Page updated:
May 19, 2021
Author: Curtis Mobley
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Solving the Radiative Transfer Equation
The 1D, timeindependent RTEs are linear twopoint boundary value problems. That is to say, there are boundary conditions describing the radiance at the top and bottom of the atmosphere or ocean (the two spatial points), and the propagation of radiance within the medium—between the boundaries—is governed by the linear integrodiﬀerential RTE. In the oceanographic setting, the upper boundary condition speciﬁes the sky radiance incident onto the sea surface. The lower boundary condition speciﬁes how the sea bottom reﬂects the downwelling radiance. (The ”sea bottom” can be a physical bottom, or the deepest depth in the water column at which the RTE needs to be solved to obtain the radiances above that depth.) In the vectorlevel equations, the sky radiance speciﬁcation is in terms of Stokes vectors; for the scalar RTE the sky input is speciﬁed by the unpolarized sky radiance.
Two point boundary value problems in general do not have solutions. Consider a simple example:
$$\frac{dy}{dx}={x}^{2}$$ 
with the twopoint boundary conditions
$$y=0\phantom{\rule{1em}{0ex}}at\phantom{\rule{1em}{0ex}}x=0\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}y=2\phantom{\rule{1em}{0ex}}at\phantom{\rule{1em}{0ex}}x=1\phantom{\rule{0.3em}{0ex}}.$$ 
What is the function $y\left(x\right)$ that satisﬁes the diﬀerential equation and the two boundary conditions? Integration gives $y={x}^{3}\u22153$, which satisﬁes the ﬁrst boundary condition but not the second. Thus the problem as stated has no solution; i.e., there is no such function $y\left(x\right)$. This equation however does have a solution for the two boundary conditions
$$y=0\phantom{\rule{1em}{0ex}}at\phantom{\rule{1em}{0ex}}x=0\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}y=\frac{1}{3}\phantom{\rule{1em}{0ex}}at\phantom{\rule{1em}{0ex}}x=1\phantom{\rule{0.3em}{0ex}}.$$ 
In the case of the radiative transfer equation and boundary conditions on the radiance at the sea surface and sea bottom (when properly formulated), the equation does have a solution. We know a solution must exist simply because light propagates in the ocean! However, ﬁnding that solution is quite diﬃcult. Even the scalar radiative transfer equation (SRTE; Eq. 3 of the The Scalar Radiative Transfer Equation page) considered here is quite diﬃcult to solve.
Exact Analytical Solutions
Exact analytical (i.e., pencil and paper) solutions of the SRTE can be obtained only for very simple situations, such as no scattering. There is no function (that anyone has ever found) that gives
where $f$ is some function in which you can “plug in” the absorption coeﬃcient, VSF, and other parameters and get back the radiance. This is true even for very simple situations such as homogeneous water with isotropic scattering. Even the conceptually simple geometry of an isotropically emitting point light source in an inﬁnite homogeneous ocean is unsolved. (A very complicated solution for the the scalar irradiance ${E}_{o}\left(r\right)$ around an isotropically emitting point source with isotropic scattering does exist; see Davison and Sykes (1957), eq. 5.25 or Mobley (1996), Section 9.2.) This may seem surprising because other pointsource problems, such as the electric ﬁled around a point charge or the gravitational ﬁeld around a point mass, are often very simple. The diﬀerence with optics lies in the complications caused by scattering within the medium surrounding the point source (which do not exist for problems like the gravitational ﬁeld around a point mass).
Beer’s Law
If there is no scattering, the SRTE reduces to just
This is a linear, ﬁrst order, ordinary diﬀerential equation, which is easily solved (see any text on diﬀerential equations, e.g., Rainville (1964), page 36). If the medium is homogeneous, so that the absorption coeﬃcient and source function do not depend on distance $r$, the solution of Eq. (1) is (dropping the wavelength and direction arguments for brevity)
$$L\left(r\right)=L\left(0\right){e}^{ar}+\frac{S}{a}\left[1{e}^{ar}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}},$$  (2) 
where $L\left(0\right)$ is the initial radiance at distance $r=0$. In sourcefree water, $S=0$ and the solution is a simple exponential decay of the initial radiance with distance:
$$L\left(r\right)=L\left(0\right){e}^{ar}\phantom{\rule{0.3em}{0ex}},$$  (3) 
This result is known as Beer’s Law, Lambert’s Law, Bouguer’s Law, or some hyphenated combination of these names. An equivalent result was derived in another way in the discussion on the Measuring IOPs page; see Eq. (3) of that page.
The exponential decay of light through a medium was ﬁrst reported in graphical form by Bouguer (1729) (his Fig. 4) based on observations of candles made by eye. Lambert (1760) placed Bouguer’s graphical result in the mathematical form an of exponential decay with distance. Beer (1852) found that for a ﬁxed distance, the transmitted light decreased as an exponential function of the concentration of salts in solution. It thus seems that Bouguer should be credited with the original understanding of the exponential decrease of light when traveling through a material medium, and Lambert gets credit for putting Bouguer’s results into the mathematical form seen in Eq. (3). It is then “Lambert’s law” if you use Eq. (3) to predict the attenuation of light as a function of distance for a given substance, and it is “Beer’s law” if you use Eq. (3) to predict the attenuation of light as a function of concentration for a given distance. To give full credit to everyone, it should be called the “BouguerLambertBeer law.” It seems that “Beer’s law” or the “LambertBeer law” seems to be most common in optical oceanography.
Note that for great distances, $r\to \infty $, the radiance depends only on the source function and the absorption coeﬃcient: $L\to S\u2215a$. This result for the asymptotic behavior of the radiance with distance also holds when scattering is present, as shown on the page for An Analytical Asymptotic Solution for Internal Sources.
Approximate Analytical Solutions
A number of approximate analytical solutions to the SRTE can be derived after simplifying the SRTE in various ways. One of these approximate solutions is the Singlescattering Approximation (SSA). This solution is developed by assuming that the water is homogeneous, the sea surface is level, the sun is a point source in a black sky, there are no internal sources, and only singlescattering of photons is considered. The Quasisinglescattering Approximation (QSSA) is a further simpliﬁcation of the SSA. These are discussed in detail in Level 2 and in Gordon (1994). A much more complicated solution including secondorder scattering (i.e., including photons that have been scattered twice) is developed in Walker (1994), Section 26.
These approximate solutions of the SRTE are useful for isolating the main factors inﬂuencing underwater radiances. However, the solutions depend on various simplifying assumptions and the predicted radiances are typically accurate to a few tens of percent at best, and can be oﬀ by an order of magnitude.
Numerical Solutions
If accurate solutions of the vector or scalar RTE are to be obtained for realistic oceanic conditions, numerical methods must be used. Many such methods have been developed. Some of these solution techniques have been tailored to speciﬁc environments, such as stellar or planetary atmospheres, and are not commonly used in oceanography. The numerical methods most commonly employed in oceanographic radiative transfer, and their salient characteristics, can be summarized as follows:
 Monte Carlo

