The LIDORT and VLIDORT Linearized Scalar and Vector Discrete Ordinate Radiative Transfer Models: Updates in the Last 10 Years

Abstract

It has been 10 years since the last major review paper on the LIDORT and VLIDORT radiative transfer models; this paper appeared in Light Scattering Reviews, Volume 3 (Spurr in Light scattering reviews. Springer, Berlin, 2008), hereinafter referenced as [R1]).

Appendices

Appendix A. RTE Solutions for VLIDORT

1.1 A.1 Azimuthal Separation

In applications involving randomly oriented particles with one plane of symmetry, the scattering matrix ${\mathbf{F}}\left(\Theta \right)$ in Eq. (2.5) has six independent entries:

$$ {\mathbf{F}}\left(\Theta \right) = \left( {\begin{array}{*{20}c} {a_{1} \left(\Theta \right)} & {b_{1} \left(\Theta \right)} & 0 & 0 \\ {b_{1} \left(\Theta \right)} & {a_{2} \left(\Theta \right)} & 0 & 0 \\ 0 & 0 & {a_{3} \left(\Theta \right)} & {b_{2} \left(\Theta \right)} \\ 0 & 0 & { - b_{2} \left(\Theta \right)} & {a_{4} \left(\Theta \right)} \\ \end{array} } \right). $$

(A.1)

For this form of the scattering matrix, one can develop expansions of these functions in terms of a set of generalized spherical functions $P_{mn}^{l} \left( {\cos\Theta } \right)$ (Mishchenko et al. 2006):

$$ \begin{aligned} a_{1} \left(\Theta \right) & = \sum\limits_{l = 0}^{M} {\beta_{l} P_{00}^{l} \left( {\cos\Theta } \right)} ; \\ a_{2} \left(\Theta \right) + a_{3} \left(\Theta \right) & = \sum\limits_{l = 0}^{M} {\left( {\alpha_{l} + \zeta_{l} } \right)P_{2,2}^{l} \left( {\cos\Theta } \right)} ; \\ a_{2} \left(\Theta \right) - a_{3} \left(\Theta \right) & = \sum\limits_{l = 0}^{M} {\left( {\alpha_{l} - \zeta_{l} } \right)P_{2, - 2}^{l} \left( {\cos\Theta } \right)} ; \\ a_{4} \left(\Theta \right) & = \sum\limits_{l = 0}^{M} {\delta_{l} P_{00}^{l} \left( {\cos\Theta } \right)} \\ \end{aligned} $$

(A.2)

$$ b_{1} \left(\Theta \right) = \sum\limits_{l = 0}^{M} {\gamma_{l} P_{02}^{l} \left( {\cos\Theta } \right)} ; \quad b_{2} \left(\Theta \right) = - \sum\limits_{l = 0}^{M} {\epsilon_{l} P_{02}^{l} \left( {\cos\Theta } \right)} $$

(A.3)

The (1,1) entry in ${\mathbf{F}}\left(\Theta \right)$ is just the phase function and satisfies the normalization condition:

$$ \frac{1}{2}\int\limits_{0}^{\pi } {} a_{1} \left(\Theta \right){ \sin }\Theta d\Theta = 1. $$

(A.4)

The set of six expansion coefficients (“Greek constants”) $\left\{ {\alpha_{l} ,\beta_{l} ,\gamma_{l} ,\delta_{l} ,\varepsilon_{l} ,\zeta_{l} } \right\}$ are key inputs to VLIDORT, and the number of terms $M$ in these expansions depends on the desired level of numerical accuracy. Here, $\left\{ {\beta_{l} } \right\}$ are the phase function expansion coefficients as used in the scalar RTE. These “Greek constants” specify the polarized-light single-scattering law, and there are a number of efficient analytical techniques for their computation, not only for spherical particles (see for example de Rooij and van der Stap 1984) but also for randomly-oriented homogeneous and inhomogeneous non-spherical particles and aggregated scatterers (Hovenier et al. 2004; Mackowski and Mishchenko 1996; Mishchenko and Travis 1998).

To proceed with the RTE solution, it is necessary to make Fourier decompositions (in terms of the cosine and sine of the relative azimuth angle between incident and scattered light directions) of the phase matrix and the Stokes vector in order to separate the azimuthal dependence.

$$ {\mathbf{I}}\left( {x,\mu ,\phi - \phi^{{\prime }} } \right) = \frac{1}{2}\sum\limits_{m = 0}^{M} {\left( {2 - \delta_{m,0} } \right){\varvec{\Phi}}_{m} \left( {\phi - \phi^{{\prime }} } \right){\mathbf{I}}_{m} \left( {x,\mu } \right)} ; $$

(A.5)

$$ {\varvec{\Pi}}\left( {\mu ,\phi ,\mu^{{\prime }} ,\phi^{{\prime }} } \right) = \frac{1}{2}\sum\limits_{m = 0}^{M} {\left( {2 - \delta_{m,0} } \right)\left[ {{\mathbf{C}}_{m} \left( {\mu ,\mu^{{\prime }} } \right){ \cos } m\left( {\phi - \phi^{{\prime }} } \right) + {\mathbf{S}}_{m} \left( {\mu ,\mu^{{\prime }} } \right){ \sin } m\left( {\phi - \phi^{{\prime }} } \right)} \right]} ; $$

(A.6)

Here, ${\varvec{\Phi}}_{m} \left( \phi \right) \equiv Diag\left[ {\cos m\phi ,\cos m\phi ,\sin m\phi ,\sin m\phi } \right]$. We follow the formulation for azimuthal separation of the scattering matrix developed by Siewert (1982), Vestrucci and Siewert (1984). Most vector radiative transfer models now follow this work. Accordingly, we write:

$$ {\mathbf{C}}_{m} \left( {\mu ,\mu^{{\prime }} } \right) = {\mathbf{A}}_{m} \left( {\mu ,\mu^{{\prime }} } \right) + {\mathbf{DA}}_{m} \left( {\mu ,\mu^{{\prime }} } \right){\mathbf{D}};\, {\mathbf{S}}_{m} \left( {\mu ,\mu^{{\prime }} } \right) = {\mathbf{A}}_{m} \left( {\mu ,\mu^{{\prime }} } \right){\mathbf{D}} - {\mathbf{DA}}_{m} \left( {\mu ,\mu^{{\prime }} } \right); $$

(A.7)

$$ {\mathbf{A}}_{m} \left( {\mu ,\mu^{{\prime }} } \right) = \sum\limits_{l = m}^{M} {{\mathbf{P}}_{l}^{m} \left( \mu \right){\mathbf{B}}_{l} {\mathbf{P}}_{l}^{m} \left( {\mu^{{\prime }} } \right)} ;\,{\mathbf{D}} = Diag\left[ {1,1, - 1, - 1} \right];\varvec{ } $$

(A.8)

$$ {\mathbf{B}}_{l} = \left( {\begin{array}{*{20}c} {\beta_{l} } & {\gamma_{l} } & 0 & 0 \\ {\gamma_{l} } & {\alpha_{l} } & 0 & 0 \\ 0 & 0 & {\delta_{l} } & { - \varepsilon_{l} } \\ 0 & 0 & {\varepsilon_{l} } & {\zeta_{l} } \\ \end{array} } \right); {\mathbf{P}}_{l}^{m} \left( \mu \right) = \left( {\begin{array}{*{20}c} {P_{l}^{m} \left( \mu \right)} & {\gamma_{l} } & 0 & 0 \\ {\gamma_{l} } & {R_{l}^{m} \left( \mu \right)} & { - T_{l}^{m} \left( \mu \right)} & 0 \\ 0 & { - T_{l}^{m} \left( \mu \right)} & {R_{l}^{m} \left( \mu \right)} & 0 \\ 0 & 0 & 0 & {P_{l}^{m} \left( \mu \right)} \\ \end{array} } \right). $$

(A.9)

The “Greek matrices” contain the spherical-function expansion coefficients, while the matrices ${\mathbf{P}}_{l}^{m} \left( \mu \right)$ contain the associated Legendre functions $P_{l}^{m} \left( \mu \right)$ and the functions $R_{l}^{m} \left( \mu \right)$ and $T_{l}^{m} \left( \mu \right)$ which are closely related to the generalized spherical functions $P_{mn}^{l} \left( \mu \right)$ (for details, see for example Siewert 2000b).

