1. Accurate implementation of the source wavelet in finite-difference modeling
Fang Liu and Arthur Weglein
Houston, Texas
May 12th, 2006
Annual report:
page

2. Why ?
Many seismic modeling procedures emphasis the travel time of the events, rather than the full wave-form.
Information beyond correct travel-time, especially accurate implement of the source wavelet.
Inverse scattering series preferred actual wave-fields.

3. Outline Introduction Source implementation
A very simple quality procedure
Physical meaning of discrete model
Issues of free surface
Conclusion

4. Introduction Forward modeling problem: differential equation
Velocity model
Source wavelet
Very well-known unsolvable problem with analytic solution

5. Finite difference approximation partial derivative over time
Previous time
Current time
Future time
(Second order in time)

6. Finite difference approximation spatial partial derivatives
(Fourth order in space)

7. Finite difference approximation of LHS of wave-equation
How about the RHS ?

8. Source implementation
Issues of the source:
A source invisible to the grids
Grid points

9. Issues of the source term
A source visible to the grids
Grid points
But the wave-field at the source point will be singular

10. Convert the source problem to an initial problem
Initial problem without source
The above 2 problem will produce the same response because the first one will evolve into the second.
After certain time, the source become very weak and hence negligible.
But the second problem is much easier because it contains no source. And the source is implemented by a distribution of grids rather than in the single source point, source can be flexibly put between grids.

11. How the source is implemented
wave-fields
at previous
time
wave-fields
at future
time
current
wave-fields

12. Advantages The new problem is much simpler to solve.
Source can be flexibly put on the grids or in between.
Singular source is implemented by a regular distribution of points.

13. Procedures to propagate forward in time
Previous step
Next step (future time)
Current step (current time)

14. Modeling parameters
We want the digitization interval the smaller the better.
But computer set the lower-limit for them.
Lower limit of the digitization interval constrains the frequency content of the source wavelet.

15. A simple quality control procedure For the modeling parameters
Construct a model with the same set of velocity contrast but no lateral variation
Run forward modeling with point source
Summing all receivers together
Compare with analytic solution
1500(m/s)
1500(m/s)
2000(m/s)
2000(m/s)
1500(m/s)
1500(m/s)

16. Summing over receivers
Define:
Summing all receivers
We have an equivalent 1D wave-equation:

17. Source issue for a 3rd party program
Geological model
The receiver line
(depth=25 m)
Source depth=0m
The first reflector (depth = 497.5m)
The second reflector
(depth = 897.5m)

18. It was given maximal freedom to decide parameters
Model designed very big to minimize boundary artifacts
We only specify :
Give the standard package the maximal freedom to decide
Velocity contrast set to be small to minimize other issues

19. Sample shot gather and the sum

20. What’s the problem ? Waveform of each event: 1500(m/s) 2000(m/s)
Positive and negative half should be symmetrical
1500(m/s)
Size of
first
Peak
Size of
second
Peak
Events
Difference
Direct wave
30.274
0.93 %
First
primary
-3.719
4.941
32.86 %
Second
primary
35.59 %
3.574
-4.846

21. Our results for the same parameters
First
primary
-7420
7414
0.08 %
Second
primary
0.05 %
7414
-4.846

22. Example 1 : geological model and modeling parameters
Source and receivers line (depth=0m)
997.5m
1497.5m

23. Example 1 : pre-calculated wave-fields

24. Example 1 : Summing of all receivers
Sum from finite-difference data
Analytic integral from equation (14)

25. First experiment with Ricker wavelet (without free-surface)
(1) Pre-calculated wave field
0 m
1500 (m/s)
997.5 m
R1 =
3200 (m/s) m
R2’ =
6100 (m/s)
Field pre

26. My own experiment with Ricker wavelet
Recovering zero frequency by integrating twice
Problem only for
large time

27. Physical meaning of discrete model
Velocity model within computer memory
Where is the reflector?
Interpretation above is very important for free-surface implementation.

28. Analytic using z1=997.5 (m)
Sum from finite-difference data
Analytic using z1=995 (m)
Analytic using z1=1000 (m)

29. Experiment with Ricker wavelet
with free-surface
Location of free-surface is critical because it determine the location of the image source, and affect the pre-calculated wave-field 0: 1:
Δx=Δz=5
free-surface (2.5 m) 2: 3:
79:
water bottom (397.5 m)
Water-bottom reflection coef:

30. Ricker wavelet with dominant freq=17 HZ
Correct free-surface (level=2.5m)

31. Ricker wavelet with dominant freq=17 HZ Summing all traces together
Still a little high-frequency noise
But better than 17 hz

32. Ricker wavelet with dominant freq=17 HZ
Amplitude check
Primary+ghost
5.57(cm)
2nd multiple+ghost
0.7 (cm)
1st multiple+ghost
-2 (cm)

33. In the plane-wave world, amplitude of each events have very simple relations
Image source
Free-surface
Source
Water-bottom
Amplitude (Primary)
=-R
Amplitude (1st free-surface multiple)
Amplitude (1st free-surface multiple)
=-R
Amplitude (2nd free-surface multiple)

34. Ricker wavelet with dominant freq=17 HZ Wrong free-surface (level=0m)
Shot gather looks very similar as before

35. Ricker wavelet with dominant freq=17 HZ Summing all traces together
But the result as summing all traces together is totally unreasonable

36. Ricker wavelet with dominant freq=15 HZ

37. Ricker wavelet with dominant freq=17 HZ Summing all traces together
High-frequency noise a little weaker

38. Conclusions Accurate implementation of the source wavelet.
Simple quality-control for modeling parameters.
Physical meaning of the digitized model.
Implementation of the free-surface.

39. Acknowledgements M-OSRP members M-OSRP sponsors
NSF-CMG award DMS
DOE Basic Sciences award DE-FG02-05ER15697