Fourier Expansion of the Stokes Vector and Phase Matrix

It will prove useful to expand the azimuthal dependence of the Stokes vector and phase matrix in a Fourier cosine and sine series in order to reduce the number of variables treated at any one time. Thus,

  

$${\b Z}\angdep {\b Z}^{c0}(\tau;\mu,\mu') + 2 \sum_{m=1}^{M} [{\b Z}^{cm}(\mu,\mu') \cos{m(\phi-\phi')}$$ = (7.12)

$$+ {\b Z}^{sm}(\tau; \mu,\mu') \sin{m(\phi-\phi')}]$$

$${\b I}(\tau;\mu,\phi) {\b I}^{c0}(\tau;\mu) + 2 \sum_{m=1}^{M} [{\b I}^{cm}(\tau;\mu) \cos{m\phi}$$ = (7.13)

$$+ {\b I}^{sm}(\tau;\mu) \sin{m\phi}]$$

where m is the harmonic index and M is the highest harmonic used in the expansion. In general an infinite number of terms is required to exactly represent the the phase matrix. In practice, however, the summation is truncated at M and its value is dependent upon the nature of the scatterers. For isotropic scatterers, which have azimuthally independent phase matrix elements, only the m=0 term is required for an exact series representation. Rayleigh scatterers require terms up to m=2. The number required for Mie scattering will be highly dependent upon the size to wavelength ratio. For example, $\alpha\,(=2\pi r/\lambda)=2$ (where $\alpha$ is the size parameter as defined in Section 2.4.1), terms up to m=8 will suffice while for $\alpha=10$ terms up to m=16 are required. In addition, if only the irradiance or mean radiance are desired, then only the m=0 term need be computed as these are azimuthally independent quantities.

The azimuthally independent phase matrix coefficients must be calculated. This is done by making use of the orthogonality of sines and cosines. For example,

$${\bf Z}^{cm}(\tau;\mu;\mu') = \int_0^{2 \pi} {\bf Z}(\tau;\mu,\phi;\mu',\phi') \cos{m(\phi-\phi')} d(\phi-\phi') \\$$ (7.14)

with an analogous expression for odd coefficients. Many Fourier coefficients of the various phase matrix elements will end up being zero as all elements are either even or odd functions of $(\phi-\phi')$. The elements of ${\b Z}$ are even or odd according to,

$$\left[ \begin{array}{cccc} e & e & o & - \\ e & e & o & o \\ o & o & e & e \\ - & o & e & e \\ \end{array} \right]$$ (7.15)

where e denotes an even function and o an odd function.

Upon substituting these expansions into the plane-parallel radiative transfer equation, rewritten with the source term shown explicitly,

$$\mu \frac{d{\b I}(\tau;\mu,\phi)}{d\tau} = {\b I}(\tau;\mu,\p... ...tau;\mu,\phi;\mu',\phi') {\b I}(\tau;\mu',\phi') d\mu' d\phi'$$ (7.16)

the following set of differential equations relating the expansion coefficients for a given harmonic are obtained,

   

$$\mu \frac{d {\b I}^{c0}(\tau;\mu)}{d \tau} {\b I}^{c0}(\tau;\mu) - \frac{\tilde{\omega}(\tau)}{2} \int_{-1}^1 {\bf Z}^{c0}(\tau;\mu;\mu') {\bf I}^{c0}(\tau;\mu') d\mu'$$ = (7.17)

$$\mu \frac{d {\b I}^{cm}(\tau;\mu)}{d \tau} {\b I}^{cm}(\tau;\mu) -$$ = (7.18)

$$\frac{\tilde{\omega}(\tau)}{2} \int_{-1}^1 [{\bf Z}^{cm}(\tau;\mu... ...I}^{cm}(\tau;\mu') - {\bf Z}^{sm}(\tau;\mu;\mu') {\bf I}^{sm}(\tau;\mu')] d\mu'$$

$$\mu \frac{d {\b I}^{sm}(\tau;\mu)}{d \tau} {\b I}^{sm}(\tau;\mu) -$$ = (7.19)

$$\frac{\tilde{\omega}(\tau)}{2} \int_{-1}^1 [{\bf Z}^{sm}(\tau;\mu... ...I}^{cm}(\tau;\mu') + {\bf Z}^{cm}(\tau;\mu;\mu') {\bf I}^{sm}(\tau;\mu')] d\mu'$$

where $m=1,2,\ldots$ The single radiative transfer equation has been decomposed into 2M+1 azimuthally independent equations. Also, Equations (4.18) and (4.19) are coupled in ${\b I}^{cm}$ and ${\b I}^{sm}$.

Equations (4.17)-(4.19) have been recast in integral form below through the use of the integrating factor $e^{\tau/\mu}$. Only the downward streams are given,

$${\b I}^{c0}(\tau;-\mu) {\b I}^{c0}(0;\mu) e^{-\tau/\mu} + \int_0^{\tau} {\b J}^{c0}(\tau;\mu) e^{-(\tau-\tau')/\mu} \, \frac{d \tau'}{\mu}$$ = (7.20)

$${\b I}^{cm}(\tau;-\mu) {\b I}^{cm}(0;\mu) e^{-\tau/\mu} + \int_0^{\tau} {\b J}^{cm}(\tau;\mu) e^{-(\tau-\tau')/\mu} \, \frac{d \tau'}{\mu}$$ = (7.21)

$${\b I}^{sm}(\tau;-\mu) {\b I}^{sm}(0;\mu) e^{-\tau/\mu} + \int_0^{\tau} {\b J}^{sm}(\tau;\mu) e^{-(\tau-\tau')/\mu} \, \frac{d \tau'}{\mu}$$ = (7.22)

and the upward streams may be likewise determined from equations (4.17)-(4.19) using the integrating factor $e^{-\tau/\mu}$. Also, the following definitions have been used,

$${\b J}^{c0}(\tau;\mu) \equiv \frac{\tilde{\omega}(\tau)}{2} \int_{-1}^1 {\bf Z}^{c0}(\tau;\mu;\mu') {\bf I}^{c0}(\tau;\mu') d\mu'$$ (7.23)

$${\b J}^{cm}(\tau;\mu) \equiv \frac{\tilde{\omega}(\tau)}{2} \int_{-1}^1 [{\bf Z}^{cm}(\tau;\mu... ...u') - {\bf Z}^{sm}(\tau;\mu;\mu') {\bf I}^{sm}(\tau;\mu')] d\mu' \,\,\,\,\,\,\,$$ (7.24)

$${\b J}^{sm}(\tau;\mu) \equiv \frac{\tilde{\omega}(\tau)}{2} \int_{-1}^1 [{\bf Z}^{sm}(\tau;\mu... ...u') + {\bf Z}^{cm}(\tau;\mu;\mu') {\bf I}^{sm}(\tau;\mu')] d\mu' \,\,\,\,\,\,\,$$ (7.25)

in order to simplify these and future equations.