannot be simplified to a 2-D case. Further, benchmark comparisons with other methods such as Monte Carlo are required to validate the approach and examine the conditions where it is necessary.

3-D PHASE-FUNCTION NORMALIZATION PART I 205 In Part I of this study, the impacts of phase-function normalization on the conservation of scattered energy as well as the preservation of phase-function asymmetry factor and overall shape are discussed for various phase-function approximations in 3-D discrete-ordinates solution of the ERT. Deviations in the conservation of scattered energy and alterations in scattering effect after various orders of directional discretization when no phase-function normalization is implemented are presented, in detail, to determine when normalization is necessary. In addition, discrepancies in the discretized phase-function asymmetry factor after application of both the previously published normalization technique and the new technique of Hunter and Guo are analyzed. Results are presented for various DOM quadrature sets to investigate the impact of the number of discrete directions. Finally, the impact of the normalization technique on the conservation of out-scattered energy and the asymmetry factor when ballistic incidence is present is examined. In Part II of this study, the impact of proper phase-function normalization on both steady-state and ultrafast radiative transfer solutions in highly anisotropic scattering media will be investigated, and benchmark comparisons with Monte Carlo simulations will be performed. DISCRETIZATION OF THE ERT The time-dependent ERT of diffuse gray radiation intensity I can be written as follows [1]: 1 c qiðr;^s; tþ qt þ ^s rir;^s; ð t Þ ¼ ðr a þ r s ÞIr;^s; ð tþþr a I b ðr; tþ þ r Z s 4p 4p Ir;^s ð 0 ; tþuð^s 0 ;^sþ dx 0 ð1þ where the terms on the left-hand side represent both the temporal and spatial gradients of radiative intensity; the first term on the right-hand side accounts for intensity attenuation due to absorption and scattering, the second term accounts for contribution from blackbody emission, and the third term represents intensity augmentation due to in-scattering of radiative energy from direction ^s 0 into ^s. Using the TDOM, Eq. (1) can be expanded for a general 3-D enclosure, in Cartesian coordinates, into a set of simultaneous partial differential equations in any discrete direction ^s l : 1 qi l l l qi qi þ ml þ gl c qt qx qy þ qi l nl qz ¼ ðr a þ r sm ÞI l þ S l l ¼ 1; 2;...; M ð2aþ 0 S l ¼ r a I b þ r s B @ 4p X M l 0 ¼1 l 0 6¼l w l0 U l0l I l0 þ X B r sm ¼ r s 1 1 4p wl U ll 1 I B U lb lc A ð2bþ ð2cþ

206 B. HUNTER AND Z. GUO where the in-scattering integral in Eq. (1) has been replaced by quadrature summations and M is the total number of discrete directions. In these equations, m ¼ sinh cos/, g ¼ sinh sin/, and n ¼ cosh are the direction cosines corresponding to the x, y, and z directions, respectively; and h and / are the polar and azimuthal angles defining radiation direction ^s l. The first summation term in Eq. (2b) corresponds to the in-scattering of diffuse radiative intensity. In this term, w l0 is the DOM quadrature weight corresponding to radiation direction ^s l0, and U l0l is the diffuse scattering phase function between directions ^s l0 and ^s l. The second summation represents the in-scattering of any ballistic radiation, which becomes crucial in applications involving irradiation of collimated=focused laser or solar incidence. In this term, I B is the radiative heat flux of the ballistic radiation for a given location, and U lb l is the ballistic scattering phase function between the direction of ballistic incidence ^s lb and discrete direction ^s l. In order to increase the computational efficiency of the DOM for strongly scattering media, Eq. (2a) follows the treatment of Chai et al. [25], in which the forward-scattering term is removed from the phase-function summation and treated as transmission. Using this treatment, a modified scattering coefficient r sm is defined by Eq. (2c). In general, the choice of quadrature scheme for the DOM is arbitrary. However, the chosen discrete directions and corresponding weighting factors should satisfy the zeroth, first, and second moments as described in standard texts [1, 2] and thus not repeated here. The most commonly implemented quadrature is the S N quadrature scheme, where the total number of discrete directions M ¼ N(N þ 2). Currently, the highest quadrature available in the literature is S 16. Phase-Function Normalization It is well recognized that the discretization of the continuous angular variation using the DOM must accurately satisfy the conservation of scattered energy in the system. This energy conservation is not guaranteed by the three moment constraints in the formation of a quadrature scheme if scattering is not isotropic, and thus an additional constraint must be implemented. This can be done via phase-function normalization, such that 1 X M ~U l0 l w l ¼ 1 l 0 ¼ 1; 2;...; M ð3aþ 4p l¼1 in which ~ U l0 l is the discrete value of the normalized phase function. For cases involving scattering of ballistic radiation, the following constraint must also be satisfied for any ballistic direction: 1 X M ~U lb l w l ¼ 1 ð3bþ 4p l¼1 It should be noticed that the ballistic direction ^s lb possibly may not coincide with any discrete direction in the DOM scheme used.

