How could one use a GIS (or other appropriate system) to calculate the change in mean ocean levels across the world if all the land below sea level were flooded (Death Valley, Caspian Depression, Afar Depression for example)?
This is a volume question, rather than one of cartography. Here goes for a back of an envelope calculation:
Of the two hundred and fifty five recognized countries or protectorates in the world, 33 have land below sea level. Most of these are only a few meters down (ref: geology.com).
The approximate surface area of the sea is 360,000,000 km^2. So, taking an highly unscientific approach of averaging the depth of the depressions we get a mean depth of approximately 73m (though the true average depth is likely to be considerably less given the non-uniform nature of the depressions, their respective areas etc).
So to reduce the sea level by 1m the total area of these depressions needs to be around 5,000,000 km^2. This is around half the total land area of the USA. I don't have the figures to hand but given that these area would probably all fit into an area considerably smaller than Texas - I estimate that flooding land which is below sea level will reduce the sea level by exactly 'not very much at all' (give or take a millimeter - and very much less than 1m in any event).
I suggest that you calculate a more accurate answer and report back. You could derive a better approximate answer by getting the golbal SRTM data. Global coastline data are also available, so you should be able to delineate the areas easily and calculate their volume. May I suggest you use GRASS GIS or ARC for this?
If the question is about defence against sea level rise, the alternative to flooding land, could be to dredge the sea, dumping the trailings in shallow coastal areas (I'm being a little tongue in cheek here) - thus deepening the sea and building new land/coastal defences at the same time. Now that poses another very nice 'cut-and-fill' GIS question as a counterpoint to the original question, one that you could use Arc or GRASS (probably also SAGA) to answer.
I was looking at the effects of flooding the Qattara depression in Egypt as a sunday activity, to see if it would lower today's sea level to a noticeable degree. I'm not a geographer or a mathematician, but this is what I came up with:
There are 360,000,000,000,000 sq meters of ocean on earth. Make those cubic meters and you have the total volume of the uppermost meter of water at 360,000,000,000,000 cubic meters.
The Qattara depression has an average depth of 60 meters, with an area of 19,605 sq km (19,605,000,000 sq meters), so it has a volume of 1,176,300,000,000 cubic meters. This is about 0.3% of what we would need to lower sea level by one meter. This may sound discouraging until you take into account the fact that water will not only fill the depression, but also the desert areas around it in the form of water vapor, rivers and ponds further uphill. This is also only one depression among ten or fifteen more of comparable size around the world.
If we could lower sea level by even 10 or 15 centimeters, we would be able to reverse rising sea levels by 30 to 50 times the recorded yearly rate of rise in 2013, and likely bring agriculture and economy to parts of the world that mankind hasn't been able to use for millenia.