1. AVO inversion of multi-component data for P and S impedance
Faranak Mahmoudian
Gary F. Margrave

2. Outline Weighted stack PP only, PS only, Joint inversion
Low-frequency restoration
Synthetics examples
Real log example
Inversion of noisy data
We will see what the weighted stack is.
Also the application of weighted stack for PP only, PS only, Joint inversion.
We will discus why and how we include the low-frequency trend to the inversion results.
Then we will look at some synthetics and real log examples.
and at the end we will see how inversion methods perform in noisy data.

3. PP inversion Smith and Gidlow (1987) A weighted stack!
Based on the Aki-Richards approximation for Rpp
Smith and Gidlow (1987) showed that Aki and Richards approximation for Rpp, can be inverted (by least-square inversion of PP data) to get the Δα/α or Δβ/β in the form of a weighted stacking scheme. That Wpp is a function of incident P angle and velocity model not the data itself.
A weighted stack!
α: P-wave velocity, β: S-wave velocity, θ:P-wave incident angle Weights are functions of θ, and velocity model not the data itself

4. Joint PP and PS Inversion
Stewart (1990)
Based on the Aki-Richards approximation for Rpp and Rps
In 1990 Stewart derived the extension of the Smith-Gidlow procedure incorporating both PP and PS data. He derived a sort of this equation. There are two sets of weights, Wpp and Wps that are function of incident P and reflected S angles and velocity model
θ: P-wave incident angle, φ: S-wave reflected angle
I: P-wave impedance, J: S-wave impedance

5. First application of Joint Inversion Larsen and Margrave (1998)
Compressional Impedance, Bottom of Glauconitic channel
The first application of joint inversion to real data, was ***** by Larson and Margrave (1998). This figure shows a the time slice of P-wave impedance at the bottom of Glauconitic channel. They showed there is better a continuity here in joint inversion that resolves channel better than PP inversion only.
3 km
P-P (1C)
Joint P-P and P-S (3C)

6. Implementation Converting PP section (CIP gather) to depth
Correlating PP and PS sections
Converting PS section
Velocity model
Ray-tracing
To get angles
Calculate the weights
+ Low-frequency trend
Weighted stack
I and J
We wrote a code in MATLAB, that is designed to have a PP and PS synthetic as well as velocity model as input.
additionally PP data or PS data alone can be used as input , resulting in a PP inversion only or PS inversion only.
This figure shows a schematic algorithm of code. With a smoothed background velocity, we ray traced, to get the angles required to calculate the weights.
The combination of PP and PS data in an stack requires that two data types be correlated in time or depth. So we converted PP and PS data to depth.
Once we did that the weighted stack can be done. The inversion result is DelI and DelJ. And at last by including low-freq trend to to Del I or Delj, P and S impedance is calculated.

7. Why Low-frequency restoration?
Missing low-freq trend is a common characteristic of seismic data
Band-limited sources
Band-pass filters in processing
Lindseth (1979) :
Inversion of band-limited data leads to band-limited impedances.
low-freq content contains much of the essential velocity information.
But why the low-frequency is important? Missing low-freq trend is a common characteristic of seismic data that is mostly due to Band-limited sources also applying Band-pass filters during the processing. The low-frequency restoration is an old idea by Lindseth (1979). He stated that Inversion of band-limited data leads to band-limited impedances. While low-freq content contains much of the essential velocity information.

8. After Lindseth (1979). 6-250 0-5 Hertz 0-250 time velocity
sec
In this figure the importance of low-frequency component, is shown clearly. This trace shows a velocity log with a freq band width of hz. as we can see, if we take out the 0-5 components, the rest including components, doesn’t have the gross velocity information.
0-250
6-250
0-5 Hertz
After Lindseth (1979).

9. Blimp combines seismic and well log in frequency domain.
Low-Frequency restoration
Band limited Impedance, BLIMP
Ferguson and Margrave (1996)
Impedance with low-freq.
Inversion
Anyway, we restored the low-freq trend to inversion result, using function, BLIMP, band-limited impedance, found in CREWES MATLAB library.
how Blimp works? Blimp estimates impedance (I) from a reflectivity (DelI/I), using a well-log to provide the low-frequency components.
Blimp combines seismic and well log in frequency domain.
Low frequencies from well logs + higher frequencies from seismic  broadband response.