3-D PHASE-FUNCTION NORMALIZATION PART I 207 If scattering is isotropic, the conservation of scattered energy, as well as asymmetry factor, is automatic, due to the moment constraints satisfied by the DOM quadratures. As scattering transitions to an anisotropic nature, it is necessary to examine when and where the conservation of scattered energy and asymmetry factor breaks down. To ensure that scattered energy is conserved after directional discretization, one old technique described by Kim and Lee [19] is as follows: ~U l0 l ¼ U l 0l! 1 X M 1 U l0l w l 4p l¼1 ð4þ While the normalization via Eq. (4) was fashioned to explicitly conserve scattered energy, recent works [21 24] have shown that application of this approach significantly alters both the asymmetry factor and the overall shape of anisotropically scattering phase functions, leading to vast inaccuracies in radiative transfer predictions. For any given discrete direction ^s l0, the phase-function asymmetry factor g, i.e., the average cosine of the scattering angle, must be retained after directional discretization and normalization of phase function: 1 X M ~U l0 l cos H l 0l w l ¼ g l 0 ¼ 1; 2;...; M ð5aþ 4p l¼1 1 X M ~U lb l cos H l Bl w l ¼ g ð5bþ 4p l¼1 where H is the scattering angle. Equation (5b) is needed when ballistic incidence is presented. It is essential that the scattered energy and asymmetry factor conservation conditions of Eqs. (3) and (5) are accurately realized in order to attain accurate radiative transfer solutions. In order to correct the issues, Hunter and Guo [22] recently proposed a new normalization technique, in which both scattered energy and asymmetry factor conservation conditions are satisfied. The phase function is normalized in the following manner: ~U l0 l ¼ð1 þ A l 0l ÞU l0 l ð6þ where the normalization parameters A l0 l are determined such that ~U l0 l satisfies Eqs. (3a) and (5a) simultaneously. In addition, the normalized phase function satisfies directional symmetry, i.e., ~U l0 l ¼ ~U ll0. The linear system comprised of Eq. (6) and Eqs. (3a) and(5a) contains 2M equations and M(M þ 1)=2 unknowns, meaning that there are infinitely many solutions. The desired solution for the normalization parameters A l0 l is that which has the minimum norm, which can be determined using pseudo-inversion. Using this technique, the normalization parameters which will accurately conserve both scattered energy and the asymmetry factor simultaneously can be readily determined. As previously noted, for problems involving ballistic

208 B. HUNTER AND Z. GUO incidence, the ballistic scattering phase function U lbl must also satisfy both scattered energy and asymmetry factor conservation, i.e., Eqs. (3b) and (5b). However, normalization of the ballistic scattering phase function is independent of the normalization of the diffuse scattering phase function U l0l, except when the direction of ballistic incidence ^s lb corresponds directly to one of the discrete quadrature directions. Now, let s look back the physical reality of the modified scattering coefficient defined by Eq. (2c). Since it is an attenuation term in Eq. (2a), its value should never be negative; otherwise, it becomes a false source that augments intensity in the medium. Thus, a necessary condition using the non-normalized DOM is 1 1 4p wl U ll 0 l ¼ 1; 2;...; M ð7þ This condition can be rearranged as U ll 4p w l l ¼ 1; 2;...; M ð8þ This provides a necessary limiting condition for the value of the forward peak of the phase function based on the DOM quadrature weights. Furthermore, for lower-order quadrature (where the weighting factors are larger), the limiting condition will be highly restrictive. Mathematically, Eq. (7) is automatically satisfied when the conservation of the scattered energy constraint of Eq. (3a) is satisfied, i.e., after the application of phase-function normalization to conserve scattered energy. However, investigation of Eq. (8) will provide insight as to exactly when the problem becomes unrealistic, and when extra treatment must be implemented to eliminate the appearance of the false source in the discretized governing equations. The Mie scattering phase function U is generally a highly oscillatory function, which can be expressed as an infinite series of Legendre polynomials, as follows: UH ð Þ ¼ 1 þ X1 i¼1 C i P i ðcos HÞ where H is the scattering angle between radiation directions ^s 0 and ^s, and the coefficients C i are determined via Mie theory. While exact numerical implementation of the Mie phase function is possible, it is common to approximate the scattering phase function as a truncated Legendre series: ð9þ U L ðhþ ¼ 1 þ XN i¼1 C i P i ðcos HÞ ð10þ where N is the chosen order of approximation. Here we name such an approximate function as the Legendre phase function. For N ¼ 1, Eq. (10) reduces to the so-called linear-anisotropic (LA) approximation.