This azimuth separation process yields the following RTE for the Fourier component ${\mathbf{I}}_{m} \left( {x,\mu } \right)$:

$$ \mu \frac{{d{\mathbf{I}}_{m} \left( {x,\mu } \right)}}{dx} = - {\mathbf{I}}_{m} \left( {x,\mu } \right) + \frac{\omega }{2}\sum\limits_{l = m}^{M} {{\mathbf{P}}_{l}^{m} \left( \mu \right){\mathbf{B}}_{l} } \int\limits_{ - 1}^{1} {{\mathbf{P}}_{l}^{m} \left( {\mu^{{\prime }} } \right){\mathbf{I}}_{m} \left( {x,\mu^{{\prime }} } \right)d\mu^{{\prime }} + {\mathbf{Q}}_{m} \left( {x,\mu } \right)} . $$

(A.10)

For the solar source term, with solar direction $\left\{ { - \mu_{0} ,\phi_{0} } \right\}$, we have.

$$ {\mathbf{Q}}_{m}^{ \odot } \left( {x,\mu } \right) = \frac{\omega }{2}\sum\limits_{l = m}^{M} {{\mathbf{P}}_{l}^{m} \left( \mu \right){\mathbf{B}}_{l} {\mathbf{P}}_{l}^{m} \left( { - \mu_{0} } \right){\mathbf{F}}_{ \odot } { \exp }\left[ { - \tau_{ \odot } \left( {x,\mu } \right)} \right],} $$

(A.11)

in terms of the TOA solar flux ${\mathbf{F}}_{ \odot } = \left[ {F_{ \odot } ,0,0,0} \right]^{\text{T}}$ and solar beam attenuation ${ \exp }\left[ { - \tau_{ \odot } \left( {x,\mu_{0} } \right)} \right]$, where $\tau_{ \odot } \left( {x,\mu_{0} } \right)$ is the beam optical depth in a spherical-shell atmosphere.

1.2 A.2 Homogeneous Solutions in VLIDORT

1.2.1 A.2.1 Eigenvalue Solutions

In the discrete-ordinate solution method, we solve for each Fourier component in Eq. (A.10) by first finding the homogeneous solutions (without the solar or thermal source terms). With the familiar discrete ordinate quadrature with stream directions and weights $\left\{ { \pm \mu_{i} ,c_{i} } \right\}$ for $i = 1, \ldots N_{d}$, where $N_{d}$ is the number of discrete ordinates in the polar half space. The resulting vector RTE for these streams is then

$$ \begin{aligned} \pm \mu_{i} \frac{{d{\mathbf{I}}_{m} \left( {x, \pm \mu_{i} } \right)}}{dx} \pm {\mathbf{I}}_{m} \left( {x, \pm \mu_{i} } \right) & = \frac{\omega }{2}\sum\limits_{l = m}^{M} {{\mathbf{P}}_{l}^{m} \left( { \pm \mu_{i} } \right){\mathbf{B}}_{l} } \sum\limits_{j = 1}^{{N_{d} }} {\left[ {{\mathbf{P}}_{l}^{m} \left( {\mu_{j} } \right){\mathbf{I}}_{m} \left( {x,\mu_{j} } \right)} \right.} \\ & \quad + \left. {{\mathbf{P}}_{l}^{m} \left( { - \mu_{j} } \right){\mathbf{I}}_{m} \left( {x, - \mu_{j} } \right)} \right] \\ \end{aligned} $$

(A.12)

There are $8N_{d}$ coupled first-order linear differential equations for ${\mathbf{I}}_{m} \left( {x, \pm \mu_{i} } \right)$ in this system, which is solved by eigenvalue methods, using the ansatz:

$$ {\mathbf{I}}_{\alpha } \left( {x, \pm \mu_{i} } \right) = {\mathbf{W}}_{\alpha } \left( { \pm \mu_{i} } \right)\exp \left[ { - k_{\alpha } x} \right]. $$

(A.13)

We then define the following two vectors of rank $4N_{d}$:

$$ {\mathbb{W}}_{\alpha }^{ \pm } = \left[ {{\mathbf{W}}_{\alpha }^{\text{T}} \left( { \pm \mu_{1} } \right),{\mathbf{W}}_{\alpha }^{\text{T}} \left( { \pm \mu_{2} } \right), \ldots ,{\mathbf{W}}_{\alpha }^{\text{T}} \left( { \pm \mu_{{N_{d} }} } \right)} \right]^{\text{T}} . $$

(A.14)

The system is decoupled using sum and difference vectors ${\mathbb{X}}_{\alpha } = {\mathbb{W}}_{\alpha }^{ + } + {\mathbb{W}}_{\alpha }^{ - }$ and ${\mathbb{Y}}_{\alpha } = {\mathbb{W}}_{\alpha }^{ + } - {\mathbb{W}}_{\alpha }^{ - }$, and the order is then be reduced from $8N_{d}$ to $4N_{d}$. The result is an eigenproblem for the collection of separation constants and associated solution vectors $\left\{ {k_{\alpha } ,{\mathbb{X}}_{\alpha } } \right\}$, where index $\alpha = 1, \ldots ,4N_{d}$ labels the eigensolutions. The eigenmatrix for this system is constructed from the optical property inputs $\left\{ {\omega ,{\mathbf{B}}_{l} } \right\}$ and combination products of matrices ${\mathbf{P}}_{l}^{m} \left( { \pm \mu_{i} } \right)$. The eigenproblem is Siewert (2000b):

$$ {\mathbb{X}}_{\alpha }^{*} {\mathbb{G}} = k_{\alpha }^{2} {\mathbb{X}}_{\alpha }^{*} ; {{\mathbb{G}}{\mathbb{X}}}_{\alpha } = k_{\alpha }^{2} {\mathbb{X}}_{\alpha } ; $$

(A.15a)

$$ {\mathbb{G}} = {\mathbb{F}}^{ + } {\mathbb{F}}^{ - } ; {\mathbb{F}}^{ \pm } = \left[ {{\mathbb{E}} - \frac{\omega }{2}\sum\limits_{l = m}^{M} {{\mathbb{P}}\left( {l,m} \right){\mathbf{B}}_{l} {\mathbf{A}}^{ \pm } {\mathbb{P}}^{\text{T}} \left( {l,m} \right){\mathbb{C}}} } \right]{\mathbb{M}}^{ - 1} ; $$

(A.15b)

$$ {\mathbb{P}}\left( {l,m} \right) = Diag\left[ {{\mathbf{P}}_{l}^{m} \left( {\mu_{1} } \right),{\mathbf{P}}_{l}^{m} \left( {\mu_{2} } \right), \ldots ,{\mathbf{P}}_{l}^{m} \left( {\mu_{{N_{d} }} } \right)} \right]^{\text{T}} ; $$

(A.15c)

$$ {\mathbb{M}} = Diag\left[ {\mu_{1} {\mathbf{E}},\mu_{2} {\mathbf{E}}, \ldots ,\mu_{{N_{d} }} {\mathbf{E}}} \right];\varvec{ }{\mathbb{C}}\varvec{ } = Diag\left[ {c_{1} {\mathbf{E}},c_{2} {\mathbf{E}}, \ldots ,c_{{N_{d} }} {\mathbf{E}}} \right]. $$

(A.15d)

Here, $\varvec{ }{\mathbf{A}}^{ \pm } = {\mathbf{E}} \pm \left( { - 1} \right)^{l - m} {\mathbf{D}};{\mathbf{E}} = Diag\left[ {1,1,1,1} \right];\varvec{ }{\mathbf{D}} = Diag\left[ {1,1, - 1, - 1} \right]$, ${\mathbb{E}}$ is the $4N_{d} \times 4N_{d}$ identity matrix; ${\mathbb{X}}_{\alpha }^{*}$ and ${\mathbb{X}}_{\alpha }$ are the left and right eigenvectors respectively, with ${\mathbb{X}}_{\alpha }^{*}$ the conjugate transpose of ${\mathbb{X}}_{\alpha }$. The link between ${\mathbb{X}}_{\alpha }$ and solution vectors ${\mathbb{W}}_{\alpha }^{ \pm }$ comes through the auxiliary equations:

$$ {\mathbb{W}}_{\alpha }^{ \pm } = \frac{1}{2}{\mathbb{M}}^{ - 1} \left[ {{\mathbb{E}} \pm \frac{1}{{k_{\alpha } }}{\mathbb{F}}^{ + } } \right]{\mathbb{X}}_{\alpha } . $$

(A.16)

Eigenvalues occur in pairs $\left\{ { \pm k_{\alpha } } \right\}$. Left and right eigenvectors share the same spectrum of eigenvalues. As noted by Siewert (2000b), both complex- and real-variable eigensolutions may be present in the full Stokes 4-vector case (rank $4N_{d}$). Eigensolutions may be determined numerically with the complex-variable eigensolver DGEEV from the LAPACK suite (Anderson et al. 1995). DGEEV generates both right and left eigenvectors, which have unit moduli. In the scalar case (no polarization, solutions only for the I-component of the Stokes vector), the eigensystem has rank $N_{d}$, the eigenmatrix is symmetric and all eigensolutions are real-valued. In this case, the eigensolver module ASYMTX (Stamnes et al. 1988) may be used. ASYMTX is an adaptation of the LAPACK routine for real-valued problems; it delivers only the right eigenvectors. For vector solutions neglecting circular polarization (Stokes I, Q and U only), complex eigensolutions are absent, and one may then use the faster ASYMTX routine.

