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Propagation of tsunamis in deep water (>50 m, Shuto, 1991) obeys linear dynamics. For a linear dynamics system, any response of the system to arbitrary external forcing can be expressed as a linear combination of the Green’s functions. The Green’s functions can be pre-calculated for fixed physical parameters and geometry of the system, whereas the precalculation implies a great time saving, which is essential when a catastrophe occurs and an immediate solution is required. A Green’s function has traditionally been calculated as a domain wise solution to a unitimpulse acting at a grid point. Such a Green’s function may be referred as a single-source Green’s function (SSGF). However a tsunami is rarely triggered at a single source point. A reasonable extension is therefore to pre-calculate a set of such SSGFs corresponding to a group of source points in a pre-assumed tsunami source region. Nevertheless when a future tsunami happens outside of the pre-assumed source region, the pre-calculated SSGFs will not be helpful. Xu (2007) proposed a new type of Green’s function, an all-source Green’s function (ASGF). An ASGF focuses on a receiver point, regarding all the model grid points as potential sources. The computational cost for an ASGF is the same as that for an SSGF. However an ASGF provides thorough preparedness for a point of interest (POI) against all possible sources. Tsunamis from different sources may arrive at a POI at different times and may have different maximum wave amplitudes within certain period. Xu showed that with an ASGF one could make an arrival time map and a map of relative maximum amplitudes for a POI. Xu further demonstrated that with the ASGFs, one could also establish a real-time simulation system, which would give tsunami arrival time series at one or more POIs immediately. This chapter will describe in details the ASGF in terms of its numerical constructions and its applications to tsunami problems. It will first present the governing equations and boundary conditions. It will move on to grid auto-generations from a realistic topographic The All-Source Green’s Function of Linear Shallow Water Dynamic System: Its Numerical Constructions and Applications to Tsunami Problems 25

The All-Source Green's Function of Linear Shallow Water Dynamic System: Its Numerical Constructions and Applications to Tsunami Problems