r andom n oise in s eismic d ata : t ypes, o rigins, e stimation, and r emoval principle...
TRANSCRIPT
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
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 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
ttnt
nt dxxpxXE )()(
Autocorrelation function
Autocovariance function
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!
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
Backward Run: Starting from λT+1|T = 0 and i = T, T-1, …, 0
The desired estimate is