The Boundary Element Method for atmospheric scattering

1
The Boundary Element Method for atmospheric
scattering

• Problem how do we calculate the scattering
  pattern from complex particles (ice aggregates,
  aerosol...)?

2
The slow way...
By

• Discretize Maxwells curl equations directly
• This is the Finite Difference Time Domain method
  (very expensive in 3D)

Ez
Ez
Bx
Bx
By
Ez
Ez

• A sphere (or circle in 2D)

Scattered field (total - incident)
Refractive index Total Ez field
Many more animations at www.met.rdg.ac.uk/swrhgnr
j/maxwell (interferometer, diffraction grating,
dish antenna, clear-air radar)
3
The Boundary Element Method

• Active research in Maths Dept
• Steve Langdon, Simon Chandler-Wilde, Timo Bechte
• Mostly applied to acoustic problems
• Applicable to EM scattering (but more complicated
  due to polarization)
• Only one paper has applied it to a meteorological
  problem!
• First step if the source is continuous, we can
  represent the electric field in time harmonic
  form
• So we want to find the complex number E(x)
  everywhere in space (represented by position
  vector x) that represents the amplitude and phase
  of the electric field

4
Greens representation formula

• Need to solve an integral equation
• As every point on the surface depends on every
  other point, this boils down to solving a matrix
  problem

. Point on surface y
Surface s
Source at x0 (could be at infinity)
. Point x
5
Green functions look like this

• Inside the object

• Outside the object
• Simply the scattering from point on the surface y
  to point x elsewhere

6
Scattering from a circle n 1.5

• Easy to calculate the far-field scattering
  pattern, which is what we want in meteorology

7
Scattering from an absorbing square
8
Source need not be a plane wave
9
Outlook

• Potentially very efficient as need only
  discretize the surface of an object, rather than
  the entire volume
• Number of elements goes as size2 not size3
• Still need 10 points per wavelength
• If all the surfaces are flat, it might be
  possible to represent electric field on each
  surface by a 2D Fourier series, requiring only 2
  coefficients per wavelength
• 5x5 25 times fewer points
• In 3D, need to use more complicated formula for
  all three components of the electric field
• Rather complicated to code up...