By the way, When we say a vertical-over-a-ground plane, we are not talking about a "ground plane antenna". That commonly refers to a 1/4 wave radiating element and four "ground plane" radials.
But this is no better than a vertical dipole when you model it in free space. Yes, the radials make a good match to 50 Ohm coax, but do not simulate an infinite ground plane.. The antenna pattern is still just a donut with half the energy going into the ground.
So here is the question. If one put another set of ground radials 1/4 wave below the initial radials and tuned them longer to act like real 180 degree phase reversals (like reflectors on a yagi), would this then make the antenna really pick up the missing 3 dBi that otherwise goes into the ground?
Bob