Analysis: Propagation of Thermal Radiation

Note: This page is highly mathematical.

Let’s look at the equation which governs the propagation of thermal radiation through a partially-transparent medium without significant scattering.

This topic is also addressed in Chapter 8 of Petty, Grant (2006). A First Course in Radiation Physics (2nd ed.). Madison, WI: Sundog Publishing. Petty’s presentation is in some ways more elegant than and preferable to what I present below (which I produced prior to seeing Petty’s presentation). See also Wallace, J. M.; Hobbs, P. V. (2006). Atmospheric Science (2 ed.). Academic Press. ISBN 978-0-12-732951-2.

Table of Contents

Propagation of radiation intensity along direction of travel

Let L_{\nu,\Omega} be the intensity1More technically, this quantity is called spectral radiance. (W sr-1 m-2 Hz-1) of thermal radiation at frequency \nu propagating through the medium. It is assumed that the intensity entering the medium is uniform over a plane where the radiation enters the medium. Then, the rate of change in the intensity relative to the distance traveled, s, is given by the Schwartzchild equation for radiative transfer:

(1)   \begin{equation*} \deriv{L_{\nu,\Omega}}{s} = -n \sigma_\nu L_{\nu,\Omega} + n \sigma_\nu B_{\nu,\Omega}(T) \end{equation*}

Here, n is the numerical density of absorbing/emitting molecules (molecules/m3), \sigma_\nu is the absorption cross section of each molecule (m2), T is the temperature of the medium (Kelvin), and B_{\nu,\Omega}(T) is the Planck spectrum that would be emitted by a radiating black-body.

The first term reflects absorption, while the second term reflects spontaneous thermal emissions within the medium. It has been assumed that molecular collisions are sufficiently frequent that local thermal equilibrium holds, which results in the rate of stimulated emissions being negligible. (This is the case in Earth’s troposphere.)

The solution to the Schwartzchild equation can be expressed as:

(2)   \begin{equation*} L_{\nu,\Omega}(s) = L_{\nu,\Omega}(0) - G_{\nu,\Omega}(s) \end{equation*}


(3)   \begin{equation*} G_{\nu,\Omega}(s) = L_{\nu,\Omega}(0)\cdot \left[ 1 - Y_{\nu,\Omega}(0,s) \right] \; - \int_0^s  B_{\nu,\Omega}(T(s^\prime)) \,W_{\nu,\Omega}(s^\prime, s) \,\dd s^\prime \end{equation*}

(4)   \begin{equation*} \tau_{\nu,\Omega}(s^{\prime}, s) = \int_{s^{\prime}}^s n(s^{\prime\prime}) \,\sigma_\nu(s^{\prime\prime}) \,\dd s^{\prime\prime} \end{equation*}

(5)   \begin{equation*} Y_{\nu,\Omega}(s^\prime, s) = \exp\left[ -\tau_{\nu,\Omega}(s^{\prime}, s) \right] \end{equation*}

(6)   \begin{equation*} W_{\nu,\Omega}(s^\prime, s) = \begin{cases} n(s^\prime) \,\sigma_\nu(s^\prime) \, Y_{\nu,\Omega}(s^\prime, s), & \text{if } s^\prime \le s \\ 0, & \text{if } s^\prime > s \end{cases} \end{equation*}

The newly introduced quantities are:

  • G_{\nu,\Omega}(s) — The net amount by which the incident intensity is reduced after propagating for a distance s, i.e., the amount that is absorbed without this loss being compensated for by a comparable amount of emissions from within the medium.
  • \tau_{\nu,\Omega}(s^{\prime}, s) — The optical thickness associated with propagation from s^\prime to s.
  • Y_{\nu,\Omega}(s^\prime, s) — The directional transmittance (fraction transmitted) associated with propagation from s^\prime to s.
  • W_{\nu,\Omega}(s^\prime, s) — The intensity emissions weighting function, i.e., a weight factor for black-body emissions at different points along the propagation path. (Defining the function W_{\nu,\Omega}(s^\prime, s) as 0 when s^\prime > s means that we could formally integrate over s^\prime from 0 to \infty instead of from 0 to \s.)