3-D PHASE-FUNCTION NORMALIZATION PART I 209 Another commonly implemented phase-function approximation, due to its ability to capture the strong forward-scattering peak inherent in highly anisotropic scattering media, is the Henyey-Greenstein (HG) phase-function approximation, whose analytic form is as follows: 1 g 2 U HG ðhþ ¼ ½1 þ g 2 2g cosðhþš 1:5 ð11þ IMPACT OF PHASE-FUNCTION NORMALIZATION Prior to investigating the direct impact of normalization on radiative transfer predictions (as will be presented in Part II of this study), it is essential to examine the necessity of the new normalization technique and where and when it should be implemented. An understanding of this will shed light on errors in DOM radiative transfer solutions when normalization is not properly applied. First, conservation of scattered energy after directional discretization with and without normalization for diffuse radiation is investigated to determine the necessity for further normalization for the HG, Legendre and LA phase-function approximations at various levels of prescribed asymmetry factor. Second, changes in asymmetry factor with and without normalization in diffuse radiation are examined to determine the importance of realizing the conservation of asymmetry factor constraint after directional discretization. Finally, the impact of phase function normalization on the scattering of the ballistic radiation is discussed. First, we examined the LA phase-function approximation. It was found that scattered energy and asymmetry factor were effectively conserved with no phasefunction normalization necessary, regardless of the order of directional quadrature (S 4, S 8, and S 12 ) or the value of asymmetry factor (g 1=3 for LA approximation in nature). This is due to the linear behavior, as the linear-anisotropic phase function does not exhibit the sharp forward-scattering peak or oscillatory behavior seen in the HG and Legendre approximations. Since no normalization was necessary, the details are omitted, for brevity. For a thorough examination of various types of scattering media, the HG asymmetry factor is varied in the range 0.1 g 0.995. For the Legendre case, seven different phase-function profiles are considered, with asymmetry factors g ¼ 0.4000, 0.4856, 0.6697, 0.7693, 0.8189, 0.8453, and 0.9273. The Mie coefficients C i for the 3-term g ¼ 0.4000, 9-term g ¼ 0.6697, and 13-term g ¼ 0.84534 expansions are presented by Kim and Lee [26], while the Mie coefficients for the 34-term g ¼ 0.4856, 36-term g ¼ 0.7693, 26-term g ¼ 0.8189, and 27-term g ¼ 0.9273 expansions are presented by Lee and Buckius [27]. Results are presented for three DOM quadratures sets: S 4, S 8, and S 12 (24, 80, and 168 total discrete directions). While DOM quadrature up to 288 discrete directions (S 16 ) is available in the literature, complex 3-D radiative problems usually require fewer discrete directions to achieve better computational efficiency. For example, the low-order S 4 quadrature set is commonly implemented to describe radiation processes in 3-D combustion furnaces. It was found that, in all cases, results generated with the S 16 quadrature differed minimally from

