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IUGG XXIV General Assembly July 2-13, 2007 Perugia, Italy
(S) - IASPEI - International Association of Seismology and Physics of the Earth's
Interior
JSS002 Poster presentation 1786
Restoration of Tsunami Incident by Remote Measurements of the Surface
Water Oscillations
Dr. Tatiana Voronina
JSS002 Inst.of Comput. Math.&Mathem. Geophysics SD RAS IASPEI
During the past few years, the tsunami events that occurred in the Pacific and the Indian Oceans had
caused to turn to on the inverse tsunami problem. The inversion problem to infer the initial sea
perturbation is considered as a usual ill-posed problem of the hydrodynamic inversion of tsunami tidegage
records. Mathematically, this problem is formulated as inverse problem of mathematical physics
for restoration of the initial water displacement in the source area by the water level oscillations
observed at a number of points distributed in the ocean. We assume that the time dependency of the
source function to be described by the Heavyside function . The forward problem, i.e. the calculation of
synthetic tide-gage records from the initial water elevation field, is based on a linear shallow-water
system of differential equations in the rectangular coordinates. This system is approximated by the
explicit-implicit finite difference scheme on a uniform rectangular grid, so the system of the linear
algebraic equations is obtained. The ill-posed inverse restoration problem is regularized by means of the
least square inversion using the truncated SVD approach. In this method, the inverse operator is
regularized with the help of its restriction on the subspace spanned on a finite sample of the first right
singular vectors. The so-called r-solution is a result of the numerical process. The quality of the solution
obtained is evaluated as relative errors (in $L_2$-norm) in restoration of the source function. The
results are fairly satisfactory, if the receivers have a good azimuthally coverage with respect to source
area. It is necessary to recognize that the results obtained strongly depend on the signal-to-noise ratio
due to the ill-posedness of the problem. As a model of initial water displacement we used displacement
representing the bottom deformation due to the typical tsunamigenic earthquakes with reverse dip-slip
or low-angle trust mechanisms. This paper represents the algorithm of the solution of the inverse
problem dealing with an arbitrary bottom topography having some basic morphological features typical
of the island arc regions with inner and outer boundaries. We have shown that to attain a reasonable
quality of the source restoration in this case we need, at least seven records distributed over the space
domain and their azimuthally coverage plays the key role in obtaining the satisfactory results of
inversion. The algorithm proposed enables us to evaluate the potential of a certain observation system
in restoring the initial tsunami waveform.
Keywords: ill posed inverse problem, numerical modelihng, regularization