I’m not arguing with the idea most of the signal gets to the ground, but, I’m worried that you get specifics quite wrong. Planes don’t want a reflection from 50

I’m not arguing with the idea most of the signal gets to the ground, but, I’m worried that you get specifics quite wrong. Planes don’t want a reflection from 50% of the sphere. They want it from directly below. Other land is further from the plane and obscures the measurement. But they’re not broadcasting to the whole sphere either, because the radar instrument is not an isotropic radiator. That antenna has gain. It’s louder / more sensitive in a specific direction. Probably by a lot. You can’t make perfect antennas, but you don’t need any signal at all from most of the sphere, so you can gain many decibels. The angle of the beam you do have is probably fixed, and being fixed would imply that the area swept by the core of the beam is indeed proportional to the square of the altitude. But I don’t think two trips through an n^2 distance is n^4 unless I’m missing some of the physics. Shouldn’t it be (2n)^2 and this still proportional to the square of the altitude not to the hypercube? (And affected by scattering and bad reflectivity from the terrain, which seems likely to be independent of altitude?) This is different for detecting random entities in free space where most of the first n^2 trip really is lost. So it’s like you say but for different reasons??!?!