Let's consider the 1-D elastic system shown in Figure ,
which can either represent a discrete system or the discrete representation of
a continuous system. Differently from the system in Cunha (1991), this
system represents the masses and springs as a single entity.
In the case of a discrete system, the equation of motion for point *i* is

(1) |

(2) |

Figure 1

On the other hand, if we consider the system as continuous, the continuous equation to be discretized is

(3) |

Using equation (1) as the basis for the discretization
process offers some advantages over the usual methods.
The differential operator can be applied in a single
stage (without the intermediate evaluation of strain components),
allowing for the design of a more efficient algorithm.
In addition, the differentiation of the model parameters is completely
separated from the differentiation of the wavefield components, and
since the model parameters do not change with time their derivatives
need to be computed only once. This independence of the derivatives of the
elastic constants relative to the wavefield has an extra advantage when
a *blocky model* (one with sharp boundaries) is to be modeled.
In such models, a short two-point operator can be used for the elastic
derivatives since *large operators will not give the right answer at
the boundaries because blocky models are aliased*.
Finally, the operation can be implemented in a non-staggered grid, since
it is executed in one stage and all derivatives are centered.

12/18/1997