Random Noise in Seismic Data: Types, Origins, Estimation, and Removal

1. Random Noise in Seismic Data:Types, Origins, Estimation, andRemoval 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 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, or

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