The Boundary Element The Boundary Element MethodMethod

for atmospheric scattering for atmospheric scattering• Problem: how do we calculate

the scattering pattern from complex particles (ice aggregates, aerosol...)?

The slow way...The slow way...• Discretize Maxwell’s curl equations directly

• This is the Finite Difference Time Domain method (very expensive in 3D)

Refractive index Total Ez field Scattered field(total − incident)

Many more animations at www.met.rdg.ac.uk/~swrhgnrj/maxwell(interferometer, diffraction grating, dish antenna, clear-air radar…)

BE

2

2

n

c

tE

B

t

• A sphere (or circle in 2D)

Ez Ez

Ez Ez

Bx Bx

By

By

The Boundary Element The Boundary Element MethodMethod

• 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

tieEtE )(),( xx

Green’s representation Green’s representation formulaformula• 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

)()(

),()(

)(),(),()( 0 y

y

xyy

yxyxxx ds

n

GE

n

EGGEs

Electric field at

point x...

...equals the incident wave from source at point x0...

...plus the integral over the surface of the object..

...of a function of the scattering from the surface at point y to the point x.

Surface sSource at x0

(could be at infinity)

. Point on surface y

. Point x

• Inside the object

Green functions look like thisGreen functions look like this

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

elsewhere ),( xyG

Scattering from a circle Scattering from a circle n n =1.5=1.5

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

Scattering from an absorbing Scattering from an absorbing squaresquare

Source need not be a plane Source need not be a plane wavewave

OutlookOutlook• 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...