A COMPARISON OF TWO STREAM APPROXIMATION FOR THE DISCRETE ORDINATE. METHOD AND THE SOS METHOD. Weizhen Hou1,3 Qiu Yin2 Hua ...
A COMPARISON OF TWO STREAM APPROXIMATION FOR THE DISCRETE ORDINATE METHOD AND THE SOS METHOD Weizhen Hou1,3 Qiu Yin2 Hua Xu1,3 Li Li1 Zhenghua Chen1 1. Institute of Remote Sensing Applications Chinese Academy of Sciences, Beijing 100101, China; 2. Shanghai Center for Satellite Remote Sensing Applications, Shanghai 201100, China; 3. Graduate School of Chinese Academy of Sciences, Beijing 100049, China ABSTRACT The two-stream discrete ordinates method and the twostream successive orders of scattering method are compared, and the key features of two methods are discussed. Based on the convergence characteristics of successive scattering, we use a semi-empirical model to improve the computing efficiency in the SOS method. Using the delta-M method, we investigate the effect of the two two-stream method for non-absorbing and absorbing case respectively. The 32stream DISORT is used as the benchmark for assessments of the relative accuracy of the two methods investigated. With the comparisons for the accuracy of flux, the results of the two two-stream methods are almost the same in general and the absorbing media lead to larger errors of flux compared with that of the non-absorbing case. Index Terms—Approximation methods, differential equations, planetary atmospheres, remote sensing, scattering 1. INTRODUCTION The radiative transfer process is key issue for the fundamentals of atmospheric radiation. The physical process of radiative transfer is described by a differentialintegral equation. The azimuthally independent radiative transfer equation is Z 1 dI (W , P ) P I (W , P ) ³ I (W , P ' )P ( P , P ' )d P ' dW 2 1 (1) Z W F0 P ( P , P0 ) exp( ) P0 4S where I (W , P ) is the azimuthally averaged intensity at optical depth W and for the direction of the cosine of the zenith angle P , P0 is the cosine of the solar zenith angle, Z is the single scattering albedo, F0 is the incident solar
irrandiance, and P( P , P ' ) is the azimuthally averaged phase Thanks for funding by Important National Science & Technology Specific Projects of 2009ZX07527-006-2 and National Key Technology R&D Program of 2008BAC34B02
978-1-4244-9566-5/10/$26.00 ©2010 IEEE
3580
function. Due to the complexity of this equation, it is generally solved numerically rather than analytically. Among the available approximation methods, the two stream approximation is the most feasible in a computationally efficient manner. The two-stream approximation is one the simplest techniques used to solve the radiative transfer equation. This method divides the multiple scattering contributions into two components, and then the upward and downward fluxes are calculated by solving two coupled differential equations [1]. In the next section the discrete ordinates method is discussed. In section 3, the convergence characteristics of successive scattering are discussed and a semi-empirical model is used to reduce the computational burden of the two-stream SOS method. In section 4, the accuracy of the two two-stream methods is analyzed and compared. Finally, we present our conclusions in section5. 2. THE DISCRETE ORDINATES METHOD
The discrete ordinates method is one of the most frequently used techniques to solve the radiative transfer equation. It was originally developed by Chandrasekhar to study the transfer of radiation in planetary atmospheres, and has become a useful tool in calculating radiation in atmospheres containing aerosols and clouds. This method solves the raditive transfer equation via the discretization of the basic equation, representing it as a set of 2N coupled differential equations. They are solved using Eigen analysis [2]. The discrete ordinate method for radiative transfer in vertically inhomogeneous layered media, as embodied in a FORTRAN computer code, is called DISORT. In particular, a multiple scattering code which uses only two streams, to be henceforth called TWOSTR. DISORT and TWOSTR, which are extremely well tested and documented, are available for download from the NASA Goddard website ftp://climate1.gsfc.nasa.gov/wiscombe/Multiple_Scatt/. In the two-stream approximation, the azimuthally independent phase function is expanded to two terms P ( P , P ' ) 1 3 g PP ' (2) and the single node Gaussian quadrature is use to decompose an integration as
IGARSS 2010
Fig. 1. Intensity distribution along optical depth of each scattering ( P0 0.9 , g 0.75 , Z 1 ).
³
1
1
with P1
f ( P )d P
f ( P1 ) f ( P1 )
Fig. 2. Ratio of I n / I n 1 as a function of scattering order at different optical depths ( g 0.75 , Z 1 ).
(3)
3 / 3 . Hence, we can obtain two equations for
I I (W , P1 ) and I I (W , P1 ) corresponding to the two stream radiative intensities in up and down hemispheres:
dI (W )
P1
dW dI (W )
I (W ) a I (W ) a I (W ) S exp(
W P0
P1
dW
I (W ) a I (W ) a I (W ) S exp(
I (W , P )
)
W P0
number of sub-layers for an optically thick medium, particularly when the single scattering albedo tends to 1 [3]. The solution of radiative transfer for the SOS method is expressed as a summation of contributions of all successive orders of scattering N
¦I
n
(W , P )
(5)
n 1
(4) )
where ar Z (1 r g ) / 2 , S r F0Z (1 r 3g P1 P0 ) / 4S . Then, analytic solutions to the two equations can be obtained.
