Invariant Imbedding Theory for the Vector Radiative Transfer Equation

Abstract

This chapter develops invariant imbedding theory as needed to solve the time-independent vector (polarized) radiative transfer equation (VRTE) for a plane-parallel water body bounded by a wind-blown sea surface and a reflective bottom.

A Fourier Analysis of Discrete Functions

The invariant imbedding solution algorithm for the directionally discretized VRTE employs a Fourier decomposition of the equations in azimuthal direction. This appendix collects for reference various results for the Fourier decomposition of discrete functions of azimuthal angle, as needed in Sect. 1.5.

1.1.1 A.1 Functions of One Azimuthal Angle

Let $f_v \equiv f(\phi _v)$ be any discrete function of the azimuthal angle $\phi$, i.e., $f_v$ is defined only at the discrete values $\phi _v = (v-1)\varDelta \phi$, for $v = 1, 2, ..., 2N$. Here $\varDelta \phi = 2\pi /2N$ is the angular width of a quad as shown in Fig. 1.5; $\varDelta \phi$ is the same for each quad. Let $\delta _k$ be the Kronecker delta function defined by

$$\begin{aligned} \delta _k \equiv {\left\{ \begin{array}{ll} ~1 &{} \text {if}\,\,k = 0 \\ ~0 &{} \text {if}\,\,k \ne 0 \end{array}\right. } ~~~\mathrm{or~equivalently}~~~ \delta _{k-\ell } \equiv {\left\{ \begin{array}{ll} ~1 &{} \text {if}\,\,k = \ell \\ ~0 &{} \text {if}\,\,k \ne \ell \end{array}\right. } \end{aligned}$$

(1.187)

where k and $\ell$ are integers.

The discrete orthogonality relations for sines and cosines can then be written

$$\begin{aligned} \sum _{s=1}^{2N} \cos (k \phi _s) \cos (\ell \phi _s) =&{\left\{ \begin{array}{ll} ~0 &{} \text {if}\,\,k \ne \ell \\ ~N &{} \text {if}\,\,k = \ell \,\,and\,\,\ell = 1, 2,..., N-1 \\ 2N &{} \text {if}\,\,k = \ell \,\,and\,\,\ell = 0 or \ell = N \end{array}\right. } \nonumber \\ =&~ N (\delta _{k+\ell } + \delta _{k - \ell } + \delta _{k + \ell - 2 N} ) \end{aligned}$$

(1.188)

$$\begin{aligned} \sum _{s=1}^{2N} \sin (k \phi _s) \sin (\ell \phi _s) =&{\left\{ \begin{array}{ll} 0 &{} \text {if}\,\,k \ne \ell \\ N &{} \text {if}\,\,k = \ell \,\,and\,\,\ell = 1, 2,..., N-1 \\ 0 &{} \text {if}\,\,k = \ell \,\,and\,\,\ell = 0\,\,or \ell = N \end{array}\right. } \nonumber \\ =&~ N (\delta _{k-\ell } - \delta _{k + \ell } - \delta _{k + \ell - 2 N} ) \end{aligned}$$

(1.189)

and

$$\begin{aligned} \sum _{s=1}^{2N} \cos (k \phi _s) \sin (\ell \phi _s) = 0 \quad \text {for all}\,\,k\,\,\text {and}\,\,\ell . \end{aligned}$$

(1.190)

A discrete function of one azimuthal angle (e.g., a discretized Stokes vector $\underline{S}(\zeta ,u,v,j)$ with the depth $\zeta$, polar angle u, and wavelength j being held constant) has the Fourier spectral decomposition

$$\begin{aligned} f(v) = \sum _{\ell = 0}^{N} \left[ \hat{f}_1 (\ell ) \cos (\ell \phi _v) + \hat{f}_2 (\ell ) \sin (\ell \phi _v) \right] \end{aligned}$$

(1.191)

where $v = 1, 2, ..., 2N$. The notation $\hat{f}$ denotes a Fourier amplitude of the corresponding physical variable f; subscript 1 denotes cosine amplitudes and subscript 2 denotes sine amplitudes. The cosine amplitudes $\hat{f}_1 (\ell )$ are obtained by multiplying Eq. (1.191) by $\cos (k \phi _v)$, summing over v, and applying the orthogonality relations to get

$$\begin{aligned} \hat{f}_1 (\ell ) = \frac{1}{\epsilon _l} \sum _{v=1}^{2N} f(v) \cos (\ell \phi _v) \quad \text {for } \ell = 0,1,...,N \end{aligned}$$

