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... Discretize Maxwells curl equations directly This is the Finite Difference Time Domain method (very expensive in 3D) Refractive index Total E z field Scattered field (total incident) Many more animations at www.met.rdg.ac.uk/~swrhgnrj/maxwell (interferometer, diffraction grating, dish antenna, clear-air radar…) A sphere (or circle in 2D) EzEz EzEz EzEz EzEz BxBx BxBx ByBy ByBy

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 Electric field at point x......equals the incident wave from source at point x 0......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 s Source at x 0 (could be at infinity). Point on surface y. Point x

5. Inside the object Green functions look like this 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 size 2 not size 3 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...