Interpreting the initial result

A weighted average

It may be further verified that:

(7)   \begin{equation*} Y_{\nu,\Omega}(0, s) \,+\, \int_0^s W_{\nu,\Omega}(s^\prime, s) \,\dd s^\prime  = 1 \end{equation*}

This allows us to write:

(8)   \begin{equation*} L_{\nu,\Omega}(s) = \exw{L_{\nu,\Omega}(0),\;  B_{\nu,\Omega}(T(s^\prime)}{W_{\nu,\Omega}}_{s^\prime}  \end{equation*}

(9)   \begin{equation*} G_{\nu,\Omega}(s) = L_{\nu,\Omega}(0) \;-\; \exw{L_{\nu,\Omega}(0),\;  B_{\nu,\Omega}(T({s^\prime})}{W_{\nu,\Omega}}_{s^\prime}  = \exw{0,\;  \left(L_{\nu,\Omega}(0) - B_{\nu,\Omega}(T({s^\prime})\right)}{W_{\nu,\Omega}}_{s^\prime}  \end{equation*}

where the weighted average \exw{f_0,\;f({s^\prime})}{W}_{s^\prime} is defined as

(10)   \begin{equation*} \exw{f_0,\;f({s^\prime})}{W}_{s^\prime} = f_0 \; \left[1 - \int_0^\infty W_s(s^\prime, s) \,\dd s^\prime \right] + \int_0^\infty  f({s^\prime} ) \,W_s(s^\prime, s) \,\dd s^\prime \end{equation*}

This allows us to interpret L_{\nu,\Omega}(s) as being a weighted average of thermal emissions entering the medium and emitted inside it, with Y_{\nu,\Omega}(0,s) and W_{\nu,\Omega}(s^\prime, s) specifying the weighting to be used in computing the average.

Note that, as one might expect from a weighted average:

(11)   \begin{equation*} \exw{1,\;1}{W}_{s^\prime} = 1 \end{equation*}

Effect of uniform temperature

Suppose that we assume the incident thermal radiation at frequency \nu has equal intensity in all directions, so that it is equivalent to the thermal radiation from a black-body at temperature, T_{b,\nu}:

(12)   \begin{equation*} L_{\nu,\Omega}(0) = B_{\nu,\Omega}(T_{b,\nu}) \end{equation*}

Then, if the temperature of the medium uniformly equals T_{b,\nu}, then B_{\nu,\Omega}(T(s^\prime)) becomes uniformly B_{\nu,\Omega}(T_{b,\nu}), and we find:

(13)   \begin{equation*} G_{\nu,\Omega}(s) = \exw{0,\;  \left(B_{\nu,\Omega}(T_{b,\nu}) - B_{\nu,\Omega}(T_{b,\nu})\right)}{W_{\nu,\Omega}}_s = 0 \end{equation*}

In the absence of temperature changes within the medium, the exiting intensity will be the same as what entered. For the net spectral radiance exiting the medium to be less than what entered, the overall weighted mean temperature must be lower than the brightness temperature of the entering radiation.

Effect of increasing concentration

What is the effect of increasing the concentration of the absorbing and emitting molecules? This would be expressed as a proportionate increase in the number density, n, at all locations within the medium, and a corresponding increase in absorptance. Examining the formula for the weight, W_{\nu,\Omega}(s^\prime, s), it is apparent that, in response to an increase in concentration:

  • the entering radiation will be deemphasized;
  • emissions earlier along the propagation path will be deemphasized;
  • emissions further along the propagation path will become more heavily weighted.

Propagation of radiation flux

The spectral radiance involves radiation traveling in many different directions. The directions of travel may be specified using spherical coordinates, \eta, \phi, where \eta is the zenith angle and \phi is the azimuthal angle.

Let L_\nu(x) be the flux2Technically, this is the spectral flux density Even more rigorously, the amount entering is called the spectral irradiance, while the amount leaving is called the spectral radiosity—or the spectral exitance, if it all the radiation is associated with thermal emissions. Since I’m looking a propagation, the radiation is leaving one section of medium and entering the next. So, I’m using the more generic term, spectral flux density, and simplifying this to simply “flux” to be less formal. of thermal radiation (W m-2 Hz-1) at frequency \nu passing through a plane a distance x from the plane where the radiation entered the medium. (The coordinate x is a generic coordinate, which when applied to the atmosphere will be equal to the altitude, z, for upwelling radiation, and related to -z for downwelling radiation.)

The flux is related to the intensity by an integration over all directions within a hemisphere, i.e., all directions with at least some component of travel in the same direction. In particular:

(14)   \begin{equation*} L_\nu(x) = \int_0^\frac{\pi}{2} \int_{-\pi}^{\pi} L_{\nu,\Omega}(\eta, \phi)\;\dd\phi \; \sin\eta \cos\eta \;\dd\eta \end{equation*}

The factor \cos\eta is needed to translate flux in the direction of travel to flux relative to the plane where the flux density is being measured. The factor \sin\eta is required to reflect integration over the hemisphere.

