Wave Equation Modeling
The propagation of seismic waves in an inhomogeneous media is mathematically represented by the wave equation. This section describes the spatial dimensions of the model, the finite difference approximation, and the computational aspects of the seismic modeling problem.
The spatial derivative has three forms. In one dimension the wave equation is expressed in Equation 2.2. The media velocity C is a constant for homogeneous models and varies for inhomogeneous models. The two dimensional wave equation is Equation 2.3, and the three dimensional is Equation 2.4. The time derivative in two dimensions can be augmented to include the 2.5D amplitude corrections Liner 1991 as shown in Equation 2.5.
A significant new result is the 2.5D stability condition which exhibits a time dependent nature unlike the 2D and 3D stability conditions which are constant, Bording (1992). The applications of acoustic wave equations include exploding reflector modeling and reverse time migration, Bording (1994). Shot record modeling, exploding reflector modeling, and reverse time migration algorithms Bording (1993) and (1994) have been developed using these wave equations.
Given a source function, derivative approximations, and appropriate boundary conditions, it is possible to model waves. Figure 2.1 is an example of a propagating wave in a four layer model. The model is a rectangular box with the source placed at the center. The model velocity is different for each layer. The top layer has velocity of 2000 m/s, the next layer is 2400 m/s and the remaining layers alternate between 2000 and 2400. An example of the boundary condition method is presented in Section 2.5.
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Seismic Modeling and Imaging with the Complete Wave Equation
“Seismic modeling and imaging of the earth's subsurface are complex and difficult computational tasks. The authors present general numerical methods based on the complete wave equation for solving these important seismic exploration problems.”