Issuing warnings takes minutes but processing the data before deciding to place the warning can take tens of minutes or a few hours depending on the location of the sea sensors in relation to the Earthquake location. They have to make sure a tsunami event has taken place before issuing a warning, if false warnings are issued any warnings in the future may be ignored.
isis
Many tsunamis only travel in one direction (although not in this case), the Tsunami in PNG a few years back is an example of a local event. A lot of complex anaylsis needs to be done to predict how a Tsunami will behave and travel.
ps
I should mention that I worked as a geophysicist for 17 years and helped setup seismic monitoring stations in the UK.
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