Returning to the full Stokes 4-vector case, the complete homogeneous solution in one layer is then:

$$ {\mathbb{I}}^{ + } \left( x \right) = {\mathbb{D}}^{ + } \sum\limits_{\alpha = 1}^{{4N_{d} }} {\left\{ {{\mathfrak{L}}_{\alpha } {\mathbb{W}}_{\alpha }^{ + } \exp \left[ { - k_{\alpha } x} \right] + {\mathfrak{M}}_{\alpha } {\mathbb{W}}_{\alpha }^{ - } \exp \left[ { - k_{\alpha } \left( {\Delta - x} \right)} \right]} \right\}} ; $$

(A.17a)

$$ {\mathbb{I}}^{ - } \left( x \right) = {\mathbb{D}}^{ - } \sum\limits_{\alpha = 1}^{{4N_{d} }} {\left\{ {{\mathfrak{L}}_{\alpha } {\mathbb{W}}_{\alpha }^{ - } \exp \left[ { - k_{\alpha } x} \right] + {\mathfrak{M}}_{\alpha } {\mathbb{W}}_{\alpha }^{ + } \exp \left[ { - k_{\alpha } \left( {{\Delta } - x} \right)} \right]} \right\}} ; $$

(A.17b)

Here, ${\mathbb{D}}^{ - } = Diag\left[ {{\mathbf{D}},{\mathbf{D}}, \ldots {\mathbf{D}}} \right]$, and ${\mathbb{D}}^{ + } = {\mathbb{E}}$; these matrices arise from application of symmetry relations (Siewert 2000b). The use of optical thickness $\Delta - x$ in the second exponential ensures that solutions remain bounded (Stamnes and Conklin 1984). The quantities $\left\{ {{\mathfrak{L}}_{\alpha } ,{\mathfrak{M}}_{\alpha } } \right\}$ are the constants of integration; determined by application of the boundary conditions and solution of the resulting boundary-value problem.

In the Stokes 4-vector case, some contributions to ${\mathbb{I}}^{ \pm } \left( x \right)$ will be complex, some real. It is understood that we compute the real parts of any complex variable expressions. Thus for example if $\left\{ {k_{\alpha } ,{\mathbb{W}}_{\alpha }^{ - } } \right\}$ is a complex eigensolution with associated (complex) integration constant ${\mathfrak{L}}_{\alpha }$, the real part of the solution will be:

$$ {\text{Re}}\left[ {{\mathfrak{L}}_{\alpha } {\mathbb{W}}_{\alpha }^{ - } e^{{ - k_{\alpha } x}} } \right] = {\text{Re}}\left[ {{\mathfrak{L}}_{\alpha } } \right]{\text{Re}}\left[ {{\mathbb{W}}_{\alpha }^{ - } e^{{ - k_{\alpha } x}} } \right] - {\text{Im}}\left[ {{\mathfrak{L}}_{\alpha } } \right]{\text{Im}}\left[ {{\mathbb{W}}_{\alpha }^{ - } e^{{ - k_{\alpha } x}} } \right]. $$

(A.18)

From a bookkeeping standpoint, one must keep count of the number of real and complex solutions, and treat them separately in the numerical implementation. For clarity of exposition, we have not made an explicit separation of complex variables, and it will be clear from the context whether real or complex variables are under consideration.

1.2.2 A.2.2 Linearization of the Eigenvalue Solutions

For a single layer, we require derivatives of the eigensolution $\left\{ {k_{\alpha } ,{\mathbb{W}}_{\alpha }^{ \pm } } \right\}$ with respect to some atmospheric variable $\xi$ in that layer. The starting point for the differentiation is the set of linearized optical properties ${\mathcal{V}} \equiv \xi \partial \Delta /\partial \xi ;{ \mathcal{U}} \equiv \xi \partial\upomega/\partial \xi ; {\boldsymbol{{\mathcal{Z}}}}_{l} \equiv \xi \partial {\mathbf{B}}_{l} /\partial \xi$, that is, normalized partial derivatives of the set of IOPs $\left\{ {\Delta ,\upomega, {\mathbf{B}}_{l} } \right\}$. The eigensolution depends only on the quantities $\left\{ {\upomega, {\mathbf{B}}_{l} } \right\}$. Applying the linearization operator ${\mathcal{L}} \equiv \xi \partial /\partial \xi$ to the eigenmatrix, we find

$$ {\mathcal{L}}\left( {\mathbb{G}} \right) = { \mathcal{L}}\left( {{\mathbb{F}}^{ + } } \right){\mathbb{F}}^{ - } + {\mathbb{F}}^{ + } {\mathcal{L}}\left( {{\mathbb{F}}^{ - } } \right); $$

(A.19a)

$$ {\mathcal{L}}\left( {{\mathbb{F}}^{ \pm } } \right) = - \left[ {\frac{1}{2}\left\{ {{\mathcal{U}{\mathbb{P}}}_{lm} {\mathbf{B}}_{l} + {\omega {\mathbb{P}}}_{lm} {\mathbf{\mathcal{Z}}}_{l} } \right\}{\mathbf{A}}^{ \pm } {\mathbb{P}}_{lm}^{\text{T}} {\mathbb{C}}} \right]{\mathbb{M}}^{ - 1} . $$

(A.19b)

Linearization treatments are different for the full Stokes 4-vector case, and the scalar and Stokes 3-vector situations.

4-vector case. We apply linearization to both the left and right eigensystems:

$$ {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha }^{*} } \right){\mathbb{G}} + {\mathbb{X}}_{\alpha }^{*} {\mathcal{L}}\left( {\mathbb{G}} \right) = 2k_{\alpha } {\mathcal{L}}\left( {k_{\alpha } } \right){\mathbb{X}}_{\alpha }^{*} + k_{\alpha }^{2} {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha }^{*} } \right); $$

(A.20a)

$$ {{\mathbb{G}}\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) + {\mathcal{L}}\left( {\mathbb{G}} \right){\mathbb{X}}_{\alpha } = 2k_{\alpha } {\mathcal{L}}\left( {k_{\alpha } } \right){\mathbb{X}}_{\alpha } + k_{\alpha }^{2} {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right). $$

(A.20b)

We form a dot product $\otimes$ by pre-multiplying the second of these equations by the transpose vector ${\mathbb{X}}_{\alpha }^{*}$:

$$ {\mathbb{X}}_{\alpha }^{*} \otimes {{\mathbb{G}}\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) + {\mathbb{X}}_{\alpha }^{*} \otimes {\mathcal{L}}\left( {\mathbb{G}} \right){\mathbb{X}}_{\alpha } = 2k_{\alpha } {\mathcal{L}}\left( {k_{\alpha } } \right){\mathbb{X}}_{\alpha }^{*} \otimes {\mathbb{X}}_{\alpha } + k_{\alpha }^{2} {\mathbb{X}}_{\alpha }^{*} \otimes {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right). $$

(A.21)

Using the relation ${\mathbb{X}}_{\alpha }^{*} \otimes {{\mathbb{G}}\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) = {\mathbb{X}}_{\alpha }^{*} {\mathbb{G}} \otimes {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) = k_{\alpha }^{2} {\mathbb{X}}_{\alpha }^{*} \otimes {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right)$, we find that

$$ y_{\alpha } \equiv {\mathcal{L}}\left( {k_{\alpha } } \right) = \frac{{{\mathbb{X}}_{\alpha }^{*} \otimes {\mathcal{L}}\left( {\mathbb{G}} \right){\mathbb{X}}_{\alpha } }}{{2k_{\alpha } {\mathbb{X}}_{\alpha }^{*} \otimes {\mathbb{X}}_{\alpha } }}. $$

(A.22)

This is the linearization of the separation constants. Next, we substitute this result in Eq. (A.20a) to obtain the following linear algebra problem for each eigensolution linearization:

$$ {\mathbb{H}}_{\alpha } {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) = {\mathbb{C}}_{\alpha } ;\quad {\mathbb{H}}_{\alpha } = {\mathbb{G}} - k_{\alpha }^{2} {\mathbb{E}};\quad {\mathbb{C}}_{\alpha } = 2k_{\alpha } y_{\alpha } {\mathbb{X}}_{\alpha } - {\mathcal{L}}\left( {\mathbb{G}} \right){\mathbb{X}}_{\alpha } . $$

(A.23)

For real eigensolutions, this system has rank $4N_{d}$, and for complex solutions, the rank is $8N_{d}$.

Implementation of this system of equations “as is” is not possible due to the degeneracy of the eigenproblem, and we need additional constraints to find the unique solution for ${ \mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right)$. The treatment for real and complex solutions is different.

• For the real-valued eigensolutions, the unit-modulus eigenvector normalization is ${\mathbb{X}}_{\alpha } \otimes {\mathbb{X}}_{\alpha } = 1$ in dot-product notation. Linearizing, this yields one equation:

$$ {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) \otimes {\mathbb{X}}_{\alpha } + {\mathbb{X}}_{\alpha } \otimes {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) = 0. $$

(A.24)

The solution procedure uses $4N_{d} - 1$ equations from Eq. (A.23), along with Eq. (A.24) to form a slightly modified linear system of rank $4N_{d}$. This system is then solved by standard means using the DGETRF and DGETRS LU-decomposition routines from the LAPACK suite.

