1. Adriana Citlali Ramírez
Progressing the analysis of the phase and amplitude prediction properties of the inverse scattering internal multiple attenuation algorithm
Adriana Citlali Ramírez
Arthur B. Weglein
Dept. of Physics
University of Houston
April 2005

2. Plan for this talk
Extend an earlier analysis of the inverse scattering internal multiple attenuator done by Weglein et al. (2003).
Determine the efficacy of the algorithm to attenuate 1st order internal multiples without knowing or determining the velocity or any properties inside the medium.
Using an analytic data example
Determine the efficacy of the attenuator towards its stated objective, i.e. precisely how effective is this temporal prediction and amplitude reduction process

3. Background
There are many methods that have been developed to deal with internal multiples that are often effective.
Procedures that assume the earth is 1D, or that a reasonable velocity estimate is obtainable can fail in deep water and/or in a complex multidimensional earth.
The inverse scattering series processing methods can accommodate a complex multi-dimensional earth, without requiring any subsurface information
See e.g. Verschuur (1991), Berkhout et al (1997), Weglein (1999) and Weglein and Dragoset (2005)
Removing multiples is not a new idea, and}
; those distinct algorithms for free surface and internal multiples were the first comprehensive methods to address that challenge, and remain the high-water mark of effectiveness.

4. Provide Prerequisites:
Inverse Scattering series:
Provide Prerequisites:
Wavelet estimation
Deghosting
Data extrapolation
Z. Guo, PhD Thesis, 2005.
Z. Guo, Weglein, T.H. Tan, Annual Report: 2005
See also references by Weglein & Secrest, H. Tan, A. Osen, L.Amundsen.
Provide prerequisites, e.g., wavelet, deghosting, data extrapolation, at least as complete as the methods they are meant to serve
J. Zhang and Weglein, Annual Report: 2005.
F. Miranda, PhD Thesis, 2004.

5. Identify “Task-specific” subseries: Free-surface multiple removal.
How do we solve the Inverse Scattering series?
Identify “Task-specific” subseries:
Free-surface multiple removal.
P. Carvalho, PhD Thesis, 1992.
Internal multiple removal.
F. Araújo, PhD Thesis, 1994.
K. Matson, PhD Thesis, 1997.
B. Nita and Weglein, Annual Report, 2005.
Weglein et al., Geophysics, 1997.
Provide prerequisites, e.g., wavelet, deghosting, data extrapolation, at least as complete as the methods they are meant to serve
Imaging in depth.
Parameter estimation.
S. Shaw, PhD Thesis 2005.
F. Liu et al., Annual Report, 2005.
H. Zhang and Weglein, Annual Report, 2005.
Weglein et al., Inverse Problems (2003).

6. Internal multiple attenuation algorithm – sub-events
1st Order internal multiple = 2 upward and 1 downward reflection
Primary = one upward reflection

7. Objectives Detemine the effectiveness of the attenuation algorithm.
Progress the insight and understanding of the internal multiple attenuator for 1st order internal multiples.
Present the results of an earlier analysis in Weglein et al.(2003).
Extend the analysis to:
1D model with two interfaces
Three interfaces
An arbitrary number of interfaces
In the next section we first present the results of an earlier analysis in Weglein et
al (2003)[17], in a 1D model with two interfaces, to determine the effectiveness
of the attenuation algorithm. We then progress the insight and understanding of
the internal multiple attenuator by extending the analysis to first three and then
an arbitrary number of interfaces. These analytic examples further illustrate
the ability of the inverse scattering algorithm to attenuate first order internal
multiples without knowing or determining the velocity or any properties inside
the medium.
See Weglein et al, Topical Review: Inverse scattering series and seismic exploration, Inverse Problems (2003).

8. Objectives
For a 1D earth and a normal incidence wave the internal multiple attenuator is:
In the next section we first present the results of an earlier analysis in Weglein et
al (2003)[17], in a 1D model with two interfaces, to determine the effectiveness
of the attenuation algorithm. We then progress the insight and understanding of
the internal multiple attenuator by extending the analysis to first three and then
an arbitrary number of interfaces. These analytic examples further illustrate
the ability of the inverse scattering algorithm to attenuate first order internal
multiples without knowing or determining the velocity or any properties inside
the medium.
See Weglein et al, Topical Review: Inverse scattering series and seismic exploration, Inverse Problems (2003).

9. 1
b1(k)

10. 1st Analytic Example
The data used in this example consisted of two primaries created by a normal incident spike wave :
See Weglein et al, Topical Review: Inverse scattering series and seismic exploration, Inverse Problems (2003).

11. The substitution of b1(z) into:
gives the following prediction of a first order internal multiple:

12. True Internal Multiple:
b3(t) has the precise time and an attenuated amplitude by a factor of :

13. Analytic example with 3 interfaces
The data used in this example consists of three primaries created by a normal incident spike wave :

14. We substitute b1(z) into:

17. attenuated amplitude by a factor of :

18. Analytic example with n interfaces and layer velocities of cn
Definitions:
Reflection data from a normal incident spike wave
We obtain the result:

19. Attenuation factor of the predicted internal multiple=
j=1
j=2
T01R2T10 x R1 x
T01R2T10 =
T01T T01R2R1R2T10 *
True amplitude

20. Attenuation factor of the predicted internal multiple=
j=1
j=2
j=3
T01T12R3T21T10 x R1 x
T01T12R3T21T10 =
T01T T01T12R3T21R1T12R3T21T10 *
True amplitude

21. Attenuation factor of the predicted internal multiple=
j=1
j=2
j=3
T01T12R3T21T10 x
T01R1T10 x
T01T12R3T21T10 =
T12T21*(T01T10)2* T01T12R3T21R1T12R3T21T10
True amplitude

22. Conclusions The algorithm predicts the correct arrival time.
The difference between complete elimination and the attenuation, that the algorithm provides, resides in the attenuator having extra powers of transmission coefficients for all the interfaces down to and including the depth of the shallowest downward reflection.
The predicted amplitude differs from the actual in a precise and well understood manner, and the residual multiple after attenuation is always with the same sign as the original.