development of the linearized vector radiative transfer...

RT SOLUTIONS, Inc. 9 Channing Street, Cambridge, MA 02138.

Final Report O3SAF-VS, 2005

Development of the Linearized Vector Radiative Transfer Model VLIDORT

(Part 2)

Robert J. D. Spurr

Director RT Solutions, Inc.

9 Channing Street, Cambridge, MA 02138

31 December 2005

Co-Investigators

Dr. Jukka Kujanpaa Ozone and UV Radiation Research,

Finnish Meteorological Institute, PO.Box 503, 00101 Helsinki, Finland

Dr. Roeland Van Oss Royal Dutch Meteorological Institute

P. O. Box 201 3730 AE de Bilt, The Netherlands

Contractual Tarja Riihisaari

Ozone and UV Radiation Research, Finnish Meteorological Institute, Vuorikatu 19, P.O.BOX 503, FIN-00101 Helsinki, Finland

RT Solutions, Inc. Tel: +1 617 492 1183 9 Channing Street, Cambridge, MA 02138, USA Fax: +1 617 354 3415 Email: [email protected] EIN: 20 1995227 Citizens Bank, 6 JFK Street, Cambridge MA 02138, USA Routeing 211070175, Account 1139673456

O3SAF-VS 2005 Final Report VLIDORT Development, Part 2

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1. Introduction This is the Final Report (FR) for the 2005 O3 SAF Visiting Scientist (VS) grant for the second-stage development of the vector radiative transfer model VLIDORT. The first-stage development of VLIDORT was completed as part of the 2004 VS grant [1]. Following this work in 2005, a number of improvements and additions were made to the code, and these were described in the 2005 O3 SAF VS Mid-Term Report [2]. Previous VS activity by PI Robert Spurr has contributed in a major way to the LIDORT development; this includes LIDORT Versions 2.1 and 2.3 (VS 2000 and VS 2001, see [3-4]), LIDORT-RRS and LIDORT V2.4 (VS 2002 [5]), LIDORT 2.2+ (VS 2003 [6]).

In 2004, VLIDORT was validated against Siewert’s benchmark results [7] and older Rayleigh atmosphere results [8]; see [1] for details. The model was prepared for first release to FMI and two other users at the beginning of October 2004, following a visit by R. Spurr to FMI. The first O3 SAF application for the GOME-2 project was directed at polarization sensitivity studies on the UV index algorithm. The second-stage VLIDORT development is the subject of the VS 2005 activity, and the work is divided into two parts. The first (part A) has already been completed and it was concerned with the implementation of additional single-scatter and sphericity options in the VLIDORT model, and the introduction of a number of performance improvements. This work was summarized in the Mid-term report [2].

Part B of the VS 2005 work concerns the development of a linearization facility for VLIDORT. This work involves the complete differentiation of the polarized radiative transfer scattering theory in a multilayer atmosphere, with the result that VLIDORT can now simultaneously generate analytic atmospheric and surface property weighting functions alongside regular Stokes vector radiation fields. In general, the linearization of the vector model follows methods developed for the scalar LIDORT code [3-5], though there are some notable differences in the treatment of the homogeneous solutions of the vector radiative transfer equation (RTE).

Part B is covered in this Final Report, which is organized as follows. Section 2 describes the linearization of the vector RTE solution in the absence of the solar scattering source term, with particular attention being paid to the presence of complex variables. This section represents the key new work in the linearization. Section 3 describes the linearization of the remaining steps in the solution of the radiation field: the particular solutions of the RTE due to the solar scattering terms, the boundary value problem for the solution of the discrete ordinate field, and the post-processing function for the generation of weighting functions at arbitrary viewing geometry and optical thickness. In Section 4, we present some issues concerning the VLIDORT package. In Section 5 we give some examples, including two studies pf ozone profile weighting functions for wavelengths in the Huggins bands. In particular, we examine the error induced by the neglect of polarization in the computation of weighting functions.

The first public release (including the linearization facility) of VLIDORT Version 2.0 will follow shortly after the dissemination of this report. This release of the code will be accompanied by a User’s Guide and a GNU-type public license, and will be available from the RT SOLUTIONS Web site (currently under construction).


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2. Linearization of the Homogeneous Vector RTE 2.1. Homogeneous RTE: eigenproblem reduction We consider first the vector radiative transfer equation (RTE) in a single layer n. The optical properties describing the layer are its optical thickness ∆n, the layer total single scatter albedo ωn, and the 4x4 matrix Βn of expansion coefficients for the Legendre polynomial expansions of scattering quantities pertinent to polarized light scatter. As with the scalar code, the RTE is solved first by a separation into Fourier cosine and sine series in the relative azimuth. For each Fourier term m, the multiple scatter integral over the upper and lower polar direction half spaces is approximated by a double Gaussian quadrature scheme. The resulting vector RTE for Fourier component m in the absence of any solar source terms is then:

∑∑=

−+

=

±±

−+±=±±N

jj

mljj

mlj

L

mlnli

ml

ni

ii xxx

dxxd

1)()()()()(

2)(

)(µµµ

ωµ PIPIBPI

I (1)

Here, N is the number of discrete ordinate streams in the half-hemisphere, and with 4 Stokes vector components, there are 8N equations for the radiation field at the discrete ordinates ±µ

)(xi±I

i, i = 1, …N. Each 4-vector I consists of Stokes components I, Q, U, V [9], and x is the optical thickness measured from the top of the layer. The other matrix quantities are:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−=

nlnl

nlnl

nlnl

nlnl

nl

δεες

αγγβ

0000

0000

B ; (2)

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

=

)(0000)()(00)()(0000)(

)(

µµµµµ

µ

µ

ml

ml

ml

ml

ml

ml

ml

PRTTR

P

P

nlB is the matrix of “Greek constants” that is the input scattering law for the vector RTE. The matrix contains entries of normalized Legendre functions and similar quantities (for details, see [7]).

