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Published byLeah Robertson Modified over 4 years ago

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The Boundary Element Method for atmospheric scattering Problem: how do we calculate the scattering pattern from complex particles (ice aggregates, aerosol...)?

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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

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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

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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

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Inside the object Green functions look like this Outside the object –Simply the scattering from point on the surface y to point x elsewhere

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Scattering from a circle n =1.5 Easy to calculate the far-field scattering pattern, which is what we want in meteorology

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Scattering from an absorbing square

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Source need not be a plane wave

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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...

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c = 300.000 km/sec I F = I 0 x (cosθ) 2.

c = 300.000 km/sec I F = I 0 x (cosθ) 2.

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