1. Microwave Emission and
Scattering of Foam
Based on Monte Carlo Simulations of
Dense Media

2. What has been proposed in the paper
• Model of microwave emissivity
There are empirical models of microwave emissivity available like that by William and
Wilheit. But these models do not take into account the physical microstructure of foam
and the foam layer thickness.
Hence we use Monte Carlo simulations of solutions of Maxwell’s equations of densely
packed coated particles to analyze the microwave emission
Using the above, the absorption, scattering and
extinction coefficients have been calculated and then the
DMRT (Dense Media Radiative Transfer)theory has been used to calculate the
emissivity.
• To model the foam, we use the FCC structure.
( face centered cubic) to simulate high density packing
Lattice points on the faces of the cube and on the corners
Total 4 lattice points: ( 1/8 * 8 + ½ * 6)
Atomic packing factor: 0.740 (highest possible for any
lattice)

3. Description of foam
Void fraction: 80% to 90% in most cases
Assumed that the foam is composed of spherical
bubbles with fcc structure has a fractional volume of
74%
Let N be the number of coated particles, and the jth
coated particle is of inner radii bj and outer radius aj.
If the total volume of the foam is V, the fractional
volume of coated particles is
Video micrograph of the bubble structure
Radii
structure of a
Fractional volume of sea water bubble

4. Absorption and extinction based on
independent scattering
Absorption
For an incident field of the electric field inside the shell
at r vector distance is
And similarly for fields in the x and y directions

5. Absorption and extinction based on
independent scattering
Where
er is the relative permittivity of the medium
a : represent the outer radius of the coated particle
b : represents the inner radius of the coated particle.

6. Absorption and extinction based on
independent scattering
Absorption
For a combined electric field :
We simply sum up the earlier equations
Power absorbed
Where
Angular frequency
Imaginary part of
permittivity
Volume of coated particle
Consider N coated particles in a volume V.
According to independent scattering
assumption the absorption and
scattering of N particles is the sum of
the individual particle’s absorption and
scattering

7. Absorption and extinction based on
independent scattering
Absorption
The absorption coefficient is the
absorption cross section per unit
volume of the collection of particles
Where n is the free space wave
impedance
Scattering
Calculating the scattering coefficient
Requires Integration of the scattered
intensity over all solid angles. It is
the scattering cross section per unit volume
Where is the relative permittivity of
Coated particle

8. Monte Carlo Simulations and DMRT
theory
Consider thermal emission from a layered medium with coated particles
embedded in a background medium of air, as indicated in Figure. The layer
consists of coated particles (region 1), and covers a half space of ocean (region
2). Next figure shows the collection of coated particles. In the Monte Carlo
simulations, we consider the absorption and scattering of particles collectively
by solving Maxwell’s equations. The scattering coefficient and absorption
coefficient are defined respectively as scattering cross section per unit volume
and absorption cross section per unit volume.

9. Monte Carlo Simulations and DMRT
theory
In Monte Carlo simulations, we consider the absorption
and scattering of N particles collectively by solving
Maxwell’s Equations.
A volume integral equation is used to solve Maxwell’s
equation for the N particles. Let the internal field in the
sea-water coating region of particle j be
The Maxwell equation for the collection of particles

10. Monte Carlo Simulations and DMRT
theory
Then the following steps are carried out:
1. We expand the internal field in the coating region of particle j into three basis
functions.
2. We apply the galerkin’s method to write them into a linear system of equations.
3. We make the small particle assumption and simplify it and get the scattering
coefficient as
And effective propagation constant as

11. Numerical simulations of emissivity and
comparison with experimental measurements
Now, we illustrate the numerical results of the emissivity based on a model of
coated particles in a fcc structure. The absorption rate, scattering rate, and effective
permittivity are first calculated using Monte Carlo simulation. Subsequently, these
parameters are used to compute the emissivity.
Vertical polarization; radius of coated air hosrizontal polarization; radius of coated
bubble = 1.0 mm air bubble = 1.0 mm
As the size of the bubbles increases, the scattering coefficient increases, and the
albedo also increases. The increase in albedo causes the corresponding bightness
temperatures to decrease.

12. Numerical simulations of emissivity and
comparison with experimental measurements
The above table shows the parameters
calculated from monte carlo
simulations for the for the two graphs
shows in prev slide
And on the right emissivity as a
function of thickness of the foam layer.

13. Numerical simulations of emissivity and
comparison with experimental measurements
Emissivity at 10.8 and 36.5 GHz at vertical and horizontal polarization as a
function of thickness of foam layer for different radii of bubble.
Observation angle 53 degrees; radius Observation angle 53 degrees; radius
of air coated bubble = 1.0 mm of air coated bubble = 0.5 mm

14. Numerical simulations of emissivity and
comparison with experimental measurements
Comparison of experimental results and that obtained theoretically by the DMRT
theory at two different frequencies.

15. Conclusion
1. We apply Monte Carlo simulations and dense-media radiative
transfer theory to analyze the microwave emissivity and
scattering of foam on a seawater surface.
2. We model the foam as densely packed air bubbles with a thin coating of seawater.
Numerical simulations show the polarization and frequency dependencies
of emissivity on microstructure properties such as foam layer thickness and the size of
foam air bubbles. The results of numerical simulations are in good agreement with
experimental measurements.