)(µmlP )(µm

lP

Solutions for these homogeneous equations have the form , and we use sum and difference fields to reduce the order of this equation from 8N to 4N. If we define vector , then substitution of this ansatz in the RTE yields an eigenproblem for a collection of separation constants k

kxex −±± = WI )(

−+ += WWXα and the associated 4N-vector solutions Xα.

The 4Nx4N eigenmatrix Γn is constructed from the optical property inputs ωn and Βnl and products of associated Legendre polynomials . The details need not concern us here; see for example Siewert [7] for a derivation. This eigenproblem may have complex roots, and the form of Γ

)( jml µP

n is such that both the left and right eigenvectors share the same spectrum of eigenvalues k2. Dropping the layer index n, the eigenproblem is:

XΓX ˆˆ 2k=∗ and (3) 2kXXΓ =∗


4

Here, the (^) symbol indicates the conjugate transpose of the appropriate 4N vector. Solutions may be determined from the LAPACK eigensolver DGEEV. This returns eigenvalues and left and right eigenvectors. In the scalar DISORT [10] and LIDORT models, the eigensolver module ASYMTX is used for this problem. ASYMTX is a modification of the LAPACK routine that deals only with real roots: for the scalar case, the eigenmatrix Γ is symmetric and there are no complex roots. Without the requirement to find complex roots, ASYMTX is on average about twice as fast as DGEEV. ASYMTX only returns the right eigenvectors. For more discussion on the choice of Eigensolver, see section 4.2.

2.2. Linearization of the eigenproblem In order to find derivatives of the multiple scatter radiance fields with respect to some variable ξ in layer n, we first find the derivatives of the above eigenvectors and separation constants. The basic input to the RTE is the set of linearized optical properties [L(∆n), L(ωn), L(Βn)], where L(*) indicates the normalized derivative ξ∂(*)/∂ξ.

The eigenmatrix Γn is a linear function of the single scatter albedo ωn and the Greek matrix of Legendre expansion coefficients Βn, and its linearization is easy to establish from chain-rule differentiation. This linearized matrix L(Γn) is a real matrix.

In order to proceed further with the linearization, we will make use of both equations (3) and some results using the left (adjoint) eigensolutions. Differentiating (3), we find that

)ˆ(ˆ)(2)(ˆ)ˆ( 2 XXΓXΓX LLLL kkk +=∗+∗ (4a)

)()(2)()( 2 XXXΓXΓ LLLL kkk +=∗+∗ (4b)

We now form a dot product by pre-multiplying (4b) with the transpose vector : X

⟩∗⟨−⟩⟨=⟩∗⟨−⟩⟨ )(,ˆ)(,ˆ)(,ˆ,ˆ)(2 2 XΓXXXXΓXXX LLLL kkk (5)

However, we see that from the definition in Eqs. (3), and hence the right hand side of (5) is identically zero. We thus have:

⟩⟨=⟩∗⟨=⟩∗⟨ )(,ˆ)(,ˆ)(,ˆ 2 XXXΓXXΓX LLL k

⟩⟨⟩∗⟨

=XX

XΓX,ˆ2)(,ˆ

)(k

k LL (6)

This result is familiar from adjoint perturbation theory. For complex separation constants, we evaluate both the real and imaginary parts of L(k). Once we have established the linearization of the separation constant k from (6), we substitute back in (4b) to obtain the following 4Nx4N linear algebra problem:

(7) CXΗ =∗ )(L

where , EΓH 2k−= XΓXC ∗−= )()(2 LL kk and E the identity matrix. Implementation of this equation “as is” is not possible due to the degeneracy of the eigenproblem, and we must impose additional constraints to find the unique solution for L(X). The treatment for real and complex solutions is different.


5

Real solutions. Here, we use the dot-product normalization . Linearization yields the single equation:

1.ˆ =⟩⟨ XX

0XXXX =∗+∗ )(ˆ)ˆ( LL (8)

The solution procedure uses 4N−1 equations from the system in Eq. (7), along with the single Eq. (8) to form a slightly modified linear system of rank 4N. This system is then solved by standard means using the DGETRF and DGETRS LU-decomposition routines from the LAPACK suite.

This procedure was not used for LIDORT. This is because of the lack of the adjoint solution from ASYMTX. In other words, Eq. (6) is absent, so the linearization L(k) must be determined from scratch using another method. The solution for LIDORT was to use the complete set of 4N Eqs. (7) in addition to the constraint Eq. (8) to form a 4N+1 system for the unknowns L(k) and L(X); see Spurr [3] for details. See also the discussion below in section 4.2.

Complex solutions. In this case, Eq. (7) is a complex problem for both the real and imaginary parts of the linearized eigenvectors. There are 8N equations in all. There are two constraint conditions implied by the degeneracy. The first is Eq. (8), where now the conjugate transpose solutions must be used in the complex variable treatment. The second condition is imposed by the DGEEV normalization: for the component of the eigenvector with the largest real value, the corresponding imaginary part is set to zero. Thus for an eigenvector with label a, if Xam = maxRe[Xaj] for j = 1,.. 4N, then Im[Xam] = 0. In this case, it is also true that L(Im[Xam]) = 0. This is the second condition.

The solution procedure is then (a) to take the 8N system in Eq. (7) and strike out the row and column in matrix H corresponding to the quantity Im[Xam] (which is set to zero), and strike out the corresponding row in the right-hand vector C; and (b) in the resulting 8N−1 system, replace one of the rows with the constraint Eq. (8). The desired linearization then follows by solving the resulting linear algebra system with DGETRF and DGETRS from LAPACK.

This is the most crucial step in the linearization process, and the only one that is notably different from its scalar equivalent. For the remaining linearizations, we follow closely the principles laid down for the LIDORT scalar model [3-5].