210 B. HUNTER AND Z. GUO those generated with the S 12 quadrature, so results for S 16 have been omitted from the following results, for brevity. Scattered Energy Conservation Before examining conservation of scattered energy, it is important to determine where the condition of Eq. (8) is violated, in order to provide a preliminary range where normalization is required to retain the physical meaning of the modified scattering. For the HG phase-function approximation, Eq. (8) can be rewritten as follows: 1 g 2 4p ð1 þ g 2 1:5 2gÞ maxðw l Þ Use of the maximum weighting factor for a given quadrature scheme will determine the maximum value of g for which the inequality will be satisfied. Table 1 examines the maximum value of HG asymmetry factor g below which Eq. (7) will be satisfied for various quadrature sets, ranging from S 2 (8 discrete directions) to S 16 (288 discrete directions). The maximum value of weighting factor w l,as well as the maximum possible value of diffuse phase function U ll, is also listed. In general, as the number of directions increases, the maximum value of the weighting factor decreases, leading to a larger maximum value of asymmetry factor where Eq. (12) will be satisfied. Examination of this condition provides a sufficient condition for phase-function normalization. It is important to restate that this condition will be automatically satisfied, regardless of DOM quadrature or asymmetry factor, when both the old and new normalization techniques are implemented. Figure 1 examines the conservation of scattered energy (represented by a unity value) for diffuse radiation versus prescribed HG phase-function asymmetry factor for the S 4, S 8, and S 12 quadrature sets without implementation of phase-function normalization. The appearance of three separate conservation values for S 8 and five values for S 12 in Figure 1 stems from the discrete direction weight w l, as the S 8 and S 12 quadratures have three and five distinct weighting factors, respectively. For the S 4 quadrature, all directional weights are equivalent. As the number of discrete directions increases, conservation is better achieved for higher values of asymmetry ð12þ Table 1. Examination of maximum HG asymmetry factor below which Eq. (7) is satisfied for various DOM quadratures S N No. of directions Max. weight factor Max. U 00 Max. g 2 8 1.570796 8.0000 0.5586 4 24 0.523599 24.000 0.7614 6 48 0.362647 34.652 0.7738 8 80 0.461718 27.217 0.7467 10 120 0.148395 84.682 0.8521 12 168 0.111154 113.05 0.8713 14 224 0.097659 128.68 0.8792 16 288 0.085065 147.73 0.8870

3-D PHASE-FUNCTION NORMALIZATION PART I 211 Figure 1. Examination of conservation of scattered energy versus prescribed HG phase-function asymmetry factor using DOM S 4, S 8, and S 12 quadratures without application of normalization (color figure available online). factor. As seen in the inset of Figure 1, scattered energy values start to deviate from unity quickly for the S 4 quadrature set, reaching differences of 0.97% and 1.77% for g ¼ 0.4 and 0.45. For the S 8 quadrature, deviations manifest slightly less rapidly, with a maximum deviation of 1.07% appearing for g ¼ 0.45. As the quadrature increases to S 12, the maximum deviation goes to 2.74% at g ¼ 0.75. For all three quadratures, deviations become extreme for highly anisotropic scattering, a common feature for turbid tissues. In general, normalization should be implemented for g 0.35, 0.40, and 0.60 for the S 4, S 8,andS 12 quadratures, respectively, in order to accurately conserve scattered energy. Comparing results to those in Table 1, it is seen that actual normalization is required for much lower values of g to conserve scattered energy. Figure 2 examines the conservation of scattered energy without application of normalization versus prescribed Legendre phase-function asymmetry factor. For all three quadratures, scattered-energy is conserved within 0.0001% for g ¼ 0.4000, and within 0.75% for g ¼ 0.6697. However, significant deviations in scattered-energy conservation are witnessed for the remaining Legendre phase functions. For the S 4 quadrature, deviations range from 64.26% for g ¼ 0.4856 to a maximum of 372.5% for g ¼ 0.8189. Results are similar for the S 8 quadrature set. When the quadrature is increased to S 12, accurate convergence is realized for g ¼ 0.8453, in addition to g ¼ 0.4000 and 0.6697, and a maximum deviation of 24.58% occurs for g ¼ 0.8189. For the Legendre phase function, errors in conservation of scattered energy do not follow the strictly increasing pattern observed for the HG phase function. The maximum error in prenormalized scattered energy occurs for g ¼ 0.8189 for all quadratures. In addition, the phase function with g ¼ 0.4856 displays a very large error.

