2 The Modeling Domain and Wave Equations

The acoustic wave equation in two dimensions relates the spatial derivatives to the time derivative. Using the acoustic approximation the wave equation is derived in Section 4. The dimensionality of the equation requires a media velocity , the sound speed. In the two dimensional model of Figure 1 the surface coordinate is and the depth coordinate is with the positive axis down, a tradition in the oil industry. This half space is defined for and for a bounded .

The pressure wave field is and the seismic source is . The inhomogeneous constant density two dimensional wave equation is

This equation is hyperbolic and the inhomogeneity is due to the variable model velocities. The seismic source is applied at or near the surface. Sources are sometimes buried to improve the signal strength and to reduce noise generated by the near surface. Receivers are placed in a line parallel to the expected principal dip of the subsurface. This is done to reduce the energy scattering by out of plane reflectors.

The rest of this chapter is organized in the following manner:

• The first order wave equation is presented.
• The wave equation is developed from first principles.
• Model results for a one dimensional problem are used to illustrate the reflection coefficient.
• A matrix formulation for computing difference weights is reviewed.
• The seismic source is presented as a numerical method to produce realistic seismic waves with a finite duration in time and limited in bandwidth.
• A geological model is defined and two methods are presented which reduce unwanted reflections from the model boundaries.
• The exploding reflector model is introduced as a method to generate a synthetic seismogram.
• The ``reverse'' time imaging concept is presented to process the synthetic seismic data.
• The seismic modeling and imaging codes are introduced with operating instructions and several example models.

The seismic data are too complex to visualize as numbers so visualization tools are provided to help in understanding the modeling and imaging process.