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Parameters Characterizing the Radiative Properties of Aerosols

From Haywood and Shine (1995), we can define the parameters affecting the contribution of aerosols to radiative forcing, RF, expressed in units of Watts per square meter (W-m-2), applied to the troposphere. The troposphere is of particular importance because it is the layer of the atmosphere in which we live, and in which our weather and climate changes occur. It extends from the ground up to around 10 km above sea level. Vertical motion in the troposphere is sufficient to lift fine aerosols and increase their time aloft. The Greek word trope, which pertains to turning, and later took on a meaning having to do with changes, refers to vertical convection or overturning and probably to the resulting altitudinal changes in temperature and water vapor content which do not occur in the lower altitudes of the overlying stratosphere, where little vertical motion occurs. The term was apparently coined by the French meteorologist Philippe Teisserenc de Bort, whose pioneering use of balloons to explore the vertical structure of the atmosphere led to the distinction of the troposphere from the stratosphere.

The changes in tropospheric radiative forcing RF since preindustrial times (defined by the Intergovernmental Panel on Climate Change as before year 1750 C.E.) are summarized in Figure TS 6 of the Technical Summary of the 5th Assessment Report of the Intergovernmental Panel on Climate Change (Stocker et al., 2013). Of all the influences on climate portrayed in the graph, the largest error terms by far apply to aerosols. We know, however, that aerosols can be important contributors to changes in radiative forcing, via scattering and absorption of incoming solar radiation.

Incoming solar radiation is strongly absorbed in wavelengths up to about 0.35 micrometers; that absorbed energy drives the oxygen-ozone cycle in the upper stratosphere. From 0.35 to about 0.9 micrometers most of the energy passes through the atmosphere to the earth's surface as visible light, but for wavelengths greater than about 0.79 micrometer some absorption bands occur, mostly due to water vapor and carbon dioxide.

Aerosols can absorb or scatter incoming solar radiation that reaches the troposphere, thereby affecting the troposphere's radiation budget. A list of the aerosol-related parameters involved in radiative forcing in the troposphere is given by Haywood and Shine and reproduced by Ogren (1998). A change in radiative forcing to the troposphere, ΔF, in W-m-2, due to aerosols can be written as:


ΔF is comparable to the radiative perturbations listed in Figure TS 6 of Stocker et al. (2013).
D = daylight fraction which is defined geometrically from the Earth's orbital parameters.
So is amount of incoming radiation intercepted by a flat surface of unit area perpendicular to the sun's rays at the top of the atmosphere. The dependence of other terms on solar zenith angle account for the earth's curvature.
Tat is the transmissivity of the atmosphere above the troposphere; this term accounts for any absorption of incoming solar radiation in the overlying layers of the atmosphere.
Ac is the fractional cloud cover. Note that a change in cloud condensation nuclei may change this parameter.
Rs is the percentage of incoming solar radiation reflected from the earth's surface.
δ is the aerosol optical depth.
o is the single scattering albedo.
is the average aerosol upscatter fraction, or fraction of radiation scattered upward by the aerosol. It is equal to the amount of backscattered radiation when the sun is directly overhead.

For comparisons with the radiative forcing diagrams given in Stocker et al. (2013) all of these parameters are expressed in terms of their departures from their values in year 1750, which is taken as the "pre-industrial" or "natural" value.

The last three terms require some further explanation.

δ Aerosol optical depth, or aerosol optical thickness of a medium, is a measure of transparency through a thickness, s, of the medium. It is related to the extinction coefficient, γ, which is the fraction of light intensity I, depleted per meter of depth in the aerosol layer.

- γ ds = dI

Values of extinction coefficient (sometimes called the attenuation coefficient) in the atmosphere are frequently given in inverse megameters (e.g., fractional value/106m).

Integrating both sides of the above equation gives:

- γs = ln (I/I0), which is the optical depth, or optical thickness, δ, and s/δ is the mean free path of a photon.

The extinction coefficient and the optical depth each have components due to scattering and to absorption. Typical values of optical depth (from both components combined) in the cloud-free atmosphere are around 0.1 to 0.3 except just downwind of deserts and large industrial areas where they often approach 1.0.