• For complex-valued eigensolutions, Eq. (A.23) is a complex-variable system for both the real and imaginary parts of the linearized eigenvectors. There are $8N_{d}$ equations in all, but now we require two constraint conditions to remove the eigenproblem arbitrariness. The first is Eq. (24). The second condition is imposed by the following normalization from the LAPACK DGEEV routine for solution of complex-valued eigenproblems: for that element of a (complex-valued) eigenvector which has the largest real value, the corresponding imaginary part is always set to zero. Thus, for eigenvector ${\mathbb{X}}_{\alpha }$ with components ${\text{X}}_{j} \in {\mathbb{X}}_{\alpha } , j = 1,2, \ldots 4N_{d}$, if ${\text{X}}_{q}$ satisfies condition ${\text{Re}}[{\text{X}}_{q} ] = \max_{{j = 1,2, \ldots 4N_{d} }} \left\{ {{\text{Re}}[{\text{X}}_{j} ]} \right\}$, then ${\text{Im}}[{\text{X}}_{q} ] = 0$, and consequently ${\mathcal{L}}\left( {{\text{Im}}[{\text{X}}_{q} ]} \right) = 0$. This is the second condition. The solution procedure is then: (1) in Eq. (A.24) to strike out the qth row and column in matrix ${\mathbb{H}}_{\alpha }$ for which ${\text{Im}}[{\text{X}}_{q} ]$ is zero, and the qth column in vector ${\mathbb{C}}_{\alpha }$; and (2) in the resulting system of rank $8N_{d} - 1$, replace one of the rows with the normalization constraint Eq. (A.24). ${\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right)$ is then the solution of the resulting linear system.

• Scalar and 3-vector case. Here the (real-valued) eigensolutions are obtained using eigensolver ASYMTX—this has no adjoint solution, so there is no determination of ${\mathcal{L}}\left( {k_{\alpha } } \right)$ as in Eq. (A.22). Instead, we solve for variables $\left\{ {{\mathcal{L}}\left( {k_{\alpha } } \right),{\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right)} \right\}$ using ${\mathbb{H}}_{\alpha } {\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right) = 2k_{\alpha } {\mathcal{L}}\left( {k_{\alpha } } \right){\mathbb{X}}_{\alpha } - {\mathcal{L}}\left( {\mathbb{G}} \right){\mathbb{X}}_{\alpha }$ from above, plus the normalization condition to form a joint system of rank $3N_{d} + 1$ (vector) or rank $N_{d} + 1$ (scalar).

Having derived the linearizations $\left\{ {{\mathcal{L}}\left( {k_{\alpha } } \right),{\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right)} \right\}$, we complete this section by differentiating the auxiliary result in Eq. (A.16) to establish ${ \mathcal{L}}\left( {{\mathbb{W}}_{\alpha }^{ \pm } } \right)$:

$$ {\mathcal{L}}\left( {{\mathbb{W}}_{\alpha }^{ \pm } } \right) = \frac{1}{2}{\mathbb{M}}^{ - 1} \left[ { \mp \frac{{{\mathcal{L}}\left( {k_{\alpha } } \right)}}{{k_{\alpha }^{2} }}{\mathbb{F}}^{ + } \pm \frac{1}{{k_{\alpha } }}{\mathcal{L}}\left( {{\mathbb{F}}^{ + } } \right)} \right]{\mathbb{X}}_{\alpha } + \frac{1}{2}{\mathbb{M}}^{ - 1} \left[ {{\mathbb{E}} \pm \frac{1}{{k_{\alpha } }}{\mathbb{F}}^{ + } } \right]{\mathcal{L}}\left( {{\mathbb{X}}_{\alpha } } \right). $$

(A.25)

Finally, we have linearizations of the transmittance derivatives in Eq. (A.17a):

$$ {\mathcal{L}}\left( {\exp \left[ { - k_{\alpha } x} \right]} \right) = - \left[ {x{\mathcal{L}}\left( {k_{\alpha } } \right) + k_{\alpha } {\mathcal{L}}\left( x \right)} \right]\exp \left[ { - k_{\alpha } x} \right]. $$

(A.26)

Since the partial layer optical thickness $x$ is proportional to the total layer optical depth $\Delta$ in an optically uniform layer, we have ${\mathcal{L}}\left( x \right) = x/\Delta {\mathcal{L}}\left( \Delta \right) = x/\Delta {\mathcal{V}}$ in terms of the basic linearized optical property input ${\mathcal{V}} \equiv \xi \partial \Delta /\partial \xi$.

1.3 A.3 Particular Integral Solar Solutions in VLIDORT

In VLIDORT, particular integrals for both solar and thermal sources in the vector RTE are established using traditional substitution methods, rather than the Green’s function approach which is used in LIDORT. This is mainly for bookkeeping reasons associated with the use of complex and real variables. In this section we discuss solutions for the solar sources using this method.

1.3.1 A.3.1 Solar-Beam Sources—Substitution Method

For the solar source term with direction $\left\{ { - \mu_{0} ,\phi_{0} } \right\}$, we have from Eq. (A.11) above, the following source terms in the discrete ordinate directions:

$$ {\mathbf{Q}}_{nm}^{ \odot } \left( {x, \pm \mu_{i} } \right) = \frac{{\omega_{n} }}{2}\sum\limits_{l = m}^{M} {{\mathbf{P}}_{l}^{m} \left( { \pm \mu_{i} } \right){\mathbf{B}}_{nl} {\mathbf{P}}_{l}^{m} \left( { - \mu_{0} } \right){\mathbf{F}}_{ \odot } {\text{T}}_{n} { \exp }\left[ { - \lambda_{n} x} \right]} . $$

(A.27)

Here, we have kept the layer index explicit, and $x$ denotes the vertical optical thickness as measured from the top of layer $n$; we used the pseudo-spherical treatment of solar beam attenuation (Sect. 3.4). The exponential form for the beam attenuation allows us to write the particular solution in the form:

$$ {\mathbf{I}}_{nm}^{ \odot } \left( {x, \pm \mu_{i} } \right) = {\mathbf{Z}}_{n} \left( { \pm \mu_{i} } \right){\text{T}}_{n} { \exp }\left[ { - \lambda_{n} x} \right], $$

(A.28)

and by analogy with the homogeneous case, we may define the following vectors of rank $4N_{d}$:

$$ {\mathbb{Z}}_{n}^{ \pm } = \left[ {{\mathbf{Z}}_{n}^{\text{T}} \left( { \pm \mu_{1} } \right),{\mathbf{Z}}_{n}^{\text{T}} \left( { \pm \mu_{2} } \right), \ldots ,{\mathbf{Z}}_{n}^{\text{T}} \left( { \pm \mu_{{N_{d} }} } \right)} \right]^{\text{T}} . $$

(A.29)

We decouple the equations using sum and difference vectors ${\mathbb{R}}_{n} = {\mathbb{Z}}_{n}^{ + } + {\mathbb{Z}}_{n}^{ - }$ and ${\mathbb{S}}_{n} = {\mathbb{Z}}_{n}^{ + } - {\mathbb{Z}}_{n}^{ - }$, and the order is reduced from $8N_{d}$ to $4N_{d}$. We obtain the following linear-algebra problem of rank $4N_{d}$:

$$ {\mathbb{H}}_{{\varvec{ n}}} {\mathbb{R}}_{n} = {\mathbb{B}}_{\varvec{n}} ; \quad {\mathbb{H}}_{\varvec{n}} = \lambda_{n}^{2} {\mathbb{E}} - {\mathbb{G}}_{n} ; \quad {\mathbb{B}}_{\varvec{n}} = \left[ {{\mathbb{F}}_{n}^{ - } {\mathbb{Q}}_{n}^{ + } + \frac{1}{{\lambda_{n} }}{\mathbb{Q}}_{n}^{ - } } \right]{\mathbb{M}}^{ - 1} ; $$

(A.30)

$$ {\mathbb{Q}}_{n}^{ \pm } = \omega_{n} \sum\nolimits_{l = m}^{M} {{\mathbb{P}}_{ \odot } \left( {l,m} \right){\mathbf{B}}_{nl} {\mathbf{A}}^{ \pm } {\mathbb{P}}_{ \odot }^{\text{T}} \left( {l,m} \right){\mathbb{M}}^{ - 1} ;} $$

(A.31a)

$$ {\mathbb{P}}_{ \odot } \left( {l,m} \right) = Diag\left[ {{\mathbf{P}}_{l}^{m} \left( { - \mu_{0} } \right),{\mathbf{P}}_{l}^{m} \left( { - \mu_{0} } \right), \ldots ,{\mathbf{P}}_{l}^{m} \left( { - \mu_{0} } \right)} \right]^{\text{T}} . $$

(A.31b)

Matrices ${\mathbb{F}}_{n}^{ - }$ and ${\mathbb{G}}_{n}$ were defined in Eqs. (A.15).

This system is solved using the LU-decomposition modules DGETRF and DGETRS from LAPACK; the formal solution is ${\mathbb{X}}_{n}^{ \odot } = {\mathbb{H}}_{n}^{ - 1} {\mathbb{B}}_{\varvec{n}}$. The particular integral is completed with the auxiliary equations:

$$ {\mathbb{Z}}_{n}^{ \pm } = \frac{1}{2}{\mathbb{M}}^{ - 1} \left[ {{\mathbb{E}} \pm \frac{1}{{\lambda_{n} }}{\mathbb{F}}_{n}^{ + } } \right]{\mathbb{R}}_{n} . $$

(A.32)

In the scalar LIDORT model, this system has rank $N_{d}$. In the vector model, the particular solution consists only of real variables.

