Combined Henyey–Greenstein and Rayleigh phase function

Quanhua Liu and Fuzhong Weng

The phase function is an important parameter that affects the distribution of scattered radiation. InRayleigh scattering, a scatterer is approximated by a dipole, and its phase function is analytically relatedto the scattering angle. For the Henyey–Greenstein (HG) approximation, the phase function preserves onlythe correct asymmetry factor (i.e., the first moment), which is essentially important for anisotropic scat-tering. When the HG function is applied to small particles, it produces a significant error in radiance. Inaddition, the HG function is applied only for an intensity radiative transfer. We develop a combined HG andRayleigh (HG–Rayleigh) phase function. The HG phase function plays the role of modulator extending theapplication of the Rayleigh phase function for small asymmetry scattering. The HG–Rayleigh phase func-tion guarantees the correct asymmetry factor and is valid for a polarization radiative transfer. It approachesthe Rayleigh phase function for small particles. Thus the HG–Rayleigh phase function has wider applica-tions for both intensity and polarimetric radiative transfers. For microwave radiative transfer modeling inthis study, the largest errors in the brightness temperature calculations for weak asymmetry scattering aregenerally below 0.02 K by using the HG–Rayleigh phase function. The errors can be much larger, in the 1–3K range, if the Rayleigh and HG functions are applied separately. © 2006 Optical Society of America

OCIS codes: 010.0010, 010.1310, 200.0200.

1. Introduction

In satellite retrieval systems1,2 and data assimilationprocesses,3,4 accurate and computationally efficient ra-diative transfer models are needed for radiances andassociated derivatives (i.e., Jacobians). For radiativetransfer in a scattering and absorbing atmosphere, thephase function plays an important role5 by affectingthe mainly angular distribution of scattered radiances.For small scatterers, the Rayleigh6 phase function iswidely applied in intensity and polarization calcula-tions. It is normally valid for molecular scattering invisible and ultraviolet wavelength ranges and for non-precipitation cloud scatterings at low frequencies ofthe microwave range. The Henyey–Greenstein7 (HG)phase function is often applied in visible and infraredranges in which the asymmetry factor (i.e., the firstmoment) of the phase function represents the asym-metry forward and backward scattering. In general,the actual phase function differs from the Rayleigh and

HG phase functions. We propose an analytical phasefunction and investigate its accuracy for microwavebrightness temperature calculations.

2. Radiative Transfer Model

The radiative transfer model used here is the ad-vanced doubling-adding (ADA) method.8 The ADAmodel applied the exact analytical expression for thesource function, which much simplified the originaldoubling-adding (DA) method.9 Comparisons amongthe DA, vectorized discrete-ordinate radiative trans-fer (VDISORT),10 and the ADA within the commu-nity radiative transfer model (CRTM) frameworkshowed that all three models achieved the same ac-curacy. The ADA was implemented into the CRTMdeveloped at the Joint Center for Satellite Data As-similation (JCSDA), Camp Springs, Maryland. Tofurther take into account the residual polarizationfrom atmospheric scattering, the ADA needs to beextended for the polarization effect. The extension iscarried out by using a 2 � 2 phase matrix calculatedby general spherical harmonic functions. Use of thegeneral spherical harmonic functions avoids time-consuming integration over azimuthal angles whenthe phase matrix is multiplied by rotational matri-ces.11

3. Henyey–Greenstein–Rayleigh Phase Function

In this study we develop an analytical phase functioncalled the HG and Rayleigh phase function for small

Q. Liu (Quanhua.Liu@noaa.gov) is with the Joint Center for Sat-ellite Data Assimilation, Camp Springs, Maryland 20746, and QSSGroup, Incorporated, Camp Springs Maryland 20746. F. Weng iswith the National Oceanic and Atmospheric Administration�National Satellite Data and Information Service, Office of Researchand Applications, Camp Springs, Maryland 20746.

Received 24 February 2006; revised 20 April 2006; accepted 17May 2006; posted 18 May 2006 (Doc. ID 68439).