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3. Linearization of the Complete Vector Radiation Field 3.1. Linearizing the particular solutions The particular integral solutions of the vector RTE arise from the solutions of Eq. (1) with the additional term:

)exp()()(2

),( 0 xTx nnm

lnl

L

mli

mli

m λµµωµ −−±=± ∑=

DFPBPQ (9)

This represents primary scattering by the solar beam, with flux F = F,0,0,0, and solar zenith angle cosine 0µ− . Here, Tn is the transmittance to the top of layer n, and in the pseudo-spherical treatment of solar beam attenuation, λn is the average secant in layer n. In the plane-parallel approximation, 0/1 µλ −=n . Details of the pseudo-spherical formulation may be found in [3]. The matrix D = diag1,1,0,0 and the other matrices have been defined in section 2.1.

Continuing, we note that particular solutions have the form . In solving Eq. (1) with source term Eq. (9), the order of the discrete ordinate equations for the particular solution may again be reduced from 8N to 4N (see for example [4] for the scalar case). We define the vector ; we have deliberately retained the layer indexing now. Substituting this second ansatz in the RTE to yields the following linear algebra problem:

)exp()( xλx nn −= ±± ΦI

−+ += nnn ΦΦZ

nnn CZΗ ~~ =∗ (10)

where EΓΗ 2~nnn λ−= , with E the identity matrix. Vector nC~ is constructed from the

optical property inputs, the solar beam transmittance and average secant for this layer, and the appropriate combinations of Legendre functions relevant for light rays scattered out of the solar beam into the discrete ordinate directions. Again the details need not concern us here (for the scalar case, see Spurr [3] for example). The computations are very similar to those for the scalar case. [VLIDORT does not use the Green’s function solution methodology as developed for the slab problem by Siewert [7], mainly for reasons of algebraic simplicity].

For the linearization, we note that Eq. (10) is differentiable with respect to atmospheric variables ξp in all layers p ≥ n. Differentiation yields a related linear algebra problem:

nnpnpnpn ZΗCZH ∗−=∗ )~()~()(~ LLL (11)

This has the same matrix as in the original equation (10), but a different source vector. The solution is then found by back-substitution, given that the inverse of the matrix has already been established in the solution of the original problem for Zn in Eq. (10). The linearizations of the right hand quantities may be found by differentiation, bearing in mind some simple rules. The principles are exactly the same as those for the scalar problem, and we are not going to give details here. We merely note that the particular solutions are all real; there are no complications with complex variables.


7

3.2. Linearizing the boundary value problem The Stokes vector solutions for all discrete angles in layer n may be written:

(12) [ ] xn

Nxnk

nnxk

nnnneeex λ

α

ααα

ααα

−±

=

−−−±± +∗+∗= ∑ WXMXLI1

)()( ∆m

Here, are homogeneous solutions derived from the eigenproblem, with separation constants k

±αnX

α, and are the particular solutions corresponding to the solar source term with exponential dependence on the average secant λ

±nW

n. Quantities and comprise the matrices of

αnL αnMconstants of integration for the homogeneous solutions, and

these are determined by the imposition of three boundary conditions:

I. No diffuse downwelling radiation at TOA: for n = 1. 0)0( =↓nI

II. A surface reflection condition relating the upwelling and downwelling radiation fields at the bottom of the atmosphere: for n = N)()( nnnn ∆∆ ↓↑ ∗= IRI A (the number of layers in the atmosphere), and reflection matrix R; ∆n is the optical thickness of the lowest layer of the atmosphere.

III. Continuity of the upwelling and downwelling radiation fields at intermediate boundaries: for n = 2, … N)0()( 11

±−

±− = nnn II ∆ A.

For the homogeneous solutions, we need both the real eigensolutions and the real parts of the complex eigensolutions when writing down these boundary conditions. Application of these conditions yields (for a multi-layer atmosphere) a large sparse tri-diagonal banded linear matrix algebra system [3-4, 10] for the quantities and . This system has rank 8NxN

αnL αnMA and consists only of real variables. It may be written in the

symbolic form:

QYP ~~ =∗ (13)

The solution proceeds by LU-decomposition using the LAPACK routine DGBTRF to find the inverse , and the final answer is then obtained by back-substitution (using DGBTRS). The back-substitution routine may be used many times once the inverse has been established. For the slab problem (single layer medium), boundary condition (III) is absent, and the associated linear problem is not banded and may be solved using the DGETRF and DGETRS routines (as employed for the particular solutions in sections 2.2). Linearizing Eq. (13) with respect to a variable ξ

1~ −P

p in layer p, we obtain:

YPQYP ~)~()~()(~ ∗−=∗ pnpp LLL (14)

We notice that this is the same linearization problem, but now with a different source vector on the right hand side. Since we already have the inverse obtained by DGETRF or DGBTRF during the determination of the original boundary value problem, it is straightforward to apply back-substitution (by calling DGETRS or DGBTRS) to obtain the linearization of the boundary value constants.

1~ −P


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3.3. Post-processed solutions in VLIDORT and their linearizations So far, we have established the solutions and linearizations of the discrete ordinate multiple scatter Stokes vector field. The well-known source function integration technique is used to determine solutions at off-quadrature polar directions µ and at arbitrary optical thickness values in the multilayer medium. The mathematical exposition of this technique may be found in the literature (see for example [11]), and we will only note down the principal results. The methodology follows closely that used in the linearization of the scalar LIDORT code, so long as we remember with the Stokes-vector formulation to use the real parts of any complex solutions in the derivations.

The upwelling solution in layer n at direction µ for optical thickness x (as measured from the top of the layer) is given by

( ) ),()()(),(),(),( /)( µµµµµµ µ xxex nnnnx

nn++++−−++ +++= EQDHII ∆∆ (15)

This comprises a transmission term (the first), and three contributions which together constitute the layer source term. The first is due to the homogeneous solutions and has the form

[ ]∑=

−+−++++ ∗+∗=N

nnnnn xxx1

),()(),()(),(α

αααααα µµµµµ nn HH XMXLH (16)

∑∑=

±

=

± ±=N

jinj

mlnl

L

ml

mln

1)()()(

2)( µµµωµ αα XPBPX m (17a)

α

µ

α µµ

αα

n

xkxk

keeex

nnnn

+−

=−−−−

++

1),(

/)(∆∆

nH (17b)

α

µ

α µµ

α

n

xkx

keex

nnn

−−

=−−−−

−+

1),(

/)()( ∆∆

nH (17c)

Here, are the homogeneous solutions defined at user-defined angles, and are the homogeneous solution multipliers for the upwelling field. Similar

expressions exist for the downwelling field. Some of the solution vectors, integration constants and multipliers will contain complex variables, and we must remember to take the real part when computing the source term Eq. (16).

