The Phase Function in the Tertiary Scattering (I)

Abstract

Take any three points F, T and E in the earth's atmosphere, and a point O'on its surface. Define a system of rectangular coordinates X1, Y1 and Z1 with its center at F, with X1 axis drawn towards the Sun. Resolve the direct insolation reaching F into two plane polarized rays; the one travels to X1 and oscillates in Z1 direction, which is denoted by (1), the other travels to X1 and oscillates in Y1, which is denoted by (2). The primary scattered ray generated at F when (1) encounters one air particle at F, travels in FT direction and oscillates in a direction normal to it. FT direction is denoted by X2 and the direction of oscillation by Z2 and determined by X1, Y1, Z1 and the positions of F and T. This scattered ray is denoted by E1.
The primary scattered ray generated at F when (2) encounters one air particle at F, travels in FT direction and oscillates in a direction normal to FT. This direction is denoted by Z2' and determined by X1, Y1, Z1, F, and T. This scattered ray is denoted by E1'.
E1 travels in FT direction and is scattered at T when it encounters one air particle there. This secondary scattered ray is denoted by E2 and its direction of oscillation by Z3. In the same way, the secondary scattered ray which is caused by the scattering of E1' at T, is denoted by E2' and its direction of oscillation by Z3'. Furthermore, E2 and E2' are scattered at E and the resulting tertiary scattered rays E3, reach O'.
If ω1 ω1', ω2, ω2', ω3 and ω3' are respectively the angle between FT and Z1, FT and Y1, TE and Z2, TE and Z2', EO' and Z3, EO' and Z3', then the phase function in the tertiary scattering is expressed by Π3n=1sin2ωn+3Πn=1sin2ωn'.