1. High Resolution AVO NMObc
High Resolution AVO NMO
Jon Downton
Larry Lines

2. Theme
This paper demonstrates a methodology to produce high-resolution AVO reflectivity attribute estimates similar to sparse spike deconvolution.
The AVO estimate is performed prior to NMO thus avoiding NMO stretch distortions and incorporating differential tuning into the forward model.
Approach similar to Downton and Lines (2002) but better able to handle tuning

3. Message
High-resolution AVO NMO is better able to estimate reflectors undergoing NMO stretch and differential tuning than traditional AVO performed on a sample-by-sample basis on NMO corrected gathers
Provides framework to do elastic impedance inversion similar to poststack sparse spike impedance inversion

4. Methodology
Theory for new algorithm developed using Bayesian formulism
Instead of typical assumption of Gaussian priors will assume long tailed distribution
Linear operator based on AVO NMO model presented by Downton and Lines (2002) assuming some input wavelet
Constraints analogous to the high-resolution Radon transform (Sacchi and Ulrych, 1995)
Similar to the poststack sparse spike impedance inversion presented by Sacchi (1999) and Trad (2002)
Inversion is nonlinear solved by iterative CG

5. Outline High-Resolution AVO Prestack Inversion Theory
Example 1: model with NMO stretch and differential tuning
Example 2: well log model
Example 3: Data example
Prestack Inversion
Example 4

6. Theory: Convolutional model
Use convolutional model as basis of AVO NMO inversion scheme
Assumes locally, earth is composed of a series of flat, homogenous, isotropic layers
Approximation of Zoeppritz equation used to model how reflectivity changes as a function of offset
Ray tracing is done to map relationship between angle of incidence and offset
Transmission losses, converted waves and multiples are not incorporated in this model and so must be addressed through pre-processing
Gain corrections such as spherical divergence, absorption, directivity and array corrections can be incorporated in this model but are not considered for brevity and simplicity

7. Theory: Convolutional model
AVO NMO formulation (Downton and Lines, 2002)
where W is the wavelet, Nk is the NMO operator, F and G are diagonal matrices containing AVO weights, rp is the P-impedance reflectivity series, rf is the fluid stack reflectivity series, and dk is the data at offset k
This model may be generalized for more offsets and interfaces as the linear equation
Choose this particular parameterization since uncorrelated

8. Theory: constraints Need to regularize problem
Problem is typically underdetermined or ill-conditioned due to the
Band-limited nature of the data
Differential tuning in the NMO operator
Need to regularize problem
Choose a weighting function that will treat certain reflection coefficients as being more reliable than others
Choosing a long tailed a priori distribution will lead to such a weighting function
Good results were obtained with Huber, L1, and Cauchy norms
For illustration purposes show weighting function that arises from L1 norm

9. Theory: constraints
Sacchi (1999) shows that for the case when p=1 this leads to a prior of the form
where J is the objective function and Q is a diagonal matrix defined as
where sm is the L1 standard deviation of the reflectivity and e is some user chosen threshold.
Matrix Q must be calculated iteratively in a bootstrap fashion

10. Theory: Nonlinear inversion
The Likelihood function may be combined with the a priori probability function using Bayes’ theorem. The most likely solution leads to the nonlinear equation
where Q is regularization parameter, L is the linear operator and
=Lm-d
two sources of nonlinearity
the estimate of the regularization parameter m
the calculation of Q
Solve in iterative fashion
Q improves the condition number significantly

11. Theory: Numerical Solution
Inverse problem is quite large
Most efficiently solved using iterative techniques such as conjugate gradient
Solving inverse problem requires two nested loops
Inner loop
conjugate gradient algorithm is used to estimate reflectivity using the previously calculated values of m and Q
maximum number of conjugate gradient iterations is used as a trade-off parameter between resolution and reliablity (Skewchuk, 1994 )
Outer loop
After solving for the reflectivity the estimate of m and the matrix Q is updated

12. Outline High-Resolution AVO Prestack Inversion Theory
Example 1: model with NMO stretch and differential tuning
Example 2: well log model
Example 3: Data example
Prestack Inversion
Example 4

13. NMO Stretch and Differential tuning
Dong (1999) quantified this error for the intercept A and the Gradient B
where f is a simple function
Conclusions
Most of error is in the gradient term
Size of error is dependent on B/A ratio
Small for class II (A=0)
Small for class I (A<<B)
Potentially large for class III and IV (A>B)
Reflectors on mudrock trend error is small (A<<B)