)(µα±nX

),( µα x±+nH

The other two layer source term contributions come from the diffuse and direct solar source scattering respectively. All variables are real numbers, and the relevant quantities are:

∑∑=

+

=

+ −=N

jjnj

mlnl

L

ml

mln

1)()()(

2)( µµµωµ ZPBPD (18a)

DFPBPQ )()(2

)2()( 0

0 µµδω

µ −−

= ∑=

+ mlnl

L

mli

ml

mn (18b)


9

n

xx

n

nnnn eeeTxµλ

µµλλ

+−

=−−−−

+

1),(

/)(∆∆

nE (18c)

These expressions have similar counterparts in the scalar code (see for example [3]).

The point here is that all quantities are known in terms of the basic optical property inputs to VLIDORT ∆n, ωn, Βnl, and the homogeneous solutions, particular solutions and boundary value integration constants of the previous sections.

Linearizations. Derivatives of all these expressions may be determined by chain-rule differentiation with respect to any variable ξn in layer n. The end points of the chain rule are the linearized optical property inputs Ln(∆n), Ln(ωn), Ln(Βnl). For linearization of the homogeneous post-processing source term in layer n, there is no dependency on any quantities outside of layer n; in other words, for p ≠ n. At the end, we require the real part of this linearization. The particular solution post-processing source terms in layer n depend additionally on optical thickness values in all layers above and equal to n through the presence of the average secant and the solar beam transmittances, so there will be cross-layer linearizations. However the chain-rule differentiation method is the same, and requires a careful exercise in algebraic manipulation. All linearizations of the particular solutions are real.

0)],([ ≡+ µxnp HL

The above expressions [Eqs. (17) and (18)] for the multipliers in have appeared a number of times in the literature. The linearizations were discussed in [3] and [4], and we need only make two remarks here. Firstly, the real and complex homogeneous solution multipliers are treated separately, with the real part of the complex variable result to be used in the final reckoning. Second, the multipliers for the particular solutions are exactly the same as those for the scalar model, so software developed for LIDORT for these quantities was used directly in VLIDORT.

2.4. Linearization of the exact single scatter solutions In part A of the VS 2005 work with VLIDORT [2], we introduced an exact single-scatter computation based on the Nakajima-Tanaka (N-T) correction developed for the scalar discrete ordinate models [12]. When this correction is used, VLIDORT is executed in “multiple-scatter mode”, with the post-processing calculation of the single scatter contribution suppressed in favor of the more accurate N-T single scatter computation which uses all possible phase matrix information without the truncation inherent in the use of the discrete ordinate quadrature approximation. With no N-T correction, VLIDORT’s regular internal single scatter computation necessarily uses a truncated subset of the complete scatter-matrix information, the number of usable Legendre coefficients being limited to 2N−1 for N discrete ordinate streams. The implementation in VLIDORT is very similar to that in the scalar model. Confining our attention to a single layer, the (upwelling) post-processed solution in stream direction µ is

)()()()( µµµµ µτexactbottomtop e SMII ++= − . (19)

Here, M(µ) is the multiple scatter layer source term from VLIDORT and Sexact(µ) is the single scatter layer source term taken from a dedicated module which uses all possible


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scattering matrix information. Similar expressions can be written for the downwelling field. We may write:

⊕−= FΨS ),,(),(E~

)( n, 00

0

4φφµµ

πµµω

µ nn

exactn . (20)

Here F⊕ is the solar flux vector, nω~ is the (delta-M scaled) single scattering albedo in layer n, En(µ, µ0) is the source function multiplier obtained by integrating exponential transmittance factors over the layer optical thickness (Eq. (18c)), and Ψn is the scattering matrix which depends linearly on the Greek matrix Bn of Legendre expansion coefficients.

The application of chain-rule differentiation to Eq. (20) will yield the linearization of the exact single scatter correction term. Linearization of the multiplier En(µ, µ 0) has already been established in the previous section on the post-processing differentiation. Since the phase matrix Ψn is a multiple expansion in terms of Legendre functions, its linearization is straightforward to write down in terms of the inputs . ][BnnL

The delta-M scaling is an important part of the Nakajima-Tanaka correction [12]. When this scaling has been applied, optical thickness values and single scattering albedos used in this result have been scaled by the delta-M approximation, while the Greek coefficients are used in their unscaled form. We therefore need linearizations of the delta-M scaling truncation factors, and these may be established from the basic definitions (see [2], section 2.1).

Note. In VLIDORT the Nakajima-Tanaka single scatter correction is now done before the main discrete ordinate calculations of the diffuse field. The single scatter radiation is computed at all output optical depths and then added to the Fourier m = 0 component of the diffuse field in the absence of the truncated single-scatter contribution. The linearized exact single scatter corrections will also be added at this stage to the weighting functions. VLIDORT will converge a little more quickly, since higher-order Fourier components of the diffuse fields are comparatively smaller when compared with the total fields [M. Christi, private communication, see also [13]).