0003-6935/06/287475-05$15.00/0© 2006 Optical Society of America

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asymmetry scattering. The HG–Rayleigh phase func-tion is a normalized product of the Rayleigh phasefunction and the HG phase function with a modifiedasymmetry factor (G). The modified asymmetry fac-tor is related to the original asymmetry factor (g) ofthe detailed phase function through the followingderivations. For intensity radiative transfer, the HG–Rayleigh phase function can be written as

P��, g� � C Rayleigh���HG��, G�g��

�C38 �1 � cos2 ��

�12

1 � G2�g��1 � G2�g� � 2G�g�cos ��3�2. (1)

By using a normalization condition

�0

�

P��, g�sin �d� � 1, (2)

and preserving the first moment

�0

�

cos �P��, g�sin �d� � g, (3)

it gives

C �4

2 � G2, (4)

G �59 g � 3�1

2 �109 g �

250729 g3� ��

� 3��� �12 �10

9 g �250729 g3, (5)

where

� � 12 ��

109 g �

250729 g3�2

� 13 �4 �

2527 g2�3

. (6)

For a fast radiative transfer calculation, the phasefunction is often reproduced from a few Legendreexpansions. Using the relationship of Legendre func-tions,

�2n � 1�cos �Pn � �n � 1�Pn�1 � nPn�1, (7)

the HG–Rayleigh phase function can also be ex-panded as

P��, g� �4

2 � G2

38 �1 � cos2 ��HG��, G�

�3

4 � 2G2 �n�0

n�n � 1�2n � 1 Gn�2

��n � 2��n � 1�

2n � 3 Gn�2 ��n � 1�2

2n � 3 Gn

�5n2 � 12n � 1 Gn�Pn�cos ��. (8)

The modified asymmetry factor G can be calculatedfrom the original asymmetry factor g in Eqs. (5) and(6). Figure 1 shows g � G against g. It may be note-worthy to point out that G is an asymmetry factorused in the HG part of the HG–Rayleigh func-tion rather than the asymmetry factor of theHG–Rayleigh function. The reason for the difference(g � G) is that the angular distribution of theHG–Rayleigh function is different from the distribu-tion of the HG function.

For polarimetric radiation, the HG–Rayleigh scat-tering matrix can be written as

P��, g� �4

2 � G2

12

1 � G2�g��1 � G2�g� � 2G�g�cos ��3�2

38

� 1 � cos2 � �1 � cos2 � 0 0

�1 � cos2 � 1 � cos2 � 0 00 0 cos � 00 0 0 cos �

�(9)

for the Stokes vector �I, Q, U, V�t. The phase matrixis then the product of the scattering matrix and as-sociated rotational matrices.

4. Result

This section compares the Rayleigh, HG, and HG–Rayleigh functions with the phase functions computedwith the finite-difference time domain method.12 Thelatter phase functions were provided by the Liougroup at the University of California at Los Anglesand are referred to as original phase functions here-after. In the Liou et al.13 studies, the phase functionat 183 GHz was derived for randomly oriented icecrystals with a maximum size of 300 �m and a widthof 100 �m. The volume equivalent radius of a sphereis 90 �m and the ratio of the particle size to themicrowave length is small. Although the particle sizeis larger than the mean particle size of ice clouds, it istypical in a microwave application since microwaveradiation is sensitive to large ice particles. The valueof the asymmetry factor of the phase function is

Fig. 1. Difference between the original (g) and modified (G) asym-metry factor.

7476 APPLIED OPTICS � Vol. 45, No. 28 � 1 October 2006

0.027, which represents a weak asymmetry scatter-ing. It is sufficient to use four streams for scatteringin the radiative transfer calculation. As shown in Fig.2(a), the phase element for the intensity (see theblack curve) has larger forward but smaller backwardparts in comparison with Rayleigh scattering (see theyellow curve). The original phase function is substan-tially different from the HG phase function for thesame asymmetry factor (see the red curve). This in-dicates again that the HG phase function is not ap-propriate for weak asymmetry scattering. By usingthe modified G from Eq. (5), the HG–Rayleigh can becomputed from Eq. (1). The HG–Rayleigh phase func-tion (see the blue line) agrees extremely well with theoriginal phase function. The phase element for linearpolarization is given in Fig. 2(b). The curves of theoriginal, the Rayleigh, and the HG–Rayleigh phasefunctions in Fig. 2(b) are almost the same. The resultusing the HG–Rayleigh phase function is very en-couraging.

