Two basic observables in the wave equation are the wave field (eg. stress, displacement) and the medium properties (stiffnesses, velocities). If the frequency content of the wave field and the material properties is similar, the same type of derivative operators can be used for both. Then (4) should be used rather than (6). The necessary condition is that model and wave field roughness are of the same scale. For example, if the medium properties vary smoothly in space then a fairly long derivative operator can be adequate to approximate numerically the analytical derivative. However, if the medium is varying very rapidly, the same derivative operator may not be adequate. As an extreme example, realistic modeling wave propagation in a random medium is affected by the length of the finite difference operator. A long operator implicitly averages medium properties. To describe local effects a much shorter operator would be necessary, but for the same prescribed spacing, that operator gives a less accurate result. A layered model is an extreme case. The spatial variability in the horizontal direction is zero, the spatial variability in depth occurs in discontinuous steps. Applying equation (4) with conventional finite difference operators to such a model would implicitly assume a continuous and band limited medium. Equation (6) allows to ``separate'' the medium from the wave field. We can adapt the derivative operators to the properties of the observables they are operating on.

11/17/1997