212 B. HUNTER AND Z. GUO Figure 2. Examination of conservation of scattered energy versus prescribed Legendre phase-function asymmetry factor using DOM S 4, S 8, and S 12 quadratures without application of normalization (color figure available online). This phenomenon stems from both the exact shape of the Legendre phase-function and the number of terms in the expansion. For the HG phase function, the exact phase-function shape is tied directly to the phase-function asymmetry factor. For the Legendre phase function, both the number of terms and the magnitude of the higher-order-term coefficients have a large impact on the exact shape of the phase function, including the severity and number of oscillations present. Since the asymmetry factor is determined solely by the coefficient C 1 (g ¼ C 1 =3), two phase functions with identical asymmetry factors could have vastly different shapes and oscillatory behavior due to differences in the higher-order terms or the actual number of terms in the expansion. It is observed, in general, that the discrepancies for the 3-, 9-, and 13-term expansions are less than those seen for the more intricate expansions of 26-, 27-, and 34-terms. The results from Figures 1 and 2 indicate the necessity of phase function normalization to conserve scattered energy. Both the old normalization technique using Eq. (4) and the new technique using Eq. (6) are mathematically devised to accurately conserve scattered energy. Hence, focus is placed solely on asymmetry-factor conservation from this point forward. Asymmetry Factor Conservation Figure 3 examines the conservation of HG asymmetry factor versus prescribed asymmetry factor for the three quadrature sets without phase-function normalization. Conservation of the asymmetry factor was calculated by averaging the directional values of the summation in Eq. (5a), and then dividing the average by the

3-D PHASE-FUNCTION NORMALIZATION PART I 213 Figure 3. Examination of conservation of HG asymmetry factor using DOM S 4, S 8, and S 12 quadratures without application of normalization (color figure available online). prescribed asymmetry factor, where a ratio of unity indicates accurate conservation. For all quadratures, similar correlations to those seen for scattered-energy conservation in Figure 1 are observed. As g increases, there is a strict increase in the deviation from true conservation of the asymmetry factor, and a refinement of the angular quadrature results in reduction of conservation deviations at all prescribed asymmetry factors. For the S 4 quadrature, deviations in asymmetry-factor conservation manifest more quickly than deviations in scattered energy. For example, at g ¼ 0.40, the deviation from asymmetry-factor conservation is 3.29%, as opposed to the 0.97% difference seen for scattered energy. In addition, a visible discrepancy is seen for g ¼ 0.20, indicating that normalization is necessary for this asymmetry factor to accurately conserve the asymmetry factor. For the S 8 and S 12 quadratures, deviations of greater than 1% are first attained for g ¼ 0.60 and 0.75, respectively. As the asymmetry factor increases to g ¼ 0.90, extreme deviations in the discretized asymmetry factor are seen, with a deviation of 70.3% occurring for the S 12 quadrature. The breakdown of asymmetry-factor preservation in directional discretization alters substantially the scattering properties of the medium, inducing another kind of false scattering, which is a term originally used to describe numerical scattering diffusion due to spatial discretization [28]. Such false scattering is due to ray effect as well. Combining the results from Figures 1 and 3, normalization of HG phase functions should be implemented for g 0.20, 0.40, and 0.60 for the S 4, S 8, and S 12 quadratures, respectively, in order to accurately conserve both scattered energy and the asymmetry factor. Figures 4a and 4b examine the conservation of the prescribed asymmetry factor for the seven Legendre phase functions without application of normalization for the

214 B. HUNTER AND Z. GUO Figure 4. Examination of the conservation of Legendre asymmetry factor versus cosine of polar angle using DOM without application of normalization: (a) S 4 and (b) S 12 quadratures (color figure available online). S 4 and S 12 quadratures, respectively. Conservation ratios determined using Eq. (5a) are examined for each individual radiation direction ^s l0, and are plotted against the cosine of polar angle h. For the S 4 quadrature, changes in polar angle do not affect the conservation of the asymmetry factor, due to the fact that all directional weighting factors are identical. Accurate conservation is observed for the g ¼ 0.4000 expansion. Larger discrepancies occur for the remaining phase functions. The degree of deviation corresponds to the deviations in scattered energy seen in Figure 2. For the S 12 quadrature, conservation ratios show a dependence on the polar angle of the discrete direction, due to the fact that the discrete weighting factors w l0 vary with discrete direction angle for this quadrature set. The asymmetry factor is accurately conserved, for all polar angles, for the g ¼ 0.4000, 0.6697, and 0.8453 phase functions, conforming to the results seen in Figure 2 for scattered energy. The largest deviation occurs for g ¼ 0.8189, reaching a maximum of 30.76%. The results from Figures 2 and 4 indicate that normalization for the S 4 scheme is necessary to conserve both scattered energy and the asymmetry factor for all the phase functions except g ¼ 0.4000. For the S 12 quadrature, normalization is required for g ¼ 0.4856, 0.7693, 0.8189, and 0.9273. Results for the S 8 quadrature are not shown, but normalization is required for all phase functions except g ¼ 0.4000 and 0.6697 for this quadrature. Figures 5a and 5b examine the conservation of the HG and Legendre phasefunction asymmetry factors, respectively, after application of either the old normalization technique or the new technique. When the new normalization technique of Hunter and Guo is applied, the asymmetry factor is accurately conserved after directional discretization for all phase functions at any prescribed asymmetry factor for all three previously mentioned DOM quadrature sets. The angular false scattering can then be eliminated via the new normalization technique. When the old normalization technique is applied, however, it is seen that asymmetry factor is not accurately conserved after discretization for highly anisotropic scattering.

