random noise in seismic data: types, origins, estimation, and removal
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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RANDOM NOISE IN SEISMIC DATA:TYPES, ORIGINS, ESTIMATION, ANDREMOVALPrinciple Investigator: Dr. Tareq Y. Al-Naffouri
Co-Investigators:
Ahmed Abdul Quadeer
Babar Hasan Khan
Ahsan Ali
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ACKNOWLEDGEMENTS
Saudi Aramco
Schlumberger
SRAK
KFUPM
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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
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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
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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
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TOOLS USED IN STOCHASTIC PROCESS? Statistical averages - Ensemble
ttnt
nt dxxpxXE )()(
Autocorrelation function
Autocovariance function
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LINEAR ESTIMATION TECHNIQUES FOR NOISE REMOVAL
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LINEAR MODEL Consider the linear model
Mathematically,
In Matrix form,
or
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LEAST SQUARES & MINIMUM MEAN SQUARES ESTIMATION
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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!
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JOINT CHANNEL AND DATA RECOVERY
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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
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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?
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UTILIZING STRUCTURE TO ENHANCE PERFORMANCE
Channel constraints: Sparsity Time variation
Data Constraints Finite alphabet constraint Transmit precoding Pilots
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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
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FORWARD BACKWARD KALMAN FILTER
Estimates the sequence h0, h1, …, hn optimally given the observation y0, y1, …, yn.
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FORWARD BACKWARD KALMAN FILTER
Forward Run:
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FORWARD BACKWARD KALMAN FILTER
Backward Run: Starting from λT+1|T = 0 and i = T, T-1, …, 0
The desired estimate is
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COMPARISON OVER OSTBC MIMO-OFDM SYSTEM
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USE OF RANDOM MATRIX THEORY FOR SEISMIC SIGNAL PROCESSING
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INTRODUCTION TO RANDOM MATRIX THEORY
Wishart Matrix
PDF of the eigenvalues
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EXAMPLE: ESTIMATION OF POWER AND THE NUMBER OF SOURCES
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COVARIANCE MATRIX AND ITS ESTIMATE
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EIGEN VALUES OF CX
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FREE PROBABILITY THEORY
R-Transform
S-Transform
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??
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APPROXIMATION OF CX
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CONCLUSIONS
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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