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The Vector Radiative Transfer Equation

Page updated: Jan 3, 2014
Principal author: Curtis Mobley
 

When describing polarized light at the phenomenological level of radiative transfer theory, the radiance is specified by the 4-component Stokes column vector $ {\bf\underline S} = [I,Q,U,V]^{\rm T}$, where the superscript T indicates the transpose (on this page, underlined bold font indicates 4 component vectors or 4 x 4 matrices). Vector component I is the total radiance, as is measured by a detector that is not sensitive to polarization. (L is usually used for radiance in oceanography, but I is customary in polarization studies involving Stokes vectors.) Q and U describe the state of linear polarization in a coordinate system chosen for the problem at hand, and V describes the state of circular polarization. Unpolarized radiance has the Stokes vector $ [I,0,0,0]^{\rm T}$, with I being the radiance in $ \rm {W~m^{-2}~sr^{-1}~nm^{-1}}$.

In oceanic optics, it is convenient to pick a coordinate system with the $ \bf x$ and $ \bf y$ axes lying in the mean sea surface and the $ \bf z$ axis normal to the mean sea surface (bold font not underlined indicates unit vectors specifying directions in Euclidean 3-space). In HydroLight, for example, the $ \bf x$ axis points in the downwind direction and $ \bf z$ points downward into the water. Figure 1 shows such a coordinate system. The red arrow in the figure indicates the direction $ \bf r'$ of propagation of an unscattered beam of light; the beam direction is specified by polar and azimuthal angles $ (\theta ', \phi')$ in the $ (\bf x, \bf y, \bf z)$ coordinate system. The shaded plane defined by $ \bf z$ and $ \bf r'$ is called the meridian plane. $ \bf h'$ is a unit vector that is normal to the meridian plane and thus parallel to the mean sea surface, i.e., "horizontal." Direction $ \bf v' = \bf r' \times \bf h'$ lies in the meridian place and thus is "vertical" to the mean sea surface when $ \phi'= 90 \deg$. Stokes vectors are usually specified with reference to the meridian plane and the "horizontal-vertical" $ (\bf h', \bf v', \bf r')$ coordinate system of Fig. 1. If the light is linearly polarized with the electric field vector E vibrating in the meridian plane as illustrated by the blue arrows in Fig. 1, then the light is said to be "vertically" polarized. The corresponding Stokes vector is $ {\bf\underline S '} = [1,-1,0,0]^{\rm T}$, if the magnitude of the total radiance is $ \it I = \rm {1~W~m^{-2}~sr^{-1}~nm^{-1}}$.

Figure: Fig. 1. Coordinate systems used in specifying the Stokes vector. The meridian plane is shaded. The electric field E and Stokes vector $ \bf \underline \it S '$ represent vertically polarized light propagating in the $ \bf r'$ direction.
Image 894446e6723b472960a2fe2b104cc5c1

Because the electric field describing a propagating wave at the level of Maxwell's equations is a vector, and the corresponding Stokes representation of the radiance is a 4-component column vector, care must be taken when referring these vectors to different coordinate systems when light is scattered. When light is scattered from incident direction $ \bf r'$ to a final direction $ \bf r$, the meridian planes are different. Thus the initial and final Stokes vectors are specified in different coordinate systems. In addition, the 4x4 Mueller matrix M, which describes scattering in the Stokes-vector representation, is specified in the scattering plane defined the by incident and final directions $ \bf r'$ and $ \bf r$, respectively.

Figure 2 shows the coordinate systems needed to define scattering of polarized radiance. The $ (\bf h', \bf v', \bf r')$ coordinate system is the same as in Fig. 1. The $ (\bf h, \bf v, \bf r)$ coordinate system is similarly defined for the meridian plane of the light scattered into direction $ (\theta, \phi)$. The scattering plane is shown shaded. The scattering angle $ \psi$ can be computed from

$\displaystyle \cos \psi = \bf {r '} \cdot \bf { r} = \cos{\theta'} \cos \theta + \sin{\theta'} \sin \theta \cos (\phi' - \phi )\,\, .$ (1)