3-D PHASE-FUNCTION NORMALIZATION PART I 215 Figure 5. Comparison of discretized and prescribed asymmetry factors after application of either old or new normalization technique using DOM S 4, S 8, and S 12 quadratures: (a) HG phase function and (b) Legendre phase function (color figure available online). As seen in Figure 5a for the HG phase function, the discretized asymmetry factor overpredicts the prescribed value by 1.19% at g ¼ 0.30 for the S 4 quadrature using the old normalization technique, with the error in the asymmetry factor reaching 14.82% for g ¼ 0.80 (resulting in a discretized asymmetry factor of 0.9186). For the S 8 quadrature using the old normalization, an overprediction of 1% is not reached until g ¼ 0.70, with the maximum difference of 6.48% occurring for g ¼ 0.90. For the S 12 quadrature using the old normalization, a maximum difference of 4.24% is seen for g ¼ 0.90, with discrepancies first reaching 1% for g ¼ 0.80. As scattering becomes extremely anisotropic (g > 0.90), the percentage differences decrease steadily for all three quadratures using the old technique. This is due to the fact that while the discretized asymmetry factors overpredict prescribed values for all g, the discretized asymmetry factor is mathematically limited to a maximum of unity. Thus, as the prescribed asymmetry factor approaches unity, the percentage overprediction of the discretized value strongly decreases. While these differences may seem insignificant, radiative transfer calculations in strongly scattering media can be vastly affected by even minor changes in asymmetry factor when g is close to unity. This is because change in scattering effect is manifested in the difference in (1 g), according to the isotropic scaling law [29, 30]. We will see such an effect in Part II of this study. As seen in Figure 5b for the Legendre phase function, when the old normalization technique is applied, the asymmetry factor is effectively conserved for all DOM quadrature sets for g ¼ 0.4000. For g ¼ 0.6697, the asymmetry factor is effectively conserved within 0.03% for both S 8 and S 12, but a 1.29% difference (leading to a discretized value of 0.6784) is observed for S 4. The largest differences in discretized asymmetry factor for S 8 and S 12 occur for g ¼ 0.8189 (corresponding to the largest discrepancies in scattered-energy conservation), reaching values of 6.88% and 2.64%, respectively. Figure 6a and 6b plot discretized HG phase-function values versus scattering angle cosine for g ¼ 0.90 using the old and new normalization techniques,

216 B. HUNTER AND Z. GUO Figure 6. Comparison of discretized HG phase-function values versus cosine of scattering angle with prescribed g ¼ 0.9 using DOM S 4, S 8, and S 12 quadrature schemes after application of (a) old normalization technique and (b) new normalization technique (color figure available online). respectively. Prescribed values are included in each figure as a reference. When the old normalization technique is implemented, the phase-function profiles are drastically skewed in comparison with the prescribed values, corresponding to alteration of the asymmetry factor as seen in Figure 5a. For the S 4, S 8,andS 12 quadratures, the overall asymmetry factor is altered from g ¼ 0.90 to g ¼ 0.9894, 0.9583, and 0.9382, respectively, after the old normalization is applied. When the new normalization technique is applied, the shape of the prescribed HG phase function is accurately conserved for all quadratures, as seen in Figure 6b. Figures 7a and 7b plot discretized Legendre 26-term g ¼ 0.8189 phase-function values versus scattering-angle cosine using the old and new normalization Figure 7. Comparison of discretized Legendre phase-function value versus cosine of scattering angle with prescribed g ¼ 0.8189 using DOM S 4, S 8, and S 12 quadrature schemes after application of (a) old normalization technique and (b) new normalization technique (color figure available online).

