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Linearization of a pseudo-spherical vector radiative transfer model
Journal of Quantitative Spectroscopy & Radiative Transfer 85 (2004) 251 – 283
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Linearization of a pseudo-spherical vector radiative transfer model Holger H. Walter∗ , Jochen Landgraf, Otto P. Hasekamp National Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands Received 9 January 2003; accepted 7 May 2003
Abstract We present a linearization of pseudo-spherical vector radiative transfer using the forward-adjoint perturbation theory. The method, which is based on the usual forward and the adjoint formulation of radiative transfer, is extended to radiative transfer problems with internal radiation sources. This allows one to linearize vector radiative transfer in pseudo-spherical approximation. The model PS-LIRA is developed based on this approach. Here, the calculation of the direct beam as well as the integration along the line of sight are performed for a full spherical geometry. The multiply scattered radiation is calculated in a plane-parallel atmosphere. The proposed model provides the intensity vector as well as its derivatives at the top of the atmosphere for solar zenith angles (SZA) up to 90:0◦ and for o;-nadir viewing geometries ranging from limb to limb. In its scalar reduction the developed radiative transfer model is veri
1. Introduction Satellite measurements of backscattered sunlight play an important role in monitoring the chemical composition of the Earth’s atmosphere. For example, the Global Ozone Monitoring Experiment (GOME), launched in 1995 on board of ESA’s ERS-2 satellite, observes a range of atmospheric ∗
Corresponding author. E-mail address: [email protected] (H.H. Walter).
0022-4073/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0022-4073(03)00228-0
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trace constituents with an emphasis on global ozone distributions. GOME is a nadir viewing spectrometer that measures the solar radiation scattered by the atmosphere in the ultraviolet (UV) and visible spectral region (240 –790 nm). Likewise, the Solar Backscatter Ultraviolet (SBUV/2) instrument, currently on board NASA’s NOAA-16 polar orbiting satellite, measures the ultraviolet sunlight scattered by the Earth’s atmosphere at several wavelengths ranging from 252 to 340 nm. Measurements at the shortest eight wavelengths are used to estimate ozone vertical pro
(1)
Here, the N -dimensional state vector x represents the trace gas pro
where Oxk is the kth element of Ox = x − x0 . Substitution of Eq. (2) into Eq. (1) and neglecting the higher-order terms O(Ox2 ) yields the atmospheric state vector x iteratively using standard inversion techniques [1–4]. Thus, the retrieval of trace gas pro
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instruments like GOME and SCIAMACHY with a relatively narrow
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of the intensity vector at moderate SZA. In order to perform retrievals from measurements at low sun and for instruments with a wide
(3)
where T indicates the transposed vector. In the remainder of the paper we will use the notation Ii (i = 1; : : : ; 4) for the four Stokes parameters. Given a radiative transfer model which simulates the spectral intensity vector Itop at TOA in the viewing direction of the instrument, the transport of light from the entrance slit to the detector plane of the instrument may be described by the Mueller matrix approach (see e.g. [24]). The forward model at wavelength j that simulates the radiance measured by the jth detector pixel is given by F( j ) =
4
M1i ( j )Iitop ( j );
(4)
i=1
where the M1i are the corresponding elements of the 4 × 4 Mueller matrix M characterising the transmission properties of the instrument. Thus, the derivatives of the forward model in Eq. (2) can be calculated in a straightforward manner from the corresponding derivatives of the Stokes parameters Iitop with respect to the elements xk of the atmospheric state vector x, viz 4
@I top @F ( j ) = M1i ( j ) i ( j ): @xk @xk i=1
(5)
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Here we neglect any e;ect of the instrument slit and the sampling of the signal along the detector pixel. In this paper we present an e;ective approach to calculate the derivatives @Iitop =@xk for pseudo-spherical radiative transfer. 2.2. Linearized radiative transfer in a plane-parallel atmosphere The forward-adjoint perturbation theory, proposed by Marchuk [20] and later independently by Box et al. [21,25,26], provides an elegant and eScient tool to linearize atmospheric radiative transfer. This is demonstrated by several authors, e.g. Ustinov [27–29], Rozanov et al. [30], Landgraf et al. [22,31] and Hasekamp and Landgraf [23]. This section gives a brief overview of the method developed for horizontally homogeneous, plane-parallel atmospheres, for further details we refer to the papers themselves. For a linearization the RTE has to be solved in its forward and adjoint formulation. The forward formulation of polarized radiative transfer [23] in a plane-parallel atmosphere is given by the operator equation ˆ =S LI with the radiation source S, the total vector intensity
(6)
(7)
Here, E is the 4 × 4 unity matrix and e and s are the extinction and scattering coeScients, respectively. The altitude is denoted as z. = (; ’) describes the propagation direction of the radiation, given by the cosine of the zenith angle ( ¡ 0 for downward directed radiation and ¿ 0 for upward directed radiation) and the azimuth angle ’, where the azimuth angle is measured clockwise when looking downward. Furthermore, d ˜ = d ˜ d ’˜ and is the Dirac-delta function with ˜ ( − )=(− )(’− ˜ ’). ˜ represents the Heaviside step function. The
(8)
In the UV and visible part of the spectrum thermal emission can be neglected and the radiation source S is given by unpolarized sunlight that illuminates the TOA: S(z; ) = 0 (z − ztop )( − 0 )F0 e1 ;
(9)
where ztop is the height of the model atmosphere, 0 = (−0 ; ’0 ) indicates the direction of the solar beam and F0 represents the extraterrestrial Nux. ei (i = 1; : : : ; 4) is the unity vector in the direction of the ith component of the intensity vector.
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In addition to Eq. (6), the intensity vector has to ful
for ¡ 0;
I(zbot ; ) = [0; 0; 0; 0]T
for ¿ 0;
(10)
where zbot denotes the bottom of the atmosphere. The solution of Eq. (6) gives the intensity vector
(11)
where R is the corresponding response vector function [20,21]. The inner product of two arbitrary vector functions a and b is de
(13)
where the response function R acts as a source for the adjoint vector intensity
(15)
ˆ x0 : OE = −Ix†0 |OLI
(16)
with Here, the change of the transport operator OLˆ due to the perturbation in the atmospheric state is given by ˆ ˆ 0 ): OLˆ = L(x) − L(x
(17)
The source S and the response function R are not a;ected by the perturbation Ox. For the case that the state vectors x and x0 di;er only in their kth component by a small amount Oxk a comparison
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of Eq. (15) with a corresponding
(18)
In order to calculate the derivatives of the Stokes parameters Iitop in Eq. (5) we interprete those as four radiative e;ects Ei , de
(19)
with i = 1; : : : ; 4 and corresponding response functions Ri (z; ) = (z − ztop )( − v )ei ;
(20)
where v = (v ; ’v ) denotes the viewing direction of the instrument. This provides the expressions @Iitop 1 † ˆ x0 (x0 ) = − I (Ri )|OLI @xk Oxk x0
(21)
for the four derivatives. Here, Ix†0 (Ri ) is the solution of Eq. (13) with the adjoint source Ri . Thus, in order to calculate the derivatives we have to solve the forward radiative transfer problem as well as the adjoint transport problem for the four di;erent adjoint sources Ri . Due to the analytical form of Eq. (21) the perturbation theory approach combines an accurate calculation of the derivatives with low computational costs compared to a
(22)
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The direct part of the intensity
(23)
where the radiation source S is given by Eq. (9) and the transport operator corresponds to the streaming term in Eq. (7), @ ˆ + e (z) E ◦ : Ldir = (24) @z The direct intensity
(25)
with the plane-parallel transmission function T (z; ) = e− s (z; ) ;
(26)
where s (z; ) describes the slant optical depth at altitude z and in direction in the model atmosphere: 1 ztop d z e (z ): (27) s (z; ) = z The transport equation (6) can be rewritten in a corresponding RTE for the di;use intensity
(28)