Microwave brightness temperature simulationswere performed for the Special Sensor Microwave Im-ager Sounder (SSMIS) and NOAA-18 Advanced Micro-wave Sounder Unit A (AMSUA)/Microwave HumiditySounder (MHS) channels by using the Rayleigh, HG,HG–Rayleigh, and original phase functions, respec-

tively. NOAA-18 was successfully launched on 20 May2005. The SSMIS onboard the Defense MeteorologicalSatellite Program (DMSP) F16 satellite was success-fully launched on 18 October 2003. It was the firstconical scanning sensor with both imager and sounderchannels on the same platform. In the following nu-merical experiment, we chose the first atmosphericprofile from the 52 Europeon Center for MediumRange Weather Forecasting (ECMWF) profiles used inthe optical path transmittance (OPTRAN) study. Aone-layer rain cloud was located near 600 hPa. Thetotal water content of the cloud was 1 mm. The reasonfor choosing the rain cloud is that rain clouds producesignificant residual polarization. We used the phasefunctions of Liou et al.13 to investigate the resultingerror from using the HG–Rayleigh, Rayleigh, and HGphase functions. A Lambertian surface with an emis-sivity value of unity is used. The local zenith anglewas 53° for the conical scan sensor SSMIS and 0° forthe cross scan sensor AMSUA�MHS. Figures 2(c) and2(d) show the differences in the brightness tempera-tures at the top of the atmosphere using the HG–Rayleigh (blue curve), the Rayleigh (yellow curve),and the HG (red curve) in comparison with using theoriginal phase function. The errors for those channelssensitive to the atmosphere above the cloud are neg-

Fig. 2. (a) Comparisons of the phase function, (b) linear polarization of the phase function, (c) difference in the brightness temperaturefor NOAA-18 AMSUA�MHS channels and (d) differences for SSMIS using the original (black curve), the HG–Rayleigh (blue curve), theRayleigh (yellow curve), and the HG phase functions (red curve).

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ligible. The maximum errors are approximately 0.02,1.1, and 3.2 K for the HG–Rayleigh, Rayleigh, andHG phase functions, respectively. The oscillation ofthe error from using the HG phase function in Fig.2(d) is due to the fact that the HG phase function isunpolarized and by using the HG phase function thecomputed vertically polarized brightness tempera-ture is underestimated while the horizontally polar-ized brightness temperature is overestimated. Theerrors basically reflect the difference of their phaseelement for the intensity to the original value. Theerrors in the polarization difference from using theHG–Rayleigh and the Rayleigh are very small. Thegood agreement of the polarization differences usingthe approximated phase functions is because thephase element for the linear polarization agrees well[see Fig. 2(b)].

Since the HG–Rayleigh phase function is a modu-lated Rayleigh phase function, it departs from thereal phase function for increasing asymmetry scat-tering. We investigated a moderate asymmetry scat-tering. The phase function was again computed at183 GHz using the finite-difference time domainmethod12 for a larger ice crystal with a volume equiv-alent radius of 276 �m. The asymmetry factor was0.271, ten times larger than the value (0.027) of theprevious case. Eight streams were used for the scat-tering in the radiative transfer calculation. The HG–Rayleigh phase function again agreed with theoriginal phase function. The Rayleigh phase functionwas significantly different from the original phasefunction. The HG phase function was relatively closeto the original phase function in comparison to thethe Rayleigh phase function. The differences in

phase functions were also reflected in the bright-ness temperature calculation for the same atmo-spheric profile used in the above. Table 1 shows thecalculated brightness temperature for NOAA-18AMSUA�MHS channels using the original, HG–Rayleigh, Rayleigh, and HG phase functions. Themaximum errors are approximately about 0.7, 11,and 1.5 K for using the HG–Rayleigh, Rayleigh, andHG phase functions, respectively. Table 2 shows asimilar comparison for SSMIS channels. The maxi-mum error using the HG phase function increaseswhen it is applied to SSMIS polarized channels (e.g.,channels 12–18 in Table 2).

