Tsunami Forecasting and Warning Improvement Act of 2013 - Amends the Tsunami Warning and Education Act to direct the Administrator of the National Oceanic and Atmospheric Administration (NOAA), through the National Weather Service, to establish or maintain a Caribbean Tsunami Warning Center in Puerto Rico. (The responsibilities of tsunami warning centers include monitoring and evaluating earthquakes, disseminating forecasts and tsunami warning bulletins to federal, state, and local governments as well as the public, and ensuring information sharing between forecasters and emergency management officials.)