a few questions

I'll try to answer your questions (in reverse order):

An eon is an indeterminate (but large) geologic time span. There have been three eons so far in the history of the Earth: from oldest to youngest, the Hadean, Archean, Proterozoic, and Phanerozoic. There's a lot more on Wikipedia's page on Eons.

Center of mass is the point in a body where all of its mass can be assumed to be concentrated, so the motions of the entire body can be described by the motions of this point through space and the rotation of the body around this point.

Center of gravity is the point in a body through which the resultant of all of the gravitational forces on the body passes. So, as far as gravity goes, you could replace the entire object by this single point (of the same mass) and it would behave in the same way as the whole. In a uniform gravitational field (as at most places on Earth, if the object is small enough) COG is the same as COM. However if the g-field is non uniform then the COG will differ somewhat from the COM. Again, you can check Wikipedia for more information

As for Ground Penetrating Radar: I'll assume that you know basically how "normal " radar works: a transmitter sends out pulses of microwave energy and a reciever listens for the echoes of those pulses as they bounce off of objects. The time between transmission and reception tells you how far away the object is.

Ground Penetrating Radar works the same way - by transmitting microwave pulses into the ground and listening for the reflections. The different types of soil etc. will reflect, absorb, and scatter microwaves in different ways and a map of the underground structures can be constructed from the resulting signal. There's a good page on how GPR works and how to interpret the results on this USDA-Natural Resources Conservation Service page. There's a more detailed description of different types of GPR here. GPR has been used for mapping subsurface water, finding hazardous waste, and finding buried minefields, among many other uses.

As for BrainWave's response (above), I think it was Richard Feynman that said (pardon me if I don't get the quote just right) "If you can't explain something to a layman, then you don't really understand it."

Nice reply!

However, you cited four names for three eons in Earth's history. ;-)

Oops! Thanks for the catch. One site I looked at said "three," but there have actually been four (three "in the past" and the one we're in now). I fixed part of my post but obviously not all of it: an advertisement for the benefits of careful proofreading!