 based on conceptually simple physics that mimics how nature absorbs and scatters idealized light rays
 completely general; can solve timedependent and 3D problems with arbitrary geometry
 sophisticated mathematical ”tricks” can be used to speed up the calculations
 easy to program
 computed radiances have statistical errors, which can be reduced by tracing more initial rays (requiring longer computer times)
 computer run times can be extremely slow for some problems (e.g., solving the RTE to large optical depths; run times increase exponentially with optical depth)
 Discrete Ordinates

 highly mathematical
 diﬃcult to program
 does not handle highly peaked scattering phase functions well
 models the medium as a stack of homogeneous layers
 is fast for irradiance calculations and homogeneous water, but can be slow for radiances or if many layers are needed to resolve depthdependent IOPs
 Invariant Imbedding

 highly mathematical
 diﬃcult to program
 can solve only 1D problems (the one dimension being the depth in optical oceanography)
 includes all orders of multiple scattering
 computed radiances do not have statistical errors
 is extremely fast (run times increase linearly with optical depth)
Because of their simplicity and generality, Monte Carlo methods are widely used to solve RTEs in ﬁelds as diverse as oceanography, atmospheric sciences, astronomy, medical physics, and nuclear engineering. The tradeoﬀ for their simplicity and generality is long computer run times for many problems. Nevertheless, Monte Carlo methods warrant a chapter of their own. The scattering phase functions for atmospheric aerosols are not as highly peaked at very small scattering angles as are those for oceanic particles. Discrete ordinates can handle aerosol phase functions well and is often used in atmospheric optics, but is not much used for underwater calculations because of the need to resolve highly peaked phase functions and to have many layers if the IOPs vary greatly with depth. Invariant imbedding is the solution technique used by the HydroLight numerical model, which is widely used in oceanography. Each of these techniques gives the same answer for the same inputs and boundary conditions for the RTE, for problems where all three techniques are applicable, as can be seen in the model comparison study of Mobley et al. (1993). They diﬀer only in their internal mathematics and the resulting computer run times, and welldebugged and validated computer programs exist for each. In this sense, solving the RTE in the oceanographic setting can be considered a “solved problem.”