with the di;use radiation source Sdi; (z; ) =
s (z) A Z(z; 0 ; )T (z; 0 )F0 e1 + (z)0 T (z; 0 )()||F0 e1 : 4
(29)
Here, the
(30)
with the response function 1 s (z) T AT Z (z; v ; )T (z; v )ei + (z)T (z; v )(−)||i1 ei : R˜ i (z; ) = v 4
(31)
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In this case the inner product in Eq. (30) represents the line of sight integration of the scattering source function s (z) T d Z(z; v ; )I(z; ) (32) Ji (z; v ) = e 4 i 4 for the ith Stokes parameter. The second term in Eq. (31) describes the contribution of light reNected at a Lambertian surface, which only takes place for the
(33)
where I˜† is the corresponding adjoint vector intensity
(34)
ˆ di; ; x0 + I˜†x |OSdi; + OR˜ i |Idi; ; x0 + OR˜ i |Idir; x0 + R˜ i; x0 |OIdir ; OEi = −I˜†x0 |OLI 0
(35)
with
where the subscript x0 denotes the unperturbed quantity. (See also Appendix A for a derivation of Eq. (35).) For an in
(36)
where Sdi; ; x0 represents the derivative of Sdi; with respect to the component xk of the state vector. Analogous expressions hold for the perturbations OR˜ i and OIdir with the derivatives R˜ i; x0 and Idir; x0 , ˜ respectively. The use of Eq. (36) and of the corresponding expressions for ORi and OIdir in Eq. (35) yields the
(37)
Here and in the remainder of this paper the subscript x0 is omitted. Finally, a comparison of Eq. (34) with a corresponding Taylor expansion of the radiative e;ect Ei around x0 yields again the analytical expressions for the derivatives of Ei . Taking into account that the radiative e;ects can be identi
(38)
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So far, Eq. (38) solves the identical problem as Eq. (21). However, the use of internal forward and adjoint radiation sources Sdi; and R˜ i leads to four additional inner products in Eq. (38). These additional terms describe the impact of the perturbed internal sources as well as the impact of the perturbed direct radiation on the radiative e;ect Ei . In contrast to Eq. (38), the simplicity of Eq. (21) becomes evident. However, Eq. (38) is needed in order to include the e;ects of a spherical atmosphere in the calculation of the derivatives of the Stokes parameters. 3.2. Linearized radiative transfer in a pseudo-spherical atmosphere The pseudo-spherical approximation can be seen as a spherical modi
solar irradiation
line of sight VZA
k=1
SZA
∆v1
∆s11
k=2 ∆s12
∆v2 ∆s22 ∆v3 ∆z1
∆z2
∆z3
∆s33 j=3
∆s23 j=2
∆s13
k=3
j=1
Fig. 1. Schematic overview of the geometry used for the calculation of radiative transfer in a spherical shell atmosphere consisting of N = 3 layers. The graph shows the direct radiation and the line of sight employed in the calculation of the single scattered part of the intensity vector in the viewing direction. The slant path elements Osjk and Ovj can be calculated using simple geometrical relationships. The viewing zenith angle VZA and the SZA are de
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261
used geometry. In a pseudo-spherical atmosphere the local SZA along the ray paths as well as the local viewing zenith angles (VZA) along the line of sight depend on height, and we have to replace the slant optical depth in Eq. (27) by ztop 1 s (z; ) = d z (39) (z ); ) e (z z where (z) denotes the cosine of the height-dependent zenith angle. For a model atmosphere which consists out of N spherical shells, the slant optical depth along the direct beam and along the line of sight can be approximated by the Chapman function [13,37]. Simple geometrical considerations lead to the following expressions for the corresponding Chapman functions at the lower boundary zk (k = 1; : : : ; N ) of a model layer k: ch(zk ; 0 ) =
k O j ; 0 (zj ) j=1
k O j ch(zk ; v ) = v (zj ) j=1
(40)
(41)
with 0 (zj ) ≈
Ozj ; Osjk
(42)
v (zj ) ≈
Ozj : Ovj
(43)
Here, O j = (zj ) − (zj−1 ) is the vertical optical thickness of a model layer j, Osjk is the slant path length element of the solar ray and Ovj describes the corresponding path length of the line of sight in this model layer (see Fig. 1). Ozj = zj−1 − zj represents the geometrical thickness of the same layer. The direction of the solar beam 0 = (−0 ; ’0 ) as well as the viewing direction v = (v ; ’v ) are de
(44)
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Substitution of Eq. (44) in Eqs. (25) and (29) gives the direct intensity
(45)
and of the approximated v , Eq. (43), in Eq. (31) provides the spherical response function R˜ sph; i , which describes the integration of the scattering source function along a spherical line of sight. This approach ensures that the contribution of singly scattered photons to the radiative e;ect Ei is treated in a full spherical manner. For multiply scattered photons the spherical geometry can be taken into account in both, the initializing single scattering
4. Calculation of derivatives with respect to trace gas pro(les 4.1. Calculation of the derivatives In the context of a trace gas pro
(47)
k Here, zk −1 and zk represent the top and the bottom height of the model layer k, $abs is the absorption cross section of the considered trace gas species and is assumed to be constant in the model layer.
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Substitution of Eq. (47) in Eq. (46) yields the derivatives zk N zn @Iitop †T k † T = $abs d z d I˜ (z; )Idi; (z; ) − d z d I˜ (z; )Ssph (z; ) @xk zk −1 4 n=1 zn−1 4 −
N n=1
−
−
zn − 1
N n=1
zn
zn − 1
N n=1
zn
zn
zn − 1
4
4
4
˜ Tsph; i (z; )Idi; (z; ) d z d R ˜ Tsph; i (z; )Idir; sph (z; ) d z d R ˜ Tsph; i (z; )Idir; d z d R sph (z; );
(48)
where we made use of the discretization of the model atmosphere in N homogeneous layers. Here, z0 = ztop and zN = zbot . In order to evaluate the perturbation integrals in Eq. (48) a numerical scheme is required to solve the forward and the adjoint RTE (28) and Eq. (33), respectively. Due to the di;erent form of the forward and adjoint transport operators Lˆ and Lˆ † one may assume that two di;erent numerical schemes are needed. However, with the substitution of the adjoint vector intensity
(49)
the adjoint formulation, Eq. (33), transforms to the corresponding pseudo-forward formulation Lˆ & ) = S&
(50)
with S& (z; ) = R˜ sph; i (z; − ) as a new radiation source [21]. The transport operator Lˆ & in Eq. (50) ˜ ) corresponds to the forward transport operator Lˆ from Eq. (7), except that the phase matrix Z(z; ; ˜ is replaced by QZ(z; ; )Q with Q=diag[1; 1; 1; −1] [38]. Eq. (50) is subject to the same boundary conditions as Eq. (6) in Section 2.2. For further details we refer to Hasekamp and Landgraf [23]. With the transformation above it is now possible to use the same solution technique for the forward as well as for the pseudo-forward RTE. Next the substitution of Eq. (49) in Eq. (48) allows one to calculate the derivatives @Iitop =@xk . We use the Fourier expansion presented by de Haan et al. [39] for the integration over the azimuth angle. See [23] for the use in the context of the forward-adjoint perturbation theory. This Fourier expansion is described in detail in Appendix B. With the Fourier expansion for Idi; , ), Ssph , Rsph; i and Idir; sph as well as a double Gaussian quadrature of order 2S to perform the remaining integration over , we obtain the following expressions for the derivatives: ∞ @Iitop =− cos m(’v − ’0 )(2 − 0m )[)mi + *im + +im ]; (51) @xk m=0 for i = 1; 2 and ∞ @Iitop =− sin m(’v − ’0 )(2 − 0m )[)mi − *im + +im ]; @xk m=0
(52)
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for i = 3; 4. The perturbation integrals are given by zk S m k m aj d z)mT (z; −j )+Idi; (z; j ) )i = −2$abs
and *im = 2
S
aj
N n=1
j=−S j =0
+
N
zn
zn − 1
n=1
(53)
zk − 1
j=−S j =0
zn
zn − 1
d z )mT (z; −j )+Smsph (z; j )
m d z R˜ msph;T i (z; j )+Idi; (z; j )
(54)
and +im =
N n=1
zn
zn − 1
d z R˜ msph;T i (z; 0 )Tsph (z; 0 )F0 e1 +
N n=1
zn
zn − 1
d z R˜ mT sph; i (z; 0 )Tsph (z; 0 )F0 e1 :
(55)
In Eqs. (53) and (54), the aj are the Gaussian weights and the j are the Gaussian streams. Here, positive indices j = 1; : : : ; S indicate upward directed streams and corresponding negative indices m indicate downward directed streams. Furthermore, + = diag[1; 1; −1; −1], and Idi; , )m , Smsph and R˜ msph; i denote the Fourier coeScients of the corresponding vector functions. In Eqs. (53)–(55) the derivatives of the spherical radiation source Smsph , of the spherical response m function R˜ msph; i and of the direct beam Idir; sph , equivalent to the derivative of the spherical transmission , with respect to an atmospheric trace gas pro
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265
theory, forms the main component of the linearized pseudo-spherical vector radiative transfer model PS-LIRA.