Figure: Fig. 2. Coordinate systems needed for specification of the scattering process. The scattering plane is shaded.
Image 4e1a76c9a27796a8b60a92203eac0337

The Stokes vector must be specified in a coordinate system defined by the scattering plane before the Mueller matrix can be applied. The $ (\bf {s', p',r'})$ coordinate system of the incident light in the scattering plane is defined with $ \bf s'$ being perpendicular (senkrecht, in German) to the scattering plane, and $ \bf p' = \bf r'\times \bf s'$ being parallel to the scattering plane, as shown in Fig. 2. The Stokes vector $ \bf\underline S$ of the scattered light obtained from application of the Mueller matrix to the rotated incident vector $ \bf\underline S'$ is specified in the $ (\bf {s, p, r})$ scattering plane system, but $ \bf\underline S$ is measured in the $ (\bf {h, v, r})$ meridian plane system. The steps necessary to describe the scattering process are thus

  • Specify the Stokes vector $ \bf\underline S'$ of the incident radiance in the $ (\bf {h', v', r'})$ coordinate system defined by the incident meridian plane, where $ \bf\underline S'$ is measured.
  • Rotate the $ (\bf {h', v', r'})$ coordinate system into the $ (\bf {s', p',r'})$ coordinate system of the scattering plane, to obtain the components of $ \bf\underline S'$ in the scattering plane, where the Mueller matrix is specified.
  • Apply the Mueller matrix $ \bf\underline M(\psi)$ to scatter $ \bf\underline S'$ through scattering angle $ \psi$ into $ \bf\underline S$ in the scattering plane system $ (\bf s, \bf p, \bf r)$.
  • Rotate the $ (\bf {s, p, r})$ system into the final meridian plane $ (\bf {h, v, r})$, where the scattered Stokes vector $ \bf\underline S$ is measured.

Define a rotation through a positive angle as being a counterclockwise rotation about the beam direction, when looking into the beam. Thus the initial rotation from $ (\bf {h', v', r'})$ into $ (\bf {s', p',r'})$ is a rotation through a positive angle $ \alpha'$, and the final rotation from $ (\bf {s, p, r})$ into $ (\bf {h, v, r})$ is a rotation through a negative angle $ - \alpha$, as illustrated in Fig. 2. Spherical trigonometry gives these angles as (van de Hulst, 1980), p 499)

$\displaystyle \sin \alpha' ~=$ $\displaystyle ~\sin \theta \sin (\phi' - \phi ) / \sin \psi$ (2)
$\displaystyle \sin \alpha ~=$ $\displaystyle ~\sin \theta' \sin (\phi' - \phi ) / \sin \psi \,\, .$ (3)

With this definition of a positive angle rotation, the 4x4 rotation matrix $ \bf\underline R (\alpha ')$ that transforms $ \bf\underline S'$ from the initial meridian plane into the scattering plane is (e.g., Mischenko et al., 2002, Eq. 1.97)

$\displaystyle {\bf\underline R}(\alpha' )=\left (
 \begin{array}{cccc} 
 1 & 0 & 0 & 0 \\ 
 0 & \cos 2 \alpha' & - \sin 2 \alpha' & 0 \\ 
 0 & \sin 2\alpha' & \cos 2 \alpha' & 0 \\ 
 0 & 0 & 0 & 1
 \end{array}
 \right )\,\, .$ (4)

[Other rotation angle conventions are common in the literature. Mischenko et al. (2002) define a positive rotation as being clockwise looking along the beam, which is equivalent to the present counterclockwise looking into the beam. However, Bohren and Huffman, (1983, their Eq. 2.83), Lenoble, (1993, her Eq. 14.24), and Kattawar and Adams (1989, their Eq. 8) define a positive rotation as being counterclockwise looking along the beam, or clockwise looking into the beam. Those authors thus derive a Stokes vector rotation matrix that is the transpose of Eq. (4).]