(1.192)

where

$$\begin{aligned} \epsilon _{\ell } = N (1 + \delta _{2 \ell } + \delta _{2 \ell - 2 N}) = {\left\{ \begin{array}{ll} 2N &{} \text {if}\,\,\ell = 0\,\,or\,\,\ell = N \\ ~N &{} \text {if}\,\,\ell = 1, 2,..., N-1 \end{array}\right. } \end{aligned}$$

(1.193)

Note that the $\hat{f}_1 (0)$ amplitude is just the average value of f(v). The sine amplitudes $\hat{f}_2 (\ell )$ are obtained in a similar way by multiplying (1.191) by $\sin (k \phi _v)$:

$$\begin{aligned} \hat{f}_2 (\ell ) = \frac{1}{\gamma _l} \sum _{v=1}^{2N} f(v) \cos (\ell \phi _v) \quad \text {for } \ell = 1,...,N-1 \end{aligned}$$

(1.194)

where

$$\begin{aligned} \gamma _{\ell } = N (1 - \delta _{2 \ell } - \delta _{2 \ell - 2 N}) = {\left\{ \begin{array}{ll} 0 &{} \text {if}\,\,\ell = 0\,\,or\,\,\ell = N \\ N &{} \text {if}\,\,\ell = 1, 2,..., N-1 \end{array}\right. } \end{aligned}$$

(1.195)

Note that values of $\gamma _0 = \gamma _N = 0$ do not occur in Eq. (1.194) for the sine amplitudes.

If f(v) is a constant f then $\hat{f}_1(0) = f$ and all other cosine and sine components are zero. This is the case for a polar cap radiance, which has no azimuthal dependence. In general, the 2N values of f(v) are determined exactly by the $N + 1$ cosine amplitudes and the $N - 1$ sine amplitudes; the information content of the physical and Fourier representations is the same. In the Fourier decomposition of discrete functions all terms must be included; there can be no truncation of these summations.

1.1.2 A.2 Functions of the Difference of Two Azimuthal Angles

Let $g_{\mathrm{cos}}(v,s) = g[\cos (\phi _v - \phi _s)]$ be a discrete cosine function of $\phi _v -\phi _s$, where $\phi _v$ and $\phi _s$ are two azimuthal angles and $v, s = 1, ..., 2N$. Then $g_{\mathrm{cos}}(v,s)$ has the Fourier expansion (L&W 8.24)

$$\begin{aligned} g_{\mathrm{cos}}(v,s) = \sum _{k = 0}^{N} \hat{g}_{1} (k) \cos [k ( \phi _v - \phi _s) ] \,. \end{aligned}$$

(1.196)

Multiplying this equation by $\cos (\ell \phi _v)$, summing over v, expanding the cosine, and using the orthogonality relations gives (L&W 8.25)

$$\begin{aligned} \hat{g}_1(\ell ) = \frac{1}{\epsilon _{\ell } \cos (\ell \phi _s)} \sum _{v = 1}^{2 N} g_{\mathrm{cos}}(v,s) \cos \ell \phi _v) \quad \text {for } \ell = 0,...,N \,. \end{aligned}$$

(1.197)

Similarly, let $g_{\mathrm{sin}}(v,s) = g[\sin (\phi _v - \phi _s)]$ be a discrete sine function of $\phi _v - \phi _s$. Then $g_{\mathrm{sin}}(v,s)$ has the Fourier expansion

$$\begin{aligned} g_{\mathrm{sin}}(v,s) = \sum _{k = 1}^{N-1} \hat{g}_{2} (k) \sin [k ( \phi _v - \phi _s) ] \,. \end{aligned}$$

(1.198)

Multiplying this equation by $\sin (\ell \phi _v)$, summing over v, expanding the sine, and using the orthogonality relations gives

$$\begin{aligned} \hat{g}_2(\ell ) = \frac{1}{\gamma _{\ell } \cos (\ell \phi _s)} \sum _{v = 1}^{2 N} g_{\mathrm{sin}}(v,s) \sin \ell \phi _v) \quad \text {for } \ell = 1,...,N-1 \,. \end{aligned}$$