2.5. Additional linearizations BRDF and surface property linearization. In Part A, we reported on the implementation in VLIDORT of the scalar 3-kernel BRDF scheme as used in LIDORT V2.4 [5], in which the BRDF ),,( φφµµρ ′−′ is specified as a linear combination of up to 3 semi-empirical kernel functions:

∑=

′−′=′−′3

1);,,(),,(

kkkkA Wφφµµφφµµρ Ω (21)

Here, Ak are the kernel amplitudes, and for each kernel there is a vector of parameters Wk which specify the shape (for instance W is just the wind-speed for the Cox-Munk glint kernel). The 1, 1 elements of kernels Ωk are fully specified by the normal scalar functions as used in [5]. The BRDF implementation works through the development of a


11

Fourier cosine series expansion of the above kernel functions, and we assume that a cosine series still applies for the vector development (this is a valid assumption with sunlight). Linearization of this BRDF scheme was reported on in [5], and a mechanism developed for the generation of surface property weighting functions with respect to the linear kernel coefficients and also to the non-linear kernel parameters. This linearization scheme has been taken over in its entirety for VLIDORT, and we need only make one remark here concerning post-processing of the radiation field in the presence of BRDF surfaces.

In Part A [2] it was noted that the field at the bottom of the atmosphere (BOA) is expressed as the sum of diffuse and direct components for each Fourier term, and that it is possible to replace the direct part of the reflected beam using a precise set of BRDF kernels rather than their truncated forms. Thus an option for an exact “direct beam (DB) correction” (akin to the exact N-T single scatter correction) was implemented in VLIDORT. This DB correction is done before the main discrete ordinate diffuse field calculation, and the result is added to the Fourier m = 0 field before the convergence in Fourier cosine is examined. The only additional requirement is for an exact computation of the derivatives of this DB correction with respect to the surface kernel factors and parameters. For atmospheric weighting functions, the solar beam transmittance that forms part of the DB correction also needs to be differentiated with respect to variables ξp varying in layer p, but this is also straightforward.

Enhanced Sphericity option (VLIDORT 2.0+). This option was discussed in Part A of the VS 2005 work, and the enhanced sphericity extension refers to the treatment of both the incident solar beam and the line-of-sight path for a curved atmosphere. This option has been designed only for the correction of the TOA radiation field in remote sensing applications. An exposition of this method can be found in [3-4] and [6] and in the VS 2003 report. The principles are exactly the same for VLIDORT – the only difference is in the vectorization of the exact single scatter correction for varying geometry along the line of sight. Examples of this correction were presented in Part A of the VS 2005 work [2]. Linearization of this single scatter correction follows the outline given in section 2.4 above.


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4. Computational Issues 4.1. Performance enhancements introduced in Version 2.0 (Part A) Vector codes are computationally slower than their scalar versions, nominally by a factor of ~16 (for a matrix treatment of Stokes 4-vectors); however the use of complex eigen-solvers also slows the models down. To counteract this deficit, the codes can be made faster by implementing time saving devices, and these are especially useful in UV Index applications for which there is a single cloud layer in a otherwise Rayleigh atmosphere. In the VS 2005 MTR [2] the following two performance enhancements were described:

1. Solution Saving. This avoids RTE solution computation for layers with no scattering (e.g. Rayleigh conditions for Fourier m > 2), in which the RTE is transmittance-only;

2. BVP Telescoping. For a multi-layer atmosphere with one block of contiguous layers with scattering sources, the boundary-value problem can be “telescoped” so that it is only necessary to make a matrix-algebra computation of constants of integration for the block of active scattering layers. Constants of integration in the non-active layers (no scattering sources) are then found by transmittance.

For layers with no scattering, it is straightforward to write down linearizations of the transmittance solutions, and the principles already outlined for the BVP linearization in section 2.2 apply equally to the telescoped problem. Details are not given here, but the complete linearization of these two performance options is part of the VLIDORT model.

4.2. Eigensolver Usage In section 2.1, we have already noted differences between the LAPACK solver DGEEV and the condensed version ASYMTX as used in LIDORT and DISORT. DGEEV must be used for any layers with scattering by aerosols or clouds, since there will be complex roots in this case. ASYMTX only deals with real symmetric eigenmatrices. Linearization of the homogeneous solutions from DGEEV uses adjoint theory and has some subtleties; adjoint solutions are not available for ASYMTX.

It turns out that, aside from additional elements down the diagonal, the eigenmatrix Γn in layer n consists of blocks of 4x4 matrices of the form , where the P and B

)()( Tjlmnlilm µµ PBP

nl matrices were defined in Eq. (2) in section 2.1, iµ are the discrete ordinates, and the ‘T’ superscript denotes matrix transpose. Since P and PT are symmetric, then Γn will be symmetric if Βnl is. Thus Γn will be symmetric if the Greek constants εl in Βnl are zero for all values of l. This is a special case that is satisfied by the Rayleigh scattering law, but in general this is not true for scattering with aerosols and clouds.

For aerosols and clouds we require the complex eigensolver DGEEV from LAPACK, but for Rayleigh scattering we can use the faster “real-only” ASYMTX package. Our policy in VLIDORT will be to retain both eigensolvers and use them as appropriate – if any of the Greek constants εl in Βnl is non-zero for a given scattering layer, then we will choose the complex eigensolver in that layer. The use of ASYMTX and its linearization for Rayleigh layers will represent a considerable saving in processing time. For an


13

application with a few particulate layers in an otherwise Rayleigh-scattering atmosphere, both eigensolvers will be required.

4.3. Input optical property requirements In this section we give a brief introduction to the input requirements for VLIDORT, in particular the determination of optical property inputs (including linearized quantities). It is already clear that for a Stokes vector computation using VLIDORT, we require the input set ∆n, ωn, Bnl for each layer n, where ∆n is the total optical thickness, ωn the total single scatter albedo, and Bnl the set of Greek matrices specifying the total scattering law. As an example, we consider Rayleigh scattering by air molecules; trace gas absorption, and scattering and extinction by aerosols. Dropping the layer index, if the Rayleigh scattering optical depth is δRay and trace gas absorption optical thickness αgas, then the total optical property inputs are given by:

aerRaygas τδα ++=∆ ; ∆

Rayaer δδω

+= ;

aerRay

aerlaerRaylRayl δδ

δδ+

+= ,, BB

B . (22)

The Rayleigh Greek matrix coefficients are tabulated here:

αl βl γl δl εl ζll=0 0 1 0 0 0 0 l=1 0 0 0

ρρ

+−

2)21(3 0 0

l=2 ρρ

+−

2)1(6

ρρ

+−

2)1(

ρρ

+−

−2

)1(6 0 0 0

For zero depolarization ratio, the only surviving Greek constants are: β0 = 1.0, β2 = 0.5, α2 = 3.0, γ2 = −√6/2 and δ1 = 1.5. Aerosol quantities must in general be derived from a suitable particle scattering model (Mie calculations, T-matrix methods, etc.).

