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Radiative Transfer Theory

Deriving the Radiative Transfer Equation

Page updated: Dec 15, 2016
Principal author: Curtis Mobley
 
The Light and Radiometry chapter showed how various physical and mathematical quantities such as energy and solid angle can be combined to describe light in terms of radiance. The Overview chapter showed how various IOPs can be used to describe the optical properties of the medium through which light propagates. We now build on that knowledge to derive an equation-the radiative transfer equation-that relates how a medium affects a beam of light traveling through the medium. This derivation can be done in a mathematically rigorous way, starting even from the first principles of quantum electrodynamics, but a much simpler heuristic approach will get us to the same final equation.

Radiative Processes

The radiative transfer equation, commonly called the RTE, expresses conservation of energy written for a collimated beam of radiance traveling through an absorbing, scattering and emitting medium. We thus begin by considering the various processes that can occur when a photon interacts with an atom or molecule.

The photon may be annihilated, leaving the atom or molecule in an excited state with higher internal (electronic, vibrational, or rotational) energy. All or part of the absorbed photon's energy may be subsequently converted into thermal (kinetic) or chemical energy (manifested, for example, in the formation of new chemical compounds). The annihilation of a photon and conversion of its energy into a nonradiant form is called absorption. If the molecule almost immediately (on a femtosecond or shorter time scale) returns to its original internal energy state by emitting a new photon of the same energy as the absorbed photon (but probably traveling in a different direction from the original photon), the process is called elastic scattering. Because of the extremely short time required for these events, elastic scattering can reasonably be thought of as the photon interacting with the molecule and simply "changing direction" without an exchange of energy with the scattering molecule.

The excited molecule also may emit a photon of lower energy (longer wavelength) than the incident photon. The molecule thus remains in an intermediate excited state and may at a later time emit another photon and return to its original state, or the retained energy may be converted to thermal or chemical energy. Indeed, if the molecule is initially in an excited state, it may absorb the incident photon and then emit a photon of greater energy (shorter wavelength) than the absorbed photon, thereby returning to a lower energy state. In either case the scattered (emitted) photon has a different wavelength than the incident (absorbed) photon, and the processes is called inelastic scattering. An important example of this process in the ocean is Raman scattering by water molecules. Fluorescence is an absorption and re-emission process that occurs on a time scale of $ 10^{-11}$ to $ 10^{-8}~\rm {sec}$ . If the re-emission requires longer than about $ 10^{-8}~\rm {sec}$ , the process is usually called phosphorescence. The physical and chemical processes that lead to the vastly different times scales of Raman scattering vs. fluorescence vs. phosphorescence are much different. The distinctions between the very short time scale of Raman "scattering" vs. the longer time scale of fluorescence "absorption and re-emission" do not concern us in the derivation of the time-independent RTE. However, the terminology has evolved somewhat differently, e.g., Raman scattering usually refers to "incident" and "scattered" wavelengths, whereas fluorescence usually refers to "excitation" and "emission" wavelengths.

The reverse process to absorption is also possible, as when chemical energy is converted into light; this process is called emission. An example of this is bioluminescence, in which an organism converts part of the energy from a chemical reaction into light.

In order to formulate the RTE, it is convenient to imagine light in the form of many beams of photons of various wavelengths coursing in all directions through each point of a water body. We then consider a single one of these beams, which is traveling in some direction $ (\theta, \phi)$ and has wavelength $ \lambda$ . This beam and the processes affecting it are illustrated in Fig. 1.

Figure: Fig. 1. Illustration of a single beam of radiance and the processes that affect it as it propagates a distance $ \Delta r$ .
Image f45c3624a49a31224925fbb4e3bb67c7

Now think of all the ways in which that beam's population of photons, i.e. its radiance, can be decreased or increased. Bearing in mind the preceding comments, the following six processes are both necessary and sufficient to write down an energy balance equation for a beam of photons on the phenomenological level:

  1. loss of photons from the beam through annihilation of photons and conversion of radiant energy to nonradiant energy (absorption)

  2. loss of photons from the beam through scattering to other directions without change in wavelength (elastic scattering)

  3. loss of photons from the beam through scattering (perhaps to other directions) with change in wavelength (inelastic scattering)

  4. gain of photons by the beam through scattering from other directions without change in wavelength (elastic scattering)

  5. gain of photons by the beam through scattering (perhaps from other directions) with a change in wavelength (inelastic scattering)

  6. gain of photons by the beam through creation of photons by conversion of nonradiant energy into radiant energy (emission)

Next we must mathematically express how these six processes change the radiance as the beam travels a short distance $ \Delta r$ in passing through a small volume $ \Delta V$ of water, which is represented by the blue rectangle of Fig. 1.