14. NMO stretch AVO distortions
Offset (m)
Generated simple model
Generated Class III and IV anomalies
Most reflections follow mudrock trend
Density follows Gardner relationship
Convolution model
Reflectivity generated by Aki and Richards equation
Ricker wavelet
32.5 Hz central frequency
S/N ratio = 4
Constant velocity so simple angle to offset relationships
Maximum offset is 4X target depth to give angles out to 65 degrees
Use large angles to highlight distortions
Time (s)

15. Traditional Flow: Error due to NMO
Offset (m)
Time (s)
Theoretical errors introduced by NMO correction
Systematic error biases S-reflectivity estimate
Input (generated
with no moveout)
NMO corrected data
Difference

16. Traditional Flow (45 degree )
Time (s)
Extraction to 45 degrees
estimate
actual

17. High Resolution AVO NMO
Offset (m)
Offset (m)
Time (s)
Input
Reconstruction
5X difference

18. High Resolution AVO NMO
Time (s)
45 degree results
S-reflectivity is more noisy than P-reflectivity due to noise amplification resulting in fatter cross-plot
Summarize AVO NMO results
estimate
actual
Filtered

19. High Resolution AVO NMO
Time (s)
estimate
actual
filtered
unfiltered

20. Outline High-Resolution AVO Prestack Inversion Theory
Example 1: model with NMO stretch and differential tuning
Example 2: well log model
Example 3: Data example
Prestack Inversion
Example 4

21. Example 2: Well log synthetic
Offset (m)
Offset (m)
Generated synthetic based on well log
Most reflections follow mudrock trend but significant deviations at Charlie Lake and Halfway
Convolution model
Reflectivity generated by Zoeppritz equation
Wavelet
10/14 – 90/110 Hz
S/N ratio = 4
AVO inversion done using angles 0 – 35 degrees

22. Example 2
Time (s)
estimate
actual
filtered
unfiltered

23. Example 2
Time (s)
estimate
actual
filtered
unfiltered

24. Outline High-Resolution AVO Prestack Inversion Theory
Example 1: model with NMO stretch and differential tuning
Example 2: well log model
Example 3: Colony data example
Prestack Inversion
Example 4

25. Constraints even improve zero offset P-impedance section
Gas
Gas
Describe wells
Constraints even improve zero offset P-impedance section

26. Xscl tscl noband Describe wells
Gas
Gas
Describe wells
Constraints even improve zero offset P-impedance section
Xscl tscl noband

27. Data example
Offset (m)
Time (s)
Input
Reconstruction
Difference

28. Outline High-Resolution AVO Prestack Inversion Theory
Example 1: model with NMO stretch and differential tuning
Example 2: well log model
Example 3: Data example
Prestack Inversion
Example 4

29. Theory: well log constraints
Relationship between reflectivity r and impedance x
Write a series of constraints that specify that the average estimated impedance from the seismic over some time interval must be equal the average impedance of the well log over that same interval with Sw variance
Pm=w
where P is the linear operator for integration and w is the well log data
This leads to an equation of the form
which can be solved in an iterative fashion similar to before

30. Example 4 Time (s) Generated simple model Processing sequence
Offset (m)
Generated simple model
Geologic model
sparse reflectivity
most reflections follow mudrock trend
class III anomaly
Density follows Gardner relationship
Convolution model
Reflectivity generated by the Zoeppritz equation
Wavelet
5/10-40/50 Hz
S/N ratio = 4
Processing sequence
High-resolution AVO NMO inversion
Convert sparse spike reflectivity to impedance
Time (s)

31. Example 4 Offset (m) Time (s) Input Reconstruction Difference

32. Example 4
Time (s)
estimate
actual
filtered
unfiltered

33. Conclusions High-resolution AVO NMO
is able to resolve reflectors undergoing NMO stretch and differential tuning
is more reliable than the estimates provided by the traditional AVO analysis that is performed on a sample-by-sample basis on NMO corrected gathers
greater reliability is due to the classic trade-off between resolution and reliability
a few sparse reflectivity values are estimated with greater certainty than the dense reflectivity at every time sample as in the traditional AVO analysis.
Provides framework to do elastic impedance inversion similar to poststack sparse spike impedance inversion
Algorithm is fast
only a few iterations of Conjugate Gradient are needed to get satisfactory results

34. Acknowledgements
The authors would like to thank Core Lab, N.S.E.R.C. and the C.R.E.W.E.S. sponsors for funding this research.