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.