Translations:Layer-cake model/4/en

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The main purpose of a seismic signal model is to explain the seismic-wave propagation phenomenon. The most valuable models are three dimensional. Such models tend to be numerical because the mathematics of a theoretical 3D model is much too involved to produce closed-form solutions except in the simplest cases. The most pronounced variations in the earth layering are usually along the vertical direction, so a 1D vertical model often is adequate. The foremost 1D model — the so-called stratified, or layered-earth, or layer-cake model — is mathematically identical to the lattice model for electric transmission lines. The model is also mathematically identical both to the acoustic tube model used in speech processing and to the thin-film model used in optics. In such a 1D model, the earth is sliced mathematically into many thin horizontal layers that are normal to the vertical z direction. Such a theoretical division of the earth into thin layers produces a stratified medium characterized by the interfaces between the layers. Thus, a vector of the discrete amplitudes of the signal thus can represent a digital signal. One vector results if the amplitude of the signal is measured by a geophone in terms of particle velocity. Another vector results if the amplitude of the signal is measured by a hydrophone in terms of pressure. The inner (or dot) product of these two vectors gives the energy of the signal. In the basic model treated here, there is no allowance for dissipation of kinetic energy into heat. Thus, all the source energy imparted into the body can be accounted for, over time, in terms of the resulting elastic wave motion.