Time Scales: Towards Extending the Finite Difference Technique for Non-homogeneous Grids

Abstract

Tremendous progress in seismology over last years is greatly due to availability of high quality seismic waveforms. Their availability prompts the new mathematical and numerical algorithms for their more detailed analysis. This analysis usually takes a form of the inverse problems—an estimation of physical parameters from seismic waveforms called the full waveform inversion (FWI). No matter which inversion algorithm is used, the FWI technique requires precise modeling of synthetic seismograms for a given lithological model. This is by no means a trivial task from the algorithmic point of view, as it requires solving (usually numerically) the wave equation describing propagation of seismic waves in complex 3D media, taking into account such effects as spatial heterogeneities of media properties, anisotropy, and energy attenuation, to name a few. Although many numerical algorithms have been developed to handle this task, there is still a need for further development as there is no single universal approach equally good for all tasks in hand. In this chapter, the possibility of using the Time Scale Calculus formalism to advance the synthetic seismograms calculation is discussed. This modern approach developed the late 1990s with the aim of unifying analytical and numerical calculations provides the very promising basement for developing new computational methods for seismological, or more general geophysical applications. In this chapter we review the basic elements of the Time Scale Calculus keeping in mind its application in seismology but also we extend the initial concept of Hilger’s derivative towards the backward-type and central-type derivatives using the unified approach and compare their properties for various time scales. Using these results we define the second order differential operators (laplacians) and provide explicit formulas for different time scales. Finally, the formalism of time scales is used for solving 1D linear, acoustic wave equation for a velocity model with large velocity discontinuities. Based on this simple example we demonstrate that even in such a simple case using an extension of the classical finite difference schemata towards irregular grid leads to a significant improvement of computational efficiency.