The black-body spectral radiance, B_{\nu,\Omega}(T), is the same in all directions. This makes it simple to calculate the analogously-defined hemispheric black-body flux, B_\nu(T):

(15)   \begin{equation*} B_\nu(T) = B_{\nu,\Omega}(T) \int_0^\frac{\pi}{2} \int_{-\pi}^{\pi} \dd\phi \; \sin\eta \cos\eta \;\,\dd\eta = \pi\, B_{\nu,\Omega}(T) \end{equation*}

Let us assume that the intensity entering the medium, L_{\nu,\Omega}(0), is also uniform in all directions. This leads to:

(16)   \begin{equation*} L_\nu(0) = \pi\,L_{\nu,\Omega}(0) \end{equation*}

Now, let’s work on transforming our solution for intensity into a solution for flux. Recognizing that z = s\,\cos\eta, integrals with respect to \dd s become integrals with respect to (1/\cos\eta)\,\dd z. Integrating both sides of the equation with respect to \phi is simple, since we assume all quantities to be independent of \phi. Finally, we integrate both sides of the equation for L_{\nu,\Omega}(s) with respect to \sin\eta \cos\eta \;\,\dd\eta. This procedure yields:

(17)   \begin{equation*} L_\nu(x) = L_\nu(0) - G_\nu(x) \end{equation*}


(18)   \begin{equation*} G_\nu(x) = L_\nu(0)\cdot\left[1-Y_{\nu}(0,x)\right] - \int_0^x  B_{\nu}(T(x^\prime)) \,W_{\nu}(x^\prime, x) \:\dd x^\prime \end{equation*}

(19)   \begin{equation*} \tau_{\nu}(x^{\prime}, x) = \int_{x^{\prime}}^x n(x^{\prime\prime}) \,\sigma_\nu(x^{\prime\prime}) \,\dd x^{\prime\prime} \end{equation*}

(20)   \begin{equation*} Y_{\nu,\eta}(x^\prime, x) = 2\,\sin\eta \,\cos\eta\;\exp\left[ -\frac{\tau_{\nu}(x^{\prime}, z)}{\cos\eta} \right] \end{equation*}

(21)   \begin{equation*} Y_{\nu}(x^\prime, x) = \int_0^\frac{\pi}{2} Y_{\nu,\eta}(x^\prime, x)\;\dd\eta \end{equation*}

(22)   \begin{equation*} W_{\nu}(x^\prime, x) = \begin{cases} n(x^\prime) \,\sigma_\nu(x^\prime) \, \int_0^\frac{\pi}{2}  \frac{Y_{\nu,\eta}(x^{\prime}, x)}{\cos\eta}\;\dd\eta , & \text{if } x^\prime \le x \\ 0, & \text{if } x^\prime > x \end{cases} \end{equation*}

We could also write:

(23)   \begin{equation*} L_{\nu}(x) = \exw{L_{\nu}(0),\;  B_{\nu}(T({x^\prime}))}{W_{\nu}}_{x^\prime} \end{equation*}

(24)   \begin{equation*} G_{\nu}(x) = L_{\nu}(0) \;-\; \exw{L_{\nu}(0),\;  B_{\nu}(T(x^\prime))}{W_{\nu,\Omega}}_{x^\prime} = \exw{0,\;  \left(L_{\nu}(0) - B_{\nu}(T({x^\prime})\right)}{W_{\nu}}_{x^\prime} \end{equation*}

The key quantities here (for a particular frequency \nu) may be identified as:

  • G_{\nu}(x) — The net amount by which the incident flux is reduced at a plane a distance z beyond the initial plane, i.e., the amount that is absorbed without this loss being compensated for by a comparable amount of emissions from within the medium. When considering the propagation of thermal radiation flux upward from Earth’s surface to TOA, G_{\nu}(x) is the Greenhouse effect at frequency \nu. (This quantity G_{\nu}(x) isn’t particularly meaningful for propagation in other directions.)
  • n(x) \,\sigma_\nu(x) — The absorption coefficient at location x.
  • \tau_{\nu}(x^{\prime}, x) — The optical thickness associated with propagation from x^\prime to x for radiation propagating in a direction normal to the plane in which flux is measured.
  • Y_{\nu}(x^\prime, x) — The spectral hemispherical transmittance (fraction transmitted) associated with propagation of flux from x^\prime to x.
  • W_{\nu}(x^\prime, x) — The flux emissions weighting function, i.e., a weight factor for black-body emissions at planes.

