1. Random Noise in Seismic Data: Types, Origins, Estimation, and Removal
Principle Investigator: Dr. Tareq Y. Al-Naffouri
Co-Investigators:
Ahmed Abdul Quadeer
Babar Hasan Khan
Ahsan Ali

2. Acknowledgements
Saudi Aramco
Schlumberger
SRAK
KFUPM

3. Outline Introduction A breif overview of Noise and Stochastic Process
Linear Estimation Techniques for Noise Removal
Least Squares
Minimum-Mean Squares
Expectation Maximization
Kalman Filter
Random Matrix Theory
Conclusion

4. Introduction
Seismic exploration has undergone a digital revolution – advancement of computers and digital signal processing
Seismic signals from underground are weak and mostly distorted – noise!
The aim of this presentation – provide an overview of some very constructive concepts of statistical signal processing to seismic exploration

5. What is a Stochastic Process?
What is Noise?
Noise simply means unwanted signal
Common Types of Noise:
Binary and binomial noise
Gaussian noise
Impulsive noise
What is a Stochastic Process?
Broadly – processes which change with time
Stochastic – no specific patterns

6. Tools Used in Stochastic Process?
Statistical averages - Ensemble
Autocorrelation function
Autocovariance function

7. Linear Estimation Techniques for Noise Removal

8. Linear Model Consider the linear model Mathematically, In Matrix form,

9. Least Squares & Minimum Mean Squares Estimation

10. Least Squares & Minimum Mean Squares Estimation
Advantages:
Linear in the observation y.
MMSE estimates blindly given the joint 2nd order statistics of h and y.
Problem: X is generally not known!
Solution: Joint Estimation!

11. Joint Channel and Data Recovery

12. Expectation Maximization Algorithm
One way to recover both X and h is to do so jointly.
Assume we have an initial estimate of h then X can be estimated using least squares from
The estimate can in turn be used to obtain refined estimate of h
The procedure goes on iterating between x and h

13. Expectation Maximization Algorithm
Problems:
Where do we obtain the initial estimate of h from?
How could we guarantee that the iterative procedure will consistently yield better estimates?

14. Utilizing Structure To Enhance Performance
Channel constraints:
Sparsity
Time variation
Data Constraints
Finite alphabet constraint
Transmit precoding
Pilots

15. Kalman Filter
A filtering technique which uses a set of mathematical equations that provide efficient and recursive computational means to estimate the state of a process.
The recursions minimize the mean squared error.
Consider a state space model

16. Forward Backward Kalman Filter
Estimates the sequence h0, h1, …, hn optimally given the observation y0, y1, …, yn.

17. Forward Backward Kalman Filter
Forward Run:

18. Forward Backward Kalman Filter
Backward Run: Starting from λT+1|T = 0 and i = T, T-1, …, 0
The desired estimate is

19. Comparison Over OSTBC MIMO-OFDM System

20. Use of Random Matrix Theory for Seismic Signal Processing

21. Introduction To Random Matrix Theory
Wishart Matrix
PDF of the eigenvalues

22. Example: Estimation of power and the number of sources

23. Covariance Matrix and its Estimate

24. Eigen Values of Cx

25. Free Probability Theory
R-Transform
S-Transform

26. ??

27. Approximation of Cx

28. Conclusions

29. The Ideas presented here are commonly used in Digital Communication
But when applied to seismic signal processing can produce valuable results, with of course some modifications
For Example: Kalman Filter, Random Matrix Theory