3-D PHASE-FUNCTION NORMALIZATION PART I 217 techniques, respectively. As with the HG phase function, the discretized phasefunction values are significantly skewed when the old normalization technique is implemented. The overall asymmetry factor is altered from prescribed g ¼ 0.8189 to g ¼ 0.9647, 0.8753, and 0.8406 for the S 4, S 8,andS 12 quadratures, respectively. For lower quadratures, it is observed that not only is the overall asymmetry factor skewed, the limited number of directions is not able to fully capture the strong oscillatory nature of the Legendre phase function inherent from the higher-order terms in the expansion. As the quadrature increases, the shape is more accurately retained after normalization. This helps to explain the discrepancies in the conservation quantities seen in Figures 2 and 5b, as phase functions with strong oscillatory behavior require a higher quadrature to fully capture the exact behavior. When the new normalization technique is applied as shown in Figure 7b, the overall asymmetry factor is accurately conserved for all quadratures, although the exact behavior once again cannot be exactly represented by lower quadrature sets. Impact on Out-Scattering of Ballistic Incidence Normalization of the phase function of ballistic radiation scattering is independent of the normalization of the phase function for diffuse radiation, except in the case where the direction of ballistic incidence corresponds directly to one of the quadrature discrete directions. It is necessary to ensure that ballistic outscattered energy and the asymmetry factor are conserved accurately after directional discretization, in addition to the conservation of the diffuse counterpart. Figures 8a and 8b examine the conservation of ballistic out-scattered energy [Eq. (3b)] and asymmetry factor [(Eq. (5b)], respectively, versus the prescribed HG asymmetry-factor for the three quadrature sets without application of Figure 8. Examination of conservation of (a) ballistic out-scattered energy and (b) ballistic phase-function asymmetry factor versus prescribed HG asymmetry factor for various quadrature sets and ballistic polar angles (color figure available online).

218 B. HUNTER AND Z. GUO phase-function normalization. In addition, the direction of ballistic incidence is varied in order to gauge the effect of incident direction on scattered energy and asymmetry-factor conservation. While ballistic incidence does not have to be restricted to a single collimated direction (i.e., a cone of ballistic incidence), collimated incidence was investigated here in order to present the findings more clearly. The azimuthal angle of incidence / i is kept constant at 0, while the polar angle of incidence h i is taken as 0 (normal incident), 10, 30, 45, and 75. For the S 4 quadrature, significant deviations (1% or greater) in the conservation of ballistic out-scattered energy in Figure 8a start to appear for g 0.30, while deviations for the S 8 and S 12 quadratures are significant for g 0.45 and 0.75, respectively. As the asymmetry factor increases further, scattered energy becomes totally nonconserved for all quadratures and incident angles. In addition, the angle of ballistic incidence greatly affects the conservation of scattered energy. For example, for the S 4 quadrature at g ¼ 0.50, the deviation from energy conservation is 6.34% for h i ¼ 0, but only 0.152% for h i ¼ 30. For the S 8 quadrature at the same asymmetry factor, deviations are 1.70% and 0.0004% for the same two incident angles, respectively. The differences in out-scattered-energy conservation for varying ballistic incident directions stem from the approximation of the continuous angular variation with a discrete quadrature set. As the number of directions increases, the impact of ballistic incident direction decreases due to improved approximation of the continuous angular variation. As seen in Figure 8b for the S 4 with a prescribed asymmetry factor of g ¼ 0.15, deviations in asymmetry factor reach 1.30%, 1.11%, 0.076%, 0.330%, and 0.890% for h i ¼ 0, 15, 30, 45, and 75, respectively, showing again that the angle of incidence greatly affects the conservation value. For the S 8 and S 12 quadratures, maximum deviations of 1% or greater first appear for g 0.40 and g 0.75. Combining the results in Figures 8a and 8b, it is determined that normalization of the phase function of ballistic radiation is required for g 0.15, 0.40, and 0.75 for the three quadrature sets, respectively, when the HG phase function approximation is considered. When the old normalization technique is applied to the phase function of ballistic radiation, out-scattered energy is certainly conserved for all g. However, it is necessary to examine the conservation of the asymmetry factor. Figures 9a and 9b investigate the conservation of the ballistic asymmetry factor after applying the old normalization for varying prescribed HG asymmetry factors, quadrature sets, and ballistic incident angles. First, conservation was examined for a ballistic incident azimuthal angle of / i ¼ 0 in Figure 9a. Extreme deviations from asymmetry-factor conservation are witnessed for all quadrature sets and polar angles as scattering becomes more strongly anisotropic. For the S 4 quadrature, deviations of 1% are witnessed for g as small as 0.15. For S 8 and S 12, deviations of 1% and greater first appear for g ¼ 0.45 and 0.80, respectively. The impact of incident polar angle is large for lower-order quadratures. For S 4, deviations in asymmetry factor range from an overprediction of 5.51% to an underprediction of 13.2% for g ¼ 0.70, depending on the polar angle. As quadrature increases, these differences are reduced. For the results in Figure 9b, the azimuthal angle of incidence / i has been changed to 45. While similar trends are seen as compared to Figure 9a, the change in azimuthal angle has greatly affected asymmetry conservation at the various polar angles. The largest change can be seen for the S 12 quadrature. For g ¼ 0.90, the deviations from