Cloud optical depth is a measure of attenuation of the light passing through the atmosphere due to the scattering and absorption by cloud droplets (instead of aerosols); typical values are much higher than for aerosols, ranging around 10-60. Otherwise it is the same as the aerosol optical depth discussed above, i.e., the integrated extinction coefficient over a thickness, s, within a vertical column of unit-area cross section.

o Single scattering albedo is the probability that, given an interaction between a photon and a particle, the photon will be scattered rather than absorbed. Quantitatively, this is the ratio of total scattering to extinction (extinction = absorption + scattering).

o = δs ,
(δs + δa)

where δs and δa refer, respectively, to the components of optical depth due to scattering and absorption.

Scattering of light occurs via reflection, refraction and diffraction, all of which depend on the wavelength of radiation encountered. The net result of these processes is some scattering in all directions, but the relatively larger particles (diameters of about 1 micrometer or more) scatter incoming solar radiation primarily in the forward (downward) directions. Single-scattering refers to an assumption of only one scattering event, rather than multiple scattering. The assumption is good in most atmospheric conditions, where particles are far apart relative to their sizes. The value of o for a purely scattering aerosol is 1, while very strong absorbers (e.g., elemental carbon) have values around 0.3.

Some other parameters are frequently given in conjunction with o and δ. Among the most common are the Angstrom coefficient, the phase function, and the asymmetry parameter.

Angstrom Coefficient: Optical depth varies with the amount of scattering and of absorption which, in turn, are functions of wavelength. Plotting the optical depth at 2 or more different wavelengths on a logarithmic scale will provide a slope from which the optical depths for other wavelengths can be calculated. The slope is known as the Angstrom coefficient. The variation of scattering with wavelength provides information on particle size; the angstrom coefficient is around 2 for scattering dominated by sub-micrometer particles, and around zero where the scattering is dominated by particles greater than 1 micrometer in diameter.

Mathematically, the Angstrom exponent, α can be computed from optical depths, δ1 and δ2, at wavelengths, λ1 and λ2, respectively:

and the optical depth on any wavelength may be found from the relation:

where λ0 is usually taken to be 1 micrometer.

Phase function and asymmetry parameter: The phase function describes the angular distribution of scattered light, thereby indicating the preferred scattering direction (forward or backward) for light encountering an aerosol particle. The phase function is defined as the ratio of the energy scattered over a solid angle in a given direction to the average scattered energy in all directions.

The mean product of the phase function values and their respective cosines is the asymmetry parameter. Its value approaches +1 for scattering more strongly in the forward direction, and it approaches -1 for scattering more strongly in the direction from which the radiant energy came. Values approaching zero indicate scattering by small particles (Rayleigh scattering), which is the same in all directions.

Average aerosol upscatter is the fraction of scattered light that is scattered upward with respect to the earth's surface. It depends on sun angle, and is equal to the backscattered fraction, β, when the sun is directly overhead. It also depends on aerosol size; larger particles scatter a higher fraction of light in the forward direction.

General Comments

Additional information about the parameters involved in aerosol studies can be found at the following sites:

Many of these parameters depend, one way or another, on the sun angle, which is known for any given point and time at any specified location. Mathematics integrates the changing geometry over all sun angles to obtain average contributions to radiative forcing for a specified location for all daylengths over a year.

These parameters are also depend on the wavelength of the incident radiation. Therefore it is common to see subscripts r, g, and b indicating red, green, and blue waves respectively. At other times the wavelengths are simply specified, for example as 450 nm (blue) 550 nm (green) and 700 nm (red).

Aerosols can provide surfaces for condensation of water vapor; these aerosols are called cloud condensation nuclei (CCN). The amount of condensation affects the radiative properties of the aerosols and depends on the relative humidity of the surrounding air.


  • Haywood, J.M. and K.P. Shine, 1995. The effect of anthropogenic sulfate and soot aerosol on the clear sky planetary radiation budget. Geophys. Res. Lett. 22(5): 603-606.
  • Stocker, T.F., D. Qin, G.-K. Plattner, L.V. Alexander, S.K. Allen, N.L. Bindoff, F.-M. Bréon, J.A. Church, U. Cubasch, S. Emori, P. Forster, P. Friedlingstein, N. Gillett, J.M. Gregory, D.L. Hartmann, E. Jansen, B. Kirtman, R. Knutti, K. Krishna Kumar, P. Lemke, J. Marotzke, V. Masson-Delmotte, G.A. Meehl, I.I. Mokhov, S. Piao, V. Ramaswamy, D. Randall, M. Rhein, M. Rojas, C. Sabine, D. Shindell, L.D. Talley, D.G. Vaughan and S.-P. Xie, 2013: Technical Summary. In: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assess¬ment Report of the Intergovernmental Panel on Climate Change [Stocker, T.F., D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.