Validation Report of the Vector Radiative Transfer Model MOMO

Transcription

1 Validation Report of the Vector Radiative Transfer Model MOMO André Hollstein, Jürgen Fischer, René Preusker December 2, 2010 Abstract The radiative transfer model MOMO is able to calculate the vector light field in a plane parallel atmosphere ocean system (AOS) bounded by a rough (wind blown) surface. The vector version of this model is based on the former scalar version of MOMO[1]. In this document we describe our efforts to validate the new version of the model. We describe our internal physical consistency checks, comparison with published tables and a comparison with other radiative transfer models. Contents 1 Validation Fourier Expansion of Phase Matrices Validation Using Tables Vijay et. al Kokhanovsky et. al Conservation of Flux Other Models Conclusion 12 1 Validation 1.1 Fourier Expansion of Phase Matrices The scattering function for two given directions µ s and µ o can be calculated by rotating the scattering matrix Z(θ) so that the incident and reflected Stokes vectors lie in the plane of scattering. These rotation can be expressed by multiplying Z with the two rotation matrices L: Z (µ s, µ o, ϕ) = L (π σ s,o ) Z (θ) L ( σ o,s ). (1) The Fourier coefficient matrix Z can then be derived by expanding Z in a Fourier series: Z(µ s, µ o, i) = π 0 ( 12 cos (iϕ) 1 dϕz(µ s, µ o, ϕ) 2 sin (iϕ) 1 2 sin (iϕ) 1 2 cos (iϕ) With the following definition of the rotation angles and rotation matrices: 1 ). (2)

2 cos (σ s,o ) = µ s µ o cos(θ) 1 µo sin(theta) (3) cos(θ) = µ s µ o + 1 µ s 2 1 µ 2 o cos(ϕ) (4) L (µ s, µ o ) = 0 cos(2α) sin(2α) 0 0 sin(2α) cos(2α) 0. (5) The Rayleigh phase matrix is due to its simple analytic form especially suited for the validation of this part of MOMO. We use two independent approaches to verify the accuracy of the program. First we derive analytic expressions of the Fourier coefficients for special cases and secondly we compare the results of our program with an independent implementation in Mathematica. The Rayleigh phase matrix without depolarization is defined as: ( 3 cos 2 (θ) + 1 ) ( 3 cos 2 (θ) 1 ) 0 0 Z (θ) = ( 4 cos 2 (θ) 1 ) ( 3 cos 2 (θ) + 1 ) cos(θ) cos(θ) 2 For the analytic approach we use the Z 2,2 element of the Fourier coefficient matrix. This element leads to simple analytic expressions and is also affected by the rotation matrices L. We will discuss two scattering cases: Z 2,2 (µ, ±µ, i) = π 0 dϕ 3 (1 4 cos(iϕ) 4µ 2 + µ 4 + cos(ϕ) (±2µ 2 2µ 4 + cos(ϕ) ( 1 + µ 2) )) 2 These integrals can be solved analytically and lead to the following results: { 9 Z(µ, ±µ, i = 1, 2, 3) = 4 π ( µ 2 1 ) 2 3 (, 2 πµ2 µ 2 1 ), 3 8 π ( µ ) } 2 These expressions can be evaluated to any numerical precision and have been used to verify the preprocessor and its numeric precision. The numeric precession is mainly limited by the number of sampling points of the phase matrices for the numeric evaluation of the Fourier integrals. The Rayleigh phase function behaves nicely with respect to numeric integration hence the the accuracy shown here represents an lower bound to the accuracy. For real world applications phase matrices are given in form of numeric tables and the true values of the Fourier coefficients is in principle not known. In Table (1) we show the mean difference and mean quotient of the results of out program and the true values. The results are shown for three sampling point numbers ranging from 101 to These results show that the agreement between the numerical obtained results is very well and tend to vanish when we are increasing the number of sampling points. Other cases have been tested with independent integration methods in Mathematica and have lead to similar results. This makes us very confident that our preprocessing program delivers the right numbers. (6) (7) (8) 2