We can interpret the flux exiting the medium, L_\nu(x), as being the average of thermal emissions entering the medium and emitted inside it:

(25)   \begin{equation*} L_{\nu}(x) = \exw{L_{\nu}(0),\;  B_{\nu}(T(x^\prime))}{W_{\nu}}_{x^\prime} \end{equation*}

Application to Atmosphere

Upwelling radiation

Upwelling thermal radiation at a given frequency at altitude z is given by:

(26)   \begin{equation*} L_\nu^\uparrow(z) =  \exw{L^\uparrow_{\nu}(0),\;  B_{\nu}(T({z^\prime}))}{W^\uparrow_{\nu}(z^\prime, z)}_{z^\prime} \end{equation*}

where W^\uparrow_{\nu}(z^\prime, z) is the emissions weighting function for upwelling radiation, which identical the the previously defined W(z^\prime, z).

For emissions at the top of the atmosphere, one can set z equal to Z_\mathrm{toa} (or perhaps to \infty).

The Greenhouse effect at a given frequency, G_\nu, is given by:

(27)   \begin{equation*} G_\nu =  \exw{0,\;  \left(L^\uparrow(0) - B_{\nu}(T({z^\prime})\right)}{W^\uparrow_{\nu}(z^\prime, z)}_{z^\prime}  \end{equation*}

where the emissions weighting function for upwelling radiation is evaluated for z equal to Z_\mathrm{toa} (or perhaps to \infty).

Downwelling radiation

For the the radiation flux propagating downward, we take x = Z_\mathrm{toa} -z. Z_\mathrm{toa} is at top-of-atmosphere altitude such that it’s assumed there is no significant difference between using that as an integration limit and integrating to \infty. Then, given prior results, and given the absence of any downwelling longwave radiation entering the top of the atmosphere, the downward flux is given by:

(28)   \begin{equation*} L_\nu^\downarrow(x) = \int_0^x  B_{\nu}(T(x^\prime)) \,W_{\nu}(x^\prime, w) \:\dd x^\prime \end{equation*}

When we change variable to express this in terms of z, that will change the sign of the expression on the right, but that can be fixed by exchanging the limits on the integral over \dd z^\prime. Something similar happens with the integral used to compute \tau_{\nu}(x^{\prime}, x); that can again be fixed by exchanging the limits of integration. To account for that, we define a downwelling emissions weighting function given by:

(29)   \begin{equation*} W^\downarrow_{\nu}(z^\prime, z) = W_{\nu}(z, z^\prime) \end{equation*}

This allows us to write:

(30)   \begin{equation*} L_\nu^\downarrow(z) = \int_z^{Z_\mathrm{toa}}  B_{\nu}(T(z^\prime)) \,W^\downarrow_{\nu}(z^\prime, z) \:\dd z^\prime \end{equation*}

This can also be written:

(31)   \begin{equation*} L_\nu^\downarrow(z) = \exw{0,\;  B_{\nu}(T({z^\prime}))}{W^\downarrow_{\nu}}_{z^\prime} \end{equation*}

For downwelling radiation at the surface, one would set z = 0.

Heat flux

The upward heat flux at a given frequency is given by:

(32)   \begin{equation*} Q_\nu = L_\nu^\uparrow(z) - L_\nu^\downarrow(z) \end{equation*}

The net rate of radiation heat transfer into the atmosphere is given by -\deriv{Q_\nu}{z}.3See, for example, “Thermal Radiation Heat Transfer” by Siegel & Howell, equation (14-15).

Spectral integration

Let L(z) be the flux integrated over all frequencies:

(33)   \begin{equation*} L(z) = \int_0^\infty L_\nu(z)\;\dd\nu =  \int_0^\infty \exw{L_{\nu}(0),\;  B_{\nu}(T(z))}{W_{\nu}}_z\;\dd\nu \end{equation*}

(34)   \begin{equation*} L(z) = L(0) - G(z) \end{equation*}

(35)   \begin{equation*} G(z) = \int_0^\infty L_\nu(0)\,A_{\nu}(0,z) \;\dd\nu \;\:- \int_0^\infty \int_0^z  B_{\nu}(T(z^\prime)) \: W_{\nu}(z^\prime, z) \,\dd z^\prime\,\dd \nu \end{equation*}

Normalized Greenhouse effect

Let us define the normalized Greenhouse effect, \nghe, as:

(36)   \begin{equation*} \nghe = \frac{G}{L^\uparrow(0)} \end{equation*}

Spectral normalized Greenhouse effect

Let us define the spectral normalized Greenhouse effect, \nghe_\nu, for frequency \nu as:

(37)   \begin{equation*} \nghe_{\nu} =A_{\nu}(0,\infty) - \int_0^\infty  \frac{B_{\nu}(T(z^\prime))}{ L^\uparrow_\nu(0)} \,W^\uparrow_{\nu}(z^\prime, z) \:\dd z^\prime \end{equation*}

