Derive the Wave Equation
The two most important physical laws underlying the wave equation are Newton’s second law of motion and Hooke’s Law. Use these to derive the wave equation for sound waves in a thin rod (1D Wave Equation).
Derivation of Derivatives, Finite differencing of the Wave Equation. Second derivatives are the essence of the wave equation. In order to compute these second derivatives we need to relate the data to the derivatives. The Taylor Series does this for us. Apply the Taylor Series and develop the expressions for f(h) and f(−h) to compute the f″(0).
The rule of thumb for avoiding grid dispersion using second order methods is to use 10 grid points per wavelength.
A nifty idea for suppressing multiples in reverse-time migration is to make the reflectors transparent. The variable density wave equation is used because the density can be manipulated to make the reflection coefficient zero or very small.
For those more mathematically inclined, the iteration matrix represented by the finite difference operator has eigenvalues. Violation of the stability criteria causes an eigen-value greater than one, and any number greater than one raised to a large power quickly goes to infinity.
Model Definition Sketch in the box below a rather simple geological model. Assume you are going to generate a set of synthetic shot records. Place the model dimensions in the upper right hand corner and the lower left hand corner. Pencil in the interval velocity values for each layer.
Figures & Tables
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.”