10. SYNTH ( MATLAB CREWES library)
Synthetics example 1
Velocity model
---- P-wave velocity
---- S-wave velocity
---- Density Vs ρ Vp
Several synthetics models were used to test the inversion program. We present the inversion results of two sample models here.
This is a simple velocity model. For a depth range 600 to Red curve is P-wave velocity, green curve is S-wave velocity and blue curve is density log. We used SYNTH software (CREWES MATLAB library).to generate PP and PS section in time.
SYNTH ( MATLAB CREWES library)

11. Initial Ricker-100 wavelet With few low-frequency component
Correlating PP and PS section
Both PP and PS synthetics have the same initial zero-phase (100 HZ ricker) wavelet and the same offset range from 0 to 1000 m. Here is PP and PS data, which converted to depth. they have primaries only (no multiples), does not include transmission losses or spherical spreading. but NMO is removed.
Initial Ricker-100 wavelet
With few low-frequency component

12. P-wave reflectivity Red curve: Reflectivity after low-freq restoration
This figure shows the ΔI/I as estimated from PP, PS only and joint inversion. The blue curve is true ΔI/I calculated directly from the log model and red curve is estimated ΔI/I. as we can see the depth of each reflector as well as the magnitude of reflectivity, is adequately determined.
Red curve: Reflectivity after low-freq restoration
Blue curve: True reflectivity from logs

13. S-wave reflectivity Red curve: Reflectivity after low-freq restoration
Also this is the ΔJ/J as estimated from PP, PS only and joint inversion.
Now we are going to show you the magnificent impedance estimation.
Red curve: Reflectivity after low-freq restoration
Blue curve: True reflectivity from logs

14. P-wave Impedance, I
Here is the estimated P-wave impedance results.
Black curve : Impedance before low-freq restoration
Red curve: Impedance after low-freq restoration
Blue curve: True Impedance from log
This figure is a very good example showing significant impedance improvement by low-freq restoration.
Black curve : Impedance before low-freq restoration
Red curve: Impedance after low-freq restoration
Blue curve: True Impedance from log

15. S-wave Impedance, J
Again the S-wave impedance. And red curve shows a very significant impedance improvement by low-freq restoration.
Black curve : Impedance before low-freq restoration
Red curve: Impedance after low-freq restoration
Blue curve: True Impedance from log

16. Synthetics example 2 Velocity modelVs ρ Vp
---- P-wave velocity
---- S-wave velocity
---- Density
This is the second example that we will examine the inversion result for. There is a contrast about 15 meters here.
SYNTH ( MATLAB CREWES library),

17. Zero-phase wavelet 5-10-80-100 With ample low-frequency
PP and PS synthetics section
For the above model, the initial wavelet for generating the PP and PS synthetics, is a zero-phase wavelet with the band-width of
Zero-phase wavelet
With ample low-frequency

18. P-wave Impedance, I
Here is the estimated P-wave impedance from all three methods. The black curve is impedance before low-frequency restoration and red curve is impedance after low-frequency restoration. And blue curve is true impedance.
Although in this example the synthetics had already ample low-frequency content. However the low-frequency restoration has improved the impedance estimation.
Black curve : Impedance before low-freq restoration
Red curve: Impedance after low-freq restoration
Blue curve: True Impedance from log

19. Real log from Blackfoot field, owned and operated by Encana, south-eastern Alberta, Canada
Real-log example
Velocity model
---- P-wave velocity
---- S-wave velocity
---- Density Vs ρ Vp
As the last example the inversion results for this real log model is examined. The example comes from Blackfoot Field, owned and operated by Encana, in south-eastern Alberta, Canada.