(38)   \begin{equation*} \nghe_{\nu} = \int_0^\infty  \frac{\left((L^\uparrow_\nu(0) - B_{\nu}(T(z^\prime))\right)}{ L^\uparrow_\nu(0)} \,W^\uparrow_{\nu}(z^\prime, z) \:\dd z^\prime \end{equation*}

(39)   \begin{equation*} \nghe_{\nu} =  \exw{0,\;\left(1 -  \frac{B_{\nu}(T(z^\prime))}{ L^\uparrow_\nu(0)}\right)}{W^\uparrow_{\nu}}_{z^\prime} \end{equation*}


(40)   \begin{equation*} G = \int_0^\infty \nghe_{\nu}\; L^\uparrow_{\nu}(0)\; \dd\nu \end{equation*}

This may be used to compute the normalized Greenhouse effect as:

(41)   \begin{equation*} \nghe = \frac{ \int_0^\infty \nghe_{\nu}\; L^\uparrow_{\nu}(0)\; \dd\nu}{\int_0^\infty L^\uparrow_{\nu}(0)\; \dd\nu} = \exw{\nghe_{\nu}}{L^\uparrow_{\nu}(0)}_\nu \end{equation*}

In other words, \nghe is the average of \nghe_{\nu} over frequency as weighted by the flux emmited by the surface, L^\uparrow_{\nu}(0).

Optional change of variables of reflect density changes

Let’s see what happens to our equations if we define a new variable, u, by

(42)   \begin{equation*} u = \frac{1}{n(0)} \int_0^z n(z^\prime)\, \dd z^\prime \end{equation*}


(43)   \begin{equation*} \deriv{u}{z} = \frac{n(z)}{n(0)}  \end{equation*}

The variable u essentially hides any changes in density. While u has units of meters, it’s not measuring ordinary distance. Instead, each time radiation propagates 1 meter as measured by u, it will have passed by as many molecules as it would have passed in traveling 1 meter at the surface. When working with u as our distance measure, it’s as if the density of molecules remains constant.

Rewriting our flux equation in terms of u yields:

(44)   \begin{equation*} L_\nu(u) = L_\nu(0) - G_\nu(u) \end{equation*}


(45)   \begin{equation*} G_\nu(u) = L_\nu(0)\cdot\left[1-Y_{\nu}(0,u)\right] - \int_0^u  B_{\nu}(T(u^\prime)) \,\tilde{W}_{\nu}(u^\prime, u) \:\dd u^\prime \end{equation*}

(46)   \begin{equation*} \tau_{\nu}(u^{\prime}, u) = n(0)\,\int_{u^{\prime}}^u \sigma_\nu(u^{\prime\prime}) \,\dd u^{\prime\prime} \end{equation*}

(47)   \begin{equation*} Y_{\nu,\eta}(u^\prime, u) = 2\,\sin\eta \,\cos\eta\;\exp\left[ -\frac{\tau_{\nu}(u^{\prime}, u)}{\cos\eta} \right] \end{equation*}

(48)   \begin{equation*} Y_{\nu}(u^\prime, u) = \int_0^\frac{\pi}{2} Y_{\nu,\eta}(u^\prime, u)\;\dd\eta \end{equation*}

(49)   \begin{equation*} \tilde{W}_{\nu}(u^\prime, u) = n(0) \,\sigma_\nu(u^\prime) \, \int_0^\frac{\pi}{2}  \frac{Y_{\nu,\eta}(u^{\prime}, u)}{\cos\eta}\;\dd\eta  \end{equation*}

This transformation is most likely to be useful if the absorption cross section, \sigma_\nu, doesn’t depend on altitude, i.e., doesn’t depend on z or u. In that case, \tau_{\nu}(u^{\prime}, u) becomes:

(50)   \begin{equation*} \tau_{\nu}(u^{\prime}, u) = n(0)\,\sigma_\nu\cdot(u - u^{\prime}) \end{equation*}

It may or may not be worthwhile to do calculations and analyses using u instead of z.

Addressing simplifications

This analysis relied on assumptions that are only valid in the troposphere. However, it is possible to generalize this sort of analysis to address propagation throughout the atmosphere.

More general radiation energy transfer equations which include scattering are derived in Chapter 14 of the book “Thermal Radiation Heat Transfer” by Siegel and Howell.


It’s possible to make some simplifying approximations. For example, see this discussion in this blog post of the diffusivity approximation, which eliminates the need to explicitly integrate over angles.