For the linearized inputs with respect to a parameter ξ for which require weighting functions, we define normalized quantities:

ξ

ξφξ ∂∂

=∆

∆;

ξω

ωξϕξ ∂

∂= ;

ξξ

ξ ∂∂

= l

ll

BB

Ψ , . (23)

These may be established by differentiating the definitions in Eq. (22). In Section 5 below we give some applications of these formulae.

4.4. Overview of the VLIDORT Package It has been necessary with the linearization to introduce another layer of superstructure to the VLIDORT numerical package, and it is appropriate here to summarize the organization of the VLIDORT software. Much greater detail may be found in the VLIDORT 2.0 User’s Guide which accompanies this report.

The directory structure is summarized in Figure 1. From the parent directory, there are 5 main subdirectories. The “Lv2_environment” subdirectory contains the calling programs for VLIDORT, the ‘makefile’, all executables and shell scripts, input configuration files


14

to read control options and any user-defined atmospheric setups (that is, data files) containing pre-calculated optical property inputs.

Parent Directory

Figure 1. Directory structure for the VLIDORT 2.0 installation package.

All VLIDORT source code has been collected in one subdirectory (“sourcecode”). In the “includes” subdirectory, variables stored in F77 common blocks are classified into 10 files with the extension “.VARS”, with 5 of these files containing variables required for the radiation field, and an additional 5 files needed when computation of Jacobians is done. The “LAPACK” directory contains auxiliary numerical modules from the LAPACK suite of numerical software packages. Object files for the VLIDORT code are stored in a separate directory “OBJECTS” for convenience.

The division into “sourcecode” and “includes” directories follows the same bookkeeping system introduced for LIDORT Version 3.0. Further, the include file-names follow the same bookkeeping as that employed for files in LIDORT, and there is an equivalence in the naming of many variables. Thus users of the LIDORT code will find many things that are familiar in the vector code.

Postscript. Following some feedback from Jukaa Kujanpaa, memory requirements have been considerably improved by the removal of an integer masking array for the boundary value problem. This array has been replaced by an integer function. A number of time-saving operations have also been introduced.

4.5. Benchmarking of VLIDORT The original VLIDORT code was validated against the slab-problem results of Siewert [7], and for Rayleigh atmospheres, against the Coulson-Dave-Sekera tables [8]. These validations were reported in [1]. Siewert solved his slab problem [7] using discrete ordinate methods, so one would expect the validation to be precise since VLIDORT was to some extent modeled on the formalism in [7].

However, an additional benchmarking for VLIDORT has been done against the results of Garcia and Siewert [14] for another slab problem, this time with albedo 0.1 (the original slab problem [7] used a dark lower surface). With VLIDORT set to calculate using only 20 discrete ordinate streams in the half space, tables 3-10 in [14] were reproduced to

Lv2_environment “makefile”

input/data files executables output files

includes Pars file Vars files

sourcecode: F77 subroutines

OBJECTS Object files

LAPACK Auxiliary Modules


15

within 1 digit of six significant figures. This result is noteworthy because the radiative transfer computations in [14] were done using a completely different radiative transfer methodology (the so-called F-N method).

For Jacobians, it is only necessary to validate the derivative by using a finite difference estimate (ratio of the small change in the Stokes vector induced by a small change in a parameter in one layer):

δξδ

ξξIIK ≈

∂∂

≡ . (24)

Verification of each stage of the linearization may also be done in this way – a lengthy and tedious process that requires a lot of care – not recommended for Users.

VLIDORT has a number of options for Jacobians. Finite difference testing was carried for the following sets of basic tests.

1. Slab problem, plane parallel, one layer, arbitrary optical depth and user angles, upwelling and downwelling. This is the basic test, done using “Problem II” in [14] (see above).

2. Slab problem divided into a number of optically identical layers. This tests the multi-layer facility in the model, as the invariance principle dictates that results should reproduce those for the single-layer slab.

3. Multilayer atmosphere with Rayleigh and Aerosol, Pseudo-spherical approximation. Scalar results validate against LIDORT Version 3.0.

4. Delta-M approximation and Nakajima-Tanaka corrections validated for all cases.

5. Performance options [2]. Solution Saving validated for the linearized code for atmospheric weighting functions.

6. All surface BRDF weighting function options validated for scalar case against LIDORT. Lambertian albedo Jacobian validated in Vector case.

Still to do at the time of writing are the following tests.

1. The single scatter correction for VLIDORT 2.0+ still needs to be linearized and tested. This is easy to accomplish; once done, the linearization of the complete sequence of VLIDORT 2.0+ calculation will need to be tested.

2. Linearization of the BVP telescoping performance option has been completed but not yet tested. Further, the BVP option is currently confined to one active layer, and an extension will be made for the situation with one or more contiguous active layers – methodology will follow the LIDORT 3.0 work done already.

3. Flux and Mean intensity linearizations need to be tested.


16

5. Weighting Function Examples 5.1. Weighting functions for the Benchmark results The first results were obtained using the slab problem of Siewert [7]. This is a single layer, plane-parallel optically uniform medium with total optical thickness ∆ = 1.0, single scatter albedo ω = 0.973527, and with an 11-moment Greek-matrix scattering law for prolate spheroids (for the table of Greek-matrix coefficients, see [7]). For a Lambertian albedo of zero, this problem was used as one of the benchmarks for verifying the first versions of VLIDORT [2].