1.3.2 A.3.2 Linearized Solar-Beam Sources—Substitution Method

For this linearization, the most important consideration is the presence of cross-derivatives: in a fully illuminated atmosphere, the particular solution in layer $n$ is differentiable with respect to atmospheric variables $\xi_{p}$ in all layers $p \le n$. For the solar beam attenuation, transmittance ${\text{T}}_{n}$ depends on variables $\xi_{p}$ in layers $p < n$, while average secant $\lambda_{n}$ depends on variables $\xi_{p}$ in layers $p \le n$. Linearization of the average-secant parameterization is in Sect. 3.4. It follows that the solution vectors ${\mathbb{Z}}_{n}^{ \pm }$ will also depend on $\xi_{p}$ for $p \le n$, so their linearizations will contain cross-derivatives. Finally, we note that the eigenmatrix ${\mathbb{G}}_{n}$ is constructed from optical properties only defined in layer $n$, so that ${\mathcal{L}}_{p} \left( {{\mathbb{G}}_{n} } \right) = 0, \forall p \ne n$.

Differentiation of Eqs. (A.30) and A.31a) yields a related linear problem:

$$ {\mathbb{H}}_{{\varvec{ n}}} {\mathcal{L}}_{p} \left( {{\mathbb{R}}_{n} } \right) = {\mathbb{B}}_{np}^{{\prime }} = {\mathcal{L}}_{p} \left( {{\mathbb{B}}_{\varvec{n}} } \right) - {\mathcal{L}}_{p} \left( {{\mathbb{H}}_{{\varvec{ n}}} } \right){\mathbb{R}}_{n} ; $$

(A.33a)

$$ {\mathcal{L}}_{p} \left( {{\mathbb{H}}_{{\varvec{ n}}} } \right) = - \delta_{np} {\mathcal{L}}_{p} \left( {{\mathbb{G}}_{n} } \right) + 2\lambda_{n} {\mathcal{L}}_{p} \left( {\lambda_{n} } \right){\mathbb{E}}; $$

(A.33b)

$$ {\mathcal{L}}_{p} \left( {{\mathbb{B}}_{\varvec{n}} } \right) = \delta_{np} \left[ {{\mathcal{L}}_{n} \left( {{\mathbb{F}}_{n}^{ - } } \right){\mathbb{Q}}_{n}^{ + } + {\mathbb{F}}_{n}^{ - } {\mathcal{L}}_{n} \left( {{\mathbb{Q}}_{n}^{ + } } \right) + \frac{1}{{\lambda_{n} }}{\mathcal{L}}_{n} \left( {{\mathbb{Q}}_{n}^{ - } } \right)} \right]{\mathbb{M}}^{ - 1} - \frac{{{\mathcal{L}}_{p} \left( {\lambda_{n} } \right)}}{{\lambda_{n}^{2} }}{\mathbb{Q}}_{n}^{ - } {\mathbb{M}}^{ - 1} ; $$

(A.33c)

$$ {\mathcal{L}}_{n} \left( {{\mathbb{Q}}_{n}^{ \pm } } \right) = \sum\nolimits_{l = m}^{M} {\left[ {{\mathcal{U}}_{n} {\mathbb{P}}_{ \odot } \left( {l,m} \right){\mathbf{B}}_{nl} + \omega_{n} {\mathbb{P}}_{ \odot } \left( {l,m} \right){\boldsymbol{\mathcal{Z}}}_{nl} } \right]} {\mathbf{A}}^{ \pm } {\mathbb{P}}_{ \odot }^{\text{T}} \left( {l,m} \right){\mathbb{M}}^{ - 1} . $$

(A.33d)

In Eq. (A.33c), the quantity ${\mathcal{L}}_{n} \left( {{\mathbb{F}}_{n}^{ - } } \right)$ comes from Eq. (A.19b). This linear system has the same matrix ${\mathbb{H}}_{{\varvec{ }n}}$, but with a different source vector ${\mathbb{B}}_{np}^{{\prime }}$. The solution is then found by back-substitution, given that the inverse of ${\mathbb{H}}_{{\varvec{ }n}}$ has already been established when solving for ${\mathbb{R}}_{n}$. Thus, ${\mathcal{L}}_{p} \left( {{\mathbb{R}}_{n} } \right) = {\mathbb{H}}_{n}^{ - 1} {\mathbb{B}}_{n}^{{\prime }}$. Linearization of the particular integral is then completed through differentiation of the auxiliary equations (A.32):

$$ {\mathcal{L}}_{p} \left( {{\mathbb{Z}}_{n}^{ \pm } } \right) = \frac{1}{2}{\mathbb{M}}^{ - 1} \left[ {{\mathbb{E}} \pm \frac{1}{{\lambda_{n} }}{\mathbb{F}}_{n}^{ + } } \right]{\mathcal{L}}_{p} \left( {{\mathbb{R}}_{n} } \right) \pm \frac{1}{{2\lambda_{n}^{2} }}{\mathbb{M}}^{ - 1} \left[ {\lambda_{n} \delta_{np} {\mathcal{L}}_{n} \left( {{\mathbb{F}}_{n}^{ + } } \right) - {\mathcal{L}}_{p} \left( {\lambda_{n} } \right){\mathbb{F}}_{n}^{ + } } \right]{\mathbb{R}}_{n} . $$

(A.34)

Appendix B. BRDF Kernel Functions

2.1 B.1 Ocean Glitter Kernels

2.1.1 B.1.1 Cox-Munk Glint Reflectance

For ocean glitter, we use the well-known geometric-optics regime for a single rough-surface redistribution of incident light, in which the reflection function is governed by Fresnel reflectance and takes the form (Jin et al. 2006):

$$ \rho_{CM} \left( {\mu ,\mu^{{\prime }} ,\varphi - \varphi^{{\prime }} ,m,\sigma^{2} } \right) = \frac{{F\left( {m,\theta_{r} } \right)}}{{\mu \mu^{{\prime }} \left| {\gamma_{r} } \right|^{4} }}P\left( {\gamma_{r} ,\sigma^{2} } \right)D\left( {\mu ,\mu^{{\prime }} ,\sigma^{2} } \right) $$

(B.1)

Here, $\sigma^{2}$ is the slope-squared variance (also known as the mean-slope-square) of the Gaussian probability function $P\left( {\gamma_{r} ,\sigma^{2} } \right)$ which has argument $\gamma_{r}$ (the polar direction of the reflected beam), $D\left( {\mu ,\mu^{{\prime }} ,\sigma^{2} } \right)$ is a shadow function correction (Sancer 1969).

$F\left( {m,\theta_{r} } \right)$ is the scalar Fresnel reflection for incident angle $\theta_{r} = \frac{1}{2}\theta_{SCAT}$ and relative refractive index $m$. The scattering angle $\theta_{SCAT}$ is determined in the usual manner from the incident and reflected directional cosines $\mu^{{\prime }}$ and $\mu$, and the relative azimuth $\varphi - \varphi^{{\prime }}$.

The two non-linear parameters characterizing this kernel are $\left\{ {m,\sigma^{2} } \right\}$. The probability and shadow functions are given by:

$$ \begin{aligned} P\left( {\alpha ,\sigma^{2} } \right) & = \frac{1}{{\pi \sigma^{2} }}{ \exp } \left[ { - \frac{{\alpha^{2} }}{{\sigma^{2} \left( {1 - \alpha^{2} } \right)}}} \right].\quad D\left( {\alpha ,\beta ,\sigma^{2} } \right) = \frac{1}{{1 +\Lambda \left( {\alpha ,\sigma^{2} } \right) +\Lambda \left( {\beta ,\sigma^{2} } \right)}}; \\\Lambda \left( {\alpha ,\sigma^{2} } \right) & = \frac{1}{2}\left\{ {\sqrt {\frac{{\left( {1 - \alpha^{2} } \right)}}{\pi }} \frac{\sigma }{\alpha }{ \exp } \left[ { - \frac{{\alpha^{2} }}{{\sigma^{2} \left( {1 - \alpha^{2} } \right)}}} \right] - erfc\left[ { - \frac{\alpha }{{\sigma \sqrt {\left( {1 - \alpha^{2} } \right)} }}} \right]} \right\}. \\ \end{aligned} $$

(B.2)

The variance is commonly related to the wind speed $W$ in (m/s) through the empirical relation $\sigma^{2} = 0.003 + 0.00512W$ (Cox and Munk 1954a). A typical value for $m$ is 1.33.

For the linearization, the key parameter is the wind-speed (or equivalently, the mean slope-square) The probability function is easily differentiated with respect to $\sigma^{2}$. Indeed, we have:

$$ \frac{{\partial P\left( {\alpha ,\sigma^{2} } \right)}}{{\partial \sigma^{2} }} = \frac{{P\left( {\alpha ,\sigma^{2} } \right)}}{{\sigma^{4} }}\left[ {\frac{{\alpha^{2} }}{{\left( {1 - \alpha^{2} } \right)}} - \sigma^{2} } \right]. $$

(B.3)

Again, the shadow function can be differentiated analytically with respect to $\sigma^{2}$ in a straightforward manner, once we recall that the derivative of the error function is Gaussian. We thus have linearized BRDF quantities prepared for (V)LIDORT, which will then be able to generate analytic weighting functions with respect to the wind speed.