20. PP and PS synthetics section
To simulate the real data, in this example PP and PS synthetics were generated with different input wavelets. The zero-phase wavelet is used for PP synthetics and zero-phase wavelet is used for PS synthetics.
Zero-phase wavelet
Zero-phase wavelet
PP and PS data with different frequency content

21. Impedance, Joint inversion
Here is the estimated P-wave and S-wave impedance from joint inversion. The black curve is impedance before low-frequency restoration and red curve is impedance after low-frequency restoration. And blue curve is true impedance. We can see that the red curve is a very good estimate of the Blue curve.
Black curve : Impedance before low-freq restoration
Red curve: Impedance after low-freq restoration
Blue curve: True Impedance from log

22. PP inversion only in presence of noise
Because any given seismic recording has some amount of noise, we added a randon noise to the data with signal to noise ration of 2. Here is result from PP inversion only. Black curve is estimated reflectivity from noisy data and red curve is estimated reflectivity from noise-free data. we can see that PP inversion only has perfect result in DeltaI/I estimation but not for deltaj/j.
Black curve : Impedance with noise
Red curve: Impedance without noise Signal to noise ratio = 2
PP only not good for ΔJ/J

23. PS inversion only in presence of noise
Here is result from PS inversion only. We can see that PS inversion is pretty good in Delta J/J estimation but not for deltaI/I.
Black curve : Impedance with noise
Red curve: Impedance without noise Signal to noise ratio = 2
PS only not good for ΔI/I

24. Joint inversion in presence of noise
and this slide shows the joint inversion result in presence of noise. This slide shows very clearly that the joint inversion method has a very good impedance estimation for noisy data. This effect was the result of duplicating the data fold input into estimation of P and S reflectivity. Lets have a look at deltaJ/J from joint inversion and PP only inversion
Black curve : Impedance with noise
Red curve: Impedance without noise Signal to noise ratio = 2
Joint inversion
best method

25. PP inversion only in presence of noise
Lets compare the deltaJ/J estoimation from joint invertion to PP inversion only. We can see that delatJ/J from joint has less frequency that delta J/J from pp inversion only. Also this effect can be seen in real data inversion.
Black curve : Impedance with noise
Red curve: Impedance without noise Signal to noise ratio = 2
PP only not good for ΔJ/J

26. Application of joint inversion
Zhang and Margrave (2003)
Well
Seismic
Well
Seismic
Here is the application of joint inversion to real data from Zhang and Margrave (2003). The left-hand side plot shows the deltaj/J from joint inversion and the right hand side deltaj/J from PP inversion only. Also here delta j/J from joint has less frequency comparing to PP inversion result.
ΔJ/J
Joint inversion
ΔJ/J
PP inversion only

27. Stacking weights, ΔJ/J estimation
In joint inversion, PS data have more influence on ΔJ/J estimation than PP data do.

28. Stacking weights, ΔI/I estimation
In joint inversion, PP data have more influence on ΔI/I estimation than do PS data.

29. Conclusion Presenting MATLAB routine for Joint inversion.
Including the low-frequency trend can provide the impedance estimation significantly better.
In presence of random noise, the joint inversion was significantly more accurate than other methods
In joint inversion, PS data have more influence on ΔJ/J or J estimation than do PP data.
In joint inversion, PP data have a more influence on ΔI/I or J estimation than do PS data.
We presented
We showed that including …
We showed that in present of noise
We saw that

30. Acknowledgment
The authors gratefully acknowledge Kevin Hall, Jeff Thurston and Brooke Berard, the support of the CREWES sponsors, and discussions with CREWES staff and students.

31. References
Ferguson R. J., Margrave G. F., 1996, A simple algorithm for band-limited impedance inversion, CREWES Research Report, 8, 1-10.
Larson J.A., 1999, AVO inversion by simultaneous PP and PS inversion, Master thesis at the University of Calgary.
Lindseth R. O., 1979, Synthetic sonic logs – a process for stratigraphic interpretation, Geophysics, 44, 3-26.
Margrave, G. F., Stewart R.R. and Larsen, J. A., 2001, Joint PP and PS seismic inversion, The Leading Edge,
Smith, G. C., and Gidlow, P. M., 1987, Weighted stacking for rock property estimation and detection of gas: Geophysical Prospecting, 35,
Stewart, R. R., 1990, Joint P and P-SV inversion: CREWES Research Report, 2,