Numerical modeling of waves for a tsunami early warning system

Numerical modeling of waves for a tsunami early warning system
obtained by multiplying N ′ for the Fourier Transform of the unknown source
term, indicated as S(ω):
N (x, y, ω) =S(ω)N ′ (x, y, ω) . (3.49)
Let assume that at one point P of the computational domain the elevation
of the surface ηP is available, i.e. the free water surface registration. Then at
that point the transformed variable NP can be easily calculated. Equation
(3.49) can be inverted to obtain the source term S(ω):
S(ω) = NP (ω)
N ′ P (ω)
(3.50)
where N ′ P is the result of the unit source term computations at the point
P. Of course this procedure is easy to apply using only the surface elevation at
one point and when an identical source term, which in principle is a complex
number, applies to all the generation areas/boundaries. This implies for
example that the waves are generated with the same height and phase.
If the records at more than one point are available, two alternative uses
can be made of the data. On the one hand it can be assumed that the source
term is identical for all the generation areas/boundaries, and an optimization
procedure can be used to find the value that best fits the data. On the other
hand it can be assumed that each of the generation area/boundary has its
own value of the source term and it is possible to write a linear system to
be solved for these unknown source terms. Alternatively an over-determined
system (the number of records available is greater than the number of source
terms to be found) can be solved by means of an optimization procedure.
A further practical point of interest is that the discrete Fourier Transform
is used and a finite set of equations is obtained, representing a finite time
interval in the frequency domain. The integral transform of the data used
to generate the waves (time series of ηP , h, u I ) is carried out using the Fast
Fourier Transform. Each of the resulting field equations in the frequency
domain (3.29) or (3.48) is solved using an available mild slope equation solver
based on the finite element method (for details see Beltrami et al., 2001;
Bellotti et al., 2003). The inverse transform of N is finally carried out using
the Inverse Fast Fourier Transform.
As far as the length of the time interval to be considered is concerned
it should be kept in mind that when solving partial differential equations
using the discrete Fourier Transform the solution is obtained for a finite time
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