Figure 2. Jacobians for the Aerosol slab problem

In the present case, we calculate results for a variety of solar zenith angles, for a view angle of 10° and a relative azimuth of 0°. For this problem, two Jacobians were defined, one with respect to the optical thickness of the medium, the other with respect to its single scatter albedo. From Section 4.3, the linearized optical property inputs Eq. (23) are

1=∆φ , 0=∆ϕ , and for optical thickness, and 0, =∆lΨ 0=ωφ , 1=ωϕ , and 0, =ωlΨ for single scatter albedo.

Figure 2 shows some results for this case. In the top panel, the effect of including polarization is shown by contrasting the total intensity weighting function for the


17

upwelling radiation field at the top of the slab. The lower panel shows weighting functions P′ defined for the degree of linear polarization P:

IVUQ

P222 ++

= ; ( )

IIP

PIVVUUQQP

′−

′+′+′=′

2 (25)

Here, the Stokes vector is I = I,Q,U,V, and VLIDORT generates this as well as the weighting function output I′ = I′,Q′,U′,V′ where the prime symbol indicates parameter derivatives. In this case, both the polarization and its Jacobian are small because of the generally depolarizing effect of the aerosols.

5.2. Ozone profile Jacobians in the Huggins Bands It is well known that the neglect of polarization in the calculation of backscatter radiances in a Rayleigh atmosphere can lead to errors of up to 10%, depending in particular on the geometrical configuration (incident and scattering directions) [15-16]. Related calculations for profile weighting functions can of course be done using finite differences as in Eq. (24), but it is far superior to use the linearization facility of VLIDORT.

Figure 3. Ozone profile weighting functions at 325 nm.

For this example, we used a 22-layer stratified atmosphere with an ozone profile taken from a northern hemisphere mid-latitude region. Rayleigh scattering coefficients and depolarization ratios are taken from [17]. The surface was assumed to be Lambertian with albedo 0.05. Ozone cross-sections were taken from the GOME FM data set [18].


18

Calculations were done at a wavelength of 325 nm. In the absence of aerosol or cloud scattering, the layer optical property inputs are given by:

RayOC δσ += 3∆ ; ∆Rayδ

ω = ; Rayll ,BB = . (26)

Here, C is the partial column of ozone in any given layer. For ozone profile Jacobians, we require the derivatives in Eq. (23) as inputs, taken with respect to C. These are:

∆

∆∆

3OC

CC

C σφ =

∂∂

≡ ; ∆

3OC

CC

C σωω

ϕ −=∂∂

≡ ; 0, =∂∂

≡ξ

ξξ

l

ll

BB

Ψ . (27)

Figure 3 shows some results for three different solar zenith angles, in a nadir viewing scenario. The left panel shows the relative differences (in percentage) in the ozone profile weighting functions for total intensity I between the scalar and vector calculations. The right panel shows the weighting functions for the second Stokes component Q. Of interest here is the change of sign from stratosphere to troposphere for the lowest solar angle (penetrating furthest into the troposphere). The Q-component sensitivity indicates that there could be additional information available for ozone profile retrieval for a polarizing instrument such as GOME-2 measuring light parallel and perpendicular to the plane of the slit [that is, GOME-2 measures ½(Ι ± Q)].

The use of polarization measurements of backscatter to improve ozone profile retrievals is a profitable area of study for GOME 2, and VLIDORT would be ideally suited for this task, as it can generate Jacobians for all scenarios including those with optically thick layers. Indeed the retrieval of ozone profiles has already been looked at for ground-based polarization skylight measurements in a new synthetic study based on VLIDORT [21].

5.3. Ozone profile Jacobians with aerosol layer between 2-4 km In this example, we include some particulate matter in the atmosphere.

We retain the same ozone profile as in the previous atmosphere, and we assume a Rayleigh atmosphere as before, but now with an optically uniform aerosol layer placed between 2 and 4 km. The ozone and Rayleigh information is taken from the same source as above. For the clouds, we run the Mie program [19] to generate the necessary expansion coefficients. In this case the aerosol single scatter albedo was 0.91, and there were some 109 scattering law expansion coefficients, down to a threshold of 10-5. The input optical properties are given by Eq. (22) with 3Ogas Cσα = , the derivative optical property inputs by Eq. (27).

This time we perform all calculations for a viewing zenith angle of 35° and for a relative azimuth of 90°. A default value of 6 discrete ordinates was used for most calculations. Without any single scatter correction, this means that only 11 Fourier components can be used in the truncated calculations. We also did some runs with the single scatter correction turned on, using 79 Fourier moments for the exact single scatter, and we also ran the VLIDORT with 40 discrete ordinates to check the single scatter computations.


19

Figure 4. Ozone profile weighting functions, with some aerosols

Again we are interested in the scalar-vector difference, and this time we show differences in the Stokes vector total intensity as well as the weighting functions. Figure 4 shows the scalar/vector difference in the total intensity ozone profile Jacobians (top diagram), while the lower panels show the Q-component and U-component profile Jacobians. All plots are for the same three solar zenith angles as in the previous Figure. The large differences in the intensity weighting functions illustrate the need for a vector model.

A second set of runs was done to test the delta-M scaling approximation and the use of the Nakajima-Tanaka (NT) single scattering correction in the weighting function context. The delta-M scaling and NT corrections for VLIDORT were introduced in the mid-term report for the VS 2005 work [2]. Both options have now successfully been linearized and tested by finite-difference verification.

Without the NT correction, and with 6 discrete ordinates, the solutions did not converge after the maximum number of Fourier components (11), even with the delta-M scaling turned on. Only 6 Fourier components were required for the calculation with NT correction – a considerable saving of effort which comes about by the procedure of having the single scatter calculation done first before the main VLIDORT diffuse field computation. Figure 5 show differences in ozone profile weighting functions with and without the NT correction including delta-M scaling, for the I-component of the Jacobian (left panel) and the Q and U components (right panel). Results for the total intensity case


20

show similar dependencies and values for the scalar-only calculation without polarization. The results for the Q and U components are completely new.

