Finite Difference Solution to the
Acoustic Wave Equation
Figure 1. Seismograms and snapshot associated with a line source placed
just above the 2-layer medium.
Objective: Calculate synthetic seismograms by a 2-2 FD solution to the
wave equation. Learn how space and time sampling increments affect the output.
Skill Learned: Implementation, design and construction of synthetic
seismograms by finite-difference solutions to the wave equation.
- Download the codes fd.m and plt.m and type
"fd". This will generate the synthetic seismograms and snapshots
for a 2-layer medium.
- From the snapshots estimate
the wavelengths of the waves in the top and bottom layers (Use the zoom
facility in MATLAB). Compare these wavelengths to the theoretical
estimates from lambda=c/f..
- Change ABCs so they are turned on for
all sides except for the top boundary..make the top a free surface. Repeat
simulations. Describe the changes in snapshots compared to previous
simulation. Fix the leftside ABC so it absorbs by changing a sign in the
ABC equation for the left side boundary
(Clayton, R. and Enquist, B., 1977, Absorbing Boundary Conditions for Acoustic and Elastic Wave Equations, Bull. Seis. Soc. Am., 67:6, 1529--1540 ) .
- Identify the direct, refraction, and reflection waves in the snapshots and
the seismograms. Estimate their apparent velocities. Explain why the
reflection waves moveout with the slowest apparent velocities.
- Raise the frequency of the
source by 50% and repeat the simulations. Comment on loss of accuracy and
- Choose a dt value that
violates stability. Rerun simulations. What happens?
- The head
wave arrival is extremely weak, as predicted by theory. However, a
diving wave that gets trapped in a thin interface just below the
refracting interface can boost up the amplitude. Adjust the velocity model
so there is a thin layer (10 points thick) just below the original
interface. make sure the velocity is the average between layer 1 and 2.
Now rerun the fd.m code. Is there a difference in head wave amplitudes?
Change models until you get a satisfactory amplitude.