5. Discussion

The proposed analytical phase function is applicablefor radiance and polarization simulations for smallasymmetry scattering. The HG–Rayleigh phase func-tions can reproduce accurate phase elements for bothintensity and linear polarization. Comparisons forthe intensity, polarization difference, and the phasefunctions themselves show that the HG–Rayleighphase function achieves the best agreement with theoriginal values. The analytical HG–Rayleigh phasefunctions may be used in theoretical and educationalstudies in radiative transfer.

We have shown that the HG–Rayleigh phase func-tion is applicable for small and moderate asymmetryfactors. It can be used in most microwave radiativetransfer calculations. The error in the brightnesstemperature calculation could increase with an in-

Table 1. Comparisons of the Brightness Temperaturesa

AMSUA�MHS Original HG–Ray Rayleigh HG

1 255.389 254.748 245.808 256.8522 250.562 249.835 239.350 252.2523 252.028 251.421 243.038 253.6944 255.206 254.876 250.585 256.1795 252.088 251.951 250.218 252.5236 240.981 240.958 240.670 241.0667 229.276 229.273 229.230 229.2898 218.134 218.134 218.130 218.1359 207.062 207.062 207.062 207.062

10 213.336 213.336 213.336 213.33611 223.611 223.611 223.611 223.61112 232.424 232.424 232.424 232.42413 240.966 240.966 240.966 240.96614 251.866 251.866 251.866 251.86615 250.961 250.258 240.272 252.69416 250.957 250.254 240.267 252.69017 255.049 254.448 246.222 256.65918 258.250 258.250 258.250 258.25019 264.221 264.212 264.100 264.25120 264.660 264.459 261.937 265.202

aCalculated using the original phase function (original),the HG–Rayleigh function (HG–Ray), the Rayleigh function(Rayleigh), and the HG function for NOAA-18 AMSUA�MHSsensors.

Table 2. Comparisons of the Brightness Temperaturesa

SSMIS Original HG–Ray Rayleigh HG

1 247.585 247.680 241.087 246.10102 251.267 251.250 248.593 250.3033 244.588 244.569 243.847 244.2344 231.130 231.127 231.058 231.0935 211.190 211.190 211.190 211.1906 208.775 208.775 208.775 208.7757 218.934 218.934 218.934 218.9348 246.398 246.353 239.389 249.1249 262.792 262.788 261.284 263.442

10 261.894 261.894 261.865 261.90911 256.183 256.183 256.183 256.18312 250.969 250.928 244.210 253.16713 254.433 254.578 247.708 253.16714 253.011 253.136 246.244 251.61815 238.084 238.017 229.038 241.07916 242.769 242.999 233.768 241.07917 245.355 245.541 237.074 243.69818 240.760 240.700 232.377 243.69819 248.013 248.013 248.013 248.01320 238.729 238.729 238.729 238.72921 247.819 247.819 247.819 247.81922 256.000 256.000 256.000 256.00023 241.307 241.307 241.307 241.30724 231.341 231.341 231.341 231.341

aCalculated using the original phase function (Original), theHG–Rayleigh function (HG–Ray), the Rayleigh function (Ray-leigh), and the HG function for SSMIS sensors.

7478 APPLIED OPTICS � Vol. 45, No. 28 � 1 October 2006

crease of the asymmetry factor. For strong forwardscattering, delta truncation to remove the forwardpeak is necessary.

The authors thank the Liou group at the Univer-sity of California, Los Angeles for providing the orig-inal phase functions. The authors also thank JerrySullivan for his valuable suggestions and corrections.This study was funded at the Joint Center for Satel-lite Data Assimilation. The views expressed in thispublication are those of the authors and do not nec-essarily represent those of the National Oceanic andAtmospheric Administration.

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