The ocean is deep.
Last week, there was some halchal regarding the purported discovery of the lost city of Atlantis by users of Google Earth, which has recently added bathymetry data. Here is Google's write-up about it. (By the way, that write-up also describes the Loihi Seamount, which is a Hawaiian 'almost' island.)
So what was this so-called discovery? It was an artifact of the estimated depth. High quality ocean depth data can be collected by boats using echosounding, but this data acquisition is only local to where the boat is, and it is expensive to cover large areas. Lower quality data can be collected using remote sensing satellites. The artifact was that in the places there were boat measurements, a lower depth was estimated than in places that there weren't boat measurements.
An estimation procedure ought to assimilate the various data available and perhaps also incorporate a prior model so as not to leave artifacts that people might think are Atlantis. As described in the related paper, it is not an easy problem. However, perhaps benefits could be gained by utilizing methods such as those described in this and this.
Speaking of measuring the ocean and boats, after a talk by Marco Duarte a couple of weeks ago, my officemate Matt Johnson was saying how it would be a great demonstration of the power of compressed sensing (CS) if you made a version of the game Battleship in which one player could take CS measurements and exploit structured sparsity, while the other player played normally. The CS player would win, either every time or with overwhelming probability --- I'm not sure exactly which.
An interesting statement in the Atlantis write-up is: "If there really are little green men hiding somewhere, the ocean's not a bad place to do it. Mars, Venus, the moon, and even some asteroids are mapped at far higher resolution than our own oceans (the global map of Mars is about 250 times as accurate as the global map of our own ocean)." It is easier to figure out relief above water than underwater, but the opposite is true for figuring out what is in the crust. Doing large seismological surveys under the ocean with boats is easier than doing it on land with heavy trucks, as I've come to learn from Richard Sears, who sits two doors down from my office.
The ocean is deep and the crust is thick.