Let's say that we're going to seriously shield a manned space station (to the tune of 1-2 m thickness) in LEO so that radiation doses are a reasonable multiple of background on Earth. Let's also assume that we have freedom of attitude selection, so we can have one direction constantly pointing toward the Earth.
Could you omit the Earth-facing wall because cosmic rays won't come from below? Or maybe the radiation will scatter back up? What would an optimized shielding envelope look like?
I don't think I'd completely eliminate the Earth facing board, but I would reduce the shielding. The shielding should be stronger for the side pointing away from Earth, and be reduced as one gets to a side closer to Earth. Basically, this is a geometry problem. You can still hit the spacecraft from the sides of the Earth, even though cosmic rays will not be transmitted through the Earth itself. Thus, for any position not on the surface of a body, there is still a chance of some radiation coming through. The closer to Earth, the less radiation would spill through.
However, it's worth noting that Earth's magnetic field does a pretty reasonable job at removing such radiation. They also move the direction somewhat differently. In some cases, they might even come in from below (Although I'd have to actually do the math, which I am not at the moment). Still, I suspect more shielding would be needed on the sides, and less the side pointed directly to Earth and directly away from Earth, due to the magnetic field.