The matrix multiplications involved in scattering $ \bf\underline S'$ into $ \bf\underline S$ are thus

$\displaystyle {\bf\underline S} = {\bf\underline R} (- \alpha )\, {\bf\underline M} (\psi )\,{\bf\underline R} (\alpha' )\,{\bf\underline S '} \,\, .$ (5)

These preliminaries allow us to write (in analogy with the scalar RTE), the 1D, time-independent, vector (or polarized) radiative transfer equation (VRTE) as

$\displaystyle \cos \theta \frac{d {\bf\underline S} (\theta,\phi)}{d z}~=$ $\displaystyle - c \, \bf\underline {S} (\theta, \phi)$    
$\displaystyle +$ $\displaystyle \int_0^{2 \pi} \int_0^{\pi} {\bf\underline R}(-\alpha)\, {\bf\underline M} (\psi)\,{\bf\underline R}(\alpha ')\, {\bf\underline S'} (\theta ', \phi') \sin \theta ' d \theta ' d \phi '$ (6)
$\displaystyle +$ $\displaystyle {\bf\underline \Sigma} (\theta, \phi) \,\,.$    

Here c is the beam attenuation coefficient. $ \bf\underline \Sigma$ is any internal source of polarized or unpolarized radiance in direction $ (\theta, \phi)$, with units of radiance per unit distance, i.e. $ \rm W~m^{-3}~sr^{-1}~nm^{-1}$. The Stokes vectors, $ \bf\underline{M}$, c, and $ \bf\underline \Sigma$ generally depend on depth and wavelength, but those arguments are not shown here for brevity. It should be remembered that (just as for the volume scattering function) each element M(i,j) of a Mueller matrix is also a function of the scattering angle $ \psi, 0 \leq \psi \leq \pi$, and that the functional form of these elements is determined by the type of scattering particle. In the integration over all incident directions, the angles $ \psi, \alpha ', \rm {~and~} \alpha$ are obtained from Eqs. (1-3) for the given $ (\theta ', \phi', \theta, \phi)$.

The M(1,1) element of $ \bf\underline{M}$ is the volume scattering function for unpolarized radiance, $ \beta$. Mueller matrices $ \bf\underline{M}$ are often presented as reduced Mueller matrices $ \bf\underline {\tilde{M}}$ in which each element of $ \bf\underline{M}$ has been divided by the M(1,1) element. This amounts to "factoring out" the volume scattering function for unpolarized radiance. Thus, for example, the Mueller matrix for Rayleigh scattering can be written as

$\displaystyle \bf\underline M _{\rm {Ray}}(\psi, \lambda) = \beta _{\rm {Ray}}(\psi, \lambda)
 \left (
 \begin{array}{cccc}
 1 & -{\frac{\sin ^{2} \psi }{1 + \cos ^{2} \psi }} & 0 & 0 \\ 
 -{\frac{\sin ^{2} \psi }{1 + \cos ^{2} \psi }} & 1 & 0 & 0 \\ 
 0 & 0 &{\frac{2 \cos \psi }{1 + \cos ^{2} \psi }} & 0 \\ 
 0 & 0 & 0 &{\frac{2 \cos \psi }{1 + \cos ^{2} \psi }}
 \end{array} 
 \right )$ (7)

or

$\displaystyle {\bf\underline M} _{\rm Ray} (\psi , \lambda ) = \beta _{\rm Ray} (\psi , \lambda ) \, {\bf\underline {\tilde{M}}} _{\rm Ray} (\psi )\,\, .$ (8)

Here

   
$ \bf\underline {\tilde{M}}_{\rm Ray} (\psi )$ is the $ 4 \times 4$ reduced Mueller matrix for Rayleigh scattering seen in Eq. (7).
   
$ \beta _{\rm Ray} (\psi ,\lambda )~=~b _{\rm Ray} (\lambda ) \frac{3}{16\pi } (1 +\cos 2 \psi )$ is the Rayleigh volume scattering function for unpolarized radiance [with units of $ \rm m^{-1}~sr^{-1}$].
   
$ b_{\rm Ray}(\lambda)$ is the Rayleigh scattering coefficient [units of $ \rm m^{-1}$], which has a $ \lambda^{-4}$ wavelength dependence.
   