3. THE SOS METHOD The successive order of scattering (SOS) method, by computing the contribution of each scattering order, is physically straightforward and is easier to analyze the importance and characteristics of scattering-absorption processes. Since the SOS method traces the photons for each scattering event, the inhomogeneous structure of the medium as well as gaseous absorption process can be incorporated in terms of integration along the photon path. However, substantial computational burden due to a slow convergence with a high single scattering albedo and a large
3581
Here, N is the scattering number and I n / I H , H is a given precision value. In the two-stream approximation, P r 3 / 3 , as the direction of the two stream radiative intensities in up and down hemispheres. For the conservative scattering case of the single scattering albedo Z 1 , we investigate the effect of the two-stream SOS methods by the delta-M method in the homogenous layer, which has a Henyey-Greenstein phase function with the asymmetry factor g 0.75 [4]. For each scattering order, we compute the results of intensity distributions upward at the top of the atmosphere (TOA) and downward at surface, using the two-stream SOS method with a small incremental optical depth of 0.025. Fig. 1 shows the intensity distribution along optical depth of each scattering while P0 0.9 . Fig. 2 shows the ratio of I n / I n 1 as a function of scattering order at different optical depths. From Fig. 2, the ratio of two successive scattering radiances asymptotically approaches a constant. The fundamental reason is that photons lose the memory of their origin after
Fig. 3. Value of b ( g
0.75 , Z 1 ).
Fig. 6. Relative errors of two-stream discrete ordinates method for the flux ( g 0.75 , Z 1 ). Fig. 4. Comparison of N and Nc ( g
0.75 , Z 1 ).
where b I Nc / I Nc 1 . Fig. 3 shows value of b as a function of optical depths. Fig. 4 shows the comparison of N and Nc , where N is the scattering number of H 1% and Nc is the scattering order when the ratio of I n / I n 1 approaches a constant. Therefore, by using the semi-empirical model, the computational time can be significantly reduced without calculating higher order of scattering.
4. COMPARISON RESULTS
Fig. 5. Comparison flux upward at TOA and down at surface ( P0 0.9 , g 0.75 , Z 1 ). several scattering events and thus the radiation field asymptotically reaches a diffusion equilibrium state. Therefore, to accelerate the convergence to reduce the computational burden of the SOS method, a semi-empirical model can be used [3]. The summation of Eq. (5) can be truncated at the nth order of scattering when the ratio approaches a constant. The reminder can be replaced by a geometric series as I (W , P )
Nc 1
¦I
n
(W , P ) I Nc (W , P ) / (1 b)
(6)
n 1
3582
To analyze the accuracy of the two two-stream methods, we consider the case of anisotropic scattering by delta-M method, using a Henyey-Greenstein phase function with the asymmetry factor g 0.75 . We use the code TWOSTR to compute the flux as the results of the two-stream discrete ordinates method. Meanwhile, we use 32-stream DISORT as the benchmark model to compute the flux upward at TOA and downward at surface. For the conservative scattering case of the single scattering albedo Z 1 without absorption, the comparison of flux is shown in Fig. 5, where Fd means the flux down at surface and Fu means the flux up at surface. Further, Fig. 6 shows the relative errors of two-stream discrete ordinates method, and Fig. 7 shows the relative errors of two-stream SOS method. Compare Fig. 7 with Fig. 6, we can see that though the results shown in Fig. 8 differ slightly from those
Fig. 7. Relative errors of two-stream SOS method for the flux ( g 0.75 , Z 1 ).
Fig. 8. Relative errors of two-stream discrete ordinates method for the flux ( g 0.75 , Z 0.9 ).
in Fig.7, the results of the two two-stream methods are almost the same in general. For the flux downward at surface, the results are generally more accurate at smaller solar zenith angles. When the cosine of the solar zenith angle P0 is bigger than 0.3, the relative error is smaller than 10%, particularly in the region of P0 between 0.4 and 1, the relative error is smaller than 5%. Besides, the thicker of the optical depth, the smaller of the relative error. For the flux upward at TOA, the results are generally more accurate in the region of thicker optical depths. When the optical depth is bigger than 2, the relative error is smaller than 10%, especially in the region of optical depth between 8 and 14, the relative error is smaller than 1%. Y 0.9 is used as an example of non-conservative scattering case with absorption, as well as g 0.75 . The relative errors of the two-stream discrete ordinates method are illustrated in Fig. 8. The relative errors of two-stream SOS method are not shown. Just as in the case of conservative scattering, the results of the two two-stream methods are almost the same in general. Fig. 8 shows that the absorbing media lead to larger errors of flux in the most of the region, compared with that of the non-absorbing case.
In this study, we make a comparison of two stream approximation for the discrete ordinate method and the successive orders of scattering method. Based on the convergence characteristics of successive scattering, we use a semi-empirical model to improve the computing efficiency in the SOS method. The delta-M method is also used in the two methods. With the comparisons for the accuracy of flux, the results of the two two-stream methods are almost the same in general and the absorbing media lead to larger errors of flux compared with that of the non-absorbing case.
6. REFERENCES [1] P. Lu, H. Hang and J.N. Li, “A comparison of two-stream DISORT and Eddington radiative transfer schemes in a real atmospheric profile”, JQSRT, vol. 110, pp. 129-138, 2009. [2] K. Stamnes, S.C. Tsay, W. Wiscombe and K. Jayaweera, “Numerically stable algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media”, Appl. Opt., vol. 27, pp. 2502-2509, 1988. [3] M.Z. Duan and Q.L. Min, “A semi-analytic technique to speed up successive order of scattering model for optically thick media”, JQSRT, vol. 95, pp. 21-32, 2005. [4] W.J. Wiscombe, “The delta-M method: Rapid yet accurate radiative flux calculations for strongly asymmetric phase functions”, J. Atmos. Sci., vol. 34, no. 9, pp. l408-l422, 1977.
5. CONCLUSION
3583