5. Veri(cation 5.1. Veri8cation of the forward model In this section we investigate the accuracy of the pseudo-spherical approach given in Section 3.2. The RTE (28) is solved using the Gauss–Seidel iteration technique. As there is no other pseudo-spherical vector radiative transfer model freely available which can serve as a reference, we show a comparison between the scalar version of our radiative transfer model and the pseudo-spherical DISORT [11,13] code. Furthermore, the pseudo-spherical DISORT code does not provide the possibility to integrate the scattering source function along a spherical line of sight. Therefore, all comparisons in this section are carried out for a plane-parallel line of sight integration. Fig. 2 shows the relative di;erence between the reNectance calculated with the pseudo-spherical DISORT code and the pseudo-spherical PS-LIRA code for a SZA of 89:0◦ and two VZA of 0:0◦ and 79:0◦ , corresponding to a nadir measurement and an o;-nadir measurement, respectively. The overall deviation is smaller than 0.005%. The largest di;erences occur at longer wavelengths and are due to the di;erent treatment of the multiple scattering in both models (see also [22]). However,
Fig. 2. Relative di;erence of the reNected intensity at TOA calculated with the scalar PS-LIRA code and the pseudo-spherical version of DISORT [11,13] for the spectral range 290 –322 nm. The reNectance is calculated for a clear-sky atmosphere including ozone absorption and Rayleigh scattering using an 8-stream discretization of the di;use intensity
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the di;erences are much smaller than e.g. uncertainties in the measurements and errors in the input parameters for radiative transfer. Therefore, we assume these di;erences to be negligible in our retrieval approach. 5.2. Veri8cation of the derivatives To investigate the accuracy of the derivatives of the di;erent Stokes parameters in Eqs. (51) and (52), we choose ozone as the relevant trace gas species to be retrieved. As reference we use the
which converges to the derivative for small perturbations in the ozone number density Oxk . The perturbed and the unperturbed Stokes parameters at TOA Iitop (xk + Oxk ) and Iitop (xk ) in Eq. (56) are calculated with the forward mode of our radiative transfer model using the Gauss–Seidel iteration approach. A relative perturbation in the number density of Oxk = 10−5 xk throughout the atmosphere is suScient for a reliable determination of @Iitop =@xk . Figs. 3–5 show the derivatives @Iitop =@xk calculated with the pseudo-spherical perturbation theory approach and their relative di;erences to the
Fig. 3. Left panel: derivatives of the Stokes parameters I1top (solid line), I2top (dashed line) and I3top (dot-dashed line) normalized to the incoming solar Nux F0 with respect to the ozone density at di;erent altitude levels. The derivatives are calculated with the pseudo-spherical perturbation theory approach. The computations are performed at = 299 nm for clear sky conditions described in the caption of Fig. 2. The SZA is 89:0◦ , the VZA is 79:0◦ and the relative azimuth angle PHI is 60:0◦ . A Lambertian ground albedo of 0:1 is used. Right panel: Relative di;erence between computations using the pseudo-spherical perturbation theory, Eqs. (51) and (52), and the
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Fig. 4. Same as Fig. 3 but for a wavelength of = 312 nm.
Fig. 5. Same as Fig. 3 but for a wavelength of = 322 nm.
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299, 312 and 322 nm, respectively. The overall deviation of the pseudo-spherical perturbation theory approach to the
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269
method [1,2], where a side constraint is added in the minimization of the least-squares condition. The minimization of the norm of the
(57)
Here, the matrix A˜ is called the averaging kernel [4] and ex is the error in the retrieved pro
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Fig. 6. Left panel: Comparison of a retrieved ozone pro
6.2. The impact of plane-parallel radiative transfer on ozone pro8le retrieval In order to study the importance of pseudo-spherical radiative transfer for ozone pro
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271
Fig. 7. Averaging kernel A˜ corresponding to the retrieved ozone pro
Fig. 8. Upper panel: pseudo-spherical simulation of the reNectance r (solid line) for a SZA of 80:0◦ , a VZA of 0:0◦ and a relative azimuth angle PHI of 60:0◦ in the wavelength range 290 –322 nm and a spectral sampling of 0:2 nm. Lower panel: relative di;erence in the simulated reNectance using either the plane-parallel (PP) radiative transfer model (solid line) or the spherically corrected radiative transfer model using the Kasten and Young approximation (KY) (dotted line).
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Fig. 9. Same as Fig. 6 but using the plane-parallel version of the radiative transfer model (PP) in the retrieval of the ozone pro
In addition, we have investigated the use of a modi
(58)
with . = 90:0 − SZA, a = 0:50572, b = 6:07995 and c = 1:6364 [42]. As this spherical correction is developed for homogeneous atmospheres, it was originally not meant for atmospheres with a realistic vertical distribution of ozone. However, as indicated in Fig. 8 it provides a signi
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273
Fig. 10. Same as Fig. 9 but using the Kasten and Young (KY) approximation to correct the retrieval of ozone pro
Furthermore, we investigated the e;ect of a large VZA on the retrieval of ozone pro
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Fig. 11. Same as Fig. 8 but for a SZA of 50:0◦ and a VZA of 50:0◦ .
Fig. 12. Same as Fig. 10 but for a SZA of 50:0◦ and a VZA of 50:0◦ .
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ozone pro
7. Conclusions We have presented a linearization of pseudo-spherical vector radiative transfer in which the needed derivatives are calculated using the forward-adjoint radiative perturbation theory. For this purpose the perturbation theory approach is extended to pseudo-spherical atmospheres taking the perturbation of internal sources into account. It is the
Acknowledgements We would like to thank A. Maurellis for comments on an earlier version of this paper. This research was supported by SRON under project number 6430-SCIARALI (GO-2).
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Appendix A. Perturbation of the radiative e4ect In this appendix, we provide a derivation of Eq. (35), which describes the linearization of radiative transfer employing internal sources. The radiation e;ect Ei is given by Ei (x) = R˜ i; x |Ix = R˜ i; x |Idi; ; x + Idir; x ;
(A.1)
where the subscript x denotes the atmospheric state. Thus, the perturbation of the radiative e;ect OEi due to a perturbation Ox = x − x0 in the atmospheric state vector can be calculated by OEi = Ei (x) − Ei (x0 ) = R˜ i; x |Idi; ; x + Idir; x − R˜ i; x0 |Idi; ; x0 + Idir; x0 :
(A.2)
For small perturbations Ox, Eq. (A.2) can be further simpli
(A.3)
Idi; ; x = Idi; ; x0 + OIdi;
(A.4)
Idir; x = Idir; x0 + OIdir :
(A.5)
and
This leads to OEi = R˜ i; x0 |OIdi; + OIdir + OR˜ i |Idi; ; x0 + Idir; x0 + OR˜ i |OIdi; + OIdir :
(A.6)
For small perturbations Ox of the atmospheric state the perturbations OR˜ i , OIdi; and OIdir are small too, and products of them, like in the last term of Eq. (A.6), lead to second-order perturbations. Therefore, OEi can be written as OEi = R˜ i; x0 |OIdi; + R˜ i; x0 |OIdir + OR˜ i |Idi; ; x0 + OR˜ i |Idir; x0 + O(Ox2 );
(A.7)
where O(Ox2 ) represents the perturbations of second and higher order. According to Eq. (33) R˜ i; x0 can be replaced by Lˆ †0 I˜†x0 which leads to OEi = Lˆ †0 I˜†x0 |OIdi; + R˜ i; x0 |OIdir + OR˜ i |Idi; ; x0 + OR˜ i |Idir; x0 + O(Ox2 ):
(A.8)
In a next step it is necessary to replace OIdi; in the
(A.9)
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Further, the perturbed forward RTE (28) in its form ˆ (Lˆ 0 + OL)(I di; ; x0 + OIdi; ) = Sdi; ; x0 + OSdi;
(A.10)