3 sampling points mean difference mean quotient Table 1: Mean differences and mean quotient of the Fourier expansion program and the analytic truth. The mean difference decreases with increasing sampling points and the mean quotient approaches unity. 1.2 Validation Using Tables Tables containing numeric values of the polarized radiation field for special cases have been published by several authors[2, 3, 4, 5, 6, 7, 8]. In this study we will use the publications from Natraj et al.. and Kokhanovsky et al.. to validate the scattering part our model. First we compare Stokes vector tables for pure Rayleigh scattering for a set of different optical thicknesses and surface albedos. Secondly we compare light fields calculated with SCIATRAN[9] for scattering by two given scattering matrices over a black surface Vijay et. al The tables for Rayleigh scattering given by Natraj et al.[7] have been calculated using a numerically more stable approach to the solution of the X and Y functions given by Chandrasekhar and Coulsen[10, 2]. We have chosen these tables since the author claims that these tables are accurate to eight decimal places and they are freely available from the authors website. The range of viewing angles in the tables is from 0 to 90. The covered solar positions range from 0 to 90. Azimuth positions range from 0 to 180. Surface albedos from 0.0 to 1.0 and Rayleigh optical depth ranging from 0.02 to 1.0. The intermediate values are given in Table (2). zenith angles: 0.02, 0.06, 0.1, 0.16, 0.2, 0.28, 0.32, 0.4, 0.52, 0.64, 0.72, 0.84, 0.92, 0.96, 0.98, 1. solar positions:.1,.2,.4,.6,.8,.92, 1 azimuth angles: 0, 30, 60, 90, 120, 150, 180 optical depth: 0.02,0.05,0.1,0.25,1.0 surface albedos: 0.,0.25,0.8 Table 2: Resolution of the Rayleigh tables. The main result of our comparison is shown in Fig. (1). In panel (a) we show absolute differences of every table entry and MOMO results for every viewing angle µ as a gray line. The black line indicates the mean value over all cases. The black dashed line represents the mean over all values. In panel (b) we show the same graph but for significant differences. We define the significant difference of two numbers a and b as the difference of the significant in Language Independent Arithmetic of a and b divided by ten to the power of the exponent of a. The resulting value indicates the number of (nonzero) equal digits of the two numbers. An over view over some explicit cases is given in Fig. (2). We can conclude that our model agrees very well with the tables. Since the zenith angles of our model are defined at the Gauss Lobatto quadrature angles we had to interpolate our 3

4 absolute differences s i, i 1,2,3 Μ φ Ω Τ (a) Absolute differences of the published tables and MOMO results. significant differences s i, i 1,2,3 Μ φ Ω Τ Μ in (b) Significant differences of the published tables and MOMO results. Significant differences indicates the number of equal digits of the compared numbers. Figure 1: Difference of MOMO calculations and Vijay tables[7]. All available cases are plotted in gray. An zenith resolved mean in black and the overall mean as number on the right scale. results to the angles given in the table. For the calculations we used 95 angles and a linear interpolation scheme to the angles given in the table. The absolute difference is in the same order as the accuracy of the Fourier expansion with the given azimuth resolution. The biggest deviations in intensity are found for viewing directions close to the horizon which is probably due to the lower density in Gaussian points in this region. 4

5 Sun at 66 MOMO Tables I rel Sun at 66 Tables O (a) Upward directed intensity at optical thickness τ = 0.25 for different ground albedos. (b) Downward directed intensity for a black surface and varying optical thickness. MOMO Tables p Angel in principle plane Sun at 66 MOMO Tables Ρ 0. Τ 1. Τ 0.5 Τ 0.25 Τ 0.15 Τ 0.1 Τ 0.05 Τ 0.02 (c) Upward directed degree of polarization at optical thickness τ = 0.25 for different ground albedos. (d) Downward directed degree of polarization for a black surface and varying optical thickness. Sun at p MOMO Tables 0.00 Angel in principle plane Ρ 0.8 Τ 1. φ 0 φ 30 φ 60 φ 90 φ 120 φ 150 φ 180 (e) Downward directed degree of polarization for optical thickness of one and ground albedo of ρ = 0.8 for varying azimuth angle. Figure 2: Overview of comparisons of radiance and degree of polarization results from MOMO calculations and Rayleigh tables[7]. 5

