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).
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.
Please click on the icon to view the equations.A linear model for observed radiances:
(a)Radiative transfer equation
(c)Selected size distribution of particles:
(d)Estimation of the total masses:
Assumptions and conditions:
(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.
(c)Retrieval of optical depth
(d)Retrieval of effective radius
Refractive index of different samples
Band4 (Real,Imaginary), Band5 (Real, Imaginary)
(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.
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.
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.