can be written as ˆ di; ; x0 + OSdi; − OLOI ˆ di; : Lˆ 0 OIdi; = −OLI
(A.11)
Substitution of Eq. (A.11) in Eq. (A.9) and again neglecting higher-order perturbation terms, results in ˆ di; ; x0 + I˜†x |OSdi; + OR˜ i |Idi; ; x0 OEi = −I˜†x |OLI 0
0
+OR˜ i |Idir; x0 + R˜ i; x0 |OIdir + O(Ox2 );
(A.12)
which is equivalent to Eq. (35) where second-order terms in the perturbations are neglected. In all inner products quantities occur which are either calculated for the unperturbed atmospheric state x0 or which can be calculated analytically. Thus, the determination of OEi becomes straightforward. Appendix B. Fourier expansion for azimuthal dependence In order to handle the integration over azimuth in Eqs. (28), (48) and (50) we use the Fourier expansion proposed by Hovenier and van der Mee [43] and de Haan et al. [39]. For its use within the context of perturbation theory see [23]. The expansion of the phase matrix is given by ∞ ˜ ) = 1 Z(z; ; (2 − 0m )[B+m (’˜ − ’)Zm (z; ; ˜ )(E + +) 2 m=0 ˜ )(E − +)]; + B−m (’˜ − ’)Zm (z; ;
(B.1)
where 0m is the Kronecker delta, + = diag[1; 1; −1; −1];
(B.2)
B+m (’) = diag[cos m’; cos m’; sin m’; sin m’]
(B.3)
B−m (’) = diag[ − sin m’; −sin m’; cos m’; cos m’]:
(B.4)
and
The mth Fourier coeScient of the phase matrix can be calculated by ∞ m m Z (z; ; ˜ ) = (−1) Pml (−)Sl (z)Pml (−); ˜
(B.5)
l=m
where Sl is the expansion coeScient matrix and Pml is the generalized spherical function matrix [39]. In the corresponding Fourier expansion of the di;use intensity vector two types of Fourier coef
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with +m (z; ) Idi;
1 = 2
1 Idi; (z; ) = 2 −m
2
0
d’ B+m (’0 − ’)Idi; (z; );
2
0
d’ B−m (’0 − ’)Idi; (z; ):
(B.7)
The pseudo-spherically modi
(B.8)
using Eqs. (B.1) and (B.7) as well as a Fourier expansion of the spherical radiation source Ssph (z; ). Thus, the transport operator in Eq. (B.8) can be written as 1 @ s (z) m m ˆ + e (z) ( − )E Z (z; ; d ˜ ˜ − ˜ ) L = @z 4 −1 A ˜ | ˜ ◦: (B.9) − 0m (z)()||(−)| m The Fourier coeScients S± sph of the radiation source Ssph are given by
S+m sph (z; ) =
1 s (z) m Z (z; 0 ; )(E + +)Tsph (z; 0 )F0 e1 2 4 + 0m
A (z)0 (z)Tsph (z; 0 )()||F0 e1
(B.10)
and m T S− sph (z; ) = [0; 0; 0; 0] :
(B.11)
+m The forward vector intensity
(B.12)
However, as the pseudo-forward source is de
1 1 s (z) mT Z (z; v ; −)(E + +)Tsph (z; v )ei 2 v (z) 4 + 0m
AT (z)Tsph (z; v )()||e1 i1 ;
m T S− & (z; ) = [0; 0; 0; 0] :
(B.13)
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279
Thus, for the corresponding pseudo-forward problems we obtain a Fourier expansion of ) containing terms of )+m only. For i = 3; 4 we obtain T S+m & (z; ) = [0; 0; 0; 0] ; m S− & (z; ) =
1 1 s (z) mT Z (z; v ; −)(E − +)Tsph (z; v )ei : 2 v (z) 4
(B.14)
Hence, for the corresponding pseudo-forward problems we obtain a Fourier expansion containing terms of )−m only. In this paper )m denotes either )+m or )−m , depending on the corresponding pseudo-forward problem. The Fourier coeScients for the spherical response function R˜ sph; i (z; ) follow immediately from Eqs. (B.13) and (B.14) with the de
(B.16)
For further details we refer to Hasekamp and Landgraf [23]. Appendix C. Calculation of derivatives with respect to trace gas pro(les m In this appendix we provide the derivatives Smsph , R˜ msph; i and Idir; sph of the Fourier coeScients of the spherical radiation source, of the spherical response function and of the direct beam with respect to an atmospheric trace gas pro
+ 0m
A (z)0 (z)Tsph (z; 0 )()||F0 e1 :
(C.1)
Here, Tsph (z; 0 ) = e−ch(z; 0 ) is the spherical transmission function, Eq. (44). The Chapman function within a model layer k is expressed by a linear interpolation in height ch(z; 0 ) = ak z + bk with coeScients ch(zk −1 ; 0 ) − ch(zk ; 0 ) ak = ; Ozk ch(zk ; 0 )zk −1 − ch(zk −1 ; 0 )zk bk = : Ozk
(C.2)
(C.3) (C.4)
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Assuming a constant scattering coeScient sk as well as a constant phase matrix Zmk within a model layer k, Eq. (C.1) can be written as Smsph (z; j ) =
1 sk m Z (0 ; j )(E + +)e−ch(z; 0 ) F0 e1 2 4 k + 0m
A (z)0 (z)e−ch(z; 0 ) ()||F0 e1
(C.5)
where zk −1 ¿ z ¿ zk . Thus, the derivative with respect to the ozone number density xk in a model layer k is given by @Smsph @xk
(z; j ) = −
@ch(z; 0 ) 1 sk m Z (0 ; j )(E + +)e−ch(z; 0 ) F0 e 1 2 4 k @xk
− 0m
A @ch(z; 0 ) (z)0 (z)e−ch(z; 0 ) ()||F0 e1 : @xk
(C.6)
The derivative of the interpolated Chapman function can be calculated in a straightforward manner using Eqs. (C.3), (C.4) and the dependence of the Chapman function, Eq. (40), at the layer boundaries on ozone. Finally, this results in @Smsph @xk
k (z; j ) = −$abs
1 sk m Oskk F0 e 1 Zk (0 ; j )(E + +)e−(ak z+bk ) (zk −1 − z) 2 4 Ozk
(C.7)
for zk −1 ¿ z ¿ zk and @Smsph @xk
1 sn m Ozk −(an z+bn ) Z (0 ; j )(E + +) e 2 4 n Ozn Oskn Oskn−1 Oskn−1 Oskn F 0 e1 × zn−1 − zn +z − Ozk Ozk Ozk Ozk
k (z; j ) = −$abs
(C.8)
for layers located below the perturbed model layer, i.e. zn−1 ¿ z ¿ zn where n ¿ k. Thus, the spherical source in all the layers n located beneath a given layer k is a;ected by a perturbation of the trace gas number density in this kth layer. At ground level, for z = zN , the derivative is @Smsph @xk
k (zN ; j ) = −$abs
A OskN 0 (zN )e−(aN zN +bN ) ()||Ozk F 0 e1 : Ozk
(C.9)
For altitudes above the perturbed model layer, i.e. z ¿ zk −1 , the derivative of the spherical radiation source with respect to the trace gas pro
k (z; j ) = −$abs
Ovk 1 1 sk mT Z (v ; j )(E + +)e−(ck z+dk ) (zk −1 − z) ei : 2 v (z) 4 k Ozk
(C.10)
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281
for zk −1 ¿ z ¿ zk . For the cosine v (z) in Eq. (C.10) we use a linear interpolation with height between the boundary values, de
(C.12)
(C.13)
For i = 3; 4 the derivatives are obtained in an analogous manner. For altitudes above the perturbed model layer, i.e. z ¿ zk −1 , the derivative of the spherical response function with respect to the trace gas pro
(C.15)
for zn−1 ¿ z ¿ zn , n ¿ k. At ground level, for z = zN , the derivative of the reNected direct beam is m @Idir; OskN sph k A 0 (zN )( − 0 )e−(aN zN +bN ) ()||Ozk (zN ; j ) = −$abs F0 e1 : (C.16) @xk Ozk Again for altitudes above the perturbed layer the derivative is always zero. References [1] Phillips P. A technique for the numerical solution of certain integral equations of the
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[6] Oikarinen L, Sihvola E, Kyrola E. Multiple scattering in limb viewing geometry. J Geophys Res 1999;104: 31261–74. [7] Caudill TR, Flittner DE, Herman BM, Torres O, McPeters RD. Evaluation of the pseudo-spherical approximation for backscattered ultraviolet radiances and ozone retrieval. J Geophys Res 1997;102:3881–90. [8] Rozanov A, Rozanov V, Burrrows JP. Combined di;erential–integral approach for the radiation
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Hovenier JW. Multiple scattering of polarized light in planetary atmospheres. Astron Astrophys 1971;13:7–29. Herman B, Browning S. A numerical solution to the equation of radiative transfer. J Atmos Sci 1965;22:559–66. Evans KF, Stephens GL. A new polarized atmospheric radiative transfer model. JQSRT 1991;46:413–23. Schulz FM, Stamnes K, Weng F. Vdisort: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer. JQSRT 1999;61:105–22. Thomas GE, Stamnes K. Radiative transfer in the atmosphere and ocean. New York: Cambridge University Press, 1999. Hovenier JW. Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles. J Atmos Sci 1969;26:488–99. de Haan JF, Bosma P, Hovenier JW. The adding method for multiple scattering calculations of polarized light. Astron Astrophys 1987;181:371–91. National Oceanic and Atmospheric Administration (NOAA). US Standard Atmosphere. Rep. NOAA-S/T76-1562. US Gov. Print. O;.: Washington, DC; 1976. Rodgers G. Inverse methods for atmospheric sounding: theory and practice. River Edge, NJ: World Sc., 2000. Kasten F, Young T. Revised optical air mass tables and approximation formula. Appl Opt 1989;28:4735–8. Hovenier JW, van der Mee CVM. Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere. Astron Astrophys 1983;128:1–16.
