**Page updated:**
April 12, 2020 **Author:** Curtis Mobley

# The SRTE: Heuristic Development

The previous page obtained the scalar radiative transfer equation (SRTE) by a rigorous sequence of steps starting with fundamental physics. That development showed the assumptions and approximations needed to obtain the SRTE, and the errors resulting from those simpliﬁcations were discussed. It can be argued that no further discussion of the SRTE is needed. However, there is perhaps still some value in presenting the heuristic derivation of the SRTE as is commonly seen in textbooks. This derivation is not rigorous (it would be rigorous only if light were unpolarized), but it does give an additional perspective on the SRTE—in particular on the interpretation of the various terms of the equation. This derivation is also of historical interest because it shows how the founding fathers of radiative transfer theory proceeded in order to obtain a governing equation (e.g., Preisendorfer (1965), page 65) before the link between fundamental physics and radiative transfer theory was ﬁrmly established.

#### Radiative Processes

To the extent that polarization can be ignored, the SRTE 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 light interacts with an atom or molecule.

The light (electromagnetic radiation) 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 radiant energy may be subsequently converted into thermal (kinetic) or chemical energy (manifested, for example, in the formation of new chemical compounds during photosynthesis). The annihilation of the light and conversion of its energy into a nonradiant form is called absorption. (See The Physics of Absorption page for further discussion of the quantum mechanics of absorption processes.) If the molecule almost immediately (on a femtosecond or shorter time scale) returns to its original internal energy state by re-emitting radiation of the same energy as the absorbed radiation (but probably traveling in a diﬀerent direction from the original radiation), 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 light interacting with the molecule and simply ”changing direction” without an exchange of energy with the scattering molecule.

The excited molecule also may emit radiation of lower energy (longer wavelength) than the incident radiation. The molecule thus remains in an intermediate excited state and may at a later time emit new radiation 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 light and then emit light of greater energy (shorter wavelength) than the absorbed light, thereby returning to a lower energy state. In either case the scattered (emitted) radiation has a diﬀerent wavelength than the incident (absorbed) radiation, 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 $1{0}^{-11}$ to $1{0}^{-8}\phantom{\rule{1em}{0ex}}sec$. If the re-emission requires longer than about $1{0}^{-8}\phantom{\rule{1em}{0ex}}sec$, 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 the total light ﬁeld as many beams of electromagnetic radiation 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 $\left(\mathit{\theta},\varphi \right)$ and has wavelength $\lambda $. This beam and the processes aﬀecting it are illustrated in Fig. 1.

Now think of all the ways in which that beam’s energy 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 light on the phenomenological level:

- Process 1
- loss of energy from the beam through annihilation of the light and conversion of radiant energy to nonradiant energy (absorption)
- Process 2
- loss of energy from the beam through scattering to other directions without change in wavelength (elastic scattering)
- Process 3
- loss of energy from the beam through scattering (perhaps to other directions) with change in wavelength (inelastic scattering)
- Process 4
- gain of energy by the beam through scattering from other directions without change in wavelength (elastic scattering)
- Process 5
- gain of energy by the beam through scattering (perhaps from other directions) with a change in wavelength (inelastic scattering)
- Process 6
- gain of energy by the beam through creation of light 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 radiance there is, the more is lost to absorption. Thus we can write

$$\begin{array}{llll}\hfill \frac{L\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)-L\left(r,\mathit{\theta},\varphi ,\lambda \right)}{\Delta r}\phantom{\rule{1em}{0ex}}=& \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{\Delta L\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)}{\Delta r}\phantom{\rule{1em}{0ex}}=& \phantom{\rule{1em}{0ex}}-a\left(r,\lambda \right)L\left(r,\mathit{\theta},\varphi ,\lambda \right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\end{array}$$

Here $\Delta L\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)$ denotes the change in L between r and $r+\Delta r$. The minus sign is necessary because the radiance decreases (energy is 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 $a\left(r,\lambda \right)$ 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 (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 $\left(\mathit{\theta},\varphi \right)$ beam direction into all other directions can be written as

where $b\left(r,\lambda \right)$ 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 $\left(\mathit{\theta},\varphi \right)$. Figure 2 shows Fig. 1 redrawn to illustrate scattering along path length $\Delta r$ from one direction $\left({\mathit{\theta}}^{\prime},{\varphi}^{\prime}\right)$ into the direction $\left(\mathit{\theta},\varphi \right)$ of interest. These incident and ﬁnal directions correspond to scattering angle $\psi $ as shown in Fig. 2.

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

Here ${I}_{s}\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)$ is the intensity exiting the scattering volume at location $r+\Delta r$ in direction $\left(\mathit{\theta},\varphi \right)$. All of this intensity is created along $\Delta r$ by scattering from direction $\left({\mathit{\theta}}^{\prime},{\varphi}^{\prime}\right)$ into $\left(\mathit{\theta},\varphi \right)$, so $\Delta {I}_{s}\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)$ = ${I}_{s}\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)$. 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:

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

where $\Delta L\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)$ 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

This equation gives the contribution to $\Delta L\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)\u2215\Delta r$ by scattering from one particular direction $\left({\mathit{\theta}}^{\prime},{\varphi}^{\prime}\right)$. However, ambient radiance may be passing through the scattering volume from all directions. We can sum up the contributions to $\Delta L\left(r+\Delta r,\mathit{\theta},\varphi ,\lambda \right)\u2215\Delta r$ from all directions by integrating the right hand side of Eq. (6) over all directions,

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 $\left(\mathit{\theta},\varphi \right)$ at wavelength $\lambda $ by inelastic scattering from other all other wavelengths ${\lambda}^{\prime}\ne \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 ${\lambda}^{\prime}$ and converted to the wavelength $\lambda $ of interest.