Linearization of the kernel with respect to refractive index m will require the partial derivative $\partial F\left( {m,\theta_{r} } \right)/\partial m$, which is easy to determine from the usual Fresnel formula; this Jacobian is less useful.

We note also that VLIDORT has a vector kernel function for sea-surface glitter reflectance, based on the specification in Mishchenko and Travis (1998).

This formulation is for a single Fresnel reflectance by wave facets. In reality, glitter is the result of many reflectances. It is possible to incorporate a correction for multiple reflectances for this glitter contribution, both for the direct-bounce term and the Fourier components. Given that the glitter maximum is typically dominated by direct reflectance of the solar beam, we confine discussion to multiple-reflectance of this term. We consider only one extra order of reflectance:

$$ R\left( {\Omega ,\Omega _{0} } \right) \cong R_{0} \left( {\Omega ,\Omega _{0} } \right) + R_{1} \left( {\Omega ,\Omega _{0} } \right); R_{1} \left( {\Omega ,\Omega _{0} } \right) = \int\limits_{0}^{2\pi } {\int\limits_{0}^{1} {R_{0} \left( {\Omega ,\Omega ^{{\prime }} } \right)R_{0} \left( {\Omega ^{{\prime }} ,\Omega _{0} } \right)d\mu^{{\prime }} d\varphi^{{\prime }} } } . $$

(B.4)

$R_{0} \left( {\Omega ,\Omega ^{{\prime }} } \right)$ is the zeroth-order reflectance from incident direction $\Omega ^{{\prime }} = \left\{ {\mu^{{\prime }} ,\varphi^{{\prime }} } \right\}$ to reflected direction $\Omega = \left\{ {\mu ,\varphi } \right\}$. The azimuthal integration is done by double Gaussian quadrature over the intervals $\left[ { - \pi ,0} \right]$ and $\left[ {0,\pi } \right]$; the polar stream integration is also done by quadrature. It is obviously possible to calculate higher orders of reflectance for all BRDF functions, but this rapidly becomes computationally prohibitive. We have found that the neglect of multiple glitter reflectances can lead to errors of 1–3% in the upwelling intensity at the top of the atmosphere, the higher figures being for larger solar zenith angles. Finally, we note that the higher-order reflectances are in turn differentiable with respect to the slope-squared parameter, so that Jacobians for the wind speed can still be determined.

2.1.2 B.1.2 Alternative Cox-Munk Glint Reflectance

The above ocean-glint reflectance option in LIDORT is based on an older implementation of the well-known Cox-Munk distribution of wave facets—this does not include any treatment of whitecaps (foam), and there is no allowance for the wind direction. Further, the CM implementation is based on a fixed real-valued refractive index for water.

We have now added an alternative Cox-Munk implementation which addresses these issues. This new-CM option is based on the glint treatment in the 6S code (Vermote et al. 1997; Kotchenova et al. 2006), and includes an empirical whitecap contribution. The latter model also includes a more recent treatment of water-leaving radiance (Morel and Gentili 2009), and the LIDORT SLEAVE supplement has been updated according to the 6S treatment. These new 6S-based options in the LIDORT BRDF and SLEAVE supplements are designed to operate in tandem. Indeed, the total surface radiance in the 6S model is given by

$$ I\left( {\mu_{1} ,\mu_{0} ,\varphi_{1} - \varphi_{0} } \right) = \left( {1 - R_{wc} } \right)S\left( {\mu_{1} ,\mu_{0} } \right) + R_{wc} + \left( {1 - R_{L} } \right) \rho_{NCM} \left( {\mu_{1} ,\mu_{0} ,\varphi_{1} - \varphi_{0} } \right), $$

(B.5)

where the water-leaving term $S\left( {\mu_{1} ,\mu_{0} } \right)$ does depend on the outgoing and incoming directions but is azimuth-independent, $\rho_{NCM}$ is the Cox-Munk glint reflectance, and $R_{wc}$ and $R_{L}$ are the empirically-derived whitecap contributions (final and Lambertian).

For LIDORT to possess this functionality for the ocean surface, the BRDF supplement must provide the second and third terms on the right-hand-side of (B.5), while the SLEAVE supplement supplies the first term. Note that both supplements require the same whitecap formulation. The derivation of the water-leaving term is done in the next section.

The Cox-Munk calculation for $\rho_{NCM}$ depends on the wind speed, wind-direction and refractive index. The latter is a complex variable that depends on the salinity of the ocean, and we use the 6S formulation. The anisotropic treatment assumes an (azimuthal) wind direction relative to the solar incident beam, and this follows the formulae developed by Cox and Munk in the 1950s.

For facet anisotropy, the wind-direction is dependent on the solar position, and it follows that any BRDF quantity will pick up additional dependence on the solar angle. It is then only possible to use this “New-CM” option with a single solar zenith angle (and hence a single wind-direction azimuth) - multiple solar beams are ruled out. See also the remark in Sect. 4.2.3 regarding use of the black-sky albedo. There is exception handling for this eventuality.

In addition, we have linearized this “New-CM” glint reflectance with respect to the wind speed. Finally, we note that this glint reflectance is scalar only, so that the above considerations apply only to the (1,1) element of the reflectance matrix in the vector BRDF supplement for VLIDORT. Polarization of this contribution is currently undergoing investigation.

2.2 B.2 Land-Surface BRDF Kernels

2.2.1 B.2.1 MODIS BRDF System

The five MODIS-type kernels (numbers 2–6 in Table 3) (Wanner et al. 1995) must be used in a linear combination with a Lambertian kernel. The kernels divide naturally into two groups: the volume scattering terms with no non-linear parameters (Ross-thin, Ross-thick) and the geometric-optics terms with 2 non-linear parameters (Li-sparse, Li-dense) or no non-linear parameters (Roujean). See Wanner et al. (1995) and Spurr (2004) for details of the kernel formulae.

In fact, it is standard practice in MODIS BRDF retrievals to use a combination of Lambertian, Ross-thick and Li-Sparse kernels, and this 3-kernel combination is common:

$$ \rho_{total} \left( {\Omega ,\Omega ^{{\prime }} } \right) = f_{iso} + f_{vol} \rho_{RossThick} \left( {\Omega ,\Omega ^{{\prime }} } \right) + f_{geo} \rho_{LiSparse} \left( {\Omega ,\Omega ^{{\prime }} } \right) $$

(B.6)

An alternative form of the Ross kernels has also been introduced—this has a better parameterization of the hot-spot effect. The Rahman and Hapke kernels (#7 or #8) were discussed in [R1].

2.2.2 B.2.2 BPDF Kernels

The 3 BPDF kernels (Maignan et al. 2009) (numbers 10–12 in Table 3) were developed as semi-empirical models for polarized land-surface bidirectional reflectances. All three kernels are based on Fresnel reflectance. The polarization effects enter through the Fresnel reflectance term; for LIDORT, we require only the scalar Fresnel reflectance.

For the BPDF “Soil” type, the scalar reflectance is:

$$ \rho_{SOIL} \left( {\mu ,\mu^{{\prime }} ,\varphi - \varphi^{{\prime }} ,m} \right) = \frac{{F\left( {m,\theta_{r} } \right)}}{{4\mu \mu^{{\prime }} }}\left( {1 - \sin \theta_{r} } \right). $$

(B.7)

As before, $F\left( {m,\theta_{r} } \right)$ is the Fresnel reflection for incident angle $\theta_{r} = \frac{1}{2}\theta_{SCAT}$ and relative refractive index $m$ (which is the sole kernel parameter). Linearization of the kernel with respect to refractive index m requires the partial derivative of $F\left( {m,\theta_{r} } \right)$.

For the BDPF “Vegetation” type, there is dependence on the leaf orientation probability $\sigma \left( \alpha \right)$ and the leaf facet projections, through use of a plagiophile distribution:

$$ \begin{aligned} \rho_{VEGN} \left( {\mu ,\mu^{{\prime }} ,\varphi - \varphi^{{\prime }} ,m} \right) & = \frac{{F\left( {m,\theta_{r} } \right)}}{{4\mu \mu^{{\prime }} }}\frac{\zeta \left( \alpha \right)}{\text{H}}\left( {1 - \sin \theta_{r} } \right); \\ \cos \alpha & = \frac{{\left( {\mu + \mu^{{\prime }} } \right)}}{{2\cos \theta_{r} }}; \quad \zeta \left( \alpha \right) = \frac{16}{\pi } \cos^{2} \alpha \,\sin \alpha ; \\ {\text{H}} & = \frac{{\mathop \sum \nolimits_{k = 0}^{3} a_{k} \mu^{k} }}{\mu } + \frac{{\mathop \sum \nolimits_{k = 0}^{3} a_{k} \mu^{{{\prime }k}} }}{{\mu^{{\prime }} }}, \\ \end{aligned} $$

(B.8)

where the leaf projection ${\text{H}}$ depends on the set of “plagiophile coefficients” $\left\{ {a_{k} } \right\}$. Again, the refractive index is the only surviving kernel parameter to be considered for linearization.