Figure 5. O3 Jacobians using the Nakajima-Tanaka correction.


21

6. Concluding Remarks The linearization of the vector model VLIDORT is now almost completely finished. All the key parts of the code have been linearized and there are only a few performance “bells and whistles” to be added. A large effort has been made to bring the code up to the level of its scalar counterpart LIDORT Version 3.0. Both codes have been streamlined and reorganized so that the inputs and outputs are consistent. A Web site for RT Solutions is currently under construction, and it will be possible to download the VLIDORT and LIDORT codes from spring 2006 once the Web site is completed.

Interest in the VLIDORT code has expanded greatly in the last half of 2005. Apart from the original application to the O3 SAF UV Index algorithm, the model has been used extensively in development of the Level 2 Algorithm for the Orbiting Carbon Observatory Project (OCO) in order to test assumptions about the neglect of polarization (see for example [20]). The author will be continuing to contribute to this development in 2006 and it is anticipated that the few remaining tasks to be done for VLIDORT Version 2.0 will be completed in the first half of 2006.

The model has also been used as part of a synthetic study to demonstrate that the information content and sensitivity of ozone profile retrieval measurements from ground-based instruments may be significantly improved if polarized skylight measurements are used [21]. The model will also be used for new computations of the TOMS AAI index in the context of aerosol retrieval [R. Martin, private communication], and will be introduced to NASA GSFC in January 2006.

Acknowledgments The author would like to thank Jukaa Kujanpaa (Finnish Meteorological Institute) and Vijay Natraj (CalTech) for some very helpful user feedback. Radiative transfer colleagues Mick Christi (Colorado State University) and Knut Stamnes (Stevens Institute for Technology) are also acknowledged for a number of stimulating discussions on vector radiative transfer aspects.

Funding for the whole VLIDORT development came from two O3 SAF Visiting Scientist Grants for 2004 and 2005. The author is grateful for the opportunity to carry out this work; it has been a satisfying adventure.


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7. References [1] Spurr, R., Development of the Linearized Vector Radiative Transfer Model

VLIDORT (Part 1), Final Report, November 2004, O3SAF-VS. [2] Spurr, R., Development of the Linearized Vector Radiative Transfer Model

VLIDORT (Part 2), Mid-Term Report, June 2005, O3SAF-VS. [3] Spurr, R.J.D., Simultaneous derivation of intensities and weighting functions in a

general pseudo-spherical discrete ordinate radiative transfer treatment, JQSRT, 75, 129-175 (2002).

[4] Van Oss, R.F, and R.J.D. Spurr, Fast and accurate 4- and 6-stream linearized discrete ordinate radiative transfer models for ozone profile retrieval, JQSRT, 75, 177-220 (2002).

[5] Spurr, R.J.D., A New Approach to the Retrieval of Surface Properties from Earthshine Measurements, JQSRT, 83, 15-46 (2004).

[6] Spurr, R.J.D., LIDORT V2PLUS: A comprehensive radiative transfer package for UV/VIS/NIR nadir remote sensing; a General Quasi-Analytic Solution, Proc. S.P.I.E. International Symposium, Remote Sensing 2003, Barcelona, Spain, September 2003.

[7] Siewert, C.E., A discrete-ordinates solution for radiative transfer models that include polarization effects, JQSRT, 64, 227-254 (2000).

[8] Coulson, K., J. Dave and D. Sekera, Tables related to radiation emerging from planetary atmosphere with Rayleigh scattering, University of California Press, Berkeley, 1960.

[9] Chandrasekhar, S., Radiative Transfer, Dover Publications Inc., New York, 1960. [10] Stamnes, K., S.-C. Tsay, W. Wiscombe, and K. Jayaweera, Numerically stable

algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media, Applied Optics, 27, 2502-2509 (1988).

[11] Stamnes, K. and G. Thomas, Radiative Transfer in the Atmosphere and Ocean, Cambridge University Press, 1999.

[12] Nakajima T, Tanaka M. Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation, JQSRT, 40, 51-69 (1988).

[13] Spurr R., and M. Christi, Linearization of the Interaction Principle: Analytic Jacobians in the Radiant model, submitted to JQSRT, 2005.

[14] Garcia, R.D.M., and C.E. Siewert, The FN method for radiative transfer models that include polarization, JQSRT, 41, 117-145 (1989).

[15] Mishchenko, M., A. Lacis, and L. Travis, Errors induced by the neglect of polarization in radiance calculations for Rayleigh scattering atmospheres, JQSRT, 51, 491-510 (1994).

[16] Lacis, A., J. Chowdhary, M. Mishchenko, and B. Cairns, Modeling errors in diffuse sky radiance: vector vs. scalar treatment, Geophys. Res. Lett, 25, 135-8 (1998).


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[17] Bodhaine, B., N. Wood, E. Dutton, and J. Slusser, On Rayleigh optical depth calculations, J. Atmos. Ocean. Tech., 16, 1854-1861 (1999).

[18] Burrows, J., A. Richter, A. Dehn. B. Deters, S. Himmelmann, S. Voigt, and J. Orphal, Atmospheric remote sensing reference data from GOME: Part 2. Temperature-dependent absorption cross-sections of O3 in the 231-794 nm range, JQSRT, 61, 509-517 (1999).

[19] de Rooij, W.A., and C.C.A.H. van der Stap, Expansion of Mie scattering matrices in generalized spherical functions, Astron. Astrophys., 131, 237-248 (1984).

[20] Natraj, V., R. Spurr, H. Boesch, Y. Jiang and Y.L. Yung, Evaluation of Errors from Neglecting Polarization in the Forward Modeling of O2 A Band measurements from Space, with Relevance to the CO2 Column Retrieval from Polarization-Sensitive Instruments, submitted to JQSRT, 2005.

[21] Guo., V., et al., Retrieval of ozone profile from ground-based measurements with polarization: A synthetic study, submitted to JQSRT, 2005.

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