Classical physics gives much attention to energy conservation and dissipation. Engineering filter theory gives much attention to causality--that there can be no response before the excitation. In geophysics we often need to ensure both causality and energy loss. We need to incorporate both, not only in theoretical derivations, but also in computations, and sometimes in computations that are discretized in time. There is a special class of mathematical functions called impedance functions that describe causal, linear disturbances in physical objects that dissipate energy.
Nature evolves forward in time. Naturally, impedance functions play a fundamental role in any modeling calculation where time evolves from past to future. Besides their use in physical modeling, impedances also find use in the depth extrapolation of waves. We geophysicists take data on the earth's surface and extrapolate downward to get information at depth. It is not the same as nature's extrapolation in time. In principle we don't require impedance functions to extrapolate in depth. But depth extrapolations made without impedance functions could exhibit growing oscillations, much like a physical system receiving energy from an external source. In fact, ``straightforward'' implementations of physical equations often exhibit unstable extrapolations. By formulating our extrapolation problems with impedance functions, we ensure stability. Of all the virtues a computational algorithm can have--stability, accuracy, clarity, generality, speed, modularity, etc.--the most important seems to be stability.
In this section we examine the theory of impedance functions,
their precise definition,
their computation in the world of discretized time,
and the rules for combining simple impedances
to get more complicated ones.
We will also examine other special functions, the
filter in their relation to the impedance filter.
Wide-angle wave extrapolation and migration in the time domain
will be formulated with impedance functions.
Rocks are unlike ``pure'' substances because they contain
irregularities at all scales.
A particularly simple impedance function will be found that
mimics the dissipation of energy in rocks,
unlike the classical equations of Newtonian viscoelasticity.