Processes 1 and 3. It is reasonable to assume that the change in radiance while traveling distance $ \Delta r$ due to absorption is proportional to the incident radiance, i.e., the more incident photons there are, the more are lost to absorption. Thus we can write

$\displaystyle \frac{L(r+\Delta r,\theta ,\phi ,\lambda ) - L(r,\theta ,\phi ,\lambda )}{\Delta r} ~=$      
$\displaystyle \frac{\Delta L(r+\Delta r,\theta ,\phi ,\lambda )}{\Delta r} ~=$ $\displaystyle ~- a(r,\lambda ) L(r,\theta ,\phi ,\lambda )\,\,.$ (1)

Here $ \Delta L(r+\Delta r, \theta, \phi, \lambda)$ denotes the change in L between r and $ r+\Delta r$ . The minus sign is necessary because the radiance decreases (photons are disappearing, so $ \Delta L$ is negative) along $ \Delta r$ . Referring back to Eq. (1) on the IOP page, it is easy to see that the present Eq. (1) is just the definition of the absorption coefficient written as a change in radiance over distance $ \Delta r$ , rather than as a change in absorptance. Thus the proportionality constant $ a(r,\lambda)$ in Eq. (1) is just the absorption coefficient as defined on the IOP page. Note that absorption at the wavelength $ \lambda$ of interest accounts both for energy converted to non-radiant form ("true" absorption) and for energy that disappears from wavelength $ \lambda$ and re-appears at a different wavelength (inelastic scattering). Either process leads to a loss of energy from the beam at wavelength $ \lambda$ .

Process 2. In a similar fashion, the loss due to elastic scattering out of the $ (\theta, \phi)$ beam direction into all other directions can be written as

$\displaystyle \frac{\Delta L(r+\Delta r,\theta ,\phi ,\lambda )}{\Delta r} ~=~- b(r,\lambda ) L(r,\theta ,\phi ,\lambda )\,\, ,$ (2)

where $ b(r,\lambda)$ is the scattering coefficient as defined on the IOP page.

Process 4. This process accounts for elastic scattering from all other directions into the beam direction $ (\theta, \phi)$ . Figure 2 shows Fig. 1 redrawn to illustrate scattering along path length $ \Delta r$ from one direction $ (\theta ', \phi ')$ into the direction $ (\theta, \phi)$ of interest. These incident and final directions correspond to scattering angle $ \psi$ as shown in Fig. 2.

Figure: Fig. 2. Illustration of a beam of radiance in direction $ (\theta ', \phi ')$ generating radiance in the direction of interest $ (\theta, \phi)$ by elastic scattering.
Image 27a61e0ba54c5d47aa147154eaeb557b

Recalling from Eq. (2) of the IOP page that one definition of the volume scattering function $ \beta$ is scattered intensity per unit incident irradiance per unit volume, we can write

$\displaystyle I _{s} (r +\Delta r, \theta , \phi , \lambda )~=~E _{i} (\theta' , \phi', \lambda ) \beta (\theta' ,\phi' \rightarrow \theta , \phi ; \lambda )\Delta V \,\, .$ (3)

Here $ I_{s}(r+\Delta r, \theta, \phi, \lambda)$ is the intensity exiting the scattering volume at location $ r+\Delta r$ in direction $ (\theta, \phi)$ . All of this intensity is created along $ \Delta r$ by scattering from direction $ (\theta ', \phi ')$ into $ (\theta, \phi)$ , so $ \Delta I_{s}(r+\Delta r, \theta, \phi, \lambda)$ = $ I_{s}(r+\Delta r, \theta, \phi, \lambda)$ . The incident irradiance $ E_i$ is computed on a surface normal to the incident beam direction, as illustrated by the dotted lines in Fig. 2. We can rewrite $ E_i$ as the incident radiance times the solid angle of the incident beam:

$\displaystyle E _{i} (\theta' ,\phi' , \lambda ) ~=~L(\theta' ,\phi' , \lambda ) \Delta \Omega (\theta' ,\phi' )\,\, .$ (4)

Next recall from the geometrical radiometry page that intensity is radiance times area. Thus the intensity created by scattering along pathlength $ \Delta r$ and exiting the scattering volume over an area $ \Delta A$ can be written as

$\displaystyle \Delta I _{s} (r+\Delta r, \theta , \phi , \lambda )~ =~ \Delta L(r+\Delta r, \theta , \phi , \lambda )\Delta A \,\, ,$ (5)

where $ \Delta L(r+\Delta r, \theta, \phi, \lambda)$ is the radiance created by scattering along $ \Delta r$ and exiting the scattering volume over a surface area $ \Delta A$ . Using Eqs. (4) and (5) in (3) and writing the scattering volume as $ \Delta V = \Delta r \Delta A$ gives