Process 6 accounts for radiance created de novo by 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 will be treated elsewhere (pages under development). For the moment, we can simply include a generic source function that represents creation of radiance along pathlength $\Delta r$ in direction $\left(\mathit{\theta},\varphi \right)$ at wavelength $\lambda $ by any inelastic scattering or emission process. Thus we write just

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

#### 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:

$$\begin{array}{llll}\hfill \frac{dL\left(r,\mathit{\theta},\varphi ,\lambda \right)}{dr}\phantom{\rule{1em}{0ex}}=& \phantom{\rule{1em}{0ex}}-\left[a\left(r,\lambda \right)+b\left(r,\lambda \right)\right]L\left(r,\mathit{\theta},\varphi ,\lambda \right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill +& \phantom{\rule{1em}{0ex}}{\int}_{0}^{2\pi}{\int}_{0}^{\pi}L\left(r,{\mathit{\theta}}^{\prime},{\varphi}^{\prime},\lambda \right)\beta \left(r;{\mathit{\theta}}^{\prime},{\varphi}^{\prime}\to \mathit{\theta},\varphi ;\lambda \right)sin{\mathit{\theta}}^{\prime}d{\mathit{\theta}}^{\prime}d{\varphi}^{\prime}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill +& \phantom{\rule{1em}{0ex}}S\left(r,\mathit{\theta},\varphi ,\lambda \right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(W\phantom{\rule{1em}{0ex}}{m}^{-3}\phantom{\rule{1em}{0ex}}s{r}^{-1}\phantom{\rule{1em}{0ex}}n{m}^{-1}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(9)}\end{array}$$

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: $dr=dz\u2215cos\mathit{\theta}$. Using this in Eq. (9), assuming that the ocean is horizontally homogeneous, and recalling that $a+b=c$, we get

$$\begin{array}{llll}\hfill cos\mathit{\theta}\frac{dL\left(z,\mathit{\theta},\varphi ,\lambda \right)}{dz}\phantom{\rule{1em}{0ex}}=& \phantom{\rule{1em}{0ex}}-c\left(z,\lambda \right)L\left(z,\mathit{\theta},\varphi ,\lambda \right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill +& \phantom{\rule{1em}{0ex}}{\int}_{0}^{2\pi}{\int}_{0}^{\pi}L\left(z,{\mathit{\theta}}^{\prime},{\varphi}^{\prime},\lambda \right)\beta \left(z;{\mathit{\theta}}^{\prime},{\varphi}^{\prime}\to \mathit{\theta},\varphi ;\lambda \right)sin{\mathit{\theta}}^{\prime}d{\mathit{\theta}}^{\prime}d{\varphi}^{\prime}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill +& \phantom{\rule{1em}{0ex}}S\left(z,\mathit{\theta},\varphi ,\lambda \right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(10)}\end{array}$$

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

$$d\zeta \phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}c\left(z,\lambda \right)dz\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.$$ | (11) |

Dividing Eq. (10) by $c\left(z,\lambda \right)$ and using (11) gives the RTE written in terms of optical depth. It is also common to use $\mu =cos\mathit{\theta}$ 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 $\stackrel{\u0303}{\beta}$. Finally, recalling the deﬁnition of the albedo of single scattering ${\omega}_{o}=b\u2215c$, we can re-write Eq. (10) as

$$\begin{array}{llll}\hfill \mu \frac{dL\left(\zeta ,\mu ,\varphi ,\lambda \right)}{d\zeta}\phantom{\rule{1em}{0ex}}=& -L\left(\zeta ,\mu ,\varphi ,\lambda \right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill +& \phantom{\rule{1em}{0ex}}{\omega}_{o}\left(\zeta ,\lambda \right){\int}_{0}^{2\pi}{\int}_{-1}^{1}L\left(\zeta ,{\mu}^{\prime},{\varphi}^{\prime},\lambda \right)\stackrel{\u0303}{\beta}\left(\zeta ;{\mu}^{\prime},{\varphi}^{\prime}\to \mu ,\varphi ;\lambda \right)d{\mu}^{\prime}d{\varphi}^{\prime}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill +& \phantom{\rule{1em}{0ex}}\frac{\phantom{\rule{1em}{0ex}}1}{c\left(\zeta ,\lambda \right)}S\left(\zeta ,\mu ,\varphi ,\lambda \right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(12)}\end{array}$$

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 ${\omega}_{o}$, phase function $\stackrel{\u0303}{\beta}$, 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 ${\omega}_{o}$ 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):

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.