1. R ANDOM N OISE IN S EISMIC D ATA : T YPES, O RIGINS, E STIMATION, AND R EMOVAL Principle Investigator: Dr. Tareq Y. Al-Naffouri Co-Investigators: Ahmed Abdul Quadeer Babar Hasan Khan Ahsan Ali

2. A CKNOWLEDGEMENTS Saudi Aramco Schlumberger SRAK KFUPM

3. O UTLINE 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. I NTRODUCTION 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. W HAT IS N OISE ? Noise simply means unwanted signal Common Types of Noise: Binary and binomial noise Gaussian noise Impulsive noise W HAT IS A S TOCHASTIC P ROCESS ? Broadly – processes which change with time Stochastic – no specific patterns

6. T OOLS U SED IN S TOCHASTIC P ROCESS ? Statistical averages - Ensemble A utocorrelation function A utocovariance function

7. L INEAR E STIMATION T ECHNIQUES FOR N OISE R EMOVAL

8. L INEAR M ODEL Consider the linear model Mathematically, In Matrix form, or

9. L EAST S QUARES & M INIMUM M EAN S QUARES E STIMATION

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

11. J OINT C HANNEL AND D ATA R ECOVERY

12. E XPECTATION M AXIMIZATION A LGORITHM 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. E XPECTATION M AXIMIZATION A LGORITHM 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. U TILIZING S TRUCTURE T O E NHANCE P ERFORMANCE Channel constraints: Sparsity Time variation Data Constraints Finite alphabet constraint Transmit precoding Pilots

15. K ALMAN F ILTER 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. F ORWARD B ACKWARD K ALMAN F ILTER Estimates the sequence h 0, h 1, …, h n optimally given the observation y 0, y 1, …, y n.

17. F ORWARD B ACKWARD K ALMAN F ILTER Forward Run:

18. F ORWARD B ACKWARD K ALMAN F ILTER Backward Run: Starting from λ T+1|T = 0 and i = T, T-1, …, 0 The desired estimate is

19. C OMPARISON O VER OSTBC MIMO- OFDM S YSTEM

20. U SE OF R ANDOM M ATRIX T HEORY FOR S EISMIC S IGNAL P ROCESSING

21. I NTRODUCTION T O R ANDOM M ATRIX T HEORY Wishart Matrix PDF of the eigenvalues

22. E XAMPLE : E STIMATION OF POWER AND THE NUMBER OF SOURCES

23. C OVARIANCE M ATRIX AND ITS E STIMATE

24. E IGEN V ALUES OF C X

25. F REE P ROBABILITY T HEORY R-Transform S-Transform

26. ??

27. A PPROXIMATION OF C X

28. C ONCLUSIONS

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