6 1.2.2 Kokhanovsky et. al In a paper from 2010 Kokhanovsky et al.[8] have published some benchmark results from their vector radiative transfer models. In the publication several authors compare the results of their programs for three cases which each other. Tabulated results from SCIATRAN are available at the website of Alexander Kokhanovsky. All three cases involve a purely scattering homogeneous layer over a black surface. As scaterers they have chosen the Rayleigh-, an aerosol- and a cloud phase matrix. Here we show the results for the two last cases since the Rayleigh case was already considered in Sec. (1.2.1). The phase matrices under consideration are shown in Fig. (3). Both phase functions are strongly peaked in the forward direction. The other phase matrix elements are feature rich and hence useful to test the accuracy of the models for given resolutions Aerosol Cloud m 1 m (a) M 1 and M 3 component m 5 m 6 Aerosol 0.10 Cloud (b) M 5 and M 6 component. Figure 3: Given benchmark phase matrices from Kokhanovsky et al..[8]. The results of MOMO calculations for the two cases together with the SCIATRAN results are shown in Fig. (5). For the two cases we show the upward- and downward directed radiance (I) and also the degree of polarization. This is done for three azimuth values (0,90,180 ). Curves for different azimuth values are shown with different gray values. The SCIATRAN results are indicated with symbols (square, circle, diamond) and the MOMO results are shown with lines. The dashed lines represent cases in which the phase functions have been truncated. In general our computations also agree well with the results from SCIATRAN. The results using the truncated phase matrices also agree reasonable well since they do not (as expected) reproduce the forward scattering peak. In Fig. (4) we show the mean absolute differences in logarithmic scale for the computations 6

7 1 Benchmark phase matrix: aerosol rayleigh cloud Truncated pase matrix: Original phase matrix: mean difference of Sciatran and MOMO for the I components (a) Intensity differences with logarithmic scale mean difference of Sciatran and MOMO for Q,U,V components (b) Q,U and V component differences with logarithmic scale. Figure 4: Differences of MOMO calculations and SCIATRAN results from[8]. with original (lines) and truncated (dashed lines) results. Concluding this comparison we can state that our results agree with the published tables for cases with more complex scattering functions. We have also shown the effects of the phase function truncation. These results can not reproduce the intensity in the forward scattering direction but can reproduce the components Q,U,V in this region. Away from the forward scattering region the SCIATRAN results can be reproduced with less computational effort. 7

8 I I I azimuth angle: Sciatran result: Original phase matrix: Truncated phase matrix: (a) Upward directed radiance for the aerosol case azimuth angle: Sciatran result: Original phase matrix: Truncated phase matrix: upwelling radiance for cloud case Μ in (b) Upward directed radiance for the cloud case. downwelling radiance for cloud case Μ in (c) Downward directed radiance for the aerosol case. (d) Downward directed radiance for the cloud case. 0.2 DOP upwelling degree of polarization for aerosol case azimuth angle: Sciatran result: Original phase matrix: Truncated phase matrix: Μ in (e) Upward directed degree of polarization for the aerosol case. DOP upwelling degree of polarization for cloud case azimuth angle: Sciatran result: Original phase matrix: Truncated phase matrix: (f) Upward directed degree of polarization for the cloud case. Μ in 0.06 upwelling degree of polarization for cloud case (g) Downward directed degree of polarization for the aerosol case. DOP (h) Downward directed degree of polarization for the cloud case. Figure 5: In detail comparisons of MOMO and SCIATRAN results. Μ in 8