www.elsevier.com/locate/jqsrt
Linearization of a pseudo-spherical vector radiative transfer model Holger H. Walter∗ , Jochen Landgraf, Otto P. Hasekamp National Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands Received 9 January 2003; accepted 7 May 2003
Abstract We present a linearization of pseudo-spherical vector radiative transfer using the forward-adjoint perturbation theory. The method, which is based on the usual forward and the adjoint formulation of radiative transfer, is extended to radiative transfer problems with internal radiation sources. This allows one to linearize vector radiative transfer in pseudo-spherical approximation. The model PS-LIRA is developed based on this approach. Here, the calculation of the direct beam as well as the integration along the line of sight are performed for a full spherical geometry. The multiply scattered radiation is calculated in a plane-parallel atmosphere. The proposed model provides the intensity vector as well as its derivatives at the top of the atmosphere for solar zenith angles (SZA) up to 90:0◦ and for o;-nadir viewing geometries ranging from limb to limb. In its scalar reduction the developed radiative transfer model is veri
1. Introduction Satellite measurements of backscattered sunlight play an important role in monitoring the chemical composition of the Earth’s atmosphere. For example, the Global Ozone Monitoring Experiment (GOME), launched in 1995 on board of ESA’s ERS-2 satellite, observes a range of atmospheric ∗
Corresponding author. E-mail address: [email protected] (H.H. Walter).
0022-4073/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0022-4073(03)00228-0
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trace constituents with an emphasis on global ozone distributions. GOME is a nadir viewing spectrometer that measures the solar radiation scattered by the atmosphere in the ultraviolet (UV) and visible spectral region (240 –790 nm). Likewise, the Solar Backscatter Ultraviolet (SBUV/2) instrument, currently on board NASA’s NOAA-16 polar orbiting satellite, measures the ultraviolet sunlight scattered by the Earth’s atmosphere at several wavelengths ranging from 252 to 340 nm. Measurements at the shortest eight wavelengths are used to estimate ozone vertical pro
(1)
Here, the N -dimensional state vector x represents the trace gas pro
where Oxk is the kth element of Ox = x − x0 . Substitution of Eq. (2) into Eq. (1) and neglecting the higher-order terms O(Ox2 ) yields the atmospheric state vector x iteratively using standard inversion techniques [1–4]. Thus, the retrieval of trace gas pro
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253
instruments like GOME and SCIAMACHY with a relatively narrow
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H.H. Walter et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 85 (2004) 251 – 283
of the intensity vector at moderate SZA. In order to perform retrievals from measurements at low sun and for instruments with a wide
(3)
where T indicates the transposed vector. In the remainder of the paper we will use the notation Ii (i = 1; : : : ; 4) for the four Stokes parameters. Given a radiative transfer model which simulates the spectral intensity vector Itop at TOA in the viewing direction of the instrument, the transport of light from the entrance slit to the detector plane of the instrument may be described by the Mueller matrix approach (see e.g. [24]). The forward model at wavelength j that simulates the radiance measured by the jth detector pixel is given by F( j ) =
4
M1i ( j )Iitop ( j );
(4)
i=1
where the M1i are the corresponding elements of the 4 × 4 Mueller matrix M characterising the transmission properties of the instrument. Thus, the derivatives of the forward model in Eq. (2) can be calculated in a straightforward manner from the corresponding derivatives of the Stokes parameters Iitop with respect to the elements xk of the atmospheric state vector x, viz 4
@I top @F ( j ) = M1i ( j ) i ( j ): @xk @xk i=1
(5)
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255
Here we neglect any e;ect of the instrument slit and the sampling of the signal along the detector pixel. In this paper we present an e;ective approach to calculate the derivatives @Iitop =@xk for pseudo-spherical radiative transfer. 2.2. Linearized radiative transfer in a plane-parallel atmosphere The forward-adjoint perturbation theory, proposed by Marchuk [20] and later independently by Box et al. [21,25,26], provides an elegant and eScient tool to linearize atmospheric radiative transfer. This is demonstrated by several authors, e.g. Ustinov [27–29], Rozanov et al. [30], Landgraf et al. [22,31] and Hasekamp and Landgraf [23]. This section gives a brief overview of the method developed for horizontally homogeneous, plane-parallel atmospheres, for further details we refer to the papers themselves. For a linearization the RTE has to be solved in its forward and adjoint formulation. The forward formulation of polarized radiative transfer [23] in a plane-parallel atmosphere is given by the operator equation ˆ =S LI with the radiation source S, the total vector intensity
(6)
(7)
Here, E is the 4 × 4 unity matrix and e and s are the extinction and scattering coeScients, respectively. The altitude is denoted as z. = (; ’) describes the propagation direction of the radiation, given by the cosine of the zenith angle ( ¡ 0 for downward directed radiation and ¿ 0 for upward directed radiation) and the azimuth angle ’, where the azimuth angle is measured clockwise when looking downward. Furthermore, d ˜ = d ˜ d ’˜ and is the Dirac-delta function with ˜ ( − )=(− )(’− ˜ ’). ˜ represents the Heaviside step function. The
(8)
In the UV and visible part of the spectrum thermal emission can be neglected and the radiation source S is given by unpolarized sunlight that illuminates the TOA: S(z; ) = 0 (z − ztop )( − 0 )F0 e1 ;
(9)
where ztop is the height of the model atmosphere, 0 = (−0 ; ’0 ) indicates the direction of the solar beam and F0 represents the extraterrestrial Nux. ei (i = 1; : : : ; 4) is the unity vector in the direction of the ith component of the intensity vector.
