ON DIFFERENTIABLE MAPPINGS ACCOMPANYING KINEMATIC MODELS OF EARTHQUAKE SOURCE. 2. ON SEISMIC SOURCE WITH ARBITRARY EXCITATION TIME
Illustration of general theory (part 1) that is known as Sato-Hirasawa model of the earthquake source is considered. It is shown that the model is too simple for described mathematical scheme, "parametrically poor". One of choices of the generalization is replacing Heaviside unit step function with a smooth distribution in narrow zone near the border, meet with mathematical difficulties of analytical calculations and computer-based evaluation of double integrals in expression for the displacement. An alternative turns out mathematically more realistic, it is to leave a spatial border sharply delineated but assume that the process continues for some time yet, this time interval is not necessarily negligible small. The model with an arbitrary excitation time retains characteristic difficulties with piecewise smooth seismogram. It implies that peak acceleration, the most essential parameter for purposes of seismic risk zoning, is not determined correctly. But on the other hand, here we have a fortunate exception in sense that it is possible to calculate displacement at a receiving point in analytical form, as a known function of model parameters. Then a new possibility is being emerged, to see to what model "well-quality" or "poor-quality", we will thus come. Operator of time integration within limits of impulse duration with a power-type weight factor converts the seismogram to a known analytical function of the parameters. This is, in the present case a complete, in sense of depending on the parameters, "recipe" of functional of "observable" proves at our disposal. Varying index of power for weight factors has got the system of equations with respect to parameters, the system analytically assignes a mapping of the set of the parameters to a set of "observables". Corresponding "map of mappings" visually reveals situations of instability in accompanying pattern of situations associated with model parameters restoration. The results can be used in computer-based processing system of seismic records and as a general mathematical approach to the categorization of models of earthquake source.