Numerical modeling of waves for a tsunami early warning system

Numerical modeling of waves for a tsunami early warning system
interval; it is assumed that the solution is periodical over that time interval
and that it repeats identically over the following and the preceding time.
Therefore the selected time interval should be long enough to allow all the
wave energy to exit the computational domain trough the open boundaries
before the time interval ends. Otherwise interference between wave energy
coming from different time intervals is obtained. If all the wave energy has
left the domain at the end of the computation, the following one has zero
initial conditions everywhere, as desired.
To save computational time the elliptic equation (3.29) or (3.48) can be
solved only for those wave frequencies for which the Fourier Transform of
the source terms used to generate the waves is significantly greater than
zero. Some wave components (especially the high frequency ones) receiving
negligible energy should not been considered and the corresponding equation
(3.29) or (3.48) not solved.
The model here presented has some features that make it suitable to be
used in early warning systems for the tsunamis. Firstly it can be applied
in two stages: one is extremely expensive from the computational point of
view, the other is very fast and can be applied in real-time. The idea is
first to define the computational domain and the areas/boundaries where
the waves are generated and to solve the elliptic equations using a unit value
of the source term. This part of the procedure may be computationally
expensive since the solution of hundreds of elliptic equations may require
hours or days. However after these computations have been performed and
the results at the points of interest have been saved, the actual computation
of the wave propagation is very fast. Once the time series of the waves to
be generated (‘direct’ approach) or the time series of the waves recorded at
some point of the domain by tidal gauges (‘indirect’ approach, as it would
be for early warning systems) are available, the source term can be quickly
calculated and the results at each point obtained by means of the inverse
Fourier transform. As known these computations, for the amount of data
commonly used for the problems studied here, take few seconds also on very
slow computers. Therefore when setting up an early warning system the big
part of the work is to define the scenarios (generation areas/boundaries),
and to perform the unit source terms computations. Then the results can
be used for the simulation in real time, forecasting the properties of the
waves at the desired points of the computational domain at an acceptable
computational cost. In the section 5.3 is presented an application of the
model which shows an example of tsunami forecasting in real time: after
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