$\displaystyle \frac{\Delta L(r+\Delta r, \theta , \phi , \lambda )}{\Delta r} =~L(\theta' ,\phi' ,\lambda ) \beta (\theta' ,\phi' \rightarrow \theta , \phi ; \lambda )\Delta \Omega (\theta' ,\phi' )\,\, .$ (6)

This equation gives the contribution to $ \Delta L(r+\Delta r, \theta, \phi, \lambda)/\Delta r$ by scattering from one particular direction $ (\theta ', \phi ')$ . However, ambient radiance may be passing through the scattering volume from all directions. We can sum up the contributions to $ \Delta L(r+\Delta r, \theta, \phi, \lambda)/\Delta r$ from all directions by integrating the right hand side of Eq. (6) over all directions,

$\displaystyle \frac{\Delta L(r+\Delta r, \theta , \phi , \lambda )}{\Delta r} = \int_{0}^{2 \pi } \int_{0}^{\pi} L(\theta' ,\phi' , \lambda ) \beta (\theta' ,\phi' \rightarrow \theta , \phi ; \lambda ) \sin \theta' d \theta' d \phi' \,\, ,$ (7)

where we have written the element of solid angle in terms the angles using Eq. (6) from the geometry page.

Processes 5 and 6. Process 5 accounts for radiance created along pathlength $ \Delta r$ in direction $ (\theta, \phi)$ at wavelength $ \lambda$ by inelastic scattering from other all other wavelengths $ \lambda '\neq \lambda$ . Each such process, such as Raman scattering by water molecules or fluorescence by chlorophyll or CDOM molecules, requires a separate mathematical formulation to specify how radiance is absorbed from an incident beam at wavelength $ \lambda '$ and converted to the wavelength $ \lambda$ of interest. Process 6 accounts for radiance created de novo by "true" emission, e.g. by bioluminescence, and each emission process again requires a separate formulation to define how the light is emitted as a function of location, direction, and wavelength. Those detailed formulations can be complex and are treated elsewhere. For the moment, we can simply include a generic source function that represents creation of radiance along pathlength $ \Delta r$ in direction $ (\theta, \phi)$ at wavelength $ \lambda$ by any inelastic scattering or emission process. Thus we write just

$\displaystyle \frac{\Delta L(r+\Delta r,\theta ,\phi ,\lambda )}{\Delta r} ~=~S(r,\theta , \phi , \lambda )\,\, ,$ (8)

without specifying the mathematical form of the source function S.

We can now sum of the various contributions to the changes in L along $ \Delta$ r. We can also take the conceptual limit of $ \Delta r \to 0$ and write

$\displaystyle \frac{d L(r,\theta ,\phi ,\lambda )}{d r} ~=~\lim_{\Delta r \to 0} \frac{\Delta L(r+\Delta r,\theta ,\phi ,\lambda )}{\Delta r} \,\, .
$

Standard Forms of the RTE

The net change in radiance due to all six radiative processes is the sum of the right hand sides of Eqs. (1), (2), (7), and (8). We thus obtain an equation relating the changes in radiance with distance along a given beam direction to the optical properties of the medium and the ambient radiance in other directions:

$\displaystyle \frac{d L (r, \theta , \phi , \lambda )}{d r} ~=$ $\displaystyle ~- [a(r,\lambda ) + b(r, \lambda )] L(r, \theta , \phi , \lambda )$    
$\displaystyle +$ $\displaystyle ~\int_{0}^{2 \pi } \int_{0}^{\pi} L(r, \theta' ,\phi' , \lambda ) \beta (r;\theta' ,\phi' \rightarrow \theta , \phi ;\lambda ) \sin \theta' d \theta' d \phi'$    
$\displaystyle +$ $\displaystyle ~S(r, \theta , \phi ,\lambda )~~~~~~~~~~ {\rm (W~m^{-3}~sr^{-1}~nm^{-1})} \,\, .$ (9)

This is one form of the RTE, written for changes in radiance along the beam path.

In oceanography, it is usually convenient to use a coordinate system with the depth z being normal to the mean sea surface and positive downward. Thus depth z is a more convenient spatial coordinate than location r along the beam path. Changes in r are related to changes in z as shown in Fig. 1: $ dr = dz/cos \theta$ . Using this in Eq. (9), assuming that the ocean is horizontally homogeneous, and recalling that $ a + b = c$ , we get

$\displaystyle \cos \theta \frac{d L (z, \theta , \phi , \lambda )}{d z} ~=$ $\displaystyle ~- c(z,\lambda )L(z, \theta , \phi , \lambda )$    
$\displaystyle +$ $\displaystyle ~\int_{0}^{2 \pi } \int_{0}^{\pi} L(z,\theta' ,\phi' , \lambda ) \beta (z;\theta' ,\phi' \rightarrow \theta , \phi ;\lambda ) \sin \theta' d \theta' d \phi'$    
$\displaystyle +$ $\displaystyle ~S(z, \theta , \phi ,\lambda ) \,\,.$ (10)

This equation expresses location as geometric depth z and the IOPs in terms of the beam attenuation c and the volume scattering function $ \beta$ .

