I caught a bit of Nova or something the other night that was about how satellites have added so much to our knowledge base. Unless I misunderstood, one of the things they can do is map the ocean floor to within something like 1/2 inch from satellite radar imagery. The radar detects differences in the surface level of the ocean to determine what lies beneath. First of all, I find this to be astounding. Why would the water over an underwater mountain be any “higher” than the surrounding water. And how much higher? Surface tension? I though water sought its own level. And how do they take swells/waves into account? I need to see the whole show again. The things the satellites can do really are incredible.
It’s due to the extra gravity of the mountain. All mass attracts all other mass according to Newton’s Laws of Graviation, which means that if you have a big concentration of mass in one area (such as an undersea mountain), the water will be pulled towards it and pile up a bit more on the surface.
You mention that water “finds its own level”, but that’s not strictly true. Really, the ocean surface will “want” to assume the shape of the geoid, which is a sort of abstract surface that’s influenced by the local strength of gravity, nearby masses, latitude (due to the centrifugal force from the Earth’s rotation), and so on. What these satellites are really doing is measuring the geoid to a very high precision; if you know where the geoid is, you can in principle reverse-engineer where the masses are that influence it.
Oh, and I believe that ocean swells and such are just averaged over; the heights they measure are just averaged over a large enough amount of time and distance that any major swells are cancelled out by the corresponding troughs and you end up with the “real” height that the water would want to be at.
The question has been well answered (yes, it is amazing!). Let me just add a plug fir a terrific new book, “Soundings,” a biography of Mary Tharpe, who was basically the first person to map the world’s ocean floors, mainly in the 1950s out of Lamont Doherty Observatory in New York state.
the resolutions for this seems very good - up to ~1/2 a foot vertically, but horizontally it is no better than anything from 10m to 10km
All in all, that kind pf precision is still pretty impressive. Averaging makes perfect sense and I though that might be the case.
So what you’re saying is just a relief contour map. I though remote sensing could actually tell you about the geology of the OF.
Well, indirectly, it can. We know well by now what various typical undersea geological structures look like just by their topography – subduction zones, rift valley spreading centers, eroded volcanoes, abyssal plains, etc. If by “geology” you mean “rock types.” we also know about the expected mineral makeup of such features. If we have further questions about a specific feature, then we do dredging of core samples at carefully chosen locations.
^
Not to mention seismic refraction. 
this is a subject I know a lot about. 
I recently worked on these data sets.
First a correction. The satellite altimeters have a 1/2 in. resolution-but a 2-3 in. accuracy of the the sea surface, not the sea floor. The seafloor is not the goal of these satellites and as we all know the radar can’t penetrate the water anyway. The bathymetry can be inferred from the altimetry as has been explained by the other posters, but it is a side-effect of the measurements. If you want to see some way cool gravity from space go to:
http://www.csr.utexas.edu/grace/
side note: NASA recently did the same thing for the Moon.
The satellite altimetry measures ocean currents since any moving fluid on a rotating earth will have a slope-the geostrophic effect. The Gulf Stream for instance is a mound of water 1 meter high in the center. Satellites measure these bumps and from that we can calculate the presence of currents and eddies-including sub-surface eddies. The bump is there even if there is no surface expression (in temperature) of the current.
It is amazing work. All sorts of effects have to be compensated for. Things like the electron density of the upper atmosphere changes the speed of light as the radar beam passes through. That has to be corrected. The satellite has to be precisely tracked since any satellite’s orbit moves up and down 2-300 meters due to changes in the earth’s gravity field, atmospheric highs and lows have to be mapped since the higher air pressure depresses the sea surface, etc.
All that said, the measurements are routinely accurate to 5-7 cm.