The basis for discretization

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  

$$\Delta x^2 \rho_i {\partial^2 u_i \over \partial t^2} = \hat... ... - (\hat{Y}_{i-1} + \hat{Y}_{i+1}) u_i + \hat{Y}_{i+1} u_{i+1},$$ (1)

where

$$\begin{eqnarray} \hat{Y}_{i-1} & = & 2 { Y_{i-1} Y_i \over Y_{i-1} + Y_i} \nonumber\\ \hat{Y}_{i+1} & = & 2 { Y_{i+1} Y_i \over Y_{i+1} + Y_i}\end{eqnarray}$$ (2)

are the equivalent Young moduli that control the interaction between adjacent units of the system.

 

spring2
Figure 1 A mass-spring model that can represent either a discrete physical system or the discretization of a continuous elastic system.

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

$$\rho {\partial^2 u \over \partial t^2} = {\partial Y {\parti... ...tial u \over \partial x} + Y {\partial^2 u \over \partial x^2}.$$ (3)

Two important related issues come up in the discretization of (3): the definition of the appropriate Y to use in each of the terms of the equation, and the choice of the differential star for the first-order derivative. If we choose the more precise (-1,1) star, the two terms of equation (3) will not be centered at the same point, while the star (-.5,0,.5) will not have the same resolution. Looking into the physics of the process gives a clue for the correct choice of Y; it is the one that is actually involved in the interaction between adjacent points (the equivalent Young modulus). It turns out that this choice also solves the problem of the differential star. Using this criterion, the discretization of (3) leads to which is identical to equation (1).

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.