3-D PHASE-FUNCTION NORMALIZATION PART I 219 Figure 9. Examination of conservation of ballistic phase-function asymmetry factor versus prescribed HG asymmetry factor for various quadratures and ballistic incident polar angles with azimuthal angles: (a) / i ¼ 0 and (b) / i ¼ 45 (color figure available online). conservation range from a 5.83% underprediction for h i ¼ 0 to a 3.55% overprediction at h i ¼ 10. Noticeable differences are also seen when comparing results for specific h i between the two azimuthal angles. For example, for the S 4, the discretized asymmetry factor for h i ¼ 75 underpredicts the prescribed value for all g for / i ¼ 45. However, the results are quite different for / i ¼ 0, in which the discretized asymmetry factor overpredicts the prescribed value for 0.10 g 0.80, and then underpredicts the prescribed value for g > 0.80. A further investigation of the impact of ballistic incident direction is presented in Figure 10, which depicts the percentage difference between discretized and prescribed phase-function asymmetry factor for the 26-term g ¼ 0.8189 phase function versus both the azimuthal and polar angles of ballistic incidence with the old normalization technique implemented for the S 8 quadrature. The polar (h i ) and azimuthal (/ i ) angles of ballistic incidence were varied between 0 h i, / i 90, as the results for the other octants are identical, due to DOM directional symmetry. The percentage difference between discretized and prescribed ballistic asymmetry factor ranges between 0.001% and 58.22%, depending on the exact direction of ballistic incidence, indicating that for certain directions of ballistic incidence, the asymmetry factor is effectively conserved after the old normalization is applied, while for other directions the skew is extreme. The average error in discretized ballistic asymmetry factor is 11.00%. For a fixed azimuthal angle at / ¼ 45, the percentage differences range between 0.185% at h ¼ 81.60 and 58.20% at h ¼ 33.70, showing that changes in azimuthal and polar angle can significantly alter the scattering properties of the ballistic incidence. As the quadrature decreases, the discrepancies in discretized and prescribed ballistic asymmetry factor increase over the range of polar and azimuthal angles, though the result is not shown here, for brevity. Conversely, as quadrature increases, the discrepancies are reduced and become more uniform.

220 B. HUNTER AND Z. GUO Figure 10. Percentage difference of discretized collimated Legendre phase-function asymmetry factor versus azimuthal and polar angles of collimated incidence with g ¼ 0.8189 using DOM S 8 quadrature (color figure available online). CONCLUSIONS In this study, the impact of a lack of conservation of phase-function asymmetry factor after directional discretization of the ERT using the DOM has been presented for various phase-function approximations. The following conclusions can be drawn: 1. Normalization is unnecessary for the LA phase function approximation, since it is not oscillatory. 2. Normalization must effectively conserve both scattered energy and the asymmetry factor. Lack of conservation of the asymmetry factor alters the scattering property and results in a second-type of false scattering. 3. For the HG phase-function approximation, errors in both scattered energy and asymmetry conservation increase as scattering becomes more strongly anisotropic. Normalization to accurately conserve these quantities simultaneously is required for g 0.20, 0.40, and 0.60 for the S 4, S 8, and S 12 quadratures, respectively. 4. For the Legendre phase-function approximation, the need for of normalization depends on the value of the asymmetry factor, the degree of oscillatory shape of the phase function, and the order of DOM quadrature. Normalization is required for highly anisotropic scattering. 5. The direction of ballistic incidence affects ballistic out-scattered energy and asymmetry-factor conservation due to the finite number of discrete directions, in particular for lower-order quadratures. The need for normalization of the phase-function for ballistic radiation observes the rule of phase-function normalization for diffuse radiation.

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