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

Acknowledgements Saudi Aramco Schlumberger SRAK KFUPM

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

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

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

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

Linear Estimation Techniques for Noise Removal

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

Least Squares & Minimum Mean Squares Estimation

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!

Joint Channel and Data Recovery

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

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?

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

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

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

Forward Backward Kalman Filter Forward Run:

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

Comparison Over OSTBC MIMO-OFDM System

Use of Random Matrix Theory for Seismic Signal Processing

Introduction To Random Matrix Theory Wishart Matrix PDF of the eigenvalues

Example: Estimation of power and the number of sources

Covariance Matrix and its Estimate

Eigen Values of Cx

Free Probability Theory R-Transform S-Transform

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Approximation of Cx

Conclusions

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