I'm looking into the 3d graphics equations for light traveling through a medium. Specifically, light traveling along a ray to the eye. One of the components is called in-scattering, where light is scattered into this ray to add light. They define a phase function phase(v', v) which describes how much light from the direction v' is scattered into the viewing ray direction v. So for a point p on the viewing ray, they compute SphericalIntegral( L_i(p,v')*phase(v',v)dv' ) where L_i is the incoming light at p from v'. So we sum all the incoming light from all directions and weight it against the probability that it scatters in the direction v. Now it seems to me that this integral gives the amount of light added along the viewing ray, but they take it one step further and say: "To get the total inscattered radiance over a path length dt, we need to multiply this by the probability of an inscattering event happening over that path length (note that this is the same probability of an outscattering event happening)." Why is this needed in the model? Doesn't the phase function determine the probability of how much light is scattered into the view ray? Is it really because some of the inscattered light has a probability of being outscattered again (which is why it mentions the probability of an inscattering event per unit length is the same as an outscattering event)?