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In addition to Eq. (6), the intensity vector has to ful
for ¡ 0;
I(zbot ; ) = [0; 0; 0; 0]T
for ¿ 0;
(10)
where zbot denotes the bottom of the atmosphere. The solution of Eq. (6) gives the intensity vector
(11)
where R is the corresponding response vector function [20,21]. The inner product of two arbitrary vector functions a and b is de
(13)
where the response function R acts as a source for the adjoint vector intensity
(15)
ˆ x0 : OE = −Ix†0 |OLI
(16)
with Here, the change of the transport operator OLˆ due to the perturbation in the atmospheric state is given by ˆ ˆ 0 ): OLˆ = L(x) − L(x
(17)
The source S and the response function R are not a;ected by the perturbation Ox. For the case that the state vectors x and x0 di;er only in their kth component by a small amount Oxk a comparison
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257
of Eq. (15) with a corresponding
(18)
In order to calculate the derivatives of the Stokes parameters Iitop in Eq. (5) we interprete those as four radiative e;ects Ei , de
(19)
with i = 1; : : : ; 4 and corresponding response functions Ri (z; ) = (z − ztop )( − v )ei ;
(20)
where v = (v ; ’v ) denotes the viewing direction of the instrument. This provides the expressions @Iitop 1 † ˆ x0 (x0 ) = − I (Ri )|OLI @xk Oxk x0
(21)
for the four derivatives. Here, Ix†0 (Ri ) is the solution of Eq. (13) with the adjoint source Ri . Thus, in order to calculate the derivatives we have to solve the forward radiative transfer problem as well as the adjoint transport problem for the four di;erent adjoint sources Ri . Due to the analytical form of Eq. (21) the perturbation theory approach combines an accurate calculation of the derivatives with low computational costs compared to a
(22)
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The direct part of the intensity
(23)
where the radiation source S is given by Eq. (9) and the transport operator corresponds to the streaming term in Eq. (7), @ ˆ + e (z) E ◦ : Ldir = (24) @z The direct intensity
(25)
with the plane-parallel transmission function T (z; ) = e− s (z; ) ;
(26)
where s (z; ) describes the slant optical depth at altitude z and in direction in the model atmosphere: 1 ztop d z e (z ): (27) s (z; ) = z The transport equation (6) can be rewritten in a corresponding RTE for the di;use intensity
(28)
with the di;use radiation source Sdi; (z; ) =
s (z) A Z(z; 0 ; )T (z; 0 )F0 e1 + (z)0 T (z; 0 )()||F0 e1 : 4
(29)
Here, the
(30)
with the response function 1 s (z) T AT Z (z; v ; )T (z; v )ei + (z)T (z; v )(−)||i1 ei : R˜ i (z; ) = v 4
(31)
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259
In this case the inner product in Eq. (30) represents the line of sight integration of the scattering source function s (z) T d Z(z; v ; )I(z; ) (32) Ji (z; v ) = e 4 i 4 for the ith Stokes parameter. The second term in Eq. (31) describes the contribution of light reNected at a Lambertian surface, which only takes place for the
(33)
where I˜† is the corresponding adjoint vector intensity
(34)
ˆ di; ; x0 + I˜†x |OSdi; + OR˜ i |Idi; ; x0 + OR˜ i |Idir; x0 + R˜ i; x0 |OIdir ; OEi = −I˜†x0 |OLI 0
(35)
with
where the subscript x0 denotes the unperturbed quantity. (See also Appendix A for a derivation of Eq. (35).) For an in
(36)
where Sdi; ; x0 represents the derivative of Sdi; with respect to the component xk of the state vector. Analogous expressions hold for the perturbations OR˜ i and OIdir with the derivatives R˜ i; x0 and Idir; x0 , ˜ respectively. The use of Eq. (36) and of the corresponding expressions for ORi and OIdir in Eq. (35) yields the
(37)
Here and in the remainder of this paper the subscript x0 is omitted. Finally, a comparison of Eq. (34) with a corresponding Taylor expansion of the radiative e;ect Ei around x0 yields again the analytical expressions for the derivatives of Ei . Taking into account that the radiative e;ects can be identi
(38)
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So far, Eq. (38) solves the identical problem as Eq. (21). However, the use of internal forward and adjoint radiation sources Sdi; and R˜ i leads to four additional inner products in Eq. (38). These additional terms describe the impact of the perturbed internal sources as well as the impact of the perturbed direct radiation on the radiative e;ect Ei . In contrast to Eq. (38), the simplicity of Eq. (21) becomes evident. However, Eq. (38) is needed in order to include the e;ects of a spherical atmosphere in the calculation of the derivatives of the Stokes parameters. 3.2. Linearized radiative transfer in a pseudo-spherical atmosphere The pseudo-spherical approximation can be seen as a spherical modi
solar irradiation
line of sight VZA
k=1
SZA
∆v1
∆s11
k=2 ∆s12
∆v2 ∆s22 ∆v3 ∆z1
∆z2
∆z3
∆s33 j=3
∆s23 j=2
∆s13
k=3
j=1
Fig. 1. Schematic overview of the geometry used for the calculation of radiative transfer in a spherical shell atmosphere consisting of N = 3 layers. The graph shows the direct radiation and the line of sight employed in the calculation of the single scattered part of the intensity vector in the viewing direction. The slant path elements Osjk and Ovj can be calculated using simple geometrical relationships. The viewing zenith angle VZA and the SZA are de
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261
used geometry. In a pseudo-spherical atmosphere the local SZA along the ray paths as well as the local viewing zenith angles (VZA) along the line of sight depend on height, and we have to replace the slant optical depth in Eq. (27) by ztop 1 s (z; ) = d z (39) (z ); ) e (z z where (z) denotes the cosine of the height-dependent zenith angle. For a model atmosphere which consists out of N spherical shells, the slant optical depth along the direct beam and along the line of sight can be approximated by the Chapman function [13,37]. Simple geometrical considerations lead to the following expressions for the corresponding Chapman functions at the lower boundary zk (k = 1; : : : ; N ) of a model layer k: ch(zk ; 0 ) =
k O j ; 0 (zj ) j=1
k O j ch(zk ; v ) = v (zj ) j=1
(40)
(41)
with 0 (zj ) ≈
Ozj ; Osjk
(42)
v (zj ) ≈
Ozj : Ovj
(43)
Here, O j = (zj ) − (zj−1 ) is the vertical optical thickness of a model layer j, Osjk is the slant path length element of the solar ray and Ovj describes the corresponding path length of the line of sight in this model layer (see Fig. 1). Ozj = zj−1 − zj represents the geometrical thickness of the same layer. The direction of the solar beam 0 = (−0 ; ’0 ) as well as the viewing direction v = (v ; ’v ) are de
(44)
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Substitution of Eq. (44) in Eqs. (25) and (29) gives the direct intensity
(45)
and of the approximated v , Eq. (43), in Eq. (31) provides the spherical response function R˜ sph; i , which describes the integration of the scattering source function along a spherical line of sight. This approach ensures that the contribution of singly scattered photons to the radiative e;ect Ei is treated in a full spherical manner. For multiply scattered photons the spherical geometry can be taken into account in both, the initializing single scattering
4. Calculation of derivatives with respect to trace gas pro(les 4.1. Calculation of the derivatives In the context of a trace gas pro
(47)
k Here, zk −1 and zk represent the top and the bottom height of the model layer k, $abs is the absorption cross section of the considered trace gas species and is assumed to be constant in the model layer.
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263
Substitution of Eq. (47) in Eq. (46) yields the derivatives zk N zn @Iitop †T k † T = $abs d z d I˜ (z; )Idi; (z; ) − d z d I˜ (z; )Ssph (z; ) @xk zk −1 4 n=1 zn−1 4 −
N n=1
−
−
zn − 1
N n=1
zn
zn − 1
N n=1
zn
zn
zn − 1
4
4
4
˜ Tsph; i (z; )Idi; (z; ) d z d R ˜ Tsph; i (z; )Idir; sph (z; ) d z d R ˜ Tsph; i (z; )Idir; d z d R sph (z; );
(48)
where we made use of the discretization of the model atmosphere in N homogeneous layers. Here, z0 = ztop and zN = zbot . In order to evaluate the perturbation integrals in Eq. (48) a numerical scheme is required to solve the forward and the adjoint RTE (28) and Eq. (33), respectively. Due to the di;erent form of the forward and adjoint transport operators Lˆ and Lˆ † one may assume that two di;erent numerical schemes are needed. However, with the substitution of the adjoint vector intensity
(49)
the adjoint formulation, Eq. (33), transforms to the corresponding pseudo-forward formulation Lˆ & ) = S&
(50)
with S& (z; ) = R˜ sph; i (z; − ) as a new radiation source [21]. The transport operator Lˆ & in Eq. (50) ˜ ) corresponds to the forward transport operator Lˆ from Eq. (7), except that the phase matrix Z(z; ; ˜ is replaced by QZ(z; ; )Q with Q=diag[1; 1; 1; −1] [38]. Eq. (50) is subject to the same boundary conditions as Eq. (6) in Section 2.2. For further details we refer to Hasekamp and Landgraf [23]. With the transformation above it is now possible to use the same solution technique for the forward as well as for the pseudo-forward RTE. Next the substitution of Eq. (49) in Eq. (48) allows one to calculate the derivatives @Iitop =@xk . We use the Fourier expansion presented by de Haan et al. [39] for the integration over the azimuth angle. See [23] for the use in the context of the forward-adjoint perturbation theory. This Fourier expansion is described in detail in Appendix B. With the Fourier expansion for Idi; , ), Ssph , Rsph; i and Idir; sph as well as a double Gaussian quadrature of order 2S to perform the remaining integration over , we obtain the following expressions for the derivatives: ∞ @Iitop =− cos m(’v − ’0 )(2 − 0m )[)mi + *im + +im ]; (51) @xk m=0 for i = 1; 2 and ∞ @Iitop =− sin m(’v − ’0 )(2 − 0m )[)mi − *im + +im ]; @xk m=0
(52)
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for i = 3; 4. The perturbation integrals are given by zk S m k m aj d z)mT (z; −j )+Idi; (z; j ) )i = −2$abs
and *im = 2
S
aj
N n=1
j=−S j =0
+
N
zn
zn − 1
n=1
(53)
zk − 1
j=−S j =0
zn
zn − 1
d z )mT (z; −j )+Smsph (z; j )
m d z R˜ msph;T i (z; j )+Idi; (z; j )
(54)
and +im =
N n=1
zn
zn − 1
d z R˜ msph;T i (z; 0 )Tsph (z; 0 )F0 e1 +
N n=1
zn
zn − 1
d z R˜ mT sph; i (z; 0 )Tsph (z; 0 )F0 e1 :
(55)
In Eqs. (53) and (54), the aj are the Gaussian weights and the j are the Gaussian streams. Here, positive indices j = 1; : : : ; S indicate upward directed streams and corresponding negative indices m indicate downward directed streams. Furthermore, + = diag[1; 1; −1; −1], and Idi; , )m , Smsph and R˜ msph; i denote the Fourier coeScients of the corresponding vector functions. In Eqs. (53)–(55) the derivatives of the spherical radiation source Smsph , of the spherical response m function R˜ msph; i and of the direct beam Idir; sph , equivalent to the derivative of the spherical transmission , with respect to an atmospheric trace gas pro
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theory, forms the main component of the linearized pseudo-spherical vector radiative transfer model PS-LIRA.