9 1.3 Conservation of Flux Conservative scattering is an energy conserving process and we can test weather our model deviates from this assumption. To investigate only the effects of scattering we calculated a number of test cases where the direct solar source is only present in the first layer. Hence the upper layer is a source of diffuse radiation that can propagate trough the system. With no internal sources and no absorption present the outward directed fluxes from a single layer must balance the inward directed fluxes from the neighbour layers. If this is not the case the layer would be a source or a sink for flux in the system: ( ) ( ) ɛ = F (τ n ) + F (τ n ) F (τ n 1 ) + F (τ n+1 ) (9) }{{}}{{} =:F out F in δ = F out F in 1 2 (F (10) out + F in ) The deficiency value ɛ should be small for all layers and can serve as a proxy for the accuracy of the program for a given resolution. For our test we use the value of δ which describes the ratio of the deficiency to the mean of the two fluxes. To test the Fourier expansion we calculate the fluxes from the zenith resolved light field at a layer boundary: F, (τ) = dµdφµl, (τ, µ, φ) (11) It is of special interest to test the atmosphere ocean interface since it is computed by an external program. We use our standard atmosphere ocean test case but deactivate the source term for all but the first layer. For layers with only molecular scattering δ is very well under 1% when choosing 30 atmospheric Gaussian points and 36 Fourier terms (for the ocean interface) and azimuthal resolved output at every 30. For the interface and the same resolution we get deficiencies below 3%. For the given zenith and azimuth resolution we found this to be a very good and reasonable agreement. 1.4 Other Models We have validated the scattering part of the model using published tables. To completely validate our model we would also need such trusted tables for well defined, possible simple, atmosphere ocean systems. To our best knowledge these haven t been published yet 1. We can test the implementation of our reflection matrix using models without ocean but such an interface as lower boundary. For this test we use the scalar COART model[12, 13] and the 6S[14, 15] model. The COART model can be operated trough an web interface 2 and and 6S can be downloaded from the homepage 3 of the authors. As test case we use 545nm and a wind speed of 7m/s. Solar zenith angle is and relative refractive index of We also compute the case for a black surface to emphasize the effects of the ocean surface. At 545nm the Rayleigh optical thickness is so we also could use the Rayleigh tables results to verify the normalization conditions of the three programs. For direct comparisons with the 1 an statement also given by Zhai et al. in 2010[11]

10 Rayleigh tables the MOMO results must be normalized with π and the 6S reflectances with µ s. The COART output is given in absolute Radiances and have been normalized to match the zenith reflectance of the 6S results. In Fig. (6) we show the top of atmosphere reflectance results from the models for the case with and without interface. The MOMO Rayleigh (black line) and the 6S Rayleigh (gray with circles) agree with each other. The results of COART (gray,small dashed), 6S (gray large dashed) and MOMO (black with diamonds) agree in their characteristics but show some signifficant numeric deviations. reflectance I MOMO AOS MOMO Ray 6S 6S Ray COART Figure 6: Comparison of the model reflectances. Solar zenith angle is 23.07, relative refractive index is 1.34 and wind speed is 7m/s. The vertical lines indicate real and opposite position of the sun. In Fig. (7) we show the top of atmosphere degree of polarization for MOMO and 6S. The Rayleigh result for MOMO (black line) and 6S (gray line) show some significant deviations. The relative difference is approximately and almost constant for all viewing directions. We found the same behavior for other solar angles. By now we don t know the origin of this deviation but will further investigate it. Since the Rayleigh results don t agree with each other we don t expect that the polarization results do. The two curves show differences that cant not be explained with a constant factor. In Fig. (8) we show a plot where we corrected the 6S results. The Rayleigh results agree after this correction but the results for the case with surface still differ. In the direction of the sun glint the MOMO results are higher that 6S. In the opposite direction the MOMO results are higher as well but the curvature of the results seem to have the wrong sign. We will have to make more research on this comparison to fully explain our results. 10

11 dop MOMO AOS MOMO Ray 6S AOS 6S Ray Figure 7: Like Fig. (6) but for the degree of polarization. dop MOMO AOS MOMO Ray 6S AOS 6S Ray Μ in Figure 8: Like Fig. (7) but with with the corrected values for 6S results. 11

12 2 Conclusion We have used published results for scattering with and without a Lambertian reflector at ground level. Partly these external results have been computed using independent methods or at least independent implementations. The comparison efforts have shown that our results and those published by other authors agree very well with each other. The validation of the sea surface is not yet final since we found deviations we still have to explain. Since no second combined atmosphere-ocean radiative transfer model which includes clear water Raman scattering is known to us, the validation of the Raman module is performed with azimuthally averaged results from the ocean model HydroLight. 12

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