Other forms of the RTE are often used. The nondimensional optical depth $ \zeta$ is defined by

$\displaystyle d \zeta ~=~c(z,\lambda ) dz\,\, .$ (11)

Dividing Eq. (10) by $ c(z,\lambda)$ and using (11) gives the RTE written in terms of optical depth. It is also common to use $ \mu = cos \theta$ as the polar angle variable. Recalling Eq. (7) of the volume scattering function page, we can factor the volume scattering function $ \beta$ into the scattering coefficient b times the scattering phase function $ \tilde{\beta}$ . Finally, recalling the definition of the albedo of single scattering $ \omega_{\rm o} = b/c$ , we can re-write Eq. (10) as

$\displaystyle \mu \frac{d L(\zeta ,\mu , \phi , \lambda )}{d \zeta } ~=$ $\displaystyle - L(\zeta ,\mu ,\phi ,\lambda )$    
$\displaystyle +$ $\displaystyle ~\omega _{\rm o} (\zeta , \lambda ) \int_{0}^{2 \pi } \int_{-1}^{1} L(\zeta ,\mu' ,\phi' , \lambda ) \tilde{\beta } (\zeta ;\mu' ,\phi' \rightarrow \mu , \phi ;\lambda ) d \mu' d\phi'$    
$\displaystyle +$ $\displaystyle ~\frac{~1}{c(\zeta ,\lambda )} S(\zeta ,\mu , \phi ,\lambda )\,\, .$ (12)

This equation now shows all quantities as a function of optical depth.

Any of Eqs. (9), (10), or (11) is called the monochromatic (1 wavelength), one-dimensional (the depth is the only spatial variable), time-independent RTE.

Form (12) of the RTE yields an important observation: In source-free (S = 0) waters, any two water bodies having the same single-scattering albedo $ \omega_{\rm o}$ , phase function $ \tilde{\beta}$ , and boundary conditions (including incident radiances) will have the same radiance distribution L at a given optical depth. This is why optical depth, albedo of single scattering, and phase function are often the preferred variables in radiative transfer theory. Note, for example, that doubling the absorption and scattering coefficients a and b leaves $ \omega_{\rm o}$ unchanged, so that the radiance remains the same for a given optical depth. However, the geometric depth corresponding to a given optical depth will different after such a change in the IOPs.

We can convert geometric depth to optical depth, or vice versa, by integrating Eq. (11):

$\displaystyle \zeta ~=~\int_{0}^{z} c({z'} , \lambda ) d{z'} ~~~~~{\rm or} ~~~~~z~=~\int_{0}^{\zeta} {\frac{d{\zeta'} }{c ({\zeta'} , \lambda )}} \,\, ,$ (13)

Note that the optical depth $ \zeta$ corresponding to a given geometric depth z is usually different for different wavelengths, because the beam attenuation c depends on wavelength. This is inconvenient for oceanographic work, so Eq. (10) is usually the preferred form of the RTE for oceanography.

We have now derived the RTE in a form adequate for much oceanographic work. Technically, the RTE is a linear integrodifferential equation because it involves both an integral and a derivative of the unknown radiance. This makes solving the equation for given IOPs and boundary conditions quite difficult. Fortunately, the radiance appears only to the first power. Nevertheless, there are almost no analytic (i.e., pencil and paper) solutions of the RTE except for trivial special cases, such as non-scattering waters. Sophisticated numerical methods therefore must be employed to solve the RTE for realistic oceanic conditions.

Finally, the development on this page has been for unpolarized light. However, scattering (either by particles within the water or by reflection and transmission by the air-water surface) induces polarization even if the incident beam is unpolarized. Thus underwater radiance in a particular direction is usually at least partially polarized. Nevertheless, the unpolarized, or scalar RTE (SRTE), derived here gives sufficiently accurate solutions for many (but not all) oceanographic applications. There are three main reasons for the utility of the SRTE in underwater optics:

  • The particles responsible for scattering within the ocean are usually much larger than the wavelength of light. Polarization by scattering is greatest for particles much smaller than the wavelength (for example, Rayleigh scattering by water molecules). The larger the particle, the less polarization is induced by scattering of unpolarized sun light.
  • Multiple scattering is almost always significant underwater. Multiple scattering tends to depolarized the radiance, i.e., reduce the overall degree of polarization induced by single scattering events.
  • Irradiances are often the radiometric quantity of interest. Irradiances are computed from integration of the radiance over direction, which tends to average out different polarizations in different directions.

However, if very accurate results are needed, or if the state of polarization itself is of interest, then a polarized or vector RTE (VRTE), must be used. The VRTE is more complicated than the SRTE developed here. The VRTE is presented in Level 2.