5. Veri(cation 5.1. Veri8cation of the forward model In this section we investigate the accuracy of the pseudo-spherical approach given in Section 3.2. The RTE (28) is solved using the Gauss–Seidel iteration technique. As there is no other pseudo-spherical vector radiative transfer model freely available which can serve as a reference, we show a comparison between the scalar version of our radiative transfer model and the pseudo-spherical DISORT [11,13] code. Furthermore, the pseudo-spherical DISORT code does not provide the possibility to integrate the scattering source function along a spherical line of sight. Therefore, all comparisons in this section are carried out for a plane-parallel line of sight integration. Fig. 2 shows the relative di;erence between the reNectance calculated with the pseudo-spherical DISORT code and the pseudo-spherical PS-LIRA code for a SZA of 89:0◦ and two VZA of 0:0◦ and 79:0◦ , corresponding to a nadir measurement and an o;-nadir measurement, respectively. The overall deviation is smaller than 0.005%. The largest di;erences occur at longer wavelengths and are due to the di;erent treatment of the multiple scattering in both models (see also [22]). However,
Fig. 2. Relative di;erence of the reNected intensity at TOA calculated with the scalar PS-LIRA code and the pseudo-spherical version of DISORT [11,13] for the spectral range 290 –322 nm. The reNectance is calculated for a clear-sky atmosphere including ozone absorption and Rayleigh scattering using an 8-stream discretization of the di;use intensity
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the di;erences are much smaller than e.g. uncertainties in the measurements and errors in the input parameters for radiative transfer. Therefore, we assume these di;erences to be negligible in our retrieval approach. 5.2. Veri8cation of the derivatives To investigate the accuracy of the derivatives of the di;erent Stokes parameters in Eqs. (51) and (52), we choose ozone as the relevant trace gas species to be retrieved. As reference we use the
which converges to the derivative for small perturbations in the ozone number density Oxk . The perturbed and the unperturbed Stokes parameters at TOA Iitop (xk + Oxk ) and Iitop (xk ) in Eq. (56) are calculated with the forward mode of our radiative transfer model using the Gauss–Seidel iteration approach. A relative perturbation in the number density of Oxk = 10−5 xk throughout the atmosphere is suScient for a reliable determination of @Iitop =@xk . Figs. 3–5 show the derivatives @Iitop =@xk calculated with the pseudo-spherical perturbation theory approach and their relative di;erences to the
Fig. 3. Left panel: derivatives of the Stokes parameters I1top (solid line), I2top (dashed line) and I3top (dot-dashed line) normalized to the incoming solar Nux F0 with respect to the ozone density at di;erent altitude levels. The derivatives are calculated with the pseudo-spherical perturbation theory approach. The computations are performed at = 299 nm for clear sky conditions described in the caption of Fig. 2. The SZA is 89:0◦ , the VZA is 79:0◦ and the relative azimuth angle PHI is 60:0◦ . A Lambertian ground albedo of 0:1 is used. Right panel: Relative di;erence between computations using the pseudo-spherical perturbation theory, Eqs. (51) and (52), and the
H.H. Walter et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 85 (2004) 251 – 283
Fig. 4. Same as Fig. 3 but for a wavelength of = 312 nm.
Fig. 5. Same as Fig. 3 but for a wavelength of = 322 nm.
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299, 312 and 322 nm, respectively. The overall deviation of the pseudo-spherical perturbation theory approach to the
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method [1,2], where a side constraint is added in the minimization of the least-squares condition. The minimization of the norm of the
(57)
Here, the matrix A˜ is called the averaging kernel [4] and ex is the error in the retrieved pro
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Fig. 6. Left panel: Comparison of a retrieved ozone pro
6.2. The impact of plane-parallel radiative transfer on ozone pro8le retrieval In order to study the importance of pseudo-spherical radiative transfer for ozone pro
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271
Fig. 7. Averaging kernel A˜ corresponding to the retrieved ozone pro
Fig. 8. Upper panel: pseudo-spherical simulation of the reNectance r (solid line) for a SZA of 80:0◦ , a VZA of 0:0◦ and a relative azimuth angle PHI of 60:0◦ in the wavelength range 290 –322 nm and a spectral sampling of 0:2 nm. Lower panel: relative di;erence in the simulated reNectance using either the plane-parallel (PP) radiative transfer model (solid line) or the spherically corrected radiative transfer model using the Kasten and Young approximation (KY) (dotted line).
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Fig. 9. Same as Fig. 6 but using the plane-parallel version of the radiative transfer model (PP) in the retrieval of the ozone pro
In addition, we have investigated the use of a modi
(58)
with . = 90:0 − SZA, a = 0:50572, b = 6:07995 and c = 1:6364 [42]. As this spherical correction is developed for homogeneous atmospheres, it was originally not meant for atmospheres with a realistic vertical distribution of ozone. However, as indicated in Fig. 8 it provides a signi
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Fig. 10. Same as Fig. 9 but using the Kasten and Young (KY) approximation to correct the retrieval of ozone pro
Furthermore, we investigated the e;ect of a large VZA on the retrieval of ozone pro
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Fig. 11. Same as Fig. 8 but for a SZA of 50:0◦ and a VZA of 50:0◦ .
Fig. 12. Same as Fig. 10 but for a SZA of 50:0◦ and a VZA of 50:0◦ .
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ozone pro
7. Conclusions We have presented a linearization of pseudo-spherical vector radiative transfer in which the needed derivatives are calculated using the forward-adjoint radiative perturbation theory. For this purpose the perturbation theory approach is extended to pseudo-spherical atmospheres taking the perturbation of internal sources into account. It is the
Acknowledgements We would like to thank A. Maurellis for comments on an earlier version of this paper. This research was supported by SRON under project number 6430-SCIARALI (GO-2).
