Electricity/magnetism-- radiative transfer

Beer Lambert Law

The Beer Lambert Law as expressed by chemists for analytical purposes considers transmittance of electromagnetic energy which can be reduced by both scattering and absorption such as what is encountered in a cloud.

The transmission of light through coffee (absorption) and milk (scattering) in petri dishes lying on an overhead projector shows the point of this, in scattering we can see that one is dark and the other bright, but in transmission both are the same.

The transmission can be expressed:

Transmission = 1 - Extinction

Extinction = scattering + absorption

Single Scattering Albedo is a parameter that emphasizes this dichotomy, and varies from 0 (absorption dominant) to 1 (scattering dominant).

Black Bodies and Passive Remote Sensing

Wiens Displacement Law describes the emission of the sun and the earth. Each behaves somewhat as a black body at a characteristic temperature, with the sun at 6000 K and the earth at 300 K.

In volcanic cloud sensing we exploit reflected solar energy for TOMS sensing, while the IR data from GOES, AVHRR and MODIS is mainly derived from the earth. A fundamental lecture review of these points is found in most remote sensing books.


Reference: Wen and Rose: Retrieval of Sizes and Total Masses of particles in Volcanic Clouds Using AVHRR Infrared Bands.

Please click on the icon to view the equations.

A linear model for observed radiances:


  1. The cloud approximates a planar homogeneous cloud layer parallel to the surface.
  2. The background surface is homogenous.
  3. The atmosphere above the cloud and between the surface and the cloud are clear window.

    (a)Radiative transfer equation

    (b)Eddington's approximation

    (c)Selected size distribution of particles:

    (d)Estimation of the total masses:

    (a) (b)

    (c) (d)


    Assumptions and conditions:

    1. Silicates are the only kind of particle in the volcanic cloud, and the refractive index is from Andesite (Pollack et al., 1973).
    2. The shape of volcanic particles is spherical , therefore Mie theory can be used to calculate extinction cross section (sigma_e), asymmetric parameter (g), and single scattering albedo.
    3. The particle size distribution, n(r), is uniform and monodisperse within each pixel.
    4. The volcanic clouds are continuous, i.e. Ac=1.
    5. Knowing surface temperature: Ts=273 K
    6. AVHRR infrared bands: Band3 (3.55-3.95 um), Band4 (10.3- 11.3 um), Band5 (11.5-12.5 um).

      (a)Cloud-top temperature retrieval model

      (b)Particle radius retrieval model

      (c)Optical depth retrieval model

      (a) (b) (c)


    (a)AVHRR band4 image of Spurr volcanic clouds The AVHRR Band 4 image of the Crater Peak/Spurr eruption cloud, August 19, 1992, 1338 GMT. The center square is the study frame, with an area of about 165 km 110 km. In the central part of the cloud shown Ac=1 for all pixels but at the edges sometimes this is not true.

    (b)Two-band temperature difference model Two-band temperature difference model at 10.8 m and 12 m. The near horizontal solid lines represent different effective radii, and the near vertical dashed lines represent the dependence of optical depth at 10.8 m with particle radius.

    (a) (b)

    (c)Retrieval of optical depth

    (d)Retrieval of effective radius

    (c) (d)


    Refractive index of different samples

    Band4 (Real,Imaginary), Band5 (Real, Imaginary)

    1. Andesite(54.15% SiO2)---- ( 2.0534, 0.60897),(1.8392, 0.13786)
    2. Basalt(53.25% SiO2)---- (2.1848 ,0.48812),(1.9051, 0.14670)
    3. Basaltic Glass(53.45% SiO2)----(2.1241,0.71211),(2.0129, 0.24037)
    4. Obsidian-little Glass, Mt. California,r hyolite(73.45% SiO2)----(2.0085,0.27476),(1.7281 , 0.18407)
    5. Obsidian-Lake County Oregon, rhyolite(76.20% SiO2)----(2.0268, 0.27882), (1.7410, 0.18462)
    6. Volcanic dust: two samples are averaged. The andesitic Irazu ash sampled during ashfall is dark grey with feldspar,and the Hawaii sample was lightly weathered vesicular basaltic glass ----(1.9700, 0.3700),(1.8000,0.18000)

      (a)refractive index of basalt The two-band temperature difference model with uniform size distribution and the Basalt (sample 2) refractive index.

      (b)rafractive index of basalt-glass The two-band temperature difference model with uniform size distribution and the Basalt glass (sample 3) refractive index.

      (c)refractive index of obsidian The two-band temperature difference model with uniform size distribution and the Obsidian (sample 4) refractive index.

      (d)refractive index of dust The two-band temperature difference model with uniform size distribution and the Volcanic dust (sample 6) refractive index.

      (a) (b)

      (c) (d)

      Pixel-scale retrieval of masses for different samples:


    Efficiency facters for differnt size distributions

    Relationship of effective radius and efficiency factors at 10.8 m for different size distributions of particles. (1) (solid line ) is associated with uniform distribution, (2) (dotted line) with gamma, and (3) (dashed line) with lognormal distribution.

    (a)lognormal distribution

    (b)gamma distribution

    (a) (b)

    Frame scale retrieval of masses for different size distributions

    Mass calculations for different refractive index and distributions. The relatively higher calculated masses is due to the assumption of lognormal size distribution of particles, and lower calculated masses is due to the assumption of the volcanic ashes only containing glassy basalt component.


    1. Volcanic clouds have negative brightness temperature differences (band 4 - band 5) only if they have a dominance of particles with radius less than 5 um.
    2. Our model works best when there is large difference between the temperature of the underlying surface and the volcanic cloud. The lowest temperature difference (band 4 - band 5) of the volcanic cloud is a linear function of the temperature difference between the underlying surface and the volcanic clouds (Ts - Tc).
    3. This method can be used to interpret volcanic clouds with a dominant effective radius between 0.8 and 4.3 m for uniform size distribution, between 0.1 and 17 um for lognormal size distribution.
    4. The mean radius and the optical depths within the test frame of a 13 hour old August 1992 Crater Peak/Spurr volcanic cloud are determined to be 2.8 m and 0.66, respectively, based on the two-band model. If lognormal size distribution is considered, the average of particle radius is about 0.62 to 0.78 m, the estimate of the mass of the volcanic cloud particles is about 42 - 56 X 10^3 tons in the frame, and about 0.24 - 0.31 106 tons in the whole cloud, which is about 0.7 - 0.9% of the total volcanic ash measured in the deposited ash blanket (36x106 tons).
    5. The mass estimate is more sensitive to the assumed ash size distribution than it is to the ash composition.