For the BPDF “NDVI” kernel, we have:

$$ \rho_{NDVI} \left( {\mu ,\mu^{{\prime }} ,\varphi - \varphi^{{\prime }} ,m,N,C} \right) = \frac{{CF\left( {m,\theta_{r} } \right)}}{{4\left( {\mu + \mu^{\prime}} \right)}}\exp \left[ { - \tan \theta_{r} } \right]\exp \left[ { - N} \right], $$

(B.9)

where there are exponential attenuation terms, one of which depends on the NDVI $N$; in this formula, the scaling factor is $C$ (nominally, this is set to 1.0). Linearizations with respect to the parameters $N$ and $C$ are easy to establish. The NDVI varies from −1 to 1 and is defined as the ratio of the difference to the sum of two radiance measurements, one in the visible and one in the infrared.

Appendix C. Taylor Series Expansions

We have already noted that certain “multipliers” arising from optical-thickness integrations which are needed to find various solutions to components of the RT field may possess instability when certain limiting conditions are in place. Taylor series expansions are then required to remove such instabilities. Although some simple expansions were used in earlier versions of the LIDORT and VLIDORT codes, the whole issue has been completely revised in recent versions of the code (since [R1] was published), and we go into some detail here. Remarks on the VLIDORT implementation are made where appropriate.

3.1 C.1 Multipliers for the Intensity Field

We first look at the homogeneous-solution and primary-scatter downwelling multipliers, and the Green’s function (downwelling) multiplier for the diffuse source term at discrete-ordinate polar cosines. These are respectively:

$$ H_{j}^{ \downarrow } = \frac{{\left( {e^{{ - \Delta k_{j} }} - e^{{ - \Delta \mu^{ - 1} }} } \right)}}{{\mu^{ - 1} - k_{j} }}; F^{ \downarrow } = \frac{{\left( {e^{{ - \Delta \mu^{ - 1} }} - e^{ - \Delta \lambda } } \right)}}{{\lambda - \mu^{ - 1} }}; G_{j}^{ \downarrow } = \frac{{\left( {e^{{ - \Delta k_{j} }} - e^{ - \Delta \lambda } } \right)}}{{\lambda - k_{j} }}. $$

(C.1)

As before, $\lambda$ is the layer average secant corresponding to spherical-shell attenuation of the solar beam, $k_{j}$ is the separation constant corresponding to discrete ordinate stream $j$ arising from solution of the homogeneous RTE eigenproblem, $\mu$ is the cosine of the line-of-sight viewing polar angle, and $\Delta$ is the layer total optical depth. [Layer index $n$ is suppressed for clarity].

For polarized RT with VLIDORT, some of the separation constants $k_{j}$ may be complex variables; but $\lambda$ and $\mu$ are always real-valued. Taylor series expansions involving $k_{j}$ are only applicable for real values.

We consider the limiting cases $\left| {\lambda - k_{j} } \right| \to 0, \left| {\mu^{ - 1} - k_{j} } \right| \to 0$ or $\left| {\lambda - \mu^{ - 1} } \right| \to 0$. Writing $\epsilon$ for any of these quantities, if the order of the Taylor-series expansion is $M$, then we neglect terms $O\left( {\epsilon^{M + 1} } \right)$. Considering first the multiplier $G_{j}^{ \downarrow }$ from Eq. (C.1), then if $\epsilon = \lambda - k_{j}$, we have $e^{{ - \Delta k_{j} }} = We^{\epsilon \Delta } \approx W{\mathbf{z}}_{M + 1} \left( \Delta \right)$, where we have written $W = e^{ - \Delta \lambda }$, and we have used the notation ${\mathbf{z}}_{M + 1} \left( x \right) = \sum\nolimits_{m = 0}^{M + 1} {z_{m} \left( x \right)\epsilon^{m} }$ to approximate the exponential $e^{\epsilon \Delta }$, so that the coefficients are $z_{0} \left( x \right) = 1, z_{m} \left( x \right) = x^{m} /m!$ for $0 < m \le M + 1$. Applying the expansion, we find:

$$ G_{j}^{ \downarrow } \approx \frac{{W\left( {{\mathbf{z}}_{M + 1} \left( \Delta \right) - 1} \right)}}{\epsilon } = \Delta W{\mathbf{z}}_{M}^{*} \left( \Delta \right). $$

(C.2)

Here, the new coefficients are $z_{0}^{*} = 1, z_{m}^{*} = \Delta^{m} /\left( {m + 1} \right)!$ for $0 < m \le M$. Note that we need to expand first to order $M + 1$ to ensure that the final expression is defined to order $M$.

Next we look at those two multipliers for the Green’s function post-processed field which may be susceptible to instability through appearance of the term $\lambda - k_{j}$ in the denominator:

$$ M_{j}^{ \uparrow \uparrow } = \frac{{H_{j}^{ \uparrow } - F^{ \uparrow } }}{{\lambda - k_{j} }};\quad M_{j}^{ \downarrow \downarrow } = \frac{{H_{j}^{ \downarrow } - F^{ \downarrow } }}{{\lambda - k_{j} }}, $$

(C.3)

where $H_{j}^{ \downarrow }$ and $F^{ \downarrow }$ have been defined in Eq. (1), and

$$ H_{j}^{ \uparrow } = \frac{{\left( {1 - e^{{ - \Delta k_{j} }} e^{{ - \Delta \mu^{ - 1} }} } \right)}}{{\mu^{ - 1} + k_{j} }}; F^{ \uparrow } = \frac{{\left( {1 - e^{{ - \Delta \mu^{ - 1} }} e^{ - \Delta \lambda } } \right)}}{{\lambda + \mu^{ - 1} }}. $$

(C.4)

Looking at $M_{j}^{ \uparrow \uparrow }$ in Eq. (C.3), we set $k_{j} = \lambda - \epsilon$, and find:

$$ M_{j}^{ \uparrow \uparrow } \approx \frac{Y}{\epsilon }\left[ {\left( {1 - WU{\mathbf{z}}_{M + 1} \left( \Delta \right)} \right){\mathbf{a}}_{M + 1} \left( Y \right) - \left( {1 - WU} \right)} \right], $$

(C.5)

where $U = e^{{ - \Delta \mu^{ - 1} }} ,Y = \left( {\lambda + \mu^{ - 1} } \right)^{ - 1}$, and series ${\mathbf{a}}_{M + 1} \left( x \right)$ has coefficients $a_{0} \left( x \right) = 1, a_{m} \left( x \right) = x^{m}$ for $0 < m \le M + 1$. Since $a_{0} = z_{0} = 1$, it is apparent that the $\left( {1 - WU} \right)$ contributions will fall out, and the result can then be written:

$$ M_{j}^{ \uparrow \uparrow } \approx Y\left[ {Y{\mathbf{a}}_{M} \left( Y \right) - WU{\mathbf{c}}_{M} \left( {\Delta ,Y} \right)} \right]. $$

(C.6)

Here ${\mathbf{c}}_{M} \left( {\Delta ,Y} \right)$ has coefficients $c_{0} = 1$, and $c_{m} \left( {\Delta ,Y} \right) = \sum\nolimits_{p = 0}^{m} {z_{p} \left( \Delta \right)a_{m - p} \left( Y \right)}$ for $0 < m \le M$. We have found that use of these series coefficients is convenient for computation, allowing us to generate expressions to any order of accuracy without complicated algebraic expressions.

The other multiplier $M_{j}^{ \downarrow \downarrow }$ in Eq. (C.3) may be treated similarly. Note that multipliers in Eq. (C.4) are numerically stable entities, along with the other Green’s function multipliers $G_{j}^{ \uparrow } , M_{j}^{ \downarrow \uparrow }$ and $M_{j}^{ \uparrow \downarrow }$, the latter three quantities being defined with denominator $\lambda + k_{j}$.

The above multipliers are required for output of the upwelling and downwelling radiance fields at layer boundaries. LIDORT has the ability to generate output at any level away from layer boundaries (the “partial-layer” option). In this case, source function integration to an optical thickness $\tau < \Delta$ in layer n will result in “partial-layer” multipliers similar to those already defined above. For example, consider the following three functions:

$$ H_{j}^{ \downarrow } \left( \tau \right) = \frac{{\left( {e^{{ - \tau k_{j} }} - e^{{ - \tau \mu^{ - 1} }} } \right)}}{{\mu^{ - 1} - k_{j} }}; F^{ \downarrow } \left( \tau \right) = \frac{{\left( {e^{{ - \tau \mu^{ - 1} }} - e^{ - \tau \lambda } } \right)}}{{\lambda - \mu^{ - 1} }}; G_{j}^{ \downarrow } \left( \tau \right) = \frac{{\left( {e^{{ - \tau k_{j} }} - e^{ - \tau \lambda } } \right)}}{{\lambda - k_{j} }}; $$

(C.7)

These are the downwelling partial-layer multipliers equivalent to those in Eq. (C.1). Taylor series expansions for these and the corresponding Green’s function multipliers have been generated in a similar fashion.

3.2 C.2 Linearized Multipliers for the Jacobian Fields

The entire LIDORT discrete ordinate solution is analytically differentiable (Spurr 2002) with respect to any atmospheric quantity, and this includes the multipliers discussed above. It is therefore necessary to develop Taylor series expansions for those linearized multipliers which are susceptible to numerical instability.