(1.199)

These expansions will be used for the elements of the phase matrix , which depend on either the cosine or the sine of $\phi _v - \phi _s$. Note that in Eqs. (1.197) and (1.199), the choice $\phi _s = \phi _1 = 0$ can be made without loss of generality. This merely anchors the difference $\phi _v - \phi _s$ to $\phi _1 = 0$. The computer code then needs to compute and store phase function elements only for the range of $v = 1,...,N+1$, which generates all discretized scattering angles $\psi$ as $\phi _v = 0 ~\mathrm{to}~ 180^\circ$. The phase matrix element $\underline{P}(r,s \rightarrow u,v)$ is then obtained from the stored value of $\underline{P}(r,1 \rightarrow u,v-s+1)$, which can be stored as a three-index array $\underline{P}(r,u,v)$.

1.1.3 A.3 Functions of Two Azimuthal Angles

Finally, let $h(s,v) = h(\phi _s,\phi _v)$ be an arbitrary function of two azimuthal angles. Then h(s, v) can be represented as

$$\begin{aligned} h(s,v) =&\sum _{k=0}^{N} \sum _{\ell =0}^{N} \hat{h}_{11}(k,\ell ) \cos (k \phi _s) \cos (\ell \phi _v) \nonumber \\ +&\sum _{k=0}^{N} \sum _{\ell =0}^{N} \hat{h}_{12}(k,\ell ) \cos (k \phi _s) \sin (\ell \phi _v) \nonumber \\ +&\sum _{k=0}^{N} \sum _{\ell =0}^{N} \hat{h}_{21}(k,\ell ) \sin (k \phi _s) \cos (\ell \phi _v) \nonumber \\ +&\sum _{k=0}^{N} \sum _{\ell =0}^{N} \hat{h}_{22}(k,\ell ) \sin (k \phi _s) \sin (\ell \phi _v) \,. \end{aligned}$$

(1.200)

To find $\hat{h}_{11}$, multiply Eq. (1.200) by $\cos (k ' \phi _s) \cos (\ell ' \phi _v)$, sum over s and v, and apply the orthogonality relations. The other three amplitudes are found in an analogous manner. The results are

$$\begin{aligned} \hat{h}_{11}(k,\ell ) =&~ \frac{1}{\epsilon _k \epsilon _{\ell }} \sum _{s=1}^{2N} \sum _{v=1}^{2N} h(s,v) \cos (k \phi _s) \cos (\ell \phi _v) \,, \nonumber \\ \hat{h}_{12}(k,\ell ) =&~ \frac{1}{\epsilon _k \gamma _{\ell }} \sum _{s=1}^{2N} \sum _{v=1}^{2N} h(s,v) \cos (k \phi _s) \sin (\ell \phi _v) \,, \nonumber \\ \hat{h}_{21}(k,\ell ) =&~ \frac{1}{\gamma _k \epsilon _{\ell }} \sum _{s=1}^{2N} \sum _{v=1}^{2N} h(s,v) \sin (k \phi _s) \cos (\ell \phi _v) \,, \nonumber \\ \hat{h}_{22}(k,\ell ) =&~ \frac{1}{\gamma _k \gamma _{\ell }} \sum _{s=1}^{2N} \sum _{v=1}^{2N} h(s,v) \sin (k \phi _s) \sin (\ell \phi _v) \,. \end{aligned}$$

(1.201)

The arbitrary zero sine amplitudes $\hat{f}_2 (0)$ and $\hat{f}_2 (N)$ have their counterparts here:

$$\begin{aligned} \hat{h}_{12}(k,0) =&~\hat{h}_{12}(k,N) = 0 \quad \mathrm{for\ } k = 0,...,N \,, \nonumber \\ \hat{h}_{21}(0,\ell ) =&~ \hat{h}_{21}(N,\ell ) = 0 \quad \mathrm{for\ } \ell = 0,...,N \,, \nonumber \\ \hat{h}_{22}(0,0) =&~\hat{h}_{22}(0,N) = \hat{h}_{22}(N,0) = \hat{h}_{22}(N,N) = 0 \,. \end{aligned}$$

(1.202)

These special cases allow the exclusion of any k or $\ell$ values in Eq. (1.201) that would result in division by zero resulting from the $\gamma _k$ and $\gamma _{\ell }$ factors.