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Appendix A. Perturbation of the radiative e4ect In this appendix, we provide a derivation of Eq. (35), which describes the linearization of radiative transfer employing internal sources. The radiation e;ect Ei is given by Ei (x) = R˜ i; x |Ix = R˜ i; x |Idi; ; x + Idir; x ;
(A.1)
where the subscript x denotes the atmospheric state. Thus, the perturbation of the radiative e;ect OEi due to a perturbation Ox = x − x0 in the atmospheric state vector can be calculated by OEi = Ei (x) − Ei (x0 ) = R˜ i; x |Idi; ; x + Idir; x − R˜ i; x0 |Idi; ; x0 + Idir; x0 :
(A.2)
For small perturbations Ox, Eq. (A.2) can be further simpli
(A.3)
Idi; ; x = Idi; ; x0 + OIdi;
(A.4)
Idir; x = Idir; x0 + OIdir :
(A.5)
and
This leads to OEi = R˜ i; x0 |OIdi; + OIdir + OR˜ i |Idi; ; x0 + Idir; x0 + OR˜ i |OIdi; + OIdir :
(A.6)
For small perturbations Ox of the atmospheric state the perturbations OR˜ i , OIdi; and OIdir are small too, and products of them, like in the last term of Eq. (A.6), lead to second-order perturbations. Therefore, OEi can be written as OEi = R˜ i; x0 |OIdi; + R˜ i; x0 |OIdir + OR˜ i |Idi; ; x0 + OR˜ i |Idir; x0 + O(Ox2 );
(A.7)
where O(Ox2 ) represents the perturbations of second and higher order. According to Eq. (33) R˜ i; x0 can be replaced by Lˆ †0 I˜†x0 which leads to OEi = Lˆ †0 I˜†x0 |OIdi; + R˜ i; x0 |OIdir + OR˜ i |Idi; ; x0 + OR˜ i |Idir; x0 + O(Ox2 ):
(A.8)
In a next step it is necessary to replace OIdi; in the
(A.9)
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Further, the perturbed forward RTE (28) in its form ˆ (Lˆ 0 + OL)(I di; ; x0 + OIdi; ) = Sdi; ; x0 + OSdi;
(A.10)
can be written as ˆ di; ; x0 + OSdi; − OLOI ˆ di; : Lˆ 0 OIdi; = −OLI
(A.11)
Substitution of Eq. (A.11) in Eq. (A.9) and again neglecting higher-order perturbation terms, results in ˆ di; ; x0 + I˜†x |OSdi; + OR˜ i |Idi; ; x0 OEi = −I˜†x |OLI 0
0
+OR˜ i |Idir; x0 + R˜ i; x0 |OIdir + O(Ox2 );
(A.12)
which is equivalent to Eq. (35) where second-order terms in the perturbations are neglected. In all inner products quantities occur which are either calculated for the unperturbed atmospheric state x0 or which can be calculated analytically. Thus, the determination of OEi becomes straightforward. Appendix B. Fourier expansion for azimuthal dependence In order to handle the integration over azimuth in Eqs. (28), (48) and (50) we use the Fourier expansion proposed by Hovenier and van der Mee [43] and de Haan et al. [39]. For its use within the context of perturbation theory see [23]. The expansion of the phase matrix is given by ∞ ˜ ) = 1 Z(z; ; (2 − 0m )[B+m (’˜ − ’)Zm (z; ; ˜ )(E + +) 2 m=0 ˜ )(E − +)]; + B−m (’˜ − ’)Zm (z; ;
(B.1)
where 0m is the Kronecker delta, + = diag[1; 1; −1; −1];
(B.2)
B+m (’) = diag[cos m’; cos m’; sin m’; sin m’]
(B.3)
B−m (’) = diag[ − sin m’; −sin m’; cos m’; cos m’]:
(B.4)
and
The mth Fourier coeScient of the phase matrix can be calculated by ∞ m m Z (z; ; ˜ ) = (−1) Pml (−)Sl (z)Pml (−); ˜
(B.5)
l=m
where Sl is the expansion coeScient matrix and Pml is the generalized spherical function matrix [39]. In the corresponding Fourier expansion of the di;use intensity vector two types of Fourier coef
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with +m (z; ) Idi;
1 = 2
1 Idi; (z; ) = 2 −m
2
0
d’ B+m (’0 − ’)Idi; (z; );
2
0
d’ B−m (’0 − ’)Idi; (z; ):
(B.7)
The pseudo-spherically modi
(B.8)
using Eqs. (B.1) and (B.7) as well as a Fourier expansion of the spherical radiation source Ssph (z; ). Thus, the transport operator in Eq. (B.8) can be written as 1 @ s (z) m m ˆ + e (z) ( − )E Z (z; ; d ˜ ˜ − ˜ ) L = @z 4 −1 A ˜ | ˜ ◦: (B.9) − 0m (z)()||(−)| m The Fourier coeScients S± sph of the radiation source Ssph are given by
S+m sph (z; ) =
1 s (z) m Z (z; 0 ; )(E + +)Tsph (z; 0 )F0 e1 2 4 + 0m
A (z)0 (z)Tsph (z; 0 )()||F0 e1
(B.10)
and m T S− sph (z; ) = [0; 0; 0; 0] :
(B.11)
+m The forward vector intensity
(B.12)
However, as the pseudo-forward source is de
1 1 s (z) mT Z (z; v ; −)(E + +)Tsph (z; v )ei 2 v (z) 4 + 0m
AT (z)Tsph (z; v )()||e1 i1 ;
m T S− & (z; ) = [0; 0; 0; 0] :
(B.13)
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Thus, for the corresponding pseudo-forward problems we obtain a Fourier expansion of ) containing terms of )+m only. For i = 3; 4 we obtain T S+m & (z; ) = [0; 0; 0; 0] ; m S− & (z; ) =
1 1 s (z) mT Z (z; v ; −)(E − +)Tsph (z; v )ei : 2 v (z) 4
(B.14)
Hence, for the corresponding pseudo-forward problems we obtain a Fourier expansion containing terms of )−m only. In this paper )m denotes either )+m or )−m , depending on the corresponding pseudo-forward problem. The Fourier coeScients for the spherical response function R˜ sph; i (z; ) follow immediately from Eqs. (B.13) and (B.14) with the de
(B.16)
For further details we refer to Hasekamp and Landgraf [23]. Appendix C. Calculation of derivatives with respect to trace gas pro(les m In this appendix we provide the derivatives Smsph , R˜ msph; i and Idir; sph of the Fourier coeScients of the spherical radiation source, of the spherical response function and of the direct beam with respect to an atmospheric trace gas pro
+ 0m
A (z)0 (z)Tsph (z; 0 )()||F0 e1 :
(C.1)
Here, Tsph (z; 0 ) = e−ch(z; 0 ) is the spherical transmission function, Eq. (44). The Chapman function within a model layer k is expressed by a linear interpolation in height ch(z; 0 ) = ak z + bk with coeScients ch(zk −1 ; 0 ) − ch(zk ; 0 ) ak = ; Ozk ch(zk ; 0 )zk −1 − ch(zk −1 ; 0 )zk bk = : Ozk
(C.2)
(C.3) (C.4)
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Assuming a constant scattering coeScient sk as well as a constant phase matrix Zmk within a model layer k, Eq. (C.1) can be written as Smsph (z; j ) =
1 sk m Z (0 ; j )(E + +)e−ch(z; 0 ) F0 e1 2 4 k + 0m
A (z)0 (z)e−ch(z; 0 ) ()||F0 e1
(C.5)
where zk −1 ¿ z ¿ zk . Thus, the derivative with respect to the ozone number density xk in a model layer k is given by @Smsph @xk
(z; j ) = −
@ch(z; 0 ) 1 sk m Z (0 ; j )(E + +)e−ch(z; 0 ) F0 e 1 2 4 k @xk
− 0m
A @ch(z; 0 ) (z)0 (z)e−ch(z; 0 ) ()||F0 e1 : @xk
(C.6)
The derivative of the interpolated Chapman function can be calculated in a straightforward manner using Eqs. (C.3), (C.4) and the dependence of the Chapman function, Eq. (40), at the layer boundaries on ozone. Finally, this results in @Smsph @xk
k (z; j ) = −$abs
1 sk m Oskk F0 e 1 Zk (0 ; j )(E + +)e−(ak z+bk ) (zk −1 − z) 2 4 Ozk
(C.7)
for zk −1 ¿ z ¿ zk and @Smsph @xk
1 sn m Ozk −(an z+bn ) Z (0 ; j )(E + +) e 2 4 n Ozn Oskn Oskn−1 Oskn−1 Oskn F 0 e1 × zn−1 − zn +z − Ozk Ozk Ozk Ozk
k (z; j ) = −$abs
(C.8)
for layers located below the perturbed model layer, i.e. zn−1 ¿ z ¿ zn where n ¿ k. Thus, the spherical source in all the layers n located beneath a given layer k is a;ected by a perturbation of the trace gas number density in this kth layer. At ground level, for z = zN , the derivative is @Smsph @xk
k (zN ; j ) = −$abs
A OskN 0 (zN )e−(aN zN +bN ) ()||Ozk F 0 e1 : Ozk
(C.9)
For altitudes above the perturbed model layer, i.e. z ¿ zk −1 , the derivative of the spherical radiation source with respect to the trace gas pro
k (z; j ) = −$abs
Ovk 1 1 sk mT Z (v ; j )(E + +)e−(ck z+dk ) (zk −1 − z) ei : 2 v (z) 4 k Ozk
(C.10)
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for zk −1 ¿ z ¿ zk . For the cosine v (z) in Eq. (C.10) we use a linear interpolation with height between the boundary values, de
(C.12)
(C.13)
For i = 3; 4 the derivatives are obtained in an analogous manner. For altitudes above the perturbed model layer, i.e. z ¿ zk −1 , the derivative of the spherical response function with respect to the trace gas pro
(C.15)
for zn−1 ¿ z ¿ zn , n ¿ k. At ground level, for z = zN , the derivative of the reNected direct beam is m @Idir; OskN sph k A 0 (zN )( − 0 )e−(aN zN +bN ) ()||Ozk (zN ; j ) = −$abs F0 e1 : (C.16) @xk Ozk Again for altitudes above the perturbed layer the derivative is always zero. References [1] Phillips P. A technique for the numerical solution of certain integral equations of the
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[6] Oikarinen L, Sihvola E, Kyrola E. Multiple scattering in limb viewing geometry. J Geophys Res 1999;104: 31261–74. [7] Caudill TR, Flittner DE, Herman BM, Torres O, McPeters RD. Evaluation of the pseudo-spherical approximation for backscattered ultraviolet radiances and ozone retrieval. J Geophys Res 1997;102:3881–90. [8] Rozanov A, Rozanov V, Burrrows JP. Combined di;erential–integral approach for the radiation
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