We start with partial derivatives $\dot{k}_{j} \equiv \partial k_{j} /\partial \xi ,\dot{\lambda } \equiv \partial \lambda /\partial \xi$ and $\dot{\Delta } \equiv \partial \Delta /\partial \xi$ with respect to some atmospheric quantity $\xi$ defined in layer $n$ (index is not explicit here). Since the layer optical depth $\Delta$ is an intrinsic optical property, its derivative $\dot{\Delta }$ is also an intrinsic input. The viewing angle cosine $\mu$ has no derivative. Note that $\dot{k}_{j} = 0$ and $\dot{\Delta } = 0$ for a profile atmospheric quantity $\xi_{m }$ defined in layer $m \ne n$ (no cross-layer derivatives); on the other hand, the average secant $\lambda$ will have cross-layer derivatives for layers $m < n$, thanks to solar beam attenuation through the atmosphere. For the third multiplier in Eq. (C.1), the derivative is

$$ \frac{{\partial G_{j}^{ \downarrow } }}{\partial \xi } = \frac{{ - e^{{ - \Delta k_{j} }} \left( {\dot{k}_{j} \Delta + k_{j} \dot{\Delta }} \right) + e^{ - \Delta \lambda } \left( {\dot{\lambda }\Delta + \lambda \dot{\Delta }} \right) - G_{j}^{ \downarrow } \left( {\dot{\lambda } - \dot{k}_{j} } \right)}}{{\lambda - k_{j} }}. $$

(C.8)

Expanding Eq. (C.8) as a Taylor series with $k_{j} = \lambda - \epsilon$, and using the series-coefficient notation developed in Sect. C.1, we find:

$$ \frac{{ \partial G_{j}^{ \downarrow } }}{\partial \xi } \approx \frac{{ - \left( {\dot{k}_{j} \Delta + \lambda \dot{\Delta } + \epsilon \dot{\Delta }} \right)W{\mathbf{z}}_{M + 1} + \left( {\dot{\lambda }\Delta + \lambda \dot{\Delta }} \right)W - \left( {\dot{\lambda } - \dot{k}_{j} } \right)\Delta W{\mathbf{z}}_{M + 1}^{*} }}{\epsilon }. $$

(C.9)

Again, $W = e^{ - \Delta \lambda }$, and the series ${\mathbf{z}}_{M + 1}$ and ${\mathbf{z}}_{M + 1}^{*}$ have argument ${\Delta }$ and were defined in Sect. C.1. Since $z_{0} = z_{0}^{*} = 1$, the lowest-order terms in the numerator of Eq. (C.9) will fall out, and we are left with:

$$ \frac{{\partial G_{j}^{ \downarrow } }}{\partial \xi } \approx - W\left[ {\dot{\Delta }{\mathbf{z}}_{M} + \left( {\dot{k}_{j} \Delta + \lambda \dot{\Delta }} \right)\Delta {\mathbf{z}}_{M}^{*} + \left( {\dot{\lambda } - \dot{k}_{j} } \right)\Delta^{2} {\mathbf{z}}_{M}^{\dag } } \right], $$

(C.10)

where the third series ${\mathbf{z}}_{M}^{\dag }$ has coefficients $z_{0}^{\dag } = 1/2$ and $z_{m}^{\dag } = \Delta^{m} /\left( {m + 2} \right)!$ for $0 < m \le M$. Note that the presence of the series ${\mathbf{z}}_{M}^{\dag }$ to order $M$ implies that the original series must be computed to order $M + 2$; that is, we require ${\mathbf{z}}_{M + 2}$.

Linearization of the Green’s function multipliers in Eq. (C.3) follows similar considerations. We give one example; explicit differentiation of $M_{j}^{ \uparrow \uparrow }$ yields

$$ \frac{{\partial M_{j}^{ \uparrow \uparrow } }}{\partial \xi } = \frac{{\dot{H}_{j}^{ \uparrow } - \dot{F}^{ \uparrow } - \left( {\dot{\lambda } - \dot{k}_{j} } \right)M_{j}^{ \uparrow \uparrow } }}{{\lambda - k_{j} }}; $$

(C.11)

$$ \dot{H}_{j}^{ \uparrow } = \frac{{Ue^{{ - \Delta k_{j} }} \left[ {\dot{\Delta }\left( {\mu^{ - 1} + k_{j} } \right) + \dot{k}_{j} \Delta } \right] - \dot{k}_{j} H_{j}^{ \uparrow } }}{{\mu^{ - 1} + k_{j} }}; \dot{F}^{ \uparrow } = \frac{{UW\left[ {\dot{\Delta }\left( {\mu^{ - 1} + \lambda } \right) + \dot{\lambda }\Delta } \right] - \dot{\lambda }F^{ \uparrow } }}{{\lambda + \mu^{ - 1} }}. $$

(C.12)

Expanding in the usual manner and employing results established already, we find

$$ \dot{H}_{j}^{ \uparrow } \approx \left[ {UW\left( {\dot{k}_{j} \Delta + \dot{\Delta }Y^{ - 1} - \epsilon \dot{\Delta }} \right){\mathbf{z}}_{N} - \dot{k}_{j} Y\left( {1 - WU{\mathbf{z}}_{N} } \right){\mathbf{a}}_{N} } \right]Y{\mathbf{a}}_{N} ; $$

(C.13)

$$ \dot{F}^{ \uparrow } = \left[ {UW\left( {\dot{\lambda }\Delta + \dot{\Delta }Y^{ - 1} } \right) - \dot{\lambda }Y\left( {1 - WU} \right)} \right]Y; $$

(C.14)

where now $N = M + 2$ terms in the series have been retained in the expansions. Eq. (C.11) implies that we also require the original multiplier $M_{j}^{ \uparrow \uparrow }$ expanded to order $M + 1$, that is,

$$ M_{j}^{ \uparrow \uparrow } \approx Y\left[ {Y{\mathbf{a}}_{M + 1} \left( Y \right) - WU{\mathbf{c}}_{M + 1} \left( {\Delta ,Y} \right)} \right] $$

(C.15)

should be used in Eq. (C.11). Putting together the above three equations, and canceling out the lowest-order terms in the expansions, we find after some algebra that

$$ \frac{{\partial M_{j}^{ \uparrow \uparrow } }}{\partial \xi } \approx Y\left[ {WU{\mathbf{S}}_{M}^{\left( 1 \right)} - {\mathbf{S}}_{M}^{\left( 2 \right)} - {\mathbf{S}}_{M}^{\left( 3 \right)} } \right], $$

(C.16)

where the three series ${\mathbf{S}}_{M}^{\left( q \right)} = \sum\nolimits_{m = 0}^{M} {s_{m}^{\left( q \right)} \epsilon^{m } }$ have coefficients

$$ s_{m}^{\left( 1 \right)} = \left( {\dot{\Delta }Y^{ - 1} + \dot{k}_{j} \Delta } \right)c_{m + 1} \left( {\Delta ,Y} \right) - \dot{\Delta }c_{m} \left( {\Delta ,Y} \right); $$

(C.17a)

$$ s_{m}^{\left( 2 \right)} = Y\dot{k}_{j} \left[ {b_{m + 1} \left( Y \right) + WUd_{m + 1} \left( {\Delta ,Y} \right)} \right]; $$

(C.17b)

$$ s_{m}^{\left( 3 \right)} = \left( {\dot{\lambda } - \dot{k}_{j} } \right)\left[ {b_{m + 2} \left( Y \right) + WUc_{m + 2} \left( {\Delta ,Y} \right)} \right]. $$

(C.17c)

Subsidiary coefficients for $0 < m \le M$ are given by $a_{m} = Y^{m}$ (series expansion of $\left( {1 - \epsilon Y} \right)^{ - 1}$), and $b_{m} = \left( {m + 1} \right)Y^{m}$ (series expansion of $\left( {1 - \epsilon Y} \right)^{ - 2}$). We also have the product coefficients $c_{m} \left( {\Delta ,Y} \right) = \sum\nolimits_{p = 0}^{m} {z_{p} \left( \Delta \right)a_{m - p} \left( Y \right)}$ and $d_{m} \left( {\Delta ,Y} \right) = \sum\nolimits_{p = 0}^{m} {z_{p} \left( \Delta \right)b_{m - p} \left( Y \right)}$ obtained from the first exponential series ${\mathbf{z}}\left( \Delta \right)$ which approximates $e^{\Delta \epsilon }$.

Derivatives of the other multipliers subject to possible instability may be obtained similarly, and there are also derivatives of the partial-layer multipliers to be considered.

Software for all these instability cases has been written for LIDORT and VLIDORT - in both models, the order $M$ controls the accuracy of the Taylor series expansions. The other parameter controlling the use of these limiting-case calculations is the “small-number” value $\epsilon$. LIDORT and VLIDORT use double-precision floating-point arithmetic. With this in mind, we have chosen $\varepsilon = 10^{ - 3}$ as the default, after testing linearized multiplier accuracies obtained by running the model with and without the instability corrections. In practice $M = 3$ provides more than sufficient accuracy for this choice of $\epsilon$.