Generalities on tsunami forecast and modeling

Tsunami waves, like most waves, are characterized by three phases: generation, propagation and effects in the arrival area. A unified theory able to incorporate all these three phases to equivalent accuracy and under identical assumption would be highly desirable. Unfortunately this is not the current case: tsunami theories approximate or eliminate some aspects of the problem, hopefully unimportant, to reach some computable results. As a matter of fact during tsunami phenomena some previous unimportant quantities become important and vice-versa. Present day theories dictate separate formulation for tsunami generation, propagation and disposition. Many different approaches had been used to envisage this problem. Piston-like approach to the tsunami generation problem ( based on Laplace equation; see for instance HONDA1951 and many others), though elucidating tsunami general properties, like dispersion and geometrical attenuation, could obtain virtually any tsunami motion by selecting initial water pile of suitable shape. This indetermination in the choice of water pile parameters is worsted by the still scarce knowledge of the earthquakes physical mechanisms; so tsunami theoretical work relies on 'ad-hoc' parametrization. From the '60, kinematics description of earthquakes had been used. The dislocation model and its extensions consider parameters like strike, dip, length and width of the fault plane, slip direction, rupture velocity etc. Faulting parameters are very difficult to be estimated within this description, as they do not depend linearly from the seismogram they generate. During the same period BURRIDGE1964 developed a model based on an equivalent body force. In the equivalent force model, each multi-pole term of the force expansion depends linearly on the total seismic field, and hence on the tsunami field. Using a simple model, namely a point force derived from the divergence of a symmetric, second order moment tensor, only six parameters are needed to fit reasonably well the seismic radiation from earthquakes at regional far field distance WARD1980, WARD1981, WARD1982. In the local approach to tsunami wave modeling, an elastic half space overlaid by a water layer COMER-TG can also be used, without loss of generality instead of a spherical non-rotating Earth model ( S. Ward, cit.). Elastic bulk parameters and local gravitational field must be supplied. Although real earthquakes have finite physical dimensions and at least a line source approach should be used, in first approximation a point source having same depth, moment and mechanism gives roughly the maximum amplitude expected to the one using a line source model. A peculiar feature in tsunami radiation patterns resulting from line source descriptions, which point source approximations are not able to recover, is beaming; the direction of which is mainly perpendicular to long and thin earthquake faults, regardless of the focal mechanism characteristics. In the tsunami generation problem the point source approach puts into light four different main features, depending on the excited wave periods. The values of the upper and lower limits of the period and the frequency range between two periods depend upon the earth model chosen:

1. a long period region that deviates from (gh)$^{1/2}$ behavior, g being the gravity acceleration and h the water depth, where earth rotation and ellipticity become important, but where earthquakes are not very efficient in generating tsunami.

2. a middle period region, where shallow water approximation can be applied and tsunamis are non dispersive and propagate at velocity near (gh)$^{1/2}$.

3. a transition zone period, where tsunamis exhibit a combination of deep water and shallow water wave characteristics. This range of periods is strongly excited by submarine earthquakes.

4. a short period region in which propagation is independent of water depth, but poorly excited by earthquakes.

The propagation characteristics of each normal mode are strongly dependent upon the media in the volume in which the energy is mostly concentrated, and for tsunamis this media is the ocean. The excitation characteristics of each mode are on the contrary determined by the amplitude of the eigenfunction in the source volume, which for tsunamis is the solid earth. In fact, sub-oceanic structures, while probably having little influence on tsunami propagation, could have great influence on tsunami generation. To give a rough idea, WARD1980 suggests that ``the variation in tsunami excitation, solely due to the Earth structure may be around 20% even in similar Earth models``.