Equations (1.200)–(1.202) are applicable to the air-water surface transfer functions. However, the symmetries of those functions result in considerable simplification. In particular, all $\hat{h}_{12}(k,\ell )$ and $\hat{h}_{21}(k,\ell )$ are zero for the surface transfer functions.

The equations of Sect. 1.5.3 contain sums over k and $\ell$ having the form

$$\begin{aligned} S1 = \sum _{l=0}^{N} \sum _{k=0}^{N} f(k) g(\ell ) \left[ \sum _{s=1}^{2N} \cos [k(\phi _v - \phi _s)] \cos (\ell \phi _s) \right] \end{aligned}$$

where f(k) and $g(\ell )$ are discrete functions of k and $\ell$ for $k, \ell = 0,...,N$. Application of the the trigonometric formula for the cosine of the difference of two angles and the previous equations reduces this to

$$\begin{aligned} S1 =&\sum _{l=0}^{N} \sum _{k=0}^{N} f(k) g(\ell )~ \Bigg \{ \left[ \sum _{s=1}^{2N} \cos (k \phi _s) \cos (\ell \phi _s) \right] \cos (k \phi _v) \nonumber \\ +&\left[ \sum _{s=1}^{2N} \sin (k \phi _s) \cos (\ell \phi _s) \right] \sin (k \phi _v) \Bigg \} \nonumber \\ =&\sum _{l=0}^{N} \sum _{k=0}^{N} f(k) g(\ell )~ N (\delta _{k+\ell } + \delta _{k - \ell } + \delta _{k + \ell - 2 N} ) \cos (k \phi _v) \nonumber \\ =&\sum _{l=0}^{N} f(\ell ) g(\ell )~ N (1 + \delta _{2\ell } + \delta _{2\ell - 2 N} ) \cos (\ell \phi _v) \nonumber \\ =&\sum _{l=0}^{N} \epsilon _{\ell } f(\ell ) g(\ell ) \cos (\ell \phi _v) \,. \end{aligned}$$

(1.203)

The same process gives

$$\begin{aligned} S2 =&\sum _{l=0}^{N} \sum _{k=0}^{N} f(k) g(\ell ) \left[ \sum _{s=1}^{2N} \cos [k(\phi _v - \phi _s)] \sin (\ell \phi _s) \right] \nonumber \\ =&\sum _{l=0}^{N} \sum _{k=0}^{N} f(k) g(\ell ) ~N (\delta _{k-\ell } - \delta _{k + \ell } - \delta _{k + \ell - 2 N} ) \sin (k \phi _v) \nonumber \\ =&\sum _{l=0}^{N} f(\ell ) g(\ell ) ~N (1 - \delta _{2\ell } - \delta _{2\ell - 2 N} ) \sin (\ell \phi _v) \nonumber \\ =&\sum _{l=0}^{N} \gamma _{\ell } f(\ell ) g(\ell ) \sin (\ell \phi _v) \,. \end{aligned}$$

(1.204)

Likewise

$$\begin{aligned} S3 =&\sum _{l=0}^{N} \sum _{k=0}^{N} f(k) g(\ell ) \left[ \sum _{s=1}^{2N} \sin [k(\phi _v - \phi _s)] \cos (\ell \phi _s) \right] \nonumber \\ =&\sum _{l=0}^{N} \epsilon _{\ell } f(\ell ) g(\ell ) \sin (\ell \phi _v) \,, \end{aligned}$$

(1.205)

and

$$\begin{aligned} S4 =&\sum _{l=0}^{N} \sum _{k=0}^{N} f(k) g(\ell ) \left[ \sum _{s=1}^{2N} \sin [k(\phi _v - \phi _s)] \sin (\ell \phi _s) \right] \nonumber \\ =&- \sum _{l=0}^{N} \gamma _{\ell } f(\ell ) g(\ell ) \cos (\ell \phi _v) \,. \end{aligned}$$

(1.206)

Note the minus sign in the last equation.

These results for the Fourier decomposition of discrete functions of one or two azimuthal angles provide all of the tools necessary for converting the discretized physical-space VRTE into discretized equations in Fourier space .