Synthetic seismograms using the state‐space approach

We develop time‐domain state‐space models for lossless layered media which are described by the wave equation and boundary conditions. We develop state‐space models for two cases: (1) source and sensor at the surface, and (2) source and sensor in the first layer. Our models are for nonequal one‐way traveltimes; hence, they are more general than most existing models of layered media which are usually for layers of equal one‐way traveltimes. A notable exception to this is the work of Wuenschel (1960); however, most of the useful results even in his paper are developed only for the uniform traveltime case. Our state‐space model treat all of the equations that describe a layered‐media system together in the time domain. Earlier approaches (e.g., Wuenschel, 1960; Robinson, 1968) recursively connect adjacent layers by means of frequency‐domain relationships. We refer to our state equations as “causal functional equations.” They actually represent a new class of equations. Why are we interested in a different class of models for what appears to be a well‐studied system? As is well known, there is a vast literature associated with systems which are described by time‐domain state‐space models. Most recent results in estimation and identification theories, for example, require a state‐space model. These time‐domain techniques have proven very beneficial outside of the geophysics field and we feel should also be beneficial in the geophysics field. In fact, our ultimate objective is to apply those theories to the layered‐media problem; but, to do so, of course, requires state‐space models—hence, this paper.