$ \tilde{\beta }_{\rm Ray} (\psi )= \frac{3}{16 \pi} (1 + \cos 2 \psi )$ is the Rayleigh phase function for unpolarized radiance [with units of $ \rm sr^{-1}$]. This phase function satisfies the normalization condition $ 2 \pi \int _{0}^{\pi } \tilde{\beta }_{\rm Ray} (\psi ) \sin \psi d \psi = 1$ .

The VRTE can thus be written as

$\displaystyle \cos \theta \frac{d {\bf\underline S} (\theta,\phi)}{d z}~=$ $\displaystyle - c \, {\bf\underline S} (\theta, \phi)$    
$\displaystyle +$ $\displaystyle \int_0^{2 \pi} \int_0^{\pi} \beta ( \psi ) \, {\bf\underline R}(-\alpha)\, {\bf\underline {\tilde{M}}} (\psi)\,{\bf\underline R}(\alpha ')\, {\bf\underline S'} (\theta ', \phi') \sin \theta ' d \theta ' d \phi '$ (9)
$\displaystyle +$ $\displaystyle {\bf\underline \Sigma} (\theta, \phi) \,\,.$    

The product of the rotation and reduced Mueller matrices is often written as

$\displaystyle {\bf\underline R}(-\alpha)\, {\bf\underline {\tilde{M}}} (\psi)\,{\bf\underline R}(\alpha ') = {\bf\underline {\tilde{P}}} (\theta' ,\phi' \rightarrow \theta , \phi ) = {\bf\underline {\tilde{P}}} (\psi )\,\, ,$ (10)

where $ {\bf\underline {\tilde{P}}} (\psi )$ is called the phase matrix in analogy to the phase function $ \tilde {\beta} (\psi)$ of scalar theory. The VRTE is then

$\displaystyle \cos \theta \frac{d {\bf\underline S} (\theta,\phi)}{d z} =$ $\displaystyle - c \, {\bf\underline S} (\theta, \phi)$    
$\displaystyle +$ $\displaystyle \int_0^{2 \pi} \int_0^{\pi} \beta ( \psi ) \, {\bf\underline {\tilde{P}}} (\psi )\, {\bf\underline S'} (\theta ', \phi') \sin \theta ' d \theta ' d \phi '$ (11)
$\displaystyle +$ $\displaystyle {\bf\underline \Sigma} (\theta, \phi) \,\,.$    

Note that if $ {\bf\underline {\tilde{P}}} (1,1) = 1$ and all other elements are 0, then Eq. (11) reduces to the scalar RTE.

Mueller matrices for spherical particles larger than those of the Rayleigh domain (particle size $ \ll \lambda$) can be computed by Mie theory. However, there are almost no actual measurements of Mueller matrices for ocean waters. Figure 3 shows one of the few actual data sets, acquired by Voss and Fry (1984) in Atlantic waters. It should be noted that the reduced Mueller matrix for ocean waters is similar, but not identical, to that for Rayleigh scattering.

Figure: Fig. 3. The top panel is a graphical display of the 16 elements of the reduced Mueller matrix for Rayleigh scattering, as given analytically in Eq. (*). The bottom panel is the reduced Mueller matrix for ocean water as measured by Voss and Fry (1984). The 0 to 180 range of each plot shows the functional dependence on the scattering angle $ \psi$ from 0 to 180 degrees.
Image 95e5218dc8b1a4477d4ece3bd2b6f627

The similarities seen in Fig. 3 between the reduced Muller matrices for Rayleigh scattering and for ocean waters suggests that for some (but probably not all) purposes it may be reasonable to use a Mueller matrix that combines a VSF for unpolarized scattering by oceanic particles and the reduced Mueller matrix for Rayleigh scattering, which has a convenient analytic form. With this approximation, $ \beta (\psi ) \, \bf\underline {\tilde{P}} (\psi )$ in Eq. (11) would be replaced by $ \beta (\psi ) \, \bf\underline {\tilde{P}}_{\rm Ray} (\psi)$ . Whether or not this is an acceptable simplification for general oceanographic calculations awaits measurements of in-water Mueller matrices for a variety of oceanic particles and subsequent comparison of numerical computations based on those measurements with those based on the simplification.